Notation by work
Realization Calculus
All notation · Glossary · Source coverage
6672 symbol spellings in 22425 mathematical passages across 146 selected source files. Subscripts and superscripts have their own entries alongside their base symbols.
The same symbol can have several meanings here. A passage classified as a definition may define another part of its expression. Read the linked context before assigning a meaning.
All 6672 entries are available below.
*Notation table · 442 passages
Search aliases: *
Notation table · atlas-pages.tex, line 27
W_α^*W_α · A projector on the original signal space · A scalar packet energy
Exact mathematical passage
W_\alpha^*W_\alpha
+Notation table · 2446 passages
Search aliases: +
Notation table · 06_models.tex, line 182
J_ℓ+1,ℓ · exact coordinate refinement map
Exact mathematical passage
J_{\ell+1,\ell}
+^mNotation table · 5 passages
Search aliases: +^m
Notation table · Paper_V_Traditional_Final.tex, line 1717
ϕ_m,τ · the truncated power packet (t-τ)_+^m/m!
Exact mathematical passage
(t-\tau)_+^m/m!
=Notation table · 7513 passages
Search aliases: =
Notation table · 01_orientation.tex, line 61
G=(Q,I) · Typed grammar · A quiver with finite vertex set and a two-sided ideal of source and target compatible relations.
Exact mathematical passage
\mathfrak G=(Q,I)
ANotation table · 1430 passages
Search aliases: A
Notation table · 01_orientation.tex, line 65
Hull_A(S) · Generated hull · The submodule AS, or its declared topological closure.
Exact mathematical passage
\Hull_A(S)
\ANotation table · 66 passages
Search aliases: A mathcal
Notation table · 01_orientation.tex, line 63
A_X · Realized algebra · The image of the grammar algebra in End_𝕜(|X|).
Exact mathematical passage
\A_X
\mathcal ANotation table · 49 passages
Search aliases: A mathcal
Notation table · 01_orientation.tex, line 69
A_ P · Analysis · The map x↦(P_α x)_α.
Exact mathematical passage
\mathcal A_{\Pcal}
\mathcal A_{\Pcal}Notation table · 14 passages
Search aliases: A_Pcal A_ P mathcal Pcal mathcal mathcal
Notation table · 01_orientation.tex, line 69
A_ P · Analysis · The map x↦(P_α x)_α.
Exact mathematical passage
\mathcal A_{\Pcal}
\A_XNotation table · 7 passages
Search aliases: A_X A mathcal
Notation table · 01_orientation.tex, line 63
A_X · Realized algebra · The image of the grammar algebra in End_𝕜(|X|).
Exact mathematical passage
\A_X
\AcalNotation table · 47 passages
Search aliases: Acal A mathcal
Notation table · Paper_V_Traditional_Final.tex, line 1712
A_L, S_L · lattice analysis and interleaving synthesis
Exact mathematical passage
\Acal_L,\Scal_L
\Acal_LNotation table · 6 passages
Search aliases: Acal_L A_L Acal mathcal
Notation table · Paper_V_Traditional_Final.tex, line 1712
A_L, S_L · lattice analysis and interleaving synthesis
Exact mathematical passage
\Acal_L,\Scal_L
\bbkNotation table · 117 passages
Search aliases: bbk bbk 𝕜 Bbbk
Notation table · 01_orientation.tex, line 62
Λ_ G · Grammar algebra · The quotient path algebra 𝕜 Q/I.
Exact mathematical passage
\bbk Q/I
CNotation table · 1062 passages
Search aliases: C
Notation table · 06_models.tex, line 183
C_ℓ,ℓ+1 · exact coordinate regrouping from fine to coarse
Exact mathematical passage
C_{\ell,\ell+1}
cNotation table · 772 passages
Search aliases: c
Notation table · 06_models.tex, line 191
Δ · linear refinement defect for one step CT_f-T_cC
Exact mathematical passage
CT_f-T_cC
C_{\ell,\ell+1}Notation table · 1 passage
Search aliases: C_ℓ,ℓ+1 ell ell
Notation table · 06_models.tex, line 183
C_ℓ,ℓ+1 · exact coordinate regrouping from fine to coarse
Exact mathematical passage
C_{\ell,\ell+1}
\cANotation table · 33 passages
Search aliases: cA A mathcal
Notation table · 06_models.tex, line 181
A_ℓ, S_ℓ · packet analysis and synthesis maps
Exact mathematical passage
\cA_\ell,\cS_\ell
\cA_\ellNotation table · 2 passages
Search aliases: cA_ℓ A_ℓ cA ell mathcal ell
Notation table · 06_models.tex, line 181
A_ℓ, S_ℓ · packet analysis and synthesis maps
Exact mathematical passage
\cA_\ell,\cS_\ell
\cSNotation table · 23 passages
Search aliases: cS S mathcal
Notation table · 06_models.tex, line 181
A_ℓ, S_ℓ · packet analysis and synthesis maps
Exact mathematical passage
\cA_\ell,\cS_\ell
\cS_\ellNotation table · 1 passage
Search aliases: cS_ℓ S_ℓ cS ell mathcal ell
Notation table · 06_models.tex, line 181
A_ℓ, S_ℓ · packet analysis and synthesis maps
Exact mathematical passage
\cA_\ell,\cS_\ell
DNotation table · 797 passages
Search aliases: D
Notation table · atlas-pages.tex, line 24
D · One channel-index delay · Necessarily one original-sample delay
Exact mathematical passage
D
dNotation table · 985 passages
Search aliases: d
Notation table · Paper_V_Traditional_Final.tex, line 1709
L≤ Z^d · a finite index sublattice
Exact mathematical passage
L\leq\ZZ^d
ENotation table · 699 passages
Search aliases: E
Notation table · 06_models.tex, line 190
s_π,Λ(E) · separation weight of a rational projector on the chosen lattice, when a clearing exponent exists
Exact mathematical passage
s_{\pi,\Lambda}(E)
eNotation table · 1024 passages
Search aliases: e
Notation table · 06_models.tex, line 186
w(e) · positive filtration weight of a transfer edge
Exact mathematical passage
w(e)
\EndNotation table · 57 passages
Search aliases: End operatorname
Notation table · 01_orientation.tex, line 63
A_X · Realized algebra · The image of the grammar algebra in End_𝕜(|X|).
Exact mathematical passage
\End_{\bbk}(|X|)
\End_{\bbk}Notation table · 12 passages
Search aliases: End_bbk End_𝕜 End bbk operatorname Bbbk
Notation table · 01_orientation.tex, line 63
A_X · Realized algebra · The image of the grammar algebra in End_𝕜(|X|).
Exact mathematical passage
\End_{\bbk}(|X|)
fNotation table · 710 passages
Search aliases: f
Notation table · 06_models.tex, line 191
Δ · linear refinement defect for one step CT_f-T_cC
Exact mathematical passage
CT_f-T_cC
f_rNotation table · 9 passages
Search aliases: f_r
Notation table · atlas-pages.tex, line 23
f_r[q] · The value f[Nq+r] · An average of neighboring samples
Exact mathematical passage
f_r[q]
\mathfrak GNotation table · 14 passages
Search aliases: G mathfrak
Notation table · 01_orientation.tex, line 61
G=(Q,I) · Typed grammar · A quiver with finite vertex set and a two-sided ideal of source and target compatible relations.
Exact mathematical passage
\mathfrak G=(Q,I)
\GamNotation table · 12 passages
Search aliases: Gam Γ Gamma
Notation table · Paper_V_Traditional_Final.tex, line 1475
Γ · reflection coefficient · Smith-disk coordinate
Exact mathematical passage
\Gam
HNotation table · 702 passages
Search aliases: H
Notation table · Paper_V_Traditional_Final.tex, line 1716
H_τ · the step packet 1_[τ,1]
Exact mathematical passage
H_\tau
hNotation table · 502 passages
Search aliases: h
Notation table · Paper_V_Traditional_Final.tex, line 1718
h · the grid spacing in the recovery theorems
Exact mathematical passage
h
H_\tauNotation table · 11 passages
Search aliases: H_τ tau
Notation table · Paper_V_Traditional_Final.tex, line 1716
H_τ · the step packet 1_[τ,1]
Exact mathematical passage
H_\tau
\HullNotation table · 33 passages
Search aliases: Hull operatorname
Notation table · 01_orientation.tex, line 65
Hull_A(S) · Generated hull · The submodule AS, or its declared topological closure.
Exact mathematical passage
\Hull_A(S)
\Hull_ANotation table · 9 passages
Search aliases: Hull_A Hull operatorname
Notation table · 01_orientation.tex, line 65
Hull_A(S) · Generated hull · The submodule AS, or its declared topological closure.
Exact mathematical passage
\Hull_A(S)
INotation table · 928 passages
Search aliases: I
Notation table · 01_orientation.tex, line 61
G=(Q,I) · Typed grammar · A quiver with finite vertex set and a two-sided ideal of source and target compatible relations.
Exact mathematical passage
\mathfrak G=(Q,I)
I_\ellNotation table · 3 passages
Search aliases: I_ℓ ell
Notation table · 06_models.tex, line 179
I_ℓ · packet address set at level ℓ
Exact mathematical passage
I_\ell
\operatorname{inputs}Notation table · 5 passages
Search aliases: inputs operatorname
Notation table · visual_atlas.tex, line 36
t,inputs(t) · A finite operation tree and its syntactically used input labels. · pagerefV20
Exact mathematical passage
t,\operatorname{inputs}(t)
JNotation table · 496 passages
Search aliases: J
Notation table · 06_models.tex, line 182
J_ℓ+1,ℓ · exact coordinate refinement map
Exact mathematical passage
J_{\ell+1,\ell}
jNotation table · 1437 passages
Search aliases: j
Notation table · 06_models.tex, line 185
P_N,j · residue or phase selector for address jpmod N
Exact mathematical passage
P_{N,j}
J_{\ell+1,\ell}Notation table · 1 passage
Search aliases: J_ℓ+1,ℓ ell ell
Notation table · 06_models.tex, line 182
J_ℓ+1,ℓ · exact coordinate refinement map
Exact mathematical passage
J_{\ell+1,\ell}
LNotation table · 1008 passages
Search aliases: L
Notation table · Paper_V_Traditional_Final.tex, line 1709
L≤ Z^d · a finite index sublattice
Exact mathematical passage
L\leq\ZZ^d
MNotation table · 831 passages
Search aliases: M
Notation table · 01_orientation.tex, line 64
S⊆ M · Seed family · The vectors from which the operator orbit is generated.
Exact mathematical passage
S\subseteq M
mNotation table · 1522 passages
Search aliases: m
Notation table · Paper_V_Traditional_Final.tex, line 1714
Q_m · a bounded linear approximation operator acting on the smooth part
Exact mathematical passage
Q_m
NNotation table · 1736 passages
Search aliases: N
Notation table · 06_models.tex, line 185
P_N,j · residue or phase selector for address jpmod N
Exact mathematical passage
P_{N,j}
\oneNotation table · 32 passages
Search aliases: one 1 bm one 1 mathbf
Notation table · Paper_V_Traditional_Final.tex, line 1716
H_τ · the step packet 1_[τ,1]
Exact mathematical passage
\one_{[\tau,1]}
\one_{[\tau,1]}Notation table · 4 passages
Search aliases: one_[τ,1] 1_[τ,1] one tau mathbf tau
Notation table · Paper_V_Traditional_Final.tex, line 1716
H_τ · the step packet 1_[τ,1]
Exact mathematical passage
\one_{[\tau,1]}
PNotation table · 1428 passages
Search aliases: P
Notation table · 01_orientation.tex, line 67
P=(P_α) · Packet system · Pairwise annihilating idempotents summing to the identity.
Exact mathematical passage
\Pcal=(P_\alpha)
P_{N,j}Notation table · 30 passages
Search aliases: P_N,j
Notation table · 06_models.tex, line 185
P_N,j · residue or phase selector for address jpmod N
Exact mathematical passage
P_{N,j}
P_{N,r}Notation table · 18 passages
Search aliases: P_N,r
Notation table · atlas-pages.tex, line 22
P_N,rf · A masked signal on the original lattice · The shorter decimated channel
Exact mathematical passage
P_{N,r}f
P_rNotation table · 17 passages
Search aliases: P_r
Notation table · Paper_V_Traditional_Final.tex, line 1711
P_r · the coordinate projector onto the coset r+L
Exact mathematical passage
P_r
P_\alphaNotation table · 248 passages
Search aliases: P_α alpha
Notation table · 01_orientation.tex, line 67
P=(P_α) · Packet system · Pairwise annihilating idempotents summing to the identity.
Exact mathematical passage
\Pcal=(P_\alpha)
P_\betaNotation table · 92 passages
Search aliases: P_β beta
Notation table · 01_orientation.tex, line 71
T_βα · Transfer block · The source and target block P_β TP_α:X_α→ X_β.
Exact mathematical passage
P_\beta TP_\alpha:X_\alpha\to X_\beta
P_{\ell,\alpha}Notation table · 6 passages
Search aliases: P_ℓ,α ell alpha
Notation table · 06_models.tex, line 178
P_ℓ,α · packet projector at refinement level ℓ
Exact mathematical passage
P_{\ell,\alpha}
P_{\ell,\beta}Notation table · 1 passage
Search aliases: P_ℓ,β ell beta
Notation table · 06_models.tex, line 184
T^(ℓ)_βα · transfer block P_ℓ,βTP_ℓ,α
Exact mathematical passage
P_{\ell,\beta}TP_{\ell,\alpha}
\PcalNotation table · 182 passages
Search aliases: Pcal P mathcal
Notation table · 01_orientation.tex, line 67
P=(P_α) · Packet system · Pairwise annihilating idempotents summing to the identity.
Exact mathematical passage
\Pcal=(P_\alpha)
\pmodNotation table · 199 passages
Search aliases: pmod
Notation table · 06_models.tex, line 185
P_N,j · residue or phase selector for address jpmod N
Exact mathematical passage
j\pmod N
\psrankNotation table · 9 passages
Search aliases: psrank operatorname
Notation table · 01_orientation.tex, line 73
psrank · Rank for seeds confined to individual packets · The minimal number of generating seeds each supported in a single packet.
Exact mathematical passage
\psrank
QNotation table · 610 passages
Search aliases: Q
Notation table · 01_orientation.tex, line 61
G=(Q,I) · Typed grammar · A quiver with finite vertex set and a two-sided ideal of source and target compatible relations.
Exact mathematical passage
\mathfrak G=(Q,I)
qNotation table · 839 passages
Search aliases: q
Notation table · atlas-pages.tex, line 23
f_r[q] · The value f[Nq+r] · An average of neighboring samples
Exact mathematical passage
f_r[q]
Q_mNotation table · 42 passages
Search aliases: Q_m
Notation table · Paper_V_Traditional_Final.tex, line 1714
Q_m · a bounded linear approximation operator acting on the smooth part
Exact mathematical passage
Q_m
RNotation table · 951 passages
Search aliases: R
Notation table · Paper_V_Traditional_Final.tex, line 1710
R · a set of representatives for Z^d/L
Exact mathematical passage
R
rNotation table · 1608 passages
Search aliases: r
Notation table · Paper_V_Traditional_Final.tex, line 1711
P_r · the coordinate projector onto the coset r+L
Exact mathematical passage
P_r
SNotation table · 508 passages
Search aliases: S
Notation table · 01_orientation.tex, line 64
S⊆ M · Seed family · The vectors from which the operator orbit is generated.
Exact mathematical passage
S\subseteq M
\mathcal SNotation table · 29 passages
Search aliases: S mathcal
Notation table · 01_orientation.tex, line 70
S_ P · Synthesis · The map (x_α)↦∑_α x_α.
Exact mathematical passage
\mathcal S_{\Pcal}
sNotation table · 1137 passages
Search aliases: s
Notation table · 06_models.tex, line 189
Z_s,t(t) · weighted transfer-zeta function from vertex s to vertex t
Exact mathematical passage
Z_{s,t}(t)
\mathcal S_{\Pcal}Notation table · 16 passages
Search aliases: S_Pcal S_ P mathcal Pcal mathcal mathcal
Notation table · 01_orientation.tex, line 70
S_ P · Synthesis · The map (x_α)↦∑_α x_α.
Exact mathematical passage
\mathcal S_{\Pcal}
s_{\pi,\Lambda}Notation table · 3 passages
Search aliases: s_π,Λ pi Lambda
Notation table · 06_models.tex, line 190
s_π,Λ(E) · separation weight of a rational projector on the chosen lattice, when a clearing exponent exists
Exact mathematical passage
s_{\pi,\Lambda}(E)
\ScalNotation table · 23 passages
Search aliases: Scal S mathcal
Notation table · Paper_V_Traditional_Final.tex, line 1712
A_L, S_L · lattice analysis and interleaving synthesis
Exact mathematical passage
\Acal_L,\Scal_L
\Scal_LNotation table · 2 passages
Search aliases: Scal_L S_L Scal mathcal
Notation table · Paper_V_Traditional_Final.tex, line 1712
A_L, S_L · lattice analysis and interleaving synthesis
Exact mathematical passage
\Acal_L,\Scal_L
\srankNotation table · 46 passages
Search aliases: srank operatorname
Notation table · 01_orientation.tex, line 66
srank_A(M) · Seed rank · The smallest cardinality of a generating family. A finite value is the minimum number of seeds.
Exact mathematical passage
\srank_A(M)
\srank_ANotation table · 24 passages
Search aliases: srank_A srank operatorname
Notation table · 01_orientation.tex, line 66
srank_A(M) · Seed rank · The smallest cardinality of a generating family. A finite value is the minimum number of seeds.
Exact mathematical passage
\srank_A(M)
\succNotation table · 4 passages
Search aliases: succ
Notation table · 06_models.tex, line 180
βsuccα · fine packet β is a child of coarse packet α
Exact mathematical passage
\beta\succ\alpha
\SuppNotation table · 29 passages
Search aliases: Supp operatorname
Notation table · 01_orientation.tex, line 72
Supp_ P(x) · Packet support · The packet indices on which x has a nonzero coordinate.
Exact mathematical passage
\Supp_{\Pcal}(x)
\Supp_{\Pcal}Notation table · 6 passages
Search aliases: Supp_Pcal Supp_ P Supp Pcal operatorname mathcal
Notation table · 01_orientation.tex, line 72
Supp_ P(x) · Packet support · The packet indices on which x has a nonzero coordinate.
Exact mathematical passage
\Supp_{\Pcal}(x)
TNotation table · 900 passages
Search aliases: T
Notation table · 01_orientation.tex, line 71
T_βα · Transfer block · The source and target block P_β TP_α:X_α→ X_β.
Exact mathematical passage
T_{\beta\alpha}
tNotation table · 1444 passages
Search aliases: t
Notation table · 06_models.tex, line 188
A(t) · polynomial weighted adjacency matrix
Exact mathematical passage
A(t)
T^{(\ell)}_{\beta\alpha}Notation table · 1 passage
Search aliases: T^(ℓ)_βα ell beta alpha
Notation table · 06_models.tex, line 184
T^(ℓ)_βα · transfer block P_ℓ,βTP_ℓ,α
Exact mathematical passage
T^{(\ell)}_{\beta\alpha}
T_cNotation table · 30 passages
Search aliases: T_c
Notation table · 06_models.tex, line 191
Δ · linear refinement defect for one step CT_f-T_cC
Exact mathematical passage
CT_f-T_cC
T_fNotation table · 30 passages
Search aliases: T_f
Notation table · 06_models.tex, line 191
Δ · linear refinement defect for one step CT_f-T_cC
Exact mathematical passage
CT_f-T_cC
T_{\beta\alpha}Notation table · 31 passages
Search aliases: T_βα beta alpha
Notation table · 01_orientation.tex, line 71
T_βα · Transfer block · The source and target block P_β TP_α:X_α→ X_β.
Exact mathematical passage
T_{\beta\alpha}
WNotation table · 210 passages
Search aliases: W
Notation table · 06_models.tex, line 187
W(γ) · cumulative weight of a transfer word or graph walk
Exact mathematical passage
W(\gamma)
wNotation table · 376 passages
Search aliases: w
Notation table · 06_models.tex, line 186
w(e) · positive filtration weight of a transfer edge
Exact mathematical passage
w(e)
W_\alphaNotation table · 5 passages
Search aliases: W_α alpha
Notation table · atlas-pages.tex, line 26
W_α · The addressed Haar analysis operator · An edge detector by itself
Exact mathematical passage
W_\alpha
W_\alpha^*Notation table · 3 passages
Search aliases: W_α^* alpha
Notation table · atlas-pages.tex, line 27
W_α^*W_α · A projector on the original signal space · A scalar packet energy
Exact mathematical passage
W_\alpha^*W_\alpha
\widehat jNotation table · 38 passages
Search aliases: widehat j widehat
Notation table · Paper_V_Traditional_Final.tex, line 1715
j,widehat j · the true and estimated singular packets
Exact mathematical passage
j,\widehat j
XNotation table · 911 passages
Search aliases: X
Notation table · 01_orientation.tex, line 63
A_X · Realized algebra · The image of the grammar algebra in End_𝕜(|X|).
Exact mathematical passage
\A_X
xNotation table · 1511 passages
Search aliases: x
Notation table · 01_orientation.tex, line 69
A_ P · Analysis · The map x↦(P_α x)_α.
Exact mathematical passage
x\mapsto(P_\alpha x)_\alpha
X_\alphaNotation table · 65 passages
Search aliases: X_α alpha
Notation table · 01_orientation.tex, line 68
X_α · Packet space · The range P_α X.
Exact mathematical passage
X_\alpha
x_\alphaNotation table · 16 passages
Search aliases: x_α alpha
Notation table · 01_orientation.tex, line 70
S_ P · Synthesis · The map (x_α)↦∑_α x_α.
Exact mathematical passage
(x_\alpha)\mapsto\sum_\alpha x_\alpha
X_\betaNotation table · 16 passages
Search aliases: X_β beta
Notation table · 01_orientation.tex, line 71
T_βα · Transfer block · The source and target block P_β TP_α:X_α→ X_β.
Exact mathematical passage
P_\beta TP_\alpha:X_\alpha\to X_\beta
ZNotation table · 211 passages
Search aliases: Z
Notation table · 06_models.tex, line 189
Z_s,t(t) · weighted transfer-zeta function from vertex s to vertex t
Exact mathematical passage
Z_{s,t}(t)
zNotation table · 749 passages
Search aliases: z
Notation table · atlas-pages.tex, line 30
Γ,z · Reflection coefficient and normalized impedance · Coefficients that can be linearly interchanged
Exact mathematical passage
\Gamma,z
Z_{s,t}Notation table · 4 passages
Search aliases: Z_s,t
Notation table · 06_models.tex, line 189
Z_s,t(t) · weighted transfer-zeta function from vertex s to vertex t
Exact mathematical passage
Z_{s,t}(t)
\ZZNotation table · 84 passages
Search aliases: ZZ ZZ Z mathbb
Notation table · Paper_V_Traditional_Final.tex, line 1709
L≤ Z^d · a finite index sublattice
Exact mathematical passage
L\leq\ZZ^d
\ZZ^dNotation table · 32 passages
Search aliases: ZZ^d Z^d ZZ mathbb
Notation table · Paper_V_Traditional_Final.tex, line 1709
L≤ Z^d · a finite index sublattice
Exact mathematical passage
L\leq\ZZ^d
|Notation table · 289 passages
Search aliases: |
Notation table · 01_orientation.tex, line 63
A_X · Realized algebra · The image of the grammar algebra in End_𝕜(|X|).
Exact mathematical passage
\End_{\bbk}(|X|)
\alphaNotation table · 740 passages
Search aliases: α alpha
Notation table · 01_orientation.tex, line 67
P=(P_α) · Packet system · Pairwise annihilating idempotents summing to the identity.
Exact mathematical passage
\Pcal=(P_\alpha)
\betaNotation table · 339 passages
Search aliases: β beta
Notation table · 01_orientation.tex, line 71
T_βα · Transfer block · The source and target block P_β TP_α:X_α→ X_β.
Exact mathematical passage
T_{\beta\alpha}
\GammaNotation table · 151 passages
Search aliases: Γ Gamma
Notation table · atlas-pages.tex, line 30
Γ,z · Reflection coefficient and normalized impedance · Coefficients that can be linearly interchanged
Exact mathematical passage
\Gamma,z
\gammaNotation table · 245 passages
Search aliases: γ gamma
Notation table · 06_models.tex, line 187
W(γ) · cumulative weight of a transfer word or graph walk
Exact mathematical passage
W(\gamma)
\DeltaNotation table · 370 passages
Search aliases: Δ Delta
Notation table · 06_models.tex, line 191
Δ · linear refinement defect for one step CT_f-T_cC
Exact mathematical passage
\Delta
\deltaNotation table · 285 passages
Search aliases: δ delta
Notation table · atlas-pages.tex, line 21
midrule S,δ_0 · Sample shift and unit impulse · A finite truncation of every ℓ^2 signal
Exact mathematical passage
S,\delta_0
\delta_0Notation table · 26 passages
Search aliases: δ_0 delta
Notation table · atlas-pages.tex, line 21
midrule S,δ_0 · Sample shift and unit impulse · A finite truncation of every ℓ^2 signal
Exact mathematical passage
S,\delta_0
\LambdaNotation table · 210 passages
Search aliases: Λ Lambda
Notation table · 01_orientation.tex, line 62
Λ_ G · Grammar algebra · The quotient path algebra 𝕜 Q/I.
Exact mathematical passage
\Lambda_{\mathfrak G}
\Lambda_{\mathfrak G}Notation table · 10 passages
Search aliases: Λ_ G Lambda mathfrak
Notation table · 01_orientation.tex, line 62
Λ_ G · Grammar algebra · The quotient path algebra 𝕜 Q/I.
Exact mathematical passage
\Lambda_{\mathfrak G}
\piNotation table · 410 passages
Search aliases: π pi
Notation table · 06_models.tex, line 190
s_π,Λ(E) · separation weight of a rational projector on the chosen lattice, when a clearing exponent exists
Exact mathematical passage
s_{\pi,\Lambda}(E)
\tauNotation table · 247 passages
Search aliases: τ tau
Notation table · Paper_V_Traditional_Final.tex, line 1716
H_τ · the step packet 1_[τ,1]
Exact mathematical passage
H_\tau
\phiNotation table · 59 passages
Search aliases: ϕ phi
Notation table · Paper_V_Traditional_Final.tex, line 1717
ϕ_m,τ · the truncated power packet (t-τ)_+^m/m!
Exact mathematical passage
\phi_{m,\tau}
\phi_{m,\tau}Notation table · 8 passages
Search aliases: ϕ_m,τ phi tau
Notation table · Paper_V_Traditional_Final.tex, line 1717
ϕ_m,τ · the truncated power packet (t-τ)_+^m/m!
Exact mathematical passage
\phi_{m,\tau}
\ellNotation table · 346 passages
Search aliases: ℓ ell
Notation table · 06_models.tex, line 178
P_ℓ,α · packet projector at refinement level ℓ
Exact mathematical passage
P_{\ell,\alpha}
\ell^2Notation table · 36 passages
Search aliases: ℓ^2 ell
Notation table · Paper_V_Traditional_Final.tex, line 1503
[source reference] · HilbertPackets · Complex ℓ^2 impulse expansion, finite-support identity, unitary shifts, and closed shift-generated span.
Exact mathematical passage
\ell^2
\toNotation table · 386 passages
Search aliases: → to
Notation table · 01_orientation.tex, line 71
T_βα · Transfer block · The source and target block P_β TP_α:X_α→ X_β.
Exact mathematical passage
P_\beta TP_\alpha:X_\alpha\to X_\beta
\mapstoNotation table · 50 passages
Search aliases: ↦ mapsto
Notation table · 01_orientation.tex, line 69
A_ P · Analysis · The map x↦(P_α x)_α.
Exact mathematical passage
x\mapsto(P_\alpha x)_\alpha
\sumNotation table · 1172 passages
Search aliases: ∑ sum
Notation table · 01_orientation.tex, line 70
S_ P · Synthesis · The map (x_α)↦∑_α x_α.
Exact mathematical passage
(x_\alpha)\mapsto\sum_\alpha x_\alpha
\sum_\alphaNotation table · 61 passages
Search aliases: ∑_α sum alpha
Notation table · 01_orientation.tex, line 70
S_ P · Synthesis · The map (x_α)↦∑_α x_α.
Exact mathematical passage
(x_\alpha)\mapsto\sum_\alpha x_\alpha
\leqNotation table · 505 passages
Search aliases: ≤ leq
Notation table · Paper_V_Traditional_Final.tex, line 1709
L≤ Z^d · a finite index sublattice
Exact mathematical passage
L\leq\ZZ^d
\subseteqNotation table · 147 passages
Search aliases: ⊆ subseteq
Notation table · 01_orientation.tex, line 64
S⊆ M · Seed family · The vectors from which the operator orbit is generated.
Exact mathematical passage
S\subseteq M
\mathfrak ADefinition passage · 13 passages
Search aliases: A mathfrak
Definition passage · Paper_IV_Traditional_Final.tex, line 787
Let A_N^+= C[ξ]/(ξ^N-1) and A_N^-= C[ξ]/(ξ^N+1). Every element has a unique representative of degree below N.
Exact mathematical passage
\mathfrak A_N^+=\CC[\xi]/(\xi^N-1)
aDefinition passage · 1740 passages
Search aliases: a
Definition passage: Typed operator grammar · 02_foundations.tex, line 48
(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a∈ Q_1).
Exact mathematical passage
X_a:X_{s(a)}\longrightarrow X_{t(a)} \qquad(a\in Q_1)
a^2Definition passage · 7 passages
Search aliases: a^2
Definition passage: Divided square · 03_nonlinear_coordinates.tex, line 665
[Divided square] A divided square on degree one elements is a map q satisfying q(x+y)=q(x)+q(y)+[x,y], q(ax)=a^2q(x), whenever the expressions are defined.
Exact mathematical passage
q(x+y)=q(x)+q(y)+[x,y], \qquad q(ax)=a^2q(x),
A^{m-1}Definition passage · 2 passages
Search aliases: A^m-1
Definition passage: Krylov space · 01_manuscript.tex, line 1096
[Krylov space] For a linear operator A, a vector v, and a positive integer m, define K_m(A,v) = span {v,Av,…,A^m-1v}.
Exact mathematical passage
\Kcal_m(A,v) = \operatorname{span} \{v,Av,\ldots,A^{m-1}v\}.
A^rDefinition passage · 3 passages
Search aliases: A^r
Definition passage: Grade respecting operation · 03_nonlinear_coordinates.tex, line 573
[Grade respecting operation] An r multilinear map μ:A^r→ A respects the grading when μ(A_g_1,…,A_g_r) ⊆ A_g_1+⋯+g_r.
Exact mathematical passage
\mu:A^r\to A
A^{\oplus r}Definition passage · 9 passages
Search aliases: A^⊕ r oplus
Definition passage: Seed evaluation map · 02_foundations.tex, line 218
ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.
Exact mathematical passage
\varepsilon_S:A^{\oplus r}\longrightarrow M, \qquad (a_1,\ldots,a_r)\longmapsto\sum_{j=1}^{r}a_js_j.
a_0Definition passage · 40 passages
Search aliases: a_0
Definition passage: Connection transport along a path · 02_connections_obstructions.tex, line 266
[Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transport.ReversibleConnection.indexedPathTransport
Exact mathematical passage
p=(a_0,a_1,\ldots,a_n)
a_1Definition passage · 44 passages
Search aliases: a_1
Definition passage: Seed evaluation map · 02_foundations.tex, line 218
ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.
Exact mathematical passage
\varepsilon_S:A^{\oplus r}\longrightarrow M, \qquad (a_1,\ldots,a_r)\longmapsto\sum_{j=1}^{r}a_js_j.
\A_CDefinition passage · 5 passages
Search aliases: A_C A mathcal
Definition passage: Packet pure seed rank · 03_packets_and_graphs.tex, line 1020
[Packet pure seed rank] A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).
Exact mathematical passage
P_C:=\sum_{\alpha\in C}P_\alpha, \qquad X_C:=P_CX, \qquad \A_C:=P_C\A P_C,
A_{g_1}Definition passage · 1 passage
Search aliases: A_g_1
Definition passage: Grade respecting operation · 03_nonlinear_coordinates.tex, line 574
[Grade respecting operation] An r multilinear map μ:A^r→ A respects the grading when μ(A_g_1,…,A_g_r) ⊆ A_g_1+⋯+g_r.
Exact mathematical passage
\mu(A_{g_1},\ldots,A_{g_r}) \subseteq A_{g_1+\cdots+g_r}.
A_{g_1+\cdots+g_r}Definition passage · 1 passage
Search aliases: A_g_1+⋯+g_r cdots
Definition passage: Grade respecting operation · 03_nonlinear_coordinates.tex, line 574
[Grade respecting operation] An r multilinear map μ:A^r→ A respects the grading when μ(A_g_1,…,A_g_r) ⊆ A_g_1+⋯+g_r.
Exact mathematical passage
\mu(A_{g_1},\ldots,A_{g_r}) \subseteq A_{g_1+\cdots+g_r}.
A_{g_r}Definition passage · 1 passage
Search aliases: A_g_r
Definition passage: Grade respecting operation · 03_nonlinear_coordinates.tex, line 574
[Grade respecting operation] An r multilinear map μ:A^r→ A respects the grading when μ(A_g_1,…,A_g_r) ⊆ A_g_1+⋯+g_r.
Exact mathematical passage
\mu(A_{g_1},\ldots,A_{g_r}) \subseteq A_{g_1+\cdots+g_r}.
a_jDefinition passage · 24 passages
Search aliases: a_j
Definition passage: Seed evaluation map · 02_foundations.tex, line 218
ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.
Exact mathematical passage
\varepsilon_S:A^{\oplus r}\longrightarrow M, \qquad (a_1,\ldots,a_r)\longmapsto\sum_{j=1}^{r}a_js_j.
a_nDefinition passage · 62 passages
Search aliases: a_n
Definition passage: Unconditional packet sum space · 03_packets_and_graphs.tex, line 359
[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .
Exact mathematical passage
\|x\|_u := \sup_{F\Subset\mathbb N} \sup_{|a_n|\leq1} \left\|\sum_{n\in F}a_nx_n\right\|.
a_{n-1}Definition passage · 4 passages
Search aliases: a_n-1
Definition passage: Connection transport along a path · 02_connections_obstructions.tex, line 267
[Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transport.ReversibleConnection.indexedPathTransport
Exact mathematical passage
\Gamma_p = \Gamma_{a_n\leftarrow a_{n-1}}\cdots \Gamma_{a_1\leftarrow a_0}, \qquad \Gamma_{(a_0)}=\id_{C_{a_0}}.
\mathfrak A_N^+Definition passage · 3 passages
Search aliases: A_N^+ mathfrak
Definition passage · Paper_IV_Traditional_Final.tex, line 787
Let A_N^+= C[ξ]/(ξ^N-1) and A_N^-= C[ξ]/(ξ^N+1). Every element has a unique representative of degree below N.
Exact mathematical passage
\mathfrak A_N^+=\CC[\xi]/(\xi^N-1)
\mathfrak A_N^-Definition passage · 3 passages
Search aliases: A_N^- mathfrak
Definition passage · Paper_IV_Traditional_Final.tex, line 791
Let A_N^+= C[ξ]/(ξ^N-1) and A_N^-= C[ξ]/(ξ^N+1). Every element has a unique representative of degree below N.
Exact mathematical passage
\mathfrak A_N^-=\CC[\xi]/(\xi^N+1).
a_qDefinition passage · 24 passages
Search aliases: a_q
Definition passage: Sobolev step packet class · Paper_V_Traditional_Final.tex, line 2522
[Sobolev step packet class] Fix r≥1, 1≤ p≤∞, and an integer Q≥0. The class F_r,p(Q) consists of functions f=s+∑_q=1^Qa_qϕ_0,τ_q with s∈ W^r,p(0,1) and 0<τ_1<⋯<τ_Q<1. No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.
Exact mathematical passage
f=s+\sum_{q=1}^{Q}a_q\phi_{0,\tau_q}
a_rDefinition passage · 7 passages
Search aliases: a_r
Definition passage: Seed evaluation map · 02_foundations.tex, line 218
ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.
Exact mathematical passage
\varepsilon_S:A^{\oplus r}\longrightarrow M, \qquad (a_1,\ldots,a_r)\longmapsto\sum_{j=1}^{r}a_js_j.
A_{v_1v_0}Definition passage · 1 passage
Search aliases: A_v_1v_0
Definition passage: Path transport · Paper_XI_Traditional_Final.tex, line 1621
[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.
Exact mathematical passage
A_\gamma=A_{v_m v_{m-1}}\cdots A_{v_1v_0}, \qquad B_\gamma=B_{v_m v_{m-1}}\cdots B_{v_1v_0}.
A_{v_m v_{m-1}}Definition passage · 1 passage
Search aliases: A_v_m v_m-1
Definition passage: Path transport · Paper_XI_Traditional_Final.tex, line 1621
[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.
Exact mathematical passage
A_\gamma=A_{v_m v_{m-1}}\cdots A_{v_1v_0}, \qquad B_\gamma=B_{v_m v_{m-1}}\cdots B_{v_1v_0}.
A_\gammaDefinition passage · 11 passages
Search aliases: A_γ gamma
Definition passage: Path transport · Paper_XI_Traditional_Final.tex, line 1621
[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.
Exact mathematical passage
A_\gamma=A_{v_m v_{m-1}}\cdots A_{v_1v_0}, \qquad B_\gamma=B_{v_m v_{m-1}}\cdots B_{v_1v_0}.
\absDefinition passage · 70 passages
Search aliases: abs
Definition passage: Normalized Gauss phase · Paper_XI_Traditional_Final.tex, line 746
[Normalized Gauss phase] For nontrivial χ, define γ(χ,ψ) = q^-1/2G(χ,ψ). Then absγ(χ,ψ)=1.
Exact mathematical passage
\abs{\gamma(\chi,\psi)}=1
\mathrm{abs}Definition passage · 12 passages
Search aliases: abs mathrm
Definition passage: Realization functor and preservation profile · 04_transport.tex, line 384
[Realization functor and preservation profile] Let SeedSys_abs be an abstract category of seed systems and let C_D be a domain category. A realization functor is a functor Ev_D:SeedSys_abs⟶ C_D with a declared preservation profile specifying which of the following structures are preserved: 0, I, finite sums, composition, biproducts,
Exact mathematical passage
\mathbf{SeedSys}_{\mathrm{abs}}
\Acal_{\Pcal}Definition passage · 14 passages
Search aliases: Acal_Pcal A_ P Acal Pcal mathcal mathcal
Definition passage: Packet system · 02_coordinates.tex, line 25
ket_system Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.
Exact mathematical passage
\Acal_{\Pcal}x=(P_\alpha x)_\alpha, \qquad \Scal_{\Pcal}(x_\alpha)_\alpha=\sum_\alpha x_\alpha.
\Acal_{\Qcal}Definition passage · 11 passages
Search aliases: Acal_Qcal A_ Q Acal Qcal mathcal mathcal
Definition passage: Coordinate change across levels · 02_coordinates.tex, line 84
[Coordinate change across levels] For two packet systems P and Q on X, define J_ Q← P = A_ Q S_ P:X^ P⟶ X^ Q and C_ P← Q = A_ P S_ Q:X^ Q⟶ X^ P. When Q refines P, the first map resolves each coarse coordinate into its children and the second adds the children back to their parent.
Exact mathematical passage
J_{\Qcal\leftarrow\Pcal} = \Acal_{\Qcal}\Scal_{\Pcal}:X^{\Pcal}\longrightarrow X^{\Qcal}
\algDefinition passage · 2 passages
Search aliases: alg operatorname
Definition passage: Transfer graph of a family generating the algebra · 03_packets_and_graphs.tex, line 919
[Transfer graph of a family generating the algebra] Let P=(P_α)_α∈ I be a finite packet system on X with every P_α∈ A. Let G⊆ A satisfy A=alg_𝕜({P_α:α∈ I}∪ G), where alg_𝕜(E) denotes the unital subalgebra over 𝕜 generated by E. The transfer graph Γ( G, P) has vertex set I and an arrow α⟶β when P_β GP_α≠0 for at least one G∈ G. Reachability includes paths of length zero.
Exact mathematical passage
\A=\alg_{\bbk}\bigl(\{P_\alpha:\alpha\in I\}\cup\mathcal G\bigr),
\alg_{\bbk}Definition passage · 2 passages
Search aliases: alg_bbk alg_𝕜 alg bbk operatorname Bbbk
Definition passage: Transfer graph of a family generating the algebra · 03_packets_and_graphs.tex, line 919
[Transfer graph of a family generating the algebra] Let P=(P_α)_α∈ I be a finite packet system on X with every P_α∈ A. Let G⊆ A satisfy A=alg_𝕜({P_α:α∈ I}∪ G), where alg_𝕜(E) denotes the unital subalgebra over 𝕜 generated by E. The transfer graph Γ( G, P) has vertex set I and an arrow α⟶β when P_β GP_α≠0 for at least one G∈ G. Reachability includes paths of length zero.
Exact mathematical passage
\A=\alg_{\bbk}\bigl(\{P_\alpha:\alpha\in I\}\cup\mathcal G\bigr),
\AnnDefinition passage · 9 passages
Search aliases: Ann operatorname
Definition passage: Support and near faithfulness · Paper_XII_Traditional_Final.tex, line 581
[Support and near faithfulness] Let R be a commutative ring and let M be an R module. Its support is Supp_R(M) = { p∈Spec R:M_ p≠0}. If R is Noetherian and M is finite, then M is nearly faithful when Ann_R(M) is a nilpotent ideal.
Exact mathematical passage
\Ann_R(M)
\Ann_RDefinition passage · 9 passages
Search aliases: Ann_R Ann operatorname
Definition passage: Support and near faithfulness · Paper_XII_Traditional_Final.tex, line 581
[Support and near faithfulness] Let R be a commutative ring and let M be an R module. Its support is Supp_R(M) = { p∈Spec R:M_ p≠0}. If R is Noetherian and M is finite, then M is nearly faithful when Ann_R(M) is a nilpotent ideal.
Exact mathematical passage
\Ann_R(M)
\ArfDefinition passage · 5 passages
Search aliases: Arf operatorname
Definition passage: Arf invariant · Paper_XI_Traditional_Final.tex, line 1489
[Arf invariant] Choose a symplectic basis (e_i,f_i)_i=1^r. Define Arf(q) = ∑_i=1^rq(e_i)q(f_i) ∈ F_2.
Exact mathematical passage
\Arf(q) = \sum_{i=1}^{r}q(e_i)q(f_i) \in\FF_2.
\operatorname{Aut}Definition passage · 5 passages
Search aliases: Aut operatorname
Definition passage: Noncanonical connection normalized at each chart · 02_connections_obstructions.tex, line 13
[Noncanonical connection normalized at each chart] A minimal connection is a family Γ_b← a∈Aut_R(M) (a,b∈ B) with only Γ_a← a=id_M. Neither inversion on reversed edges nor the cocycle law is a field of the structure. statusleanthreeKomorebi.PaperIII.Transport.Connection
Exact mathematical passage
\Gamma_{b\leftarrow a}\in\operatorname{Aut}_R(M) \qquad(a,b\in B)
\operatorname{Aut}_RDefinition passage · 3 passages
Search aliases: Aut_R operatorname
Definition passage: Noncanonical connection normalized at each chart · 02_connections_obstructions.tex, line 13
[Noncanonical connection normalized at each chart] A minimal connection is a family Γ_b← a∈Aut_R(M) (a,b∈ B) with only Γ_a← a=id_M. Neither inversion on reversed edges nor the cocycle law is a field of the structure. statusleanthreeKomorebi.PaperIII.Transport.Connection
Exact mathematical passage
\Gamma_{b\leftarrow a}\in\operatorname{Aut}_R(M) \qquad(a,b\in B)
BDefinition passage · 760 passages
Search aliases: B
Definition passage: Packet system surjective along edges · 03_packets_and_graphs.tex, line 1058
[Packet system surjective along edges] The module resolved into packets is surjective along edges relative to Γ( G, P) when, for every graph edge α→β, there exists B∈ A such that the pure block P_β BP_α:X_α↠ X_β is surjective.
Exact mathematical passage
B\in\A
\BDefinition passage · 13 passages
Search aliases: B mathcal
Definition passage: Packet compatible operator morphism · 04_transport.tex, line 50
[Packet compatible operator morphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):= ∑_σ(α)=jP_α. An empty fiber is interpreted as the zero projector.
Exact mathematical passage
(Y,\B,\Qcal)
\mathcal BDefinition passage · 36 passages
Search aliases: B mathcal
Definition passage: Packet blocks · Paper_VI_Traditional_Final.tex, line 2063
[Packet blocks] For B∈ B(X), define B_βα=P_β BP_α. If A∈ B(X) and T(t)= e^tA, write T_βα(t)=P_β T(t)P_α.
Exact mathematical passage
B\in\mathcal B(X)
bDefinition passage · 849 passages
Search aliases: b
Definition passage: Packet graph and scalar edge system · 01_finite_coordinates.tex, line 66
[Packet graph and scalar edge system] For a,b∈ V and r∈ R, define packetMatrix(a,b;r)_yx = r, · y=b and x=a,
Exact mathematical passage
a,b\in V
B^*Definition passage · 27 passages
Search aliases: B^*
Definition passage: Feshbach map and self energy · 02_strengthened.tex, line 1502
[Feshbach map and self energy] For z∉Spec D, define Σ(z)=B(z-D)^-1B^* and F_P(z)=z-A-Σ(z).
Exact mathematical passage
\Sigma(z)=B(z-D)^{-1}B^*
B_{v_1v_0}Definition passage · 1 passage
Search aliases: B_v_1v_0
Definition passage: Path transport · Paper_XI_Traditional_Final.tex, line 1621
[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.
Exact mathematical passage
A_\gamma=A_{v_m v_{m-1}}\cdots A_{v_1v_0}, \qquad B_\gamma=B_{v_m v_{m-1}}\cdots B_{v_1v_0}.
B_{v_m v_{m-1}}Definition passage · 1 passage
Search aliases: B_v_m v_m-1
Definition passage: Path transport · Paper_XI_Traditional_Final.tex, line 1621
[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.
Exact mathematical passage
A_\gamma=A_{v_m v_{m-1}}\cdots A_{v_1v_0}, \qquad B_\gamma=B_{v_m v_{m-1}}\cdots B_{v_1v_0}.
B_{\beta\alpha}Definition passage · 14 passages
Search aliases: B_βα beta alpha
Definition passage: Packet blocks · Paper_VI_Traditional_Final.tex, line 2064
[Packet blocks] For B∈ B(X), define B_βα=P_β BP_α. If A∈ B(X) and T(t)= e^tA, write T_βα(t)=P_β T(t)P_α.
Exact mathematical passage
B_{\beta\alpha}=P_\beta BP_\alpha.
B_\gammaDefinition passage · 2 passages
Search aliases: B_γ gamma
Definition passage: Path transport · Paper_XI_Traditional_Final.tex, line 1621
[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.
Exact mathematical passage
A_\gamma=A_{v_m v_{m-1}}\cdots A_{v_1v_0}, \qquad B_\gamma=B_{v_m v_{m-1}}\cdots B_{v_1v_0}.
b_\rhoDefinition passage · 5 passages
Search aliases: b_ρ rho
Definition passage: One break symplectic envelope · Paper_XI_Traditional_Final.tex, line 1090
[One break symplectic envelope] The one break symplectic envelope is V_ρ=A/rad b_ρ.
Exact mathematical passage
\boxed{ V_\rho=A/\rad b_\rho. }
\bar xDefinition passage · 32 passages
Search aliases: bar x bar
Definition passage: Residual seed system · Paper_X_Block27_Amended.tex, line 162
[Residual seed system] For bar x∈ M(k), define the residual fiber M_bar x(A) = {x∈ M(A):red_A(x)=bar x}. The point bar x is the residual root. The family of all nonempty fibers is the residual special fiber, and the formal deformation problems above them form the residual forest.
Exact mathematical passage
\bar x\in\mathcal M(k)
\bar\rhoDefinition passage · 16 passages
Search aliases: barρ bar rho
Definition passage: Admissible residual transport · Paper_X_Block27_Amended.tex, line 652
[Admissible residual transport] An admissible transport from barρ to barρ' consists of T:barρ⟼barρ' together with natural maps T_A:Def_barρ(A)⟶Def_barρ'(A) for every coefficient algebra A, compatible with base change and reduction. When the transport is reversible, it is an isomorphism in the residual transport groupoid. One-way operations give a directed category.
Exact mathematical passage
\bar\rho
\bar\rho'Definition passage · 3 passages
Search aliases: barρ' bar rho
Definition passage: Admissible residual transport · Paper_X_Block27_Amended.tex, line 652
[Admissible residual transport] An admissible transport from barρ to barρ' consists of T:barρ⟼barρ' together with natural maps T_A:Def_barρ(A)⟶Def_barρ'(A) for every coefficient algebra A, compatible with base change and reduction. When the transport is reversible, it is an isomorphism in the residual transport groupoid. One-way operations give a directed category.
Exact mathematical passage
\bar\rho'
\BcalDefinition passage · 74 passages
Search aliases: Bcal B mathcal
Definition passage: Complete multiplicative filtration · 03_weighted.tex, line 6
[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
\Bcal
\Bcal_{ab}Definition passage · 1 passage
Search aliases: Bcal_ab B_ab Bcal mathcal
Definition passage: Two sided operator packet · 02_strengthened.tex, line 949
[Two sided operator packet] For a complete orthogonal family (P_a), define P_ab(X)=P_aXP_b. The range B_ab=P_aEnd( H)P_b is the operator packet with output address a and input address b.
Exact mathematical passage
\Bcal_{ab}=P_a\End(\Hcal)P_b
\Bcal_{N,j}Definition passage · 10 passages
Search aliases: Bcal_N,j B_N,j Bcal mathcal
Definition passage · Paper_IV_Traditional_Final.tex, line 459
For λ∈ C and 0≤ j<N, define E_N,j(λ z) = B_N,j(e^λ z) = ∑_n≥0frac(λ z)^Nn+j(Nn+j)!.
Exact mathematical passage
\Efun_{N,j}(\lambda z) = \Bcal_{N,j}(e^{\lambda z}) = \sum_{n\geq0}\frac{(\lambda z)^{Nn+j}}{(Nn+j)!}.
\operatorname{binary}Definition passage · 1 passage
Search aliases: binary operatorname
Definition passage: Finite grammar for operation trees · 02_connections_obstructions.tex, line 772
[Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the finite formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No algebraic, continuity, or differentiability law is imposed on these semantics. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.OperationTree; Kom…
Exact mathematical passage
\begin{aligned} t ::={}&\operatorname{input}(i) \mid\operatorname{constant}(k) \mid\operatorname{unary}(u,t)\\ &\mid\operatorname{binary}(b,t_1,t_2). \end{aligned}
\bmodDefinition passage · 73 passages
Search aliases: bmod
Definition passage · Paper_IV_Traditional_Final.tex, line 1801
For θ∈ R, define h_N(θ) = max_0≤ k<N cos(θ+frac(2k+1)πN). The tie directions are T_N = {frac2π kN:0≤ k<N} bmod 2π.
Exact mathematical passage
\mathfrak T_N = \left\{\frac{2\pi k}{N}:0\leq k<N\right\} \quad\bmod 2\pi.
\mathrm{boundary}Definition passage · 1 passage
Search aliases: boundary mathrm
Definition passage: Canonical theorem record · 02_foundations.tex, line 17
[Canonical theorem record] A canonical theorem record is the tuple (ID,status,hypotheses,statement, dependencies,proof,exports,boundary). A later chapter cites the stable record ID. A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this chapter.
Exact mathematical passage
(\mathrm{ID},\mathrm{status},\mathrm{hypotheses},\mathrm{statement}, \mathrm{dependencies},\mathrm{proof},\mathrm{exports},\mathrm{boundary}).
\bulletDefinition passage · 21 passages
Search aliases: bullet
Definition passage: Mapping cone · 03_nonlinear_coordinates.tex, line 1124
[Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).
Exact mathematical passage
D:C^\bullet\to E^\bullet
\bytesDefinition passage · 1 passage
Search aliases: bytes operatorname
Definition passage: Arithmetic intensity · 01_manuscript.tex, line 389
[Arithmetic intensity] When the modeled byte traffic is strictly positive, the modeled arithmetic intensity is I = fracflopbytes.
Exact mathematical passage
\Ical = \frac{\flop}{\bytes}.
\mathbf CDefinition passage · 3 passages
Search aliases: C mathbf
Definition passage: Realization functor and preservation profile · 04_transport.tex, line 384
[Realization functor and preservation profile] Let SeedSys_abs be an abstract category of seed systems and let C_D be a domain category. A realization functor is a functor Ev_D:SeedSys_abs⟶ C_D with a declared preservation profile specifying which of the following structures are preserved: 0, I, finite sums, composition, biproducts,
Exact mathematical passage
\mathbf C_D
\mathfrak cDefinition passage · 6 passages
Search aliases: c mathfrak
Definition passage: Universal closure obstruction · 02_strengthened.tex, line 2182
[Universal closure obstruction] Let q_ c:End( H)⟶End( H)/ c be the quotient map. Define Ob_ab^rm cl = q_ c([C_a,C_b]).
Exact mathematical passage
q_{\mathfrak c}:\End(\Hcal)\longrightarrow\End(\Hcal)/\mathfrak c
C^\bulletDefinition passage · 16 passages
Search aliases: C^bullet bullet
Definition passage: Mapping cone · 03_nonlinear_coordinates.tex, line 1124
[Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).
Exact mathematical passage
D:C^\bullet\to E^\bullet
C^{n+1}Definition passage · 3 passages
Search aliases: C^n+1
Definition passage: Mapping cone · 03_nonlinear_coordinates.tex, line 1125
[Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).
Exact mathematical passage
\Cone(D)^n=E^n\oplus C^{n+1}
C_aDefinition passage · 43 passages
Search aliases: C_a
Definition passage: Normalized reversible connection · 02_connections_obstructions.tex, line 119
[Normalized reversible connection] A normalized reversible connection is a family of linear equivalences Γ_b← a:C_axrightarrow ∼ C_b (a,b∈ B) such that Γ_a← a=id, Γ_a← b=Γ_b← a^-1. No cocycle equation is included in the structure. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection
Exact mathematical passage
\Gamma_{b\leftarrow a}:C_a\xrightarrow{\;\sim\;}C_b \qquad(a,b\in B)
C_{a_0}Definition passage · 1 passage
Search aliases: C_a_0
Definition passage: Connection transport along a path · 02_connections_obstructions.tex, line 267
[Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transport.ReversibleConnection.indexedPathTransport
Exact mathematical passage
\Gamma_p = \Gamma_{a_n\leftarrow a_{n-1}}\cdots \Gamma_{a_1\leftarrow a_0}, \qquad \Gamma_{(a_0)}=\id_{C_{a_0}}.
C_bDefinition passage · 12 passages
Search aliases: C_b
Definition passage: Normalized reversible connection · 02_connections_obstructions.tex, line 119
[Normalized reversible connection] A normalized reversible connection is a family of linear equivalences Γ_b← a:C_axrightarrow ∼ C_b (a,b∈ B) such that Γ_a← a=id, Γ_a← b=Γ_b← a^-1. No cocycle equation is included in the structure. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection
Exact mathematical passage
\Gamma_{b\leftarrow a}:C_a\xrightarrow{\;\sim\;}C_b \qquad(a,b\in B)
C_cDefinition passage · 10 passages
Search aliases: C_c
Definition passage: Cocycle defect · 02_connections_obstructions.tex, line 176
[Cocycle defect] For a,b,c∈ B, define the additive cocycle defect Δ_c,b,a = Γ_c← bΓ_b← a -Γ_c← a ∈Hom_R(C_a,C_c). statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.cocycleDefect
Exact mathematical passage
\Delta_{c,b,a} = \Gamma_{c\leftarrow b}\Gamma_{b\leftarrow a} -\Gamma_{c\leftarrow a} \in\Hom_R(C_a,C_c).
\mathbf C_DDefinition passage · 2 passages
Search aliases: C_D mathbf
Definition passage: Realization functor and preservation profile · 04_transport.tex, line 384
[Realization functor and preservation profile] Let SeedSys_abs be an abstract category of seed systems and let C_D be a domain category. A realization functor is a functor Ev_D:SeedSys_abs⟶ C_D with a declared preservation profile specifying which of the following structures are preserved: 0, I, finite sums, composition, biproducts,
Exact mathematical passage
\mathbf C_D
c_{e,k}Definition passage · 1 passage
Search aliases: c_e,k
Definition passage: Discrete precision option · 01_manuscript.tex, line 1422
[Discrete precision option] For edge e, a discrete precision option k has a measured or modeled cost c_e,k and a nonnegative integer error weight w_e,k. Exactly one option must be selected on every edge.
Exact mathematical passage
c_{e,k}
C_jDefinition passage · 25 passages
Search aliases: C_j
Definition passage: Projective limit space · Paper_VI_Traditional_Final.tex, line 5292
[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.
Exact mathematical passage
(X_j,C_j)
C_{m,n}Definition passage · 8 passages
Search aliases: C_m,n
Definition passage: Reconstructed fine evolution · Paper_VI_Traditional_Final.tex, line 2537
[Reconstructed fine evolution] For a semigroup T_n(t) on X_n, define the evolution seen from level m by U_m,n(t)=C_m,nT_n(t)R_n,m.
Exact mathematical passage
U_{m,n}(t)=C_{m,n}T_n(t)R_{n,m}.
C_{N,j}Definition passage · 53 passages
Search aliases: C_N,j
Definition passage · Paper_IV_Traditional_Final.tex, line 1404
or 0≤ j<N, define Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)!. Then E_N,j(z) = fracz^jj!Φ_N,j(z^N). Define the residue-section reciprocal function C_N,j(z) = Φ_N,j(z^N)^-1 = fracz^j/j! E_N,j(z). For j>0, the last quotient at z=0 is interpreted by its removable continuation, namely the first expression in the display. Write C_N,j(z) = ∑_n≥0C_N,j(n)fracz^Nn(Nn)!.
Exact mathematical passage
\Cfun_{N,j}(z) = \sum_{n\geq0}C_{N,j}(n)\frac{z^{Nn}}{(Nn)!}.
C_{N,j,n}Definition passage · 7 passages
Search aliases: C_N,j,n
Definition passage · Paper_IV_Traditional_Final.tex, line 1439
For r∈ C, define formal coefficients C_N,j,n(r) by C_N,j(z)^r = ∑_n≥0C_N,j,n(r)fracz^Nn(Nn)!. On a simply connected region containing 0 and avoiding the zeros of Φ_N,j(z^N), interpret this power as e^rL, where L is the holomorphic logarithm of C_N,j normalized by L(0)=0.
Exact mathematical passage
C_{N,j,n}(r)
C_{\Pcal\leftarrow\Qcal}Definition passage · 6 passages
Search aliases: C_Pcal←Qcal C_ P← Q Pcal leftarrow Qcal mathcal leftarrow mathcal
Definition passage: Coordinate change across levels · 02_coordinates.tex, line 90
[Coordinate change across levels] For two packet systems P and Q on X, define J_ Q← P = A_ Q S_ P:X^ P⟶ X^ Q and C_ P← Q = A_ P S_ Q:X^ Q⟶ X^ P. When Q refines P, the first map resolves each coarse coordinate into its children and the second adds the children back to their parent.
Exact mathematical passage
C_{\Pcal\leftarrow\Qcal} = \Acal_{\Pcal}\Scal_{\Qcal}:X^{\Qcal}\longrightarrow X^{\Pcal}.
C_XDefinition passage · 6 passages
Search aliases: C_X
Definition passage: Nonlinear refinement defect · 03_nonlinear_coordinates.tex, line 737
[Nonlinear refinement defect] The pointwise defect of F is Δ_F(x) = C_YF_f(x)-F_c(C_Xx). The two models are exactly compatible when Δ_F vanishes everywhere.
Exact mathematical passage
\Delta_F(x) = C_YF_f(x)-F_c(C_Xx).
C_YDefinition passage · 6 passages
Search aliases: C_Y
Definition passage: Nonlinear refinement defect · 03_nonlinear_coordinates.tex, line 737
[Nonlinear refinement defect] The pointwise defect of F is Δ_F(x) = C_YF_f(x)-F_c(C_Xx). The two models are exactly compatible when Δ_F vanishes everywhere.
Exact mathematical passage
\Delta_F(x) = C_YF_f(x)-F_c(C_Xx).
C_\GammaDefinition passage · 6 passages
Search aliases: C_Γ Gamma
Definition passage: Linear tube bound · Paper_V_Traditional_Final.tex, line 2833
[Linear tube bound] A boundary Γ has a linear tube bound if there are constants C_Γ and δ_0>0 such that μ(N_δ(Γ)) ≤ C_Γδ for 0<δ<δ_0.
Exact mathematical passage
C_\Gamma
c_\rhoDefinition passage · 2 passages
Search aliases: c_ρ rho
Definition passage: Clifford cocycle class · Paper_XI_Traditional_Final.tex, line 1065
[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.
Exact mathematical passage
[c_\rho]\in H^2(A,\CC^{\times})
C_{\ell,r}Definition passage · 1 passage
Search aliases: C_ℓ,r ell
Definition passage: Finite quantum refinement tower · 02_strengthened.tex, line 2348
[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is recorded by the corresponding operator norm defects.
Exact mathematical passage
C_{\ell,r}
C_\inftyDefinition passage · 6 passages
Search aliases: C_∞ infty
Definition passage: Derived patching datum · Paper_XII_Traditional_Final.tex, line 864
[Derived patching datum] A derived patching datum consists of a Noetherian ring R_∞, an ideal a, a perfect complex C_∞ over R_∞, and identifications R≅ R_∞/ a, C≃ C_∞ mathbinmathop⊗limits_R_∞^ LR.
Exact mathematical passage
C_\infty
\cA_bDefinition passage · 3 passages
Search aliases: cA_b A_b cA mathcal
Definition passage: Canonical exact coordinate transport · 01_finite_coordinates.tex, line 472
[Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRefinementEquiv
Exact mathematical passage
J^X_{b\leftarrow a} = \cA_b\cS_a: \Coord(\cP^a)\longrightarrow\Coord(\cP^b).
\calHDefinition passage · 4 passages
Search aliases: calH H mathcal
Definition passage: Schrödinger operators · Paper_XI_Traditional_Final.tex, line 811
[Schrödinger operators] On H= C^X, define (ρ(x,ξ,z)f)(u) = ψ(z+ξ(u)+frac12ξ(x))f(u+x).
Exact mathematical passage
\calH=\CC^X
\calXDefinition passage · 4 passages
Search aliases: calX X mathcal
Definition passage: One break Clifford datum · Paper_XI_Traditional_Final.tex, line 1012
[One break Clifford datum] A one break Clifford datum consists of finite groups and representations Ntriangleleft G, A=G/N, Π:G→GL( X), ρ:N→GL(W), with the following properties.
Exact mathematical passage
N\triangleleft G, \qquad A=G/N, \qquad \Pi:G\to\GL(\calX), \qquad \rho:N\to\GL(W),
\CCDefinition passage · 99 passages
Search aliases: CC C mathbb
Definition passage · Paper_IV_Traditional_Final.tex, line 458
For λ∈ C and 0≤ j<N, define E_N,j(λ z) = B_N,j(e^λ z) = ∑_n≥0frac(λ z)^Nn+j(Nn+j)!.
Exact mathematical passage
\lambda\in\CC
\CC^NDefinition passage · 1 passage
Search aliases: CC^N C^N CC mathbb
Definition passage · Paper_IV_Traditional_Final.tex, line 824
Let J_N be the signed cyclic shift on the standard basis e_0,…,e_N-1 of C^N, defined by J_Ne_j=e_j+1 (0≤ j<N-1), J_Ne_N-1=-e_0. Then J_N^N=-I.
Exact mathematical passage
\CC^N
\CC^{n\times r}Definition passage · 3 passages
Search aliases: CC^n× r C^n× r CC times mathbb times
Definition passage: Block diagonal plus low rank operator · 01_manuscript.tex, line 307
[Block diagonal plus low rank operator] A block diagonal plus low rank operator has the form A=D+UV^*, where D=⊕_α∈ ID_α, U,V∈ C^n× r. The abbreviation BDLR will be used below.
Exact mathematical passage
D=\bigoplus_{\alpha\in I}D_\alpha, \qquad U,V\in\CC^{n\times r}.
\CC^XDefinition passage · 2 passages
Search aliases: CC^X C^X CC mathbb
Definition passage: Schrödinger operators · Paper_XI_Traditional_Final.tex, line 811
[Schrödinger operators] On H= C^X, define (ρ(x,ξ,z)f)(u) = ψ(z+ξ(u)+frac12ξ(x))f(u+x).
Exact mathematical passage
\calH=\CC^X
\CC^{\times}Definition passage · 9 passages
Search aliases: CC^× C^× CC times mathbb times
Definition passage: Clifford cocycle class · Paper_XI_Traditional_Final.tex, line 1065
[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.
Exact mathematical passage
[c_\rho]\in H^2(A,\CC^{\times})
\CcalDefinition passage · 11 passages
Search aliases: Ccal C mathcal
Definition passage: Code compression obstruction · 02_strengthened.tex, line 1277
[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.
Exact mathematical passage
q_P:\End(\Ccal)\longrightarrow\End(\Ccal)/(\CC I_{\Ccal})
\CfunDefinition passage · 15 passages
Search aliases: Cfun C mathsf
Definition passage · Paper_IV_Traditional_Final.tex, line 1394
For 0≤ j<N, define Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)!. Then E_N,j(z) = fracz^jj!Φ_N,j(z^N). Define the residue-section reciprocal function C_N,j(z) = Φ_N,j(z^N)^-1 = fracz^j/j! E_N,j(z). For j>0, the last quotient at z=0 is interpreted by its removable continuation, namely the first expression in the display. Write C_N,j(z) = ∑_n≥0C_N,j(n)fracz^Nn(Nn)!.
Exact mathematical passage
\Cfun_{N,j}(z) = \Phi_{N,j}(z^N)^{-1} = \frac{z^j/j!}{\Efun_{N,j}(z)}.
\Cfun_{N,j}Definition passage · 11 passages
Search aliases: Cfun_N,j C_N,j Cfun mathsf
Definition passage · Paper_IV_Traditional_Final.tex, line 1394
For 0≤ j<N, define Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)!. Then E_N,j(z) = fracz^jj!Φ_N,j(z^N). Define the residue-section reciprocal function C_N,j(z) = Φ_N,j(z^N)^-1 = fracz^j/j! E_N,j(z). For j>0, the last quotient at z=0 is interpreted by its removable continuation, namely the first expression in the display. Write C_N,j(z) = ∑_n≥0C_N,j(n)fracz^Nn(Nn)!.
Exact mathematical passage
\Cfun_{N,j}(z) = \Phi_{N,j}(z^N)^{-1} = \frac{z^j/j!}{\Efun_{N,j}(z)}.
\cHDefinition passage · 18 passages
Search aliases: cH H mathcal
Definition passage: Finite quantum seed system · 01_finite_core.tex, line 42
[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltonian when dynamics are present, and τ specifies the topology used in a later completion.
Exact mathematical passage
\mathfrak Q= \left( \cH, \psi_0, \cO, \cR, \{P_{\ell,\alpha}\}, H, \tau \right),
\operatorname{ch}Definition passage · 3 passages
Search aliases: ch operatorname
Definition passage: Complete packet pruning · Paper_V_Traditional_Final.tex, line 900
[Complete packet pruning] Let T be a finite rooted refinement tree. Every node v carries a projector P_v, the root satisfies P_∅=I, and every internal node satisfies P_v=∑_u∈ch(v)P_u, P_uP_w=0 (u≠ w in ch(v)). A set Λ of nodes is a complete pruning if it is an antichain and every root-to-terminal path meets Λ exactly once.
Exact mathematical passage
P_v=\sum_{u\in\operatorname{ch}(v)}P_u, \qquad P_uP_w=0\quad(u\ne w\text{ in }\operatorname{ch}(v)).
\operatorname{child}Definition passage · 5 passages
Search aliases: child operatorname
Definition passage: Unitary packet tree · Paper_V_Traditional_Final.tex, line 2335
[Unitary packet tree] A finite unitary packet tree consists of Hilbert spaces H_v at its vertices and, at every internal vertex v, a unitary map U_v: H_v⟶⊕_w∈child(v) H_w. An admissible pruning is a set of vertices that meets every root to terminal path exactly once.
Exact mathematical passage
U_v:\Hcal_v\longrightarrow\bigoplus_{w\in\operatorname{child}(v)}\Hcal_w.
\ClDefinition passage · 6 passages
Search aliases: Cl operatorname
Definition passage: Directed seed rank · Paper_X_Block27_Amended.tex, line 667
[Directed seed rank] For a directed transport graph G, define srank_G = min{|S|:Cl_G(S)=V(G)}, where Cl_G(S) is the set of vertices reachable from S.
Exact mathematical passage
\srank_G = \min\{|S|:\Cl_G(S)=V(G)\},
\clDefinition passage · 4 passages
Search aliases: cl
Definition passage: Generated hull · 02_foundations.tex, line 162
[Generated hull] For S⊆ X, define Hull_ A(S):=∑_s∈ S A s. In the topological regime define Hull_ A,τ(S):=clHull_ A(S)^ τ. For a singleton S={s}, write Hull_ A(s).
Exact mathematical passage
\Hull_{\A,\tau}(S):=\cl{\Hull_{\A}(S)}^{\,\tau}.
\Cl_GDefinition passage · 2 passages
Search aliases: Cl_G Cl operatorname
Definition passage: Directed seed rank · Paper_X_Block27_Amended.tex, line 667
[Directed seed rank] For a directed transport graph G, define srank_G = min{|S|:Cl_G(S)=V(G)}, where Cl_G(S) is the set of vertices reachable from S.
Exact mathematical passage
\srank_G = \min\{|S|:\Cl_G(S)=V(G)\},
\cODefinition passage · 5 passages
Search aliases: cO O mathcal
Definition passage: Finite quantum seed system · 01_finite_core.tex, line 42
[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltonian when dynamics are present, and τ specifies the topology used in a later completion.
Exact mathematical passage
\mathfrak Q= \left( \cH, \psi_0, \cO, \cR, \{P_{\ell,\alpha}\}, H, \tau \right),
\cokerDefinition passage · 30 passages
Search aliases: coker operatorname
Definition passage: Linear correction datum · 03_nonlinear_coordinates.tex, line 912
[Linear correction datum] A linear correction datum consists of a homomorphism D:U⟶ V and a defect r∈ V. A correction is an element u∈ U satisfying Du=-r. The obstruction class is Ob_D(r)=[r]∈coker D=V/im D.
Exact mathematical passage
\Ob_D(r)=[r]\in\coker D=V/\im D.
\CompatDefinition passage · 10 passages
Search aliases: Compat C mathfrak
Definition passage: Common arithmetic spine · Paper_X_Block27_Amended.tex, line 761
[Common arithmetic spine] Let M be a number field. A common spine C consists of: item characteristic polynomials P_v(X)∈ M[X] at almost all places; item coefficient-prime independent weight and determinant data; item prescribed local Weil--Deligne or inertial packet labels at the ramified places. For each finite place λ of M, let X_λ,m( C) be the finite set of admissible objects modulo ϖ_λ^m realizing the reduction of this spine.
Exact mathematical passage
\Compat
\ConeDefinition passage · 7 passages
Search aliases: Cone operatorname
Definition passage: Mapping cone · 03_nonlinear_coordinates.tex, line 1125
[Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).
Exact mathematical passage
\Cone(D)^n=E^n\oplus C^{n+1}
\operatorname{constant}Definition passage · 1 passage
Search aliases: constant operatorname
Definition passage: Finite grammar for operation trees · 02_connections_obstructions.tex, line 772
[Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the finite formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No algebraic, continuity, or differentiability law is imposed on these semantics. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.OperationTree; Kom…
Exact mathematical passage
\begin{aligned} t ::={}&\operatorname{input}(i) \mid\operatorname{constant}(k) \mid\operatorname{unary}(u,t)\\ &\mid\operatorname{binary}(b,t_1,t_2). \end{aligned}
\CoordDefinition passage · 3 passages
Search aliases: Coord operatorname
Definition passage: Canonical exact coordinate transport · 01_finite_coordinates.tex, line 472
[Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRefinementEquiv
Exact mathematical passage
J^X_{b\leftarrow a} = \cA_b\cS_a: \Coord(\cP^a)\longrightarrow\Coord(\cP^b).
\cosDefinition passage · 55 passages
Search aliases: cos
Definition passage · Paper_IV_Traditional_Final.tex, line 1794
For θ∈ R, define h_N(θ) = max_0≤ k<N cos(θ+frac(2k+1)πN). The tie directions are T_N = {frac2π kN:0≤ k<N} bmod 2π.
Exact mathematical passage
h_N(\theta) = \max_{0\leq k<N} \cos\left(\theta+\frac{(2k+1)\pi}{N}\right).
\cPDefinition passage · 5 passages
Search aliases: cP P mathcal
Definition passage: Canonical exact coordinate transport · 01_finite_coordinates.tex, line 472
[Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRefinementEquiv
Exact mathematical passage
J^X_{b\leftarrow a} = \cA_b\cS_a: \Coord(\cP^a)\longrightarrow\Coord(\cP^b).
\cP^aDefinition passage · 5 passages
Search aliases: cP^a P^a cP mathcal
Definition passage: Canonical exact coordinate transport · 01_finite_coordinates.tex, line 472
[Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRefinementEquiv
Exact mathematical passage
J^X_{b\leftarrow a} = \cA_b\cS_a: \Coord(\cP^a)\longrightarrow\Coord(\cP^b).
\cP^bDefinition passage · 2 passages
Search aliases: cP^b P^b cP mathcal
Definition passage: Canonical exact coordinate transport · 01_finite_coordinates.tex, line 472
[Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRefinementEquiv
Exact mathematical passage
J^X_{b\leftarrow a} = \cA_b\cS_a: \Coord(\cP^a)\longrightarrow\Coord(\cP^b).
\cRDefinition passage · 6 passages
Search aliases: cR R mathcal
Definition passage: Finite quantum seed system · 01_finite_core.tex, line 42
[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltonian when dynamics are present, and τ specifies the topology used in a later completion.
Exact mathematical passage
\mathfrak Q= \left( \cH, \psi_0, \cO, \cR, \{P_{\ell,\alpha}\}, H, \tau \right),
\cS_aDefinition passage · 5 passages
Search aliases: cS_a S_a cS mathcal
Definition passage: Canonical exact coordinate transport · 01_finite_coordinates.tex, line 472
[Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRefinementEquiv
Exact mathematical passage
J^X_{b\leftarrow a} = \cA_b\cS_a: \Coord(\cP^a)\longrightarrow\Coord(\cP^b).
\cWDefinition passage · 4 passages
Search aliases: cW W mathcal
Definition passage: Recursive word expansion · 01_finite_coordinates.tex, line 213
[Recursive word expansion] Define W_0=I, W_n+1=T W_n. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordExpansion
Exact mathematical passage
\cW_0=I, \qquad \cW_{n+1}=T\cW_n.
\cW_0Definition passage · 1 passage
Search aliases: cW_0 W_0 cW mathcal
Definition passage: Recursive word expansion · 01_finite_coordinates.tex, line 213
[Recursive word expansion] Define W_0=I, W_n+1=T W_n. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordExpansion
Exact mathematical passage
\cW_0=I, \qquad \cW_{n+1}=T\cW_n.
\cW_nDefinition passage · 4 passages
Search aliases: cW_n W_n cW mathcal
Definition passage: Recursive word expansion · 01_finite_coordinates.tex, line 213
[Recursive word expansion] Define W_0=I, W_n+1=T W_n. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordExpansion
Exact mathematical passage
\cW_0=I, \qquad \cW_{n+1}=T\cW_n.
\cW_{n+1}Definition passage · 2 passages
Search aliases: cW_n+1 W_n+1 cW mathcal
Definition passage: Recursive word expansion · 01_finite_coordinates.tex, line 213
[Recursive word expansion] Define W_0=I, W_n+1=T W_n. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordExpansion
Exact mathematical passage
\cW_0=I, \qquad \cW_{n+1}=T\cW_n.
D^NDefinition passage · 2 passages
Search aliases: D^N
Definition passage: Hurwitz integral series · strengthening.tex, line 48
[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.
Exact mathematical passage
D=\frac{d}{dt},\qquad L_\lambda=\frac{D^N-\lambda}{p}.
D^rDefinition passage · 11 passages
Search aliases: D^r
Definition passage: Packet jet · 03_nonlinear_coordinates.tex, line 220
[Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.
Exact mathematical passage
J^{r}_{\gamma\mid\alpha_1,\ldots,\alpha_r}F(x) = Q_\gamma D^rF(x) \big|_{X_{\alpha_1}\times\cdots\times X_{\alpha_r}}.
d_CDefinition passage · 2 passages
Search aliases: d_C
Definition passage: Mapping cone · 03_nonlinear_coordinates.tex, line 1129
[Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).
Exact mathematical passage
d_{\Cone}(e,c)=(d_Ee+Dc,-d_Cc).
d_{\Cone}Definition passage · 2 passages
Search aliases: d_Cone Cone operatorname
Definition passage: Mapping cone · 03_nonlinear_coordinates.tex, line 1129
[Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).
Exact mathematical passage
d_{\Cone}(e,c)=(d_Ee+Dc,-d_Cc).
d_EDefinition passage · 3 passages
Search aliases: d_E
Definition passage: Mapping cone · 03_nonlinear_coordinates.tex, line 1129
[Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).
Exact mathematical passage
d_{\Cone}(e,c)=(d_Ee+Dc,-d_Cc).
d_LDefinition passage · 5 passages
Search aliases: d_L
Definition passage: Heat trace and spectral dimension · 02_strengthened.tex, line 1961
[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).
Exact mathematical passage
d_L(t) =-2\frac{\dd\log Z_L(t)}{\dd\log t} = 2t \frac{\Tr(L\e^{-tL})}{\Tr(\e^{-tL})}.
D_\alphaDefinition passage · 17 passages
Search aliases: D_α alpha
Definition passage: Block diagonal plus low rank operator · 01_manuscript.tex, line 307
[Block diagonal plus low rank operator] A block diagonal plus low rank operator has the form A=D+UV^*, where D=⊕_α∈ ID_α, U,V∈ C^n× r. The abbreviation BDLR will be used below.
Exact mathematical passage
D=\bigoplus_{\alpha\in I}D_\alpha, \qquad U,V\in\CC^{n\times r}.
\DcalDefinition passage · 10 passages
Search aliases: Dcal D mathcal
Definition passage: Packet dephasing · 02_strengthened.tex, line 992
[Packet dephasing] The packet dephasing map is D_ P(X)=∑_aP_aXP_a.
Exact mathematical passage
\Dcal_{\Pcal}(X)=\sum_aP_aXP_a.
\Dcal_{\Pcal}Definition passage · 7 passages
Search aliases: Dcal_Pcal D_ P Dcal Pcal mathcal mathcal
Definition passage: Packet dephasing · 02_strengthened.tex, line 992
[Packet dephasing] The packet dephasing map is D_ P(X)=∑_aP_aXP_a.
Exact mathematical passage
\Dcal_{\Pcal}(X)=\sum_aP_aXP_a.
\ddDefinition passage · 82 passages
Search aliases: dd d mathrm
Definition passage: Heat trace and spectral dimension · 02_strengthened.tex, line 1961
[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).
Exact mathematical passage
d_L(t) =-2\frac{\dd\log Z_L(t)}{\dd\log t} = 2t \frac{\Tr(L\e^{-tL})}{\Tr(\e^{-tL})}.
\dd^dDefinition passage · 1 passage
Search aliases: dd^d d^d dd mathrm
Definition passage: Einstein infrared criterion · 02_strengthened.tex, line 2558
[Einstein infrared criterion] A quantum geometric tower meets an Einstein infrared criterion only if an effective action is controlled at long wavelengths and has an expansion whose leading geometric terms take the form Γ_rm eff[g] = ∫ sqrtabs g ( Λ + frac116π GR + controlled higher order terms ) d^dx. Here the volume coefficient includes the sign and normalization written explicitly in the cosmological term in Part I. The expansion must respect the limiting gauge identities and must include an …
Exact mathematical passage
\Gamma_{\rm eff}[g] = \int \sqrt{\abs g} \left( \Lambda + \frac{1}{16\pi G}R + \text{controlled higher order terms} \right) \dd^dx.
\DefDefinition passage · 2 passages
Search aliases: Def operatorname
Definition passage: Admissible residual transport · Paper_X_Block27_Amended.tex, line 657
[Admissible residual transport] An admissible transport from barρ to barρ' consists of T:barρ⟼barρ' together with natural maps T_A:Def_barρ(A)⟶Def_barρ'(A) for every coefficient algebra A, compatible with base change and reduction. When the transport is reversible, it is an isomorphism in the residual transport groupoid. One-way operations give a directed category.
Exact mathematical passage
\mathcal T_A:\Def_{\bar\rho}(A)\longrightarrow\Def_{\bar\rho'}(A)
\Def_{\bar\rho}Definition passage · 2 passages
Search aliases: Def_barρ Def bar rho operatorname bar rho
Definition passage: Admissible residual transport · Paper_X_Block27_Amended.tex, line 657
[Admissible residual transport] An admissible transport from barρ to barρ' consists of T:barρ⟼barρ' together with natural maps T_A:Def_barρ(A)⟶Def_barρ'(A) for every coefficient algebra A, compatible with base change and reduction. When the transport is reversible, it is an isomorphism in the residual transport groupoid. One-way operations give a directed category.
Exact mathematical passage
\mathcal T_A:\Def_{\bar\rho}(A)\longrightarrow\Def_{\bar\rho'}(A)
\Def_{\bar\rho'}Definition passage · 1 passage
Search aliases: Def_barρ' Def bar rho operatorname bar rho
Definition passage: Admissible residual transport · Paper_X_Block27_Amended.tex, line 657
[Admissible residual transport] An admissible transport from barρ to barρ' consists of T:barρ⟼barρ' together with natural maps T_A:Def_barρ(A)⟶Def_barρ'(A) for every coefficient algebra A, compatible with base change and reduction. When the transport is reversible, it is an isomorphism in the residual transport groupoid. One-way operations give a directed category.
Exact mathematical passage
\mathcal T_A:\Def_{\bar\rho}(A)\longrightarrow\Def_{\bar\rho'}(A)
\mathrm{dependencies}Definition passage · 1 passage
Search aliases: dependencies mathrm
Definition passage: Canonical theorem record · 02_foundations.tex, line 17
[Canonical theorem record] A canonical theorem record is the tuple (ID,status,hypotheses,statement, dependencies,proof,exports,boundary). A later chapter cites the stable record ID. A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this chapter.
Exact mathematical passage
(\mathrm{ID},\mathrm{status},\mathrm{hypotheses},\mathrm{statement}, \mathrm{dependencies},\mathrm{proof},\mathrm{exports},\mathrm{boundary}).
\mathrm{der}Definition passage · 2 passages
Search aliases: der mathrm
Definition passage · Paper_IV_Traditional_Final.tex, line 2925
Let q be a positive integer. Choose logarithms of the simple zeros of D_N with arguments in a fixed bounded interval. Define Z_N,q^der(s) = ∑_substack D_N(ρ)=0 D_N'(ρ)≠0 frace^-sLogρ D_N'(ρ)^q.
Exact mathematical passage
\mathfrak Z_{N,q}^{\mathrm{der}}(s) = \sum_{\substack{\Dfun_N(\rho)=0\\ \Dfun_N'(\rho)\ne0}} \frac{e^{-s\Log\rho}}{\Dfun_N'(\rho)^q}.
\detDefinition passage · 46 passages
Search aliases: det
Definition passage: Odd finite Weil phase · Paper_XI_Traditional_Final.tex, line 1308
[Odd finite Weil phase] For fixed point free g, define Φ_p(g,V) = (frac-1p)^r (fracdet(g-I)p).
Exact mathematical passage
\Phi_p(g,V) = \left(\frac{-1}{p}\right)^r \left(\frac{\det(g-I)}{p}\right).
\DfunDefinition passage · 114 passages
Search aliases: Dfun D mathsf
Definition passage · Paper_IV_Traditional_Final.tex, line 606
For 0≤ j<N, define D_N,j(z) = ∑_n≥0(-1)^nfracz^Nn+j(Nn+j)!. The distinguished signed denominator is D_N(z)= D_N,0(z).
Exact mathematical passage
\Dfun_{N,j}(z) = \sum_{n\geq0}(-1)^n\frac{z^{Nn+j}}{(Nn+j)!}.
\Dfun_NDefinition passage · 75 passages
Search aliases: Dfun_N D_N Dfun mathsf
Definition passage · Paper_IV_Traditional_Final.tex, line 612
For 0≤ j<N, define D_N,j(z) = ∑_n≥0(-1)^nfracz^Nn+j(Nn+j)!. The distinguished signed denominator is D_N(z)= D_N,0(z).
Exact mathematical passage
\Dfun_N(z)=\Dfun_{N,0}(z).
\Dfun_N'Definition passage · 17 passages
Search aliases: Dfun_N' D_N' Dfun mathsf
Definition passage · Paper_IV_Traditional_Final.tex, line 1262
Define the transfer tangent T_N(z) =-frac D_N'(z) D_N(z). Near the origin, its convergent block expansion has the form T_N(z) = ∑_n≥0T_N(n) fracz^Nn+N-1(Nn+N-1)!.
Exact mathematical passage
\Tfun_N(z) =-\frac{\Dfun_N'(z)}{\Dfun_N(z)}.
\Dfun_{N,0}Definition passage · 4 passages
Search aliases: Dfun_N,0 D_N,0 Dfun mathsf
Definition passage · Paper_IV_Traditional_Final.tex, line 612
For 0≤ j<N, define D_N,j(z) = ∑_n≥0(-1)^nfracz^Nn+j(Nn+j)!. The distinguished signed denominator is D_N(z)= D_N,0(z).
Exact mathematical passage
\Dfun_N(z)=\Dfun_{N,0}(z).
\Dfun_{N,j}Definition passage · 8 passages
Search aliases: Dfun_N,j D_N,j Dfun mathsf
Definition passage · Paper_IV_Traditional_Final.tex, line 606
For 0≤ j<N, define D_N,j(z) = ∑_n≥0(-1)^nfracz^Nn+j(Nn+j)!. The distinguished signed denominator is D_N(z)= D_N,0(z).
Exact mathematical passage
\Dfun_{N,j}(z) = \sum_{n\geq0}(-1)^n\frac{z^{Nn+j}}{(Nn+j)!}.
\dimDefinition passage · 54 passages
Search aliases: dim
Definition passage: Branching dimension · Paper_X_Block27_Amended.tex, line 612
[Branching dimension] When the limit exists, define the finite-level counting exponent dim_seed(bar x) = lim_m→∞ fraclog_q#X_m(bar x)m. A limsup may be used in general.
Exact mathematical passage
\dim_{\mathrm{seed}}(\bar x) = \lim_{m\to\infty} \frac{\log_q\#X_m(\bar x)}{m}.
\dim_{\mathrm{seed}}Definition passage · 2 passages
Search aliases: dim_seed dim mathrm
Definition passage: Branching dimension · Paper_X_Block27_Amended.tex, line 612
[Branching dimension] When the limit exists, define the finite-level counting exponent dim_seed(bar x) = lim_m→∞ fraclog_q#X_m(bar x)m. A limsup may be used in general.
Exact mathematical passage
\dim_{\mathrm{seed}}(\bar x) = \lim_{m\to\infty} \frac{\log_q\#X_m(\bar x)}{m}.
\eDefinition passage · 198 passages
Search aliases: e mathrm
Definition passage: Packet blocks · Paper_VI_Traditional_Final.tex, line 2067
[Packet blocks] For B∈ B(X), define B_βα=P_β BP_α. If A∈ B(X) and T(t)= e^tA, write T_βα(t)=P_β T(t)P_α.
Exact mathematical passage
T(t)=\e^{tA}
E'Definition passage · 44 passages
Search aliases: E'
Definition passage: Ordinary completion datum at a point · Paper_XII_Traditional_Final.tex, line 1641
[Ordinary completion datum at a point] An ordinary completion datum at a characteristic zero point x:R→ E' consists of the following objects. item A represented global deformation ring R with declared local quotient conditions and universal representation ρ_R. item An ordinary patching datum (R_∞, a,M_∞,R,H) whose finite level module H is an automorphic Hecke carrier. item Full patched support Supp_R_∞(M_∞)=Spec R_∞. item An isomorphism R≅ T with the Hecke algebra acting on H. The R action on H …
Exact mathematical passage
x:R\to E'
e^{-s\log|\lambda|}Definition passage · 1 passage
Search aliases: e^-slog|λ| log lambda
Definition passage · Paper_IV_Traditional_Final.tex, line 2754
Define Z_N^abs(s) = ∑_ρ∈ Z( D_N)|ρ|^-s and Z_N^red(s) = ∑_λ∈Λ_N|λ|^-s, with multiplicities. Each term means e^-slog|ρ| or e^-slog|λ|.
Exact mathematical passage
e^{-s\log|\lambda|}
e^{-s\log|\rho|}Definition passage · 1 passage
Search aliases: e^-slog|ρ| log rho
Definition passage · Paper_IV_Traditional_Final.tex, line 2754
Define Z_N^abs(s) = ∑_ρ∈ Z( D_N)|ρ|^-s and Z_N^red(s) = ∑_λ∈Λ_N|λ|^-s, with multiplicities. Each term means e^-slog|ρ| or e^-slog|λ|.
Exact mathematical passage
e^{-s\log|\rho|}
e^{-s\Log\lambda}Definition passage · 1 passage
Search aliases: e^-sLogλ Log lambda operatorname lambda
Definition passage · Paper_IV_Traditional_Final.tex, line 2907
Choose a logarithm for each member of the finite exceptional part of Λ_N. Use the real logarithm for the eventual positive sequence. Define Z_N^orb(s) = ∑_λ∈Λ_Ne^-sLogλ. This series is initially defined on Re s>1/N.
Exact mathematical passage
\mathfrak Z_N^{\mathrm{orb}}(s) = \sum_{\lambda\in\Lambda_N}e^{-s\Log\lambda}.
e^{-s\Log\rho}Definition passage · 2 passages
Search aliases: e^-sLogρ Log rho operatorname rho
Definition passage · Paper_IV_Traditional_Final.tex, line 2925
Let q be a positive integer. Choose logarithms of the simple zeros of D_N with arguments in a fixed bounded interval. Define Z_N,q^der(s) = ∑_substack D_N(ρ)=0 D_N'(ρ)≠0 frace^-sLogρ D_N'(ρ)^q.
Exact mathematical passage
\mathfrak Z_{N,q}^{\mathrm{der}}(s) = \sum_{\substack{\Dfun_N(\rho)=0\\ \Dfun_N'(\rho)\ne0}} \frac{e^{-s\Log\rho}}{\Dfun_N'(\rho)^q}.
\e^{-tL}Definition passage · 7 passages
Search aliases: e^-tL e mathrm
Definition passage: Heat trace and spectral dimension · 02_strengthened.tex, line 1957
[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).
Exact mathematical passage
Z_L(t)=\Tr(\e^{-tL})
E^\bulletDefinition passage · 2 passages
Search aliases: E^bullet bullet
Definition passage: Mapping cone · 03_nonlinear_coordinates.tex, line 1124
[Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).
Exact mathematical passage
D:C^\bullet\to E^\bullet
E^nDefinition passage · 5 passages
Search aliases: E^n
Definition passage: Mapping cone · 03_nonlinear_coordinates.tex, line 1125
[Mapping cone] Let D:C^bullet→ E^bullet be a cochain map. Its mapping cone is Cone(D)^n=E^n⊕ C^n+1 with differential d_Cone(e,c)=(d_Ee+Dc,-d_Cc).
Exact mathematical passage
\Cone(D)^n=E^n\oplus C^{n+1}
e^{rL}Definition passage · 1 passage
Search aliases: e^rL
Definition passage · Paper_IV_Traditional_Final.tex, line 1445
For r∈ C, define formal coefficients C_N,j,n(r) by C_N,j(z)^r = ∑_n≥0C_N,j,n(r)fracz^Nn(Nn)!. On a simply connected region containing 0 and avoiding the zeros of Φ_N,j(z^N), interpret this power as e^rL, where L is the holomorphic logarithm of C_N,j normalized by L(0)=0.
Exact mathematical passage
e^{rL}
\e^{tA}Definition passage · 22 passages
Search aliases: e^tA e mathrm
Definition passage: Packet blocks · Paper_VI_Traditional_Final.tex, line 2067
[Packet blocks] For B∈ B(X), define B_βα=P_β BP_α. If A∈ B(X) and T(t)= e^tA, write T_βα(t)=P_β T(t)P_α.
Exact mathematical passage
T(t)=\e^{tA}
e^{xz}Definition passage · 5 passages
Search aliases: e^xz
Definition passage · Paper_IV_Traditional_Final.tex, line 1484
Define polynomials E_m^(N,j,r)(x) by C_N,j(z)^re^xz = ∑_m≥0E_m^(N,j,r)(x)fracz^mm!.
Exact mathematical passage
\Cfun_{N,j}(z)^re^{xz} = \sum_{m\geq0}E_m^{(N,j,r)}(x)\frac{z^m}{m!}.
e^{\lambda z}Definition passage · 3 passages
Search aliases: e^λ z lambda
Definition passage · Paper_IV_Traditional_Final.tex, line 459
For λ∈ C and 0≤ j<N, define E_N,j(λ z) = B_N,j(e^λ z) = ∑_n≥0frac(λ z)^Nn+j(Nn+j)!.
Exact mathematical passage
\Efun_{N,j}(\lambda z) = \Bcal_{N,j}(e^{\lambda z}) = \sum_{n\geq0}\frac{(\lambda z)^{Nn+j}}{(Nn+j)!}.
e_0Definition passage · 56 passages
Search aliases: e_0
Definition passage · Paper_IV_Traditional_Final.tex, line 824
Let J_N be the signed cyclic shift on the standard basis e_0,…,e_N-1 of C^N, defined by J_Ne_j=e_j+1 (0≤ j<N-1), J_Ne_N-1=-e_0. Then J_N^N=-I.
Exact mathematical passage
e_0,\ldots,e_{N-1}
e_1Definition passage · 80 passages
Search aliases: e_1
Definition passage: Weighted transfer series · 03_weighted.tex, line 390
W(γ)=n u_i v_j∏_ν=1^|γ|m(e_ν). Here γ=(e_1,…,e_|γ|) is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the endpoint vectors weight the start and end of each walk, rather than only selecting subsets of vertices.
Exact mathematical passage
\gamma=(e_1,\ldots,e_{|\gamma|})
E_{2m-b}Definition passage · 1 passage
Search aliases: E_2m-b
Definition passage: Odd-length source and inverse coefficients · 20_arithmetic.tex, line 761
[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd. Primality is needed only to interpret the result as a prime-adic congruence ladder.
Exact mathematical passage
\operatorname{sech}^2t=\sum_{m\ge0}\gamma_m\frac{t^{2m}}{(2m)!}, \qquad \gamma_m=\sum_{b=0}^{2m}\binom{2m}{b}E_bE_{2m-b},
E_a^*Definition passage · 25 passages
Search aliases: E_a^*
Definition passage: Approximate Knill--Laflamme defect · 02_strengthened.tex, line 1423
[Approximate Knill--Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.
Exact mathematical passage
\varepsilon_{\rm KL} = \max_{a,b} \inf_{c\in\CC} \norm{PE_a^*E_bP-cP}.
E_bDefinition passage · 22 passages
Search aliases: E_b
Definition passage: Odd-length source and inverse coefficients · 20_arithmetic.tex, line 761
[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd. Primality is needed only to interpret the result as a prime-adic congruence ladder.
Exact mathematical passage
\operatorname{sech}^2t=\sum_{m\ge0}\gamma_m\frac{t^{2m}}{(2m)!}, \qquad \gamma_m=\sum_{b=0}^{2m}\binom{2m}{b}E_bE_{2m-b},
E_d^{(N,j)}Definition passage · 2 passages
Search aliases: E_d^(N,j)
Definition passage: Congruential Euler coefficients · 20_arithmetic.tex, line 15
[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the multiples of N vanish.
Exact mathematical passage
\label{eq:arith-euler-egf} \sum_{d\ge0}E_d^{(N,j)}\frac{t^d}{d!} =\left(\sum_{n\ge0}\frac{t^{Nn}}{(Nn+j)!}\right)^{-1}.
e_iDefinition passage · 15 passages
Search aliases: e_i
Definition passage: Arf invariant · Paper_XI_Traditional_Final.tex, line 1488
[Arf invariant] Choose a symplectic basis (e_i,f_i)_i=1^r. Define Arf(q) = ∑_i=1^rq(e_i)q(f_i) ∈ F_2.
Exact mathematical passage
(e_i,f_i)_{i=1}^{r}
e_jDefinition passage · 21 passages
Search aliases: e_j
Definition passage · Paper_IV_Traditional_Final.tex, line 825
Let J_N be the signed cyclic shift on the standard basis e_0,…,e_N-1 of C^N, defined by J_Ne_j=e_j+1 (0≤ j<N-1), J_Ne_N-1=-e_0. Then J_N^N=-I.
Exact mathematical passage
J_Ne_j=e_{j+1} \quad (0\leq j<N-1), \qquad J_Ne_{N-1}=-e_0.
e_{j+1}Definition passage · 4 passages
Search aliases: e_j+1
Definition passage · Paper_IV_Traditional_Final.tex, line 825
Let J_N be the signed cyclic shift on the standard basis e_0,…,e_N-1 of C^N, defined by J_Ne_j=e_j+1 (0≤ j<N-1), J_Ne_N-1=-e_0. Then J_N^N=-I.
Exact mathematical passage
J_Ne_j=e_{j+1} \quad (0\leq j<N-1), \qquad J_Ne_{N-1}=-e_0.
E_m^{(N,j,r)}Definition passage · 9 passages
Search aliases: E_m^(N,j,r)
Definition passage · Paper_IV_Traditional_Final.tex, line 1483
Define polynomials E_m^(N,j,r)(x) by C_N,j(z)^re^xz = ∑_m≥0E_m^(N,j,r)(x)fracz^mm!.
Exact mathematical passage
E_m^{(N,j,r)}(x)
e_nDefinition passage · 17 passages
Search aliases: e_n
Definition passage: Composable word and execution operator · 01_finite_coordinates.tex, line 173
[Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordOp
Exact mathematical passage
w=(e_1,\ldots,e_n)
e_{N-1}Definition passage · 3 passages
Search aliases: e_N-1
Definition passage · Paper_IV_Traditional_Final.tex, line 824
Let J_N be the signed cyclic shift on the standard basis e_0,…,e_N-1 of C^N, defined by J_Ne_j=e_j+1 (0≤ j<N-1), J_Ne_N-1=-e_0. Then J_N^N=-I.
Exact mathematical passage
e_0,\ldots,e_{N-1}
E_{Nn}^{(N,j)}Definition passage · 7 passages
Search aliases: E_Nn^(N,j)
Definition passage: Congruential Euler coefficients · 20_arithmetic.tex, line 21
[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the multiples of N vanish.
Exact mathematical passage
U_{N,j}(n)=E_{Nn}^{(N,j)},\qquad C_{N,j}(n)=\frac{U_{N,j}(n)}{j!}.
e_rDefinition passage · 6 passages
Search aliases: e_r
Definition passage: Composable word and execution operator · 01_finite_coordinates.tex, line 174
[Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordOp
Exact mathematical passage
\tgt(e_r)=\src(e_{r+1}) \qquad (1\le r<n).
e_{r+1}Definition passage · 1 passage
Search aliases: e_r+1
Definition passage: Composable word and execution operator · 01_finite_coordinates.tex, line 174
[Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordOp
Exact mathematical passage
\tgt(e_r)=\src(e_{r+1}) \qquad (1\le r<n).
e_{|\gamma|}Definition passage · 1 passage
Search aliases: e_|γ| gamma
Definition passage: Weighted transfer series · 03_weighted.tex, line 390
W(γ)=n u_i v_j∏_ν=1^|γ|m(e_ν). Here γ=(e_1,…,e_|γ|) is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the endpoint vectors weight the start and end of each walk, rather than only selecting subsets of vertices.
Exact mathematical passage
\gamma=(e_1,\ldots,e_{|\gamma|})
E_\alphaDefinition passage · 13 passages
Search aliases: E_α alpha
Definition passage: Lattice packet data · 04_arithmetic.tex, line 239
[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.
Exact mathematical passage
E_\alpha
e_\nuDefinition passage · 1 passage
Search aliases: e_ν nu
Definition passage: Weighted transfer series · 03_weighted.tex, line 386
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|γ|m(e_ν). Here γ=(e_1,…,e_|γ|) is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the en…
Exact mathematical passage
a_n=\sum_{i,j}\ \sum_{\substack{\gamma:i\leadsto j\\W(\gamma)=n}} u_i v_j\prod_{\nu=1}^{|\gamma|}m(e_\nu).
\EfunDefinition passage · 21 passages
Search aliases: Efun E mathsf
Definition passage · Paper_IV_Traditional_Final.tex, line 459
For λ∈ C and 0≤ j<N, define E_N,j(λ z) = B_N,j(e^λ z) = ∑_n≥0frac(λ z)^Nn+j(Nn+j)!.
Exact mathematical passage
\Efun_{N,j}(\lambda z) = \Bcal_{N,j}(e^{\lambda z}) = \sum_{n\geq0}\frac{(\lambda z)^{Nn+j}}{(Nn+j)!}.
\Efun_{N,0}Definition passage · 3 passages
Search aliases: Efun_N,0 E_N,0 Efun mathsf
Definition passage · Paper_IV_Traditional_Final.tex, line 1636
Set H_N(z)= E_N,0(z) = ∑_n≥0fracz^Nn(Nn)!. For u∈ C, define symupΘ_N,n(u) by H_N(uz)^-1H_N(z)^-u = ∑_n≥0symupΘ_N,n(u)fracz^Nn(Nn)!. This is first a formal identity at the origin. Analytically, it is interpreted on a common simply connected zero-free region with the branch of log H_N normalized by log H_N(0)=0.
Exact mathematical passage
H_N(z)=\Efun_{N,0}(z) = \sum_{n\geq0}\frac{z^{Nn}}{(Nn)!}.
\Efun_{N,j}Definition passage · 17 passages
Search aliases: Efun_N,j E_N,j Efun mathsf
Definition passage · Paper_IV_Traditional_Final.tex, line 459
For λ∈ C and 0≤ j<N, define E_N,j(λ z) = B_N,j(e^λ z) = ∑_n≥0frac(λ z)^Nn+j(Nn+j)!.
Exact mathematical passage
\Efun_{N,j}(\lambda z) = \Bcal_{N,j}(e^{\lambda z}) = \sum_{n\geq0}\frac{(\lambda z)^{Nn+j}}{(Nn+j)!}.
\evDefinition passage · 1 passage
Search aliases: ev operatorname
Definition passage: Typed operation tree and evaluation · 02_connections_obstructions.tex, line 848
[Typed operation tree and evaluation] A typed operation tree is recursively either a leaf of declared color or a node labeled by σ, with one correctly colored child for each input slot of σ. Evaluation is recursive: ev(node(σ;t_1,…,t_r(σ))) = Φ_σ(ev(t_1),…,ev(t_r(σ))). statusroadmapCandidate API: leannameKomorebi.PaperIII.NonlinearObstructions.ColoredOperationTree and leannameColoredOperationTree.eval. This is stronger than the compiled uncolored unary--binary grammar.
Exact mathematical passage
\ev(\operatorname{node}(\sigma;t_1,\ldots,t_{r(\sigma)})) = \Phi_\sigma(\ev(t_1),\ldots,\ev(t_{r(\sigma)})).
\mathsf{Ev}Definition passage · 1 passage
Search aliases: Ev mathsf
Definition passage: Realization functor and preservation profile · 04_transport.tex, line 385
[Realization functor and preservation profile] Let SeedSys_abs be an abstract category of seed systems and let C_D be a domain category. A realization functor is a functor Ev_D:SeedSys_abs⟶ C_D with a declared preservation profile specifying which of the following structures are preserved: 0, I, finite sums, composition, biproducts,
Exact mathematical passage
\mathsf{Ev}_D:\mathbf{SeedSys}_{\mathrm{abs}}\longrightarrow\mathbf C_D
\mathsf{Ev}_DDefinition passage · 1 passage
Search aliases: Ev_D mathsf
Definition passage: Realization functor and preservation profile · 04_transport.tex, line 385
[Realization functor and preservation profile] Let SeedSys_abs be an abstract category of seed systems and let C_D be a domain category. A realization functor is a functor Ev_D:SeedSys_abs⟶ C_D with a declared preservation profile specifying which of the following structures are preserved: 0, I, finite sums, composition, biproducts,
Exact mathematical passage
\mathsf{Ev}_D:\mathbf{SeedSys}_{\mathrm{abs}}\longrightarrow\mathbf C_D
\operatorname{eval}Definition passage · 1 passage
Search aliases: eval operatorname
Definition passage: Packet operator representation · 01_manuscript.tex, line 92
B_βα:X_α⟶ X_β. Local blocks are identified with their ambient extensions by inclusion and projection. In particular, the notation P_β AP_α denotes either the ambient block or its restriction X_α→ X_β according to the displayed domain, and ι_β B_βαπ_α is the corresponding ambient extension. Its evaluation is eval(IR) = ∑_(β,α)∈ E ι_β B_βαπ_α, where ι_β:X_β→ X is the coordinate inclusion.
Exact mathematical passage
\operatorname{eval}(\IR) = \sum_{(\beta,\alpha)\in E} \iota_\beta B_{\beta\alpha}\pi_\alpha,
\expDefinition passage · 49 passages
Search aliases: exp
Definition passage · Paper_IV_Traditional_Final.tex, line 958
For r∈ C, define S_N^(r)(z) = D_N(z)^-r = exp(-rlog D_N(z)). Write S_N^(r)(z) = ∑_n≥0P_N,n(r)fracz^Nn(Nn)!. Define the logarithmic coefficients κ_N,m by -logΔ_N(w) = ∑_m≥1κ_N,mfracw^mm.
Exact mathematical passage
\Sfun_N^{(r)}(z) = \Dfun_N(z)^{-r} = \exp\bigl(-r\log\Dfun_N(z)\bigr).
\expnDefinition passage · 4 passages
Search aliases: expn exp operatorname
Definition passage: Lattice packet data · 04_arithmetic.tex, line 261
mbda_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.
Exact mathematical passage
\expn_\pi(G)
\expn_\piDefinition passage · 4 passages
Search aliases: expn_π exp_π expn pi operatorname pi
Definition passage: Lattice packet data · 04_arithmetic.tex, line 261
mbda_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.
Exact mathematical passage
\expn_\pi(G)
\mathrm{exports}Definition passage · 1 passage
Search aliases: exports mathrm
Definition passage: Canonical theorem record · 02_foundations.tex, line 17
[Canonical theorem record] A canonical theorem record is the tuple (ID,status,hypotheses,statement, dependencies,proof,exports,boundary). A later chapter cites the stable record ID. A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this chapter.
Exact mathematical passage
(\mathrm{ID},\mathrm{status},\mathrm{hypotheses},\mathrm{statement}, \mathrm{dependencies},\mathrm{proof},\mathrm{exports},\mathrm{boundary}).
FDefinition passage · 900 passages
Search aliases: F
Definition passage: Unconditional packet sum space · 03_packets_and_graphs.tex, line 359
[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .
Exact mathematical passage
\|x\|_u := \sup_{F\Subset\mathbb N} \sup_{|a_n|\leq1} \left\|\sum_{n\in F}a_nx_n\right\|.
F^0Definition passage · 9 passages
Search aliases: F^0
Definition passage: Complete multiplicative filtration · 03_weighted.tex, line 7
[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
\Bcal=F^0\Bcal\supseteq F^1\Bcal\supseteq F^2\Bcal\supseteq\cdots
F^1Definition passage · 6 passages
Search aliases: F^1
Definition passage: Complete multiplicative filtration · 03_weighted.tex, line 7
[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
\Bcal=F^0\Bcal\supseteq F^1\Bcal\supseteq F^2\Bcal\supseteq\cdots
F^2Definition passage · 3 passages
Search aliases: F^2
Definition passage: Complete multiplicative filtration · 03_weighted.tex, line 7
[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
\Bcal=F^0\Bcal\supseteq F^1\Bcal\supseteq F^2\Bcal\supseteq\cdots
F^LDefinition passage · 16 passages
Search aliases: F^L
Definition passage: Complete multiplicative filtration · 03_weighted.tex, line 11
[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
F^r\Bcal\,F^s\Bcal\subseteq F^{r+s}\Bcal, \qquad \bigcap_{L\geq0}F^L\Bcal=0,
F^mDefinition passage · 34 passages
Search aliases: F^m
Definition passage: Depth obstruction · 04_correction_transport.tex, line 247
[Depth obstruction] The depth m obstruction of x_m is Ob_m(x_m) = [Φ(x_m)] ∈ cokeroverline L_m, where Φ(x_m) is first viewed in F^mY/F^m+1Y.
Exact mathematical passage
F^mY/F^{m+1}Y
F^{m+1}Definition passage · 15 passages
Search aliases: F^m+1
Definition passage: Depth obstruction · 04_correction_transport.tex, line 247
[Depth obstruction] The depth m obstruction of x_m is Ob_m(x_m) = [Φ(x_m)] ∈ cokeroverline L_m, where Φ(x_m) is first viewed in F^mY/F^m+1Y.
Exact mathematical passage
F^mY/F^{m+1}Y
F^rDefinition passage · 8 passages
Search aliases: F^r
Definition passage: Complete multiplicative filtration · 03_weighted.tex, line 11
[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
F^r\Bcal\,F^s\Bcal\subseteq F^{r+s}\Bcal, \qquad \bigcap_{L\geq0}F^L\Bcal=0,
F^{r+s}Definition passage · 1 passage
Search aliases: F^r+s
Definition passage: Complete multiplicative filtration · 03_weighted.tex, line 11
[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
F^r\Bcal\,F^s\Bcal\subseteq F^{r+s}\Bcal, \qquad \bigcap_{L\geq0}F^L\Bcal=0,
F^sDefinition passage · 1 passage
Search aliases: F^s
Definition passage: Complete multiplicative filtration · 03_weighted.tex, line 11
[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
F^r\Bcal\,F^s\Bcal\subseteq F^{r+s}\Bcal, \qquad \bigcap_{L\geq0}F^L\Bcal=0,
F^{w(e)}Definition passage · 5 passages
Search aliases: F^w(e)
Definition passage: Complete multiplicative filtration · 03_weighted.tex, line 21
with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
N_e\in F^{w(e)}\Bcal.
f_{ab}^{\phantom{ab}c}Definition passage · 6 passages
Search aliases: f_ab^phantomabc phantom
Definition passage: Anomaly free continuum constraint system · 02_strengthened.tex, line 2540
[Anomaly free continuum constraint system] A continuum constraint system consists of a Hilbert space, a common invariant dense domain D, and operators C_a on D. It is anomaly free for prescribed structure coefficients when [C_a,C_b]ψ = i∑_cf_ab^phantomabcC_cψ for every ψ∈ D, and when the associated gauge evolution is defined on the physical quotient or physical subspace.
Exact mathematical passage
[C_a,C_b]\psi = \ii\sum_cf_{ab}^{\phantom{ab}c}C_c\psi
F_cDefinition passage · 13 passages
Search aliases: F_c
Definition passage: Nonlinear refinement defect · 03_nonlinear_coordinates.tex, line 737
[Nonlinear refinement defect] The pointwise defect of F is Δ_F(x) = C_YF_f(x)-F_c(C_Xx). The two models are exactly compatible when Δ_F vanishes everywhere.
Exact mathematical passage
\Delta_F(x) = C_YF_f(x)-F_c(C_Xx).
f_dDefinition passage · 1 passage
Search aliases: f_d
Definition passage: Hurwitz integral series · strengthening.tex, line 42
[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.
Exact mathematical passage
\mathcal H_p=\left\{\sum_{d\geq0}f_d\frac{t^d}{d!}: f_d\in\mathbb Z_p\right\} \subset\mathbb Q_p[[t]].
F_fDefinition passage · 13 passages
Search aliases: F_f
Definition passage: Nonlinear refinement defect · 03_nonlinear_coordinates.tex, line 737
[Nonlinear refinement defect] The pointwise defect of F is Δ_F(x) = C_YF_f(x)-F_c(C_Xx). The two models are exactly compatible when Δ_F vanishes everywhere.
Exact mathematical passage
\Delta_F(x) = C_YF_f(x)-F_c(C_Xx).
f_iDefinition passage · 13 passages
Search aliases: f_i
Definition passage: Arf invariant · Paper_XI_Traditional_Final.tex, line 1488
[Arf invariant] Choose a symplectic basis (e_i,f_i)_i=1^r. Define Arf(q) = ∑_i=1^rq(e_i)q(f_i) ∈ F_2.
Exact mathematical passage
(e_i,f_i)_{i=1}^{r}
f_{n,m}Definition passage · 2 passages
Search aliases: f_n,m
Definition passage: Stable core · Paper_XII_Traditional_Final.tex, line 1196
[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).
Exact mathematical passage
f_{n,m}:V_n\to V_m
F_PDefinition passage · 26 passages
Search aliases: F_P
Definition passage: Feshbach map and self energy · 02_strengthened.tex, line 1506
[Feshbach map and self energy] For z∉Spec D, define Σ(z)=B(z-D)^-1B^* and F_P(z)=z-A-Σ(z).
Exact mathematical passage
F_P(z)=z-A-\Sigma(z).
F_{\Qcal\leftarrow\Pcal}Definition passage · 7 passages
Search aliases: F_Qcal←Pcal F_ Q← P Qcal leftarrow Pcal mathcal leftarrow mathcal
Definition passage: Nonlinear packet coordinate map · 03_nonlinear_coordinates.tex, line 106
[Nonlinear packet coordinate map] Let P be a packet system on X and let Q be a packet system on Y. For any map F:X→ Y, define F_ Q← P = A_ QF S_ P. Thus F_ Q← P((x_α)_α) = ( Q_γ F(∑_α x_α) )_γ.
Exact mathematical passage
F_{\Qcal\leftarrow\Pcal} = \Acal_{\Qcal}F\Scal_{\Pcal}.
f_{s(a)}Definition passage · 1 passage
Search aliases: f_s(a)
Definition passage: Typed operator grammar · 02_foundations.tex, line 53
linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a∈ Q_1).
Exact mathematical passage
f_{t(a)}X_a=Y_af_{s(a)} \qquad(a\in Q_1).
f_{t(a)}Definition passage · 1 passage
Search aliases: f_t(a)
Definition passage: Typed operator grammar · 02_foundations.tex, line 53
linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a∈ Q_1).
Exact mathematical passage
f_{t(a)}X_a=Y_af_{s(a)} \qquad(a\in Q_1).
f_vDefinition passage · 1 passage
Search aliases: f_v
Definition passage: Typed operator grammar · 02_foundations.tex, line 52
l on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a∈ Q_1).
Exact mathematical passage
f_v:X_v\to Y_v
\FcalDefinition passage · 7 passages
Search aliases: Fcal F mathcal
Definition passage: Sobolev step packet class · Paper_V_Traditional_Final.tex, line 2521
[Sobolev step packet class] Fix r≥1, 1≤ p≤∞, and an integer Q≥0. The class F_r,p(Q) consists of functions f=s+∑_q=1^Qa_qϕ_0,τ_q with s∈ W^r,p(0,1) and 0<τ_1<⋯<τ_Q<1. No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.
Exact mathematical passage
\Fcal_{r,p}(Q)
\Fcal_{r,p}Definition passage · 1 passage
Search aliases: Fcal_r,p F_r,p Fcal mathcal
Definition passage: Sobolev step packet class · Paper_V_Traditional_Final.tex, line 2521
[Sobolev step packet class] Fix r≥1, 1≤ p≤∞, and an integer Q≥0. The class F_r,p(Q) consists of functions f=s+∑_q=1^Qa_qϕ_0,τ_q with s∈ W^r,p(0,1) and 0<τ_1<⋯<τ_Q<1. No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.
Exact mathematical passage
\Fcal_{r,p}(Q)
\FFDefinition passage · 74 passages
Search aliases: FF F mathbb
Definition passage: Quadratic refinement · Paper_XI_Traditional_Final.tex, line 1470
[Quadratic refinement] A quadratic refinement of b is a map q:V→ F_2 satisfying q(x+y)-q(x)-q(y)=b(x,y).
Exact mathematical passage
q:V\to\FF_2
\FF_2Definition passage · 13 passages
Search aliases: FF_2 F_2 FF mathbb
Definition passage: Quadratic refinement · Paper_XI_Traditional_Final.tex, line 1470
[Quadratic refinement] A quadratic refinement of b is a map q:V→ F_2 satisfying q(x+y)-q(x)-q(y)=b(x,y).
Exact mathematical passage
q:V\to\FF_2
\operatorname{Fin}Definition passage · 4 passages
Search aliases: Fin operatorname
Definition passage: Finitary colored operation signature · 02_connections_obstructions.tex, line 831
[Finitary colored operation signature] A finitary colored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a dependent inductive typed tree. No such declaration is part of the finite algebraic completion contract.
Exact mathematical passage
\operatorname{in}_\sigma:\operatorname{Fin}(r(\sigma))\to V
\flopDefinition passage · 4 passages
Search aliases: flop operatorname
Definition passage: Arithmetic intensity · 01_manuscript.tex, line 389
[Arithmetic intensity] When the modeled byte traffic is strictly positive, the modeled arithmetic intensity is I = fracflopbytes.
Exact mathematical passage
\Ical = \frac{\flop}{\bytes}.
\frakaDefinition passage · 23 passages
Search aliases: fraka a mathfrak
Definition passage: Ordinary patching datum · Paper_XII_Traditional_Final.tex, line 730
[Ordinary patching datum] An ordinary patching datum consists of a Noetherian ring R_∞, an ideal a⊆ R_∞, a finite R_∞ module M_∞, and identifications R≅ R_∞/ a, M≅ M_∞/ a M_∞. In automorphy lifting applications, R_∞ is a patched deformation ring and M_∞ is a patched automorphic module. The definition itself contains no automorphy assertion.
Exact mathematical passage
\fraka\subseteq R_\infty
\frakpDefinition passage · 42 passages
Search aliases: frakp p mathfrak
Definition passage: Support and near faithfulness · Paper_XII_Traditional_Final.tex, line 576
[Support and near faithfulness] Let R be a commutative ring and let M be an R module. Its support is Supp_R(M) = { p∈Spec R:M_ p≠0}. If R is Noetherian and M is finite, then M is nearly faithful when Ann_R(M) is a nilpotent ideal.
Exact mathematical passage
\Supp_R(M) = \{\frakp\in\Spec R:M_{\frakp}\neq0\}.
GDefinition passage · 388 passages
Search aliases: G
Definition passage: Transfer graph of a family generating the algebra · 03_packets_and_graphs.tex, line 927
l=(P_α)_α∈ I be a finite packet system on X with every P_α∈ A. Let G⊆ A satisfy A=alg_𝕜({P_α:α∈ I}∪ G), where alg_𝕜(E) denotes the unital subalgebra over 𝕜 generated by E. The transfer graph Γ( G, P) has vertex set I and an arrow α⟶β when P_β GP_α≠0 for at least one G∈ G. Reachability includes paths of length zero.
Exact mathematical passage
P_\beta GP_\alpha\neq0
\mathcal GDefinition passage · 10 passages
Search aliases: G mathcal
Definition passage: Transfer graph of a family generating the algebra · 03_packets_and_graphs.tex, line 918
[Transfer graph of a family generating the algebra] Let P=(P_α)_α∈ I be a finite packet system on X with every P_α∈ A. Let G⊆ A satisfy A=alg_𝕜({P_α:α∈ I}∪ G), where alg_𝕜(E) denotes the unital subalgebra over 𝕜 generated by E. The transfer graph Γ( G, P) has vertex set I and an arrow α⟶β when P_β GP_α≠0 for at least one G∈ G. Reachability includes paths of length zero.
Exact mathematical passage
\mathcal G\subseteq\A
gDefinition passage · 367 passages
Search aliases: g
Definition passage: Separation weight · 04_arithmetic.tex, line 186
[Separation weight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG preserves Λ. The minimal Lean theorem needs only the lattice action of u^-1. The bundled wrapper for lattice units records both u and u^-1. A nonzero s…
Exact mathematical passage
v,g
g_1Definition passage · 6 passages
Search aliases: g_1
Definition passage: Grade respecting operation · 03_nonlinear_coordinates.tex, line 574
[Grade respecting operation] An r multilinear map μ:A^r→ A respects the grading when μ(A_g_1,…,A_g_r) ⊆ A_g_1+⋯+g_r.
Exact mathematical passage
\mu(A_{g_1},\ldots,A_{g_r}) \subseteq A_{g_1+\cdots+g_r}.
G_ADefinition passage · 4 passages
Search aliases: G_A
Definition passage: Support graph · 01_manuscript.tex, line 138
[Support graph] The support graph G_A has vertex set I. It has a directed edge α⟶β when P_β AP_α≠0.
Exact mathematical passage
G_A
G_aDefinition passage · 12 passages
Search aliases: G_a
Definition passage: Gauge change · 02_connections_obstructions.tex, line 312
[Gauge change] A gauge is a family of automorphisms G_a∈Aut_R(C_a). It acts on a connection by Γ^G_b← a = G_bΓ_b← aG_a^-1. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.Gauge; Komorebi.PaperIII.Transport.ReversibleConnection.gauge
Exact mathematical passage
G_a\in\operatorname{Aut}_R(C_a)
G_a^{-1}Definition passage · 11 passages
Search aliases: G_a^-1
Definition passage: Gauge change · 02_connections_obstructions.tex, line 314
[Gauge change] A gauge is a family of automorphisms G_a∈Aut_R(C_a). It acts on a connection by Γ^G_b← a = G_bΓ_b← aG_a^-1. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.Gauge; Komorebi.PaperIII.Transport.ReversibleConnection.gauge
Exact mathematical passage
\Gamma^G_{b\leftarrow a} = G_b\Gamma_{b\leftarrow a}G_a^{-1}.
G_bDefinition passage · 6 passages
Search aliases: G_b
Definition passage: Gauge change · 02_connections_obstructions.tex, line 314
[Gauge change] A gauge is a family of automorphisms G_a∈Aut_R(C_a). It acts on a connection by Γ^G_b← a = G_bΓ_b← aG_a^-1. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.Gauge; Komorebi.PaperIII.Transport.ReversibleConnection.gauge
Exact mathematical passage
\Gamma^G_{b\leftarrow a} = G_b\Gamma_{b\leftarrow a}G_a^{-1}.
g_rDefinition passage · 3 passages
Search aliases: g_r
Definition passage: Grade respecting operation · 03_nonlinear_coordinates.tex, line 574
[Grade respecting operation] An r multilinear map μ:A^r→ A respects the grading when μ(A_g_1,…,A_g_r) ⊆ A_g_1+⋯+g_r.
Exact mathematical passage
\mu(A_{g_1},\ldots,A_{g_r}) \subseteq A_{g_1+\cdots+g_r}.
G_u^{-1}Definition passage · 1 passage
Search aliases: G_u^-1
Definition passage: Gauge change · 04_correction_transport.tex, line 653
[Gauge change] A gauge is a family G_v∈GL(X). It changes the edge transports to T'_vu=G_vT_vuG_u^-1.
Exact mathematical passage
T'_{vu}=G_vT_{vu}G_u^{-1}.
G_vDefinition passage · 2 passages
Search aliases: G_v
Definition passage: Gauge change · 04_correction_transport.tex, line 652
[Gauge change] A gauge is a family G_v∈GL(X). It changes the edge transports to T'_vu=G_vT_vuG_u^-1.
Exact mathematical passage
G_v\in\operatorname{GL}(X)
g_vDefinition passage · 5 passages
Search aliases: g_v
Definition passage: Decision gap · 01_manuscript.tex, line 1694
[Decision gap] At an internal node v, let g_v = absK_λ(v)-S_λ(v). This is the gap between the keep and split scores after the child subproblems have been solved.
Exact mathematical passage
g_v = \abs{K_\lambda(v)-S_\lambda(v)}.
G_{\Lambda,\Pcal}Definition passage · 12 passages
Search aliases: G_Λ,Pcal G_Λ, P Lambda Pcal Lambda mathcal
Definition passage: Lattice packet data · 04_arithmetic.tex, line 246
[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.
Exact mathematical passage
G_{\Lambda,\Pcal} = \Lambda/\Lambda_{\mathrm{split}}.
\GLDefinition passage · 23 passages
Search aliases: GL operatorname
Definition passage: One break Clifford datum · Paper_XI_Traditional_Final.tex, line 1012
[One break Clifford datum] A one break Clifford datum consists of finite groups and representations Ntriangleleft G, A=G/N, Π:G→GL( X), ρ:N→GL(W), with the following properties.
Exact mathematical passage
N\triangleleft G, \qquad A=G/N, \qquad \Pi:G\to\GL(\calX), \qquad \rho:N\to\GL(W),
\operatorname{GL}Definition passage · 19 passages
Search aliases: GL operatorname
Definition passage: Gauge change · 04_correction_transport.tex, line 652
[Gauge change] A gauge is a family G_v∈GL(X). It changes the edge transports to T'_vu=G_vT_vuG_u^-1.
Exact mathematical passage
G_v\in\operatorname{GL}(X)
\mathcal HDefinition passage · 26 passages
Search aliases: H mathcal
Definition passage: Hurwitz integral series · strengthening.tex, line 42
[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.
Exact mathematical passage
\mathcal H_p=\left\{\sum_{d\geq0}f_d\frac{t^d}{d!}: f_d\in\mathbb Z_p\right\} \subset\mathbb Q_p[[t]].
H^*Definition passage · 9 passages
Search aliases: H^*
Definition passage: Finite quantum seed system · 01_finite_core.tex, line 54
inition:001 A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltonian when dynamics are present, and τ specifies the topology used in a later completion.
Exact mathematical passage
H=H^*
H^2Definition passage · 4 passages
Search aliases: H^2
Definition passage: Clifford cocycle class · Paper_XI_Traditional_Final.tex, line 1065
[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.
Exact mathematical passage
[c_\rho]\in H^2(A,\CC^{\times})
H_{c,b,a}Definition passage · 12 passages
Search aliases: H_c,b,a
Definition passage: Triangle holonomy · 02_connections_obstructions.tex, line 192
[Triangle holonomy] The triangle holonomy based at a is H_c,b,a = Γ_a← c Γ_c← b Γ_b← a ∈Aut_R(C_a). It traverses a→ b→ c→ a. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.triangleHolonomy; Komorebi.PaperIII.Transport.ReversibleConnection.triangleObstruction
Exact mathematical passage
H_{c,b,a} = \Gamma_{a\leftarrow c} \Gamma_{c\leftarrow b} \Gamma_{b\leftarrow a} \in\operatorname{Aut}_R(C_a).
H_NDefinition passage · 20 passages
Search aliases: H_N
Definition passage · Paper_IV_Traditional_Final.tex, line 1636
Set H_N(z)= E_N,0(z) = ∑_n≥0fracz^Nn(Nn)!. For u∈ C, define symupΘ_N,n(u) by H_N(uz)^-1H_N(z)^-u = ∑_n≥0symupΘ_N,n(u)fracz^Nn(Nn)!. This is first a formal identity at the origin. Analytically, it is interpreted on a common simply connected zero-free region with the branch of log H_N normalized by log H_N(0)=0.
Exact mathematical passage
H_N(z)=\Efun_{N,0}(z) = \sum_{n\geq0}\frac{z^{Nn}}{(Nn)!}.
h_NDefinition passage · 6 passages
Search aliases: h_N
Definition passage · Paper_IV_Traditional_Final.tex, line 1794
For θ∈ R, define h_N(θ) = max_0≤ k<N cos(θ+frac(2k+1)πN). The tie directions are T_N = {frac2π kN:0≤ k<N} bmod 2π.
Exact mathematical passage
h_N(\theta) = \max_{0\leq k<N} \cos\left(\theta+\frac{(2k+1)\pi}{N}\right).
\mathcal H_pDefinition passage · 12 passages
Search aliases: H_p mathcal
Definition passage: Hurwitz integral series · strengthening.tex, line 42
[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.
Exact mathematical passage
\mathcal H_p=\left\{\sum_{d\geq0}f_d\frac{t^d}{d!}: f_d\in\mathbb Z_p\right\} \subset\mathbb Q_p[[t]].
h_{rs}Definition passage · 5 passages
Search aliases: h_rs
Definition passage: Packet transfer kernel · Paper_V_Traditional_Final.tex, line 1919
[Packet transfer kernel] For output residue r∈ R and input residue s∈ R, define a kernel on L by h_rs(m)=h(r-s+m), m∈ L.
Exact mathematical passage
h_{rs}(m)=h(r-s+m), \qquad m\in L.
h_\thetaDefinition passage · 4 passages
Search aliases: h_θ theta
Definition passage: Bridge datum · Paper_XI_Traditional_Final.tex, line 1197
item a one break Galois envelope (V_ρ,b_ρ) item a type side quotient (V_θ,h_θ) item an equivariant symplectic isometry ι:V_ρ⟶ V_θ for a finite cyclic stabilizer C=⟨ c⟩ item the same nontrivial central character on the two Heisenberg groups item when p=2, compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensions lifting ι.
Exact mathematical passage
(V_\theta,h_\theta)
H_\ellDefinition passage · 10 passages
Search aliases: H_ℓ ell
Definition passage: Finite quantum refinement tower · 02_strengthened.tex, line 2348
[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is recorded by the corresponding operator norm defects.
Exact mathematical passage
H_\ell
\hboxDefinition passage · 9 passages
Search aliases: hbox
Definition passage: Within packet information loss · Paper_VI_Traditional_Final.tex, line 3560
[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.
Exact mathematical passage
\eta_i(p) = \lambda_i-\overline\lambda_\alpha(p) \quad\hbox{when }i\in I_\alpha.
\HcalDefinition passage · 68 passages
Search aliases: Hcal H mathcal
Definition passage: Unitary packet tree · Paper_V_Traditional_Final.tex, line 2334
[Unitary packet tree] A finite unitary packet tree consists of Hilbert spaces H_v at its vertices and, at every internal vertex v, a unitary map U_v: H_v⟶⊕_w∈child(v) H_w. An admissible pruning is a set of vertices that meets every root to terminal path exactly once.
Exact mathematical passage
\Hcal_v
\Hcal_vDefinition passage · 3 passages
Search aliases: Hcal_v H_v Hcal mathcal
Definition passage: Unitary packet tree · Paper_V_Traditional_Final.tex, line 2334
[Unitary packet tree] A finite unitary packet tree consists of Hilbert spaces H_v at its vertices and, at every internal vertex v, a unitary map U_v: H_v⟶⊕_w∈child(v) H_w. An admissible pruning is a set of vertices that meets every root to terminal path exactly once.
Exact mathematical passage
\Hcal_v
\Hcal_wDefinition passage · 1 passage
Search aliases: Hcal_w H_w Hcal mathcal
Definition passage: Unitary packet tree · Paper_V_Traditional_Final.tex, line 2335
[Unitary packet tree] A finite unitary packet tree consists of Hilbert spaces H_v at its vertices and, at every internal vertex v, a unitary map U_v: H_v⟶⊕_w∈child(v) H_w. An admissible pruning is a set of vertices that meets every root to terminal path exactly once.
Exact mathematical passage
U_v:\Hcal_v\longrightarrow\bigoplus_{w\in\operatorname{child}(v)}\Hcal_w.
\Hcal_\ellDefinition passage · 5 passages
Search aliases: Hcal_ℓ H_ℓ Hcal ell mathcal ell
Definition passage: Finite quantum refinement tower · 02_strengthened.tex, line 2344
[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is recorded by the corresponding operator norm defects.
Exact mathematical passage
\Hcal_\ell
\Hcal_{\ell+1}Definition passage · 3 passages
Search aliases: Hcal_ℓ+1 H_ℓ+1 Hcal ell mathcal ell
Definition passage: Finite quantum refinement tower · 02_strengthened.tex, line 2345
[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is recorded by the corresponding operator norm defects.
Exact mathematical passage
R_\ell:\Hcal_\ell\longrightarrow\Hcal_{\ell+1},
\HomDefinition passage · 10 passages
Search aliases: Hom operatorname
Definition passage: Transfer block · 03_packets_and_graphs.tex, line 495
[Transfer block] For T∈End_𝕜(X) and a finite packet system P, define T_βα:=P_β TP_α ∈Hom_𝕜(X_α,X_β).
Exact mathematical passage
T_{\beta\alpha}:=P_\beta TP_\alpha \in\Hom_{\bbk}(X_\alpha,X_\beta).
\Hom_{\bbk}Definition passage · 2 passages
Search aliases: Hom_bbk Hom_𝕜 Hom bbk operatorname Bbbk
Definition passage: Transfer block · 03_packets_and_graphs.tex, line 495
[Transfer block] For T∈End_𝕜(X) and a finite packet system P, define T_βα:=P_β TP_α ∈Hom_𝕜(X_α,X_β).
Exact mathematical passage
T_{\beta\alpha}:=P_\beta TP_\alpha \in\Hom_{\bbk}(X_\alpha,X_\beta).
\Hom_NDefinition passage · 1 passage
Search aliases: Hom_N Hom operatorname
Definition passage: One break Clifford datum · Paper_XI_Traditional_Final.tex, line 1032
The multiplicity space is M_ρ=Hom_N(W, X).
Exact mathematical passage
M_\rho=\Hom_N(W,\calX).
\Hom_RDefinition passage · 1 passage
Search aliases: Hom_R Hom operatorname
Definition passage: Cocycle defect · 02_connections_obstructions.tex, line 176
[Cocycle defect] For a,b,c∈ B, define the additive cocycle defect Δ_c,b,a = Γ_c← bΓ_b← a -Γ_c← a ∈Hom_R(C_a,C_c). statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.cocycleDefect
Exact mathematical passage
\Delta_{c,b,a} = \Gamma_{c\leftarrow b}\Gamma_{b\leftarrow a} -\Gamma_{c\leftarrow a} \in\Hom_R(C_a,C_c).
\Hull_{\A}Definition passage · 10 passages
Search aliases: Hull_A Hull_ A Hull A operatorname mathcal
Definition passage: Generated hull · 02_foundations.tex, line 158
[Generated hull] For S⊆ X, define Hull_ A(S):=∑_s∈ S A s. In the topological regime define Hull_ A,τ(S):=clHull_ A(S)^ τ. For a singleton S={s}, write Hull_ A(s).
Exact mathematical passage
\Hull_{\A}(S):=\sum_{s\in S}\A s.
\Hull_{\A,\tau}Definition passage · 1 passage
Search aliases: Hull_A,τ Hull_ A,τ Hull A tau operatorname mathcal tau
Definition passage: Generated hull · 02_foundations.tex, line 162
[Generated hull] For S⊆ X, define Hull_ A(S):=∑_s∈ S A s. In the topological regime define Hull_ A,τ(S):=clHull_ A(S)^ τ. For a singleton S={s}, write Hull_ A(s).
Exact mathematical passage
\Hull_{\A,\tau}(S):=\cl{\Hull_{\A}(S)}^{\,\tau}.
\Hull_{\B}Definition passage · 7 passages
Search aliases: Hull_B Hull_ B Hull B operatorname mathcal
Definition passage: Morphism of seeded systems · 04_transport.tex, line 80
[Morphism of seeded systems] A morphism (X, A,S, P)⟶(Y, B,S', Q) is a packet compatible operator morphism satisfying the seed condition Φ(S)⊆Hull_ B(S'). In the topological category all maps and actions are required to be continuous and the seed condition uses the topological hull when declared.
Exact mathematical passage
\Phi(S)\subseteq\Hull_{\B}(S').
\mathrm{hypotheses}Definition passage · 1 passage
Search aliases: hypotheses mathrm
Definition passage: Canonical theorem record · 02_foundations.tex, line 17
[Canonical theorem record] A canonical theorem record is the tuple (ID,status,hypotheses,statement, dependencies,proof,exports,boundary). A later chapter cites the stable record ID. A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this chapter.
Exact mathematical passage
(\mathrm{ID},\mathrm{status},\mathrm{hypotheses},\mathrm{statement}, \mathrm{dependencies},\mathrm{proof},\mathrm{exports},\mathrm{boundary}).
iDefinition passage · 890 passages
Search aliases: i
Definition passage: Weighted transfer series · 03_weighted.tex, line 374
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j
Exact mathematical passage
A(t)_{ij} = \sum_{e:i\to j}m(e)t^{w(e)}.
I_{\Ccal}Definition passage · 1 passage
Search aliases: I_Ccal I_ C Ccal mathcal
Definition passage: Code compression obstruction · 02_strengthened.tex, line 1277
[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.
Exact mathematical passage
q_P:\End(\Ccal)\longrightarrow\End(\Ccal)/(\CC I_{\Ccal})
I_XDefinition passage · 20 passages
Search aliases: I_X
Definition passage: Finite packet system · 03_packets_and_graphs.tex, line 17
[Finite packet system] A finite packet system on a vector space X over 𝕜 is a family P=(P_α)_α∈ I⊆End_𝕜(X) satisfying P_α^2=P_α, P_α P_β=0 (α≠β), ∑_α∈ IP_α=I_X. Set X_α=P_α X. Zero packet spaces may be removed without changing any identity. No orthogonality is assumed unless stated.
Exact mathematical passage
P_\alpha^2=P_\alpha, \qquad P_\alpha P_\beta=0\quad(\alpha\neq\beta), \qquad \sum_{\alpha\in I}P_\alpha=I_X.
I_\alphaDefinition passage · 27 passages
Search aliases: I_α alpha
Definition passage: Within packet information loss · Paper_VI_Traditional_Final.tex, line 3560
[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.
Exact mathematical passage
\eta_i(p) = \lambda_i-\overline\lambda_\alpha(p) \quad\hbox{when }i\in I_\alpha.
\IcalDefinition passage · 2 passages
Search aliases: Ical I mathcal
Definition passage: Arithmetic intensity · 01_manuscript.tex, line 389
[Arithmetic intensity] When the modeled byte traffic is strictly positive, the modeled arithmetic intensity is I = fracflopbytes.
Exact mathematical passage
\Ical = \frac{\flop}{\bytes}.
\idDefinition passage · 31 passages
Search aliases: id mathrm
Definition passage: Noncanonical connection normalized at each chart · 02_connections_obstructions.tex, line 18
[Noncanonical connection normalized at each chart] A minimal connection is a family Γ_b← a∈Aut_R(M) (a,b∈ B) with only Γ_a← a=id_M. Neither inversion on reversed edges nor the cocycle law is a field of the structure. statusleanthreeKomorebi.PaperIII.Transport.Connection
Exact mathematical passage
\Gamma_{a\leftarrow a}=\id_M.
\mathrm{ID}Definition passage · 1 passage
Search aliases: ID mathrm
Definition passage: Canonical theorem record · 02_foundations.tex, line 17
[Canonical theorem record] A canonical theorem record is the tuple (ID,status,hypotheses,statement, dependencies,proof,exports,boundary). A later chapter cites the stable record ID. A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this chapter.
Exact mathematical passage
(\mathrm{ID},\mathrm{status},\mathrm{hypotheses},\mathrm{statement}, \mathrm{dependencies},\mathrm{proof},\mathrm{exports},\mathrm{boundary}).
\id_{C_a}Definition passage · 4 passages
Search aliases: id_C_a id mathrm
Definition passage: Holonomy obstruction and packet residuals · 02_connections_obstructions.tex, line 472
[Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy within one packet. Off-diagonal blocks detect packet mixing induced by holonomy. statusleanthreeKomorebi.PaperIII.HolonomyBlocks.PacketResolution; Komorebi…
Exact mathematical passage
\Ob_{c,b,a}=H_{c,b,a}-\id_{C_a}, \qquad \Ob^{\mu\nu}_{c,b,a} = \Pi^a_\mu\Ob_{c,b,a}\Pi^a_\nu.
\id_{C_{a_0}}Definition passage · 1 passage
Search aliases: id_C_a_0 id mathrm
Definition passage: Connection transport along a path · 02_connections_obstructions.tex, line 267
[Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transport.ReversibleConnection.indexedPathTransport
Exact mathematical passage
\Gamma_p = \Gamma_{a_n\leftarrow a_{n-1}}\cdots \Gamma_{a_1\leftarrow a_0}, \qquad \Gamma_{(a_0)}=\id_{C_{a_0}}.
\id_MDefinition passage · 2 passages
Search aliases: id_M id mathrm
Definition passage: Noncanonical connection normalized at each chart · 02_connections_obstructions.tex, line 18
[Noncanonical connection normalized at each chart] A minimal connection is a family Γ_b← a∈Aut_R(M) (a,b∈ B) with only Γ_a← a=id_M. Neither inversion on reversed edges nor the cocycle law is a field of the structure. statusleanthreeKomorebi.PaperIII.Transport.Connection
Exact mathematical passage
\Gamma_{a\leftarrow a}=\id_M.
\iiDefinition passage · 95 passages
Search aliases: ii i mathrm
Definition passage: Anomaly free continuum constraint system · 02_strengthened.tex, line 2540
[Anomaly free continuum constraint system] A continuum constraint system consists of a Hilbert space, a common invariant dense domain D, and operators C_a on D. It is anomaly free for prescribed structure coefficients when [C_a,C_b]ψ = i∑_cf_ab^phantomabcC_cψ for every ψ∈ D, and when the associated gauge evolution is defined on the physical quotient or physical subspace.
Exact mathematical passage
[C_a,C_b]\psi = \ii\sum_cf_{ab}^{\phantom{ab}c}C_c\psi
\imDefinition passage · 49 passages
Search aliases: im operatorname
Definition passage: Realized operator algebra · 02_foundations.tex, line 117
The realized action and faithful reduction. [Realized operator algebra] For a realization X, the realized operator algebra is A_X:=im (Λ_ G⟶End_𝕜(|X|)). It records the actual operators in the realization and may be a proper quotient of the abstract grammar algebra.
Exact mathematical passage
\A_X:=\im\!\left(\Lambda_{\mathfrak G}\longrightarrow\End_{\bbk}(|X|)\right).
\operatorname{in}Definition passage · 2 passages
Search aliases: in operatorname
Definition passage: Finitary colored operation signature · 02_connections_obstructions.tex, line 831
[Finitary colored operation signature] A finitary colored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a dependent inductive typed tree. No such declaration is part of the finite algebraic completion contract.
Exact mathematical passage
\operatorname{in}_\sigma:\operatorname{Fin}(r(\sigma))\to V
\operatorname{in}_\sigmaDefinition passage · 2 passages
Search aliases: in_σ operatorname sigma
Definition passage: Finitary colored operation signature · 02_connections_obstructions.tex, line 831
[Finitary colored operation signature] A finitary colored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a dependent inductive typed tree. No such declaration is part of the finite algebraic completion contract.
Exact mathematical passage
\operatorname{in}_\sigma:\operatorname{Fin}(r(\sigma))\to V
\infDefinition passage · 3 passages
Search aliases: inf
Definition passage: Approximate Knill--Laflamme defect · 02_strengthened.tex, line 1423
[Approximate Knill--Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.
Exact mathematical passage
\varepsilon_{\rm KL} = \max_{a,b} \inf_{c\in\CC} \norm{PE_a^*E_bP-cP}.
\inf_{c\in\CC}Definition passage · 2 passages
Search aliases: inf_c∈CC inf_c∈ C inf in CC inf in mathbb
Definition passage: Approximate Knill--Laflamme defect · 02_strengthened.tex, line 1423
[Approximate Knill--Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.
Exact mathematical passage
\varepsilon_{\rm KL} = \max_{a,b} \inf_{c\in\CC} \norm{PE_a^*E_bP-cP}.
\operatorname{input}Definition passage · 1 passage
Search aliases: input operatorname
Definition passage: Finite grammar for operation trees · 02_connections_obstructions.tex, line 772
[Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the finite formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No algebraic, continuity, or differentiability law is imposed on these semantics. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.OperationTree; Kom…
Exact mathematical passage
\begin{aligned} t ::={}&\operatorname{input}(i) \mid\operatorname{constant}(k) \mid\operatorname{unary}(u,t)\\ &\mid\operatorname{binary}(b,t_1,t_2). \end{aligned}
\IRDefinition passage · 2 passages
Search aliases: IR mathsf
Definition passage: Packet operator representation · 01_manuscript.tex, line 80
[Packet operator representation] A finite packet operator representation consists of IR=( P,E,(B_βα)_(β,α)∈ E), where E⊆ I× I and B_βα:X_α⟶ X_β. Local blocks are identified with their ambient extensions by inclusion and projection. In particular, the notation P_β AP_α denotes either the ambient block or its restriction X_α→ X_β according to the displayed domain, and ι_β B_βαπ_α is the corresponding ambient extension. Its evaluation is eval(IR) = ∑_(β,α)∈ E ι_β B_βαπ_α, where ι_β:X_β→ X is the co…
Exact mathematical passage
\IR=(\Pcal,E,(B_{\beta\alpha})_{(\beta,\alpha)\in E}),
\operatorname{IsSolution}Definition passage · 2 passages
Search aliases: IsSolution operatorname
Definition passage: Solution and obstruction packet for a candidate · 02_connections_obstructions.tex, line 409
[Solution and obstruction packet for a candidate] A candidate x is a solution when IsSolution_r(x) Longleftrightarrow ∀ p, r(x,p)=0. An obstruction packet for x is a pair consisting of a named packet p and a certificate r(x,p)≠0. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.IsSolution; Komorebi.PaperIII.NonlinearObstructions.ObstructionPacket
Exact mathematical passage
\operatorname{IsSolution}_r(x) \quad\Longleftrightarrow\quad \forall p,\ r(x,p)=0.
\operatorname{IsSolution}_rDefinition passage · 2 passages
Search aliases: IsSolution_r operatorname
Definition passage: Solution and obstruction packet for a candidate · 02_connections_obstructions.tex, line 409
[Solution and obstruction packet for a candidate] A candidate x is a solution when IsSolution_r(x) Longleftrightarrow ∀ p, r(x,p)=0. An obstruction packet for x is a pair consisting of a named packet p and a certificate r(x,p)≠0. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.IsSolution; Komorebi.PaperIII.NonlinearObstructions.ObstructionPacket
Exact mathematical passage
\operatorname{IsSolution}_r(x) \quad\Longleftrightarrow\quad \forall p,\ r(x,p)=0.
J^{r}_{\gamma\mid\alpha_1,\ldots,\alpha_r}Definition passage · 4 passages
Search aliases: J^r_γmidα_1,…,α_r gamma mid alpha ldots alpha
Definition passage: Packet jet · 03_nonlinear_coordinates.tex, line 220
[Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.
Exact mathematical passage
J^{r}_{\gamma\mid\alpha_1,\ldots,\alpha_r}F(x) = Q_\gamma D^rF(x) \big|_{X_{\alpha_1}\times\cdots\times X_{\alpha_r}}.
J^X_{b\leftarrow a}Definition passage · 8 passages
Search aliases: J^X_b← a leftarrow
Definition passage: Canonical exact coordinate transport · 01_finite_coordinates.tex, line 472
[Canonical exact coordinate transport] For two exact packet systems on the same ambient module, define J^X_b← a = A_b S_a: Coord( P^a)⟶Coord( P^b). The arrow points from the source coordinate system a to the target coordinate system b. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates; Komorebi.PaperII.NestedRefinement.coordinateRefinementEquiv
Exact mathematical passage
J^X_{b\leftarrow a} = \cA_b\cS_a: \Coord(\cP^a)\longrightarrow\Coord(\cP^b).
J_{b\leftarrow a}Definition passage · 12 passages
Search aliases: J_b← a leftarrow
Definition passage: Calibration and flatness · 02_connections_obstructions.tex, line 141
[Calibration and flatness] When the C_a are exact packet coordinates of one ambient module, the connection is canonically calibrated if Γ_b← a=J_b← a for every a,b. It is flat if Γ_c← bΓ_b← a = Γ_c← a for every a,b,c. The canonical connection is flat. A general flat connection is proved in [source reference] to be related to it by a gauge change. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.CocycleAt; Komorebi.PaperIII.Transport.ReversibleConnection.Flat; Komorebi.PaperIII.Tra…
Exact mathematical passage
\Gamma_{b\leftarrow a}=J_{b\leftarrow a} \quad\text{for every }a,b.
J_NDefinition passage · 17 passages
Search aliases: J_N
Definition passage · Paper_IV_Traditional_Final.tex, line 824
Let J_N be the signed cyclic shift on the standard basis e_0,…,e_N-1 of C^N, defined by J_Ne_j=e_j+1 (0≤ j<N-1), J_Ne_N-1=-e_0. Then J_N^N=-I.
Exact mathematical passage
J_N
J_N^NDefinition passage · 2 passages
Search aliases: J_N^N
Definition passage · Paper_IV_Traditional_Final.tex, line 831
Let J_N be the signed cyclic shift on the standard basis e_0,…,e_N-1 of C^N, defined by J_Ne_j=e_j+1 (0≤ j<N-1), J_Ne_N-1=-e_0. Then J_N^N=-I.
Exact mathematical passage
J_N^N=-I
J_{\Qcal\leftarrow\Pcal}Definition passage · 6 passages
Search aliases: J_Qcal←Pcal J_ Q← P Qcal leftarrow Pcal mathcal leftarrow mathcal
Definition passage: Coordinate change across levels · 02_coordinates.tex, line 84
[Coordinate change across levels] For two packet systems P and Q on X, define J_ Q← P = A_ Q S_ P:X^ P⟶ X^ Q and C_ P← Q = A_ P S_ Q:X^ Q⟶ X^ P. When Q refines P, the first map resolves each coarse coordinate into its children and the second adds the children back to their parent.
Exact mathematical passage
J_{\Qcal\leftarrow\Pcal} = \Acal_{\Qcal}\Scal_{\Pcal}:X^{\Pcal}\longrightarrow X^{\Qcal}
KDefinition passage · 597 passages
Search aliases: K
Definition passage: Lattice packet data · 04_arithmetic.tex, line 239
[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.
Exact mathematical passage
K
kDefinition passage · 1060 passages
Search aliases: k
Definition passage: Finite grammar for operation trees · 02_connections_obstructions.tex, line 772
[Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the finite formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No algebraic, continuity, or differentiability law is imposed on these semantics. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.OperationTree; Kom…
Exact mathematical passage
\begin{aligned} t ::={}&\operatorname{input}(i) \mid\operatorname{constant}(k) \mid\operatorname{unary}(u,t)\\ &\mid\operatorname{binary}(b,t_1,t_2). \end{aligned}
K_\lambdaDefinition passage · 9 passages
Search aliases: K_λ lambda
Definition passage: Decision gap · 01_manuscript.tex, line 1694
[Decision gap] At an internal node v, let g_v = absK_λ(v)-S_λ(v). This is the gap between the keep and split scores after the child subproblems have been solved.
Exact mathematical passage
g_v = \abs{K_\lambda(v)-S_\lambda(v)}.
\KcalDefinition passage · 25 passages
Search aliases: Kcal K mathcal
Definition passage: Krylov space · 01_manuscript.tex, line 1096
[Krylov space] For a linear operator A, a vector v, and a positive integer m, define K_m(A,v) = span {v,Av,…,A^m-1v}.
Exact mathematical passage
\Kcal_m(A,v) = \operatorname{span} \{v,Av,\ldots,A^{m-1}v\}.
\Kcal_mDefinition passage · 2 passages
Search aliases: Kcal_m K_m Kcal mathcal
Definition passage: Krylov space · 01_manuscript.tex, line 1096
[Krylov space] For a linear operator A, a vector v, and a positive integer m, define K_m(A,v) = span {v,Av,…,A^m-1v}.
Exact mathematical passage
\Kcal_m(A,v) = \operatorname{span} \{v,Av,\ldots,A^{m-1}v\}.
\KerDefinition passage · 46 passages
Search aliases: Ker operatorname
Definition passage: Canonical lumping obstruction · Paper_VI_Traditional_Final.tex, line 2854
[Canonical lumping obstruction] For a linear fine transition operator K, define Ob_C(K) = (CK)|_Ker C ∈ B(Ker C, R^m).
Exact mathematical passage
\Ob_C(K) = (CK)|_{\Ker C} \in \mathcal B(\Ker C,\RR^m).
\mathbf LDefinition passage · 9 passages
Search aliases: L mathbf
Definition passage: Derived support · Paper_XII_Traditional_Final.tex, line 813
[Derived support] Let R be a commutative ring and let C be a perfect complex of R modules. Its derived support is Supp_R(C) = { p∈Spec R: Cmathbinmathop⊗limits_R^ Lκ( p) not≃0 }.
Exact mathematical passage
\Supp_R(C) = \left\{ \frakp\in\Spec R: C\mathbin{\mathop{\otimes}\limits_R^{\mathbf L}}\kappa(\frakp) \not\simeq0 \right\}.
lDefinition passage · 19 passages
Search aliases: l
Definition passage: Universal closure obstruction · 02_strengthened.tex, line 2186
[Universal closure obstruction] Let q_ c:End( H)⟶End( H)/ c be the quotient map. Define Ob_ab^rm cl = q_ c([C_a,C_b]).
Exact mathematical passage
\Ob_{ab}^{\rm cl} = q_{\mathfrak c}([C_a,C_b]).
L_eDefinition passage · 21 passages
Search aliases: L_e
Definition passage: Packet graph and scalar edge system · 01_finite_coordinates.tex, line 77
0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.edgeOp; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp
Exact mathematical passage
L_e=\operatorname{packetMatrix}(\src(e),\tgt(e);c(e)), \qquad T=\sum_{e\in E}L_e.
L_{e_1}Definition passage · 4 passages
Search aliases: L_e_1
Definition passage: Composable word and execution operator · 01_finite_coordinates.tex, line 179
[Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordOp
Exact mathematical passage
L_w=L_{e_n}\cdots L_{e_1}, \qquad L_{\varnothing}=I.
L_{e_n}Definition passage · 4 passages
Search aliases: L_e_n
Definition passage: Composable word and execution operator · 01_finite_coordinates.tex, line 179
[Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordOp
Exact mathematical passage
L_w=L_{e_n}\cdots L_{e_1}, \qquad L_{\varnothing}=I.
L_wDefinition passage · 5 passages
Search aliases: L_w
Definition passage: Composable word and execution operator · 01_finite_coordinates.tex, line 179
[Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordOp
Exact mathematical passage
L_w=L_{e_n}\cdots L_{e_1}, \qquad L_{\varnothing}=I.
L_\lambdaDefinition passage · 3 passages
Search aliases: L_λ lambda
Definition passage: Hurwitz integral series · strengthening.tex, line 48
[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.
Exact mathematical passage
D=\frac{d}{dt},\qquad L_\lambda=\frac{D^N-\lambda}{p}.
L_\lambda^kDefinition passage · 1 passage
Search aliases: L_λ^k lambda
Definition passage: Hurwitz integral series · strengthening.tex, line 51
[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.
Exact mathematical passage
L_\lambda^k A\in\mathcal H_p
L_{\varnothing}Definition passage · 1 passage
Search aliases: L_∅ varnothing
Definition passage: Composable word and execution operator · 01_finite_coordinates.tex, line 179
[Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordOp
Exact mathematical passage
L_w=L_{e_n}\cdots L_{e_1}, \qquad L_{\varnothing}=I.
\leadstoDefinition passage · 8 passages
Search aliases: leadsto
Definition passage: Weighted transfer series · 03_weighted.tex, line 386
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|γ|m(e_ν). Here γ=(e_1,…,e_|γ|) is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the en…
Exact mathematical passage
a_n=\sum_{i,j}\ \sum_{\substack{\gamma:i\leadsto j\\W(\gamma)=n}} u_i v_j\prod_{\nu=1}^{|\gamma|}m(e_\nu).
\limDefinition passage · 7 passages
Search aliases: lim
Definition passage: Branching dimension · Paper_X_Block27_Amended.tex, line 612
[Branching dimension] When the limit exists, define the finite-level counting exponent dim_seed(bar x) = lim_m→∞ fraclog_q#X_m(bar x)m. A limsup may be used in general.
Exact mathematical passage
\dim_{\mathrm{seed}}(\bar x) = \lim_{m\to\infty} \frac{\log_q\#X_m(\bar x)}{m}.
\lim_{m\to\infty}Definition passage · 1 passage
Search aliases: lim_m→∞ lim to infty
Definition passage: Branching dimension · Paper_X_Block27_Amended.tex, line 612
[Branching dimension] When the limit exists, define the finite-level counting exponent dim_seed(bar x) = lim_m→∞ fraclog_q#X_m(bar x)m. A limsup may be used in general.
Exact mathematical passage
\dim_{\mathrm{seed}}(\bar x) = \lim_{m\to\infty} \frac{\log_q\#X_m(\bar x)}{m}.
\mathrm{loc}Definition passage · 11 passages
Search aliases: loc mathrm
Definition passage: Global deformation datum · Paper_XII_Traditional_Final.tex, line 215
m A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_R^locR^loc,τ, R^loc,τ:=ma…
Exact mathematical passage
R^{\mathrm{loc}}:=\wideotimes_{v\in S,\OO}R_v^\square \longrightarrow R^\square.
\LogDefinition passage · 3 passages
Search aliases: Log operatorname
Definition passage · Paper_IV_Traditional_Final.tex, line 2907
Choose a logarithm for each member of the finite exceptional part of Λ_N. Use the real logarithm for the eventual positive sequence. Define Z_N^orb(s) = ∑_λ∈Λ_Ne^-sLogλ. This series is initially defined on Re s>1/N.
Exact mathematical passage
\mathfrak Z_N^{\mathrm{orb}}(s) = \sum_{\lambda\in\Lambda_N}e^{-s\Log\lambda}.
\logDefinition passage · 127 passages
Search aliases: log
Definition passage · Paper_IV_Traditional_Final.tex, line 958
For r∈ C, define S_N^(r)(z) = D_N(z)^-r = exp(-rlog D_N(z)). Write S_N^(r)(z) = ∑_n≥0P_N,n(r)fracz^Nn(Nn)!. Define the logarithmic coefficients κ_N,m by -logΔ_N(w) = ∑_m≥1κ_N,mfracw^mm.
Exact mathematical passage
\Sfun_N^{(r)}(z) = \Dfun_N(z)^{-r} = \exp\bigl(-r\log\Dfun_N(z)\bigr).
\log_qDefinition passage · 1 passage
Search aliases: log_q log
Definition passage: Branching dimension · Paper_X_Block27_Amended.tex, line 612
[Branching dimension] When the limit exists, define the finite-level counting exponent dim_seed(bar x) = lim_m→∞ fraclog_q#X_m(bar x)m. A limsup may be used in general.
Exact mathematical passage
\dim_{\mathrm{seed}}(\bar x) = \lim_{m\to\infty} \frac{\log_q\#X_m(\bar x)}{m}.
\longleftarrowDefinition passage · 2 passages
Search aliases: longleftarrow
Definition passage: Stable core · Paper_XII_Traditional_Final.tex, line 1192
[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).
Exact mathematical passage
V_1\longleftarrow V_2\longleftarrow V_3\longleftarrow\cdots
\LongleftrightarrowDefinition passage · 55 passages
Search aliases: Longleftrightarrow
Definition passage: Solution and obstruction packet for a candidate · 02_connections_obstructions.tex, line 409
[Solution and obstruction packet for a candidate] A candidate x is a solution when IsSolution_r(x) Longleftrightarrow ∀ p, r(x,p)=0. An obstruction packet for x is a pair consisting of a named packet p and a certificate r(x,p)≠0. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.IsSolution; Komorebi.PaperIII.NonlinearObstructions.ObstructionPacket
Exact mathematical passage
\operatorname{IsSolution}_r(x) \quad\Longleftrightarrow\quad \forall p,\ r(x,p)=0.
\mathcal MDefinition passage · 16 passages
Search aliases: M mathcal
Definition passage: Residual seed system · Paper_X_Block27_Amended.tex, line 162
[Residual seed system] For bar x∈ M(k), define the residual fiber M_bar x(A) = {x∈ M(A):red_A(x)=bar x}. The point bar x is the residual root. The family of all nonempty fibers is the residual special fiber, and the formal deformation problems above them form the residual forest.
Exact mathematical passage
\bar x\in\mathcal M(k)
\mathfrak MDefinition passage · 14 passages
Search aliases: M mathfrak
Definition passage: Support relation and support space · 03_packets_and_graphs.tex, line 792
Support patterns, preorders, and triangular structure. [Support relation and support space] For a relation R⊆ I× I, with (β,α) interpreted as an allowed transfer α→β, define M_R := {T∈End(X):T_βα=0 whenever (β,α)∉ R}.
Exact mathematical passage
\mathfrak M_R := \{T\in\End(X):T_{\beta\alpha}=0\text{ whenever }(\beta,\alpha)\notin R\}.
M^rDefinition passage · 1 passage
Search aliases: M^r
Definition passage: Seed evaluation map · 02_foundations.tex, line 217
gives a canonical map from a finite free module. The generated hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.
Exact mathematical passage
S=(s_1,\ldots,s_r)\in M^r
\mathcal M_{\bar x}Definition passage · 5 passages
Search aliases: M_bar x mathcal bar
Definition passage: Residual seed system · Paper_X_Block27_Amended.tex, line 163
[Residual seed system] For bar x∈ M(k), define the residual fiber M_bar x(A) = {x∈ M(A):red_A(x)=bar x}. The point bar x is the residual root. The family of all nonempty fibers is the residual special fiber, and the formal deformation problems above them form the residual forest.
Exact mathematical passage
\mathcal M_{\bar x}(A) = \{x\in\mathcal M(A):\red_A(x)=\bar x\}.
M_{\frakp}Definition passage · 10 passages
Search aliases: M_frakp M_ p frakp mathfrak
Definition passage: Support and near faithfulness · Paper_XII_Traditional_Final.tex, line 576
[Support and near faithfulness] Let R be a commutative ring and let M be an R module. Its support is Supp_R(M) = { p∈Spec R:M_ p≠0}. If R is Noetherian and M is finite, then M is nearly faithful when Ann_R(M) is a nilpotent ideal.
Exact mathematical passage
\Supp_R(M) = \{\frakp\in\Spec R:M_{\frakp}\neq0\}.
\mathfrak M_RDefinition passage · 11 passages
Search aliases: M_R mathfrak
Definition passage: Support relation and support space · 03_packets_and_graphs.tex, line 792
Support patterns, preorders, and triangular structure. [Support relation and support space] For a relation R⊆ I× I, with (β,α) interpreted as an allowed transfer α→β, define M_R := {T∈End(X):T_βα=0 whenever (β,α)∉ R}.
Exact mathematical passage
\mathfrak M_R := \{T\in\End(X):T_{\beta\alpha}=0\text{ whenever }(\beta,\alpha)\notin R\}.
M_\rhoDefinition passage · 7 passages
Search aliases: M_ρ rho
Definition passage: One break Clifford datum · Paper_XI_Traditional_Final.tex, line 1032
The multiplicity space is M_ρ=Hom_N(W, X).
Exact mathematical passage
M_\rho=\Hom_N(W,\calX).
M_\inftyDefinition passage · 8 passages
Search aliases: M_∞ infty
Definition passage: Ordinary patching datum · Paper_XII_Traditional_Final.tex, line 730
[Ordinary patching datum] An ordinary patching datum consists of a Noetherian ring R_∞, an ideal a⊆ R_∞, a finite R_∞ module M_∞, and identifications R≅ R_∞/ a, M≅ M_∞/ a M_∞. In automorphy lifting applications, R_∞ is a patched deformation ring and M_∞ is a patched automorphic module. The definition itself contains no automorphy assertion.
Exact mathematical passage
M_\infty
\mathbinDefinition passage · 17 passages
Search aliases: mathbin
Definition passage: Derived support · Paper_XII_Traditional_Final.tex, line 813
[Derived support] Let R be a commutative ring and let C be a perfect complex of R modules. Its derived support is Supp_R(C) = { p∈Spec R: Cmathbinmathop⊗limits_R^ Lκ( p) not≃0 }.
Exact mathematical passage
\Supp_R(C) = \left\{ \frakp\in\Spec R: C\mathbin{\mathop{\otimes}\limits_R^{\mathbf L}}\kappa(\frakp) \not\simeq0 \right\}.
\mathopDefinition passage · 9 passages
Search aliases: mathop
Definition passage: Derived support · Paper_XII_Traditional_Final.tex, line 813
[Derived support] Let R be a commutative ring and let C be a perfect complex of R modules. Its derived support is Supp_R(C) = { p∈Spec R: Cmathbinmathop⊗limits_R^ Lκ( p) not≃0 }.
Exact mathematical passage
\Supp_R(C) = \left\{ \frakp\in\Spec R: C\mathbin{\mathop{\otimes}\limits_R^{\mathbf L}}\kappa(\frakp) \not\simeq0 \right\}.
\maxDefinition passage · 84 passages
Search aliases: max
Definition passage: Lattice packet data · 04_arithmetic.tex, line 258
et Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.
Exact mathematical passage
s_\Lambda(\Pcal)=\max\bigl(\{0\}\cup\{s_\Lambda(E_\alpha):\alpha\in I\}\bigr).
\max_{0\leq k<N}Definition passage · 1 passage
Search aliases: max_0≤ k<N max leq
Definition passage · Paper_IV_Traditional_Final.tex, line 1794
For θ∈ R, define h_N(θ) = max_0≤ k<N cos(θ+frac(2k+1)πN). The tie directions are T_N = {frac2π kN:0≤ k<N} bmod 2π.
Exact mathematical passage
h_N(\theta) = \max_{0\leq k<N} \cos\left(\theta+\frac{(2k+1)\pi}{N}\right).
\max_{a,b}Definition passage · 1 passage
Search aliases: max_a,b max
Definition passage: Approximate Knill--Laflamme defect · 02_strengthened.tex, line 1423
[Approximate Knill--Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.
Exact mathematical passage
\varepsilon_{\rm KL} = \max_{a,b} \inf_{c\in\CC} \norm{PE_a^*E_bP-cP}.
\midDefinition passage · 128 passages
Search aliases: mid
Definition passage: Finite grammar for operation trees · 02_connections_obstructions.tex, line 772
[Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the finite formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No algebraic, continuity, or differentiability law is imposed on these semantics. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.OperationTree; Kom…
Exact mathematical passage
\begin{aligned} t ::={}&\operatorname{input}(i) \mid\operatorname{constant}(k) \mid\operatorname{unary}(u,t)\\ &\mid\operatorname{binary}(b,t_1,t_2). \end{aligned}
\minDefinition passage · 134 passages
Search aliases: min
Definition passage: Seed rank · 02_foundations.tex, line 373
[Seed rank] For a unital left module M over A, define srank_A(M):=min{|S|:AS=M}. For a topological module (X,τ), define srank_ A,τ(X):=min{|S|:cl A S^ τ=X}. Set srank_A(0)=0. When the value is finite, it is the minimal number of module generators.
Exact mathematical passage
\srank_A(M):=\min\{|S|:AS=M\}.
\mathrm{mov}Definition passage · 1 passage
Search aliases: mov mathrm
Definition passage: Moving phase · Paper_XI_Traditional_Final.tex, line 1403
[Moving phase] Under the hypotheses of [source reference], define Φ_p^mov(g,V) = Φ_p(g|_im(g-I),im(g-I)).
Exact mathematical passage
\Phi_p^{\mathrm{mov}}(g,V) = \Phi_p\left(g|_{\im(g-I)},\im(g-I)\right).
\NDefinition passage · 62 passages
Search aliases: N mathbb
Definition passage: Finitary colored operation signature · 02_connections_obstructions.tex, line 829
[Finitary colored operation signature] A finitary colored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a dependent inductive typed tree. No such declaration is part of the finite algebraic completion contract.
Exact mathematical passage
r(\sigma)\in\N
\mathbb NDefinition passage · 12 passages
Search aliases: N mathbb
Definition passage: Unconditional packet sum space · 03_packets_and_graphs.tex, line 359
[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .
Exact mathematical passage
\|x\|_u := \sup_{F\Subset\mathbb N} \sup_{|a_n|\leq1} \left\|\sum_{n\in F}a_nx_n\right\|.
nDefinition passage · 1807 passages
Search aliases: n
Definition passage: Unconditional packet sum space · 03_packets_and_graphs.tex, line 354
[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .
Exact mathematical passage
(X_n)_{n\geq1}
N_eDefinition passage · 7 passages
Search aliases: N_e
Definition passage: Complete multiplicative filtration · 03_weighted.tex, line 20
ef:filtered_algebra A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
N_e
n_\alphaDefinition passage · 11 passages
Search aliases: n_α alpha
Definition passage: Packet layout · 01_manuscript.tex, line 70
[Packet layout] A packet layout is a finite family of finite dimensional Hilbert spaces P=(X_α)_α∈ I with direct sum X=⊕_α∈ IX_α. Write n_α=dim X_α, n=∑_α∈ In_α. The coordinate inclusion and projection are ι_α:X_α→ X and π_α:X→ X_α. The ambient coordinate projector is P_α=ι_απ_α.
Exact mathematical passage
n_\alpha=\dim X_\alpha, \qquad n=\sum_{\alpha\in I}n_\alpha.
N_\deltaDefinition passage · 6 passages
Search aliases: N_δ delta
Definition passage: Linear tube bound · Paper_V_Traditional_Final.tex, line 2834
[Linear tube bound] A boundary Γ has a linear tube bound if there are constants C_Γ and δ_0>0 such that μ(N_δ(Γ)) ≤ C_Γδ for 0<δ<δ_0.
Exact mathematical passage
\mu(N_\delta(\Gamma)) \leq C_\Gamma\delta
\operatorname{node}Definition passage · 1 passage
Search aliases: node operatorname
Definition passage: Typed operation tree and evaluation · 02_connections_obstructions.tex, line 848
[Typed operation tree and evaluation] A typed operation tree is recursively either a leaf of declared color or a node labeled by σ, with one correctly colored child for each input slot of σ. Evaluation is recursive: ev(node(σ;t_1,…,t_r(σ))) = Φ_σ(ev(t_1),…,ev(t_r(σ))). statusroadmapCandidate API: leannameKomorebi.PaperIII.NonlinearObstructions.ColoredOperationTree and leannameColoredOperationTree.eval. This is stronger than the compiled uncolored unary--binary grammar.
Exact mathematical passage
\ev(\operatorname{node}(\sigma;t_1,\ldots,t_{r(\sigma)})) = \Phi_\sigma(\ev(t_1),\ldots,\ev(t_{r(\sigma)})).
\normDefinition passage · 205 passages
Search aliases: norm
Definition passage: Approximate Knill--Laflamme defect · 02_strengthened.tex, line 1423
[Approximate Knill--Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.
Exact mathematical passage
\varepsilon_{\rm KL} = \max_{a,b} \inf_{c\in\CC} \norm{PE_a^*E_bP-cP}.
\notDefinition passage · 13 passages
Search aliases: not
Definition passage: Visibility weight · Paper_X_Block27_Amended.tex, line 476
[Visibility weight] For two integral packet coordinates a,b∈ O, define their separation depth w(a,b)=min{m≥1:anot≡ bpmodϖ^m}, when such an m exists. A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.
Exact mathematical passage
w(a,b)=\min\{m\ge1:a\not\equiv b\pmod{\varpi^m}\},
\ObDefinition passage · 43 passages
Search aliases: Ob operatorname
Definition passage: Holonomy obstruction and packet residuals · 02_connections_obstructions.tex, line 472
[Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy within one packet. Off-diagonal blocks detect packet mixing induced by holonomy. statusleanthreeKomorebi.PaperIII.HolonomyBlocks.PacketResolution; Komorebi…
Exact mathematical passage
\Ob_{c,b,a}=H_{c,b,a}-\id_{C_a}, \qquad \Ob^{\mu\nu}_{c,b,a} = \Pi^a_\mu\Ob_{c,b,a}\Pi^a_\nu.
\obDefinition passage · 1 passage
Search aliases: ob o mathfrak
Definition passage: Code compression obstruction · 02_strengthened.tex, line 1285
[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.
Exact mathematical passage
\ob_P(A) = PAP- \frac{\Tr(PAP)}{k}P.
\Ob^{\mu\nu}_{c,b,a}Definition passage · 5 passages
Search aliases: Ob^μν_c,b,a Ob mu nu operatorname mu nu
Definition passage: Holonomy obstruction and packet residuals · 02_connections_obstructions.tex, line 472
[Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy within one packet. Off-diagonal blocks detect packet mixing induced by holonomy. statusleanthreeKomorebi.PaperIII.HolonomyBlocks.PacketResolution; Komorebi…
Exact mathematical passage
\Ob_{c,b,a}=H_{c,b,a}-\id_{C_a}, \qquad \Ob^{\mu\nu}_{c,b,a} = \Pi^a_\mu\Ob_{c,b,a}\Pi^a_\nu.
\Ob_{ab}^{\rm cl}Definition passage · 2 passages
Search aliases: Ob_ab^rm cl Ob rm operatorname rm
Definition passage: Universal closure obstruction · 02_strengthened.tex, line 2186
[Universal closure obstruction] Let q_ c:End( H)⟶End( H)/ c be the quotient map. Define Ob_ab^rm cl = q_ c([C_a,C_b]).
Exact mathematical passage
\Ob_{ab}^{\rm cl} = q_{\mathfrak c}([C_a,C_b]).
\Ob_CDefinition passage · 3 passages
Search aliases: Ob_C Ob operatorname
Definition passage: Canonical lumping obstruction · Paper_VI_Traditional_Final.tex, line 2854
[Canonical lumping obstruction] For a linear fine transition operator K, define Ob_C(K) = (CK)|_Ker C ∈ B(Ker C, R^m).
Exact mathematical passage
\Ob_C(K) = (CK)|_{\Ker C} \in \mathcal B(\Ker C,\RR^m).
\Ob_{c,b,a}Definition passage · 4 passages
Search aliases: Ob_c,b,a Ob operatorname
Definition passage: Holonomy obstruction and packet residuals · 02_connections_obstructions.tex, line 472
[Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy within one packet. Off-diagonal blocks detect packet mixing induced by holonomy. statusleanthreeKomorebi.PaperIII.HolonomyBlocks.PacketResolution; Komorebi…
Exact mathematical passage
\Ob_{c,b,a}=H_{c,b,a}-\id_{C_a}, \qquad \Ob^{\mu\nu}_{c,b,a} = \Pi^a_\mu\Ob_{c,b,a}\Pi^a_\nu.
\Ob_DDefinition passage · 4 passages
Search aliases: Ob_D Ob operatorname
Definition passage: Linear correction datum · 03_nonlinear_coordinates.tex, line 912
[Linear correction datum] A linear correction datum consists of a homomorphism D:U⟶ V and a defect r∈ V. A correction is an element u∈ U satisfying Du=-r. The obstruction class is Ob_D(r)=[r]∈coker D=V/im D.
Exact mathematical passage
\Ob_D(r)=[r]\in\coker D=V/\im D.
\Ob_mDefinition passage · 3 passages
Search aliases: Ob_m Ob operatorname
Definition passage: Depth obstruction · 04_correction_transport.tex, line 240
[Depth obstruction] The depth m obstruction of x_m is Ob_m(x_m) = [Φ(x_m)] ∈ cokeroverline L_m, where Φ(x_m) is first viewed in F^mY/F^m+1Y.
Exact mathematical passage
\Ob_m(x_m) = [\Phi(x_m)] \in \coker\overline L_m,
\Ob_PDefinition passage · 2 passages
Search aliases: Ob_P Ob operatorname
Definition passage: Code compression obstruction · 02_strengthened.tex, line 1281
[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.
Exact mathematical passage
\Ob_P(A)=q_P(PAP).
\ob_PDefinition passage · 1 passage
Search aliases: ob_P o_P ob mathfrak
Definition passage: Code compression obstruction · 02_strengthened.tex, line 1285
[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.
Exact mathematical passage
\ob_P(A) = PAP- \frac{\Tr(PAP)}{k}P.
\OcalDefinition passage · 41 passages
Search aliases: Ocal O mathcal
Definition passage: Visibility weight · Paper_X_Block27_Amended.tex, line 475
[Visibility weight] For two integral packet coordinates a,b∈ O, define their separation depth w(a,b)=min{m≥1:anot≡ bpmodϖ^m}, when such an m exists. A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.
Exact mathematical passage
a,b\in\Ocal
\mathrm{odd}Definition passage · 5 passages
Search aliases: odd mathrm
Definition passage: Odd-length source and inverse coefficients · 20_arithmetic.tex, line 767
rse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd. Primality is needed only to interpret the result as a prime-adic congruence ladder.
Exact mathematical passage
T^{\mathrm{odd}}_{p,M}(n)=\sum_{m=0}^{\min(M,n)} \binom{2n}{2m}\gamma_m p^{2m}S_{2(n-m)}(p).
\one_{r+L}Definition passage · 1 passage
Search aliases: one_r+L 1_r+L one mathbf
Definition passage: Lattice packet · Paper_V_Traditional_Final.tex, line 1770
[Lattice packet] For r∈ R, define the coordinate projector (P_rx)(n)= 1_r+L(n)x(n). The family (P_r)_r∈ R is called the packet system associated with L and R.
Exact mathematical passage
(P_rx)(n)=\one_{r+L}(n)x(n).
\OODefinition passage · 112 passages
Search aliases: OO O mathcal
Definition passage: Separation weight · 04_arithmetic.tex, line 179
[Separation weight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG preserves Λ. The minimal Lean theorem needs only the lattice action of u^-1. The bundled wrapper for lattice units records both u and u^-1. A nonzero s…
Exact mathematical passage
\OO
\mathrm{orb}Definition passage · 4 passages
Search aliases: orb mathrm
Definition passage · Paper_IV_Traditional_Final.tex, line 2907
Choose a logarithm for each member of the finite exceptional part of Λ_N. Use the real logarithm for the eventual positive sequence. Define Z_N^orb(s) = ∑_λ∈Λ_Ne^-sLogλ. This series is initially defined on Re s>1/N.
Exact mathematical passage
\mathfrak Z_N^{\mathrm{orb}}(s) = \sum_{\lambda\in\Lambda_N}e^{-s\Log\lambda}.
\operatorname{out}Definition passage · 2 passages
Search aliases: out operatorname
Definition passage: Finitary colored operation signature · 02_connections_obstructions.tex, line 832
[Finitary colored operation signature] A finitary colored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a dependent inductive typed tree. No such declaration is part of the finite algebraic completion contract.
Exact mathematical passage
\operatorname{out}(\sigma)\in V
\overline LDefinition passage · 26 passages
Search aliases: overline L overline
Definition passage: Depth obstruction · 04_correction_transport.tex, line 240
[Depth obstruction] The depth m obstruction of x_m is Ob_m(x_m) = [Φ(x_m)] ∈ cokeroverline L_m, where Φ(x_m) is first viewed in F^mY/F^m+1Y.
Exact mathematical passage
\Ob_m(x_m) = [\Phi(x_m)] \in \coker\overline L_m,
\overline L_mDefinition passage · 8 passages
Search aliases: overline L_m overline
Definition passage: Depth obstruction · 04_correction_transport.tex, line 240
[Depth obstruction] The depth m obstruction of x_m is Ob_m(x_m) = [Φ(x_m)] ∈ cokeroverline L_m, where Φ(x_m) is first viewed in F^mY/F^m+1Y.
Exact mathematical passage
\Ob_m(x_m) = [\Phi(x_m)] \in \coker\overline L_m,
\overline\lambdaDefinition passage · 13 passages
Search aliases: overlineλ overline lambda
Definition passage: Within packet information loss · Paper_VI_Traditional_Final.tex, line 3560
[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.
Exact mathematical passage
\eta_i(p) = \lambda_i-\overline\lambda_\alpha(p) \quad\hbox{when }i\in I_\alpha.
\overline\lambda_\alphaDefinition passage · 12 passages
Search aliases: overlineλ_α overline lambda alpha
Definition passage: Within packet information loss · Paper_VI_Traditional_Final.tex, line 3560
[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.
Exact mathematical passage
\eta_i(p) = \lambda_i-\overline\lambda_\alpha(p) \quad\hbox{when }i\in I_\alpha.
pDefinition passage · 1971 passages
Search aliases: p
Definition passage: Connection transport along a path · 02_connections_obstructions.tex, line 266
[Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transport.ReversibleConnection.indexedPathTransport
Exact mathematical passage
p=(a_0,a_1,\ldots,a_n)
p^{2m}Definition passage · 7 passages
Search aliases: p^2m
Definition passage: Odd-length source and inverse coefficients · 20_arithmetic.tex, line 767
rse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd. Primality is needed only to interpret the result as a prime-adic congruence ladder.
Exact mathematical passage
T^{\mathrm{odd}}_{p,M}(n)=\sum_{m=0}^{\min(M,n)} \binom{2n}{2m}\gamma_m p^{2m}S_{2(n-m)}(p).
p^aDefinition passage · 33 passages
Search aliases: p^a
Definition passage: Hurwitz integral series · strengthening.tex, line 47
[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.
Exact mathematical passage
N=p^a
p^LDefinition passage · 86 passages
Search aliases: p^L
Definition passage: Differences and valuation conventions · 20_arithmetic.tex, line 110
put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ν_p(u)=v_p(u) for u>0 and set ν_p(0)=0. This finite convention makes the displayed exponents meaningful at st…
Exact mathematical passage
x\in p^L\Z_p
P_aDefinition passage · 138 passages
Search aliases: P_a
Definition passage: Two sided operator packet · 02_strengthened.tex, line 944
[Two sided operator packet] For a complete orthogonal family (P_a), define P_ab(X)=P_aXP_b. The range B_ab=P_aEnd( H)P_b is the operator packet with output address a and input address b.
Exact mathematical passage
(P_a)
P_bDefinition passage · 66 passages
Search aliases: P_b
Definition passage: Two sided operator packet · 02_strengthened.tex, line 945
[Two sided operator packet] For a complete orthogonal family (P_a), define P_ab(X)=P_aXP_b. The range B_ab=P_aEnd( H)P_b is the operator packet with output address a and input address b.
Exact mathematical passage
\Pcal_{ab}(X)=P_aXP_b.
P_CDefinition passage · 6 passages
Search aliases: P_C
Definition passage: Packet pure seed rank · 03_packets_and_graphs.tex, line 1020
[Packet pure seed rank] A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).
Exact mathematical passage
P_C:=\sum_{\alpha\in C}P_\alpha, \qquad X_C:=P_CX, \qquad \A_C:=P_C\A P_C,
p_iDefinition passage · 49 passages
Search aliases: p_i
Definition passage: Relative information · Paper_VI_Traditional_Final.tex, line 3639
[Relative information] For probability vectors q and p, define D(qVert p) = ∑_iq_ilogfracq_ip_i. The convention is 0log0=0. A term with q_i>0 and p_i=0 has value +∞.
Exact mathematical passage
D(q\Vert p) = \sum_iq_i\log\frac{q_i}{p_i}.
P_{N,n}Definition passage · 23 passages
Search aliases: P_N,n
Definition passage · Paper_IV_Traditional_Final.tex, line 966
For r∈ C, define S_N^(r)(z) = D_N(z)^-r = exp(-rlog D_N(z)). Write S_N^(r)(z) = ∑_n≥0P_N,n(r)fracz^Nn(Nn)!. Define the logarithmic coefficients κ_N,m by -logΔ_N(w) = ∑_m≥1κ_N,mfracw^mm.
Exact mathematical passage
\Sfun_N^{(r)}(z) = \sum_{n\geq0}P_{N,n}(r)\frac{z^{Nn}}{(Nn)!}.
P_uDefinition passage · 1 passage
Search aliases: P_u
Definition passage: Complete packet pruning · Paper_V_Traditional_Final.tex, line 900
[Complete packet pruning] Let T be a finite rooted refinement tree. Every node v carries a projector P_v, the root satisfies P_∅=I, and every internal node satisfies P_v=∑_u∈ch(v)P_u, P_uP_w=0 (u≠ w in ch(v)). A set Λ of nodes is a complete pruning if it is an antichain and every root-to-terminal path meets Λ exactly once.
Exact mathematical passage
P_v=\sum_{u\in\operatorname{ch}(v)}P_u, \qquad P_uP_w=0\quad(u\ne w\text{ in }\operatorname{ch}(v)).
P_vDefinition passage · 23 passages
Search aliases: P_v
Definition passage: Common arithmetic spine · Paper_X_Block27_Amended.tex, line 763
[Common arithmetic spine] Let M be a number field. A common spine C consists of: item characteristic polynomials P_v(X)∈ M[X] at almost all places; item coefficient-prime independent weight and determinant data; item prescribed local Weil--Deligne or inertial packet labels at the ramified places. For each finite place λ of M, let X_λ,m( C) be the finite set of admissible objects modulo ϖ_λ^m realizing the reduction of this spine.
Exact mathematical passage
P_v(X)\in M[X]
P_wDefinition passage · 1 passage
Search aliases: P_w
Definition passage: Complete packet pruning · Paper_V_Traditional_Final.tex, line 900
[Complete packet pruning] Let T be a finite rooted refinement tree. Every node v carries a projector P_v, the root satisfies P_∅=I, and every internal node satisfies P_v=∑_u∈ch(v)P_u, P_uP_w=0 (u≠ w in ch(v)). A set Λ of nodes is a complete pruning if it is an antichain and every root-to-terminal path meets Λ exactly once.
Exact mathematical passage
P_v=\sum_{u\in\operatorname{ch}(v)}P_u, \qquad P_uP_w=0\quad(u\ne w\text{ in }\operatorname{ch}(v)).
P_\alpha^2Definition passage · 4 passages
Search aliases: P_α^2 alpha
Definition passage: Finite packet system · 03_packets_and_graphs.tex, line 17
[Finite packet system] A finite packet system on a vector space X over 𝕜 is a family P=(P_α)_α∈ I⊆End_𝕜(X) satisfying P_α^2=P_α, P_α P_β=0 (α≠β), ∑_α∈ IP_α=I_X. Set X_α=P_α X. Zero packet spaces may be removed without changing any identity. No orthogonality is assumed unless stated.
Exact mathematical passage
P_\alpha^2=P_\alpha, \qquad P_\alpha P_\beta=0\quad(\alpha\neq\beta), \qquad \sum_{\alpha\in I}P_\alpha=I_X.
P_{\sigma^{-1}(j)}Definition passage · 5 passages
Search aliases: P_σ^-1(j) sigma
Definition passage: Packet compatible operator morphism · 04_transport.tex, line 59
orphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):= ∑_σ(α)=jP_α. An empty fiber is interpreted as the zero projector.
Exact mathematical passage
Q_j\Phi = \Phi P_{\sigma^{-1}(j)}, \qquad P_{\sigma^{-1}(j)}:= \sum_{\sigma(\alpha)=j}P_\alpha.
P_{\ell,a}Definition passage · 3 passages
Search aliases: P_ℓ,a ell
Definition passage: Finite quantum refinement tower · 02_strengthened.tex, line 2348
[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is recorded by the corresponding operator norm defects.
Exact mathematical passage
P_{\ell,a}
P_{\varnothing}Definition passage · 2 passages
Search aliases: P_∅ varnothing
Definition passage: Complete packet pruning · Paper_V_Traditional_Final.tex, line 899
[Complete packet pruning] Let T be a finite rooted refinement tree. Every node v carries a projector P_v, the root satisfies P_∅=I, and every internal node satisfies P_v=∑_u∈ch(v)P_u, P_uP_w=0 (u≠ w in ch(v)). A set Λ of nodes is a complete pruning if it is an antichain and every root-to-terminal path meets Λ exactly once.
Exact mathematical passage
P_{\varnothing}=I
\operatorname{packetMatrix}Definition passage · 4 passages
Search aliases: packetMatrix operatorname
Definition passage: Packet graph and scalar edge system · 01_finite_coordinates.tex, line 67
[Packet graph and scalar edge system] For a,b∈ V and r∈ R, define packetMatrix(a,b;r)_yx = r, · y=b and x=a, 0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem; Komorebi.PaperIII.EdgeOperators.PacketGr…
Exact mathematical passage
\operatorname{packetMatrix}(a,b;r)_{yx} = \begin{cases} r,&y=b\text{ and }x=a,\\ 0,&\text{otherwise}. \end{cases}
\Pcal_{ab}Definition passage · 4 passages
Search aliases: Pcal_ab P_ab Pcal mathcal
Definition passage: Two sided operator packet · 02_strengthened.tex, line 945
[Two sided operator packet] For a complete orthogonal family (P_a), define P_ab(X)=P_aXP_b. The range B_ab=P_aEnd( H)P_b is the operator packet with output address a and input address b.
Exact mathematical passage
\Pcal_{ab}(X)=P_aXP_b.
\perpbigDefinition passage · 4 passages
Search aliases: perpbig mathop⊥limits mathop perp limits
Definition passage: Orthogonal break decomposition · Paper_XI_Traditional_Final.tex, line 1752
[Orthogonal break decomposition] An orthogonal break decomposition is a symplectic direct sum V=mathop⊥limits_s∈ SV_s that is preserved by the declared tame stabilizer. Each V_s is the finite symplectic packet assigned to one break.
Exact mathematical passage
V=\perpbig_{s\in S}V_s
\perpbig_{s\in S}Definition passage · 1 passage
Search aliases: perpbig_s∈ S mathop⊥limits_s∈ S perpbig in mathop perp limits in
Definition passage: Orthogonal break decomposition · Paper_XI_Traditional_Final.tex, line 1752
[Orthogonal break decomposition] An orthogonal break decomposition is a symplectic direct sum V=mathop⊥limits_s∈ SV_s that is preserved by the declared tame stabilizer. Each V_s is the finite symplectic packet assigned to one break.
Exact mathematical passage
V=\perpbig_{s\in S}V_s
\mathrm{proof}Definition passage · 1 passage
Search aliases: proof mathrm
Definition passage: Canonical theorem record · 02_foundations.tex, line 17
[Canonical theorem record] A canonical theorem record is the tuple (ID,status,hypotheses,statement, dependencies,proof,exports,boundary). A later chapter cites the stable record ID. A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this chapter.
Exact mathematical passage
(\mathrm{ID},\mathrm{status},\mathrm{hypotheses},\mathrm{statement}, \mathrm{dependencies},\mathrm{proof},\mathrm{exports},\mathrm{boundary}).
\mathsf{PSC}Definition passage · 1 passage
Search aliases: PSC mathsf
Definition passage: Chapter I export object · 04_transport.tex, line 486
[Chapter I export object] The canonical Chapter I export is PSC_I= ( c typed grammars and realized operator algebras, seed systems, free presentations, and seed rank, module-top, semisimple, and single-operator formulas, finite and unconditional packet coordinates, transfer blocks and support algebras, reachability and packet-pure source generation, morphisms, invariant exact sequences, quotients, completions, realization functors and equational theorem transport ).
Exact mathematical passage
\boxed{ \mathsf{PSC}_I= \left( \begin{array}{c} \text{typed grammars and realized operator algebras},\\ \text{seed systems, free presentations, and seed rank},\\ \text{module-top, semisimple, and single-operator formulas},\\ \text{finite and unconditional packet coordinates},\\ \text{transfer blocks and support algebras},\\ \text{reachability and packet-pure source generation},\\ \text{morphisms, invariant exact sequences, quotients, completions},\\ \text{realization functors and equational theorem transport} \end{array} \right). }
\mathsf{PSC}_IDefinition passage · 1 passage
Search aliases: PSC_I mathsf
Definition passage: Chapter I export object · 04_transport.tex, line 486
[Chapter I export object] The canonical Chapter I export is PSC_I= ( c typed grammars and realized operator algebras, seed systems, free presentations, and seed rank, module-top, semisimple, and single-operator formulas, finite and unconditional packet coordinates, transfer blocks and support algebras, reachability and packet-pure source generation, morphisms, invariant exact sequences, quotients, completions, realization functors and equational theorem transport ).
Exact mathematical passage
\boxed{ \mathsf{PSC}_I= \left( \begin{array}{c} \text{typed grammars and realized operator algebras},\\ \text{seed systems, free presentations, and seed rank},\\ \text{module-top, semisimple, and single-operator formulas},\\ \text{finite and unconditional packet coordinates},\\ \text{transfer blocks and support algebras},\\ \text{reachability and packet-pure source generation},\\ \text{morphisms, invariant exact sequences, quotients, completions},\\ \text{realization functors and equational theorem transport} \end{array} \right). }
\psrank_{\A,\Pcal}Definition passage · 5 passages
Search aliases: psrank_A,Pcal psrank_ A, P psrank A Pcal operatorname mathcal mathcal
Definition passage: Packet pure seed rank · 03_packets_and_graphs.tex, line 1014
[Packet pure seed rank] A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).
Exact mathematical passage
\psrank_{\A,\Pcal}(X) := \min\{|S|:S\text{ is packet-pure and }\A S=X\}.
\psrank_{\A_C,\Pcal|_C}Definition passage · 1 passage
Search aliases: psrank_A_C,Pcal|_C psrank_ A_C, P|_C psrank A Pcal operatorname mathcal mathcal
Definition passage: Packet pure seed rank · 03_packets_and_graphs.tex, line 1028
f:directed-seed-rank A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).
Exact mathematical passage
r_C:=\psrank_{\A_C,\Pcal|_C}(X_C).
\mathbb QDefinition passage · 30 passages
Search aliases: Q mathbb
Definition passage: Hurwitz integral series · strengthening.tex, line 42
[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.
Exact mathematical passage
\mathcal H_p=\left\{\sum_{d\geq0}f_d\frac{t^d}{d!}: f_d\in\mathbb Z_p\right\} \subset\mathbb Q_p[[t]].
\mathfrak QDefinition passage · 11 passages
Search aliases: Q mathfrak
Definition passage: Finite quantum seed system · 01_finite_core.tex, line 42
[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltonian when dynamics are present, and τ specifies the topology used in a later completion.
Exact mathematical passage
\mathfrak Q= \left( \cH, \psi_0, \cO, \cR, \{P_{\ell,\alpha}\}, H, \tau \right),
q^{-1}Definition passage · 4 passages
Search aliases: q^-1
Definition passage: Monodromy packet chain · Paper_XI_Traditional_Final.tex, line 254
[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.
Exact mathematical passage
q^{-1}
q^{-1/2}Definition passage · 1 passage
Search aliases: q^-1/2
Definition passage: Normalized Gauss phase · Paper_XI_Traditional_Final.tex, line 741
[Normalized Gauss phase] For nontrivial χ, define γ(χ,ψ) = q^-1/2G(χ,ψ). Then absγ(χ,ψ)=1.
Exact mathematical passage
\gamma(\chi,\psi) = q^{-1/2}G(\chi,\psi).
q_{\mathfrak c}Definition passage · 2 passages
Search aliases: q_ c mathfrak
Definition passage: Universal closure obstruction · 02_strengthened.tex, line 2182
[Universal closure obstruction] Let q_ c:End( H)⟶End( H)/ c be the quotient map. Define Ob_ab^rm cl = q_ c([C_a,C_b]).
Exact mathematical passage
q_{\mathfrak c}:\End(\Hcal)\longrightarrow\End(\Hcal)/\mathfrak c
Q_0Definition passage · 8 passages
Search aliases: Q_0
Definition passage: Typed operator grammar · 02_foundations.tex, line 39
[Typed operator grammar] Let Q=(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying …
Exact mathematical passage
Q=(Q_0,Q_1,s,t)
Q_1Definition passage · 6 passages
Search aliases: Q_1
Definition passage: Typed operator grammar · 02_foundations.tex, line 39
[Typed operator grammar] Let Q=(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying …
Exact mathematical passage
Q=(Q_0,Q_1,s,t)
q_iDefinition passage · 14 passages
Search aliases: q_i
Definition passage: Relative information · Paper_VI_Traditional_Final.tex, line 3639
[Relative information] For probability vectors q and p, define D(qVert p) = ∑_iq_ilogfracq_ip_i. The convention is 0log0=0. A term with q_i>0 and p_i=0 has value +∞.
Exact mathematical passage
D(q\Vert p) = \sum_iq_i\log\frac{q_i}{p_i}.
Q_jDefinition passage · 35 passages
Search aliases: Q_j
Definition passage: Packet compatible operator morphism · 04_transport.tex, line 59
orphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):= ∑_σ(α)=jP_α. An empty fiber is interpreted as the zero projector.
Exact mathematical passage
Q_j\Phi = \Phi P_{\sigma^{-1}(j)}, \qquad P_{\sigma^{-1}(j)}:= \sum_{\sigma(\alpha)=j}P_\alpha.
\mathbb Q_pDefinition passage · 13 passages
Search aliases: Q_p mathbb
Definition passage: Hurwitz integral series · strengthening.tex, line 42
[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.
Exact mathematical passage
\mathcal H_p=\left\{\sum_{d\geq0}f_d\frac{t^d}{d!}: f_d\in\mathbb Z_p\right\} \subset\mathbb Q_p[[t]].
q_PDefinition passage · 2 passages
Search aliases: q_P
Definition passage: Code compression obstruction · 02_strengthened.tex, line 1277
[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.
Exact mathematical passage
q_P:\End(\Ccal)\longrightarrow\End(\Ccal)/(\CC I_{\Ccal})
Q_\alphaDefinition passage · 24 passages
Search aliases: Q_α alpha
Definition passage: Direct rotation of nearby packet systems · 04_correction_transport.tex, line 893
[Direct rotation of nearby packet systems] Let (P_α)_α∈ I and (Q_α)_α∈ I be complete orthogonal packet systems on the same Hilbert space. Put S( Q, P)=∑_α∈ IQ_α P_α. When S( Q, P) is invertible, define U( Q, P) = S( Q, P) (S( Q, P)^*S( Q, P))^-1/2.
Exact mathematical passage
(Q_\alpha)_{\alpha\in I}
Q_\betaDefinition passage · 30 passages
Search aliases: Q_β beta
Definition passage: Refinement · 02_coordinates.tex, line 34
[Refinement] Let P=(P_α)_α∈ I and Q=(Q_β)_β∈ J be packet systems on the same module. We say that Q refines P when there is a surjection ρ:J⟶ I such that P_α=∑_ρ(β)=αQ_β for every α∈ I. The index β is then called a child of ρ(β).
Exact mathematical passage
\Qcal=(Q_\beta)_{\beta\in J}
Q_\gammaDefinition passage · 29 passages
Search aliases: Q_γ gamma
Definition passage: Nonlinear packet coordinate map · 03_nonlinear_coordinates.tex, line 112
[Nonlinear packet coordinate map] Let P be a packet system on X and let Q be a packet system on Y. For any map F:X→ Y, define F_ Q← P = A_ QF S_ P. Thus F_ Q← P((x_α)_α) = ( Q_γ F(∑_α x_α) )_γ.
Exact mathematical passage
F_{\Qcal\leftarrow\Pcal}((x_\alpha)_\alpha) = \left( Q_\gamma F\left(\sum_\alpha x_\alpha\right) \right)_\gamma.
\QcalDefinition passage · 70 passages
Search aliases: Qcal Q mathcal
Definition passage: Packet compatible operator morphism · 04_transport.tex, line 50
[Packet compatible operator morphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):= ∑_σ(α)=jP_α. An empty fiber is interpreted as the zero projector.
Exact mathematical passage
(Y,\B,\Qcal)
\mathbb RDefinition passage · 8 passages
Search aliases: R mathbb
Definition passage · Paper_IV_Traditional_Final.tex, line 1793
For θ∈ R, define h_N(θ) = max_0≤ k<N cos(θ+frac(2k+1)πN). The tie directions are T_N = {frac2π kN:0≤ k<N} bmod 2π.
Exact mathematical passage
\theta\in\mathbb R
R^{\mathbf L}Definition passage · 2 passages
Search aliases: R^ L mathbf
Definition passage: Derived support · Paper_XII_Traditional_Final.tex, line 813
[Derived support] Let R be a commutative ring and let C be a perfect complex of R modules. Its derived support is Supp_R(C) = { p∈Spec R: Cmathbinmathop⊗limits_R^ Lκ( p) not≃0 }.
Exact mathematical passage
\Supp_R(C) = \left\{ \frakp\in\Spec R: C\mathbin{\mathop{\otimes}\limits_R^{\mathbf L}}\kappa(\frakp) \not\simeq0 \right\}.
R^{\mathrm{loc}}Definition passage · 10 passages
Search aliases: R^loc mathrm
Definition passage: Global deformation datum · Paper_XII_Traditional_Final.tex, line 215
m A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_R^locR^loc,τ, R^loc,τ:=ma…
Exact mathematical passage
R^{\mathrm{loc}}:=\wideotimes_{v\in S,\OO}R_v^\square \longrightarrow R^\square.
R^{\mathrm{loc},\tau}Definition passage · 3 passages
Search aliases: R^loc,τ mathrm tau
Definition passage: Global deformation datum · Paper_XII_Traditional_Final.tex, line 220
every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_R^locR^loc,τ, R^loc,τ:=mathbinwidehat⊗_v∈ S, OR_v^τ_v.
Exact mathematical passage
R^{\tau}:=R^\square\wideotimes_{R^{\mathrm{loc}}}R^{\mathrm{loc},\tau}, \qquad R^{\mathrm{loc},\tau}:=\wideotimes_{v\in S,\OO}R_v^{\tau_v}.
R^\squareDefinition passage · 7 passages
Search aliases: R^square square
Definition passage: Global deformation datum · Paper_XII_Traditional_Final.tex, line 214
l deformation datum] A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_R^locR…
Exact mathematical passage
R^\square
R^{\tau}Definition passage · 3 passages
Search aliases: R^τ tau
Definition passage: Global deformation datum · Paper_XII_Traditional_Final.tex, line 220
every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_R^locR^loc,τ, R^loc,τ:=mathbinwidehat⊗_v∈ S, OR_v^τ_v.
Exact mathematical passage
R^{\tau}:=R^\square\wideotimes_{R^{\mathrm{loc}}}R^{\mathrm{loc},\tau}, \qquad R^{\mathrm{loc},\tau}:=\wideotimes_{v\in S,\OO}R_v^{\tau_v}.
r_CDefinition passage · 6 passages
Search aliases: r_C
Definition passage: Packet pure seed rank · 03_packets_and_graphs.tex, line 1028
f:directed-seed-rank A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).
Exact mathematical passage
r_C:=\psrank_{\A_C,\Pcal|_C}(X_C).
R_{n,m}Definition passage · 11 passages
Search aliases: R_n,m
Definition passage: Reconstructed fine evolution · Paper_VI_Traditional_Final.tex, line 2537
[Reconstructed fine evolution] For a semigroup T_n(t) on X_n, define the evolution seen from level m by U_m,n(t)=C_m,nT_n(t)R_n,m.
Exact mathematical passage
U_{m,n}(t)=C_{m,n}T_n(t)R_{n,m}.
R_v^\squareDefinition passage · 2 passages
Search aliases: R_v^square square
Definition passage: Global deformation datum · Paper_XII_Traditional_Final.tex, line 212
[Global deformation datum] A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_…
Exact mathematical passage
R_v^\square
R_v^{\tau_v}Definition passage · 2 passages
Search aliases: R_v^τ_v tau
Definition passage: Global deformation datum · Paper_XII_Traditional_Final.tex, line 213
[Global deformation datum] A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_…
Exact mathematical passage
R_v^{\tau_v}
r_{\alpha,i}Definition passage · 10 passages
Search aliases: r_α,i alpha
Definition passage: Within packet information loss · Paper_VI_Traditional_Final.tex, line 3567
[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.
Exact mathematical passage
\sum_{i\in I_\alpha}r_{\alpha,i}\eta_i(p)=0.
R_\ellDefinition passage · 16 passages
Search aliases: R_ℓ ell
Definition passage: Finite quantum refinement tower · 02_strengthened.tex, line 2345
[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is recorded by the corresponding operator norm defects.
Exact mathematical passage
R_\ell:\Hcal_\ell\longrightarrow\Hcal_{\ell+1},
R_\inftyDefinition passage · 17 passages
Search aliases: R_∞ infty
Definition passage: Ordinary patching datum · Paper_XII_Traditional_Final.tex, line 729
[Ordinary patching datum] An ordinary patching datum consists of a Noetherian ring R_∞, an ideal a⊆ R_∞, a finite R_∞ module M_∞, and identifications R≅ R_∞/ a, M≅ M_∞/ a M_∞. In automorphy lifting applications, R_∞ is a patched deformation ring and M_∞ is a patched automorphic module. The definition itself contains no automorphy assertion.
Exact mathematical passage
R_\infty
\radDefinition passage · 2 passages
Search aliases: rad operatorname
Definition passage: One break symplectic envelope · Paper_XI_Traditional_Final.tex, line 1090
[One break symplectic envelope] The one break symplectic envelope is V_ρ=A/rad b_ρ.
Exact mathematical passage
\boxed{ V_\rho=A/\rad b_\rho. }
\ReDefinition passage · 22 passages
Search aliases: Re
Definition passage · Paper_IV_Traditional_Final.tex, line 2912
Choose a logarithm for each member of the finite exceptional part of Λ_N. Use the real logarithm for the eventual positive sequence. Define Z_N^orb(s) = ∑_λ∈Λ_Ne^-sLogλ. This series is initially defined on Re s>1/N.
Exact mathematical passage
\Re s>1/N
\mathrm{red}Definition passage · 12 passages
Search aliases: red mathrm
Definition passage · Paper_IV_Traditional_Final.tex, line 2749
Define Z_N^abs(s) = ∑_ρ∈ Z( D_N)|ρ|^-s and Z_N^red(s) = ∑_λ∈Λ_N|λ|^-s, with multiplicities. Each term means e^-slog|ρ| or e^-slog|λ|.
Exact mathematical passage
\mathfrak Z_N^{\mathrm{red}}(s) = \sum_{\lambda\in\Lambda_N}|\lambda|^{-s},
\redDefinition passage · 4 passages
Search aliases: red operatorname
Definition passage: Residual seed system · Paper_X_Block27_Amended.tex, line 163
[Residual seed system] For bar x∈ M(k), define the residual fiber M_bar x(A) = {x∈ M(A):red_A(x)=bar x}. The point bar x is the residual root. The family of all nonempty fibers is the residual special fiber, and the formal deformation problems above them form the residual forest.
Exact mathematical passage
\mathcal M_{\bar x}(A) = \{x\in\mathcal M(A):\red_A(x)=\bar x\}.
\red_ADefinition passage · 3 passages
Search aliases: red_A red operatorname
Definition passage: Residual seed system · Paper_X_Block27_Amended.tex, line 163
[Residual seed system] For bar x∈ M(k), define the residual fiber M_bar x(A) = {x∈ M(A):red_A(x)=bar x}. The point bar x is the residual root. The family of all nonempty fibers is the residual special fiber, and the formal deformation problems above them form the residual forest.
Exact mathematical passage
\mathcal M_{\bar x}(A) = \{x\in\mathcal M(A):\red_A(x)=\bar x\}.
\rmDefinition passage · 68 passages
Search aliases: rm
Definition passage: Approximate Knill--Laflamme defect · 02_strengthened.tex, line 1423
[Approximate Knill--Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.
Exact mathematical passage
\varepsilon_{\rm KL} = \max_{a,b} \inf_{c\in\CC} \norm{PE_a^*E_bP-cP}.
\RRDefinition passage · 26 passages
Search aliases: RR R mathbb
Definition passage: Canonical lumping obstruction · Paper_VI_Traditional_Final.tex, line 2854
[Canonical lumping obstruction] For a linear fine transition operator K, define Ob_C(K) = (CK)|_Ker C ∈ B(Ker C, R^m).
Exact mathematical passage
\Ob_C(K) = (CK)|_{\Ker C} \in \mathcal B(\Ker C,\RR^m).
\RR^mDefinition passage · 1 passage
Search aliases: RR^m R^m RR mathbb
Definition passage: Canonical lumping obstruction · Paper_VI_Traditional_Final.tex, line 2854
[Canonical lumping obstruction] For a linear fine transition operator K, define Ob_C(K) = (CK)|_Ker C ∈ B(Ker C, R^m).
Exact mathematical passage
\Ob_C(K) = (CK)|_{\Ker C} \in \mathcal B(\Ker C,\RR^m).
\mathfrak SDefinition passage · 2 passages
Search aliases: S mathfrak
Definition passage: Linear seed system · 02_foundations.tex, line 148
Seeds, addressed hulls, and natural presentations. [Linear seed system] A linear seed system is a tuple S=(X, A,S, P,τ), where X is a unital left module over A, A is fixed independently of the target element, S=(s_1,…,s_r) is a finite distinguished seed family, P is either empty or a declared packet system, and τ is either the algebraic convention or a specified Hausdorff topology in which the declared operator action is continuous.
Exact mathematical passage
\mathfrak S=(X,\A,S,\Pcal,\tau),
S'Definition passage · 16 passages
Search aliases: S'
Definition passage: Morphism of seeded systems · 04_transport.tex, line 76
[Morphism of seeded systems] A morphism (X, A,S, P)⟶(Y, B,S', Q) is a packet compatible operator morphism satisfying the seed condition Φ(S)⊆Hull_ B(S'). In the topological category all maps and actions are required to be continuous and the seed condition uses the topological hull when declared.
Exact mathematical passage
(X,\A,S,\Pcal)\longrightarrow(Y,\B,S',\Qcal)
S^*Definition passage · 25 passages
Search aliases: S^*
Definition passage: Operator charge · 02_strengthened.tex, line 295
[Operator charge] An operator A∈End( H) has cyclic charge q∈ Z/N Z when SAS^*=ω^qA.
Exact mathematical passage
SAS^*=\omega^qA.
s_1Definition passage · 20 passages
Search aliases: s_1
Definition passage: Linear seed system · 02_foundations.tex, line 151
Seeds, addressed hulls, and natural presentations. [Linear seed system] A linear seed system is a tuple S=(X, A,S, P,τ), where X is a unital left module over A, A is fixed independently of the target element, S=(s_1,…,s_r) is a finite distinguished seed family, P is either empty or a declared packet system, and τ is either the algebraic convention or a specified Hausdorff topology in which the declared operator action is continuous.
Exact mathematical passage
S=(s_1,\ldots,s_r)
S_{2(n-m)}Definition passage · 2 passages
Search aliases: S_2(n-m)
Definition passage: Odd-length source and inverse coefficients · 20_arithmetic.tex, line 767
rse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd. Primality is needed only to interpret the result as a prime-adic congruence ladder.
Exact mathematical passage
T^{\mathrm{odd}}_{p,M}(n)=\sum_{m=0}^{\min(M,n)} \binom{2n}{2m}\gamma_m p^{2m}S_{2(n-m)}(p).
S_dDefinition passage · 1 passage
Search aliases: S_d
Definition passage: Odd-length source and inverse coefficients · 20_arithmetic.tex, line 757
[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd. Primality is needed only to interpret the result as a prime-adic congruence ladder.
Exact mathematical passage
S_d(p)=\sum_{a=0}^{p-1}(-1)^a(2a+1)^d.
s_jDefinition passage · 6 passages
Search aliases: s_j
Definition passage: Seed evaluation map · 02_foundations.tex, line 218
ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.
Exact mathematical passage
\varepsilon_S:A^{\oplus r}\longrightarrow M, \qquad (a_1,\ldots,a_r)\longmapsto\sum_{j=1}^{r}a_js_j.
S_mDefinition passage · 18 passages
Search aliases: S_m
Definition passage: Dyadic truncation · 20_arithmetic.tex, line 880
[Dyadic truncation] Define β_m by the formal unit-series identity ∑_m≥0β_m z^2m=(fraczsinh z)^2, β_0=1, where the quotient means the inverse of the unit series (sinh z/z)^2. Define A(n)=frac3^2n+2-14(2n+1)(2n+2), S_m(n)=β_m2^2mfrac(2n)!(2n-2m)!A(n-m) (0≤ m≤ n), and T_M(n)=∑_m=0^min(M,n)S_m(n) (M,n∈ N).
Exact mathematical passage
A(n)=\frac{3^{2n+2}-1}{4(2n+1)(2n+2)},\qquad S_m(n)=\beta_m2^{2m}\frac{(2n)!}{(2n-2m)!}A(n-m) \quad(0\le m\le n),
s_rDefinition passage · 14 passages
Search aliases: s_r
Definition passage: Linear seed system · 02_foundations.tex, line 151
Seeds, addressed hulls, and natural presentations. [Linear seed system] A linear seed system is a tuple S=(X, A,S, P,τ), where X is a unital left module over A, A is fixed independently of the target element, S=(s_1,…,s_r) is a finite distinguished seed family, P is either empty or a declared packet system, and τ is either the algebraic convention or a specified Hausdorff topology in which the declared operator action is continuous.
Exact mathematical passage
S=(s_1,\ldots,s_r)
S_\lambdaDefinition passage · 4 passages
Search aliases: S_λ lambda
Definition passage: Decision gap · 01_manuscript.tex, line 1694
[Decision gap] At an internal node v, let g_v = absK_λ(v)-S_λ(v). This is the gap between the keep and split scores after the child subproblems have been solved.
Exact mathematical passage
g_v = \abs{K_\lambda(v)-S_\lambda(v)}.
s_\LambdaDefinition passage · 11 passages
Search aliases: s_Λ Lambda
Definition passage: Lattice packet data · 04_arithmetic.tex, line 252
et V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.
Exact mathematical passage
s_\Lambda(E_\alpha) = \min\{s\geq0:\pi^sE_\alpha\Lambda\subseteq\Lambda\}.
\Scal_{\Pcal}Definition passage · 13 passages
Search aliases: Scal_Pcal S_ P Scal Pcal mathcal mathcal
Definition passage: Packet system · 02_coordinates.tex, line 25
ket_system Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.
Exact mathematical passage
\Acal_{\Pcal}x=(P_\alpha x)_\alpha, \qquad \Scal_{\Pcal}(x_\alpha)_\alpha=\sum_\alpha x_\alpha.
\Scal_{\Qcal}Definition passage · 9 passages
Search aliases: Scal_Qcal S_ Q Scal Qcal mathcal mathcal
Definition passage: Coordinate change across levels · 02_coordinates.tex, line 90
[Coordinate change across levels] For two packet systems P and Q on X, define J_ Q← P = A_ Q S_ P:X^ P⟶ X^ Q and C_ P← Q = A_ P S_ Q:X^ Q⟶ X^ P. When Q refines P, the first map resolves each coarse coordinate into its children and the second adds the children back to their parent.
Exact mathematical passage
C_{\Pcal\leftarrow\Qcal} = \Acal_{\Pcal}\Scal_{\Qcal}:X^{\Qcal}\longrightarrow X^{\Pcal}.
\operatorname{sech}Definition passage · 32 passages
Search aliases: sech operatorname
Definition passage: Odd-length source and inverse coefficients · 20_arithmetic.tex, line 760
[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd. Primality is needed only to interpret the result as a prime-adic congruence ladder.
Exact mathematical passage
\operatorname{sech}^2
\operatorname{sech}^2Definition passage · 10 passages
Search aliases: sech^2 operatorname
Definition passage: Odd-length source and inverse coefficients · 20_arithmetic.tex, line 760
[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd. Primality is needed only to interpret the result as a prime-adic congruence ladder.
Exact mathematical passage
\operatorname{sech}^2
\mathrm{seed}Definition passage · 2 passages
Search aliases: seed mathrm
Definition passage: Branching dimension · Paper_X_Block27_Amended.tex, line 612
[Branching dimension] When the limit exists, define the finite-level counting exponent dim_seed(bar x) = lim_m→∞ fraclog_q#X_m(bar x)m. A limsup may be used in general.
Exact mathematical passage
\dim_{\mathrm{seed}}(\bar x) = \lim_{m\to\infty} \frac{\log_q\#X_m(\bar x)}{m}.
\mathbf{SeedSys}Definition passage · 3 passages
Search aliases: SeedSys mathbf
Definition passage: Realization functor and preservation profile · 04_transport.tex, line 384
[Realization functor and preservation profile] Let SeedSys_abs be an abstract category of seed systems and let C_D be a domain category. A realization functor is a functor Ev_D:SeedSys_abs⟶ C_D with a declared preservation profile specifying which of the following structures are preserved: 0, I, finite sums, composition, biproducts,
Exact mathematical passage
\mathbf{SeedSys}_{\mathrm{abs}}
\mathbf{SeedSys}_{\mathrm{abs}}Definition passage · 2 passages
Search aliases: SeedSys_abs mathbf mathrm
Definition passage: Realization functor and preservation profile · 04_transport.tex, line 384
[Realization functor and preservation profile] Let SeedSys_abs be an abstract category of seed systems and let C_D be a domain category. A realization functor is a functor Ev_D:SeedSys_abs⟶ C_D with a declared preservation profile specifying which of the following structures are preserved: 0, I, finite sums, composition, biproducts,
Exact mathematical passage
\mathbf{SeedSys}_{\mathrm{abs}}
\SfunDefinition passage · 10 passages
Search aliases: Sfun S mathsf
Definition passage · Paper_IV_Traditional_Final.tex, line 958
For r∈ C, define S_N^(r)(z) = D_N(z)^-r = exp(-rlog D_N(z)). Write S_N^(r)(z) = ∑_n≥0P_N,n(r)fracz^Nn(Nn)!. Define the logarithmic coefficients κ_N,m by -logΔ_N(w) = ∑_m≥1κ_N,mfracw^mm.
Exact mathematical passage
\Sfun_N^{(r)}(z) = \Dfun_N(z)^{-r} = \exp\bigl(-r\log\Dfun_N(z)\bigr).
\Sfun_N^{(r)}Definition passage · 9 passages
Search aliases: Sfun_N^(r) S_N^(r) Sfun mathsf
Definition passage · Paper_IV_Traditional_Final.tex, line 958
For r∈ C, define S_N^(r)(z) = D_N(z)^-r = exp(-rlog D_N(z)). Write S_N^(r)(z) = ∑_n≥0P_N,n(r)fracz^Nn(Nn)!. Define the logarithmic coefficients κ_N,m by -logΔ_N(w) = ∑_m≥1κ_N,mfracw^mm.
Exact mathematical passage
\Sfun_N^{(r)}(z) = \Dfun_N(z)^{-r} = \exp\bigl(-r\log\Dfun_N(z)\bigr).
\sinhDefinition passage · 36 passages
Search aliases: sinh
Definition passage: Dyadic truncation · 20_arithmetic.tex, line 874
[Dyadic truncation] Define β_m by the formal unit-series identity ∑_m≥0β_m z^2m=(fraczsinh z)^2, β_0=1, where the quotient means the inverse of the unit series (sinh z/z)^2. Define A(n)=frac3^2n+2-14(2n+1)(2n+2), S_m(n)=β_m2^2mfrac(2n)!(2n-2m)!A(n-m) (0≤ m≤ n), and T_M(n)=∑_m=0^min(M,n)S_m(n) (M,n∈ N).
Exact mathematical passage
\sum_{m\ge0}\beta_m z^{2m}=\left(\frac{z}{\sinh z}\right)^2, \qquad \beta_0=1,
\operatorname{span}Definition passage · 11 passages
Search aliases: span operatorname
Definition passage: Finite quantum seed system · 02_strengthened.tex, line 93
[Finite quantum seed system] A finite quantum seed system is a triple ( H, A,S), where H is a finite dimensional Hilbert space, A⊆End( H) is a unital star subalgebra, and S⊆ H is a finite set. Its generated hull is A S = span{Aψ:A∈ A, ψ∈ S}. The seed set is cyclic when A S= H. The quantum seed rank is the least cardinality of a cyclic seed set.
Exact mathematical passage
\Acal S = \operatorname{span}\{A\psi:A\in\Acal,\ \psi\in S\}.
\SpecDefinition passage · 45 passages
Search aliases: Spec operatorname
Definition passage: Support and near faithfulness · Paper_XII_Traditional_Final.tex, line 576
[Support and near faithfulness] Let R be a commutative ring and let M be an R module. Its support is Supp_R(M) = { p∈Spec R:M_ p≠0}. If R is Noetherian and M is finite, then M is nearly faithful when Ann_R(M) is a nilpotent ideal.
Exact mathematical passage
\Supp_R(M) = \{\frakp\in\Spec R:M_{\frakp}\neq0\}.
\mathrm{split}Definition passage · 7 passages
Search aliases: split mathrm
Definition passage: Lattice packet data · 04_arithmetic.tex, line 240
[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.
Exact mathematical passage
\Lambda_\alpha=\Lambda\cap V_\alpha, \qquad \Lambda_{\mathrm{split}}=\bigoplus_\alpha\Lambda_\alpha,
\squareDefinition passage · 11 passages
Search aliases: square
Definition passage: Global deformation datum · Paper_XII_Traditional_Final.tex, line 212
[Global deformation datum] A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_…
Exact mathematical passage
R_v^\square
\srank_{\A,\tau}Definition passage · 1 passage
Search aliases: srank_A,τ srank_ A,τ srank A tau operatorname mathcal tau
Definition passage: Seed rank · 02_foundations.tex, line 377
[Seed rank] For a unital left module M over A, define srank_A(M):=min{|S|:AS=M}. For a topological module (X,τ), define srank_ A,τ(X):=min{|S|:cl A S^ τ=X}. Set srank_A(0)=0. When the value is finite, it is the minimal number of module generators.
Exact mathematical passage
\srank_{\A,\tau}(X):=\min\{|S|:\cl{\A S}^{\,\tau}=X\}.
\srank_GDefinition passage · 2 passages
Search aliases: srank_G srank operatorname
Definition passage: Directed seed rank · Paper_X_Block27_Amended.tex, line 667
[Directed seed rank] For a directed transport graph G, define srank_G = min{|S|:Cl_G(S)=V(G)}, where Cl_G(S) is the set of vertices reachable from S.
Exact mathematical passage
\srank_G = \min\{|S|:\Cl_G(S)=V(G)\},
\srcDefinition passage · 19 passages
Search aliases: src operatorname
Definition passage: Packet graph and scalar edge system · 01_finite_coordinates.tex, line 77
0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.edgeOp; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp
Exact mathematical passage
L_e=\operatorname{packetMatrix}(\src(e),\tgt(e);c(e)), \qquad T=\sum_{e\in E}L_e.
\mathrm{statement}Definition passage · 1 passage
Search aliases: statement mathrm
Definition passage: Canonical theorem record · 02_foundations.tex, line 17
[Canonical theorem record] A canonical theorem record is the tuple (ID,status,hypotheses,statement, dependencies,proof,exports,boundary). A later chapter cites the stable record ID. A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this chapter.
Exact mathematical passage
(\mathrm{ID},\mathrm{status},\mathrm{hypotheses},\mathrm{statement}, \mathrm{dependencies},\mathrm{proof},\mathrm{exports},\mathrm{boundary}).
\mathrm{status}Definition passage · 1 passage
Search aliases: status mathrm
Definition passage: Canonical theorem record · 02_foundations.tex, line 17
[Canonical theorem record] A canonical theorem record is the tuple (ID,status,hypotheses,statement, dependencies,proof,exports,boundary). A later chapter cites the stable record ID. A domain specialization may add domain hypotheses, but it does not repeat a generic proof from this chapter.
Exact mathematical passage
(\mathrm{ID},\mathrm{status},\mathrm{hypotheses},\mathrm{statement}, \mathrm{dependencies},\mathrm{proof},\mathrm{exports},\mathrm{boundary}).
\SubsetDefinition passage · 2 passages
Search aliases: Subset
Definition passage: Unconditional packet sum space · 03_packets_and_graphs.tex, line 359
[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .
Exact mathematical passage
\|x\|_u := \sup_{F\Subset\mathbb N} \sup_{|a_n|\leq1} \left\|\sum_{n\in F}a_nx_n\right\|.
\supDefinition passage · 37 passages
Search aliases: sup
Definition passage: Unconditional packet sum space · 03_packets_and_graphs.tex, line 359
[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .
Exact mathematical passage
\|x\|_u := \sup_{F\Subset\mathbb N} \sup_{|a_n|\leq1} \left\|\sum_{n\in F}a_nx_n\right\|.
\sup_{F\Subset\mathbb N}Definition passage · 2 passages
Search aliases: sup_FSubset N sup Subset mathbb
Definition passage: Unconditional packet sum space · 03_packets_and_graphs.tex, line 359
[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .
Exact mathematical passage
\|x\|_u := \sup_{F\Subset\mathbb N} \sup_{|a_n|\leq1} \left\|\sum_{n\in F}a_nx_n\right\|.
\sup_{|a_n|\leq1}Definition passage · 2 passages
Search aliases: sup_|a_n|≤1 sup leq
Definition passage: Unconditional packet sum space · 03_packets_and_graphs.tex, line 359
[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .
Exact mathematical passage
\|x\|_u := \sup_{F\Subset\mathbb N} \sup_{|a_n|\leq1} \left\|\sum_{n\in F}a_nx_n\right\|.
\Supp_RDefinition passage · 19 passages
Search aliases: Supp_R Supp operatorname
Definition passage: Support and near faithfulness · Paper_XII_Traditional_Final.tex, line 576
[Support and near faithfulness] Let R be a commutative ring and let M be an R module. Its support is Supp_R(M) = { p∈Spec R:M_ p≠0}. If R is Noetherian and M is finite, then M is nearly faithful when Ann_R(M) is a nilpotent ideal.
Exact mathematical passage
\Supp_R(M) = \{\frakp\in\Spec R:M_{\frakp}\neq0\}.
\Supp_{R_\infty}Definition passage · 3 passages
Search aliases: Supp_R_∞ Supp infty operatorname infty
Definition passage: Ordinary completion datum at a point · Paper_XII_Traditional_Final.tex, line 1650
datum at a characteristic zero point x:R→ E' consists of the following objects. item A represented global deformation ring R with declared local quotient conditions and universal representation ρ_R. item An ordinary patching datum (R_∞, a,M_∞,R,H) whose finite level module H is an automorphic Hecke carrier. item Full patched support Supp_R_∞(M_∞)=Spec R_∞. item An isomorphism R≅ T with the Hecke algebra acting on H. The R action on H supplied by the patched control isomorphism is the restriction…
Exact mathematical passage
\Supp_{R_\infty}(M_\infty)=\Spec R_\infty.
\mathcal TDefinition passage · 21 passages
Search aliases: T mathcal
Definition passage: Admissible residual transport · Paper_X_Block27_Amended.tex, line 653
[Admissible residual transport] An admissible transport from barρ to barρ' consists of T:barρ⟼barρ' together with natural maps T_A:Def_barρ(A)⟶Def_barρ'(A) for every coefficient algebra A, compatible with base change and reduction. When the transport is reversible, it is an isomorphism in the residual transport groupoid. One-way operations give a directed category.
Exact mathematical passage
\mathcal T:\bar\rho\longmapsto\bar\rho'
\mathfrak TDefinition passage · 5 passages
Search aliases: T mathfrak
Definition passage · Paper_IV_Traditional_Final.tex, line 1801
For θ∈ R, define h_N(θ) = max_0≤ k<N cos(θ+frac(2k+1)πN). The tie directions are T_N = {frac2π kN:0≤ k<N} bmod 2π.
Exact mathematical passage
\mathfrak T_N = \left\{\frac{2\pi k}{N}:0\leq k<N\right\} \quad\bmod 2\pi.
T'_{vu}Definition passage · 1 passage
Search aliases: T'_vu
Definition passage: Gauge change · 04_correction_transport.tex, line 653
[Gauge change] A gauge is a family G_v∈GL(X). It changes the edge transports to T'_vu=G_vT_vuG_u^-1.
Exact mathematical passage
T'_{vu}=G_vT_{vu}G_u^{-1}.
t^{2m}Definition passage · 3 passages
Search aliases: t^2m
Definition passage: Odd-length source and inverse coefficients · 20_arithmetic.tex, line 761
[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd. Primality is needed only to interpret the result as a prime-adic congruence ladder.
Exact mathematical passage
\operatorname{sech}^2t=\sum_{m\ge0}\gamma_m\frac{t^{2m}}{(2m)!}, \qquad \gamma_m=\sum_{b=0}^{2m}\binom{2m}{b}E_bE_{2m-b},
t^dDefinition passage · 24 passages
Search aliases: t^d
Definition passage: Congruential Euler coefficients · 20_arithmetic.tex, line 15
[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the multiples of N vanish.
Exact mathematical passage
\label{eq:arith-euler-egf} \sum_{d\ge0}E_d^{(N,j)}\frac{t^d}{d!} =\left(\sum_{n\ge0}\frac{t^{Nn}}{(Nn+j)!}\right)^{-1}.
T^mDefinition passage · 5 passages
Search aliases: T^m
Definition passage: Seed zeta function · Paper_X_Block27_Amended.tex, line 531
[Seed zeta function] For a residual root bar x with finite levels, define Z_bar x(T) = ∑_m≥1#X_m(bar x)T^m. When a weighted chart is specified, write Z_ w(T).
Exact mathematical passage
\boxed{ Z_{\bar x}(T) = \sum_{m\ge1}\#X_m(\bar x)T^m. }
t^nDefinition passage · 21 passages
Search aliases: t^n
Definition passage: Weighted transfer series · 03_weighted.tex, line 380
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j
Exact mathematical passage
Z_{u,v}(t) = \sum_{n\geq0}a_nt^n,
t^{Nn}Definition passage · 10 passages
Search aliases: t^Nn
Definition passage: Congruential Euler coefficients · 20_arithmetic.tex, line 15
[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the multiples of N vanish.
Exact mathematical passage
\label{eq:arith-euler-egf} \sum_{d\ge0}E_d^{(N,j)}\frac{t^d}{d!} =\left(\sum_{n\ge0}\frac{t^{Nn}}{(Nn+j)!}\right)^{-1}.
T^{\mathrm{odd}}_{p,M}Definition passage · 2 passages
Search aliases: T^odd_p,M mathrm
Definition passage: Odd-length source and inverse coefficients · 20_arithmetic.tex, line 767
rse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd. Primality is needed only to interpret the result as a prime-adic congruence ladder.
Exact mathematical passage
T^{\mathrm{odd}}_{p,M}(n)=\sum_{m=0}^{\min(M,n)} \binom{2n}{2m}\gamma_m p^{2m}S_{2(n-m)}(p).
t^{w(e)}Definition passage · 3 passages
Search aliases: t^w(e)
Definition passage: Weighted transfer series · 03_weighted.tex, line 374
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j
Exact mathematical passage
A(t)_{ij} = \sum_{e:i\to j}m(e)t^{w(e)}.
t_1Definition passage · 14 passages
Search aliases: t_1
Definition passage: Finite grammar for operation trees · 02_connections_obstructions.tex, line 772
[Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the finite formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No algebraic, continuity, or differentiability law is imposed on these semantics. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.OperationTree; Kom…
Exact mathematical passage
\begin{aligned} t ::={}&\operatorname{input}(i) \mid\operatorname{constant}(k) \mid\operatorname{unary}(u,t)\\ &\mid\operatorname{binary}(b,t_1,t_2). \end{aligned}
t_2Definition passage · 1 passage
Search aliases: t_2
Definition passage: Finite grammar for operation trees · 02_connections_obstructions.tex, line 772
[Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the finite formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No algebraic, continuity, or differentiability law is imposed on these semantics. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.OperationTree; Kom…
Exact mathematical passage
\begin{aligned} t ::={}&\operatorname{input}(i) \mid\operatorname{constant}(k) \mid\operatorname{unary}(u,t)\\ &\mid\operatorname{binary}(b,t_1,t_2). \end{aligned}
T_aDefinition passage · 26 passages
Search aliases: T_a
Definition passage: Clifford cocycle class · Paper_XI_Traditional_Final.tex, line 1061
[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.
Exact mathematical passage
T_aT_b=c(a,b)T_{a+b}.
\mathcal T_ADefinition passage · 1 passage
Search aliases: T_A mathcal
Definition passage: Admissible residual transport · Paper_X_Block27_Amended.tex, line 657
[Admissible residual transport] An admissible transport from barρ to barρ' consists of T:barρ⟼barρ' together with natural maps T_A:Def_barρ(A)⟶Def_barρ'(A) for every coefficient algebra A, compatible with base change and reduction. When the transport is reversible, it is an isomorphism in the residual transport groupoid. One-way operations give a directed category.
Exact mathematical passage
\mathcal T_A:\Def_{\bar\rho}(A)\longrightarrow\Def_{\bar\rho'}(A)
T_{a+b}Definition passage · 2 passages
Search aliases: T_a+b
Definition passage: Clifford cocycle class · Paper_XI_Traditional_Final.tex, line 1061
[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.
Exact mathematical passage
T_aT_b=c(a,b)T_{a+b}.
T_bDefinition passage · 4 passages
Search aliases: T_b
Definition passage: Clifford cocycle class · Paper_XI_Traditional_Final.tex, line 1061
[Clifford cocycle class] Choose scalars c(a,b) such that T_aT_b=c(a,b)T_a+b. The cohomology class [c_ρ]∈ H^2(A, C^×) is the Clifford cocycle class of the datum.
Exact mathematical passage
T_aT_b=c(a,b)T_{a+b}.
T_MDefinition passage · 12 passages
Search aliases: T_M
Definition passage: Dyadic truncation · 20_arithmetic.tex, line 886
n] Define β_m by the formal unit-series identity ∑_m≥0β_m z^2m=(fraczsinh z)^2, β_0=1, where the quotient means the inverse of the unit series (sinh z/z)^2. Define A(n)=frac3^2n+2-14(2n+1)(2n+2), S_m(n)=β_m2^2mfrac(2n)!(2n-2m)!A(n-m) (0≤ m≤ n), and T_M(n)=∑_m=0^min(M,n)S_m(n) (M,n∈ N).
Exact mathematical passage
\label{eq:arith-tm} T_M(n)=\sum_{m=0}^{\min(M,n)}S_m(n)\qquad(M,n\in\N).
T_NDefinition passage · 13 passages
Search aliases: T_N
Definition passage · Paper_IV_Traditional_Final.tex, line 1267
Define the transfer tangent T_N(z) =-frac D_N'(z) D_N(z). Near the origin, its convergent block expansion has the form T_N(z) = ∑_n≥0T_N(n) fracz^Nn+N-1(Nn+N-1)!.
Exact mathematical passage
\Tfun_N(z) = \sum_{n\geq0}T_N(n) \frac{z^{Nn+N-1}}{(Nn+N-1)!}.
T_nDefinition passage · 18 passages
Search aliases: T_n
Definition passage: Reconstructed fine evolution · Paper_VI_Traditional_Final.tex, line 2536
[Reconstructed fine evolution] For a semigroup T_n(t) on X_n, define the evolution seen from level m by U_m,n(t)=C_m,nT_n(t)R_n,m.
Exact mathematical passage
T_n(t)
\mathfrak T_NDefinition passage · 2 passages
Search aliases: T_N mathfrak
Definition passage · Paper_IV_Traditional_Final.tex, line 1801
For θ∈ R, define h_N(θ) = max_0≤ k<N cos(θ+frac(2k+1)πN). The tie directions are T_N = {frac2π kN:0≤ k<N} bmod 2π.
Exact mathematical passage
\mathfrak T_N = \left\{\frac{2\pi k}{N}:0\leq k<N\right\} \quad\bmod 2\pi.
t_{r(\sigma)}Definition passage · 1 passage
Search aliases: t_r(σ) sigma
Definition passage: Typed operation tree and evaluation · 02_connections_obstructions.tex, line 848
[Typed operation tree and evaluation] A typed operation tree is recursively either a leaf of declared color or a node labeled by σ, with one correctly colored child for each input slot of σ. Evaluation is recursive: ev(node(σ;t_1,…,t_r(σ))) = Φ_σ(ev(t_1),…,ev(t_r(σ))). statusroadmapCandidate API: leannameKomorebi.PaperIII.NonlinearObstructions.ColoredOperationTree and leannameColoredOperationTree.eval. This is stronger than the compiled uncolored unary--binary grammar.
Exact mathematical passage
\ev(\operatorname{node}(\sigma;t_1,\ldots,t_{r(\sigma)})) = \Phi_\sigma(\ev(t_1),\ldots,\ev(t_{r(\sigma)})).
T_{vu}Definition passage · 6 passages
Search aliases: T_vu
Definition passage: Gauge change · 04_correction_transport.tex, line 653
[Gauge change] A gauge is a family G_v∈GL(X). It changes the edge transports to T'_vu=G_vT_vuG_u^-1.
Exact mathematical passage
T'_{vu}=G_vT_{vu}G_u^{-1}.
T_\ellDefinition passage · 18 passages
Search aliases: T_ℓ ell
Definition passage: Finite quantum refinement tower · 02_strengthened.tex, line 2348
[Finite quantum refinement tower] A finite quantum refinement tower consists of Hilbert spaces H_ℓ, isometries R_ℓ: H_ℓ⟶ H_ℓ+1, self adjoint Hamiltonians H_ℓ, packet projectors P_ℓ,a, constraints C_ℓ,r, and positive transfer operators T_ℓ. Exact compatibility means that every declared object intertwines with R_ℓ. Approximate compatibility is recorded by the corresponding operator norm defects.
Exact mathematical passage
T_\ell
\TfunDefinition passage · 13 passages
Search aliases: Tfun T mathsf
Definition passage · Paper_IV_Traditional_Final.tex, line 1262
Define the transfer tangent T_N(z) =-frac D_N'(z) D_N(z). Near the origin, its convergent block expansion has the form T_N(z) = ∑_n≥0T_N(n) fracz^Nn+N-1(Nn+N-1)!.
Exact mathematical passage
\Tfun_N(z) =-\frac{\Dfun_N'(z)}{\Dfun_N(z)}.
\Tfun_NDefinition passage · 10 passages
Search aliases: Tfun_N T_N Tfun mathsf
Definition passage · Paper_IV_Traditional_Final.tex, line 1262
Define the transfer tangent T_N(z) =-frac D_N'(z) D_N(z). Near the origin, its convergent block expansion has the form T_N(z) = ∑_n≥0T_N(n) fracz^Nn+N-1(Nn+N-1)!.
Exact mathematical passage
\Tfun_N(z) =-\frac{\Dfun_N'(z)}{\Dfun_N(z)}.
\tgtDefinition passage · 19 passages
Search aliases: tgt operatorname
Definition passage: Packet graph and scalar edge system · 01_finite_coordinates.tex, line 77
0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.edgeOp; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp
Exact mathematical passage
L_e=\operatorname{packetMatrix}(\src(e),\tgt(e);c(e)), \qquad T=\sum_{e\in E}L_e.
\ThetafunDefinition passage · 10 passages
Search aliases: Thetafun symupΘ symup Theta
Definition passage · Paper_IV_Traditional_Final.tex, line 1641
Set H_N(z)= E_N,0(z) = ∑_n≥0fracz^Nn(Nn)!. For u∈ C, define symupΘ_N,n(u) by H_N(uz)^-1H_N(z)^-u = ∑_n≥0symupΘ_N,n(u)fracz^Nn(Nn)!. This is first a formal identity at the origin. Analytically, it is interpreted on a common simply connected zero-free region with the branch of log H_N normalized by log H_N(0)=0.
Exact mathematical passage
\Thetafun_{N,n}(u)
\Thetafun_{N,n}Definition passage · 7 passages
Search aliases: Thetafun_N,n symupΘ_N,n Thetafun symup Theta
Definition passage · Paper_IV_Traditional_Final.tex, line 1641
Set H_N(z)= E_N,0(z) = ∑_n≥0fracz^Nn(Nn)!. For u∈ C, define symupΘ_N,n(u) by H_N(uz)^-1H_N(z)^-u = ∑_n≥0symupΘ_N,n(u)fracz^Nn(Nn)!. This is first a formal identity at the origin. Analytically, it is interpreted on a common simply connected zero-free region with the branch of log H_N normalized by log H_N(0)=0.
Exact mathematical passage
\Thetafun_{N,n}(u)
\TrDefinition passage · 49 passages
Search aliases: Tr Tr operatorname
Definition passage: Code compression obstruction · 02_strengthened.tex, line 1285
[Code compression obstruction] Let q_P:End( C)⟶End( C)/( C I_ C) be the quotient map. Define Ob_P(A)=q_P(PAP). Equivalently, one may use the traceless representative o_P(A) = PAP- fracTr(PAP)kP.
Exact mathematical passage
\ob_P(A) = PAP- \frac{\Tr(PAP)}{k}P.
\triangleleftDefinition passage · 3 passages
Search aliases: triangleleft
Definition passage: Typed operator grammar · 02_foundations.tex, line 43
[Typed operator grammar] Let Q=(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying …
Exact mathematical passage
I\triangleleft \bbk Q
\TTDefinition passage · 60 passages
Search aliases: TT T mathbb
Definition passage: Automorphic Hecke datum · Paper_XII_Traditional_Final.tex, line 1294
[Automorphic Hecke datum] An automorphic Hecke datum consists of the following objects. item A commutative O algebra T. item A finite T module H, or a perfect T complex C, obtained from an automorphic or cohomological carrier. item For every unramified place v outside a finite set, a polynomial P_v(X)∈ T[X]. item A declared characteristic zero locus on which a nonzero field fiber is known to be classical and is known to carry an attached Galois representation whose Frobenius polynomial at v is t…
Exact mathematical passage
\TT
UDefinition passage · 562 passages
Search aliases: U
Definition passage: Finite grammar for operation trees · 02_connections_obstructions.tex, line 770
[Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the finite formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t)
Exact mathematical passage
U
uDefinition passage · 660 passages
Search aliases: u
Definition passage: Unconditional packet sum space · 03_packets_and_graphs.tex, line 355
[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .
Exact mathematical passage
\bigoplus_{n\geq1}^{u}X_n
u^{-1}Definition passage · 5 passages
Search aliases: u^-1
Definition passage: Separation weight · 04_arithmetic.tex, line 187
[Separation weight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG preserves Λ. The minimal Lean theorem needs only the lattice action of u^-1. The bundled wrapper for lattice units records both u and u^-1. A nonzero s…
Exact mathematical passage
u^{-1}
u_iDefinition passage · 6 passages
Search aliases: u_i
Definition passage: Weighted transfer series · 03_weighted.tex, line 386
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|γ|m(e_ν). Here γ=(e_1,…,e_|γ|) is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the en…
Exact mathematical passage
a_n=\sum_{i,j}\ \sum_{\substack{\gamma:i\leadsto j\\W(\gamma)=n}} u_i v_j\prod_{\nu=1}^{|\gamma|}m(e_\nu).
U_{m,n}Definition passage · 8 passages
Search aliases: U_m,n
Definition passage: Reconstructed fine evolution · Paper_VI_Traditional_Final.tex, line 2537
[Reconstructed fine evolution] For a semigroup T_n(t) on X_n, define the evolution seen from level m by U_m,n(t)=C_m,nT_n(t)R_n,m.
Exact mathematical passage
U_{m,n}(t)=C_{m,n}T_n(t)R_{n,m}.
U_{N,j}Definition passage · 8 passages
Search aliases: U_N,j
Definition passage: Congruential Euler coefficients · 20_arithmetic.tex, line 21
[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the multiples of N vanish.
Exact mathematical passage
U_{N,j}(n)=E_{Nn}^{(N,j)},\qquad C_{N,j}(n)=\frac{U_{N,j}(n)}{j!}.
U_vDefinition passage · 15 passages
Search aliases: U_v
Definition passage: Unitary packet tree · Paper_V_Traditional_Final.tex, line 2335
[Unitary packet tree] A finite unitary packet tree consists of Hilbert spaces H_v at its vertices and, at every internal vertex v, a unitary map U_v: H_v⟶⊕_w∈child(v) H_w. An admissible pruning is a set of vertices that meets every root to terminal path exactly once.
Exact mathematical passage
U_v:\Hcal_v\longrightarrow\bigoplus_{w\in\operatorname{child}(v)}\Hcal_w.
\UcalDefinition passage · 16 passages
Search aliases: Ucal U mathcal
Definition passage: Direct rotation of nearby packet systems · 04_correction_transport.tex, line 898
[Direct rotation of nearby packet systems] Let (P_α)_α∈ I and (Q_α)_α∈ I be complete orthogonal packet systems on the same Hilbert space. Put S( Q, P)=∑_α∈ IQ_α P_α. When S( Q, P) is invertible, define U( Q, P) = S( Q, P) (S( Q, P)^*S( Q, P))^-1/2.
Exact mathematical passage
\Ucal(\Qcal,\Pcal) = S(\Qcal,\Pcal) \big(S(\Qcal,\Pcal)^*S(\Qcal,\Pcal)\big)^{-1/2}.
\operatorname{unary}Definition passage · 1 passage
Search aliases: unary operatorname
Definition passage: Finite grammar for operation trees · 02_connections_obstructions.tex, line 772
[Finite grammar for operation trees] For types of input labels I, constants K, unary symbols U, and binary symbols B, the finite formal grammar has four constructors: t ::= · input(i) midconstant(k) midunary(u,t) · midbinary(b,t_1,t_2). Given an arbitrary value type Y, evaluation takes completely explicit maps I→ Y, K→ Y, U→ Y→ Y, B→ Y→ Y→ Y. No algebraic, continuity, or differentiability law is imposed on these semantics. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.OperationTree; Kom…
Exact mathematical passage
\begin{aligned} t ::={}&\operatorname{input}(i) \mid\operatorname{constant}(k) \mid\operatorname{unary}(u,t)\\ &\mid\operatorname{binary}(b,t_1,t_2). \end{aligned}
VDefinition passage · 490 passages
Search aliases: V
Definition passage: Separation weight · 04_arithmetic.tex, line 193
rojector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG preserves Λ. The minimal Lean theorem needs only the lattice action of u^-1. The bundled wrapper for lattice units records both u and u^-1. A nonzero scalar in the fraction field need not…
Exact mathematical passage
\boxed{\pi^sE=u^{-1}VG}
vDefinition passage · 794 passages
Search aliases: v
Definition passage: Typed operator grammar · 02_foundations.tex, line 47
r] Let Q=(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a…
Exact mathematical passage
(X_v)_{v\in Q_0}
V^*Definition passage · 27 passages
Search aliases: V^*
Definition passage: Block diagonal plus low rank operator · 01_manuscript.tex, line 303
[Block diagonal plus low rank operator] A block diagonal plus low rank operator has the form A=D+UV^*, where D=⊕_α∈ ID_α, U,V∈ C^n× r. The abbreviation BDLR will be used below.
Exact mathematical passage
A=D+UV^*,
V_0Definition passage · 12 passages
Search aliases: V_0
Definition passage: Monodromy packet chain · Paper_XI_Traditional_Final.tex, line 246
[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.
Exact mathematical passage
V_0,\ldots,V_{m-1}
v_0Definition passage · 49 passages
Search aliases: v_0
Definition passage: Path transport · Paper_XI_Traditional_Final.tex, line 1617
[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.
Exact mathematical passage
\gamma=(v_0\to v_1\to\cdots\to v_m),
V_1Definition passage · 8 passages
Search aliases: V_1
Definition passage: Stable core · Paper_XII_Traditional_Final.tex, line 1192
[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).
Exact mathematical passage
V_1\longleftarrow V_2\longleftarrow V_3\longleftarrow\cdots
v_1Definition passage · 10 passages
Search aliases: v_1
Definition passage: Path transport · Paper_XI_Traditional_Final.tex, line 1617
[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.
Exact mathematical passage
\gamma=(v_0\to v_1\to\cdots\to v_m),
V_2Definition passage · 7 passages
Search aliases: V_2
Definition passage: Stable core · Paper_XII_Traditional_Final.tex, line 1192
[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).
Exact mathematical passage
V_1\longleftarrow V_2\longleftarrow V_3\longleftarrow\cdots
V_3Definition passage · 2 passages
Search aliases: V_3
Definition passage: Stable core · Paper_XII_Traditional_Final.tex, line 1192
[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).
Exact mathematical passage
V_1\longleftarrow V_2\longleftarrow V_3\longleftarrow\cdots
V_jDefinition passage · 20 passages
Search aliases: V_j
Definition passage: Monodromy packet chain · Paper_XI_Traditional_Final.tex, line 247
[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.
Exact mathematical passage
V=\bigoplus_{j=0}^{m-1}V_j.
v_jDefinition passage · 19 passages
Search aliases: v_j
Definition passage: Weighted transfer series · 03_weighted.tex, line 386
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|γ|m(e_ν). Here γ=(e_1,…,e_|γ|) is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the en…
Exact mathematical passage
a_n=\sum_{i,j}\ \sum_{\substack{\gamma:i\leadsto j\\W(\gamma)=n}} u_i v_j\prod_{\nu=1}^{|\gamma|}m(e_\nu).
V_{j-1}Definition passage · 2 passages
Search aliases: V_j-1
Definition passage: Monodromy packet chain · Paper_XI_Traditional_Final.tex, line 251
[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.
Exact mathematical passage
N(V_j)\subseteq V_{j-1}
V_mDefinition passage · 8 passages
Search aliases: V_m
Definition passage: Stable core · Paper_XII_Traditional_Final.tex, line 1196
[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).
Exact mathematical passage
f_{n,m}:V_n\to V_m
v_mDefinition passage · 4 passages
Search aliases: v_m
Definition passage: Path transport · Paper_XI_Traditional_Final.tex, line 1617
[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.
Exact mathematical passage
\gamma=(v_0\to v_1\to\cdots\to v_m),
V_{m-1}Definition passage · 1 passage
Search aliases: V_m-1
Definition passage: Monodromy packet chain · Paper_XI_Traditional_Final.tex, line 246
[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.
Exact mathematical passage
V_0,\ldots,V_{m-1}
v_{m-1}Definition passage · 1 passage
Search aliases: v_m-1
Definition passage: Path transport · Paper_XI_Traditional_Final.tex, line 1621
[Path transport] For an oriented path γ=(v_0→ v_1→⋯→ v_m), define A_γ=A_v_m v_m-1⋯ A_v_1v_0, B_γ=B_v_m v_m-1⋯ B_v_1v_0. For a loop based at r, these are the based holonomies.
Exact mathematical passage
A_\gamma=A_{v_m v_{m-1}}\cdots A_{v_1v_0}, \qquad B_\gamma=B_{v_m v_{m-1}}\cdots B_{v_1v_0}.
V_nDefinition passage · 4 passages
Search aliases: V_n
Definition passage: Stable core · Paper_XII_Traditional_Final.tex, line 1196
[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).
Exact mathematical passage
f_{n,m}:V_n\to V_m
v_pDefinition passage · 42 passages
Search aliases: v_p
Definition passage: Differences and valuation conventions · 20_arithmetic.tex, line 106
[Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ν_p(u)=v_p(u) for u>0 and set ν_p(0)=0…
Exact mathematical passage
v_p(0)=+\infty
V_sDefinition passage · 8 passages
Search aliases: V_s
Definition passage: Orthogonal break decomposition · Paper_XI_Traditional_Final.tex, line 1752
[Orthogonal break decomposition] An orthogonal break decomposition is a symplectic direct sum V=mathop⊥limits_s∈ SV_s that is preserved by the declared tame stabilizer. Each V_s is the finite symplectic packet assigned to one break.
Exact mathematical passage
V=\perpbig_{s\in S}V_s
V_\alphaDefinition passage · 10 passages
Search aliases: V_α alpha
Definition passage: Lattice packet data · 04_arithmetic.tex, line 236
[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.
Exact mathematical passage
V=\bigoplus_{\alpha\in I}V_\alpha
V_\thetaDefinition passage · 8 passages
Search aliases: V_θ theta
Definition passage: Bridge datum · Paper_XI_Traditional_Final.tex, line 1197
item a one break Galois envelope (V_ρ,b_ρ) item a type side quotient (V_θ,h_θ) item an equivariant symplectic isometry ι:V_ρ⟶ V_θ for a finite cyclic stabilizer C=⟨ c⟩ item the same nontrivial central character on the two Heisenberg groups item when p=2, compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensions lifting ι.
Exact mathematical passage
(V_\theta,h_\theta)
V_\rhoDefinition passage · 6 passages
Search aliases: V_ρ rho
Definition passage: One break symplectic envelope · Paper_XI_Traditional_Final.tex, line 1090
[One break symplectic envelope] The one break symplectic envelope is V_ρ=A/rad b_ρ.
Exact mathematical passage
\boxed{ V_\rho=A/\rad b_\rho. }
\valDefinition passage · 2 passages
Search aliases: val operatorname
Definition passage: Separation weight · 04_arithmetic.tex, line 189
ight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG preserves Λ. The minimal Lean theorem needs only the lattice action of u^-1. The bundled wrapper for lattice units records both u and u^-1. A nonzero scalar in the f…
Exact mathematical passage
s_{\pi,\Lambda}(E)\le s=\val_\pi(r).
\val_\piDefinition passage · 2 passages
Search aliases: val_π val pi operatorname pi
Definition passage: Separation weight · 04_arithmetic.tex, line 189
ight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG preserves Λ. The minimal Lean theorem needs only the lattice action of u^-1. The bundled wrapper for lattice units records both u and u^-1. A nonzero scalar in the f…
Exact mathematical passage
s_{\pi,\Lambda}(E)\le s=\val_\pi(r).
\varprojlimDefinition passage · 8 passages
Search aliases: varprojlim
Definition passage: Complete multiplicative filtration · 03_weighted.tex, line 17
[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
\Bcal\longrightarrow\varprojlim_L\Bcal/F^L\Bcal
\varprojlim_LDefinition passage · 1 passage
Search aliases: varprojlim_L varprojlim
Definition passage: Complete multiplicative filtration · 03_weighted.tex, line 17
[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
\Bcal\longrightarrow\varprojlim_L\Bcal/F^L\Bcal
\VertDefinition passage · 9 passages
Search aliases: Vert
Definition passage: Relative information · Paper_VI_Traditional_Final.tex, line 3639
[Relative information] For probability vectors q and p, define D(qVert p) = ∑_iq_ilogfracq_ip_i. The convention is 0log0=0. A term with q_i>0 and p_i=0 has value +∞.
Exact mathematical passage
D(q\Vert p) = \sum_iq_i\log\frac{q_i}{p_i}.
\bm wDefinition passage · 4 passages
Search aliases: w bm
Definition passage: Seed zeta function · Paper_X_Block27_Amended.tex, line 538
[Seed zeta function] For a residual root bar x with finite levels, define Z_bar x(T) = ∑_m≥1#X_m(bar x)T^m. When a weighted chart is specified, write Z_ w(T).
Exact mathematical passage
Z_{\bm w}(T)
w^mDefinition passage · 12 passages
Search aliases: w^m
Definition passage · Paper_IV_Traditional_Final.tex, line 972
For r∈ C, define S_N^(r)(z) = D_N(z)^-r = exp(-rlog D_N(z)). Write S_N^(r)(z) = ∑_n≥0P_N,n(r)fracz^Nn(Nn)!. Define the logarithmic coefficients κ_N,m by -logΔ_N(w) = ∑_m≥1κ_N,mfracw^mm.
Exact mathematical passage
-\log\Delta_N(w) = \sum_{m\geq1}\kappa_{N,m}\frac{w^m}{m}.
w^nDefinition passage · 20 passages
Search aliases: w^n
Definition passage · Paper_IV_Traditional_Final.tex, line 723
Define the reduced denominator Δ_N(w) = ∑_n≥0(-1)^nfracw^n(Nn)!. Then D_N(z)=Δ_N(z^N).
Exact mathematical passage
\Delta_N(w) = \sum_{n\geq0}(-1)^n\frac{w^n}{(Nn)!}.
W^{r,p}Definition passage · 11 passages
Search aliases: W^r,p
Definition passage: Sobolev step packet class · Paper_V_Traditional_Final.tex, line 2525
[Sobolev step packet class] Fix r≥1, 1≤ p≤∞, and an integer Q≥0. The class F_r,p(Q) consists of functions f=s+∑_q=1^Qa_qϕ_0,τ_q with s∈ W^r,p(0,1) and 0<τ_1<⋯<τ_Q<1. No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.
Exact mathematical passage
s\in W^{r,p}(0,1)
w_{e,k}Definition passage · 1 passage
Search aliases: w_e,k
Definition passage: Discrete precision option · 01_manuscript.tex, line 1422
[Discrete precision option] For edge e, a discrete precision option k has a measured or modeled cost c_e,k and a nonnegative integer error weight w_e,k. Exactly one option must be selected on every edge.
Exact mathematical passage
w_{e,k}
\wideotimesDefinition passage · 5 passages
Search aliases: wideotimes mathbinwidehat⊗ mathbin widehat otimes
Definition passage: Global deformation datum · Paper_XII_Traditional_Final.tex, line 215
m A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_R^locR^loc,τ, R^loc,τ:=ma…
Exact mathematical passage
R^{\mathrm{loc}}:=\wideotimes_{v\in S,\OO}R_v^\square \longrightarrow R^\square.
\wideotimes_{R^{\mathrm{loc}}}Definition passage · 2 passages
Search aliases: wideotimes_R^loc mathbinwidehat⊗_R^loc wideotimes mathrm mathbin widehat otimes mathrm
Definition passage: Global deformation datum · Paper_XII_Traditional_Final.tex, line 220
every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_R^locR^loc,τ, R^loc,τ:=mathbinwidehat⊗_v∈ S, OR_v^τ_v.
Exact mathematical passage
R^{\tau}:=R^\square\wideotimes_{R^{\mathrm{loc}}}R^{\mathrm{loc},\tau}, \qquad R^{\mathrm{loc},\tau}:=\wideotimes_{v\in S,\OO}R_v^{\tau_v}.
\wideotimes_{v\in S,\OO}Definition passage · 2 passages
Search aliases: wideotimes_v∈ S,OO mathbinwidehat⊗_v∈ S, O wideotimes in OO mathbin widehat otimes in mathcal
Definition passage: Global deformation datum · Paper_XII_Traditional_Final.tex, line 215
m A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_R^locR^loc,τ, R^loc,τ:=ma…
Exact mathematical passage
R^{\mathrm{loc}}:=\wideotimes_{v\in S,\OO}R_v^\square \longrightarrow R^\square.
X^{\Pcal}Definition passage · 5 passages
Search aliases: X^Pcal X^ P Pcal mathcal
Definition passage: Packet system · 02_coordinates.tex, line 19
[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.
Exact mathematical passage
X_\alpha=P_\alpha X, \qquad X^{\Pcal}=\bigoplus_{\alpha\in I}X_\alpha.
X^{\Qcal}Definition passage · 3 passages
Search aliases: X^Qcal X^ Q Qcal mathcal
Definition passage: Coordinate change across levels · 02_coordinates.tex, line 84
[Coordinate change across levels] For two packet systems P and Q on X, define J_ Q← P = A_ Q S_ P:X^ P⟶ X^ Q and C_ P← Q = A_ P S_ Q:X^ Q⟶ X^ P. When Q refines P, the first map resolves each coarse coordinate into its children and the second adds the children back to their parent.
Exact mathematical passage
J_{\Qcal\leftarrow\Pcal} = \Acal_{\Qcal}\Scal_{\Pcal}:X^{\Pcal}\longrightarrow X^{\Qcal}
X_aDefinition passage · 4 passages
Search aliases: X_a
Definition passage: Typed operator grammar · 02_foundations.tex, line 48
(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a∈ Q_1).
Exact mathematical passage
X_a:X_{s(a)}\longrightarrow X_{t(a)} \qquad(a\in Q_1)
X_CDefinition passage · 10 passages
Search aliases: X_C
Definition passage: Packet pure seed rank · 03_packets_and_graphs.tex, line 1020
[Packet pure seed rank] A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).
Exact mathematical passage
P_C:=\sum_{\alpha\in C}P_\alpha, \qquad X_C:=P_CX, \qquad \A_C:=P_C\A P_C,
X_{\operatorname{in}_\sigma(q)}Definition passage · 1 passage
Search aliases: X_in_σ(q) operatorname sigma
Definition passage: Finitary colored operation signature · 02_connections_obstructions.tex, line 835
lored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a dependent inductive typed tree. No such declaration is part of the finite algebraic completion contract.
Exact mathematical passage
\Phi_\sigma: \prod_{q\in\operatorname{Fin}(r(\sigma))}X_{\operatorname{in}_\sigma(q)} \longrightarrow X_{\operatorname{out}(\sigma)}.
X_jDefinition passage · 15 passages
Search aliases: X_j
Definition passage: Projective limit space · Paper_VI_Traditional_Final.tex, line 5292
[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.
Exact mathematical passage
(X_j,C_j)
x_jDefinition passage · 23 passages
Search aliases: x_j
Definition passage: Projective limit space · Paper_VI_Traditional_Final.tex, line 5293
[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.
Exact mathematical passage
X_\infty = \left\{ (x_j)_{j\geq0}\in\prod_{j\geq0}X_j: C_jx_{j+1}=x_j\text{ for every }j \right\}.
X_{j+1}Definition passage · 2 passages
Search aliases: X_j+1
Definition passage: Projective limit space · Paper_VI_Traditional_Final.tex, line 5292
[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.
Exact mathematical passage
C_j:X_{j+1}\to X_j
x_{j+1}Definition passage · 2 passages
Search aliases: x_j+1
Definition passage: Projective limit space · Paper_VI_Traditional_Final.tex, line 5293
[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.
Exact mathematical passage
X_\infty = \left\{ (x_j)_{j\geq0}\in\prod_{j\geq0}X_j: C_jx_{j+1}=x_j\text{ for every }j \right\}.
X_mDefinition passage · 8 passages
Search aliases: X_m
Definition passage: Seed zeta function · Paper_X_Block27_Amended.tex, line 531
[Seed zeta function] For a residual root bar x with finite levels, define Z_bar x(T) = ∑_m≥1#X_m(bar x)T^m. When a weighted chart is specified, write Z_ w(T).
Exact mathematical passage
\boxed{ Z_{\bar x}(T) = \sum_{m\ge1}\#X_m(\bar x)T^m. }
x_mDefinition passage · 41 passages
Search aliases: x_m
Definition passage: Depth obstruction · 04_correction_transport.tex, line 239
[Depth obstruction] The depth m obstruction of x_m is Ob_m(x_m) = [Φ(x_m)] ∈ cokeroverline L_m, where Φ(x_m) is first viewed in F^mY/F^m+1Y.
Exact mathematical passage
x_m
X_nDefinition passage · 17 passages
Search aliases: X_n
Definition passage: Unconditional packet sum space · 03_packets_and_graphs.tex, line 354
[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .
Exact mathematical passage
(X_n)_{n\geq1}
x_nDefinition passage · 18 passages
Search aliases: x_n
Definition passage: Unconditional packet sum space · 03_packets_and_graphs.tex, line 358
[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .
Exact mathematical passage
x=(x_n)
X_{\operatorname{out}(\sigma)}Definition passage · 1 passage
Search aliases: X_out(σ) operatorname sigma
Definition passage: Finitary colored operation signature · 02_connections_obstructions.tex, line 835
lored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a dependent inductive typed tree. No such declaration is part of the finite algebraic completion contract.
Exact mathematical passage
\Phi_\sigma: \prod_{q\in\operatorname{Fin}(r(\sigma))}X_{\operatorname{in}_\sigma(q)} \longrightarrow X_{\operatorname{out}(\sigma)}.
X_{s(a)}Definition passage · 1 passage
Search aliases: X_s(a)
Definition passage: Typed operator grammar · 02_foundations.tex, line 48
(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a∈ Q_1).
Exact mathematical passage
X_a:X_{s(a)}\longrightarrow X_{t(a)} \qquad(a\in Q_1)
X_{t(a)}Definition passage · 1 passage
Search aliases: X_t(a)
Definition passage: Typed operator grammar · 02_foundations.tex, line 48
(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a∈ Q_1).
Exact mathematical passage
X_a:X_{s(a)}\longrightarrow X_{t(a)} \qquad(a\in Q_1)
X_vDefinition passage · 10 passages
Search aliases: X_v
Definition passage: Typed operator grammar · 02_foundations.tex, line 47
r] Let Q=(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a…
Exact mathematical passage
(X_v)_{v\in Q_0}
X_{\alpha_1}Definition passage · 2 passages
Search aliases: X_α_1 alpha
Definition passage: Packet jet · 03_nonlinear_coordinates.tex, line 220
[Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.
Exact mathematical passage
J^{r}_{\gamma\mid\alpha_1,\ldots,\alpha_r}F(x) = Q_\gamma D^rF(x) \big|_{X_{\alpha_1}\times\cdots\times X_{\alpha_r}}.
X_{\alpha_r}Definition passage · 2 passages
Search aliases: X_α_r alpha
Definition passage: Packet jet · 03_nonlinear_coordinates.tex, line 220
[Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.
Exact mathematical passage
J^{r}_{\gamma\mid\alpha_1,\ldots,\alpha_r}F(x) = Q_\gamma D^rF(x) \big|_{X_{\alpha_1}\times\cdots\times X_{\alpha_r}}.
X_{\lambda,1}Definition passage · 5 passages
Search aliases: X_λ,1 lambda
Definition passage: Finite companion forest · Paper_XII_Traditional_Final.tex, line 1431
[Finite companion forest] Fix a coefficient place λ of L. A finite companion forest for the spine is an inverse system of finite sets X_λ,1 longleftarrow X_λ,2 longleftarrow X_λ,3 longleftarrow⋯ whose points are admissible realizations modulo λ^m, with edges given by reduction. A complete realization is admissible for this forest when all its finite reductions belong to the displayed levels; the empty level obstruction below concerns this declared admissibility problem.
Exact mathematical passage
X_{\lambda,1} \longleftarrow X_{\lambda,2} \longleftarrow X_{\lambda,3} \longleftarrow\cdots
X_{\lambda,2}Definition passage · 1 passage
Search aliases: X_λ,2 lambda
Definition passage: Finite companion forest · Paper_XII_Traditional_Final.tex, line 1431
[Finite companion forest] Fix a coefficient place λ of L. A finite companion forest for the spine is an inverse system of finite sets X_λ,1 longleftarrow X_λ,2 longleftarrow X_λ,3 longleftarrow⋯ whose points are admissible realizations modulo λ^m, with edges given by reduction. A complete realization is admissible for this forest when all its finite reductions belong to the displayed levels; the empty level obstruction below concerns this declared admissibility problem.
Exact mathematical passage
X_{\lambda,1} \longleftarrow X_{\lambda,2} \longleftarrow X_{\lambda,3} \longleftarrow\cdots
X_{\lambda,3}Definition passage · 1 passage
Search aliases: X_λ,3 lambda
Definition passage: Finite companion forest · Paper_XII_Traditional_Final.tex, line 1431
[Finite companion forest] Fix a coefficient place λ of L. A finite companion forest for the spine is an inverse system of finite sets X_λ,1 longleftarrow X_λ,2 longleftarrow X_λ,3 longleftarrow⋯ whose points are admissible realizations modulo λ^m, with edges given by reduction. A complete realization is admissible for this forest when all its finite reductions belong to the displayed levels; the empty level obstruction below concerns this declared admissibility problem.
Exact mathematical passage
X_{\lambda,1} \longleftarrow X_{\lambda,2} \longleftarrow X_{\lambda,3} \longleftarrow\cdots
X_{\lambda,m}Definition passage · 8 passages
Search aliases: X_λ,m lambda
Definition passage: Common arithmetic spine · Paper_X_Block27_Amended.tex, line 768
ic spine] Let M be a number field. A common spine C consists of: item characteristic polynomials P_v(X)∈ M[X] at almost all places; item coefficient-prime independent weight and determinant data; item prescribed local Weil--Deligne or inertial packet labels at the ramified places. For each finite place λ of M, let X_λ,m( C) be the finite set of admissible objects modulo ϖ_λ^m realizing the reduction of this spine.
Exact mathematical passage
X_{\lambda,m}(\Compat)
X_\inftyDefinition passage · 9 passages
Search aliases: X_∞ infty
Definition passage: Projective limit space · Paper_VI_Traditional_Final.tex, line 5293
[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.
Exact mathematical passage
X_\infty = \left\{ (x_j)_{j\geq0}\in\prod_{j\geq0}X_j: C_jx_{j+1}=x_j\text{ for every }j \right\}.
\xrightarrowDefinition passage · 21 passages
Search aliases: xrightarrow
Definition passage: Normalized reversible connection · 02_connections_obstructions.tex, line 119
[Normalized reversible connection] A normalized reversible connection is a family of linear equivalences Γ_b← a:C_axrightarrow ∼ C_b (a,b∈ B) such that Γ_a← a=id, Γ_a← b=Γ_b← a^-1. No cocycle equation is included in the structure. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection
Exact mathematical passage
\Gamma_{b\leftarrow a}:C_a\xrightarrow{\;\sim\;}C_b \qquad(a,b\in B)
YDefinition passage · 253 passages
Search aliases: Y
Definition passage: Typed operator grammar · 02_foundations.tex, line 52
k G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a∈ Q_1).
Exact mathematical passage
f:X\to Y
yDefinition passage · 497 passages
Search aliases: y
Definition passage: Packet graph and scalar edge system · 01_finite_coordinates.tex, line 67
[Packet graph and scalar edge system] For a,b∈ V and r∈ R, define packetMatrix(a,b;r)_yx = r, · y=b and x=a, 0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem; Komorebi.PaperIII.EdgeOperators.PacketGr…
Exact mathematical passage
\operatorname{packetMatrix}(a,b;r)_{yx} = \begin{cases} r,&y=b\text{ and }x=a,\\ 0,&\text{otherwise}. \end{cases}
Y_aDefinition passage · 1 passage
Search aliases: Y_a
Definition passage: Typed operator grammar · 02_foundations.tex, line 53
linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a∈ Q_1).
Exact mathematical passage
f_{t(a)}X_a=Y_af_{s(a)} \qquad(a\in Q_1).
Y_mDefinition passage · 8 passages
Search aliases: Y_m
Definition passage: Stable core · Paper_XII_Traditional_Final.tex, line 1197
[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).
Exact mathematical passage
Y_m = \bigcap_{n\geq m}\im(f_{n,m}).
Y_vDefinition passage · 1 passage
Search aliases: Y_v
Definition passage: Typed operator grammar · 02_foundations.tex, line 52
l on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a∈ Q_1).
Exact mathematical passage
f_v:X_v\to Y_v
Y_\gammaDefinition passage · 5 passages
Search aliases: Y_γ gamma
Definition passage: Packet jet · 03_nonlinear_coordinates.tex, line 227
[Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.
Exact mathematical passage
X_{\alpha_1}\times\cdots\times X_{\alpha_r} \longrightarrow Y_\gamma.
\ZDefinition passage · 191 passages
Search aliases: Z mathbb Z mathbf
Definition passage: Differences and valuation conventions · 20_arithmetic.tex, line 110
put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ν_p(u)=v_p(u) for u>0 and set ν_p(0)=0. This finite convention makes the displayed exponents meaningful at st…
Exact mathematical passage
x\in p^L\Z_p
\mathbb ZDefinition passage · 187 passages
Search aliases: Z mathbb
Definition passage: Complete multiplicative filtration · 03_weighted.tex, line 20
bra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
w(e)\in\mathbb Z_{>0}
\mathfrak ZDefinition passage · 20 passages
Search aliases: Z mathfrak
Definition passage · Paper_IV_Traditional_Final.tex, line 2743
Define Z_N^abs(s) = ∑_ρ∈ Z( D_N)|ρ|^-s and Z_N^red(s) = ∑_λ∈Λ_N|λ|^-s, with multiplicities. Each term means e^-slog|ρ| or e^-slog|λ|.
Exact mathematical passage
\mathfrak Z_N^{\mathrm{abs}}(s) = \sum_{\rho\in\Zcal(\Dfun_N)}|\rho|^{-s}
z^{2m}Definition passage · 3 passages
Search aliases: z^2m
Definition passage: Dyadic truncation · 20_arithmetic.tex, line 874
[Dyadic truncation] Define β_m by the formal unit-series identity ∑_m≥0β_m z^2m=(fraczsinh z)^2, β_0=1, where the quotient means the inverse of the unit series (sinh z/z)^2. Define A(n)=frac3^2n+2-14(2n+1)(2n+2), S_m(n)=β_m2^2mfrac(2n)!(2n-2m)!A(n-m) (0≤ m≤ n), and T_M(n)=∑_m=0^min(M,n)S_m(n) (M,n∈ N).
Exact mathematical passage
\sum_{m\ge0}\beta_m z^{2m}=\left(\frac{z}{\sinh z}\right)^2, \qquad \beta_0=1,
z^jDefinition passage · 9 passages
Search aliases: z^j
Definition passage · Paper_IV_Traditional_Final.tex, line 1388
For 0≤ j<N, define Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)!. Then E_N,j(z) = fracz^jj!Φ_N,j(z^N). Define the residue-section reciprocal function C_N,j(z) = Φ_N,j(z^N)^-1 = fracz^j/j! E_N,j(z). For j>0, the last quotient at z=0 is interpreted by its removable continuation, namely the first expression in the display. Write C_N,j(z) = ∑_n≥0C_N,j(n)fracz^Nn(Nn)!.
Exact mathematical passage
\Efun_{N,j}(z) = \frac{z^j}{j!}\Phi_{N,j}(z^N).
z^mDefinition passage · 9 passages
Search aliases: z^m
Definition passage · Paper_IV_Traditional_Final.tex, line 1484
Define polynomials E_m^(N,j,r)(x) by C_N,j(z)^re^xz = ∑_m≥0E_m^(N,j,r)(x)fracz^mm!.
Exact mathematical passage
\Cfun_{N,j}(z)^re^{xz} = \sum_{m\geq0}E_m^{(N,j,r)}(x)\frac{z^m}{m!}.
z^NDefinition passage · 35 passages
Search aliases: z^N
Definition passage · Paper_IV_Traditional_Final.tex, line 729
Define the reduced denominator Δ_N(w) = ∑_n≥0(-1)^nfracw^n(Nn)!. Then D_N(z)=Δ_N(z^N).
Exact mathematical passage
\Dfun_N(z)=\Delta_N(z^N).
z^nDefinition passage · 17 passages
Search aliases: z^n
Definition passage: Residue selector · 02_coordinates.tex, line 370
[Residue selector] Let R be a commutative ring and let F(z)=∑_n≥0a_nz^n∈ R[[z]]. For N≥1 and 0≤ j<N, define (P_N,jF)(z) = ∑_n≡ jpmod Na_nz^n.
Exact mathematical passage
F(z)=\sum_{n\geq0}a_nz^n\in R[[z]].
z^{Nn}Definition passage · 10 passages
Search aliases: z^Nn
Definition passage · Paper_IV_Traditional_Final.tex, line 966
For r∈ C, define S_N^(r)(z) = D_N(z)^-r = exp(-rlog D_N(z)). Write S_N^(r)(z) = ∑_n≥0P_N,n(r)fracz^Nn(Nn)!. Define the logarithmic coefficients κ_N,m by -logΔ_N(w) = ∑_m≥1κ_N,mfracw^mm.
Exact mathematical passage
\Sfun_N^{(r)}(z) = \sum_{n\geq0}P_{N,n}(r)\frac{z^{Nn}}{(Nn)!}.
z^{Nn+j}Definition passage · 5 passages
Search aliases: z^Nn+j
Definition passage · Paper_IV_Traditional_Final.tex, line 606
For 0≤ j<N, define D_N,j(z) = ∑_n≥0(-1)^nfracz^Nn+j(Nn+j)!. The distinguished signed denominator is D_N(z)= D_N,0(z).
Exact mathematical passage
\Dfun_{N,j}(z) = \sum_{n\geq0}(-1)^n\frac{z^{Nn+j}}{(Nn+j)!}.
z^{Nn+N-1}Definition passage · 2 passages
Search aliases: z^Nn+N-1
Definition passage · Paper_IV_Traditional_Final.tex, line 1267
Define the transfer tangent T_N(z) =-frac D_N'(z) D_N(z). Near the origin, its convergent block expansion has the form T_N(z) = ∑_n≥0T_N(n) fracz^Nn+N-1(Nn+N-1)!.
Exact mathematical passage
\Tfun_N(z) = \sum_{n\geq0}T_N(n) \frac{z^{Nn+N-1}}{(Nn+N-1)!}.
Z_{\bm w}Definition passage · 2 passages
Search aliases: Z_ w bm
Definition passage: Seed zeta function · Paper_X_Block27_Amended.tex, line 538
[Seed zeta function] For a residual root bar x with finite levels, define Z_bar x(T) = ∑_m≥1#X_m(bar x)T^m. When a weighted chart is specified, write Z_ w(T).
Exact mathematical passage
Z_{\bm w}(T)
\mathbb Z_{>0}Definition passage · 1 passage
Search aliases: Z_>0 mathbb
Definition passage: Complete multiplicative filtration · 03_weighted.tex, line 20
bra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
w(e)\in\mathbb Z_{>0}
Z_{\bar x}Definition passage · 1 passage
Search aliases: Z_bar x bar
Definition passage: Seed zeta function · Paper_X_Block27_Amended.tex, line 531
[Seed zeta function] For a residual root bar x with finite levels, define Z_bar x(T) = ∑_m≥1#X_m(bar x)T^m. When a weighted chart is specified, write Z_ w(T).
Exact mathematical passage
\boxed{ Z_{\bar x}(T) = \sum_{m\ge1}\#X_m(\bar x)T^m. }
Z_LDefinition passage · 4 passages
Search aliases: Z_L
Definition passage: Heat trace and spectral dimension · 02_strengthened.tex, line 1957
[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).
Exact mathematical passage
Z_L(t)=\Tr(\e^{-tL})
\mathfrak Z_{N,q}^{\mathrm{der}}Definition passage · 2 passages
Search aliases: Z_N,q^der mathfrak mathrm
Definition passage · Paper_IV_Traditional_Final.tex, line 2925
Let q be a positive integer. Choose logarithms of the simple zeros of D_N with arguments in a fixed bounded interval. Define Z_N,q^der(s) = ∑_substack D_N(ρ)=0 D_N'(ρ)≠0 frace^-sLogρ D_N'(ρ)^q.
Exact mathematical passage
\mathfrak Z_{N,q}^{\mathrm{der}}(s) = \sum_{\substack{\Dfun_N(\rho)=0\\ \Dfun_N'(\rho)\ne0}} \frac{e^{-s\Log\rho}}{\Dfun_N'(\rho)^q}.
\mathfrak Z_N^{\mathrm{abs}}Definition passage · 6 passages
Search aliases: Z_N^abs mathfrak mathrm
Definition passage · Paper_IV_Traditional_Final.tex, line 2743
Define Z_N^abs(s) = ∑_ρ∈ Z( D_N)|ρ|^-s and Z_N^red(s) = ∑_λ∈Λ_N|λ|^-s, with multiplicities. Each term means e^-slog|ρ| or e^-slog|λ|.
Exact mathematical passage
\mathfrak Z_N^{\mathrm{abs}}(s) = \sum_{\rho\in\Zcal(\Dfun_N)}|\rho|^{-s}
\mathfrak Z_N^{\mathrm{orb}}Definition passage · 2 passages
Search aliases: Z_N^orb mathfrak mathrm
Definition passage · Paper_IV_Traditional_Final.tex, line 2907
Choose a logarithm for each member of the finite exceptional part of Λ_N. Use the real logarithm for the eventual positive sequence. Define Z_N^orb(s) = ∑_λ∈Λ_Ne^-sLogλ. This series is initially defined on Re s>1/N.
Exact mathematical passage
\mathfrak Z_N^{\mathrm{orb}}(s) = \sum_{\lambda\in\Lambda_N}e^{-s\Log\lambda}.
\mathfrak Z_N^{\mathrm{red}}Definition passage · 6 passages
Search aliases: Z_N^red mathfrak mathrm
Definition passage · Paper_IV_Traditional_Final.tex, line 2749
Define Z_N^abs(s) = ∑_ρ∈ Z( D_N)|ρ|^-s and Z_N^red(s) = ∑_λ∈Λ_N|λ|^-s, with multiplicities. Each term means e^-slog|ρ| or e^-slog|λ|.
Exact mathematical passage
\mathfrak Z_N^{\mathrm{red}}(s) = \sum_{\lambda\in\Lambda_N}|\lambda|^{-s},
\Z_pDefinition passage · 56 passages
Search aliases: Z_p Z mathbb Z_p Z mathbf
Definition passage: Differences and valuation conventions · 20_arithmetic.tex, line 110
put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ν_p(u)=v_p(u) for u>0 and set ν_p(0)=0. This finite convention makes the displayed exponents meaningful at st…
Exact mathematical passage
x\in p^L\Z_p
\mathbb Z_pDefinition passage · 34 passages
Search aliases: Z_p mathbb
Definition passage: Hurwitz integral series · strengthening.tex, line 42
[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.
Exact mathematical passage
\mathcal H_p=\left\{\sum_{d\geq0}f_d\frac{t^d}{d!}: f_d\in\mathbb Z_p\right\} \subset\mathbb Q_p[[t]].
Z_{u,v}Definition passage · 5 passages
Search aliases: Z_u,v
Definition passage: Weighted transfer series · 03_weighted.tex, line 380
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j
Exact mathematical passage
Z_{u,v}(t) = \sum_{n\geq0}a_nt^n,
\ZcalDefinition passage · 8 passages
Search aliases: Zcal Z mathcal
Definition passage · Paper_IV_Traditional_Final.tex, line 2743
Define Z_N^abs(s) = ∑_ρ∈ Z( D_N)|ρ|^-s and Z_N^red(s) = ∑_λ∈Λ_N|λ|^-s, with multiplicities. Each term means e^-slog|ρ| or e^-slog|λ|.
Exact mathematical passage
\mathfrak Z_N^{\mathrm{abs}}(s) = \sum_{\rho\in\Zcal(\Dfun_N)}|\rho|^{-s}
|^{-s}Definition passage · 3 passages
Search aliases: |^-s
Definition passage · Paper_IV_Traditional_Final.tex, line 2743
Define Z_N^abs(s) = ∑_ρ∈ Z( D_N)|ρ|^-s and Z_N^red(s) = ∑_λ∈Λ_N|λ|^-s, with multiplicities. Each term means e^-slog|ρ| or e^-slog|λ|.
Exact mathematical passage
\mathfrak Z_N^{\mathrm{abs}}(s) = \sum_{\rho\in\Zcal(\Dfun_N)}|\rho|^{-s}
|_CDefinition passage · 1 passage
Search aliases: |_C
Definition passage: Packet pure seed rank · 03_packets_and_graphs.tex, line 1028
f:directed-seed-rank A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).
Exact mathematical passage
r_C:=\psrank_{\A_C,\Pcal|_C}(X_C).
|_{\im(g-I)}Definition passage · 1 passage
Search aliases: |_im(g-I) im operatorname
Definition passage: Moving phase · Paper_XI_Traditional_Final.tex, line 1403
[Moving phase] Under the hypotheses of [source reference], define Φ_p^mov(g,V) = Φ_p(g|_im(g-I),im(g-I)).
Exact mathematical passage
\Phi_p^{\mathrm{mov}}(g,V) = \Phi_p\left(g|_{\im(g-I)},\im(g-I)\right).
|_{\Ker C}Definition passage · 1 passage
Search aliases: |_Ker C Ker operatorname
Definition passage: Canonical lumping obstruction · Paper_VI_Traditional_Final.tex, line 2854
[Canonical lumping obstruction] For a linear fine transition operator K, define Ob_C(K) = (CK)|_Ker C ∈ B(Ker C, R^m).
Exact mathematical passage
\Ob_C(K) = (CK)|_{\Ker C} \in \mathcal B(\Ker C,\RR^m).
|_{X_{\alpha_1}\times\cdots\times X_{\alpha_r}}Definition passage · 1 passage
Search aliases: |_X_α_1×⋯× X_α_r alpha times cdots times alpha
Definition passage: Packet jet · 03_nonlinear_coordinates.tex, line 220
[Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.
Exact mathematical passage
J^{r}_{\gamma\mid\alpha_1,\ldots,\alpha_r}F(x) = Q_\gamma D^rF(x) \big|_{X_{\alpha_1}\times\cdots\times X_{\alpha_r}}.
\timesDefinition passage · 308 passages
Search aliases: × times
Definition passage: Support relation and support space · 03_packets_and_graphs.tex, line 791
Support patterns, preorders, and triangular structure. [Support relation and support space] For a relation R⊆ I× I, with (β,α) interpreted as an allowed transfer α→β, define M_R := {T∈End(X):T_βα=0 whenever (β,α)∉ R}.
Exact mathematical passage
R\subseteq I\times I
\alpha_1Definition passage · 16 passages
Search aliases: α_1 alpha
Definition passage: Packet jet · 03_nonlinear_coordinates.tex, line 220
[Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.
Exact mathematical passage
J^{r}_{\gamma\mid\alpha_1,\ldots,\alpha_r}F(x) = Q_\gamma D^rF(x) \big|_{X_{\alpha_1}\times\cdots\times X_{\alpha_r}}.
\alpha_rDefinition passage · 13 passages
Search aliases: α_r alpha
Definition passage: Packet jet · 03_nonlinear_coordinates.tex, line 220
[Packet jet] For 1≤ r≤ m, define J^r_γmidα_1,…,α_rF(x) = Q_γ D^rF(x) |_X_α_1×⋯× X_α_r. This is a continuous multilinear map X_α_1×⋯× X_α_r ⟶ Y_γ.
Exact mathematical passage
J^{r}_{\gamma\mid\alpha_1,\ldots,\alpha_r}F(x) = Q_\gamma D^rF(x) \big|_{X_{\alpha_1}\times\cdots\times X_{\alpha_r}}.
\beta_0Definition passage · 5 passages
Search aliases: β_0 beta
Definition passage: Dyadic truncation · 20_arithmetic.tex, line 874
[Dyadic truncation] Define β_m by the formal unit-series identity ∑_m≥0β_m z^2m=(fraczsinh z)^2, β_0=1, where the quotient means the inverse of the unit series (sinh z/z)^2. Define A(n)=frac3^2n+2-14(2n+1)(2n+2), S_m(n)=β_m2^2mfrac(2n)!(2n-2m)!A(n-m) (0≤ m≤ n), and T_M(n)=∑_m=0^min(M,n)S_m(n) (M,n∈ N).
Exact mathematical passage
\sum_{m\ge0}\beta_m z^{2m}=\left(\frac{z}{\sinh z}\right)^2, \qquad \beta_0=1,
\beta_mDefinition passage · 18 passages
Search aliases: β_m beta
Definition passage: Dyadic truncation · 20_arithmetic.tex, line 873
[Dyadic truncation] Define β_m by the formal unit-series identity ∑_m≥0β_m z^2m=(fraczsinh z)^2, β_0=1, where the quotient means the inverse of the unit series (sinh z/z)^2. Define A(n)=frac3^2n+2-14(2n+1)(2n+2), S_m(n)=β_m2^2mfrac(2n)!(2n-2m)!A(n-m) (0≤ m≤ n), and T_M(n)=∑_m=0^min(M,n)S_m(n) (M,n∈ N).
Exact mathematical passage
\beta_m
\Gamma^G_{b\leftarrow a}Definition passage · 1 passage
Search aliases: Γ^G_b← a Gamma leftarrow
Definition passage: Gauge change · 02_connections_obstructions.tex, line 314
[Gauge change] A gauge is a family of automorphisms G_a∈Aut_R(C_a). It acts on a connection by Γ^G_b← a = G_bΓ_b← aG_a^-1. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.Gauge; Komorebi.PaperIII.Transport.ReversibleConnection.gauge
Exact mathematical passage
\Gamma^G_{b\leftarrow a} = G_b\Gamma_{b\leftarrow a}G_a^{-1}.
\Gamma_{(a_0)}Definition passage · 1 passage
Search aliases: Γ_(a_0) Gamma
Definition passage: Connection transport along a path · 02_connections_obstructions.tex, line 267
[Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transport.ReversibleConnection.indexedPathTransport
Exact mathematical passage
\Gamma_p = \Gamma_{a_n\leftarrow a_{n-1}}\cdots \Gamma_{a_1\leftarrow a_0}, \qquad \Gamma_{(a_0)}=\id_{C_{a_0}}.
\gamma_0Definition passage · 2 passages
Search aliases: γ_0 gamma
Definition passage: Odd-length source and inverse coefficients · 20_arithmetic.tex, line 766
-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd. Primality is needed only to interpret the result as a prime-adic congruence ladder.
Exact mathematical passage
\gamma_0=1
\Gamma_{a_1\leftarrow a_0}Definition passage · 1 passage
Search aliases: Γ_a_1← a_0 Gamma leftarrow
Definition passage: Connection transport along a path · 02_connections_obstructions.tex, line 267
[Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transport.ReversibleConnection.indexedPathTransport
Exact mathematical passage
\Gamma_p = \Gamma_{a_n\leftarrow a_{n-1}}\cdots \Gamma_{a_1\leftarrow a_0}, \qquad \Gamma_{(a_0)}=\id_{C_{a_0}}.
\Gamma_{a_n\leftarrow a_{n-1}}Definition passage · 1 passage
Search aliases: Γ_a_n← a_n-1 Gamma leftarrow
Definition passage: Connection transport along a path · 02_connections_obstructions.tex, line 267
[Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transport.ReversibleConnection.indexedPathTransport
Exact mathematical passage
\Gamma_p = \Gamma_{a_n\leftarrow a_{n-1}}\cdots \Gamma_{a_1\leftarrow a_0}, \qquad \Gamma_{(a_0)}=\id_{C_{a_0}}.
\Gamma_{a\leftarrow a}Definition passage · 3 passages
Search aliases: Γ_a← a Gamma leftarrow
Definition passage: Noncanonical connection normalized at each chart · 02_connections_obstructions.tex, line 18
[Noncanonical connection normalized at each chart] A minimal connection is a family Γ_b← a∈Aut_R(M) (a,b∈ B) with only Γ_a← a=id_M. Neither inversion on reversed edges nor the cocycle law is a field of the structure. statusleanthreeKomorebi.PaperIII.Transport.Connection
Exact mathematical passage
\Gamma_{a\leftarrow a}=\id_M.
\Gamma_{a\leftarrow b}Definition passage · 2 passages
Search aliases: Γ_a← b Gamma leftarrow
Definition passage: Normalized reversible connection · 02_connections_obstructions.tex, line 124
[Normalized reversible connection] A normalized reversible connection is a family of linear equivalences Γ_b← a:C_axrightarrow ∼ C_b (a,b∈ B) such that Γ_a← a=id, Γ_a← b=Γ_b← a^-1. No cocycle equation is included in the structure. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection
Exact mathematical passage
\Gamma_{a\leftarrow a}=\id, \qquad \Gamma_{a\leftarrow b}=\Gamma_{b\leftarrow a}^{-1}.
\Gamma_{a\leftarrow c}Definition passage · 7 passages
Search aliases: Γ_a← c Gamma leftarrow
Definition passage: Triangle holonomy · 02_connections_obstructions.tex, line 192
[Triangle holonomy] The triangle holonomy based at a is H_c,b,a = Γ_a← c Γ_c← b Γ_b← a ∈Aut_R(C_a). It traverses a→ b→ c→ a. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.triangleHolonomy; Komorebi.PaperIII.Transport.ReversibleConnection.triangleObstruction
Exact mathematical passage
H_{c,b,a} = \Gamma_{a\leftarrow c} \Gamma_{c\leftarrow b} \Gamma_{b\leftarrow a} \in\operatorname{Aut}_R(C_a).
\Gamma_{b\leftarrow a}Definition passage · 17 passages
Search aliases: Γ_b← a Gamma leftarrow
Definition passage: Noncanonical connection normalized at each chart · 02_connections_obstructions.tex, line 13
[Noncanonical connection normalized at each chart] A minimal connection is a family Γ_b← a∈Aut_R(M) (a,b∈ B) with only Γ_a← a=id_M. Neither inversion on reversed edges nor the cocycle law is a field of the structure. statusleanthreeKomorebi.PaperIII.Transport.Connection
Exact mathematical passage
\Gamma_{b\leftarrow a}\in\operatorname{Aut}_R(M) \qquad(a,b\in B)
\Gamma_{b\leftarrow a}^{-1}Definition passage · 2 passages
Search aliases: Γ_b← a^-1 Gamma leftarrow
Definition passage: Normalized reversible connection · 02_connections_obstructions.tex, line 124
[Normalized reversible connection] A normalized reversible connection is a family of linear equivalences Γ_b← a:C_axrightarrow ∼ C_b (a,b∈ B) such that Γ_a← a=id, Γ_a← b=Γ_b← a^-1. No cocycle equation is included in the structure. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection
Exact mathematical passage
\Gamma_{a\leftarrow a}=\id, \qquad \Gamma_{a\leftarrow b}=\Gamma_{b\leftarrow a}^{-1}.
\Gamma_{c\leftarrow a}Definition passage · 4 passages
Search aliases: Γ_c← a Gamma leftarrow
Definition passage: Calibration and flatness · 02_connections_obstructions.tex, line 146
[Calibration and flatness] When the C_a are exact packet coordinates of one ambient module, the connection is canonically calibrated if Γ_b← a=J_b← a for every a,b. It is flat if Γ_c← bΓ_b← a = Γ_c← a for every a,b,c. The canonical connection is flat. A general flat connection is proved in [source reference] to be related to it by a gauge change. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.CocycleAt; Komorebi.PaperIII.Transport.ReversibleConnection.Flat; Komorebi.PaperIII.Tra…
Exact mathematical passage
\Gamma_{c\leftarrow b}\Gamma_{b\leftarrow a} = \Gamma_{c\leftarrow a} \quad\text{for every }a,b,c.
\Gamma_{c\leftarrow b}Definition passage · 7 passages
Search aliases: Γ_c← b Gamma leftarrow
Definition passage: Calibration and flatness · 02_connections_obstructions.tex, line 146
[Calibration and flatness] When the C_a are exact packet coordinates of one ambient module, the connection is canonically calibrated if Γ_b← a=J_b← a for every a,b. It is flat if Γ_c← bΓ_b← a = Γ_c← a for every a,b,c. The canonical connection is flat. A general flat connection is proved in [source reference] to be related to it by a gauge change. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.CocycleAt; Komorebi.PaperIII.Transport.ReversibleConnection.Flat; Komorebi.PaperIII.Tra…
Exact mathematical passage
\Gamma_{c\leftarrow b}\Gamma_{b\leftarrow a} = \Gamma_{c\leftarrow a} \quad\text{for every }a,b,c.
\gamma_mDefinition passage · 11 passages
Search aliases: γ_m gamma
Definition passage: Odd-length source and inverse coefficients · 20_arithmetic.tex, line 760
[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd. Primality is needed only to interpret the result as a prime-adic congruence ladder.
Exact mathematical passage
\gamma_m
\Gamma_pDefinition passage · 3 passages
Search aliases: Γ_p Gamma
Definition passage: Connection transport along a path · 02_connections_obstructions.tex, line 267
[Connection transport along a path] For a vertex sequence p=(a_0,a_1,…,a_n), define Γ_p = Γ_a_n← a_n-1⋯ Γ_a_1← a_0, Γ_(a_0)=id_C_a_0. If a_n=a_0, then Γ_p is the holonomy of the loop p. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport; Komorebi.PaperIII.Transport.ReversibleConnection.VertexPath; Komorebi.PaperIII.Transport.ReversibleConnection.indexedPathTransport
Exact mathematical passage
\Gamma_p = \Gamma_{a_n\leftarrow a_{n-1}}\cdots \Gamma_{a_1\leftarrow a_0}, \qquad \Gamma_{(a_0)}=\id_{C_{a_0}}.
\Gamma_{\rm eff}Definition passage · 1 passage
Search aliases: Γ_rm eff Gamma rm
Definition passage: Einstein infrared criterion · 02_strengthened.tex, line 2558
[Einstein infrared criterion] A quantum geometric tower meets an Einstein infrared criterion only if an effective action is controlled at long wavelengths and has an expansion whose leading geometric terms take the form Γ_rm eff[g] = ∫ sqrtabs g ( Λ + frac116π GR + controlled higher order terms ) d^dx. Here the volume coefficient includes the sign and normalization written explicitly in the cosmological term in Part I. The expansion must respect the limiting gauge identities and must include an …
Exact mathematical passage
\Gamma_{\rm eff}[g] = \int \sqrt{\abs g} \left( \Lambda + \frac{1}{16\pi G}R + \text{controlled higher order terms} \right) \dd^dx.
\Delta_{c,b,a}Definition passage · 8 passages
Search aliases: Δ_c,b,a Delta
Definition passage: Cocycle defect · 02_connections_obstructions.tex, line 176
[Cocycle defect] For a,b,c∈ B, define the additive cocycle defect Δ_c,b,a = Γ_c← bΓ_b← a -Γ_c← a ∈Hom_R(C_a,C_c). statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.cocycleDefect
Exact mathematical passage
\Delta_{c,b,a} = \Gamma_{c\leftarrow b}\Gamma_{b\leftarrow a} -\Gamma_{c\leftarrow a} \in\Hom_R(C_a,C_c).
\Delta_FDefinition passage · 12 passages
Search aliases: Δ_F Delta
Definition passage: Nonlinear refinement defect · 03_nonlinear_coordinates.tex, line 737
[Nonlinear refinement defect] The pointwise defect of F is Δ_F(x) = C_YF_f(x)-F_c(C_Xx). The two models are exactly compatible when Δ_F vanishes everywhere.
Exact mathematical passage
\Delta_F(x) = C_YF_f(x)-F_c(C_Xx).
\Delta_hDefinition passage · 7 passages
Search aliases: Δ_h Delta
Definition passage: Differences and valuation conventions · 20_arithmetic.tex, line 101
[Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ν_p(u)=v_p(u) for u>0 and set ν_p(0)=0…
Exact mathematical passage
\Delta_h f(n)=f(n+h)-f(n),\qquad \Delta_h^0 f(n)=f(n),\qquad \Delta_h^{k+1}f(n)=\Delta_h^k f(n+h)-\Delta_h^k f(n).
\Delta_h^0Definition passage · 2 passages
Search aliases: Δ_h^0 Delta
Definition passage: Differences and valuation conventions · 20_arithmetic.tex, line 101
[Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ν_p(u)=v_p(u) for u>0 and set ν_p(0)=0…
Exact mathematical passage
\Delta_h f(n)=f(n+h)-f(n),\qquad \Delta_h^0 f(n)=f(n),\qquad \Delta_h^{k+1}f(n)=\Delta_h^k f(n+h)-\Delta_h^k f(n).
\Delta_h^kDefinition passage · 8 passages
Search aliases: Δ_h^k Delta
Definition passage: Differences and valuation conventions · 20_arithmetic.tex, line 101
[Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ν_p(u)=v_p(u) for u>0 and set ν_p(0)=0…
Exact mathematical passage
\Delta_h f(n)=f(n+h)-f(n),\qquad \Delta_h^0 f(n)=f(n),\qquad \Delta_h^{k+1}f(n)=\Delta_h^k f(n+h)-\Delta_h^k f(n).
\Delta_h^{k+1}Definition passage · 1 passage
Search aliases: Δ_h^k+1 Delta
Definition passage: Differences and valuation conventions · 20_arithmetic.tex, line 101
[Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ν_p(u)=v_p(u) for u>0 and set ν_p(0)=0…
Exact mathematical passage
\Delta_h f(n)=f(n+h)-f(n),\qquad \Delta_h^0 f(n)=f(n),\qquad \Delta_h^{k+1}f(n)=\Delta_h^k f(n+h)-\Delta_h^k f(n).
\Delta_NDefinition passage · 51 passages
Search aliases: Δ_N Delta
Definition passage · Paper_IV_Traditional_Final.tex, line 723
Define the reduced denominator Δ_N(w) = ∑_n≥0(-1)^nfracw^n(Nn)!. Then D_N(z)=Δ_N(z^N).
Exact mathematical passage
\Delta_N(w) = \sum_{n\geq0}(-1)^n\frac{w^n}{(Nn)!}.
\epsilonDefinition passage · 55 passages
Search aliases: ϵ epsilon
Definition passage: Candidate representation tree · 01_manuscript.tex, line 1505
[Candidate representation tree] A candidate representation tree is a finite rooted tree. Every node v has κ_v≥0, ϵ_v≥0, representing the cost and error of keeping v as one leaf. Every internal node also has a split overhead σ_v≥0. An admissible pruning contains exactly one selected node on every root to terminal path.
Exact mathematical passage
\kappa_v\geq0, \qquad \epsilon_v\geq0,
\varepsilonDefinition passage · 185 passages
Search aliases: ε varepsilon
Definition passage: Seed evaluation map · 02_foundations.tex, line 218
ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.
Exact mathematical passage
\varepsilon_S:A^{\oplus r}\longrightarrow M, \qquad (a_1,\ldots,a_r)\longmapsto\sum_{j=1}^{r}a_js_j.
\varepsilon_{\rm KL}Definition passage · 4 passages
Search aliases: ε_rm KL varepsilon rm
Definition passage: Approximate Knill--Laflamme defect · 02_strengthened.tex, line 1423
[Approximate Knill--Laflamme defect] For a finite error family, define the following defect. When the family is empty, assign the maximum the value zero. ε_rm KL = max_a,b inf_c∈ C normPE_a^*E_bP-cP.
Exact mathematical passage
\varepsilon_{\rm KL} = \max_{a,b} \inf_{c\in\CC} \norm{PE_a^*E_bP-cP}.
\varepsilon_SDefinition passage · 9 passages
Search aliases: ε_S varepsilon
Definition passage: Seed evaluation map · 02_foundations.tex, line 218
ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.
Exact mathematical passage
\varepsilon_S:A^{\oplus r}\longrightarrow M, \qquad (a_1,\ldots,a_r)\longmapsto\sum_{j=1}^{r}a_js_j.
\epsilon_vDefinition passage · 12 passages
Search aliases: ϵ_v epsilon
Definition passage: Candidate representation tree · 01_manuscript.tex, line 1505
[Candidate representation tree] A candidate representation tree is a finite rooted tree. Every node v has κ_v≥0, ϵ_v≥0, representing the cost and error of keeping v as one leaf. Every internal node also has a split overhead σ_v≥0. An admissible pruning contains exactly one selected node on every root to terminal path.
Exact mathematical passage
\kappa_v\geq0, \qquad \epsilon_v\geq0,
\etaDefinition passage · 132 passages
Search aliases: η eta
Definition passage: Within packet information loss · Paper_VI_Traditional_Final.tex, line 3560
[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.
Exact mathematical passage
\eta_i(p) = \lambda_i-\overline\lambda_\alpha(p) \quad\hbox{when }i\in I_\alpha.
\eta_iDefinition passage · 6 passages
Search aliases: η_i eta
Definition passage: Within packet information loss · Paper_VI_Traditional_Final.tex, line 3560
[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.
Exact mathematical passage
\eta_i(p) = \lambda_i-\overline\lambda_\alpha(p) \quad\hbox{when }i\in I_\alpha.
\thetaDefinition passage · 136 passages
Search aliases: θ theta
Definition passage · Paper_IV_Traditional_Final.tex, line 1793
For θ∈ R, define h_N(θ) = max_0≤ k<N cos(θ+frac(2k+1)πN). The tie directions are T_N = {frac2π kN:0≤ k<N} bmod 2π.
Exact mathematical passage
\theta\in\mathbb R
\iotaDefinition passage · 38 passages
Search aliases: ι iota
Definition passage: Bridge datum · Paper_XI_Traditional_Final.tex, line 1199
item a one break Galois envelope (V_ρ,b_ρ) item a type side quotient (V_θ,h_θ) item an equivariant symplectic isometry ι:V_ρ⟶ V_θ for a finite cyclic stabilizer C=⟨ c⟩ item the same nontrivial central character on the two Heisenberg groups item when p=2, compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensions lifting ι.
Exact mathematical passage
\iota:V_\rho\longrightarrow V_\theta
\iota_\alphaDefinition passage · 3 passages
Search aliases: ι_α iota alpha
Definition passage: Packet layout · 01_manuscript.tex, line 75
[Packet layout] A packet layout is a finite family of finite dimensional Hilbert spaces P=(X_α)_α∈ I with direct sum X=⊕_α∈ IX_α. Write n_α=dim X_α, n=∑_α∈ In_α. The coordinate inclusion and projection are ι_α:X_α→ X and π_α:X→ X_α. The ambient coordinate projector is P_α=ι_απ_α.
Exact mathematical passage
\iota_\alpha:X_\alpha\to X
\iota_\betaDefinition passage · 4 passages
Search aliases: ι_β iota beta
Definition passage: Packet operator representation · 01_manuscript.tex, line 91
IR=( P,E,(B_βα)_(β,α)∈ E), where E⊆ I× I and B_βα:X_α⟶ X_β. Local blocks are identified with their ambient extensions by inclusion and projection. In particular, the notation P_β AP_α denotes either the ambient block or its restriction X_α→ X_β according to the displayed domain, and ι_β B_βαπ_α is the corresponding ambient extension. Its evaluation is eval(IR) = ∑_(β,α)∈ E ι_β B_βαπ_α, where ι_β:X_β→ X is the coordinate inclusion.
Exact mathematical passage
\iota_\beta B_{\beta\alpha}\pi_\alpha
\kappaDefinition passage · 129 passages
Search aliases: κ kappa
Definition passage: Packet coordinate condition number · 03_packets_and_graphs.tex, line 285
[Packet coordinate condition number] Fix a finite product norm on ⊕_α X_α equivalent to the maximum norm, as in [source reference]. Define κ( P):= A_ P S_ P .
Exact mathematical passage
\kappa(\Pcal):= \|\mathcal A_{\Pcal}\|\, \|\mathcal S_{\Pcal}\|.
\kappa_{N,m}Definition passage · 18 passages
Search aliases: κ_N,m kappa
Definition passage · Paper_IV_Traditional_Final.tex, line 971
For r∈ C, define S_N^(r)(z) = D_N(z)^-r = exp(-rlog D_N(z)). Write S_N^(r)(z) = ∑_n≥0P_N,n(r)fracz^Nn(Nn)!. Define the logarithmic coefficients κ_N,m by -logΔ_N(w) = ∑_m≥1κ_N,mfracw^mm.
Exact mathematical passage
\kappa_{N,m}
\kappa_vDefinition passage · 15 passages
Search aliases: κ_v kappa
Definition passage: Candidate representation tree · 01_manuscript.tex, line 1505
[Candidate representation tree] A candidate representation tree is a finite rooted tree. Every node v has κ_v≥0, ϵ_v≥0, representing the cost and error of keeping v as one leaf. Every internal node also has a split overhead σ_v≥0. An admissible pruning contains exactly one selected node on every root to terminal path.
Exact mathematical passage
\kappa_v\geq0, \qquad \epsilon_v\geq0,
\lambdaDefinition passage · 535 passages
Search aliases: λ lambda
Definition passage · Paper_IV_Traditional_Final.tex, line 458
For λ∈ C and 0≤ j<N, define E_N,j(λ z) = B_N,j(e^λ z) = ∑_n≥0frac(λ z)^Nn+j(Nn+j)!.
Exact mathematical passage
\lambda\in\CC
\lambda^{-m}Definition passage · 1 passage
Search aliases: λ^-m lambda
Definition passage · Paper_IV_Traditional_Final.tex, line 2438
For m≥1, define σ_N,m = ∑_λ∈Λ_Nλ^-m, with multiplicities.
Exact mathematical passage
\sigma_{N,m} = \sum_{\lambda\in\Lambda_N}\lambda^{-m},
\lambda^mDefinition passage · 3 passages
Search aliases: λ^m lambda
Definition passage: Finite companion forest · Paper_XII_Traditional_Final.tex, line 1439
[Finite companion forest] Fix a coefficient place λ of L. A finite companion forest for the spine is an inverse system of finite sets X_λ,1 longleftarrow X_λ,2 longleftarrow X_λ,3 longleftarrow⋯ whose points are admissible realizations modulo λ^m, with edges given by reduction. A complete realization is admissible for this forest when all its finite reductions belong to the displayed levels; the empty level obstruction below concerns this declared admissibility problem.
Exact mathematical passage
\lambda^m
\lambda_1Definition passage · 8 passages
Search aliases: λ_1 lambda
Definition passage: Heat trace and spectral dimension · 02_strengthened.tex, line 1956
[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).
Exact mathematical passage
\lambda_1,\ldots,\lambda_N
\lambda_iDefinition passage · 47 passages
Search aliases: λ_i lambda
Definition passage: Within packet information loss · Paper_VI_Traditional_Final.tex, line 3560
[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.
Exact mathematical passage
\eta_i(p) = \lambda_i-\overline\lambda_\alpha(p) \quad\hbox{when }i\in I_\alpha.
\Lambda_NDefinition passage · 23 passages
Search aliases: Λ_N Lambda
Definition passage · Paper_IV_Traditional_Final.tex, line 2438
For m≥1, define σ_N,m = ∑_λ∈Λ_Nλ^-m, with multiplicities.
Exact mathematical passage
\sigma_{N,m} = \sum_{\lambda\in\Lambda_N}\lambda^{-m},
\lambda_NDefinition passage · 11 passages
Search aliases: λ_N lambda
Definition passage: Heat trace and spectral dimension · 02_strengthened.tex, line 1956
[Heat trace and spectral dimension] Let the eigenvalues of L be λ_1,…,λ_N, with at least one zero eigenvalue. Define Z_L(t)=Tr( e^-tL) and d_L(t) =-2frac dlog Z_L(t) dlog t = 2t fracTr(L e^-tL)Tr( e^-tL).
Exact mathematical passage
\lambda_1,\ldots,\lambda_N
\Lambda_{\mathrm{split}}Definition passage · 7 passages
Search aliases: Λ_split Lambda mathrm
Definition passage: Lattice packet data · 04_arithmetic.tex, line 240
[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.
Exact mathematical passage
\Lambda_\alpha=\Lambda\cap V_\alpha, \qquad \Lambda_{\mathrm{split}}=\bigoplus_\alpha\Lambda_\alpha,
\Lambda_\alphaDefinition passage · 7 passages
Search aliases: Λ_α Lambda alpha
Definition passage: Lattice packet data · 04_arithmetic.tex, line 240
[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.
Exact mathematical passage
\Lambda_\alpha=\Lambda\cap V_\alpha, \qquad \Lambda_{\mathrm{split}}=\bigoplus_\alpha\Lambda_\alpha,
\muDefinition passage · 155 passages
Search aliases: μ mu
Definition passage: Holonomy obstruction and packet residuals · 02_connections_obstructions.tex, line 472
[Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy within one packet. Off-diagonal blocks detect packet mixing induced by holonomy. statusleanthreeKomorebi.PaperIII.HolonomyBlocks.PacketResolution; Komorebi…
Exact mathematical passage
\Ob_{c,b,a}=H_{c,b,a}-\id_{C_a}, \qquad \Ob^{\mu\nu}_{c,b,a} = \Pi^a_\mu\Ob_{c,b,a}\Pi^a_\nu.
\nuDefinition passage · 75 passages
Search aliases: ν nu
Definition passage: Weighted transfer series · 03_weighted.tex, line 386
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|γ|m(e_ν). Here γ=(e_1,…,e_|γ|) is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the en…
Exact mathematical passage
a_n=\sum_{i,j}\ \sum_{\substack{\gamma:i\leadsto j\\W(\gamma)=n}} u_i v_j\prod_{\nu=1}^{|\gamma|}m(e_\nu).
\nu_pDefinition passage · 4 passages
Search aliases: ν_p nu
Definition passage: Differences and valuation conventions · 20_arithmetic.tex, line 111
d Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ν_p(u)=v_p(u) for u>0 and set ν_p(0)=0. This finite convention makes the displayed exponents meaningful at step zero, where all positive-order differe…
Exact mathematical passage
\nu_p(u)=v_p(u)
\xiDefinition passage · 41 passages
Search aliases: ξ xi
Definition passage · Paper_IV_Traditional_Final.tex, line 787
Let A_N^+= C[ξ]/(ξ^N-1) and A_N^-= C[ξ]/(ξ^N+1). Every element has a unique representative of degree below N.
Exact mathematical passage
\mathfrak A_N^+=\CC[\xi]/(\xi^N-1)
\xi^NDefinition passage · 2 passages
Search aliases: ξ^N xi
Definition passage · Paper_IV_Traditional_Final.tex, line 787
Let A_N^+= C[ξ]/(ξ^N-1) and A_N^-= C[ξ]/(ξ^N+1). Every element has a unique representative of degree below N.
Exact mathematical passage
\mathfrak A_N^+=\CC[\xi]/(\xi^N-1)
\PiDefinition passage · 27 passages
Search aliases: Π Pi
Definition passage: Holonomy obstruction and packet residuals · 02_connections_obstructions.tex, line 472
[Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy within one packet. Off-diagonal blocks detect packet mixing induced by holonomy. statusleanthreeKomorebi.PaperIII.HolonomyBlocks.PacketResolution; Komorebi…
Exact mathematical passage
\Ob_{c,b,a}=H_{c,b,a}-\id_{C_a}, \qquad \Ob^{\mu\nu}_{c,b,a} = \Pi^a_\mu\Ob_{c,b,a}\Pi^a_\nu.
\varpiDefinition passage · 29 passages
Search aliases: ϖ varpi
Definition passage: Visibility weight · Paper_X_Block27_Amended.tex, line 476
[Visibility weight] For two integral packet coordinates a,b∈ O, define their separation depth w(a,b)=min{m≥1:anot≡ bpmodϖ^m}, when such an m exists. A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.
Exact mathematical passage
w(a,b)=\min\{m\ge1:a\not\equiv b\pmod{\varpi^m}\},
\Pi^a_\muDefinition passage · 8 passages
Search aliases: Π^a_μ Pi mu
Definition passage: Holonomy obstruction and packet residuals · 02_connections_obstructions.tex, line 472
[Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy within one packet. Off-diagonal blocks detect packet mixing induced by holonomy. statusleanthreeKomorebi.PaperIII.HolonomyBlocks.PacketResolution; Komorebi…
Exact mathematical passage
\Ob_{c,b,a}=H_{c,b,a}-\id_{C_a}, \qquad \Ob^{\mu\nu}_{c,b,a} = \Pi^a_\mu\Ob_{c,b,a}\Pi^a_\nu.
\Pi^a_\nuDefinition passage · 5 passages
Search aliases: Π^a_ν Pi nu
Definition passage: Holonomy obstruction and packet residuals · 02_connections_obstructions.tex, line 472
[Holonomy obstruction and packet residuals] Define Ob_c,b,a=H_c,b,a-id_C_a, Ob^μν_c,b,a = Π^a_μOb_c,b,aΠ^a_ν. The map Ob^μν_c,b,a is the obstruction block, or packet residual, from input packet ν to output packet μ. If it is nonzero, then (μ,ν) and that nonvanishing proof form an obstruction packet certificate. Diagonal blocks detect holonomy within one packet. Off-diagonal blocks detect packet mixing induced by holonomy. statusleanthreeKomorebi.PaperIII.HolonomyBlocks.PacketResolution; Komorebi…
Exact mathematical passage
\Ob_{c,b,a}=H_{c,b,a}-\id_{C_a}, \qquad \Ob^{\mu\nu}_{c,b,a} = \Pi^a_\mu\Ob_{c,b,a}\Pi^a_\nu.
\varpi^mDefinition passage · 12 passages
Search aliases: ϖ^m varpi
Definition passage: Visibility weight · Paper_X_Block27_Amended.tex, line 476
[Visibility weight] For two integral packet coordinates a,b∈ O, define their separation depth w(a,b)=min{m≥1:anot≡ bpmodϖ^m}, when such an m exists. A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.
Exact mathematical passage
w(a,b)=\min\{m\ge1:a\not\equiv b\pmod{\varpi^m}\},
\pi^sDefinition passage · 24 passages
Search aliases: π^s pi
Definition passage: Separation weight · 04_arithmetic.tex, line 181
[Separation weight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG preserves Λ. The minimal Lean theorem needs only the lattice action of u^-1. The bundled wrapper for lattice units records both u and u^-1. A nonzero s…
Exact mathematical passage
s_{\pi,\Lambda}(E) = \min\{s\ge0:\pi^sE(\Lambda)\subseteq\Lambda\}.
\pi_\alphaDefinition passage · 5 passages
Search aliases: π_α pi alpha
Definition passage: Packet layout · 01_manuscript.tex, line 75
gindefinition[Packet layout] A packet layout is a finite family of finite dimensional Hilbert spaces P=(X_α)_α∈ I with direct sum X=⊕_α∈ IX_α. Write n_α=dim X_α, n=∑_α∈ In_α. The coordinate inclusion and projection are ι_α:X_α→ X and π_α:X→ X_α. The ambient coordinate projector is P_α=ι_απ_α.
Exact mathematical passage
\pi_\alpha:X\to X_\alpha
\varpi_\lambda^mDefinition passage · 1 passage
Search aliases: ϖ_λ^m varpi lambda
Definition passage: Common arithmetic spine · Paper_X_Block27_Amended.tex, line 771
at consists of: item characteristic polynomials P_v(X)∈ M[X] at almost all places; item coefficient-prime independent weight and determinant data; item prescribed local Weil--Deligne or inertial packet labels at the ramified places. For each finite place λ of M, let X_λ,m( C) be the finite set of admissible objects modulo ϖ_λ^m realizing the reduction of this spine.
Exact mathematical passage
\varpi_\lambda^m
\rhoDefinition passage · 335 passages
Search aliases: ρ rho
Definition passage: Refinement · 02_coordinates.tex, line 35
[Refinement] Let P=(P_α)_α∈ I and Q=(Q_β)_β∈ J be packet systems on the same module. We say that Q refines P when there is a surjection ρ:J⟶ I such that P_α=∑_ρ(β)=αQ_β for every α∈ I. The index β is then called a child of ρ(β).
Exact mathematical passage
\rho:J\longrightarrow I
\rho_RDefinition passage · 11 passages
Search aliases: ρ_R rho
Definition passage: Ordinary completion datum at a point · Paper_XII_Traditional_Final.tex, line 1643
[Ordinary completion datum at a point] An ordinary completion datum at a characteristic zero point x:R→ E' consists of the following objects. item A represented global deformation ring R with declared local quotient conditions and universal representation ρ_R. item An ordinary patching datum (R_∞, a,M_∞,R,H) whose finite level module H is an automorphic Hecke carrier. item Full patched support Supp_R_∞(M_∞)=Spec R_∞. item An isomorphism R≅ T with the Hecke algebra acting on H. The R action on H …
Exact mathematical passage
\rho_R
\rho_xDefinition passage · 38 passages
Search aliases: ρ_x rho
Definition passage: Ordinary completion datum at a point · Paper_XII_Traditional_Final.tex, line 1658
phism. item An automorphic Hecke datum ( T,H,(P_v)) and a pointwise classicality and Galois realization theorem satisfying all the inputs of [source reference] for the nonzero fiber at x: an occurring classical automorphic representation with a continuous semisimple attached representation; equality of the characteristic polynomials of both that representation and ρ_x=x∘ρ_R with x(P_v(X)) on the same declared Chebotarev dense union of Frobenius conjugacy classes; and the standard semisimplificat…
Exact mathematical passage
\rho_x=x\circ\rho_R
\SigmaDefinition passage · 40 passages
Search aliases: Σ Sigma
Definition passage: Finitary colored operation signature · 02_connections_obstructions.tex, line 828
[Finitary colored operation signature] A finitary colored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a dependent inductive typed tree. No such declaration is part of the finite algebraic completion contract.
Exact mathematical passage
\Sigma
\sigmaDefinition passage · 145 passages
Search aliases: σ sigma
Definition passage: Packet compatible operator morphism · 04_transport.tex, line 51
[Packet compatible operator morphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):= ∑_σ(α)=jP_α. An empty fiber is interpreted as the zero projector.
Exact mathematical passage
(\Phi,\varphi,\sigma)
\sigma^{-1}Definition passage · 7 passages
Search aliases: σ^-1 sigma
Definition passage: Packet compatible operator morphism · 04_transport.tex, line 59
orphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):= ∑_σ(α)=jP_α. An empty fiber is interpreted as the zero projector.
Exact mathematical passage
Q_j\Phi = \Phi P_{\sigma^{-1}(j)}, \qquad P_{\sigma^{-1}(j)}:= \sum_{\sigma(\alpha)=j}P_\alpha.
\sigma_{N,m}Definition passage · 8 passages
Search aliases: σ_N,m sigma
Definition passage · Paper_IV_Traditional_Final.tex, line 2438
For m≥1, define σ_N,m = ∑_λ∈Λ_Nλ^-m, with multiplicities.
Exact mathematical passage
\sigma_{N,m} = \sum_{\lambda\in\Lambda_N}\lambda^{-m},
\sigma_vDefinition passage · 11 passages
Search aliases: σ_v sigma
Definition passage: Candidate representation tree · 01_manuscript.tex, line 1510
[Candidate representation tree] A candidate representation tree is a finite rooted tree. Every node v has κ_v≥0, ϵ_v≥0, representing the cost and error of keeping v as one leaf. Every internal node also has a split overhead σ_v≥0. An admissible pruning contains exactly one selected node on every root to terminal path.
Exact mathematical passage
\sigma_v\geq0
\tau_1Definition passage · 4 passages
Search aliases: τ_1 tau
Definition passage: Sobolev step packet class · Paper_V_Traditional_Final.tex, line 2526
[Sobolev step packet class] Fix r≥1, 1≤ p≤∞, and an integer Q≥0. The class F_r,p(Q) consists of functions f=s+∑_q=1^Qa_qϕ_0,τ_q with s∈ W^r,p(0,1) and 0<τ_1<⋯<τ_Q<1. No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.
Exact mathematical passage
0<\tau_1<\cdots<\tau_Q<1.
\tau_QDefinition passage · 2 passages
Search aliases: τ_Q tau
Definition passage: Sobolev step packet class · Paper_V_Traditional_Final.tex, line 2526
[Sobolev step packet class] Fix r≥1, 1≤ p≤∞, and an integer Q≥0. The class F_r,p(Q) consists of functions f=s+∑_q=1^Qa_qϕ_0,τ_q with s∈ W^r,p(0,1) and 0<τ_1<⋯<τ_Q<1. No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.
Exact mathematical passage
0<\tau_1<\cdots<\tau_Q<1.
\tau_qDefinition passage · 15 passages
Search aliases: τ_q tau
Definition passage: Sobolev step packet class · Paper_V_Traditional_Final.tex, line 2522
[Sobolev step packet class] Fix r≥1, 1≤ p≤∞, and an integer Q≥0. The class F_r,p(Q) consists of functions f=s+∑_q=1^Qa_qϕ_0,τ_q with s∈ W^r,p(0,1) and 0<τ_1<⋯<τ_Q<1. No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.
Exact mathematical passage
f=s+\sum_{q=1}^{Q}a_q\phi_{0,\tau_q}
\tau_vDefinition passage · 2 passages
Search aliases: τ_v tau
Definition passage: Global deformation datum · Paper_XII_Traditional_Final.tex, line 213
[Global deformation datum] A global deformation datum consists of the following objects. item A residual representation barρ. item For every v∈ S, a framed local deformation ring R_v^square. item A quotient R_v^τ_v representing the chosen local condition whenever such a quotient is known to exist. item A global deformation ring R^square with a continuous local map R^loc:=mathbinwidehat⊗_v∈ S, OR_v^square ⟶ R^square. item The induced global ring with local conditions R^τ:=R^squaremathbinwidehat⊗_…
Exact mathematical passage
R_v^{\tau_v}
\PhiDefinition passage · 233 passages
Search aliases: Φ Phi
Definition passage: Witness for seed rank · 02_foundations.tex, line 634
Quotient witnesses for lower bounds. [Witness for seed rank] A surjective intertwiner Φ:M↠ N is a witness for seed rank of order r when srank(N)=r. It certifies srank(M)≥ r.
Exact mathematical passage
\Phi:M\twoheadrightarrow N
\varphiDefinition passage · 59 passages
Search aliases: φ varphi
Definition passage: Packet compatible operator morphism · 04_transport.tex, line 51
[Packet compatible operator morphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):= ∑_σ(α)=jP_α. An empty fiber is interpreted as the zero projector.
Exact mathematical passage
(\Phi,\varphi,\sigma)
\phi_{0,\tau_q}Definition passage · 1 passage
Search aliases: ϕ_0,τ_q phi tau
Definition passage: Sobolev step packet class · Paper_V_Traditional_Final.tex, line 2522
[Sobolev step packet class] Fix r≥1, 1≤ p≤∞, and an integer Q≥0. The class F_r,p(Q) consists of functions f=s+∑_q=1^Qa_qϕ_0,τ_q with s∈ W^r,p(0,1) and 0<τ_1<⋯<τ_Q<1. No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.
Exact mathematical passage
f=s+\sum_{q=1}^{Q}a_q\phi_{0,\tau_q}
\Phi_{N,j}Definition passage · 16 passages
Search aliases: Φ_N,j Phi
Definition passage · Paper_IV_Traditional_Final.tex, line 1382
For 0≤ j<N, define Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)!. Then E_N,j(z) = fracz^jj!Φ_N,j(z^N). Define the residue-section reciprocal function C_N,j(z) = Φ_N,j(z^N)^-1 = fracz^j/j! E_N,j(z). For j>0, the last quotient at z=0 is interpreted by its removable continuation, namely the first expression in the display. Write C_N,j(z) = ∑_n≥0C_N,j(n)fracz^Nn(Nn)!.
Exact mathematical passage
\Phi_{N,j}(w) = j!\sum_{n\geq0}\frac{w^n}{(Nn+j)!}.
\Phi_pDefinition passage · 8 passages
Search aliases: Φ_p Phi
Definition passage: Odd finite Weil phase · Paper_XI_Traditional_Final.tex, line 1308
[Odd finite Weil phase] For fixed point free g, define Φ_p(g,V) = (frac-1p)^r (fracdet(g-I)p).
Exact mathematical passage
\Phi_p(g,V) = \left(\frac{-1}{p}\right)^r \left(\frac{\det(g-I)}{p}\right).
\Phi_p^{\mathrm{mov}}Definition passage · 1 passage
Search aliases: Φ_p^mov Phi mathrm
Definition passage: Moving phase · Paper_XI_Traditional_Final.tex, line 1403
[Moving phase] Under the hypotheses of [source reference], define Φ_p^mov(g,V) = Φ_p(g|_im(g-I),im(g-I)).
Exact mathematical passage
\Phi_p^{\mathrm{mov}}(g,V) = \Phi_p\left(g|_{\im(g-I)},\im(g-I)\right).
\Phi_\sigmaDefinition passage · 3 passages
Search aliases: Φ_σ Phi sigma
Definition passage: Finitary colored operation signature · 02_connections_obstructions.tex, line 835
lored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a dependent inductive typed tree. No such declaration is part of the finite algebraic completion contract.
Exact mathematical passage
\Phi_\sigma: \prod_{q\in\operatorname{Fin}(r(\sigma))}X_{\operatorname{in}_\sigma(q)} \longrightarrow X_{\operatorname{out}(\sigma)}.
\chiDefinition passage · 85 passages
Search aliases: χ chi
Definition passage: Normalized Gauss phase · Paper_XI_Traditional_Final.tex, line 740
[Normalized Gauss phase] For nontrivial χ, define γ(χ,ψ) = q^-1/2G(χ,ψ). Then absγ(χ,ψ)=1.
Exact mathematical passage
\chi
\psiDefinition passage · 183 passages
Search aliases: ψ psi
Definition passage: Normalized Gauss phase · Paper_XI_Traditional_Final.tex, line 741
[Normalized Gauss phase] For nontrivial χ, define γ(χ,ψ) = q^-1/2G(χ,ψ). Then absγ(χ,ψ)=1.
Exact mathematical passage
\gamma(\chi,\psi) = q^{-1/2}G(\chi,\psi).
\psi_0Definition passage · 18 passages
Search aliases: ψ_0 psi
Definition passage: Finite quantum seed system · 01_finite_core.tex, line 42
[Finite quantum seed system] A finite quantum seed system is a tuple Q= ( H, ψ_0, O, R, {P_ℓ,α}, H, τ ), where H is a finite-dimensional Hilbert space, ψ_0∈ H is a nonzero seed, O is the declared operator grammar, R is a collection of algebraic or physical constraints, {P_ℓ,α} is an exact packet system at each refinement level, H=H^* is a Hamiltonian when dynamics are present, and τ specifies the topology used in a later completion.
Exact mathematical passage
\mathfrak Q= \left( \cH, \psi_0, \cO, \cR, \{P_{\ell,\alpha}\}, H, \tau \right),
\omegaDefinition passage · 123 passages
Search aliases: ω omega
Definition passage: Operator charge · 02_strengthened.tex, line 295
[Operator charge] An operator A∈End( H) has cyclic charge q∈ Z/N Z when SAS^*=ω^qA.
Exact mathematical passage
SAS^*=\omega^qA.
\omega^qDefinition passage · 3 passages
Search aliases: ω^q omega
Definition passage: Operator charge · 02_strengthened.tex, line 295
[Operator charge] An operator A∈End( H) has cyclic charge q∈ Z/N Z when SAS^*=ω^qA.
Exact mathematical passage
SAS^*=\omega^qA.
\ldotsDefinition passage · 261 passages
Search aliases: … ldots
Definition passage: Linear seed system · 02_foundations.tex, line 151
Seeds, addressed hulls, and natural presentations. [Linear seed system] A linear seed system is a tuple S=(X, A,S, P,τ), where X is a unital left module over A, A is fixed independently of the target element, S=(s_1,…,s_r) is a finite distinguished seed family, P is either empty or a declared packet system, and τ is either the algebraic convention or a specified Hausdorff topology in which the declared operator action is continuous.
Exact mathematical passage
S=(s_1,\ldots,s_r)
\leftarrowDefinition passage · 87 passages
Search aliases: ← leftarrow
Definition passage: Coordinate change across levels · 02_coordinates.tex, line 84
[Coordinate change across levels] For two packet systems P and Q on X, define J_ Q← P = A_ Q S_ P:X^ P⟶ X^ Q and C_ P← Q = A_ P S_ Q:X^ Q⟶ X^ P. When Q refines P, the first map resolves each coarse coordinate into its children and the second adds the children back to their parent.
Exact mathematical passage
J_{\Qcal\leftarrow\Pcal} = \Acal_{\Qcal}\Scal_{\Pcal}:X^{\Pcal}\longrightarrow X^{\Qcal}
\twoheadrightarrowDefinition passage · 16 passages
Search aliases: ↠ twoheadrightarrow
Definition passage: Witness for seed rank · 02_foundations.tex, line 634
Quotient witnesses for lower bounds. [Witness for seed rank] A surjective intertwiner Φ:M↠ N is a witness for seed rank of order r when srank(N)=r. It certifies srank(M)≥ r.
Exact mathematical passage
\Phi:M\twoheadrightarrow N
\forallDefinition passage · 8 passages
Search aliases: ∀ forall
Definition passage: Solution and obstruction packet for a candidate · 02_connections_obstructions.tex, line 409
[Solution and obstruction packet for a candidate] A candidate x is a solution when IsSolution_r(x) Longleftrightarrow ∀ p, r(x,p)=0. An obstruction packet for x is a pair consisting of a named packet p and a certificate r(x,p)≠0. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.IsSolution; Komorebi.PaperIII.NonlinearObstructions.ObstructionPacket
Exact mathematical passage
\operatorname{IsSolution}_r(x) \quad\Longleftrightarrow\quad \forall p,\ r(x,p)=0.
\varnothingDefinition passage · 21 passages
Search aliases: ∅ varnothing
Definition passage: Composable word and execution operator · 01_finite_coordinates.tex, line 179
[Composable word and execution operator] A finite word w=(e_1,…,e_n) is composable when tgt(e_r)=src(e_r+1) (1≤ r<n). Set L_w=L_e_n⋯ L_e_1, L_∅=I. This definition records the written order of execution even though matrix composition is read right to left. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Composable; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.wordOp
Exact mathematical passage
L_w=L_{e_n}\cdots L_{e_1}, \qquad L_{\varnothing}=I.
\inDefinition passage · 1383 passages
Search aliases: ∈ in
Definition passage: Typed operator grammar · 02_foundations.tex, line 47
r] Let Q=(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a…
Exact mathematical passage
(X_v)_{v\in Q_0}
\notinDefinition passage · 29 passages
Search aliases: ∉ notin
Definition passage: Support relation and support space · 03_packets_and_graphs.tex, line 792
Support patterns, preorders, and triangular structure. [Support relation and support space] For a relation R⊆ I× I, with (β,α) interpreted as an allowed transfer α→β, define M_R := {T∈End(X):T_βα=0 whenever (β,α)∉ R}.
Exact mathematical passage
\mathfrak M_R := \{T\in\End(X):T_{\beta\alpha}=0\text{ whenever }(\beta,\alpha)\notin R\}.
\prodDefinition passage · 93 passages
Search aliases: ∏ prod
Definition passage: Weighted transfer series · 03_weighted.tex, line 386
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|γ|m(e_ν). Here γ=(e_1,…,e_|γ|) is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the en…
Exact mathematical passage
a_n=\sum_{i,j}\ \sum_{\substack{\gamma:i\leadsto j\\W(\gamma)=n}} u_i v_j\prod_{\nu=1}^{|\gamma|}m(e_\nu).
\prod_{j\geq0}Definition passage · 1 passage
Search aliases: ∏_j≥0 prod geq
Definition passage: Projective limit space · Paper_VI_Traditional_Final.tex, line 5293
[Projective limit space] Let (X_j,C_j) be a sequence of Banach spaces and bounded linear maps C_j:X_j+1→ X_j. Its projective limit is X_∞ = { (x_j)_j≥0∈∏_j≥0X_j: C_jx_j+1=x_j for every j }. We equip X_∞ with the subspace topology inherited from the product topology.
Exact mathematical passage
X_\infty = \left\{ (x_j)_{j\geq0}\in\prod_{j\geq0}X_j: C_jx_{j+1}=x_j\text{ for every }j \right\}.
\prod_{q\in\operatorname{Fin}(r(\sigma))}Definition passage · 1 passage
Search aliases: ∏_q∈Fin(r(σ)) prod in operatorname sigma
Definition passage: Finitary colored operation signature · 02_connections_obstructions.tex, line 835
lored signature consists of: item a finite type V of colors; item a type Σ of operation symbols; item an arity r(σ)∈ N for every σ∈Σ; item input colors in_σ:Fin(r(σ))→ V; item an output color out(σ)∈ V. A realization assigns a space X_v to each color and a map Φ_σ: ∏_q∈Fin(r(σ))X_in_σ(q) ⟶ X_out(σ). statusroadmapThe intended Lean target is a dependent inductive typed tree. No such declaration is part of the finite algebraic completion contract.
Exact mathematical passage
\Phi_\sigma: \prod_{q\in\operatorname{Fin}(r(\sigma))}X_{\operatorname{in}_\sigma(q)} \longrightarrow X_{\operatorname{out}(\sigma)}.
\prod_{\nu=1}^{|\gamma|}Definition passage · 1 passage
Search aliases: ∏_ν=1^|γ| prod nu gamma
Definition passage: Weighted transfer series · 03_weighted.tex, line 386
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|γ|m(e_ν). Here γ=(e_1,…,e_|γ|) is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the en…
Exact mathematical passage
a_n=\sum_{i,j}\ \sum_{\substack{\gamma:i\leadsto j\\W(\gamma)=n}} u_i v_j\prod_{\nu=1}^{|\gamma|}m(e_\nu).
\sum_{(\beta,\alpha)\in E}Definition passage · 6 passages
Search aliases: ∑_(β,α)∈ E sum beta alpha in
Definition passage: Packet operator representation · 01_manuscript.tex, line 92
B_βα:X_α⟶ X_β. Local blocks are identified with their ambient extensions by inclusion and projection. In particular, the notation P_β AP_α denotes either the ambient block or its restriction X_α→ X_β according to the displayed domain, and ι_β B_βαπ_α is the corresponding ambient extension. Its evaluation is eval(IR) = ∑_(β,α)∈ E ι_β B_βαπ_α, where ι_β:X_β→ X is the coordinate inclusion.
Exact mathematical passage
\operatorname{eval}(\IR) = \sum_{(\beta,\alpha)\in E} \iota_\beta B_{\beta\alpha}\pi_\alpha,
\sum_aDefinition passage · 72 passages
Search aliases: ∑_a sum
Definition passage: Packet dephasing · 02_strengthened.tex, line 992
[Packet dephasing] The packet dephasing map is D_ P(X)=∑_aP_aXP_a.
Exact mathematical passage
\Dcal_{\Pcal}(X)=\sum_aP_aXP_a.
\sum_{a=0}^{p-1}Definition passage · 4 passages
Search aliases: ∑_a=0^p-1 sum
Definition passage: Odd-length source and inverse coefficients · 20_arithmetic.tex, line 757
[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd. Primality is needed only to interpret the result as a prime-adic congruence ladder.
Exact mathematical passage
S_d(p)=\sum_{a=0}^{p-1}(-1)^a(2a+1)^d.
\sum_{b=0}^{2m}Definition passage · 1 passage
Search aliases: ∑_b=0^2m sum
Definition passage: Odd-length source and inverse coefficients · 20_arithmetic.tex, line 761
[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd. Primality is needed only to interpret the result as a prime-adic congruence ladder.
Exact mathematical passage
\operatorname{sech}^2t=\sum_{m\ge0}\gamma_m\frac{t^{2m}}{(2m)!}, \qquad \gamma_m=\sum_{b=0}^{2m}\binom{2m}{b}E_bE_{2m-b},
\sum_cDefinition passage · 9 passages
Search aliases: ∑_c sum
Definition passage: Anomaly free continuum constraint system · 02_strengthened.tex, line 2540
[Anomaly free continuum constraint system] A continuum constraint system consists of a Hilbert space, a common invariant dense domain D, and operators C_a on D. It is anomaly free for prescribed structure coefficients when [C_a,C_b]ψ = i∑_cf_ab^phantomabcC_cψ for every ψ∈ D, and when the associated gauge evolution is defined on the physical quotient or physical subspace.
Exact mathematical passage
[C_a,C_b]\psi = \ii\sum_cf_{ab}^{\phantom{ab}c}C_c\psi
\sum_{d\ge0}Definition passage · 7 passages
Search aliases: ∑_d≥0 sum ge
Definition passage: Congruential Euler coefficients · 20_arithmetic.tex, line 15
[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the multiples of N vanish.
Exact mathematical passage
\label{eq:arith-euler-egf} \sum_{d\ge0}E_d^{(N,j)}\frac{t^d}{d!} =\left(\sum_{n\ge0}\frac{t^{Nn}}{(Nn+j)!}\right)^{-1}.
\sum_{d\geq0}Definition passage · 3 passages
Search aliases: ∑_d≥0 sum geq
Definition passage: Hurwitz integral series · strengthening.tex, line 42
[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.
Exact mathematical passage
\mathcal H_p=\left\{\sum_{d\geq0}f_d\frac{t^d}{d!}: f_d\in\mathbb Z_p\right\} \subset\mathbb Q_p[[t]].
\sum_{e:i\to j}Definition passage · 2 passages
Search aliases: ∑_e:i→ j sum to
Definition passage: Weighted transfer series · 03_weighted.tex, line 374
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j
Exact mathematical passage
A(t)_{ij} = \sum_{e:i\to j}m(e)t^{w(e)}.
\sum_{e\in E}Definition passage · 9 passages
Search aliases: ∑_e∈ E sum in
Definition passage: Packet graph and scalar edge system · 01_finite_coordinates.tex, line 77
0, · otherwise. An edge system is a coefficient function c:E→ R. Its elementary edge operator and total transfer operator are L_e=packetMatrix(src(e),tgt(e);c(e)), T=∑_e∈ EL_e. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph; Komorebi.PaperIII.EdgeOperators.PacketGraph.packetMatrix; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.edgeOp; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp
Exact mathematical passage
L_e=\operatorname{packetMatrix}(\src(e),\tgt(e);c(e)), \qquad T=\sum_{e\in E}L_e.
\sum_iDefinition passage · 44 passages
Search aliases: ∑_i sum
Definition passage: Relative information · Paper_VI_Traditional_Final.tex, line 3639
[Relative information] For probability vectors q and p, define D(qVert p) = ∑_iq_ilogfracq_ip_i. The convention is 0log0=0. A term with q_i>0 and p_i=0 has value +∞.
Exact mathematical passage
D(q\Vert p) = \sum_iq_i\log\frac{q_i}{p_i}.
\sum_{i,j}Definition passage · 4 passages
Search aliases: ∑_i,j sum
Definition passage: Weighted transfer series · 03_weighted.tex, line 386
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|γ|m(e_ν). Here γ=(e_1,…,e_|γ|) is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the en…
Exact mathematical passage
a_n=\sum_{i,j}\ \sum_{\substack{\gamma:i\leadsto j\\W(\gamma)=n}} u_i v_j\prod_{\nu=1}^{|\gamma|}m(e_\nu).
\sum_{i=1}^{r}Definition passage · 1 passage
Search aliases: ∑_i=1^r sum
Definition passage: Arf invariant · Paper_XI_Traditional_Final.tex, line 1489
[Arf invariant] Choose a symplectic basis (e_i,f_i)_i=1^r. Define Arf(q) = ∑_i=1^rq(e_i)q(f_i) ∈ F_2.
Exact mathematical passage
\Arf(q) = \sum_{i=1}^{r}q(e_i)q(f_i) \in\FF_2.
\sum_{i\in I_\alpha}Definition passage · 8 passages
Search aliases: ∑_i∈ I_α sum in alpha
Definition passage: Within packet information loss · Paper_VI_Traditional_Final.tex, line 3567
[Within packet information loss] For a prior p, define η_i(p) = λ_i-overlineλ_α(p) hboxwhen i∈ I_α. Then ∑_i∈ I_αr_α,iη_i(p)=0.
Exact mathematical passage
\sum_{i\in I_\alpha}r_{\alpha,i}\eta_i(p)=0.
\sum_{j=1}^{r}Definition passage · 1 passage
Search aliases: ∑_j=1^r sum
Definition passage: Seed evaluation map · 02_foundations.tex, line 218
ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.
Exact mathematical passage
\varepsilon_S:A^{\oplus r}\longrightarrow M, \qquad (a_1,\ldots,a_r)\longmapsto\sum_{j=1}^{r}a_js_j.
\sum_{m=0}^{\min(M,n)}Definition passage · 2 passages
Search aliases: ∑_m=0^min(M,n) sum min
Definition passage: Odd-length source and inverse coefficients · 20_arithmetic.tex, line 767
rse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd. Primality is needed only to interpret the result as a prime-adic congruence ladder.
Exact mathematical passage
T^{\mathrm{odd}}_{p,M}(n)=\sum_{m=0}^{\min(M,n)} \binom{2n}{2m}\gamma_m p^{2m}S_{2(n-m)}(p).
\sum_{m\ge0}Definition passage · 4 passages
Search aliases: ∑_m≥0 sum ge
Definition passage: Odd-length source and inverse coefficients · 20_arithmetic.tex, line 761
[Odd-length source and inverse coefficients] For a positive odd integer p, define S_d(p)=∑_a=0^p-1(-1)^a(2a+1)^d. Let γ_m be the integral Hurwitz coefficients of sech^2: sech^2t=∑_m≥0γ_mfract^2m(2m)!, γ_m=∑_b=0^2mbinom2mbE_bE_2m-b, where E_b=0 for odd b. Then γ_0=1. Define T^odd_p,M(n)=∑_m=0^min(M,n) binom2n2mγ_m p^2mS_2(n-m)(p). Here p need only be odd. Primality is needed only to interpret the result as a prime-adic congruence ladder.
Exact mathematical passage
\operatorname{sech}^2t=\sum_{m\ge0}\gamma_m\frac{t^{2m}}{(2m)!}, \qquad \gamma_m=\sum_{b=0}^{2m}\binom{2m}{b}E_bE_{2m-b},
\sum_{m\geq0}Definition passage · 8 passages
Search aliases: ∑_m≥0 sum geq
Definition passage · Paper_IV_Traditional_Final.tex, line 1484
Define polynomials E_m^(N,j,r)(x) by C_N,j(z)^re^xz = ∑_m≥0E_m^(N,j,r)(x)fracz^mm!.
Exact mathematical passage
\Cfun_{N,j}(z)^re^{xz} = \sum_{m\geq0}E_m^{(N,j,r)}(x)\frac{z^m}{m!}.
\sum_{m\ge1}Definition passage · 5 passages
Search aliases: ∑_m≥1 sum ge
Definition passage: Seed zeta function · Paper_X_Block27_Amended.tex, line 531
[Seed zeta function] For a residual root bar x with finite levels, define Z_bar x(T) = ∑_m≥1#X_m(bar x)T^m. When a weighted chart is specified, write Z_ w(T).
Exact mathematical passage
\boxed{ Z_{\bar x}(T) = \sum_{m\ge1}\#X_m(\bar x)T^m. }
\sum_{m\geq1}Definition passage · 13 passages
Search aliases: ∑_m≥1 sum geq
Definition passage · Paper_IV_Traditional_Final.tex, line 972
For r∈ C, define S_N^(r)(z) = D_N(z)^-r = exp(-rlog D_N(z)). Write S_N^(r)(z) = ∑_n≥0P_N,n(r)fracz^Nn(Nn)!. Define the logarithmic coefficients κ_N,m by -logΔ_N(w) = ∑_m≥1κ_N,mfracw^mm.
Exact mathematical passage
-\log\Delta_N(w) = \sum_{m\geq1}\kappa_{N,m}\frac{w^m}{m}.
\sum_nDefinition passage · 11 passages
Search aliases: ∑_n sum
Definition passage: Unconditional packet sum space · 03_packets_and_graphs.tex, line 358
[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .
Exact mathematical passage
\sum_nx_n
\sum_{n\in F}Definition passage · 7 passages
Search aliases: ∑_n∈ F sum in
Definition passage: Unconditional packet sum space · 03_packets_and_graphs.tex, line 359
[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .
Exact mathematical passage
\|x\|_u := \sup_{F\Subset\mathbb N} \sup_{|a_n|\leq1} \left\|\sum_{n\in F}a_nx_n\right\|.
\sum_{n\equiv j\pmod N}Definition passage · 1 passage
Search aliases: ∑_n≡ jpmod N sum equiv pmod
Definition passage: Residue selector · 02_coordinates.tex, line 374
[Residue selector] Let R be a commutative ring and let F(z)=∑_n≥0a_nz^n∈ R[[z]]. For N≥1 and 0≤ j<N, define (P_N,jF)(z) = ∑_n≡ jpmod Na_nz^n.
Exact mathematical passage
(P_{N,j}F)(z) = \sum_{n\equiv j\pmod N}a_nz^n.
\sum_{n\ge0}Definition passage · 28 passages
Search aliases: ∑_n≥0 sum ge
Definition passage: Congruential Euler coefficients · 20_arithmetic.tex, line 15
[Congruential Euler coefficients] For N≥1 and 0≤ j<N, define rational coefficients by the formal identity ∑_d≥0E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The reciprocal exists because its denominator has constant coefficient 1/j!≠0. Set U_N,j(n)=E_Nn^(N,j), C_N,j(n)=fracU_N,j(n)j!. Thus C_N,j(0)=1 and U_N,j(0)=j!. All coefficients outside the multiples of N vanish.
Exact mathematical passage
\label{eq:arith-euler-egf} \sum_{d\ge0}E_d^{(N,j)}\frac{t^d}{d!} =\left(\sum_{n\ge0}\frac{t^{Nn}}{(Nn+j)!}\right)^{-1}.
\sum_{n\geq0}Definition passage · 37 passages
Search aliases: ∑_n≥0 sum geq
Definition passage: Residue selector · 02_coordinates.tex, line 370
[Residue selector] Let R be a commutative ring and let F(z)=∑_n≥0a_nz^n∈ R[[z]]. For N≥1 and 0≤ j<N, define (P_N,jF)(z) = ∑_n≡ jpmod Na_nz^n.
Exact mathematical passage
F(z)=\sum_{n\geq0}a_nz^n\in R[[z]].
\sum_{q=1}^{Q}Definition passage · 6 passages
Search aliases: ∑_q=1^Q sum
Definition passage: Sobolev step packet class · Paper_V_Traditional_Final.tex, line 2522
[Sobolev step packet class] Fix r≥1, 1≤ p≤∞, and an integer Q≥0. The class F_r,p(Q) consists of functions f=s+∑_q=1^Qa_qϕ_0,τ_q with s∈ W^r,p(0,1) and 0<τ_1<⋯<τ_Q<1. No lower bound on the jump amplitudes is required for representation. Recovery results later impose such a bound.
Exact mathematical passage
f=s+\sum_{q=1}^{Q}a_q\phi_{0,\tau_q}
\sum_{\substack{\Dfun_N(\rho)=0\\ \Dfun_N'(\rho)\ne0}}Definition passage · 1 passage
Search aliases: ∑_substackDfun_N(ρ)=0 Dfun_N'(ρ)≠0 ∑_substack D_N(ρ)=0 D_N'(ρ)≠0 sum substack Dfun rho Dfun rho ne sum substack mathsf rho mathsf rho ne
Definition passage · Paper_IV_Traditional_Final.tex, line 2925
Let q be a positive integer. Choose logarithms of the simple zeros of D_N with arguments in a fixed bounded interval. Define Z_N,q^der(s) = ∑_substack D_N(ρ)=0 D_N'(ρ)≠0 frace^-sLogρ D_N'(ρ)^q.
Exact mathematical passage
\mathfrak Z_{N,q}^{\mathrm{der}}(s) = \sum_{\substack{\Dfun_N(\rho)=0\\ \Dfun_N'(\rho)\ne0}} \frac{e^{-s\Log\rho}}{\Dfun_N'(\rho)^q}.
\sum_{\substack{\gamma:i\leadsto j\\W(\gamma)=n}}Definition passage · 1 passage
Search aliases: ∑_substackγ:ileadsto j W(γ)=n sum substack gamma leadsto W gamma
Definition passage: Weighted transfer series · 03_weighted.tex, line 386
[Weighted transfer series] Let G be a finite directed multigraph. Give every edge e a positive integer weight w(e) and a scalar multiplicity m(e) in a field bbk. Define the polynomial adjacency matrix A(t)_ij = ∑_e:i→ jm(e)t^w(e). For column vectors u and v, define Z_u,v(t) = ∑_n≥0a_nt^n, where a_n=∑_i,j ∑_substackγ:ileadsto j W(γ)=n u_i v_j∏_ν=1^|γ|m(e_ν). Here γ=(e_1,…,e_|γ|) is a directed walk. Vertices and edges may repeat, and the empty walk has weight zero and multiplicity one. Thus the en…
Exact mathematical passage
a_n=\sum_{i,j}\ \sum_{\substack{\gamma:i\leadsto j\\W(\gamma)=n}} u_i v_j\prod_{\nu=1}^{|\gamma|}m(e_\nu).
\sum_{s\in S}Definition passage · 1 passage
Search aliases: ∑_s∈ S sum in
Definition passage: Generated hull · 02_foundations.tex, line 158
[Generated hull] For S⊆ X, define Hull_ A(S):=∑_s∈ S A s. In the topological regime define Hull_ A,τ(S):=clHull_ A(S)^ τ. For a singleton S={s}, write Hull_ A(s).
Exact mathematical passage
\Hull_{\A}(S):=\sum_{s\in S}\A s.
\sum_{u\in\operatorname{ch}(v)}Definition passage · 3 passages
Search aliases: ∑_u∈ch(v) sum in operatorname
Definition passage: Complete packet pruning · Paper_V_Traditional_Final.tex, line 900
[Complete packet pruning] Let T be a finite rooted refinement tree. Every node v carries a projector P_v, the root satisfies P_∅=I, and every internal node satisfies P_v=∑_u∈ch(v)P_u, P_uP_w=0 (u≠ w in ch(v)). A set Λ of nodes is a complete pruning if it is an antichain and every root-to-terminal path meets Λ exactly once.
Exact mathematical passage
P_v=\sum_{u\in\operatorname{ch}(v)}P_u, \qquad P_uP_w=0\quad(u\ne w\text{ in }\operatorname{ch}(v)).
\sum_{\alpha\in C}Definition passage · 2 passages
Search aliases: ∑_α∈ C sum alpha in
Definition passage: Packet pure seed rank · 03_packets_and_graphs.tex, line 1020
[Packet pure seed rank] A vector is packet pure when it is zero or lies in one packet space X_α, equivalently when its packet support has cardinality at most one. Define psrank_ A, P(X) := min{|S|:S is packet-pure and A S=X}. For C⊆ I, put P_C:=∑_α∈ CP_α, X_C:=P_CX, A_C:=P_C A P_C, and define the local packet pure rank r_C:=psrank_ A_C, P|_C(X_C).
Exact mathematical passage
P_C:=\sum_{\alpha\in C}P_\alpha, \qquad X_C:=P_CX, \qquad \A_C:=P_C\A P_C,
\sum_{\alpha\in I}Definition passage · 10 passages
Search aliases: ∑_α∈ I sum alpha in
Definition passage: Finite packet system · 03_packets_and_graphs.tex, line 17
[Finite packet system] A finite packet system on a vector space X over 𝕜 is a family P=(P_α)_α∈ I⊆End_𝕜(X) satisfying P_α^2=P_α, P_α P_β=0 (α≠β), ∑_α∈ IP_α=I_X. Set X_α=P_α X. Zero packet spaces may be removed without changing any identity. No orthogonality is assumed unless stated.
Exact mathematical passage
P_\alpha^2=P_\alpha, \qquad P_\alpha P_\beta=0\quad(\alpha\neq\beta), \qquad \sum_{\alpha\in I}P_\alpha=I_X.
\sum_{\lambda\in\Lambda_N}Definition passage · 9 passages
Search aliases: ∑_λ∈Λ_N sum lambda in Lambda
Definition passage · Paper_IV_Traditional_Final.tex, line 2438
For m≥1, define σ_N,m = ∑_λ∈Λ_Nλ^-m, with multiplicities.
Exact mathematical passage
\sigma_{N,m} = \sum_{\lambda\in\Lambda_N}\lambda^{-m},
\sum_{\rho(\beta)=\alpha}Definition passage · 2 passages
Search aliases: ∑_ρ(β)=α sum rho beta alpha
Definition passage: Refinement · 02_coordinates.tex, line 39
[Refinement] Let P=(P_α)_α∈ I and Q=(Q_β)_β∈ J be packet systems on the same module. We say that Q refines P when there is a surjection ρ:J⟶ I such that P_α=∑_ρ(β)=αQ_β for every α∈ I. The index β is then called a child of ρ(β).
Exact mathematical passage
P_\alpha=\sum_{\rho(\beta)=\alpha}Q_\beta
\sum_{\rho\in\Zcal(\Dfun_N)}Definition passage · 1 passage
Search aliases: ∑_ρ∈Zcal(Dfun_N) ∑_ρ∈ Z( D_N) sum rho in Zcal Dfun sum rho in mathcal mathsf
Definition passage · Paper_IV_Traditional_Final.tex, line 2743
Define Z_N^abs(s) = ∑_ρ∈ Z( D_N)|ρ|^-s and Z_N^red(s) = ∑_λ∈Λ_N|λ|^-s, with multiplicities. Each term means e^-slog|ρ| or e^-slog|λ|.
Exact mathematical passage
\mathfrak Z_N^{\mathrm{abs}}(s) = \sum_{\rho\in\Zcal(\Dfun_N)}|\rho|^{-s}
\sum_{\sigma(\alpha)=j}Definition passage · 2 passages
Search aliases: ∑_σ(α)=j sum sigma alpha
Definition passage: Packet compatible operator morphism · 04_transport.tex, line 59
orphism] Let (X, A, P) and (Y, B, Q) be packet resolved operator modules with packet index sets I and J. A packet compatible operator morphism is a triple (Φ,φ,σ) consisting of a unital algebra homomorphism φ: A→ B, a linear map Φ:X→ Y satisfying Φ(Ax)=φ(A)Φ(x), and a map σ:I→ J satisfying Q_jΦ = Φ P_σ^-1(j), P_σ^-1(j):= ∑_σ(α)=jP_α. An empty fiber is interpreted as the zero projector.
Exact mathematical passage
Q_j\Phi = \Phi P_{\sigma^{-1}(j)}, \qquad P_{\sigma^{-1}(j)}:= \sum_{\sigma(\alpha)=j}P_\alpha.
\circDefinition passage · 56 passages
Search aliases: ∘ circ
Definition passage: Ordinary completion datum at a point · Paper_XII_Traditional_Final.tex, line 1658
phism. item An automorphic Hecke datum ( T,H,(P_v)) and a pointwise classicality and Galois realization theorem satisfying all the inputs of [source reference] for the nonzero fiber at x: an occurring classical automorphic representation with a continuous semisimple attached representation; equality of the characteristic polynomials of both that representation and ρ_x=x∘ρ_R with x(P_v(X)) on the same declared Chebotarev dense union of Frobenius conjugacy classes; and the standard semisimplificat…
Exact mathematical passage
\rho_x=x\circ\rho_R
\inftyDefinition passage · 236 passages
Search aliases: ∞ infty
Definition passage: Differences and valuation conventions · 20_arithmetic.tex, line 106
[Differences and valuation conventions] For a sequence f and h,k,n∈ N, put Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f(n)=f(n), Δ_h^k+1f(n)=Δ_h^k f(n+h)-Δ_h^k f(n). We use the extended valuation v_p(0)=+∞ in mathematical statements. Thus v_p(x)≥ L, including negative integers L, means x=0 or the ordinary valuation of the nonzero rational x is at least L. For integral coefficients and L≥0, it is equivalent to x∈ p^L Z_p. In expressions involving a step multiplier we write ν_p(u)=v_p(u) for u>0 and set ν_p(0)=0…
Exact mathematical passage
v_p(0)=+\infty
\capDefinition passage · 39 passages
Search aliases: ∩ cap
Definition passage: Lattice packet data · 04_arithmetic.tex, line 240
[Lattice packet data] Let V=⊕_α∈ IV_α be a finite direct sum over K, with packet projectors E_α. Let Λ⊆ V be an O lattice. Put Λ_α=Λ∩ V_α, Λ_split=⊕_αΛ_α, and define the gluing module G_Λ, P = Λ/Λ_split. The separation weight of E_α is s_Λ(E_α) = min{s≥0:π^sE_αΛ⊆Λ}. The global separation weight is s_Λ( P)=max({0}∪{s_Λ(E_α):α∈ I}). For a finitely generated torsion O module G, write exp_π(G) for the least s such that π^sG=0.
Exact mathematical passage
\Lambda_\alpha=\Lambda\cap V_\alpha, \qquad \Lambda_{\mathrm{split}}=\bigoplus_\alpha\Lambda_\alpha,
\cupDefinition passage · 13 passages
Search aliases: ∪ cup
Definition passage: Transfer graph of a family generating the algebra · 03_packets_and_graphs.tex, line 919
[Transfer graph of a family generating the algebra] Let P=(P_α)_α∈ I be a finite packet system on X with every P_α∈ A. Let G⊆ A satisfy A=alg_𝕜({P_α:α∈ I}∪ G), where alg_𝕜(E) denotes the unital subalgebra over 𝕜 generated by E. The transfer graph Γ( G, P) has vertex set I and an arrow α⟶β when P_β GP_α≠0 for at least one G∈ G. Reachability includes paths of length zero.
Exact mathematical passage
\A=\alg_{\bbk}\bigl(\{P_\alpha:\alpha\in I\}\cup\mathcal G\bigr),
\intDefinition passage · 100 passages
Search aliases: ∫ int
Definition passage: Einstein infrared criterion · 02_strengthened.tex, line 2558
[Einstein infrared criterion] A quantum geometric tower meets an Einstein infrared criterion only if an effective action is controlled at long wavelengths and has an expansion whose leading geometric terms take the form Γ_rm eff[g] = ∫ sqrtabs g ( Λ + frac116π GR + controlled higher order terms ) d^dx. Here the volume coefficient includes the sign and normalization written explicitly in the cosmological term in Part I. The expansion must respect the limiting gauge identities and must include an …
Exact mathematical passage
\Gamma_{\rm eff}[g] = \int \sqrt{\abs g} \left( \Lambda + \frac{1}{16\pi G}R + \text{controlled higher order terms} \right) \dd^dx.
\simDefinition passage · 17 passages
Search aliases: ∼ sim
Definition passage: Normalized reversible connection · 02_connections_obstructions.tex, line 119
[Normalized reversible connection] A normalized reversible connection is a family of linear equivalences Γ_b← a:C_axrightarrow ∼ C_b (a,b∈ B) such that Γ_a← a=id, Γ_a← b=Γ_b← a^-1. No cocycle equation is included in the structure. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection
Exact mathematical passage
\Gamma_{b\leftarrow a}:C_a\xrightarrow{\;\sim\;}C_b \qquad(a,b\in B)
\simeqDefinition passage · 30 passages
Search aliases: ≃ simeq
Definition passage: Derived support · Paper_XII_Traditional_Final.tex, line 813
[Derived support] Let R be a commutative ring and let C be a perfect complex of R modules. Its derived support is Supp_R(C) = { p∈Spec R: Cmathbinmathop⊗limits_R^ Lκ( p) not≃0 }.
Exact mathematical passage
\Supp_R(C) = \left\{ \frakp\in\Spec R: C\mathbin{\mathop{\otimes}\limits_R^{\mathbf L}}\kappa(\frakp) \not\simeq0 \right\}.
\congDefinition passage · 111 passages
Search aliases: ≅ cong
Definition passage: Ordinary patching datum · Paper_XII_Traditional_Final.tex, line 731
[Ordinary patching datum] An ordinary patching datum consists of a Noetherian ring R_∞, an ideal a⊆ R_∞, a finite R_∞ module M_∞, and identifications R≅ R_∞/ a, M≅ M_∞/ a M_∞. In automorphy lifting applications, R_∞ is a patched deformation ring and M_∞ is a patched automorphic module. The definition itself contains no automorphy assertion.
Exact mathematical passage
R\cong R_\infty/\fraka, \qquad M\cong M_\infty/\fraka M_\infty.
\neDefinition passage · 186 passages
Search aliases: ≠ ne
Definition passage: Packet system · 02_coordinates.tex, line 12
[Packet system] Let X be a module over a unital ring. A finite packet system on X is a family P=(P_α)_α∈ I ⊆End(X) of nonzero idempotents such that P_α P_β=0 when α≠β, ∑_α∈ IP_α=I_X. Write X_α=P_α X, X^ P=⊕_α∈ IX_α. The analysis and synthesis maps are A_ Px=(P_α x)_α, S_ P(x_α)_α=∑_α x_α. They are inverse isomorphisms.
Exact mathematical passage
P_\alpha P_\beta=0 \quad\text{when }\alpha\ne\beta, \qquad \sum_{\alpha\in I}P_\alpha=I_X.
\neqDefinition passage · 60 passages
Search aliases: ≠ neq
Definition passage: Finite packet system · 03_packets_and_graphs.tex, line 17
[Finite packet system] A finite packet system on a vector space X over 𝕜 is a family P=(P_α)_α∈ I⊆End_𝕜(X) satisfying P_α^2=P_α, P_α P_β=0 (α≠β), ∑_α∈ IP_α=I_X. Set X_α=P_α X. Zero packet spaces may be removed without changing any identity. No orthogonality is assumed unless stated.
Exact mathematical passage
P_\alpha^2=P_\alpha, \qquad P_\alpha P_\beta=0\quad(\alpha\neq\beta), \qquad \sum_{\alpha\in I}P_\alpha=I_X.
\equivDefinition passage · 200 passages
Search aliases: ≡ equiv
Definition passage: Residue selector · 02_coordinates.tex, line 374
[Residue selector] Let R be a commutative ring and let F(z)=∑_n≥0a_nz^n∈ R[[z]]. For N≥1 and 0≤ j<N, define (P_N,jF)(z) = ∑_n≡ jpmod Na_nz^n.
Exact mathematical passage
(P_{N,j}F)(z) = \sum_{n\equiv j\pmod N}a_nz^n.
\leDefinition passage · 329 passages
Search aliases: ≤ le
Definition passage: Separation weight · 04_arithmetic.tex, line 189
ight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG preserves Λ. The minimal Lean theorem needs only the lattice action of u^-1. The bundled wrapper for lattice units records both u and u^-1. A nonzero scalar in the f…
Exact mathematical passage
s_{\pi,\Lambda}(E)\le s=\val_\pi(r).
\geDefinition passage · 396 passages
Search aliases: ≥ ge
Definition passage: Separation weight · 04_arithmetic.tex, line 181
[Separation weight] For a rational projector E and a specified O lattice Λ, provided at least one clearing exponent exists, define s_π,Λ(E) = min{s≥0:π^sE(Λ)⊆Λ}. If the evaluated certificate arises from integral v,g and r=uπ^s with u^-1 preserving the lattice, then [source reference] gives the upper bound s_π,Λ(E)≤ s=val_π(r). Indeed, π^sE=u^-1VG preserves Λ. The minimal Lean theorem needs only the lattice action of u^-1. The bundled wrapper for lattice units records both u and u^-1. A nonzero s…
Exact mathematical passage
s_{\pi,\Lambda}(E) = \min\{s\ge0:\pi^sE(\Lambda)\subseteq\Lambda\}.
\geqDefinition passage · 524 passages
Search aliases: ≥ geq
Definition passage: Witness for seed rank · 02_foundations.tex, line 634
Quotient witnesses for lower bounds. [Witness for seed rank] A surjective intertwiner Φ:M↠ N is a witness for seed rank of order r when srank(N)=r. It certifies srank(M)≥ r.
Exact mathematical passage
\srank(M)\geq r
\subsetDefinition passage · 27 passages
Search aliases: ⊂ subset
Definition passage: Hurwitz integral series · strengthening.tex, line 42
[Hurwitz integral series] For a prime p, let H_p={∑_d≥0f_dfract^dd!: f_d∈ Z_p} ⊂ Q_p[[t]]. For N=p^a and λ∈ Z_p, put D=fracddt, L_λ=fracD^N-λp. A series A is λ-flat if L_λ^k A∈ H_p for every k≥0.
Exact mathematical passage
\mathcal H_p=\left\{\sum_{d\geq0}f_d\frac{t^d}{d!}: f_d\in\mathbb Z_p\right\} \subset\mathbb Q_p[[t]].
\supseteqDefinition passage · 4 passages
Search aliases: ⊇ supseteq
Definition passage: Complete multiplicative filtration · 03_weighted.tex, line 7
[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
\Bcal=F^0\Bcal\supseteq F^1\Bcal\supseteq F^2\Bcal\supseteq\cdots
\bigoplusDefinition passage · 87 passages
Search aliases: ⊕ bigoplus
Definition passage: Realization on a single space · 02_foundations.tex, line 63
[Realization on a single space] For a typed realization X, define its realization on a single space by |X|:=⊕_v∈ Q_0X_v. Every vertex idempotent acts as a coordinate projector on |X|, and every path acts as a block operator from its source space to its target space.
Exact mathematical passage
|X|:=\bigoplus_{v\in Q_0}X_v.
\oplusDefinition passage · 71 passages
Search aliases: ⊕ oplus
Definition passage: Seed evaluation map · 02_foundations.tex, line 218
ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.
Exact mathematical passage
\varepsilon_S:A^{\oplus r}\longrightarrow M, \qquad (a_1,\ldots,a_r)\longmapsto\sum_{j=1}^{r}a_js_j.
\bigoplus_{j=0}^{m-1}Definition passage · 1 passage
Search aliases: ⊕_j=0^m-1 bigoplus
Definition passage: Monodromy packet chain · Paper_XI_Traditional_Final.tex, line 247
[Monodromy packet chain] Let V_0,…,V_m-1 be copies of the space of τ. Put V=⊕_j=0^m-1V_j. A monodromy packet chain is a nilpotent operator N with N(V_j)⊆ V_j-1 for j>0, with N(V_0)=0, together with a Frobenius operator whose conjugation scales N by q^-1.
Exact mathematical passage
V=\bigoplus_{j=0}^{m-1}V_j.
\bigoplus_{n\geq1}^{u}Definition passage · 3 passages
Search aliases: ⊕_n≥1^u bigoplus geq
Definition passage: Unconditional packet sum space · 03_packets_and_graphs.tex, line 355
[Unconditional packet sum space] Let (X_n)_n≥1 be closed subspaces of a real or complex Banach space X. Define ⊕_n≥1^uX_n to be the space of sequences x=(x_n) with x_n∈ X_n for which ∑_nx_n converges unconditionally in X. Equip it with x _u := sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nx_n .
Exact mathematical passage
\bigoplus_{n\geq1}^{u}X_n
\bigoplus_{v\in Q_0}Definition passage · 1 passage
Search aliases: ⊕_v∈ Q_0 bigoplus in
Definition passage: Realization on a single space · 02_foundations.tex, line 63
[Realization on a single space] For a typed realization X, define its realization on a single space by |X|:=⊕_v∈ Q_0X_v. Every vertex idempotent acts as a coordinate projector on |X|, and every path acts as a block operator from its source space to its target space.
Exact mathematical passage
|X|:=\bigoplus_{v\in Q_0}X_v.
\bigoplus_{w\in\operatorname{child}(v)}Definition passage · 1 passage
Search aliases: ⊕_w∈child(v) bigoplus in operatorname
Definition passage: Unitary packet tree · Paper_V_Traditional_Final.tex, line 2335
[Unitary packet tree] A finite unitary packet tree consists of Hilbert spaces H_v at its vertices and, at every internal vertex v, a unitary map U_v: H_v⟶⊕_w∈child(v) H_w. An admissible pruning is a set of vertices that meets every root to terminal path exactly once.
Exact mathematical passage
U_v:\Hcal_v\longrightarrow\bigoplus_{w\in\operatorname{child}(v)}\Hcal_w.
\bigoplus_\alphaDefinition passage · 12 passages
Search aliases: ⊕_α bigoplus alpha
Definition passage: Packet coordinate condition number · 03_packets_and_graphs.tex, line 284
[Packet coordinate condition number] Fix a finite product norm on ⊕_α X_α equivalent to the maximum norm, as in [source reference]. Define κ( P):= A_ P S_ P .
Exact mathematical passage
\bigoplus_\alpha X_\alpha
\bigoplus_{\alpha\in I}Definition passage · 15 passages
Search aliases: ⊕_α∈ I bigoplus alpha in
Definition passage: Packet analysis and synthesis · 03_packets_and_graphs.tex, line 72
[Packet analysis and synthesis] For a finite packet system, define A_ P:X⟶⊕_α∈ IX_α, x⟼(P_α x)_α, and S_ P:⊕_α∈ IX_α⟶ X, (x_α)_α⟼∑_α x_α.
Exact mathematical passage
\mathcal A_{\Pcal}:X\longrightarrow\bigoplus_{\alpha\in I}X_\alpha, \qquad x\longmapsto(P_\alpha x)_\alpha,
\otimesDefinition passage · 140 passages
Search aliases: ⊗ otimes
Definition passage: Derived support · Paper_XII_Traditional_Final.tex, line 813
[Derived support] Let R be a commutative ring and let C be a perfect complex of R modules. Its derived support is Supp_R(C) = { p∈Spec R: Cmathbinmathop⊗limits_R^ Lκ( p) not≃0 }.
Exact mathematical passage
\Supp_R(C) = \left\{ \frakp\in\Spec R: C\mathbin{\mathop{\otimes}\limits_R^{\mathbf L}}\kappa(\frakp) \not\simeq0 \right\}.
\bigcapDefinition passage · 9 passages
Search aliases: ⋂ bigcap
Definition passage: Complete multiplicative filtration · 03_weighted.tex, line 11
[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
F^r\Bcal\,F^s\Bcal\subseteq F^{r+s}\Bcal, \qquad \bigcap_{L\geq0}F^L\Bcal=0,
\bigcap_{L\geq0}Definition passage · 2 passages
Search aliases: ⋂_L≥0 bigcap geq
Definition passage: Complete multiplicative filtration · 03_weighted.tex, line 11
[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
F^r\Bcal\,F^s\Bcal\subseteq F^{r+s}\Bcal, \qquad \bigcap_{L\geq0}F^L\Bcal=0,
\bigcap_{n\geq m}Definition passage · 1 passage
Search aliases: ⋂_n≥ m bigcap geq
Definition passage: Stable core · Paper_XII_Traditional_Final.tex, line 1197
[Stable core] Let V_1longleftarrow V_2longleftarrow V_3longleftarrow⋯ be an inverse system of finite dimensional vector spaces. For n≥ m, write f_n,m:V_n→ V_m. The stable core at level m is Y_m = ⋂_n≥ mim(f_n,m).
Exact mathematical passage
Y_m = \bigcap_{n\geq m}\im(f_{n,m}).
\cdotsDefinition passage · 124 passages
Search aliases: ⋯ cdots
Definition passage: Complete multiplicative filtration · 03_weighted.tex, line 7
[Complete multiplicative filtration] A complete filtered algebra is a unital algebra B with a decreasing sequence of two-sided ideals B=F^0 B⊇ F^1 B⊇ F^2 B⊇⋯ such that F^r B F^s B⊆ F^r+s B, ⋂_L≥0F^L B=0, and the natural map B⟶varprojlim_L B/F^L B is an isomorphism. A transfer term N_e has a declared positive integer weight w(e)∈ Z_>0 when N_e∈ F^w(e) B.
Exact mathematical passage
\Bcal=F^0\Bcal\supseteq F^1\Bcal\supseteq F^2\Bcal\supseteq\cdots
\langleDefinition passage · 47 passages
Search aliases: ⟨ langle
Definition passage: Bridge datum · Paper_XI_Traditional_Final.tex, line 1202
item a one break Galois envelope (V_ρ,b_ρ) item a type side quotient (V_θ,h_θ) item an equivariant symplectic isometry ι:V_ρ⟶ V_θ for a finite cyclic stabilizer C=⟨ c⟩ item the same nontrivial central character on the two Heisenberg groups item when p=2, compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensions lifting ι.
Exact mathematical passage
C=\langle c\rangle
\rangleDefinition passage · 47 passages
Search aliases: ⟩ rangle
Definition passage: Bridge datum · Paper_XI_Traditional_Final.tex, line 1202
item a one break Galois envelope (V_ρ,b_ρ) item a type side quotient (V_θ,h_θ) item an equivariant symplectic isometry ι:V_ρ⟶ V_θ for a finite cyclic stabilizer C=⟨ c⟩ item the same nontrivial central character on the two Heisenberg groups item when p=2, compatible quadratic refinements and an isomorphism of the resulting Heisenberg central extensions lifting ι.
Exact mathematical passage
C=\langle c\rangle
\longrightarrowDefinition passage · 156 passages
Search aliases: ⟶ longrightarrow
Definition passage: Typed operator grammar · 02_foundations.tex, line 48
(Q_0,Q_1,s,t) be a quiver with finite vertex set. A typed operator grammar over 𝕜 is a pair G=(Q,I), where Itriangleleft 𝕜 Q is an ideal on both sides generated by linear combinations over 𝕜 of paths having common source and common target. Write Λ_ G:=𝕜 Q/I. A realization of G is a family (X_v)_v∈ Q_0 and linear maps X_a:X_s(a)⟶ X_t(a) (a∈ Q_1) that annihilate every relation in I. A morphism f:X→ Y of realizations is a family of linear maps f_v:X_v→ Y_v satisfying f_t(a)X_a=Y_af_s(a) (a∈ Q_1).
Exact mathematical passage
X_a:X_{s(a)}\longrightarrow X_{t(a)} \qquad(a\in Q_1)
\longmapstoDefinition passage · 35 passages
Search aliases: ⟼ longmapsto
Definition passage: Seed evaluation map · 02_foundations.tex, line 218
ed hull is exactly its image, and the relations among addressed seeds form the kernel. This converts orbit generation into a presentation problem. Finite free covers. [Seed evaluation map] Let A be a unital ring, let M be a left module over A, and let S=(s_1,…,s_r)∈ M^r. Define the linear map over A ε_S:A^⊕ r⟶ M, (a_1,…,a_r)⟼∑_j=1^ra_js_j.
Exact mathematical passage
\varepsilon_S:A^{\oplus r}\longrightarrow M, \qquad (a_1,\ldots,a_r)\longmapsto\sum_{j=1}^{r}a_js_j.
\mathbf1Introduction passage · 26 passages
Search aliases: 1 mathbf
Introduction passage · 10_foundations.tex, line 129
In characteristic p, factorial division is unavailable in general, but the coefficient operation remains meaningful over every commutative ring: (f⋆ g)_n=∑_r=0^nbinom nr f_r g_n-r, (D^d f)_n=f_n+d, 1_n=1, · n=0, 0, · n>0. Here ⋆ is Hurwitz multiplication. The symbol D^d denotes a coefficient shift, which agrees with formal differentiation after characteristic-zero encoding.
Exact mathematical passage
(f\star g)_n=\sum_{r=0}^{n}\binom nr f_r g_{n-r},\qquad (D^d f)_n=f_{n+d},\qquad \mathbf1_n=\begin{cases}1,&n=0,\\0,&n>0.\end{cases}
\mathbf1_nIntroduction passage · 2 passages
Search aliases: 1_n mathbf
Introduction passage · 10_foundations.tex, line 129
In characteristic p, factorial division is unavailable in general, but the coefficient operation remains meaningful over every commutative ring: (f⋆ g)_n=∑_r=0^nbinom nr f_r g_n-r, (D^d f)_n=f_n+d, 1_n=1, · n=0, 0, · n>0. Here ⋆ is Hurwitz multiplication. The symbol D^d denotes a coefficient shift, which agrees with formal differentiation after characteristic-zero encoding.
Exact mathematical passage
(f\star g)_n=\sum_{r=0}^{n}\binom nr f_r g_{n-r},\qquad (D^d f)_n=f_{n+d},\qquad \mathbf1_n=\begin{cases}1,&n=0,\\0,&n>0.\end{cases}
A^*Introduction passage · 9 passages
Search aliases: A^*
Introduction passage · monograph_atlas.tex, line 20
vpage2The objects on the pageAN ATLAS OF TYPES AND CONVENTIONSChapter VII: Definition II.2.1, Definition II.7.1. The same letter can stand for different kinds of data in different calculations. Keep the following types fixed while reading a formula. In this appendix, vectors are columns, operators compose from right to left, and A^* denotes the conjugate transpose. linewidth@p.13linewidthYY@ Symbol · Object · What it records
Exact mathematical passage
A^*
A^2Introduction passage · 17 passages
Search aliases: A^2
Introduction passage · Paper_X_Block27_Amended.tex, line 2109
To prove this, apply the preceding averaging correction with both representations equal to ρ. This produces an exact equivariant matrix A with the prescribed reduction at depth k, so that A^2-A∈(π)^kM_n(R). Averaging alone need not preserve idempotence. Use the integral polynomial f(A)=3A^2-2A^3. The identities f(A)-A=(1-2A)(A^2-A), f(A)^2-f(A)=(A^2-A)^2(4A^2-4A-3) show that successive corrections preserve the retained reduction and improve the idempotence defect. In these formulas scalar consta…
Exact mathematical passage
A^2-A\in(\pi)^kM_n(R)
A^3Introduction passage · 3 passages
Search aliases: A^3
Introduction passage · Paper_X_Block27_Amended.tex, line 2111
To prove this, apply the preceding averaging correction with both representations equal to ρ. This produces an exact equivariant matrix A with the prescribed reduction at depth k, so that A^2-A∈(π)^kM_n(R). Averaging alone need not preserve idempotence. Use the integral polynomial f(A)=3A^2-2A^3. The identities f(A)-A=(1-2A)(A^2-A), f(A)^2-f(A)=(A^2-A)^2(4A^2-4A-3) show that successive corrections preserve the retained reduction and improve the idempotence defect. In these formulas scalar consta…
Exact mathematical passage
f(A)=3A^2-2A^3.
A_+Introduction passage · 4 passages
Search aliases: A_+
Introduction passage · 03_supplement_05.tex, line 71
A channel on the whole ambient space. Put Q=I-∑_r∈ A_+V_rV_r^*. The recovery is R(σ)=∑_r∈ A_+V_r^*σ V_r+Qσ Q. The complement is an orthogonal projector and ∑_r∈ A_+V_rV_r^*+Q^*Q=I. Hence the recovery is trace preserving. Its finite Kraus form also proves positivity after amplification by the identity on every finite auxiliary space. The complement term matters: the map is defined as a channel even outside the syndrome ranges.
Exact mathematical passage
Q=I-\sum_{r\in A_+}V_rV_r^*
A_0Introduction passage · 11 passages
Search aliases: A_0
Introduction passage · 02_supplement_04.tex, line 91
The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;x_j,·)_u. Induction, the Kronecker entry formula, and finite-sum distributivity prove Apply_k(A;x)=M_k(A)x. Thus correctness is not inferred merely fr…
Exact mathematical passage
M_0=[1],\qquad M_{k+1}=A_0\otimes M_k(A_1,\ldots,A_k).
A_1Introduction passage · 18 passages
Search aliases: A_1
Introduction passage · 02_supplement_04.tex, line 91
The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;x_j,·)_u. Induction, the Kronecker entry formula, and finite-sum distributivity prove Apply_k(A;x)=M_k(A)x. Thus correctness is not inferred merely fr…
Exact mathematical passage
M_0=[1],\qquad M_{k+1}=A_0\otimes M_k(A_1,\ldots,A_k).
A_aIntroduction passage · 5 passages
Search aliases: A_a
Introduction passage · 03_supplement_06.tex, line 38
The mutually inverse local maps. Write T_a(x)=T(P(x),P(a)). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a ≅ U_a× E_a is a local product homeomorphism that preserves the base point.
Exact mathematical passage
A_a(x):\mathcal E_x\longrightarrow\mathcal E_a, \quad w\longmapsto P(a)T_a(x)^{-1}w.
A_cIntroduction passage · 49 passages
Search aliases: A_c
Introduction passage · atlas-pages.tex, line 570
vpage25Naturality also transports derivativesHOLOMORPHIC CALCULUS / A RECTANGULAR COMPARISONChapter VI, Part I: Theorem 15.1. Exact intertwining passes to the specified functional calculus. Consider a fine operator, a coarse operator, and a rectangular map A_f=mat2 · 1 · 0 0 · 2 · 0 0 · 0 · 9, A_c=mat2 · 1 0 · 2, C_r=mat1 · 0 · 0 0 · 1 · 0. Direct multiplication gives C_rA_f=A_cC_r. For F(z)=1/(1+z), which is defined at both spectra, compute the matrices by inversion: F(A_f)=mat1/3 · -1/9 · 0
Exact mathematical passage
A_f=\mat{2&1&0\\0&2&0\\0&0&9},\qquad A_c=\mat{2&1\\0&2},\qquad C_r=\mat{1&0&0\\0&1&0}.
A_fIntroduction passage · 39 passages
Search aliases: A_f
Introduction passage · atlas-pages.tex, line 570
vpage25Naturality also transports derivativesHOLOMORPHIC CALCULUS / A RECTANGULAR COMPARISONChapter VI, Part I: Theorem 15.1. Exact intertwining passes to the specified functional calculus. Consider a fine operator, a coarse operator, and a rectangular map A_f=mat2 · 1 · 0 0 · 2 · 0 0 · 0 · 9, A_c=mat2 · 1 0 · 2, C_r=mat1 · 0 · 0 0 · 1 · 0. Direct multiplication gives C_rA_f=A_cC_r. For F(z)=1/(1+z), which is defined at both spectra, compute the matrices by inversion: F(A_f)=mat1/3 · -1/9 · 0
Exact mathematical passage
A_f=\mat{2&1&0\\0&2&0\\0&0&9},\qquad A_c=\mat{2&1\\0&2},\qquad C_r=\mat{1&0&0\\0&1&0}.
a_iIntroduction passage · 40 passages
Search aliases: a_i
Introduction passage · 02_foundations.tex, line 544
dinal, and choose an index set I of cardinality r. Choose minimal generating families for M and N, index them by subsets of I, and fill any unused positions with zeros. Write the resulting families as (m_i)_i∈ I and (n_i)_i∈ I. For any m∈ M and n∈ N, their finite expressions involve a common finite subset F⊆ I after zero coefficients have been added. Thus m=∑_i∈ Fa_im_i, n=∑_i∈ Fb_in_i, and (m,n)=∑_i∈ F(a_i,b_i)(m_i,n_i). The family of paired vectors generates and has cardinality at most r. The …
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m=\sum_{i\in F}a_im_i, \qquad n=\sum_{i\in F}b_in_i,
A_kIntroduction passage · 7 passages
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Introduction passage · 02_supplement_04.tex, line 91
The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;x_j,·)_u. Induction, the Kronecker entry formula, and finite-sum distributivity prove Apply_k(A;x)=M_k(A)x. Thus correctness is not inferred merely fr…
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M_0=[1],\qquad M_{k+1}=A_0\otimes M_k(A_1,\ldots,A_k).
A_{QQ}Introduction passage · 32 passages
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Introduction passage · Paper_VI_Traditional_Final.tex, line 4579
If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every integer n≥0, after increasing M_β to cover the finitely many initial values. Write t=n+r, where 0≤ r<1. Then norm e^tB≤ M_βexp(norm B+max(0,-β)) e^β t. Thus…
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B=A_{QQ}
A_rIntroduction passage · 3 passages
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Introduction passage · 72_recovered.tex, line 2041
Let c=a*b be the ordinary Cauchy convolution. Write A_r, B_r, and C_r for the phase ordinary generating functions.
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A_r
A_xIntroduction passage · 28 passages
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Introduction passage · continuation_01.tex, line 20
Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer modules. It factors through reduction to R_q^d. The cokernel below is presented directly using its integer-domain image.
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r_x=F(x)/q^L\in\mathbb Z^e, \qquad A_x=DF(x):\mathbb Z^d\longrightarrow\mathbb Z^e, \qquad J_x=\rho_q\circ A_x.
A_{\gamma\beta}Introduction passage · 1 passage
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Introduction passage · 01_manuscript.tex, line 154
[Block composition] Let A,B∈End(X). Then (AB)_γα = ∑_β∈ I A_γβB_βα. Consequently, supp(AB) ⊆ supp(A)∘supp(B), where relational composition is defined by (γ,α)∈ E∘ F when some β satisfies (γ,β)∈ E, (β,α)∈ F.
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\boxed{ (AB)_{\gamma\alpha} = \sum_{\beta\in I} A_{\gamma\beta}B_{\beta\alpha}. }
\absnIntroduction passage · 8 passages
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Introduction passage · residue_continuation_body.tex, line 23
Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding counting block. Its norm function scales by t^d under a positive real dilation t. The full-paper ledger remains 17 checked claims and 3 partial claims.
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\Nm(x)=\absn{n(x)}
\operatorname{Anc}Introduction passage · 2 passages
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Introduction passage · 02_supplement_05.tex, line 8
For each vertex, define its strict ancestor set Anc(v)={u:there is a positive-length path uleadsto v}. For every edge u→ v, the proof establishes Anc(u)subsetneqAnc(v). The inclusion follows by appending the edge. It is strict because u is an ancestor of v but cannot be its own strict ancestor in an acyclic graph. Cardinality therefore decreases when following predecessors. This constructs a well-founded relation on which the level recursion is defined: ℓ(v)=1+max_u→ vℓ(u), max∅=0.
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\operatorname{Anc}(v)=\{u:\text{there is a positive-length path }u\leadsto v\}.
\operatorname{Apply}Introduction passage · 2 passages
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Introduction passage · 02_supplement_04.tex, line 95
The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;x_j,·)_u. Induction, the Kronecker entry formula, and finite-sum distributivity prove Apply_k(A;x)=M_k(A)x. Thus correctness is not inferred merely fr…
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y_{i,u}=\sum_{j=0}^{d-1}(A_0)_{ij}\, \operatorname{Apply}_{k}(A_1,\ldots,A_k;x_{j,\cdot})_u.
\operatorname{Apply}_kIntroduction passage · 1 passage
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Introduction passage · 02_supplement_04.tex, line 100
otimes M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;x_j,·)_u. Induction, the Kronecker entry formula, and finite-sum distributivity prove Apply_k(A;x)=M_k(A)x. Thus correctness is not inferred merely from agreeing dimension or storage formulas.
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\boxed{\operatorname{Apply}_k(A;x)=M_k(A)x.}
\operatorname{Apply}_{k}Introduction passage · 1 passage
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Introduction passage · 02_supplement_04.tex, line 95
The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;x_j,·)_u. Induction, the Kronecker entry formula, and finite-sum distributivity prove Apply_k(A;x)=M_k(A)x. Thus correctness is not inferred merely fr…
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y_{i,u}=\sum_{j=0}^{d-1}(A_0)_{ij}\, \operatorname{Apply}_{k}(A_1,\ldots,A_k;x_{j,\cdot})_u.
b_0Introduction passage · 13 passages
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Introduction passage · visual_atlas.tex, line 428
1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.
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g=b_0+yb_1
B_1Introduction passage · 19 passages
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Introduction passage · strengthening.tex, line 411
Let B_m denote the Bernoulli numbers with B_1=-1/2. The identity obtained by substituting u=e^t-1 into log(1+u)/u is B_m=∑_r=0^m(-1)^rfracr!r+1 {m
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B_1=-1/2
b_1Introduction passage · 23 passages
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Introduction passage · visual_atlas.tex, line 428
1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.
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g=b_0+yb_1
B_4^sIntroduction passage · 1 passage
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Introduction passage · supplement_11.tex, line 78
Write H_int for the subring of Q_2[[t]] whose factorial-scaled coefficients belong to Z_2. Flatness means B_4^sF∈ H_int for every s. The proof of multiplicative and reciprocal closure is induction on s, using the full product correction. It separately proves that the actual reciprocal has integral Hurwitz coefficients when F(0)=1.
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B_4^sF\in\mathcal H_{\mathrm{int}}
B_aIntroduction passage · 24 passages
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Introduction passage · 03_supplement_06.tex, line 34
The mutually inverse local maps. Write T_a(x)=T(P(x),P(a)). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a ≅ U_a× E_a is a local product homeomorphism that preserves the base point.
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B_a(x):\mathcal E_a\longrightarrow\mathcal E_x, \quad v\longmapsto T_a(x)v,
b_iIntroduction passage · 5 passages
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Introduction passage · 02_foundations.tex, line 544
dinal, and choose an index set I of cardinality r. Choose minimal generating families for M and N, index them by subsets of I, and fill any unused positions with zeros. Write the resulting families as (m_i)_i∈ I and (n_i)_i∈ I. For any m∈ M and n∈ N, their finite expressions involve a common finite subset F⊆ I after zero coefficients have been added. Thus m=∑_i∈ Fa_im_i, n=∑_i∈ Fb_in_i, and (m,n)=∑_i∈ F(a_i,b_i)(m_i,n_i). The family of paired vectors generates and has cardinality at most r. The …
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m=\sum_{i\in F}a_im_i, \qquad n=\sum_{i\in F}b_in_i,
B_jIntroduction passage · 18 passages
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Introduction passage · strengthening.tex, line 1089
[Exact image and residue enumeration] Let widetilde C_j denote the dyadic Newton interpolation. The map T_j(x)=fracwidetilde C_j(x)-12^b_j is a bijective isometry Z_2→ Z_2, and widetilde C_j( Z_2)=1+2^b_j Z_2, j!widetilde C_j( Z_2)=j!+2^B_j Z_2. For r>B_j, one least period modulo 2^r visits every residue in j!+2^B_j Z/2^r Z exactly once. For r≤ B_j, there is one residue and the least row period is one.
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\widetilde C_j(\mathbb Z_2)=1+2^{b_j}\mathbb Z_2, \qquad j!\widetilde C_j(\mathbb Z_2)=j!+2^{B_j}\mathbb Z_2. \label{eq:four-interpolation-images-final}
b_jIntroduction passage · 18 passages
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Introduction passage · strengthening.tex, line 1085
[Exact image and residue enumeration] Let widetilde C_j denote the dyadic Newton interpolation. The map T_j(x)=fracwidetilde C_j(x)-12^b_j is a bijective isometry Z_2→ Z_2, and widetilde C_j( Z_2)=1+2^b_j Z_2, j!widetilde C_j( Z_2)=j!+2^B_j Z_2. For r>B_j, one least period modulo 2^r visits every residue in j!+2^B_j Z/2^r Z exactly once. For r≤ B_j, there is one residue and the least row period is one.
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T_j(x)=\frac{\widetilde C_j(x)-1}{2^{b_j}}
B_mIntroduction passage · 12 passages
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Introduction passage · Paper_IV_Traditional_Final.tex, line 1621
For (N,j)=(2,0), C_2,0(z)=sech z. For (N,j)=(2,1), C_2,1(z)=frac zsinh z. The first case is a scaled and translated Euler family. In fact, if E_m(x) denotes the classical Euler polynomial, then E_m^(2,0,1)(x) = 2^mE_m(fracx+12). If B_m(x) denotes the classical Bernoulli polynomial, the second case has the exact identification E_m^(2,1,1)(x) = 2^mB_m(fracx+12), obtained by substituting t=2z in t e^yt/(e^t-1).
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B_m(x)
b_{N,n}Introduction passage · 6 passages
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Introduction passage · Paper_IV_Traditional_Final.tex, line 1170
Let D_n denote the displayed determinant, with D_0=1. Expansion along the last row gives D_n = ∑_k=1^n frac(n-1)!(n-k)!p_kD_n-k. Here the cofactor sign is canceled by the product of the negative superdiagonal entries -(n-k+1),…,-(n-1). Thus D_n/n! satisfies the same recurrence and initial value as b_N,n(r).
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b_{N,n}(r)
B_rIntroduction passage · 19 passages
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Introduction passage · 72_recovered.tex, line 2041
Let c=a*b be the ordinary Cauchy convolution. Write A_r, B_r, and C_r for the phase ordinary generating functions.
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B_r
\bar MIntroduction passage · 1 passage
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Introduction passage · atlas-pages.tex, line 746
More generally, for a surjective aggregation map C, a linear map M descends exactly if CM vanishes on ker C. The descended map is defined by bar M(Cx)=CMx. The kernel condition makes this independent of the representative x.
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\bar M(Cx)=CMx
\bbCIntroduction passage · 21 passages
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Introduction passage · Paper_V_Traditional_Final.tex, line 265
Let ℓ^2( Z; C), abbreviated ℓ^2( Z), be the Hilbert space of square-summable complex discrete signals. The inner product is conjugate-linear in its first argument. A superscript * on a bounded operator denotes its Hilbert adjoint. Write δ_n for the unit impulse at address n and let the bilateral shift be (Sx)[n]=x[n-1]. Then S^nδ_0=δ_n. The primitive signal seed is therefore s=δ_0, the address operators are the shift words S^n, and the completion is the ℓ^2 norm.
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\ell^2(\bbZ;\bbC)
\bbZIntroduction passage · 20 passages
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Introduction passage · Paper_V_Traditional_Final.tex, line 265
Let ℓ^2( Z; C), abbreviated ℓ^2( Z), be the Hilbert space of square-summable complex discrete signals. The inner product is conjugate-linear in its first argument. A superscript * on a bounded operator denotes its Hilbert adjoint. Write δ_n for the unit impulse at address n and let the bilateral shift be (Sx)[n]=x[n-1]. Then S^nδ_0=δ_n. The primitive signal seed is therefore s=δ_0, the address operators are the shift words S^n, and the completion is the ℓ^2 norm.
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\ell^2(\bbZ;\bbC)
\CIntroduction passage · 24 passages
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Introduction passage · 03_supplement_03.tex, line 31
Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feasible seed counts. It proves both attainment and minimality. For an empty block family the finite supremum is defined as 0, matching the zero module.
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\mathcal A=\bigoplus_i\bigl(M_{d_i}(\C)\otimes I_{m_i}\bigr),\qquad d_i>0,
C^{\mathsf T}Introduction passage · 1 passage
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Introduction passage · 03_supplement_03.tex, line 74
The actual partial trace and its support. The reduced matrix is defined by tracing the addressed index of the density matrix of the pure state: (ρ_B)_jℓ=∑_iψ_ijoverlineψ_iℓ. For the coefficient convention used here, the exact Gram identity is ρ_B=C^ T(C^ T)^*, rankρ_B=rank C=r(ψ). The transpose is important. With this convention, writing C^*C instead would conjugate the reduced matrix. The Lean source checks the entrywise identity and positivity of the resulting matrix.
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\rho_B=C^{\mathsf T}(C^{\mathsf T})^*, \qquad \rank\rho_B=\rank C=r(\psi).
C^*Introduction passage · 5 passages
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Introduction passage · 03_supplement_03.tex, line 78
al trace and its support The reduced matrix is defined by tracing the addressed index of the density matrix of the pure state: (ρ_B)_jℓ=∑_iψ_ijoverlineψ_iℓ. For the coefficient convention used here, the exact Gram identity is ρ_B=C^ T(C^ T)^*, rankρ_B=rank C=r(ψ). The transpose is important. With this convention, writing C^*C instead would conjugate the reduced matrix. The Lean source checks the entrywise identity and positivity of the resulting matrix.
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C^*C
C^{-1}Introduction passage · 11 passages
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Introduction passage · residue_continuation_body.tex, line 108
Let [J]=C^-1 in Cl( O), and let G(J) denote the actual fundamental-cone generator set. For every predicate P: R→Prop and every real X, the norm-preserving correspondence gives · #{x∈ G(J): N(x)≤ X N J, P( N(x))}
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[J]=C^{-1}
C^NIntroduction passage · 21 passages
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Introduction passage · supplement_06.tex, line 69
Principal targets. targetCompanion.orbit_of_recurrence targetCompanion.companion_jump, Companion.source_companion_jump targetCompanion.packet_step, Companion.packet_advance targetCompanion.packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det(XI-D)=X^d+∑_a=0^d-1c_aX^a be its actual characteristic polynomial. The implementation reuses the pinned Mathlib Cayley--Hamilton theorem, expands pol…
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D=C^N
c_0Introduction passage · 15 passages
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Introduction passage · 03_packets_and_graphs.tex, line 1116
[Unweighted source component count] Assume the module resolved into packets is packet propagating and X_α≠0 for every α∈ I. If c_0 denotes the number of source strongly connected components, then psrank_ A, P(X)=c_0. [source reference]
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c_0
c_aIntroduction passage · 5 passages
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Introduction passage · supplement_06.tex, line 70
Principal targets. targetCompanion.orbit_of_recurrence targetCompanion.companion_jump, Companion.source_companion_jump targetCompanion.packet_step, Companion.packet_advance targetCompanion.packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det(XI-D)=X^d+∑_a=0^d-1c_aX^a be its actual characteristic polynomial. The implementation reuses the pinned Mathlib Cayley--Hamilton theorem, expands pol…
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\chi_D(X)=\det(XI-D)=X^d+\sum_{a=0}^{d-1}c_aX^a
c_iIntroduction passage · 15 passages
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Introduction passage · conditional_completion.tex, line 15
Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality of numerical traces. The selected family ⊕_iρ_i,λ^⊕ m_i has a split equivariant inclusion in the carrier. The finite trace sample uniquely determines…
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c_i
C_rIntroduction passage · 9 passages
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Introduction passage · 72_recovered.tex, line 2041
Let c=a*b be the ordinary Cauchy convolution. Write A_r, B_r, and C_r for the phase ordinary generating functions.
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C_r
\mathrm{car}Introduction passage · 1 passage
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Introduction passage · conditional_completion.tex, line 16
Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality of numerical traces. The selected family ⊕_iρ_i,λ^⊕ m_i has a split equivariant inclusion in the carrier. The finite trace sample uniquely determines…
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V_{\mathrm{car},\lambda}\simeq\bigoplus_i\rho_{i,\lambda}^{\oplus c_i}, \qquad U_{0,\lambda}\simeq\bigoplus_i\rho_{i,\lambda}^{\oplus m_i}.
\CC^2Introduction passage · 7 passages
Search aliases: CC^2 C^2 CC mathbb
Introduction passage · monograph_atlas.tex, line 712
al values. At integer time n≥0, T^n=diag(1,2^-n,4^-n), P_0=diag(1,0,0), normT^n-P_0=2^-n. The gap Δ=log2 determines the rate e^-nΔ exactly in this example. xfigtransfer_ground headStrict positivity is an essential finite hypothesis For T'=diag(1,0), the operator is positive semidefinite, but there is no finite self-adjoint H' on all of C^2 with T'=e^-H': every finite eigenvalue of H' exponentiates to a strictly positive value. One may restrict to the positive support, but that changes the space …
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\CC^2
\CC^nIntroduction passage · 4 passages
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Introduction passage · Paper_VI_Traditional_Final.tex, line 3019
Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.
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\CC^n
\Cfun_{2,0}Introduction passage · 1 passage
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Introduction passage · Paper_IV_Traditional_Final.tex, line 1608
For (N,j)=(2,0), C_2,0(z)=sech z. For (N,j)=(2,1), C_2,1(z)=frac zsinh z. The first case is a scaled and translated Euler family. In fact, if E_m(x) denotes the classical Euler polynomial, then E_m^(2,0,1)(x) = 2^mE_m(fracx+12). If B_m(x) denotes the classical Bernoulli polynomial, the second case has the exact identification E_m^(2,1,1)(x) = 2^mB_m(fracx+12), obtained by substituting t=2z in t e^yt/(e^t-1).
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\Cfun_{2,0}(z)=\sech z.
\Cfun_{2,1}Introduction passage · 1 passage
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Introduction passage · Paper_IV_Traditional_Final.tex, line 1612
For (N,j)=(2,0), C_2,0(z)=sech z. For (N,j)=(2,1), C_2,1(z)=frac zsinh z. The first case is a scaled and translated Euler family. In fact, if E_m(x) denotes the classical Euler polynomial, then E_m^(2,0,1)(x) = 2^mE_m(fracx+12). If B_m(x) denotes the classical Bernoulli polynomial, the second case has the exact identification E_m^(2,1,1)(x) = 2^mB_m(fracx+12), obtained by substituting t=2z in t e^yt/(e^t-1).
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\Cfun_{2,1}(z)=\frac z{\sinh z}.
\operatorname{CNOT}Introduction passage · 1 passage
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Introduction passage · 01_finite_core.tex, line 337
[Bell seed circuit] In this example only, H denotes the Hadamard gate, not a Hamiltonian. Starting from ket00, (H⊗ I)ket00 = fracket00+ket10sqrt2, and then CNOT(H⊗ I)ket00 = fracket00+ket11sqrt2. The parity packets show that the final state lies entirely in the even sector, while the intermediate circuit path passes through both even and odd addresses.
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\operatorname{CNOT}(H\otimes I)\ket{00} = \frac{\ket{00}+\ket{11}}{\sqrt2}.
\coshIntroduction passage · 35 passages
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Introduction passage · 72_recovered.tex, line 66
The odd and dyadic Euler formulas share a common reconstruction framework. For each prime p we define an even source series H_p and an inverse kernel K_p satisfying K_p(t)H_p(t)=secht. At an odd prime, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). At the dyadic prime, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t). The coefficient convolution is identical in form, but the kernels have different integral weights. This is the correct sense in which the local theories unify.
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H_p(t)=\cosh^2(pt)\operatorname{sech}t, \qquad K_p(t)=\operatorname{sech}^2(pt).
\cosh^2Introduction passage · 8 passages
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Introduction passage · 72_recovered.tex, line 66
The odd and dyadic Euler formulas share a common reconstruction framework. For each prime p we define an even source series H_p and an inverse kernel K_p satisfying K_p(t)H_p(t)=secht. At an odd prime, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). At the dyadic prime, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t). The coefficient convolution is identical in form, but the kernels have different integral weights. This is the correct sense in which the local theories unify.
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H_p(t)=\cosh^2(pt)\operatorname{sech}t, \qquad K_p(t)=\operatorname{sech}^2(pt).
\cosh_1Introduction passage · 1 passage
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Introduction passage · supplement_02.tex, line 36
r pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-series ring Q[[t]]. Define S_n(p)=∑_a=0^p-1(-1)^a(2a+1)^n∈ Z, F_p(t)=∑_n≥0S_n(p)fract^nn!. The formal exponential is Mathlib's mcodePowerSeries.exp. Scalar substitution is its actual mcoderescale homomorphism. We define exp_a(t)=e^at, cosh_a(t)=frace^at+e^-at2, sech(t)=cosh_1(t)^-1. The constant term of cosh_a is proved to be one, so the reciprocal used here is a unit inverse.
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\exp_a(t)=e^{at},\qquad \cosh_a(t)=\frac{e^{at}+e^{-at}}2,\qquad \sech(t)=\cosh_1(t)^{-1}.
\cosh_aIntroduction passage · 3 passages
Search aliases: cosh_a cosh
Introduction passage · supplement_02.tex, line 36
r pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-series ring Q[[t]]. Define S_n(p)=∑_a=0^p-1(-1)^a(2a+1)^n∈ Z, F_p(t)=∑_n≥0S_n(p)fract^nn!. The formal exponential is Mathlib's mcodePowerSeries.exp. Scalar substitution is its actual mcoderescale homomorphism. We define exp_a(t)=e^at, cosh_a(t)=frace^at+e^-at2, sech(t)=cosh_1(t)^-1. The constant term of cosh_a is proved to be one, so the reciprocal used here is a unit inverse.
Exact mathematical passage
\exp_a(t)=e^{at},\qquad \cosh_a(t)=\frac{e^{at}+e^{-at}}2,\qquad \sech(t)=\cosh_1(t)^{-1}.
D^2Introduction passage · 10 passages
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Introduction passage · atlas-pages.tex, line 76
lter becomes a matrix of actual delaysFIR TRANSFER / A FOUR-TAP EXAMPLEChapter V: Theorem 3.1. Take the causal filter h=(1,2,-1,1), with taps at 0,1,2,3, acting by y[n]=∑_qh[q]f[n-q]. Resolve the input into even and odd channels. Let D denote one delay in the channel index. vfigfir_transfer headReading one matrix entry The odd-input contribution to the even output is (2D+D^2)f_1. The tap at original delay one reaches the preceding odd channel sample. The tap at delay three reaches the odd channe…
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(2D+D^2)f_1
D^aIntroduction passage · 2 passages
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Introduction passage · supplement_06.tex, line 74
packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det(XI-D)=X^d+∑_a=0^d-1c_aX^a be its actual characteristic polynomial. The implementation reuses the pinned Mathlib Cayley--Hamilton theorem, expands polynomial evaluation into a finite sum and proves ∑_a=0^dχ_D[a]D^a=0, χ_D[d]=1. Here χ_D[a] denotes the coefficient at index a. The identities hold over any commutative coefficient ring. The …
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\sum_{a=0}^{d}\chi_D[a]D^a=0, \qquad \chi_D[d]=1.
D^dIntroduction passage · 5 passages
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Introduction passage · 10_foundations.tex, line 129
In characteristic p, factorial division is unavailable in general, but the coefficient operation remains meaningful over every commutative ring: (f⋆ g)_n=∑_r=0^nbinom nr f_r g_n-r, (D^d f)_n=f_n+d, 1_n=1, · n=0, 0, · n>0. Here ⋆ is Hurwitz multiplication. The symbol D^d denotes a coefficient shift, which agrees with formal differentiation after characteristic-zero encoding.
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(f\star g)_n=\sum_{r=0}^{n}\binom nr f_r g_{n-r},\qquad (D^d f)_n=f_{n+d},\qquad \mathbf1_n=\begin{cases}1,&n=0,\\0,&n>0.\end{cases}
D_0Introduction passage · 19 passages
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Introduction passage · Paper_IV_Traditional_Final.tex, line 1161
Let D_n denote the displayed determinant, with D_0=1. Expansion along the last row gives D_n = ∑_k=1^n frac(n-1)!(n-k)!p_kD_n-k. Here the cofactor sign is canceled by the product of the negative superdiagonal entries -(n-k+1),…,-(n-1). Thus D_n/n! satisfies the same recurrence and initial value as b_N,n(r).
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D_0=1
D_{[X,Y]}Introduction passage · 1 passage
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Introduction passage · 01_finite_core.tex, line 617
textitCurvature. Differentiating P^2=P gives P(dP(X))P=0. Write D_Xs=ds(X). For ambient smooth sections, D_XD_Ys-D_YD_Xs-D_[X,Y]s=0. Expansion of the projected derivatives therefore leaves R(X,Y)s=P(dP(X))D_Ys-P(dP(Y))D_Xs. Since Ps=s, differentiation gives D_Ys=(dP(Y))s+PD_Ys. The terms with a middle P vanish. Thus R(X,Y)s=P(dP(X)dP(Y)-dP(Y)dP(X))P s. This expression is alternating and bilinear in the tangent vectors, and it preserves the range. The ordinary chain rule and the transformation la…
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D_XD_Ys-D_YD_Xs-D_{[X,Y]}s=0.
D_iIntroduction passage · 10 passages
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Introduction passage · 02_foundations.tex, line 740
[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]
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A=\prod_{i=1}^{q}M_{d_i}(D_i)
d_iIntroduction passage · 41 passages
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Introduction passage · 02_foundations.tex, line 740
[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]
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A=\prod_{i=1}^{q}M_{d_i}(D_i)
D_i^{d_i}Introduction passage · 1 passage
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Introduction passage · 02_foundations.tex, line 743
[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]
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S_i=D_i^{d_i}
D_MIntroduction passage · 4 passages
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Introduction passage · 03_supplement_06.tex, line 61
A connection on the constructed bundle. The global formula is defined using the actual manifold derivative: ∇_Xs(x)=P(x) D_Ms(x)[X(x)]. Here D_M is represented by pathmvfderiv. A second definition, pathGlobalConnection.covariant, has values in the actual fiber: ∇ s(x):T_xM⟶ E_x. It is not only an ambient expression asserted to land in the range.
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\nabla_Xs(x)=P(x)\,D_Ms(x)[X(x)].
D_nIntroduction passage · 3 passages
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Introduction passage · Paper_IV_Traditional_Final.tex, line 1161
Let D_n denote the displayed determinant, with D_0=1. Expansion along the last row gives D_n = ∑_k=1^n frac(n-1)!(n-k)!p_kD_n-k. Here the cofactor sign is canceled by the product of the negative superdiagonal entries -(n-k+1),…,-(n-1). Thus D_n/n! satisfies the same recurrence and initial value as b_N,n(r).
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D_n
D_{n-k}Introduction passage · 1 passage
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Introduction passage · Paper_IV_Traditional_Final.tex, line 1162
Let D_n denote the displayed determinant, with D_0=1. Expansion along the last row gives D_n = ∑_k=1^n frac(n-1)!(n-k)!p_kD_n-k. Here the cofactor sign is canceled by the product of the negative superdiagonal entries -(n-k+1),…,-(n-1). Thus D_n/n! satisfies the same recurrence and initial value as b_N,n(r).
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D_n = \sum_{k=1}^{n} \frac{(n-1)!}{(n-k)!}p_kD_{n-k}.
D_TIntroduction passage · 11 passages
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Introduction passage · 40_regularity.tex, line 17
Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives are the derivatives of the evaluation map.
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J=D_YF(a)\in M_e(\mathbb K),\qquad K=D_TF(a)\in M_{e\times k}(\mathbb K). \label{eq:regularity-blocks}
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Introduction passage · 02_supplement_05.tex, line 42
Forward substitution with an actual stage evaluator. textitSource item 6.5; module TriangularSolve Use the graph and levels constructed on the previous page. A vertex v carries an arbitrary vector space X_v over a field. The spaces may have different dimensions. Let D_v:X_voverset∼⟶X_v, T_vu:X_u⟶ X_v be an invertible diagonal map and linear transfer maps. The operator being solved is defined by its actual block action: (Ax)_v=D_vx_v+∑_u→ vT_vux_u. No solution, solve equation or precomputed topol…
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D_v:X_v\overset{\sim}{\longrightarrow}X_v, \qquad T_{vu}:X_u\longrightarrow X_v
D_XIntroduction passage · 3 passages
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Introduction passage · 01_finite_core.tex, line 616
textitCurvature. Differentiating P^2=P gives P(dP(X))P=0. Write D_Xs=ds(X). For ambient smooth sections, D_XD_Ys-D_YD_Xs-D_[X,Y]s=0. Expansion of the projected derivatives therefore leaves R(X,Y)s=P(dP(X))D_Ys-P(dP(Y))D_Xs. Since Ps=s, differentiation gives D_Ys=(dP(Y))s+PD_Ys. The terms with a middle P vanish. Thus R(X,Y)s=P(dP(X)dP(Y)-dP(Y)dP(X))P s. This expression is alternating and bilinear in the tangent vectors, and it preserves the range. The ordinary chain rule and the transformation la…
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D_Xs=ds(X)
D_YIntroduction passage · 16 passages
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Introduction passage · 40_regularity.tex, line 17
Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives are the derivatives of the evaluation map.
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J=D_YF(a)\in M_e(\mathbb K),\qquad K=D_TF(a)\in M_{e\times k}(\mathbb K). \label{eq:regularity-blocks}
d_\lambdaIntroduction passage · 6 passages
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Introduction passage · 02_strengthened.tex, line 203
Let G be a finite group and let U:G⟶ U( H) be a unitary representation. Write widehat G for the set of irreducible unitary representations. For λ∈widehat G, let d_λ and χ_λ denote its dimension and character.
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d_\lambda
\Dfun_N^{(N)}Introduction passage · 1 passage
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Introduction passage · Paper_IV_Traditional_Final.tex, line 1363
Write D_N=e^g locally. Then g'=- T_N. Divide the equation D_N^(N)=- D_N by D_N and use the defining identity for Y_N.
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\Dfun_N^{(N)}=-\Dfun_N
\Dfun_N^{-r}Introduction passage · 5 passages
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Introduction passage · Paper_IV_Traditional_Final.tex, line 2615
For an entire function f, let Z(f) denote its multiset of nonzero zeros. Let r be a positive integer. Then D_N^-r is meromorphic and a simple zero of D_N becomes a pole of order r.
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\Dfun_N^{-r}
\diagIntroduction passage · 60 passages
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Introduction passage · monograph_atlas.tex, line 705
sfer determines its HamiltonianSTRICT POSITIVITY AND THE GROUND GAPChapter VII: Theorem II.10.2, Proposition II.10.5. On HH= C^3, take the strictly positive contraction T=diag(1,1/2,1/4), H=-log T=diag(0,log2,2log2). At unit Euclidean time, T=e^-H. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integer time n≥0, T^n=diag(1,2^-n,4^-n), P_0=diag(1,0,0), normT^n-P_0=2^-n. The gap Δ=log2 determines the rate e^-nΔ exactly in this example. xfigtransfer_…
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T^n=\diag(1,2^{-n},4^{-n}),\qquad P_0=\diag(1,0,0),\qquad\norm{T^n-P_0}=2^{-n}.
\dim_{\FF_p}Introduction passage · 1 passage
Search aliases: dim_FF_p dim_ F_p dim FF dim mathbb
Introduction passage · Paper_XI_Traditional_Final.tex, line 2159
For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i…
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\begin{array}{c|c|c} & p\text{ odd} & p=2\\ \hline \dim_{\FF_p}(J^1/H^1) & p^d-1 & 2^d-2\\ \Swan(\pi)&p^{d-1}(p+1)h_d&3\cdot2^{d-1}h_d \end{array}
\dot UIntroduction passage · 17 passages
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Introduction passage · 03_supplement_02.tex, line 90
Fix a real time t and a finite complex coordinate space. Let U(s) be a unitary matrix curve, differentiable entrywise at t, and let ψ(s) be differentiable componentwise at t. Write dot U and dotψ for those derivatives. Suppose idotψ=Hψ(t). Only the value of the Hamiltonian at the chosen time is needed. No derivative of H is assumed.
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\dot U
\dot\psiIntroduction passage · 4 passages
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Introduction passage · 03_supplement_02.tex, line 90
Fix a real time t and a finite complex coordinate space. Let U(s) be a unitary matrix curve, differentiable entrywise at t, and let ψ(s) be differentiable componentwise at t. Write dot U and dotψ for those derivatives. Suppose idotψ=Hψ(t). Only the value of the Hamiltonian at the chosen time is needed. No derivative of H is assumed.
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\dot\psi
\bm EIntroduction passage · 7 passages
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Introduction passage · Paper_V_Traditional_Final.tex, line 498
[Perfect-reconstruction criterion] For d∈ Z, such a bank reconstructs every input up to a delay of d channel indices exactly when the following Laurent-polynomial identity holds: R(z) E(z)=z^-dI_N. If the right side is c z^-dI_N with c≠0, reconstruction holds up to gain c and channel delay d. After interleaving, this is a delay of Nd original samples. A negative delay denotes an advance.
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\boxed{ \bm R(z)\bm E(z)=z^{-d}I_N. }
\mathcal EIntroduction passage · 38 passages
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Introduction passage · 03_supplement_06.tex, line 34
The mutually inverse local maps. Write T_a(x)=T(P(x),P(a)). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a ≅ U_a× E_a is a local product homeomorphism that preserves the base point.
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B_a(x):\mathcal E_a\longrightarrow\mathcal E_x, \quad v\longmapsto T_a(x)v,
E^{(N,j)}_{Nn}Introduction passage · 1 passage
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Introduction passage · supplement_07.tex, line 184
Normalization is also an explicit implication. Using the existing definition E^(N,j)_Nn=j!C_N,j(n), the source proves that multiplication by j! commutes with every iterated difference. If the unnormalized difference has precision A, the normalized difference has precision A-v_p(j!). Precision is defined as x=0 hboxor v_p(x)≥ A, so zero and negative factorial-loss thresholds are handled correctly. For j<p, the loss is zero.
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E^{(N,j)}_{Nn}=j!C_{N,j}(n)
e^{-at}Introduction passage · 3 passages
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Introduction passage · supplement_02.tex, line 36
r pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-series ring Q[[t]]. Define S_n(p)=∑_a=0^p-1(-1)^a(2a+1)^n∈ Z, F_p(t)=∑_n≥0S_n(p)fract^nn!. The formal exponential is Mathlib's mcodePowerSeries.exp. Scalar substitution is its actual mcoderescale homomorphism. We define exp_a(t)=e^at, cosh_a(t)=frace^at+e^-at2, sech(t)=cosh_1(t)^-1. The constant term of cosh_a is proved to be one, so the reciprocal used here is a unit inverse.
Exact mathematical passage
\exp_a(t)=e^{at},\qquad \cosh_a(t)=\frac{e^{at}+e^{-at}}2,\qquad \sech(t)=\cosh_1(t)^{-1}.
e^{-H}Introduction passage · 1 passage
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Introduction passage · monograph_atlas.tex, line 704
vpage36A positive transfer determines its HamiltonianSTRICT POSITIVITY AND THE GROUND GAPChapter VII: Theorem II.10.2, Proposition II.10.5. On HH= C^3, take the strictly positive contraction T=diag(1,1/2,1/4), H=-log T=diag(0,log2,2log2). At unit Euclidean time, T=e^-H. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integer time n≥0, T^n=diag(1,2^-n,4^-n), P_0=diag(1,0,0), normT^n-P_0=2^-n. The gap Δ=log2 determines the rate e^-nΔ exactly in this …
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T=e^{-H}
e^{-H'}Introduction passage · 1 passage
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Introduction passage · monograph_atlas.tex, line 712
integer time n≥0, T^n=diag(1,2^-n,4^-n), P_0=diag(1,0,0), normT^n-P_0=2^-n. The gap Δ=log2 determines the rate e^-nΔ exactly in this example. xfigtransfer_ground headStrict positivity is an essential finite hypothesis For T'=diag(1,0), the operator is positive semidefinite, but there is no finite self-adjoint H' on all of C^2 with T'=e^-H': every finite eigenvalue of H' exponentiates to a strictly positive value. One may restrict to the positive support, but that changes the space on which the l…
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T'=e^{-H'}
e^{-i(t-s)H}Introduction passage · 1 passage
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Introduction passage · 04_alternative_derivations.tex, line 12
ive conclusions are included there. Duhamel proof of leakage from a fixed packet. Let H=H^* on a finite-dimensional complex Hilbert space, P=P^*=P^2, Q=I-P, and t∈ R. [Alternative proof of the commutator leakage bound] Because QP=0, Qe^-itHP=Q[e^-itH,P]P. The Duhamel identity gives [e^-itH,P]=-i∫_0^t e^-i(t-s)H[H,P]e^-isH ds. Both unitary factors have norm one, so taking norms and using the integration interval's length |t| yields Qe^-itHP ≤ |t| [H,P] . Proof of projected curvature with exterior…
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[e^{-itH},P]=-i\int_0^t e^{-i(t-s)H}[H,P]e^{-isH}\,ds.
e^{-isH}Introduction passage · 3 passages
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Introduction passage · 04_alternative_derivations.tex, line 12
ive conclusions are included there. Duhamel proof of leakage from a fixed packet. Let H=H^* on a finite-dimensional complex Hilbert space, P=P^*=P^2, Q=I-P, and t∈ R. [Alternative proof of the commutator leakage bound] Because QP=0, Qe^-itHP=Q[e^-itH,P]P. The Duhamel identity gives [e^-itH,P]=-i∫_0^t e^-i(t-s)H[H,P]e^-isH ds. Both unitary factors have norm one, so taking norms and using the integration interval's length |t| yields Qe^-itHP ≤ |t| [H,P] . Proof of projected curvature with exterior…
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[e^{-itH},P]=-i\int_0^t e^{-i(t-s)H}[H,P]e^{-isH}\,ds.
e^{-itH}Introduction passage · 7 passages
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Introduction passage · 04_alternative_derivations.tex, line 12
ive conclusions are included there. Duhamel proof of leakage from a fixed packet. Let H=H^* on a finite-dimensional complex Hilbert space, P=P^*=P^2, Q=I-P, and t∈ R. [Alternative proof of the commutator leakage bound] Because QP=0, Qe^-itHP=Q[e^-itH,P]P. The Duhamel identity gives [e^-itH,P]=-i∫_0^t e^-i(t-s)H[H,P]e^-isH ds. Both unitary factors have norm one, so taking norms and using the integration interval's length |t| yields Qe^-itHP ≤ |t| [H,P] . Proof of projected curvature with exterior…
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[e^{-itH},P]=-i\int_0^t e^{-i(t-s)H}[H,P]e^{-isH}\,ds.
e^{-n\Delta}Introduction passage · 1 passage
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Introduction passage · monograph_atlas.tex, line 709
HH= C^3, take the strictly positive contraction T=diag(1,1/2,1/4), H=-log T=diag(0,log2,2log2). At unit Euclidean time, T=e^-H. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integer time n≥0, T^n=diag(1,2^-n,4^-n), P_0=diag(1,0,0), normT^n-P_0=2^-n. The gap Δ=log2 determines the rate e^-nΔ exactly in this example. xfigtransfer_ground headStrict positivity is an essential finite hypothesis For T'=diag(1,0), the operator is positive semidefinite, b…
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e^{-n\Delta}
e^{at}Introduction passage · 2 passages
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Introduction passage · supplement_02.tex, line 36
r pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-series ring Q[[t]]. Define S_n(p)=∑_a=0^p-1(-1)^a(2a+1)^n∈ Z, F_p(t)=∑_n≥0S_n(p)fract^nn!. The formal exponential is Mathlib's mcodePowerSeries.exp. Scalar substitution is its actual mcoderescale homomorphism. We define exp_a(t)=e^at, cosh_a(t)=frace^at+e^-at2, sech(t)=cosh_1(t)^-1. The constant term of cosh_a is proved to be one, so the reciprocal used here is a unit inverse.
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\exp_a(t)=e^{at},\qquad \cosh_a(t)=\frac{e^{at}+e^{-at}}2,\qquad \sech(t)=\cosh_1(t)^{-1}.
\e^BIntroduction passage · 1 passage
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Introduction passage · Paper_VI_Traditional_Final.tex, line 4579
If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every integer n≥0, after increasing M_β to cover the finitely many initial values. Write t=n+r, where 0≤ r<1. Then norm e^tB≤ M_βexp(norm B+max(0,-β)) e^β t. Thus…
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r(\e^B)=\e^{s(B)}<\e^\beta
e^gIntroduction passage · 3 passages
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Introduction passage · Paper_IV_Traditional_Final.tex, line 1363
Write D_N=e^g locally. Then g'=- T_N. Divide the equation D_N^(N)=- D_N by D_N and use the defining identity for Y_N.
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\Dfun_N=e^g
\e^{nB}Introduction passage · 1 passage
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Introduction passage · Paper_VI_Traditional_Final.tex, line 4579
ginproof If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every integer n≥0, after increasing M_β to cover the finitely many initial values. Write t=n+r, where 0≤ r<1. Then norm e^tB≤ M_βexp(norm B+max(0,-β)) e^…
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\norm{\e^{nB}}\leq M_\beta\e^{\beta n}
\e^{s(B)}Introduction passage · 1 passage
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Introduction passage · Paper_VI_Traditional_Final.tex, line 4579
If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every integer n≥0, after increasing M_β to cover the finitely many initial values. Write t=n+r, where 0≤ r<1. Then norm e^tB≤ M_βexp(norm B+max(0,-β)) e^β t. Thus…
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r(\e^B)=\e^{s(B)}<\e^\beta
e^tIntroduction passage · 14 passages
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Introduction passage · Paper_IV_Traditional_Final.tex, line 1629
=frac zsinh z. The first case is a scaled and translated Euler family. In fact, if E_m(x) denotes the classical Euler polynomial, then E_m^(2,0,1)(x) = 2^mE_m(fracx+12). If B_m(x) denotes the classical Bernoulli polynomial, the second case has the exact identification E_m^(2,1,1)(x) = 2^mB_m(fracx+12), obtained by substituting t=2z in t e^yt/(e^t-1).
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t e^{yt}/(e^t-1)
e^{yt}Introduction passage · 1 passage
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Introduction passage · Paper_IV_Traditional_Final.tex, line 1629
=frac zsinh z. The first case is a scaled and translated Euler family. In fact, if E_m(x) denotes the classical Euler polynomial, then E_m^(2,0,1)(x) = 2^mE_m(fracx+12). If B_m(x) denotes the classical Bernoulli polynomial, the second case has the exact identification E_m^(2,1,1)(x) = 2^mB_m(fracx+12), obtained by substituting t=2z in t e^yt/(e^t-1).
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t e^{yt}/(e^t-1)
\e^\betaIntroduction passage · 1 passage
Search aliases: e^β e beta mathrm beta
Introduction passage · Paper_VI_Traditional_Final.tex, line 4579
If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every integer n≥0, after increasing M_β to cover the finitely many initial values. Write t=n+r, where 0≤ r<1. Then norm e^tB≤ M_βexp(norm B+max(0,-β)) e^β t. Thus…
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r(\e^B)=\e^{s(B)}<\e^\beta
\e^{\beta n}Introduction passage · 1 passage
Search aliases: e^β n e beta mathrm beta
Introduction passage · Paper_VI_Traditional_Final.tex, line 4579
ginproof If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every integer n≥0, after increasing M_β to cover the finitely many initial values. Write t=n+r, where 0≤ r<1. Then norm e^tB≤ M_βexp(norm B+max(0,-β)) e^…
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\norm{\e^{nB}}\leq M_\beta\e^{\beta n}
E_aIntroduction passage · 36 passages
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Introduction passage · 02_strengthened.tex, line 1485
[Correctable observable algebras] Let Φ be a finite completely positive trace preserving channel with Kraus operators (E_a). A finite dimensional unital star subalgebra M⊆ PEnd( H)P is correctable when a completely positive trace preserving recovery R satisfies P(Φ^*∘ R^*)(X)P=X for every X∈ M, where stars on maps denote Heisenberg adjoints. This holds exactly when every compressed product PE_a^*E_bP belongs to the commutant M' on the code. The scalar code criterion is the special case M=PEnd( H…
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(E_a)
\mathcal E_aIntroduction passage · 5 passages
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Introduction passage · 03_supplement_06.tex, line 34
The mutually inverse local maps. Write T_a(x)=T(P(x),P(a)). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a ≅ U_a× E_a is a local product homeomorphism that preserves the base point.
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B_a(x):\mathcal E_a\longrightarrow\mathcal E_x, \quad v\longmapsto T_a(x)v,
E_hIntroduction passage · 2 passages
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Introduction passage · 10_foundations.tex, line 271
Write E_h f(n)=f(n+h) for the shift operator. The unit shift is I+Δ_1. Expanding its nth power and evaluating at zero gives reconstruction. Composing difference operators gives the coefficient shift and applying reconstruction to Δ_1^s f gives the third identity. The binomial expansion of (E_h-I)^s gives the explicit stencil. Binomial coefficients vanish when their lower index exceeds n, which proves the interpolation claim. Equality of every point of the stencil proves equality of differences. …
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E_h f(n)=f(n+h)
E_{ij}Introduction passage · 2 passages
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Introduction passage · visual_atlas.tex, line 249
vpage10Which zero patterns survive multiplication?SUPPORT ALGEBRAS / A UNIVERSAL TESTMain text: [source reference]. All packet spaces are nonzero Suppose every matrix supported in a specified pattern is allowed. Then multiplication can create a new entry whenever two allowed edges compose. Let E_ij denote the matrix that sends e_j to e_i and kills the other basis vectors.
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E_{ij}
E_jIntroduction passage · 39 passages
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Introduction passage · supplement_10.tex, line 5
ntCheckpoint scope. This complete source body retains the status of its dated original supplement. Earlier pending or partial-coverage labels are historical. The final core records their resolution. Scope and final statement. Let p be a prime, let m>0 divide p-1, put N=mp and h=(p-1)/m, and let 0≤ j<N. The original rational packet is defined by ∑_q=0^nbinomNn+jNqC_N,j(q)=δ_n0, E_j(n)=j! C_N,j(n). For every n,k,u∈ N, the new formal result establishes E_j(n)∈ Z_p, Δ_uh^kE_j(n)∈ p^min(k,mn) Z_p. He…
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\sum_{q=0}^{n}\binom{Nn+j}{Nq}C_{N,j}(q)=\delta_{n0}, \qquad E_j(n)=j!\,C_{N,j}(n).
e_{j+1\bmod8}Introduction passage · 1 passage
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Introduction passage · visual_atlas.tex, line 103
vpage5The operator changes coordinates tooMODEL A / COHERENCE AND NATURALITYChapter II, mathematical text: Theorem 3.5, Theorem 3.8 Use a finite periodic carrier X=QQ^8 for this example. Let Se_j=e_j+1bmod8 and let T=I+2S+S^2. This is a concrete cyclic model, not an unannounced boundary convention for the finitely supported signal on Plate II.20. Write R_N for the permutation that groups coordinates in increasing residue order modulo N.
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Se_j=e_{j+1\bmod8}
E_mIntroduction passage · 6 passages
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Introduction passage · Paper_IV_Traditional_Final.tex, line 1615
For (N,j)=(2,0), C_2,0(z)=sech z. For (N,j)=(2,1), C_2,1(z)=frac zsinh z. The first case is a scaled and translated Euler family. In fact, if E_m(x) denotes the classical Euler polynomial, then E_m^(2,0,1)(x) = 2^mE_m(fracx+12). If B_m(x) denotes the classical Bernoulli polynomial, the second case has the exact identification E_m^(2,1,1)(x) = 2^mB_m(fracx+12), obtained by substituting t=2z in t e^yt/(e^t-1).
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E_m(x)
E_m^{(2,0,1)}Introduction passage · 1 passage
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Introduction passage · Paper_IV_Traditional_Final.tex, line 1616
For (N,j)=(2,0), C_2,0(z)=sech z. For (N,j)=(2,1), C_2,1(z)=frac zsinh z. The first case is a scaled and translated Euler family. In fact, if E_m(x) denotes the classical Euler polynomial, then E_m^(2,0,1)(x) = 2^mE_m(fracx+12). If B_m(x) denotes the classical Bernoulli polynomial, the second case has the exact identification E_m^(2,1,1)(x) = 2^mB_m(fracx+12), obtained by substituting t=2z in t e^yt/(e^t-1).
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E_m^{(2,0,1)}(x) = 2^mE_m\left(\frac{x+1}{2}\right).
E_m^{(2,1,1)}Introduction passage · 1 passage
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Introduction passage · Paper_IV_Traditional_Final.tex, line 1623
daries For (N,j)=(2,0), C_2,0(z)=sech z. For (N,j)=(2,1), C_2,1(z)=frac zsinh z. The first case is a scaled and translated Euler family. In fact, if E_m(x) denotes the classical Euler polynomial, then E_m^(2,0,1)(x) = 2^mE_m(fracx+12). If B_m(x) denotes the classical Bernoulli polynomial, the second case has the exact identification E_m^(2,1,1)(x) = 2^mB_m(fracx+12), obtained by substituting t=2z in t e^yt/(e^t-1).
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E_m^{(2,1,1)}(x) = 2^mB_m\left(\frac{x+1}{2}\right),
e_{n+h}^{(m,p)}Introduction passage · 3 passages
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Introduction passage · 72_recovered.tex, line 55
Fix an odd prime p, a divisor mmid p-1, and put h=fracp-1m. The reduced standard window inverse is defined by ∑_q=0^n binommnmqe_q^(m,p) = δ_n0 in F_p. We prove that e_n+h^(m,p)=e_n^(m,p) (n≥1). The result first follows from subgroup symmetry in the finite Hurwitz algebra. A second proof shows that the standard window operator preserves the finite space of sequences that are h periodic after their constant term. The two arguments expose different parts of the same finite character structure.
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e_{n+h}^{(m,p)}=e_n^{(m,p)} \qquad(n\geq1).
e_{n+uh}^{(m,p)}Introduction passage · 1 passage
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Introduction passage · supplement_07.tex, line 109
The positive-tail period. Write h=(p-1)/m. For n≥1, evaluate U^pf_m=Uf_m at the nonnegative degree mn-1: f_m(mn+p-1)=f_m(mn). Since mn+p-1=m(n+h), the identified sequence satisfies e_n+h^(m,p)=e_n^(m,p) (n≥1). Induction gives e_n+uh^(m,p)=e_n^(m,p) for every u≥0. This proves a period, not necessarily the least positive period. The proof also includes p=2, where m=1 and h=1.
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e_{n+uh}^{(m,p)}=e_n^{(m,p)}
e_n^{(m,p)}Introduction passage · 10 passages
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Introduction passage · 72_recovered.tex, line 55
Fix an odd prime p, a divisor mmid p-1, and put h=fracp-1m. The reduced standard window inverse is defined by ∑_q=0^n binommnmqe_q^(m,p) = δ_n0 in F_p. We prove that e_n+h^(m,p)=e_n^(m,p) (n≥1). The result first follows from subgroup symmetry in the finite Hurwitz algebra. A second proof shows that the standard window operator preserves the finite space of sequences that are h periodic after their constant term. The two arguments expose different parts of the same finite character structure.
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e_{n+h}^{(m,p)}=e_n^{(m,p)} \qquad(n\geq1).
e_q^{(m,p)}Introduction passage · 2 passages
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Introduction passage · 72_recovered.tex, line 47
Fix an odd prime p, a divisor mmid p-1, and put h=fracp-1m. The reduced standard window inverse is defined by ∑_q=0^n binommnmqe_q^(m,p) = δ_n0 in F_p. We prove that e_n+h^(m,p)=e_n^(m,p) (n≥1). The result first follows from subgroup symmetry in the finite Hurwitz algebra. A second proof shows that the standard window operator preserves the finite space of sequences that are h periodic after their constant term. The two arguments expose different parts of the same finite character structure.
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\sum_{q=0}^{n} \binom{mn}{mq}e_q^{(m,p)} = \delta_{n0} \quad\text{in }\FF_p.
\mathcal E_xIntroduction passage · 5 passages
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Introduction passage · 03_supplement_06.tex, line 34
The mutually inverse local maps. Write T_a(x)=T(P(x),P(a)). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a ≅ U_a× E_a is a local product homeomorphism that preserves the base point.
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B_a(x):\mathcal E_a\longrightarrow\mathcal E_x, \quad v\longmapsto T_a(x)v,
\exp_aIntroduction passage · 1 passage
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Introduction passage · supplement_02.tex, line 36
r pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-series ring Q[[t]]. Define S_n(p)=∑_a=0^p-1(-1)^a(2a+1)^n∈ Z, F_p(t)=∑_n≥0S_n(p)fract^nn!. The formal exponential is Mathlib's mcodePowerSeries.exp. Scalar substitution is its actual mcoderescale homomorphism. We define exp_a(t)=e^at, cosh_a(t)=frace^at+e^-at2, sech(t)=cosh_1(t)^-1. The constant term of cosh_a is proved to be one, so the reciprocal used here is a unit inverse.
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\exp_a(t)=e^{at},\qquad \cosh_a(t)=\frac{e^{at}+e^{-at}}2,\qquad \sech(t)=\cosh_1(t)^{-1}.
f^*Introduction passage · 1 passage
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Introduction passage · Paper_XII_Traditional_Final.tex, line 1739
eorem supplies an ordinary finite module or a perfect complex with full patched support, together with the coefficient quotient and the module or derived control isomorphism. Write M for the resulting finite level carrier over R. item Comparison gate. Let H be a finite ordinary Hecke module or a perfect Hecke complex over T. Supply a ring map f:R→ T satisfying (f^*)^-1(Supp_R(M))⊆Supp_ T(H). Use ordinary or derived support according to the carrier. Actual scalar extension satisfies this inclusio…
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(f^*)^{-1}\bigl(\Supp_R(M)\bigr)\subseteq\Supp_{\TT}(H).
F_0Introduction passage · 15 passages
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Introduction passage · 72_recovered.tex, line 2892
[Fibonacci residue packets] Let F_0=0, F_1=1, and F_n+2=F_n+1+F_n. Let L_N denote the Lucas number. For every N≥1 and every address j, F_N(k+2)+j = L_NF_N(k+1)+j -(-1)^NF_Nk+j.
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F_0=0
f_0Introduction passage · 12 passages
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Introduction passage · atlas-pages.tex, line 62
of residues 0 and 2 modulo four. Therefore the coarse even mask is the sum of two disjoint fine masks: P_2,0=P_4,0+P_4,2. vfigrefinement platecaptionThe common horizontal axis is the original sample index. Empty cells denote masked-out addresses. There are no interpolated samples between the displayed cells. headWhy the coordinate convention matters The decimated channel f_0[q]=f[2q] has its own index q. At modulus four the descendants are f[4q] and f[4q+2]. These shorter arrays must be interlea…
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f_0[q]=f[2q]
F_1Introduction passage · 12 passages
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Introduction passage · 72_recovered.tex, line 2638
Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod p. Let J_F(x_L) denote the Jacobian matrix and write overline J_F(x_L) for its reduction modulo p.
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F=(F_1,\ldots,F_r):\Zp^d\longrightarrow\Zp^r
f_1Introduction passage · 14 passages
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Introduction passage · atlas-pages.tex, line 76
lter becomes a matrix of actual delaysFIR TRANSFER / A FOUR-TAP EXAMPLEChapter V: Theorem 3.1. Take the causal filter h=(1,2,-1,1), with taps at 0,1,2,3, acting by y[n]=∑_qh[q]f[n-q]. Resolve the input into even and odd channels. Let D denote one delay in the channel index. vfigfir_transfer headReading one matrix entry The odd-input contribution to the even output is (2D+D^2)f_1. The tap at original delay one reaches the preceding odd channel sample. The tap at delay three reaches the odd channe…
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(2D+D^2)f_1
F_eIntroduction passage · 2 passages
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Introduction passage · continuation_01.tex, line 14
Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer modules. It factors through reduction to R_q^d. The cokernel below is presented directly using its integer-domain image.
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F=(F_1,\ldots,F_e)\in\mathbb Z[X_1,\ldots,X_d]^e, \quad q\ge1,\quad L\ge1, \quad x\in\mathbb Z^d, \quad F(x)\equiv0\pmod{q^L}.
F_iIntroduction passage · 24 passages
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Introduction passage · 72_recovered.tex, line 2641
Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod p. Let J_F(x_L) denote the Jacobian matrix and write overline J_F(x_L) for its reduction modulo p.
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F_i\in\Z_p[X_1,\ldots,X_d]
f_jIntroduction passage · 16 passages
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Introduction passage · Paper_VI_Traditional_Final.tex, line 3013
Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.
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(P_\mathcal Qf)_i = \frac{\sum_{j\in B(i)}\pi_jf_j} {\sum_{j\in B(i)}\pi_j}.
f_mIntroduction passage · 8 passages
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Introduction passage · supplement_07.tex, line 101
The positive-tail period. Write h=(p-1)/m. For n≥1, evaluate U^pf_m=Uf_m at the nonnegative degree mn-1: f_m(mn+p-1)=f_m(mn). Since mn+p-1=m(n+h), the identified sequence satisfies e_n+h^(m,p)=e_n^(m,p) (n≥1). Induction gives e_n+uh^(m,p)=e_n^(m,p) for every u≥0. This proves a period, not necessarily the least positive period. The proof also includes p=2, where m=1 and h=1.
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U^pf_m=Uf_m
F_nIntroduction passage · 6 passages
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Introduction passage · 72_recovered.tex, line 2892
[Fibonacci residue packets] Let F_0=0, F_1=1, and F_n+2=F_n+1+F_n. Let L_N denote the Lucas number. For every N≥1 and every address j, F_N(k+2)+j = L_NF_N(k+1)+j -(-1)^NF_Nk+j.
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F_{n+2}=F_{n+1}+F_n
F_{N(k+1)+j}Introduction passage · 3 passages
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Introduction passage · 72_recovered.tex, line 2893
[Fibonacci residue packets] Let F_0=0, F_1=1, and F_n+2=F_n+1+F_n. Let L_N denote the Lucas number. For every N≥1 and every address j, F_N(k+2)+j = L_NF_N(k+1)+j -(-1)^NF_Nk+j.
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\boxed{ F_{N(k+2)+j} = L_NF_{N(k+1)+j} -(-1)^NF_{Nk+j}. }
F_{N(k+2)+j}Introduction passage · 3 passages
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Introduction passage · 72_recovered.tex, line 2893
[Fibonacci residue packets] Let F_0=0, F_1=1, and F_n+2=F_n+1+F_n. Let L_N denote the Lucas number. For every N≥1 and every address j, F_N(k+2)+j = L_NF_N(k+1)+j -(-1)^NF_Nk+j.
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\boxed{ F_{N(k+2)+j} = L_NF_{N(k+1)+j} -(-1)^NF_{Nk+j}. }
F_{n+1}Introduction passage · 3 passages
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Introduction passage · 72_recovered.tex, line 2892
[Fibonacci residue packets] Let F_0=0, F_1=1, and F_n+2=F_n+1+F_n. Let L_N denote the Lucas number. For every N≥1 and every address j, F_N(k+2)+j = L_NF_N(k+1)+j -(-1)^NF_Nk+j.
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F_{n+2}=F_{n+1}+F_n
F_{n+2}Introduction passage · 1 passage
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Introduction passage · 72_recovered.tex, line 2892
[Fibonacci residue packets] Let F_0=0, F_1=1, and F_n+2=F_n+1+F_n. Let L_N denote the Lucas number. For every N≥1 and every address j, F_N(k+2)+j = L_NF_N(k+1)+j -(-1)^NF_Nk+j.
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F_{n+2}=F_{n+1}+F_n
f_{n+d}Introduction passage · 1 passage
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Introduction passage · 10_foundations.tex, line 129
In characteristic p, factorial division is unavailable in general, but the coefficient operation remains meaningful over every commutative ring: (f⋆ g)_n=∑_r=0^nbinom nr f_r g_n-r, (D^d f)_n=f_n+d, 1_n=1, · n=0, 0, · n>0. Here ⋆ is Hurwitz multiplication. The symbol D^d denotes a coefficient shift, which agrees with formal differentiation after characteristic-zero encoding.
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(f\star g)_n=\sum_{r=0}^{n}\binom nr f_r g_{n-r},\qquad (D^d f)_n=f_{n+d},\qquad \mathbf1_n=\begin{cases}1,&n=0,\\0,&n>0.\end{cases}
F_{Nk+j}Introduction passage · 5 passages
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Introduction passage · 72_recovered.tex, line 2893
[Fibonacci residue packets] Let F_0=0, F_1=1, and F_n+2=F_n+1+F_n. Let L_N denote the Lucas number. For every N≥1 and every address j, F_N(k+2)+j = L_NF_N(k+1)+j -(-1)^NF_Nk+j.
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\boxed{ F_{N(k+2)+j} = L_NF_{N(k+1)+j} -(-1)^NF_{Nk+j}. }
F_pIntroduction passage · 4 passages
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Introduction passage · supplement_02.tex, line 31
rence theorem remain pending. No continuum convergence or analytic summability theorem is inferred from the formal power-series equalities in this block. The historical numerical study is not being relabelled as a fresh experiment. pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-series ring Q[[t]]. Define S_n(p)=∑_a=0^p-1(-1)^a(2a+1)^n∈ Z, F_p(t)=∑_n≥0S_n(p)fract^nn!. The formal exponential is Mathlib's mcodePowerSeries.exp. S…
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S_n(p)=\sum_{a=0}^{p-1}(-1)^a(2a+1)^n\in\Z, \qquad F_p(t)=\sum_{n\ge0}S_n(p)\frac{t^n}{n!}.
F_rIntroduction passage · 16 passages
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Introduction passage · 72_recovered.tex, line 2638
Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod p. Let J_F(x_L) denote the Jacobian matrix and write overline J_F(x_L) for its reduction modulo p.
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F=(F_1,\ldots,F_r):\Zp^d\longrightarrow\Zp^r
\FF_pIntroduction passage · 30 passages
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Introduction passage · 72_recovered.tex, line 47
Fix an odd prime p, a divisor mmid p-1, and put h=fracp-1m. The reduced standard window inverse is defined by ∑_q=0^n binommnmqe_q^(m,p) = δ_n0 in F_p. We prove that e_n+h^(m,p)=e_n^(m,p) (n≥1). The result first follows from subgroup symmetry in the finite Hurwitz algebra. A second proof shows that the standard window operator preserves the finite space of sequences that are h periodic after their constant term. The two arguments expose different parts of the same finite character structure.
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\sum_{q=0}^{n} \binom{mn}{mq}e_q^{(m,p)} = \delta_{n0} \quad\text{in }\FF_p.
\FF_p^+Introduction passage · 4 passages
Search aliases: FF_p^+ F_p^+ FF mathbb
Introduction passage · 72_recovered.tex, line 175
Let p be prime. Write F_p^+ for the additive group of the finite field.
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\FF_p^+
\FF_p^rIntroduction passage · 3 passages
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Introduction passage · 72_recovered.tex, line 2522
Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.
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\rho_L(x_L) = \frac{Mx_L-b}{p^L} \pmod p \in\FF_p^r.
\FF_p^{\times}Introduction passage · 1 passage
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Introduction passage · Paper_XI_Traditional_Final.tex, line 1303
[Fixed point free Weil trace] Under the declared normalization, Trω(g) = (frac-1p)^r (fracdet(g-I)p). Here (frac·p) denotes the quadratic character of F_p^×.
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\FF_p^{\times}
\frakmIntroduction passage · 24 passages
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Introduction passage · Paper_XII_Traditional_Final.tex, line 274
Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient topology on S/L is precisely this image ideal topology.
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\fraka=\frakm_{\OO}S+\sum_i\frakm_{R_i}S, \qquad L=\sum_i I_iS.
\frakm_{\OO}Introduction passage · 1 passage
Search aliases: frakm_OO m_ O frakm OO mathfrak mathcal
Introduction passage · Paper_XII_Traditional_Final.tex, line 274
Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient topology on S/L is precisely this image ideal topology.
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\fraka=\frakm_{\OO}S+\sum_i\frakm_{R_i}S, \qquad L=\sum_i I_iS.
\frakm_{R_i}Introduction passage · 1 passage
Search aliases: frakm_R_i m_R_i frakm mathfrak
Introduction passage · Paper_XII_Traditional_Final.tex, line 274
Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient topology on S/L is precisely this image ideal topology.
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\fraka=\frakm_{\OO}S+\sum_i\frakm_{R_i}S, \qquad L=\sum_i I_iS.
g'Introduction passage · 5 passages
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Introduction passage · Paper_IV_Traditional_Final.tex, line 1363
Write D_N=e^g locally. Then g'=- T_N. Divide the equation D_N^(N)=- D_N by D_N and use the defining identity for Y_N.
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g'=-\Tfun_N
G_{J,\chi}Introduction passage · 3 passages
Search aliases: G_J,χ chi
Introduction passage · residue_continuation_body.tex, line 117
· =#μ(K) #{I≠0:[I]=C, N I≤ X, P( N I N J)}. The same norm-preserving equivalence carries arbitrary complex weights. In particular, write G_J,χ(Y) for the sum of χ( N(x)bmod m) over these generators with N(x)≤ Y, and T_C,χ(X) for the corresponding sum over ideals in class C. For multiplicative χ preserving 0 and 1, the exact identity is G_J,χ(X N J) =#μ(K) χ( N Jbmod m) T_C,χ(X). Both signs are included before applying this full roots-of-unity factor. A root of unity can interchange the sign port…
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G_{J,\chi}(Y)
g_{n-r}Introduction passage · 2 passages
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Introduction passage · 10_foundations.tex, line 129
In characteristic p, factorial division is unavailable in general, but the coefficient operation remains meaningful over every commutative ring: (f⋆ g)_n=∑_r=0^nbinom nr f_r g_n-r, (D^d f)_n=f_n+d, 1_n=1, · n=0, 0, · n>0. Here ⋆ is Hurwitz multiplication. The symbol D^d denotes a coefficient shift, which agrees with formal differentiation after characteristic-zero encoding.
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(f\star g)_n=\sum_{r=0}^{n}\binom nr f_r g_{n-r},\qquad (D^d f)_n=f_{n+d},\qquad \mathbf1_n=\begin{cases}1,&n=0,\\0,&n>0.\end{cases}
G_{\Q}Introduction passage · 9 passages
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Introduction passage · Paper_X_Block27_Amended.tex, line 2402
The hyperref[book:x-construction-21]fixed-quadratic-packet interpretation completes this restricted arithmetic interpretation and its conditional completion. The packet is defined by one fixed quotient. It is not the collection of all quadratic characters of G_ Q. No local ramification criterion, Hodge--Tate weight, Weil--Deligne attachment, or cross-coefficient local invariant was established by the hyperref[book:x-construction-21]fixed-quadratic-packet interpretation. The full manuscript still…
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G_{\Q}
H'Introduction passage · 3 passages
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Introduction passage · monograph_atlas.tex, line 712
positive spectral values. At integer time n≥0, T^n=diag(1,2^-n,4^-n), P_0=diag(1,0,0), normT^n-P_0=2^-n. The gap Δ=log2 determines the rate e^-nΔ exactly in this example. xfigtransfer_ground headStrict positivity is an essential finite hypothesis For T'=diag(1,0), the operator is positive semidefinite, but there is no finite self-adjoint H' on all of C^2 with T'=e^-H': every finite eigenvalue of H' exponentiates to a strictly positive value. One may restrict to the positive support, but that cha…
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H'
H^1Introduction passage · 30 passages
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Introduction passage · Paper_XI_Traditional_Final.tex, line 2159
For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i…
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\begin{array}{c|c|c} & p\text{ odd} & p=2\\ \hline \dim_{\FF_p}(J^1/H^1) & p^d-1 & 2^d-2\\ \Swan(\pi)&p^{d-1}(p+1)h_d&3\cdot2^{d-1}h_d \end{array}
H_0Introduction passage · 22 passages
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Introduction passage · Paper_V_Traditional_Final.tex, line 622
Write H_0,H_1 for the literal low and detail operators. The preceding energy identity implies H_bx _2≤ x _2, so both have bounded Hilbert adjoints. At depth zero the only packet operator is the identity. If the energy identity holds at depth ℓ, apply the one-level identity to each W_ℓ,αx and sum. This proves it at depth ℓ+1.
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H_0,H_1
H_1Introduction passage · 23 passages
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Introduction passage · Paper_V_Traditional_Final.tex, line 622
Write H_0,H_1 for the literal low and detail operators. The preceding energy identity implies H_bx _2≤ x _2, so both have bounded Hilbert adjoints. At depth zero the only packet operator is the identity. If the energy identity holds at depth ℓ, apply the one-level identity to each W_ℓ,αx and sum. This proves it at depth ℓ+1.
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H_0,H_1
H_2Introduction passage · 18 passages
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Introduction passage · 72_recovered.tex, line 72
The odd and dyadic Euler formulas share a common reconstruction framework. For each prime p we define an even source series H_p and an inverse kernel K_p satisfying K_p(t)H_p(t)=secht. At an odd prime, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). At the dyadic prime, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t). The coefficient convolution is identical in form, but the kernels have different integral weights. This is the correct sense in which the local theories unify.
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H_2(t)=\frac{\sinh(2t)\sinh t}{2t^2}, \qquad K_2(t)=\frac{4t^2}{\sinh^2(2t)}.
H_{2N}Introduction passage · 3 passages
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Introduction passage · 02_supplement_04.tex, line 129
1 · -1, and the full family is defined by H_1=[1] and H_2N=H_2⊗ H_N =H_N · H_N H_N · -H_N. The two-point orthogonality identity is checked explicitly. Kronecker multiplication and induction then prove H_NH_N^T=NI, N=2^k. The ordinary transpose is sufficient because all constructed entries are real, even though the scalar field is C.
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H_{2N}=H_2\otimes H_N =\begin{pmatrix}H_N&H_N\\H_N&-H_N\end{pmatrix}.
H_bIntroduction passage · 1 passage
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Introduction passage · Paper_V_Traditional_Final.tex, line 622
Write H_0,H_1 for the literal low and detail operators. The preceding energy identity implies H_bx _2≤ x _2, so both have bounded Hilbert adjoints. At depth zero the only packet operator is the identity. If the energy identity holds at depth ℓ, apply the one-level identity to each W_ℓ,αx and sum. This proves it at depth ℓ+1.
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\|H_bx\|_2\le\|x\|_2
h_dIntroduction passage · 10 passages
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Introduction passage · Paper_XI_Traditional_Final.tex, line 2159
For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i…
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\begin{array}{c|c|c} & p\text{ odd} & p=2\\ \hline \dim_{\FF_p}(J^1/H^1) & p^d-1 & 2^d-2\\ \Swan(\pi)&p^{d-1}(p+1)h_d&3\cdot2^{d-1}h_d \end{array}
h_iIntroduction passage · 16 passages
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Introduction passage · Paper_XI_Traditional_Final.tex, line 2171
i) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i=1^d(p^2i-p^2i-2)h_i · · (p odd), a(π^vee×π) · =N^2-1+∑_i=1^d3·4^i-1h_i · · (p=2). These identities do not…
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\begin{aligned} a(\pi^\vee\times\pi) &=p^{2d}-1+\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i &&(p\text{ odd}),\\ a(\pi^\vee\times\pi) &=N^2-1+\sum_{i=1}^d3\cdot4^{i-1}h_i &&(p=2). \end{aligned}
\mathcal H_{\mathrm{int}}Introduction passage · 2 passages
Search aliases: H_int mathcal mathrm
Introduction passage · supplement_11.tex, line 78
Write H_int for the subring of Q_2[[t]] whose factorial-scaled coefficients belong to Z_2. Flatness means B_4^sF∈ H_int for every s. The proof of multiplicative and reciprocal closure is induction on s, using the full product correction. It separately proves that the actual reciprocal has integral Hurwitz coefficients when F(0)=1.
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\mathcal H_{\mathrm{int}}
h_nIntroduction passage · 11 passages
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Introduction passage · Paper_IV_Traditional_Final.tex, line 2371
[Complete symmetric function form] For a positive integer r, let each eigenvalue λ^-1 of K_N be repeated r times. If h_n denotes the complete homogeneous symmetric function of degree n in this absolutely summable multiset, then fracP_N,n(r)(Nn)!=h_n. For arbitrary complex r, the Bell polynomial formula is the polynomial continuation obtained by multiplying every power sum by r.
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h_n
H_pIntroduction passage · 10 passages
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Introduction passage · 72_recovered.tex, line 61
The odd and dyadic Euler formulas share a common reconstruction framework. For each prime p we define an even source series H_p and an inverse kernel K_p satisfying K_p(t)H_p(t)=secht. At an odd prime, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). At the dyadic prime, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t). The coefficient convolution is identical in form, but the kernels have different integral weights. This is the correct sense in which the local theories unify.
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H_p
H_{\rm phase}Introduction passage · 1 passage
Search aliases: H_rm phase rm
Introduction passage · visual_atlas.tex, line 487
If w denotes one delay on a phase sequence, the filter becomes y_rm even y_rm odd = underbrace1+w · 2w 2 · 1+w_H_rm phase(w) e o. The cross-phase delay is not optional. A unit shift sends an odd sample to the next even position, while it sends an even sample to the odd position in the same phase block.
Exact mathematical passage
\begin{pmatrix}y_{\rm even}\\y_{\rm odd}\end{pmatrix} = \underbrace{\begin{pmatrix}1+w&2w\\2&1+w\end{pmatrix}}_{H_{\rm phase}(w)} \begin{pmatrix}e\\o\end{pmatrix}.
\hlineIntroduction passage · 5 passages
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Introduction passage · Paper_XI_Traditional_Final.tex, line 2159
For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i…
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\begin{array}{c|c|c} & p\text{ odd} & p=2\\ \hline \dim_{\FF_p}(J^1/H^1) & p^d-1 & 2^d-2\\ \Swan(\pi)&p^{d-1}(p+1)h_d&3\cdot2^{d-1}h_d \end{array}
I_0Introduction passage · 9 passages
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Introduction passage · 03_packets_and_graphs.tex, line 997
For s∈ S, write s=∑_α∈ I_0P_α s. If β is unreachable from I_0, then for every A∈ A, P_β As = ∑_α∈ I_0P_β AP_α s =0 by [source reference]. The same holds for every finite linear combination of orbit vectors.
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s=\sum_{\alpha\in I_0}P_\alpha s
I_AIntroduction passage · 4 passages
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Introduction passage · 03_supplement_05.tex, line 146
An arbitrary finite reference system. For any finite reference index set A, the amplified channel is defined using the actual Kronecker Kraus operators, widetilde N_E(ρ) =∑_a(I_A⊗ E_a)ρ(I_A⊗ E_a)^*. Its reference blocks are proved equal to the original channel applied blockwise. If (I_A⊗ P)ρ(I_A⊗ P)=ρ, then every reference block is code supported. Applying the checked correction theorem for operators to those blocks gives widetilde R widetilde N_E(ρ)=ρ. This includes entangled density matrices s…
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\widetilde{\mathcal N}_E(\rho) =\sum_a(I_A\otimes E_a)\rho(I_A\otimes E_a)^*.
I_iIntroduction passage · 7 passages
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Introduction passage · Paper_XII_Traditional_Final.tex, line 274
Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient topology on S/L is precisely this image ideal topology.
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\fraka=\frakm_{\OO}S+\sum_i\frakm_{R_i}S, \qquad L=\sum_i I_iS.
I_{m_i}Introduction passage · 3 passages
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Introduction passage · 03_supplement_03.tex, line 31
Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feasible seed counts. It proves both attainment and minimality. For an empty block family the finite supremum is defined as 0, matching the zero module.
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\mathcal A=\bigoplus_i\bigl(M_{d_i}(\C)\otimes I_{m_i}\bigr),\qquad d_i>0,
I_NIntroduction passage · 19 passages
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Introduction passage · Paper_V_Traditional_Final.tex, line 498
[Perfect-reconstruction criterion] For d∈ Z, such a bank reconstructs every input up to a delay of d channel indices exactly when the following Laurent-polynomial identity holds: R(z) E(z)=z^-dI_N. If the right side is c z^-dI_N with c≠0, reconstruction holds up to gain c and channel delay d. After interleaving, this is a delay of Nd original samples. A negative delay denotes an advance.
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\boxed{ \bm R(z)\bm E(z)=z^{-d}I_N. }
\mathrm{int}Introduction passage · 2 passages
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Introduction passage · supplement_11.tex, line 78
Write H_int for the subring of Q_2[[t]] whose factorial-scaled coefficients belong to Z_2. Flatness means B_4^sF∈ H_int for every s. The proof of multiplicative and reciprocal closure is induction on s, using the full product correction. It separately proves that the actual reciprocal has integral Hurwitz coefficients when F(0)=1.
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\mathcal H_{\mathrm{int}}
\ipIntroduction passage · 24 passages
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Introduction passage · Paper_VI_Traditional_Final.tex, line 3020
Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.
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\ip{f}{g}_\pi = \sum_i\pi_if_i\overline{g_i}.
\IrrIntroduction passage · 2 passages
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Introduction passage · 02_foundations.tex, line 886
[Primary decomposition for polynomial modules] Every module X over 𝕜[t] with finite dimension over the coefficient field decomposes as X=⊕_p∈Irr(𝕜[t])X_p, where Irr(𝕜[t]) denotes the monic irreducible polynomials, only finitely many components are nonzero, and every nonzero X_p is a finite direct sum of modules 𝕜[t]/(p^e).
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X=\bigoplus_{p\in\Irr(\bbk[t])}X_p,
J^{-1}Introduction passage · 30 passages
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Introduction passage · continuation_11.tex, line 34
bseteq K^k, V⊆ K^e such that t_0∈ U, y_0∈ V, ψ(t_0)=y_0, ψ(U)⊆ V. The map is analytic at every point of U, and its graph describes all roots in the product neighborhood: ∀ t∈ U ∀ y∈ V, F(y,t)=0 Longleftrightarrow y=ψ(t). At the base parameter it has the strict Fréchet derivative Dψ(t_0)=-J^-1K. Consequently it is differentiable on U. The map is defined on the ambient parameter space, but its root, analyticity and uniqueness conclusions are restricted to the specified neighborhood. Uniqueness is …
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\boxed{D\psi(t_0)=-J^{-1}K.}
J^1Introduction passage · 21 passages
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Introduction passage · Paper_XI_Traditional_Final.tex, line 2159
For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i…
Exact mathematical passage
\begin{array}{c|c|c} & p\text{ odd} & p=2\\ \hline \dim_{\FF_p}(J^1/H^1) & p^d-1 & 2^d-2\\ \Swan(\pi)&p^{d-1}(p+1)h_d&3\cdot2^{d-1}h_d \end{array}
J_FIntroduction passage · 3 passages
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Introduction passage · 72_recovered.tex, line 2652
rrow Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod p. Let J_F(x_L) denote the Jacobian matrix and write overline J_F(x_L) for its reduction modulo p.
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J_F(x_L)
J_xIntroduction passage · 40 passages
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Introduction passage · continuation_01.tex, line 20
Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer modules. It factors through reduction to R_q^d. The cokernel below is presented directly using its integer-domain image.
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r_x=F(x)/q^L\in\mathbb Z^e, \qquad A_x=DF(x):\mathbb Z^d\longrightarrow\mathbb Z^e, \qquad J_x=\rho_q\circ A_x.
J_yIntroduction passage · 6 passages
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Introduction passage · Paper_V_Traditional_Final.tex, line 923
Let y denote the observed data. For a complete pruning Λ, consider any finite criterion J_y(Λ) = R_y(Λ) +η L_y(Λ) +τ K(Λ), where R_y is a representation cost, L_y is a cross-packet leakage cost, and K is a tree-complexity penalty. Here R_y,L_y,K take finite real values, and η,τ are fixed real weights (usually nonnegative in applications). No additivity is needed for the first theorem.
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J_y(\Lambda) = R_y(\Lambda) +\eta L_y(\Lambda) +\tau K(\Lambda),
\mathbb KIntroduction passage · 23 passages
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Introduction passage · 40_regularity.tex, line 10
Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives are the derivatives of the evaluation map.
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\mathbb K
\mathbb K^eIntroduction passage · 8 passages
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Introduction passage · 40_regularity.tex, line 11
Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives are the derivatives of the evaluation map.
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(Y,T)\in\mathbb K^e\times\mathbb K^k
\mathbb K^kIntroduction passage · 9 passages
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Introduction passage · 40_regularity.tex, line 11
Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives are the derivatives of the evaluation map.
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(Y,T)\in\mathbb K^e\times\mathbb K^k
K_2Introduction passage · 12 passages
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Introduction passage · 72_recovered.tex, line 72
The odd and dyadic Euler formulas share a common reconstruction framework. For each prime p we define an even source series H_p and an inverse kernel K_p satisfying K_p(t)H_p(t)=secht. At an odd prime, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). At the dyadic prime, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t). The coefficient convolution is identical in form, but the kernels have different integral weights. This is the correct sense in which the local theories unify.
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H_2(t)=\frac{\sinh(2t)\sinh t}{2t^2}, \qquad K_2(t)=\frac{4t^2}{\sinh^2(2t)}.
K_NIntroduction passage · 7 passages
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Introduction passage · Paper_IV_Traditional_Final.tex, line 2371
[Complete symmetric function form] For a positive integer r, let each eigenvalue λ^-1 of K_N be repeated r times. If h_n denotes the complete homogeneous symmetric function of degree n in this absolutely summable multiset, then fracP_N,n(r)(Nn)!=h_n. For arbitrary complex r, the Bell polynomial formula is the polynomial continuation obtained by multiplying every power sum by r.
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K_N
K_pIntroduction passage · 10 passages
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Introduction passage · 72_recovered.tex, line 61
The odd and dyadic Euler formulas share a common reconstruction framework. For each prime p we define an even source series H_p and an inverse kernel K_p satisfying K_p(t)H_p(t)=secht. At an odd prime, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). At the dyadic prime, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t). The coefficient convolution is identical in form, but the kernels have different integral weights. This is the correct sense in which the local theories unify.
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K_p
\kerIntroduction passage · 73 passages
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Introduction passage · 30_lifting.tex, line 164
The next obstruction concerns compatibility between individually attainable corrections. Its algebraic form works over an arbitrary commutative ring. Let U,V,W be R-modules and let J:U→ V, K:U→ W be linear. Define C(J,K)=U/(ker J+ker K). An attainable value Ju has label [u]∈ C(J,K). This is independent of the choice of its preimage because ker J is killed in the quotient. Similarly, Kv has label [v]. Denote these maps by λ_J:im J→ C(J,K) and λ_K:im K→ C(J,K).
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\label{eq:common-quotient} C(J,K)=U/(\ker J+\ker K).
\ketIntroduction passage · 70 passages
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Introduction passage · 01_finite_core.tex, line 157
[One seed for the computational basis] For an integer n≥0, let ψ_0=ket0^⊗ n and let X_j denote the Pauli X operator on qubit j. Then { X_1^a_1⋯ X_n^a_nψ_0: (a_1,…,a_n)∈{0,1}^n } is the complete computational basis. Hence the seed is cyclic under the algebra generated by the X_j.
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\psi_0=\ket{0}^{\otimes n}
\mathcal LIntroduction passage · 62 passages
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Introduction passage · continuation_08.tex, line 47
Every nonempty L_v has a checked free and transitive action of ker J. The action is defined without choosing a base lift. Choosing one valid solved digit then gives an explicit bijection L_v≅ker J. In particular, for every positive modulus, o_x(v)=0 Longrightarrow # L_v=#ker J. These results permit composite q, the degenerate modulus q=1, and zero dimensions. Neither J nor the full reduced Jacobian is assumed surjective.
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\mathcal L_v
L_NIntroduction passage · 10 passages
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Introduction passage · 72_recovered.tex, line 2892
[Fibonacci residue packets] Let F_0=0, F_1=1, and F_n+2=F_n+1+F_n. Let L_N denote the Lucas number. For every N≥1 and every address j, F_N(k+2)+j = L_NF_N(k+1)+j -(-1)^NF_Nk+j.
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L_N
\mathcal L_vIntroduction passage · 9 passages
Search aliases: L_v mathcal
Introduction passage · continuation_08.tex, line 47
Every nonempty L_v has a checked free and transitive action of ker J. The action is defined without choosing a base lift. Choosing one valid solved digit then gives an explicit bijection L_v≅ker J. In particular, for every positive modulus, o_x(v)=0 Longrightarrow # L_v=#ker J. These results permit composite q, the degenerate modulus q=1, and zero dimensions. Neither J nor the full reduced Jacobian is assumed surjective.
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\mathcal L_v
L_yIntroduction passage · 8 passages
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Introduction passage · Paper_V_Traditional_Final.tex, line 923
Let y denote the observed data. For a complete pruning Λ, consider any finite criterion J_y(Λ) = R_y(Λ) +η L_y(Λ) +τ K(Λ), where R_y is a representation cost, L_y is a cross-packet leakage cost, and K is a tree-complexity penalty. Here R_y,L_y,K take finite real values, and η,τ are fixed real weights (usually nonnegative in applications). No additivity is needed for the first theorem.
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J_y(\Lambda) = R_y(\Lambda) +\eta L_y(\Lambda) +\tau K(\Lambda),
\lceilIntroduction passage · 23 passages
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Introduction passage · 03_supplement_03.tex, line 35
Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feasible seed counts. It proves both attainment and minimality. For an empty block family the finite supremum is defined as 0, matching the zero module.
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\boxed{\srank_{\mathcal A}(\mathcal H)=\max_i\left\lceil\frac{m_i}{d_i}\right\rceil.}
\LongrightarrowIntroduction passage · 18 passages
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Introduction passage · continuation_08.tex, line 51
Every nonempty L_v has a checked free and transitive action of ker J. The action is defined without choosing a base lift. Choosing one valid solved digit then gives an explicit bijection L_v≅ker J. In particular, for every positive modulus, o_x(v)=0 Longrightarrow # L_v=#ker J. These results permit composite q, the degenerate modulus q=1, and zero dimensions. Neither J nor the full reduced Jacobian is assumed surjective.
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o_x(v)=0\quad\Longrightarrow\quad\#\mathcal L_v=\#\ker J.
M_0Introduction passage · 1 passage
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Introduction passage · 02_supplement_04.tex, line 91
The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;x_j,·)_u. Induction, the Kronecker entry formula, and finite-sum distributivity prove Apply_k(A;x)=M_k(A)x. Thus correctness is not inferred merely fr…
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M_0=[1],\qquad M_{k+1}=A_0\otimes M_k(A_1,\ldots,A_k).
M_{d_i}Introduction passage · 8 passages
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Introduction passage · 02_foundations.tex, line 740
[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]
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A=\prod_{i=1}^{q}M_{d_i}(D_i)
M_eIntroduction passage · 7 passages
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Introduction passage · 40_regularity.tex, line 17
Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives are the derivatives of the evaluation map.
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J=D_YF(a)\in M_e(\mathbb K),\qquad K=D_TF(a)\in M_{e\times k}(\mathbb K). \label{eq:regularity-blocks}
M_{e\times k}Introduction passage · 4 passages
Search aliases: M_e× k times
Introduction passage · 40_regularity.tex, line 17
Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives are the derivatives of the evaluation map.
Exact mathematical passage
J=D_YF(a)\in M_e(\mathbb K),\qquad K=D_TF(a)\in M_{e\times k}(\mathbb K). \label{eq:regularity-blocks}
m_iIntroduction passage · 50 passages
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Introduction passage · 02_foundations.tex, line 543
The coordinate projections give the lower bound. Let r be the maximum on the right, regarded as a cardinal, and choose an index set I of cardinality r. Choose minimal generating families for M and N, index them by subsets of I, and fill any unused positions with zeros. Write the resulting families as (m_i)_i∈ I and (n_i)_i∈ I. For any m∈ M and n∈ N, their finite expressions involve a common finite subset F⊆ I after zero coefficients have been added. Thus m=∑_i∈ Fa_im_i, n=∑_i∈ Fb_in_i, and (m,n)…
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(m_i)_{i\in I}
M_kIntroduction passage · 5 passages
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Introduction passage · 02_supplement_04.tex, line 91
The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;x_j,·)_u. Induction, the Kronecker entry formula, and finite-sum distributivity prove Apply_k(A;x)=M_k(A)x. Thus correctness is not inferred merely fr…
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M_0=[1],\qquad M_{k+1}=A_0\otimes M_k(A_1,\ldots,A_k).
M_{k+1}Introduction passage · 1 passage
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Introduction passage · 02_supplement_04.tex, line 91
The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;x_j,·)_u. Induction, the Kronecker entry formula, and finite-sum distributivity prove Apply_k(A;x)=M_k(A)x. Thus correctness is not inferred merely fr…
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M_0=[1],\qquad M_{k+1}=A_0\otimes M_k(A_1,\ldots,A_k).
M_nIntroduction passage · 6 passages
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Introduction passage · Paper_X_Block27_Amended.tex, line 2109
To prove this, apply the preceding averaging correction with both representations equal to ρ. This produces an exact equivariant matrix A with the prescribed reduction at depth k, so that A^2-A∈(π)^kM_n(R). Averaging alone need not preserve idempotence. Use the integral polynomial f(A)=3A^2-2A^3. The identities f(A)-A=(1-2A)(A^2-A), f(A)^2-f(A)=(A^2-A)^2(4A^2-4A-3) show that successive corrections preserve the retained reduction and improve the idempotence defect. In these formulas scalar consta…
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A^2-A\in(\pi)^kM_n(R)
M_{r,d}Introduction passage · 1 passage
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Introduction passage · 72_recovered.tex, line 2512
Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.
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M\in M_{r,d}(\Zp), \qquad b\in\Zp^r.
M_yIntroduction passage · 6 passages
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Introduction passage · visual_atlas.tex, line 425
headMultiplication by y transfers between the lanes In the ordered coefficient coordinates (a,b), M_y=mat0 · x^3-x 1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.
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M_y=\mat{0&x^3-x\\1&0},\qquad M_y^2=(x^3-x)I.
M_y^2Introduction passage · 2 passages
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Introduction passage · visual_atlas.tex, line 425
headMultiplication by y transfers between the lanes In the ordered coefficient coordinates (a,b), M_y=mat0 · x^3-x 1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.
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M_y=\mat{0&x^3-x\\1&0},\qquad M_y^2=(x^3-x)I.
M_\betaIntroduction passage · 3 passages
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Introduction passage · Paper_VI_Traditional_Final.tex, line 4579
ginproof If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every integer n≥0, after increasing M_β to cover the finitely many initial values. Write t=n+r, where 0≤ r<1. Then norm e^tB≤ M_βexp(norm B+max(0,-β)) e^…
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\norm{\e^{nB}}\leq M_\beta\e^{\beta n}
\matIntroduction passage · 77 passages
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Introduction passage · visual_atlas.tex, line 425
headMultiplication by y transfers between the lanes In the ordered coefficient coordinates (a,b), M_y=mat0 · x^3-x 1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.
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M_y=\mat{0&x^3-x\\1&0},\qquad M_y^2=(x^3-x)I.
\mathscrIntroduction passage · 24 passages
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Introduction passage · root_scope.tex, line 91
Let Z_2⟨ x⟩ denote the restricted power series: their integral coefficients tend to zero dyadically. Extend A by mathscr A(x)=frac9^x+1-18(x+1)(2x+1), mathscr A(-1)=-fraclog98. The apparent singularity at x=-1 is removable. Since v_2(log9)=3, its convergent expansion is mathscr A(x)=fraclog9/81+2x ∑_h≥0frac(log9)^h(h+1)!(x+1)^h. It follows coefficientwise that mathscr A∈ Z_2⟨ x⟩^×, mathscr A≡1pmod2, and mathscr A(N)=A(N).
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\mathscr A(x)=\frac{9^{x+1}-1}{8(x+1)(2x+1)}, \qquad \mathscr A(-1)=-\frac{\log9}{8}.
\max_iIntroduction passage · 8 passages
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Introduction passage · 03_supplement_03.tex, line 35
Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feasible seed counts. It proves both attainment and minimality. For an empty block family the finite supremum is defined as 0, matching the zero module.
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\boxed{\srank_{\mathcal A}(\mathcal H)=\max_i\left\lceil\frac{m_i}{d_i}\right\rceil.}
\max_{u\to v}Introduction passage · 1 passage
Search aliases: max_u→ v max to
Introduction passage · 02_supplement_05.tex, line 16
r every edge u→ v, the proof establishes Anc(u)subsetneqAnc(v). The inclusion follows by appending the edge. It is strict because u is an ancestor of v but cannot be its own strict ancestor in an acyclic graph. Cardinality therefore decreases when following predecessors. This constructs a well-founded relation on which the level recursion is defined: ℓ(v)=1+max_u→ vℓ(u), max∅=0.
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\boxed{\ell(v)=1+\max_{u\to v}\ell(u),\qquad \max\varnothing=0.}
\McalIntroduction passage · 4 passages
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Introduction passage · 02_strengthened.tex, line 1485
[Correctable observable algebras] Let Φ be a finite completely positive trace preserving channel with Kraus operators (E_a). A finite dimensional unital star subalgebra M⊆ PEnd( H)P is correctable when a completely positive trace preserving recovery R satisfies P(Φ^*∘ R^*)(X)P=X for every X∈ M, where stars on maps denote Heisenberg adjoints. This holds exactly when every compressed product PE_a^*E_bP belongs to the commutant M' on the code. The scalar code criterion is the special case M=PEnd( H…
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\Mcal\subseteq P\End(\Hcal)P
\Mcal'Introduction passage · 1 passage
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Introduction passage · 02_strengthened.tex, line 1485
hannel with Kraus operators (E_a). A finite dimensional unital star subalgebra M⊆ PEnd( H)P is correctable when a completely positive trace preserving recovery R satisfies P(Φ^*∘ R^*)(X)P=X for every X∈ M, where stars on maps denote Heisenberg adjoints. This holds exactly when every compressed product PE_a^*E_bP belongs to the commutant M' on the code. The scalar code criterion is the special case M=PEnd( H)P [source reference].
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\Mcal'
N^2Introduction passage · 16 passages
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Introduction passage · Paper_XI_Traditional_Final.tex, line 2171
i) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i=1^d(p^2i-p^2i-2)h_i · · (p odd), a(π^vee×π) · =N^2-1+∑_i=1^d3·4^i-1h_i · · (p=2). These identities do not…
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\begin{aligned} a(\pi^\vee\times\pi) &=p^{2d}-1+\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i &&(p\text{ odd}),\\ a(\pi^\vee\times\pi) &=N^2-1+\sum_{i=1}^d3\cdot4^{i-1}h_i &&(p=2). \end{aligned}
n_iIntroduction passage · 6 passages
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Introduction passage · 02_foundations.tex, line 543
The coordinate projections give the lower bound. Let r be the maximum on the right, regarded as a cardinal, and choose an index set I of cardinality r. Choose minimal generating families for M and N, index them by subsets of I, and fill any unused positions with zeros. Write the resulting families as (m_i)_i∈ I and (n_i)_i∈ I. For any m∈ M and n∈ N, their finite expressions involve a common finite subset F⊆ I after zero coefficients have been added. Thus m=∑_i∈ Fa_im_i, n=∑_i∈ Fb_in_i, and (m,n)…
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(n_i)_{i\in I}
\NmIntroduction passage · 17 passages
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Introduction passage · residue_continuation_body.tex, line 21
Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding counting block. Its norm function scales by t^d under a positive real dilation t. The full-paper ledger remains 17 checked claims and 3 partial claims.
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\Nm I=\#(\OO/I)
\Nm_{K/\mathbb Q}Introduction passage · 1 passage
Search aliases: Nm_K/ Q N_K/ Q Nm mathbb operatorname mathbb
Introduction passage · residue_continuation_body.tex, line 22
Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding counting block. Its norm function scales by t^d under a positive real dilation t. The full-paper ledger remains 17 checked claims and 3 partial claims.
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n(x)=\Nm_{K/\mathbb Q}(x)\in\ZZ
\mathcal OIntroduction passage · 29 passages
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Introduction passage · Paper_X_Block27_Amended.tex, line 1257
General finite-field Frobenius comparison. Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a nonzero maximal ideal above (p), and the induced residue map is injective.
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\mathcal O_L\to\mathcal A
oIntroduction passage · 89 passages
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Introduction passage · visual_atlas.tex, line 487
If w denotes one delay on a phase sequence, the filter becomes y_rm even y_rm odd = underbrace1+w · 2w 2 · 1+w_H_rm phase(w) e o. The cross-phase delay is not optional. A unit shift sends an odd sample to the next even position, while it sends an even sample to the odd position in the same phase block.
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\begin{pmatrix}y_{\rm even}\\y_{\rm odd}\end{pmatrix} = \underbrace{\begin{pmatrix}1+w&2w\\2&1+w\end{pmatrix}}_{H_{\rm phase}(w)} \begin{pmatrix}e\\o\end{pmatrix}.
\mathcal O_LIntroduction passage · 7 passages
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Introduction passage · Paper_X_Block27_Amended.tex, line 1257
General finite-field Frobenius comparison. Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a nonzero maximal ideal above (p), and the induced residue map is injective.
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\mathcal O_L\to\mathcal A
o_xIntroduction passage · 16 passages
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Introduction passage · continuation_08.tex, line 51
Every nonempty L_v has a checked free and transitive action of ker J. The action is defined without choosing a base lift. Choosing one valid solved digit then gives an explicit bijection L_v≅ker J. In particular, for every positive modulus, o_x(v)=0 Longrightarrow # L_v=#ker J. These results permit composite q, the degenerate modulus q=1, and zero dimensions. Neither J nor the full reduced Jacobian is assumed surjective.
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o_x(v)=0\quad\Longrightarrow\quad\#\mathcal L_v=\#\ker J.
\one_{[\tau,\infty)}Introduction passage · 1 passage
Search aliases: one_[τ,∞) 1_[τ,∞) one tau infty mathbf tau infty
Introduction passage · Paper_V_Traditional_Final.tex, line 3186
The adjacent difference detector is local in the sample domain. Wavelet coefficients and higher differences give related multiscale packets. In the Haar subsections the signal is defined on R. Here the step convention is H_τ(t)= 1_[τ,∞)(t), whose restriction to (0,1) agrees with the preceding interval step. A chosen extension beyond the observation interval must use this half-line step convention, and the smooth extension must satisfy the stated Lipschitz hypothesis on every displayed wavelet su…
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H_\tau(t)=\one_{[\tau,\infty)}(t)
\OO_F^\timesIntroduction passage · 1 passage
Search aliases: OO_F^× O_F^× OO times mathcal times
Introduction passage · dyadic_character.tex, line 371
Let omega_F be any smooth quasicharacter trivial on U_F 1. Since O_F^×=U_F^1, it is determined by its value omega_F(t). The actual maximal type extends to K times J. Concretely, (K times J)/J is infinite cyclic, generated by a K-uniformizer. An intertwiner for its conjugation action on eta gives an extension Lambda_0. Denote its central character by omega_0.
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\OO_F^\times=U_F^1
\OO_KIntroduction passage · 2 passages
Search aliases: OO_K O_K OO mathcal
Introduction passage · residue_continuation_body.tex, line 21
Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding counting block. Its norm function scales by t^d under a positive real dilation t. The full-paper ledger remains 17 checked claims and 3 partial claims.
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\OO=\OO_K
\overline JIntroduction passage · 5 passages
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Introduction passage · 72_recovered.tex, line 2652
tbfEditorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod p. Let J_F(x_L) denote the Jacobian matrix and write overline J_F(x_L) for its reduction modulo p.
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\overline J_F(x_L)
\overline J_FIntroduction passage · 5 passages
Search aliases: overline J_F overline
Introduction passage · 72_recovered.tex, line 2652
tbfEditorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod p. Let J_F(x_L) denote the Jacobian matrix and write overline J_F(x_L) for its reduction modulo p.
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\overline J_F(x_L)
\overline MIntroduction passage · 12 passages
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Introduction passage · 72_recovered.tex, line 2529
Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.
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\overline M
\overline{g_i}Introduction passage · 2 passages
Search aliases: overlineg_i overline
Introduction passage · Paper_VI_Traditional_Final.tex, line 3020
Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.
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\ip{f}{g}_\pi = \sum_i\pi_if_i\overline{g_i}.
\overline{\Q}Introduction passage · 7 passages
Search aliases: overlineQ overline Q overline mathbf
Introduction passage · Paper_X_Block27_Amended.tex, line 1255
General finite-field Frobenius comparison. Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a nonzero maximal ideal above (p), and the induced residue map is injective.
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j:L\hookrightarrow\overline{\Q}
\overline{\psi_{i\ell}}Introduction passage · 1 passage
Search aliases: overlineψ_iℓ overline psi ell
Introduction passage · 03_supplement_03.tex, line 70
The actual partial trace and its support. The reduced matrix is defined by tracing the addressed index of the density matrix of the pure state: (ρ_B)_jℓ=∑_iψ_ijoverlineψ_iℓ. For the coefficient convention used here, the exact Gram identity is ρ_B=C^ T(C^ T)^*, rankρ_B=rank C=r(ψ). The transpose is important. With this convention, writing C^*C instead would conjugate the reduced matrix. The Lean source checks the entrywise identity and positivity of the resulting matrix.
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(\rho_B)_{j\ell}=\sum_i\psi_{ij}\overline{\psi_{i\ell}}.
\oversetIntroduction passage · 1 passage
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Introduction passage · 02_supplement_05.tex, line 42
Forward substitution with an actual stage evaluator. textitSource item 6.5; module TriangularSolve Use the graph and levels constructed on the previous page. A vertex v carries an arbitrary vector space X_v over a field. The spaces may have different dimensions. Let D_v:X_voverset∼⟶X_v, T_vu:X_u⟶ X_v be an invertible diagonal map and linear transfer maps. The operator being solved is defined by its actual block action: (Ax)_v=D_vx_v+∑_u→ vT_vux_u. No solution, solve equation or precomputed topol…
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D_v:X_v\overset{\sim}{\longrightarrow}X_v, \qquad T_{vu}:X_u\longrightarrow X_v
\mathfrak PIntroduction passage · 10 passages
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Introduction passage · Paper_X_Block27_Amended.tex, line 1259
:x-construction-27 Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a nonzero maximal ideal above (p), and the induced residue map is injective.
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\mathfrak P\subset\mathcal O_L
\mathfrak pIntroduction passage · 2 passages
Search aliases: p mathfrak
Introduction passage · rank2.tex, line 23
Write R=rad(A), U_A^a=1+R^a for atextgreater=1, and U_K^a=1+ p_K^a. Let theta be a full simple character. Since q is odd and the residue field is F2, the actual maximal simple type group and character groups coincide:
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U_K^a=1+\mathfrak p_K^a
p^{(p-1)x+j}Introduction passage · 2 passages
Search aliases: p^(p-1)x+j
Introduction passage · 72_recovered.tex, line 1881
The analytic function is defined by its unit source series. The expression p^(p-1)x+j in the interpolation identity is evaluated only at nonnegative integers. No analytic function x↦ p^x on all of Z_p is used.
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p^{(p-1)x+j}
P^2Introduction passage · 17 passages
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Introduction passage · 01_finite_core.tex, line 616
textitCurvature. Differentiating P^2=P gives P(dP(X))P=0. Write D_Xs=ds(X). For ambient smooth sections, D_XD_Ys-D_YD_Xs-D_[X,Y]s=0. Expansion of the projected derivatives therefore leaves R(X,Y)s=P(dP(X))D_Ys-P(dP(Y))D_Xs. Since Ps=s, differentiation gives D_Ys=(dP(Y))s+PD_Ys. The terms with a middle P vanish. Thus R(X,Y)s=P(dP(X)dP(Y)-dP(Y)dP(X))P s. This expression is alternating and bilinear in the tangent vectors, and it preserves the range. The ordinary chain rule and the transformation la…
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P^2=P
p^{2d}Introduction passage · 2 passages
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Introduction passage · Paper_XI_Traditional_Final.tex, line 2171
i) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i=1^d(p^2i-p^2i-2)h_i · · (p odd), a(π^vee×π) · =N^2-1+∑_i=1^d3·4^i-1h_i · · (p=2). These identities do not…
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\begin{aligned} a(\pi^\vee\times\pi) &=p^{2d}-1+\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i &&(p\text{ odd}),\\ a(\pi^\vee\times\pi) &=N^2-1+\sum_{i=1}^d3\cdot4^{i-1}h_i &&(p=2). \end{aligned}
p^{2i}Introduction passage · 2 passages
Search aliases: p^2i
Introduction passage · Paper_XI_Traditional_Final.tex, line 2171
i) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i=1^d(p^2i-p^2i-2)h_i · · (p odd), a(π^vee×π) · =N^2-1+∑_i=1^d3·4^i-1h_i · · (p=2). These identities do not…
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\begin{aligned} a(\pi^\vee\times\pi) &=p^{2d}-1+\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i &&(p\text{ odd}),\\ a(\pi^\vee\times\pi) &=N^2-1+\sum_{i=1}^d3\cdot4^{i-1}h_i &&(p=2). \end{aligned}
p^{2i-2}Introduction passage · 2 passages
Search aliases: p^2i-2
Introduction passage · Paper_XI_Traditional_Final.tex, line 2171
i) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i=1^d(p^2i-p^2i-2)h_i · · (p odd), a(π^vee×π) · =N^2-1+∑_i=1^d3·4^i-1h_i · · (p=2). These identities do not…
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\begin{aligned} a(\pi^\vee\times\pi) &=p^{2d}-1+\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i &&(p\text{ odd}),\\ a(\pi^\vee\times\pi) &=N^2-1+\sum_{i=1}^d3\cdot4^{i-1}h_i &&(p=2). \end{aligned}
p^dIntroduction passage · 6 passages
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Introduction passage · Paper_XI_Traditional_Final.tex, line 2159
For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i…
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\begin{array}{c|c|c} & p\text{ odd} & p=2\\ \hline \dim_{\FF_p}(J^1/H^1) & p^d-1 & 2^d-2\\ \Swan(\pi)&p^{d-1}(p+1)h_d&3\cdot2^{d-1}h_d \end{array}
p^{d-1}Introduction passage · 7 passages
Search aliases: p^d-1
Introduction passage · Paper_XI_Traditional_Final.tex, line 2159
For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i…
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\begin{array}{c|c|c} & p\text{ odd} & p=2\\ \hline \dim_{\FF_p}(J^1/H^1) & p^d-1 & 2^d-2\\ \Swan(\pi)&p^{d-1}(p+1)h_d&3\cdot2^{d-1}h_d \end{array}
p^eIntroduction passage · 2 passages
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Introduction passage · 02_foundations.tex, line 889
classicalinput[Primary decomposition for polynomial modules] Every module X over 𝕜[t] with finite dimension over the coefficient field decomposes as X=⊕_p∈Irr(𝕜[t])X_p, where Irr(𝕜[t]) denotes the monic irreducible polynomials, only finitely many components are nonzero, and every nonzero X_p is a finite direct sum of modules 𝕜[t]/(p^e).
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\bbk[t]/(p^e)
p^{\min(k,mn)}Introduction passage · 7 passages
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Introduction passage · 20_arithmetic.tex, line 193
[All-address differences on the divisor branch] Let p be prime, m>0, mmid p-1, N=mp, and 0≤ j<N. Write h=(p-1)/m. For every u,k,n∈ N, U_N,j(n) · ∈ Z_p, Δ_uh^kU_N,j(n) · ∈ p^min(k,mn) Z_p, v_p(Δ_uh^kC_N,j(n)) · ≥min(k,mn)-v_p(j!). The factorial correction is part of the normalized statement, including addresses j≥ p.
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U_{N,j}(n)&\in\Z_p,\label{eq:arith-divisor-integral}\\ \Delta_{uh}^{k}U_{N,j}(n)&\in p^{\min(k,mn)}\Z_p,\label{eq:arith-divisor-bound}\\ v_p\bigl(\Delta_{uh}^{k}C_{N,j}(n)\bigr) &\ge\min(k,mn)-v_p(j!).\label{eq:arith-divisor-normalized}
p^xIntroduction passage · 1 passage
Search aliases: p^x
Introduction passage · 72_recovered.tex, line 1881
The analytic function is defined by its unit source series. The expression p^(p-1)x+j in the interpolation identity is evaluated only at nonnegative integers. No analytic function x↦ p^x on all of Z_p is used.
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x\mapsto p^x
P_\mathcalIntroduction passage · 6 passages
Search aliases: P_ mathcal
Introduction passage · Paper_VI_Traditional_Final.tex, line 3013
Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.
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(P_\mathcal Qf)_i = \frac{\sum_{j\in B(i)}\pi_jf_j} {\sum_{j\in B(i)}\pi_j}.
P_0Introduction passage · 33 passages
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Introduction passage · monograph_atlas.tex, line 705
sfer determines its HamiltonianSTRICT POSITIVITY AND THE GROUND GAPChapter VII: Theorem II.10.2, Proposition II.10.5. On HH= C^3, take the strictly positive contraction T=diag(1,1/2,1/4), H=-log T=diag(0,log2,2log2). At unit Euclidean time, T=e^-H. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integer time n≥0, T^n=diag(1,2^-n,4^-n), P_0=diag(1,0,0), normT^n-P_0=2^-n. The gap Δ=log2 determines the rate e^-nΔ exactly in this example. xfigtransfer_…
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T^n=\diag(1,2^{-n},4^{-n}),\qquad P_0=\diag(1,0,0),\qquad\norm{T^n-P_0}=2^{-n}.
P_{2,0}Introduction passage · 4 passages
Search aliases: P_2,0
Introduction passage · atlas-pages.tex, line 56
vpage4Refine packets on a common latticeEXACT REFINEMENT / WHAT CAN BE ADDEDChapter V: Theorem 2.3. An even sample belongs to exactly one of residues 0 and 2 modulo four. Therefore the coarse even mask is the sum of two disjoint fine masks: P_2,0=P_4,0+P_4,2. vfigrefinement platecaptionThe common horizontal axis is the original sample index. Empty cells denote masked-out addresses. There are no interpolated samples between the displayed cells. headWhy the coordinate convention matters The decima…
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P_{2,0}=P_{4,0}+P_{4,2}.
P_{4,0}Introduction passage · 2 passages
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Introduction passage · atlas-pages.tex, line 56
vpage4Refine packets on a common latticeEXACT REFINEMENT / WHAT CAN BE ADDEDChapter V: Theorem 2.3. An even sample belongs to exactly one of residues 0 and 2 modulo four. Therefore the coarse even mask is the sum of two disjoint fine masks: P_2,0=P_4,0+P_4,2. vfigrefinement platecaptionThe common horizontal axis is the original sample index. Empty cells denote masked-out addresses. There are no interpolated samples between the displayed cells. headWhy the coordinate convention matters The decima…
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P_{2,0}=P_{4,0}+P_{4,2}.
P_{4,2}Introduction passage · 5 passages
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Introduction passage · atlas-pages.tex, line 56
vpage4Refine packets on a common latticeEXACT REFINEMENT / WHAT CAN BE ADDEDChapter V: Theorem 2.3. An even sample belongs to exactly one of residues 0 and 2 modulo four. Therefore the coarse even mask is the sum of two disjoint fine masks: P_2,0=P_4,0+P_4,2. vfigrefinement platecaptionThe common horizontal axis is the original sample index. Empty cells denote masked-out addresses. There are no interpolated samples between the displayed cells. headWhy the coordinate convention matters The decima…
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P_{2,0}=P_{4,0}+P_{4,2}.
p_kIntroduction passage · 3 passages
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Introduction passage · Paper_IV_Traditional_Final.tex, line 1162
Let D_n denote the displayed determinant, with D_0=1. Expansion along the last row gives D_n = ∑_k=1^n frac(n-1)!(n-k)!p_kD_n-k. Here the cofactor sign is canceled by the product of the negative superdiagonal entries -(n-k+1),…,-(n-1). Thus D_n/n! satisfies the same recurrence and initial value as b_N,n(r).
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D_n = \sum_{k=1}^{n} \frac{(n-1)!}{(n-k)!}p_kD_{n-k}.
\mathfrak p_K^aIntroduction passage · 1 passage
Search aliases: p_K^a mathfrak
Introduction passage · rank2.tex, line 23
Write R=rad(A), U_A^a=1+R^a for atextgreater=1, and U_K^a=1+ p_K^a. Let theta be a full simple character. Since q is odd and the residue field is F2, the actual maximal simple type group and character groups coincide:
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U_K^a=1+\mathfrak p_K^a
P_{N,k}Introduction passage · 7 passages
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Introduction passage · Paper_V_Traditional_Final.tex, line 475
[Operator-block form] If H denotes convolution by h, the same identity is H=∑_k,sP_N,kHP_N,s. Thus the familiar polyphase matrix is exactly the transfer-block matrix of the seed calculus.
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H=\sum_{k,s}P_{N,k}HP_{N,s}.
P_{N,s}Introduction passage · 6 passages
Search aliases: P_N,s
Introduction passage · Paper_V_Traditional_Final.tex, line 475
[Operator-block form] If H denotes convolution by h, the same identity is H=∑_k,sP_N,kHP_N,s. Thus the familiar polyphase matrix is exactly the transfer-block matrix of the seed calculus.
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H=\sum_{k,s}P_{N,k}HP_{N,s}.
P_\alpha^*Introduction passage · 7 passages
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Introduction passage · 01_finite_core.tex, line 5
Conventions. Unless the zero space is explicitly included as a boundary case, quantum state spaces are nonzero complex Hilbert spaces of finite dimension. A complete packet family is a finite collection satisfying P_α^*=P_α, P_α P_β=δ_αβP_α, ∑_α P_α=I. All operator norms in the analytic estimates are induced by the norm on the Hilbert space. The estimates do not use the entrywise maximum norm or the Frobenius norm. We write [A,B]=AB-BA and use units with ℏ=1. Parameter derivatives are real deriv…
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P_\alpha^*=P_\alpha,\qquad P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha,\qquad \sum_\alpha P_\alpha=I.
\mathrm{Prop}Introduction passage · 1 passage
Search aliases: Prop mathrm
Introduction passage · residue_continuation_body.tex, line 109
Let [J]=C^-1 in Cl( O), and let G(J) denote the actual fundamental-cone generator set. For every predicate P: R→Prop and every real X, the norm-preserving correspondence gives · #{x∈ G(J): N(x)≤ X N J, P( N(x))}
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P:\RR\to\mathrm{Prop}
\QIntroduction passage · 62 passages
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Introduction passage · 72_recovered.tex, line 2641
Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod p. Let J_F(x_L) denote the Jacobian matrix and write overline J_F(x_L) for its reduction modulo p.
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\Q_p
\mathcal QIntroduction passage · 19 passages
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Introduction passage · Paper_VI_Traditional_Final.tex, line 3012
Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.
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\mathcal Q
Q^*Introduction passage · 9 passages
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Introduction passage · 03_supplement_05.tex, line 76
A channel on the whole ambient space. Put Q=I-∑_r∈ A_+V_rV_r^*. The recovery is R(σ)=∑_r∈ A_+V_r^*σ V_r+Qσ Q. The complement is an orthogonal projector and ∑_r∈ A_+V_rV_r^*+Q^*Q=I. Hence the recovery is trace preserving. Its finite Kraus form also proves positivity after amplification by the identity on every finite auxiliary space. The complement term matters: the map is defined as a channel even outside the syndrome ranges.
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\sum_{r\in A_+}V_rV_r^*+Q^*Q=I.
q^LIntroduction passage · 46 passages
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Introduction passage · continuation_01.tex, line 14
Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer modules. It factors through reduction to R_q^d. The cokernel below is presented directly using its integer-domain image.
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F=(F_1,\ldots,F_e)\in\mathbb Z[X_1,\ldots,X_d]^e, \quad q\ge1,\quad L\ge1, \quad x\in\mathbb Z^d, \quad F(x)\equiv0\pmod{q^L}.
\Q_2Introduction passage · 7 passages
Search aliases: Q_2 Q mathbb Q_2 Q mathbf
Introduction passage · supplement_11.tex, line 78
Write H_int for the subring of Q_2[[t]] whose factorial-scaled coefficients belong to Z_2. Flatness means B_4^sF∈ H_int for every s. The proof of multiplicative and reciprocal closure is induction on s, using the full product correction. It separately proves that the actual reciprocal has integral Hurwitz coefficients when F(0)=1.
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\Q_2[[t]]
\Q_pIntroduction passage · 22 passages
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Introduction passage · 72_recovered.tex, line 2641
Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod p. Let J_F(x_L) denote the Jacobian matrix and write overline J_F(x_L) for its reduction modulo p.
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\Q_p
\QpIntroduction passage · 21 passages
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Introduction passage · 00_scope.tex, line 19
Throughout, N={0,1,2,…}. When p is prime, Z_p denotes the p-adic integers and Q_p their fraction field. We normalize v_p(p)=1. For a rational or p-adic quantity, a displayed lower valuation bound is understood with the mathematical convention v_p(0)=+∞. The corresponding formal predicates are zero-safe: the quantity is zero or its valuation meets the stated bound. An exact finite valuation assertion requires a nonzero quantity, and this is stated where used. Congruence in Z_p means ideal divisib…
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\Qp
\QQIntroduction passage · 53 passages
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Introduction passage · visual_atlas.tex, line 428
1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.
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\QQ[x]
\QQ^8Introduction passage · 1 passage
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Introduction passage · visual_atlas.tex, line 103
vpage5The operator changes coordinates tooMODEL A / COHERENCE AND NATURALITYChapter II, mathematical text: Theorem 3.5, Theorem 3.8 Use a finite periodic carrier X=QQ^8 for this example. Let Se_j=e_j+1bmod8 and let T=I+2S+S^2. This is a concrete cyclic model, not an unannounced boundary convention for the finitely supported signal on Plate II.20. Write R_N for the permutation that groups coordinates in increasing residue order modulo N.
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X=\QQ^8
\bm RIntroduction passage · 6 passages
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Introduction passage · Paper_V_Traditional_Final.tex, line 498
[Perfect-reconstruction criterion] For d∈ Z, such a bank reconstructs every input up to a delay of d channel indices exactly when the following Laurent-polynomial identity holds: R(z) E(z)=z^-dI_N. If the right side is c z^-dI_N with c≠0, reconstruction holds up to gain c and channel delay d. After interleaving, this is a delay of Nd original samples. A negative delay denotes an advance.
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\boxed{ \bm R(z)\bm E(z)=z^{-d}I_N. }
\mathcal RIntroduction passage · 23 passages
Search aliases: R mathcal
Introduction passage · 03_supplement_05.tex, line 72
A channel on the whole ambient space. Put Q=I-∑_r∈ A_+V_rV_r^*. The recovery is R(σ)=∑_r∈ A_+V_r^*σ V_r+Qσ Q. The complement is an orthogonal projector and ∑_r∈ A_+V_rV_r^*+Q^*Q=I. Hence the recovery is trace preserving. Its finite Kraus form also proves positivity after amplification by the identity on every finite auxiliary space. The complement term matters: the map is defined as a channel even outside the syndrome ranges.
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\boxed{\mathcal R(\sigma)=\sum_{r\in A_+}V_r^*\sigma V_r+Q\sigma Q.}
R^aIntroduction passage · 1 passage
Search aliases: R^a
Introduction passage · rank2.tex, line 22
Write R=rad(A), U_A^a=1+R^a for atextgreater=1, and U_K^a=1+ p_K^a. Let theta be a full simple character. Since q is odd and the residue field is F2, the actual maximal simple type group and character groups coincide:
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U_A^a=1+R^a
r_1Introduction passage · 10 passages
Search aliases: r_1
Introduction passage · extended_pages.tex, line 59
(r_1,r_2)=(1,3), the aggregated objectives are J_rm keep=4+2+3·7=27, J_rm split=2+8+3·2=16. The exact shared optimum is split. Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the declared one-pruning problem. headA hand-computable decision threshold Fix r_1=1 and vary r_2=r≥0. Keeping costs 6+7r and splitting costs 10+2r. Splitting is strictly cheaper when r>4/5. Its first integer reu…
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r_1=1
r_2Introduction passage · 6 passages
Search aliases: r_2
Introduction passage · extended_pages.tex, line 59
, the aggregated objectives are J_rm keep=4+2+3·7=27, J_rm split=2+8+3·2=16. The exact shared optimum is split. Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the declared one-pruning problem. headA hand-computable decision threshold Fix r_1=1 and vary r_2=r≥0. Keeping costs 6+7r and splitting costs 10+2r. Splitting is strictly cheaper when r>4/5. Its first integer reuse win is r=1. …
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r_2=r\ge0
R_iIntroduction passage · 6 passages
Search aliases: R_i
Introduction passage · Paper_XII_Traditional_Final.tex, line 273
Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient topology on S/L is precisely this image ideal topology.
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S=\bigotimes_{i=1}^{s}R_i
R_NIntroduction passage · 15 passages
Search aliases: R_N
Introduction passage · visual_atlas.tex, line 107
vpage5The operator changes coordinates tooMODEL A / COHERENCE AND NATURALITYChapter II, mathematical text: Theorem 3.5, Theorem 3.8 Use a finite periodic carrier X=QQ^8 for this example. Let Se_j=e_j+1bmod8 and let T=I+2S+S^2. This is a concrete cyclic model, not an unannounced boundary convention for the finitely supported signal on Plate II.20. Write R_N for the permutation that groups coordinates in increasing residue order modulo N.
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R_N
R_qIntroduction passage · 2 passages
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Introduction passage · continuation_01.tex, line 19
Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer modules. It factors through reduction to R_q^d. The cokernel below is presented directly using its integer-domain image.
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R_q=\mathbb Z/q\mathbb Z
R_q^dIntroduction passage · 1 passage
Search aliases: R_q^d
Introduction passage · continuation_01.tex, line 27
et ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer modules. It factors through reduction to R_q^d. The cokernel below is presented directly using its integer-domain image.
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R_q^d
R_q^eIntroduction passage · 3 passages
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Introduction passage · continuation_01.tex, line 26
F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer modules. It factors through reduction to R_q^d. The cokernel below is presented directly using its integer-domain image.
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R_q^e
r_xIntroduction passage · 12 passages
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Introduction passage · continuation_01.tex, line 20
Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer modules. It factors through reduction to R_q^d. The cokernel below is presented directly using its integer-domain image.
Exact mathematical passage
r_x=F(x)/q^L\in\mathbb Z^e, \qquad A_x=DF(x):\mathbb Z^d\longrightarrow\mathbb Z^e, \qquad J_x=\rho_q\circ A_x.
R_yIntroduction passage · 3 passages
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Introduction passage · Paper_V_Traditional_Final.tex, line 923
Let y denote the observed data. For a complete pruning Λ, consider any finite criterion J_y(Λ) = R_y(Λ) +η L_y(Λ) +τ K(Λ), where R_y is a representation cost, L_y is a cross-packet leakage cost, and K is a tree-complexity penalty. Here R_y,L_y,K take finite real values, and η,τ are fixed real weights (usually nonnegative in applications). No additivity is needed for the first theorem.
Exact mathematical passage
J_y(\Lambda) = R_y(\Lambda) +\eta L_y(\Lambda) +\tau K(\Lambda),
\rankIntroduction passage · 34 passages
Search aliases: rank operatorname
Introduction passage · 03_supplement_03.tex, line 74
The actual partial trace and its support. The reduced matrix is defined by tracing the addressed index of the density matrix of the pure state: (ρ_B)_jℓ=∑_iψ_ijoverlineψ_iℓ. For the coefficient convention used here, the exact Gram identity is ρ_B=C^ T(C^ T)^*, rankρ_B=rank C=r(ψ). The transpose is important. With this convention, writing C^*C instead would conjugate the reduced matrix. The Lean source checks the entrywise identity and positivity of the resulting matrix.
Exact mathematical passage
\rho_B=C^{\mathsf T}(C^{\mathsf T})^*, \qquad \rank\rho_B=\rank C=r(\psi).
\RcalIntroduction passage · 28 passages
Search aliases: Rcal R mathcal
Introduction passage · 02_strengthened.tex, line 1485
[Correctable observable algebras] Let Φ be a finite completely positive trace preserving channel with Kraus operators (E_a). A finite dimensional unital star subalgebra M⊆ PEnd( H)P is correctable when a completely positive trace preserving recovery R satisfies P(Φ^*∘ R^*)(X)P=X for every X∈ M, where stars on maps denote Heisenberg adjoints. This holds exactly when every compressed product PE_a^*E_bP belongs to the commutant M' on the code. The scalar code criterion is the special case M=PEnd( H…
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\Rcal
\Rcal^*Introduction passage · 1 passage
Search aliases: Rcal^* R^* Rcal mathcal
Introduction passage · 02_strengthened.tex, line 1485
[Correctable observable algebras] Let Φ be a finite completely positive trace preserving channel with Kraus operators (E_a). A finite dimensional unital star subalgebra M⊆ PEnd( H)P is correctable when a completely positive trace preserving recovery R satisfies P(Φ^*∘ R^*)(X)P=X for every X∈ M, where stars on maps denote Heisenberg adjoints. This holds exactly when every compressed product PE_a^*E_bP belongs to the commutant M' on the code. The scalar code criterion is the special case M=PEnd( H…
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P(\Phi^*\circ\Rcal^*)(X)P=X
\rceilIntroduction passage · 23 passages
Search aliases: rceil
Introduction passage · 03_supplement_03.tex, line 35
Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feasible seed counts. It proves both attainment and minimality. For an empty block family the finite supremum is defined as 0, matching the zero module.
Exact mathematical passage
\boxed{\srank_{\mathcal A}(\mathcal H)=\max_i\left\lceil\frac{m_i}{d_i}\right\rceil.}
\operatorname{Re}Introduction passage · 21 passages
Search aliases: Re operatorname
Introduction passage · Paper_VI_Traditional_Final.tex, line 4579
If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every integer n≥0, after increasing M_β to cover the finitely many initial values. Write t=n+r, where 0≤ r<1. Then norm e^tB≤ M_βexp(norm B+max(0,-β)) e^β t. Thus…
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s(B)=\max\operatorname{Re}\Spec(B)
S^2Introduction passage · 4 passages
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Introduction passage · visual_atlas.tex, line 104
vpage5The operator changes coordinates tooMODEL A / COHERENCE AND NATURALITYChapter II, mathematical text: Theorem 3.5, Theorem 3.8 Use a finite periodic carrier X=QQ^8 for this example. Let Se_j=e_j+1bmod8 and let T=I+2S+S^2. This is a concrete cyclic model, not an unannounced boundary convention for the finitely supported signal on Plate II.20. Write R_N for the permutation that groups coordinates in increasing residue order modulo N.
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T=I+2S+S^2.
S^nIntroduction passage · 5 passages
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Introduction passage · Paper_V_Traditional_Final.tex, line 270
Let ℓ^2( Z; C), abbreviated ℓ^2( Z), be the Hilbert space of square-summable complex discrete signals. The inner product is conjugate-linear in its first argument. A superscript * on a bounded operator denotes its Hilbert adjoint. Write δ_n for the unit impulse at address n and let the bilateral shift be (Sx)[n]=x[n-1]. Then S^nδ_0=δ_n. The primitive signal seed is therefore s=δ_0, the address operators are the shift words S^n, and the completion is the ℓ^2 norm.
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S^n\delta_0=\delta_n.
S_iIntroduction passage · 1 passage
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Introduction passage · 02_foundations.tex, line 743
[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]
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S_i=D_i^{d_i}
S_i^{\oplus d_i}Introduction passage · 1 passage
Search aliases: S_i^⊕ d_i oplus
Introduction passage · 02_foundations.tex, line 744
[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]
Exact mathematical passage
\boxed{A\cong\bigoplus_{i=1}^{q}S_i^{\oplus d_i}.}
S_nIntroduction passage · 10 passages
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Introduction passage · supplement_02.tex, line 31
rence theorem remain pending. No continuum convergence or analytic summability theorem is inferred from the formal power-series equalities in this block. The historical numerical study is not being relabelled as a fresh experiment. pageheading1. The actual odd-power generating functionLemma 4.1; manuscript page 10 Work in the rational formal power-series ring Q[[t]]. Define S_n(p)=∑_a=0^p-1(-1)^a(2a+1)^n∈ Z, F_p(t)=∑_n≥0S_n(p)fract^nn!. The formal exponential is Mathlib's mcodePowerSeries.exp. S…
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S_n(p)=\sum_{a=0}^{p-1}(-1)^a(2a+1)^n\in\Z, \qquad F_p(t)=\sum_{n\ge0}S_n(p)\frac{t^n}{n!}.
\sechIntroduction passage · 16 passages
Search aliases: sech operatorname
Introduction passage · Paper_IV_Traditional_Final.tex, line 1608
For (N,j)=(2,0), C_2,0(z)=sech z. For (N,j)=(2,1), C_2,1(z)=frac zsinh z. The first case is a scaled and translated Euler family. In fact, if E_m(x) denotes the classical Euler polynomial, then E_m^(2,0,1)(x) = 2^mE_m(fracx+12). If B_m(x) denotes the classical Bernoulli polynomial, the second case has the exact identification E_m^(2,1,1)(x) = 2^mB_m(fracx+12), obtained by substituting t=2z in t e^yt/(e^t-1).
Exact mathematical passage
\Cfun_{2,0}(z)=\sech z.
\sinh^2Introduction passage · 11 passages
Search aliases: sinh^2 sinh
Introduction passage · 72_recovered.tex, line 72
The odd and dyadic Euler formulas share a common reconstruction framework. For each prime p we define an even source series H_p and an inverse kernel K_p satisfying K_p(t)H_p(t)=secht. At an odd prime, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). At the dyadic prime, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t). The coefficient convolution is identical in form, but the kernels have different integral weights. This is the correct sense in which the local theories unify.
Exact mathematical passage
H_2(t)=\frac{\sinh(2t)\sinh t}{2t^2}, \qquad K_2(t)=\frac{4t^2}{\sinh^2(2t)}.
\srank_{\mathcal A}Introduction passage · 1 passage
Search aliases: srank_ A srank mathcal operatorname mathcal
Introduction passage · 03_supplement_03.tex, line 35
Simultaneous generation in a direct sum. For the stipulated block representation A=⊕_i(M_d_i( C)⊗ I_m_i), d_i>0, the proof establishes that k seeds generate all blocks simultaneously exactly when m_i≤ kd_i for every i. It follows that srank_ A( H)=max_ilceilfracm_id_irceil. The final Lean proposition is an codeIsLeast statement over the set of feasible seed counts. It proves both attainment and minimality. For an empty block family the finite supremum is defined as 0, matching the zero module.
Exact mathematical passage
\boxed{\srank_{\mathcal A}(\mathcal H)=\max_i\left\lceil\frac{m_i}{d_i}\right\rceil.}
\subsetneqIntroduction passage · 2 passages
Search aliases: subsetneq
Introduction passage · 02_supplement_05.tex, line 12
For each vertex, define its strict ancestor set Anc(v)={u:there is a positive-length path uleadsto v}. For every edge u→ v, the proof establishes Anc(u)subsetneqAnc(v). The inclusion follows by appending the edge. It is strict because u is an ancestor of v but cannot be its own strict ancestor in an acyclic graph. Cardinality therefore decreases when following predecessors. This constructs a well-founded relation on which the level recursion is defined: ℓ(v)=1+max_u→ vℓ(u), max∅=0.
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\operatorname{Anc}(u)\subsetneq\operatorname{Anc}(v).
\suppIntroduction passage · 6 passages
Search aliases: supp operatorname
Introduction passage · 01_manuscript.tex, line 163
[Block composition] Let A,B∈End(X). Then (AB)_γα = ∑_β∈ I A_γβB_βα. Consequently, supp(AB) ⊆ supp(A)∘supp(B), where relational composition is defined by (γ,α)∈ E∘ F when some β satisfies (γ,β)∈ E, (β,α)∈ F.
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\supp(AB) \subseteq \supp(A)\circ\supp(B),
\Supp_{\TT}Introduction passage · 3 passages
Search aliases: Supp_TT Supp_ T Supp TT operatorname mathbb
Introduction passage · Paper_XII_Traditional_Final.tex, line 1739
eorem supplies an ordinary finite module or a perfect complex with full patched support, together with the coefficient quotient and the module or derived control isomorphism. Write M for the resulting finite level carrier over R. item Comparison gate. Let H be a finite ordinary Hecke module or a perfect Hecke complex over T. Supply a ring map f:R→ T satisfying (f^*)^-1(Supp_R(M))⊆Supp_ T(H). Use ordinary or derived support according to the carrier. Actual scalar extension satisfies this inclusio…
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(f^*)^{-1}\bigl(\Supp_R(M)\bigr)\subseteq\Supp_{\TT}(H).
\SwanIntroduction passage · 4 passages
Search aliases: Swan operatorname
Introduction passage · Paper_XI_Traditional_Final.tex, line 2159
For the same candidates the constructions give c|c|c · p odd · p=2 dim_ F_p(J^1/H^1) · p^d-1 · 2^d-2 Swan(π) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i…
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\begin{array}{c|c|c} & p\text{ odd} & p=2\\ \hline \dim_{\FF_p}(J^1/H^1) & p^d-1 & 2^d-2\\ \Swan(\pi)&p^{d-1}(p+1)h_d&3\cdot2^{d-1}h_d \end{array}
\mathsf TIntroduction passage · 71 passages
Search aliases: T mathsf
Introduction passage · 03_supplement_03.tex, line 74
The actual partial trace and its support. The reduced matrix is defined by tracing the addressed index of the density matrix of the pure state: (ρ_B)_jℓ=∑_iψ_ijoverlineψ_iℓ. For the coefficient convention used here, the exact Gram identity is ρ_B=C^ T(C^ T)^*, rankρ_B=rank C=r(ψ). The transpose is important. With this convention, writing C^*C instead would conjugate the reduced matrix. The Lean source checks the entrywise identity and positivity of the resulting matrix.
Exact mathematical passage
\rho_B=C^{\mathsf T}(C^{\mathsf T})^*, \qquad \rank\rho_B=\rank C=r(\psi).
T'Introduction passage · 8 passages
Search aliases: T'
Introduction passage · monograph_atlas.tex, line 712
lidean time, T=e^-H. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integer time n≥0, T^n=diag(1,2^-n,4^-n), P_0=diag(1,0,0), normT^n-P_0=2^-n. The gap Δ=log2 determines the rate e^-nΔ exactly in this example. xfigtransfer_ground headStrict positivity is an essential finite hypothesis For T'=diag(1,0), the operator is positive semidefinite, but there is no finite self-adjoint H' on all of C^2 with T'=e^-H': every finite eigenvalue of H' exponentia…
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T'=\diag(1,0)
t^2Introduction passage · 50 passages
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Introduction passage · 72_recovered.tex, line 72
The odd and dyadic Euler formulas share a common reconstruction framework. For each prime p we define an even source series H_p and an inverse kernel K_p satisfying K_p(t)H_p(t)=secht. At an odd prime, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). At the dyadic prime, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t). The coefficient convolution is identical in form, but the kernels have different integral weights. This is the correct sense in which the local theories unify.
Exact mathematical passage
H_2(t)=\frac{\sinh(2t)\sinh t}{2t^2}, \qquad K_2(t)=\frac{4t^2}{\sinh^2(2t)}.
T^nIntroduction passage · 14 passages
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Introduction passage · monograph_atlas.tex, line 705
sfer determines its HamiltonianSTRICT POSITIVITY AND THE GROUND GAPChapter VII: Theorem II.10.2, Proposition II.10.5. On HH= C^3, take the strictly positive contraction T=diag(1,1/2,1/4), H=-log T=diag(0,log2,2log2). At unit Euclidean time, T=e^-H. The self-adjoint logarithm is unique because the logarithm acts on positive spectral values. At integer time n≥0, T^n=diag(1,2^-n,4^-n), P_0=diag(1,0,0), normT^n-P_0=2^-n. The gap Δ=log2 determines the rate e^-nΔ exactly in this example. xfigtransfer_…
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T^n=\diag(1,2^{-n},4^{-n}),\qquad P_0=\diag(1,0,0),\qquad\norm{T^n-P_0}=2^{-n}.
t_0Introduction passage · 63 passages
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Introduction passage · 40_regularity.tex, line 12
Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives are the derivatives of the evaluation map.
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F\in\mathbb K[Y_1,\ldots,Y_e,T_1,\ldots,T_k]^e, \qquad a=(y_0,t_0),
T_1Introduction passage · 12 passages
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Introduction passage · 40_regularity.tex, line 12
Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives are the derivatives of the evaluation map.
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F\in\mathbb K[Y_1,\ldots,Y_e,T_1,\ldots,T_k]^e, \qquad a=(y_0,t_0),
T_{C,\chi}Introduction passage · 3 passages
Search aliases: T_C,χ chi
Introduction passage · residue_continuation_body.tex, line 119
· =#μ(K) #{I≠0:[I]=C, N I≤ X, P( N I N J)}. The same norm-preserving equivalence carries arbitrary complex weights. In particular, write G_J,χ(Y) for the sum of χ( N(x)bmod m) over these generators with N(x)≤ Y, and T_C,χ(X) for the corresponding sum over ideals in class C. For multiplicative χ preserving 0 and 1, the exact identity is G_J,χ(X N J) =#μ(K) χ( N Jbmod m) T_C,χ(X). Both signs are included before applying this full roots-of-unity factor. A root of unity can interchange the sign port…
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T_{C,\chi}(X)
T_jIntroduction passage · 14 passages
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Introduction passage · strengthening.tex, line 1085
[Exact image and residue enumeration] Let widetilde C_j denote the dyadic Newton interpolation. The map T_j(x)=fracwidetilde C_j(x)-12^b_j is a bijective isometry Z_2→ Z_2, and widetilde C_j( Z_2)=1+2^b_j Z_2, j!widetilde C_j( Z_2)=j!+2^B_j Z_2. For r>B_j, one least period modulo 2^r visits every residue in j!+2^B_j Z/2^r Z exactly once. For r≤ B_j, there is one residue and the least row period is one.
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T_j(x)=\frac{\widetilde C_j(x)-1}{2^{b_j}}
T_kIntroduction passage · 7 passages
Search aliases: T_k
Introduction passage · 40_regularity.tex, line 12
Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives are the derivatives of the evaluation map.
Exact mathematical passage
F\in\mathbb K[Y_1,\ldots,Y_e,T_1,\ldots,T_k]^e, \qquad a=(y_0,t_0),
T_xIntroduction passage · 1 passage
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Introduction passage · 03_supplement_06.tex, line 65
A connection on the constructed bundle. The global formula is defined using the actual manifold derivative: ∇_Xs(x)=P(x) D_Ms(x)[X(x)]. Here D_M is represented by pathmvfderiv. A second definition, pathGlobalConnection.covariant, has values in the actual fiber: ∇ s(x):T_xM⟶ E_x. It is not only an ambient expression asserted to land in the range.
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\nabla s(x):T_xM\longrightarrow\mathcal E_x.
U^pIntroduction passage · 9 passages
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Introduction passage · supplement_07.tex, line 101
The positive-tail period. Write h=(p-1)/m. For n≥1, evaluate U^pf_m=Uf_m at the nonnegative degree mn-1: f_m(mn+p-1)=f_m(mn). Since mn+p-1=m(n+h), the identified sequence satisfies e_n+h^(m,p)=e_n^(m,p) (n≥1). Induction gives e_n+uh^(m,p)=e_n^(m,p) for every u≥0. This proves a period, not necessarily the least positive period. The proof also includes p=2, where m=1 and h=1.
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U^pf_m=Uf_m
U_{0,\lambda}Introduction passage · 1 passage
Search aliases: U_0,λ lambda
Introduction passage · conditional_completion.tex, line 16
Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality of numerical traces. The selected family ⊕_iρ_i,λ^⊕ m_i has a split equivariant inclusion in the carrier. The finite trace sample uniquely determines…
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V_{\mathrm{car},\lambda}\simeq\bigoplus_i\rho_{i,\lambda}^{\oplus c_i}, \qquad U_{0,\lambda}\simeq\bigoplus_i\rho_{i,\lambda}^{\oplus m_i}.
U_aIntroduction passage · 10 passages
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Introduction passage · 03_supplement_06.tex, line 33
The mutually inverse local maps. Write T_a(x)=T(P(x),P(a)). For x∈ U_a, B_a(x): E_a⟶ E_x, v⟼ T_a(x)v, A_a(x): E_x⟶ E_a, w⟼ P(a)T_a(x)^-1w. The intertwining identity and its inverse counterpart show that the displayed targets are correct. Lean proves both inverse laws, continuity, linearity in the fiber and smooth dependence on the base. Thus E|_U_a ≅ U_a× E_a is a local product homeomorphism that preserves the base point.
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x\in U_a
U_A^aIntroduction passage · 2 passages
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Introduction passage · rank2.tex, line 22
Write R=rad(A), U_A^a=1+R^a for atextgreater=1, and U_K^a=1+ p_K^a. Let theta be a full simple character. Since q is odd and the residue field is F2, the actual maximal simple type group and character groups coincide:
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U_A^a=1+R^a
U_F^1Introduction passage · 17 passages
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Introduction passage · dyadic_character.tex, line 371
Let omega_F be any smooth quasicharacter trivial on U_F 1. Since O_F^×=U_F^1, it is determined by its value omega_F(t). The actual maximal type extends to K times J. Concretely, (K times J)/J is infinite cyclic, generated by a K-uniformizer. An intertwiner for its conjugation action on eta gives an extension Lambda_0. Denote its central character by omega_0.
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\OO_F^\times=U_F^1
U_K^aIntroduction passage · 1 passage
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Introduction passage · rank2.tex, line 23
Write R=rad(A), U_A^a=1+R^a for atextgreater=1, and U_K^a=1+ p_K^a. Let theta be a full simple character. Since q is odd and the residue field is F2, the actual maximal simple type group and character groups coincide:
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U_K^a=1+\mathfrak p_K^a
v^TIntroduction passage · 10 passages
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Introduction passage · pages.tex, line 5
n 3, Appendix F. A dense array, a factored operator, and a compiled execution plan are different objects. They may denote the same linear map, but they need not store the same data, return the same output, or incur the same costs. This guide illustrates those distinctions with calculations. vfigthree_views headThree models to carry through the paper The first is a six-dimensional operator A=D+uv^T. Every entry of A is nonzero, yet three 2×2 diagonal blocks and two thin vectors suffice to apply i…
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A=D+uv^T
v_2Introduction passage · 136 passages
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Introduction passage · root_scope.tex, line 96
Let Z_2⟨ x⟩ denote the restricted power series: their integral coefficients tend to zero dyadically. Extend A by mathscr A(x)=frac9^x+1-18(x+1)(2x+1), mathscr A(-1)=-fraclog98. The apparent singularity at x=-1 is removable. Since v_2(log9)=3, its convergent expansion is mathscr A(x)=fraclog9/81+2x ∑_h≥0frac(log9)^h(h+1)!(x+1)^h. It follows coefficientwise that mathscr A∈ Z_2⟨ x⟩^×, mathscr A≡1pmod2, and mathscr A(N)=A(N).
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v_2(\log9)=3
V_{\mathrm{car},\lambda}Introduction passage · 1 passage
Search aliases: V_car,λ mathrm lambda
Introduction passage · conditional_completion.tex, line 16
Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality of numerical traces. The selected family ⊕_iρ_i,λ^⊕ m_i has a split equivariant inclusion in the carrier. The finite trace sample uniquely determines…
Exact mathematical passage
V_{\mathrm{car},\lambda}\simeq\bigoplus_i\rho_{i,\lambda}^{\oplus c_i}, \qquad U_{0,\lambda}\simeq\bigoplus_i\rho_{i,\lambda}^{\oplus m_i}.
V_rIntroduction passage · 10 passages
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Introduction passage · 03_supplement_05.tex, line 71
A channel on the whole ambient space. Put Q=I-∑_r∈ A_+V_rV_r^*. The recovery is R(σ)=∑_r∈ A_+V_r^*σ V_r+Qσ Q. The complement is an orthogonal projector and ∑_r∈ A_+V_rV_r^*+Q^*Q=I. Hence the recovery is trace preserving. Its finite Kraus form also proves positivity after amplification by the identity on every finite auxiliary space. The complement term matters: the map is defined as a channel even outside the syndrome ranges.
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Q=I-\sum_{r\in A_+}V_rV_r^*
V_r^*Introduction passage · 7 passages
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Introduction passage · 03_supplement_05.tex, line 71
A channel on the whole ambient space. Put Q=I-∑_r∈ A_+V_rV_r^*. The recovery is R(σ)=∑_r∈ A_+V_r^*σ V_r+Qσ Q. The complement is an orthogonal projector and ∑_r∈ A_+V_rV_r^*+Q^*Q=I. Hence the recovery is trace preserving. Its finite Kraus form also proves positivity after amplification by the identity on every finite auxiliary space. The complement term matters: the map is defined as a channel even outside the syndrome ranges.
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Q=I-\sum_{r\in A_+}V_rV_r^*
\veeIntroduction passage · 10 passages
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Introduction passage · Paper_XI_Traditional_Final.tex, line 2171
i) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i=1^d(p^2i-p^2i-2)h_i · · (p odd), a(π^vee×π) · =N^2-1+∑_i=1^d3·4^i-1h_i · · (p=2). These identities do not…
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\begin{aligned} a(\pi^\vee\times\pi) &=p^{2d}-1+\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i &&(p\text{ odd}),\\ a(\pi^\vee\times\pi) &=N^2-1+\sum_{i=1}^d3\cdot4^{i-1}h_i &&(p=2). \end{aligned}
w'Introduction passage · 2 passages
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Introduction passage · atlas-pages.tex, line 428
1-w · 1+w. vfigprojective_poles headThe action is defined before a ratio is taken The determinant is w, so the matrix cannot send a nonzero pair to zero. Direct multiplication gives A(w)A(w')=A(ww'), A(1)=I. For w=i, the orbit through zero is [0:1] ⟼ [-i:1] ⟼ [1:0] ⟼ [i:1] ⟼ [0:1]. The third point is a legitimate projective point, even though a finite impedance formula has a zero denominator there. The second iterate is A(-1), which swaps the two coordinates. Thus a half turn exchanges 0 and ∞ a…
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A(w)A(w')=A(ww'),\qquad A(1)=I.
W_{\ell,\alpha}Introduction passage · 5 passages
Search aliases: W_ℓ,α ell alpha
Introduction passage · Paper_V_Traditional_Final.tex, line 622
Write H_0,H_1 for the literal low and detail operators. The preceding energy identity implies H_bx _2≤ x _2, so both have bounded Hilbert adjoints. At depth zero the only packet operator is the identity. If the energy identity holds at depth ℓ, apply the one-level identity to each W_ℓ,αx and sum. This proves it at depth ℓ+1.
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W_{\ell,\alpha}x
\WalkIntroduction passage · 3 passages
Search aliases: Walk operatorname
Introduction passage · 01_finite_coordinates.tex, line 274
Write Walk_n(i,j) for the finite set of composable edge tuples of length n whose first source is i and whose final target is j. For n=0, it has one empty walk when i=j and is empty otherwise. Assign the empty walk the coefficient 1.
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\Walk_n(i,j)
\Walk_nIntroduction passage · 3 passages
Search aliases: Walk_n Walk operatorname
Introduction passage · 01_finite_coordinates.tex, line 274
Write Walk_n(i,j) for the finite set of composable edge tuples of length n whose first source is i and whose final target is j. For n=0, it has one empty walk when i=j and is empty otherwise. Assign the empty walk the coefficient 1.
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\Walk_n(i,j)
\wedgeIntroduction passage · 13 passages
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Introduction passage · 04_alternative_derivations.tex, line 22
eld and extend the projected connection to differential forms with values in the range bundle by ∇ω=P dω. Thus ∇^2 denotes the exterior covariant square, including the Lie bracket term in the exterior derivative when evaluated on vector fields. [Alternative proof of the curvature formula] For a section s with Ps=s, ∇^2s=P d(P ds)=P(dP)wedge ds. Since ds=d(Ps)=(dP)s+Pds and P(dP)P=0, the surviving term is P(dPwedge dP)Ps. Restricted low-mode proof. Let h≠0, x=kh/2, and 0≤ x≤1. [Alternative proof …
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\nabla^2s=P\,d(P\,ds)=P(dP)\wedge ds.
\widehat GIntroduction passage · 5 passages
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Introduction passage · 02_strengthened.tex, line 203
Let G be a finite group and let U:G⟶ U( H) be a unitary representation. Write widehat G for the set of irreducible unitary representations. For λ∈widehat G, let d_λ and χ_λ denote its dimension and character.
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\widehat G
\widetilde CIntroduction passage · 15 passages
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Introduction passage · strengthening.tex, line 1083
[Exact image and residue enumeration] Let widetilde C_j denote the dyadic Newton interpolation. The map T_j(x)=fracwidetilde C_j(x)-12^b_j is a bijective isometry Z_2→ Z_2, and widetilde C_j( Z_2)=1+2^b_j Z_2, j!widetilde C_j( Z_2)=j!+2^B_j Z_2. For r>B_j, one least period modulo 2^r visits every residue in j!+2^B_j Z/2^r Z exactly once. For r≤ B_j, there is one residue and the least row period is one.
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\widetilde C_j
\widetilde C_jIntroduction passage · 4 passages
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Introduction passage · strengthening.tex, line 1083
[Exact image and residue enumeration] Let widetilde C_j denote the dyadic Newton interpolation. The map T_j(x)=fracwidetilde C_j(x)-12^b_j is a bijective isometry Z_2→ Z_2, and widetilde C_j( Z_2)=1+2^b_j Z_2, j!widetilde C_j( Z_2)=j!+2^B_j Z_2. For r>B_j, one least period modulo 2^r visits every residue in j!+2^B_j Z/2^r Z exactly once. For r≤ B_j, there is one residue and the least row period is one.
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\widetilde C_j
\widetilde{\mathcal N}Introduction passage · 2 passages
Search aliases: widetilde N widetilde mathcal
Introduction passage · 03_supplement_05.tex, line 146
An arbitrary finite reference system. For any finite reference index set A, the amplified channel is defined using the actual Kronecker Kraus operators, widetilde N_E(ρ) =∑_a(I_A⊗ E_a)ρ(I_A⊗ E_a)^*. Its reference blocks are proved equal to the original channel applied blockwise. If (I_A⊗ P)ρ(I_A⊗ P)=ρ, then every reference block is code supported. Applying the checked correction theorem for operators to those blocks gives widetilde R widetilde N_E(ρ)=ρ. This includes entangled density matrices s…
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\widetilde{\mathcal N}_E(\rho) =\sum_a(I_A\otimes E_a)\rho(I_A\otimes E_a)^*.
\widetilde{\mathcal N}_EIntroduction passage · 2 passages
Search aliases: widetilde N_E widetilde mathcal
Introduction passage · 03_supplement_05.tex, line 146
An arbitrary finite reference system. For any finite reference index set A, the amplified channel is defined using the actual Kronecker Kraus operators, widetilde N_E(ρ) =∑_a(I_A⊗ E_a)ρ(I_A⊗ E_a)^*. Its reference blocks are proved equal to the original channel applied blockwise. If (I_A⊗ P)ρ(I_A⊗ P)=ρ, then every reference block is code supported. Applying the checked correction theorem for operators to those blocks gives widetilde R widetilde N_E(ρ)=ρ. This includes entangled density matrices s…
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\widetilde{\mathcal N}_E(\rho) =\sum_a(I_A\otimes E_a)\rho(I_A\otimes E_a)^*.
\widetilde{\mathcal R}Introduction passage · 1 passage
Search aliases: widetilde R widetilde mathcal
Introduction passage · 03_supplement_05.tex, line 155
channel is defined using the actual Kronecker Kraus operators, widetilde N_E(ρ) =∑_a(I_A⊗ E_a)ρ(I_A⊗ E_a)^*. Its reference blocks are proved equal to the original channel applied blockwise. If (I_A⊗ P)ρ(I_A⊗ P)=ρ, then every reference block is code supported. Applying the checked correction theorem for operators to those blocks gives widetilde R widetilde N_E(ρ)=ρ. This includes entangled density matrices supported on the combined reference and code space. In fact, the identity holds for every j…
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\boxed{\widetilde{\mathcal R}\,\widetilde{\mathcal N}_E(\rho)=\rho.}
x^2Introduction passage · 47 passages
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Introduction passage · visual_atlas.tex, line 434
1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.
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(x^2+y)(x-y)=x+(x-x^2)y.
x^3Introduction passage · 18 passages
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Introduction passage · visual_atlas.tex, line 425
headMultiplication by y transfers between the lanes In the ordered coefficient coordinates (a,b), M_y=mat0 · x^3-x 1 · 0, M_y^2=(x^3-x)I. The entries denote multiplication maps on Q[x]. For f=a_0+ya_1 and g=b_0+yb_1, fg=(a_0b_0+(x^3-x)a_1b_1) +y(a_0b_1+a_1b_0). In particular, (x^2+y)(x-y)=x+(x-x^2)y.
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M_y=\mat{0&x^3-x\\1&0},\qquad M_y^2=(x^3-x)I.
X^aIntroduction passage · 1 passage
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Introduction passage · supplement_06.tex, line 70
Principal targets. targetCompanion.orbit_of_recurrence targetCompanion.companion_jump, Companion.source_companion_jump targetCompanion.packet_step, Companion.packet_advance targetCompanion.packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det(XI-D)=X^d+∑_a=0^d-1c_aX^a be its actual characteristic polynomial. The implementation reuses the pinned Mathlib Cayley--Hamilton theorem, expands pol…
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\chi_D(X)=\det(XI-D)=X^d+\sum_{a=0}^{d-1}c_aX^a
X^dIntroduction passage · 2 passages
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Introduction passage · supplement_06.tex, line 70
Principal targets. targetCompanion.orbit_of_recurrence targetCompanion.companion_jump, Companion.source_companion_jump targetCompanion.packet_step, Companion.packet_advance targetCompanion.packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det(XI-D)=X^d+∑_a=0^d-1c_aX^a be its actual characteristic polynomial. The implementation reuses the pinned Mathlib Cayley--Hamilton theorem, expands pol…
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\chi_D(X)=\det(XI-D)=X^d+\sum_{a=0}^{d-1}c_aX^a
X_1Introduction passage · 44 passages
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Introduction passage · 72_recovered.tex, line 2641
Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod p. Let J_F(x_L) denote the Jacobian matrix and write overline J_F(x_L) for its reduction modulo p.
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F_i\in\Z_p[X_1,\ldots,X_d]
X_1^{a_1}Introduction passage · 4 passages
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Introduction passage · 01_finite_core.tex, line 161
[One seed for the computational basis] For an integer n≥0, let ψ_0=ket0^⊗ n and let X_j denote the Pauli X operator on qubit j. Then { X_1^a_1⋯ X_n^a_nψ_0: (a_1,…,a_n)∈{0,1}^n } is the complete computational basis. Hence the seed is cyclic under the algebra generated by the X_j.
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\left\{ X_1^{a_1}\cdots X_n^{a_n}\psi_0: (a_1,\ldots,a_n)\in\{0,1\}^n \right\}
X_dIntroduction passage · 20 passages
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Introduction passage · 72_recovered.tex, line 2641
Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod p. Let J_F(x_L) denote the Jacobian matrix and write overline J_F(x_L) for its reduction modulo p.
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F_i\in\Z_p[X_1,\ldots,X_d]
x_{j,\cdot}Introduction passage · 1 passage
Search aliases: x_j,· cdot
Introduction passage · 02_supplement_04.tex, line 95
The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;x_j,·)_u. Induction, the Kronecker entry formula, and finite-sum distributivity prove Apply_k(A;x)=M_k(A)x. Thus correctness is not inferred merely fr…
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y_{i,u}=\sum_{j=0}^{d-1}(A_0)_{ij}\, \operatorname{Apply}_{k}(A_1,\ldots,A_k;x_{j,\cdot})_u.
x_LIntroduction passage · 34 passages
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Introduction passage · 72_recovered.tex, line 2517
Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.
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x_L\in\Zp^d
X_n^{a_n}Introduction passage · 3 passages
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Introduction passage · 01_finite_core.tex, line 161
[One seed for the computational basis] For an integer n≥0, let ψ_0=ket0^⊗ n and let X_j denote the Pauli X operator on qubit j. Then { X_1^a_1⋯ X_n^a_nψ_0: (a_1,…,a_n)∈{0,1}^n } is the complete computational basis. Hence the seed is cyclic under the algebra generated by the X_j.
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\left\{ X_1^{a_1}\cdots X_n^{a_n}\psi_0: (a_1,\ldots,a_n)\in\{0,1\}^n \right\}
X_pIntroduction passage · 11 passages
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Introduction passage · 02_foundations.tex, line 886
[Primary decomposition for polynomial modules] Every module X over 𝕜[t] with finite dimension over the coefficient field decomposes as X=⊕_p∈Irr(𝕜[t])X_p, where Irr(𝕜[t]) denotes the monic irreducible polynomials, only finitely many components are nonzero, and every nonzero X_p is a finite direct sum of modules 𝕜[t]/(p^e).
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X=\bigoplus_{p\in\Irr(\bbk[t])}X_p,
X_QIntroduction passage · 7 passages
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Introduction passage · Paper_VI_Traditional_Final.tex, line 4579
If X_Q={0}, the memory kernel and self-energy are both zero, so the identity is immediate. Otherwise write B=A_QQ, which is a bounded operator on the hidden Banach space, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every integer n≥0, after increasing M_β to cover the finitely many initial values. Write t=n+r, where 0≤ r<1. Then norm e^tB≤ M_βexp(norm B+max(0,-β)) e^β t. Thus…
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X_Q=\{0\}
X_uIntroduction passage · 1 passage
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Introduction passage · 02_supplement_05.tex, line 42
Forward substitution with an actual stage evaluator. textitSource item 6.5; module TriangularSolve Use the graph and levels constructed on the previous page. A vertex v carries an arbitrary vector space X_v over a field. The spaces may have different dimensions. Let D_v:X_voverset∼⟶X_v, T_vu:X_u⟶ X_v be an invertible diagonal map and linear transfer maps. The operator being solved is defined by its actual block action: (Ax)_v=D_vx_v+∑_u→ vT_vux_u. No solution, solve equation or precomputed topol…
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D_v:X_v\overset{\sim}{\longrightarrow}X_v, \qquad T_{vu}:X_u\longrightarrow X_v
x_uIntroduction passage · 2 passages
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Introduction passage · 02_supplement_05.tex, line 47
iangularSolve Use the graph and levels constructed on the previous page. A vertex v carries an arbitrary vector space X_v over a field. The spaces may have different dimensions. Let D_v:X_voverset∼⟶X_v, T_vu:X_u⟶ X_v be an invertible diagonal map and linear transfer maps. The operator being solved is defined by its actual block action: (Ax)_v=D_vx_v+∑_u→ vT_vux_u. No solution, solve equation or precomputed topological certificate is assumed.
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(Ax)_v=D_vx_v+\sum_{u\to v}T_{vu}x_u.
x_vIntroduction passage · 4 passages
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Introduction passage · 02_supplement_05.tex, line 47
iangularSolve Use the graph and levels constructed on the previous page. A vertex v carries an arbitrary vector space X_v over a field. The spaces may have different dimensions. Let D_v:X_voverset∼⟶X_v, T_vu:X_u⟶ X_v be an invertible diagonal map and linear transfer maps. The operator being solved is defined by its actual block action: (Ax)_v=D_vx_v+∑_u→ vT_vux_u. No solution, solve equation or precomputed topological certificate is assumed.
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(Ax)_v=D_vx_v+\sum_{u\to v}T_{vu}x_u.
y_0Introduction passage · 34 passages
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Introduction passage · 40_regularity.tex, line 12
Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives are the derivatives of the evaluation map.
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F\in\mathbb K[Y_1,\ldots,Y_e,T_1,\ldots,T_k]^e, \qquad a=(y_0,t_0),
Y_1Introduction passage · 11 passages
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Introduction passage · 40_regularity.tex, line 12
Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives are the derivatives of the evaluation map.
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F\in\mathbb K[Y_1,\ldots,Y_e,T_1,\ldots,T_k]^e, \qquad a=(y_0,t_0),
Y_eIntroduction passage · 4 passages
Search aliases: Y_e
Introduction passage · 40_regularity.tex, line 12
Let K be a complete nontrivially normed field. Let e,k∈ N, and write the variables as (Y,T)∈ K^e× K^k. For F∈ K[Y_1,…,Y_e,T_1,…,T_k]^e, a=(y_0,t_0), define the matrices of evaluated formal partial derivatives J=D_YF(a)∈ M_e( K), K=D_TF(a)∈ M_e× k( K). Rows index equations and columns index variables. We first establish that these formal derivatives are the derivatives of the evaluation map.
Exact mathematical passage
F\in\mathbb K[Y_1,\ldots,Y_e,T_1,\ldots,T_k]^e, \qquad a=(y_0,t_0),
y_{i,u}Introduction passage · 1 passage
Search aliases: y_i,u
Introduction passage · 02_supplement_04.tex, line 95
The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;x_j,·)_u. Induction, the Kronecker entry formula, and finite-sum distributivity prove Apply_k(A;x)=M_k(A)x. Thus correctness is not inferred merely fr…
Exact mathematical passage
y_{i,u}=\sum_{j=0}^{d-1}(A_0)_{ij}\, \operatorname{Apply}_{k}(A_1,\ldots,A_k;x_{j,\cdot})_u.
Y_NIntroduction passage · 4 passages
Search aliases: Y_N
Introduction passage · Paper_IV_Traditional_Final.tex, line 1363
Write D_N=e^g locally. Then g'=- T_N. Divide the equation D_N^(N)=- D_N by D_N and use the defining identity for Y_N.
Exact mathematical passage
Y_N
y_{\rm even}Introduction passage · 2 passages
Search aliases: y_rm even rm
Introduction passage · visual_atlas.tex, line 487
If w denotes one delay on a phase sequence, the filter becomes y_rm even y_rm odd = underbrace1+w · 2w 2 · 1+w_H_rm phase(w) e o. The cross-phase delay is not optional. A unit shift sends an odd sample to the next even position, while it sends an even sample to the odd position in the same phase block.
Exact mathematical passage
\begin{pmatrix}y_{\rm even}\\y_{\rm odd}\end{pmatrix} = \underbrace{\begin{pmatrix}1+w&2w\\2&1+w\end{pmatrix}}_{H_{\rm phase}(w)} \begin{pmatrix}e\\o\end{pmatrix}.
y_{\rm odd}Introduction passage · 2 passages
Search aliases: y_rm odd rm
Introduction passage · visual_atlas.tex, line 487
If w denotes one delay on a phase sequence, the filter becomes y_rm even y_rm odd = underbrace1+w · 2w 2 · 1+w_H_rm phase(w) e o. The cross-phase delay is not optional. A unit shift sends an odd sample to the next even position, while it sends an even sample to the odd position in the same phase block.
Exact mathematical passage
\begin{pmatrix}y_{\rm even}\\y_{\rm odd}\end{pmatrix} = \underbrace{\begin{pmatrix}1+w&2w\\2&1+w\end{pmatrix}}_{H_{\rm phase}(w)} \begin{pmatrix}e\\o\end{pmatrix}.
z^{-d}Introduction passage · 6 passages
Search aliases: z^-d
Introduction passage · Paper_V_Traditional_Final.tex, line 498
[Perfect-reconstruction criterion] For d∈ Z, such a bank reconstructs every input up to a delay of d channel indices exactly when the following Laurent-polynomial identity holds: R(z) E(z)=z^-dI_N. If the right side is c z^-dI_N with c≠0, reconstruction holds up to gain c and channel delay d. After interleaving, this is a delay of Nd original samples. A negative delay denotes an advance.
Exact mathematical passage
\boxed{ \bm R(z)\bm E(z)=z^{-d}I_N. }
\mathbb Z^dIntroduction passage · 13 passages
Search aliases: Z^d mathbb
Introduction passage · continuation_01.tex, line 14
Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer modules. It factors through reduction to R_q^d. The cokernel below is presented directly using its integer-domain image.
Exact mathematical passage
F=(F_1,\ldots,F_e)\in\mathbb Z[X_1,\ldots,X_d]^e, \quad q\ge1,\quad L\ge1, \quad x\in\mathbb Z^d, \quad F(x)\equiv0\pmod{q^L}.
\mathbb Z^eIntroduction passage · 4 passages
Search aliases: Z^e mathbb
Introduction passage · continuation_01.tex, line 20
Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer modules. It factors through reduction to R_q^d. The cokernel below is presented directly using its integer-domain image.
Exact mathematical passage
r_x=F(x)/q^L\in\mathbb Z^e, \qquad A_x=DF(x):\mathbb Z^d\longrightarrow\mathbb Z^e, \qquad J_x=\rho_q\circ A_x.
\Z_2Introduction passage · 33 passages
Search aliases: Z_2 Z mathbb Z_2 Z mathbf
Introduction passage · supplement_11.tex, line 78
Write H_int for the subring of Q_2[[t]] whose factorial-scaled coefficients belong to Z_2. Flatness means B_4^sF∈ H_int for every s. The proof of multiplicative and reciprocal closure is induction on s, using the full product correction. It separately proves that the actual reciprocal has integral Hurwitz coefficients when F(0)=1.
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\Z_2
\mathbb Z_2Introduction passage · 59 passages
Search aliases: Z_2 mathbb
Introduction passage · root_scope.tex, line 89
Let Z_2⟨ x⟩ denote the restricted power series: their integral coefficients tend to zero dyadically. Extend A by mathscr A(x)=frac9^x+1-18(x+1)(2x+1), mathscr A(-1)=-fraclog98. The apparent singularity at x=-1 is removable. Since v_2(log9)=3, its convergent expansion is mathscr A(x)=fraclog9/81+2x ∑_h≥0frac(log9)^h(h+1)!(x+1)^h. It follows coefficientwise that mathscr A∈ Z_2⟨ x⟩^×, mathscr A≡1pmod2, and mathscr A(N)=A(N).
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\mathbb Z_2\langle x\rangle
\ZpIntroduction passage · 66 passages
Search aliases: Zp Z_p mathbb
Introduction passage · 00_scope.tex, line 18
Throughout, N={0,1,2,…}. When p is prime, Z_p denotes the p-adic integers and Q_p their fraction field. We normalize v_p(p)=1. For a rational or p-adic quantity, a displayed lower valuation bound is understood with the mathematical convention v_p(0)=+∞. The corresponding formal predicates are zero-safe: the quantity is zero or its valuation meets the stated bound. An exact finite valuation assertion requires a nonzero quantity, and this is stated where used. Congruence in Z_p means ideal divisib…
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\Zp
\Zp^dIntroduction passage · 11 passages
Search aliases: Zp^d Z_p^d Zp mathbb
Introduction passage · 72_recovered.tex, line 2517
Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.
Exact mathematical passage
x_L\in\Zp^d
\Zp^rIntroduction passage · 3 passages
Search aliases: Zp^r Z_p^r Zp mathbb
Introduction passage · 72_recovered.tex, line 2512
Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.
Exact mathematical passage
M\in M_{r,d}(\Zp), \qquad b\in\Zp^r.
\cdotIntroduction passage · 53 passages
Search aliases: · cdot
Introduction passage · Paper_XI_Traditional_Final.tex, line 1303
[Fixed point free Weil trace] Under the declared normalization, Trω(g) = (frac-1p)^r (fracdet(g-I)p). Here (frac·p) denotes the quadratic character of F_p^×.
Exact mathematical passage
(\frac{\cdot}{p})
\Delta_1Introduction passage · 10 passages
Search aliases: Δ_1 Delta
Introduction passage · 10_foundations.tex, line 272
Write E_h f(n)=f(n+h) for the shift operator. The unit shift is I+Δ_1. Expanding its nth power and evaluating at zero gives reconstruction. Composing difference operators gives the coefficient shift and applying reconstruction to Δ_1^s f gives the third identity. The binomial expansion of (E_h-I)^s gives the explicit stencil. Binomial coefficients vanish when their lower index exceeds n, which proves the interpolation claim. Equality of every point of the stencil proves equality of differences. …
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I+\Delta_1
\Delta_1^sIntroduction passage · 15 passages
Search aliases: Δ_1^s Delta
Introduction passage · 10_foundations.tex, line 274
Write E_h f(n)=f(n+h) for the shift operator. The unit shift is I+Δ_1. Expanding its nth power and evaluating at zero gives reconstruction. Composing difference operators gives the coefficient shift and applying reconstruction to Δ_1^s f gives the third identity. The binomial expansion of (E_h-I)^s gives the explicit stencil. Binomial coefficients vanish when their lower index exceeds n, which proves the interpolation claim. Equality of every point of the stencil proves equality of differences. …
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\Delta_1^s f
\delta_nIntroduction passage · 5 passages
Search aliases: δ_n delta
Introduction passage · Paper_V_Traditional_Final.tex, line 265
Let ℓ^2( Z; C), abbreviated ℓ^2( Z), be the Hilbert space of square-summable complex discrete signals. The inner product is conjugate-linear in its first argument. A superscript * on a bounded operator denotes its Hilbert adjoint. Write δ_n for the unit impulse at address n and let the bilateral shift be (Sx)[n]=x[n-1]. Then S^nδ_0=δ_n. The primitive signal seed is therefore s=δ_0, the address operators are the shift words S^n, and the completion is the ℓ^2 norm.
Exact mathematical passage
\delta_n
\delta_{n0}Introduction passage · 14 passages
Search aliases: δ_n0 delta
Introduction passage · 72_recovered.tex, line 47
Fix an odd prime p, a divisor mmid p-1, and put h=fracp-1m. The reduced standard window inverse is defined by ∑_q=0^n binommnmqe_q^(m,p) = δ_n0 in F_p. We prove that e_n+h^(m,p)=e_n^(m,p) (n≥1). The result first follows from subgroup symmetry in the finite Hurwitz algebra. A second proof shows that the standard window operator preserves the finite space of sequences that are h periodic after their constant term. The two arguments expose different parts of the same finite character structure.
Exact mathematical passage
\sum_{q=0}^{n} \binom{mn}{mq}e_q^{(m,p)} = \delta_{n0} \quad\text{in }\FF_p.
\Delta_uIntroduction passage · 1 passage
Search aliases: Δ_u Delta
Introduction passage · 00_scope.tex, line 32
Write Δ_u f(n)=f(n+u)-f(n) and use Δ_u^k for its k-fold iterate. Euler coefficients indexed by ordinary degree and by packet row are distinguished explicitly. A packet displacement by u rows is a degree displacement by Nu in a packet of width N.
Exact mathematical passage
\Delta_u f(n)=f(n+u)-f(n)
\Delta_u^kIntroduction passage · 3 passages
Search aliases: Δ_u^k Delta
Introduction passage · 00_scope.tex, line 32
Write Δ_u f(n)=f(n+u)-f(n) and use Δ_u^k for its k-fold iterate. Euler coefficients indexed by ordinary degree and by packet row are distinguished explicitly. A packet displacement by u rows is a degree displacement by Nu in a packet of width N.
Exact mathematical passage
\Delta_u^k
\Delta_{uh}^{k}Introduction passage · 8 passages
Search aliases: Δ_uh^k Delta
Introduction passage · 20_arithmetic.tex, line 193
[All-address differences on the divisor branch] Let p be prime, m>0, mmid p-1, N=mp, and 0≤ j<N. Write h=(p-1)/m. For every u,k,n∈ N, U_N,j(n) · ∈ Z_p, Δ_uh^kU_N,j(n) · ∈ p^min(k,mn) Z_p, v_p(Δ_uh^kC_N,j(n)) · ≥min(k,mn)-v_p(j!). The factorial correction is part of the normalized statement, including addresses j≥ p.
Exact mathematical passage
U_{N,j}(n)&\in\Z_p,\label{eq:arith-divisor-integral}\\ \Delta_{uh}^{k}U_{N,j}(n)&\in p^{\min(k,mn)}\Z_p,\label{eq:arith-divisor-bound}\\ v_p\bigl(\Delta_{uh}^{k}C_{N,j}(n)\bigr) &\ge\min(k,mn)-v_p(j!).\label{eq:arith-divisor-normalized}
\delta_{\alpha\beta}Introduction passage · 7 passages
Search aliases: δ_αβ delta alpha beta
Introduction passage · 01_finite_core.tex, line 5
Conventions. Unless the zero space is explicitly included as a boundary case, quantum state spaces are nonzero complex Hilbert spaces of finite dimension. A complete packet family is a finite collection satisfying P_α^*=P_α, P_α P_β=δ_αβP_α, ∑_α P_α=I. All operator norms in the analytic estimates are induced by the norm on the Hilbert space. The estimates do not use the entrywise maximum norm or the Frobenius norm. We write [A,B]=AB-BA and use units with ℏ=1. Parameter derivatives are real deriv…
Exact mathematical passage
P_\alpha^*=P_\alpha,\qquad P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha,\qquad \sum_\alpha P_\alpha=I.
\lambda^{-1}Introduction passage · 5 passages
Search aliases: λ^-1 lambda
Introduction passage · Paper_IV_Traditional_Final.tex, line 2371
[Complete symmetric function form] For a positive integer r, let each eigenvalue λ^-1 of K_N be repeated r times. If h_n denotes the complete homogeneous symmetric function of degree n in this absolutely summable multiset, then fracP_N,n(r)(Nn)!=h_n. For arbitrary complex r, the Bell polynomial formula is the polynomial continuation obtained by multiplying every power sum by r.
Exact mathematical passage
\lambda^{-1}
\lambda_JIntroduction passage · 1 passage
Search aliases: λ_J lambda
Introduction passage · 30_lifting.tex, line 170
s over an arbitrary commutative ring. Let U,V,W be R-modules and let J:U→ V, K:U→ W be linear. Define C(J,K)=U/(ker J+ker K). An attainable value Ju has label [u]∈ C(J,K). This is independent of the choice of its preimage because ker J is killed in the quotient. Similarly, Kv has label [v]. Denote these maps by λ_J:im J→ C(J,K) and λ_K:im K→ C(J,K).
Exact mathematical passage
\lambda_J:\im J\to C(J,K)
\lambda_KIntroduction passage · 1 passage
Search aliases: λ_K lambda
Introduction passage · 30_lifting.tex, line 170
ring. Let U,V,W be R-modules and let J:U→ V, K:U→ W be linear. Define C(J,K)=U/(ker J+ker K). An attainable value Ju has label [u]∈ C(J,K). This is independent of the choice of its preimage because ker J is killed in the quotient. Similarly, Kv has label [v]. Denote these maps by λ_J:im J→ C(J,K) and λ_K:im K→ C(J,K).
Exact mathematical passage
\lambda_K:\im K\to C(J,K)
\pi^\veeIntroduction passage · 3 passages
Search aliases: π^vee pi vee
Introduction passage · Paper_XI_Traditional_Final.tex, line 2171
i) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i=1^d(p^2i-p^2i-2)h_i · · (p odd), a(π^vee×π) · =N^2-1+∑_i=1^d3·4^i-1h_i · · (p=2). These identities do not…
Exact mathematical passage
\begin{aligned} a(\pi^\vee\times\pi) &=p^{2d}-1+\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i &&(p\text{ odd}),\\ a(\pi^\vee\times\pi) &=N^2-1+\sum_{i=1}^d3\cdot4^{i-1}h_i &&(p=2). \end{aligned}
\pi_iIntroduction passage · 9 passages
Search aliases: π_i pi
Introduction passage · Paper_VI_Traditional_Final.tex, line 3012
Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.
Exact mathematical passage
\pi_i
\pi_jIntroduction passage · 4 passages
Search aliases: π_j pi
Introduction passage · Paper_VI_Traditional_Final.tex, line 3013
Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.
Exact mathematical passage
(P_\mathcal Qf)_i = \frac{\sum_{j\in B(i)}\pi_jf_j} {\sum_{j\in B(i)}\pi_j}.
\rho_BIntroduction passage · 5 passages
Search aliases: ρ_B rho
Introduction passage · 03_supplement_03.tex, line 70
The actual partial trace and its support. The reduced matrix is defined by tracing the addressed index of the density matrix of the pure state: (ρ_B)_jℓ=∑_iψ_ijoverlineψ_iℓ. For the coefficient convention used here, the exact Gram identity is ρ_B=C^ T(C^ T)^*, rankρ_B=rank C=r(ψ). The transpose is important. With this convention, writing C^*C instead would conjugate the reduced matrix. The Lean source checks the entrywise identity and positivity of the resulting matrix.
Exact mathematical passage
(\rho_B)_{j\ell}=\sum_i\psi_{ij}\overline{\psi_{i\ell}}.
\rho_iIntroduction passage · 3 passages
Search aliases: ρ_i rho
Introduction passage · conditional_completion.tex, line 15
Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality of numerical traces. The selected family ⊕_iρ_i,λ^⊕ m_i has a split equivariant inclusion in the carrier. The finite trace sample uniquely determines…
Exact mathematical passage
\rho_i
\rho_{i,\lambda}^{\oplus c_i}Introduction passage · 1 passage
Search aliases: ρ_i,λ^⊕ c_i rho lambda oplus
Introduction passage · conditional_completion.tex, line 16
Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality of numerical traces. The selected family ⊕_iρ_i,λ^⊕ m_i has a split equivariant inclusion in the carrier. The finite trace sample uniquely determines…
Exact mathematical passage
V_{\mathrm{car},\lambda}\simeq\bigoplus_i\rho_{i,\lambda}^{\oplus c_i}, \qquad U_{0,\lambda}\simeq\bigoplus_i\rho_{i,\lambda}^{\oplus m_i}.
\rho_{i,\lambda}^{\oplus m_i}Introduction passage · 2 passages
Search aliases: ρ_i,λ^⊕ m_i rho lambda oplus
Introduction passage · conditional_completion.tex, line 16
Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality of numerical traces. The selected family ⊕_iρ_i,λ^⊕ m_i has a split equivariant inclusion in the carrier. The finite trace sample uniquely determines…
Exact mathematical passage
V_{\mathrm{car},\lambda}\simeq\bigoplus_i\rho_{i,\lambda}^{\oplus c_i}, \qquad U_{0,\lambda}\simeq\bigoplus_i\rho_{i,\lambda}^{\oplus m_i}.
\rho_LIntroduction passage · 4 passages
Search aliases: ρ_L rho
Introduction passage · 72_recovered.tex, line 2522
Let p be a prime. Let M∈ M_r,d( Z_p), b∈ Z_p^r. Suppose that x_L∈ Z_p^d satisfies Mx_L≡ bpmod p^L for some L≥1. Define the reduced residual ρ_L(x_L) = fracMx_L-bp^L pmod p ∈ F_p^r. Write overline M for the reduction of M modulo p.
Exact mathematical passage
\rho_L(x_L) = \frac{Mx_L-b}{p^L} \pmod p \in\FF_p^r.
\rho_L^FIntroduction passage · 3 passages
Search aliases: ρ_L^F rho
Introduction passage · 72_recovered.tex, line 2646
Let F=(F_1,…,F_r): Z_p^d⟶ Z_p^r be a polynomial map. Editorial coefficient clarification. Here polynomial means F_i∈ Z_p[X_1,…,X_d], not merely an integer-valued polynomial with coefficients in Q_p. This makes the Jacobian and Taylor remainder integral and their reduction well typed. Suppose F(x_L)∈ p^L Z_p^r. Define ρ_L^F(x_L) = fracF(x_L)p^L pmod p. Let J_F(x_L) denote the Jacobian matrix and write overline J_F(x_L) for its reduction modulo p.
Exact mathematical passage
\rho_L^F(x_L) = \frac{F(x_L)}{p^L} \pmod p.
\rho_qIntroduction passage · 7 passages
Search aliases: ρ_q rho
Introduction passage · continuation_01.tex, line 19
Let F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, q≥1, L≥1, x∈ Z^d, F(x)≡0pmodq^L. Write R_q= Z/q Z, let ρ_q denote coordinatewise reduction, and put r_x=F(x)/q^L∈ Z^e, A_x=DF(x): Z^d⟶ Z^e, J_x=ρ_q∘ A_x. Here A_x is computed from the actual formal partial derivatives of the input polynomials. The Lean map J_x has domain Z^d and codomain R_q^e, viewed as integer modules. It factors through reduction to R_q^d. The cokernel below is presented directly using its integer-domain image.
Exact mathematical passage
\rho_q
\tau_mIntroduction passage · 22 passages
Search aliases: τ_m tau
Introduction passage · Paper_IV_Traditional_Final.tex, line 2065
sufficiently large zero in the fundamental sector into one of these disks. The zero sets of G and D_N coincide because their prefactor is nonzero. Each disk is invariant under complex conjugation, and D_N has real coefficients. The conjugate of its unique zero is therefore the same zero. It is real, positive for large m, and simple by the derivative bound above. Denote it by τ_m.
Exact mathematical passage
\tau_m
\Phi^*Introduction passage · 1 passage
Search aliases: Φ^* Phi
Introduction passage · 02_strengthened.tex, line 1485
[Correctable observable algebras] Let Φ be a finite completely positive trace preserving channel with Kraus operators (E_a). A finite dimensional unital star subalgebra M⊆ PEnd( H)P is correctable when a completely positive trace preserving recovery R satisfies P(Φ^*∘ R^*)(X)P=X for every X∈ M, where stars on maps denote Heisenberg adjoints. This holds exactly when every compressed product PE_a^*E_bP belongs to the commutant M' on the code. The scalar code criterion is the special case M=PEnd( H…
Exact mathematical passage
P(\Phi^*\circ\Rcal^*)(X)P=X
\chi_DIntroduction passage · 5 passages
Search aliases: χ_D chi
Introduction passage · supplement_06.tex, line 70
Principal targets. targetCompanion.orbit_of_recurrence targetCompanion.companion_jump, Companion.source_companion_jump targetCompanion.packet_step, Companion.packet_advance targetCompanion.packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det(XI-D)=X^d+∑_a=0^d-1c_aX^a be its actual characteristic polynomial. The implementation reuses the pinned Mathlib Cayley--Hamilton theorem, expands pol…
Exact mathematical passage
\chi_D(X)=\det(XI-D)=X^d+\sum_{a=0}^{d-1}c_aX^a
\chi_\lambdaIntroduction passage · 2 passages
Search aliases: χ_λ chi lambda
Introduction passage · 02_strengthened.tex, line 203
Let G be a finite group and let U:G⟶ U( H) be a unitary representation. Write widehat G for the set of irreducible unitary representations. For λ∈widehat G, let d_λ and χ_λ denote its dimension and character.
Exact mathematical passage
\chi_\lambda
\psi_{ij}Introduction passage · 3 passages
Search aliases: ψ_ij psi
Introduction passage · 03_supplement_03.tex, line 70
The actual partial trace and its support. The reduced matrix is defined by tracing the addressed index of the density matrix of the pure state: (ρ_B)_jℓ=∑_iψ_ijoverlineψ_iℓ. For the coefficient convention used here, the exact Gram identity is ρ_B=C^ T(C^ T)^*, rankρ_B=rank C=r(ψ). The transpose is important. With this convention, writing C^*C instead would conjugate the reduced matrix. The Lean source checks the entrywise identity and positivity of the resulting matrix.
Exact mathematical passage
(\rho_B)_{j\ell}=\sum_i\psi_{ij}\overline{\psi_{i\ell}}.
\OmegaIntroduction passage · 43 passages
Search aliases: Ω Omega
Introduction passage · Paper_VI_Traditional_Final.tex, line 3012
Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.
Exact mathematical passage
\Omega=\{1,\ldots,n\}
\hbarIntroduction passage · 1 passage
Search aliases: ℏ hbar
Introduction passage · 01_finite_core.tex, line 10
of finite dimension. A complete packet family is a finite collection satisfying P_α^*=P_α, P_α P_β=δ_αβP_α, ∑_α P_α=I. All operator norms in the analytic estimates are induced by the norm on the Hilbert space. The estimates do not use the entrywise maximum norm or the Frobenius norm. We write [A,B]=AB-BA and use units with ℏ=1. Parameter derivatives are real derivatives. A matrix X is supported on a code projector P when PXP=X.
Exact mathematical passage
\hbar=1
\ell^pIntroduction passage · 9 passages
Search aliases: ℓ^p ell
Introduction passage · Paper_V_Traditional_Final.tex, line 1725
Let 1≤ p≤∞. For a countable set G, write ℓ^p(G) for the usual complex sequence space. For an integer d≥1, when G= Z^d, translation by a∈ Z^d is (T_ax)(n)=x(n-a). Let δ_0 be the impulse at the origin.
Exact mathematical passage
\ell^p(G)
\hookrightarrowIntroduction passage · 11 passages
Search aliases: ↪ hookrightarrow
Introduction passage · Paper_X_Block27_Amended.tex, line 1255
General finite-field Frobenius comparison. Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a nonzero maximal ideal above (p), and the induced residue map is injective.
Exact mathematical passage
j:L\hookrightarrow\overline{\Q}
\nablaIntroduction passage · 22 passages
Search aliases: ∇ nabla
Introduction passage · 03_supplement_06.tex, line 61
A connection on the constructed bundle. The global formula is defined using the actual manifold derivative: ∇_Xs(x)=P(x) D_Ms(x)[X(x)]. Here D_M is represented by pathmvfderiv. A second definition, pathGlobalConnection.covariant, has values in the actual fiber: ∇ s(x):T_xM⟶ E_x. It is not only an ambient expression asserted to land in the range.
Exact mathematical passage
\nabla_Xs(x)=P(x)\,D_Ms(x)[X(x)].
\nabla^2Introduction passage · 3 passages
Search aliases: ∇^2 nabla
Introduction passage · 04_alternative_derivations.tex, line 19
rms and using the integration interval's length |t| yields Qe^-itHP ≤ |t| [H,P] . Proof of projected curvature with exterior forms. Let P be a smooth idempotent field and extend the projected connection to differential forms with values in the range bundle by ∇ω=P dω. Thus ∇^2 denotes the exterior covariant square, including the Lie bracket term in the exterior derivative when evaluated on vector fields. [Alternative proof of the curvature formula] For a section s with Ps=s, ∇^2s=P d(P ds)=P(dP)…
Exact mathematical passage
\nabla^2
\nabla_XIntroduction passage · 11 passages
Search aliases: ∇_X nabla
Introduction passage · 03_supplement_06.tex, line 61
A connection on the constructed bundle. The global formula is defined using the actual manifold derivative: ∇_Xs(x)=P(x) D_Ms(x)[X(x)]. Here D_M is represented by pathmvfderiv. A second definition, pathGlobalConnection.covariant, has values in the actual fiber: ∇ s(x):T_xM⟶ E_x. It is not only an ambient expression asserted to land in the range.
Exact mathematical passage
\nabla_Xs(x)=P(x)\,D_Ms(x)[X(x)].
\prod_{i=1}^{q}Introduction passage · 4 passages
Search aliases: ∏_i=1^q prod
Introduction passage · 02_foundations.tex, line 740
[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]
Exact mathematical passage
A=\prod_{i=1}^{q}M_{d_i}(D_i)
\sum_{a=0}^{d}Introduction passage · 2 passages
Search aliases: ∑_a=0^d sum
Introduction passage · supplement_06.tex, line 74
packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det(XI-D)=X^d+∑_a=0^d-1c_aX^a be its actual characteristic polynomial. The implementation reuses the pinned Mathlib Cayley--Hamilton theorem, expands polynomial evaluation into a finite sum and proves ∑_a=0^dχ_D[a]D^a=0, χ_D[d]=1. Here χ_D[a] denotes the coefficient at index a. The identities hold over any commutative coefficient ring. The …
Exact mathematical passage
\sum_{a=0}^{d}\chi_D[a]D^a=0, \qquad \chi_D[d]=1.
\sum_{a=0}^{d-1}Introduction passage · 3 passages
Search aliases: ∑_a=0^d-1 sum
Introduction passage · supplement_06.tex, line 70
Principal targets. targetCompanion.orbit_of_recurrence targetCompanion.companion_jump, Companion.source_companion_jump targetCompanion.packet_step, Companion.packet_advance targetCompanion.packet_refinement, Companion.packet_zero_step pageheading2. From the characteristic polynomial to a recurrenceTheorem 9.1, continued Put D=C^N and let χ_D(X)=det(XI-D)=X^d+∑_a=0^d-1c_aX^a be its actual characteristic polynomial. The implementation reuses the pinned Mathlib Cayley--Hamilton theorem, expands pol…
Exact mathematical passage
\chi_D(X)=\det(XI-D)=X^d+\sum_{a=0}^{d-1}c_aX^a
\sum_{h\ge0}Introduction passage · 1 passage
Search aliases: ∑_h≥0 sum ge
Introduction passage · root_scope.tex, line 97
Let Z_2⟨ x⟩ denote the restricted power series: their integral coefficients tend to zero dyadically. Extend A by mathscr A(x)=frac9^x+1-18(x+1)(2x+1), mathscr A(-1)=-fraclog98. The apparent singularity at x=-1 is removable. Since v_2(log9)=3, its convergent expansion is mathscr A(x)=fraclog9/81+2x ∑_h≥0frac(log9)^h(h+1)!(x+1)^h. It follows coefficientwise that mathscr A∈ Z_2⟨ x⟩^×, mathscr A≡1pmod2, and mathscr A(N)=A(N).
Exact mathematical passage
\mathscr A(x)=\frac{\log9/8}{1+2x} \sum_{h\ge0}\frac{(\log9)^h}{(h+1)!}(x+1)^h. \label{eq:root:source}
\sum_{i=1}^dIntroduction passage · 6 passages
Search aliases: ∑_i=1^d sum
Introduction passage · Paper_XI_Traditional_Final.tex, line 2171
i) · p^d-1(p+1)h_d · 3·2^d-1h_d and torsion number one. Here Swan denotes the Swan conductor, and a below denotes the ordinary pair conductor. An epsilon exponent for a specified additive character is a different quantity. The separate progress paper, Sections 22--24, reconciles these conventions and the actual-order hypotheses. Their self-pair conductor formulas are a(π^vee×π) · =p^2d-1+∑_i=1^d(p^2i-p^2i-2)h_i · · (p odd), a(π^vee×π) · =N^2-1+∑_i=1^d3·4^i-1h_i · · (p=2). These identities do not…
Exact mathematical passage
\begin{aligned} a(\pi^\vee\times\pi) &=p^{2d}-1+\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i &&(p\text{ odd}),\\ a(\pi^\vee\times\pi) &=N^2-1+\sum_{i=1}^d3\cdot4^{i-1}h_i &&(p=2). \end{aligned}
\sum_{i\in F}Introduction passage · 2 passages
Search aliases: ∑_i∈ F sum in
Introduction passage · 02_foundations.tex, line 544
dinal, and choose an index set I of cardinality r. Choose minimal generating families for M and N, index them by subsets of I, and fill any unused positions with zeros. Write the resulting families as (m_i)_i∈ I and (n_i)_i∈ I. For any m∈ M and n∈ N, their finite expressions involve a common finite subset F⊆ I after zero coefficients have been added. Thus m=∑_i∈ Fa_im_i, n=∑_i∈ Fb_in_i, and (m,n)=∑_i∈ F(a_i,b_i)(m_i,n_i). The family of paired vectors generates and has cardinality at most r. The …
Exact mathematical passage
m=\sum_{i\in F}a_im_i, \qquad n=\sum_{i\in F}b_in_i,
\sum_{j=0}^{d-1}Introduction passage · 2 passages
Search aliases: ∑_j=0^d-1 sum
Introduction passage · 02_supplement_04.tex, line 95
The represented matrix is defined recursively using the actual Kronecker product: M_0=[1], M_k+1=A_0⊗ M_k(A_1,…,A_k). The algorithm is defined independently. On each slice with fixed first coordinate, it recursively contracts the trailing modes. It then applies A_0 to the resulting first-coordinate fibers: y_i,u=∑_j=0^d-1(A_0)_ij Apply_k(A_1,…,A_k;x_j,·)_u. Induction, the Kronecker entry formula, and finite-sum distributivity prove Apply_k(A;x)=M_k(A)x. Thus correctness is not inferred merely fr…
Exact mathematical passage
y_{i,u}=\sum_{j=0}^{d-1}(A_0)_{ij}\, \operatorname{Apply}_{k}(A_1,\ldots,A_k;x_{j,\cdot})_u.
\sum_{j\in B(i)}Introduction passage · 2 passages
Search aliases: ∑_j∈ B(i) sum in
Introduction passage · Paper_VI_Traditional_Final.tex, line 3013
Let Ω={1,…,n} have strictly positive probabilities π_i. For a partition Q, let B(i) denote the block containing i. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j ∑_j∈ B(i)π_j. Equip C^n with ipfg_π = ∑_iπ_if_ioverlineg_i.
Exact mathematical passage
(P_\mathcal Qf)_i = \frac{\sum_{j\in B(i)}\pi_jf_j} {\sum_{j\in B(i)}\pi_j}.
\sum_{k,s}Introduction passage · 1 passage
Search aliases: ∑_k,s sum
Introduction passage · Paper_V_Traditional_Final.tex, line 475
[Operator-block form] If H denotes convolution by h, the same identity is H=∑_k,sP_N,kHP_N,s. Thus the familiar polyphase matrix is exactly the transfer-block matrix of the seed calculus.
Exact mathematical passage
H=\sum_{k,s}P_{N,k}HP_{N,s}.
\sum_{k=1}^{n}Introduction passage · 9 passages
Search aliases: ∑_k=1^n sum
Introduction passage · Paper_IV_Traditional_Final.tex, line 1162
Let D_n denote the displayed determinant, with D_0=1. Expansion along the last row gives D_n = ∑_k=1^n frac(n-1)!(n-k)!p_kD_n-k. Here the cofactor sign is canceled by the product of the negative superdiagonal entries -(n-k+1),…,-(n-1). Thus D_n/n! satisfies the same recurrence and initial value as b_N,n(r).
Exact mathematical passage
D_n = \sum_{k=1}^{n} \frac{(n-1)!}{(n-k)!}p_kD_{n-k}.
\sum_qIntroduction passage · 10 passages
Search aliases: ∑_q sum
Introduction passage · atlas-pages.tex, line 73
vpage5A filter becomes a matrix of actual delaysFIR TRANSFER / A FOUR-TAP EXAMPLEChapter V: Theorem 3.1. Take the causal filter h=(1,2,-1,1), with taps at 0,1,2,3, acting by y[n]=∑_qh[q]f[n-q]. Resolve the input into even and odd channels. Let D denote one delay in the channel index. vfigfir_transfer headReading one matrix entry The odd-input contribution to the even output is (2D+D^2)f_1. The tap at original delay one reaches the preceding odd channel sample. The tap at delay three reaches the …
Exact mathematical passage
y[n]=\sum_qh[q]f[n-q]
\sum_{q=0}^{n}Introduction passage · 20 passages
Search aliases: ∑_q=0^n sum
Introduction passage · 72_recovered.tex, line 47
Fix an odd prime p, a divisor mmid p-1, and put h=fracp-1m. The reduced standard window inverse is defined by ∑_q=0^n binommnmqe_q^(m,p) = δ_n0 in F_p. We prove that e_n+h^(m,p)=e_n^(m,p) (n≥1). The result first follows from subgroup symmetry in the finite Hurwitz algebra. A second proof shows that the standard window operator preserves the finite space of sequences that are h periodic after their constant term. The two arguments expose different parts of the same finite character structure.
Exact mathematical passage
\sum_{q=0}^{n} \binom{mn}{mq}e_q^{(m,p)} = \delta_{n0} \quad\text{in }\FF_p.
\sum_{r=0}^mIntroduction passage · 4 passages
Search aliases: ∑_r=0^m sum
Introduction passage · strengthening.tex, line 414
Let B_m denote the Bernoulli numbers with B_1=-1/2. The identity obtained by substituting u=e^t-1 into log(1+u)/u is B_m=∑_r=0^m(-1)^rfracr!r+1 {m r}, where the braces are Stirling numbers of the second kind. Each 2r!/(r+1) is dyadically integral. Indeed, if s=v_2(r+1)>0, then r≥2^s-1 and v_2(r!)≥ v_2((2^s-1)!)=2^s-1-s≥ s-1. The last inequality is 2^s≥2s. The case s=0 is immediate. Consequently 2B_m∈ Z_2 (m≥0). Thus the transfer factors below require no appeal to the von Staudt--Clausen theorem.
Exact mathematical passage
B_m=\sum_{r=0}^m(-1)^r\frac{r!}{r+1} \left\{\begin{matrix}m\\r\end{matrix}\right\}, \label{eq:bernoulli-stirling-denominator}
\sum_{r=0}^{n}Introduction passage · 15 passages
Search aliases: ∑_r=0^n sum
Introduction passage · 10_foundations.tex, line 129
In characteristic p, factorial division is unavailable in general, but the coefficient operation remains meaningful over every commutative ring: (f⋆ g)_n=∑_r=0^nbinom nr f_r g_n-r, (D^d f)_n=f_n+d, 1_n=1, · n=0, 0, · n>0. Here ⋆ is Hurwitz multiplication. The symbol D^d denotes a coefficient shift, which agrees with formal differentiation after characteristic-zero encoding.
Exact mathematical passage
(f\star g)_n=\sum_{r=0}^{n}\binom nr f_r g_{n-r},\qquad (D^d f)_n=f_{n+d},\qquad \mathbf1_n=\begin{cases}1,&n=0,\\0,&n>0.\end{cases}
\sum_{r\in A_+}Introduction passage · 3 passages
Search aliases: ∑_r∈ A_+ sum in
Introduction passage · 03_supplement_05.tex, line 71
A channel on the whole ambient space. Put Q=I-∑_r∈ A_+V_rV_r^*. The recovery is R(σ)=∑_r∈ A_+V_r^*σ V_r+Qσ Q. The complement is an orthogonal projector and ∑_r∈ A_+V_rV_r^*+Q^*Q=I. Hence the recovery is trace preserving. Its finite Kraus form also proves positivity after amplification by the identity on every finite auxiliary space. The complement term matters: the map is defined as a channel even outside the syndrome ranges.
Exact mathematical passage
Q=I-\sum_{r\in A_+}V_rV_r^*
\sum_{u\to v}Introduction passage · 3 passages
Search aliases: ∑_u→ v sum to
Introduction passage · 02_supplement_05.tex, line 47
iangularSolve Use the graph and levels constructed on the previous page. A vertex v carries an arbitrary vector space X_v over a field. The spaces may have different dimensions. Let D_v:X_voverset∼⟶X_v, T_vu:X_u⟶ X_v be an invertible diagonal map and linear transfer maps. The operator being solved is defined by its actual block action: (Ax)_v=D_vx_v+∑_u→ vT_vux_u. No solution, solve equation or precomputed topological certificate is assumed.
Exact mathematical passage
(Ax)_v=D_vx_v+\sum_{u\to v}T_{vu}x_u.
\sum_{\alpha\in I_0}Introduction passage · 2 passages
Search aliases: ∑_α∈ I_0 sum alpha in
Introduction passage · 03_packets_and_graphs.tex, line 997
For s∈ S, write s=∑_α∈ I_0P_α s. If β is unreachable from I_0, then for every A∈ A, P_β As = ∑_α∈ I_0P_β AP_α s =0 by [source reference]. The same holds for every finite linear combination of orbit vectors.
Exact mathematical passage
s=\sum_{\alpha\in I_0}P_\alpha s
\sum_{\beta\in I}Introduction passage · 3 passages
Search aliases: ∑_β∈ I sum beta in
Introduction passage · 01_manuscript.tex, line 154
[Block composition] Let A,B∈End(X). Then (AB)_γα = ∑_β∈ I A_γβB_βα. Consequently, supp(AB) ⊆ supp(A)∘supp(B), where relational composition is defined by (γ,α)∈ E∘ F when some β satisfies (γ,β)∈ E, (β,α)∈ F.
Exact mathematical passage
\boxed{ (AB)_{\gamma\alpha} = \sum_{\beta\in I} A_{\gamma\beta}B_{\beta\alpha}. }
\int_0^tIntroduction passage · 39 passages
Search aliases: ∫_0^t int
Introduction passage · 04_alternative_derivations.tex, line 12
ive conclusions are included there. Duhamel proof of leakage from a fixed packet. Let H=H^* on a finite-dimensional complex Hilbert space, P=P^*=P^2, Q=I-P, and t∈ R. [Alternative proof of the commutator leakage bound] Because QP=0, Qe^-itHP=Q[e^-itH,P]P. The Duhamel identity gives [e^-itH,P]=-i∫_0^t e^-i(t-s)H[H,P]e^-isH ds. Both unitary factors have norm one, so taking norms and using the integration interval's length |t| yields Qe^-itHP ≤ |t| [H,P] . Proof of projected curvature with exterior…
Exact mathematical passage
[e^{-itH},P]=-i\int_0^t e^{-i(t-s)H}[H,P]e^{-isH}\,ds.
\bigoplus_iIntroduction passage · 11 passages
Search aliases: ⊕_i bigoplus
Introduction passage · conditional_completion.tex, line 16
Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality of numerical traces. The selected family ⊕_iρ_i,λ^⊕ m_i has a split equivariant inclusion in the carrier. The finite trace sample uniquely determines…
Exact mathematical passage
V_{\mathrm{car},\lambda}\simeq\bigoplus_i\rho_{i,\lambda}^{\oplus c_i}, \qquad U_{0,\lambda}\simeq\bigoplus_i\rho_{i,\lambda}^{\oplus m_i}.
\bigoplus_{i=1}^{q}Introduction passage · 3 passages
Search aliases: ⊕_i=1^q bigoplus
Introduction passage · 02_foundations.tex, line 744
[Multiplicities in the regular module] Let A=∏_i=1^qM_d_i(D_i) and let S_i=D_i^d_i denote the natural simple left module for the ith factor, extended by zero across the other factors. Then, as a left module over A, A≅⊕_i=1^qS_i^⊕ d_i. [source reference]
Exact mathematical passage
\boxed{A\cong\bigoplus_{i=1}^{q}S_i^{\oplus d_i}.}
\bigoplus_{p\in\Irr(\bbk[t])}Introduction passage · 1 passage
Search aliases: ⊕_p∈Irr(bbk[t]) ⊕_p∈Irr(𝕜[t]) bigoplus in Irr bbk bigoplus in operatorname Bbbk
Introduction passage · 02_foundations.tex, line 886
[Primary decomposition for polynomial modules] Every module X over 𝕜[t] with finite dimension over the coefficient field decomposes as X=⊕_p∈Irr(𝕜[t])X_p, where Irr(𝕜[t]) denotes the monic irreducible polynomials, only finitely many components are nonzero, and every nonzero X_p is a finite direct sum of modules 𝕜[t]/(p^e).
Exact mathematical passage
X=\bigoplus_{p\in\Irr(\bbk[t])}X_p,
\bigotimesIntroduction passage · 3 passages
Search aliases: ⊗ bigotimes
Introduction passage · Paper_XII_Traditional_Final.tex, line 273
Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient topology on S/L is precisely this image ideal topology.
Exact mathematical passage
S=\bigotimes_{i=1}^{s}R_i
\bigotimes_{i=1}^{s}Introduction passage · 2 passages
Search aliases: ⊗_i=1^s bigotimes
Introduction passage · Paper_XII_Traditional_Final.tex, line 273
Write S=⊗_i=1^sR_i for the ordinary tensor product. Its ideal of definition is a= m_ OS+∑_i m_R_iS, L=∑_i I_iS. The scalar term retains the base adic topology when the family is empty. The ordinary quotient theorem gives a canonical algebra isomorphism S/L≅⊗_i=1^s(R_i/I_i), carrying the image of a to the defining ideal on the right. The quotient topology on S/L is precisely this image ideal topology.
Exact mathematical passage
S=\bigotimes_{i=1}^{s}R_i
\otimes_{\Lambda,\epsilon}Introduction passage · 5 passages
Search aliases: ⊗_Λ,ϵ otimes Lambda epsilon
Introduction passage · Paper_XII_Traditional_Final.tex, line 525
For M⊗_Λ,ϵ O, we use a right Λ module M. The left module convention is O⊗_Λ,ϵM. Throughout, a M denotes the augmentation relation submodule: it is generated by ma for right modules and by am for left modules, with a∈ a. This convention also applies to nonabelian groups in [source reference]. For abelian groups, both tensor orders agree over the commutative group algebra.
Exact mathematical passage
M\otimes_{\Lambda,\epsilon}\OO
\starIntroduction passage · 16 passages
Search aliases: ⋆ star
Introduction passage · 10_foundations.tex, line 129
In characteristic p, factorial division is unavailable in general, but the coefficient operation remains meaningful over every commutative ring: (f⋆ g)_n=∑_r=0^nbinom nr f_r g_n-r, (D^d f)_n=f_n+d, 1_n=1, · n=0, 0, · n>0. Here ⋆ is Hurwitz multiplication. The symbol D^d denotes a coefficient shift, which agrees with formal differentiation after characteristic-zero encoding.
Exact mathematical passage
(f\star g)_n=\sum_{r=0}^{n}\binom nr f_r g_{n-r},\qquad (D^d f)_n=f_{n+d},\qquad \mathbf1_n=\begin{cases}1,&n=0,\\0,&n>0.\end{cases}
\operatorname*Use in source · 1 passage
Search aliases: * operatorname
Use in source · Paper_V_Traditional_Final.tex, line 937
[label=(roman*)] item A global minimizer Λ_y^*∈*argmin_Λ∈ P( T)J_y(Λ) always exists. item If the minimizer is unique and at least one competing pruning exists, its gap Γ_y = min_Λ≠Λ_y^* (J_y(Λ)-J_y(Λ_y^*)) is attained and strictly positive. If the pruning set is a singleton, its only element minimizes every criterion. No real-valued minimum over an empty competitor set is introduced. item In the case with competitors, if new data y' satisfy sup_Λ∈ P( T) |J_y'(Λ)-J_y(Λ)| <fracΓ_y2, then Λ_y'^*=Λ_…
Exact mathematical passage
\Lambda_y^*\in\operatorname*{argmin}_{\Lambda\in\mathfrak P(\mathcal T)}J_y(\Lambda)
+^{m-\ell}Use in source · 1 passage
Search aliases: +^m-ℓ ell
Use in source · Paper_V_Traditional_Final.tex, line 2560
For 0≤ ℓ≤ m, the classical derivative of ϕ_m,τ on the two open sides is ϕ_m,τ^(ℓ)(t) = frac(t-τ)_+^m-ℓ(m-ℓ)!. This derivative has zero jump when ℓ<m and a unit jump when ℓ=m. Derivatives of order larger than m vanish on both open sides. It follows that subtracting the term a_q,mϕ_m,τ_q removes exactly the jump in the derivative of order m without changing the other classical trace jumps. Hence s_f has matching traces through order r-1.
Exact mathematical passage
\phi_{m,\tau}^{(\ell)}(t) = \frac{(t-\tau)_+^{m-\ell}}{(m-\ell)!}.
\bm1Use in source · 8 passages
Search aliases: 1 bm
Use in source · Paper_VI_Traditional_Final.tex, line 555
Choose λ≥max_i(-Q_ii). Then B=Q+λ I has nonnegative entries, so e^tB is entrywise nonnegative by its power series. Hence e^tQ=e^-λ te^tB is nonnegative. Since 1^TQ=0, differentiation gives fracddt 1^Te^tQ= 1^TQe^tQ=0, so 1^Te^tQ= 1^T.
Exact mathematical passage
\bm 1^TQ=0
\bm1^TUse in source · 8 passages
Search aliases: 1^T bm
Use in source · Paper_VI_Traditional_Final.tex, line 555
Choose λ≥max_i(-Q_ii). Then B=Q+λ I has nonnegative entries, so e^tB is entrywise nonnegative by its power series. Hence e^tQ=e^-λ te^tB is nonnegative. Since 1^TQ=0, differentiation gives fracddt 1^Te^tQ= 1^TQe^tQ=0, so 1^Te^tQ= 1^T.
Exact mathematical passage
\bm 1^TQ=0
\mathbf1_{[\tau,1]}Use in source · 1 passage
Search aliases: 1_[τ,1] mathbf tau
Use in source · atlas-pages.tex, line 578
vpage31Derivative jumps have their own packetsTRACE DATA / A STEP AND A RAMPChapter V: Theorem E7.2, Corollary E7.3. On [0,1], let H_τ=1_[τ,1] and take f(t)=t^2+2H_1/3(t)-3(t-2/3)_+. The function has a value jump 2 at 1/3 and a first-derivative jump -3 at 2/3. Its packet and remainder are j_f(t)=2ϕ_0,1/3(t)-3ϕ_1,2/3(t), s_f(t)=t^2. vfig[.92]ext_trace platecaptionThe displayed curves are exact polynomial pieces. The solid jump is not connected across its discontinuity. The dashed curve is the glo…
Exact mathematical passage
H_\tau=\mathbf1_{[\tau,1]}
\mathbf1_{[\tau,\infty)}Use in source · 1 passage
Search aliases: 1_[τ,∞) mathbf tau infty
Use in source · atlas-pages.tex, line 698
vpage37Haar sees a jump only through its position in the supportSCALE SEPARATION / A QUANTIFIED THRESHOLDChapter V: Theorem E10.1, Theorem E10.2, Corollary E10.3. Use a normalized Haar wavelet on an available interval of length L on the real line, positive on its left half and negative on its right half. Here H_τ is the half-line step 1_[τ,∞), so there is no artificial second endpoint jump. If u is the relative jump position in the support, then |⟨ aH_τ,ψ⟩| =|a|sqrt Lmin{u,1-u}. The coefficient …
Exact mathematical passage
\mathbf1_{[\tau,\infty)}
\mathbf1_{d\equiv r\ (N)}Use in source · 1 passage
Search aliases: 1_d≡ r (N) mathbf equiv
Use in source · supplement_09.tex, line 83
Modules. mcodeCyclotomicFiltration, mcodePrimeBlockPolynomial, and mcodeCyclicAmiceCoefficients. 2. Connection to the actual section. An integral coordinate estimate becomes relevant only after connecting it to the manuscript's generating functions. For a residue r∈ Z/N Z, define H_r(t)=∑_d≥01_d≡ r (N)fract^dd!, F_r(T)=H_r(log(1+T)). The implementation uses the actual formal logarithm and substitution in Q[[T]]. It proves H_r'=H_r-1, (1+T)F_r'=F_r-1, and hence the coefficient recurrence (d+1)coe…
Exact mathematical passage
H_r(t)=\sum_{d\ge0}\mathbf1_{d\equiv r\ (N)}\frac{t^d}{d!}, \qquad F_r(T)=H_r(\log(1+T)).
\mathbf1_{n_j\text{ even}}Use in source · 1 passage
Search aliases: 1_n_j even mathbf text
Use in source · dyadic_synthesis.tex, line 133
The dyadic floors cannot be replaced by the odd-prime dimension formula. At stage j, the centralizer grade has dimension 2^d-j. Its contribution to J^1/H^1 is (2^d-j+1-2^d-j) 1_n_j even. Every n_j for j<d is even, whereas n_d=q_1 is odd and contributes zero. The sum is exactly N-2. BK nondegeneracy makes this a symplectic space. A whole source packet of d planes fits if and only if N-2≥2d, equivalently d≥3.
Exact mathematical passage
\bigl(2^{d-j+1}-2^{d-j}\bigr)\,\mathbf1_{n_j\text{ even}}.
\mathbf1_{n\equiv r\, (\mathrm{mod}\,N)}Use in source · 1 passage
Search aliases: 1_n≡ r (mod N) mathbf equiv mathrm
Use in source · Paper_V_Traditional_Final.tex, line 316
Fix an integer N≥1. For 0≤ r<N, define (P_N,rx)[n] = 1_n≡ r (mod N)x[n]. The corresponding decimated channel is x_r[q]=x[Nq+r].
Exact mathematical passage
(P_{N,r}x)[n] = \mathbf 1_{n\equiv r\, (\mathrm{mod}\,N)}x[n].
\mathbf1_{r=0}Use in source · 2 passages
Search aliases: 1_r=0 mathbf
Use in source · supplement_09.tex, line 96
For a residue r∈ Z/N Z, define H_r(t)=∑_d≥01_d≡ r (N)fract^dd!, F_r(T)=H_r(log(1+T)). The implementation uses the actual formal logarithm and substitution in Q[[T]]. It proves H_r'=H_r-1, (1+T)F_r'=F_r-1, and hence the coefficient recurrence (d+1)coefTd+1F_r =coefTdF_r-1-dcoefTdF_r. The initial values are F_r(0)=1_r=0.
Exact mathematical passage
F_r(0)=\mathbf1_{r=0}
\mathbf1_{s=t,\,n=0}Use in source · 2 passages
Search aliases: 1_s=t, n=0 mathbf
Use in source · 03_weighted.tex, line 499
[Automatic finite-graph transfer-zeta rationality] Assume the vertex and edge sets are finite and every edge has positive integer weight. For every n, exact enumeration of edge tuples of lengths at most n, followed by endpoint and total-weight filtering, returns a_n(s,t). These coefficients obey the recurrence obtained by removing the first edge a_n(s,t) = 1_s=t, n=0 +∑_substacke: source(e)=s w(e)≤ n a_n-w(e)(target(e),t). Let T(t)=I-A(t), Q(t)=det T(t), P_s,t(t)=adj(T(t))_s,t. Then Q(0)=1, henc…
Exact mathematical passage
a_n(s,t) = \mathbf 1_{s=t,\,n=0} +\sum_{\substack{e:\,\operatorname{source}(e)=s\\w(e)\le n}} a_{n-w(e)}(\operatorname{target}(e),t).
\mathbf1_{\widehat\Omega}Use in source · 6 passages
Search aliases: 1_widehatΩ mathbf widehat Omega
Use in source · Paper_V_Traditional_Final.tex, line 707
Let (X,Σ,μ) be a measure space, with measurable masks Ω,widehatΩ⊆ X and scalar amplitudes a,widehat a∈ C (real amplitudes are a special case). Suppose a scalar jump packet is j=a 1_Ω, and an estimated packet is widehat j=widehat a 1_widehatΩ.
Exact mathematical passage
\widehat j=\widehat a\mathbf 1_{\widehat\Omega}.
\mathbf1_{\{|t|<1/4\}}Use in source · 1 passage
Search aliases: 1_{|t|<1/4} mathbf
Use in source · atlas-pages.tex, line 266
vpage16Why a Fourier cutoff cannot reproduce this jump pointwiseA CONTINUOUS PULSE / EXACT ENDPOINT VALUESChapter V: Section 6, Section 8. Use a separate period-one example on [-1/2,1/2): j(t)= 1_{|t|<1/4}, S_mj(t)=tfrac12+frac2π∑_n=1^mfracsin(π n/2)ncos(2π nt). This is the exact finite Fourier formula. The drawn curves sample it numerically. The endpoint calculation below is symbolic. vfigfourier_jump headOne value is fixed at every cutoff At t=±1/4, every summand vanishes because sin(π n/2)cos…
Exact mathematical passage
j(t)=\mathbf 1_{\{|t|<1/4\}},\qquad S_mj(t)=\tfrac12+\frac2\pi\sum_{n=1}^{m}\frac{\sin(\pi n/2)}{n}\cos(2\pi nt).
\mathbf1_\OmegaUse in source · 3 passages
Search aliases: 1_Ω mathbf Omega
Use in source · Paper_V_Traditional_Final.tex, line 742
Write a 1_Ω-widehat a 1_widehatΩ =(a-widehat a) 1_Ω +widehat a( 1_Ω- 1_widehatΩ). Apply the triangle inequality and use 1_Ω- 1_widehatΩ _L^p =μ(ΩtrianglewidehatΩ)^1/p. For the supremum estimate, |1_Ω-1_widehatΩ|≤1 pointwise, and this function vanishes everywhere when the symmetric difference is empty. The same pointwise bound also bounds the essential supremum. Nonempty null symmetric differences need not contribute to the essential norm. The stated bound remains valid in that case and need not …
Exact mathematical passage
|\mathbf1_\Omega-\mathbf1_{\widehat\Omega}|\le1
\mathbf1_{\Omega}Use in source · 3 passages
Search aliases: 1_Ω mathbf Omega
Use in source · Paper_V_Traditional_Final.tex, line 703
Let (X,Σ,μ) be a measure space, with measurable masks Ω,widehatΩ⊆ X and scalar amplitudes a,widehat a∈ C (real amplitudes are a special case). Suppose a scalar jump packet is j=a 1_Ω, and an estimated packet is widehat j=widehat a 1_widehatΩ.
Exact mathematical passage
j=a\mathbf 1_{\Omega},
\mathbf1_{\Omega_q}Use in source · 1 passage
Search aliases: 1_Ω_q mathbf Omega
Use in source · Paper_V_Traditional_Final.tex, line 1295
[Edge-packet approximation class] There should exist natural function spaces in which a signal is represented as f=s+∑_q a_q1_Ω_q, with separate norms for smooth approximation, boundary geometry, packet amplitude, and packet identification. A projector-resolved approximation theory on such spaces should yield rates that do not mix smooth truncation error with discontinuity-location error.
Exact mathematical passage
f=s+\sum_q a_q\mathbf1_{\Omega_q},
\mathbf1_{\Omega\triangle\widehat\Omega\ne\varnothing}Use in source · 1 passage
Search aliases: 1_ΩtrianglewidehatΩ≠∅ mathbf Omega triangle widehat Omega ne varnothing
Use in source · Paper_V_Traditional_Final.tex, line 723
orms, j-widehat j _L^p ≤ |a-widehat a| μ(Ω)^1/p + |widehat a| μ(ΩtrianglewidehatΩ)^1/p. If μ(Ω) and μ(ΩtrianglewidehatΩ) are finite, all terms in this displayed jump bound are finite and it is an ordinary real-valued L^p inequality. For the pointwise supremum, and consequently also the essential L^∞ seminorm, j-widehat j _∞ ≤ |a-widehat a|+|widehat a| 1_ΩtrianglewidehatΩ≠∅.
Exact mathematical passage
\|j-\widehat j\|_{\infty} \le |a-\widehat a|+|\widehat a|\, \mathbf 1_{\Omega\triangle\widehat\Omega\ne\varnothing}.
\bm aUse in source · 1 passage
Search aliases: a bm
Use in source · 02_strengthened.tex, line 920
The Pauli identity ( a·σ)( b·σ) =( a· b)I + i( a× b)·σ gives - iTr(P[∂_θ P,∂_ϕ P]) =frac12 n· (∂_θ n×∂_ϕ n) =frac12sinθ. Integrating the displayed curvature over the sphere and applying the stated normalization gives one.
Exact mathematical passage
(\bm a\cdot\bm\sigma)(\bm b\cdot\bm\sigma) =(\bm a\cdot\bm b)I +\ii(\bm a\times\bm b)\cdot\bm\sigma
\mathbf aUse in source · 1 passage
Search aliases: a mathbf
Use in source · 01_finite_core.tex, line 651
P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)·σ. Using Tr(P a·σ)= n· a therefore yields F_θϕ =frac12 n·(∂_θ nmathbin×∂_ϕ n) =frac12sinθ. The scalar sign changes if the opposite convention for A is chosen. This explicit calculation is not a separate entry among the 19 items in the Lean register.
Exact mathematical passage
\Tr(P\,\mathbf a\cdot\boldsymbol\sigma)=\mathbf n\cdot\mathbf a
\mathrm AUse in source · 4 passages
Search aliases: A mathrm
Use in source · atlas-pages.tex, line 90
Any other complete packet family still has exact block convolution. For the coordinate groups A and B, P_b e^(s+t)QP_a=∑_c∈{ A, B}(P_b e^sQP_c)(P_c e^tQP_a). The intermediate packet cannot generally be omitted. Only a reducing family eliminates all cross terms.
Exact mathematical passage
P_b e^{(s+t)Q}P_a=\sum_{c\in\{\mathrm A,\mathrm B\}}(P_b e^{sQ}P_c)(P_c e^{tQ}P_a).
a'Use in source · 7 passages
Search aliases: a'
Use in source · 71_native.tex, line 238
R(a,b), R(a',b), R(a',b') Longrightarrow R(a,b')
Exact mathematical passage
R(a,b),\ R(a',b),\ R(a',b')\quad\Longrightarrow\quad R(a,b')
A^{(1)}Use in source · 1 passage
Search aliases: A^(1)
Use in source · 01_manuscript.tex, line 1579
[Shared compilation for an operator family] Let operators A^(1),…,A^(m) use the same candidate tree. Give operator j a nonnegative reuse weight r_j. Replace the local keep cost and error by κ_v = κ_v^rm setup + ∑_j=1^m r_jκ_v^(j) and ϵ_v = ∑_j=1^m r_jϵ_v^(j). Then [source reference] gives the exact optimal shared pruning for the additive weighted objective.
Exact mathematical passage
A^{(1)},\ldots,A^{(m)}
A^{(m)}Use in source · 1 passage
Search aliases: A^(m)
Use in source · 01_manuscript.tex, line 1579
[Shared compilation for an operator family] Let operators A^(1),…,A^(m) use the same candidate tree. Give operator j a nonnegative reuse weight r_j. Replace the local keep cost and error by κ_v = κ_v^rm setup + ∑_j=1^m r_jκ_v^(j) and ϵ_v = ∑_j=1^m r_jϵ_v^(j). Then [source reference] gives the exact optimal shared pruning for the additive weighted objective.
Exact mathematical passage
A^{(1)},\ldots,A^{(m)}
a^{(m)}Use in source · 5 passages
Search aliases: a^(m)
Use in source · 72_recovered.tex, line 259
[Multiple selector] Define a^(m)∈ H( F_p) by a_n^(m) = 1, · mmid n,
Exact mathematical passage
a^{(m)}\in\Hcal(\FF_p)
\mathfrak A^{(\ell)}_{AB}Use in source · 1 passage
Search aliases: A^(ℓ)_AB mathfrak ell
Use in source · 01_finite_core.tex, line 1223
[Anomaly-free refinement] For a specified nontrivial quantum-geometric constraint model, there exists a coherent packet tower with nonzero physical spaces, compatible physical projectors and exact operator closure A^(ℓ)_AB=0 at every finite level. The compatible physical compression then vanishes as a consequence, not as a substitute for closure. The substantive research target is the operator condition. The earlier condition stated only in terms of compression was too weak. A choice of model, d…
Exact mathematical passage
\mathfrak A^{(\ell)}_{AB}=0
A^{-1}Use in source · 16 passages
Search aliases: A^-1
Use in source · 04_correction_transport.tex, line 540
[Nonlinear Schur derivative] At (u_0,v_0), DR = D_uF_P - D_vF_P A^-1D_uF_Q.
Exact mathematical passage
DR = D_uF_P - D_vF_P A^{-1}D_uF_Q.
a^{-1}Use in source · 5 passages
Search aliases: a^-1
Use in source · Paper_X_Block27_Amended.tex, line 1284
The choice of absolute prime and Frobenius lift is also controlled. Let ( Q_u,γ_u) and ( Q_v,γ_v) be any two actual arithmetic witnesses at the same rational prime. Transitivity on primes of A above (p) gives a∈ G_ Q with Q_v=a Q_u, γ_v^-1aγ_u a^-1∈ I_ Q_v. Consequently, any representation trivial on I_ Q_v has the same trace on the two witnesses. For representations factoring through L, triviality on this actual absolute inertia is derived whenever p∉ S_L. The restrictions of all absolute witne…
Exact mathematical passage
\mathfrak Q_v=a\mathfrak Q_u,\qquad \gamma_v^{-1}a\gamma_u a^{-1}\in I_{\mathfrak Q_v}.
a^{-2}Use in source · 1 passage
Search aliases: a^-2
Use in source · strengthening.tex, line 730
ties give sinh t-sin t =2asinh(at)cosh(bt)+2bcosh(at)sinh(bt), cosh t-cos t=2sinh(at)sinh(bt). Consequently D_3(t):=G_4,3(t)F_2(t) =frac32(fraccoth(at)at +fraccoth(bt)bt). This expression is a power series: its two polar terms cancel because a^-2+b^-2=0. Its constant is one, only degrees divisible by four survive, and the Bernoulli expansion gives D_3(t)=∑_i≥0d_ifract^4i(4i)!, d_i=3(-frac14)^i frac2^4i+2B_4i+2(4i+2)(4i+1). Again v_2(d_i)≥2i, and the inverse coefficients h_i have the same bound. …
Exact mathematical passage
a^{-2}+b^{-2}=0
A^{2k}Use in source · 1 passage
Search aliases: A^2k
Use in source · visual_atlas.tex, line 260
In this particular model, A^2=8I. Therefore v_2(A^2k)=3k, v_2(A^2k+1)=3k+1, where the matrix valuation means the minimum entry valuation. Three geometric terms leave a residual of valuation four and do not suffice modulo 32. Four terms leave valuation six and do suffice.
Exact mathematical passage
v_2(A^{2k})=3k,\qquad v_2(A^{2k+1})=3k+1,
A^{2k+1}Use in source · 1 passage
Search aliases: A^2k+1
Use in source · visual_atlas.tex, line 260
In this particular model, A^2=8I. Therefore v_2(A^2k)=3k, v_2(A^2k+1)=3k+1, where the matrix valuation means the minimum entry valuation. Three geometric terms leave a residual of valuation four and do not suffice modulo 32. Four terms leave valuation six and do suffice.
Exact mathematical passage
v_2(A^{2k})=3k,\qquad v_2(A^{2k+1})=3k+1,
A^4Use in source · 1 passage
Search aliases: A^4
Use in source · visual_atlas.tex, line 376
vpage15Walk counts make an endpoint cyclicGRAPH MODEL / THE PATH CERTIFICATEMain text: Worked graph model. Endpoint Krylov proof for every path size Let A be the adjacency matrix of the path with five vertices. The (i,j) entry of K=[e_1,Ae_1,A^2e_1,A^3e_1,A^4e_1] counts walks of length j-1 from vertex 1 to vertex i.
Exact mathematical passage
K=[e_1,Ae_1,A^2e_1,A^3e_1,A^4e_1]
A^{b_1}Use in source · 2 passages
Search aliases: A^b_1
Use in source · 04_correction_transport.tex, line 766
[Abelian cohomology class] When the structure group A is abelian, gauge classes of transports are naturally H^1(Γ,A) ≅ Hom(H_1(Γ, Z),A). After a basis of H_1(Γ, Z) is chosen, this group is identified with A^b_1.
Exact mathematical passage
A^{b_1}
A^{h+1}Use in source · 1 passage
Search aliases: A^h+1
Use in source · 02_supplement_01.tex, line 60
Nilpotence certificate. If no walk of length h+1 exists between any pair of packets, then A^h+1=0. Every block vanishes, and reconstruction forces the whole operator to vanish. A second theorem derives the required absence of walks from a supplied level map ℓ:I⟶{0,…,h}, A_bc≠0 Longrightarrow ℓ(c)<ℓ(b). It proves that a length-n walk increases the level by at least n. This certifies the acyclic case with a bounded height. This block does not implement or verify an algorithm that constructs a topo…
Exact mathematical passage
\boxed{A^{h+1}=0.}
A^jUse in source · 11 passages
Search aliases: A^j
Use in source · pages.tex, line 292
With A=diag(1,2), the vectors v and Av=(1,2)^T are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors as extra address operators separates the coordinates even when A=I_2. Both changes enlarge the addressing power, but in different ways. headWhen invariance is only approximate If A=D+K with D packet diagonal, then A^j-D^j ≤ jM^j-1 K , M=max{ A , D }. This controls departure from the decoupled iterate. It is not a claim that the di…
Exact mathematical passage
\|A^j-D^j\|\le jM^{j-1}\|K\|, \qquad M=\max\{\|A\|,\|D\|\}.
a^jUse in source · 1 passage
Search aliases: a^j
Use in source · Paper_X_Block27_Amended.tex, line 1575
Finite-exponential reconstruction and a cubic inverse. Let K be a characteristic-zero field and let (r,N) be supplied group data with nilpotent N and r(g)N=ν(g)Nr(g). For a supplied scalar map satisfying t(gh)=t(g)+ν(g)t(h), the formalization constructs ρ(g)=exp(t(g)N)r(g), exp(aN)=∑_j=0^d-1fraca^jj!N^j (N^d=0). Lean proves the representation laws, actual finite-copy and split-map compatibility, and joint continuity from continuity of N, r and t with the stated continuous vector-space operations…
Exact mathematical passage
\rho(g)=\exp(t(g)N)r(g),\qquad \exp(aN)=\sum_{j=0}^{d-1}\frac{a^j}{j!}N^j\quad(N^d=0).
A^{j-1-k}Use in source · 1 passage
Search aliases: A^j-1-k
Use in source · 01_manuscript.tex, line 1169
Use the telescoping identity A^j-D^j = ∑_k=0^j-1 A^j-1-kKD^k. Each term has norm at most M^j-1normK. The vector D^jv lies in the direct sum of local Krylov spaces.
Exact mathematical passage
A^j-D^j = \sum_{k=0}^{j-1} A^{j-1-k}KD^k.
a^kUse in source · 1 passage
Search aliases: a^k
Use in source · supplement_11.tex, line 54
Define a Kummer sequence over a commutative ring by P_a(f) Longleftrightarrow a^kmidΔ_1^k f(n) for all k,n. The exact discrete product rule Δ_1^s(fg)(n)=∑_i=0^sbinom si Δ_1^i f(n) Δ_1^s-ig(n+i) is proved by induction. It proves closure under products. Applied to fg=1, the same formula isolates Δ_1^s g(n)f(n+s). Induction and multiplication by g(n+s) prove closure under actual pointwise inverses.
Exact mathematical passage
\PP_a(f)\quad\Longleftrightarrow\quad a^k\mid\Delta_1^k f(n)\ \text{for all }k,n.
A^mUse in source · 3 passages
Search aliases: A^m
Use in source · Paper_VI_Traditional_Final.tex, line 420
The hypothesis makes A block diagonal. Every power A^m is block diagonal with respect to the same projectors, and therefore so is the exponential series.
Exact mathematical passage
A^m
A^{m+n}Use in source · 1 passage
Search aliases: A^m+n
Use in source · Paper_VI_Traditional_Final.tex, line 343
Both exponentials are absolutely convergent matrix power series. Since every power of A commutes with every other power, the Cauchy product gives e^sAe^tA =∑_m,n≥0fracs^mt^nm!n!A^m+n =∑_r≥0frac(s+t)^rr!A^r. Termwise differentiation is valid in finite dimension and gives the derivative identities.
Exact mathematical passage
e^{sA}e^{tA} =\sum_{m,n\ge0}\frac{s^mt^n}{m!n!}A^{m+n} =\sum_{r\ge0}\frac{(s+t)^r}{r!}A^r.
A^NUse in source · 2 passages
Search aliases: A^N
Use in source · 03_weighted.tex, line 328
W(γ)<LA_γ. Lean proves directly that q_L(S^pr_L,N) equals the mapped geometric inverse whenever q_L(A^N)=0, and hence is a two-sided inverse of q_L(I+A), where A=∑_eA_e=D^-1H. With the supplied two-sided inverse D^-1 of D, this gives the explicit two-sided inverse q_L (S^pr_L,ND^-1) of q_L(D+H). If every edge weight is at least m and L≤ mN, then the nilpotence condition follows from the filtration data because m|γ|≤ W(γ). For m>0, the executable choice N=lceil L/mrceil satisfies L≤ mN. This elem…
Exact mathematical passage
q_L(A^N)=0
A^nUse in source · 2 passages
Search aliases: A^n
Use in source · 02_supplement_01.tex, line 54
F_n+1(b,a) · =∑_c: A_bc≠0A_bcF_n(c,a). The checked identity is P_bA^nP_a=F_n(b,a) (n≥0). Its proof is induction using the exact composition theorem, not an assumed support recurrence. When no active walk exists, the corresponding contribution sum is zero, and hence the actual block of A^n vanishes.
Exact mathematical passage
\boxed{P_bA^nP_a=F_n(b,a)\qquad(n\ge0).}
A^{n-1}Use in source · 1 passage
Search aliases: A^n-1
Use in source · 05_worked_models.tex, line 349
…,s_r, 𝕜^n=span_𝕜{p(A_P_n)s_j:p∈𝕜[t], 1≤ j≤ r} }. Over C, the familiar spectral description has the distinct eigenvalues λ_k=2cosfrackπn+1, 1≤ k≤ n, and the normalized eigenvectors have endpoint coordinate proportional to sinfrackπn+1≠0. There is also a combinatorial proof valid over every field. Let K_n=[e_1,Ae_1,…,A^n-1e_1]. Its (i,j) entry counts walks of length j-1 from the first vertex to vertex i. This entry is zero when i>j, while the diagonal entry is one because there is a unique shorte…
Exact mathematical passage
K_n=[e_1,Ae_1,\ldots,A^{n-1}e_1].
a^{p^L}Use in source · 1 passage
Search aliases: a^p^L
Use in source · 10_foundations.tex, line 364
[Finite unit coordinates] Let p be prime, L≥0, and a∈( Z/p^L+1 Z)^×. Define ω_L(a)=a^p^L, ⟨ a⟩_L=aω_L(a)^-1. Then a=ω_L(a)⟨ a⟩_L, ω_L(a)^p-1=1, ⟨ a⟩_L≡1pmod p. The two factors are uniquely determined by these properties and are compatible with reduction to lower precision. For a∈( Z/2^L+2 Z)^×, there is instead a unique decomposition a=ϵ(a)⟨ a⟩_4, ϵ(a)∈{1,-1}, ⟨ a⟩_4≡1pmod4.
Exact mathematical passage
\omega_L(a)=a^{p^L},\qquad \langle a\rangle_L=a\omega_L(a)^{-1}.
a^{p^{\ell-1}}Use in source · 1 passage
Search aliases: a^p^ℓ-1 ell
Use in source · supplement_05.tex, line 81
ffPadic_coe, .standard_window_integral targetWindowIndependence.residue_independence, .reduced_unique targetWindowIndependence.two_window_addresses_congruent pageheading2. Explicit finite prime-power decompositionProposition 6.2; odd-prime branch Let p be prime, ℓ≥1, and a∈( Z/p^ℓ Z)^×. Define actual units in the same ring by ω_ℓ(a)=a^p^ℓ-1, ⟨ a⟩_ℓ=a ω_ℓ(a)^-1. The Lean source writes ℓ=L+1, so zero precision is excluded without truncated natural-number subtraction.
Exact mathematical passage
\boxed{\omega_\ell(a)=a^{p^{\ell-1}},\qquad \langle a\rangle_\ell=a\,\omega_\ell(a)^{-1}.}
a^{p^{\ell-1}(p-1)}Use in source · 1 passage
Search aliases: a^p^ℓ-1(p-1) ell
Use in source · supplement_05.tex, line 89
The root equation and residue are derived. Euler's unit-group exponent gives ω_ℓ(a)^p-1 =a^p^ℓ-1(p-1)=a^φ(p^ℓ)=1. Reduction to the actual field F_p commutes with powers. Its Frobenius identity gives overlineω_ℓ(a) =bar a^p^ℓ-1=bar a. Consequently, a=ω_ℓ(a)⟨ a⟩_ℓ, ω_ℓ(a)^p-1=1, overline⟨ a⟩_ℓ=1. The manuscript's finite existence conclusion is obtained directly. The proof does not assume a Hensel lift or an already supplied root representative.
Exact mathematical passage
\omega_\ell(a)^{p-1} =a^{p^{\ell-1}(p-1)}=a^{\varphi(p^\ell)}=1.
A^qUse in source · 1 passage
Search aliases: A^q
Use in source · Paper_V_Traditional_Final.tex, line 1076
[Cyclic product law] P_N,k(AB) = ∑_i+j≡ k (mod N)A_iB_j. For q∈ N, the qth-power identity is P_N,k(A^q) = ∑_i_1+⋯+i_q≡ k A_i_1⋯ A_i_q.
Exact mathematical passage
P_{N,k}(A^q) = \sum_{i_1+\cdots+i_q\equiv k} A_{i_1}\cdots A_{i_q}.
A^{\mathrm{symp}}Use in source · 2 passages
Search aliases: A^symp mathrm
Use in source · Paper_XI_Traditional_Final.tex, line 983
[Radical free envelope] For an arbitrary scalar projective action of A, the quotient A^symp=A/rad b_c is the maximal quotient detected by noncommuting projective intertwiners. If A is an elementary abelian p group, then A^symp is a symplectic F_p space.
Exact mathematical passage
\boxed{ A^{\mathrm{symp}}=A/\rad b_c }
A^TUse in source · 8 passages
Search aliases: A^T
Use in source · pages.tex, line 107
vpage7A transpose, not an adjointKRONECKER ACTION AND ARRAY ORDERChapter VIII: Theorem 4.5, Corollary 4.6. With column-major vectorization, the tensor identity is (A⊗ B)vec(X)=vec(BXA^T). Here the transpose records index order, not a Hermitian inner product. Use the explicitly complex matrix A=mat1 · i
Exact mathematical passage
(A\otimes B)\operatorname{vec}(X)=\operatorname{vec}(BXA^T).
a^TUse in source · 12 passages
Search aliases: a^T
Use in source · Paper_VI_Traditional_Final.tex, line 1593
Let A be a finite subgenerator: its off-diagonal entries are nonnegative and its column sums are nonpositive. These sign conditions alone allow nonabsorbing states. Eventual absorption is a separate hypothesis. Let p_0 be an initial transient-state distribution. Define p(t)=e^tAp_0, S(t)=1^Tp(t). The exit-rate row vector is a^T=-1^TA≥0.
Exact mathematical passage
a^T=-\bm1^TA\ge0.
A^\alphaUse in source · 8 passages
Search aliases: A^α alpha
Use in source · Paper_VI_Traditional_Final.tex, line 1194
Let A≥0 act on a fine Hilbert space, let B≥0 act on a coarse Hilbert space, and let C be a bounded coarsening map. Set D=CA-BC, D_α=CA^α-B^α C, 0<α<1.
Exact mathematical passage
D=CA-BC, \qquad D_\alpha=CA^\alpha-B^\alpha C, \qquad 0<\alpha<1.
a^{\varphi(p^\ell)}Use in source · 1 passage
Search aliases: a^φ(p^ℓ) varphi ell
Use in source · supplement_05.tex, line 89
The root equation and residue are derived. Euler's unit-group exponent gives ω_ℓ(a)^p-1 =a^p^ℓ-1(p-1)=a^φ(p^ℓ)=1. Reduction to the actual field F_p commutes with powers. Its Frobenius identity gives overlineω_ℓ(a) =bar a^p^ℓ-1=bar a. Consequently, a=ω_ℓ(a)⟨ a⟩_ℓ, ω_ℓ(a)^p-1=1, overline⟨ a⟩_ℓ=1. The manuscript's finite existence conclusion is obtained directly. The proof does not assume a Hensel lift or an already supplied root representative.
Exact mathematical passage
\omega_\ell(a)^{p-1} =a^{p^{\ell-1}(p-1)}=a^{\varphi(p^\ell)}=1.
\A^{\oplus r}Use in source · 1 passage
Search aliases: A^⊕ r A oplus mathcal oplus
Use in source · 02_foundations.tex, line 336
Topological completion of a presentation. [Completion of a dense free presentation] Let X be a complete Hausdorff locally convex space, let A⊆End(X) act continuously, and let S=(s_1,…,s_r). Put E:= A^⊕ r/N_S algebraically and let j:E→ X be the injective map induced by ε_S. Give E the locally convex topology transported from j(E), equivalently the topology generated by widetilde p(e):=p(j(e)) for the continuous seminorms p on X. Every a∈ A acts continuously on E in the transported topology and th…
Exact mathematical passage
E:=\A^{\oplus r}/N_S
A^{\oplus2}Use in source · 1 passage
Search aliases: A^⊕2 oplus
Use in source · 02_foundations.tex, line 701
[title=A local nilpotent algebra] Let A=𝕜[ε]/(ε^3) and M=A^⊕2. The radical is J=(ε), and M/JM≅𝕜^2. Therefore two seeds are necessary and sufficient. The extra nilpotent layers ε M and ε^2M are generated automatically once the two top directions are seeded.
Exact mathematical passage
M=A^{\oplus2}
A_{-c}Use in source · 1 passage
Search aliases: A_-c
Use in source · 20_arithmetic.tex, line 525
d product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [source reference] proves that multiplication adds charges. If FG=1, the same formula yields A_-cG=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G'). Induction, starting from the independently established integrality of the reciprocal G, proves that inversion negates the charge. Finally A_c+2=A_c-1, so increasing …
Exact mathematical passage
A_{-c}G=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G').
\mathcal A_0Use in source · 5 passages
Search aliases: A_0 mathcal
Use in source · strengthening.tex, line 339
Let Γ_R(P,Q) be the R-subalgebra generated by all γ_m(P) and γ_r(Q). Adjoin E, F with E^2=1+2P, F^2=1+2Q, and call the resulting algebra A_0. Its elements are finite expressions. No completion is being presumed. It is generated as a Γ_R(P,Q)-module by 1, E, F, E F. The functional maps this module to R. For an even basis term this is [source reference] and independence. For a basis term with an odd Euler factor the moment is zero, since P and Q are even polynomials in their respective variables.
Exact mathematical passage
\mathcal A_0
a_{0,1}Use in source · 1 passage
Search aliases: a_0,1
Use in source · 20_arithmetic.tex, line 672
The first Newton coefficients, derived from [source reference], are a_0,1=-2, a_1,1=-frac65, a_2,1=-frac1615, a_3,1=-frac3635. Their exact valuations are c_j. Theorem~[source reference] gives v_2(a_j,k)≥ k for all orders. To obtain the stronger logarithmic envelope required by Lemma~[source reference], only a finite initial segment needs improvement. The recurrence proves the following exact valuations: cll Address j · Further orders k · v_2(a_j,k) in that order
Exact mathematical passage
a_{0,1}=-2,\quad a_{1,1}=-\frac65,\quad a_{2,1}=-\frac{16}{15},\quad a_{3,1}=-\frac{36}{35}.
A_{00}Use in source · 1 passage
Search aliases: A_00
Use in source · 02_supplement_01.tex, line 80
-1 · -1, A^2=0. The coordinate block A_00 is nonzero, and a length-two active walk from zero to zero exists. Nevertheless the corresponding block of A^2 is zero. This is an exact counterexample to replacing support inclusion by equality.
Exact mathematical passage
A_{00}
A_0^{-1}Use in source · 1 passage
Search aliases: A_0^-1
Use in source · dyadic_synthesis.tex, line 358
There is also a recovery from compact data explicitly recording that root. With A_0=Λ(ϖ_K) and X=c - Ind_J^Gη, define (Tf)(g)=A_0^-1f(ϖ_Kg). Then X/(T-1)X≃π. After imposing the fixed central character, N/(F^× J) is cyclic of order N, and the finite average N^-1∑_i=0^N-1T^i selects this same extension. These operators act on complex representation spaces, so their scalar division by N is valid. This is a different recovery from the Block159 projector onto a sector of η. The exact kernel and finit…
Exact mathematical passage
(Tf)(g)=A_0^{-1}f(\varpi_Kg)
a_0^{-1}Use in source · 1 passage
Search aliases: a_0^-1
Use in source · strengthening.tex, line 64
Products preserve H_p because the coefficient at factorial degree d is (AB)_d=∑_i=0^dbinom di a_i b_d-i. The derivative D acts by the coefficient shift (DA)_d=a_d+1. Consequently every nonnegative power of D preserves the lattice. If A∈ H_p has unit constant term, its reciprocal belongs to the same lattice: with B=A^-1, b_0=a_0^-1, b_d=-a_0^-1∑_i=1^dbinom di a_i b_d-i. If A is supported at degrees divisible by N, this recurrence also proves that B has the same support.
Exact mathematical passage
b_0=a_0^{-1},\qquad b_d=-a_0^{-1}\sum_{i=1}^d\binom di a_i b_{d-i}. \label{eq:hurwitz-reciprocal-recurrence}
a_0^{[M]}Use in source · 2 passages
Search aliases: a_0^[M]
Use in source · 72_recovered.tex, line 1977
For a sequence a=(a_n)_n≥0 and M≥2, define a_r^[M](q)=a_Mq+r (0≤ r<M). The analysis map is P_Ma = (a_0^[M],…,a_M-1^[M]). Synthesis interleaves the branches by the rule a_Mq+r=a_r^[M](q).
Exact mathematical passage
\mathcal P_Ma = \left(a_0^{[M]},\ldots,a_{M-1}^{[M]}\right).
a_{1,1}Use in source · 1 passage
Search aliases: a_1,1
Use in source · 20_arithmetic.tex, line 672
The first Newton coefficients, derived from [source reference], are a_0,1=-2, a_1,1=-frac65, a_2,1=-frac1615, a_3,1=-frac3635. Their exact valuations are c_j. Theorem~[source reference] gives v_2(a_j,k)≥ k for all orders. To obtain the stronger logarithmic envelope required by Lemma~[source reference], only a finite initial segment needs improvement. The recurrence proves the following exact valuations: cll Address j · Further orders k · v_2(a_j,k) in that order
Exact mathematical passage
a_{0,1}=-2,\quad a_{1,1}=-\frac65,\quad a_{2,1}=-\frac{16}{15},\quad a_{3,1}=-\frac{36}{35}.
a_{1,g_1}Use in source · 1 passage
Search aliases: a_1,g_1
Use in source · 03_nonlinear_coordinates.tex, line 583
[Graded packet convolution] If a_j=∑_g∈ Ga_j,g and μ respects the grading, then P_kμ(a_1,…,a_r) = ∑_g_1+⋯+g_r=k μ(a_1,g_1,…,a_r,g_r).
Exact mathematical passage
P_k\mu(a_1,\ldots,a_r) = \sum_{g_1+\cdots+g_r=k} \mu(a_{1,g_1},\ldots,a_{r,g_r}).
A_1^sUse in source · 1 passage
Search aliases: A_1^s
Use in source · 20_arithmetic.tex, line 541
Packing g_j into degrees divisible by four gives K_4,j. Its source property is precisely charge-one flatness. Inversion gives charge -1, and the charge shift restores charge one for the reciprocal P_4,j. If dec(F)(d)=d![t^d]F, the exact coefficient identity is dec(A_1^sF)(d) =2^-sΔ_4^sdec(F)(d). At d=4n, packing turns the right side into 2^-sΔ_1^sC_j(n). This extraction requires no negative degree and therefore holds at every n, including zero. It proves the unit-step bound leanKomorebiPaperIX.D…
Exact mathematical passage
\operatorname{dec}(A_1^sF)(d) =2^{-s}\Delta_4^s\operatorname{dec}(F)(d).
A_2Use in source · 5 passages
Search aliases: A_2
Use in source · visual_atlas.tex, line 337
6 · 7 · 8 · 9 · 10, choose A_1 from columns 1,2 of W, A_2 from columns 3,4, and A_3 with first column equal to column 5 of W and second column zero. Then A_1V_1+A_2V_2+A_3V_3=W. The same construction works for arbitrary entries of W, so the lower bound is attained: srank_M_2( Q)M_2×5( Q)=lceil5/2rceil=3.
Exact mathematical passage
A_2
\mathcal A_2Use in source · 5 passages
Search aliases: A_2 mathcal
Use in source · visual_atlas.tex, line 87
Define X_N^rm pkt=⊕_j P_N,jX, A_Nx=(P_N,jx)_j, S_N(x_j)_j=∑_jx_j. Completeness and mutual annihilation give both inverse laws. Consequently, J= A_4 S_2, C= A_2 S_4, CJ=I, JC=I. The larger list of fine packets has smaller individual packet spaces.
Exact mathematical passage
J=\mathcal A_4\mathcal S_2,\qquad C=\mathcal A_2\mathcal S_4, \qquad CJ=I,\quad JC=I.
a_2Use in source · 12 passages
Search aliases: a_2
Use in source · visual_atlas.tex, line 311
From QZ_0,1=2t^2 one obtains a_0=a_1=0, a_2=2, a_n=a_n-1+2a_n-3 (n≥3). The numerator is responsible for the initial exception. Omitting it would incorrectly force the entire series to vanish.
Exact mathematical passage
a_0=a_1=0,\quad a_2=2,\qquad a_n=a_{n-1}+2a_{n-3}\quad(n\ge3).
a_{2,1}Use in source · 1 passage
Search aliases: a_2,1
Use in source · 20_arithmetic.tex, line 672
The first Newton coefficients, derived from [source reference], are a_0,1=-2, a_1,1=-frac65, a_2,1=-frac1615, a_3,1=-frac3635. Their exact valuations are c_j. Theorem~[source reference] gives v_2(a_j,k)≥ k for all orders. To obtain the stronger logarithmic envelope required by Lemma~[source reference], only a finite initial segment needs improvement. The recurrence proves the following exact valuations: cll Address j · Further orders k · v_2(a_j,k) in that order
Exact mathematical passage
a_{0,1}=-2,\quad a_{1,1}=-\frac65,\quad a_{2,1}=-\frac{16}{15},\quad a_{3,1}=-\frac{36}{35}.
A_3Use in source · 4 passages
Search aliases: A_3
Use in source · visual_atlas.tex, line 337
6 · 7 · 8 · 9 · 10, choose A_1 from columns 1,2 of W, A_2 from columns 3,4, and A_3 with first column equal to column 5 of W and second column zero. Then A_1V_1+A_2V_2+A_3V_3=W. The same construction works for arbitrary entries of W, so the lower bound is attained: srank_M_2( Q)M_2×5( Q)=lceil5/2rceil=3.
Exact mathematical passage
A_3
a_3Use in source · 2 passages
Search aliases: a_3
Use in source · monograph_atlas.tex, line 44
vpage3One seed, eight addressesCYCLICITY DEPENDS ON THE GRAMMARChapter VII: Proposition II.2.5. Start with ψ_0=ket000 and allow the three bit flip operators X_1,X_2,X_3. For a binary address a=(a_1,a_2,a_3), put W_a=X_1^a_1X_2^a_2X_3^a_3. W_aψ_0=keta_1a_2a_3. The drawing shows the actual eight states reached by these words. Dashed edges connect the two squares by changing the first bit. The other edges change one of the remaining bits. vfigaddresses headTwo different meanings of ``reach'' The ei…
Exact mathematical passage
a=(a_1,a_2,a_3)
A_{3,1}Use in source · 1 passage
Search aliases: A_3,1
Use in source · 72_recovered.tex, line 1277
[Exact change between local source charts] Editorial range correction. This comparison holds for the even source subsequences, equivalently the evenized generating functions H_p. It does not identify their full odd-degree source values. For p=3, the original odd source has A_3,1=S_1(3)=3, whereas every odd dyadic source coefficient is zero. The even kernel L_3K_2 cannot change that odd coefficient. Read the coefficient consequence below only at even indices. The displayed identity H_p=L_pK_ℓ H_ℓ…
Exact mathematical passage
A_{3,1}=S_1(3)=3
a_{3,1}Use in source · 1 passage
Search aliases: a_3,1
Use in source · 20_arithmetic.tex, line 672
The first Newton coefficients, derived from [source reference], are a_0,1=-2, a_1,1=-frac65, a_2,1=-frac1615, a_3,1=-frac3635. Their exact valuations are c_j. Theorem~[source reference] gives v_2(a_j,k)≥ k for all orders. To obtain the stronger logarithmic envelope required by Lemma~[source reference], only a finite initial segment needs improvement. The recurrence proves the following exact valuations: cll Address j · Further orders k · v_2(a_j,k) in that order
Exact mathematical passage
a_{0,1}=-2,\quad a_{1,1}=-\frac65,\quad a_{2,1}=-\frac{16}{15},\quad a_{3,1}=-\frac{36}{35}.
A_4Use in source · 5 passages
Search aliases: A_4
Use in source · Paper_IV_Traditional_Final.tex, line 2208
[Exact fourth-order strings] For N=4, D_4(z) = cos(frac zsqrt2) cosh(frac zsqrt2). The positive zeros are exactly τ_m = sqrt2π(m+frac12) = A_4(m+frac12). Rotation by the fourth roots of unity gives the real and imaginary zero strings. In this case the error term in [source reference] is identically zero.
Exact mathematical passage
\tau_m = \sqrt2\pi\left(m+\frac12\right) = A_4\left(m+\frac12\right).
\mathcal A_4Use in source · 2 passages
Search aliases: A_4 mathcal
Use in source · visual_atlas.tex, line 87
Define X_N^rm pkt=⊕_j P_N,jX, A_Nx=(P_N,jx)_j, S_N(x_j)_j=∑_jx_j. Completeness and mutual annihilation give both inverse laws. Consequently, J= A_4 S_2, C= A_2 S_4, CJ=I, JC=I. The larger list of fine packets has smaller individual packet spaces.
Exact mathematical passage
J=\mathcal A_4\mathcal S_2,\qquad C=\mathcal A_2\mathcal S_4, \qquad CJ=I,\quad JC=I.
A_4^{-4s}Use in source · 1 passage
Search aliases: A_4^-4s
Use in source · Paper_IV_Traditional_Final.tex, line 2872
[Exact zeta models in orders two and four] For N=2, Z_2^red(s) = π^-2sζ(2s,frac12) and Z_2^abs(s) = 2π^-sζ(s,frac12). For N=4, put A_4=sqrt2π. Then Z_4^red(s) = A_4^-4sζ(4s,frac12) and Z_4^abs(s) = 4A_4^-sζ(s,frac12). In these two cases the entire correction terms in [source reference] vanish identically.
Exact mathematical passage
\mathfrak Z_4^{\mathrm{red}}(s) = A_4^{-4s}\zeta\left(4s,\frac12\right)
A_4^{-s}Use in source · 1 passage
Search aliases: A_4^-s
Use in source · Paper_IV_Traditional_Final.tex, line 2878
[Exact zeta models in orders two and four] For N=2, Z_2^red(s) = π^-2sζ(2s,frac12) and Z_2^abs(s) = 2π^-sζ(s,frac12). For N=4, put A_4=sqrt2π. Then Z_4^red(s) = A_4^-4sζ(4s,frac12) and Z_4^abs(s) = 4A_4^-sζ(s,frac12). In these two cases the entire correction terms in [source reference] vanish identically.
Exact mathematical passage
\mathfrak Z_4^{\mathrm{abs}}(s) = 4A_4^{-s}\zeta\left(s,\frac12\right).
A_4^4Use in source · 3 passages
Search aliases: A_4^4
Use in source · Paper_IV_Traditional_Final.tex, line 2893
For N=2, the reduced zeros are π^2(m+frac12)^2, m≥0. For N=4, [source reference] gives the reduced zeros A_4^4(m+frac12)^4, m≥0. Summing their negative powers proves the reduced formulas. The absolute formulas follow from Z_N^abs(s) = N Z_N^red(s/N).
Exact mathematical passage
A_4^4\left(m+\frac12\right)^4, \qquad m\geq0.
\mathcal A_aUse in source · 1 passage
Search aliases: A_a mathcal
Use in source · visual_atlas.tex, line 28
A_a, S_a · Inverse analysis and synthesis maps, which may be maps of sets. · pagerefV10
Exact mathematical passage
\mathcal A_a,\mathcal S_a
\mathcal A_{ab}Use in source · 3 passages
Search aliases: A_ab mathcal
Use in source · 03_supplement_01.tex, line 189
A constraint diagnostic that vanishes automatically. Section 12 defines the physical projector using the exact common kernel of the constraints. In finite dimensions this gives C_aP=0 for every a. Absorb the factor i and any structure coefficient into f_c, and write A_ab=C_aC_b-C_bC_a-∑_cf_cC_c. [Exact common-kernel compression] If C_cP=0 for every c, then A_abP=0, P A_abP=0. No idempotence or adjoint property of P is needed for this implication. Associativity gives A_abP=C_a(C_bP)-C_b(C_aP)-∑_c…
Exact mathematical passage
\mathcal A_{ab}=C_aC_b-C_bC_a-\sum_cf_cC_c.
\mathfrak A_{AB}Use in source · 3 passages
Search aliases: A_AB mathfrak
Use in source · 01_finite_core.tex, line 1199
At one finite level, let widehat C_A be a finite family of matrix constraints, with fixed scalar structure coefficients, and let Π^phys be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients f_AB^ C, define the anomaly operator A_AB = [widehat C_A,widehat C_B] -i∑_C f_AB^ Cwidehat C_C. Its physical packet is Π^phys A_ABΠ^phys. In this exact finite setting, widehat C_CΠ^phys=0 for every C. Associativity gives A_ABΠ^phys=0, Π^phys…
Exact mathematical passage
\boxed{ \mathfrak A_{AB} = [\widehat C_A,\widehat C_B] -i\sum_C f_{AB}^{\ \ C}\widehat C_C. }
\mathfrak A_{ab}Use in source · 2 passages
Search aliases: A_ab mathfrak
Use in source · 02_strengthened.tex, line 2206
If structure coefficients f_ab^phantomabc are prescribed, define the fixed structure anomaly A_ab = [C_a,C_b] - i∑_cf_ab^phantomabcC_c. Exact realization of the prescribed algebra is the stronger condition A_ab=0.
Exact mathematical passage
\mathfrak A_{ab} = [C_a,C_b] -\ii\sum_cf_{ab}^{\phantom{ab}c}C_c.
A_bUse in source · 1 passage
Search aliases: A_b
Use in source · 03_supplement_06.tex, line 49
Smooth overlap maps and rank. The coordinate transition is g_ab(x)=A_b(x)B_a(x). It is smooth on U_a∩ U_b, agrees with the actual change between local homeomorphisms, and satisfies g_aa=I, g_bcg_ab=g_ac. A continuous linear equivalence between E_a and E_x also gives local constancy of fiber dimension. No single constant rank across disconnected components is assumed.
Exact mathematical passage
g_{ab}(x)=A_b(x)B_a(x)
a_{B^s f}Use in source · 1 passage
Search aliases: a_B^s f
Use in source · supplement_08.tex, line 91
The earlier coefficient criterion is recovered exactly. Define the literal coefficient defect (δ a)(d)=a(d+p^2)-a(d+p). The new source proves, for every s,d, a_B^s f(d)=p^-s(δ^s a_f)(d). For an integral coefficient x, it proves the bridge p^-sx∈ Z_p Longleftrightarrow p^smid x in Z_p. Thus the differential definition is equivalent to the existing coefficient-level Dwork criterion. Division occurs in Q_p, while the assertion after division is membership in Z_p. It is not divisibility in the field…
Exact mathematical passage
a_{B^s f}(d)=p^{-s}(\delta^s a_f)(d).
A_{ba}Use in source · 7 passages
Search aliases: A_ba
Use in source · 02_strengthened.tex, line 12
The elementary coordinate object is a complete family of orthogonal projectors P_aP_b=δ_abP_a, ∑_a∈ IP_a=I. Every vector has packet coordinates P_aψ. Every operator has transfer blocks A_ba=P_bAP_a. These identities are simple. Their value appears when the same coordinates are used across representation theory, circuits, channels, error correction, effective dynamics, and geometric limits.
Exact mathematical passage
A_{ba}=P_bAP_a.
A_{bc}Use in source · 4 passages
Search aliases: A_bc
Use in source · 02_supplement_01.tex, line 23
Composition. Inserting the projector identity between factors yields (AB)_ba=∑_c A_bcB_ca. If the left side is nonzero, at least one intermediate c has both A_bc≠0 and B_ca≠0. Consequently supp(AB)⊆supp(A)∘supp(B). No equality of supports is asserted.
Exact mathematical passage
\boxed{(AB)_{ba}=\sum_c A_{bc}B_{ca}.}
A_CUse in source · 3 passages
Search aliases: A_C
Use in source · residue_continuation_body.tex, line 138
Ordinary class characters: an unconditional power saving. The preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl( O)→ C, define the actual finite ideal sum S_w(X)=∑_substackI≠0
Exact mathematical passage
A_C(X)=c_KX+O_{K,C}(X^{1-1/d}),\qquad c_K=\frac{2^{r_1}(2\pi)^{r_2}R_K}{\#\mu(K)\sqrt{\absn{\Delta_K}}}.
A_{c+2}Use in source · 1 passage
Search aliases: A_c+2
Use in source · 20_arithmetic.tex, line 530
atives preserve H_2 and commute with A_c. Induction on the number of iterations in [source reference] proves that multiplication adds charges. If FG=1, the same formula yields A_-cG=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G'). Induction, starting from the independently established integrality of the reciprocal G, proves that inversion negates the charge. Finally A_c+2=A_c-1, so increasing a charge by two preserves flatness. These are leanKomorebiPaperIX.ChargedDworkClosure.B_mul, leanKomorebiPaperIX.Cha…
Exact mathematical passage
A_{c+2}=A_c-1
A_{c+d}Use in source · 1 passage
Search aliases: A_c+d
Use in source · 20_arithmetic.tex, line 517
For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [source reference] proves that multiplication adds charges. If FG=1, the same formula yields A_-cG=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G'). Induction, starting…
Exact mathematical passage
\label{eq:arith-charged-product} A_{c+d}(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'.
A_{c-1}Use in source · 3 passages
Search aliases: A_c-1
Use in source · 20_arithmetic.tex, line 216
Set H_c(t)=∑_n≥0fract^Nn+c(Nn+c)!, A_c(t)=∑_n≥0fract^Nn(Nn+c)!, R_c(t)=fracA_c-1(t)A_c(t) (1≤ c<N). The denominator of this last quotient has nonzero constant coefficient. The section identities give R_cH_c=tH_c-1, R_c(0)=c, j! P_N,j= P_N,0∏_c=1^jR_c. Cancellation of consecutive source sections proves the product identity and ties the ratios directly to the family under study.
Exact mathematical passage
H_c(t)=\sum_{n\ge0}\frac{t^{Nn+c}}{(Nn+c)!},\qquad A_c(t)=\sum_{n\ge0}\frac{t^{Nn}}{(Nn+c)!},\qquad R_c(t)=\frac{A_{c-1}(t)}{A_c(t)}\quad(1\le c<N).
A_c^sUse in source · 1 passage
Search aliases: A_c^s
Use in source · 20_arithmetic.tex, line 515
For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [source reference] proves that multiplication adds charges. If FG=1, the same formula yields A_-cG=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G'). Induction, starting…
Exact mathematical passage
A_c^s
A_{C_n}Use in source · 1 passage
Search aliases: A_C_n
Use in source · 05_worked_models.tex, line 372
Two adjacent vertex vectors generate the adjacency module of every simple cycle. Over C, a Fourier quotient built from a primitive root of unity supplies a matching scalar obstruction of dimension two. Hence, for every n≥3, two is the least finite number of adjacency seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any f…
Exact mathematical passage
{ 2=\min\left\{r\in\N: \exists s_1,\ldots,s_r,\quad \C^n=\Span_{\C}\{p(A_{C_n})s_j:p\in\C[t],\ 1\le j\le r\} \right\}. }
A_dUse in source · 1 passage
Search aliases: A_d
Use in source · 20_arithmetic.tex, line 517
For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [source reference] proves that multiplication adds charges. If FG=1, the same formula yields A_-cG=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G'). Induction, starting…
Exact mathematical passage
\label{eq:arith-charged-product} A_{c+d}(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'.
a_dUse in source · 8 passages
Search aliases: a_d
Use in source · 20_arithmetic.tex, line 231
At the singular addresses c=ap, 1≤ a<m, use the weighted ring W_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping the falling factorial (x)_d=∏_i=0^d-1(x-i) into consecutive blocks of length p then gives an integral residue-coordinate factor p^v_p(lfloor d/prfloor!)…
Exact mathematical passage
\mathcal W_p=\left\{\sum_{d\ge0}a_dT^d: p^{\lfloor d/p\rfloor}a_d\in\Z_p\right\}.
a_{d+1}Use in source · 5 passages
Search aliases: a_d+1
Use in source · 20_arithmetic.tex, line 272
To deduce flatness from the weighted estimate, use Θ=(1+T)d/dT. Both Θ and (Θ^p^2-Θ^p)/p preserve W_p. For the first operator, its coefficient formula is (d+1)a_d+1+da_d. Whenever the floor weight rises, the factor d+1 provides the missing p. For the second operator, the proof uses its full Leibniz correction and the scalar congruence p^2mid b^p^2-b^p. Weighted monomials factor as a_dT^d=(p^lfloor d/prfloora_d) (T^p/p)^lfloor d/prfloorT^dbmod p. The operator preserves these monomials and hence f…
Exact mathematical passage
(d+1)a_{d+1}+da_d
a_{d+pk}Use in source · 1 passage
Search aliases: a_d+pk
Use in source · 20_arithmetic.tex, line 333
Here is the coefficient extraction explicitly. If a_d=d![t^d]F, then d![t^d]B_p^kF=p^-kΔ_p(p-1)^ka_d+pk. For F=j! P_N,j and k≤ mn, take d=mpn-pk. Support in multiples of mp turns this into p^-kΔ_h^kU_N,j(n), so flatness gives divisibility by p^k. For k>mn, expand Δ_h^k-mn applied to Δ_h^mnU_N,j(n). Every shifted row is at least n, so each summand is divisible by p^mn. Finally Δ_uh=((I+Δ_h)^u-I), whose kth power for u>0 is an integral combination of powers of Δ_h of order at least k. The case u=0…
Exact mathematical passage
d![t^d]B_p^kF=p^{-k}\Delta_{p(p-1)}^ka_{d+pk}.
a_d^2Use in source · 1 passage
Search aliases: a_d^2
Use in source · atlas-pages.tex, line 205
. For actual Haar packets, set P_v=W_v^*W_v. Parent-to-child conservation gives P_v=P_v0+P_v1. Replacing unselected parents by their children until the complete pruning is reached proves ∑_v∈ΛP_v=I. This is why the coefficients at different depths still form an exact coordinate system. headAn exhaustive finite check At depth three the pruning count obeys a_0=1 and a_d+1=1+a_d^2, giving 1,2,5,26. The supplied suite checks all 26 covering conditions and all 26 actual 16×16 projector-sum identities…
Exact mathematical passage
a_{d+1}=1+a_d^2
A_eUse in source · 17 passages
Search aliases: A_e
Use in source · 03_weighted.tex, line 48
[Weighted inverse] Let B be a complete filtered algebra. Let D∈ B be invertible with D,D^-1∈ F^0 B. Let E be a finite set and suppose T=D+∑_e∈ EN_e, N_e∈ F^w(e) B, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 = ∑_γ(-1)^|γ|A_γ D^-1. The series converges in the filtration, and the term indexed by γ belongs to F^W(γ) B.
Exact mathematical passage
A_e=D^{-1}N_e.
A_{e_1}Use in source · 2 passages
Search aliases: A_e_1
Use in source · 03_weighted.tex, line 52
[Weighted inverse] Let B be a complete filtered algebra. Let D∈ B be invertible with D,D^-1∈ F^0 B. Let E be a finite set and suppose T=D+∑_e∈ EN_e, N_e∈ F^w(e) B, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 = ∑_γ(-1)^|γ|A_γ D^-1. The series converges in the filtration, and the term indexed by γ belongs to F^W(γ) B.
Exact mathematical passage
A_\gamma=A_{e_r}\cdots A_{e_1}, \qquad W(\gamma)=\sum_{j=1}^rw(e_j).
A_{e_r}Use in source · 2 passages
Search aliases: A_e_r
Use in source · 03_weighted.tex, line 52
[Weighted inverse] Let B be a complete filtered algebra. Let D∈ B be invertible with D,D^-1∈ F^0 B. Let E be a finite set and suppose T=D+∑_e∈ EN_e, N_e∈ F^w(e) B, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 = ∑_γ(-1)^|γ|A_γ D^-1. The series converges in the filtration, and the term indexed by γ belongs to F^W(γ) B.
Exact mathematical passage
A_\gamma=A_{e_r}\cdots A_{e_1}, \qquad W(\gamma)=\sum_{j=1}^rw(e_j).
a_fUse in source · 4 passages
Search aliases: a_f
Use in source · supplement_08.tex, line 66
pageheading2. The actual integral Hurwitz latticeConnecting differential expressions to coefficient divisibility The specialization works inside Q_p[[t]] and uses factorial-scaled coefficients a_f(d)=d![t^d]f, I_p={f∈ Q_p[[t]]:a_f(d)∈ Z_p for every d}. The source constructs I_p as a subring. Actual exponential-series multiplication gives a_fg(d)=∑_r=0^dbinom dr a_f(r)a_g(d-r), a_Uf(d)=a_f(d+1). These identities prove multiplication and derivative stability of the lattice. Membership does not req…
Exact mathematical passage
a_f(d)=d![t^d]f,\qquad I_p=\{f\in\Q_p[[t]]:a_f(d)\in\Z_p\text{ for every }d\}.
a_{f^{-1}}Use in source · 1 passage
Search aliases: a_f^-1
Use in source · supplement_08.tex, line 79
The reciprocal is integral. For f∈ I_p with constant coefficient one, the usual power-series reciprocal is used. Its inverse equation yields the exact coefficient recurrence a_f^-1(d)=δ_d0-∑_r=0^d-1 binom dr a_f^-1(r)a_f(d-r). Strong induction proves that all of these coefficients are integral. Combining this construction with the preceding all-order theorem proves flatness of the actual reciprocal. No flatness or integrality of that reciprocal is assumed in the normalized-series specialization.
Exact mathematical passage
a_{f^{-1}}(d)=\delta_{d0}-\sum_{r=0}^{d-1} \binom dr a_{f^{-1}}(r)a_f(d-r).
a_{fg}Use in source · 1 passage
Search aliases: a_fg
Use in source · supplement_08.tex, line 71
pageheading2. The actual integral Hurwitz latticeConnecting differential expressions to coefficient divisibility The specialization works inside Q_p[[t]] and uses factorial-scaled coefficients a_f(d)=d![t^d]f, I_p={f∈ Q_p[[t]]:a_f(d)∈ Z_p for every d}. The source constructs I_p as a subring. Actual exponential-series multiplication gives a_fg(d)=∑_r=0^dbinom dr a_f(r)a_g(d-r), a_Uf(d)=a_f(d+1). These identities prove multiplication and derivative stability of the lattice. Membership does not req…
Exact mathematical passage
a_{fg}(d)=\sum_{r=0}^{d}\binom dr a_f(r)a_g(d-r), \qquad a_{Uf}(d)=a_f(d+1).
A_{\frakp}Use in source · 1 passage
Search aliases: A_frakp A_ p frakp mathfrak
Use in source · Paper_XII_Traditional_Final.tex, line 624
If M=0, the statement follows from the usual convention Fitt_0(0)=R. Assume M≠0, and choose a finite presentation R^mxrightarrowAR^n⟶ M⟶0 with n≥1. The zeroth Fitting ideal is generated by the n by n minors of A. After localization at a prime p, the module M_ p vanishes exactly when A_ p is surjective. A map onto the free module R_ p^n is surjective exactly when one maximal minor is a unit. Thus M_ p=0 exactly when Fitt_0^R(M)not⊆ p.
Exact mathematical passage
A_{\frakp}
A_GUse in source · 5 passages
Search aliases: A_G
Use in source · 05_worked_models.tex, line 274
Let G be a nonempty finite graph with adjacency matrix A_G, and regard 𝕜^V(G) as a module over 𝕜[A_G]. The general result applies directly to a spectral certificate. For a symmetric real adjacency matrix, the classical spectral theorem supplies such data.
Exact mathematical passage
A_G
A_gUse in source · 1 passage
Search aliases: A_g
Use in source · 03_nonlinear_coordinates.tex, line 567
Let G be a finite abelian group and let A=⊕_g∈ GA_g be a G graded module.
Exact mathematical passage
A=\bigoplus_{g\in G}A_g
a_gUse in source · 2 passages
Search aliases: a_g
Use in source · supplement_08.tex, line 71
pageheading2. The actual integral Hurwitz latticeConnecting differential expressions to coefficient divisibility The specialization works inside Q_p[[t]] and uses factorial-scaled coefficients a_f(d)=d![t^d]f, I_p={f∈ Q_p[[t]]:a_f(d)∈ Z_p for every d}. The source constructs I_p as a subring. Actual exponential-series multiplication gives a_fg(d)=∑_r=0^dbinom dr a_f(r)a_g(d-r), a_Uf(d)=a_f(d+1). These identities prove multiplication and derivative stability of the lattice. Membership does not req…
Exact mathematical passage
a_{fg}(d)=\sum_{r=0}^{d}\binom dr a_f(r)a_g(d-r), \qquad a_{Uf}(d)=a_f(d+1).
a_{H_ax+j}Use in source · 1 passage
Search aliases: a_H_ax+j
Use in source · 72_recovered.tex, line 2393
, v_0,w∈ Z_p^d, H>0, and a,j,n,r∈ N. These are the hypotheses of the two corrected matrix statements used by the retained proof. Without integral observation vectors or positive L, the preceding counterexamples invalidate the claimed precision. Assume C^H=I+p^LB. Put H_a=Hp^a. Then C^H_a=I+p^L+aB_a for an integral matrix B_a. Every refined packet b_a,j(x)=a_H_ax+j has Mahler coefficients satisfying v_p(c_a,j,r)≥(L+a)r. Equivalently, Δ_H_a^ ra_n ≡0 pmod p^(L+a)r.
Exact mathematical passage
b_{a,j}(x)=a_{H_ax+j}
A_{HH}Use in source · 1 passage
Search aliases: A_HH
Use in source · 01_manuscript.tex, line 1071
[Schur support inclusion] Split the packets into retained indices R and hidden indices H. Suppose the hidden block A_HH is invertible. The retained Schur complement is S = A_RR - A_RHA_HH^-1A_HR. A retained block (β,α) can be active only if it was already active in A_RR, or if there is a directed walk from α through hidden packets to β. The internal hidden path may have length zero, so entrance and exit may use the same hidden packet.
Exact mathematical passage
A_{HH}
A_{HH}^{-1}Use in source · 4 passages
Search aliases: A_HH^-1
Use in source · pages.tex, line 266
1 · 1 · 2. The third coordinate is hidden. Its invertible block is [2], so eliminating it produces S=A_RR-A_RHA_HH^-1A_HR =mat3/2 · -1/2 -1/2 · 3/2. vfigschur_fill headThe hidden route has a sign The retained coordinates had no direct coupling, yet both touch the hidden coordinate. The resulting correction couples them by -1/2. This is fill: a new active retained block produced by an eliminated route.
Exact mathematical passage
S=A_{RR}-A_{RH}A_{HH}^{-1}A_{HR} =\mat{3/2&-1/2\\-1/2&3/2}.
A_{HR}Use in source · 5 passages
Search aliases: A_HR
Use in source · pages.tex, line 266
1 · 1 · 2. The third coordinate is hidden. Its invertible block is [2], so eliminating it produces S=A_RR-A_RHA_HH^-1A_HR =mat3/2 · -1/2 -1/2 · 3/2. vfigschur_fill headThe hidden route has a sign The retained coordinates had no direct coupling, yet both touch the hidden coordinate. The resulting correction couples them by -1/2. This is fill: a new active retained block produced by an eliminated route.
Exact mathematical passage
S=A_{RR}-A_{RH}A_{HH}^{-1}A_{HR} =\mat{3/2&-1/2\\-1/2&3/2}.
a_{Hx+j}Use in source · 2 passages
Search aliases: a_Hx+j
Use in source · 72_recovered.tex, line 1681
means the matrix-binomial interpolation. With p=3, C=4, L=1, w=1 and v_0=1/3, one has Δ_1a_0=1, which is not in 3 Z_3. The printed claim without these restrictions is therefore not a certificate as written. The full original proof follows under the corrected hypotheses. Assume C^H=I+p^LB with B∈ M_d( Z_p). For every address j and every x∈ Z_p, a_Hx+j = ∑_r≥0 p^Lrbinom xr w^TC^jB^rv_0. The series converges on Z_p, and Δ_H^ra_n ≡0 pmod p^Lr.
Exact mathematical passage
\boxed{ a_{Hx+j} = \sum_{r\geq0} p^{Lr}\binom xr w^TC^jB^rv_0. }
A_iUse in source · 15 passages
Search aliases: A_i
Use in source · visual_atlas.tex, line 513
vpage21Tensor products require every cross termOPERATIONS ON SEED RANK / A RECTANGLE, NOT A DIAGONALMain text: [source reference]. Algebraic tensor products If (x_i) generates X under A and (y_j) generates Y under B, then the candidate tensor seeds are all x_i⊗ y_j. For finite expansions, x=∑_i A_ix_i, y=∑_j B_jy_j, bilinearity gives the required double sum: x⊗ y=∑_i∑_j(A_i⊗ B_j)(x_i⊗ y_j).
Exact mathematical passage
x=\sum_i A_ix_i,\qquad y=\sum_j B_jy_j,
A_{i+j}Use in source · 1 passage
Search aliases: A_i+j
Use in source · 03_nonlinear_coordinates.tex, line 597
[Cyclic product formula] For a cyclically graded algebra A=⊕_j∈ Z/N ZA_j, A_iA_j⊆ A_i+j, one has P_k(ab)= ∑_i+j=ka_ib_j.
Exact mathematical passage
A=\bigoplus_{j\in\mathbb Z/N\mathbb Z}A_j, \qquad A_iA_j\subseteq A_{i+j},
a_{i,1}Use in source · 1 passage
Search aliases: a_i,1
Use in source · Paper_X_Block27_Amended.tex, line 304
[Smooth formal branch count] If R_bar x≅ O[[t_1,…,t_d]] as complete local O-algebras, then #X_m(bar x)=q^d(m-1). Fix a uniformizer and a section s:k→ O of reduction, with s(0)=0. Every O-valued branch is encoded by d positive-depth digit streams t_i=s(a_i,1)ϖ+s(a_i,2)ϖ^2+⋯, a_i,j∈ k,
Exact mathematical passage
t_i=s(a_{i,1})\varpi+s(a_{i,2})\varpi^2+\cdots, \qquad a_{i,j}\in k,
a_{i,2}Use in source · 1 passage
Search aliases: a_i,2
Use in source · Paper_X_Block27_Amended.tex, line 304
[Smooth formal branch count] If R_bar x≅ O[[t_1,…,t_d]] as complete local O-algebras, then #X_m(bar x)=q^d(m-1). Fix a uniformizer and a section s:k→ O of reduction, with s(0)=0. Every O-valued branch is encoded by d positive-depth digit streams t_i=s(a_i,1)ϖ+s(a_i,2)ϖ^2+⋯, a_i,j∈ k,
Exact mathematical passage
t_i=s(a_{i,1})\varpi+s(a_{i,2})\varpi^2+\cdots, \qquad a_{i,j}\in k,
a_{i,j}Use in source · 3 passages
Search aliases: a_i,j
Use in source · Paper_X_Block27_Amended.tex, line 304
[Smooth formal branch count] If R_bar x≅ O[[t_1,…,t_d]] as complete local O-algebras, then #X_m(bar x)=q^d(m-1). Fix a uniformizer and a section s:k→ O of reduction, with s(0)=0. Every O-valued branch is encoded by d positive-depth digit streams t_i=s(a_i,1)ϖ+s(a_i,2)ϖ^2+⋯, a_i,j∈ k,
Exact mathematical passage
t_i=s(a_{i,1})\varpi+s(a_{i,2})\varpi^2+\cdots, \qquad a_{i,j}\in k,
A_i^jUse in source · 1 passage
Search aliases: A_i^j
Use in source · 02_supplement_03.tex, line 97
Exact and approximate packet Krylov spaces. textitSource items 7.2 and 7.3 Write K_m(A,v)=span{v,Av,…,A^m-1v}. For a complete finite projector family P_i, suppose A commutes with every P_i. Let A_i=P_iAP_i. Lean proves the actual orbit identity A^jv=∑_i A_i^jP_iv and therefore K_m(A,v)⊆ ⊕_i K_m(A_i,P_iv). The case j=0 uses ∑_iP_i=I. The zeroth local power is the ambient identity, applied to P_iv, rather than an incorrectly substituted local projector power.
Exact mathematical passage
A^jv=\sum_i A_i^jP_iv
A_{i_1}Use in source · 1 passage
Search aliases: A_i_1
Use in source · Paper_V_Traditional_Final.tex, line 1076
[Cyclic product law] P_N,k(AB) = ∑_i+j≡ k (mod N)A_iB_j. For q∈ N, the qth-power identity is P_N,k(A^q) = ∑_i_1+⋯+i_q≡ k A_i_1⋯ A_i_q.
Exact mathematical passage
P_{N,k}(A^q) = \sum_{i_1+\cdots+i_q\equiv k} A_{i_1}\cdots A_{i_q}.
A_{i_q}Use in source · 1 passage
Search aliases: A_i_q
Use in source · Paper_V_Traditional_Final.tex, line 1076
[Cyclic product law] P_N,k(AB) = ∑_i+j≡ k (mod N)A_iB_j. For q∈ N, the qth-power identity is P_N,k(A^q) = ∑_i_1+⋯+i_q≡ k A_i_1⋯ A_i_q.
Exact mathematical passage
P_{N,k}(A^q) = \sum_{i_1+\cdots+i_q\equiv k} A_{i_1}\cdots A_{i_q}.
A_jUse in source · 25 passages
Search aliases: A_j
Use in source · 03_nonlinear_coordinates.tex, line 597
[Cyclic product formula] For a cyclically graded algebra A=⊕_j∈ Z/N ZA_j, A_iA_j⊆ A_i+j, one has P_k(ab)= ∑_i+j=ka_ib_j.
Exact mathematical passage
A=\bigoplus_{j\in\mathbb Z/N\mathbb Z}A_j, \qquad A_iA_j\subseteq A_{i+j},
A_{j+1}Use in source · 5 passages
Search aliases: A_j+1
Use in source · Paper_VI_Traditional_Final.tex, line 5275
[Summable generator defects] Suppose C_j are contractions and (T_j(t)) are contraction semigroups with bounded generators A_j. Set δ_j=normC_jA_j+1-A_jC_j. Then normC_ℓ,mT_m(t)-T_ℓ(t)C_ℓ,m ≤ t∑_j=ℓ^m-1δ_j. If ∑_j≥0δ_j<∞, the right side tends to zero as ℓ→∞, uniformly in m>ℓ and uniformly for t in a fixed bounded interval.
Exact mathematical passage
\delta_j=\norm{C_jA_{j+1}-A_jC_j}.
A_{j,e}Use in source · 1 passage
Search aliases: A_j,e
Use in source · 02_supplement_03.tex, line 75
Packetwise mixed precision. At each stage let A_j=∑_e A_j,e, widehat A_j=∑_e(A_j,e+Δ_j,e), Δ_j,e=Δ_j,eP_j,a(e). The previous packet certificate supplies the uniform operator bound normwidehat A_j-A_j≤ δ_j:=∑_enormΔ_j,enormP_j,a(e). The new theorem substitutes these δ_j into the actual product-chain estimate. The resulting vector error is at most the operator certificate times normx. A vector-specific packet bound is not silently reused as an operator norm.
Exact mathematical passage
A_j=\sum_e A_{j,e},\qquad \widehat A_j=\sum_e(A_{j,e}+\Delta_{j,e}),\qquad \Delta_{j,e}=\Delta_{j,e}P_{j,a(e)}.
a_{j,g}Use in source · 1 passage
Search aliases: a_j,g
Use in source · 03_nonlinear_coordinates.tex, line 582
[Graded packet convolution] If a_j=∑_g∈ Ga_j,g and μ respects the grading, then P_kμ(a_1,…,a_r) = ∑_g_1+⋯+g_r=k μ(a_1,g_1,…,a_r,g_r).
Exact mathematical passage
a_j=\sum_{g\in G}a_{j,g}
a_{j,k}Use in source · 2 passages
Search aliases: a_j,k
Use in source · 20_arithmetic.tex, line 677
The first Newton coefficients, derived from [source reference], are a_0,1=-2, a_1,1=-frac65, a_2,1=-frac1615, a_3,1=-frac3635. Their exact valuations are c_j. Theorem~[source reference] gives v_2(a_j,k)≥ k for all orders. To obtain the stronger logarithmic envelope required by Lemma~[source reference], only a finite initial segment needs improvement. The recurrence proves the following exact valuations: cll Address j · Further orders k · v_2(a_j,k) in that order
Exact mathematical passage
v_2(a_{j,k})\ge k
A_{j-1}Use in source · 1 passage
Search aliases: A_j-1
Use in source · 01_manuscript.tex, line 887
Use the telescoping identity widehat A_m⋯widehat A_1 -A_m⋯ A_1 = ∑_j=1^m widehat A_m⋯widehat A_j+1 E_j A_j-1⋯ A_1. Take norms term by term.
Exact mathematical passage
\widehat A_m\cdots\widehat A_1 -A_m\cdots A_1 = \sum_{j=1}^m \widehat A_m\cdots\widehat A_{j+1} E_j A_{j-1}\cdots A_1.
A_{j\ell}Use in source · 1 passage
Search aliases: A_jℓ ell
Use in source · 01_manuscript.tex, line 606
The (i,j) entry of BXA^T is ∑_k=1^n∑_ℓ=1^m B_ikX_kℓA_jℓ. This is the coordinate formula for (A⊗ B)vec(X). Multiplying first by B and then by A^T gives the stated count.
Exact mathematical passage
\sum_{k=1}^{n}\sum_{\ell=1}^{m} B_{ik}X_{k\ell}A_{j\ell}.
A_KUse in source · 3 passages
Search aliases: A_K
Use in source · 02_foundations.tex, line 595
[Monotonicity under extension of scalars] Let K/𝕜 be a field extension, let A be a unital algebra over 𝕜, and let M be a finitely generated left module over A. Set A_K:=K⊗_𝕜A, M_K:=K⊗_𝕜M. Then srank_A_K(M_K)≤srank_A(M). Every generating family S⊆ M extends to the generating family 1⊗ S⊆ M_K. [source reference] ExportsExtension of the coefficient field in realizations from arithmetic, quantum theory, and representation theory.
Exact mathematical passage
A_K:=K\otimes_{\bbk}A, \qquad M_K:=K\otimes_{\bbk}M.
a_kUse in source · 15 passages
Search aliases: a_k
Use in source · 03_weighted.tex, line 441
In the recurrence, set a_k=0 when k<0. This is the coefficient convention for the formal power series Z_u,v(t) and makes the first recurrence equations meaningful even when n<d.
Exact mathematical passage
a_k=0
A_{k+1}Use in source · 1 passage
Search aliases: A_k+1
Use in source · 02_supplement_04.tex, line 148
Exact arithmetic count. Each pair of slices forms one sum and one difference at every trailing coordinate. The event type therefore gives A_0=0, A_k+1=2A_k+2^k+1. The closed form and its equivalent logarithmic expression are both checked: A_k=k2^k=Nlog_2N. The logarithm in the formal count is the natural-number binary logarithm, with its power-of-two identity proved by Mathlib. The case N=1 has zero arithmetic events.
Exact mathematical passage
A_0=0,\qquad A_{k+1}=2A_k+2^{k+1}.
a_{k+1}Use in source · 1 passage
Search aliases: a_k+1
Use in source · 02_supplement_06.tex, line 141
For any nonnegative decreasing sequence a_i, weights with these properties satisfy ∑_i a_iw_i≥∑_i>ka_i. This finite weighted inequality is itself proved. For k<d, use μ=a_k+1 and compare the summands to tail weights v_i=0 for i≤ k and v_i=1 otherwise: a_iv_i+μ(w_i-v_i)≤ a_iw_i. For a retained coordinate the inequality uses a_i≥μ and w_i≥0. For a tail coordinate it uses a_i≤μ and w_i≤1. Summation and ∑ w_i≥∑ v_i give the result.
Exact mathematical passage
\mu=a_{k+1}
a_{k+i}Use in source · 1 passage
Search aliases: a_k+i
Use in source · 20_arithmetic.tex, line 628
Finite Newton reconstruction at an arbitrary row gives Δ_1^kf(n)=∑_i=0^nbinom ni a_k+i. The monotonicity of lfloorlog_2krfloor therefore preserves the higher-coefficient bound after translation. In addition, Δ_1f(n)-a_1∈2^c+1 Z_2, so the first difference is nonzero of exact valuation c.
Exact mathematical passage
\Delta_1^kf(n)=\sum_{i=0}^{n}\binom ni a_{k+i}.
A_{k-1}Use in source · 1 passage
Search aliases: A_k-1
Use in source · 02_supplement_04.tex, line 88
Arbitrary tensor factors. textitSource item 4.6; module StructuredTensor Let A_0,…,A_k-1 be square factors of order d. Tensor coordinates are represented by a nested product of k copies of {0,…,d-1}. The empty product is a singleton. Lean proves that the coordinate type has cardinality d^k.
Exact mathematical passage
A_0,\ldots,A_{k-1}
A_{ki}Use in source · 1 passage
Search aliases: A_ki
Use in source · visual_atlas.tex, line 164
vfig[.98]rectangular_paths platecaptionThe lower board expands one final row into all six choices of intermediate indices for each input packet. The entries are C_0jB_jkA_ki, with k∈{0,1,2} and j∈{0,1}. Zero terms remain visible.
Exact mathematical passage
C_{0j}B_{jk}A_{ki}
A_mUse in source · 9 passages
Search aliases: A_m
Use in source · 02_strengthened.tex, line 366
[Tensor selection rule] If A_a has local charge q_a, then A_1⊗⋯⊗ A_m has total charge q_1+⋯+q_m.
Exact mathematical passage
A_1\otimes\cdots\otimes A_m
a_mUse in source · 24 passages
Search aliases: a_m
Use in source · Paper_IV_Traditional_Final.tex, line 239
For a power series F(z)=∑_m≥ 0a_mz^m and a positive integer N, the coefficients may be separated according to the residue of m modulo N. Root of unity averaging gives exact projection operators for this separation. This elementary construction is usually called a multisection or a root filter. It appears throughout the classical theory of generating functions and special functions. Background on multisections, hypergeometric functions, and related coefficient systems may be found in [source refe…
Exact mathematical passage
F(z)=\sum_{m\geq 0}a_mz^m
a_{M(Nq+s)+r}Use in source · 1 passage
Search aliases: a_M(Nq+s)+r
Use in source · 72_recovered.tex, line 2031
Direct substitution gives (a_r^[M])_s^[N](q) = a_M(Nq+s)+r. Iterate this identity along the mixed radix path.
Exact mathematical passage
\left(a_r^{[M]}\right)_s^{[N]}(q) = a_{M(Nq+s)+r}.
a_{M(q+1)}Use in source · 1 passage
Search aliases: a_M(q+1)
Use in source · 72_recovered.tex, line 2094
For r<M-1, (Ea)_Mq+r=a_Mq+r+1=a_r+1^[M](q). For r=M-1, (Ea)_Mq+M-1=a_M(q+1)=E_qa_0^[M](q). After M shifts every phase has advanced by one coarse index.
Exact mathematical passage
(Ea)_{Mq+M-1}=a_{M(q+1)}=E_qa_0^{[M]}(q).
A_{m-1}Use in source · 1 passage
Search aliases: A_m-1
Use in source · 01_manuscript.tex, line 937
Set D_j=widehat P_j-P_j. Then D_j=A_jD_j-1+E_j, D_0=0. Repeated substitution gives D_m = ∑_j=1^m A_mA_m-1⋯ A_j+1E_j. Taking norms yields the result.
Exact mathematical passage
D_m = \sum_{j=1}^{m} A_mA_{m-1}\cdots A_{j+1}E_j.
a_{m-1}Use in source · 6 passages
Search aliases: a_m-1
Use in source · Paper_X_Block27_Amended.tex, line 1972
The residual module partitions coordinates modulo p, verifies compatibility x_m+1≡ x_mpmodp^m, and reconstructs the digit a_m-1 by a_m-1=lfloorfracx_mp^m-1rfloorpmod p.
Exact mathematical passage
a_{m-1}
a_{M-1}^{[M]}Use in source · 1 passage
Search aliases: a_M-1^[M]
Use in source · 72_recovered.tex, line 1977
For a sequence a=(a_n)_n≥0 and M≥2, define a_r^[M](q)=a_Mq+r (0≤ r<M). The analysis map is P_Ma = (a_0^[M],…,a_M-1^[M]). Synthesis interleaves the branches by the rule a_Mq+r=a_r^[M](q).
Exact mathematical passage
\mathcal P_Ma = \left(a_0^{[M]},\ldots,a_{M-1}^{[M]}\right).
a_{\min}Use in source · 15 passages
Search aliases: a_min min
Use in source · Paper_V_Traditional_Final.tex, line 1692
The principal recovery theorem uses adjacent sample contrasts. Let b_h=Mh+2ε be the largest contrast that can be produced by a Lipschitz background and bounded noise in one grid cell. If a_min>2b_h, then every threshold between b_h and a_min-b_h detects exactly the cells containing jumps. The jump locations are recovered within h/2, and each amplitude error is at most b_h. The probability theorem replaces the deterministic noise term by 2σsqrtlog(2n/δ). Two half shifted Haar grids provide a seco…
Exact mathematical passage
a_{\min}>2b_h,
a_{Mq+r}Use in source · 5 passages
Search aliases: a_Mq+r
Use in source · 10_foundations.tex, line 485
The recovered Annals text develops a broader polyphase vocabulary. We explain it through elementary identities with explicit algebraic derivations. These identities have not been separately formalized in Lean for this consolidation. For a sequence a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient and remainder give A(z)=∑_r=0^M-1z^rA_r(z^M), (a_r^[M])_s^[N]=a_r+Ms^[MN]. For ordinary Cauchy multiplicati…
Exact mathematical passage
a_r^{[M]}(q)=a_{Mq+r}
a_{Mq+r+1}Use in source · 1 passage
Search aliases: a_Mq+r+1
Use in source · 72_recovered.tex, line 2090
For r<M-1, (Ea)_Mq+r=a_Mq+r+1=a_r+1^[M](q). For r=M-1, (Ea)_Mq+M-1=a_M(q+1)=E_qa_0^[M](q). After M shifts every phase has advanced by one coarse index.
Exact mathematical passage
(Ea)_{Mq+r}=a_{Mq+r+1}=a_{r+1}^{[M]}(q).
A_NUse in source · 5 passages
Search aliases: A_N
Use in source · Paper_IV_Traditional_Final.tex, line 1858
The zero strings are governed by two adjacent exponents. Put a_N=fracπ N, s_N=sin a_N, c_N=cos a_N, and A_N=fracπs_N, t_m=A_N(m+frac12). For N≥3, also put γ_N=c_N-cos(3a_N)>0.
Exact mathematical passage
A_N=\frac\pi{s_N}, \qquad t_m=A_N\left(m+\frac12\right).
A_nUse in source · 24 passages
Search aliases: A_n
Use in source · 72_recovered.tex, line 484
[Exact coefficient transport] Let A(t)=∑_n≥0A_nfract^nn!, S(t)=H(t)A(t). Then S_n = ∑_r=0^n binom nr h_rA_n-r. The inverse coordinate formula is A_n = ∑_r=0^n binom nr k_rS_n-r. Both transforms are lower triangular with diagonal one.
Exact mathematical passage
A(t)=\sum_{n\geq0}A_n\frac{t^n}{n!}, \qquad S(t)=H(t)A(t).
\mathcal A_NUse in source · 2 passages
Search aliases: A_N mathcal
Use in source · visual_atlas.tex, line 27
A_N, S_N · Split into range-restricted packet vectors, then sum them. · A permutation/interleaving of eight coordinates.
Exact mathematical passage
\mathcal A_N,\mathcal S_N
a_NUse in source · 23 passages
Search aliases: a_N
Use in source · Paper_IV_Traditional_Final.tex, line 1850
The zero strings are governed by two adjacent exponents. Put a_N=fracπ N, s_N=sin a_N, c_N=cos a_N, and A_N=fracπs_N, t_m=A_N(m+frac12). For N≥3, also put γ_N=c_N-cos(3a_N)>0.
Exact mathematical passage
a_N=\frac\pi N, \qquad s_N=\sin a_N, \qquad c_N=\cos a_N,
A_{n+1}Use in source · 1 passage
Search aliases: A_n+1
Use in source · atlas-pages.tex, line 716
vpage31A tail bound states what refinement buysSUMMABLE DEFECTS / A SCALAR COMPLETIONChapter VI, Part II: Theorem E4.5 and Corollary E4.7. This example has identity connecting maps. Take X_n= R, C_n=R_n=1, and a_n=1-2^-n, A_n=-a_n, T_n(t)=e^-a_nt (n≥0). Each semigroup is contractive for t≥0. The exact adjacent generator-defect magnitude is |A_n+1-A_n|=2^-(n+1), ∑_j=n^∞|A_j+1-A_j|=2^-n. vfigext_tails platecaptionThe bars show the certified tail 2^-n at horizon T=1. They are bounds on a whole time…
Exact mathematical passage
|A_{n+1}-A_n|=2^{-(n+1)},\qquad \sum_{j=n}^{\infty}|A_{j+1}-A_j|=2^{-n}.
a_{n+1}Use in source · 5 passages
Search aliases: a_n+1
Use in source · 02_connections_obstructions.tex, line 292
Induct on the path length. Appending a_n+1 left-multiplies the inductive expression by Γ_a_n+1← a_n, and flatness replaces the two factors by Γ_a_n+1← a_0. For a loop, normalization gives Γ_a_0← a_0=id.
Exact mathematical passage
a_{n+1}
A_{N,j}Use in source · 10 passages
Search aliases: A_N,j
Use in source · Paper_IV_Traditional_Final.tex, line 553
[Hypergeometric form] Let A_N,j = { fracj+mN mathrel: 1≤ m≤ N and j+m≠ N }. Then E_N,j(λ z) = frac(λ z)^jj! _0F_N-1 ( -
Exact mathematical passage
A_{N,j} = \left\{ \frac{j+m}{N} \mathrel{:} 1\leq m\leq N \text{ and } j+m\ne N \right\}.
A_{N,n}Use in source · 3 passages
Search aliases: A_N,n
Use in source · Paper_IV_Traditional_Final.tex, line 1086
[A sequence of binomial type] Define A_N,n(r) = fracn!(Nn)!P_N,n(r). Then A_N,n(r+s) = ∑_q=0^n binom nq A_N,q(r)A_N,n-q(s). Thus (A_N,n)_n≥0 is a polynomial sequence of binomial type.
Exact mathematical passage
A_{N,n}(r) = \frac{n!}{(Nn)!}P_{N,n}(r).
A_{N,n-q}Use in source · 1 passage
Search aliases: A_N,n-q
Use in source · Paper_IV_Traditional_Final.tex, line 1092
[A sequence of binomial type] Define A_N,n(r) = fracn!(Nn)!P_N,n(r). Then A_N,n(r+s) = ∑_q=0^n binom nq A_N,q(r)A_N,n-q(s). Thus (A_N,n)_n≥0 is a polynomial sequence of binomial type.
Exact mathematical passage
A_{N,n}(r+s) = \sum_{q=0}^{n} \binom nq A_{N,q}(r)A_{N,n-q}(s).
A_{N,q}Use in source · 1 passage
Search aliases: A_N,q
Use in source · Paper_IV_Traditional_Final.tex, line 1092
[A sequence of binomial type] Define A_N,n(r) = fracn!(Nn)!P_N,n(r). Then A_N,n(r+s) = ∑_q=0^n binom nq A_N,q(r)A_N,n-q(s). Thus (A_N,n)_n≥0 is a polynomial sequence of binomial type.
Exact mathematical passage
A_{N,n}(r+s) = \sum_{q=0}^{n} \binom nq A_{N,q}(r)A_{N,n-q}(s).
a_{n-3}Use in source · 1 passage
Search aliases: a_n-3
Use in source · visual_atlas.tex, line 311
From QZ_0,1=2t^2 one obtains a_0=a_1=0, a_2=2, a_n=a_n-1+2a_n-3 (n≥3). The numerator is responsible for the initial exception. Omitting it would incorrectly force the entire series to vanish.
Exact mathematical passage
a_0=a_1=0,\quad a_2=2,\qquad a_n=a_{n-1}+2a_{n-3}\quad(n\ge3).
a_{n-d}Use in source · 1 passage
Search aliases: a_n-d
Use in source · 03_weighted.tex, line 428
[Recurrence for path counts] Write Z_u,v(t)=fracP(t)Q(t), Q(t)=1+q_1t+⋯+q_dt^d. Use the convention deg 0=-∞. Then, for every nonnegative integer n>deg P, a_n+q_1a_n-1+⋯+q_da_n-d=0.
Exact mathematical passage
a_n+q_1a_{n-1}+\cdots+q_da_{n-d}=0.
a_{n-k}Use in source · 1 passage
Search aliases: a_n-k
Use in source · 03_weighted.tex, line 564
[Linear recurrence] With the canonical polynomials from [source reference], write Q(t)=∑_kq_kt^k, Z_s,t(t)=∑_n≥0a_n(s,t)t^n. Then the walk counts satisfy the recurrence with constant coefficients determined by Q after the finite initial range contributed by P_s,t, with deg 0=-∞: ∑_k=0^nq_k a_n-k(s,t)=0 (n>deg P_s,t).
Exact mathematical passage
\boxed{ \sum_{k=0}^{n}q_k a_{n-k}(s,t)=0 \qquad\bigl(n>\deg P_{s,t}\bigr). }
A_{n-r}Use in source · 1 passage
Search aliases: A_n-r
Use in source · 72_recovered.tex, line 490
[Exact coefficient transport] Let A(t)=∑_n≥0A_nfract^nn!, S(t)=H(t)A(t). Then S_n = ∑_r=0^n binom nr h_rA_n-r. The inverse coordinate formula is A_n = ∑_r=0^n binom nr k_rS_n-r. Both transforms are lower triangular with diagonal one.
Exact mathematical passage
\boxed{ S_n = \sum_{r=0}^{n} \binom nr h_rA_{n-r}. }
a_{n-r}Use in source · 2 passages
Search aliases: a_n-r
Use in source · 10_foundations.tex, line 95
[Constructed unit-triangular coordinates] Let h: N→ K satisfy h_0=1, write H= E(h) and H^-1= E(k), and define U_Ha= D(H E(a)). Then U_H is an equivalence, with inverse U_H^-1a= D(H^-1 E(a)). Its coefficient formulas are (U_Ha)_n=∑_r=0^nbinom nr h_r a_n-r, (U_H^-1a)_n=∑_r=0^nbinom nr k_r a_n-r. The resulting triangular matrix has diagonal entries 1. The transported selectors widehat P_N,j=U_HP_N,jU_H^-1 are idempotent and pairwise orthogonal, and their sum is the identity.
Exact mathematical passage
(U_Ha)_n=\sum_{r=0}^{n}\binom nr h_r a_{n-r},\qquad (U_H^{-1}a)_n=\sum_{r=0}^{n}\binom nr k_r a_{n-r}.
a_{n-w(e)}Use in source · 2 passages
Search aliases: a_n-w(e)
Use in source · 03_weighted.tex, line 499
[Automatic finite-graph transfer-zeta rationality] Assume the vertex and edge sets are finite and every edge has positive integer weight. For every n, exact enumeration of edge tuples of lengths at most n, followed by endpoint and total-weight filtering, returns a_n(s,t). These coefficients obey the recurrence obtained by removing the first edge a_n(s,t) = 1_s=t, n=0 +∑_substacke: source(e)=s w(e)≤ n a_n-w(e)(target(e),t). Let T(t)=I-A(t), Q(t)=det T(t), P_s,t(t)=adj(T(t))_s,t. Then Q(0)=1, henc…
Exact mathematical passage
a_n(s,t) = \mathbf 1_{s=t,\,n=0} +\sum_{\substack{e:\,\operatorname{source}(e)=s\\w(e)\le n}} a_{n-w(e)}(\operatorname{target}(e),t).
a_n^{(m)}Use in source · 1 passage
Search aliases: a_n^(m)
Use in source · 72_recovered.tex, line 260
[Multiple selector] Define a^(m)∈ H( F_p) by a_n^(m) = 1, · mmid n, 0, · mnmid n.
Exact mathematical passage
a_n^{(m)} = \begin{cases} 1,&m\mid n,\\ 0,&m\nmid n. \end{cases}
A_N^{-Ns}Use in source · 2 passages
Search aliases: A_N^-Ns
Use in source · Paper_IV_Traditional_Final.tex, line 2779
[Meromorphic continuation of the symmetry-orbit zero zeta] Put A_N=fracπsin(π/N). There is an entire function Q_N^red such that Z_N^red(s) = A_N^-Nsζ(Ns,frac12)+Q_N^red(s) on Re s>1/N. The right-hand side gives a meromorphic continuation to C with one simple pole at s=1/N. Its residue is fracsin(π/N)Nπ.
Exact mathematical passage
\mathfrak Z_N^{\mathrm{red}}(s) = A_N^{-Ns}\zeta\left(Ns,\frac12\right)+Q_N^{\mathrm{red}}(s)
A_N^{-s}Use in source · 1 passage
Search aliases: A_N^-s
Use in source · Paper_IV_Traditional_Final.tex, line 2791
There is an entire function Q_N^abs such that, initially on Re s>1, Z_N^abs(s) = N A_N^-sζ(s,frac12)+Q_N^abs(s). Thus Z_N^abs continues meromorphically to C with one simple pole at s=1, of residue fracNsin(π/N)π.
Exact mathematical passage
\mathfrak Z_N^{\mathrm{abs}}(s) = N A_N^{-s}\zeta\left(s,\frac12\right)+Q_N^{\mathrm{abs}}(s).
A_n^{1/2}Use in source · 2 passages
Search aliases: A_n^1/2
Use in source · atlas-pages.tex, line 621
at each level is not summable across levelsREFINEMENT LIMITS / WHAT THE FINITE MODELS DO NOT PROVEChapter VI, Part I: Corollary 10.5 and Section 17. Summability and uniform control are additional data. For each n≥1, compare the two scalar operators A_n=0 and B_n=1/n^2 through the identity. Their ordinary defects have sizes 1/n^2, while their square-root defects have sizes 1/n: ∑_n|A_n-B_n|<∞, ∑_n|A_n^1/2-B_n^1/2|=∞. These are independent scalar comparisons, not a constructed inverse system. They…
Exact mathematical passage
\sum_n|A_n-B_n|<\infty, \qquad \sum_n|A_n^{1/2}-B_n^{1/2}|=\infty.
A_pUse in source · 1 passage
Search aliases: A_p
Use in source · Paper_X_Block27_Amended.tex, line 1793
For each supplied required packet p, let A_p be its finite-dimensional carrier over a characteristic-zero field E_p, and let e_p be the actual joint spectral projector for its finite list of split commuting conditions. No order on E_p is assumed. The source notation Tr(e_p)>0 means that the unique natural number n_p with Tr(e_p)=(n_p:E_p) is positive. The checked projector formula identifies n_p with the dimension of the actual joint eigenspace.
Exact mathematical passage
A_p
A_{p,2m}Use in source · 1 passage
Search aliases: A_p,2m
Use in source · 72_recovered.tex, line 1188
[Unified local Euler transfer] For every prime p and every even n≥0, E_n = ∑_r=0^n/2 binom n2r κ_p,r A_p,n-2r. If H_p(t) = ∑_m≥0A_p,2mfract^2m(2m)! and K_p(t) = ∑_r≥0κ_p,rfract^2r(2r)!, then K_p(t)H_p(t)=secht. For odd p, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). For p=2, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t).
Exact mathematical passage
H_p(t) = \sum_{m\geq0}A_{p,2m}\frac{t^{2m}}{(2m)!}
A_{p,n}Use in source · 3 passages
Search aliases: A_p,n
Use in source · 72_recovered.tex, line 1156
For a prime p, define A_p,n = S_n(p), · p odd, R_n, · p=2 and n even, 0, · p=2 and n odd, and κ_p,r = (-1)^rZ_2r+1p^2r, · p odd,
Exact mathematical passage
A_{p,n} = \begin{cases} S_n(p),&p\text{ odd},\\ R_n,&p=2\text{ and }n\text{ even},\\ 0,&p=2\text{ and }n\text{ odd}, \end{cases}
A_{p,n-2r}Use in source · 1 passage
Search aliases: A_p,n-2r
Use in source · 72_recovered.tex, line 1177
[Unified local Euler transfer] For every prime p and every even n≥0, E_n = ∑_r=0^n/2 binom n2r κ_p,r A_p,n-2r. If H_p(t) = ∑_m≥0A_p,2mfract^2m(2m)! and K_p(t) = ∑_r≥0κ_p,rfract^2r(2r)!, then K_p(t)H_p(t)=secht. For odd p, H_p(t)=cosh^2(pt)secht, K_p(t)=sech^2(pt). For p=2, H_2(t)=fracsinh(2t)sinh t2t^2, K_2(t)=frac4t^2sinh^2(2t).
Exact mathematical passage
\boxed{ E_n = \sum_{r=0}^{n/2} \binom n{2r} \kappa_{p,r} A_{p,n-2r}. }
a_{p-1}Use in source · 1 passage
Search aliases: a_p-1
Use in source · odd_tower.tex, line 58
Then v_E(z)=pa-qb=1, so z is a uniformizer and E=L(z). Its monic minimal polynomial f(X)=X^p+a_p-1X^p-1+⋯+a_0 is Eisenstein. It is an explicit polynomial: compute the product over the p conjugates
Exact mathematical passage
f(X)=X^p+a_{p-1}X^{p-1}+\cdots+a_0
A_{P_n}Use in source · 1 passage
Search aliases: A_P_n
Use in source · 05_worked_models.tex, line 331
For every field 𝕜 and every n≥1, an endpoint of P_n is a cyclic vector for the adjacency operator. Consequently one is the least finite number of adjacency seeds: 1=min{r∈ N: ∃ s_1,…,s_r, 𝕜^n=span_𝕜{p(A_P_n)s_j:p∈𝕜[t], 1≤ j≤ r} }. Over C, the familiar spectral description has the distinct eigenvalues λ_k=2cosfrackπn+1, 1≤ k≤ n, and the normalized eigenvectors have endpoint coordinate proportional to sinfrackπn+1≠0. There is also a combinatorial proof valid over every field. Let K_n=[e_1,Ae_1,…,A…
Exact mathematical passage
{ 1=\min\left\{r\in\N: \exists s_1,\ldots,s_r,\quad \bbk^n=\Span_{\bbk}\{p(A_{P_n})s_j:p\in\bbk[t],\ 1\le j\le r\} \right\}. }
A_{p_u}Use in source · 1 passage
Search aliases: A_p_u
Use in source · Paper_XI_Traditional_Final.tex, line 1694
[Exact edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD_eA_p_u^-1.
Exact mathematical passage
D_e = U_rA_{p_v}^{-1}A_eA_{p_u} - B_{p_v}^{-1}B_eB_{p_u}U_r.
A_{p_u}^{-1}Use in source · 4 passages
Search aliases: A_p_u^-1
Use in source · Paper_XI_Traditional_Final.tex, line 1702
ct edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD_eA_p_u^-1.
Exact mathematical passage
\boxed{ U_vA_e-B_eU_u = B_{p_v}D_eA_{p_u}^{-1}. }
A_{p_v}^{-1}Use in source · 2 passages
Search aliases: A_p_v^-1
Use in source · Paper_XI_Traditional_Final.tex, line 1694
[Exact edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD_eA_p_u^-1.
Exact mathematical passage
D_e = U_rA_{p_v}^{-1}A_eA_{p_u} - B_{p_v}^{-1}B_eB_{p_u}U_r.
A_{PP}Use in source · 11 passages
Search aliases: A_PP
Use in source · Paper_VI_Traditional_Final.tex, line 1487
Let X=P X⊕ Q X and write A= A_PP · A_PQ A_QP · A_QQ .
Exact mathematical passage
A= \begin{pmatrix} A_{PP}&A_{PQ}\\ A_{QP}&A_{QQ} \end{pmatrix}.
A_{PQ}Use in source · 18 passages
Search aliases: A_PQ
Use in source · Paper_VI_Traditional_Final.tex, line 1487
Let X=P X⊕ Q X and write A= A_PP · A_PQ A_QP · A_QQ .
Exact mathematical passage
A= \begin{pmatrix} A_{PP}&A_{PQ}\\ A_{QP}&A_{QQ} \end{pmatrix}.
a_{q,m}Use in source · 3 passages
Search aliases: a_q,m
Use in source · Paper_V_Traditional_Final.tex, line 2538
egintheorem[Higher order singular packet decomposition] Let 0=τ_0<τ_1<⋯<τ_Q<τ_Q+1=1. Suppose the restriction of f to every interval (τ_q,τ_q+1) belongs to W^r,p, and choose the one-dimensional Sobolev representatives whose derivatives through order r-1 have one-sided traces at the endpoints. Define a_q,m = jumpf^(m)_τ_q = f^(m)(τ_q+)-f^(m)(τ_q-) and j_f(t) = ∑_q=1^Q∑_m=0^r-1a_q,mϕ_m,τ_q(t). Then s_f=f-j_f has matching one-sided derivatives through order r-1 at every breakpoint and belongs to W^r…
Exact mathematical passage
a_{q,m} = \jump{f^{(m)}}_{\tau_q} = f^{(m)}(\tau_q+)-f^{(m)}(\tau_q-)
A_{QP}Use in source · 19 passages
Search aliases: A_QP
Use in source · Paper_VI_Traditional_Final.tex, line 1487
Let X=P X⊕ Q X and write A= A_PP · A_PQ A_QP · A_QQ .
Exact mathematical passage
A= \begin{pmatrix} A_{PP}&A_{PQ}\\ A_{QP}&A_{QQ} \end{pmatrix}.
A_{QQ}^kUse in source · 2 passages
Search aliases: A_QQ^k
Use in source · Paper_VI_Traditional_Final.tex, line 4599
sure] Assume x_Q(0)=0 and d=dim X_Q<∞. The following conditions are equivalent. item For every retained initial state, P e^tAP = e^tA_PPP for all t≥0. item Σ_P(z)=0 for every sufficiently large z. item K(t)=0 for every t≥0. item If d=dim X_Q, then A_PQA_QQ^kA_QP=0 for 0≤ k<d.
Exact mathematical passage
A_{PQ}A_{QQ}^kA_{QP}=0
A_RUse in source · 2 passages
Search aliases: A_R
Use in source · Paper_X_Block27_Amended.tex, line 2335
Let R be any commutative ring and B any commutative R-algebra. The coordinate ring A_R=R[X]/(X^2-1) represents precisely this specified packet: Hom_R-alg(A_R,B) ≃{x∈ B:x^2=1} ≃Hom(C_2,B^×) ≃ P_q(B). The first map evaluates the coordinate. Its inverse is the quotient universal property. A root is a unit with itself as inverse and determines a character by its value on the generator of C_2. Finally pull back along q. All inverse identities and naturality under coefficient homomorphisms are proved.…
Exact mathematical passage
A_R=R[X]/(X^2-1)
a_{r+1}^{[M]}Use in source · 1 passage
Search aliases: a_r+1^[M]
Use in source · 72_recovered.tex, line 2090
For r<M-1, (Ea)_Mq+r=a_Mq+r+1=a_r+1^[M](q). For r=M-1, (Ea)_Mq+M-1=a_M(q+1)=E_qa_0^[M](q). After M shifts every phase has advanced by one coarse index.
Exact mathematical passage
(Ea)_{Mq+r}=a_{Mq+r+1}=a_{r+1}^{[M]}(q).
a_{r+Ms}^{[MN]}Use in source · 2 passages
Search aliases: a_r+Ms^[MN]
Use in source · 10_foundations.tex, line 488
polyphase vocabulary. We explain it through elementary identities with explicit algebraic derivations. These identities have not been separately formalized in Lean for this consolidation. For a sequence a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient and remainder give A(z)=∑_r=0^M-1z^rA_r(z^M), (a_r^[M])_s^[N]=a_r+Ms^[MN]. For ordinary Cauchy multiplication, the phase-r product has a term without a …
Exact mathematical passage
A(z)=\sum_{r=0}^{M-1}z^rA_r(z^M),\qquad \bigl(a_r^{[M]}\bigr)_s^{[N]}=a_{r+Ms}^{[MN]}.
a_{r,g_r}Use in source · 1 passage
Search aliases: a_r,g_r
Use in source · 03_nonlinear_coordinates.tex, line 583
[Graded packet convolution] If a_j=∑_g∈ Ga_j,g and μ respects the grading, then P_kμ(a_1,…,a_r) = ∑_g_1+⋯+g_r=k μ(a_1,g_1,…,a_r,g_r).
Exact mathematical passage
P_k\mu(a_1,\ldots,a_r) = \sum_{g_1+\cdots+g_r=k} \mu(a_{1,g_1},\ldots,a_{r,g_r}).
A_r^{[M]}Use in source · 2 passages
Search aliases: A_r^[M]
Use in source · 72_recovered.tex, line 1993
[Exact discrete polyphase reconstruction] The analysis map P_M is a bijection between sequences and ordered families of M sequences. If A(z)=∑_n≥0a_nz^n and A_r^[M](w)=∑_q≥0a_Mq+rw^q, then A(z)=∑_r=0^M-1z^rA_r^[M](z^M).
Exact mathematical passage
A_r^{[M]}(w)=\sum_{q\geq0}a_{Mq+r}w^q,
a_r^{[M]}Use in source · 6 passages
Search aliases: a_r^[M]
Use in source · 10_foundations.tex, line 485
The recovered Annals text develops a broader polyphase vocabulary. We explain it through elementary identities with explicit algebraic derivations. These identities have not been separately formalized in Lean for this consolidation. For a sequence a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient and remainder give A(z)=∑_r=0^M-1z^rA_r(z^M), (a_r^[M])_s^[N]=a_r+Ms^[MN]. For ordinary Cauchy multiplicati…
Exact mathematical passage
a_r^{[M]}(q)=a_{Mq+r}
a_r^TUse in source · 3 passages
Search aliases: a_r^T
Use in source · Paper_VI_Traditional_Final.tex, line 5039
[Competing risk packets] Suppose the exit row splits into nonnegative cause rows a^T = ∑_r=1^q a_r^T. Define f_r(t)=a_r^T e^tAp_0. Then f(t)=∑_r=1^qf_r(t). The cause specific hazard is h_r(t)=fracf_r(t)S(t), and h(t)=∑_r=1^qh_r(t). If the spectrum of A lies in the open left half-plane, the eventual probability of cause r is ∫_0^∞ f_r(t) d t = a_r^T(-A)^-1p_0.
Exact mathematical passage
a^T = \sum_{r=1}^q a_r^T.
A_{ra}Use in source · 3 passages
Search aliases: A_ra
Use in source · 03_supplement_05.tex, line 87
Deriving the Gram matrix from recovery. Suppose a finite recovery that preserves trace corrects every operator supported on the code. To derive the Knill--Laflamme condition without assuming a syndrome decomposition, the proof expands the composite channel into the finite Kraus family A_ra=R_rE_aP. For every ambient matrix ρ, correction on the code implies ∑_r,a A_raρ A_ra^*=Pρ P. This equality on all matrices follows by applying correction to the sandwich supported on the code, Pρ P, using the …
Exact mathematical passage
A_{ra}=R_rE_aP.
A_{ra}^*Use in source · 1 passage
Search aliases: A_ra^*
Use in source · 03_supplement_05.tex, line 91
Deriving the Gram matrix from recovery Suppose a finite recovery that preserves trace corrects every operator supported on the code. To derive the Knill--Laflamme condition without assuming a syndrome decomposition, the proof expands the composite channel into the finite Kraus family A_ra=R_rE_aP. For every ambient matrix ρ, correction on the code implies ∑_r,a A_raρ A_ra^*=Pρ P. This equality on all matrices follows by applying correction to the sandwich supported on the code, Pρ P, using the p…
Exact mathematical passage
\sum_{r,a} A_{ra}\rho A_{ra}^*=P\rho P.
A_{RH}Use in source · 5 passages
Search aliases: A_RH
Use in source · pages.tex, line 266
1 · 1 · 2. The third coordinate is hidden. Its invertible block is [2], so eliminating it produces S=A_RR-A_RHA_HH^-1A_HR =mat3/2 · -1/2 -1/2 · 3/2. vfigschur_fill headThe hidden route has a sign The retained coordinates had no direct coupling, yet both touch the hidden coordinate. The resulting correction couples them by -1/2. This is fill: a new active retained block produced by an eliminated route.
Exact mathematical passage
S=A_{RR}-A_{RH}A_{HH}^{-1}A_{HR} =\mat{3/2&-1/2\\-1/2&3/2}.
a_{\rm root}Use in source · 1 passage
Search aliases: a_rm root rm
Use in source · atlas-pages.tex, line 566
apter V: Proposition E6.4. Suppose the root split has nonnegative squared-norm bounds A=1/2 and B=2. Retain the left child. Split the right child with bounds 3/4 and 5/4, then retain both grandchildren. vfig[.85]ext_tree headStart with one at every retained node The left subtree has bounds a_0=b_0=1. The right subtree has a_1=3/4 and b_1=5/4. The recursive certificate gives a_rm root=tfrac12min{1,tfrac34}=tfrac38, b_rm root=2max{1,tfrac54}=tfrac52. Consequently the total leaf coefficient energy …
Exact mathematical passage
a_{\rm root}=\tfrac12\min\{1,\tfrac34\}=\tfrac38, \qquad b_{\rm root}=2\max\{1,\tfrac54\}=\tfrac52.
A_{RR}Use in source · 4 passages
Search aliases: A_RR
Use in source · pages.tex, line 266
1 · 1 · 2. The third coordinate is hidden. Its invertible block is [2], so eliminating it produces S=A_RR-A_RHA_HH^-1A_HR =mat3/2 · -1/2 -1/2 · 3/2. vfigschur_fill headThe hidden route has a sign The retained coordinates had no direct coupling, yet both touch the hidden coordinate. The resulting correction couples them by -1/2. This is fill: a new active retained block produced by an eliminated route.
Exact mathematical passage
S=A_{RR}-A_{RH}A_{HH}^{-1}A_{HR} =\mat{3/2&-1/2\\-1/2&3/2}.
A_TUse in source · 3 passages
Search aliases: A_T
Use in source · Paper_X_Block27_Amended.tex, line 1632
[Exact finite occupancy trace] The operator e_T is the projector onto the joint eigenspace A_T = {v:T_iv=a_iv for every i}. Moreover, Tr(e_T)=rank(e_T)=dim A_T∈ Z_≥0. The trace is identified with its unique nonnegative integer preimage under Z↪ E. Thus Tr(e_T)>0 refers to that integer, not to an ordering of an arbitrary field E. The finite packet is occupied exactly when this integer is positive. In positive characteristic the field-valued trace can vanish for a nonzero projector. The rank test,…
Exact mathematical passage
A_T = \{v:T_iv=a_iv\text{ for every }i\}.
A_uUse in source · 2 passages
Search aliases: A_u
Use in source · 10_foundations.tex, line 494
a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient and remainder give A(z)=∑_r=0^M-1z^rA_r(z^M), (a_r^[M])_s^[N]=a_r+Ms^[MN]. For ordinary Cauchy multiplication, the phase-r product has a term without a carry and a term with one carry: C_r(w)=∑_u+v=rA_u(w)B_v(w) +w∑_u+v=r+MA_u(w)B_v(w), 0≤ u,v<M. Adding the two phase addresses gives the formula. It describes ordinary Cauchy multiplication. Hurwitz convo…
Exact mathematical passage
C_r(w)=\sum_{u+v=r}A_u(w)B_v(w) +w\sum_{u+v=r+M}A_u(w)B_v(w),\qquad 0\leq u,v<M.
a_{Uf}Use in source · 1 passage
Search aliases: a_Uf
Use in source · supplement_08.tex, line 71
pageheading2. The actual integral Hurwitz latticeConnecting differential expressions to coefficient divisibility The specialization works inside Q_p[[t]] and uses factorial-scaled coefficients a_f(d)=d![t^d]f, I_p={f∈ Q_p[[t]]:a_f(d)∈ Z_p for every d}. The source constructs I_p as a subring. Actual exponential-series multiplication gives a_fg(d)=∑_r=0^dbinom dr a_f(r)a_g(d-r), a_Uf(d)=a_f(d+1). These identities prove multiplication and derivative stability of the lattice. Membership does not req…
Exact mathematical passage
a_{fg}(d)=\sum_{r=0}^{d}\binom dr a_f(r)a_g(d-r), \qquad a_{Uf}(d)=a_f(d+1).
A_vUse in source · 4 passages
Search aliases: A_v
Use in source · Paper_V_Traditional_Final.tex, line 2378
[Bounds for nonunitary local splits] Suppose every internal node has an analysis map U_v and constants 0≤ A_v≤ B_v<∞ satisfying A_v x ^2 ≤ ∑_w∈child(v) (U_vx)_w ^2 ≤ B_v x ^2. Define recursively a(v)= 1, · v retained,
Exact mathematical passage
0\leq A_v\leq B_v<\infty
A_{vu}Use in source · 1 passage
Search aliases: A_vu
Use in source · 04_correction_transport.tex, line 712
[Fundamental holonomy intertwining criterion] The following conditions are equivalent. item There is a family of isomorphisms U_v:V_0→ W_0 with the prescribed root value U_r and satisfying U_vA_vu=B_vuU_u for every oriented edge u→ v. item For every fundamental loop γ based at r, U_rHol_A(γ)=Hol_B(γ)U_r. item The same identity holds for every loop based at r. When these conditions hold, the family (U_v) is uniquely determined by U_r.
Exact mathematical passage
U_vA_{vu}=B_{vu}U_u
A_{vu}^{-1}Use in source · 1 passage
Search aliases: A_vu^-1
Use in source · 04_correction_transport.tex, line 729
Conversely, choose a spanning tree. Starting from U_r, propagate U_v along the unique tree path by the rule U_v=B_vuU_uA_vu^-1 on every tree edge. This is the only possible extension satisfying the tree edge equations. An edge outside the tree closes one fundamental loop. Its intertwining equation holds exactly when U_r intertwines the two corresponding fundamental holonomies. The second condition therefore supplies every remaining edge equation. Uniqueness follows from propagation along the tre…
Exact mathematical passage
U_v=B_{vu}U_uA_{vu}^{-1}
A_wUse in source · 4 passages
Search aliases: A_w
Use in source · 01_finite_core.tex, line 802
We call a recovery complete when it is a Kraus map R(X)=∑_jK_jXK_j^*, ∑_jK_j^*K_j=I. The family is exactly correctable when one such recovery works for every coherent error combination A_w=∑_aw_aE_a: for every w there is a real t_w≥0, independent of the logical input, such that PXP=X Longrightarrow R(A_wXA_w^*)=t_wX. The coefficient is an unnormalized weight. It is not necessarily a probability for an arbitrary matrix A_w.
Exact mathematical passage
A_w=\sum_aw_aE_a
A_w^*Use in source · 2 passages
Search aliases: A_w^*
Use in source · 01_finite_core.tex, line 803
We call a recovery complete when it is a Kraus map R(X)=∑_jK_jXK_j^*, ∑_jK_j^*K_j=I. The family is exactly correctable when one such recovery works for every coherent error combination A_w=∑_aw_aE_a: for every w there is a real t_w≥0, independent of the logical input, such that PXP=X Longrightarrow R(A_wXA_w^*)=t_wX. The coefficient is an unnormalized weight. It is not necessarily a probability for an arbitrary matrix A_w.
Exact mathematical passage
PXP=X\quad\Longrightarrow\quad \cR(A_wXA_w^*)=t_wX.
A_{z_n}Use in source · 2 passages
Search aliases: A_z_n
Use in source · continuation_03.tex, line 47
The one-step Taylor theorem already supplies the exact correction equation b_n+A_z_nt=0, b_n=F(z_n)/p^L+nbmod p. Surjectivity supplies a correcting digit for every residual b_n. The chosen correction gives an actual new integer point z_n+1, using the proved nonlinear Taylor remainder.
Exact mathematical passage
b_n+A_{z_n}t=0,\qquad b_n=F(z_n)/p^{L+n}\bmod p.
A_{z_{n+1}}Use in source · 1 passage
Search aliases: A_z_n+1
Use in source · continuation_03.tex, line 56
Every coordinate change is divisible by p^L+n, hence by p. Each formal partial derivative is itself an integer polynomial, so its value is unchanged modulo p. Therefore A_z_n+1=A_z_n=A_x. The proof assumes no separate surjectivity or solvability statement at later levels. A dependent recursion chooses each next point from the proved one-step existence theorem. It produces one compatible sequence, rather than unrelated witnesses at different precisions.
Exact mathematical passage
A_{z_{n+1}}=A_{z_n}=A_x.
A_\alphaUse in source · 2 passages
Search aliases: A_α alpha
Use in source · 01_manuscript.tex, line 1122
[Invariant packet Krylov decomposition] Assume [A,P_α]=0 for every packet. If v=∑_α v_α, v_α=P_α v, then A^jv = ∑_α A_α^jv_α for every j≥0, where A_α=P_α AP_α. Hence K_m(A,v) ⊆ ⊕_α K_m(A_α,v_α).
Exact mathematical passage
A_\alpha=P_\alpha AP_\alpha.
A_\alpha^jUse in source · 1 passage
Search aliases: A_α^j alpha
Use in source · 01_manuscript.tex, line 1116
[Invariant packet Krylov decomposition] Assume [A,P_α]=0 for every packet. If v=∑_α v_α, v_α=P_α v, then A^jv = ∑_α A_α^jv_α for every j≥0, where A_α=P_α AP_α. Hence K_m(A,v) ⊆ ⊕_α K_m(A_α,v_α).
Exact mathematical passage
A^jv = \sum_\alpha A_\alpha^jv_\alpha
A_{\alpha_1\alpha_0}Use in source · 1 passage
Search aliases: A_α_1α_0 alpha alpha
Use in source · 01_manuscript.tex, line 964
[Path support of powers] For every positive integer m, P_γ A^mP_α is the sum over directed walks (vertices may repeat) α=α_0 ⟶ α_1 ⟶⋯⟶ α_m=γ in G_A, with contribution A_α_mα_m-1 ⋯ A_α_1α_0. If no such path exists, the block vanishes.
Exact mathematical passage
A_{\alpha_m\alpha_{m-1}} \cdots A_{\alpha_1\alpha_0}.
A_{\alpha_m\alpha_{m-1}}Use in source · 1 passage
Search aliases: A_α_mα_m-1 alpha alpha
Use in source · 01_manuscript.tex, line 964
[Path support of powers] For every positive integer m, P_γ A^mP_α is the sum over directed walks (vertices may repeat) α=α_0 ⟶ α_1 ⟶⋯⟶ α_m=γ in G_A, with contribution A_α_mα_m-1 ⋯ A_α_1α_0. If no such path exists, the block vanishes.
Exact mathematical passage
A_{\alpha_m\alpha_{m-1}} \cdots A_{\alpha_1\alpha_0}.
A_{\alpha\alpha}Use in source · 1 passage
Search aliases: A_αα alpha alpha
Use in source · Paper_VI_Traditional_Final.tex, line 2190
thm:reducing_packets The following conditions are equivalent. item AP_α=P_α A for every α. item A_βα=0 whenever β≠α. item T_βα(t)=0 whenever β≠α and t∈ R. item Every packet space P_α X is invariant under T(t) for every t. When these conditions hold, T(t)|_P_α X = e^tA_αα.
Exact mathematical passage
T(t)|_{P_\alpha X} = \e^{tA_{\alpha\alpha}}.
A_{\alpha\gamma}Use in source · 1 passage
Search aliases: A_αγ alpha gamma
Use in source · Paper_VI_Traditional_Final.tex, line 2199
The first two conditions are equivalent because the packet resolution gives AP_α=∑_β A_βα, P_α A=∑_γ A_αγ. A block diagonal generator has block diagonal powers, so the second condition implies the third. Differentiating the off-diagonal blocks at zero proves the converse. The equivalence with invariance follows from the definition of the blocks. The final identity follows by restricting the exponential series.
Exact mathematical passage
AP_\alpha=\sum_\beta A_{\beta\alpha}, \qquad P_\alpha A=\sum_\gamma A_{\alpha\gamma}.
A_{\beta\alpha}Use in source · 7 passages
Search aliases: A_βα beta alpha
Use in source · Paper_VI_Traditional_Final.tex, line 2185
[Reducing packet criterion] The following conditions are equivalent. item AP_α=P_α A for every α. item A_βα=0 whenever β≠α. item T_βα(t)=0 whenever β≠α and t∈ R. item Every packet space P_α X is invariant under T(t) for every t. When these conditions hold, T(t)|_P_α X = e^tA_αα.
Exact mathematical passage
A_{\beta\alpha}=0
A_{\beta\alpha}^*Use in source · 1 passage
Search aliases: A_βα^* beta alpha
Use in source · 01_manuscript.tex, line 193
[Adjoint and tensor semantics] The adjoint representation has blocks (A^*)_αβ=A_βα^*. If A and B are packet resolved on X and Y, then (A⊗ B)_(β,δ),(α,γ) = A_βα⊗ B_δγ.
Exact mathematical passage
(A^*)_{\alpha\beta}=A_{\beta\alpha}^*.
A_{\beta\gamma}Use in source · 1 passage
Search aliases: A_βγ beta gamma
Use in source · Paper_VI_Traditional_Final.tex, line 2093
[Block differential system] The packet blocks are differentiable in operator norm and satisfy frac d d tT_βα(t) = ∑_γ∈ I A_βγT_γα(t) = ∑_γ∈ I T_βγ(t)A_γα. Their initial values are T_βα(0)=δ_βαP_α.
Exact mathematical passage
\frac{\dd}{\dd t}T_{\beta\alpha}(t) = \sum_{\gamma\in I} A_{\beta\gamma}T_{\gamma\alpha}(t) = \sum_{\gamma\in I} T_{\beta\gamma}(t)A_{\gamma\alpha}.
A_{\gamma\alpha}Use in source · 1 passage
Search aliases: A_γα gamma alpha
Use in source · Paper_VI_Traditional_Final.tex, line 2093
[Block differential system] The packet blocks are differentiable in operator norm and satisfy frac d d tT_βα(t) = ∑_γ∈ I A_βγT_γα(t) = ∑_γ∈ I T_βγ(t)A_γα. Their initial values are T_βα(0)=δ_βαP_α.
Exact mathematical passage
\frac{\dd}{\dd t}T_{\beta\alpha}(t) = \sum_{\gamma\in I} A_{\beta\gamma}T_{\gamma\alpha}(t) = \sum_{\gamma\in I} T_{\beta\gamma}(t)A_{\gamma\alpha}.
a_{\lambda}Use in source · 2 passages
Search aliases: a_λ lambda
Use in source · 05_worked_models.tex, line 282
[Certified graph seed number] Let 𝕜 be a field and let Λ be a nonempty finite set. Suppose there are pairwise distinct scalars a_λ∈𝕜, multiplicities m_λ∈ N, and an eigenbasis {b_λ,j:λ∈Λ, 1≤ j≤ m_λ} of 𝕜^V(G) satisfying A_Gb_λ,j=a_λb_λ,j. If m_λ≤ R for all λ and m_λ_*=R for some λ_*, then R is the least natural number of adjacency seeds: R=min{r∈ N: l there exist s_1,…,s_r with
Exact mathematical passage
a_{\lambda}\in\bbk
a_{\rho,r}Use in source · 2 passages
Search aliases: a_ρ,r rho
Use in source · Paper_IV_Traditional_Final.tex, line 2645
_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_ρ,ℓ binomm+ℓ-1ℓ-1 ρ^-m + O_R_*(m!R_*^-m). The leading Laurent coefficient is a_ρ,r = (-ρ D_N'(ρ))^-r.
Exact mathematical passage
a_{\rho,r} = (-\rho\Dfun_N'(\rho))^{-r}.
a_{\rho,\ell}Use in source · 2 passages
Search aliases: a_ρ,ℓ rho ell
Use in source · Paper_IV_Traditional_Final.tex, line 2622
[Nearest-pole coefficient expansion] Let r∈ Z_>0. Suppose the zeros of D_N of least modulus R are simple and form the finite set Z_R. Suppose the next zero modulus is R_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_ρ,ℓ binomm+ℓ-1ℓ-1 ρ^-m + O_R_*(m!R_*^-m). The leading Laurent coefficient is a_ρ,r = (-ρ D_N'(ρ))^-r.
Exact mathematical passage
\Dfun_N(z)^{-r} = \sum_{\ell=1}^{r} a_{\rho,\ell} \left(1-\frac z\rho\right)^{-\ell} + O(1).
A_\ellUse in source · 9 passages
Search aliases: A_ℓ ell
Use in source · Paper_VI_Traditional_Final.tex, line 1329
[Fractional refinement towers] Let D_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ in a bounded nonnegative self-adjoint tower, with sup_ℓnormC_ℓ≤ M. If ∑_ℓnormD_ℓ^α<∞, then the fractional defects C_ℓ A_ℓ+1^α-A_ℓ^αC_ℓ have summable operator norms. If the tower has a uniform gap m>0, it is enough that ∑_ℓnormD_ℓ<∞. On every bounded time interval, the series of corresponding fractional-semigroup defect norms converges uniformly. The maps may have different domain and codomain spaces. No operator sum across unrelated spaces…
Exact mathematical passage
D_\ell=C_\ell A_{\ell+1}-A_\ell C_\ell
a_\ellUse in source · 5 passages
Search aliases: a_ℓ ell
Use in source · Paper_IV_Traditional_Final.tex, line 518
At the origin, the derivative matrix is diagonal with entries 1,λ,…,λ^N-1. For an explicit determinant proof, write each basis function as a finite sum of exponentials with exponents a_ℓ=λω^ℓ. Its derivative matrix factors as W(z)=V diag(e^a_ℓ z) C, V_kℓ=a_ℓ^k, where C is the constant matrix of Fourier weights. Thus det W(z)=e^z∑_ℓ a_ℓdet W(0)=det W(0), because N≥2 implies ∑_ℓω^ℓ=0. The product of the diagonal initial values is the stated power of λ.
Exact mathematical passage
a_\ell=\lambda\omega^\ell
A_{\ell+1}Use in source · 4 passages
Search aliases: A_ℓ+1 ell
Use in source · Paper_VI_Traditional_Final.tex, line 1329
[Fractional refinement towers] Let D_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ in a bounded nonnegative self-adjoint tower, with sup_ℓnormC_ℓ≤ M. If ∑_ℓnormD_ℓ^α<∞, then the fractional defects C_ℓ A_ℓ+1^α-A_ℓ^αC_ℓ have summable operator norms. If the tower has a uniform gap m>0, it is enough that ∑_ℓnormD_ℓ<∞. On every bounded time interval, the series of corresponding fractional-semigroup defect norms converges uniformly. The maps may have different domain and codomain spaces. No operator sum across unrelated spaces…
Exact mathematical passage
D_\ell=C_\ell A_{\ell+1}-A_\ell C_\ell
\mathfrak A_{\ell+1,ab}Use in source · 1 passage
Search aliases: A_ℓ+1,ab mathfrak ell
Use in source · 02_strengthened.tex, line 2279
[Exact constraint transport] Assume C_ℓ+1,aR_ℓ=R_ℓ C_ℓ,a for every a. Then [C_ℓ+1,a,C_ℓ+1,b]R_ℓ = R_ℓ[C_ℓ,a,C_ℓ,b]. If the same structure coefficients are used at both levels, then A_ℓ+1,abR_ℓ = R_ℓ A_ℓ,ab. The isometry maps coarse physical states into fine physical states.
Exact mathematical passage
\mathfrak A_{\ell+1,ab}R_\ell = R_\ell\mathfrak A_{\ell,ab}.
A_{\ell+1}^{\alpha}Use in source · 1 passage
Search aliases: A_ℓ+1^α ell alpha
Use in source · Paper_VI_Traditional_Final.tex, line 1334
[Fractional refinement towers] Let D_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ in a bounded nonnegative self-adjoint tower, with sup_ℓnormC_ℓ≤ M. If ∑_ℓnormD_ℓ^α<∞, then the fractional defects C_ℓ A_ℓ+1^α-A_ℓ^αC_ℓ have summable operator norms. If the tower has a uniform gap m>0, it is enough that ∑_ℓnormD_ℓ<∞. On every bounded time interval, the series of corresponding fractional-semigroup defect norms converges uniformly. The maps may have different domain and codomain spaces. No operator sum across unrelated spaces…
Exact mathematical passage
C_\ell A_{\ell+1}^{\alpha}-A_\ell^{\alpha}C_\ell
\mathfrak A_{\ell,ab}Use in source · 1 passage
Search aliases: A_ℓ,ab mathfrak ell
Use in source · 02_strengthened.tex, line 2279
[Exact constraint transport] Assume C_ℓ+1,aR_ℓ=R_ℓ C_ℓ,a for every a. Then [C_ℓ+1,a,C_ℓ+1,b]R_ℓ = R_ℓ[C_ℓ,a,C_ℓ,b]. If the same structure coefficients are used at both levels, then A_ℓ+1,abR_ℓ = R_ℓ A_ℓ,ab. The isometry maps coarse physical states into fine physical states.
Exact mathematical passage
\mathfrak A_{\ell+1,ab}R_\ell = R_\ell\mathfrak A_{\ell,ab}.
A_{\ell,n}Use in source · 1 passage
Search aliases: A_ℓ,n ell
Use in source · 72_recovered.tex, line 1278
. For p=3, the original odd source has A_3,1=S_1(3)=3, whereas every odd dyadic source coefficient is zero. The even kernel L_3K_2 cannot change that odd coefficient. Read the coefficient consequence below only at even indices. The displayed identity H_p=L_pK_ℓ H_ℓ and its proof remain valid. For any two primes p and ℓ, the source sequences A_p,n and A_ℓ,n are related by an exact unit lower triangular transformation obtained by composing the inverse kernel L_p with the reconstruction kernel K_ℓ.
Exact mathematical passage
A_{\ell,n}
a_\ell^kUse in source · 1 passage
Search aliases: a_ℓ^k ell
Use in source · Paper_IV_Traditional_Final.tex, line 520
At the origin, the derivative matrix is diagonal with entries 1,λ,…,λ^N-1. For an explicit determinant proof, write each basis function as a finite sum of exponentials with exponents a_ℓ=λω^ℓ. Its derivative matrix factors as W(z)=V diag(e^a_ℓ z) C, V_kℓ=a_ℓ^k, where C is the constant matrix of Fourier weights. Thus det W(z)=e^z∑_ℓ a_ℓdet W(0)=det W(0), because N≥2 implies ∑_ℓω^ℓ=0. The product of the diagonal initial values is the stated power of λ.
Exact mathematical passage
W(z)=V\,\operatorname{diag}(e^{a_\ell z})\,C, \qquad V_{k\ell}=a_\ell^k,
A_\ell^{\alpha}Use in source · 1 passage
Search aliases: A_ℓ^α ell alpha
Use in source · Paper_VI_Traditional_Final.tex, line 1334
[Fractional refinement towers] Let D_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ in a bounded nonnegative self-adjoint tower, with sup_ℓnormC_ℓ≤ M. If ∑_ℓnormD_ℓ^α<∞, then the fractional defects C_ℓ A_ℓ+1^α-A_ℓ^αC_ℓ have summable operator norms. If the tower has a uniform gap m>0, it is enough that ∑_ℓnormD_ℓ<∞. On every bounded time interval, the series of corresponding fractional-semigroup defect norms converges uniformly. The maps may have different domain and codomain spaces. No operator sum across unrelated spaces…
Exact mathematical passage
C_\ell A_{\ell+1}^{\alpha}-A_\ell^{\alpha}C_\ell
A_\inftyUse in source · 2 passages
Search aliases: A_∞ infty
Use in source · Paper_VI_Traditional_Final.tex, line 2491
If T_ℓ(t)= e^tA_ℓ, the generators are bounded, and sup_ℓnormA_ℓ<∞, C_ℓ A_ℓ+1=A_ℓ C_ℓ, then the continuity condition holds and A_∞(x_ℓ)_ℓ=(A_ℓ x_ℓ)_ℓ is the bounded generator of T_∞.
Exact mathematical passage
A_\infty(x_\ell)_\ell=(A_\ell x_\ell)_\ell
\AAUse in source · 9 passages
Search aliases: AA AA A mathcal
Use in source · visual_atlas.tex, line 19
[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…
Exact mathematical passage
4pt] \textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt] \textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology? \end{tabularx} A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. \takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''} \vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}} Most of the chapter can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix. {\small \begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule \textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule $X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\ $S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\ $\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\ $\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\ $P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\ $X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\ $T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\ $F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\ $U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\ $\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\ $\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule \end{tabularx}} \head{Read products from right to left} In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column. \head{Where to begin} For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}. \vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$} Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define \[ Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0, \qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.
\Acal'Use in source · 1 passage
Search aliases: Acal' A' Acal mathcal
Use in source · 02_strengthened.tex, line 863
[Nonabelian packet connection] The one form with matrix values A=V^* d V is skew adjoint. Under a frame change V'=VG, where G(x)∈ U(r), A' = G^* A G+G^* d G. Its curvature F= d A+ Awedge A satisfies F=V^*( d Pwedge d P)V.
Exact mathematical passage
\Acal' = G^*\Acal G+G^*\dd G.
\Acal^*Use in source · 1 passage
Search aliases: Acal^* A^* Acal mathcal
Use in source · 02_strengthened.tex, line 879
Differentiate V^*V=I to obtain A^*=- A. The gauge transformation law follows by differentiating VG. Since P=VV^* and (I-P) d V= d P V, direct expansion gives d A+ Awedge A = d V^*(I-P)wedge d V = V^*( d Pwedge d P)V.
Exact mathematical passage
\Acal^*=-\Acal
\Acal_EUse in source · 2 passages
Search aliases: Acal_E A_E Acal mathcal
Use in source · Paper_V_Traditional_Final.tex, line 2139
[Frame bounds from the polyphase symbol] Assume that A I_N ≤ E( e^ iω)^*E( e^ iω) ≤ B I_N for almost every ω, where 0<A≤ B<∞. Then the analysis bank satisfies A x _2^2 ≤ A_E x _2^2 ≤ B x _2^2. The pointwise matrix R_rm can( e^ iω) = (E^*E)^-1E^* is a bounded left inverse and gives the canonical dual synthesis bank.
Exact mathematical passage
A\|x\|_2^2 \leq \|\Acal_E x\|_2^2 \leq B\|x\|_2^2.
\Acal_L^{-1}Use in source · 2 passages
Search aliases: Acal_L^-1 A_L^-1 Acal mathcal
Use in source · Paper_V_Traditional_Final.tex, line 1934
[Exact convolution transfer law] Let y=h*x. Then y_r(ℓ) = ∑_s∈ R∑_m∈ Lh_rs(m)x_s(ℓ-m). Equivalently, A_LH A_L^-1 = (H_rs)_r,s∈ R, where H_rs is convolution on L by h_rs.
Exact mathematical passage
\Acal_LH\Acal_L^{-1} = (H_{rs})_{r,s\in R},
\Acal_{\Lcal}Use in source · 2 passages
Search aliases: Acal_Lcal A_ L Acal Lcal mathcal mathcal
Use in source · Paper_V_Traditional_Final.tex, line 2343
[Reconstruction on a pruned packet tree] Every admissible pruning L defines a unitary analysis map A_ L: H_rm root ⟶ ⊕_v∈ L H_v. Its inverse is obtained by applying the adjoints of the local splits from the leaves toward the root. In particular, x ^2 = ∑_v∈ L x_v ^2.
Exact mathematical passage
\Acal_{\Lcal}:\Hcal_{\rm root} \longrightarrow \bigoplus_{v\in\Lcal}\Hcal_v.
\Acal_{\Lcal}^{-1}Use in source · 1 passage
Search aliases: Acal_Lcal^-1 A_ L^-1 Acal Lcal mathcal mathcal
Use in source · Paper_V_Traditional_Final.tex, line 2370
The synthesis map is unitary by [source reference]. Hence widehat x-x = A_ L^-1e has the same norm as e.
Exact mathematical passage
\widehat x-x =\Acal_{\Lcal}^{-1}e
\Acal_{\Rcal}Use in source · 1 passage
Search aliases: Acal_Rcal A_ R Acal Rcal mathcal mathcal
Use in source · 03_nonlinear_coordinates.tex, line 146
The identities S_ P A_ P=I_X and S_ Q A_ Q=I_Y give S_ QF_ Q← P A_ P = S_ Q A_ QF S_ P A_ P =F. For composition, G_ R← QF_ Q← P = A_ RG S_ Q A_ QF S_ P = A_ R(G∘ F) S_ P.
Exact mathematical passage
G_{\Rcal\leftarrow\Qcal}F_{\Qcal\leftarrow\Pcal} = \Acal_{\Rcal}G\Scal_{\Qcal}\Acal_{\Qcal}F\Scal_{\Pcal} = \Acal_{\Rcal}(G\circ F)\Scal_{\Pcal}.
\Acal_{\Ucal}Use in source · 1 passage
Search aliases: Acal_Ucal A_ U Acal Ucal mathcal mathcal
Use in source · 03_nonlinear_coordinates.tex, line 170
[Change of packet coordinates] Let P and P' be packet systems on X. Let Q and Q' be packet systems on Y. For any two packet systems U and V on the same space, write J_ U← V = A_ U S_ V. Then F_ Q'← P' = J_ Q'← Q F_ Q← P J_ P← P'.
Exact mathematical passage
J_{\Ucal\leftarrow\Vcal} = \Acal_{\Ucal}\Scal_{\Vcal}.
\mathrm{act}Use in source · 6 passages
Search aliases: act mathrm
Use in source · root_scope.tex, line 374
ion K_r(Z)=2^-178G_r(10202+2^15Z). The complete coefficient theorem gives K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z), a_r,b_r∈ Z_2, u_r∈ Z_2^×, R_r∈ Z_2⟨ Z⟩. Its half derivative is a dyadic isometry. It therefore has a unique critical point c_r∈ Z_2. Set Φ_r^act=K_r(c_r). The earlier independent-undepletion notation for this same scalar is Φ_r(2^2r-25). Excluding a multiple root on the surviving chart is exactly the assertion Φ_r^act≠0.
Exact mathematical passage
\Phi_r^{\mathrm{act}}=K_r(c_r)
\AdUse in source · 4 passages
Search aliases: Ad operatorname
Use in source · 02_strengthened.tex, line 1093
Let S be unitary and N≥1 an integer with S^N=I, and define the conjugation action Ad_S(X)=SXS^*.
Exact mathematical passage
\Ad_S(X)=SXS^*.
\adUse in source · 9 passages
Search aliases: ad operatorname
Use in source · Paper_X_Block27_Amended.tex, line 339
[Tangent and obstruction spaces] For a framed representation problem, first-order lifts are described by Z^1(G,adbarρ). For an unframed Schur problem, tangent directions are described by H^1(G,adbarρ). Relations in a presentation of a deformation ring map into H^2(G,adbarρ), which contains the obstruction classes.
Exact mathematical passage
Z^1(G,\ad\bar\rho)
\ad^0Use in source · 6 passages
Search aliases: ad^0 ad operatorname
Use in source · Paper_XII_Traditional_Final.tex, line 351
[Adequate image] Let G⊆GL(V) act absolutely irreducibly over a finite field k of characteristic p. Under the standard definition, adequacy imposes vanishing conditions on H^1(G,k) and H^1(G,ad^0V), a scalar condition, and a generalized eigenspace detection condition for every irreducible submodule of ad^0V.
Exact mathematical passage
H^1(G,\ad^0V)
\Ad_SUse in source · 3 passages
Search aliases: Ad_S Ad operatorname
Use in source · 02_strengthened.tex, line 1093
Let S be unitary and N≥1 an integer with S^N=I, and define the conjugation action Ad_S(X)=SXS^*.
Exact mathematical passage
\Ad_S(X)=SXS^*.
\Ad_{U(g)}Use in source · 1 passage
Search aliases: Ad_U(g) Ad operatorname
Use in source · 02_strengthened.tex, line 1166
Let Ω=∑_iketi⊗keti. Then J(Φ)=(Φ⊗id)(ketΩbraΩ). The vector Ω is invariant under U⊗overline U. Covariance therefore implies the displayed Choi invariance. Conversely, the displayed transformation is the Choi matrix of Ad_U(g)∘Φ∘Ad_U(g)^*. The Choi correspondence is injective, so invariance implies that this map equals Φ, which is covariance.
Exact mathematical passage
\Ad_{U(g)}\circ\Phi\circ\Ad_{U(g)^*}.
\Ad_{U(g)^*}Use in source · 1 passage
Search aliases: Ad_U(g)^* Ad operatorname
Use in source · 02_strengthened.tex, line 1166
Let Ω=∑_iketi⊗keti. Then J(Φ)=(Φ⊗id)(ketΩbraΩ). The vector Ω is invariant under U⊗overline U. Covariance therefore implies the displayed Choi invariance. Conversely, the displayed transformation is the Choi matrix of Ad_U(g)∘Φ∘Ad_U(g)^*. The Choi correspondence is injective, so invariance implies that this map equals Φ, which is covariance.
Exact mathematical passage
\Ad_{U(g)}\circ\Phi\circ\Ad_{U(g)^*}.
\operatorname{adj}Use in source · 5 passages
Search aliases: adj operatorname
Use in source · 03_weighted.tex, line 412
ce every entry of A(t) has zero constant term, the inverse exists in the formal power series ring and (I-A(t))^-1=∑_r≥0A(t)^r. The (i,j) entry of A(t)^r is the sum of the monomials attached to all paths of length r from i to j. Summing over r records paths of every length. Multiplication by the endpoint vectors gives the stated series. The adjugate formula gives (I-A(t))^-1 = fracadj(I-A(t))det(I-A(t)), so every entry is rational.
Exact mathematical passage
(I-A(t))^{-1} = \frac{\operatorname{adj}(I-A(t))}{\det(I-A(t))},
\operatorname{analyze}Use in source · 1 passage
Search aliases: analyze operatorname
Use in source · 06_models.tex, line 214
The verified contract is synthesize(analyze(x,N))=x. The Lean counterpart uses packet coordinates restricted to projector ranges and proves both inverse laws. The finite Python routine checks the concrete array realization.
Exact mathematical passage
\operatorname{synthesize}(\operatorname{analyze}(x,N))=x
\operatorname{Ann}Use in source · 8 passages
Search aliases: Ann operatorname
Use in source · 02_foundations.tex, line 286
[Presentation with one seed] For s∈ M, the annihilator Ann_A(s):={a∈ A:as=0} is a left ideal and A/Ann_A(s)≅Hull_A(s). [source reference]
Exact mathematical passage
\operatorname{Ann}_A(s):=\{a\in A:as=0\}
\operatorname{Ann}_AUse in source · 8 passages
Search aliases: Ann_A operatorname
Use in source · 02_foundations.tex, line 286
[Presentation with one seed] For s∈ M, the annihilator Ann_A(s):={a∈ A:as=0} is a left ideal and A/Ann_A(s)≅Hull_A(s). [source reference]
Exact mathematical passage
\operatorname{Ann}_A(s):=\{a\in A:as=0\}
\argUse in source · 2 passages
Search aliases: arg
Use in source · Paper_IV_Traditional_Final.tex, line 1871
If z=x+iy lies in the sector |arg z|≤ a_N, has x≥ R, and is a zero of D_N, then |y|≤ Y.
Exact mathematical passage
|\arg z|\leq a_N
\arraybackslashUse in source · 1 passage
Search aliases: arraybackslash
Use in source · visual_atlas.tex, line 19
[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…
Exact mathematical passage
4pt] \textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt] \textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology? \end{tabularx} A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. \takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''} \vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}} Most of the chapter can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix. {\small \begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule \textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule $X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\ $S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\ $\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\ $\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\ $P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\ $X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\ $T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\ $F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\ $U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\ $\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\ $\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule \end{tabularx}} \head{Read products from right to left} In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column. \head{Where to begin} For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}. \vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$} Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define \[ Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0, \qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.
\ArtUse in source · 1 passage
Search aliases: Art operatorname
Use in source · Paper_XI_Traditional_Final.tex, line 693
[Conductor of a nontrivial tame atom] Let τ be an irreducible tame Weil representation of dimension n>1. Then Swan(τ)=0, dimτ^I_K=0, Art(τ)=n.
Exact mathematical passage
\Swan(\tau)=0, \qquad \dim\tau^{I_K}=0, \qquad \Art(\tau)=n.
\operatorname{Art}Use in source · 1 passage
Search aliases: Art operatorname
Use in source · Paper_XI_Traditional_Final.tex, line 1829
The dimension and orbit follow from [source reference]. The cycle polynomial is [source reference]. For rank greater than one, the conductor is [source reference]. In rank one, the Swan conductor is zero, and the Artin conductor is zero for trivial inertia and one otherwise, by the formula Art=dim V-dim V^I_K+Sw. The local polynomial is [source reference]. Tensor refinement is [source reference]. The Gauss sum is its defining finite sum. For a nontrivial multiplicative character its unit phase i…
Exact mathematical passage
\operatorname{Art}=\dim V-\dim V^{I_K}+\operatorname{Sw}
\operatorname{aug}Use in source · 1 passage
Search aliases: aug operatorname
Use in source · 72_recovered.tex, line 210
[Augmentation criterion] Let x=∑_u∈ F_pc_u[u] ∈ F_p[ F_p^+]. Then x is a unit if and only if aug(x) = ∑_uc_u is nonzero. If J is the augmentation ideal, then J^p=0.
Exact mathematical passage
\operatorname{aug}(x) = \sum_uc_u
\operatorname{Aut}_KUse in source · 1 passage
Search aliases: Aut_K operatorname
Use in source · Paper_X_Block27_Amended.tex, line 2476
The inertia theorem uses actual residue actions. For a valuation subring A of an extension field L/K, let D_A be its decomposition subgroup in Aut_K(L) and put I_A=ker(D_A⟶Aut(k_A)), where k_A is the residue field. If 2 is a unit in A, every member of I_A fixes every y∈ L with y^2=-1. Indeed y^-1=-y, so y belongs to A and is a unit there. An automorphism sends y to y or -y. Their residues are distinct because 2 and y are units, whereas inertia fixes residues. The second possibility is excluded.
Exact mathematical passage
\operatorname{Aut}_K(L)
\bm BUse in source · 1 passage
Search aliases: B bm
Use in source · Paper_V_Traditional_Final.tex, line 507
Finite convolution represents Laurent-polynomial multiplication: composing shifts adds their delays, and distributivity handles finite sums. Thus the matrix identity implies the asserted action. Conversely, let B= R E-cz^-dI_N and suppose its action vanishes on every channel sequence. Insert an impulse at channel s, address zero, with every other channel zero. Output channel k reads the coefficients of the Laurent polynomial B_ks at all addresses. All of these coefficients vanish, so B_ks=0 for …
Exact mathematical passage
\bm B=\bm R\bm E-cz^{-d}I_N
\bm bUse in source · 1 passage
Search aliases: b bm
Use in source · 02_strengthened.tex, line 920
The Pauli identity ( a·σ)( b·σ) =( a· b)I + i( a× b)·σ gives - iTr(P[∂_θ P,∂_ϕ P]) =frac12 n· (∂_θ n×∂_ϕ n) =frac12sinθ. Integrating the displayed curvature over the sphere and applying the stated normalization gives one.
Exact mathematical passage
(\bm a\cdot\bm\sigma)(\bm b\cdot\bm\sigma) =(\bm a\cdot\bm b)I +\ii(\bm a\times\bm b)\cdot\bm\sigma
\mathrm BUse in source · 4 passages
Search aliases: B mathrm
Use in source · atlas-pages.tex, line 90
Any other complete packet family still has exact block convolution. For the coordinate groups A and B, P_b e^(s+t)QP_a=∑_c∈{ A, B}(P_b e^sQP_c)(P_c e^tQP_a). The intermediate packet cannot generally be omitted. Only a reducing family eliminates all cross terms.
Exact mathematical passage
P_b e^{(s+t)Q}P_a=\sum_{c\in\{\mathrm A,\mathrm B\}}(P_b e^{sQ}P_c)(P_c e^{tQ}P_a).
\mathrm bUse in source · 3 passages
Search aliases: b mathrm
Use in source · 03_packets_and_graphs.tex, line 835
[Bounded support patterns closed under *] Let X be Hilbert, let the packet projectors be orthogonal, assume X_α≠0 for every α∈ I, and put M_R^ b:= M_R∩ B(X). Then M_R^ b is closed under adjoints if and only if R is symmetric. Hence M_R^ b is a unital * subalgebra of B(X) if and only if R is an equivalence relation. [source reference]
Exact mathematical passage
\mathfrak M_R^{\mathrm b}:=\mathfrak M_R\cap\mathcal B(X).
b'Use in source · 6 passages
Search aliases: b'
Use in source · 71_native.tex, line 238
R(a,b), R(a',b), R(a',b') Longrightarrow R(a,b')
Exact mathematical passage
R(a,b),\ R(a',b),\ R(a',b')\quad\Longrightarrow\quad R(a,b')
b^{(m,p)}Use in source · 4 passages
Search aliases: b^(m,p)
Use in source · 72_recovered.tex, line 300
Since the subgroup average has augmentation one, it has a unique Hurwitz inverse. Write b^(m,p) = (a^(m))^-1.
Exact mathematical passage
b^{(m,p)} = (a^{(m)})^{-1}.
B^{-1}Use in source · 4 passages
Search aliases: B^-1
Use in source · visual_atlas.tex, line 356
Write B=1 · -1 0 · 2, AB=Bdiag(1,-1), E_+=Bdiag(1,0)B^-1. The index-two eigenlattice explains the denominator in B^-1 and in the projector. Over the localized lattice Λ_p=ZZ_(p)^2, the number 2 is a unit for every odd prime p. Hence the local denominator weight is zero there and one at p=2.
Exact mathematical passage
B=\begin{pmatrix}1&-1\\0&2\end{pmatrix},\qquad AB=B\diag(1,-1),\qquad E_+=B\diag(1,0)B^{-1}.
b^{-2}Use in source · 1 passage
Search aliases: b^-2
Use in source · strengthening.tex, line 730
ties give sinh t-sin t =2asinh(at)cosh(bt)+2bcosh(at)sinh(bt), cosh t-cos t=2sinh(at)sinh(bt). Consequently D_3(t):=G_4,3(t)F_2(t) =frac32(fraccoth(at)at +fraccoth(bt)bt). This expression is a power series: its two polar terms cancel because a^-2+b^-2=0. Its constant is one, only degrees divisible by four survive, and the Bernoulli expansion gives D_3(t)=∑_i≥0d_ifract^4i(4i)!, d_i=3(-frac14)^i frac2^4i+2B_4i+2(4i+2)(4i+1). Again v_2(d_i)≥2i, and the inverse coefficients h_i have the same bound. …
Exact mathematical passage
a^{-2}+b^{-2}=0
B^2Use in source · 3 passages
Search aliases: B^2
Use in source · root_scope.tex, line 462
One sufficient new arithmetic input would be an odd-denominator projective quadratic A Z^2+2BZ+C≡ A W_r(Z)pmod2^P, A odd, 0<|B^2-AC|<2^P, at a proved precision for every admitted ordinary r. Then (B^2-AC)/A^2≡δ_rpmod2^P cannot vanish modulo 2^P. No depth-independent theorem supplying this inequality has been established. The tested recurrence and exterior-contact identities retain balancing terms and do not force a contradiction. Finite exponential models approximate the actual target at every d…
Exact mathematical passage
A Z^2+2BZ+C\equiv A\mathcal W_r(Z)\pmod{2^P},\qquad A\text{ odd}, \qquad 0<|B^2-AC|<2^P,
\mathcal B^2Use in source · 2 passages
Search aliases: B^2 mathcal
Use in source · strengthening.tex, line 624
The variable B=2Z+1 has transform e^tfrac2te^2t-1=fractsinh t. Its odd moments vanish, and P=( B^2-1)/4=Z(Z+1) has the divided-power moments of Lemma~[source reference]. Introduce an independent copy D and Q=( D^2-1)/4. The exact factorization F_2(t)=fract^2cosh t-cos t =fracatsinh(at)fracbtsinh(bt) shows that U=a B+b D satisfies L(U^4n)=C_2(n). Direct expansion gives H_2:=fracU^4-14 =P+Q-P^2-Q^2+6PQ+i B D(P-Q).
Exact mathematical passage
P=(\mathcal B^2-1)/4=Z(Z+1)
b^2Use in source · 7 passages
Search aliases: b^2
Use in source · visual_atlas.tex, line 207
vfig[.97]weighted_word_cutoff platecaptionAll nonzero retained terms of the signed inverse sum, followed by the first discarded nonzero word. The empty word contributes I. Other words of weight below five can still be zero because a^2=b^2=0.
Exact mathematical passage
a^2=b^2=0
B^{[j]}Use in source · 1 passage
Search aliases: B^[j]
Use in source · odd_tower.tex, line 254
The field degrees and critical levels of B^[j] are exactly
Exact mathematical passage
B^{[j]}
b^pUse in source · 2 passages
Search aliases: b^p
Use in source · 20_arithmetic.tex, line 275
To deduce flatness from the weighted estimate, use Θ=(1+T)d/dT. Both Θ and (Θ^p^2-Θ^p)/p preserve W_p. For the first operator, its coefficient formula is (d+1)a_d+1+da_d. Whenever the floor weight rises, the factor d+1 provides the missing p. For the second operator, the proof uses its full Leibniz correction and the scalar congruence p^2mid b^p^2-b^p. Weighted monomials factor as a_dT^d=(p^lfloor d/prfloora_d) (T^p/p)^lfloor d/prfloorT^dbmod p. The operator preserves these monomials and hence f…
Exact mathematical passage
p^2\mid b^{p^2}-b^p
b^{p^2}Use in source · 2 passages
Search aliases: b^p^2
Use in source · 20_arithmetic.tex, line 275
To deduce flatness from the weighted estimate, use Θ=(1+T)d/dT. Both Θ and (Θ^p^2-Θ^p)/p preserve W_p. For the first operator, its coefficient formula is (d+1)a_d+1+da_d. Whenever the floor weight rises, the factor d+1 provides the missing p. For the second operator, the proof uses its full Leibniz correction and the scalar congruence p^2mid b^p^2-b^p. Weighted monomials factor as a_dT^d=(p^lfloor d/prfloora_d) (T^p/p)^lfloor d/prfloorT^dbmod p. The operator preserves these monomials and hence f…
Exact mathematical passage
p^2\mid b^{p^2}-b^p
B^rUse in source · 3 passages
Search aliases: B^r
Use in source · 72_recovered.tex, line 1681
means the matrix-binomial interpolation. With p=3, C=4, L=1, w=1 and v_0=1/3, one has Δ_1a_0=1, which is not in 3 Z_3. The printed claim without these restrictions is therefore not a certificate as written. The full original proof follows under the corrected hypotheses. Assume C^H=I+p^LB with B∈ M_d( Z_p). For every address j and every x∈ Z_p, a_Hx+j = ∑_r≥0 p^Lrbinom xr w^TC^jB^rv_0. The series converges on Z_p, and Δ_H^ra_n ≡0 pmod p^Lr.
Exact mathematical passage
\boxed{ a_{Hx+j} = \sum_{r\geq0} p^{Lr}\binom xr w^TC^jB^rv_0. }
B^sUse in source · 2 passages
Search aliases: B^s
Use in source · supplement_08.tex, line 46
All orders, without a circular closure assumption. Let I⊂ A be a subring stable under U. Define Bounded_r(f)iff B^s f∈ I (0≤ s≤ r), Flat(f)iff B^s f∈ I (s≥0). The definitions contain no product or inverse closure requirement. Since B commutes with U, derivatives of a bounded element remain bounded. Induction on r in the displayed product rule then proves closure under multiplication. The finite middle sums use the lower-order induction hypothesis, not the desired next-order conclusion.
Exact mathematical passage
\operatorname{Bounded}_r(f)\iff B^s f\in I\ (0\le s\le r), \qquad \operatorname{Flat}(f)\iff B^s f\in I\ (s\ge0).
B^\timesUse in source · 2 passages
Search aliases: B^× times
Use in source · Paper_X_Block27_Amended.tex, line 2325
Fix the continuous quotient q:G_ Q⟶ C_2 obtained by the action on a chosen square root i of -1 in the algebraic closure. Its surjectivity and the equivalence q(σ)=1 Longleftrightarrow σ(i)=i are proved using the existing actual quadratic Galois construction. For a commutative ring B, define the arithmetic packet P_q(B)= {χ:G_ Q⟶ B^×midker q⊆kerχ}, where the maps are group homomorphisms. Unique descent through the actual surjective map q is derived, rather than supplied. Every member has finite i…
Exact mathematical passage
\mathcal P_q(B)= \{\chi:G_{\Q}\longrightarrow B^\times\mid\ker q\subseteq\ker\chi\},
B^\alphaUse in source · 7 passages
Search aliases: B^α alpha
Use in source · Paper_VI_Traditional_Final.tex, line 1194
Let A≥0 act on a fine Hilbert space, let B≥0 act on a coarse Hilbert space, and let C be a bounded coarsening map. Set D=CA-BC, D_α=CA^α-B^α C, 0<α<1.
Exact mathematical passage
D=CA-BC, \qquad D_\alpha=CA^\alpha-B^\alpha C, \qquad 0<\alpha<1.
B_{\mathrm F}^{2}Use in source · 1 passage
Search aliases: B_ F^2 mathrm
Use in source · 02_supplement_06.tex, line 66
For the Frobenius norm, trace cyclicity proves left and right isometric invariance. The proof also establishes the projection identity normB-QQ^*B_ F^2 =norm B_ F^2-normQ^*B_ F^2. Nonnegativity of the left side gives left compression contractivity. The right version follows by adjoints. No norm-preservation or contractivity hypothesis is inserted.
Exact mathematical passage
\norm{B-QQ^*B}_{\mathrm F}^{2} =\norm B_{\mathrm F}^{2}-\norm{Q^*B}_{\mathrm F}^{2}.
B_0Use in source · 3 passages
Search aliases: B_0
Use in source · 20_arithmetic.tex, line 80
[The Bernoulli boundary address] At N=2, the address j=1 gives a second classical family. Define the Bernoulli numbers by B(z)=fracze^z-1=∑_r≥0B_rfracz^rr!, B_0=1, B_1=-frac12. All quotients here are regular formal series. The identity 2B(z)-B(2z)=fraczsinh z =∑_n≥0C_2,1(n)fracz^2n(2n)! follows by putting the two fractions on a common denominator and using [source reference]. Comparing even coefficients yields C_2,1(n)=(2-2^2n)B_2n (n∈ N). This classical derivation restores the boundary example …
Exact mathematical passage
B(z)=\frac{z}{e^z-1}=\sum_{r\ge0}B_r\frac{z^r}{r!}, \qquad B_0=1,\quad B_1=-\frac12.
b_{0,r}Use in source · 1 passage
Search aliases: b_0,r
Use in source · supplement_09.tex, line 99
Independently, let b_d,r be the residue-r coordinate of (x)_d in R_N. Actual group-ring multiplication gives b_0,r=1_r=0, b_d+1,r=b_d,r-1-d b_d,r. Induction comparing these two recurrences proves d!coefTdH_r(log(1+T))=b_d,r. Thus the cyclic coefficient is not an unrelated stand-in for a section coefficient. For the canonical label 0≤ c<N, the formal section is also proved equal to the original mcodeExponentialSection.allowbreakmcodeoriginalSection used in the existing Euler project.
Exact mathematical passage
b_{0,r}=\mathbf1_{r=0},\qquad b_{d+1,r}=b_{d,r-1}-d\,b_{d,r}.
B_2Use in source · 6 passages
Search aliases: B_2
Use in source · strengthening.tex, line 1013
[Exact four-address first differences] For every n≥0, H>0, and j∈{0,1,2,3}, v_2(C_j(n+H)-C_j(n)) · =v_2(H)+b_j, · (b_0,b_1,b_2,b_3) · =(1,1,4,2), v_2(E_4(n+H)^(4,j)-E_4n^(4,j)) · =v_2(H)+B_j, · (B_0,B_1,B_2,B_3) · =(1,1,5,3). Consequently, the least positive row period modulo 2^r is exactly P_j(r)=2^max(0,r-B_j) (r≥1). The corresponding displacement in the original Euler subscript is 4P_j(r).
Exact mathematical passage
v_2(C_j(n+H)-C_j(n))&=v_2(H)+b_j, &(b_0,b_1,b_2,b_3)&=(1,1,4,2),\label{eq:rank-four-normalized-isometry}\\ v_2(E_{4(n+H)}^{(4,j)}-E_{4n}^{(4,j)})&=v_2(H)+B_j, &(B_0,B_1,B_2,B_3)&=(1,1,5,3). \label{eq:rank-four-original-isometry}
b_2Use in source · 10 passages
Search aliases: b_2
Use in source · Paper_IV_Traditional_Final.tex, line 1251
[title=A coefficient calculation at order two] Write Δ_2(w)^-r=∑_n≥0b_n(r)w^n. Since Δ_2(w)=1-frac w2+fracw^224+O(w^3), -logΔ_2(w)=frac w2+fracw^212+O(w^3), the first logarithmic coefficients are κ_2,1=1/2 and κ_2,2=1/6. Newton's recurrence gives b_1(r)=frac r2, 2b_2(r)=r(frac r4+frac16). Restoring the factorial normalization yields P_2,1(r)=r and P_2,2(r)=r(3r+2). At r=1, the second value is five. No zero-location assumption is needed for this finite calculation. modelfigureparameter_polynomial…
Exact mathematical passage
b_1(r)=\frac r2,\qquad 2b_2(r)=r\left(\frac r4+\frac16\right).
B_{2m}Use in source · 8 passages
Search aliases: B_2m
Use in source · 20_arithmetic.tex, line 952
[Exact weighted inverse coefficient] Let κ_m=β_m2^2m(2m)!. For every m≥1, κ_m=-2^4m(2m-1)B_2m≠0, v_2(κ_m)=4m-1, where B_2m is the usual even Bernoulli number.
Exact mathematical passage
\kappa_m=-2^{4m}(2m-1)B_{2m}\ne0,\qquad v_2(\kappa_m)=4m-1,
B_{2n}Use in source · 2 passages
Search aliases: B_2n
Use in source · 20_arithmetic.tex, line 91
oulli numbers by B(z)=fracze^z-1=∑_r≥0B_rfracz^rr!, B_0=1, B_1=-frac12. All quotients here are regular formal series. The identity 2B(z)-B(2z)=fraczsinh z =∑_n≥0C_2,1(n)fracz^2n(2n)! follows by putting the two fractions on a common denominator and using [source reference]. Comparing even coefficients yields C_2,1(n)=(2-2^2n)B_2n (n∈ N). This classical derivation restores the boundary example from the original manuscript. No dedicated Lean bridge for this Bernoulli specialization was identified i…
Exact mathematical passage
C_{2,1}(n)=(2-2^{2n})B_{2n}\qquad(n\in\N).
B_{2r}Use in source · 4 passages
Search aliases: B_2r
Use in source · 72_recovered.tex, line 1082
[Dyadic inverse coefficients] For r≥1, β_r = - frac2^2r(2r-1)B_2r(2r)!. Consequently, v_2(β_r)=s_2(r)-1. Every R_n is a dyadic unit.
Exact mathematical passage
\boxed{ \beta_r = - \frac{2^{2r}(2r-1)B_{2r}}{(2r)!}. }
B_3Use in source · 3 passages
Search aliases: B_3
Use in source · atlas-pages.tex, line 184
r=exp (r∑_m≥1κ_N,mfracw^mm), the coefficient of w^n is a polynomial of degree n in r. Bell polynomials collect the possible ways logarithmic terms contribute total degree n. vfigbell_partitions For N=2, the first three logarithmic coefficients are κ_1=frac12, κ_2=frac16, κ_3=frac115. The third Bell polynomial is B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3. Substituting x_1=rκ_1, x_2=rκ_2, x_3=2rκ_3 and multiplying by 6!/3! gives P_2,3(r)=15r^3+30r^2+16r. At r=1 this is 61. At r=2 it is 272. At r=0 it van…
Exact mathematical passage
B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3
b_3Use in source · 5 passages
Search aliases: b_3
Use in source · atlas-pages.tex, line 163
is a triangular solveCOEFFICIENT RECONSTRUCTIONChapter IV: Theorem 7.3, Theorem 7.6. Start in the reduced variable with Δ_2(w)=1-frac w2+fracw^224-fracw^3720+⋯. Write 1/Δ_2(w)=∑ b_nw^n. Matching the product with 1 gives a unit lower-triangular system. At each row there is exactly one new unknown. vfigtriangular_inverse The first three equations are b_1-tfrac12=0, b_2-tfrac12b_1+tfrac124=0, b_3-tfrac12b_2+tfrac124b_1-tfrac1720=0. They yield b_1=1/2, b_2=5/24, and b_3=61/720. Multiplication by (2n…
Exact mathematical passage
b_1-\tfrac12=0,\qquad b_2-\tfrac12b_1+\tfrac1{24}=0, \qquad b_3-\tfrac12b_2+\tfrac1{24}b_1-\tfrac1{720}=0.
B_4Use in source · 1 passage
Search aliases: B_4
Use in source · supplement_11.tex, line 73
3. The dyadic differential operator and reciprocity. The operator required here is B_4=fracD^8-D^42, D=fracddt. It is not the preceding p=2 specialization (D^4-D^2)/2. It is also not an ordinary derivation. The new algebra module proves the actual product formula, including the middle derivative sums whose coefficients are binom 8i/2 and binom 4i/2.
Exact mathematical passage
\boxed{B_4=\frac{D^8-D^4}{2},\qquad D=\frac{d}{dt}.}
b_4Use in source · 2 passages
Search aliases: b_4
Use in source · worked_newton_values.tex, line 1
(b_0,b_1,b_2,b_3,b_4)=(1,-2,72,-33872,60511872).
Exact mathematical passage
(b_0,b_1,b_2,b_3,b_4)=(1,-2,72,-33872,60511872).
\mathcal B_{4,2}Use in source · 1 passage
Search aliases: B_4,2 mathcal
Use in source · atlas-pages.tex, line 53
r rotated arguments and attach the phase i^-jℓ: ( B_4,jF)(z)=frac14∑_ℓ=0^3i^-jℓF(i^ℓ z). A monomial a_mz^m is multiplied by the finite average of i^(m-j)ℓ. Matching phases all vote 1. Nonmatching phases cancel exactly. vfigroot_filter headAn example that can be checked coefficient by coefficient Take F(z)=∑_m=0^12(m+1)z^m. Then B_4,2F=3z^2+7z^6+11z^10. Applying the same selector again changes nothing. Applying a different selector gives zero. Summing the four selected polynomials returns the ori…
Exact mathematical passage
\mathcal B_{4,2}F=3z^2+7z^6+11z^{10}.
\mathcal B_{4,j}Use in source · 1 passage
Search aliases: B_4,j mathcal
Use in source · atlas-pages.tex, line 46
vpage3A root filter is an exact voteCYCLIC PROJECTIONChapter IV: Theorem 3.1, Proposition 3.2. For N=4, write ω=i. To extract residue j, evaluate the function at four rotated arguments and attach the phase i^-jℓ: ( B_4,jF)(z)=frac14∑_ℓ=0^3i^-jℓF(i^ℓ z). A monomial a_mz^m is multiplied by the finite average of i^(m-j)ℓ. Matching phases all vote 1. Nonmatching phases cancel exactly. vfigroot_filter headAn example that can be checked coefficient by coefficient Take F(z)=∑_m=0^12(m+1)z^m. Then B_4,2…
Exact mathematical passage
(\mathcal B_{4,j}F)(z)=\frac14\sum_{\ell=0}^{3}i^{-j\ell}F(i^\ell z).
B_{4i+2}Use in source · 2 passages
Search aliases: B_4i+2
Use in source · strengthening.tex, line 452
G_4,1(t)F_0(t) =frac12(fractanh(at)at +fractanh(bt)bt). For example, its numerator follows from sinh t+sin t =2bsinh(at)cosh(bt)+2acosh(at)sinh(bt), while cosh t+cos t=2cosh(at)cosh(bt). Only degrees divisible by four survive. Its actual Hurwitz packets are D_1(t)=∑_i≥0d_ifract^4i(4i)!, d_i=(-frac14)^i frac2^4i+2(2^4i+2-1)B_4i+2(4i+2)(4i+1). The constant is one. Since v_2(4i+2)=1, the elementary Bernoulli bound shows v_2(d_i)≥2i. Hurwitz inversion therefore gives D_1(t)^-1=∑_i≥0h_ifract^4i(4i)!,…
Exact mathematical passage
D_1(t)=\sum_{i\geq0}d_i\frac{t^{4i}}{(4i)!},\qquad d_i=\left(-\frac14\right)^i \frac{2^{4i+2}(2^{4i+2}-1)B_{4i+2}}{(4i+2)(4i+1)}. \label{eq:address-one-packets}
b_{a,j}Use in source · 1 passage
Search aliases: b_a,j
Use in source · 72_recovered.tex, line 2393
, v_0,w∈ Z_p^d, H>0, and a,j,n,r∈ N. These are the hypotheses of the two corrected matrix statements used by the retained proof. Without integral observation vectors or positive L, the preceding counterexamples invalidate the claimed precision. Assume C^H=I+p^LB. Put H_a=Hp^a. Then C^H_a=I+p^L+aB_a for an integral matrix B_a. Every refined packet b_a,j(x)=a_H_ax+j has Mahler coefficients satisfying v_p(c_a,j,r)≥(L+a)r. Equivalently, Δ_H_a^ ra_n ≡0 pmod p^(L+a)r.
Exact mathematical passage
b_{a,j}(x)=a_{H_ax+j}
B_{a-1}Use in source · 3 passages
Search aliases: B_a-1
Use in source · strengthening.tex, line 1143
ents from the reciprocal recurrence are a_1=-g(1)-1, a_2=binom2NNg(1)^2-g(2)+2g(1)+1. The first has valuation exactly one. Also binom2NN≡6pmod8 for every N=2^a≥2. To see this, write B_a=binom2^a+12^a =2∏_h=1^2^a-1(1+frac2^ah). The even-h factors give B_a-1. The odd-h factors are one modulo four for a≥2. Since B_a-1 has valuation one, B_a≡ B_a-1pmod8, and B_1=6. Therefore a_2≡6-1+2+1=0pmod8. The global Mahler bound covers all k≥3, completing the criterion with b=1. The period and image deductions…
Exact mathematical passage
B_{a-1}
b_a^{\phantom a c}Use in source · 1 passage
Search aliases: b_a^phantom a c phantom
Use in source · 02_strengthened.tex, line 2241
[Invariance of the physical space] Let H=H^*. Suppose [H,C_a] = ∑_cb_a^phantom a cC_c for every a. Then H preserves H_phys, and so does e^- i tH for every real t.
Exact mathematical passage
[H,C_a] = \sum_cb_a^{\phantom a c}C_c
B_bUse in source · 1 passage
Search aliases: B_b
Use in source · Paper_VI_Traditional_Final.tex, line 5376
Stationarity follows from overline Qoverlineπ =overline QCπ =CQπ=0. For reversibility, use the aggregate rate formula: overline Q_aboverlineπ_b · =∑_j∈ B_bπ_j∑_i∈ B_aQ_ij · =∑_i∈ B_a∑_j∈ B_bπ_iQ_ji · =overline Q_baoverlineπ_a.
Exact mathematical passage
\begin{aligned} \overline Q_{ab}\overline\pi_b &=\sum_{j\in B_b}\pi_j\sum_{i\in B_a}Q_{ij}\\ &=\sum_{i\in B_a}\sum_{j\in B_b}\pi_iQ_{ji}\\ &=\overline Q_{ba}\overline\pi_a. \end{aligned}
\mathcal B_{\bar\lambda}Use in source · 2 passages
Search aliases: B_barλ mathcal bar lambda
Use in source · atlas-pages.tex, line 227
For strictly positive fine and coarse likelihoods, and at least two coarse blocks, the manuscript proves the all-prior criterion C B_λ(p)= B_barλ(Cp) hboxfor every p Longleftrightarrow λ_i=cbarλ_a (i∈ I_a) for one global c>0. The global scale is invisible after normalization. A different arbitrary scale for each packet is not invisible: it changes posterior odds.
Exact mathematical passage
C\mathcal B_\lambda(p)=\mathcal B_{\bar\lambda}(Cp)\quad\hbox{for every }p \quad\Longleftrightarrow\quad \lambda_i=c\bar\lambda_a\quad(i\in I_a)
b_cUse in source · 11 passages
Search aliases: b_c
Use in source · Paper_XI_Traditional_Final.tex, line 923
Let A be a finite abelian group, and let c:A× A→ C^× be a normalized two cocycle. The twisted group algebra C^c[A] has basis (u_a)_a∈ A and multiplication u_au_b=c(a,b)u_a+b. Define the commutator bicharacter b_c(a,b)=c(a,b)c(b,a)^-1.
Exact mathematical passage
b_c(a,b)=c(a,b)c(b,a)^{-1}.
B_{ca}Use in source · 2 passages
Search aliases: B_ca
Use in source · 02_supplement_01.tex, line 23
Composition. Inserting the projector identity between factors yields (AB)_ba=∑_c A_bcB_ca. If the left side is nonzero, at least one intermediate c has both A_bc≠0 and B_ca≠0. Consequently supp(AB)⊆supp(A)∘supp(B). No equality of supports is asserted.
Exact mathematical passage
\boxed{(AB)_{ba}=\sum_c A_{bc}B_{ca}.}
B_DUse in source · 1 passage
Search aliases: B_D
Use in source · supplement_10.tex, line 82
p^2. Replacing F by this finite prefix proves the result for the entire formal series, without an analytic convergence assumption: B_Θ W_p⊆ W_p, B_Θ^s W_p⊆ W_p (s≥0). Modules. mcodeWeightedAmiceRing, mcodeFrobeniusSquare, mcodeThetaWeighted. 2. Actual substitution and singular higher differences. Let E(F)(t)=F(e^t-1), D=fracddt, B_D=fracD^p^2-D^pp. All substitutions use Mathlib's actual formal exponential, logarithm and substitution maps. The chain rule gives D E(F)= E(Θ F), B_D^s E(F)= E(B_Θ^sF…
Exact mathematical passage
\mathcal E(F)(t)=F(e^t-1),\qquad D=\frac{d}{dt},\qquad B_D=\frac{D^{p^2}-D^p}{p}.
b_dUse in source · 1 passage
Search aliases: b_d
Use in source · strengthening.tex, line 64
Products preserve H_p because the coefficient at factorial degree d is (AB)_d=∑_i=0^dbinom di a_i b_d-i. The derivative D acts by the coefficient shift (DA)_d=a_d+1. Consequently every nonnegative power of D preserves the lattice. If A∈ H_p has unit constant term, its reciprocal belongs to the same lattice: with B=A^-1, b_0=a_0^-1, b_d=-a_0^-1∑_i=1^dbinom di a_i b_d-i. If A is supported at degrees divisible by N, this recurrence also proves that B has the same support.
Exact mathematical passage
b_0=a_0^{-1},\qquad b_d=-a_0^{-1}\sum_{i=1}^d\binom di a_i b_{d-i}. \label{eq:hurwitz-reciprocal-recurrence}
b_{d+1,r}Use in source · 1 passage
Search aliases: b_d+1,r
Use in source · supplement_09.tex, line 99
Independently, let b_d,r be the residue-r coordinate of (x)_d in R_N. Actual group-ring multiplication gives b_0,r=1_r=0, b_d+1,r=b_d,r-1-d b_d,r. Induction comparing these two recurrences proves d!coefTdH_r(log(1+T))=b_d,r. Thus the cyclic coefficient is not an unrelated stand-in for a section coefficient. For the canonical label 0≤ c<N, the formal section is also proved equal to the original mcodeExponentialSection.allowbreakmcodeoriginalSection used in the existing Euler project.
Exact mathematical passage
b_{0,r}=\mathbf1_{r=0},\qquad b_{d+1,r}=b_{d,r-1}-d\,b_{d,r}.
b_{d,r}Use in source · 3 passages
Search aliases: b_d,r
Use in source · supplement_09.tex, line 98
Independently, let b_d,r be the residue-r coordinate of (x)_d in R_N. Actual group-ring multiplication gives b_0,r=1_r=0, b_d+1,r=b_d,r-1-d b_d,r. Induction comparing these two recurrences proves d!coefTdH_r(log(1+T))=b_d,r. Thus the cyclic coefficient is not an unrelated stand-in for a section coefficient. For the canonical label 0≤ c<N, the formal section is also proved equal to the original mcodeExponentialSection.allowbreakmcodeoriginalSection used in the existing Euler project.
Exact mathematical passage
b_{d,r}
b_{d,r-1}Use in source · 1 passage
Search aliases: b_d,r-1
Use in source · supplement_09.tex, line 99
Independently, let b_d,r be the residue-r coordinate of (x)_d in R_N. Actual group-ring multiplication gives b_0,r=1_r=0, b_d+1,r=b_d,r-1-d b_d,r. Induction comparing these two recurrences proves d!coefTdH_r(log(1+T))=b_d,r. Thus the cyclic coefficient is not an unrelated stand-in for a section coefficient. For the canonical label 0≤ c<N, the formal section is also proved equal to the original mcodeExponentialSection.allowbreakmcodeoriginalSection used in the existing Euler project.
Exact mathematical passage
b_{0,r}=\mathbf1_{r=0},\qquad b_{d+1,r}=b_{d,r-1}-d\,b_{d,r}.
b_{d-i}Use in source · 2 passages
Search aliases: b_d-i
Use in source · strengthening.tex, line 57
Products preserve H_p because the coefficient at factorial degree d is (AB)_d=∑_i=0^dbinom di a_i b_d-i. The derivative D acts by the coefficient shift (DA)_d=a_d+1. Consequently every nonnegative power of D preserves the lattice. If A∈ H_p has unit constant term, its reciprocal belongs to the same lattice: with B=A^-1, b_0=a_0^-1, b_d=-a_0^-1∑_i=1^dbinom di a_i b_d-i. If A is supported at degrees divisible by N, this recurrence also proves that B has the same support.
Exact mathematical passage
(AB)_d=\sum_{i=0}^d\binom di a_i b_{d-i}.
B_D^sUse in source · 3 passages
Search aliases: B_D^s
Use in source · supplement_10.tex, line 87
(s≥0). Modules. mcodeWeightedAmiceRing, mcodeFrobeniusSquare, mcodeThetaWeighted. 2. Actual substitution and singular higher differences. Let E(F)(t)=F(e^t-1), D=fracddt, B_D=fracD^p^2-D^pp. All substitutions use Mathlib's actual formal exponential, logarithm and substitution maps. The chain rule gives D E(F)= E(Θ F), B_D^s E(F)= E(B_Θ^sF). The implementation also proves the inverse-substitution identity E(f(log(1+T)))=f(t).
Exact mathematical passage
D\mathcal E(F)=\mathcal E(\Theta F),\qquad B_D^s\mathcal E(F)=\mathcal E(B_\Theta^sF).
B_{dc}Use in source · 1 passage
Search aliases: B_dc
Use in source · 02_supplement_01.tex, line 33
Adjoint and tensor products. With an involution and self-adjoint projectors, the formalization proves (A_ba)^*=(A^*)_ab. For actual finite matrices and actual Kronecker products, it proves (P_b⊗ Q_d)(A⊗ B)(P_a⊗ Q_c) =A_ba⊗ B_dc. The self-adjointness hypothesis is explicit in the adjoint theorem. The tensor theorem is not a statement about an uninterpreted tensor symbol.
Exact mathematical passage
(P_b\otimes Q_d)(A\otimes B)(P_a\otimes Q_c) =A_{ba}\otimes B_{dc}.
B_eUse in source · 5 passages
Search aliases: B_e
Use in source · Paper_XI_Traditional_Final.tex, line 1608
Let Γ be a connected finite graph. At every vertex v, place finite dimensional complex vector spaces V_v and W_v of the same dimension. For every oriented edge e:u→ v, let A_e:V_u⟶ V_v, B_e:W_u⟶ W_v be invertible. The reversed edge carries the inverse map.
Exact mathematical passage
A_e:V_u\longrightarrow V_v, \qquad B_e:W_u\longrightarrow W_v
b_hUse in source · 29 passages
Search aliases: b_h
Use in source · Paper_V_Traditional_Final.tex, line 1688
The principal recovery theorem uses adjacent sample contrasts. Let b_h=Mh+2ε be the largest contrast that can be produced by a Lipschitz background and bounded noise in one grid cell. If a_min>2b_h, then every threshold between b_h and a_min-b_h detects exactly the cells containing jumps. The jump locations are recovered within h/2, and each amplitude error is at most b_h. The probability theorem replaces the deterministic noise term by 2σsqrtlog(2n/δ). Two half shifted Haar grids provide a seco…
Exact mathematical passage
b_h=Mh+2\varepsilon
b_{h,\delta}Use in source · 7 passages
Search aliases: b_h,δ delta
Use in source · Paper_V_Traditional_Final.tex, line 3106
[Probability recovery bound] Fix 0<δ<1 and define t_n,δ = 2σ sqrtlog(frac2nδ) and b_h,δ=Mh+t_n,δ. With probability at least 1-δ, every contrast satisfies the conclusions of [source reference] with b_h replaced by b_h,δ.
Exact mathematical passage
b_{h,\delta}=Mh+t_{n,\delta}.
B_iUse in source · 6 passages
Search aliases: B_i
Use in source · root_scope.tex, line 223
For 0≤ i<j, let d_i=v_2(i!)+v_2((j-1-i)!), k_i=1+lfloorlog_2max(i,j-1-i)rfloor, B_i=i+2^k_i Z_2, t_i=d_i+k_i. At j=1 use d_0=k_0=t_0=0. The balls B_i partition Z_2. Indeed a residue represented twice among 0,…,j-1 splits into its two children. Every other represented residue remains one ball. On each such ball, P_j(i+2^k_iy)=2^t_iyQ_i(y), Q_i∈ Z_2⟨ y⟩^×, Q_i≡1pmod2.
Exact mathematical passage
B_i=i+2^{k_i}\mathbb Z_2,\qquad t_i=d_i+k_i.
B_{ik}Use in source · 1 passage
Search aliases: B_ik
Use in source · 01_manuscript.tex, line 606
The (i,j) entry of BXA^T is ∑_k=1^n∑_ℓ=1^m B_ikX_kℓA_jℓ. This is the coordinate formula for (A⊗ B)vec(X). Multiplying first by B and then by A^T gives the stated count.
Exact mathematical passage
\sum_{k=1}^{n}\sum_{\ell=1}^{m} B_{ik}X_{k\ell}A_{j\ell}.
B_{j,k}Use in source · 6 passages
Search aliases: B_j,k
Use in source · Paper_VI_Traditional_Final.tex, line 4650
mory of a finite hidden system] Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt ∑_k=0^ν_j-1 fract^kk!B_j,k and Σ_P(z) = ∑_j=1^r ∑_k=0^ν_j-1 fracB_j,k(z-μ_j)^k+1. Suppose Reμ_j<0 for every j. If J⊆{1,…,r} and K_J retains the compl…
Exact mathematical passage
B_{j,k} = A_{PQ}N_j^kQ_jA_{QP} \qquad (0\leq k<\nu_j).
B_{j,\nu_j-1}Use in source · 1 passage
Search aliases: B_j,ν_j-1 nu
Use in source · Paper_VI_Traditional_Final.tex, line 4782
[Pole order and hidden Jordan depth] Every finite hidden block produces a strictly proper rational self-energy. At an eigenvalue μ_j, its pole order is at most the size of the largest Jordan block of A_QQ at μ_j. Equality holds when the corresponding highest coefficient B_j,ν_j-1 is nonzero.
Exact mathematical passage
B_{j,\nu_j-1}
B_j^rUse in source · 1 passage
Search aliases: B_j^r
Use in source · Paper_X_Block27_Amended.tex, line 381
For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet are cochain maps, so they induce maps on these genuine quotients.
Exact mathematical passage
Z_j^r,B_j^r,H_j^r
B_{ji}Use in source · 3 passages
Search aliases: B_ji
Use in source · 05_worked_models.tex, line 549
[Table specification for transfer blocks] Given finite projector systems (P_i)_i∈ I and (Q_j)_j∈ J and a linear map T, form the table B_ji=Q_jTP_i. Its assembly reconstructs the input map exactly, ∑_j∈ J∑_i∈ IB_ji=T, and its reported support relation is exactly {(i,j):B_ji≠0}. The latter equality is the specification of the transfer graph output. It does not claim that Lean extracts a routine that decides the edge list for every coefficient field.
Exact mathematical passage
B_{ji}=Q_jTP_i.
B_{jk}Use in source · 1 passage
Search aliases: B_jk
Use in source · visual_atlas.tex, line 164
vfig[.98]rectangular_paths platecaptionThe lower board expands one final row into all six choices of intermediate indices for each input packet. The entries are C_0jB_jkA_ki, with k∈{0,1,2} and j∈{0,1}. Zero terms remain visible.
Exact mathematical passage
C_{0j}B_{jk}A_{ki}
B_kUse in source · 2 passages
Search aliases: B_k
Use in source · 03_nonlinear_coordinates.tex, line 353
di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n] · = ∑_π={B_1,…,B_k} ∑_γ_1,…,γ_k J^k_δmidγ_1,…,γ_kF(Gx) [ G_γ_1,B_1(xmid h),…, G_γ_k,B_k(xmid h) ]. The first sum runs over all set partitions of {1,…,n}.
Exact mathematical passage
\begin{aligned} &J^{n}_{\delta\mid\alpha_1,\ldots,\alpha_n}(F\circ G)(x) [h_1,\ldots,h_n]\\ &= \sum_{\pi=\{B_1,\ldots,B_k\}} \sum_{\gamma_1,\ldots,\gamma_k} J^{k}_{\delta\mid\gamma_1,\ldots,\gamma_k}F(Gx) \big[ G_{\gamma_1,B_1}(x\mid h),\ldots, G_{\gamma_k,B_k}(x\mid h) \big]. \end{aligned}
b_kUse in source · 8 passages
Search aliases: b_k
Use in source · 10_foundations.tex, line 411
[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalization hypothesis is required.
Exact mathematical passage
b_k=x_{Nk+j}
b_{k+1}Use in source · 1 passage
Search aliases: b_k+1
Use in source · 10_foundations.tex, line 412
[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalization hypothesis is required.
Exact mathematical passage
b_k=C^j(C^N)^kx_0,\qquad b_{k+1}=C^Nb_k.
b_{k+d}Use in source · 1 passage
Search aliases: b_k+d
Use in source · 10_foundations.tex, line 416
[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalization hypothesis is required.
Exact mathematical passage
b_{k+d}=-\sum_{r=0}^{d-1}q_r b_{k+r}.
b_{k+r}Use in source · 1 passage
Search aliases: b_k+r
Use in source · 10_foundations.tex, line 416
[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalization hypothesis is required.
Exact mathematical passage
b_{k+d}=-\sum_{r=0}^{d-1}q_r b_{k+r}.
B_{ks}Use in source · 2 passages
Search aliases: B_ks
Use in source · Paper_V_Traditional_Final.tex, line 507
multiplication: composing shifts adds their delays, and distributivity handles finite sums. Thus the matrix identity implies the asserted action. Conversely, let B= R E-cz^-dI_N and suppose its action vanishes on every channel sequence. Insert an impulse at channel s, address zero, with every other channel zero. Output channel k reads the coefficients of the Laurent polynomial B_ks at all addresses. All of these coefficients vanish, so B_ks=0 for every k,s. The representation is faithful, provin…
Exact mathematical passage
B_{ks}
B_LUse in source · 16 passages
Search aliases: B_L
Use in source · 30_lifting.tex, line 16
Unless a different coefficient ring is specified, a polynomial system is a finite list of integer polynomials. All divisibility conditions on vectors are coordinatewise. For a positive integer q, write S_q= Z/q Z and use an overline for reduction modulo q. For a prime p, write F_p=S_p, normalize |p|_p=p^-1, and put B_L(x)=x+p^L Z_p^d. An integer center is included in this notation by its natural image in Z_p^d. All dimensions below are finite and may be zero. A statement over S_q allows composit…
Exact mathematical passage
B_L(x)=x+p^L\Zp^d.
b_mUse in source · 5 passages
Search aliases: b_m
Use in source · supplement_02.tex, line 86
The inverse coefficients are integral by construction. Define the integer binomial convolution B_n=∑_r=0^nbinom nr E_rE_n-r, b_m=B_2m. The product formula for actual EGFs gives ∑_n≥0B_nfract^nn!=sech^2 t. Odd coefficients vanish by parity. In particular, every b_m is an integer. This is the integrality required by the later congruence.
Exact mathematical passage
B_n=\sum_{r=0}^{n}\binom nr E_rE_{n-r},\qquad b_m=B_{2m}.
b_{mn}^{(m,p)}Use in source · 1 passage
Search aliases: b_mn^(m,p)
Use in source · 72_recovered.tex, line 356
[Compressed inverse] Set e_n^(m,p) = b_mn^(m,p).
Exact mathematical passage
e_n^{(m,p)} = b_{mn}^{(m,p)}.
B_nUse in source · 13 passages
Search aliases: B_n
Use in source · Paper_IV_Traditional_Final.tex, line 979
The polynomial dependence on r can be read directly from the formal logarithm. We use the complete exponential Bell polynomials B_n defined by exp(∑_m≥1x_mfract^mm!) = ∑_n≥0B_n(x_1,…,x_n)fract^nn!.
Exact mathematical passage
B_n
b_nUse in source · 14 passages
Search aliases: b_n
Use in source · Paper_IV_Traditional_Final.tex, line 1245
[title=A coefficient calculation at order two] Write Δ_2(w)^-r=∑_n≥0b_n(r)w^n. Since Δ_2(w)=1-frac w2+fracw^224+O(w^3), -logΔ_2(w)=frac w2+fracw^212+O(w^3), the first logarithmic coefficients are κ_2,1=1/2 and κ_2,2=1/6. Newton's recurrence gives b_1(r)=frac r2, 2b_2(r)=r(frac r4+frac16). Restoring the factorial normalization yields P_2,1(r)=r and P_2,2(r)=r(3r+2). At r=1, the second value is five. No zero-location assumption is needed for this finite calculation. modelfigureparameter_polynomial…
Exact mathematical passage
\Delta_2(w)^{-r}=\sum_{n\geq0}b_n(r)w^n
b_{n+p-1}^{(m,p)}Use in source · 2 passages
Search aliases: b_n+p-1^(m,p)
Use in source · 72_recovered.tex, line 311
[Support and positive degree period] The inverse b^(m,p) has the following properties. item b_n^(m,p)=0 unless mmid n. item For every positive n, b_n+p-1^(m,p) = b_n^(m,p).
Exact mathematical passage
\boxed{ b_{n+p-1}^{(m,p)} = b_n^{(m,p)}. }
b_{N,0}Use in source · 1 passage
Search aliases: b_N,0
Use in source · Paper_IV_Traditional_Final.tex, line 1114
[Logarithmic Newton recurrence] Set b_N,n(r)=fracP_N,n(r)(Nn)!. Then b_N,0(r)=1 and for every n≥1, nb_N,n(r) = r∑_k=1^nκ_N,kb_N,n-k(r). Equivalently, P_N,n(r) = frac r n ∑_k=1^n frac(Nn)!(N(n-k))! κ_N,kP_N,n-k(r). If p_k=rκ_N,k, then, for n≥1, b_N,n(r) = frac1n! det p_1 · -1 · 0 · ⋯ · 0
Exact mathematical passage
b_{N,0}(r)=1
\mathcal B_{N,i}Use in source · 1 passage
Search aliases: B_N,i mathcal
Use in source · atlas-pages.tex, line 76
1+2z+3z^2+z^3, G=2+z+z^2. The table displays every product of their three selected sections. vfigcyclic_product The shaded cells contribute to residue zero: 2(1+z^3)+(2z)(z^2)+(3z^2)z=2+7z^3. Keeping only the product of the residue-zero inputs would give 2+2z^3 and lose five units of the z^3 coefficient. The mixed cells are part of the exact result. headThe general rule B_N,k(FG)= ∑_i+j≡ k (N)( B_N,iF)( B_N,jG). Every pair belongs to exactly one output residue. No term is discarded and no coeffi…
Exact mathematical passage
\mathcal B_{N,k}(FG)= \sum_{i+j\equiv k\ (N)}(\mathcal B_{N,i}F)(\mathcal B_{N,j}G).
B_{N,j}Use in source · 2 passages
Search aliases: B_N,j
Use in source · strengthening.tex, line 1163
[Small-rank complete isometry table] Every address at N∈{2,4,8} satisfies v_2(C_N,j(n+H)-C_N,j(n))=b_N,j+v_2(H). The following entries are ordered by j=0,…,N-1: c l l N · normalized shifts b_N,j · original shifts B_N,j=b_N,j+v_2(j!)
Exact mathematical passage
B_{N,j}=b_{N,j}+v_2(j!)
\mathcal B_{N,j}Use in source · 1 passage
Search aliases: B_N,j mathcal
Use in source · atlas-pages.tex, line 76
1+2z+3z^2+z^3, G=2+z+z^2. The table displays every product of their three selected sections. vfigcyclic_product The shaded cells contribute to residue zero: 2(1+z^3)+(2z)(z^2)+(3z^2)z=2+7z^3. Keeping only the product of the residue-zero inputs would give 2+2z^3 and lose five units of the z^3 coefficient. The mixed cells are part of the exact result. headThe general rule B_N,k(FG)= ∑_i+j≡ k (N)( B_N,iF)( B_N,jG). Every pair belongs to exactly one output residue. No term is discarded and no coeffi…
Exact mathematical passage
\mathcal B_{N,k}(FG)= \sum_{i+j\equiv k\ (N)}(\mathcal B_{N,i}F)(\mathcal B_{N,j}G).
b_{N,j}Use in source · 6 passages
Search aliases: b_N,j
Use in source · strengthening.tex, line 1157
[Small-rank complete isometry table] Every address at N∈{2,4,8} satisfies v_2(C_N,j(n+H)-C_N,j(n))=b_N,j+v_2(H). The following entries are ordered by j=0,…,N-1: c l l N · normalized shifts b_N,j · original shifts B_N,j=b_N,j+v_2(j!)
Exact mathematical passage
v_2(C_{N,j}(n+H)-C_{N,j}(n))=b_{N,j}+v_2(H).
\mathcal B_{N,k}Use in source · 1 passage
Search aliases: B_N,k mathcal
Use in source · atlas-pages.tex, line 76
1+2z+3z^2+z^3, G=2+z+z^2. The table displays every product of their three selected sections. vfigcyclic_product The shaded cells contribute to residue zero: 2(1+z^3)+(2z)(z^2)+(3z^2)z=2+7z^3. Keeping only the product of the residue-zero inputs would give 2+2z^3 and lose five units of the z^3 coefficient. The mixed cells are part of the exact result. headThe general rule B_N,k(FG)= ∑_i+j≡ k (N)( B_N,iF)( B_N,jG). Every pair belongs to exactly one output residue. No term is discarded and no coeffi…
Exact mathematical passage
\mathcal B_{N,k}(FG)= \sum_{i+j\equiv k\ (N)}(\mathcal B_{N,i}F)(\mathcal B_{N,j}G).
b_{N,n-k}Use in source · 1 passage
Search aliases: b_N,n-k
Use in source · Paper_IV_Traditional_Final.tex, line 1116
[Logarithmic Newton recurrence] Set b_N,n(r)=fracP_N,n(r)(Nn)!. Then b_N,0(r)=1 and for every n≥1, nb_N,n(r) = r∑_k=1^nκ_N,kb_N,n-k(r). Equivalently, P_N,n(r) = frac r n ∑_k=1^n frac(Nn)!(N(n-k))! κ_N,kP_N,n-k(r). If p_k=rκ_N,k, then, for n≥1, b_N,n(r) = frac1n! det p_1 · -1 · 0 · ⋯ · 0
Exact mathematical passage
nb_{N,n}(r) = r\sum_{k=1}^{n}\kappa_{N,k}b_{N,n-k}(r).
b_{n-k}Use in source · 1 passage
Search aliases: b_n-k
Use in source · atlas-pages.tex, line 199
vpage11Two recurrences reach the same answerNEWTON AND ADDITIONChapter IV: Theorem 7.3, Theorem 7.4, Theorem 7.6. One recurrence uses the denominator coefficients. Another uses its logarithm. With b_n(r)=P_N,n(r)/(Nn)!, the shorter logarithmic form is nb_n(r)=r∑_k=1^nκ_N,kb_n-k(r), b_0=1. The negative superdiagonal in the Newton determinant is essential. vfignewton_addition headThe additive parameter law has block weights Using the same logarithm, D_N^-rD_N^-s=D_N^-(r+s). Thus P_N,n(r+s)=∑_q=0^n…
Exact mathematical passage
nb_n(r)=r\sum_{k=1}^{n}\kappa_{N,k}b_{n-k}(r),\qquad b_0=1.
b_{n-r}Use in source · 2 passages
Search aliases: b_n-r
Use in source · 72_recovered.tex, line 34
This article begins with the Hurwitz product (amathbin⋆ b)_n = ∑_r=0^n binom nr a_rb_n-r. Over a field of characteristic zero it is multiplication of exponential generating functions. Over a finite field it realizes the additive group algebra through the exponential modes ϵ_u=(u^n)_n≥0. This finite algebra gives an exact proof of the standard window period. Hurwitz series rings are treated in [source reference]. The finite difference and umbral background may be found in [source reference].
Exact mathematical passage
(a\starh b)_n = \sum_{r=0}^{n} \binom nr a_rb_{n-r}.
b_n^{(m,p)}Use in source · 4 passages
Search aliases: b_n^(m,p)
Use in source · 72_recovered.tex, line 309
[Support and positive degree period] The inverse b^(m,p) has the following properties. item b_n^(m,p)=0 unless mmid n. item For every positive n, b_n+p-1^(m,p) = b_n^(m,p).
Exact mathematical passage
b_n^{(m,p)}=0
B_n^{1/2}Use in source · 2 passages
Search aliases: B_n^1/2
Use in source · atlas-pages.tex, line 621
at each level is not summable across levelsREFINEMENT LIMITS / WHAT THE FINITE MODELS DO NOT PROVEChapter VI, Part I: Corollary 10.5 and Section 17. Summability and uniform control are additional data. For each n≥1, compare the two scalar operators A_n=0 and B_n=1/n^2 through the identity. Their ordinary defects have sizes 1/n^2, while their square-root defects have sizes 1/n: ∑_n|A_n-B_n|<∞, ∑_n|A_n^1/2-B_n^1/2|=∞. These are independent scalar comparisons, not a constructed inverse system. They…
Exact mathematical passage
\sum_n|A_n-B_n|<\infty, \qquad \sum_n|A_n^{1/2}-B_n^{1/2}|=\infty.
B_{\mathrm{order}}Use in source · 2 passages
Search aliases: B_order mathrm
Use in source · odd_capacity.tex, line 190
The right side is the Heisenberg commutator pairing for the minimal simple stratum of gamma. Its quotient J^1(γ)/H^1(γ) is precisely U^m(A)/H^m(γ), because J^1(γ)=U^1(B_order)U^m(A), with U^1(B_order)⊂ H^1(γ). BK's nondegeneracy theorem for simple-character commutators therefore says that the form on S is nondegenerate.
Exact mathematical passage
J^1(\gamma)=U^1(B_{\mathrm{order}})U^m(A)
B_pUse in source · 5 passages
Search aliases: B_p
Use in source · 20_arithmetic.tex, line 209
Work in the Hurwitz integral ring of formal series H_p={F∈ Q_p[[t]]: d![t^d]F∈ Z_p for every d}. For D=d/dt, let B_p=(D^p^2-D^p)/p. Call F flat if B_p^sF∈ H_p for every s≥0. We will prove that j! P_N,j is flat at every address. The main difficulty is to include the addresses at which the usual unit-pivot argument fails.
Exact mathematical passage
B_p=(D^{p^2}-D^p)/p
\mathcal B_pUse in source · 2 passages
Search aliases: B_p mathcal
Use in source · supplement_10.tex, line 58
The normalized defect is not assumed to be a derivation. Its proved product rule contains the two middle Leibniz sums: B_Θ(FG)=(B_Θ F)G+F(B_Θ G)+ B_p(F,G). The coefficients of B_p are the actual integral quotients binomp^2a/p and binom pa/p. Derivative stability and this correction formula show that the elements F∈ W_p satisfying B_Θ F∈ W_p are closed under the finite ring operations.
Exact mathematical passage
B_\Theta(FG)=(B_\Theta F)G+F(B_\Theta G)+\mathcal B_p(F,G).
\mathcal B_{p,w}Use in source · 3 passages
Search aliases: B_p,w mathcal
Use in source · 02_supplement_05.tex, line 117
Exact banded addresses and modeled work. textitSource item 4.1; module BandedCount For p packets and half-bandwidth w, the actual address set is B_p,w ={(i,j)∈{0,…,p-1}^2: i≤ j+w and j≤ i+w}. The predicate is the integer version of |i-j|≤ w. The count is first defined as a double sum of indicators, then identified with the cardinality of this subtype. It is not defined by the target closed form.
Exact mathematical passage
\mathcal B_{p,w} =\{(i,j)\in\{0,\ldots,p-1\}^2: i\leq j+w\ \text{and}\ j\leq i+w\}.
B_p^kUse in source · 1 passage
Search aliases: B_p^k
Use in source · 20_arithmetic.tex, line 333
Here is the coefficient extraction explicitly. If a_d=d![t^d]F, then d![t^d]B_p^kF=p^-kΔ_p(p-1)^ka_d+pk. For F=j! P_N,j and k≤ mn, take d=mpn-pk. Support in multiples of mp turns this into p^-kΔ_h^kU_N,j(n), so flatness gives divisibility by p^k. For k>mn, expand Δ_h^k-mn applied to Δ_h^mnU_N,j(n). Every shifted row is at least n, so each summand is divisible by p^mn. Finally Δ_uh=((I+Δ_h)^u-I), whose kth power for u>0 is an integral combination of powers of Δ_h of order at least k. The case u=0…
Exact mathematical passage
d![t^d]B_p^kF=p^{-k}\Delta_{p(p-1)}^ka_{d+pk}.
B_p^rUse in source · 4 passages
Search aliases: B_p^r
Use in source · 20_arithmetic.tex, line 307
Writing q=R_c-1-(c-1) and y=R_c-c, this becomes (tD+c)y=cq+yq-y^2. On degrees divisible by p, the inverse of tD+c divides the degree-d coefficient by the unit d+c. Starting with y_0=0, form y_s+1=(tD+c)^-1(cq+y_sq-y_s^2). Flatness and support are preserved. Indeed, put E_c=tD+c and C_p=pD^p^2-D^p. The commutator is [B_p,E_c]=C_p, and iteration gives B_p^rE_cy=E_cB_p^ry+rC_pB_p^r-1y. For the inverse equation E_cy=f, induction on r makes the correction term Hurwitz-integral. Both operators preserv…
Exact mathematical passage
B_p^rE_cy=E_cB_p^ry+rC_pB_p^{r-1}y.
B_p^{r-1}Use in source · 1 passage
Search aliases: B_p^r-1
Use in source · 20_arithmetic.tex, line 307
Writing q=R_c-1-(c-1) and y=R_c-c, this becomes (tD+c)y=cq+yq-y^2. On degrees divisible by p, the inverse of tD+c divides the degree-d coefficient by the unit d+c. Starting with y_0=0, form y_s+1=(tD+c)^-1(cq+y_sq-y_s^2). Flatness and support are preserved. Indeed, put E_c=tD+c and C_p=pD^p^2-D^p. The commutator is [B_p,E_c]=C_p, and iteration gives B_p^rE_cy=E_cB_p^ry+rC_pB_p^r-1y. For the inverse equation E_cy=f, induction on r makes the correction term Hurwitz-integral. Both operators preserv…
Exact mathematical passage
B_p^rE_cy=E_cB_p^ry+rC_pB_p^{r-1}y.
B_p^sUse in source · 1 passage
Search aliases: B_p^s
Use in source · 20_arithmetic.tex, line 210
Work in the Hurwitz integral ring of formal series H_p={F∈ Q_p[[t]]: d![t^d]F∈ Z_p for every d}. For D=d/dt, let B_p=(D^p^2-D^p)/p. Call F flat if B_p^sF∈ H_p for every s≥0. We will prove that j! P_N,j is flat at every address. The main difficulty is to include the addresses at which the usual unit-pivot argument fails.
Exact mathematical passage
B_p^sF\in\mathcal H_p
B_{p_u}Use in source · 2 passages
Search aliases: B_p_u
Use in source · Paper_XI_Traditional_Final.tex, line 1694
[Exact edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD_eA_p_u^-1.
Exact mathematical passage
D_e = U_rA_{p_v}^{-1}A_eA_{p_u} - B_{p_v}^{-1}B_eB_{p_u}U_r.
B_{p_v}Use in source · 4 passages
Search aliases: B_p_v
Use in source · Paper_XI_Traditional_Final.tex, line 1702
ct edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD_eA_p_u^-1.
Exact mathematical passage
\boxed{ U_vA_e-B_eU_u = B_{p_v}D_eA_{p_u}^{-1}. }
B_{p_v}^{-1}Use in source · 1 passage
Search aliases: B_p_v^-1
Use in source · Paper_XI_Traditional_Final.tex, line 1694
[Exact edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD_eA_p_u^-1.
Exact mathematical passage
D_e = U_rA_{p_v}^{-1}A_eA_{p_u} - B_{p_v}^{-1}B_eB_{p_u}U_r.
b_{q,m}Use in source · 3 passages
Search aliases: b_q,m
Use in source · Paper_V_Traditional_Final.tex, line 2572
[Uniqueness of the trace packet] Fix the breakpoints τ_q. If f=s+∑_q=1^Q∑_m=0^r-1b_q,mϕ_m,τ_q with s∈ W^r,p(0,1), then b_q,m=jumpf^(m)_τ_q. The finite truncated power packet is therefore the unique packet supported at the declared breakpoints that removes all trace jumps through order r-1.
Exact mathematical passage
f=s+\sum_{q=1}^{Q}\sum_{m=0}^{r-1}b_{q,m}\phi_{m,\tau_q}
\mathcal B_rUse in source · 1 passage
Search aliases: B_r mathcal
Use in source · atlas-pages.tex, line 242
page11The exact information-loss vectorBAYESIAN DEFECT / MORE THAN AN ERROR BARChapter VI, Part I: Theorems 8.1 and 8.4, and Corollary 8.6. For a prior p, let P=Cp and let r_a(p) be the conditional average of the fine likelihood inside packet a. For the uniform prior and λ=(4,1,1,1), P=(1/2,1/2)^ T, r=(5/2,1)^ T. The exact coarse posterior is B_r(P)=(5/7,2/7)^ T. Suppose instead that the declared coarse likelihood is barλ=(1,1). vfigloss_vector platecaptionThe posterior error lies along the mass…
Exact mathematical passage
\mathcal B_r(P)=(5/7,2/7)^{\mathsf T}
b_rUse in source · 7 passages
Search aliases: b_r
Use in source · 03_nonlinear_coordinates.tex, line 432
Let X be a Banach space. Fix s∈ X and continuous multilinear maps B_r:X^r⟶ X, 2≤ r≤ d. Set a= s , b_r= B_r , ϕ(t)=a+∑_r=2^db_rt^r. An ordered rooted tree has leaves labeled by s and every internal vertex of arity r labeled by B_r. Its value is obtained by evaluating from the leaves toward the root.
Exact mathematical passage
a=\|s\|, \qquad b_r=\|B_r\|, \qquad \phi(t)=a+\sum_{r=2}^{d}b_rt^r.
B_r'Use in source · 1 passage
Search aliases: B_r'
Use in source · Paper_IV_Traditional_Final.tex, line 1154
Let B_r(w)=Δ_N(w)^-r = ∑_n≥0b_N,n(r)w^n. Logarithmic differentiation gives fracB_r'(w)B_r(w) = r∑_k≥1κ_N,kw^k-1. Comparison of the coefficient of w^n-1 proves the recurrence.
Exact mathematical passage
\frac{B_r'(w)}{B_r(w)} = r\sum_{k\geq1}\kappa_{N,k}w^{k-1}.
b_{r-q}Use in source · 1 passage
Search aliases: b_r-q
Use in source · supplement_02.tex, line 142
The row action is connected to the matrix entries. Reversing the finite sum and using binomial symmetry gives the actual source-index row action E_2r=∑_q=0^rU_r,q(p)S_2q(p), where U_r,q(p)= displaystylebinom2r2qb_r-qp^2(r-q), · q≤ r, 0, · q>r. The row-action equality, the diagonal identity U_r,r=1, and vanishing above the diagonal are all checked. The arithmetic weight is also explicit: p^2(r-q)mid U_r,q(p) in Z, 0≤ q≤ r. No inverse-coordinate identity, unit diagonal or weight bound is assumed f…
Exact mathematical passage
U_{r,q}(p)= \begin{cases} \displaystyle\binom{2r}{2q}b_{r-q}p^{2(r-q)},&q\le r,\\ 0,&q>r. \end{cases}
B_r^*Use in source · 2 passages
Search aliases: B_r^*
Use in source · 03_supplement_05.tex, line 53
Define the transformed errors by an actual finite sum, B_r=∑_a U_arF_a. The formal module proves the identities B_rP=B_r, B_r^*B_s=δ_rsλ_rP, ∑_rB_rρ B_r^*=∑_aF_aρ F_a^*. For ρ supported on the code, the restricted and unrestricted error sums are equal. Thus the unitary change of error coordinates preserves the actual channel on the code.
Exact mathematical passage
B_rP=B_r,\qquad B_r^*B_s=\delta_{rs}\lambda_rP, \qquad \sum_rB_r\rho B_r^*=\sum_aF_a\rho F_a^*.
b_{\rm root}Use in source · 1 passage
Search aliases: b_rm root rm
Use in source · atlas-pages.tex, line 566
apter V: Proposition E6.4. Suppose the root split has nonnegative squared-norm bounds A=1/2 and B=2. Retain the left child. Split the right child with bounds 3/4 and 5/4, then retain both grandchildren. vfig[.85]ext_tree headStart with one at every retained node The left subtree has bounds a_0=b_0=1. The right subtree has a_1=3/4 and b_1=5/4. The recursive certificate gives a_rm root=tfrac12min{1,tfrac34}=tfrac38, b_rm root=2max{1,tfrac54}=tfrac52. Consequently the total leaf coefficient energy …
Exact mathematical passage
a_{\rm root}=\tfrac12\min\{1,\tfrac34\}=\tfrac38, \qquad b_{\rm root}=2\max\{1,\tfrac54\}=\tfrac52.
B_{\rm stream}Use in source · 2 passages
Search aliases: B_rm stream rm
Use in source · pages.tex, line 412
blocks accumulate into one target, the reduction or accumulation policy is additional work. Equal matrix dimensions alone do not remove that dependency. Different accumulation types also define different shape classes under the paper's convention. headCount the data separately For three diagonal 2×2 blocks, eight-byte scalars, and q right-hand sides, the stated one-pass model moves B_rm stream=8·3(4+2q+4q)=96+144q bytes. It gives 240 bytes at q=1 and 672 bytes at q=4. It deliberately assumes no …
Exact mathematical passage
B_{\rm stream}=8\cdot3(4+2q+4q)=96+144q\text{ bytes}.
B_sUse in source · 1 passage
Search aliases: B_s
Use in source · 03_supplement_05.tex, line 53
Define the transformed errors by an actual finite sum, B_r=∑_a U_arF_a. The formal module proves the identities B_rP=B_r, B_r^*B_s=δ_rsλ_rP, ∑_rB_rρ B_r^*=∑_aF_aρ F_a^*. For ρ supported on the code, the restricted and unrestricted error sums are equal. Thus the unitary change of error coordinates preserves the actual channel on the code.
Exact mathematical passage
B_rP=B_r,\qquad B_r^*B_s=\delta_{rs}\lambda_rP, \qquad \sum_rB_r\rho B_r^*=\sum_aF_a\rho F_a^*.
B_{\mathrm{stream}}Use in source · 1 passage
Search aliases: B_stream mathrm
Use in source · 02_supplement_02.tex, line 184
Streaming. The explicit event type counts one block read, one source read, and a target read plus write for every edge. With scalar size s, its byte count is B_stream=s∑_e(m_en_e+n_eq+2m_eq). No cross-edge cache reuse is assumed. Sparse index metadata and temporary workspace remain separate.
Exact mathematical passage
B_{\mathrm{stream}}=s\sum_e(m_en_e+n_eq+2m_eq).
b_TUse in source · 3 passages
Search aliases: b_T
Use in source · 30_lifting.tex, line 867
Theorem~[source reference] gives a unique root b of F+p^M H in a+p^s+1R^e+k, with displacement depth M-s. Its last equations say exactly b_T=t_0+p^Mv=τ. Its first equations are G(b)=0. The integral first-order expression uses CbinomH(a)-v =binomA(H(a)+Kv)-p^sv. Projection onto the solved coordinates gives b_Y-y_0+p^M-sA(H(a)+Kv)∈ p^2M-3sR^e. The same inclusion identity as before converts this to [source reference].
Exact mathematical passage
b_T=t_0+p^Mv=\tau
B_vUse in source · 6 passages
Search aliases: B_v
Use in source · 10_foundations.tex, line 494
a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient and remainder give A(z)=∑_r=0^M-1z^rA_r(z^M), (a_r^[M])_s^[N]=a_r+Ms^[MN]. For ordinary Cauchy multiplication, the phase-r product has a term without a carry and a term with one carry: C_r(w)=∑_u+v=rA_u(w)B_v(w) +w∑_u+v=r+MA_u(w)B_v(w), 0≤ u,v<M. Adding the two phase addresses gives the formula. It describes ordinary Cauchy multiplication. Hurwitz convo…
Exact mathematical passage
C_r(w)=\sum_{u+v=r}A_u(w)B_v(w) +w\sum_{u+v=r+M}A_u(w)B_v(w),\qquad 0\leq u,v<M.
b_vUse in source · 2 passages
Search aliases: b_v
Use in source · 02_supplement_05.tex, line 53
Constructed solution. Well-founded recursion defines x_v=D_v^-1 (b_v-∑_u→ vT_vux_u). Applying D_v proves the original equation at v. If another vector satisfies all equations, induction over predecessors makes each previously used coordinate identical, and invertibility of D_v forces agreement at v. Thus the construction gives existence and uniqueness, and the block operator is bijective.
Exact mathematical passage
\boxed{x_v=D_v^{-1}\!\left(b_v-\sum_{u\to v}T_{vu}x_u\right).}
b_{\mathrm{val}}Use in source · 2 passages
Search aliases: b_val mathrm
Use in source · 04_arithmetic.tex, line 856
Lean proves the one-coordinate arithmetic directly over Z/p^L Z for every p>1: (∃ x∈ Z/p^L Z: p^s x=bpmodp^L) Longleftrightarrow p^min(s,L)mid b_val, where b_val is the canonical natural representative, and every admissible fiber has exactly p^min(s,L) elements. The coordinates are independent, so the product equivalence gives the displayed global count. If one coordinate fiber is empty, the global fiber is empty. Lean proves both the global solvability equivalence and the conditional count, inc…
Exact mathematical passage
\bigl(\exists x\in\mathbb Z/p^L\mathbb Z:\quad p^s x=b\pmod{p^L}\bigr) \quad\Longleftrightarrow\quad p^{\min(s,L)}\mid b_{\mathrm{val}},
B_{vu}Use in source · 2 passages
Search aliases: B_vu
Use in source · 04_correction_transport.tex, line 712
[Fundamental holonomy intertwining criterion] The following conditions are equivalent. item There is a family of isomorphisms U_v:V_0→ W_0 with the prescribed root value U_r and satisfying U_vA_vu=B_vuU_u for every oriented edge u→ v. item For every fundamental loop γ based at r, U_rHol_A(γ)=Hol_B(γ)U_r. item The same identity holds for every loop based at r. When these conditions hold, the family (U_v) is uniquely determined by U_r.
Exact mathematical passage
U_vA_{vu}=B_{vu}U_u
B_wUse in source · 2 passages
Search aliases: B_w
Use in source · residue_continuation_body.tex, line 150
Nm I≤ Xw([I]). Exact grouping by ideal class gives S_w(X)=∑_C w(C)A_C(X). If ∑_Cw(C)=0, the common leading coefficient cancels, so there is B_w≥0 such that absnS_w(X)≤ B_wX^1-1/d (X≥1). For every nontrivial character χ:Cl( O)→ C^×, the finite character sum is zero. Thus the displayed estimate applies without an additional arithmetic cancellation hypothesis. This conclusion concerns ordinary class characters. It does not identify them with the further norm-residue characters needed for Frobenius …
Exact mathematical passage
B_w\geq0
b_xUse in source · 4 passages
Search aliases: b_x
Use in source · continuation_02.tex, line 16
Let p be prime, L≥1, and F∈ Z[X_1,…,X_d]^e, x∈ Z^d, F(x)≡0pmodp^L. Write r_x=F(x)/p^L, b_x=r_xbmod p, A_x=DF(x)bmod p: F_p^d⟶ F_p^e, ρ_x=rank_ F_pA_x. The actual lift set is L_x,L= {y∈( Z/p^L+1 Z)^d: y≡ xpmodp^L, F(y)=0}, where polynomial evaluation in the displayed set is modulo p^L+1. Define O_x= F_p^e/imA_x, o_x=[b_x]. Lean proves # L_x,L= p^d-ρ_x, · o_x=0,
Exact mathematical passage
r_x=F(x)/p^L,\qquad b_x=r_x\bmod p, \qquad A_x=DF(x)\bmod p:\mathbb F_p^d\longrightarrow\mathbb F_p^e, \qquad \rho_x=\operatorname{rank}_{\mathbb F_p}A_x.
b_YUse in source · 1 passage
Search aliases: b_Y
Use in source · 30_lifting.tex, line 874
Theorem~[source reference] gives a unique root b of F+p^M H in a+p^s+1R^e+k, with displacement depth M-s. Its last equations say exactly b_T=t_0+p^Mv=τ. Its first equations are G(b)=0. The integral first-order expression uses CbinomH(a)-v =binomA(H(a)+Kv)-p^sv. Projection onto the solved coordinates gives b_Y-y_0+p^M-sA(H(a)+Kv)∈ p^2M-3sR^e. The same inclusion identity as before converts this to [source reference].
Exact mathematical passage
b_Y-y_0+p^{M-s}A(H(a)+Kv)\in p^{2M-3s}R^e.
B_\alphaUse in source · 6 passages
Search aliases: B_α alpha
Use in source · Paper_VI_Traditional_Final.tex, line 2848
Let {B_1,…,B_m} be a partition of the fine state set. Its aggregation matrix is C_α i= 1_{i∈ B_α}.
Exact mathematical passage
C_{\alpha i}=\one_{\{i\in B_\alpha\}}.
B_\betaUse in source · 3 passages
Search aliases: B_β beta
Use in source · Paper_VI_Traditional_Final.tex, line 2916
[Characterization of exact discrete lumping] Let K be column stochastic. A coarse column stochastic matrix overline K satisfying CK=overline KC exists if and only if, for every source block B_β, every pair i,j∈ B_β, and every target block B_α, ∑_r∈ B_αK_ri = ∑_r∈ B_αK_rj. When it exists, overline K is unique and overline K_αβ = ∑_r∈ B_αK_ri for any i∈ B_β.
Exact mathematical passage
B_\beta
b_\betaUse in source · 1 passage
Search aliases: b_β beta
Use in source · 01_manuscript.tex, line 1026
The equation in packet β is D_β x_β = b_β - ∑_α→βK_βαx_α. Every predecessor of β lies in an earlier level. Once those values are available, x_β is obtained from the local solve with D_β. Vertices in the same level have no directed edge between them, so their solves are independent.
Exact mathematical passage
D_\beta x_\beta = b_\beta - \sum_{\alpha\to\beta}K_{\beta\alpha}x_\alpha.
B_{\delta\gamma}Use in source · 2 passages
Search aliases: B_δγ delta gamma
Use in source · 01_manuscript.tex, line 197
[Adjoint and tensor semantics] The adjoint representation has blocks (A^*)_αβ=A_βα^*. If A and B are packet resolved on X and Y, then (A⊗ B)_(β,δ),(α,γ) = A_βα⊗ B_δγ.
Exact mathematical passage
(A\otimes B)_{(\beta,\delta),(\alpha,\gamma)} = A_{\beta\alpha}\otimes B_{\delta\gamma}.
B_\ThetaUse in source · 5 passages
Search aliases: B_Θ Theta
Use in source · supplement_10.tex, line 46
Put Θ=(1+T)fracddT, B_Θ=fracΘ^p^2-Θ^pp. The coefficient formula [T^d]Θ F=(d+1)a_d+1+da_d proves Θ W_p⊆ W_p. If the floor weight increases between d and d+1, then pmid d+1, which compensates for that increase.
Exact mathematical passage
\Theta=(1+T)\frac{d}{dT},\qquad B_\Theta=\frac{\Theta^{p^2}-\Theta^p}{p}.
B_\Theta^sUse in source · 3 passages
Search aliases: B_Θ^s Theta
Use in source · supplement_10.tex, line 75
Finally, [T^d]B_Θ F depends only on coefficients through d+p^2. Replacing F by this finite prefix proves the result for the entire formal series, without an analytic convergence assumption: B_Θ W_p⊆ W_p, B_Θ^s W_p⊆ W_p (s≥0). Modules. mcodeWeightedAmiceRing, mcodeFrobeniusSquare, mcodeThetaWeighted. 2. Actual substitution and singular higher differences. Let E(F)(t)=F(e^t-1), D=fracddt, B_D=fracD^p^2-D^pp. All substitutions use Mathlib's actual formal exponential, logarithm and substitution maps…
Exact mathematical passage
\boxed{B_\Theta\Wp\subseteq\Wp,\qquad B_\Theta^s\Wp\subseteq\Wp\ (s\ge0).}
B_\lambdaUse in source · 4 passages
Search aliases: B_λ lambda
Use in source · Paper_XII_Traditional_Final.tex, line 1486
Finally, put B_λ= O_λ[x_g,a,b:g∈Γ, a,b∈ D]. For each declared local condition index u, supply an ideal J_λ,u⊆ B_λ. Evaluation at a representation sends x_g,a,b to its (a,b) matrix entry at g. Assume that evaluation at every admissible finite representative kills J_λ,u, with polynomial coefficients reduced modulo I_λ^n. For this class based theorem, the equations must hold in every admissible frame. A condition imposed only in one chosen frame needs a separate compatible frame argument. No arbitr…
Exact mathematical passage
B_\lambda=\OO_\lambda[x_{g,a,b}:g\in\Gamma,\ a,b\in D].
\mathcal B_\lambdaUse in source · 3 passages
Search aliases: B_λ mathcal lambda
Use in source · atlas-pages.tex, line 35
B_λ · diag(λ)p/(λ^ Tp). · Nonnegative likelihood and positive total evidence.
Exact mathematical passage
\mathcal B_\lambda
b_{\lambda,j}Use in source · 2 passages
Search aliases: b_λ,j lambda
Use in source · 05_worked_models.tex, line 284
[Certified graph seed number] Let 𝕜 be a field and let Λ be a nonempty finite set. Suppose there are pairwise distinct scalars a_λ∈𝕜, multiplicities m_λ∈ N, and an eigenbasis {b_λ,j:λ∈Λ, 1≤ j≤ m_λ} of 𝕜^V(G) satisfying A_Gb_λ,j=a_λb_λ,j. If m_λ≤ R for all λ and m_λ_*=R for some λ_*, then R is the least natural number of adjacency seeds: R=min{r∈ N: l there exist s_1,…,s_r with
Exact mathematical passage
\{b_{\lambda,j}:\lambda\in\Lambda,\ 1\le j\le m_{\lambda}\}
B_\nuUse in source · 1 passage
Search aliases: B_ν nu
Use in source · 02_supplement_06.tex, line 54
ubsection:002textitModule SmallCoreNorms; arbitrary ambient competitors For rectangular isometries Q,V define L(G)=QGV^*, C(B)=Q^*BV, Q^*Q=I, V^*V=I. The following statements are proved for the actual matrix rank and actual matrix norms: C( L(G))=G, rank L(G)=rank G, rank C(B)≤rank B, norm L(G)_ν=normG_ν, norm C(B)_ν≤norm B_ν, ν∈{2, F}. For the spectral norm, the proof uses the Gram matrix identity (QG)^*(QG)=G^*G and the C^*-norm square identity, together with right-side adjoint symmetry. Compr…
Exact mathematical passage
\boxed{\norm{\mathcal L(G)}_\nu=\norm{G}_\nu, \qquad \norm{\mathcal C(B)}_\nu\leq\norm B_\nu, \quad \nu\in\{2,\mathrm F\}.}
B_\tauUse in source · 4 passages
Search aliases: B_τ tau
Use in source · 03_nonlinear_coordinates.tex, line 468
Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a to a leaf and applying the scalar map (t_1,…,t_r)↦ b_rt_1⋯ t_r at a vertex of arity r. A second induction gives ∑_τ∈ T_m B_τ(s) ≤ ∑_τ∈ T_mw(τ) =t_m, wh…
Exact mathematical passage
B_\tau(s)=B_r(B_{\tau_1}(s),\ldots,B_{\tau_r}(s)).
B_{\tau_1}Use in source · 1 passage
Search aliases: B_τ_1 tau
Use in source · 03_nonlinear_coordinates.tex, line 468
Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a to a leaf and applying the scalar map (t_1,…,t_r)↦ b_rt_1⋯ t_r at a vertex of arity r. A second induction gives ∑_τ∈ T_m B_τ(s) ≤ ∑_τ∈ T_mw(τ) =t_m, wh…
Exact mathematical passage
B_\tau(s)=B_r(B_{\tau_1}(s),\ldots,B_{\tau_r}(s)).
B_{\tau_r}Use in source · 1 passage
Search aliases: B_τ_r tau
Use in source · 03_nonlinear_coordinates.tex, line 468
Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a to a leaf and applying the scalar map (t_1,…,t_r)↦ b_rt_1⋯ t_r at a vertex of arity r. A second induction gives ∑_τ∈ T_m B_τ(s) ≤ ∑_τ∈ T_mw(τ) =t_m, wh…
Exact mathematical passage
B_\tau(s)=B_r(B_{\tau_1}(s),\ldots,B_{\tau_r}(s)).
\backslashUse in source · 1 passage
Search aliases: backslash
Use in source · Paper_XI_Traditional_Final.tex, line 581
Apply Mackey decomposition [source reference]. The double cosets W_K_nbackslash W_K/W_K_m are the orbits of simultaneous translation on Z/n Z× Z/m Z. There are d orbits, each of length L. The stabilizer of a point in each orbit is W_K_L. The second character is shifted by the chosen Frobenius representative. Summing the induced terms over the double cosets gives the decomposition.
Exact mathematical passage
W_{K_n}\backslash W_K/W_{K_m}
\bar aUse in source · 1 passage
Search aliases: bar a bar
Use in source · supplement_05.tex, line 94
The root equation and residue are derived. Euler's unit-group exponent gives ω_ℓ(a)^p-1 =a^p^ℓ-1(p-1)=a^φ(p^ℓ)=1. Reduction to the actual field F_p commutes with powers. Its Frobenius identity gives overlineω_ℓ(a) =bar a^p^ℓ-1=bar a. Consequently, a=ω_ℓ(a)⟨ a⟩_ℓ, ω_ℓ(a)^p-1=1, overline⟨ a⟩_ℓ=1. The manuscript's finite existence conclusion is obtained directly. The proof does not assume a Hensel lift or an already supplied root representative.
Exact mathematical passage
\overline{\omega_\ell(a)} =\bar a^{p^{\ell-1}}=\bar a.
\bar a^{p^{\ell-1}}Use in source · 1 passage
Search aliases: bar a^p^ℓ-1 bar ell
Use in source · supplement_05.tex, line 94
The root equation and residue are derived. Euler's unit-group exponent gives ω_ℓ(a)^p-1 =a^p^ℓ-1(p-1)=a^φ(p^ℓ)=1. Reduction to the actual field F_p commutes with powers. Its Frobenius identity gives overlineω_ℓ(a) =bar a^p^ℓ-1=bar a. Consequently, a=ω_ℓ(a)⟨ a⟩_ℓ, ω_ℓ(a)^p-1=1, overline⟨ a⟩_ℓ=1. The manuscript's finite existence conclusion is obtained directly. The proof does not assume a Hensel lift or an already supplied root representative.
Exact mathematical passage
\overline{\omega_\ell(a)} =\bar a^{p^{\ell-1}}=\bar a.
\bar bUse in source · 7 passages
Search aliases: bar b bar
Use in source · Paper_VI_Traditional_Final.tex, line 915
When sufficiency fails, the defect still has an exact finite factorization. Fix a declared coarse likelihood barλ, put P=C_π p, δ(p)=r(p)-barλ, bar b= B_barλ(P), Z_r=⟨ r(p),P⟩.
Exact mathematical passage
P=C_\pi p, \qquad \delta(p)=r(p)-\bar\lambda, \qquad \bar b=\cB_{\bar\lambda}(P), \qquad Z_r=\langle r(p),P\rangle.
\bar EUse in source · 7 passages
Search aliases: bar E bar
Use in source · Paper_X_Block27_Amended.tex, line 2093
Lifting a summand inside a supplied finite-group carrier. Let R be a commutative integral domain complete and separated for the (π)-adic filtration. Let Γ be a finite group, and let ρ:Γ→GL_n(R) be an actual integral representation. Assume that a=|Γ|≠0 in R and amidπ^e. For k>0, suppose bar E∈ M_n(R/(π)^e+k), bar E^2=bar E, barρ(g)bar E=bar Ebarρ(g) (g∈Γ). There is an actual integral matrix E satisfying E^2=E, ρ(g)E=Eρ(g), Ebmod(π)^k=bar Ebmod(π)^k. No integral summand or coherent family of quoti…
Exact mathematical passage
\bar E\in M_n\bigl(R/(\pi)^{e+k}\bigr),\qquad \bar E^2=\bar E,\qquad \bar\rho(g)\bar E=\bar E\bar\rho(g)\quad(g\in\Gamma).
\bar E^2Use in source · 1 passage
Search aliases: bar E^2 bar
Use in source · Paper_X_Block27_Amended.tex, line 2093
Lifting a summand inside a supplied finite-group carrier. Let R be a commutative integral domain complete and separated for the (π)-adic filtration. Let Γ be a finite group, and let ρ:Γ→GL_n(R) be an actual integral representation. Assume that a=|Γ|≠0 in R and amidπ^e. For k>0, suppose bar E∈ M_n(R/(π)^e+k), bar E^2=bar E, barρ(g)bar E=bar Ebarρ(g) (g∈Γ). There is an actual integral matrix E satisfying E^2=E, ρ(g)E=Eρ(g), Ebmod(π)^k=bar Ebmod(π)^k. No integral summand or coherent family of quoti…
Exact mathematical passage
\bar E\in M_n\bigl(R/(\pi)^{e+k}\bigr),\qquad \bar E^2=\bar E,\qquad \bar\rho(g)\bar E=\bar E\bar\rho(g)\quad(g\in\Gamma).
\bar E_iUse in source · 2 passages
Search aliases: bar E_i bar
Use in source · Paper_X_Block27_Amended.tex, line 2165
Simultaneous projector-family lifting and rigidity. Keep the complete domain, supplied finite-group carrier and group-order hypotheses of the hyperref[book:x-construction-18]supplied-carrier summand-lifting construction. Let J be any finite label set and k>0. Suppose (bar E_i)_i∈ J is a complete orthogonal family of equivariant idempotents modulo (π)^e+k. There exists an integral equivariant family (E_i)_i∈ J satisfying E_i^2=E_i, E_iE_j=0 (i≠ j), ∑_i∈ JE_i=1, E_ibmod(π)^k=bar E_ibmod(π)^k. The …
Exact mathematical passage
(\bar E_i)_{i\in J}
\bar KUse in source · 20 passages
Search aliases: bar K bar
Use in source · Paper_VI_Traditional_Final.tex, line 591
Let K be a column-stochastic fine transition matrix, let C aggregate fine states into coarse packets, and let bar K be a coarse transition matrix. Define Δ=CK-bar KC.
Exact mathematical passage
\bar K
\bar K^nUse in source · 9 passages
Search aliases: bar K^n bar
Use in source · Paper_VI_Traditional_Final.tex, line 598
[Exact lumpability] If Δ=0, then CK^n=bar K^nC for every n≥0. Thus fine evolution followed by coarse reconstruction equals coarse evolution from the beginning.
Exact mathematical passage
\boxed{CK^n=\bar K^nC}
\bar K^{n+1}Use in source · 1 passage
Search aliases: bar K^n+1 bar
Use in source · Paper_VI_Traditional_Final.tex, line 606
The result is immediate for n=0. If CK^n=bar K^nC, then CK^n+1=bar K^nCK=bar K^n+1C.
Exact mathematical passage
CK^{n+1}=\bar K^nCK=\bar K^{n+1}C.
\bar K^{n-1-j}Use in source · 3 passages
Search aliases: bar K^n-1-j bar
Use in source · Paper_VI_Traditional_Final.tex, line 613
[Finite-horizon defect formula] For arbitrary Δ=CK-bar KC, CK^n-bar K^nC = ∑_j=0^n-1bar K^n-1-jΔ K^j. If K and bar K are contractions in the chosen norm, then normCK^n-bar K^nC≤ nnormΔ.
Exact mathematical passage
\boxed{ CK^n-\bar K^nC = \sum_{j=0}^{n-1}\bar K^{n-1-j}\Delta K^j. }
\bar K^{n-j}Use in source · 1 passage
Search aliases: bar K^n-j bar
Use in source · Paper_VI_Traditional_Final.tex, line 628
The identity follows by telescoping: bar K^n-1-jCK^j+1-bar K^n-jCK^j =bar K^n-1-j(CK-bar KC)K^j. Summing from j=0 to n-1 leaves CK^n-bar K^nC. The norm bound follows termwise.
Exact mathematical passage
\bar K^{n-1-j}CK^{j+1}-\bar K^{n-j}CK^j =\bar K^{n-1-j}(CK-\bar KC)K^j.
\bar QUse in source · 2 passages
Search aliases: bar Q bar
Use in source · atlas-pages.tex, line 115
The continuous generator also closes: CQ=bar Q C, bar Q=-2 · 2 2 · -2. The Duhamel identity therefore gives Ce^tQ=e^tbar QC. This is a second consequence of the corresponding generator identity, not an inference that K is a fixed-time sample of Q.
Exact mathematical passage
CQ=\bar Q C,\qquad \bar Q=\begin{pmatrix}-2&2\\2&-2\end{pmatrix}.
\bar rUse in source · 7 passages
Search aliases: bar r bar
Use in source · continuation_08.tex, line 14
Let q>0, L≥1, and m,k,e≥0. Write S= Z/q Z. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0)∈ Z^m+k, F(x)≡0pmodq^L. The actual lower root supplies the unique integral residual r=q^-LF(x)∈ Z^e, bar r=rbmod q∈ S^e. Let the actual reduced Jacobian blocks be J=(D_YF)(x)bmod q:S^m→ S^e, K=(D_TF)(x)bmod q:S^k→ S^e. For a prescribed parameter digit v∈ S^k, choose its canonical integral representative widehat v. Define L_v to be the set of actual residue vectors z∈( Z/q^L+1 Z)^m+k satisfying z≡ xpmodq^L, F(z…
Exact mathematical passage
r=q^{-L}F(x)\in\mathbb Z^e,\qquad \bar r=r\bmod q\in S^e.
\bar wUse in source · 2 passages
Search aliases: bar w bar
Use in source · continuation_09.tex, line 56
Exact branching. At any stage n, put N=L+n. For every parameter digit v∈ F_p^k, let L_n,v={bar w∈( Z/p^N+1 Z)^m+k: l bar w≡ z(n)pmodp^N, F(bar w)=0, free(bar w)=t(n)+p^Nvpmodp^N+1 }. Here p^Nv means any integer representative of v, multiplied by p^N. The residue class does not depend on the representative. Then # L_n,v=p^m-e for every n and every v. Surjectivity implies e≤ m. This is a count of distinct actual next-level residue classes. It does not count integer representatives or entire infini…
Exact mathematical passage
\mathcal L_{n,v}=\left\{\bar w\in(\mathbb Z/p^{N+1}\mathbb Z)^{m+k}: \begin{array}{l} \bar w\equiv z(n)\pmod{p^N},\\ F(\bar w)=0,\\ \operatorname{free}(\bar w)=t(n)+p^Nv\pmod{p^{N+1}} \end{array}\right\}.
\bar yUse in source · 1 passage
Search aliases: bar y bar
Use in source · Paper_XI_Traditional_Final.tex, line 1158
J_θ. The quotient V_θ=J_θ^1/H_θ^1 is regarded as a finite vector space over F_p, where p is the residue characteristic. Fix a nontrivial additive character ψ_0: F_p→μ_p⊂ C^×, where μ_p is the group of p-th roots of unity. The simple-character commutator takes values in μ_p. Since ψ_0 is injective, the equation ψ_0(h_θ(bar x,bar y)) = θ([x,y]) uniquely defines the F_p-valued commutator pairing h_θ. We use the same identification of μ_p with the additive group of F_p on the Galois side when compar…
Exact mathematical passage
\psi_0\bigl(h_\theta(\bar x,\bar y)\bigr) = \theta([x,y])
\bar\kappaUse in source · 2 passages
Search aliases: barκ bar kappa
Use in source · 02_supplement_05.tex, line 222
tκ-1sqrtκ+1. Monotonicity of square root and positive denominators prove that q is monotone. Lean also proves κ_2(B)≥1, q(κ)≥0 for κ≥1, and q(κ)<1. Thus q(κ_2(B))≤ q (frac1+η1-η). For 1≤κ and e_0≥0, an explicit classical premise e_m≤2q(κ)^m e_0 and a certified upper bound κ≤barκ yield e_m≤2q(barκ)^m e_0.
Exact mathematical passage
\kappa\leq\bar\kappa
\bar\lambdaUse in source · 22 passages
Search aliases: barλ bar lambda
Use in source · Paper_VI_Traditional_Final.tex, line 881
[Bayesian refinement criterion] Assume that the partition contains at least two coarse packets and that the fine and coarse likelihoods are strictly positive. The identity C_π B_λ(p) = B_barλ(C_π p) holds for every fine prior p if and only if there exists one scalar c>0 such that λ_i=c barλ_a (i∈ I_a). Equivalently, the likelihood is constant on each coarse packet, with the coarse values determined up to the globally invisible Bayesian scaling c.
Exact mathematical passage
\boxed{ C_\pi\cB_\lambda(p) = \cB_{\bar\lambda}(C_\pi p)}
\bar\lambda_aUse in source · 7 passages
Search aliases: barλ_a bar lambda
Use in source · Paper_VI_Traditional_Final.tex, line 888
[Bayesian refinement criterion] Assume that the partition contains at least two coarse packets and that the fine and coarse likelihoods are strictly positive. The identity C_π B_λ(p) = B_barλ(C_π p) holds for every fine prior p if and only if there exists one scalar c>0 such that λ_i=c barλ_a (i∈ I_a). Equivalently, the likelihood is constant on each coarse packet, with the coarse values determined up to the globally invisible Bayesian scaling c.
Exact mathematical passage
\boxed{ \lambda_i=c\,\bar\lambda_a \qquad(i\in I_a).}
\bar\lambda_bUse in source · 1 passage
Search aliases: barλ_b bar lambda
Use in source · Paper_VI_Traditional_Final.tex, line 902
Now use priors supported on one state of packet a and one state of packet b. Equality of the two coarse posterior odds gives fracλ_aλ_b = fracbarλ_abarλ_b. This holds for every pair of coarse packets, so λ_a=cbarλ_a for one common c>0.
Exact mathematical passage
\frac{\lambda_a}{\lambda_b} = \frac{\bar\lambda_a}{\bar\lambda_b}.
\bar\lambda_{\pi(i)}Use in source · 1 passage
Search aliases: barλ_π(i) bar lambda pi
Use in source · Paper_VI_Traditional_Final.tex, line 946
tes barλ. This records the common-scale null direction at the declared coarse likelihood. The full exact defect remains homogeneous under simultaneous likelihood rescaling because the numerator and evidence denominator scale together. Moreover, C_π B_λ(p)- B_barλ(P) _1 ≤ frac2∑_aP_a|δ_a(p)| min{Z_r,Z_barλ} and ∑_aP_a|δ_a(p)| ≤ ∑_i p_i|λ_i-barλ_π(i)|.
Exact mathematical passage
\sum_aP_a|\delta_a(p)| \le \sum_i p_i\left|\lambda_i-\bar\lambda_{\pi(i)}\right|.
\BBUse in source · 4 passages
Search aliases: BB B mathcal
Use in source · visual_atlas.tex, line 312
headThis is why the independent formula uses a maximum For independent actions of A on X and B on Y, a common list of seed positions can carry the seeds for both spaces. Pad the shorter list with zeros. Independent operators then select the two components, giving srank_ A× B(X⊕ Y) =max{srank_ A(X),srank_ B(Y)}. With a shared action, this selection need not be available. The general bound is the sum.
Exact mathematical passage
\BB
\bbC^dUse in source · 1 passage
Search aliases: bbC^d C^d bbC mathbb
Use in source · 01_finite_core.tex, line 95
[Orbit dimension of one quantum seed] For integers d≥1 and m≥0, consider the action of M_d( C)⊗ I_m on C^d⊗ C^m. Let C be the coefficient matrix of a vector ψ. Then dimspan{(A⊗ I_m)ψ:A∈ M_d( C)} =d rank C. The seed is cyclic if and only if rank C=m, which is possible only when m≤ d.
Exact mathematical passage
\bbC^d\otimes\bbC^m
\bbC^{d_i}Use in source · 3 passages
Search aliases: bbC^d_i C^d_i bbC mathbb
Use in source · 01_finite_core.tex, line 86
Let a finite-dimensional unital *-subalgebra of operators act in its standard multiplicity form A ≅ ⊕_i=1^r (M_d_i( C)⊗ I_m_i) on H ≅ ⊕_i=1^r ( C^d_i⊗ C^m_i). Here every d_i is a positive integer and every multiplicity m_i is a nonnegative integer. An empty block family represents the zero space and has seed rank zero. The maximum over an empty block family is also assigned the value zero. A vector ψ_i∈ C^d_i⊗ C^m_i is identified with a d_i× m_i coefficient matrix C_i.
Exact mathematical passage
\cH \cong \bigoplus_{i=1}^r \left(\bbC^{d_i}\otimes\bbC^{m_i}\right).
\bbC^mUse in source · 1 passage
Search aliases: bbC^m C^m bbC mathbb
Use in source · 01_finite_core.tex, line 95
[Orbit dimension of one quantum seed] For integers d≥1 and m≥0, consider the action of M_d( C)⊗ I_m on C^d⊗ C^m. Let C be the coefficient matrix of a vector ψ. Then dimspan{(A⊗ I_m)ψ:A∈ M_d( C)} =d rank C. The seed is cyclic if and only if rank C=m, which is possible only when m≤ d.
Exact mathematical passage
\bbC^d\otimes\bbC^m
\bbC^{m_i}Use in source · 6 passages
Search aliases: bbC^m_i C^m_i bbC mathbb
Use in source · 01_finite_core.tex, line 86
Let a finite-dimensional unital *-subalgebra of operators act in its standard multiplicity form A ≅ ⊕_i=1^r (M_d_i( C)⊗ I_m_i) on H ≅ ⊕_i=1^r ( C^d_i⊗ C^m_i). Here every d_i is a positive integer and every multiplicity m_i is a nonnegative integer. An empty block family represents the zero space and has seed rank zero. The maximum over an empty block family is also assigned the value zero. A vector ψ_i∈ C^d_i⊗ C^m_i is identified with a d_i× m_i coefficient matrix C_i.
Exact mathematical passage
\cH \cong \bigoplus_{i=1}^r \left(\bbC^{d_i}\otimes\bbC^{m_i}\right).
\bbk^2Use in source · 10 passages
Search aliases: bbk^2 bbk bbk^2 𝕜^2 bbk Bbbk
Use in source · 01_orientation.tex, line 87
[title=R2: a matrix algebra with multiplicity] Let A=M_2(𝕜) act diagonally on M=(𝕜^2)^⊕3. One seed can contribute at most two independent multiplicity directions, because the regular module contains two copies of the simple module. Hence srank_A(M)=lceilfrac32rceil=2. This is the simplest nontrivial instance of the semisimple multiplicity formula.
Exact mathematical passage
M=(\bbk^2)^{\oplus3}
\bbk^3Use in source · 3 passages
Search aliases: bbk^3 𝕜^3 bbk Bbbk
Use in source · 02_foundations.tex, line 460
[title=A quotient can strictly lower seed rank] Let A=𝕜 act by scalars on M=𝕜^3. Then srank_A(M)=3. Quotienting by a subspace of dimension one gives a scalar module of dimension two, whose seed rank is 2. Quotient monotonicity is therefore often strict.
Exact mathematical passage
M=\bbk^3
\bbk^{d_i}Use in source · 1 passage
Search aliases: bbk^d_i 𝕜^d_i bbk Bbbk
Use in source · 02_foundations.tex, line 812
[Algebraically closed multiplicity formula] Let 𝕜 be algebraically closed, let A be a unital algebra of finite dimension over 𝕜, put J=Jac(A), and let M be a finitely generated left module over A. Assume compatible decompositions indexed by an integer q≥0: A/J≅∏_i=1^qM_d_i(𝕜), M/JM≅⊕_i=1^q(𝕜^d_i)^⊕ m_i, where the d_i are positive integers and the m_i are nonnegative integers. The first isomorphism is an isomorphism of algebras over 𝕜. The second is an isomorphism of left modules over A/J, using …
Exact mathematical passage
A/J\cong\prod_{i=1}^{q}M_{d_i}(\bbk), \qquad M/JM\cong\bigoplus_{i=1}^{q}(\bbk^{d_i})^{\oplus m_i},
\bbk^nUse in source · 2 passages
Search aliases: bbk^n 𝕜^n bbk Bbbk
Use in source · 05_worked_models.tex, line 256
Let X=𝕜^n and let A=M_n(𝕜)=End_𝕜(X), with the full matrix algebra declared before a seed or target is chosen. By [source reference], every nonzero seed is cyclic. This is the maximal linear grammar on X. It does not give a compression result. Under a smaller declared operator algebra, the same vector need not be cyclic.
Exact mathematical passage
X=\bbk^n
\bbk^{V(G)}Use in source · 3 passages
Search aliases: bbk^V(G) 𝕜^V(G) bbk Bbbk
Use in source · 05_worked_models.tex, line 275
Let G be a nonempty finite graph with adjacency matrix A_G, and regard 𝕜^V(G) as a module over 𝕜[A_G]. The general result applies directly to a spectral certificate. For a symmetric real adjacency matrix, the classical spectral theorem supplies such data.
Exact mathematical passage
\bbk^{V(G)}
\bbk^\timesUse in source · 1 passage
Search aliases: bbk^× 𝕜^× bbk times Bbbk times
Use in source · 02_connections_obstructions.tex, line 666
[A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly when vu=w. If the three coordinate systems are literally the same and canonical transport is used, then u=v=w=1 and the obstruction vanishes. Choosing …
Exact mathematical passage
u,v,w\in\bbk^\times
\bbNUse in source · 1 passage
Search aliases: bbN N mathbb
Use in source · Paper_V_Traditional_Final.tex, line 1075
[Cyclic product law] P_N,k(AB) = ∑_i+j≡ k (mod N)A_iB_j. For q∈ N, the qth-power identity is P_N,k(A^q) = ∑_i_1+⋯+i_q≡ k A_i_1⋯ A_i_q.
Exact mathematical passage
q\in\bbN
\bbPUse in source · 1 passage
Search aliases: bbP P mathbb
Use in source · Paper_VI_Traditional_Final.tex, line 575
[Symmetric random walk] Starting from δ_0 and applying the transfer T=frac12(S+S^-1) for n steps gives P(X_n=k)= 2^-nbinomn(n+k)/2 when k≡ npmod2, and zero otherwise. Hence E[X_n]=0, Var(X_n)=n. The parity packet is exact: all mass remains in the one residue class permitted by n.
Exact mathematical passage
\bbP(X_n=k)= 2^{-n}\binom{n}{(n+k)/2}
\bbRUse in source · 3 passages
Search aliases: bbR R mathbb
Use in source · Paper_VI_Traditional_Final.tex, line 334
[Finite semigroup law] For every matrix A and all s,t∈ R, e^(s+t)A=e^sAe^tA. Moreover, T(0)=I and T'(t)=AT(t)=T(t)A.
Exact mathematical passage
s,t\in\bbR
\Bcal_{c\lambda}Use in source · 1 passage
Search aliases: Bcal_cλ B_cλ Bcal lambda mathcal lambda
Use in source · Paper_VI_Traditional_Final.tex, line 3353
[Scale invariance and composition] For every c>0, B_cλ= B_λ. If all relevant evidences are positive, then B_μ( B_λ(p)) = B_λodotμ(p).
Exact mathematical passage
\Bcal_{c\lambda}=\Bcal_\lambda.
\Bcal_LUse in source · 4 passages
Search aliases: Bcal_L B_L Bcal mathcal
Use in source · Paper_VI_Traditional_Final.tex, line 5402
FloatBarrier Prior-restricted sufficiency and filtering. Let L be a strictly positive finite likelihood and let B_L(p)=Lodot p/ipLp for a probability vector p. Fix probability vectors r^(a) supported on B_a, and restrict the priors to p_i=q_a r_i^(a) for i∈ B_a, with q a coarse probability vector. [Sufficiency relative to a prior disintegration] Define overline L_a = ∑_i∈ B_aL_ir_i^(a). For every prior in the declared family, C B_L(p) = B_overline L(q).
Exact mathematical passage
\Bcal_L(p)=L\odot p/\ip{L}{p}
\Bcal_{L_y}Use in source · 1 passage
Search aliases: Bcal_L_y B_L_y Bcal mathcal
Use in source · Paper_VI_Traditional_Final.tex, line 5535
For stochastic matrices K,overline K and positive likelihoods L_y,overline L_y, put Φ_y= B_L_y∘ K and overlineΦ_y= B_overline L_y∘overline K. [Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯∘Φ_y_1 = overlineΦ_y_r∘⋯∘overlineΦ_y_1C.
Exact mathematical passage
\Phi_y=\Bcal_{L_y}\circ K
\Bcal_{N,i}Use in source · 3 passages
Search aliases: Bcal_N,i B_N,i Bcal mathcal
Use in source · Paper_IV_Traditional_Final.tex, line 383
[Root filter] If F(z)=∑_m≥0a_mz^m, then B_N,jF(z) = ∑_m≡ jbmod Na_mz^m. The operators satisfy B_N,i B_N,j=δ_ij B_N,j and ∑_j=0^N-1 B_N,j=I.
Exact mathematical passage
\Bcal_{N,i}\Bcal_{N,j}=\delta_{ij}\Bcal_{N,j}
\Bcal_{N,k}Use in source · 7 passages
Search aliases: Bcal_N,k B_N,k Bcal mathcal
Use in source · Paper_IV_Traditional_Final.tex, line 413
Every monomial selected by B_N,j has the stated covariance. Conversely, write F=∑_k B_N,kF. Replacing z by ω z gives ∑_kω^k B_N,kF = ω^j∑_k B_N,kF. Application of B_N,k gives (ω^k-ω^j) B_N,kF=0. The powers 1,ω,…,ω^N-1 are distinct, so B_N,kF=0 for k≠ j.
Exact mathematical passage
F=\sum_k\Bcal_{N,k}F
\Bcal_{\overline L}Use in source · 2 passages
Search aliases: Bcal_overline L B_overline L Bcal overline mathcal overline
Use in source · Paper_VI_Traditional_Final.tex, line 5411
ctor p. Fix probability vectors r^(a) supported on B_a, and restrict the priors to p_i=q_a r_i^(a) for i∈ B_a, with q a coarse probability vector. [Sufficiency relative to a prior disintegration] Define overline L_a = ∑_i∈ B_aL_ir_i^(a). For every prior in the declared family, C B_L(p) = B_overline L(q).
Exact mathematical passage
C\Bcal_L(p) = \Bcal_{\overline L}(q).
\Bcal_{\overline L_y}Use in source · 2 passages
Search aliases: Bcal_overline L_y B_overline L_y Bcal overline mathcal overline
Use in source · Paper_VI_Traditional_Final.tex, line 5535
For stochastic matrices K,overline K and positive likelihoods L_y,overline L_y, put Φ_y= B_L_y∘ K and overlineΦ_y= B_overline L_y∘overline K. [Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯∘Φ_y_1 = overlineΦ_y_r∘⋯∘overlineΦ_y_1C.
Exact mathematical passage
\overline\Phi_y=\Bcal_{\overline L_y}\circ\overline K
\Bcal_{\overline\lambda(p)}Use in source · 1 passage
Search aliases: Bcal_overlineλ(p) B_overlineλ(p) Bcal overline lambda mathcal overline lambda
Use in source · Paper_VI_Traditional_Final.tex, line 3484
[Exact two stage Bayesian update] For every strictly positive prior, C B_λ(p) = B_overlineλ(p)(Cp). For every packet with overlineλ_α(p)>0, the posterior conditional distribution is q_α,i = fracλ_ir_α,i overlineλ_α(p). A packet with zero average likelihood has zero posterior mass. Thus a fine Bayesian update consists of a coarse update by packet average likelihoods followed by independent local updates inside the packets that survive.
Exact mathematical passage
C\Bcal_\lambda(p) = \Bcal_{\overline\lambda(p)}(Cp).
\Bcal_{\widetilde\lambda}Use in source · 1 passage
Search aliases: Bcal_widetildeλ B_widetildeλ Bcal widetilde lambda mathcal widetilde lambda
Use in source · Paper_VI_Traditional_Final.tex, line 3629
Block constancy gives C B_widetildeλ(p) = B_ℓ(Cp). Aggregation is an ℓ^1 contraction. Apply [source reference] to the two fine posteriors and then aggregate.
Exact mathematical passage
C\Bcal_{\widetilde\lambda}(p) = \Bcal_\ell(Cp).
\Bcal_\lambdaUse in source · 13 passages
Search aliases: Bcal_λ B_λ Bcal lambda mathcal lambda
Use in source · Paper_VI_Traditional_Final.tex, line 2029
Bayesian update · B_λ · likelihood constant inside each coarse packet
Exact mathematical passage
\Bcal_\lambda
\Bcal_{\lambda^{(r)}}Use in source · 1 passage
Search aliases: Bcal_λ^(r) B_λ^(r) Bcal lambda mathcal lambda
Use in source · Paper_VI_Traditional_Final.tex, line 3861
Let K be column stochastic and let λ^(r) be the likelihood of the rth observation. The hidden Markov filter is widehat p_r=Kp_r-1, p_r= B_λ^(r)(widehat p_r). A standard account of finite hidden Markov models appears in [source reference].
Exact mathematical passage
\widehat p_r=Kp_{r-1}, \qquad p_r=\Bcal_{\lambda^{(r)}}(\widehat p_r).
\Bcal_{\lambda\odot\mu}Use in source · 1 passage
Search aliases: Bcal_λodotμ B_λodotμ Bcal lambda odot mu mathcal lambda odot mu
Use in source · Paper_VI_Traditional_Final.tex, line 3357
[Scale invariance and composition] For every c>0, B_cλ= B_λ. If all relevant evidences are positive, then B_μ( B_λ(p)) = B_λodotμ(p).
Exact mathematical passage
\Bcal_\mu(\Bcal_\lambda(p)) = \Bcal_{\lambda\odot\mu}(p).
\Bcal_\muUse in source · 3 passages
Search aliases: Bcal_μ B_μ Bcal mu mathcal mu
Use in source · Paper_VI_Traditional_Final.tex, line 3357
[Scale invariance and composition] For every c>0, B_cλ= B_λ. If all relevant evidences are positive, then B_μ( B_λ(p)) = B_λodotμ(p).
Exact mathematical passage
\Bcal_\mu(\Bcal_\lambda(p)) = \Bcal_{\lambda\odot\mu}(p).
\Bcal_\ellUse in source · 3 passages
Search aliases: Bcal_ℓ B_ℓ Bcal ell mathcal ell
Use in source · Paper_VI_Traditional_Final.tex, line 3515
[Characterization of fixed coarse sufficiency] Assume λ_i>0 for every state. The following conditions are equivalent. item There is a positive coarse likelihood ℓ∈(0,∞)^m such that C B_λ(p) = B_ℓ(Cp) for every strictly positive prior p, and the posterior conditional distribution inside every packet equals its prior conditional distribution. item The likelihood is constant on every packet. When these conditions hold, ℓ_α can be chosen as the common value of λ on I_α, and it is unique up to a comm…
Exact mathematical passage
C\Bcal_\lambda(p) = \Bcal_\ell(Cp)
\Bcal_{\ell(p)}Use in source · 1 passage
Search aliases: Bcal_ℓ(p) B_ℓ(p) Bcal ell mathcal ell
Use in source · Paper_VI_Traditional_Final.tex, line 5455
For an arbitrary prior p, put q=Cp and Z(p,L)=ipLp. Define ℓ_a(p)=∑_i∈ B_aL_ip_i/q_a when q_a>0. Choose any positive value when q_a=0. Only the product q_aℓ_a(p) enters the formulas. [Exact information loss formula] For every prior, C B_L(p) = B_ℓ(p)(q). Let overline L be any fixed positive coarse likelihood and set d(pmid L,overline L) = ∑_a=1^mq_aabsℓ_a(p)-overline L_a. Then norm C B_L(p)- B_overline L(q) _1 ≤ frac2d(pmid L,overline L) max{Z(p,L),ipoverline Lq}.
Exact mathematical passage
\boxed{ C\Bcal_L(p) = \Bcal_{\ell(p)}(q). }
\mathrm{Beta}Use in source · 2 passages
Search aliases: Beta mathrm
Use in source · atlas-pages.tex, line 305
vpage14Conjugacy tracks exponentsBETA MODEL / POSTERIOR AND EVIDENCEChapter VI, Part I: Example 9.2. A finite-parameter conjugate family, not a universal posterior model. For a small exact example, take the Beta(2,3) density ρ(θ)=12θ(1-θ)^2,qquad0≤θ≤1. Observe a specified ordered sequence with two successes and one failure. Its likelihood is θ^2(1-θ). Multiplication adds exponents, so the posterior belongs to Beta(4,4) with density 140θ^3(1-θ)^3. vfigbeta platecaptionThe dashed prior is asymmetr…
Exact mathematical passage
\mathrm{Beta}(2,3)
\operatorname{Beta}Use in source · 3 passages
Search aliases: Beta operatorname
Use in source · Paper_VI_Traditional_Final.tex, line 1072
[Beta--Binomial conjugate packet] A Beta(a,b) prior and s successes in n trials produce Beta(a+s,b+n-s). The prior exponents are additive packets: the likelihood increments them by (s,n-s). In the numerical example, (a,b)=(2,3), (s,n)=(7,10), so the posterior is Beta(9,6) with mean 0.6.
Exact mathematical passage
\operatorname{Beta}(a,b)
\bigmUse in source · 1 passage
Search aliases: bigm
Use in source · supplement_01.tex, line 135
0, · w=∞. The infinity convention expresses a vanishing coefficient, not an infinite exponent operation. For any finite index set, every term with weight at least L is divisible by p^L. A finite-sum proof yields p^L bigm| ∑_r=0^nbinom nr c_rp^[w(r)]S_n-r -∑_substack0≤ r≤ n w(r)<L binom nr c_rp^[w(r)]S_n-r. The result is specialized to actual p-adic integers R= Z_p for prime p. Divisibility there is precisely the asserted membership in p^L Z_p.
Exact mathematical passage
\boxed{p^L\ \bigm|\ \sum_{r=0}^{n}\binom nr c_rp^{[w(r)]}S_{n-r} -\sum_{\substack{0\le r\le n\\w(r)<L}} \binom nr c_rp^{[w(r)]}S_{n-r}.}
\bigsqcupUse in source · 2 passages
Search aliases: bigsqcup
Use in source · Paper_X_Block27_Amended.tex, line 199
[Residual partition] For every coefficient algebra A, M(A) = bigsqcup_bar x∈ M(k) M_bar x(A).
Exact mathematical passage
\boxed{ \mathcal M(A) = \bigsqcup_{\bar x\in\mathcal M(k)} \mathcal M_{\bar x}(A). }
\bigsqcup_{\bar x\in\mathcal M(k)}Use in source · 1 passage
Search aliases: bigsqcup_bar x∈ M(k) bigsqcup bar in mathcal
Use in source · Paper_X_Block27_Amended.tex, line 199
[Residual partition] For every coefficient algebra A, M(A) = bigsqcup_bar x∈ M(k) M_bar x(A).
Exact mathematical passage
\boxed{ \mathcal M(A) = \bigsqcup_{\bar x\in\mathcal M(k)} \mathcal M_{\bar x}(A). }
\bigsqcup_{r\in R}Use in source · 1 passage
Search aliases: bigsqcup_r∈ R bigsqcup in
Use in source · Paper_V_Traditional_Final.tex, line 1760
Let L≤ Z^d be a sublattice of finite index q. Choose a set R⊂ Z^d containing one representative from every coset in Z^d/L. Thus Z^d=bigsqcup_r∈ R(r+L). For x: Z^d→ C, its channel at r is the sequence on L x_r(ℓ)=x(r+ℓ).
Exact mathematical passage
\ZZ^d=\bigsqcup_{r\in R}(r+L).
\bottomruleUse in source · 5 passages
Search aliases: bottomrule
Use in source · visual_atlas.tex, line 19
[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…
Exact mathematical passage
4pt] \textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt] \textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology? \end{tabularx} A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. \takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''} \vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}} Most of the chapter can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix. {\small \begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule \textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule $X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\ $S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\ $\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\ $\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\ $P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\ $X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\ $T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\ $F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\ $U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\ $\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\ $\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule \end{tabularx}} \head{Read products from right to left} In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column. \head{Where to begin} For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}. \vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$} Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define \[ Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0, \qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.
\operatorname{Bounded}Use in source · 1 passage
Search aliases: Bounded operatorname
Use in source · supplement_08.tex, line 46
All orders, without a circular closure assumption. Let I⊂ A be a subring stable under U. Define Bounded_r(f)iff B^s f∈ I (0≤ s≤ r), Flat(f)iff B^s f∈ I (s≥0). The definitions contain no product or inverse closure requirement. Since B commutes with U, derivatives of a bounded element remain bounded. Induction on r in the displayed product rule then proves closure under multiplication. The finite middle sums use the lower-order induction hypothesis, not the desired next-order conclusion.
Exact mathematical passage
\operatorname{Bounded}_r(f)\iff B^s f\in I\ (0\le s\le r), \qquad \operatorname{Flat}(f)\iff B^s f\in I\ (s\ge0).
\operatorname{Bounded}_rUse in source · 1 passage
Search aliases: Bounded_r operatorname
Use in source · supplement_08.tex, line 46
All orders, without a circular closure assumption. Let I⊂ A be a subring stable under U. Define Bounded_r(f)iff B^s f∈ I (0≤ s≤ r), Flat(f)iff B^s f∈ I (s≥0). The definitions contain no product or inverse closure requirement. Since B commutes with U, derivatives of a bounded element remain bounded. Induction on r in the displayed product rule then proves closure under multiplication. The finite middle sums use the lower-order induction hypothesis, not the desired next-order conclusion.
Exact mathematical passage
\operatorname{Bounded}_r(f)\iff B^s f\in I\ (0\le s\le r), \qquad \operatorname{Flat}(f)\iff B^s f\in I\ (s\ge0).
\boxtimesUse in source · 1 passage
Search aliases: boxtimes
Use in source · 03_packets_and_graphs.tex, line 191
[Tensor product packet systems] Let P=(P_α)_α∈ I be a finite packet system on X, and let Q=(Q_γ)_γ∈ J be a finite packet system on Y. Then Pboxtimes Q := (P_α⊗ Q_γ)_(α,γ)∈ I× J is a finite packet system on the algebraic tensor product X⊗_𝕜Y, with packet spaces (P_α⊗ Q_γ)(X⊗ Y) =X_α⊗ Y_γ. Hence X⊗ Y ≅ ⊕_(α,γ)∈ I× J X_α⊗ Y_γ. [source reference] ExportsTensor refinement for quantum, signal, arithmetic, and multivariate realizations.
Exact mathematical passage
\Pcal\boxtimes\Qcal := (P_\alpha\otimes Q_\gamma)_{(\alpha,\gamma)\in I\times J}
\braUse in source · 19 passages
Search aliases: bra
Use in source · monograph_atlas.tex, line 108
vpage7A coarse packet can still contain coherenceTENSOR REFINEMENT BY TOTAL PARITYChapter VII: Theorem II.3.7, Corollary II.3.8. For three qubits, the eight fine projectors are P_a=ket abra a. Grouping addresses according to the parity of their Hamming weight gives P_rm even=P_000+P_011+P_101+P_110 =fracI+Z⊗ Z⊗ Z2, P_rm odd=I-P_rm even. This is a sum of subspaces, not an averaging of their coordinates. The rank-four even packet retains all complex amplitudes on its four children. vfigparity head…
Exact mathematical passage
P_a=\ket a\bra a
\BVUse in source · 1 passage
Search aliases: BV operatorname
Use in source · Paper_V_Traditional_Final.tex, line 3756
The higher order singular packet theorem removes finitely many trace jumps. A more complete approximation theory should connect packet norms to geometric measure quantities in BV, piecewise Sobolev spaces, and free discontinuity models. It should also determine when a redundant packet dictionary can identify the derivative order of a singularity rather than merely its location.
Exact mathematical passage
\BV
\mathbb CUse in source · 9 passages
Search aliases: C mathbb
Use in source · 03_nonlinear_coordinates.tex, line 207
Let X and Y be Banach spaces over the same field R or C and let the packet projectors on both spaces be bounded. In the complex case, derivatives are complex Fr echet derivatives and the Taylor integral below has a real parameter. Assume that F:X→ Y is m times continuously Fr echet differentiable. The same local statements hold on open domains containing the points and segments in question. For 1≤ r≤ m, the derivative D^rF(x):X^r⟶ Y is a continuous symmetric multilinear map.
Exact mathematical passage
\mathbb C
\mathcal CUse in source · 8 passages
Search aliases: C mathcal
Use in source · continuation_01.tex, line 102
Let G∈ Z[X_1,…,X_d]^f be a second fixed system with the same variables and the same lower root x. Let K_x=ρ_q∘ DG(x), and define C_x= Z^d/(ker J_x+ker K_x). This quotient is finite: it is a quotient of the finite image of J_x, or equivalently of the finite image of K_x.
Exact mathematical passage
\mathcal C_x=\mathbb Z^d/(\ker J_x+\ker K_x).
\mathfrak CUse in source · 3 passages
Search aliases: C mathfrak
Use in source · 03_packets_and_graphs.tex, line 1068
[Source component formula for packet pure seed rank] Assume I is finite, every packet space is nonzero, and the module resolved into packets is surjective along edges. Let C_0 be the set of source strongly connected components of the condensation graph of Γ( G, P). Then psrank_ A, P(X) = ∑_C∈ C_0r_C. [source reference] ExportsResidual transport forests and directed source rank calculations.
Exact mathematical passage
\mathfrak C_0
\operatorname{c\!-\!Ind}Use in source · 2 passages
Search aliases: c - Ind operatorname
Use in source · dyadic_synthesis.tex, line 344
Consequently M=K^× P is an actual open group compact modulo the center, N=K^× J is normal of index two in M, and Σ=Ind_ N^ MΛ, Σ|_P≃ E, π≃c - Ind_ M^GΣ. Restriction to P is irreducible, so Σ is irreducible. Induction in stages proves the last equality for the original Λ, including its fixed central character and noncompact root. This is an actual enlarged inducing representation. It changes neither the candidate nor its original canonical beta-extension data.
Exact mathematical passage
\Sigma=\operatorname{Ind}_{\mathcal N}^{\mathcal M}\Lambda, \qquad \Sigma|_P\simeq E, \qquad \boxed{\pi\simeq\operatorname{c\!-\!Ind}_{\mathcal M}^G\Sigma.}
\operatorname{c\!-\!Ind}_{\mathcal M}^GUse in source · 1 passage
Search aliases: c - Ind_ M^G operatorname mathcal
Use in source · dyadic_synthesis.tex, line 344
Consequently M=K^× P is an actual open group compact modulo the center, N=K^× J is normal of index two in M, and Σ=Ind_ N^ MΛ, Σ|_P≃ E, π≃c - Ind_ M^GΣ. Restriction to P is irreducible, so Σ is irreducible. Induction in stages proves the last equality for the original Λ, including its fixed central character and noncompact root. This is an actual enlarged inducing representation. It changes neither the candidate nor its original canonical beta-extension data.
Exact mathematical passage
\Sigma=\operatorname{Ind}_{\mathcal N}^{\mathcal M}\Lambda, \qquad \Sigma|_P\simeq E, \qquad \boxed{\pi\simeq\operatorname{c\!-\!Ind}_{\mathcal M}^G\Sigma.}
\operatorname{c\!-\!Ind}_J^GUse in source · 1 passage
Search aliases: c - Ind_J^G operatorname
Use in source · dyadic_synthesis.tex, line 357
There is also a recovery from compact data explicitly recording that root. With A_0=Λ(ϖ_K) and X=c - Ind_J^Gη, define (Tf)(g)=A_0^-1f(ϖ_Kg). Then X/(T-1)X≃π. After imposing the fixed central character, N/(F^× J) is cyclic of order N, and the finite average N^-1∑_i=0^N-1T^i selects this same extension. These operators act on complex representation spaces, so their scalar division by N is valid. This is a different recovery from the Block159 projector onto a sector of η. The exact kernel and finit…
Exact mathematical passage
X=\operatorname{c\!-\!Ind}_J^G\eta
c'Use in source · 2 passages
Search aliases: c'
Use in source · Paper_XI_Traditional_Final.tex, line 933
The new cocycle is c'(a,b) = λ_aλ_bλ_a+b^-1c(a,b). The scalar coboundary factor is symmetric in a and b, so it cancels from c'(a,b)c'(b,a)^-1.
Exact mathematical passage
c'(a,b) = \lambda_a\lambda_b\lambda_{a+b}^{-1}c(a,b).
c^{(p)}_{m,j}Use in source · 4 passages
Search aliases: c^(p)_m,j
Use in source · supplement_05.tex, line 49
Specializing to R= Z_p constructs c^(p)_m,j(n). Mapping to Q_p and comparing with the original rational forward substitution gives c^(p)_m,j(n)=C_mp,j(n) in Q_p. Thus every original standard-window coefficient is p-integral, for every row. Integrality is a conclusion, not an additional hypothesis on the coefficients.
Exact mathematical passage
c^{(p)}_{m,j}(n)
c^{(p)}_{m,k}Use in source · 1 passage
Search aliases: c^(p)_m,k
Use in source · supplement_05.tex, line 65
ply the residue map Independently define e^(m,p) over F_p by the unit-diagonal system ∑_q=0^nbinommnmqe^(m,p)_q=δ_n0. Its recurrence and uniqueness are checked. Applying the actual homomorphism π: Z_p→ F_p to the integral row, and then Lucas reduction, proves π(c^(p)_m,j(n))=e^(m,p)_n. For j,k<p, a separate ideal statement gives pmid c^(p)_m,j(n)-c^(p)_m,k(n) inside Z_p. The formal solver also covers m=0. The manuscript uses m≥1. No global finite-difference theorem or period assertion enters thi…
Exact mathematical passage
p\mid c^{(p)}_{m,j}(n)-c^{(p)}_{m,k}(n)\quad\text{inside }\Z_p.
c^*Use in source · 1 passage
Search aliases: c^*
Use in source · 03_supplement_05.tex, line 175
For E(c)=∑_a c_aE_a, finite-sum expansion gives R(E(c)ρ E(c)^*) =(∑_a,bc_aoverlinec_bC_ba)ρ =(c^*Cc)ρ. The formula with a double sum is the literal checked conclusion. The final notation as a quadratic form follows by renaming finite indices. It is not restricted to orthogonal errors or real coefficients.
Exact mathematical passage
\mathcal R\bigl(E(c)\rho E(c)^*\bigr) =\left(\sum_{a,b}c_a\overline{c_b}C_{ba}\right)\rho =(c^*Cc)\rho.
C^0Use in source · 4 passages
Search aliases: C^0
Use in source · 04_correction_transport.tex, line 476
[A fine class killed at the next page] Let C^0=𝕜 x and C^1=𝕜 y, with differential d(x)=y. Give C^0 the filtration F^0C^0=𝕜 x, F^1C^0=0, and give C^1 the filtration F^0C^1=F^1C^1=𝕜 y, F^2C^1=0. The associated graded differential is zero in the relevant positions, so both x and y give nonzero classes on the next page. The next differential sends the class of x to the class of y. The total complex is acyclic. Thus nonzero fine obstruction classes can cancel through a later refinement differential.
Exact mathematical passage
C^0=\bbk x
C^1Use in source · 4 passages
Search aliases: C^1
Use in source · 04_correction_transport.tex, line 476
[A fine class killed at the next page] Let C^0=𝕜 x and C^1=𝕜 y, with differential d(x)=y. Give C^0 the filtration F^0C^0=𝕜 x, F^1C^0=0, and give C^1 the filtration F^0C^1=F^1C^1=𝕜 y, F^2C^1=0. The associated graded differential is zero in the relevant positions, so both x and y give nonzero classes on the next page. The next differential sends the class of x to the class of y. The total complex is acyclic. Thus nonzero fine obstruction classes can cancel through a later refinement differential.
Exact mathematical passage
C^1=\bbk y
C^{1-\alpha}Use in source · 1 passage
Search aliases: C^1-α alpha
Use in source · Paper_VI_Traditional_Final.tex, line 1265
For small t, normCA(tI+A)^-1-B(tI+B)^-1C ≤ 2norm C, because the two positive resolvent factors are contractions. For large t, the resolvent identity gives the bound norm D/t. Splitting the integral at r=fracnorm D2norm C when C and D are nonzero yields normD_α · ≤ fracsin(πα)π ( 2norm C∫_0^r t^α-1 dt + norm D∫_r^∞ t^α-2 dt ) · = K_αnorm C^1-αnorm D^α. The zero cases are immediate.
Exact mathematical passage
\begin{aligned} \norm{D_\alpha} &\le \frac{\sin(\pi\alpha)}{\pi} \left( 2\norm C\int_0^r t^{\alpha-1}\,dt + \norm D\int_r^\infty t^{\alpha-2}\,dt \right)\\ &= K_\alpha\norm C^{1-\alpha}\norm D^\alpha. \end{aligned}
C^{12}Use in source · 1 passage
Search aliases: C^12
Use in source · 72_recovered.tex, line 2424
1 · 0 , one has C^12 =I+8 29 · 18 18 · 11 . Therefore Δ_12·2^a^ rF_n ≡0 pmod 2^(a+3)r. Every dyadic refinement doubles the step and adds one guaranteed binary digit at each Mahler depth.
Exact mathematical passage
C^{12} =I+8 \begin{pmatrix} 29&18\\ 18&11 \end{pmatrix}.
C^2Use in source · 5 passages
Search aliases: C^2
Use in source · supplement_06.tex, line 98
0 · 0) has C^2=0 and χ_C=X^2.
Exact mathematical passage
C^2=0
\mathbb C^2Use in source · 1 passage
Search aliases: C^2 mathbb
Use in source · monograph_examples.tex, line 17
model uses two coordinates to illustrate block inversion. Both the eliminated block and the full operator are invertible, and the plotted values are exact fractions. The example shows why direct compression and resolvent elimination give different results. Normalization changes the map. Let P=|0⟩⟨0| on C^2 and Z=diag(1,-1). The normalized Kraus family sqrt3/4 I, sqrt1/4 Z satisfies ∑_j K_j^*K_j=I and maps P to P. The unnormalized family sqrt3 I, Z satisfies ∑_j E_j^*E_j=4I and maps P to 4P. The …
Exact mathematical passage
\mathbb C^2
c^2Use in source · 3 passages
Search aliases: c^2
Use in source · 30_lifting.tex, line 40
[Polynomial Taylor remainder] Let R be a commutative ring, P∈ R[X_1,…,X_d], a∈ R^d, and c∈ R. There is a polynomial Q∈ R[U_1,…,U_d], with Q(0)=0, such that for every u∈ R^d, P(a+cu)=P(a)+c∑_j=1^d(∂_jP)(a)u_j+c^2Q(u). In particular, when R= Z and a,u,c are integral, subtracting the first two terms on the right from P(a+cu) gives a multiple of c^2.
Exact mathematical passage
\label{eq:polynomial-taylor} P(a+cu)=P(a)+c\sum_{j=1}^d(\partial_jP)(a)u_j+c^2Q(u).
C^{24}Use in source · 1 passage
Search aliases: C^24
Use in source · 72_recovered.tex, line 1749
1 · 0 . The matrix C is invertible modulo twenty four, and C^24-I = 24 3126 · 1932 1932 · 1194 . For every scalar coordinate of the state sequence C^nv_0, Δ_24^ ra_n ≡0 pmod 24^r. One integral jump therefore gives simultaneous congruence ladders at the primes two and three.
Exact mathematical passage
C^{24}-I = 24 \begin{pmatrix} 3126&1932\\ 1932&1194 \end{pmatrix}.
C^3Use in source · 1 passage
Search aliases: C^3
Use in source · 72_recovered.tex, line 2810
[t] centering includegraphics[width=.95linewidth,height=3.15in,keepaspectratio]../../figures/reconstructed/recurrence_packets.pdf captionNew reconstruction. Fibonacci residues F_3k+j, with F_0=0 and F_1=1. Each packet evolves by the same jump matrix C^3 and has a different initial address.
Exact mathematical passage
C^3
C^4Use in source · 1 passage
Search aliases: C^4
Use in source · Paper_VI_Traditional_Final.tex, line 4242
[Second order Poisson solution error] Suppose u∈ C^4[0,1] solves the Dirichlet problem above. Let f^h_i=f(x_i) and let u^h solve L_hu^h=f^h. Then max_iabsu^h_i-u(x_i) ≤ frach^296 normu^(4)_∞.
Exact mathematical passage
u\in C^4[0,1]
\mathbb C^dUse in source · 1 passage
Search aliases: C^d mathbb
Use in source · Paper_VI_Traditional_Final.tex, line 1655
For clarity, the finite-dimensional holomorphic calculus used below has the following canonical definition. For A acting on C^d, its generalized eigenspaces E_μ=ker(A-μ I)^d form a direct-sum decomposition over the eigenvalues. Define the unique linear operator F(A) by .F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent of a basis and of increasing the truncation index: the extra nilpotent powers vanish. It depends only on the finite Taylor jets of F at the spectrum. I…
Exact mathematical passage
\mathbb C^d
C^HUse in source · 3 passages
Search aliases: C^H
Use in source · 72_recovered.tex, line 1677
_0,w∈ Z_p^d, L≥1, and H,j,n∈ N with H>0. The value at Hx+j for x∈ Z_p means the matrix-binomial interpolation. With p=3, C=4, L=1, w=1 and v_0=1/3, one has Δ_1a_0=1, which is not in 3 Z_3. The printed claim without these restrictions is therefore not a certificate as written. The full original proof follows under the corrected hypotheses. Assume C^H=I+p^LB with B∈ M_d( Z_p). For every address j and every x∈ Z_p, a_Hx+j = ∑_r≥0 p^Lrbinom xr w^TC^jB^rv_0. The series converges on Z_p, and Δ_H^ra_n …
Exact mathematical passage
C^H=I+p^LB
C^{H_a}Use in source · 1 passage
Search aliases: C^H_a
Use in source · 72_recovered.tex, line 2389
dentEditorial hypothesis correction. Require L≥1, C,B∈ M_d( Z_p), v_0,w∈ Z_p^d, H>0, and a,j,n,r∈ N. These are the hypotheses of the two corrected matrix statements used by the retained proof. Without integral observation vectors or positive L, the preceding counterexamples invalidate the claimed precision. Assume C^H=I+p^LB. Put H_a=Hp^a. Then C^H_a=I+p^L+aB_a for an integral matrix B_a. Every refined packet b_a,j(x)=a_H_ax+j has Mahler coefficients satisfying v_p(c_a,j,r)≥(L+a)r. Equivalently,…
Exact mathematical passage
C^{H_a}=I+p^{L+a}B_a
C^{Hx+j}Use in source · 1 passage
Search aliases: C^Hx+j
Use in source · 72_recovered.tex, line 1702
Use C^Hx+j=C^j(I+p^LB)^x and the matrix binomial series. One difference in the coarse variable multiplies the series by p^LB.
Exact mathematical passage
C^{Hx+j}=C^j(I+p^LB)^x
C^iUse in source · 2 passages
Search aliases: C^i
Use in source · Paper_X_Block27_Amended.tex, line 1998
Modular Gaussian elimination computes the ranks of the differentials. For C^i-1xrightarrowd_i-1C^ixrightarrowd_iC^i+1, the finite dimension is evaluated as dim H^i = dim C^i-rank d_i-rank d_i-1. Packet subcomplexes are computed separately and checked against the total.
Exact mathematical passage
C^{i-1}\xrightarrow{d_{i-1}}C^i\xrightarrow{d_i}C^{i+1},
C^{i+1}Use in source · 1 passage
Search aliases: C^i+1
Use in source · Paper_X_Block27_Amended.tex, line 1998
Modular Gaussian elimination computes the ranks of the differentials. For C^i-1xrightarrowd_i-1C^ixrightarrowd_iC^i+1, the finite dimension is evaluated as dim H^i = dim C^i-rank d_i-rank d_i-1. Packet subcomplexes are computed separately and checked against the total.
Exact mathematical passage
C^{i-1}\xrightarrow{d_{i-1}}C^i\xrightarrow{d_i}C^{i+1},
C^{i-1}Use in source · 1 passage
Search aliases: C^i-1
Use in source · Paper_X_Block27_Amended.tex, line 1998
Modular Gaussian elimination computes the ranks of the differentials. For C^i-1xrightarrowd_i-1C^ixrightarrowd_iC^i+1, the finite dimension is evaluated as dim H^i = dim C^i-rank d_i-rank d_i-1. Packet subcomplexes are computed separately and checked against the total.
Exact mathematical passage
C^{i-1}\xrightarrow{d_{i-1}}C^i\xrightarrow{d_i}C^{i+1},
C^jUse in source · 6 passages
Search aliases: C^j
Use in source · 10_foundations.tex, line 412
[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalization hypothesis is required.
Exact mathematical passage
b_k=C^j(C^N)^kx_0,\qquad b_{k+1}=C^Nb_k.
c^mUse in source · 1 passage
Search aliases: c^m
Use in source · Paper_XI_Traditional_Final.tex, line 1255
Let C=⟨ c⟩ have order m, and let T be a lift of the projective generator. Since c^m=1, the operator T^m is scalar. Choose λ∈ C^× with λ^mT^m=I. Replacing T by λ T gives an honest cyclic representation.
Exact mathematical passage
c^m=1
\C^{m\times m}Use in source · 1 passage
Search aliases: C^m× m C times mathbb times
Use in source · 02_supplement_04.tex, line 73
Kronecker application. For A∈ C^m× m, B∈ C^n× n and X∈ C^n× m, Lean proves (A⊗ B)vec(X)=vec(BXA^T). The formal vectorization is explicitly column-major: its pair coordinate (j,i) stores X_ij. The right factor is the ordinary transpose, not the conjugate transpose. This distinction is essential over C.
Exact mathematical passage
A\in\C^{m\times m}
\C^{m\times t}Use in source · 1 passage
Search aliases: C^m× t C times mathbb times
Use in source · 02_supplement_06.tex, line 6
Compact factors from actual column spaces. textitModules CompactRangeFactors and SmallCoreTransfer Let L∈ C^m× t, R∈ C^n× t, A=LR^*. Use the ordinary Euclidean inner products on the coordinate spaces. Define S_L=range(L), r_L=dim S_L, S_R=range(R), r_R=dim S_R. The implementation constructs an orthonormal basis of each actual range subspace. Let Q_L and Q_R be the matrices of the corresponding isometric inclusions. Express the range-restricted input maps in these bases to obtain R_L,R_R.
Exact mathematical passage
L\in\C^{m\times t},\qquad R\in\C^{n\times t},\qquad A=LR^*.
C^nUse in source · 11 passages
Search aliases: C^n
Use in source · 03_nonlinear_coordinates.tex, line 1150
The class of (e,0) vanishes exactly when there are e'∈ E^n-1 and c∈ C^n with (e,0)=d_Cone(e',c). This means e=d_Ee'+Dc, d_Cc=0. Equivalently, [e]=H^n(D)[c]. The degree zero statement follows directly from the definition of the cone differential.
Exact mathematical passage
c\in C^n
\C^nUse in source · 2 passages
Search aliases: C^n C mathbb
Use in source · 05_worked_models.tex, line 372
Two adjacent vertex vectors generate the adjacency module of every simple cycle. Over C, a Fourier quotient built from a primitive root of unity supplies a matching scalar obstruction of dimension two. Hence, for every n≥3, two is the least finite number of adjacency seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any f…
Exact mathematical passage
{ 2=\min\left\{r\in\N: \exists s_1,\ldots,s_r,\quad \C^n=\Span_{\C}\{p(A_{C_n})s_j:p\in\C[t],\ 1\le j\le r\} \right\}. }
C^{n+2}Use in source · 1 passage
Search aliases: C^n+2
Use in source · supplement_06.tex, line 121
An independent Lucas sequence. Define L_0=2, L_1=1, L_n+2=L_n+L_n+1. Direct finite matrix arithmetic proves C^2=C+I. Multiplication by C^n gives C^n+2=C^n+C^n+1. Taking traces shows that tr(C^n) satisfies the same recurrence and initial values as L_n. Induction yields tr(C^N)=L_N. The implementation separately proves det C=-1, det(C^N)=(-1)^N, L_N=2F_N+1-F_N. No Binet roots or diagonalizability argument is needed.
Exact mathematical passage
C^{n+2}=C^n+C^{n+1}
C^{Nk+j}Use in source · 1 passage
Search aliases: C^Nk+j
Use in source · supplement_06.tex, line 31
pageheading1. Actual matrix orbits and arithmetic jumpsTheorem 9.1; manuscript page 22 Let R be a commutative ring, I a finite coordinate type and d=|I|. For C∈ M_I(R) and v_0∈ R^I, define orbit(C,v_0,n)=C^nv_0, u_k=packet(C,v_0,N,j,k)=C^Nk+jv_0. The implementation uses actual matrix powers and matrix--vector multiplication. A separately supplied sequence v with v_n+1=Cv_n is proved by induction to equal this power orbit. Thus the source-sequence formulation is connected to the construction.
Exact mathematical passage
\operatorname{orbit}(C,v_0,n)=C^nv_0, \qquad u_k=\operatorname{packet}(C,v_0,N,j,k)=C^{Nk+j}v_0.
\C^{n\times m}Use in source · 1 passage
Search aliases: C^n× m C times mathbb times
Use in source · 02_supplement_04.tex, line 73
Kronecker application. For A∈ C^m× m, B∈ C^n× n and X∈ C^n× m, Lean proves (A⊗ B)vec(X)=vec(BXA^T). The formal vectorization is explicitly column-major: its pair coordinate (j,i) stores X_ij. The right factor is the ordinary transpose, not the conjugate transpose. This distinction is essential over C.
Exact mathematical passage
X\in\C^{n\times m}
\C^{n\times n}Use in source · 1 passage
Search aliases: C^n× n C times mathbb times
Use in source · 02_supplement_04.tex, line 73
Kronecker application. For A∈ C^m× m, B∈ C^n× n and X∈ C^n× m, Lean proves (A⊗ B)vec(X)=vec(BXA^T). The formal vectorization is explicitly column-major: its pair coordinate (j,i) stores X_ij. The right factor is the ordinary transpose, not the conjugate transpose. This distinction is essential over C.
Exact mathematical passage
B\in\C^{n\times n}
\mathbb C^{n\times r_j}Use in source · 1 passage
Search aliases: C^n× r_j mathbb times
Use in source · 02_supplement_03.tex, line 6
Low-rank factors through a product chain. textitSource item 5.2 Let A_j=D_j+U_jV_j^*, U_j,V_j∈ C^n× r_j, with all D_j diagonal in one packet layout. Set P_m=A_m⋯ A_1, D^(m)=D_m⋯ D_1. The formalization constructs thin matrices L_m,R_m such that P_m=D^(m)+L_mR_m^*. Their column index type is a recursively constructed disjoint sum. Its cardinality is proved to equal ∑_j=1^m r_j.
Exact mathematical passage
A_j=D_j+U_jV_j^*,\qquad U_j,V_j\in\mathbb C^{n\times r_j},
\C^{n\times t}Use in source · 1 passage
Search aliases: C^n× t C times mathbb times
Use in source · 02_supplement_06.tex, line 6
Compact factors from actual column spaces. textitModules CompactRangeFactors and SmallCoreTransfer Let L∈ C^m× t, R∈ C^n× t, A=LR^*. Use the ordinary Euclidean inner products on the coordinate spaces. Define S_L=range(L), r_L=dim S_L, S_R=range(R), r_R=dim S_R. The implementation constructs an orthonormal basis of each actual range subspace. Let Q_L and Q_R be the matrices of the corresponding isometric inclusions. Express the range-restricted input maps in these bases to obtain R_L,R_R.
Exact mathematical passage
L\in\C^{m\times t},\qquad R\in\C^{n\times t},\qquad A=LR^*.
C^{p+q}Use in source · 1 passage
Search aliases: C^p+q
Use in source · 04_correction_transport.tex, line 435
[Spectral sequence of a finite filtered complex] Let C^bullet be a cochain complex with a finite decreasing filtration by subcomplexes C^bullet=F^0C^bullet⊇ F^1C^bullet⊇⋯⊇ F^N+1C^bullet=0. There is a natural spectral sequence with E_0^p,q=gr^pC^p+q, E_1^p,q=H^p+q(gr^pC^bullet), which converges to the associated graded object of H^p+q(C^bullet) with its induced filtration. A filtered cochain map induces a morphism of spectral sequences.
Exact mathematical passage
E_0^{p,q}=\gr^pC^{p+q}, \qquad E_1^{p,q}=H^{p+q}(\gr^pC^\bullet),
C^rUse in source · 3 passages
Search aliases: C^r
Use in source · 04_correction_transport.tex, line 115
[Local Lyapunov--Schmidt reduction] Let U⊆ X be an open neighborhood of zero and let F: U→ Y be a C^r map between Banach spaces with r≥1, F(0)=0, and L=DF(0). Assume there are closed decompositions X=K⊕ X_1, Y=R⊕ O, where K=Ker L, R=im L, and L_1=L|_X_1:X_1⟶ R is a Banach space isomorphism. Let π_R and π_O be the two projections on Y. Then there are neighborhoods U_K⊆ K and U_1⊆ X_1 of zero and a unique C^r map ψ:U_K⟶ U_1 with ψ(0)=0 such that π_RF(k+ψ(k))=0 for every k∈ U_K. After the neighborh…
Exact mathematical passage
C^r
\C^{r_L\times r_R}Use in source · 1 passage
Search aliases: C^r_L× r_R C times mathbb times
Use in source · 02_supplement_06.tex, line 27
Define the small core by actual multiplication, G=R_LR_R^*∈ C^r_L× r_R. Finite matrix associativity and adjoint identities give A=Q_LGQ_R^*, rank G=rank A. The rank identity follows from both multiplication inequalities and the left inverse obtained by compression. It is not inferred from agreeing dimensions.
Exact mathematical passage
G=R_LR_R^*\in\C^{r_L\times r_R}.
C^TUse in source · 1 passage
Search aliases: C^T
Use in source · Paper_VI_Traditional_Final.tex, line 5644
[Coarse Feynman--Kac transfer] Assume CQ=overline QC, and let J=C^T lift a coarse function to the corresponding block constant fine function. Suppose the potential is block constant, so VJ=Joverline V. Then e^t(Q^T-V)J = J e^t(overline Q^T-overline V) for every t≥0.
Exact mathematical passage
J=C^T
\mathbb C^\timesUse in source · 4 passages
Search aliases: C^× mathbb times
Use in source · dyadic_synthesis.tex, line 123
For the final totally ramified field, J/J^1= F_2^×=1, so J=J^1 and the canonical beta extension is η. Extend to K^× J, and choose the noncompact scalar to match ω(t). The needed N-th root exists in C^×. The compact type stays fixed. All choices precede H5. See pathcentral/DYADIC_FULL_CHARACTER167.md for the complete proof.
Exact mathematical passage
\mathbb C^\times
c^\leftarrowUse in source · 5 passages
Search aliases: c^← leftarrow
Use in source · 01_finite_coordinates.tex, line 249
For a nonempty composable word w=(e_1,…,e_n), define its reverse coefficient product c^←(w)=c(e_n)c(e_n-1)⋯ c(e_1). The reversal is essential over a noncommutative semiring.
Exact mathematical passage
c^\leftarrow(w)=c(e_n)c(e_{n-1})\cdots c(e_1).
C_0Use in source · 14 passages
Search aliases: C_0
Use in source · 02_connections_obstructions.tex, line 665
[A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly when vu=w. If the three coordinate systems are literally the same and canonical transport is used, then u=v=w=1 and the obstruction vanishes. Choosing …
Exact mathematical passage
C_0=C_1=C_2=\bbk
\mathfrak C_0Use in source · 3 passages
Search aliases: C_0 mathfrak
Use in source · 03_packets_and_graphs.tex, line 1068
[Source component formula for packet pure seed rank] Assume I is finite, every packet space is nonzero, and the module resolved into packets is surjective along edges. Let C_0 be the set of source strongly connected components of the condensation graph of Γ( G, P). Then psrank_ A, P(X) = ∑_C∈ C_0r_C. [source reference] ExportsResidual transport forests and directed source rank calculations.
Exact mathematical passage
\mathfrak C_0
C_{0,L}Use in source · 3 passages
Search aliases: C_0,L
Use in source · 05_defects.tex, line 228
[Multilevel defect decomposition] The total defect from level L to level zero is C_0,LT_L-T_0C_0,L = ∑_ℓ=0^L-1 C_0,ℓΔ_ℓ C_ℓ+1,L. For bounded maps, C_0,LT_L-T_0C_0,L ≤ ∑_ℓ=0^L-1 C_0,ℓ Δ_ℓ C_ℓ+1,L .
Exact mathematical passage
C_{0,L}T_L-T_0C_{0,L} = \sum_{\ell=0}^{L-1} C_{0,\ell}\Delta_\ell C_{\ell+1,L}.
C_{0,m}Use in source · 1 passage
Search aliases: C_0,m
Use in source · 03_nonlinear_coordinates.tex, line 860
y[Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.
Exact mathematical passage
\|C_{0,m}x_m-x_0\| \le \sum_{\ell=0}^{m-1} \|C_{0,\ell}\| \frac{\varepsilon_\ell}{1-q_\ell}.
C_{0,\ell}Use in source · 6 passages
Search aliases: C_0,ℓ ell
Use in source · 05_defects.tex, line 228
[Multilevel defect decomposition] The total defect from level L to level zero is C_0,LT_L-T_0C_0,L = ∑_ℓ=0^L-1 C_0,ℓΔ_ℓ C_ℓ+1,L. For bounded maps, C_0,LT_L-T_0C_0,L ≤ ∑_ℓ=0^L-1 C_0,ℓ Δ_ℓ C_ℓ+1,L .
Exact mathematical passage
C_{0,L}T_L-T_0C_{0,L} = \sum_{\ell=0}^{L-1} C_{0,\ell}\Delta_\ell C_{\ell+1,L}.
c_{00}Use in source · 1 passage
Search aliases: c_00
Use in source · 06_models.tex, line 5
Let x=(x_n)_n∈ Z∈ c_00( Z;R) and let the seed be δ_0 under the bilateral shifts (S^rδ_0)_n=δ_n,r. This makes the algebraic span and every displayed sum finite. The N packets are x_j[m]=x[Nm+j]. Synthesis interleaves these sequences. A linear filter with a finite impulse response that is invariant under shifts becomes an N× N matrix of transfer blocks. Refining from N to M with Nmid M splits each coarse phase into its descendants without changing the reconstructed signal. The same formulas extend…
Exact mathematical passage
x=(x_n)_{n\in\Z}\in c_{00}(\Z;R)
C_{0j}Use in source · 1 passage
Search aliases: C_0j
Use in source · visual_atlas.tex, line 164
vfig[.98]rectangular_paths platecaptionThe lower board expands one final row into all six choices of intermediate indices for each input packet. The entries are C_0jB_jkA_ki, with k∈{0,1,2} and j∈{0,1}. Zero terms remain visible.
Exact mathematical passage
C_{0j}B_{jk}A_{ki}
C_1Use in source · 18 passages
Search aliases: C_1
Use in source · 02_connections_obstructions.tex, line 665
[A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly when vu=w. If the three coordinate systems are literally the same and canonical transport is used, then u=v=w=1 and the obstruction vanishes. Choosing …
Exact mathematical passage
C_0=C_1=C_2=\bbk
c_1Use in source · 5 passages
Search aliases: c_1
Use in source · 20_arithmetic.tex, line 656
[Exact four-packet displacement law] Set (c_0,c_1,c_2,c_3)=(1,1,4,2), (d_0,d_1,d_2,d_3)=(1,1,5,3). For 0≤ j<4, n≥0, and every positive integer T, C_j(n+T)-C_j(n) · ≠0, · v_2(C_j(n+T)-C_j(n)) · =v_2(T)+c_j,
Exact mathematical passage
(c_0,c_1,c_2,c_3)=(1,1,4,2),\qquad (d_0,d_1,d_2,d_3)=(1,1,5,3).
C_2Use in source · 38 passages
Search aliases: C_2
Use in source · 02_connections_obstructions.tex, line 665
[A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly when vu=w. If the three coordinate systems are literally the same and canonical transport is used, then u=v=w=1 and the obstruction vanishes. Choosing …
Exact mathematical passage
C_0=C_1=C_2=\bbk
c_2Use in source · 4 passages
Search aliases: c_2
Use in source · 20_arithmetic.tex, line 656
[Exact four-packet displacement law] Set (c_0,c_1,c_2,c_3)=(1,1,4,2), (d_0,d_1,d_2,d_3)=(1,1,5,3). For 0≤ j<4, n≥0, and every positive integer T, C_j(n+T)-C_j(n) · ≠0, · v_2(C_j(n+T)-C_j(n)) · =v_2(T)+c_j,
Exact mathematical passage
(c_0,c_1,c_2,c_3)=(1,1,4,2),\qquad (d_0,d_1,d_2,d_3)=(1,1,5,3).
C_{2,0}Use in source · 5 passages
Search aliases: C_2,0
Use in source · atlas-pages.tex, line 244
vpage14Three similar reciprocals differNORMALIZED SECTIONSChapter IV: Definition 9.1, Theorem 9.2, Example 9.11. The signed denominator and the normalized unsigned sections produce different reciprocal families. In order two, D_2^-1=sec z, C_2,0=sech z, C_2,1=z/sinh z. Their first block-EGF coefficients make the difference visible. vfigdifferent_reciprocals headRemove the forced zero before taking a reciprocal The section E_N,j begins with z^j/j!. Define E_N,j(z)=fracz^jj!Φ_N,j(z^N), Φ_N,j(0)=1.…
Exact mathematical passage
D_2^{-1}=\sec z,\qquad C_{2,0}=\sech z,\qquad C_{2,1}=z/\sinh z.
C_{2,1}Use in source · 4 passages
Search aliases: C_2,1
Use in source · atlas-pages.tex, line 244
vpage14Three similar reciprocals differNORMALIZED SECTIONSChapter IV: Definition 9.1, Theorem 9.2, Example 9.11. The signed denominator and the normalized unsigned sections produce different reciprocal families. In order two, D_2^-1=sec z, C_2,0=sech z, C_2,1=z/sinh z. Their first block-EGF coefficients make the difference visible. vfigdifferent_reciprocals headRemove the forced zero before taking a reciprocal The section E_N,j begins with z^j/j!. Define E_N,j(z)=fracz^jj!Φ_N,j(z^N), Φ_N,j(0)=1.…
Exact mathematical passage
D_2^{-1}=\sec z,\qquad C_{2,0}=\sech z,\qquad C_{2,1}=z/\sinh z.
C_3Use in source · 4 passages
Search aliases: C_3
Use in source · 04_correction_transport.tex, line 982
Taylor's theorem in the Banach space of bounded operators gives P_α(t+h) = P_α(t)+hdot P_α(t)+E_α(t,h), E_α(t,h) ≤ C_1h^2, uniformly in t and α. Hence S( P(t+h), P(t)) = I+hK(t)+E(t,h), E(t,h) ≤ C_2h^2. Since K(t)^*=-K(t), S^*S=I+E_2(t,h), E_2(t,h) ≤ C_3h^2. For small h, the spectrum of S^*S lies in a fixed compact subinterval of (0,∞). The functional calculus for z^-1/2 gives (S^*S)^-1/2=I+E_3(t,h), E_3(t,h) ≤ C_4h^2. Multiplying the last two expansions proves the estimate.
Exact mathematical passage
S^*S=I+E_2(t,h), \qquad \|E_2(t,h)\|\le C_3h^2.
c_3Use in source · 2 passages
Search aliases: c_3
Use in source · 20_arithmetic.tex, line 656
[Exact four-packet displacement law] Set (c_0,c_1,c_2,c_3)=(1,1,4,2), (d_0,d_1,d_2,d_3)=(1,1,5,3). For 0≤ j<4, n≥0, and every positive integer T, C_j(n+T)-C_j(n) · ≠0, · v_2(C_j(n+T)-C_j(n)) · =v_2(T)+c_j,
Exact mathematical passage
(c_0,c_1,c_2,c_3)=(1,1,4,2),\qquad (d_0,d_1,d_2,d_3)=(1,1,5,3).
C_{3,0}Use in source · 1 passage
Search aliases: C_3,0
Use in source · worked_atlas.tex, line 46
At N=p=3 and j=0, put A(n)=(-1)^nC_3,0(n). The first three rows are AtlasImageodd_prime_sign The ordinary first difference at row zero is -1-1=-2. It is a 3-adic unit, so an asserted lower bound of one for that difference would be false. The twisted first difference is zero, whose mathematical valuation is +∞. At the next row, the twisted difference is 19-1=18, of valuation two.
Exact mathematical passage
A(n)=(-1)^nC_{3,0}(n)
C_4Use in source · 1 passage
Search aliases: C_4
Use in source · 04_correction_transport.tex, line 988
P_α(t)+E_α(t,h), E_α(t,h) ≤ C_1h^2, uniformly in t and α. Hence S( P(t+h), P(t)) = I+hK(t)+E(t,h), E(t,h) ≤ C_2h^2. Since K(t)^*=-K(t), S^*S=I+E_2(t,h), E_2(t,h) ≤ C_3h^2. For small h, the spectrum of S^*S lies in a fixed compact subinterval of (0,∞). The functional calculus for z^-1/2 gives (S^*S)^-1/2=I+E_3(t,h), E_3(t,h) ≤ C_4h^2. Multiplying the last two expansions proves the estimate.
Exact mathematical passage
(S^*S)^{-1/2}=I+E_3(t,h), \qquad \|E_3(t,h)\|\le C_4h^2.
C_{4,0}Use in source · 5 passages
Search aliases: C_4,0
Use in source · 20_arithmetic.tex, line 870
Return now to the classical signed Euler numbers secht=frac1cosh t =∑_n≥0E_2nfract^2n(2n)!. Thus E_2n=E_2n^(2,0) in the convention [source reference]. In particular E_0=1, E_2=-1, and E_4=5. The following construction concerns these N=2 coefficients, not C_4,0.
Exact mathematical passage
C_{4,0}
C_{4,1}Use in source · 4 passages
Search aliases: C_4,1
Use in source · supplement_11.tex, line 137
An exact arithmetic check, not an optimality claim. The independent rational recurrence gives C_4,1(1)=-1/5 and C_4,1(2)=121/45. Their difference is 26/9, of 2-adic valuation one. This agrees with the new bound at n=k=u=1. The source also contains universal instantiations of the degree-indexed theorem at all four dyadic addresses and at an odd singular address.
Exact mathematical passage
C_{4,1}(1)=-1/5
C_{4,j}Use in source · 6 passages
Search aliases: C_4,j
Use in source · 20_arithmetic.tex, line 454
For an odd prime, 2mid p-1, so the option m=2 introduces no further case. The sole remaining branch is p=2,m=2,N=4. The all-row theorem proved next gives 2^kmidΔ_1^kC_4,j(n) at every row and address. After multiplying by j!, Lemma~[source reference] gives packet period 2^r-1 and degree displacement 4·2^r-1=2^r+1=qp^r. It also gives integrality. Hence the complete statement follows. The formal degree conversion is leanKomorebiPaperIX.NishibuchiCompletion.original_degree_shift. The all-parameter d…
Exact mathematical passage
2^k\mid\Delta_1^kC_{4,j}(n)
C_6Use in source · 1 passage
Search aliases: C_6
Use in source · visual_atlas.tex, line 392
vpage16Closing a path changes the seed obstructionGRAPH MODEL / A CYCLE WITH SIX VERTICESMain text: Worked graph model. Proof of the minimum seed count for complex cycles For the adjacency matrix A of C_6, the distinct eigenvalues are 2,1,-1,-2, with multiplicities 1,2,2,1. A single vertex seed spans only four directions. Two adjacent vertex seeds span all six.
Exact mathematical passage
C_6
C_{6,0}Use in source · 1 passage
Search aliases: C_6,0
Use in source · supplement_07.tex, line 142
No automatic extra power. The original rational triangular coefficients satisfy C_6,0(1)=-1, C_6,0(2)=923, v_3(C_6,0(2)-C_6,0(1))=v_3(924)=1. A period modulo 3 therefore cannot simply be reported as a period modulo 9. This is consistent with the global theorem's proposed first-difference bound. It is not a counterexample to Theorem 7.5.
Exact mathematical passage
C_{6,0}(1)=-1,\quad C_{6,0}(2)=923,\quad v_3\bigl(C_{6,0}(2)-C_{6,0}(1)\bigr)=v_3(924)=1.
C_{6,2}Use in source · 2 passages
Search aliases: C_6,2
Use in source · supplement_08.tex, line 184
pageheading5. Examples, verification and the remaining gapThe coverage count is unchanged; the unresolved theorem now has a stronger proved part The original rational recurrence gives C_6,2(1)=-frac128, C_6,2(2)=frac425364, C_6,2(3)=-frac20397592660. Consequently, the new source proves Δ_1^2 C_6,2(1)=-frac9499591235, v_3 (Δ_1^2 C_6,2(1))=2. This attains the bound min(2,2·1)=2. A claim of a third power is rejected. The separate cubic example proves that the normalized differential defect is not a…
Exact mathematical passage
C_{6,2}(1)=-\frac1{28},\quad C_{6,2}(2)=\frac{425}{364},\quad C_{6,2}(3)=-\frac{2039759}{2660}.
C_{6,3}Use in source · 12 passages
Search aliases: C_6,3
Use in source · 20_arithmetic.tex, line 357
[Why the factorial loss is necessary] Take p=3, m=2, N=6, and j=3. The recurrence gives C_6,3(1)=-frac184, C_6,3(2)=frac7035460, C_6,3(2)-C_6,3(1)=frac64455. The last difference has 3-adic valuation zero. Here h=1, so at n=1 the order-one bound without factorial loss would require valuation at least one. The corrected normalized bound is sharp, since v_3(C_6,3(2)-C_6,3(1))=0=1-v_3(3!). Multiplication by 3! gives U_6,3(2)-U_6,3(1)=frac384455, v_3(U_6,3(2)-U_6,3(1))=1, consistent with the unnormal…
Exact mathematical passage
C_{6,3}(1)=-\frac1{84},\qquad C_{6,3}(2)=\frac{703}{5460},\qquad C_{6,3}(2)-C_{6,3}(1)=\frac{64}{455}.
C_{6,4}Use in source · 1 passage
Search aliases: C_6,4
Use in source · supplement_10.tex, line 202
w For (p,m,j)=(3,2,3), the source derives C_6,3(1)=-frac184, C_6,3(2)=frac7035460, C_6,3(3)=-frac2231997980. Consequently, Δ_1^2(3!C_6,3)(1)=-frac14647688645, v_3 (-frac14647688645)=2. This attains min(2,2·1)=2. At the regular address j=4 beyond the singular boundary, the source likewise proves Δ_1^2(4!C_6,4)(1)=-frac1027455122377375, and applies the universal exponent-two bound to this actual coefficient function.
Exact mathematical passage
\Delta_1^2\bigl(4!C_{6,4}\bigr)(1)=-\frac{102745512}{2377375},
C_{a+1}Use in source · 2 passages
Search aliases: C_a+1
Use in source · 05_defects.tex, line 221
Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.
Exact mathematical passage
C_{a,b}=C_aC_{a+1}\cdots C_{b-1},
C_{a,a}Use in source · 2 passages
Search aliases: C_a,a
Use in source · 05_defects.tex, line 224
Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.
Exact mathematical passage
C_{a,a}=I
C_{a,b}Use in source · 2 passages
Search aliases: C_a,b
Use in source · 05_defects.tex, line 221
Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.
Exact mathematical passage
C_{a,b}=C_aC_{a+1}\cdots C_{b-1},
c_{a,j,r}Use in source · 1 passage
Search aliases: c_a,j,r
Use in source · 72_recovered.tex, line 2397
ypotheses of the two corrected matrix statements used by the retained proof. Without integral observation vectors or positive L, the preceding counterexamples invalidate the claimed precision. Assume C^H=I+p^LB. Put H_a=Hp^a. Then C^H_a=I+p^L+aB_a for an integral matrix B_a. Every refined packet b_a,j(x)=a_H_ax+j has Mahler coefficients satisfying v_p(c_a,j,r)≥(L+a)r. Equivalently, Δ_H_a^ ra_n ≡0 pmod p^(L+a)r.
Exact mathematical passage
\boxed{\vp(c_{a,j,r})\geq(L+a)r.}
C_{ab}Use in source · 3 passages
Search aliases: C_ab
Use in source · 03_supplement_05.tex, line 14
Necessary and sufficient conditions. Under the stated normalization, the following equivalence is proved: ∃ R of the above finite Kraus form, [-2pt] R N_E(ρ)=ρ for every ρsucceq0, Trρ=1, Pρ P=ρ Longleftrightarrow ∃ Csucceq0, PE_a^*E_bP=C_abP for all a,b. The statement applies throughout the code space. The formalization separately proves equivalence with correction of every complex matrix supported on the code. It also proves preservation with arbitrary finite reference systems, as detailed in t…
Exact mathematical passage
\boxed{\begin{gathered} \exists\,\mathcal R\text{ of the above finite Kraus form,}\\[-2pt] \mathcal R\mathcal N_E(\rho)=\rho \quad\text{for every }\rho\succeq0,\ \Tr\rho=1,\ P\rho P=\rho \end{gathered}} \quad\Longleftrightarrow\quad \boxed{\begin{gathered} \exists C\succeq0,\\ PE_a^*E_bP=C_{ab}P\quad\text{for all }a,b. \end{gathered}}
c_{ab}Use in source · 8 passages
Search aliases: c_ab
Use in source · monograph_atlas.tex, line 598
vpage32Null Gram directions are removed before recoveryA TOTAL CHANNEL FROM SINGULAR ERROR DATAChapter VII: Theorem II.8.3, Theorem II.8.4. Let HH= C^3, with nonzero code projector P=diag(1,1,0). On the code use the error family E_1=(3/5)P, E_2=(4/5)P, and E_3=0. Its Knill--Laflamme Gram matrix is c=frac125mat9 · 12 · 0 12 · 16 · 0 0 · 0 · 0, PE_a^*E_bP=c_abP. The errors are redundant rather than three distinct syndromes. A real unitary change of error coordinates gives W=mat3/5 · 4/5 · 0
Exact mathematical passage
c=\frac1{25}\mat{9&12&0\\12&16&0\\0&0&0}, \qquad PE_a^*E_bP=c_{ab}P.
C_{ai}Use in source · 1 passage
Search aliases: C_ai
Use in source · Paper_VI_Traditional_Final.tex, line 5349
The product topology used here differs from the bounded inverse-limit norm of Part II. No uniform boundedness of coordinate generators is required by the coordinatewise statement. FloatBarrier Reversible coarse dynamics. Let Q be a finite Markov generator acting on column probability vectors, let (B_a) be a partition into nonempty blocks, and let C_ai= 1_i∈ B_a. Strong lumpability means CQ=overline Q C for a coarse Markov generator. For observables, the backward generator is Q^T. Weighted L^2 in…
Exact mathematical passage
C_{ai}=\one_{i\in B_a}
C_{b-1}Use in source · 2 passages
Search aliases: C_b-1
Use in source · 05_defects.tex, line 221
Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.
Exact mathematical passage
C_{a,b}=C_aC_{a+1}\cdots C_{b-1},
C_{ba}Use in source · 2 passages
Search aliases: C_ba
Use in source · 03_supplement_05.tex, line 169
Cross terms for the original errors. For arbitrary positive Gram data, the same recovery satisfies R(E_aρ E_b^*)=C_baρ (Pρ P=ρ). The index order ba is intentional: the defining Gram relation is PE_a^*E_bP=C_abP. The proof first derives scalar corrected Kraus composites for the constructed recovery and then uses its trace-preserving completeness to identify the original Gram entries.
Exact mathematical passage
\boxed{\mathcal R(E_a\rho E_b^*)=C_{ba}\rho\qquad(P\rho P=\rho).}
c_eUse in source · 31 passages
Search aliases: c_e
Use in source · pages.tex, line 342
vpage20Spend error where it is cheapestCONTINUOUS AND CLIPPED PRECISIONChapter VIII: Theorem 8.4, Theorem 8.5. Take sensitivities s=(1,2,4), cost coefficients c=(1,1,2), and budget τ=1. Minimizing ∑_ec_elog(1/ϵ_e) subject to ∑_es_eϵ_e=1 gives ϵ_e=fracc_e4s_e, ϵ=(1/4,1/8,1/8). vfigprecision headThe budget is weighted The three contributions to the error allowance are (1/4,1/4,1/2). Equal cost coefficients give equal weighted contributions, not equal tolerances. The third edge has twice the cost c…
Exact mathematical passage
\sum_ec_e\log(1/\epsilon_e)
c_fUse in source · 2 passages
Search aliases: c_f
Use in source · 01_manuscript.tex, line 1346
[Continuous precision allocation] Let E be a finite nonempty edge set, let s_e>0 be sensitivities and c_e>0 be costs, and let τ>0. Consider min_ε_e>0 ∑_e c_elogfrac1ε_e subject to ∑_es_eε_e=τ. The unique minimizer is ε_e = fracτ c_es_e∑_fc_f.
Exact mathematical passage
\boxed{ \eps_e = \frac{\tau c_e}{s_e\sum_fc_f}. }
C_iUse in source · 1 passage
Search aliases: C_i
Use in source · 01_finite_core.tex, line 92
lus_i=1^r ( C^d_i⊗ C^m_i). Here every d_i is a positive integer and every multiplicity m_i is a nonnegative integer. An empty block family represents the zero space and has seed rank zero. The maximum over an empty block family is also assigned the value zero. A vector ψ_i∈ C^d_i⊗ C^m_i is identified with a d_i× m_i coefficient matrix C_i.
Exact mathematical passage
C_i
c_{i-j}Use in source · 1 passage
Search aliases: c_i-j
Use in source · 02_supplement_04.tex, line 48
Toeplitz padding and column-major vectorization. textitSource items 4.4 and 4.5 Toeplitz embedding. For n>0, define T_ij= c_i-j, · j≤ i, r_j-i, · i<j. The diagonal is supplied by c_0, and the coordinate r_0 is unused. The constructed generator of length 2n is g=(c_0,…,c_n-1,0,r_n-1,…,r_1), and the padding map is the actual finite-index sum eliminator that appends n zeros.
Exact mathematical passage
T_{ij}= \begin{cases} c_{i-j},&j\le i,\\ r_{j-i},&i<j. \end{cases}
c_i^{e_i}Use in source · 1 passage
Search aliases: c_i^e_i
Use in source · strengthening.tex, line 101
e with unspecified remainders. On the tensor of the two factors use the mutually commuting operators L_λ⊗1, 1⊗ L_μ, c_iD^i⊗ D^N-i (1≤ i<N). Applying multiplication after the kth power of their sum gives exactly L_λ+μ^k(AB). Expanding it yields terms indexed by a,b,e_1,…,e_N-1≥0, with a+b+∑_i e_i=k: frack!a!b!∏_i e_i!∏_i c_i^e_i (D^α L_λ^a A)(D^β L_μ^b B), α=∑_i i e_i, β=∑_i(N-i)e_i. Every scalar is integral. If A and B have the stated flatness types, both factors in every term belong to H_p, as …
Exact mathematical passage
\frac{k!}{a!b!\prod_i e_i!}\prod_i c_i^{e_i} (D^\alpha L_\lambda^a A)(D^\beta L_\mu^b B), \quad \alpha=\sum_i i e_i,\quad \beta=\sum_i(N-i)e_i. \label{eq:translated-multinomial}
C_{ij}Use in source · 2 passages
Search aliases: C_ij
Use in source · 03_supplement_03.tex, line 45
Schmidt rank and the reduced state. Use coordinates ψ_ij in the standard product basis and let C_ij=ψ_ij. An explicit linear equivalence identifies the state vector on the product index set with its coefficient matrix. The implementation checks the actual Kronecker action: coeff((A⊗ I_B)ψ)=AC. It also proves that the transported matrix orbit equals the span of the state vectors obtained by this Kronecker action. The dimension formula therefore applies to the physical local action on the state sp…
Exact mathematical passage
C_{ij}=\psi_{ij}
c_jUse in source · 10 passages
Search aliases: c_j
Use in source · 20_arithmetic.tex, line 661
[Exact four-packet displacement law] Set (c_0,c_1,c_2,c_3)=(1,1,4,2), (d_0,d_1,d_2,d_3)=(1,1,5,3). For 0≤ j<4, n≥0, and every positive integer T, C_j(n+T)-C_j(n) · ≠0, · v_2(C_j(n+T)-C_j(n)) · =v_2(T)+c_j, U_j(n+T)-U_j(n) · ≠0, · v_2(U_j(n+T)-U_j(n)) · =v_2(T)+d_j.
Exact mathematical passage
C_j(n+T)-C_j(n)&\ne0,& v_2\bigl(C_j(n+T)-C_j(n)\bigr)&=v_2(T)+c_j,\\ U_j(n+T)-U_j(n)&\ne0,& v_2\bigl(U_j(n+T)-U_j(n)\bigr)&=v_2(T)+d_j.
C_{j+1,m}Use in source · 2 passages
Search aliases: C_j+1,m
Use in source · Paper_VI_Traditional_Final.tex, line 5253
Let C_j:X_j+1→ X_j be bounded linear maps, and put C_ℓ,m=C_ℓ⋯ C_m-1 for ℓ<m, with C_j,j=I. No common ambient space is assumed. [Multilevel telescoping identity] Let T_j be bounded operators on X_j, and set E_j=C_jT_j+1-T_jC_j. Then C_ℓ,mT_m-T_ℓ C_ℓ,m = ∑_j=ℓ^m-1 C_ℓ,jE_jC_j+1,m.
Exact mathematical passage
\boxed{ C_{\ell,m}T_m-T_\ell C_{\ell,m} = \sum_{j=\ell}^{m-1} C_{\ell,j}E_jC_{j+1,m}. }
C_{j,j}Use in source · 1 passage
Search aliases: C_j,j
Use in source · Paper_VI_Traditional_Final.tex, line 5246
Let C_j:X_j+1→ X_j be bounded linear maps, and put C_ℓ,m=C_ℓ⋯ C_m-1 for ℓ<m, with C_j,j=I. No common ambient space is assumed. [Multilevel telescoping identity] Let T_j be bounded operators on X_j, and set E_j=C_jT_j+1-T_jC_j. Then C_ℓ,mT_m-T_ℓ C_ℓ,m = ∑_j=ℓ^m-1 C_ℓ,jE_jC_j+1,m.
Exact mathematical passage
C_{j,j}=I
C_{J,K}Use in source · 2 passages
Search aliases: C_J,K
Use in source · 50_open_questions.tex, line 107
The linear compatibility theorem provides one concrete part of this program. If two local observations J:U→ V and K:U→ W are linear maps from the same correction module, define C_J,K=U/(ker J+ker K). The attainable observations im J and im K have canonical labels in this quotient. Equality of labels is equivalent to simultaneous realization by a single correction. The joint image is exactly the fiber product over C_J,K, with the proved linear universal property. The declarations are begingroupsm…
Exact mathematical passage
C_{J,K}=U/(\ker J+\ker K).
c_{j,k}Use in source · 1 passage
Search aliases: c_j,k
Use in source · 01_manuscript.tex, line 1438
ct discrete precision program] Let the total integer error budget be B. Define F_j(b) as the minimum cost for assigning precisions to the first j edges with total weight exactly b. Then F_0(0)=0, F_0(b)=+∞ for b≠0. Set F_j-1(u)=+∞ whenever u<0 or the state is unreachable. Then F_j(b) = min_k { F_j-1(b-w_j,k)+c_j,k }. The minimum of F_m(b) over 0≤ b≤ B is the exact minimum cost for the quantized model.
Exact mathematical passage
\boxed{ F_j(b) = \min_k \left\{ F_{j-1}(b-w_{j,k})+c_{j,k} \right\}. }
C_{j,m}Use in source · 1 passage
Search aliases: C_j,m
Use in source · Paper_VI_Traditional_Final.tex, line 5265
Expand each summand as C_ℓ,j+1T_j+1C_j+1,m - C_ℓ,jT_jC_j,m. The intermediate terms cancel. The first surviving term is C_ℓ,mT_m, and the last is -T_ℓ C_ℓ,m.
Exact mathematical passage
C_{\ell,j+1}T_{j+1}C_{j+1,m} - C_{\ell,j}T_jC_{j,m}.
c_{j,r}Use in source · 3 passages
Search aliases: c_j,r
Use in source · 72_recovered.tex, line 1635
[Polynomial exponential Mahler transfer] Let f(n)=P(n)u^n with P∈ Z_p[n] of degree d. Let H=Λ_p,L, fix an address j, and write Q_j(x)=P(Hx+j), y=u^H, q=y-1. Then f(Hx+j)=∑_r≥0c_j,rbinom xr, where c_j,r = u^j ∑_k=0^min(d,r) binom rk Δ^kQ_j(0)y^kq^r-k. Consequently, v_p(c_j,r)≥ Lmax(r-d,0). For R≥ d, truncation at depth R is valid modulo p^L(R+1-d) uniformly on Z_p.
Exact mathematical passage
\boxed{ f(Hx+j)=\sum_{r\geq0}c_{j,r}\binom xr, }
c_{j-k}Use in source · 4 passages
Search aliases: c_j-k
Use in source · pages.tex, line 94
4 · 3 · 2, x=(1,2,-1)^T. The diagonal pattern is Toeplitz, not circulant. To use a circular convolution, embed it into dimension six with first column g=(2,3,4,0,5,-1)^T and pad the input with three zeros. vfigtoeplitz headRead the wrapped indices For output index j<3, an input index k≤ j uses the first-column coefficient c_j-k. An input index k>j wraps to the stored first-row coefficient r_k-j. The padded input positions contribute nothing. Keeping the first three outputs therefore gives Tx=(-5…
Exact mathematical passage
c_{j-k}
c_{j-k\bmod4}Use in source · 1 passage
Search aliases: c_j-kbmod4 bmod
Use in source · pages.tex, line 73
vpage5Four coefficients, four exact frequenciesCIRCULANT MULTIPLICATIONChapter VIII: Theorem 4.3. Take c=(2,1,0,-1)^T and define C(c)_jk=c_j-kbmod4. The unitary Fourier matrix is F_jk= i^-jk/2. Its normalization is visible in the exact diagonalization: C(c)=F^*diag(2Fc)F, 2Fc=(2,2-2 i,2,2+2 i)^T. vfigcirculant headFollow an actual input For x=(1,2,-1,0)^T, dense multiplication gives (0,6,0,-2)^T. Transforming x, multiplying by the four displayed eigenvalues, and applying F^* gives exactly the sa…
Exact mathematical passage
C(c)_{jk}=c_{j-k\bmod4}
C_j^\bulletUse in source · 5 passages
Search aliases: C_j^bullet bullet
Use in source · Paper_X_Block27_Amended.tex, line 382
For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet are cochain maps, so they induce maps on these genuine quotients.
Exact mathematical passage
C_j^\bullet
C_j^rUse in source · 1 passage
Search aliases: C_j^r
Use in source · Paper_X_Block27_Amended.tex, line 375
For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet are cochain maps, so they induce maps on these genuine quotients.
Exact mathematical passage
C_j^r=\operatorname{im}P_j^r
c_{ja}Use in source · 1 passage
Search aliases: c_ja
Use in source · 03_supplement_05.tex, line 183
Every normalized channel in the same span. Suppose F_j=∑_a c_jaE_a and ∑_jF_j^*F_j=I. The decoder is not reconstructed for the F_j. Summing the coherent identities gives a scalar multiple qρ. Trace preservation of both the F channel and the recovery forces q=1 on a nonzero code. The proof uses positivity of P to justify cancellation of its trace and handles P=0 separately. Therefore R(∑_jF_jρ F_j^*)=ρ. This is pathData.recovery_span_channel, a quantified statement over arbitrary finite F familie…
Exact mathematical passage
F_j=\sum_a c_{ja}E_a
C_{jk}Use in source · 1 passage
Search aliases: C_jk
Use in source · 02_strengthened.tex, line 1903
[Reflection positive correlators] Fix a vector Ω and operators A_1,…,A_m. The matrix C_jk = ipA_jΩT^n_j+n_kA_kΩ is positive semidefinite.
Exact mathematical passage
C_{jk} = \ip{A_j\Omega}{T^{n_j+n_k}A_k\Omega}
C_kUse in source · 1 passage
Search aliases: C_k
Use in source · 03_supplement_03.tex, line 17
Several seeds: both necessity and construction. Continue under the assumption above that the number of rows is positive. For seeds C_1,…,C_k, generation means the concrete statement ∀ X∈ M_d× m( C), ∃ A_1,…,A_k∈ M_d( C): X=∑_a=1^k A_aC_a. Stack the C_a vertically into a kd× m matrix D. The source identifies the action on the stacked matrix and proves the equivalence the seeds generate Longleftrightarrow rank D=m. Thus m≤ kd is necessary. It is sufficient as well: embed the m coordinate columns i…
Exact mathematical passage
C_1,\ldots,C_k
c_KUse in source · 1 passage
Search aliases: c_K
Use in source · residue_continuation_body.tex, line 138
Ordinary class characters: an unconditional power saving. The preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl( O)→ C, define the actual finite ideal sum S_w(X)=∑_substackI≠0
Exact mathematical passage
A_C(X)=c_KX+O_{K,C}(X^{1-1/d}),\qquad c_K=\frac{2^{r_1}(2\pi)^{r_2}R_K}{\#\mu(K)\sqrt{\absn{\Delta_K}}}.
c_kUse in source · 3 passages
Search aliases: c_k
Use in source · 10_foundations.tex, line 509
For M≥1 and s,r,k∈ N, define the affine Mahler kernel by Θ_M,s(r,k)=[z^r](1+z)^s((1+z)^M-1)^k =∑_j=0^k(-1)^k-jbinom kjbinomMj+sr. It vanishes unless k≤ r≤ Mk+s. For a finite binomial expansion f(n)=∑_r=0^Dc_rbinom nr, the kth Newton coefficient of g(q)=f(Mq+s) is therefore c_k(g)=∑_r=k^min(D,Mk+s)c_rΘ_M,s(r,k). Indeed, insert the finite stencil for Δ_1^k g(0) and interchange its two finite sums. Coefficient extraction of the binomial expansion of ((1+z)^M-1)^k gives the kernel. In the binary zer…
Exact mathematical passage
c_k(g)=\sum_{r=k}^{\min(D,Mk+s)}c_r\Theta_{M,s}(r,k).
C_LUse in source · 2 passages
Search aliases: C_L
Use in source · Paper_IV_Traditional_Final.tex, line 2949
For N≥3, [source reference] gives | D_N'(ρ)|^-1 ≤ C e^-c_N|ρ| for every sufficiently large simple zero. Let L be a compact subset of the s plane. Because the chosen arguments of the zeros lie in a fixed bounded interval, there are constants C_L and M_L such that sup_s∈ L|e^-sLogρ| ≤ C_L|ρ|^M_L for every sufficiently large zero. Hence the absolute value of the corresponding summand is at most C_L'|ρ|^M_Le^-q c_N|ρ|. The linear zero counting estimate makes this majorant summable. The Weierstrass t…
Exact mathematical passage
C_L
C_L'Use in source · 1 passage
Search aliases: C_L'
Use in source · Paper_IV_Traditional_Final.tex, line 2956
-1 ≤ C e^-c_N|ρ| for every sufficiently large simple zero. Let L be a compact subset of the s plane. Because the chosen arguments of the zeros lie in a fixed bounded interval, there are constants C_L and M_L such that sup_s∈ L|e^-sLogρ| ≤ C_L|ρ|^M_L for every sufficiently large zero. Hence the absolute value of the corresponding summand is at most C_L'|ρ|^M_Le^-q c_N|ρ|. The linear zero counting estimate makes this majorant summable. The Weierstrass test gives absolute local uniform convergence …
Exact mathematical passage
C_L'|\rho|^{M_L}e^{-q c_N|\rho|}.
C_MUse in source · 3 passages
Search aliases: C_M
Use in source · Paper_IV_Traditional_Final.tex, line 1936
[Near cancellation of two exponentials] For every M>0, there are constants ε_M>0 and C_M>0 such that the following holds. If |Re w|≤ M and |1+e^w|≤ε_M, then there is a unique integer m with |w-(2m+1)π i| ≤ C_M|1+e^w|.
Exact mathematical passage
C_M>0
C_mUse in source · 7 passages
Search aliases: C_m
Use in source · Paper_X_Block27_Amended.tex, line 2365
There is also a completion theorem without chosen compatible characters. Suppose that R/I^m is finite for every m≥1, and that C_m⊆ P_q(R/I^m) are allowed subsets such that every reduction of an allowed member is allowed. Then · ∃χ∈ P_q(R), χ is continuous and χbmod I^m∈ C_m for all m≥1
Exact mathematical passage
C_m\subseteq\mathcal P_q(R/I^m)
C_{m+1}Use in source · 1 passage
Search aliases: C_m+1
Use in source · Paper_VI_Traditional_Final.tex, line 2523
Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.
Exact mathematical passage
C_{m,n}=C_mC_{m+1}\cdots C_{n-1}, \qquad R_{n,m}=R_{n-1}\cdots R_m.
C_{m,m}Use in source · 1 passage
Search aliases: C_m,m
Use in source · Paper_VI_Traditional_Final.tex, line 2529
Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.
Exact mathematical passage
C_{m,m}=I_{X_m}, \qquad R_{m,m}=I_{X_m}.
C_{m-1}Use in source · 1 passage
Search aliases: C_m-1
Use in source · Paper_VI_Traditional_Final.tex, line 5246
Let C_j:X_j+1→ X_j be bounded linear maps, and put C_ℓ,m=C_ℓ⋯ C_m-1 for ℓ<m, with C_j,j=I. No common ambient space is assumed. [Multilevel telescoping identity] Let T_j be bounded operators on X_j, and set E_j=C_jT_j+1-T_jC_j. Then C_ℓ,mT_m-T_ℓ C_ℓ,m = ∑_j=ℓ^m-1 C_ℓ,jE_jC_j+1,m.
Exact mathematical passage
C_{\ell,m}=C_\ell\cdots C_{m-1}
C_{mp,j}Use in source · 17 passages
Search aliases: C_mp,j
Use in source · 20_arithmetic.tex, line 134
[Lucas reduction and address independence] Let p be prime, m>0, and N=mp. For 0≤ j<p and n,q∈ N, binommpn+jmpq≡binommnmqpmod p. Every C_mp,j(n) belongs to Z_p. Its reduction is the unique sequence overline C_m: N→ F_p satisfying ∑_q=0^nbinommnmqoverline C_m(q)=δ_n0 in F_p. Consequently, for any two addresses 0≤ j,k<p, C_mp,j(n)≡ C_mp,k(n)pmod p (n∈ N).
Exact mathematical passage
C_{mp,j}(n)
C_{mp,k}Use in source · 1 passage
Search aliases: C_mp,k
Use in source · 20_arithmetic.tex, line 141
prop:arith-lucas-window Let p be prime, m>0, and N=mp. For 0≤ j<p and n,q∈ N, binommpn+jmpq≡binommnmqpmod p. Every C_mp,j(n) belongs to Z_p. Its reduction is the unique sequence overline C_m: N→ F_p satisfying ∑_q=0^nbinommnmqoverline C_m(q)=δ_n0 in F_p. Consequently, for any two addresses 0≤ j,k<p, C_mp,j(n)≡ C_mp,k(n)pmod p (n∈ N).
Exact mathematical passage
C_{mp,j}(n)\equiv C_{mp,k}(n)\pmod p\qquad(n\in\N).
C_nUse in source · 10 passages
Search aliases: C_n
Use in source · 05_worked_models.tex, line 372
Two adjacent vertex vectors generate the adjacency module of every simple cycle. Over C, a Fourier quotient built from a primitive root of unity supplies a matching scalar obstruction of dimension two. Hence, for every n≥3, two is the least finite number of adjacency seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any f…
Exact mathematical passage
{ 2=\min\left\{r\in\N: \exists s_1,\ldots,s_r,\quad \C^n=\Span_{\C}\{p(A_{C_n})s_j:p\in\C[t],\ 1\le j\le r\} \right\}. }
c_NUse in source · 12 passages
Search aliases: c_N
Use in source · Paper_IV_Traditional_Final.tex, line 1842
At a tie direction exactly two adjacent vertices attain the maximum. After rotation it is enough to take θ=0. With c_N=cos(π/N) and s_N=sin(π/N), the two dominant terms give frac2N e^c_Nrcos(s_Nr) and every other term is exponentially smaller. Along a sequence on which |cos(s_Nr)|≥1/2, the lower indicator bound is c_N=h_N(0). Rotational symmetry treats every tie direction.
Exact mathematical passage
c_N=\cos(\pi/N)
c_nUse in source · 3 passages
Search aliases: c_n
Use in source · Paper_XI_Traditional_Final.tex, line 410
[Möbius orbit count] The number of regular Frobenius orbits of length n is c_n(q) = frac1n ∑_dmid n μ(d) (q^n/d-1).
Exact mathematical passage
\boxed{ c_n(q) = \frac1n \sum_{d\mid n} \mu(d) \bigl(q^{n/d}-1\bigr). }
C_{N,0}Use in source · 2 passages
Search aliases: C_N,0
Use in source · strengthening.tex, line 242
[Necessary boundaries] At an odd prime the ordinary step-one difference cannot replace the sign-twisted operator in the general theorem. For j=0, C_N,0(0)=1 and C_N,0(1)=-1, so their ordinary difference is -2, a unit at every odd prime. The proof also does not extend unchanged to non-prime-power N, whose interior binomial coefficients need not be divisible by the chosen prime. Neither observation is a counterexample to Nishibuchi's original eventual congruence.
Exact mathematical passage
C_{N,0}(0)=1
C_{n,a}Use in source · 2 passages
Search aliases: C_n,a
Use in source · 02_strengthened.tex, line 2491
[Constraint closure under norm convergence] Let C_n,a→ C_a in norm for every a, and fix structure coefficients f_ab^phantomabc. If norm [C_n,a,C_n,b] - i∑_cf_ab^phantomabcC_n,c ⟶0, then [C_a,C_b] = i∑_cf_ab^phantomabcC_c.
Exact mathematical passage
C_{n,a}\to C_a
C_{n,b}Use in source · 1 passage
Search aliases: C_n,b
Use in source · 02_strengthened.tex, line 2492
[Constraint closure under norm convergence] Let C_n,a→ C_a in norm for every a, and fix structure coefficients f_ab^phantomabc. If norm [C_n,a,C_n,b] - i∑_cf_ab^phantomabcC_n,c ⟶0, then [C_a,C_b] = i∑_cf_ab^phantomabcC_c.
Exact mathematical passage
\norm{ [C_{n,a},C_{n,b}] -\ii\sum_cf_{ab}^{\phantom{ab}c}C_{n,c} } \longrightarrow0,
C_{n,c}Use in source · 1 passage
Search aliases: C_n,c
Use in source · 02_strengthened.tex, line 2492
[Constraint closure under norm convergence] Let C_n,a→ C_a in norm for every a, and fix structure coefficients f_ab^phantomabc. If norm [C_n,a,C_n,b] - i∑_cf_ab^phantomabcC_n,c ⟶0, then [C_a,C_b] = i∑_cf_ab^phantomabcC_c.
Exact mathematical passage
\norm{ [C_{n,a},C_{n,b}] -\ii\sum_cf_{ab}^{\phantom{ab}c}C_{n,c} } \longrightarrow0,
C_{N,j,n-q}Use in source · 1 passage
Search aliases: C_N,j,n-q
Use in source · Paper_IV_Traditional_Final.tex, line 1455
[Parameter-addition law for residue-section reciprocals] For each n≥1, the coefficient C_N,j,n(r) is a polynomial in r of exact degree n. Its leading coefficient is frac(Nn)!n! (-fracj!(N+j)!)^n. It satisfies C_N,j,n(r+s) = ∑_q=0^n binomNnNq C_N,j,q(r)C_N,j,n-q(s). In particular, C_N,j,n(1)=C_N,j(n).
Exact mathematical passage
C_{N,j,n}(r+s) = \sum_{q=0}^{n} \binom{Nn}{Nq} C_{N,j,q}(r)C_{N,j,n-q}(s).
C_{N,j,q}Use in source · 1 passage
Search aliases: C_N,j,q
Use in source · Paper_IV_Traditional_Final.tex, line 1455
[Parameter-addition law for residue-section reciprocals] For each n≥1, the coefficient C_N,j,n(r) is a polynomial in r of exact degree n. Its leading coefficient is frac(Nn)!n! (-fracj!(N+j)!)^n. It satisfies C_N,j,n(r+s) = ∑_q=0^n binomNnNq C_N,j,q(r)C_N,j,n-q(s). In particular, C_N,j,n(1)=C_N,j(n).
Exact mathematical passage
C_{N,j,n}(r+s) = \sum_{q=0}^{n} \binom{Nn}{Nq} C_{N,j,q}(r)C_{N,j,n-q}(s).
C_{n-1}Use in source · 2 passages
Search aliases: C_n-1
Use in source · 03_nonlinear_coordinates.tex, line 514
Assume now that a>0 and b>0. Take R= frac1-sqrt1-4ab2b. Then a+bR^2=R and 2bR<1. Apply [source reference]. The number of full ordered binary trees with n leaves is the Catalan number C_n-1, which gives the displayed majorant.
Exact mathematical passage
C_{n-1}
c_{n-1}Use in source · 3 passages
Search aliases: c_n-1
Use in source · 01_manuscript.tex, line 565
[Toeplitz circulant embedding] Let n≥1 be an integer and T∈ C^n× n be Toeplitz, with first column (c_0,c_1,…,c_n-1)^T and first row (c_0,r_1,…,r_n-1). Set g = (c_0,c_1,…,c_n-1,0,r_n-1,…,r_1)^T ∈ C^2n and append n zeros to x∈ C^n. The first n entries of the circulant product C(g)(x,0)^T equal Tx.
Exact mathematical passage
(c_0,c_1,\ldots,c_{n-1})^T
C_pUse in source · 3 passages
Search aliases: C_p
Use in source · 20_arithmetic.tex, line 305
fferentiation of the section identities gives tR_c'=R_c(1+R_c-1-R_c). Writing q=R_c-1-(c-1) and y=R_c-c, this becomes (tD+c)y=cq+yq-y^2. On degrees divisible by p, the inverse of tD+c divides the degree-d coefficient by the unit d+c. Starting with y_0=0, form y_s+1=(tD+c)^-1(cq+y_sq-y_s^2). Flatness and support are preserved. Indeed, put E_c=tD+c and C_p=pD^p^2-D^p. The commutator is [B_p,E_c]=C_p, and iteration gives B_p^rE_cy=E_cB_p^ry+rC_pB_p^r-1y. For the inverse equation E_cy=f, induction o…
Exact mathematical passage
C_p=pD^{p^2}-D^p
c_{p^ak}Use in source · 1 passage
Search aliases: c_p^ak
Use in source · 72_recovered.tex, line 2306
(q)=q has d_1=1. Thus d_1not≡ c_2pmod2. The coefficient consequence is read with integral original coefficients. Let M=p^a. Then Θ_p^a,s(r,k) ≡ binomsr-p^ak pmod p, where the binomial coefficient is zero outside its natural range. Consequently, d_s,k ≡ ∑_u=0^s binom su c_p^ak+u pmod p. In particular, d_0,k≡ c_p^akpmod p.
Exact mathematical passage
\boxed{d_{0,k}\equiv c_{p^ak}\pmod p.}
c_{p^ak+u}Use in source · 1 passage
Search aliases: c_p^ak+u
Use in source · 72_recovered.tex, line 2295
insufficient. For p=2, M=2, s=0 and f(x)=x/2, one has c_1=1/2, c_2=0, while g(q)=q has d_1=1. Thus d_1not≡ c_2pmod2. The coefficient consequence is read with integral original coefficients. Let M=p^a. Then Θ_p^a,s(r,k) ≡ binomsr-p^ak pmod p, where the binomial coefficient is zero outside its natural range. Consequently, d_s,k ≡ ∑_u=0^s binom su c_p^ak+u pmod p. In particular, d_0,k≡ c_p^akpmod p.
Exact mathematical passage
\boxed{ d_{s,k} \equiv \sum_{u=0}^{s} \binom su c_{p^ak+u} \pmod p. }
C_{p^{n_i}}Use in source · 1 passage
Search aliases: C_p^n_i
Use in source · Paper_XII_Traditional_Final.tex, line 490
[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.
Exact mathematical passage
\Delta\cong \prod_{i=1}^{r}C_{p^{n_i}}.
C_{\Pcal\leftarrow\Rcal}Use in source · 1 passage
Search aliases: C_Pcal←Rcal C_ P← R Pcal leftarrow Rcal mathcal leftarrow mathcal
Use in source · 02_coordinates.tex, line 115
herence] For any two packet systems on X, C_ P← QJ_ Q← P=I_X^ P, J_ Q← PC_ P← Q=I_X^ Q. For three packet systems P, Q, and R on X, J_ R← QJ_ Q← P = J_ R← P, C_ P← QC_ Q← R = C_ P← R. If Q refines P through ρ, then (J_ Q← Px)_β = Q_β x_ρ(β), (C_ P← Qy)_α = ∑_ρ(β)=αy_β.
Exact mathematical passage
C_{\Pcal\leftarrow\Qcal}C_{\Qcal\leftarrow\Rcal} = C_{\Pcal\leftarrow\Rcal}.
C_QUse in source · 4 passages
Search aliases: C_Q
Use in source · Paper_V_Traditional_Final.tex, line 3427
[Estimated packet Gibbs bound] Let Q_m be linear and uniformly bounded on L^p, with sup_m Q_m ≤ C_Q. For f=s+j and an estimated packet widehat j_m, define widehat f_m =Q_m(f-widehat j_m)+widehat j_m. Then f-widehat f_m _p ≤ s-Q_ms _p +(1+C_Q) j-widehat j_m _p. Consequently, smooth approximation and packet recovery may be improved independently. No claim of Gibbs removal is justified unless both terms are controlled in the norm under consideration.
Exact mathematical passage
\sup_m\|Q_m\|\leq C_Q.
C_{\Qcal\leftarrow\Rcal}Use in source · 1 passage
Search aliases: C_Qcal←Rcal C_ Q← R Qcal leftarrow Rcal mathcal leftarrow mathcal
Use in source · 02_coordinates.tex, line 115
herence] For any two packet systems on X, C_ P← QJ_ Q← P=I_X^ P, J_ Q← PC_ P← Q=I_X^ Q. For three packet systems P, Q, and R on X, J_ R← QJ_ Q← P = J_ R← P, C_ P← QC_ Q← R = C_ P← R. If Q refines P through ρ, then (J_ Q← Px)_β = Q_β x_ρ(β), (C_ P← Qy)_α = ∑_ρ(β)=αy_β.
Exact mathematical passage
C_{\Pcal\leftarrow\Qcal}C_{\Qcal\leftarrow\Rcal} = C_{\Pcal\leftarrow\Rcal}.
C_RUse in source · 4 passages
Search aliases: C_R
Use in source · 71_native.tex, line 248
for maps from A and B to a common set if and only if R is rectangular. In that case a canonical finite choice of common set is the connected-component set C_R of the bipartite graph with vertex set Asqcup B and edges given by R.
Exact mathematical passage
C_R
c_rUse in source · 41 passages
Search aliases: c_r
Use in source · 10_foundations.tex, line 215
[Filtered finite reconstruction] For every n,L∈ N and sequences c,S in that ring, · ∑_r=0^nbinom nr c_rp^[w(r)]S_n-r · ≡ ∑_substack0≤ r≤ n w(r)<L binom nr c_rp^[w(r)]S_n-rpmodp^L. Here congruence means divisibility of the difference by p^L in the coefficient ring. In Z_p, it means membership in p^L Z_p.
Exact mathematical passage
\begin{split} &\sum_{r=0}^{n}\binom nr c_rp^{[w(r)]}S_{n-r}\\ &\hspace{8mm}\equiv \sum_{\substack{0\leq r\leq n\\w(r)<L}} \binom nr c_rp^{[w(r)]}S_{n-r}\pmod{p^L}. \end{split}
C_r^*Use in source · 1 passage
Search aliases: C_r^*
Use in source · 03_supplement_02.tex, line 62
The fine and coarse families are complete orthogonal self-adjoint projector systems. In particular, C_rC_s=δ_rsC_r, ∑_r C_r=I, C_r^*=C_r. Each coarse projector is exactly the displayed sum of its fine descendants.
Exact mathematical passage
C_rC_s=\delta_{rs}C_r,\qquad \sum_r C_r=I,\qquad C_r^*=C_r.
C_sUse in source · 2 passages
Search aliases: C_s
Use in source · 03_supplement_02.tex, line 62
The fine and coarse families are complete orthogonal self-adjoint projector systems. In particular, C_rC_s=δ_rsC_r, ∑_r C_r=I, C_r^*=C_r. Each coarse projector is exactly the displayed sum of its fine descendants.
Exact mathematical passage
C_rC_s=\delta_{rs}C_r,\qquad \sum_r C_r=I,\qquad C_r^*=C_r.
c_sUse in source · 1 passage
Search aliases: c_s
Use in source · supplement_04.tex, line 54
An independently defined interpolant. For a finite coefficient list, define I_N(c)(n)=∑_r=0^Nc_rbinom nr. The source proves exact coefficient extraction: Δ_1^s I_N(c)(0)= c_s, · s≤ N, 0, · s>N. It follows that I_N(c) has the stated shifted-coefficient formula at every order, with zero coefficients beyond N. Every difference of order greater than N vanishes. Its coefficients are uniquely determined by its values at 0,…,N.
Exact mathematical passage
\Delta_1^s I_N(c)(0)= \begin{cases}c_s,&s\leq N,\\0,&s>N.\end{cases}
c_{s+j}Use in source · 5 passages
Search aliases: c_s+j
Use in source · 10_foundations.tex, line 255
[Finite Newton--Mahler identities] For every n,s,h∈ N, f(n) · =∑_r=0^nc_r(f)binom nr, c_r(Δ_1^s f) · =c_s+r(f), Δ_1^s f(n) · =∑_j=0^nc_s+j(f)binom nj, Δ_h^s f(n) · =∑_j=0^s(-1)^s-jbinom sj f(n+jh). The finite interpolant I_Nf(n)=∑_r=0^Nc_r(f)binom nr agrees with f for n≤ N. It has the same order-s unit-step difference at n whenever n+s≤ N. For arbitrary coefficients c_0,…,c_N, the corresponding binomial interpolant has exactly those coefficients and zero coefficients above N.
Exact mathematical passage
\begin{aligned} f(n)&=\sum_{r=0}^{n}c_r(f)\binom nr,\\ c_r(\Delta_1^s f)&=c_{s+r}(f),\\ \Delta_1^s f(n)&=\sum_{j=0}^{n}c_{s+j}(f)\binom nj,\\ \Delta_h^s f(n)&=\sum_{j=0}^{s}(-1)^{s-j}\binom sj f(n+jh). \end{aligned}
c_{s+r}Use in source · 1 passage
Search aliases: c_s+r
Use in source · 10_foundations.tex, line 255
[Finite Newton--Mahler identities] For every n,s,h∈ N, f(n) · =∑_r=0^nc_r(f)binom nr, c_r(Δ_1^s f) · =c_s+r(f), Δ_1^s f(n) · =∑_j=0^nc_s+j(f)binom nj, Δ_h^s f(n) · =∑_j=0^s(-1)^s-jbinom sj f(n+jh). The finite interpolant I_Nf(n)=∑_r=0^Nc_r(f)binom nr agrees with f for n≤ N. It has the same order-s unit-step difference at n whenever n+s≤ N. For arbitrary coefficients c_0,…,c_N, the corresponding binomial interpolant has exactly those coefficients and zero coefficients above N.
Exact mathematical passage
\begin{aligned} f(n)&=\sum_{r=0}^{n}c_r(f)\binom nr,\\ c_r(\Delta_1^s f)&=c_{s+r}(f),\\ \Delta_1^s f(n)&=\sum_{j=0}^{n}c_{s+j}(f)\binom nj,\\ \Delta_h^s f(n)&=\sum_{j=0}^{s}(-1)^{s-j}\binom sj f(n+jh). \end{aligned}
C_TUse in source · 1 passage
Search aliases: C_T
Use in source · 02_strengthened.tex, line 812
em[Convergence of short packet rotations] Let (P_a(t)) be twice continuously differentiable on [0,T]. For a partition 0=t_0<t_1<⋯<t_n=T, let W_π = U( P(t_n), P(t_n-1)) ⋯ U( P(t_1), P(t_0)). For sufficiently fine partitions the factors are defined, and normW_π-W(T) ≤ C_Tabsπ for a constant depending on the packet path but not on the partition. Here the size of the partition is measured by its mesh, which is the length of its largest subinterval.
Exact mathematical passage
\norm{W_\pi-W(T)} \leq C_T\abs{\pi}
c_uUse in source · 8 passages
Search aliases: c_u
Use in source · 72_recovered.tex, line 192
The product law follows from [source reference]. To prove injectivity, suppose ∑_u∈ F_pc_uϵ_u=0. Looking at the coordinates n=0,1,…,p-1 gives a Vandermonde system with matrix (u^n)_0≤ n≤ p-1, u∈ F_p. Its determinant is the product of all differences between distinct field elements, so it is nonzero. Hence every c_u vanishes.
Exact mathematical passage
\sum_{u\in\FF_p}c_u\epsilon_u=0.
c_vUse in source · 6 passages
Search aliases: c_v
Use in source · Paper_V_Traditional_Final.tex, line 978
[Dynamic-programming best packet basis] Suppose J_y(Λ)=∑_v∈Λc_v(y). For a node v, define recursively V_y(v) = c_v(y), · v terminal,
Exact mathematical passage
J_y(\Lambda)=\sum_{v\in\Lambda}c_v(y).
C_xUse in source · 1 passage
Search aliases: C_x
Use in source · monograph_atlas.tex, line 747
vpage38Transport constraints without hiding closureTHE IMAGE OF AN ISOMETRY IS NOT THE WHOLE LEVELChapter VII: Theorem II.12.2, Theorem II.12.5, Proposition II.12.6. On C^3= Cket0⊕ C^2, use C_a=0⊕σ_a for a=x,y,z. The common kernel is Cket0, and the full algebra closes: [C_x,C_y]=2 i C_z, [C_y,C_z]=2 i C_x, [C_z,C_x]=2 i C_y. These are uncompressed identities. They supply an actual closure test in the quotient by the constraint span, unlike the automatic compressed zero on the physical kernel. xf…
Exact mathematical passage
[C_x,C_y]=2\ii C_z,\qquad[C_y,C_z]=2\ii C_x,\qquad[C_z,C_x]=2\ii C_y.
\mathcal C_xUse in source · 4 passages
Search aliases: C_x mathcal
Use in source · continuation_01.tex, line 102
Let G∈ Z[X_1,…,X_d]^f be a second fixed system with the same variables and the same lower root x. Let K_x=ρ_q∘ DG(x), and define C_x= Z^d/(ker J_x+ker K_x). This quotient is finite: it is a quotient of the finite image of J_x, or equivalently of the finite image of K_x.
Exact mathematical passage
\mathcal C_x=\mathbb Z^d/(\ker J_x+\ker K_x).
C_yUse in source · 1 passage
Search aliases: C_y
Use in source · monograph_atlas.tex, line 747
vpage38Transport constraints without hiding closureTHE IMAGE OF AN ISOMETRY IS NOT THE WHOLE LEVELChapter VII: Theorem II.12.2, Theorem II.12.5, Proposition II.12.6. On C^3= Cket0⊕ C^2, use C_a=0⊕σ_a for a=x,y,z. The common kernel is Cket0, and the full algebra closes: [C_x,C_y]=2 i C_z, [C_y,C_z]=2 i C_x, [C_z,C_x]=2 i C_y. These are uncompressed identities. They supply an actual closure test in the quotient by the constraint span, unlike the automatic compressed zero on the physical kernel. xf…
Exact mathematical passage
[C_x,C_y]=2\ii C_z,\qquad[C_y,C_z]=2\ii C_x,\qquad[C_z,C_x]=2\ii C_y.
C_ZUse in source · 1 passage
Search aliases: C_Z
Use in source · 03_nonlinear_coordinates.tex, line 749
[Exact defect under composition] Let G_f:Y_f→ Z_f and G_c:Y_c→ Z_c, with coarsening C_Z:Z_f→ Z_c. Then Δ_G∘ F(x) = Δ_G(F_f(x)) + G_c(F_c(C_Xx)+Δ_F(x)) - G_c(F_c(C_Xx)). If G_c is L-Lipschitz on the relevant set, then Δ_G∘ F(x) ≤ Δ_G(F_f(x)) + L Δ_F(x) .
Exact mathematical passage
C_Z:Z_f\to Z_c
C_zUse in source · 1 passage
Search aliases: C_z
Use in source · monograph_atlas.tex, line 747
vpage38Transport constraints without hiding closureTHE IMAGE OF AN ISOMETRY IS NOT THE WHOLE LEVELChapter VII: Theorem II.12.2, Theorem II.12.5, Proposition II.12.6. On C^3= Cket0⊕ C^2, use C_a=0⊕σ_a for a=x,y,z. The common kernel is Cket0, and the full algebra closes: [C_x,C_y]=2 i C_z, [C_y,C_z]=2 i C_x, [C_z,C_x]=2 i C_y. These are uncompressed identities. They supply an actual closure test in the quotient by the constraint span, unlike the automatic compressed zero on the physical kernel. xf…
Exact mathematical passage
[C_x,C_y]=2\ii C_z,\qquad[C_y,C_z]=2\ii C_x,\qquad[C_z,C_x]=2\ii C_y.
C_{\alpha i}Use in source · 1 passage
Search aliases: C_α i alpha
Use in source · Paper_VI_Traditional_Final.tex, line 2848
Let {B_1,…,B_m} be a partition of the fine state set. Its aggregation matrix is C_α i= 1_{i∈ B_α}.
Exact mathematical passage
C_{\alpha i}=\one_{\{i\in B_\alpha\}}.
C_\alpha^\bulletUse in source · 2 passages
Search aliases: C_α^bullet alpha bullet
Use in source · 04_correction_transport.tex, line 459
[Direct packet complexes] Suppose C^bullet=⊕_α∈ IC_α^bullet is a finite direct sum of subcomplexes. Then H^n(C^bullet) ≅ ⊕_α∈ IH^n(C_α^bullet). A total cohomology class vanishes if and only if every packet class vanishes.
Exact mathematical passage
C^\bullet=\bigoplus_{\alpha\in I}C_\alpha^\bullet
C_\etaUse in source · 4 passages
Search aliases: C_η eta
Use in source · atlas-pages.tex, line 943
vpage41An operator substitution transports the defect tooFUNCTIONAL CALCULUS / AN EXACT RATIONAL TESTChapter VI, Part II: Theorem E3.4; compare Chapter VI, Part I, Theorem 15.1, and Plate VI.25. Use the operators from Plate VI.25, but perturb the rectangular comparison map: A=mat2 · 1 · 0 0 · 2 · 0 0 · 0 · 9, B=mat2 · 1 0 · 2, C_η=mat1 · 0 · η 0 · 1 · 0. The generator defect is Δ=C_η A-BC_η, with only one nonzero entry, Δ_13=7η (matrix indices begin at one). vfigext_substitution platecaptionThe …
Exact mathematical passage
A=\mat{2&1&0\\0&2&0\\0&0&9},\quad B=\mat{2&1\\0&2},\quad C_\eta=\mat{1&0&\eta\\0&1&0}.
c_{\eta u}Use in source · 1 passage
Search aliases: c_η u eta
Use in source · 72_recovered.tex, line 334
Write b^(m,p) = ∑_u∈ F_pc_uϵ_u. Invariance gives c_η u=c_u. After the change of variable u↦η u, b_n^(m,p) = η^n b_n^(m,p) for every η∈ H_m. If mnmid n, some η satisfies η^n≠1, so the coefficient vanishes.
Exact mathematical passage
c_{\eta u}=c_u
C_\piUse in source · 8 passages
Search aliases: C_π pi
Use in source · Paper_VI_Traditional_Final.tex, line 841
Let the fine state set I be partitioned into nonempty coarse packets I_a, a∈ A. The coarsening map on probability vectors is (C_π p)_a=P_a=∑_i∈ I_ap_i. For a fine likelihood λ=(λ_i)_i∈ I and a prior p, define the effective coarse likelihood r_a(p) = frac∑_i∈ I_aλ_i p_iP_a (P_a>0). Its value on a zero-mass packet is irrelevant and may be set to zero.
Exact mathematical passage
(C_\pi p)_a=P_a=\sum_{i\in I_a}p_i.
c_\chiUse in source · 1 passage
Search aliases: c_χ chi
Use in source · 72_recovered.tex, line 843
Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding term vanishes for positive n. Otherwise ((1+ζ y)^m)^h = (1+ζ y)^p-1 =1. Every summand is therefore h periodic for positive n. Thus T_m,p preserves V_h.…
Exact mathematical passage
d_q=\sum_{\chi\in\mu_h}c_\chi\chi^q.
C_\psiUse in source · 10 passages
Search aliases: C_ψ psi
Use in source · 02_strengthened.tex, line 120
[Orbit dimension of one seed] For integers d≥1, m≥0, and the action of M_d( C)⊗ I_m on C^d⊗ C^m, let C_ψ be the coefficient matrix of ψ. Then dimspan{(A⊗ I_m)ψ:A∈ M_d( C)} = drank C_ψ. The vector is cyclic exactly when rank C_ψ=m.
Exact mathematical passage
C_\psi
C_\ellUse in source · 25 passages
Search aliases: C_ℓ ell
Use in source · 05_defects.tex, line 213
Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.
Exact mathematical passage
C_\ell:X_{\ell+1}\longrightarrow X_\ell
C_{\ell+1,a}Use in source · 10 passages
Search aliases: C_ℓ+1,a ell
Use in source · 02_strengthened.tex, line 2261
Let ( H_ℓ,C_ℓ,a) and ( H_ℓ+1,C_ℓ+1,a) be two levels, and let R_ℓ: H_ℓ⟶ H_ℓ+1 be an isometry.
Exact mathematical passage
(\Hcal_{\ell+1},C_{\ell+1,a})
C_{\ell+1,b}Use in source · 3 passages
Search aliases: C_ℓ+1,b ell
Use in source · 02_strengthened.tex, line 2273
[Exact constraint transport] Assume C_ℓ+1,aR_ℓ=R_ℓ C_ℓ,a for every a. Then [C_ℓ+1,a,C_ℓ+1,b]R_ℓ = R_ℓ[C_ℓ,a,C_ℓ,b]. If the same structure coefficients are used at both levels, then A_ℓ+1,abR_ℓ = R_ℓ A_ℓ,ab. The isometry maps coarse physical states into fine physical states.
Exact mathematical passage
[C_{\ell+1,a},C_{\ell+1,b}]R_\ell = R_\ell[C_{\ell,a},C_{\ell,b}].
C_{\ell+1,L}Use in source · 3 passages
Search aliases: C_ℓ+1,L ell
Use in source · 05_defects.tex, line 228
[Multilevel defect decomposition] The total defect from level L to level zero is C_0,LT_L-T_0C_0,L = ∑_ℓ=0^L-1 C_0,ℓΔ_ℓ C_ℓ+1,L. For bounded maps, C_0,LT_L-T_0C_0,L ≤ ∑_ℓ=0^L-1 C_0,ℓ Δ_ℓ C_ℓ+1,L .
Exact mathematical passage
C_{0,L}T_L-T_0C_{0,L} = \sum_{\ell=0}^{L-1} C_{0,\ell}\Delta_\ell C_{\ell+1,L}.
C_{\ell,a}Use in source · 10 passages
Search aliases: C_ℓ,a ell
Use in source · 02_strengthened.tex, line 2261
Let ( H_ℓ,C_ℓ,a) and ( H_ℓ+1,C_ℓ+1,a) be two levels, and let R_ℓ: H_ℓ⟶ H_ℓ+1 be an isometry.
Exact mathematical passage
(\Hcal_\ell,C_{\ell,a})
C_{\ell,b}Use in source · 3 passages
Search aliases: C_ℓ,b ell
Use in source · 02_strengthened.tex, line 2273
[Exact constraint transport] Assume C_ℓ+1,aR_ℓ=R_ℓ C_ℓ,a for every a. Then [C_ℓ+1,a,C_ℓ+1,b]R_ℓ = R_ℓ[C_ℓ,a,C_ℓ,b]. If the same structure coefficients are used at both levels, then A_ℓ+1,abR_ℓ = R_ℓ A_ℓ,ab. The isometry maps coarse physical states into fine physical states.
Exact mathematical passage
[C_{\ell+1,a},C_{\ell+1,b}]R_\ell = R_\ell[C_{\ell,a},C_{\ell,b}].
C_{\ell,j}Use in source · 2 passages
Search aliases: C_ℓ,j ell
Use in source · Paper_VI_Traditional_Final.tex, line 5253
Let C_j:X_j+1→ X_j be bounded linear maps, and put C_ℓ,m=C_ℓ⋯ C_m-1 for ℓ<m, with C_j,j=I. No common ambient space is assumed. [Multilevel telescoping identity] Let T_j be bounded operators on X_j, and set E_j=C_jT_j+1-T_jC_j. Then C_ℓ,mT_m-T_ℓ C_ℓ,m = ∑_j=ℓ^m-1 C_ℓ,jE_jC_j+1,m.
Exact mathematical passage
\boxed{ C_{\ell,m}T_m-T_\ell C_{\ell,m} = \sum_{j=\ell}^{m-1} C_{\ell,j}E_jC_{j+1,m}. }
C_{\ell,j+1}Use in source · 1 passage
Search aliases: C_ℓ,j+1 ell
Use in source · Paper_VI_Traditional_Final.tex, line 5265
Expand each summand as C_ℓ,j+1T_j+1C_j+1,m - C_ℓ,jT_jC_j,m. The intermediate terms cancel. The first surviving term is C_ℓ,mT_m, and the last is -T_ℓ C_ℓ,m.
Exact mathematical passage
C_{\ell,j+1}T_{j+1}C_{j+1,m} - C_{\ell,j}T_jC_{j,m}.
C_{\ell,L}Use in source · 1 passage
Search aliases: C_ℓ,L ell
Use in source · 05_defects.tex, line 247
Insert the intermediate terms C_0,ℓT_ℓ C_ℓ,L for 0≤ℓ≤ L. Consecutive differences telescope. The difference between the terms indexed by ℓ+1 and ℓ is C_0,ℓ(C_ℓ T_ℓ+1-T_ℓ C_ℓ)C_ℓ+1,L. Summing gives the identity. The norm estimate follows from the triangle inequality.
Exact mathematical passage
C_{0,\ell}T_\ell C_{\ell,L}
C_{\ell,m}Use in source · 5 passages
Search aliases: C_ℓ,m ell
Use in source · Paper_VI_Traditional_Final.tex, line 5246
Let C_j:X_j+1→ X_j be bounded linear maps, and put C_ℓ,m=C_ℓ⋯ C_m-1 for ℓ<m, with C_j,j=I. No common ambient space is assumed. [Multilevel telescoping identity] Let T_j be bounded operators on X_j, and set E_j=C_jT_j+1-T_jC_j. Then C_ℓ,mT_m-T_ℓ C_ℓ,m = ∑_j=ℓ^m-1 C_ℓ,jE_jC_j+1,m.
Exact mathematical passage
C_{\ell,m}=C_\ell\cdots C_{m-1}
\cA^X_aUse in source · 1 passage
Search aliases: cA^X_a A^X_a cA mathcal
Use in source · 02_connections_obstructions.tex, line 713
[Exact naturality for arbitrary functions] Let A^X_a:Xxrightarrow∼X_a and A^X_b:Xxrightarrow∼X_b be exact coordinate equivalences for X, with inverses S^X_a and S^X_b. Likewise, let A^Y_r:Yxrightarrow∼Y_r and A^Y_s:Yxrightarrow∼Y_s be exact coordinate equivalences for Y, with inverses S^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity or differentiability hypothesis. Set J^X_b← a= A^X_b∘ S^X_a, J^Y_s← r= A^Y_s∘ S^Y_r, and define widehat F_r← a = A^Y_r∘ F∘ S^X_a. Then J^Y_s← r∘widehat…
Exact mathematical passage
\cA^X_a:X\xrightarrow{\sim}X_a
\cA^X_bUse in source · 2 passages
Search aliases: cA^X_b A^X_b cA mathcal
Use in source · 02_connections_obstructions.tex, line 714
[Exact naturality for arbitrary functions] Let A^X_a:Xxrightarrow∼X_a and A^X_b:Xxrightarrow∼X_b be exact coordinate equivalences for X, with inverses S^X_a and S^X_b. Likewise, let A^Y_r:Yxrightarrow∼Y_r and A^Y_s:Yxrightarrow∼Y_s be exact coordinate equivalences for Y, with inverses S^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity or differentiability hypothesis. Set J^X_b← a= A^X_b∘ S^X_a, J^Y_s← r= A^Y_s∘ S^Y_r, and define widehat F_r← a = A^Y_r∘ F∘ S^X_a. Then J^Y_s← r∘widehat…
Exact mathematical passage
\cA^X_b:X\xrightarrow{\sim}X_b
\cA^YUse in source · 1 passage
Search aliases: cA^Y A^Y cA mathcal
Use in source · 02_connections_obstructions.tex, line 752
Both sides of the naturality equation reduce to A^Y_s∘ F∘ S^X_a using S∘ A=id in X and Y. For composition, the middle S^Y∘ A^Y cancels in the same way.
Exact mathematical passage
\cS^Y\circ\cA^Y
\cA^Y_rUse in source · 3 passages
Search aliases: cA^Y_r A^Y_r cA mathcal
Use in source · 01_finite_coordinates.tex, line 511
If F:X→ Y is linear and P^a, P^b are source packet systems while Q^r, Q^s are target packet systems, put widehat F_r← a = A^Y_rF S^X_a.
Exact mathematical passage
\widehat F_{r\leftarrow a} = \cA^Y_rF\cS^X_a.
\cA^Y_sUse in source · 4 passages
Search aliases: cA^Y_s A^Y_s cA mathcal
Use in source · 01_finite_coordinates.tex, line 529
Both sides reduce to A^Y_sF S^X_a after inserting the appropriate synthesis--analysis identity. The composition law is the same calculation with the middle ambient module inserted.
Exact mathematical passage
\cA^Y_sF\cS^X_a
\cA^Z_tUse in source · 1 passage
Search aliases: cA^Z_t A^Z_t cA mathcal
Use in source · 02_connections_obstructions.tex, line 737
^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity or differentiability hypothesis. Set J^X_b← a= A^X_b∘ S^X_a, J^Y_s← r= A^Y_s∘ S^Y_r, and define widehat F_r← a = A^Y_r∘ F∘ S^X_a. Then J^Y_s← r∘widehat F_r← a = widehat F_s← b∘ J^X_b← a. If A^Z_t:Zxrightarrow∼Z_t is another exact coordinate equivalence and G:Y→ Z is any function, then exact coordinate representation also preserves composition: widehatG∘ F_t← a = widehat G_t← r∘widehat F_r← a. statusleanthreeKomorebi.PaperIII.Transport…
Exact mathematical passage
\cA^Z_t:Z\xrightarrow{\sim}Z_t
\cA^{\oplus r}Use in source · 2 passages
Search aliases: cA^⊕ r A^⊕ r cA oplus mathcal oplus
Use in source · 05_worked_models.tex, line 438
For a finite ordered seed family S=(s_1,…,s_r), let ε_S: A^⊕ r→ X be the seed evaluation map of [source reference]. Its kernel is the relation module. The free module modulo that kernel presents the generated hull by [source reference]. The invariant content is: · generated hull: Hull_ A(S),
Exact mathematical passage
\varepsilon_S:\cA^{\oplus r}\to X
\cA_aUse in source · 3 passages
Search aliases: cA_a A_a cA mathcal
Use in source · 01_finite_coordinates.tex, line 450
ordinate transport is a separate construction. Let X be an R-module, and let P^a=(P^a_i)_i∈ I_a be a finite complete family of pairwise orthogonal linear idempotents: P^a_iP^a_j=δ_ijP^a_i, ∑_i∈ I_aP^a_i=I_X. Define the exact packet coordinate module Coord( P^a) = {(x_i)_i∈ I_a:P^a_ix_i=x_i}. Its analysis and synthesis maps are A_a(x)=(P^a_ix)_i, S_a((x_i)_i)=∑_ix_i. Completeness and orthogonality give S_a A_a=id_X, A_a S_a=id_Coord( P^a).
Exact mathematical passage
\cA_a(x)=(P^a_ix)_i, \qquad \cS_a((x_i)_i)=\sum_ix_i.
\cA_cUse in source · 1 passage
Search aliases: cA_c A_c cA mathcal
Use in source · 01_finite_coordinates.tex, line 498
Insert S_b A_b=id_X: J^X_c← bJ^X_b← a = A_c S_b A_b S_a = A_c S_a =J^X_c← a. The identity statement is A_a S_a=id. Setting c=a in the cocycle gives J^X_a← bJ^X_b← a=id. Interchanging a and b gives the other composition, so the two transports are inverses.
Exact mathematical passage
J^X_{c\leftarrow b}J^X_{b\leftarrow a} =\cA_c\cS_b\cA_b\cS_a =\cA_c\cS_a =J^X_{c\leftarrow a}.
\cA_NUse in source · 4 passages
Search aliases: cA_N A_N cA mathcal
Use in source · Paper_V_Traditional_Final.tex, line 351
Define the analysis and synthesis maps A_Nx=(x_0,…,x_N-1), S_N(x_0,…,x_N-1)[Nq+r]=x_r[q].
Exact mathematical passage
\cA_Nx=(x_0,\ldots,x_{N-1}), \qquad \cS_N(x_0,\ldots,x_{N-1})[Nq+r]=x_r[q].
\cA_QUse in source · 1 passage
Search aliases: cA_Q A_Q cA mathcal
Use in source · 02_coordinates.tex, line 189
The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_Q_b,Q_a widehat T_Q_a,P_a = widehat T_Q_b,P_b J^X_P_b,P_a. The identity, cocycle, composition, inverse, and naturality laws are all proved in Lean. Th…
Exact mathematical passage
\widehat T_{Q,P}=\cA_QT\cS_P
\calH_\thetaUse in source · 1 passage
Search aliases: calH_θ H_θ calH theta mathcal theta
Use in source · Paper_XI_Traditional_Final.tex, line 1210
[Restricted Heisenberg bridge] A restricted bridge datum determines an isomorphism between the two Heisenberg representations, unique up to a nonzero scalar. More precisely, there is an invertible map U: H_ρ⟶ H_θ such that Uω_ρ(h) = ω_θ(ι_H(h))U for every Heisenberg element h, and every other such map is a scalar multiple of U.
Exact mathematical passage
U:\calH_\rho\longrightarrow\calH_\theta
\calH_\rhoUse in source · 1 passage
Search aliases: calH_ρ H_ρ calH rho mathcal rho
Use in source · Paper_XI_Traditional_Final.tex, line 1210
[Restricted Heisenberg bridge] A restricted bridge datum determines an isomorphism between the two Heisenberg representations, unique up to a nonzero scalar. More precisely, there is an invertible map U: H_ρ⟶ H_θ such that Uω_ρ(h) = ω_θ(ι_H(h))U for every Heisenberg element h, and every other such map is a scalar multiple of U.
Exact mathematical passage
U:\calH_\rho\longrightarrow\calH_\theta
\mathrm{can}Use in source · 2 passages
Search aliases: can mathrm
Use in source · Paper_V_Traditional_Final.tex, line 2165
[Coefficient noise under canonical synthesis] Under the hypotheses of [source reference], suppose the observed coefficient vector is c= A_Ex+ν. Canonical synthesis gives widehat x=R_canc and widehat x-x _2 ≤ frac1sqrt A ν _2.
Exact mathematical passage
\widehat x=R_{\mathrm{can}}c
\mathsf{Candidate}Use in source · 2 passages
Search aliases: Candidate mathsf
Use in source · 02_connections_obstructions.tex, line 397
Let Candidate, Packet, and Residual be arbitrary types, with a distinguished 0∈Residual, and let r:Candidate⟶ Packet⟶Residual be a residual function.
Exact mathematical passage
\mathsf{Candidate}
\cBUse in source · 17 passages
Search aliases: cB B mathcal
Use in source · 03_weighted.tex, line 237
Let O be a complete discrete valuation ring with uniformizer π, fraction field K, and filtration F^m B=π^m B on an O-algebra B. The finite quotient statements below do not require B itself to be complete. Suppose T=D+H, where D is invertible and the perturbation H raises filtration. The symbol K is reserved for the fraction field.
Exact mathematical passage
F^m\cB=\pi^m\cB
\cB_{\bar\lambda}Use in source · 5 passages
Search aliases: cB_barλ B_barλ cB bar lambda mathcal bar lambda
Use in source · Paper_VI_Traditional_Final.tex, line 881
[Bayesian refinement criterion] Assume that the partition contains at least two coarse packets and that the fine and coarse likelihoods are strictly positive. The identity C_π B_λ(p) = B_barλ(C_π p) holds for every fine prior p if and only if there exists one scalar c>0 such that λ_i=c barλ_a (i∈ I_a). Equivalently, the likelihood is constant on each coarse packet, with the coarse values determined up to the globally invisible Bayesian scaling c.
Exact mathematical passage
\boxed{ C_\pi\cB_\lambda(p) = \cB_{\bar\lambda}(C_\pi p)}
\cB_{\bar\lambda+\delta}Use in source · 1 passage
Search aliases: cB_barλ+δ B_barλ+δ cB bar lambda delta mathcal bar lambda delta
Use in source · Paper_VI_Traditional_Final.tex, line 955
The effective-likelihood theorem reduces the defect to B_barλ+δ(P)- B_barλ(P). Since Z_r=Z_barλ+P^ Tδ, direct subtraction gives frac1Z_r (diag(P)-bar bP^ T)δ. Also, (diag(P)-bar bP^ T)barλ =Podotbarλ-bar bZ_barλ=0. The first norm bound is the likelihood perturbation theorem applied on the coarse simplex. For a zero-mass packet, both sides of the following estimate vanish. For P_a>0, P_a|δ_a| = P_a|∑_i∈ I_afracp_iP_a (λ_i-barλ_a)| ≤ ∑_i∈ I_ap_i|λ_i-barλ_a|, and summing over a proves the second in…
Exact mathematical passage
\cB_{\bar\lambda+\delta}(P)-\cB_{\bar\lambda}(P).
\cB_{r(p)}Use in source · 2 passages
Search aliases: cB_r(p) B_r(p) cB mathcal
Use in source · Paper_VI_Traditional_Final.tex, line 855
[Exact effective-likelihood factorization] Whenever the fine evidence is positive, C_π B_λ(p) = B_r(p)(C_π p). Thus the only fine information needed for the coarse posterior, beyond the coarse prior masses, is the vector of conditional packet evidences r(p).
Exact mathematical passage
\boxed{ C_\pi\cB_\lambda(p) = \cB_{r(p)}(C_\pi p).}
\cB_\lambdaUse in source · 10 passages
Search aliases: cB_λ B_λ cB lambda mathcal lambda
Use in source · Paper_VI_Traditional_Final.tex, line 731
Let p=(p_i) be a probability vector and let λ=(λ_i) be a nonnegative likelihood with positive evidence Z_λ(p)=∑_iλ_ip_i. Define B_λ(p) = fracλodot pZ_λ(p). This map is nonlinear solely because of its scalar reconstruction factor.
Exact mathematical passage
\boxed{ \cB_\lambda(p) = \frac{\lambda\odot p}{Z_\lambda(p)}. }
\cB_{\lambda^{(r)}}Use in source · 1 passage
Search aliases: cB_λ^(r) B_λ^(r) cB lambda mathcal lambda
Use in source · Paper_VI_Traditional_Final.tex, line 1093
Let K be column stochastic and let λ^(r) be the likelihood associated with the rth observation. The filter recursion is widehat p_r=Kp_r-1, p_r= B_λ^(r)(widehat p_r).
Exact mathematical passage
\widehat p_r=Kp_{r-1}, \qquad p_r=\cB_{\lambda^{(r)}}(\widehat p_r).
\cB_{\lambda\odot\mu}Use in source · 1 passage
Search aliases: cB_λodotμ B_λodotμ cB lambda odot mu mathcal lambda odot mu
Use in source · Paper_VI_Traditional_Final.tex, line 742
[Independent-likelihood composition] For nonnegative likelihoods λ and μ, assume Z_λ(p)>0 and Z_λodotμ(p)>0. Equivalently, both successive updates have positive evidence. Then B_μ( B_λ(p)) = B_λodotμ(p).
Exact mathematical passage
\boxed{ \cB_\mu(\cB_\lambda(p)) = \cB_{\lambda\odot\mu}(p). }
\cB_\muUse in source · 2 passages
Search aliases: cB_μ B_μ cB mu mathcal mu
Use in source · Paper_VI_Traditional_Final.tex, line 742
[Independent-likelihood composition] For nonnegative likelihoods λ and μ, assume Z_λ(p)>0 and Z_λodotμ(p)>0. Equivalently, both successive updates have positive evidence. Then B_μ( B_λ(p)) = B_λodotμ(p).
Exact mathematical passage
\boxed{ \cB_\mu(\cB_\lambda(p)) = \cB_{\lambda\odot\mu}(p). }
\cCUse in source · 3 passages
Search aliases: cC C mathcal
Use in source · 01_finite_core.tex, line 792
Let P=P^*=P^2 project onto a code space C⊂ H, and let E_1,…,E_r be a finite family of complex error matrices. Neither a nonzero code nor channel normalization is assumed unless stated. The operators P E_a^*E_bP are the obstruction packets on the code. If they retain information about the logical state beyond one scalar multiple of P, the environment can distinguish logical states and exact correction is impossible.
Exact mathematical passage
\cC\subset\cH
\CC^{2n}Use in source · 1 passage
Search aliases: CC^2n C^2n CC mathbb
Use in source · 01_manuscript.tex, line 573
[Toeplitz circulant embedding] Let n≥1 be an integer and T∈ C^n× n be Toeplitz, with first column (c_0,c_1,…,c_n-1)^T and first row (c_0,r_1,…,r_n-1). Set g = (c_0,c_1,…,c_n-1,0,r_n-1,…,r_1)^T ∈ C^2n and append n zeros to x∈ C^n. The first n entries of the circulant product C(g)(x,0)^T equal Tx.
Exact mathematical passage
g = (c_0,c_1,\ldots,c_{n-1},0,r_{n-1},\ldots,r_1)^T \in\CC^{2n}
\CC^3Use in source · 6 passages
Search aliases: CC^3 C^3 CC mathbb
Use in source · monograph_atlas.tex, line 74
vpage5Why one seed is not always enoughAN EXACT MULTIPLICITY LOWER BOUNDChapter VII: Theorem II.2.3. Consider M_2( C)⊗ I_3 acting on C^2⊗ C^3. A seed is a 2×3 coefficient matrix. Its two rows span at most two directions in the three-dimensional multiplicity factor. Therefore no single seed is cyclic for this algebra. C_1=mat1 · 0 · 0
Exact mathematical passage
\CC^2\otimes\CC^3
\CC^4Use in source · 1 passage
Search aliases: CC^4 C^4 CC mathbb
Use in source · monograph_atlas.tex, line 753
l is Cket0, and the full algebra closes: [C_x,C_y]=2 i C_z, [C_y,C_z]=2 i C_x, [C_z,C_x]=2 i C_y. These are uncompressed identities. They supply an actual closure test in the quotient by the constraint span, unlike the automatic compressed zero on the physical kernel. xfigconstraint_image headExact transport holds on the coarse image Embed by the coordinate isometry R: C^3→ C^4 and set widetilde C_a=C_a⊕1. Then widetilde C_aR=RC_a, so [widetilde C_a,widetilde C_b]R=R[C_a,C_b]. The coarse physica…
Exact mathematical passage
R:\CC^3\to\CC^4
\CC^6Use in source · 1 passage
Search aliases: CC^6 C^6 CC mathbb
Use in source · monograph_atlas.tex, line 86
o groups of size at most d attains this bound. For a direct sum of such blocks, the worst block determines the required count: srank_AA(HH)=max_ilceil m_i/d_irceil. This is an invariant of the declared representation of the algebra, and depends on more than the dimension of the Hilbert space alone. If the available algebra were enlarged to the full matrix algebra on C^6, every nonzero seed would instead be cyclic.
Exact mathematical passage
\CC^6
\CC^8Use in source · 1 passage
Search aliases: CC^8 C^8 CC mathbb
Use in source · monograph_atlas.tex, line 52
2,a_3), put W_a=X_1^a_1X_2^a_2X_3^a_3. W_aψ_0=keta_1a_2a_3. The drawing shows the actual eight states reached by these words. Dashed edges connect the two squares by changing the first bit. The other edges change one of the remaining bits. vfigaddresses headTwo different meanings of ``reach'' The eight vectors are orthonormal, so their linear span is all of C^8. This proves cyclicity for the algebra generated by the flips. Preparing every normalized vector requires more than products of these fl…
Exact mathematical passage
\CC^8
\CC^cUse in source · 5 passages
Search aliases: CC^c C^c CC mathbb
Use in source · Paper_XI_Traditional_Final.tex, line 918
Let A be a finite abelian group, and let c:A× A→ C^× be a normalized two cocycle. The twisted group algebra C^c[A] has basis (u_a)_a∈ A and multiplication u_au_b=c(a,b)u_a+b. Define the commutator bicharacter b_c(a,b)=c(a,b)c(b,a)^-1.
Exact mathematical passage
\CC^c[A]
\CC^{c_\rho}Use in source · 1 passage
Search aliases: CC^c_ρ C^c_ρ CC rho mathbb rho
Use in source · Paper_XI_Traditional_Final.tex, line 1099
[Nondegenerate one break packet] If b_ρ is nondegenerate, then C^c_ρ[A] ≅ End(M_ρ), dim M_ρ=sqrtabs A.
Exact mathematical passage
\CC^{c_\rho}[A] \cong \End(M_\rho), \qquad \dim M_\rho=\sqrt{\abs A}.
\CC^dUse in source · 1 passage
Search aliases: CC^d C^d CC mathbb
Use in source · 02_strengthened.tex, line 120
[Orbit dimension of one seed] For integers d≥1, m≥0, and the action of M_d( C)⊗ I_m on C^d⊗ C^m, let C_ψ be the coefficient matrix of ψ. Then dimspan{(A⊗ I_m)ψ:A∈ M_d( C)} = drank C_ψ. The vector is cyclic exactly when rank C_ψ=m.
Exact mathematical passage
\CC^d\otimes\CC^m
\CC^{d_r}Use in source · 1 passage
Search aliases: CC^d_r C^d_r CC mathbb
Use in source · 02_strengthened.tex, line 111
Let A ≅ ⊕_r=1^s (M_d_r( C)⊗ I_m_r) act on H ≅ ⊕_r=1^s ( C^d_r⊗ C^m_r). Assume d_r,m_r≥1 for each displayed block. For an empty block family, the seed rank and the displayed finite supremum are defined as zero. A vector in one block is identified with a d_r by m_r coefficient matrix.
Exact mathematical passage
\Hcal \cong \bigoplus_{r=1}^{s} \bigl(\CC^{d_r}\otimes\CC^{m_r}\bigr).
\CC^GUse in source · 2 passages
Search aliases: CC^G C^G CC mathbb
Use in source · Paper_V_Traditional_Final.tex, line 1750
[Finite periodic signals] Let G= Z^d/L, where L≤ Z^d has finite index. The translation orbit of the impulse at the identity is a basis of C^G. In particular, one impulse generates every signal on the finite periodic lattice.
Exact mathematical passage
\CC^G
\CC^mUse in source · 2 passages
Search aliases: CC^m C^m CC mathbb
Use in source · 02_strengthened.tex, line 120
[Orbit dimension of one seed] For integers d≥1, m≥0, and the action of M_d( C)⊗ I_m on C^d⊗ C^m, let C_ψ be the coefficient matrix of ψ. Then dimspan{(A⊗ I_m)ψ:A∈ M_d( C)} = drank C_ψ. The vector is cyclic exactly when rank C_ψ=m.
Exact mathematical passage
\CC^d\otimes\CC^m
\CC^{m_r}Use in source · 2 passages
Search aliases: CC^m_r C^m_r CC mathbb
Use in source · 02_strengthened.tex, line 111
Let A ≅ ⊕_r=1^s (M_d_r( C)⊗ I_m_r) act on H ≅ ⊕_r=1^s ( C^d_r⊗ C^m_r). Assume d_r,m_r≥1 for each displayed block. For an empty block family, the seed rank and the displayed finite supremum are defined as zero. A vector in one block is identified with a d_r by m_r coefficient matrix.
Exact mathematical passage
\Hcal \cong \bigoplus_{r=1}^{s} \bigl(\CC^{d_r}\otimes\CC^{m_r}\bigr).
\CC^{m\times m}Use in source · 1 passage
Search aliases: CC^m× m C^m× m CC times mathbb times
Use in source · 01_manuscript.tex, line 587
[Kronecker modewise application] Let A∈ C^m× m, B∈ C^n× n, and X∈ C^n× m. With column major vectorization, (A⊗ B)vec(X) = vec(BXA^T). The right side has modeled cost 2n^2m+2nm^2 and stores n^2+m^2 factor entries instead of n^2m^2 entries.
Exact mathematical passage
A\in\CC^{m\times m}
\CC^{m\times r}Use in source · 1 passage
Search aliases: CC^m× r C^m× r CC times mathbb times
Use in source · 01_manuscript.tex, line 805
[Small core recompression] Let L∈ C^m× r, R∈ C^n× r. Take compact QR factorizations L=Q_LR_L, R=Q_RR_R, where the columns of Q_L and Q_R are orthonormal. Form the small core G=R_LR_R^* and let G=WΣ Z^* be an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer k≥0, retain min{k,d} columns, where d is the number of supplied thin singular directions. Tail singular values beyond d are zero. Define L_k=Q_LW_kΣ_k^1/2, R_k=Q_RZ_kΣ_k^1/2. Then L_kR_k…
Exact mathematical passage
L\in\CC^{m\times r}, \qquad R\in\CC^{n\times r}.
\CC^{n\times m}Use in source · 1 passage
Search aliases: CC^n× m C^n× m CC times mathbb times
Use in source · 01_manuscript.tex, line 587
[Kronecker modewise application] Let A∈ C^m× m, B∈ C^n× n, and X∈ C^n× m. With column major vectorization, (A⊗ B)vec(X) = vec(BXA^T). The right side has modeled cost 2n^2m+2nm^2 and stores n^2+m^2 factor entries instead of n^2m^2 entries.
Exact mathematical passage
X\in\CC^{n\times m}
\CC^{n\times n}Use in source · 2 passages
Search aliases: CC^n× n C^n× n CC times mathbb times
Use in source · 01_manuscript.tex, line 564
[Toeplitz circulant embedding] Let n≥1 be an integer and T∈ C^n× n be Toeplitz, with first column (c_0,c_1,…,c_n-1)^T and first row (c_0,r_1,…,r_n-1). Set g = (c_0,c_1,…,c_n-1,0,r_n-1,…,r_1)^T ∈ C^2n and append n zeros to x∈ C^n. The first n entries of the circulant product C(g)(x,0)^T equal Tx.
Exact mathematical passage
T\in\CC^{n\times n}
\CC^rUse in source · 2 passages
Search aliases: CC^r C^r CC mathbb
Use in source · 02_strengthened.tex, line 851
Let P(x) be a smooth rank r orthogonal projector over a parameter manifold. Choose a local isometry V(x): C^r⟶ H, V(x)V(x)^*=P(x).
Exact mathematical passage
V(x):\CC^r\longrightarrow\Hcal, \qquad V(x)V(x)^*=P(x).
\CC^\timesUse in source · 1 passage
Search aliases: CC^× C^× CC times mathbb times
Use in source · Paper_XI_Traditional_Final.tex, line 1157
Let θ be a maximal simple character for GL_n(K), with compact open subgroups H_θ^1⊆ J_θ^1⊆ J_θ. The quotient V_θ=J_θ^1/H_θ^1 is regarded as a finite vector space over F_p, where p is the residue characteristic. Fix a nontrivial additive character ψ_0: F_p→μ_p⊂ C^×, where μ_p is the group of p-th roots of unity. The simple-character commutator takes values in μ_p. Since ψ_0 is injective, the equation ψ_0(h_θ(bar x,bar y)) = θ([x,y]) uniquely defines the F_p-valued commutator pairing h_θ. We use t…
Exact mathematical passage
\psi_0:\FF_p\to\mu_p\subset\CC^\times
\Ccal_{2,0}Use in source · 1 passage
Search aliases: Ccal_2,0 C_2,0 Ccal mathcal
Use in source · 72_recovered.tex, line 664
[Euler and Bernoulli boundaries] For (N,j)=(2,0), C_2,0(t)=secht, so C_2,0(n)=E_2n. For (N,j)=(2,1), C_2,1(t)=fractsinh t, and C_2,1(n) = 2(1-2^2n-1)B_2n. The usual convention B_1=-1/2 is used. Euler and Bernoulli number conventions are reviewed in [source reference].
Exact mathematical passage
\Ccal_{2,0}(t)=\operatorname{sech}t,
\Ccal_{2,1}Use in source · 1 passage
Search aliases: Ccal_2,1 C_2,1 Ccal mathcal
Use in source · 72_recovered.tex, line 672
[Euler and Bernoulli boundaries] For (N,j)=(2,0), C_2,0(t)=secht, so C_2,0(n)=E_2n. For (N,j)=(2,1), C_2,1(t)=fractsinh t, and C_2,1(n) = 2(1-2^2n-1)B_2n. The usual convention B_1=-1/2 is used. Euler and Bernoulli number conventions are reviewed in [source reference].
Exact mathematical passage
\Ccal_{2,1}(t)=\frac{t}{\sinh t},
\Ccal_{N,j}Use in source · 2 passages
Search aliases: Ccal_N,j C_N,j Ccal mathcal
Use in source · 72_recovered.tex, line 586
[Congruential reciprocal packet] Define C_N,j(t) = fract^j/j! E_N,j(t) = ∑_n≥0 C_N,j(n) fract^Nn(Nn)!.
Exact mathematical passage
\boxed{ \Ccal_{N,j}(t) = \frac{t^j/j!}{\Ecal_{N,j}(t)} = \sum_{n\geq0} C_{N,j}(n) \frac{t^{Nn}}{(Nn)!}. }
\cDUse in source · 2 passages
Search aliases: cD D mathcal
Use in source · 01_finite_core.tex, line 691
[Block-diagonal measurement invariance] Let {P_α} be a complete packet family, and suppose every measurement effect is packet diagonal: E_k=∑_α P_α E_kP_α. Define the packet-dephased state D_P(ρ)=∑_α P_αρ P_α. Then Tr(E_kρ) = Tr(E_k D_P(ρ)) for every outcome k.
Exact mathematical passage
\cD_P(\rho)=\sum_\alpha P_\alpha\rho P_\alpha.
\cD_PUse in source · 2 passages
Search aliases: cD_P D_P cD mathcal
Use in source · 01_finite_core.tex, line 691
[Block-diagonal measurement invariance] Let {P_α} be a complete packet family, and suppose every measurement effect is packet diagonal: E_k=∑_α P_α E_kP_α. Define the packet-dephased state D_P(ρ)=∑_α P_αρ P_α. Then Tr(E_kρ) = Tr(E_k D_P(ρ)) for every outcome k.
Exact mathematical passage
\cD_P(\rho)=\sum_\alpha P_\alpha\rho P_\alpha.
\cFUse in source · 2 passages
Search aliases: cF F mathcal
Use in source · 01_finite_core.tex, line 632
For an orthogonal rank-one complex projector, use the scalar connection convention A=-i⟨ u,du⟩ in a local unit frame. The associated scalar curvature density is F_μν =-iTr(P[∂_μ P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)·σ. Using Tr(P a·σ)= n· a therefore yields F_θϕ =frac12 n·(∂_θ nmathbin×∂_ϕ n) =frac12sinθ. The scalar sign changes if the opposite convention for A i…
Exact mathematical passage
\cF_{\mu\nu} =-i\Tr\left(P[\partial_\mu P,\partial_\nu P]\right).
\cF_{\theta\phi}Use in source · 1 passage
Search aliases: cF_θϕ F_θϕ cF theta phi mathcal theta phi
Use in source · 01_finite_core.tex, line 652
I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)·σ. Using Tr(P a·σ)= n· a therefore yields F_θϕ =frac12 n·(∂_θ nmathbin×∂_ϕ n) =frac12sinθ. The scalar sign changes if the opposite convention for A is chosen. This explicit calculation is not a separate entry among the 19 items in the Lean register.
Exact mathematical passage
\boxed{\cF_{\theta\phi} =\frac12\mathbf n\cdot(\partial_\theta\mathbf n\mathbin{\times}\partial_\phi\mathbf n) =\frac12\sin\theta.}
\cF_{\mu\nu}Use in source · 1 passage
Search aliases: cF_μν F_μν cF mu nu mathcal mu nu
Use in source · 01_finite_core.tex, line 632
For an orthogonal rank-one complex projector, use the scalar connection convention A=-i⟨ u,du⟩ in a local unit frame. The associated scalar curvature density is F_μν =-iTr(P[∂_μ P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)·σ. Using Tr(P a·σ)= n· a therefore yields F_θϕ =frac12 n·(∂_θ nmathbin×∂_ϕ n) =frac12sinθ. The scalar sign changes if the opposite convention for A i…
Exact mathematical passage
\cF_{\mu\nu} =-i\Tr\left(P[\partial_\mu P,\partial_\nu P]\right).
\Cfun_{N,j}^rUse in source · 2 passages
Search aliases: Cfun_N,j^r C_N,j^r Cfun mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 1524
Multiply the block series for C_N,j^r by the exponential series and collect the coefficient of total degree m.
Exact mathematical passage
\Cfun_{N,j}^r
\mathrm{ch}Use in source · 1 passage
Search aliases: ch mathrm
Use in source · Paper_V_Traditional_Final.tex, line 362
[Analysis--synthesis identity] S_N A_N=I. On the product of the channel spaces one also has A_N S_N=I. Equip this finite product with the Hilbert norm (x_0,…,x_N-1) _ch^2 =∑_r=0^N-1 x_r _2^2. Then A_N is a linear isometric equivalence and x _2^2=∑_r=0^N-1 x_r _2^2. Euclidean division gives a unique pair (r,q) with 0≤ r<N and n=Nq+r, including negative n. The address maps n↦(nbmod N,nmathbindivN) and (r,q)↦ Nq+r are inverse. Analysis and synthesis are reindexing by this bijection. Reindexing the …
Exact mathematical passage
\|(x_0,\ldots,x_{N-1})\|_{\mathrm{ch}}^2 =\sum_{r=0}^{N-1}\|x_r\|_2^2.
\cH_AUse in source · 2 passages
Search aliases: cH_A H_A cH mathcal
Use in source · 01_finite_core.tex, line 137
[Local orbit and Schmidt rank] Let ψ∈ H_A⊗ H_B, with dim H_A=d_A≥1, and let r be its Schmidt rank. The reduced positive operator on B is ρ_B=Tr_A(ketψbraψ). Normalization of ψ does not affect its support. Then dimspan{(A⊗ I_B)ψ:A∈ M_d_A( C)} =d_A r. In particular, local operators on A generate the complete bipartite Hilbert space exactly when the reduced state on B has full support.
Exact mathematical passage
\psi\in\cH_A\otimes\cH_B
\cH_BUse in source · 2 passages
Search aliases: cH_B H_B cH mathcal
Use in source · 01_finite_core.tex, line 137
[Local orbit and Schmidt rank] Let ψ∈ H_A⊗ H_B, with dim H_A=d_A≥1, and let r be its Schmidt rank. The reduced positive operator on B is ρ_B=Tr_A(ketψbraψ). Normalization of ψ does not affect its support. Then dimspan{(A⊗ I_B)ψ:A∈ M_d_A( C)} =d_A r. In particular, local operators on A generate the complete bipartite Hilbert space exactly when the reduced state on B has full support.
Exact mathematical passage
\psi\in\cH_A\otimes\cH_B
\cH_\ellUse in source · 3 passages
Search aliases: cH_ℓ H_ℓ cH ell mathcal ell
Use in source · 01_finite_core.tex, line 1003
At resolution ℓ, define Q_ℓ = ( A_ℓ, H_ℓ, D_ℓ, {P_ℓ,α}, E_ℓ, C_ℓ, Π_ℓ^phys, T_ℓ ). In order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_ℓ: H_ℓ→ H_ℓ+1 and C_ℓ: H_ℓ+1→ H_ℓ. The left inverse relation C_ℓ E_ℓ=I alone does not imply C_ℓ=E_ℓ^* or make the embedding isometric.
Exact mathematical passage
\boxed{ \mathfrak Q_\ell = \left( \cA_\ell, \cH_\ell, D_\ell, \{P_{\ell,\alpha}\}, E_\ell, C_\ell, \Pi_\ell^{\mathrm{phys}}, T_\ell \right). }
\cH_{\ell+1}Use in source · 2 passages
Search aliases: cH_ℓ+1 H_ℓ+1 cH ell mathcal ell
Use in source · 01_finite_core.tex, line 1019
hfrak Q_ℓ = ( A_ℓ, H_ℓ, D_ℓ, {P_ℓ,α}, E_ℓ, C_ℓ, Π_ℓ^phys, T_ℓ ). In order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_ℓ: H_ℓ→ H_ℓ+1 and C_ℓ: H_ℓ+1→ H_ℓ. The left inverse relation C_ℓ E_ℓ=I alone does not imply C_ℓ=E_ℓ^* or make the embedding isometric.
Exact mathematical passage
E_\ell:\cH_\ell\to\cH_{\ell+1}
\operatorname{char}Use in source · 2 passages
Search aliases: char operatorname
Use in source · 05_worked_models.tex, line 61
Let char𝕜≠2 and let A=𝕜[x,y]/(y^2-x^3+x). The affine equation y^2=x^3-x has classical discriminant 64, which is nonzero under this hypothesis. Standard algebraic geometry then identifies its smooth projective completion as an elliptic curve [source reference]. The Lean development checks the algebraic implication 2≠0⇒64≠0, the quotient normal form, and the operator identities below. It does not formalize projective smoothness, genus, or the geometric identification of the discriminant. The packe…
Exact mathematical passage
\operatorname{char}\bbk\ne2
\mathrm{child}Use in source · 1 passage
Search aliases: child mathrm
Use in source · 02_supplement_02.tex, line 20
Recurrence. For terminal nodes, F_λ(v)=κ_v+λϵ_v. For internal nodes, F_λ(v)=min{κ_v+λϵ_v, σ_v+∑_w∈child(v)F_λ(w)}. The checked result has two parts. First, F_λ(v)≤ J_λ(L) for every legal cut of the subtree. Second, the constructed function codecompile returns a legal cut with objective exactly F_λ(v). Thus the minimum is attained, not merely bounded below.
Exact mathematical passage
\boxed{F_\lambda(v)=\min\left\{\kappa_v+\lambda\epsilon_v, \;\sigma_v+\sum_{w\in\mathrm{child}(v)}F_\lambda(w)\right\}.}
\operatorname{Child}Use in source · 3 passages
Search aliases: Child operatorname
Use in source · Paper_V_Traditional_Final.tex, line 385
Suppose N,M≥1 and Nmid M. Every coarse phase has a disjoint family of fine descendants: Child_M(N,r) = {s:0≤ s<M, s≡ rpmod N}.
Exact mathematical passage
\operatorname{Child}_{M}(N,r) = \{s:0\le s<M,\ s\equiv r\pmod N\}.
\operatorname{Child}_{M}Use in source · 1 passage
Search aliases: Child_M operatorname
Use in source · Paper_V_Traditional_Final.tex, line 385
Suppose N,M≥1 and Nmid M. Every coarse phase has a disjoint family of fine descendants: Child_M(N,r) = {s:0≤ s<M, s≡ rpmod N}.
Exact mathematical passage
\operatorname{Child}_{M}(N,r) = \{s:0\le s<M,\ s\equiv r\pmod N\}.
\cIndUse in source · 1 passage
Search aliases: cInd operatorname
Use in source · Paper_XI_Traditional_Final.tex, line 2133
At both ranks K^× normalizes P. Thus N=K^× J is normal of index two in M=K^× P, and Σ=Ind_ N^ MΛ, π≃cInd_ M^GL_N(F)Σ. Induction in stages preserves the full original Λ, its central character, and its noncompact root. See [source reference]. The full marked scalar-pushout comparison applies to these actual enlarged pairs as well as to the original higher-rank types ([source reference]). The full synthesis and scope are in [source reference].
Exact mathematical passage
\Sigma=\Ind_{\mathcal N}^{\mathcal M}\Lambda, \qquad \pi\simeq\cInd_{\mathcal M}^{\GL_N(F)}\Sigma.
\cInd_{\mathcal M}^{\GL_N(F)}Use in source · 1 passage
Search aliases: cInd_ M^GL_N(F) cInd mathcal GL operatorname mathcal operatorname
Use in source · Paper_XI_Traditional_Final.tex, line 2133
At both ranks K^× normalizes P. Thus N=K^× J is normal of index two in M=K^× P, and Σ=Ind_ N^ MΛ, π≃cInd_ M^GL_N(F)Σ. Induction in stages preserves the full original Λ, its central character, and its noncompact root. See [source reference]. The full marked scalar-pushout comparison applies to these actual enlarged pairs as well as to the original higher-rank types ([source reference]). The full synthesis and scope are in [source reference].
Exact mathematical passage
\Sigma=\Ind_{\mathcal N}^{\mathcal M}\Lambda, \qquad \pi\simeq\cInd_{\mathcal M}^{\GL_N(F)}\Sigma.
\cKUse in source · 3 passages
Search aliases: cK K mathcal
Use in source · Paper_VI_Traditional_Final.tex, line 1533
For the homogeneous system dot x=Ax, variation of constants gives x_Q(t)=e^tA_QQx_Q(0) +∫_0^te^(t-r)A_QQA_QPx_P(r) dr. Substitution yields dot x_P(t) =A_PPx_P(t)+A_PQe^tA_QQx_Q(0) +∫_0^t K(t-r)x_P(r) dr, where K(t)=A_PQe^tA_QQA_QP.
Exact mathematical passage
\dot x_P(t) =A_{PP}x_P(t)+A_{PQ}e^{tA_{QQ}}x_Q(0) +\int_0^t\cK(t-r)x_P(r)\,dr,
\mathrm{CNOT}Use in source · 1 passage
Search aliases: CNOT mathrm
Use in source · monograph_atlas.tex, line 136
le vector occupies both parity packets. The final vector occupies only the even packet. vfigbell_circuit headHow the exact path expansion works Insert P_rm even+P_rm odd=I between the two gates. Each summand records a particular intermediate packet, while the boundary projectors select its input and output packets. Summing all terms recovers the original 4×4 circuit matrix: ∑_a,b,cP_b CNOT P_a (H_rm Had⊗ I)P_c =CNOT(H_rm Had⊗ I). This expansion is an identity even though the intermediate parity …
Exact mathematical passage
\sum_{a,b,c}P_b\,\mathrm{CNOT}\,P_a (H_{\rm Had}\otimes I)P_c =\mathrm{CNOT}(H_{\rm Had}\otimes I).
\mathrm{coarse}Use in source · 2 passages
Search aliases: coarse mathrm
Use in source · 01_finite_core.tex, line 247
[Tensor/polyphase refinement] For each residue r∈ Z/N Z, define P_r^coarse = ∑_j_1+⋯+j_m≡ rpmod N P^(1)_j_1⊗⋯⊗ P^(m)_j_m. Then the P_r^coarse are complete orthogonal projectors, and every coarse packet is the exact sum of its fine descendants.
Exact mathematical passage
\boxed{ P_r^{\mathrm{coarse}} = \sum_{j_1+\cdots+j_m\equiv r\pmod N} P^{(1)}_{j_1}\otimes\cdots\otimes P^{(m)}_{j_m}. }
\codeUse in source · 1 passage
Search aliases: code
Use in source · 02_supplement_03.tex, line 136
For a predicate S on costs, the common minimum specification means: ll codenone: · no cost satisfies S, codesome(v): · S(v) and v≤ z whenever S(z). Both attainment and minimality are included. Optional minimum and optional addition are proved to implement union and independent sum of feasible sets.
Exact mathematical passage
\begin{array}{ll} \code{none}: & \text{no cost satisfies }S,\\ \code{some}(v): & S(v)\text{ and }v\leq z\text{ whenever }S(z). \end{array}
\operatorname{coe}Use in source · 1 passage
Search aliases: coe operatorname
Use in source · supplement_11.tex, line 96
Identity with the original rational coefficients. The integral lift is constructed from the decoded coefficients of the actual reciprocal. The code proves coe(e_j(n))=C_4,j(n) in Q_2, and proves that the coefficient map commutes with every difference stencil. Multiplying by j! yields the unnormalized family. The valuation result is not divisibility in a field.
Exact mathematical passage
\operatorname{coe}(e_j(n))=C_{4,j}(n)\quad\text{in }\Q_2,
\coefUse in source · 6 passages
Search aliases: coef
Use in source · supplement_09.tex, line 92
mate becomes relevant only after connecting it to the manuscript's generating functions. For a residue r∈ Z/N Z, define H_r(t)=∑_d≥01_d≡ r (N)fract^dd!, F_r(T)=H_r(log(1+T)). The implementation uses the actual formal logarithm and substitution in Q[[T]]. It proves H_r'=H_r-1, (1+T)F_r'=F_r-1, and hence the coefficient recurrence (d+1)coefTd+1F_r =coefTdF_r-1-dcoefTdF_r. The initial values are F_r(0)=1_r=0.
Exact mathematical passage
(d+1)\coef{T}{d+1}F_r =\coef{T}{d}F_{r-1}-d\coef{T}{d}F_r.
\operatorname{coeff}Use in source · 1 passage
Search aliases: coeff operatorname
Use in source · 03_supplement_03.tex, line 46
Schmidt rank and the reduced state. Use coordinates ψ_ij in the standard product basis and let C_ij=ψ_ij. An explicit linear equivalence identifies the state vector on the product index set with its coefficient matrix. The implementation checks the actual Kronecker action: coeff((A⊗ I_B)ψ)=AC. It also proves that the transported matrix orbit equals the span of the state vectors obtained by this Kronecker action. The dimension formula therefore applies to the physical local action on the state sp…
Exact mathematical passage
\operatorname{coeff}\bigl((A\otimes I_B)\psi\bigr)=AC.
\operatorname{coker}Use in source · 1 passage
Search aliases: coker operatorname
Use in source · 71_native.tex, line 134
coker(Hom_B(Ω_P/ Z⊗_P B, F_p)⟶Hom_B(I/I^2, F_p)).
Exact mathematical passage
\operatorname{coker}\left(\operatorname{Hom}_B(\Omega_{P/\mathbb Z}\otimes_P B,\mathbb F_p)\longrightarrow\operatorname{Hom}_B(I/I^2,\mathbb F_p)\right).
\commUse in source · 2 passages
Search aliases: comm
Use in source · 03_supplement_02.tex, line 146
Define C=∑_aD_aP_a(t). Then C=frac12∑_acommD_aP_a(t), C^*=-C, and, for every b, commCP_b(t)=D_b.
Exact mathematical passage
\boxed{\quad C=\frac12\sum_a\comm{D_a}{P_a(t)},\qquad C^*=-C,\quad}
\operatorname{compressed}Use in source · 1 passage
Search aliases: compressed operatorname
Use in source · 02_supplement_02.tex, line 203
Contracts, grouping, and amortization. textitSource items 2.9, 10.2, 10.4--10.6, and 11.3 Output conversion. A decoded run returns the actual pair (decode(compressed(x)), t_run(x)+t_decode(compressed(x))). Under the declared output equality, Lean proves that this returns the same explicit result and charges the conversion time. It also proves the downstream composition identity when compressed output is consumed directly. Omitting a positive decode cost strictly understates the modeled explicit-…
Exact mathematical passage
\bigl(\operatorname{decode}(\operatorname{compressed}(x)), \;t_{\mathrm{run}}(x)+t_{\mathrm{decode}}(\operatorname{compressed}(x))\bigr).
\condUse in source · 2 passages
Search aliases: cond operatorname
Use in source · 01_manuscript.tex, line 1192
[Block Jacobi condition bound] Let A=D+K act on a nonempty finite coordinate space and be Hermitian positive definite with positive definite block diagonal part D. Here D^-1/2 is the actual inverse of the positive square root of D. Suppose E=D^-1/2KD^-1/2 is Hermitian and normE_2≤η<1. Then cond_2(D^-1/2AD^-1/2) ≤ frac1+η1-η.
Exact mathematical passage
\boxed{ \cond_2(D^{-1/2}AD^{-1/2}) \leq \frac{1+\eta}{1-\eta}. }
\cond_2Use in source · 2 passages
Search aliases: cond_2 cond operatorname
Use in source · 01_manuscript.tex, line 1192
[Block Jacobi condition bound] Let A=D+K act on a nonempty finite coordinate space and be Hermitian positive definite with positive definite block diagonal part D. Here D^-1/2 is the actual inverse of the positive square root of D. Suppose E=D^-1/2KD^-1/2 is Hermitian and normE_2≤η<1. Then cond_2(D^-1/2AD^-1/2) ≤ frac1+η1-η.
Exact mathematical passage
\boxed{ \cond_2(D^{-1/2}AD^{-1/2}) \leq \frac{1+\eta}{1-\eta}. }
\coprodUse in source · 1 passage
Search aliases: coprod
Use in source · 02_supplement_02.tex, line 161
Explicit storage and operation models. textitSource items 3.1, 3.2, 3.4, 3.6, 3.8--3.10, and 4.2 For a finite collection of rectangular blocks, the scalar addresses are actual elements of coprod_e∈ E(Fin(m_e)×Fin(n_e)). The corresponding operation addresses include a right-hand-side index and one of two scalar-operation tags. Lean proves their cardinalities rather than merely defining a formula named ``cost.''
Exact mathematical passage
\coprod_{e\in E}\bigl(\operatorname{Fin}(m_e)\times\operatorname{Fin}(n_e)\bigr).
\coprod_{e\in E}Use in source · 1 passage
Search aliases: coprod_e∈ E coprod in
Use in source · 02_supplement_02.tex, line 161
Explicit storage and operation models. textitSource items 3.1, 3.2, 3.4, 3.6, 3.8--3.10, and 4.2 For a finite collection of rectangular blocks, the scalar addresses are actual elements of coprod_e∈ E(Fin(m_e)×Fin(n_e)). The corresponding operation addresses include a right-hand-side index and one of two scalar-operation tags. Lean proves their cardinalities rather than merely defining a formula named ``cost.''
Exact mathematical passage
\coprod_{e\in E}\bigl(\operatorname{Fin}(m_e)\times\operatorname{Fin}(n_e)\bigr).
\cos^2Use in source · 2 passages
Search aliases: cos^2 cos
Use in source · supplement_02.tex, line 101
Connecting to the chapter's Z_n convention. The chapter defines Z_n through sec t+tan t. The Lean construction uses Mathlib's actual formal sine and cosine series and defines sec t=(cos t)^-1, tan t=(sin t)sec t, Z_n=n. It proves the sine and cosine derivative identities, sin^2t+cos^2t=1, and (tan t)'=sec^2t. Sign substitution in the squared variable carries the hyperbolic cosine kernel to the actual cosine kernel. The same homomorphism carries their inverses and their squares…
Exact mathematical passage
\sin^2t+\cos^2t=1
\costUse in source · 11 passages
Search aliases: cost operatorname
Use in source · pages.tex, line 387
ne. Keeping either a or b costs three with error two. Splitting a costs four with zero error, and splitting b costs five with zero error. vfigtree_frontier The five complete prunings have (cost,error) pairs (1,6), (7,4), (9,2), (8,2), (10,0). At budget two the unique cheapest pruning is {a_0,a_1,b}, costing eight. It is Pareto optimal, but never minimizes cost+λerr for any λ≥0. headSee the incompatible inequalities To beat the root, that pruning would require 8+2λ≤1+6λ, hence λ≥7/4. To beat the …
Exact mathematical passage
\cost+\lambda\err
\cothUse in source · 5 passages
Search aliases: coth
Use in source · 20_arithmetic.tex, line 960
Let Q(z)=zcoth z, the regular formal series with constant coefficient one. The elementary differential identity (frac zsinh z)^2=Q(z)-zQ'(z) and the Bernoulli expansion Q(z)=1+∑_m≥1frac2^2mB_2m(2m)!z^2m give β_m=-2^2m(2m-1)B_2m/(2m)!. Multiplication by 2^2m(2m)! proves the identity for κ_m. The Bernoulli denominator theorem gives v_2(B_2m)=-1 for m>0: its denominator contains each prime q with (q-1)mid 2m exactly once, including 2[source reference]. Since 2m-1 is odd, the valuation is exactly 4m…
Exact mathematical passage
Q(z)=z\coth z
\CovUse in source · 1 passage
Search aliases: Cov operatorname
Use in source · Paper_VI_Traditional_Final.tex, line 5517
Along the Bayesian flow p_i(t)=e^tg_ip_i(0)/∑_j e^tg_jp_j(0), the following statement holds for every real observable h. [Covariance identity] For any observable h∈ R^n, frac d d t E_p(t)[h] = Cov_p(t)(h,g). In particular, frac d d t E_p(t)[g] = Var_p(t)(g)≥0.
Exact mathematical passage
\frac{\dd}{\dd t}\EE_{p(t)}[h] = \Cov_{p(t)}(h,g).
\Cov_{p(t)}Use in source · 1 passage
Search aliases: Cov_p(t) Cov operatorname
Use in source · Paper_VI_Traditional_Final.tex, line 5517
Along the Bayesian flow p_i(t)=e^tg_ip_i(0)/∑_j e^tg_jp_j(0), the following statement holds for every real observable h. [Covariance identity] For any observable h∈ R^n, frac d d t E_p(t)[h] = Cov_p(t)(h,g). In particular, frac d d t E_p(t)[g] = Var_p(t)(g)≥0.
Exact mathematical passage
\frac{\dd}{\dd t}\EE_{p(t)}[h] = \Cov_{p(t)}(h,g).
\covolUse in source · 3 passages
Search aliases: covol operatorname
Use in source · residue_continuation_body.tex, line 38
There is a constant B≥0, depending on K,M,J, such that, for every a∈ J/M and every real t≥1, |#{x∈ J:xbmod M=a, ι(x)∈ tD} -fracvol(D)q covol(L(J))t^d| ≤ B(t^d-1+1). The same constant works for all residues a.
Exact mathematical passage
\left|\#\{x\in J:x\bmod M=a,\ \iota(x)\in tD\} -\frac{\vol(D)}{q\,\covol(L(J))}t^d\right| \leq B(t^{d-1}+1).
\cQUse in source · 1 passage
Search aliases: cQ Q mathcal
Use in source · 01_finite_coordinates.tex, line 510
If F:X→ Y is linear and P^a, P^b are source packet systems while Q^r, Q^s are target packet systems, put widehat F_r← a = A^Y_rF S^X_a.
Exact mathematical passage
\cQ^r,\cQ^s
\cQ^rUse in source · 1 passage
Search aliases: cQ^r Q^r cQ mathcal
Use in source · 01_finite_coordinates.tex, line 510
If F:X→ Y is linear and P^a, P^b are source packet systems while Q^r, Q^s are target packet systems, put widehat F_r← a = A^Y_rF S^X_a.
Exact mathematical passage
\cQ^r,\cQ^s
\cQ^sUse in source · 1 passage
Search aliases: cQ^s Q^s cQ mathcal
Use in source · 01_finite_coordinates.tex, line 510
If F:X→ Y is linear and P^a, P^b are source packet systems while Q^r, Q^s are target packet systems, put widehat F_r← a = A^Y_rF S^X_a.
Exact mathematical passage
\cQ^r,\cQ^s
\crefUse in source · 1 passage
Search aliases: cref
Use in source · visual_atlas.tex, line 19
[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…
Exact mathematical passage
4pt] \textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt] \textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology? \end{tabularx} A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. \takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''} \vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}} Most of the chapter can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix. {\small \begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule \textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule $X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\ $S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\ $\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\ $\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\ $P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\ $X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\ $T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\ $F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\ $U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\ $\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\ $\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule \end{tabularx}} \head{Read products from right to left} In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column. \head{Where to begin} For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}. \vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$} Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define \[ Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0, \qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.
\cS^X_aUse in source · 6 passages
Search aliases: cS^X_a S^X_a cS mathcal
Use in source · 01_finite_coordinates.tex, line 511
If F:X→ Y is linear and P^a, P^b are source packet systems while Q^r, Q^s are target packet systems, put widehat F_r← a = A^Y_rF S^X_a.
Exact mathematical passage
\widehat F_{r\leftarrow a} = \cA^Y_rF\cS^X_a.
\cS^X_bUse in source · 1 passage
Search aliases: cS^X_b S^X_b cS mathcal
Use in source · 02_connections_obstructions.tex, line 715
[Exact naturality for arbitrary functions] Let A^X_a:Xxrightarrow∼X_a and A^X_b:Xxrightarrow∼X_b be exact coordinate equivalences for X, with inverses S^X_a and S^X_b. Likewise, let A^Y_r:Yxrightarrow∼Y_r and A^Y_s:Yxrightarrow∼Y_s be exact coordinate equivalences for Y, with inverses S^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity or differentiability hypothesis. Set J^X_b← a= A^X_b∘ S^X_a, J^Y_s← r= A^Y_s∘ S^Y_r, and define widehat F_r← a = A^Y_r∘ F∘ S^X_a. Then J^Y_s← r∘widehat…
Exact mathematical passage
\cS^X_b
\cS^YUse in source · 1 passage
Search aliases: cS^Y S^Y cS mathcal
Use in source · 02_connections_obstructions.tex, line 752
Both sides of the naturality equation reduce to A^Y_s∘ F∘ S^X_a using S∘ A=id in X and Y. For composition, the middle S^Y∘ A^Y cancels in the same way.
Exact mathematical passage
\cS^Y\circ\cA^Y
\cS^Y_rUse in source · 2 passages
Search aliases: cS^Y_r S^Y_r cS mathcal
Use in source · 02_connections_obstructions.tex, line 718
[Exact naturality for arbitrary functions] Let A^X_a:Xxrightarrow∼X_a and A^X_b:Xxrightarrow∼X_b be exact coordinate equivalences for X, with inverses S^X_a and S^X_b. Likewise, let A^Y_r:Yxrightarrow∼Y_r and A^Y_s:Yxrightarrow∼Y_s be exact coordinate equivalences for Y, with inverses S^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity or differentiability hypothesis. Set J^X_b← a= A^X_b∘ S^X_a, J^Y_s← r= A^Y_s∘ S^Y_r, and define widehat F_r← a = A^Y_r∘ F∘ S^X_a. Then J^Y_s← r∘widehat…
Exact mathematical passage
\cS^Y_r
\cS^Y_sUse in source · 1 passage
Search aliases: cS^Y_s S^Y_s cS mathcal
Use in source · 02_connections_obstructions.tex, line 718
beginproposition[Exact naturality for arbitrary functions] Let A^X_a:Xxrightarrow∼X_a and A^X_b:Xxrightarrow∼X_b be exact coordinate equivalences for X, with inverses S^X_a and S^X_b. Likewise, let A^Y_r:Yxrightarrow∼Y_r and A^Y_s:Yxrightarrow∼Y_s be exact coordinate equivalences for Y, with inverses S^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity or differentiability hypothesis. Set J^X_b← a= A^X_b∘ S^X_a, J^Y_s← r= A^Y_s∘ S^Y_r, and define widehat F_r← a = A^Y_r∘ F∘ S^X_a. Then …
Exact mathematical passage
\cS^Y_s
\cS_bUse in source · 2 passages
Search aliases: cS_b S_b cS mathcal
Use in source · 01_finite_coordinates.tex, line 497
Insert S_b A_b=id_X: J^X_c← bJ^X_b← a = A_c S_b A_b S_a = A_c S_a =J^X_c← a. The identity statement is A_a S_a=id. Setting c=a in the cocycle gives J^X_a← bJ^X_b← a=id. Interchanging a and b gives the other composition, so the two transports are inverses.
Exact mathematical passage
\cS_b\cA_b=\id_X
\cS_NUse in source · 3 passages
Search aliases: cS_N S_N cS mathcal
Use in source · Paper_V_Traditional_Final.tex, line 351
Define the analysis and synthesis maps A_Nx=(x_0,…,x_N-1), S_N(x_0,…,x_N-1)[Nq+r]=x_r[q].
Exact mathematical passage
\cA_Nx=(x_0,\ldots,x_{N-1}), \qquad \cS_N(x_0,\ldots,x_{N-1})[Nq+r]=x_r[q].
\cS_PUse in source · 1 passage
Search aliases: cS_P S_P cS mathcal
Use in source · 02_coordinates.tex, line 189
The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_Q_b,Q_a widehat T_Q_a,P_a = widehat T_Q_b,P_b J^X_P_b,P_a. The identity, cocycle, composition, inverse, and naturality laws are all proved in Lean. Th…
Exact mathematical passage
\widehat T_{Q,P}=\cA_QT\cS_P
\mathbb DUse in source · 3 passages
Search aliases: D mathbb
Use in source · Paper_V_Traditional_Final.tex, line 3658
Let H = {z∈ C:Re z>0} and define C(z)=fracz-1z+1. It maps H conformally onto the unit disk D, with inverse C^-1(w)=frac1+w1-w. For θ∈ R, let R_θ(w)= e^ iθw.
Exact mathematical passage
\mathbb D
\mathbf DUse in source · 6 passages
Search aliases: D mathbf
Use in source · 04_transport.tex, line 402
[Equational realization principle] Let F: C→ D be an additive functor between preadditive categories. Any equality of morphisms constructed from declared generating morphisms by finite sums, additive inverses, identities, zero maps, and composition is preserved by F. Scalar multiplication by a coefficient ring is also preserved when the categories are enriched over that ring and F is linear for the declared enrichment. If a proof additionally uses a chosen biproduct, kernel, cokernel, quotient, …
Exact mathematical passage
F:\mathbf C\to\mathbf D
\mathcal DUse in source · 16 passages
Search aliases: D mathcal
Use in source · 10_foundations.tex, line 78
Over a characteristic-zero field K, put E(a)=∑_n≥0fraca_nn!t^n, D(A)_n=n![t^n]A. These inverse maps between sequences and K[[t]] provide formal exponential coordinates. No analytic convergence is needed for the coefficient identities below.
Exact mathematical passage
\mathcal E(a)=\sum_{n\geq0}\frac{a_n}{n!}t^n, \qquad \mathcal D(A)_n=n![t^n]A.
\mathfrak DUse in source · 9 passages
Search aliases: D mathfrak
Use in source · 02_connections_obstructions.tex, line 889
For coordinate models of the source and target colors, an additive target coordinate space, and a chosen noncanonical connection, a pointwise node defect can be defined by D^Γ_b,a(Φ) = Γ^Y_b← a∘widehatΦ_a - widehatΦ_b∘ (∏_qΓ^X_q_b← a). This expression compares two chosen routes. With canonical exact transport in every slot it vanishes by [source reference].
Exact mathematical passage
\mathfrak D^\Gamma_{b,a}(\Phi) = \Gamma^Y_{b\leftarrow a}\circ\widehat\Phi_a - \widehat\Phi_b\circ \left(\prod_q\Gamma^{X_q}_{b\leftarrow a}\right).
D'Use in source · 5 passages
Search aliases: D'
Use in source · 03_nonlinear_coordinates.tex, line 953
[Naturality of obstruction classes] Suppose ccc U · xrightarrowD · V mathllapadownarrow · · downarrowmathrlapb U' · xrightarrowD' · V' is a commuting square. Then there is a unique induced map overline b:coker D⟶coker D' satisfying overline b Ob_D(r)=Ob_D'(br).
Exact mathematical passage
\begin{array}{ccc} U & \xrightarrow{D} & V\\ \mathllap{a}\downarrow & & \downarrow\mathrlap{b}\\ U' & \xrightarrow{D'} & V' \end{array}
D'''Use in source · 2 passages
Search aliases: D'''
Use in source · atlas-pages.tex, line 224
retain the reduced logarithmic information: T_N(z)=N∑_m≥1κ_N,mz^Nm-1, T_N(n)=N(Nn+N-1)!κ_N,n+1. For order four, κ_4,1=1/24 and κ_4,2=17/10080. These give T_4(0)=1 and T_4(1)=34, explaining the last line of the plate. headA nonlinear equation with a precise origin If D=e^g locally, then g'=-T. Differentiating e^g three times gives D'''/D=-T''+3TT'-T^3. Setting D'''/D=-1 yields the cubic Riccati identity shown above. No additional nonlinear model has been assumed. The reciprocal family also satisf…
Exact mathematical passage
D'''/D=-T''+3TT'-T^3
D^{(m)}Use in source · 4 passages
Search aliases: D^(m)
Use in source · 02_supplement_03.tex, line 10
Low-rank factors through a product chain. textitSource item 5.2 Let A_j=D_j+U_jV_j^*, U_j,V_j∈ C^n× r_j, with all D_j diagonal in one packet layout. Set P_m=A_m⋯ A_1, D^(m)=D_m⋯ D_1. The formalization constructs thin matrices L_m,R_m such that P_m=D^(m)+L_mR_m^*. Their column index type is a recursively constructed disjoint sum. Its cardinality is proved to equal ∑_j=1^m r_j.
Exact mathematical passage
P_m=A_m\cdots A_1,\qquad D^{(m)}=D_m\cdots D_1.
D^{(m-1)}Use in source · 2 passages
Search aliases: D^(m-1)
Use in source · 02_supplement_03.tex, line 19
Construction. Suppose P_m-1=D^(m-1)+L_m-1R_m-1^*. The new factors are L_m · =[D_mL_m-1+U_m(V_m^*L_m-1) U_m],
Exact mathematical passage
P_{m-1}=D^{(m-1)}+L_{m-1}R_{m-1}^*
D^*Use in source · 4 passages
Search aliases: D^*
Use in source · 03_nonlinear_coordinates.tex, line 26
A linear correction map D:U⟶ V places the obstruction to solving Du=-r in the canonical quotient coker D. This quotient has a universal property and behaves naturally under commuting squares. Mapping cones extend the same construction to cochain complexes. In a Hilbert space with closed range, the quotient is represented by the orthogonal projection onto Ker D^*. The Moore--Penrose inverse then gives the least norm correction. Relevant background appears in [source reference].
Exact mathematical passage
\Ker D^*
D^{-(r+s)}Use in source · 1 passage
Search aliases: D^-(r+s)
Use in source · atlas-pages.tex, line 238
2π i, and the power by e^-2π ir. headThree different kinds of statement A formal identity compares coefficients at the origin. A holomorphic identity for complex powers uses a common specified logarithm. A positive integer reciprocal power is globally meromorphic and needs no branch choice. These scopes cannot be exchanged without a further argument. For example, D^-rD^-s=D^-(r+s) is immediate using one fixed logarithm. Combining values built from unrelated branches can introduce additional phas…
Exact mathematical passage
D^{-r}D^{-s}=D^{-(r+s)}
D^{-1}Use in source · 55 passages
Search aliases: D^-1
Use in source · 03_weighted.tex, line 36
[Weighted inverse] Let B be a complete filtered algebra. Let D∈ B be invertible with D,D^-1∈ F^0 B. Let E be a finite set and suppose T=D+∑_e∈ EN_e, N_e∈ F^w(e) B, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 = ∑_γ(-1)^|γ|A_γ D^-1. The series converges in the filtration, and the term indexed by γ belongs to F^W(γ) B.
Exact mathematical passage
D,D^{-1}\in F^0\Bcal.
D^{-1/2}Use in source · 5 passages
Search aliases: D^-1/2
Use in source · pages.tex, line 301
vpage18Measure coupling after normalizationBLOCK JACOBI AS A GEOMETRIC CHANGEChapter VIII: Theorem 7.4, Corollary 7.6, Classical input 7.5. Let A=mat4 · 1 1 · 1, D=diag(4,1), B=D^-1/2=diag(1/2,1). Then BAB=I+E=mat1 · 1/2
Exact mathematical passage
A=\mat{4&1\\1&1},\quad D=\diag(4,1),\quad B=D^{-1/2}=\diag(1/2,1).
D^{-n-1}Use in source · 2 passages
Search aliases: D^-n-1
Use in source · 02_strengthened.tex, line 1633
[Low energy expansion] Assume D is invertible and abs znormD^-1<1. Then Σ(z) =- ∑_n=0^∞ z^nBD^-n-1B^* with norm convergence. In particular, F_P(z) = z-A+BD^-1B^*+O(abs znorm B^2normD^-1^2).
Exact mathematical passage
\Sigma(z) =- \sum_{n=0}^{\infty} z^nBD^{-n-1}B^*
D^{-r}Use in source · 1 passage
Search aliases: D^-r
Use in source · atlas-pages.tex, line 238
2π i, and the power by e^-2π ir. headThree different kinds of statement A formal identity compares coefficients at the origin. A holomorphic identity for complex powers uses a common specified logarithm. A positive integer reciprocal power is globally meromorphic and needs no branch choice. These scopes cannot be exchanged without a further argument. For example, D^-rD^-s=D^-(r+s) is immediate using one fixed logarithm. Combining values built from unrelated branches can introduce additional phas…
Exact mathematical passage
D^{-r}D^{-s}=D^{-(r+s)}
D^{-s}Use in source · 1 passage
Search aliases: D^-s
Use in source · atlas-pages.tex, line 238
2π i, and the power by e^-2π ir. headThree different kinds of statement A formal identity compares coefficients at the origin. A holomorphic identity for complex powers uses a common specified logarithm. A positive integer reciprocal power is globally meromorphic and needs no branch choice. These scopes cannot be exchanged without a further argument. For example, D^-rD^-s=D^-(r+s) is immediate using one fixed logarithm. Combining values built from unrelated branches can introduce additional phas…
Exact mathematical passage
D^{-r}D^{-s}=D^{-(r+s)}
\mathcal D^2Use in source · 2 passages
Search aliases: D^2 mathcal
Use in source · strengthening.tex, line 626
The variable B=2Z+1 has transform e^tfrac2te^2t-1=fractsinh t. Its odd moments vanish, and P=( B^2-1)/4=Z(Z+1) has the divided-power moments of Lemma~[source reference]. Introduce an independent copy D and Q=( D^2-1)/4. The exact factorization F_2(t)=fract^2cosh t-cos t =fracatsinh(at)fracbtsinh(bt) shows that U=a B+b D satisfies L(U^4n)=C_2(n). Direct expansion gives H_2:=fracU^4-14 =P+Q-P^2-Q^2+6PQ+i B D(P-Q).
Exact mathematical passage
Q=(\mathcal D^2-1)/4
d^2Use in source · 4 passages
Search aliases: d^2
Use in source · atlas-pages.tex, line 832
t t=log2, they are Z=7/3 and p=(1,2,4)/7. vfigext_cumulants platecaptionThe score values stay fixed. Their probabilities change. The rising mean score is a property of this tilted family, not an increase in the individual scores. headTwo derivatives have different statistical meanings The first derivative of log evidence is the current mean score, and the second is its variance: fracddtlog Z(t)= E_p(t)u, fracd^2dt^2log Z(t)=Var_p(t)(u)≥0. At q=2 the exact values are E u=frac107, E u^2=frac187, V…
Exact mathematical passage
\frac{d}{dt}\log Z(t)=\EE_{p(t)}u, \qquad \frac{d^2}{dt^2}\log Z(t)=\Var_{p(t)}(u)\ge0.
D^4Use in source · 3 passages
Search aliases: D^4
Use in source · 20_arithmetic.tex, line 512
For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [source reference] proves that multiplication adds charges. If FG=1, the same formula yields A_-cG=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G'). Induction, starting…
Exact mathematical passage
A_c=\frac{D^4-c}{2}.
d^4Use in source · 1 passage
Search aliases: d^4
Use in source · 01_finite_core.tex, line 1240
[Einstein infrared limit] There exist semiclassical packet states Ψ_g,ℓ such that the effective action converges to frac116π G ∫sqrt-g (R-2Λ) d^4x with controlled higher-order corrections.
Exact mathematical passage
\frac1{16\pi G} \int\sqrt{-g}\,(R-2\Lambda)\,d^4x
D^8Use in source · 1 passage
Search aliases: D^8
Use in source · supplement_11.tex, line 73
3. The dyadic differential operator and reciprocity. The operator required here is B_4=fracD^8-D^42, D=fracddt. It is not the preceding p=2 specialization (D^4-D^2)/2. It is also not an ordinary derivation. The new algebra module proves the actual product formula, including the middle derivative sums whose coefficients are binom 8i/2 and binom 4i/2.
Exact mathematical passage
\boxed{B_4=\frac{D^8-D^4}{2},\qquad D=\frac{d}{dt}.}
D^\bulletUse in source · 1 passage
Search aliases: D^bullet bullet
Use in source · Paper_X_Block27_Amended.tex, line 417
Naturality. If a cochain map F:C^bullet→ D^bullet intertwines the supplied projectors, it restricts to maps F_j:C_j^bullet→ D_j^bullet. The direct-sum decomposition commutes with H^r(F) and the family H^r(F_j). This is checked using representatives in the actual quotients, independently of any dimension computation.
Exact mathematical passage
F:C^\bullet\to D^\bullet
D^{d-r}Use in source · 1 passage
Search aliases: D^d-r
Use in source · 10_foundations.tex, line 140
[Hurwitz product rules] For d≥0, D^d(f⋆ g)=∑_r=0^dbinom dr(D^r f)⋆(D^d-rg). If R has prime characteristic p, then D^p(f⋆ g)=f⋆ D^p g+D^p f⋆ g. Over a field, if g_0=1, a sequence b is constructed recursively with b⋆ g=1. If also D^p g=Dg in characteristic p, then D^p b=Db.
Exact mathematical passage
D^d(f\star g)=\sum_{r=0}^{d}\binom dr(D^r f)\star(D^{d-r}g).
D^iUse in source · 2 passages
Search aliases: D^i
Use in source · strengthening.tex, line 75
[Translated product formula] For N=p^a, a≥1, and λ,μ∈ Z_p, L_λ+μ(AB) =(L_λ A)B+A(L_μ B) +∑_i=1^N-1c_i(D^iA)(D^N-iB), c_i=frac1pbinom Ni∈ Z. Flatness types add under multiplication.
Exact mathematical passage
L_{\lambda+\mu}(AB) =(L_\lambda A)B+A(L_\mu B) +\sum_{i=1}^{N-1}c_i(D^iA)(D^{N-i}B), \qquad c_i=\frac1p\binom Ni\in\mathbb Z. \label{eq:translated-product}
D^jUse in source · 6 passages
Search aliases: D^j
Use in source · pages.tex, line 292
With A=diag(1,2), the vectors v and Av=(1,2)^T are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors as extra address operators separates the coordinates even when A=I_2. Both changes enlarge the addressing power, but in different ways. headWhen invariance is only approximate If A=D+K with D packet diagonal, then A^j-D^j ≤ jM^j-1 K , M=max{ A , D }. This controls departure from the decoupled iterate. It is not a claim that the di…
Exact mathematical passage
\|A^j-D^j\|\le jM^{j-1}\|K\|, \qquad M=\max\{\|A\|,\|D\|\}.
D^kUse in source · 1 passage
Search aliases: D^k
Use in source · 01_manuscript.tex, line 1169
Use the telescoping identity A^j-D^j = ∑_k=0^j-1 A^j-1-kKD^k. Each term has norm at most M^j-1normK. The vector D^jv lies in the direct sum of local Krylov spaces.
Exact mathematical passage
A^j-D^j = \sum_{k=0}^{j-1} A^{j-1-k}KD^k.
d^kUse in source · 3 passages
Search aliases: d^k
Use in source · Paper_IV_Traditional_Final.tex, line 488
[Differential cycle and initial data] For 1≤ j<N, fracddz E_N,j(λ z) = λ E_N,j-1(λ z), while fracddz E_N,0(λ z) = λ E_N,N-1(λ z). Thus fracd^Ndz^N E_N,j(λ z) = λ^N E_N,j(λ z). For 0≤ k<N, .fracd^kdz^k E_N,j(λ z)|_z=0 = δ_kjλ^j. If λ≠0, the N functions E_N,j(λ z) form a basis of the solution space of y^(N)=λ^Ny.
Exact mathematical passage
\left.\frac{d^k}{dz^k}\Efun_{N,j}(\lambda z)\right|_{z=0} = \delta_{kj}\lambda^j.
d^{k+1}Use in source · 3 passages
Search aliases: d^k+1
Use in source · 01_manuscript.tex, line 615
[Equal tensor factors] For integers k,d≥0, a product of k square factors of order d, acting on a tensor with d^k entries, can be applied by k modewise contractions, reusing intermediate slices, with modeled cost 2k d^k+1 and factor storage kd^2.
Exact mathematical passage
\boxed{ 2k d^{k+1} }
d^{k+2}Use in source · 1 passage
Search aliases: d^k+2
Use in source · 02_supplement_04.tex, line 106
Operation events. The event type for k+1 modes is a disjoint union of d copies of the trailing event type and the events for the final fiber contraction. Its cardinality obeys W_0=0, W_k+1=dW_k+2d^k+2. Lean proves the closed form W_k=2kd^k+1. The factor-address type has cardinality k d^2. These statements include k=0 and d=0. With no factors, the action is the identity on the singleton coordinate type and incurs zero modeled arithmetic. With d=0 and at least one factor, the coordinate type is em…
Exact mathematical passage
W_0=0,\qquad W_{k+1}=dW_k+2d^{k+2}.
d^{k-1}Use in source · 1 passage
Search aliases: d^k-1
Use in source · 01_manuscript.tex, line 624
For k=0 there are no contractions and both modeled work and factor storage are zero. For k≥1, one mode has d^k-1 fibers of length d. Multiplying each fiber by a d by d matrix costs 2d^2, so one mode costs 2d^k+1. Sum over the k modes.
Exact mathematical passage
d^{k-1}
D^{m+1}Use in source · 2 passages
Search aliases: D^m+1
Use in source · 03_nonlinear_coordinates.tex, line 300
· +Q_γ R_m+1(x,h), where the Bochner integral remainder is R_m+1(x,h) = frac1m! ∫_0^1(1-t)^m D^m+1F(x+th)[h,…,h] dt.
Exact mathematical passage
R_{m+1}(x,h) = \frac1{m!} \int_0^1(1-t)^m D^{m+1}F(x+th)[h,\ldots,h]\,dt.
d^NUse in source · 2 passages
Search aliases: d^N
Use in source · Paper_IV_Traditional_Final.tex, line 482
[Differential cycle and initial data] For 1≤ j<N, fracddz E_N,j(λ z) = λ E_N,j-1(λ z), while fracddz E_N,0(λ z) = λ E_N,N-1(λ z). Thus fracd^Ndz^N E_N,j(λ z) = λ^N E_N,j(λ z). For 0≤ k<N, .fracd^kdz^k E_N,j(λ z)|_z=0 = δ_kjλ^j. If λ≠0, the N functions E_N,j(λ z) form a basis of the solution space of y^(N)=λ^Ny.
Exact mathematical passage
\frac{d^N}{dz^N}\Efun_{N,j}(\lambda z) = \lambda^N\Efun_{N,j}(\lambda z).
D^{N-i}Use in source · 2 passages
Search aliases: D^N-i
Use in source · strengthening.tex, line 75
[Translated product formula] For N=p^a, a≥1, and λ,μ∈ Z_p, L_λ+μ(AB) =(L_λ A)B+A(L_μ B) +∑_i=1^N-1c_i(D^iA)(D^N-iB), c_i=frac1pbinom Ni∈ Z. Flatness types add under multiplication.
Exact mathematical passage
L_{\lambda+\mu}(AB) =(L_\lambda A)B+A(L_\mu B) +\sum_{i=1}^{N-1}c_i(D^iA)(D^{N-i}B), \qquad c_i=\frac1p\binom Ni\in\mathbb Z. \label{eq:translated-product}
D^pUse in source · 7 passages
Search aliases: D^p
Use in source · 10_foundations.tex, line 144
[Hurwitz product rules] For d≥0, D^d(f⋆ g)=∑_r=0^dbinom dr(D^r f)⋆(D^d-rg). If R has prime characteristic p, then D^p(f⋆ g)=f⋆ D^p g+D^p f⋆ g. Over a field, if g_0=1, a sequence b is constructed recursively with b⋆ g=1. If also D^p g=Dg in characteristic p, then D^p b=Db.
Exact mathematical passage
D^p(f\star g)=f\star D^p g+D^p f\star g.
D^{p^2}Use in source · 3 passages
Search aliases: D^p^2
Use in source · 20_arithmetic.tex, line 209
Work in the Hurwitz integral ring of formal series H_p={F∈ Q_p[[t]]: d![t^d]F∈ Z_p for every d}. For D=d/dt, let B_p=(D^p^2-D^p)/p. Call F flat if B_p^sF∈ H_p for every s≥0. We will prove that j! P_N,j is flat at every address. The main difficulty is to include the addresses at which the usual unit-pivot argument fails.
Exact mathematical passage
B_p=(D^{p^2}-D^p)/p
D^qUse in source · 1 passage
Search aliases: D^q
Use in source · atlas-pages.tex, line 109
matrix. The same delay algebra explains both the off-diagonal phase swap and the channel powers. headSigned delays need the same care For an original-sample delay d at modulus N, y_k[n]=f_(k-d)bmod N [n+lfloorfrack-dNrfloor]. This includes advances when d<0. The finite verification checks the formula on signed sample addresses. A bank with product D^qI_N delays the original signal by Nq samples, not q samples. takeawayAlways state the unit of a delay. A phase permutation and a channel delay toge…
Exact mathematical passage
D^qI_N
D^\timesUse in source · 2 passages
Search aliases: D^× times
Use in source · 02_connections_obstructions.tex, line 61
Flatness replaces the two-edge route in D^× by the direct route, which cancels its inverse. For H, flatness first collapses a→ b→ c to a→ c, then collapses a→ c→ a to the normalized transport a→ a.
Exact mathematical passage
D^\times
D^\times_{c,b,a}Use in source · 3 passages
Search aliases: D^×_c,b,a times
Use in source · 02_connections_obstructions.tex, line 33
At a triple (a,b,c), the compiled cocycle predicate is Γ_c← bΓ_b← a =Γ_c← a. Flatness means this equality holds for every triple. The formal multiplicative relative defect and triangle holonomy are D^×_c,b,a = (Γ_c← bΓ_b← a) Γ_c← a^-1, H_c,b,a = Γ_a← c Γ_c← b Γ_b← a. The defect is based at the target c. The holonomy is based at a. statusleanthreeKomorebi.PaperIII.Transport.Connection.CocycleAt; Komorebi.PaperIII.Transport.Connection.Flat; Komorebi.PaperIII.Transport.Connection.cocycleDefect; Kom…
Exact mathematical passage
D^\times_{c,b,a} = \bigl(\Gamma_{c\leftarrow b}\Gamma_{b\leftarrow a}\bigr) \Gamma_{c\leftarrow a}^{-1}, \qquad H_{c,b,a} = \Gamma_{a\leftarrow c} \Gamma_{c\leftarrow b} \Gamma_{b\leftarrow a}.
D^\alphaUse in source · 2 passages
Search aliases: D^α alpha
Use in source · strengthening.tex, line 101
e with unspecified remainders. On the tensor of the two factors use the mutually commuting operators L_λ⊗1, 1⊗ L_μ, c_iD^i⊗ D^N-i (1≤ i<N). Applying multiplication after the kth power of their sum gives exactly L_λ+μ^k(AB). Expanding it yields terms indexed by a,b,e_1,…,e_N-1≥0, with a+b+∑_i e_i=k: frack!a!b!∏_i e_i!∏_i c_i^e_i (D^α L_λ^a A)(D^β L_μ^b B), α=∑_i i e_i, β=∑_i(N-i)e_i. Every scalar is integral. If A and B have the stated flatness types, both factors in every term belong to H_p, as …
Exact mathematical passage
\frac{k!}{a!b!\prod_i e_i!}\prod_i c_i^{e_i} (D^\alpha L_\lambda^a A)(D^\beta L_\mu^b B), \quad \alpha=\sum_i i e_i,\quad \beta=\sum_i(N-i)e_i. \label{eq:translated-multinomial}
D^\betaUse in source · 1 passage
Search aliases: D^β beta
Use in source · strengthening.tex, line 101
e with unspecified remainders. On the tensor of the two factors use the mutually commuting operators L_λ⊗1, 1⊗ L_μ, c_iD^i⊗ D^N-i (1≤ i<N). Applying multiplication after the kth power of their sum gives exactly L_λ+μ^k(AB). Expanding it yields terms indexed by a,b,e_1,…,e_N-1≥0, with a+b+∑_i e_i=k: frack!a!b!∏_i e_i!∏_i c_i^e_i (D^α L_λ^a A)(D^β L_μ^b B), α=∑_i i e_i, β=∑_i(N-i)e_i. Every scalar is integral. If A and B have the stated flatness types, both factors in every term belong to H_p, as …
Exact mathematical passage
\frac{k!}{a!b!\prod_i e_i!}\prod_i c_i^{e_i} (D^\alpha L_\lambda^a A)(D^\beta L_\mu^b B), \quad \alpha=\sum_i i e_i,\quad \beta=\sum_i(N-i)e_i. \label{eq:translated-multinomial}
\mathfrak D^\Gamma_{b,a}Use in source · 1 passage
Search aliases: D^Γ_b,a mathfrak Gamma
Use in source · 02_connections_obstructions.tex, line 889
For coordinate models of the source and target colors, an additive target coordinate space, and a chosen noncanonical connection, a pointwise node defect can be defined by D^Γ_b,a(Φ) = Γ^Y_b← a∘widehatΦ_a - widehatΦ_b∘ (∏_qΓ^X_q_b← a). This expression compares two chosen routes. With canonical exact transport in every slot it vanishes by [source reference].
Exact mathematical passage
\mathfrak D^\Gamma_{b,a}(\Phi) = \Gamma^Y_{b\leftarrow a}\circ\widehat\Phi_a - \widehat\Phi_b\circ \left(\prod_q\Gamma^{X_q}_{b\leftarrow a}\right).
D^\daggerUse in source · 5 passages
Search aliases: D^† dagger
Use in source · 04_correction_transport.tex, line 6
[Orthogonal obstruction projector] Let D:H_1→ H_2 be a bounded operator between Hilbert spaces with closed range. Let D^† be its Moore--Penrose inverse. Then P_ob=I_H_2-DD^† is the orthogonal projector onto (im D)^⊥=Ker D^*. For r∈ H_2, the following are equivalent: item Du=-r is solvable item [r]=0 in coker D item P_obr=0. When these conditions hold, -D^† r is the least norm correction. For every r, dist(r,im D)= P_obr .
Exact mathematical passage
D^\dagger
D_+Use in source · 1 passage
Search aliases: D_+
Use in source · residue_continuation_body.tex, line 84
The relation to the geometric sign is also proved. If σ ranges over real embeddings and τ over one embedding from each complex pair, then n(x)=∏_σσ(x)∏_τabsnτ(x)^2. For x≠0, the second factor is positive. Thus the sign of n(x) is exactly the sign of the real-coordinate product. Cutting D into its positive and negative portions D_+,D_- adds no boundary outside the existing frontier, and both portions retain the proved boundary atlas. They are disjoint and their union is D.
Exact mathematical passage
D_+,D_-
D_-Use in source · 1 passage
Search aliases: D_-
Use in source · residue_continuation_body.tex, line 84
The relation to the geometric sign is also proved. If σ ranges over real embeddings and τ over one embedding from each complex pair, then n(x)=∏_σσ(x)∏_τabsnτ(x)^2. For x≠0, the second factor is positive. Thus the sign of n(x) is exactly the sign of the real-coordinate product. Cutting D into its positive and negative portions D_+,D_- adds no boundary outside the existing frontier, and both portions retain the proved boundary atlas. They are disjoint and their union is D.
Exact mathematical passage
D_+,D_-
d_0Use in source · 6 passages
Search aliases: d_0
Use in source · 20_arithmetic.tex, line 656
[Exact four-packet displacement law] Set (c_0,c_1,c_2,c_3)=(1,1,4,2), (d_0,d_1,d_2,d_3)=(1,1,5,3). For 0≤ j<4, n≥0, and every positive integer T, C_j(n+T)-C_j(n) · ≠0, · v_2(C_j(n+T)-C_j(n)) · =v_2(T)+c_j,
Exact mathematical passage
(c_0,c_1,c_2,c_3)=(1,1,4,2),\qquad (d_0,d_1,d_2,d_3)=(1,1,5,3).
d_{0,k}Use in source · 1 passage
Search aliases: d_0,k
Use in source · 72_recovered.tex, line 2306
(q)=q has d_1=1. Thus d_1not≡ c_2pmod2. The coefficient consequence is read with integral original coefficients. Let M=p^a. Then Θ_p^a,s(r,k) ≡ binomsr-p^ak pmod p, where the binomial coefficient is zero outside its natural range. Consequently, d_s,k ≡ ∑_u=0^s binom su c_p^ak+u pmod p. In particular, d_0,k≡ c_p^akpmod p.
Exact mathematical passage
\boxed{d_{0,k}\equiv c_{p^ak}\pmod p.}
D_1Use in source · 9 passages
Search aliases: D_1
Use in source · 01_orientation.tex, line 56
A simple two packet model already contains all four notions. Let X=X_0⊕ X_1, D=D_0 · 0 0 · D_1, N=0 · N_01{ N_10 · 0. A finer family may split X_0 and X_1 into smaller summands. The blocks N_01 and N_10 may enter at different filtration weights. Over a fraction field, the rational packet spaces may fail to split the chosen lattice. A coarse operator on one selected summand may also fail to intertwine with the full matrix. The later sections isolate these four effects one at a time.
Exact mathematical passage
X=X_0\oplus X_1, \qquad D=\begin{pmatrix}D_0&0\\0&D_1\end{pmatrix}, \qquad N=\begin{pmatrix}0&N_{01}\\N_{10}&0\end{pmatrix}.
d_1Use in source · 6 passages
Search aliases: d_1
Use in source · 20_arithmetic.tex, line 656
[Exact four-packet displacement law] Set (c_0,c_1,c_2,c_3)=(1,1,4,2), (d_0,d_1,d_2,d_3)=(1,1,5,3). For 0≤ j<4, n≥0, and every positive integer T, C_j(n+T)-C_j(n) · ≠0, · v_2(C_j(n+T)-C_j(n)) · =v_2(T)+c_j,
Exact mathematical passage
(c_0,c_1,c_2,c_3)=(1,1,4,2),\qquad (d_0,d_1,d_2,d_3)=(1,1,5,3).
D_1^{-1}Use in source · 1 passage
Search aliases: D_1^-1
Use in source · strengthening.tex, line 467
stant is one. Since v_2(4i+2)=1, the elementary Bernoulli bound shows v_2(d_i)≥2i. Hurwitz inversion therefore gives D_1(t)^-1=∑_i≥0h_ifract^4i(4i)!, h_0=1, v_2(h_i)≥2i. This follows by induction from the inverse recurrence: for packet index i, every product d_rh_i-r has valuation at least 2i, and its binomial multiplier is integral. Because F_1=F_0D_1^-1, C_1(n)-C_0(n) =∑_i≥1h_ibinom4n4iC_0(n-i). At a natural n this is a finite sum, with i≤ n.
Exact mathematical passage
F_1=F_0D_1^{-1}
D_2Use in source · 5 passages
Search aliases: D_2
Use in source · atlas-pages.tex, line 5
: Section 2, Section 7, Section 11, Section 12. This guide follows a concrete analytic family through three descriptions: selected Taylor coefficients, a finite differential system, and the zeros of an entire function. The formulas are linked, but the evidence needed for each description is different. vfigthree_views headKeep three small models in view For N=2, the signed denominator is D_2(z)=cos z. Its reciprocal is the familiar secant series. For N=3, a third exponential perturbs the zeros of…
Exact mathematical passage
D_2(z)=\cos z
d_2Use in source · 2 passages
Search aliases: d_2
Use in source · 20_arithmetic.tex, line 656
[Exact four-packet displacement law] Set (c_0,c_1,c_2,c_3)=(1,1,4,2), (d_0,d_1,d_2,d_3)=(1,1,5,3). For 0≤ j<4, n≥0, and every positive integer T, C_j(n+T)-C_j(n) · ≠0, · v_2(C_j(n+T)-C_j(n)) · =v_2(T)+c_j,
Exact mathematical passage
(c_0,c_1,c_2,c_3)=(1,1,4,2),\qquad (d_0,d_1,d_2,d_3)=(1,1,5,3).
d_{2,\ell}Use in source · 2 passages
Search aliases: d_2,ℓ ell
Use in source · strengthening.tex, line 1257
=0,1,2,3, the dyadic interpolation has an ordinary power series widetilde C_j(x)=1+∑_ℓ≥1d_j,ℓx^ℓ, v_2(d_j,ℓ)≥ℓ+β_j. Its Taylor series converges for |x|_2<2. At addresses zero and one, v_2(d_j,ℓ)=ℓ (ℓ≥1), R_0=R_1=2. At address two, v_2(d_2,ℓ)≥ max(2ℓ+lfloorfracℓ2rfloor, ℓ+3), R_2=2^5/2. The radius at address three is at least two. Its exact radius is not asserted here. The stated exact radii at addresses zero, one, and two hold at every center in Z_2.
Exact mathematical passage
v_2(d_{2,\ell})\geq \max\left(2\ell+\left\lfloor\frac\ell2\right\rfloor, \ell+3\right), \qquad R_2=2^{5/2}. \label{eq:two-exact-taylor-radius}
D_2^{-1}Use in source · 1 passage
Search aliases: D_2^-1
Use in source · atlas-pages.tex, line 244
vpage14Three similar reciprocals differNORMALIZED SECTIONSChapter IV: Definition 9.1, Theorem 9.2, Example 9.11. The signed denominator and the normalized unsigned sections produce different reciprocal families. In order two, D_2^-1=sec z, C_2,0=sech z, C_2,1=z/sinh z. Their first block-EGF coefficients make the difference visible. vfigdifferent_reciprocals headRemove the forced zero before taking a reciprocal The section E_N,j begins with z^j/j!. Define E_N,j(z)=fracz^jj!Φ_N,j(z^N), Φ_N,j(0)=1.…
Exact mathematical passage
D_2^{-1}=\sec z,\qquad C_{2,0}=\sech z,\qquad C_{2,1}=z/\sinh z.
D_3Use in source · 4 passages
Search aliases: D_3
Use in source · atlas-pages.tex, line 95
ordinates and columns are input coordinates. The upper-triangular minus signs record crossing the signed edge while moving from the column to the row. They are not arbitrary orientation choices. headTwo checks connect the matrix and the scalar equation The initial derivative matrix is the identity. Each initial jet (y(0),y'(0),y''(0)) therefore determines a unique linear combination. Also, frac13Tr(e^zJ_3)=D_3(z), det(XI-J_3)=X^3+1. The roots of -1 appearing later in the exponential sum are alre…
Exact mathematical passage
\frac13\Tr(e^{zJ_3})=D_3(z),\qquad \det(XI-J_3)=X^3+1.
d_3Use in source · 2 passages
Search aliases: d_3
Use in source · 20_arithmetic.tex, line 656
[Exact four-packet displacement law] Set (c_0,c_1,c_2,c_3)=(1,1,4,2), (d_0,d_1,d_2,d_3)=(1,1,5,3). For 0≤ j<4, n≥0, and every positive integer T, C_j(n+T)-C_j(n) · ≠0, · v_2(C_j(n+T)-C_j(n)) · =v_2(T)+c_j,
Exact mathematical passage
(c_0,c_1,c_2,c_3)=(1,1,4,2),\qquad (d_0,d_1,d_2,d_3)=(1,1,5,3).
D_3'Use in source · 1 passage
Search aliases: D_3'
Use in source · atlas-pages.tex, line 343
vpage19A root estimate becomes a certificateEXACT RATIONAL BRACKETSChapter IV: Theorem 11.7, Theorem 11.8, Corollary 11.9. The root finder is used only to suggest intervals. Each displayed interval is accepted by exact rational arithmetic: opposite endpoint signs prove existence, and an interval bound for D_3' excludes zero throughout the interval, proving uniqueness inside it. vfigcubic_certificates headHow the certificate works Evaluate the Taylor polynomial S_90(t)=∑_n=0^90frac(-1)^nt^3n(3n)!…
Exact mathematical passage
D_3'
D_{3,0}Use in source · 2 passages
Search aliases: D_3,0
Use in source · atlas-pages.tex, line 88
vpage5One signed edge determines the cycleDIFFERENTIAL SYSTEMChapter IV: Theorem 5.3, Theorem 6.4, Corollary 6.5, Corollary 6.6. Let J_3e_0=e_1, J_3e_1=e_2, and J_3e_2=-e_0. A complete turn changes sign, so J_3^3=-I. The signed sections are the coordinates of e^zJ_3e_0. vfigsigned_shift Here D_j inside the matrix means D_3,j(z). For example, D_3,0=1-fracz^33!+fracz^66!-⋯, D_3,1=z-fracz^44!+⋯. Rows are output coordinates and columns are input coordinates. The upper-triangular minus signs record c…
Exact mathematical passage
D_{3,0}=1-\frac{z^3}{3!}+\frac{z^6}{6!}-\cdots, \qquad D_{3,1}=z-\frac{z^4}{4!}+\cdots.
D_{3,1}Use in source · 2 passages
Search aliases: D_3,1
Use in source · atlas-pages.tex, line 88
vpage5One signed edge determines the cycleDIFFERENTIAL SYSTEMChapter IV: Theorem 5.3, Theorem 6.4, Corollary 6.5, Corollary 6.6. Let J_3e_0=e_1, J_3e_1=e_2, and J_3e_2=-e_0. A complete turn changes sign, so J_3^3=-I. The signed sections are the coordinates of e^zJ_3e_0. vfigsigned_shift Here D_j inside the matrix means D_3,j(z). For example, D_3,0=1-fracz^33!+fracz^66!-⋯, D_3,1=z-fracz^44!+⋯. Rows are output coordinates and columns are input coordinates. The upper-triangular minus signs record c…
Exact mathematical passage
D_{3,0}=1-\frac{z^3}{3!}+\frac{z^6}{6!}-\cdots, \qquad D_{3,1}=z-\frac{z^4}{4!}+\cdots.
D_{3,2}Use in source · 1 passage
Search aliases: D_3,2
Use in source · atlas-pages.tex, line 104
vpage6Addition is quotient multiplicationFINITE ALGEBRAChapter IV: Proposition 5.5, Theorem 6.2. Work in C[ξ]/(ξ^3+1). Every element has coordinates in 1,ξ,ξ^2. The entire exponential has the finite-coordinate form e^zξ=D_3,0(z)+D_3,1(z)ξ+D_3,2(z)ξ^2. The multiplication table makes the carry sign unavoidable. vfigsigned_addition Multiply the coordinate forms of e^zξ and e^wξ. Their product is e^(z+w)ξ. The coefficient of 1 receives one contribution without a wrap and two contributions with a wra…
Exact mathematical passage
e^{z\xi}=D_{3,0}(z)+D_{3,1}(z)\xi+D_{3,2}(z)\xi^2.
D_{3,j}Use in source · 1 passage
Search aliases: D_3,j
Use in source · atlas-pages.tex, line 87
vpage5One signed edge determines the cycleDIFFERENTIAL SYSTEMChapter IV: Theorem 5.3, Theorem 6.4, Corollary 6.5, Corollary 6.6. Let J_3e_0=e_1, J_3e_1=e_2, and J_3e_2=-e_0. A complete turn changes sign, so J_3^3=-I. The signed sections are the coordinates of e^zJ_3e_0. vfigsigned_shift Here D_j inside the matrix means D_3,j(z). For example, D_3,0=1-fracz^33!+fracz^66!-⋯, D_3,1=z-fracz^44!+⋯. Rows are output coordinates and columns are input coordinates. The upper-triangular minus signs record c…
Exact mathematical passage
D_{3,j}(z)
D_3^{-1}Use in source · 1 passage
Search aliases: D_3^-1
Use in source · strengthening.tex, line 739
e a^-2+b^-2=0. Its constant is one, only degrees divisible by four survive, and the Bernoulli expansion gives D_3(t)=∑_i≥0d_ifract^4i(4i)!, d_i=3(-frac14)^i frac2^4i+2B_4i+2(4i+2)(4i+1). Again v_2(d_i)≥2i, and the inverse coefficients h_i have the same bound. Since F_3=F_2D_3^-1, C_3(n)-C_2(n)=∑_i≥1h_ibinom4n4iC_2(n-i). The scaled-binomial argument [source reference] applies unchanged. The basic bound for C_2 includes order zero. Its continuous extension is available from the threshold-free theo…
Exact mathematical passage
F_3=F_2D_3^{-1}
D_4Use in source · 3 passages
Search aliases: D_4
Use in source · atlas-pages.tex, line 6
n entire function. The formulas are linked, but the evidence needed for each description is different. vfigthree_views headKeep three small models in view For N=2, the signed denominator is D_2(z)=cos z. Its reciprocal is the familiar secant series. For N=3, a third exponential perturbs the zeros of a dominant cosine. For N=4, an exact factorization gives a complete zero picture: D_4(z)=cos(z/sqrt2)cosh(z/sqrt2). The examples move between these orders deliberately. Order two makes the coefficien…
Exact mathematical passage
D_4(z)=\cos(z/\sqrt2)\cosh(z/\sqrt2).
D_8Use in source · 1 passage
Search aliases: D_8
Use in source · Paper_XI_Traditional_Final.tex, line 2154
ter pair. For an independently fixed native quadratic marking, a preassigned embedding must satisfy q_W(f(v))=q_rm src(v). At equal capacity, a free embedding requires equal Arf invariant. Spare capacity for d≥4 admits either source Arf type when the embedding is free. Retaining only the central {±1} would erase the distinction between whole marked finite extensions such as D_8 and Q_8.
Exact mathematical passage
D_8
D_AUse in source · 19 passages
Search aliases: D_A
Use in source · Paper_X_Block27_Amended.tex, line 2475
The inertia theorem uses actual residue actions. For a valuation subring A of an extension field L/K, let D_A be its decomposition subgroup in Aut_K(L) and put I_A=ker(D_A⟶Aut(k_A)), where k_A is the residue field. If 2 is a unit in A, every member of I_A fixes every y∈ L with y^2=-1. Indeed y^-1=-y, so y belongs to A and is a unit there. An automorphism sends y to y or -y. Their residues are distinct because 2 and y are units, whereas inertia fixes residues. The second possibility is excluded.
Exact mathematical passage
D_A
D_aUse in source · 10 passages
Search aliases: D_a
Use in source · 72_recovered.tex, line 2754
[Packet decomposition of a lift obstruction] Suppose F_p^d = ⊕_a∈ AU_a, F_p^r = ⊕_a∈ AV_a, and suppose a linear correction map D: F_p^d→ F_p^r satisfies D(U_a)⊆ V_a for every a. Write D_a=D|_U_a. Then coker D ≅ ⊕_a∈ Acoker D_a. A residual lifts if and only if every packet obstruction vanishes.
Exact mathematical passage
D_a=D|_{U_a}
d_AUse in source · 8 passages
Search aliases: d_A
Use in source · 01_finite_core.tex, line 137
[Local orbit and Schmidt rank] Let ψ∈ H_A⊗ H_B, with dim H_A=d_A≥1, and let r be its Schmidt rank. The reduced positive operator on B is ρ_B=Tr_A(ketψbraψ). Normalization of ψ does not affect its support. Then dimspan{(A⊗ I_B)ψ:A∈ M_d_A( C)} =d_A r. In particular, local operators on A generate the complete bipartite Hilbert space exactly when the reduced state on B has full support.
Exact mathematical passage
\dim\cH_A=d_A\ge1
D_a^*Use in source · 1 passage
Search aliases: D_a^*
Use in source · 03_supplement_02.tex, line 161
The derivative product rule and uniqueness of derivatives give D_aP_b+P_aD_b=δ_abD_a. Differentiating completeness and adjoint invariance gives ∑_aD_a=0, D_a^*=D_a. These are proved derivative consequences, not additional axioms supplied in place of differentiation.
Exact mathematical passage
\sum_aD_a=0,\qquad D_a^*=D_a.
D_BUse in source · 13 passages
Search aliases: D_B
Use in source · pages.tex, line 142
vpage9The cross term has somewhere to goCLOSED LOW-RANK PRODUCT ALGEBRAChapter VIII: Theorem 5.1, Corollary 5.2. Let A=D_A+u_av_a^T be the running operator. Take D_B=diag(1,2,3,4,5,6), u_b=(1,0,1,0,1,0)^T, and v_b=(0,1,0,1,0,1)^T. Define B=D_B+u_bv_b^T. U_C=[D_Au_b+u_a(v_a^Tu_b) u_a], V_C=[v_b D_B^Tv_a]. vfigbdlr_product headWhy there are only two columns Expanding AB gives three correction terms, but two share the right factor v_b^T: D_Au_bv_b^T+u_a(v_a^Tu_b)v_b^T =(D_Au_b+u_a(v_a^Tu_b))v_b^T. …
Exact mathematical passage
D_B=\diag(1,2,3,4,5,6)
D_bUse in source · 5 passages
Search aliases: D_b
Use in source · 03_supplement_02.tex, line 150
Define C=∑_aD_aP_a(t). Then C=frac12∑_acommD_aP_a(t), C^*=-C, and, for every b, commCP_b(t)=D_b.
Exact mathematical passage
\boxed{\comm{C}{P_b(t)}=D_b.}
D_B^*Use in source · 3 passages
Search aliases: D_B^*
Use in source · 01_manuscript.tex, line 687
[Exact BDLR product] Let A=D_A+U_AV_A^*, B=D_B+U_BV_B^* use the same diagonal packet layout. Then AB=D_AD_B+U_CV_C^*, where U_C = D_AU_B+U_A(V_A^*U_B) · U_A and V_C = V_B · D_B^*V_A . The unrecompressed correction rank is at most r_A+r_B.
Exact mathematical passage
V_C = \begin{bmatrix} V_B&D_B^*V_A \end{bmatrix}.
D_B^TUse in source · 1 passage
Search aliases: D_B^T
Use in source · pages.tex, line 143
vpage9The cross term has somewhere to goCLOSED LOW-RANK PRODUCT ALGEBRAChapter VIII: Theorem 5.1, Corollary 5.2. Let A=D_A+u_av_a^T be the running operator. Take D_B=diag(1,2,3,4,5,6), u_b=(1,0,1,0,1,0)^T, and v_b=(0,1,0,1,0,1)^T. Define B=D_B+u_bv_b^T. U_C=[D_Au_b+u_a(v_a^Tu_b) u_a], V_C=[v_b D_B^Tv_a]. vfigbdlr_product headWhy there are only two columns Expanding AB gives three correction terms, but two share the right factor v_b^T: D_Au_bv_b^T+u_a(v_a^Tu_b)v_b^T =(D_Au_b+u_a(v_a^Tu_b))v_b^T. …
Exact mathematical passage
U_C=[D_Au_b+u_a(v_a^Tu_b)\ \ u_a],\qquad V_C=[v_b\ \ D_B^Tv_a].
D_cUse in source · 11 passages
Search aliases: D_c
Use in source · 03_nonlinear_coordinates.tex, line 1044
[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is a natural exact sequence 0 · ⟶Ker D_0 ⟶Ker D_f ⟶Ker D_c xrightarrowδcoker D_0
Exact mathematical passage
\begin{array}{ccccccccc} 0&\longrightarrow&U_0&\xrightarrow{i_U}&U_f&\xrightarrow{p_U}&U_c&\longrightarrow&0\\[4pt] &&\mathllap{D_0}\downarrow&&\mathllap{D_f}\downarrow&&\mathllap{D_c}\downarrow&&\\[4pt] 0&\longrightarrow&V_0&\xrightarrow{i_V}&V_f&\xrightarrow{p_V}&V_c&\longrightarrow&0. \end{array}
D_eUse in source · 3 passages
Search aliases: D_e
Use in source · Paper_XI_Traditional_Final.tex, line 1694
[Exact edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD_eA_p_u^-1.
Exact mathematical passage
D_e = U_rA_{p_v}^{-1}A_eA_{p_u} - B_{p_v}^{-1}B_eB_{p_u}U_r.
D_fUse in source · 20 passages
Search aliases: D_f
Use in source · 03_nonlinear_coordinates.tex, line 1044
[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is a natural exact sequence 0 · ⟶Ker D_0 ⟶Ker D_f ⟶Ker D_c xrightarrowδcoker D_0
Exact mathematical passage
\begin{array}{ccccccccc} 0&\longrightarrow&U_0&\xrightarrow{i_U}&U_f&\xrightarrow{p_U}&U_c&\longrightarrow&0\\[4pt] &&\mathllap{D_0}\downarrow&&\mathllap{D_f}\downarrow&&\mathllap{D_c}\downarrow&&\\[4pt] 0&\longrightarrow&V_0&\xrightarrow{i_V}&V_f&\xrightarrow{p_V}&V_c&\longrightarrow&0. \end{array}
d_{i-1}Use in source · 2 passages
Search aliases: d_i-1
Use in source · Paper_X_Block27_Amended.tex, line 1998
Modular Gaussian elimination computes the ranks of the differentials. For C^i-1xrightarrowd_i-1C^ixrightarrowd_iC^i+1, the finite dimension is evaluated as dim H^i = dim C^i-rank d_i-rank d_i-1. Packet subcomplexes are computed separately and checked against the total.
Exact mathematical passage
C^{i-1}\xrightarrow{d_{i-1}}C^i\xrightarrow{d_i}C^{i+1},
D_i^{\oplus m_i}Use in source · 1 passage
Search aliases: D_i^⊕ m_i oplus
Use in source · Paper_X_Block27_Amended.tex, line 1509
Writing Q_D(T) for the reversed local characteristic polynomial, Lean proves Q_D(T)=det(1-T r(φ)mid E(D,H)), Q_D(0)=1, Q_⊕_iD_i^⊕ m_i(T)=∏_iQ_D_i(T)^m_i. The abstract formulas do not need characteristic zero. Normality and finite dimension appear explicitly where used. Arithmetic identification of G,H,ν and φ remains separate. In the geometric-Frobenius convention, ν(φ) is the inverse residue cardinality. It is this unit, rather than the residue cardinality, that enters the earlier special-chain…
Exact mathematical passage
Q_D(T)=\det(1-T r(\varphi)\mid E(D,H)),\qquad Q_D(0)=1, \qquad Q_{\bigoplus_iD_i^{\oplus m_i}}(T)=\prod_iQ_{D_i}(T)^{m_i}.
D_jUse in source · 15 passages
Search aliases: D_j
Use in source · atlas-pages.tex, line 87
vpage5One signed edge determines the cycleDIFFERENTIAL SYSTEMChapter IV: Theorem 5.3, Theorem 6.4, Corollary 6.5, Corollary 6.6. Let J_3e_0=e_1, J_3e_1=e_2, and J_3e_2=-e_0. A complete turn changes sign, so J_3^3=-I. The signed sections are the coordinates of e^zJ_3e_0. vfigsigned_shift Here D_j inside the matrix means D_3,j(z). For example, D_3,0=1-fracz^33!+fracz^66!-⋯, D_3,1=z-fracz^44!+⋯. Rows are output coordinates and columns are input coordinates. The upper-triangular minus signs record c…
Exact mathematical passage
D_j
d_jUse in source · 13 passages
Search aliases: d_j
Use in source · 20_arithmetic.tex, line 661
[Exact four-packet displacement law] Set (c_0,c_1,c_2,c_3)=(1,1,4,2), (d_0,d_1,d_2,d_3)=(1,1,5,3). For 0≤ j<4, n≥0, and every positive integer T, C_j(n+T)-C_j(n) · ≠0, · v_2(C_j(n+T)-C_j(n)) · =v_2(T)+c_j, U_j(n+T)-U_j(n) · ≠0, · v_2(U_j(n+T)-U_j(n)) · =v_2(T)+d_j.
Exact mathematical passage
C_j(n+T)-C_j(n)&\ne0,& v_2\bigl(C_j(n+T)-C_j(n)\bigr)&=v_2(T)+c_j,\\ U_j(n+T)-U_j(n)&\ne0,& v_2\bigl(U_j(n+T)-U_j(n)\bigr)&=v_2(T)+d_j.
d_{j,m}Use in source · 1 passage
Search aliases: d_j,m
Use in source · strengthening.tex, line 1290
For j=0,1, the term m=ℓ in [source reference] has exact valuation ℓ, since s(ℓ,ℓ)=1 and the Newton valuation is exact. Every later term has valuation at least ℓ+1. Thus v_2(d_j,ℓ)=ℓ, and the coefficient radius is exactly two. At a center x_0∈ Z_2, the coefficient of y^ℓ in the translated series is ∑_m≥ℓd_j,mbinom mℓ x_0^m-ℓ. Its initial term again has exact valuation ℓ, and all later terms have at least one more bit. The exact radius is therefore two at every integral center.
Exact mathematical passage
\sum_{m\geq\ell}d_{j,m}\binom m\ell x_0^{m-\ell}. \label{eq:taylor-translation-coefficient}
d_{j,\ell}Use in source · 4 passages
Search aliases: d_j,ℓ ell
Use in source · strengthening.tex, line 1245
[Taylor coefficient bounds and radii] For j=0,1,2,3, the dyadic interpolation has an ordinary power series widetilde C_j(x)=1+∑_ℓ≥1d_j,ℓx^ℓ, v_2(d_j,ℓ)≥ℓ+β_j. Its Taylor series converges for |x|_2<2. At addresses zero and one, v_2(d_j,ℓ)=ℓ (ℓ≥1), R_0=R_1=2. At address two, v_2(d_2,ℓ)≥ max(2ℓ+lfloorfracℓ2rfloor, ℓ+3), R_2=2^5/2. The radius at address three is at least two. Its exact radius is not asserted here. The stated exact radii at addresses zero, one, and two hold at every center in Z_2.
Exact mathematical passage
\widetilde C_j(x)=1+\sum_{\ell\geq1}d_{j,\ell}x^\ell, \qquad v_2(d_{j,\ell})\geq\ell+\beta_j.
D_{j-1}Use in source · 1 passage
Search aliases: D_j-1
Use in source · 01_manuscript.tex, line 931
Set D_j=widehat P_j-P_j. Then D_j=A_jD_j-1+E_j, D_0=0. Repeated substitution gives D_m = ∑_j=1^m A_mA_m-1⋯ A_j+1E_j. Taking norms yields the result.
Exact mathematical passage
D_j=A_jD_{j-1}+E_j, \qquad D_0=0.
D_j^\bulletUse in source · 1 passage
Search aliases: D_j^bullet bullet
Use in source · Paper_X_Block27_Amended.tex, line 418
Naturality. If a cochain map F:C^bullet→ D^bullet intertwines the supplied projectors, it restricts to maps F_j:C_j^bullet→ D_j^bullet. The direct-sum decomposition commutes with H^r(F) and the family H^r(F_j). This is checked using representatives in the actual quotients, independently of any dimension computation.
Exact mathematical passage
F_j:C_j^\bullet\to D_j^\bullet
d_{ja}Use in source · 1 passage
Search aliases: d_ja
Use in source · 01_finite_core.tex, line 829
ry vector orthogonal to ψ, the quadratic form on the left is a sum of nonnegative squared absolute values and equals zero. Each L_jψ therefore lies in the span of ψ. A linear operator that preserves every line in C is scalar there: apply it to two independent vectors and their sum. The one-dimensional case is immediate, and the claim is vacuous for the zero code. Consequently K_jE_aP=d_jaP. Recovery completeness now gives PE_a^*E_bP · =∑_j (K_jE_aP)^*(K_jE_bP)
Exact mathematical passage
K_jE_aP=d_{ja}P.
d_{jb}Use in source · 1 passage
Search aliases: d_jb
Use in source · 01_finite_core.tex, line 833
he left is a sum of nonnegative squared absolute values and equals zero. Each L_jψ therefore lies in the span of ψ. A linear operator that preserves every line in C is scalar there: apply it to two independent vectors and their sum. The one-dimensional case is immediate, and the claim is vacuous for the zero code. Consequently K_jE_aP=d_jaP. Recovery completeness now gives PE_a^*E_bP · =∑_j (K_jE_aP)^*(K_jE_bP) · =(∑_joverlined_jad_jb)P. The coefficients form a Gram matrix, hence are positive se…
Exact mathematical passage
\begin{aligned} PE_a^*E_bP &=\sum_j (K_jE_aP)^*(K_jE_bP)\\ &=\left(\sum_j\overline{d_{ja}}d_{jb}\right)P. \end{aligned}
D_kUse in source · 16 passages
Search aliases: D_k
Use in source · Paper_V_Traditional_Final.tex, line 2921
me f(t) = s(t)+∑_q=1^Qa_qH_τ_q(t), H_τ(t)= 1_[τ,1](t), where s is Lipschitz with constant M, and Q≥1 is finite with nonzero jump amplitudes. The zero-jump case requires no minimum-amplitude condition. Suppose no jump lies at a sample point and no grid cell contains more than one jump. Observe y_k=f(t_k)+η_k. Define the adjacent contrast D_k=y_k+1-y_k.
Exact mathematical passage
D_k=y_{k+1}-y_k.
d_{L,a}Use in source · 2 passages
Search aliases: d_L,a
Use in source · 02_strengthened.tex, line 1999
[Heat packet decomposition] If [L,P_a]=0 for every packet, then Z_L(t)=∑_aZ_L,a(t), Z_L,a(t)=Tr(P_a e^-tL). The global spectral dimension is the average of the packet dimensions weighted by their heat traces: d_L(t) = ∑_a:Z_L,a(t)>0 fracZ_L,a(t)Z_L(t) d_L,a(t), where d_L,a(t) =2t fracTr(P_aL e^-tL)Z_L,a(t) for nonempty packet heat traces.
Exact mathematical passage
d_L(t) = \sum_{a:Z_{L,a}(t)>0} \frac{Z_{L,a}(t)}{Z_L(t)} d_{L,a}(t),
d_{L_N}Use in source · 1 passage
Search aliases: d_L_N
Use in source · 02_strengthened.tex, line 2153
[Rigorous intermediate dimension window] For every ε>0, there is a diffusion time t_ε>0 and an integer N_ε such that absd_L_N(t_ε)-d<ε for every N≥ N_ε.
Exact mathematical passage
\abs{d_{L_N}(t_\varepsilon)-d}<\varepsilon
D_mUse in source · 3 passages
Search aliases: D_m
Use in source · 01_manuscript.tex, line 937
Set D_j=widehat P_j-P_j. Then D_j=A_jD_j-1+E_j, D_0=0. Repeated substitution gives D_m = ∑_j=1^m A_mA_m-1⋯ A_j+1E_j. Taking norms yields the result.
Exact mathematical passage
D_m = \sum_{j=1}^{m} A_mA_{m-1}\cdots A_{j+1}E_j.
D_NUse in source · 12 passages
Search aliases: D_N
Use in source · atlas-pages.tex, line 31
D_N(z) · D_N,0(z), the denominator whose zeros control the signed reciprocal family.
Exact mathematical passage
D_N(z)
\mathbf D_NUse in source · 4 passages
Search aliases: D_N mathbf
Use in source · Paper_IV_Traditional_Final.tex, line 836
[Companion matrix realization] Put D_N(z) = D_N,0(z) D_N,1(z) vdots D_N,N-1(z) . Then D_N(z)=e^zJ_Ne_0 and D_N(z)=frac1Ntr(e^zJ_N). The eigenvalues of J_N are the N roots of -1, each with multiplicity one.
Exact mathematical passage
\mathbf D_N(z) = \begin{pmatrix} \Dfun_{N,0}(z)\\ \Dfun_{N,1}(z)\\ \vdots\\ \Dfun_{N,N-1}(z) \end{pmatrix}.
d_NUse in source · 3 passages
Search aliases: d_N
Use in source · monograph_atlas.tex, line 725
vpage37The dimension window has an order of limitsFINITE HEAT DATA AND THE QUANTIFIERSChapter VII: Theorem II.11.5, Corollary II.11.8. For the unit circle grid with N sites, the scaled eigenvalues are λ_N(k)=4N^2sin^2(π k/N). Put Z_N(t)=∑_ke^-tλ_N(k) and d_N(t)=2t∑_kλ_N(k)e^-tλ_N(k)/Z_N(t).
Exact mathematical passage
d_N(t)=2t\sum_k\lambda_N(k)e^{-t\lambda_N(k)}/Z_N(t)
d_nUse in source · 2 passages
Search aliases: d_n
Use in source · 72_recovered.tex, line 842
Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding term vanishes for positive n. Otherwise ((1+ζ y)^m)^h = (1+ζ y)^p-1 =1. Every summand is therefore h periodic for positive n. Thus T_m,p preserves V_h.…
Exact mathematical passage
d_n=f_n
D_N'Use in source · 2 passages
Search aliases: D_N'
Use in source · atlas-pages.tex, line 35
T_N(z) · -D_N'(z)/D_N(z), a function, not the coefficient T_N(n).
Exact mathematical passage
-D_N'(z)/D_N(z)
D_{N,0}Use in source · 1 passage
Search aliases: D_N,0
Use in source · atlas-pages.tex, line 31
D_N(z) · D_N,0(z), the denominator whose zeros control the signed reciprocal family.
Exact mathematical passage
D_{N,0}(z)
D_{N,j}Use in source · 13 passages
Search aliases: D_N,j
Use in source · atlas-pages.tex, line 30
D_N,j(z) · The same series with (-1)^n. One full derivative cycle contributes a minus sign.
Exact mathematical passage
D_{N,j}(z)
D_N^{-(r+s)}Use in source · 1 passage
Search aliases: D_N^-(r+s)
Use in source · atlas-pages.tex, line 205
rem 7.4, Theorem 7.6. One recurrence uses the denominator coefficients. Another uses its logarithm. With b_n(r)=P_N,n(r)/(Nn)!, the shorter logarithmic form is nb_n(r)=r∑_k=1^nκ_N,kb_n-k(r), b_0=1. The negative superdiagonal in the Newton determinant is essential. vfignewton_addition headThe additive parameter law has block weights Using the same logarithm, D_N^-rD_N^-s=D_N^-(r+s). Thus P_N,n(r+s)=∑_q=0^nbinomNnNqP_N,q(r)P_N,n-q(s). For N=2,n=3,r=s=1, the four weights are 1,15,15,1. The two midd…
Exact mathematical passage
D_N^{-r}D_N^{-s}=D_N^{-(r+s)}
d_n^{-1}Use in source · 1 passage
Search aliases: d_n^-1
Use in source · supplement_05.tex, line 44
First construct the integral solution. For a ring R, a triangular row system T, right-hand side b, and proved unit pivots d_n=T_nn, define s_n=d_n^-1(b_n-∑_q<nT_nqs_q). The new ring-level solver uses well-founded recursion. Its row identities and unique-solution property are proved. The source also proves compatibility of this solver with ring homomorphisms into fields.
Exact mathematical passage
s_n=d_n^{-1}\left(b_n-\sum_{q<n}T_{nq}s_q\right).
D_N^{-r}Use in source · 3 passages
Search aliases: D_N^-r
Use in source · atlas-pages.tex, line 33
P_N,n(r) · Block-EGF coefficients of D_N^-r. They are polynomials in r.
Exact mathematical passage
D_N^{-r}
D_N^{-s}Use in source · 1 passage
Search aliases: D_N^-s
Use in source · atlas-pages.tex, line 205
rem 7.4, Theorem 7.6. One recurrence uses the denominator coefficients. Another uses its logarithm. With b_n(r)=P_N,n(r)/(Nn)!, the shorter logarithmic form is nb_n(r)=r∑_k=1^nκ_N,kb_n-k(r), b_0=1. The negative superdiagonal in the Newton determinant is essential. vfignewton_addition headThe additive parameter law has block weights Using the same logarithm, D_N^-rD_N^-s=D_N^-(r+s). Thus P_N,n(r+s)=∑_q=0^nbinomNnNqP_N,q(r)P_N,n-q(s). For N=2,n=3,r=s=1, the four weights are 1,15,15,1. The two midd…
Exact mathematical passage
D_N^{-r}D_N^{-s}=D_N^{-(r+s)}
D_PUse in source · 3 passages
Search aliases: D_P
Use in source · 30_lifting.tex, line 51
Apply polynomial induction using constants, addition, and multiplication by a variable. Constants have zero remainder, and remainders add. For the multiplication step, suppose the assertion is known for P and write D_P(U)=∑_j(∂_jP)(a)U_j. Multiplication of its expansion by a_i+cU_i gives constant term P(a)a_i, linear term c(a_iD_P(U)+P(a)U_i), and remainder c^2(Q(U)(a_i+cU_i)+D_P(U)U_i). The product rule identifies the linear term with the derivative of PX_i. The displayed remainder vanishes at …
Exact mathematical passage
D_P(U)=\sum_j(\partial_jP)(a)U_j
d_pUse in source · 3 passages
Search aliases: d_p
Use in source · Paper_VI_Traditional_Final.tex, line 3409
[Likelihood perturbation] Let λ,μ≥0 have positive evidences under a fixed prior p. Set d_p(λ,μ) = ∑_iabsλ_i-μ_ip_i. Then norm B_λ(p)- B_μ(p)_1 ≤ frac2d_p(λ,μ) max{Z_λ(p),Z_μ(p)}.
Exact mathematical passage
d_p(\lambda,\mu) = \sum_i\abs{\lambda_i-\mu_i}p_i.
d_{pq}Use in source · 5 passages
Search aliases: d_pq
Use in source · Paper_IV_Traditional_Final.tex, line 882
[Entries of the fundamental matrix] For 0≤ p,q<N, let d_pq be the member of {0,…,N-1} congruent to p-q modulo N. Put ε_pq = 1, · p≥ q,
Exact mathematical passage
d_{pq}
d_qUse in source · 1 passage
Search aliases: d_q
Use in source · 72_recovered.tex, line 843
Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding term vanishes for positive n. Otherwise ((1+ζ y)^m)^h = (1+ζ y)^p-1 =1. Every summand is therefore h periodic for positive n. Thus T_m,p preserves V_h.…
Exact mathematical passage
d_q=\sum_{\chi\in\mu_h}c_\chi\chi^q.
D_rUse in source · 5 passages
Search aliases: D_r
Use in source · atlas-pages.tex, line 289
unchanged. Predicting afterward gives (3/4,1/4) instead. The order of operations expresses which time the observation concerns. vfigfilter_order platecaptionBoth routes produce valid probability vectors. Their difference is therefore not a normalization bug. It is a different conditioning problem. headKeep an unnormalized path alongside the filter For a sequence of observations, let D_r=diag(λ^(r)). The forward recursion is widehat p_r=Kp_r-1, p_r=D_rwidehat p_r/Z_r. Its unnormalized companion i…
Exact mathematical passage
D_r=\diag(\lambda^{(r)})
d_rUse in source · 16 passages
Search aliases: d_r
Use in source · root_scope.tex, line 159
[Exact classical tail valuation] For every r≥1 and N≥ r, v_2( T_r-1(N)) =4r-1+v_2binom Nr+μ_r+v_2G_r(N-r). There are unique factors G_r(X)=W_r(X)U_r(X), where W_r∈ Z_2[X] is monic of degree d_r=max J_r and U_r∈ Z_2⟨ X⟩^× satisfies U_r≡1pmod2. Consequently v_2G_r(x)=v_2W_r(x) for every x∈ Z_2, and deg W_r≤lfloorfracs_2(r)-13rfloor. Equation~[source reference], the exact weight in [source reference], and the digit-sum identity v_2binom2N2r=v_2binom Nr prove [source reference]. The binomial coeffic…
Exact mathematical passage
d_r=\max J_r
d_{r-1}Use in source · 1 passage
Search aliases: d_r-1
Use in source · Paper_X_Block27_Amended.tex, line 377
For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet are cochain maps, so they induce maps on these genuine quotients.
Exact mathematical passage
Z^r=\ker d_r,\qquad B^r=\operatorname{im}d_{r-1}\subseteq Z^r, \qquad H^r=Z^r/B^r,
D_{\rm computational}Use in source · 1 passage
Search aliases: D_rm computational rm
Use in source · monograph_atlas.tex, line 299
g Every effect in the computational basis E_a=ket abra a has the same expectation on both matrices, namely 1/2 at 00 and 11 and zero elsewhere. These measurements read only the diagonal. They cannot distinguish the pure Bell state from this incoherent mixture. headA measurement that resolves the difference Use the valid two-outcome measurement with effects E=ρ and I-E. Then Tr(Eρ)=1, Tr (E D_rm computational(ρ))=frac12. The effect itself is not diagonal in the fine computational packets. It comp…
Exact mathematical passage
\Tr(E\rho)=1,\qquad \Tr\!\left(E D_{\rm computational}(\rho)\right)=\frac12.
d_{\rm torus}Use in source · 2 passages
Search aliases: d_rm torus rm
Use in source · 02_strengthened.tex, line 2120
[Continuum torus small time dimension] Let d_rm torus(t) =-2frac dlog Z_d(t) dlog t. Then d_rm torus(t) = d+O(t^-1 e^-1/(4t)) as tdownarrow0.
Exact mathematical passage
d_{\rm torus}(t) =-2\frac{\dd\log Z_d(t)}{\dd\log t}.
d_sUse in source · 8 passages
Search aliases: d_s
Use in source · Paper_X_Block27_Amended.tex, line 946
Actual local comparisons from fixed-depth certificates. Let R be an I-adically complete separated commutative ring with finite coefficient quotients and an injective map into a field K. Fix a finite matrix rank. For finitely many local group maps ϕ_s:H_s→ G and integral template representations τ_s, fix detecting depths d_s. At precision m retain matrices P_s,m as part of the admissible object and require ρ_m(ϕ_s(h))P_s,m=P_s,mτ_s,m(h). For m≥ d_s require their determinants to remain nonzero aft…
Exact mathematical passage
d_s
d_{s,2}Use in source · 1 passage
Search aliases: d_s,2
Use in source · monograph_atlas.tex, line 454
For the Cartesian product of two identical cycles with four sites, the Laplacian eigenvalues add and the heat traces multiply: K_2=K^2. Consequently d_s,2=2d_s exactly. The product identity determines the second curve exactly from the first.
Exact mathematical passage
d_{s,2}=2d_s
d_{s,k}Use in source · 6 passages
Search aliases: d_s,k
Use in source · 72_recovered.tex, line 2212
[Mahler polyphase transfer] Let f(x)=∑_r≥0c_rbinom xr be a convergent Mahler expansion. For g_s(q)=f(Mq+s), one has g_s(q)=∑_k≥0d_s,kbinom qk, where d_s,k = ∑_r=k^Mk+s c_rΘ_M,s(r,k). Every refined coefficient uses a finite window of the original Mahler coefficients.
Exact mathematical passage
\boxed{ g_s(q)=\sum_{k\geq0}d_{s,k}\binom qk,}
d_s^{(1)}Use in source · 1 passage
Search aliases: d_s^(1)
Use in source · 01_finite_core.tex, line 1188
[Spectral dimension] The median intermediate plateaus were d_s^(1)=1.00062, d_s^(2)=2.03781. These tests recover the expected dimensions of the chosen finite geometries. They do not select a physical theory of quantum spacetime.
Exact mathematical passage
d_s^{(1)}=1.00062, \qquad d_s^{(2)}=2.03781.
d_s^{(2)}Use in source · 1 passage
Search aliases: d_s^(2)
Use in source · 01_finite_core.tex, line 1188
[Spectral dimension] The median intermediate plateaus were d_s^(1)=1.00062, d_s^(2)=2.03781. These tests recover the expected dimensions of the chosen finite geometries. They do not select a physical theory of quantum spacetime.
Exact mathematical passage
d_s^{(1)}=1.00062, \qquad d_s^{(2)}=2.03781.
D_uUse in source · 4 passages
Search aliases: D_u
Use in source · 04_correction_transport.tex, line 540
[Nonlinear Schur derivative] At (u_0,v_0), DR = D_uF_P - D_vF_P A^-1D_uF_Q.
Exact mathematical passage
DR = D_uF_P - D_vF_P A^{-1}D_uF_Q.
D_VUse in source · 12 passages
Search aliases: D_V
Use in source · Paper_VI_Traditional_Final.tex, line 4347
Let K be column stochastic. Let V be a real potential on the states and let D_V=diag(V). For functions written as columns, the backward Markov operator is K^T.
Exact mathematical passage
D_V=\diag(V).
d_vUse in source · 3 passages
Search aliases: d_v
Use in source · 72_recovered.tex, line 428
[Exponential form and minimal period] Let G_m=( F_p^×)^m. This cyclic group has order h. There are uniquely determined coefficients d_v∈ F_p, indexed by v∈ G_m, such that e_n^(m,p) = ∑_v∈ G_md_vv^n (n≥1). The minimal positive period is the least common multiple of the orders of those v for which d_v≠0. In particular, it divides h.
Exact mathematical passage
d_v\in\FF_p
D_v^{-1}Use in source · 2 passages
Search aliases: D_v^-1
Use in source · 02_supplement_05.tex, line 53
Constructed solution. Well-founded recursion defines x_v=D_v^-1 (b_v-∑_u→ vT_vux_u). Applying D_v proves the original equation at v. If another vector satisfies all equations, induction over predecessors makes each previously used coordinate identical, and invertibility of D_v forces agreement at v. Thus the construction gives existence and uniqueness, and the block operator is bijective.
Exact mathematical passage
\boxed{x_v=D_v^{-1}\!\left(b_v-\sum_{u\to v}T_{vu}x_u\right).}
D_wUse in source · 5 passages
Search aliases: D_w
Use in source · atlas-pages.tex, line 301
Here D_rK means an observation after prediction. In the Feynman--Kac example on pagerefV20, departure-state weighting has forward operator K D_w. Both conventions are legitimate once their meanings are fixed, but they cannot be substituted for one another. takeawayWrite the chronological sequence before multiplying matrices. A product can be normalized correctly and still answer the wrong time-ordered question.
Exact mathematical passage
K D_w
D_zUse in source · 5 passages
Search aliases: D_z
Use in source · 03_supplement_01.tex, line 106
-C · zI_Q-D. Set D_z=zI_Q-D, S_z=zI_P-A-BD_z^-1C. The effective Hamiltonian is H_eff(z)=A+BD_z^-1C, so S_z=zI_P-H_eff(z).
Exact mathematical passage
D_z=zI_Q-D,\qquad S_z=zI_P-A-BD_z^{-1}C.
D_z^{-1}Use in source · 2 passages
Search aliases: D_z^-1
Use in source · 03_supplement_01.tex, line 106
-C · zI_Q-D. Set D_z=zI_Q-D, S_z=zI_P-A-BD_z^-1C. The effective Hamiltonian is H_eff(z)=A+BD_z^-1C, so S_z=zI_P-H_eff(z).
Exact mathematical passage
D_z=zI_Q-D,\qquad S_z=zI_P-A-BD_z^{-1}C.
d_\alphaUse in source · 3 passages
Search aliases: d_α alpha
Use in source · 03_packets_and_graphs.tex, line 738
[Dimension count] If X is finite dimensional and d_α=dim X_α, then dimEnd_𝕜(X) = ∑_α,βd_α d_β = (∑_α d_α)^2. [source reference]
Exact mathematical passage
d_\alpha=\dim X_\alpha
D_\alpha^{-1}Use in source · 3 passages
Search aliases: D_α^-1 alpha
Use in source · 03_weighted.tex, line 94
[Block path formula] Let P=(P_α)_α∈ I be a finite packet system in a complete filtered algebra of operators. Suppose D=∑_α P_α D P_α belongs to B and is invertible on every packet. Assume that the assembled inverse D^-1=∑_α D_α^-1P_α∈ B=F^0 B, D_α=P_α DP_α, where each packet inverse is extended by zero to the other packets. Let the edge set be finite, assume w(e)≥1 for every edge, and write N=∑_e:α→βN_e, N_e=P_β N_eP_α, N_e∈ F^w(e) B. For an edge e:α→β, put A_e=(P_β DP_β)^-1N_e. Then the block f…
Exact mathematical passage
D^{-1}=\sum_\alpha D_\alpha^{-1}P_\alpha\in\Bcal=F^0\Bcal, \qquad D_\alpha=P_\alpha DP_\alpha,
D_\betaUse in source · 2 passages
Search aliases: D_β beta
Use in source · 01_manuscript.tex, line 1026
The equation in packet β is D_β x_β = b_β - ∑_α→βK_βαx_α. Every predecessor of β lies in an earlier level. Once those values are available, x_β is obtained from the local solve with D_β. Vertices in the same level have no directed edge between them, so their solves are independent.
Exact mathematical passage
D_\beta x_\beta = b_\beta - \sum_{\alpha\to\beta}K_{\beta\alpha}x_\alpha.
d_\betaUse in source · 2 passages
Search aliases: d_β beta
Use in source · 03_packets_and_graphs.tex, line 739
[Dimension count] If X is finite dimensional and d_α=dim X_α, then dimEnd_𝕜(X) = ∑_α,βd_α d_β = (∑_α d_α)^2. [source reference]
Exact mathematical passage
\dim\End_{\bbk}(X) = \sum_{\alpha,\beta}d_\alpha d_\beta = \left(\sum_\alpha d_\alpha\right)^2.
d_\etaUse in source · 2 passages
Search aliases: d_η eta
Use in source · 04_correction_transport.tex, line 396
Let L be a complete filtered differential graded Lie algebra over a ring in which two is invertible. Assume dF^i L⊆ F^i L, [F^i L,F^j L]⊆ F^i+j L. For a degree one element η, put Ω(η)=dη+frac12[η,η]. Suppose Ω(η)∈ F^m L^2 with m≥1. For every h∈ F^m L^1, Ω(η+h) ≡ Ω(η)+d_η h pmodF^m+1 L^2, d_η=d+[η,mathord·]. The obstruction to improving η by one depth is therefore the class of Ω(η) in the cokernel of gr^m(d_η): gr^m L^1 ⟶ gr^m L^2.
Exact mathematical passage
\Omega(\eta+h) \equiv \Omega(\eta)+d_\eta h \pmod{F^{m+1}\Lcal^2}, \qquad d_\eta=d+[\eta,\mathord\cdot].
d_\thetaUse in source · 3 passages
Search aliases: d_θ theta
Use in source · Paper_XI_Traditional_Final.tex, line 465
[Primitive cuspidal idempotent] Let χ_θ be the irreducible character of κ_θ, and let d_θ=χ_θ(1). Then e_θ = fracd_θabsG_n ∑_g∈ G_n χ_θ(g^-1)g is the primitive central idempotent selecting the κ_θ block in C[G_n].
Exact mathematical passage
d_\theta=\chi_\theta(1)
d_\rhoUse in source · 5 passages
Search aliases: d_ρ rho
Use in source · Paper_XI_Traditional_Final.tex, line 1837
Fix this chapter's reciprocity, additive-character, local-factor, and phase normalizations. Let A be the declared class of irreducible wild Weil atoms under consideration. For ρ∈ A, the supplied assembled finite datum d_ρ consists of the following inputs, historically denoted H1--H4: item an intrinsic finite break decomposition on the Galois side; item the declared equivariant symplectic bridges to the supplied breakwise type-side quotients; item compatible odd or dyadic central-extension and ph…
Exact mathematical passage
d_\rho
D_\ellUse in source · 5 passages
Search aliases: D_ℓ ell
Use in source · Paper_VI_Traditional_Final.tex, line 1329
[Fractional refinement towers] Let D_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ in a bounded nonnegative self-adjoint tower, with sup_ℓnormC_ℓ≤ M. If ∑_ℓnormD_ℓ^α<∞, then the fractional defects C_ℓ A_ℓ+1^α-A_ℓ^αC_ℓ have summable operator norms. If the tower has a uniform gap m>0, it is enough that ∑_ℓnormD_ℓ<∞. On every bounded time interval, the series of corresponding fractional-semigroup defect norms converges uniformly. The maps may have different domain and codomain spaces. No operator sum across unrelated spaces…
Exact mathematical passage
D_\ell=C_\ell A_{\ell+1}-A_\ell C_\ell
D_{\ell+1}Use in source · 1 passage
Search aliases: D_ℓ+1 ell
Use in source · 01_finite_core.tex, line 1038
A coherent refinement should satisfy C_ℓ E_ℓ=I and P_ℓ,α = C_ℓ ( ∑_βsuccαP_ℓ+1,β ) E_ℓ. For an operator D, define the refinement defect Δ_ℓ(D) =C_ℓ D_ℓ+1E_ℓ-D_ℓ. Exact resolution independence is Δ_ℓ(D)=0. When exact equality is too strong, the correct continuum problem is to prove convergence of the resolvents and uniform control of the effective self-energies.
Exact mathematical passage
\Delta_\ell(D) =C_\ell D_{\ell+1}E_\ell-D_\ell.
\ddot xUse in source · 4 passages
Search aliases: ddot x ddot
Use in source · monograph_atlas.tex, line 409
returns periodically. Deleting the hidden row and column would instead leave the zero Hamiltonian and predict x(t)=1. vfigmemory headEliminate exactly, not approximately Integrating the second equation with y(0)=0 gives y(t)=- i∫_0^tx(s) ds. Substitution produces dot x(t)=-∫_0^tx(s) ds. The memory kernel is the constant function one. Differentiating once more recovers ddot x=-x with x(0)=1 and dot x(0)=0, explaining the cosine solution. A nonzero hidden initial state would add a forcing term tha…
Exact mathematical passage
\ddot x=-x
\ddotsUse in source · 4 passages
Search aliases: ddots
Use in source · Paper_IV_Traditional_Final.tex, line 1131
[Logarithmic Newton recurrence] Set b_N,n(r)=fracP_N,n(r)(Nn)!. Then b_N,0(r)=1 and for every n≥1, nb_N,n(r) = r∑_k=1^nκ_N,kb_N,n-k(r). Equivalently, P_N,n(r) = frac r n ∑_k=1^n frac(Nn)!(N(n-k))! κ_N,kP_N,n-k(r). If p_k=rκ_N,k, then, for n≥1, b_N,n(r) = frac1n! det p_1 · -1 · 0 · ⋯ · 0 p_2 · p_1 · -2 · ⋯ · 0 p_3 · p_2 · p_1 · ddots · vdots vdots · vdots · ddots · ddots · -(n-1) p_n · p_n-1 · ⋯ · p_2 · p_1 .
Exact mathematical passage
b_{N,n}(r) = \frac1{n!} \det \begin{pmatrix} p_1&-1&0&\cdots&0\\ p_2&p_1&-2&\cdots&0\\ p_3&p_2&p_1&\ddots&\vdots\\ \vdots&\vdots&\ddots&\ddots&-(n-1)\\ p_n&p_{n-1}&\cdots&p_2&p_1 \end{pmatrix}.
\operatorname{dec}Use in source · 2 passages
Search aliases: dec operatorname
Use in source · 20_arithmetic.tex, line 539
Packing g_j into degrees divisible by four gives K_4,j. Its source property is precisely charge-one flatness. Inversion gives charge -1, and the charge shift restores charge one for the reciprocal P_4,j. If dec(F)(d)=d![t^d]F, the exact coefficient identity is dec(A_1^sF)(d) =2^-sΔ_4^sdec(F)(d). At d=4n, packing turns the right side into 2^-sΔ_1^sC_j(n). This extraction requires no negative degree and therefore holds at every n, including zero. It proves the unit-step bound leanKomorebiPaperIX.D…
Exact mathematical passage
\operatorname{dec}(F)(d)=d![t^d]F
\mathrm{decode}Use in source · 1 passage
Search aliases: decode mathrm
Use in source · 02_supplement_02.tex, line 203
Contracts, grouping, and amortization. textitSource items 2.9, 10.2, 10.4--10.6, and 11.3 Output conversion. A decoded run returns the actual pair (decode(compressed(x)), t_run(x)+t_decode(compressed(x))). Under the declared output equality, Lean proves that this returns the same explicit result and charges the conversion time. It also proves the downstream composition identity when compressed output is consumed directly. Omitting a positive decode cost strictly understates the modeled explicit-…
Exact mathematical passage
\bigl(\operatorname{decode}(\operatorname{compressed}(x)), \;t_{\mathrm{run}}(x)+t_{\mathrm{decode}}(\operatorname{compressed}(x))\bigr).
\operatorname{decode}Use in source · 3 passages
Search aliases: decode operatorname
Use in source · 01_manuscript.tex, line 231
[Contract conversion] Suppose one algorithm returns a compressed representation R and another returns an explicit object Y. A direct timing comparison is valid only if decode(R)=Y and the time required by decode is included, or if the downstream task consumes R directly.
Exact mathematical passage
\operatorname{decode}(R)=Y
\degUse in source · 6 passages
Search aliases: deg
Use in source · 03_weighted.tex, line 427
[Recurrence for path counts] Write Z_u,v(t)=fracP(t)Q(t), Q(t)=1+q_1t+⋯+q_dt^d. Use the convention deg 0=-∞. Then, for every nonnegative integer n>deg P, a_n+q_1a_n-1+⋯+q_da_n-d=0.
Exact mathematical passage
\deg 0=-\infty
\dephUse in source · 4 passages
Search aliases: deph D_P mathcal
Use in source · 03_supplement_01.tex, line 75
Measurements: source Theorem 7.1. For finite complex matrices, define packet dephasing by D_P(X)=∑_aP_aXP_a. The code proves idempotence and trace duality: D_P( D_P(X))= D_P(X), tr( D_P(E)ρ)=tr(E D_P(ρ)). Trace duality follows term by term from cyclicity: tr(P_aEP_aρ)=tr(EP_aρ P_a). Therefore an effect that is diagonal in the packets, D_P(E)=E, satisfies tr(Eρ)=tr(E D_P(ρ)). The proof holds for arbitrary finite complex matrices E and ρ. Density matrices and positive measurement effects are valid…
Exact mathematical passage
\deph(X)=\sum_aP_aXP_a.
\operatorname{DFT}Use in source · 3 passages
Search aliases: DFT operatorname
Use in source · 02_supplement_04.tex, line 22
Normalization and inversion. The proof identifies matrix application with the ordinary counting-measure discrete Fourier transform: Fx=n^-1/2DFT(x). It separately identifies F^*x with the appropriately scaled inverse transform. Mathlib's checked Fourier inversion theorem and the proved scalar normalization yield both identities F^*F=I, FF^*=I. No unitarity assumption is inserted.
Exact mathematical passage
Fx=n^{-1/2}\operatorname{DFT}(x).
\Dfun_1Use in source · 2 passages
Search aliases: Dfun_1 D_1 Dfun mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 752
The first denominators are D_1(z)=e^-z, D_2(z)=cos z. For N=3, D_3(z) = frac13( e^-z +2e^z/2cos(fracsqrt3z2) ). For N=4, D_4(z) = cos(frac zsqrt2) cosh(frac zsqrt2).
Exact mathematical passage
\Dfun_1(z)=e^{-z}, \qquad \Dfun_2(z)=\cos z.
\Dfun_2Use in source · 4 passages
Search aliases: Dfun_2 D_2 Dfun mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 752
The first denominators are D_1(z)=e^-z, D_2(z)=cos z. For N=3, D_3(z) = frac13( e^-z +2e^z/2cos(fracsqrt3z2) ). For N=4, D_4(z) = cos(frac zsqrt2) cosh(frac zsqrt2).
Exact mathematical passage
\Dfun_1(z)=e^{-z}, \qquad \Dfun_2(z)=\cos z.
\Dfun_3Use in source · 1 passage
Search aliases: Dfun_3 D_3 Dfun mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 758
The first denominators are D_1(z)=e^-z, D_2(z)=cos z. For N=3, D_3(z) = frac13( e^-z +2e^z/2cos(fracsqrt3z2) ). For N=4, D_4(z) = cos(frac zsqrt2) cosh(frac zsqrt2).
Exact mathematical passage
\Dfun_3(z) = \frac13\left( e^{-z} +2e^{z/2}\cos\left(\frac{\sqrt3z}{2}\right) \right).
\Dfun_4Use in source · 2 passages
Search aliases: Dfun_4 D_4 Dfun mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 767
The first denominators are D_1(z)=e^-z, D_2(z)=cos z. For N=3, D_3(z) = frac13( e^-z +2e^z/2cos(fracsqrt3z2) ). For N=4, D_4(z) = cos(frac zsqrt2) cosh(frac zsqrt2).
Exact mathematical passage
\Dfun_4(z) = \cos\left(\frac z{\sqrt2}\right) \cosh\left(\frac z{\sqrt2}\right).
\Dfun_{N,0}'Use in source · 1 passage
Search aliases: Dfun_N,0' D_N,0' Dfun mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 643
[Signed derivative cycle] For 1≤ j<N, D_N,j'= D_N,j-1, and D_N,0'=- D_N,N-1. Every signed section satisfies D_N,j^(N)=- D_N,j. The N signed sections form the basis of y^(N)=-y determined by D_N,j^(k)(0)=δ_jk for 0≤ j,k<N.
Exact mathematical passage
\Dfun_{N,0}'=-\Dfun_{N,N-1}.
\Dfun_{N,1}Use in source · 1 passage
Search aliases: Dfun_N,1 D_N,1 Dfun mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 836
[Companion matrix realization] Put D_N(z) = D_N,0(z) D_N,1(z) vdots D_N,N-1(z) . Then D_N(z)=e^zJ_Ne_0 and D_N(z)=frac1Ntr(e^zJ_N). The eigenvalues of J_N are the N roots of -1, each with multiplicity one.
Exact mathematical passage
\mathbf D_N(z) = \begin{pmatrix} \Dfun_{N,0}(z)\\ \Dfun_{N,1}(z)\\ \vdots\\ \Dfun_{N,N-1}(z) \end{pmatrix}.
\Dfun_{N,d_{pq}}Use in source · 1 passage
Search aliases: Dfun_N,d_pq D_N,d_pq Dfun mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 892
-1, · p<q. Then (e^zJ_N)_pq = ε_pq D_N,d_pq(z). Thus every entry of the fundamental matrix is one of the signed sections, with the sign recording whether the cyclic path from q to p crosses the signed edge.
Exact mathematical passage
\bigl(e^{zJ_N}\bigr)_{pq} = \varepsilon_{pq}\Dfun_{N,d_{pq}}(z).
\Dfun_{N,i}Use in source · 1 passage
Search aliases: Dfun_N,i D_N,i Dfun mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 679
[Twisted addition law] For 0≤ k<N, D_N,k(z+w) = ∑_substack0≤ i,j<N i+j≡ kbmod N (-1)^lfloor(i+j)/Nrfloor D_N,i(z) D_N,j(w).
Exact mathematical passage
\Dfun_{N,k}(z+w) = \sum_{\substack{0\leq i,j<N\\ i+j\equiv k\bmod N}} (-1)^{\lfloor(i+j)/N\rfloor} \Dfun_{N,i}(z)\Dfun_{N,j}(w).
\Dfun_{N,j}'Use in source · 1 passage
Search aliases: Dfun_N,j' D_N,j' Dfun mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 639
[Signed derivative cycle] For 1≤ j<N, D_N,j'= D_N,j-1, and D_N,0'=- D_N,N-1. Every signed section satisfies D_N,j^(N)=- D_N,j. The N signed sections form the basis of y^(N)=-y determined by D_N,j^(k)(0)=δ_jk for 0≤ j,k<N.
Exact mathematical passage
\Dfun_{N,j}'=\Dfun_{N,j-1},
\Dfun_{N,j-1}Use in source · 1 passage
Search aliases: Dfun_N,j-1 D_N,j-1 Dfun mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 639
[Signed derivative cycle] For 1≤ j<N, D_N,j'= D_N,j-1, and D_N,0'=- D_N,N-1. Every signed section satisfies D_N,j^(N)=- D_N,j. The N signed sections form the basis of y^(N)=-y determined by D_N,j^(k)(0)=δ_jk for 0≤ j,k<N.
Exact mathematical passage
\Dfun_{N,j}'=\Dfun_{N,j-1},
\Dfun_{N,j}^{(k)}Use in source · 1 passage
Search aliases: Dfun_N,j^(k) D_N,j^(k) Dfun mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 651
[Signed derivative cycle] For 1≤ j<N, D_N,j'= D_N,j-1, and D_N,0'=- D_N,N-1. Every signed section satisfies D_N,j^(N)=- D_N,j. The N signed sections form the basis of y^(N)=-y determined by D_N,j^(k)(0)=δ_jk for 0≤ j,k<N.
Exact mathematical passage
\Dfun_{N,j}^{(k)}(0)=\delta_{jk}
\Dfun_{N,j}^{(N)}Use in source · 1 passage
Search aliases: Dfun_N,j^(N) D_N,j^(N) Dfun mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 647
[Signed derivative cycle] For 1≤ j<N, D_N,j'= D_N,j-1, and D_N,0'=- D_N,N-1. Every signed section satisfies D_N,j^(N)=- D_N,j. The N signed sections form the basis of y^(N)=-y determined by D_N,j^(k)(0)=δ_jk for 0≤ j,k<N.
Exact mathematical passage
\Dfun_{N,j}^{(N)}=-\Dfun_{N,j}.
\Dfun_{N,k}Use in source · 1 passage
Search aliases: Dfun_N,k D_N,k Dfun mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 679
[Twisted addition law] For 0≤ k<N, D_N,k(z+w) = ∑_substack0≤ i,j<N i+j≡ kbmod N (-1)^lfloor(i+j)/Nrfloor D_N,i(z) D_N,j(w).
Exact mathematical passage
\Dfun_{N,k}(z+w) = \sum_{\substack{0\leq i,j<N\\ i+j\equiv k\bmod N}} (-1)^{\lfloor(i+j)/N\rfloor} \Dfun_{N,i}(z)\Dfun_{N,j}(w).
\Dfun_{N,N-1}Use in source · 3 passages
Search aliases: Dfun_N,N-1 D_N,N-1 Dfun mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 643
[Signed derivative cycle] For 1≤ j<N, D_N,j'= D_N,j-1, and D_N,0'=- D_N,N-1. Every signed section satisfies D_N,j^(N)=- D_N,j. The N signed sections form the basis of y^(N)=-y determined by D_N,j^(k)(0)=δ_jk for 0≤ j,k<N.
Exact mathematical passage
\Dfun_{N,0}'=-\Dfun_{N,N-1}.
\Dfun_N^{-q}Use in source · 1 passage
Search aliases: Dfun_N^-q D_N^-q Dfun mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 2968
The derivative weighted sum omits the finitely many zeros that might be multiple. A treatment of those zeros requires the complete principal parts of D_N^-q rather than the single derivative weight.
Exact mathematical passage
\Dfun_N^{-q}
\Dfun_N^{\prime}Use in source · 1 passage
Search aliases: Dfun_N^prime D_N^prime Dfun prime mathsf prime
Use in source · Paper_IV_Traditional_Final.tex, line 321
ic objects are introduced in the following order. First, E_N,j(z) = ∑_n≥0fracz^Nn+j(Nn+j)! is the residue j part of the exponential. Second, D_N,j(z) = ∑_n≥0(-1)^nfracz^Nn+j(Nn+j)! is the signed basis for y^(N)=-y. The section D_N= D_N,0 is the denominator used in the reciprocal family S_N^(r)= D_N^-r. Set T_N=- D_N^prime/ D_N. Equivalently, T_N is the logarithmic derivative of S_N^(1).
Exact mathematical passage
\Tfun_N=-\Dfun_N^{\prime}/\Dfun_N
\operatorname{diag}Use in source · 34 passages
Search aliases: diag operatorname
Use in source · visual_atlas.tex, line 296
vpage14Gauge changes the picture, not flatnessMODEL E / CONJUGATION AND ITS LIMITSChapter III, mathematical text: Theorem 6.2 At the base chart, change coordinates by D=diag(2,1). Transform every connection edge by the prescribed endpoint gauges. The loop transforms by conjugation, giving H^D=DHD^-1=mat1 · 2
Exact mathematical passage
D=\operatorname{diag}(2,1)
\dim_{\mathbb F_2}Use in source · 1 passage
Search aliases: dim_ F_2 dim mathbb
Use in source · dyadic_synthesis.tex, line 30
For this same constructed candidate, t(π) · =1, · Sw(π) · =3 2^d-1h_d, dim_ F_2W · =N-2, · dimη · =2^N/2-1, W=J^1/H^1, a(π^vee×π) · =N^2-1+∑_i=1^d3 4^i-1h_i. Matching these profiles does not identify the source, its endo-class, or the pair factors in original H5.
Exact mathematical passage
\begin{aligned} t(\pi)&=1,& \operatorname{Sw}(\pi)&=3\,2^{d-1}h_d,\\ \dim_{\mathbb F_2}W&=N-2,& \dim\eta&=2^{N/2-1},\qquad W=J^1/H^1,\\ a(\pi^\vee\times\pi) &=N^2-1+\sum_{i=1}^d3\,4^{i-1}h_i. \end{aligned}
\dim_{\mathbb F_p}Use in source · 2 passages
Search aliases: dim_ F_p dim mathbb
Use in source · odd_synthesis.tex, line 36
The same candidate has rank(π) · =N, · t(π) · =1, Sw(π) · =p^d-1(p+1)h_d, · dim_ F_p(J^1/H^1) · =N-1, dimη · =p^(N-1)/2, a(π^vee×π) · =p^2d-1+∑_i=1^d(p^2i-p^2i-2)h_i. These numerical identities do not identify a Weil representation, an endo-class, or the pair epsilon factors required by H5.
Exact mathematical passage
\begin{aligned} \operatorname{rank}(\pi)&=N, &t(\pi)&=1,\\ \operatorname{Sw}(\pi)&=p^{d-1}(p+1)h_d, & \dim_{\mathbb F_p}(J^1/H^1)&=N-1,\\ \dim\eta&=p^{(N-1)/2},\\ a(\pi^\vee\times\pi) &=p^{2d}-1+\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i. \end{aligned}
\dim_{\bbk}Use in source · 1 passage
Search aliases: dim_bbk dim_𝕜 dim bbk dim Bbbk
Use in source · 02_foundations.tex, line 974
[Formula for geometric multiplicity] If 𝕜 is algebraically closed, then srank_𝕜[T](X) = max_λ∈Spec(T) dim_𝕜Ker(T-λ I). [source reference]
Exact mathematical passage
\boxed{ \srank_{\bbk[T]}(X) = \max_{\lambda\in\operatorname{Spec}(T)} \dim_{\bbk}\Ker(T-\lambda I). }
\dim_{\bbk[t]/(p)}Use in source · 1 passage
Search aliases: dim_bbk[t]/(p) dim_𝕜[t]/(p) dim bbk dim Bbbk
Use in source · 02_foundations.tex, line 896
[Seed rank for a single operator] Let X have finite dimension over 𝕜 and let T∈End_𝕜(X). Regard X as a module over 𝕜[t] by t· x=Tx. For every monic irreducible p, let X_p be the primary component for p and set μ_p(T):= dim_𝕜[t]/(p)(X_p/p(T)X_p). Then srank_𝕜[T](X)=max_pμ_p(T). Moreover, μ_p(T) is the number of cyclic primary summands for p in the decomposition into elementary divisors of X_p. DependsnolinkurlPSC-I-CI-PID-PRIMARY-DECOMPOSITION, nolinkurlPSC-I-C-LOCAL-ALGEBRA-FORMULA, and nolinkur…
Exact mathematical passage
\mu_p(T):= \dim_{\bbk[t]/(p)}\bigl(X_p/p(T)X_p\bigr).
\dim_\CUse in source · 2 passages
Search aliases: dim_C dim_ C dim C dim mathbb
Use in source · 03_supplement_03.tex, line 9
A matrix X belongs to O(C) exactly when each of its rows belongs to Row(C). One direction follows by expanding AC. For the converse, choose the coefficients producing each row separately and place them in the corresponding row of A. An explicit linear equivalence therefore gives O(C)≃ Row(C)^d, dim_ C O(C)=d rank C. If d>0, equality with the full d× m coefficient space is equivalent to rank C=m. Consequently a cyclic seed requires m≤ d. The assumption that there is at least one row is essential …
Exact mathematical passage
\mathcal O(C)\simeq \operatorname{Row}(C)^d, \qquad \boxed{\dim_\C\mathcal O(C)=d\,\rank C.}
\dim_DUse in source · 1 passage
Search aliases: dim_D dim
Use in source · 02_foundations.tex, line 712
[Local algebra formula] Let A be a local ring with Jacobson radical J and residue division ring D=A/J. For every finitely generated left module M over A, srank_A(M)=dim_D(M/JM), where the dimension is that of a left vector space over D. [source reference]
Exact mathematical passage
\boxed{ \srank_A(M)=\dim_D(M/JM), }
\dim_{E'}Use in source · 1 passage
Search aliases: dim_E' dim
Use in source · Paper_XII_Traditional_Final.tex, line 1093
[Rank and trace detection in characteristic zero] Assume that E' has characteristic zero. For every maximal Hecke packet, Tr_E'(e_ m|V) = rank_E'(e_ m|V) = dim_E'e_ mV. In particular, the packet is occupied exactly when its projector has positive rank, equivalently nonzero trace.
Exact mathematical passage
\boxed{ \Tr_{E'}(e_{\frakm}|V) = \rank_{E'}(e_{\frakm}|V) = \dim_{E'}e_{\frakm}V. }
\dim_KUse in source · 1 passage
Search aliases: dim_K dim
Use in source · 05_worked_models.tex, line 565
[Exact graph seed rank from a certificate] For a finite square matrix A over a field K, define the noncomputable exact spectral quantity r_spec(A) = max_μ∈Spec_K(A) dim_Kker(A-μ I), with value zero for an empty spectrum. Suppose an exact certificate supplies a finite eigenbasis indexed by labels and multiplicities, the eigenvalue equations, a multiplicity bound attained by one label, projector polynomials that extract the padded eigenbasis generators, coverage of every spectral value by a label,…
Exact mathematical passage
r_{\mathrm{spec}}(A) = \max_{\mu\in\Spec_K(A)} \dim_K\ker(A-\mu I),
\dim_{K_p}Use in source · 1 passage
Search aliases: dim_K_p dim
Use in source · 02_foundations.tex, line 931
(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with maximal ideal generated by the class of p and residue division field K_p=𝕜[t]/(p). By [source reference], srank_R_p(X_p) = dim_K_pX_p/p(T)X_p = μ_p(T). Finally, every summand 𝕜[t]/(p^e) contributes one copy of K_p after quotienting by p, so the dimension equa…
Exact mathematical passage
\srank_{R_p}(X_p) = \dim_{K_p}X_p/p(T)X_p = \mu_p(T).
\dim_{\kappa(\frakp)}Use in source · 1 passage
Search aliases: dim_κ(frakp) dim_κ( p) dim kappa frakp dim kappa mathfrak
Use in source · Paper_XII_Traditional_Final.tex, line 629
[Higher Fitting strata] Let R be Noetherian and let M be finite. For r≥1, put μ_ p(M) = dim_κ( p) (M⊗_Rκ( p)). Then { p∈Spec R: μ_ p(M)≥ r } = V(Fitt_r-1^R(M)).
Exact mathematical passage
\mu_{\frakp}(M) = \dim_{\kappa(\frakp)} \left(M\otimes_R\kappa(\frakp)\right).
\operatorname{disc}Use in source · 1 passage
Search aliases: disc operatorname
Use in source · Paper_X_Block27_Amended.tex, line 1270
The exceptional primes are controlled by the actual field discriminant: S_L={p rational prime:pmiddisc(L)}. This set is finite. For p∉ S_L, every prime of O_L above (p) is unramified. Lean proves that the actual inertia cardinality is the ramification index. It follows that inertia is trivial and the arithmetic Frobenius at a fixed finite prime is unique as a field automorphism. Neither trivial inertia nor uniqueness is assumed in place of the arithmetic unramifiedness condition.
Exact mathematical passage
S_L=\{p\text{ rational prime}:p\mid\operatorname{disc}(L)\}.
\distUse in source · 6 passages
Search aliases: dist operatorname
Use in source · 04_correction_transport.tex, line 21
gger be its Moore--Penrose inverse. Then P_ob=I_H_2-DD^† is the orthogonal projector onto (im D)^⊥=Ker D^*. For r∈ H_2, the following are equivalent: item Du=-r is solvable item [r]=0 in coker D item P_obr=0. When these conditions hold, -D^† r is the least norm correction. For every r, dist(r,im D)= P_obr .
Exact mathematical passage
\dist(r,\im D)=\|P_{\mathrm{ob}}r\|.
\operatorname{dist}Use in source · 1 passage
Search aliases: dist operatorname
Use in source · 02_supplement_03.tex, line 124
Define the comparison space L_j+1=⊕_i K_j+1(P_iDP_i,P_iv). Packet diagonality is first shown to imply commutation. The exact theorem then proves D^jv∈ L_j+1. Using that vector as a witness in the infimum distance gives dist(A^jv, L_j+1) ≤ jM^j-1normKnormv. This statement uses Lean's actual metric infimum distance to the subspace. It does not assume that the comparison vector lies there. The block-Jacobi condition bound and the preconditioned conjugate-gradient corollary are not completed by thes…
Exact mathematical passage
\boxed{\operatorname{dist}(A^jv,\mathcal L_{j+1}) \leq jM^{j-1}\norm{K}\norm{v}.}
\mathrm{div}Use in source · 2 passages
Search aliases: div mathrm
Use in source · Paper_V_Traditional_Final.tex, line 372
one also has A_N S_N=I. Equip this finite product with the Hilbert norm (x_0,…,x_N-1) _ch^2 =∑_r=0^N-1 x_r _2^2. Then A_N is a linear isometric equivalence and x _2^2=∑_r=0^N-1 x_r _2^2. Euclidean division gives a unique pair (r,q) with 0≤ r<N and n=Nq+r, including negative n. The address maps n↦(nbmod N,nmathbindivN) and (r,q)↦ Nq+r are inverse. Analysis and synthesis are reindexing by this bijection. Reindexing the nonnegative sum ∑_n|x[n]|^2 proves the norm identity and shows that both maps p…
Exact mathematical passage
n\mapsto(n\bmod N,n\mathbin{\mathrm{div}}N)
\operatorname{Dom}Use in source · 1 passage
Search aliases: Dom operatorname
Use in source · Paper_VI_Traditional_Final.tex, line 5335
[Coordinate generators] Assume the setting of [source reference]. If x=(x_j) satisfies x_j∈Dom(A_j) and the sequence (A_jx_j) belongs to X_∞, then x lies in the generator domain of T_∞ and A_∞ x=(A_jx_j)_j≥0.
Exact mathematical passage
x_j\in\operatorname{Dom}(A_j)
\dot PUse in source · 36 passages
Search aliases: dot P dot
Use in source · 04_correction_transport.tex, line 801
Zero projections are permitted in this section and contribute zero to all generator sums. When T=0, every partition product is the empty product I. Completeness and orthogonality give P_α(t)P_β(t)=0 for α≠β, ∑_α P_α(t)=I. Define the canonical generator K(t)=∑_α dot P_α(t)P_α(t).
Exact mathematical passage
K(t)=\sum_\alpha \dot P_\alpha(t)P_\alpha(t).
\dot pUse in source · 3 passages
Search aliases: dot p dot
Use in source · Paper_VI_Traditional_Final.tex, line 1058
The semigroup law is the likelihood-composition theorem with e^sge^tg=e^(s+t)g. Differentiating the normalized exponential formula gives dot p_i=g_ip_i-p_ifrac∑_jg_je^tg_jp_j(0)∑_je^tg_jp_j(0). The quotient is ⟨ g,p(t)⟩.
Exact mathematical passage
\dot p_i=g_ip_i-p_i\frac{\sum_jg_je^{tg_j}p_j(0)}{\sum_je^{tg_j}p_j(0)}.
\dot p_0Use in source · 1 passage
Search aliases: dot p_0 dot
Use in source · atlas-pages.tex, line 277
Differentiation of the normalized flow yields the centered score equation dot p_i=(g_i-⟨ g,p⟩)p_i. For the displayed model, dot p_0=p_0(1-p_0). The negative average-score term is what keeps the total derivative zero. Omitting the derivative of the normalizer would not preserve mass.
Exact mathematical passage
\dot p_0=p_0(1-p_0)
\dot P_aUse in source · 16 passages
Search aliases: dot P_a dot
Use in source · monograph_atlas.tex, line 244
n For this complete pair of packets, define C=dot P P+dot Q Q. Exact calculation yields the constant matrix C=J, with C^*=-C and [C,P]=dot P. At t=0, the displayed small matrices allow these identities to be checked by direct multiplication. headFrom two packets to a complete family For differentiable orthogonal projectors P_a summing to the identity, the paper constructs C=∑_adot P_aP_a=frac12∑_a[dot P_a,P_a]. This single connection reproduces the motion of every packet in the family. No basis …
Exact mathematical passage
C=\sum_a\dot P_aP_a=\frac12\sum_a[\dot P_a,P_a].
\dot P_bUse in source · 9 passages
Search aliases: dot P_b dot
Use in source · 01_finite_core.tex, line 532
[Canonical packet connection] Define C(t) = frac12∑_a[dot P_a(t),P_a(t)] = ∑_adot P_a(t)P_a(t). Then C(t)^*=-C(t) and [C(t),P_b(t)]=dot P_b(t) for every packet b.
Exact mathematical passage
\boxed{ [C(t),P_b(t)]=\dot P_b(t) }
\dot p_iUse in source · 2 passages
Search aliases: dot p_i dot
Use in source · Paper_VI_Traditional_Final.tex, line 1058
The semigroup law is the likelihood-composition theorem with e^sge^tg=e^(s+t)g. Differentiating the normalized exponential formula gives dot p_i=g_ip_i-p_ifrac∑_jg_je^tg_jp_j(0)∑_je^tg_jp_j(0). The quotient is ⟨ g,p(t)⟩.
Exact mathematical passage
\dot p_i=g_ip_i-p_i\frac{\sum_jg_je^{tg_j}p_j(0)}{\sum_je^{tg_j}p_j(0)}.
\dot P_\alphaUse in source · 6 passages
Search aliases: dot P_α dot alpha
Use in source · 04_correction_transport.tex, line 801
Zero projections are permitted in this section and contribute zero to all generator sums. When T=0, every partition product is the empty product I. Completeness and orthogonality give P_α(t)P_β(t)=0 for α≠β, ∑_α P_α(t)=I. Define the canonical generator K(t)=∑_α dot P_α(t)P_α(t).
Exact mathematical passage
K(t)=\sum_\alpha \dot P_\alpha(t)P_\alpha(t).
\dot P_\betaUse in source · 5 passages
Search aliases: dot P_β dot beta
Use in source · 04_correction_transport.tex, line 810
begingroup setlengthabovedisplayskip6pt plus 2pt minus 2pt setlengthbelowdisplayskip6pt plus 2pt minus 2pt [Complete packet commutator identity] For every packet label β, dot P_β(t)=[K(t),P_β(t)]. The generator is also skew-adjoint: K(t)^*=-K(t).
Exact mathematical passage
\dot P_\beta(t)=[K(t),P_\beta(t)].
\dot QUse in source · 1 passage
Search aliases: dot Q dot
Use in source · monograph_atlas.tex, line 238
Chapter VII: Theorem II.6.1, Theorem II.6.2. Continue with P(t)=U(t)diag(1,0)U(t)^* and Q(t)=I-P(t). Differentiating P^2=P gives dot P P+Pdot P=dot P, Pdot P P=0. The second identity says that the derivative has no component entirely inside the current packet. It moves the subspace relative to its complement. vfigconnection For this complete pair of packets, define C=dot P P+dot Q Q. Exact calculation yields the constant matrix C=J, with C^*=-C and [C,P]=dot P. At t=0, the displayed small matric…
Exact mathematical passage
C=\dot P P+\dot Q Q.
\dot uUse in source · 1 passage
Search aliases: dot u dot
Use in source · Paper_VI_Traditional_Final.tex, line 4404
Conditioning over the first small time interval gives the backward equation dot u=(Q^T-D_V)u, u(0)=f. The finite system has a unique solution, namely the displayed matrix exponential. The expectation bound follows from exp(-∫_0^tV(X_s) d s) ≤ e^-v_0t. This finite argument is the matrix form of the Feynman--Kac principle originating in [source reference].
Exact mathematical passage
\dot u=(Q^T-D_V)u, \qquad u(0)=f.
\dot U^*Use in source · 2 passages
Search aliases: dot U^* dot
Use in source · 03_supplement_02.tex, line 102
The explicitly defined coordinate curve y(s)=U(s)^*ψ(s) is differentiable at t, with dot y=dot U^*ψ(t)+U(t)^*dotψ. It satisfies idot y=H_effy(t), H_eff=U(t)^*HU(t)-iU(t)^*dot U. The connection term -iU(t)^*dot U is Hermitian. If H^*=H, then H_eff^*=H_eff.
Exact mathematical passage
\dot y=\dot U^*\psi(t)+U(t)^*\dot\psi.
\dot VUse in source · 2 passages
Search aliases: dot V dot
Use in source · 04_correction_transport.tex, line 863
The continuous bounded generator has an evolution inverse. Indeed, solve dot V=-VK with V(0)=I. Differentiation gives (VU)^prime=0, hence VU=I. The product W=UV satisfies dot W=KW-WK and W(0)=I. Uniqueness of this bounded linear equation gives UV=I. Thus U is invertible.
Exact mathematical passage
\dot V=-VK
\dot WUse in source · 9 passages
Search aliases: dot W dot
Use in source · 04_correction_transport.tex, line 863
The continuous bounded generator has an evolution inverse. Indeed, solve dot V=-VK with V(0)=I. Differentiation gives (VU)^prime=0, hence VU=I. The product W=UV satisfies dot W=KW-WK and W(0)=I. Uniqueness of this bounded linear equation gives UV=I. Thus U is invertible.
Exact mathematical passage
\dot W=KW-WK
\dot W^*Use in source · 1 passage
Search aliases: dot W^* dot
Use in source · 02_strengthened.tex, line 720
Differentiate V=W^*U. Since dot W=KW and K^*=-K, dot W^*=-W^*K. Substitution gives the equation. The first term in H_rm mov is self adjoint. The operator - i W^*dot W is self adjoint because W^*dot W is skew adjoint.
Exact mathematical passage
\dot W^*=-W^*K.
\dot xUse in source · 18 passages
Search aliases: dot x dot
Use in source · Paper_VI_Traditional_Final.tex, line 1527
For the homogeneous system dot x=Ax, variation of constants gives x_Q(t)=e^tA_QQx_Q(0) +∫_0^te^(t-r)A_QQA_QPx_P(r) dr. Substitution yields dot x_P(t) =A_PPx_P(t)+A_PQe^tA_QQx_Q(0) +∫_0^t K(t-r)x_P(r) dr, where K(t)=A_PQe^tA_QQA_QP.
Exact mathematical passage
\dot x=Ax
\dot x_PUse in source · 3 passages
Search aliases: dot x_P dot
Use in source · Paper_VI_Traditional_Final.tex, line 1533
For the homogeneous system dot x=Ax, variation of constants gives x_Q(t)=e^tA_QQx_Q(0) +∫_0^te^(t-r)A_QQA_QPx_P(r) dr. Substitution yields dot x_P(t) =A_PPx_P(t)+A_PQe^tA_QQx_Q(0) +∫_0^t K(t-r)x_P(r) dr, where K(t)=A_PQe^tA_QQA_QP.
Exact mathematical passage
\dot x_P(t) =A_{PP}x_P(t)+A_{PQ}e^{tA_{QQ}}x_Q(0) +\int_0^t\cK(t-r)x_P(r)\,dr,
\dot yUse in source · 11 passages
Search aliases: dot y dot
Use in source · Paper_VI_Traditional_Final.tex, line 4802
[Finite auxiliary state realization] Every kernel of the form K(t)=B e^tDC can be represented exactly by adjoining the finite auxiliary state dot y=Dy+Cx_P, dot x_P=A_PPx_P+By, y(0)=0. A nonzero auxiliary initial state adds the explicit forcing B e^tDy(0). The self-energy of the augmented system is B(zI-D)^-1C.
Exact mathematical passage
\dot y=Dy+Cx_P, \qquad \dot x_P=A_{PP}x_P+By, \qquad y(0)=0.
\dot y_1Use in source · 1 passage
Search aliases: dot y_1 dot
Use in source · atlas-pages.tex, line 895
vpage39Two auxiliary states realize a repeated poleFINITE REALIZATION / AN EXACT DIFFERENTIAL MODELChapter VI, Part II: Corollaries E15.4 and E15.5. This is a supplied rational realization. The kernel K(t)=te^-2t from Plate VI.23 has transform 1/(z+2)^2. It is realized by two auxiliary states: dot y_1 · =-2y_1+y_2, · y_1(0) · =0, dot y_2 · =-2y_2+x, · y_2(0) · =0, dot x · =a x+y_1. · With D=mat-2 · 1
Exact mathematical passage
\begin{aligned} \dot y_1&=-2y_1+y_2,& y_1(0)&=0,\\ \dot y_2&=-2y_2+x,& y_2(0)&=0,\\ \dot x&=a x+y_1.& \end{aligned}
\dot y_2Use in source · 1 passage
Search aliases: dot y_2 dot
Use in source · atlas-pages.tex, line 895
vpage39Two auxiliary states realize a repeated poleFINITE REALIZATION / AN EXACT DIFFERENTIAL MODELChapter VI, Part II: Corollaries E15.4 and E15.5. This is a supplied rational realization. The kernel K(t)=te^-2t from Plate VI.23 has transform 1/(z+2)^2. It is realized by two auxiliary states: dot y_1 · =-2y_1+y_2, · y_1(0) · =0, dot y_2 · =-2y_2+x, · y_2(0) · =0, dot x · =a x+y_1. · With D=mat-2 · 1
Exact mathematical passage
\begin{aligned} \dot y_1&=-2y_1+y_2,& y_1(0)&=0,\\ \dot y_2&=-2y_2+x,& y_2(0)&=0,\\ \dot x&=a x+y_1.& \end{aligned}
\dot{\widetilde x}Use in source · 1 passage
Search aliases: dotwidetilde x dot widetilde
Use in source · Paper_VI_Traditional_Final.tex, line 5682
entiable solutions of the displayed equations. This direct bounded-generator estimate retains a sharper exponential constant than the general mild-semigroup estimate in Part II. [Stability of memory compression] Let x and widetilde x solve dot x(t) =Ax(t)+∫_0^t K(t-s)x(s) d s+f(t) and dotwidetilde x(t) =Awidetilde x(t) +∫_0^twidetilde K(t-s)widetilde x(s) d s +f(t) with the same initial value. Fix T>0, and assume sup_0≤ t≤ Tnormwidetilde x(t)≤ M. Set a=norm A, k=norm K_L^1(0,T), δ=norm K-widetil…
Exact mathematical passage
\dot{\widetilde x}(t) =A\widetilde x(t) +\int_0^t\widetilde\Kcal(t-s)\widetilde x(s)\dd s +f(t)
\DownarrowUse in source · 1 passage
Search aliases: Downarrow
Use in source · 05_worked_models.tex, line 627
Theorem spine at a glance. ConceptChain c free seed presentation [0.25em] Downarrow [-0.1em] minimal generator number [0.25em] Downarrow [-0.1em] reduction modulo the Jacobson radical [0.25em] Downarrow [-0.1em] semisimple or primary computation c packet idempotents [0.25em] Downarrow [-0.1em] exact analysis/synthesis [0.25em] Downarrow [-0.1em] transfer block algebra [0.25em] Downarrow [-0.1em] graphs, quotients, and realization
Exact mathematical passage
\begin{array}{c} \text{free seed presentation}\\[0.25em] \Downarrow\\[-0.1em] \text{minimal generator number}\\[0.25em] \Downarrow\\[-0.1em] \text{reduction modulo the Jacobson radical}\\[0.25em] \Downarrow\\[-0.1em] \text{semisimple or primary computation} \end{array} \qquad\qquad \begin{array}{c} \text{packet idempotents}\\[0.25em] \Downarrow\\[-0.1em] \text{exact analysis/synthesis}\\[0.25em] \Downarrow\\[-0.1em] \text{transfer block algebra}\\[0.25em] \Downarrow\\[-0.1em] \text{graphs, quotients, and realization} \end{array}
\downarrowUse in source · 12 passages
Search aliases: downarrow
Use in source · 03_nonlinear_coordinates.tex, line 953
[Naturality of obstruction classes] Suppose ccc U · xrightarrowD · V mathllapadownarrow · · downarrowmathrlapb U' · xrightarrowD' · V' is a commuting square. Then there is a unique induced map overline b:coker D⟶coker D' satisfying overline b Ob_D(r)=Ob_D'(br).
Exact mathematical passage
\begin{array}{ccc} U & \xrightarrow{D} & V\\ \mathllap{a}\downarrow & & \downarrow\mathrlap{b}\\ U' & \xrightarrow{D'} & V' \end{array}
\mathbb EUse in source · 2 passages
Search aliases: E mathbb
Use in source · Paper_V_Traditional_Final.tex, line 3091
Assume the noise variables are independent, centered, and σ sub-Gaussian: E e^λη_k ≤ e^σ^2λ^2/2.
Exact mathematical passage
\mathbb E\e^{\lambda\eta_k} \leq \e^{\sigma^2\lambda^2/2}.
e'Use in source · 4 passages
Search aliases: e'
Use in source · 03_nonlinear_coordinates.tex, line 1150
The class of (e,0) vanishes exactly when there are e'∈ E^n-1 and c∈ C^n with (e,0)=d_Cone(e',c). This means e=d_Ee'+Dc, d_Cc=0. Equivalently, [e]=H^n(D)[c]. The degree zero statement follows directly from the definition of the cone differential.
Exact mathematical passage
e'\in E^{n-1}
E'^rUse in source · 4 passages
Search aliases: E'^r
Use in source · Paper_XII_Traditional_Final.tex, line 1556
Let W = span { (t_1(ω),…,t_r(ω)):ω∈Ω } ⊆ E'^r. If W≠ E'^r, a nonzero linear functional on E'^r would vanish on every evaluation vector, giving a nontrivial linear relation among the functions. Thus W=E'^r. Choose r evaluation vectors forming a basis.
Exact mathematical passage
W = \operatorname{span} \left\{ (t_1(\omega),\ldots,t_r(\omega)):\omega\in\Omega \right\} \subseteq E'^r.
E^{\mathsf T}Use in source · 2 passages
Search aliases: E^ T mathsf
Use in source · atlas-pages.tex, line 160
)^ T becomes Ef=(2,2,26/7,26/7)^ T. vfigweighted_projection platecaptionThe bars are function values, not state probabilities. Their total need not be one. The block masses 3/10 and 7/10 determine the denominators in the averaging formula. headWhich inner product makes this an orthogonal projection? Put W=diag(π). Direct multiplication checks E^2=E, WE=E^ TW, Eones=ones. The middle equation is weighted self-adjointness. Ordinary symmetry fails: E_01=2/3 whereas E_10=1/3. Testing E=E^ T would the…
Exact mathematical passage
E^2=E,\qquad WE=E^{\mathsf T}W,\qquad E\ones=\ones.
\e^{(2a+1)t}Use in source · 1 passage
Search aliases: e^(2a+1)t e mathrm
Use in source · 72_recovered.tex, line 937
The finite geometric sum gives ∑_a=0^q-1(-1)^a e^(2a+1)t = e^tfrac1+ e^2qt1+ e^2t = e^qtcosh(qt)secht. Averaging at t and -t gives the even part.
Exact mathematical passage
\sum_{a=0}^{q-1}(-1)^a\e^{(2a+1)t} = \e^t\frac{1+\e^{2qt}}{1+\e^{2t}} = \e^{qt}\cosh(qt)\operatorname{sech}t.
e^{(2a+1)t}Use in source · 2 passages
Search aliases: e^(2a+1)t
Use in source · 20_arithmetic.tex, line 814
The exponential generating series of the source is the finite sum S_p(t)=∑_a=0^p-1(-1)^a e^(2a+1)t. Multiplication by cosh t telescopes. Since p is odd, S_p(t)=frac1+e^2pt2secht. Taking its even part gives ∑_n≥0S_2n(p)fract^2n(2n)! =cosh^2(pt)secht. Multiplication by the reciprocal sech^2(pt) yields secht. The EGF product rule now gives the reconstruction coefficient by coefficient. Replacing m by n-q produces the displayed lower triangular row. Its diagonal and weights follow from γ_0=1 and the…
Exact mathematical passage
\mathcal S_p(t)=\sum_{a=0}^{p-1}(-1)^a e^{(2a+1)t}.
e^{(2k+1)\pi i/N}Use in source · 1 passage
Search aliases: e^(2k+1)π i/N pi
Use in source · Paper_IV_Traditional_Final.tex, line 1743
We now assume N≥2. Write η_k=e^(2k+1)π i/N, 0≤ k<N. These are the vertices of a regular polygon and D_N(z)=frac1N∑_k=0^N-1e^η_kz.
Exact mathematical passage
\eta_k=e^{(2k+1)\pi i/N}, \qquad 0\leq k<N.
\e^{(a+k)T}Use in source · 1 passage
Search aliases: e^(a+k)T e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 5701
e x solve dot x(t) =Ax(t)+∫_0^t K(t-s)x(s) d s+f(t) and dotwidetilde x(t) =Awidetilde x(t) +∫_0^twidetilde K(t-s)widetilde x(s) d s +f(t) with the same initial value. Fix T>0, and assume sup_0≤ t≤ Tnormwidetilde x(t)≤ M. Set a=norm A, k=norm K_L^1(0,T), δ=norm K-widetilde K_L^1(0,T). Then sup_0≤ t≤ T normx(t)-widetilde x(t) ≤ MTδ e^(a+k)T.
Exact mathematical passage
\boxed{ \sup_{0\leq t\leq T} \norm{x(t)-\widetilde x(t)} \leq MT\delta\e^{(a+k)T}. }
\e^{(a+k)t}Use in source · 1 passage
Search aliases: e^(a+k)t e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 5720
Let e=x-widetilde x. Integrating the difference equation and changing the order of integration gives norme(t) ≤ (a+k)∫_0^tnorme(r) d r +Mtδ. Since t≤ T, Gronwall's inequality gives norme(t) ≤ MTδ e^(a+k)t. Take the supremum over [0,T].
Exact mathematical passage
\norm{e(t)} \leq MT\delta\e^{(a+k)t}.
e^{(m,p)}Use in source · 2 passages
Search aliases: e^(m,p)
Use in source · 72_recovered.tex, line 865
[A second proof of the standard window period] The reduced inverse e^(m,p) and every standard address satisfy the period in [source reference].
Exact mathematical passage
e^{(m,p)}
e^{(m,p)}_nUse in source · 1 passage
Search aliases: e^(m,p)_n
Use in source · supplement_05.tex, line 61
Then apply the residue map. Independently define e^(m,p) over F_p by the unit-diagonal system ∑_q=0^nbinommnmqe^(m,p)_q=δ_n0. Its recurrence and uniqueness are checked. Applying the actual homomorphism π: Z_p→ F_p to the integral row, and then Lucas reduction, proves π(c^(p)_m,j(n))=e^(m,p)_n. For j,k<p, a separate ideal statement gives pmid c^(p)_m,j(n)-c^(p)_m,k(n) inside Z_p. The formal solver also covers m=0. The manuscript uses m≥1. No global finite-difference theorem or period assertion en…
Exact mathematical passage
\boxed{\pi(c^{(p)}_{m,j}(n))=e^{(m,p)}_n.}
e^{(m,p)}_qUse in source · 1 passage
Search aliases: e^(m,p)_q
Use in source · supplement_05.tex, line 57
Then apply the residue map. Independently define e^(m,p) over F_p by the unit-diagonal system ∑_q=0^nbinommnmqe^(m,p)_q=δ_n0. Its recurrence and uniqueness are checked. Applying the actual homomorphism π: Z_p→ F_p to the integral row, and then Lucas reduction, proves π(c^(p)_m,j(n))=e^(m,p)_n. For j,k<p, a separate ideal statement gives pmid c^(p)_m,j(n)-c^(p)_m,k(n) inside Z_p. The formal solver also covers m=0. The manuscript uses m≥1. No global finite-difference theorem or period assertion en…
Exact mathematical passage
\sum_{q=0}^{n}\binom{mn}{mq}e^{(m,p)}_q=\delta_{n0}.
e^{(s+t)A}Use in source · 2 passages
Search aliases: e^(s+t)A
Use in source · Paper_VI_Traditional_Final.tex, line 335
[Finite semigroup law] For every matrix A and all s,t∈ R, e^(s+t)A=e^sAe^tA. Moreover, T(0)=I and T'(t)=AT(t)=T(t)A.
Exact mathematical passage
\boxed{e^{(s+t)A}=e^{sA}e^{tA}.}
e^{(s+t)g}Use in source · 1 passage
Search aliases: e^(s+t)g
Use in source · Paper_VI_Traditional_Final.tex, line 1057
The semigroup law is the likelihood-composition theorem with e^sge^tg=e^(s+t)g. Differentiating the normalized exponential formula gives dot p_i=g_ip_i-p_ifrac∑_jg_je^tg_jp_j(0)∑_je^tg_jp_j(0). The quotient is ⟨ g,p(t)⟩.
Exact mathematical passage
e^{sg}e^{tg}=e^{(s+t)g}
e^{(s+t)Q}Use in source · 1 passage
Search aliases: e^(s+t)Q
Use in source · atlas-pages.tex, line 90
Any other complete packet family still has exact block convolution. For the coordinate groups A and B, P_b e^(s+t)QP_a=∑_c∈{ A, B}(P_b e^sQP_c)(P_c e^tQP_a). The intermediate packet cannot generally be omitted. Only a reducing family eliminates all cross terms.
Exact mathematical passage
P_b e^{(s+t)Q}P_a=\sum_{c\in\{\mathrm A,\mathrm B\}}(P_b e^{sQ}P_c)(P_c e^{tQ}P_a).
\e^{(s_m-s_{m-1})D}Use in source · 1 passage
Search aliases: e^(s_m-s_m-1)D e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 2236
[Dyson expansion over packet paths] For t≥0, e^tA = · e^tD + ∑_m=1^∞ ∫_0<s_1<⋯<s_m<t e^(t-s_m)DN e^(s_m-s_m-1)D ⋯ N e^s_1D d s_1⋯ d s_m. The series converges absolutely in operator norm. After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.
Exact mathematical passage
\begin{aligned} \e^{tA} ={}&\e^{tD} + \sum_{m=1}^{\infty} \int_{0<s_1<\cdots<s_m<t} \e^{(t-s_m)D}N\e^{(s_m-s_{m-1})D} \cdots N\e^{s_1D} \dd s_1\cdots\dd s_m. \end{aligned}
e^{(t-r)A_c}Use in source · 3 passages
Search aliases: e^(t-r)A_c
Use in source · Paper_VI_Traditional_Final.tex, line 463
[Exact Duhamel refinement identity] For every t≥0, Ce^tA_f-e^tA_cC = ∫_0^t e^(t-r)A_cΔ e^rA_f dr. In particular, Δ=0 implies exact semigroup intertwining at every time.
Exact mathematical passage
\boxed{ Ce^{tA_f}-e^{tA_c}C = \int_0^t e^{(t-r)A_c}\Delta e^{rA_f}\,dr. }
e^{(t-r)A_{QQ}}Use in source · 1 passage
Search aliases: e^(t-r)A_QQ
Use in source · Paper_VI_Traditional_Final.tex, line 1528
For the homogeneous system dot x=Ax, variation of constants gives x_Q(t)=e^tA_QQx_Q(0) +∫_0^te^(t-r)A_QQA_QPx_P(r) dr. Substitution yields dot x_P(t) =A_PPx_P(t)+A_PQe^tA_QQx_Q(0) +∫_0^t K(t-r)x_P(r) dr, where K(t)=A_PQe^tA_QQA_QP.
Exact mathematical passage
x_Q(t)=e^{tA_{QQ}}x_Q(0) +\int_0^te^{(t-r)A_{QQ}}A_{QP}x_P(r)\,dr.
e^{(t-r)D}Use in source · 1 passage
Search aliases: e^(t-r)D
Use in source · atlas-pages.tex, line 907
0 · -2, B=(1,0), and C=(0,1)^ T, the feedback kernel is Be^tDC=te^-2t. vfigext_auxiliary platecaptionAn input first enters y_2, transfers to y_1, and then returns to x. The two equal decay rates produce a repeated pole through the intervening link. headElimination recovers the convolution, including its preparation Variation of constants gives y(t)=e^tDy(0)+∫_0^t e^(t-r)DCx(r) dr, so the retained equation is dot x(t)=a x(t)+Be^tDy(0) +∫_0^t (t-r)e^-2(t-r)x(r) dr. The convolution-only version is …
Exact mathematical passage
y(t)=e^{tD}y(0)+\int_0^t e^{(t-r)D}Cx(r)\,dr,
\e^{(t-s)(QAQ)|_{QX}}Use in source · 1 passage
Search aliases: e^(t-s)(QAQ)|_QX e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 2786
For x∈ PX, write p(t)=PT(t)x, q(t)=QT(t)x. The hidden component satisfies q'(t)=QAQq(t)+QAPp(t), q(0)=0. Variation of constants gives q(t)= ∫_0^t e^(t-s)(QAQ)|_QX QAPp(s) d s. Since normp(s)≤norm PnormT(s)norm x and x=Px, the first estimate follows. The contraction case is immediate.
Exact mathematical passage
q(t)= \int_0^t \e^{(t-s)(QAQ)|_{QX}} QAPp(s) \dd s.
\e^{(t-s)A_c}Use in source · 3 passages
Search aliases: e^(t-s)A_c e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 2290
[Duhamel defect identity] For every t≥0, D(t) = ∫_0^t e^(t-s)A_cΔ e^sA_f d s. If norm e^tA_f≤ M_f e^ω_f t, norm e^tA_c≤ M_c e^ω_c t, then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-s)+ω_fs d s. For contraction semigroups, norm D(t)≤ tnormΔ.
Exact mathematical passage
\mathfrak D(t) = \int_0^t \e^{(t-s)A_c}\Delta\e^{sA_f} \dd s.
\e^{(t-s)A_{QQ}}Use in source · 1 passage
Search aliases: e^(t-s)A_QQ e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 4519
For the homogeneous equation dot x=Ax, variation of constants gives x_Q(t) = e^tA_QQx_Q(0) + ∫_0^t e^(t-s)A_QQA_QPx_P(s) d s. Substitution into the retained equation gives dot x_P(t) = · A_PPx_P(t) + A_PQ e^tA_QQx_Q(0)
Exact mathematical passage
x_Q(t) = \e^{tA_{QQ}}x_Q(0) + \int_0^t \e^{(t-s)A_{QQ}}A_{QP}x_P(s) \dd s.
\e^{(t-s)D}Use in source · 1 passage
Search aliases: e^(t-s)D e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 2254
Write the variation of constants identity e^tA = e^tD + ∫_0^t e^(t-s)DN e^sA d s. Iterating the identity gives the displayed series. The norm of the term of order m is at most e^tnorm Dfrac(tnorm N)^mm!. The majorant is summable. Inserting the packet resolution around every factor N leaves only products whose consecutive packet labels are joined by nonzero transfer blocks.
Exact mathematical passage
\e^{tA} = \e^{tD} + \int_0^t\e^{(t-s)D}N\e^{sA}\dd s.
e^{(t-s)T_c}Use in source · 4 passages
Search aliases: e^(t-s)T_c
Use in source · 05_defects.tex, line 182
[Exponential defect identity] Let C:X_f→ X_c, T_f, and T_c be bounded linear maps between Banach spaces over a common field. For real t, use real or complex spaces. For complex t, use complex spaces and complex-linear maps. Then Ce^tT_f-e^tT_cC = ∫_0^t e^(t-s)T_cΔ e^sT_f ds, where the integral is taken along the straight segment from 0 to t when t is complex. For real t≥0, Ce^tT_f-e^tT_cC ≤ Δ ∫_0^t e^(t-s)T_c e^sT_f ds.
Exact mathematical passage
Ce^{tT_f}-e^{tT_c}C = \int_0^t e^{(t-s)T_c}\Delta e^{sT_f}\,ds,
\e^{(t-s_m)D}Use in source · 1 passage
Search aliases: e^(t-s_m)D e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 2236
[Dyson expansion over packet paths] For t≥0, e^tA = · e^tD + ∑_m=1^∞ ∫_0<s_1<⋯<s_m<t e^(t-s_m)DN e^(s_m-s_m-1)D ⋯ N e^s_1D d s_1⋯ d s_m. The series converges absolutely in operator norm. After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.
Exact mathematical passage
\begin{aligned} \e^{tA} ={}&\e^{tD} + \sum_{m=1}^{\infty} \int_{0<s_1<\cdots<s_m<t} \e^{(t-s_m)D}N\e^{(s_m-s_{m-1})D} \cdots N\e^{s_1D} \dd s_1\cdots\dd s_m. \end{aligned}
e^{(z+w)\xi}Use in source · 2 passages
Search aliases: e^(z+w)ξ xi
Use in source · Paper_IV_Traditional_Final.tex, line 817
Differentiation multiplies by λξ in the first algebra and by ξ in the second. Reduction of the product by the defining relation gives the derivative cycles. The identity e^(z+w)ξ=e^zξe^wξ gives the addition laws after comparison of the coefficients of 1,ξ,…,ξ^N-1.
Exact mathematical passage
e^{(z+w)\xi}=e^{z\xi}e^{w\xi}
e^{(\lambda-z)t}Use in source · 1 passage
Search aliases: e^(λ-z)t lambda
Use in source · Paper_VI_Traditional_Final.tex, line 1559
Put B=A_QQ and T_z(t)=e^-zte^tB=e^t(B-zI). The generalized-eigenspace decomposition in finite dimensions writes T_z as a finite sum of constant matrix coefficients times t^k e^(λ-z)t. This follows by expanding the nilpotent part on each generalized eigenspace. Equivalently, apply the decomposition to left multiplication by B on the matrix algebra and evaluate the exponential at I. The strict spectral inequality makes every such term absolutely integrable on (0,∞) and forces it to tend to zero. H…
Exact mathematical passage
t^k e^{(\lambda-z)t}
e^{(\mu-z)t}Use in source · 1 passage
Search aliases: e^(μ-z)t mu
Use in source · Paper_VI_Traditional_Final.tex, line 1931
General nonnormal memory (13.2). The actual damped exponential is decomposed into finitely many terms of the form t^k e^(μ-z)t. Integrability and convergence to zero follow from Reμ<Rez. The formal proof uses generalized eigenspaces of left multiplication on the matrix algebra, so nilpotent Jordan parts are included. Both inverse identities follow by integrating the two exponential derivative formulas. A continuous linear sandwich map gives the literal rectangular matrix memory integral and the …
Exact mathematical passage
t^k e^{(\mu-z)t}
E^*Use in source · 3 passages
Search aliases: E^*
Use in source · Paper_V_Traditional_Final.tex, line 2147
[Frame bounds from the polyphase symbol] Assume that A I_N ≤ E( e^ iω)^*E( e^ iω) ≤ B I_N for almost every ω, where 0<A≤ B<∞. Then the analysis bank satisfies A x _2^2 ≤ A_E x _2^2 ≤ B x _2^2. The pointwise matrix R_rm can( e^ iω) = (E^*E)^-1E^* is a bounded left inverse and gives the canonical dual synthesis bank.
Exact mathematical passage
R_{\rm can}(\e^{\ii\omega}) = (E^*E)^{-1}E^*
\e^{-(s+t)}Use in source · 1 passage
Search aliases: e^-(s+t) e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 3157
Positivity and preservation of constants follow from the integral formula. If X and Y are independent standard Gaussian variables, then e^-s ( e^-tx+sqrt1- e^-2tX ) + sqrt1- e^-2sY has the same distribution as e^-(s+t)x + sqrt1- e^-2(s+t)Z with Z standard Gaussian. This proves the semigroup law. Jensen's inequality and invariance of γ give the L^p contraction. The joint Gaussian pair ( X, e^-tX+sqrt1- e^-2tY ) is symmetric under interchange, which proves self-adjointness.
Exact mathematical passage
\e^{-(s+t)}x + \sqrt{1-\e^{-2(s+t)}}Z
e^{-(s-r)B^\alpha}Use in source · 2 passages
Search aliases: e^-(s-r)B^α alpha
Use in source · Paper_VI_Traditional_Final.tex, line 1297
[Fractional semigroup refinement] For s≥0, Ce^-sA^α-e^-sB^αC = -∫_0^s e^-(s-r)B^αD_α e^-rA^α dr. Hence normCe^-sA^α-e^-sB^αC ≤ snormD_α. Under the common gap A,B≥ mI, normCe^-sA^α-e^-sB^αC ≤ s e^-sm^αα m^α-1norm D.
Exact mathematical passage
\boxed{ Ce^{-sA^\alpha}-e^{-sB^\alpha}C = -\int_0^s e^{-(s-r)B^\alpha}D_\alpha e^{-rA^\alpha}\,dr. }
e^{-(t-r)}Use in source · 2 passages
Search aliases: e^-(t-r)
Use in source · atlas-pages.tex, line 493
vpage22Hidden state returns as memoryEXACT RETAINED DYNAMICS / NOT A SCALAR SEMIGROUPChapter VI, Part I: Theorem 13.2. The retained-state memory identity, including hidden initial data. Write the same system as x'=-x+y, y'=x-y. Solving the second equation first gives y(t)=e^-ty(0)+∫_0^t e^-(t-r)x(r) dr. Substitution is exact: x'(t)=-x(t)+e^-ty(0)+∫_0^t e^-(t-r)x(r) dr. The hidden initial state produces a separate forcing term. It must be retained even when the memory kernel is known. vfigmemory …
Exact mathematical passage
y(t)=e^{-t}y(0)+\int_0^t e^{-(t-r)}x(r)\,dr.
e^{-(t_a+t_b)H}Use in source · 1 passage
Search aliases: e^-(t_a+t_b)H
Use in source · monograph_atlas.tex, line 431
nnegative entries and column sums one. Conjugate by the unitary Z. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become -1/4. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation. headA useful finite positive kernel For this nonnegative H, vectors u_a and times t_a≥0 satisfy ∑_a,b⟨ u_a,e^-(t_a+t_b)Hu_b⟩ = ∑_a e^-t_aHu_a ^2≥0. The factorization makes positivity transparent. It remains a fini…
Exact mathematical passage
\sum_{a,b}\langle u_a,e^{-(t_a+t_b)H}u_b\rangle =\left\|\sum_a e^{-t_aH}u_a\right\|^2\ge0.
e^{-(z+2)t}Use in source · 1 passage
Search aliases: e^-(z+2)t
Use in source · atlas-pages.tex, line 529
t. The nilpotent link carries an input from the second hidden coordinate to the first. vfigjordan platecaptionThe memory kernel starts at zero, becomes positive, and peaks at t=1/2. A rule that retained the eigenvalue -2 but discarded the Jordan chain would predict the zero kernel for these entrance and exit maps. headThe transfer function sees pole order The exact transform is U(zI-J)^-1V=frac1(z+2)^2 =∫_0^∞ t e^-(z+2)t dt, Rez>-2. The repeated pole and the polynomial factor t are the same info…
Exact mathematical passage
U(zI-J)^{-1}V=\frac1{(z+2)^2} =\int_0^\infty t e^{-(z+2)t}\,dt, \qquad \operatorname{Re}z>-2.
e^{-1}Use in source · 2 passages
Search aliases: e^-1
Use in source · Paper_XII_Traditional_Final.tex, line 1389
[Derived pointwise realization] Let e:R≅ T, and let C be a perfect automorphic Hecke complex with full derived support. Choose a bounded finite projective model for C, supported in degrees [a,b]. For every characteristic zero point x:R→ E', write y=x∘ e^-1. Then some integer i∈[a,b] has H^i(Cmathbinmathop⊗limits_ T,y^ LE')≠0. Every t∈ T acts on this cohomology through y(t).
Exact mathematical passage
y=x\circ e^{-1}
\e^{-1/(4t)}Use in source · 3 passages
Search aliases: e^-1/(4t) e mathrm
Use in source · 02_strengthened.tex, line 2125
[Continuum torus small time dimension] Let d_rm torus(t) =-2frac dlog Z_d(t) dlog t. Then d_rm torus(t) = d+O(t^-1 e^-1/(4t)) as tdownarrow0.
Exact mathematical passage
d_{\rm torus}(t) = d+O\left(t^{-1}\e^{-1/(4t)}\right)
\e^{-16t\abs{k}^2}Use in source · 2 passages
Search aliases: e^-16tabsk^2 e abs mathrm abs
Use in source · 02_strengthened.tex, line 2095
ixed integer mode, [source reference] gives λ_N(k)⟶4π^2absk^2. For centered modes, the inequality sin x≥frac2xπ, 0≤ x≤fracπ2, gives λ_N(k)≥16absk^2. Also λ_N(k)≤4π^2absk^2. Extend the discrete summands by zero outside the centered residue box. The heat trace terms are dominated by e^-16tabsk^2, and the moment terms by 4π^2absk^2 e^-16tabsk^2. Both dominating sequences are summable on Z^d. Dominated convergence proves the two limits. Their quotient gives spectral dimension convergence.
Exact mathematical passage
\e^{-16t\abs{k}^2},
\e^{-2(s+t)}Use in source · 1 passage
Search aliases: e^-2(s+t) e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 3157
Positivity and preservation of constants follow from the integral formula. If X and Y are independent standard Gaussian variables, then e^-s ( e^-tx+sqrt1- e^-2tX ) + sqrt1- e^-2sY has the same distribution as e^-(s+t)x + sqrt1- e^-2(s+t)Z with Z standard Gaussian. This proves the semigroup law. Jensen's inequality and invariance of γ give the L^p contraction. The joint Gaussian pair ( X, e^-tX+sqrt1- e^-2tY ) is symmetric under interchange, which proves self-adjointness.
Exact mathematical passage
\e^{-(s+t)}x + \sqrt{1-\e^{-2(s+t)}}Z
e^{-2(s+t)}Use in source · 1 passage
Search aliases: e^-2(s+t)
Use in source · atlas-pages.tex, line 66
A point seed at state zero approaches the uniform distribution. The displayed times were selected so that every bar label is rational. This is continuous-time evolution, not a simulated trajectory. headTwo checks explain the mechanism The constant mode is unchanged, while each elapsed interval multiplies the contrast by e^-2t. Consequently two intervals multiply their contrast factors: e^-2se^-2t=e^-2(s+t), T(s)T(t)=T(s+t). For t≥0, the entries are nonnegative and every column sums to one. Diffe…
Exact mathematical passage
e^{-2s}e^{-2t}=e^{-2(s+t)},\qquad T(s)T(t)=T(s+t).
e^{-2(t-r)}Use in source · 2 passages
Search aliases: e^-2(t-r)
Use in source · atlas-pages.tex, line 142
The continuous counterpart is equally explicit. With fine generator -1, coarse generator -2, and C=1, the generator defect is 1 and e^-t-e^-2t=∫_0^t e^-2(t-r)e^-r dr. Its resolvent version is 1/(z+1)-1/(z+2)=1/((z+1)(z+2)) for z>0. takeawayA small defect does not disappear. The identity specifies how it accumulates, and the norm assumptions turn that identity into a finite-horizon certificate.
Exact mathematical passage
e^{-t}-e^{-2t}=\int_0^t e^{-2(t-r)}e^{-r}\,dr.
e^{-2\ii t}Use in source · 1 passage
Search aliases: e^-2ii t e^-2 i t ii mathrm
Use in source · monograph_atlas.tex, line 680
4/5 · 0 · 2. Its memory kernel is K(t)=frac925e^- i t+frac1625e^-2 i t, K(0)=1. Each oscillatory coefficient is positive although K(t) need not be real or pointwise nonnegative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structural identity is ∑_j,koverline v_jK(t_j-t_k)v_k = ∑_k e^ i t_kDB^*v_k ^2≥0. The exact worked check specializes to t=(0,π/2,π) and v=(1,2,-1). Its phase…
Exact mathematical passage
K(t)=\frac9{25}e^{-\ii t}+\frac{16}{25}e^{-2\ii t}, \qquad K(0)=1.
e^{-2\ii\beta\ell}Use in source · 1 passage
Search aliases: e^-2iiβℓ e^-2 iβℓ ii beta ell mathrm beta ell
Use in source · Paper_V_Traditional_Final.tex, line 1215
For a lossless transmission line, Γ_rm in=Γ_L e^-2 iβℓ. One N-phase step corresponds to a line-length increment λ/(2N), up to the chosen orientation convention. Hence the phase address has a direct electrical interpretation.
Exact mathematical passage
\Gam_{\rm in}=\Gam_L e^{-2\ii\beta\ell}.
e^{-2jt}Use in source · 1 passage
Search aliases: e^-2jt
Use in source · atlas-pages.tex, line 330
itive self-adjoint finite model and its spectral powers. Return to the four-state generator on pagerefV4 and put L=-Q. It is real symmetric and nonnegative, with orthogonal projectors P_0,P_1,P_2,P_3 and eigenvalues 0,2,4,6. Its square root is L^1/2=sqrt2P_1+2P_2+sqrt6P_3, (L^1/2)^2=L. Both heat evolutions retain these modes. Ordinary diffusion multiplies mode j by e^-2jt. Square-root diffusion multiplies it by e^-sqrt2j t. The constant mode is unchanged in both. vfigfractional platecaptionThe h…
Exact mathematical passage
e^{-2jt}
\e^{-2nt}Use in source · 1 passage
Search aliases: e^-2nt e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 3205
[Hermite packet reconstruction] Let P_n be the orthogonal projection onto C h_n. Then I=∑_n=0^∞ P_n in the strong operator topology and O_t = ∑_n=0^∞ e^-ntP_n. For f∈ L^2(γ), norm O_t f_2^2 = ∑_n=0^∞ e^-2ntnormP_nf_2^2.
Exact mathematical passage
\norm{\Ocal_t f}_2^2 = \sum_{n=0}^\infty \e^{-2nt}\norm{P_nf}_2^2.
e^{-2pt}Use in source · 1 passage
Search aliases: e^-2pt
Use in source · supplement_02.tex, line 60
Even projection is identified coefficientwise. For any formal series f, let EvenPart(f)=(f(t)+f(-t))/2. Its nth coefficient is proved to be the coefficient of f when n is even and zero otherwise. Rescaling the inverse law proves sech(-t)=sech(t). The exponential multiplication identity then gives cosh^2(pt)=frac2+e^2pt+e^-2pt4, and hence ∑_r≥0S_2r(p)fract^2r(2r)! =cosh^2(pt) sech t. Both the full and even EGF formulas are derived from the original integer sum. Neither is supplied as a hypothesis…
Exact mathematical passage
\cosh^2(pt)=\frac{2+e^{2pt}+e^{-2pt}}4,
\e^{-2s}Use in source · 1 passage
Search aliases: e^-2s e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 3148
Positivity and preservation of constants follow from the integral formula. If X and Y are independent standard Gaussian variables, then e^-s ( e^-tx+sqrt1- e^-2tX ) + sqrt1- e^-2sY has the same distribution as e^-(s+t)x + sqrt1- e^-2(s+t)Z with Z standard Gaussian. This proves the semigroup law. Jensen's inequality and invariance of γ give the L^p contraction. The joint Gaussian pair ( X, e^-tX+sqrt1- e^-2tY ) is symmetric under interchange, which proves self-adjointness.
Exact mathematical passage
\e^{-s} \left( \e^{-t}x+\sqrt{1-\e^{-2t}}X \right) + \sqrt{1-\e^{-2s}}Y
e^{-2s}Use in source · 2 passages
Search aliases: e^-2s
Use in source · atlas-pages.tex, line 66
A point seed at state zero approaches the uniform distribution. The displayed times were selected so that every bar label is rational. This is continuous-time evolution, not a simulated trajectory. headTwo checks explain the mechanism The constant mode is unchanged, while each elapsed interval multiplies the contrast by e^-2t. Consequently two intervals multiply their contrast factors: e^-2se^-2t=e^-2(s+t), T(s)T(t)=T(s+t). For t≥0, the entries are nonnegative and every column sums to one. Diffe…
Exact mathematical passage
e^{-2s}e^{-2t}=e^{-2(s+t)},\qquad T(s)T(t)=T(s+t).
e^{-2s_Ny}Use in source · 1 passage
Search aliases: e^-2s_Ny
Use in source · Paper_IV_Traditional_Final.tex, line 1913
a_N-θ. Monotonicity of cosine on [0,π], followed by reflection about π, therefore gives Re(η_ℓ z)≤ rcos(a_N+θ). The remaining vertex e^-ia_N has projection rcos(a_N-θ). Their difference is at least 2s_Nrsinθ=2s_Ny. If D_N(z)=0, divide its finite exponential sum by the term with exponent e^-ia_Nz and apply the triangle inequality: 1≤(N-1)e^-2s_Ny. Thus y≤ Y. Conjugation gives the same bound for -y on the lower half sector. Since N≥3 and s_N>0, this Y is positive.
Exact mathematical passage
1\leq(N-1)e^{-2s_Ny}.
\e^{-2t}Use in source · 5 passages
Search aliases: e^-2t e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 3121
Let γ be the standard Gaussian probability measure on R. For t≥0, define the Ornstein--Uhlenbeck transfer ( O_t f)(x) = ∫_ R f( e^-tx+sqrt1- e^-2t y ) γ( d y). Let H_n be the probabilists' Hermite polynomial defined by exp(ux-fracu^22) = ∑_n=0^∞ H_n(x)fracu^nn!, and set h_n=H_n/sqrtn!.
Exact mathematical passage
(\Ocal_t f)(x) = \int_\RR f\left( \e^{-t}x+\sqrt{1-\e^{-2t}}\,y \right) \gamma(\dd y).
e^{-2t}Use in source · 19 passages
Search aliases: e^-2t
Use in source · atlas-pages.tex, line 59
1 · -1. The total-mass direction (1,1) has eigenvalue zero. The contrast (1,-1) has eigenvalue -2. Resolving those two directions gives the full transfer, not only its first-order approximation: T(t)=e^tG=frac121+e^-2t · 1-e^-2t 1-e^-2t · 1+e^-2t. vfigclock platecaptionA point seed at state zero approaches the uniform distribution. The displayed times were selected so that every bar label is rational. This is continuous-time evolution, not a simulated trajectory. headTwo checks explain the mecha…
Exact mathematical passage
T(t)=e^{tG}=\frac12\begin{pmatrix}1+e^{-2t}&1-e^{-2t}\\1-e^{-2t}&1+e^{-2t}\end{pmatrix}.
e^{-2\gamma_Nt_m}Use in source · 1 passage
Search aliases: e^-2γ_Nt_m gamma
Use in source · Paper_IV_Traditional_Final.tex, line 2093
[Leading displacement term and derivative sign] With the notation of [source reference], τ_m = t_m+ fracR_N(t_m)2is_N + O(e^-2γ_Nt_m). The quotient R_N(t_m)/(2is_N) is real. One also has D_N'(τ_m) = (-1)^m+1 frac2s_NN e^c_Nτ_m (1+O(e^-γ_Nτ_m)).
Exact mathematical passage
\tau_m = t_m+ \frac{R_N(t_m)}{2is_N} + O(e^{-2\gamma_Nt_m}).
e^{-2\pi ir}Use in source · 1 passage
Search aliases: e^-2π ir pi
Use in source · atlas-pages.tex, line 235
gion containing 0 and normalize the logarithm to vanish there. D_N(z)^-r=exp(-rL(z)), e^L(z)=D_N(z), L(0)=0. vfigbranch_domain For D_2=cos z, the open disk |z|<π/2 is an admissible domain. A small loop around the simple zero ρ=π/2 is not contained in that disk. Analytic continuation around the zero changes the local logarithm by 2π i, and the power by e^-2π ir. headThree different kinds of statement A formal identity compares coefficients at the origin. A holomorphic identity for complex powers …
Exact mathematical passage
e^{-2\pi ir}
e^{-2\tau}Use in source · 2 passages
Search aliases: e^-2τ tau
Use in source · monograph_atlas.tex, line 422
vpage23Positive transfer is not a stochastic matrixSPECTRAL POSITIVITY IS BASIS INDEPENDENTChapter VII: Theorem II.10.1, Theorem II.10.3. For H=I-X, the two eigenvalues are zero and two. Its Euclidean transfer is T(τ)=e^-τ H =frac12mat1+q · 1-q 1-q · 1+q, q=e^-2τ. For τ≥0, the eigenvalues 1,q lie in (0,1], and multiplying two such matrices replaces q,r by qr. This is the semigroup law in spectral coordinates. vfigeuclidean At q=1/2, the displayed T happens also to have nonnegative entries and co…
Exact mathematical passage
T(\tau)=e^{-\tau H} =\frac12\mat{1+q&1-q\\1-q&1+q},\qquad q=e^{-2\tau}.
e^{-3t}Use in source · 1 passage
Search aliases: e^-3t
Use in source · atlas-pages.tex, line 850
vpage37Keep blocks by their certified contributionMEMORY COMPRESSION / AN OPTIMIZATION WITH A DEFINED COSTChapter VI, Part II: Theorem E15.1 and Corollary E15.2. Stable primary blocks. Block count is the budget. Consider the scalar memory kernel K(t)=tfrac14e^-t+(3+2t)e^-2t+tfrac13e^-3t. It has three primary blocks. The middle block carries a Jordan term and must be retained or omitted as a whole for this selection rule. Their certified integrated weights are w_1=frac14, w_2=frac32+frac22^2=2, w…
Exact mathematical passage
\mathcal K(t)=\tfrac14e^{-t}+(3+2t)e^{-2t}+\tfrac13e^{-3t}.
e^{-3t_m}Use in source · 1 passage
Search aliases: e^-3t_m
Use in source · Paper_IV_Traditional_Final.tex, line 2149
[Leading correction for N=3] For N=3, t_m=frac2πsqrt3(m+frac12) and τ_m = t_m+ frac(-1)^msqrt3e^-3t_m/2 + O(e^-3t_m).
Exact mathematical passage
\tau_m = t_m+ \frac{(-1)^m}{\sqrt3}e^{-3t_m/2} + O(e^{-3t_m}).
e^{-3t_m/2}Use in source · 3 passages
Search aliases: e^-3t_m/2
Use in source · Paper_IV_Traditional_Final.tex, line 2149
[Leading correction for N=3] For N=3, t_m=frac2πsqrt3(m+frac12) and τ_m = t_m+ frac(-1)^msqrt3e^-3t_m/2 + O(e^-3t_m).
Exact mathematical passage
\tau_m = t_m+ \frac{(-1)^m}{\sqrt3}e^{-3t_m/2} + O(e^{-3t_m}).
e^{-4t}Use in source · 1 passage
Search aliases: e^-4t
Use in source · atlas-pages.tex, line 762
nH_n for these three polynomials. Their squared norms are n! and different Hermite degrees are orthogonal. vfigext_hermite platecaptionFor f=H_1+H_2, the two bars are its separate squared-energy contributions. At t=log2, their sum is 3/8, below the spectral-gap bound 3/4. headReconstruction preserves the modes as time changes their weights The semigroup sends f=H_1+H_2 to T(t)f=e^-tH_1+e^-2tH_2, T(t)f _2^2=e^-2t+2e^-4t. Its mean is zero. The Gaussian gap bounds this by e^-2t f _2^2=3e^-2t. It ne…
Exact mathematical passage
T(t)f=e^{-t}H_1+e^{-2t}H_2,\qquad \|T(t)f\|_2^2=e^{-2t}+2e^{-4t}.
\e^{-4\pi^2t\abs{k}^2}Use in source · 3 passages
Search aliases: e^-4π^2tabsk^2 e pi abs mathrm pi abs
Use in source · 02_strengthened.tex, line 2061
[Fixed time heat trace convergence] For every t>0, ∑_k∈Λ_N^* e^-tλ_N(k) ⟶ ∑_k∈ Z^d e^-4π^2tabsk^2 as N→∞. The first spectral moments also converge: ∑_k∈Λ_N^* λ_N(k) e^-tλ_N(k) ⟶ ∑_k∈ Z^d 4π^2absk^2 e^-4π^2tabsk^2. Consequently, the discrete spectral dimensions converge to the continuum torus spectral dimension at every fixed positive time.
Exact mathematical passage
\sum_{k\in\Lambda_N^*} \e^{-t\lambda_N(k)} \longrightarrow \sum_{k\in\ZZ^d} \e^{-4\pi^2t\abs{k}^2}
e^{-4\tau}Use in source · 1 passage
Search aliases: e^-4τ tau
Use in source · monograph_atlas.tex, line 442
vpage24A dimension curve is a scale diagnosticTHE EXACT HEAT TRACE ON FOUR SITESChapter VII: Definition II.11.1, Proposition II.11.2. For the Laplacian on a cycle with four sites L=2I-S-S^*, the eigenvalues are 0,2,2,4. The normalized heat trace and its diagnostic of spectral dimension are therefore K(τ)=frac1+2e^-2τ+e^-4τ4 =frac(1+e^-2τ)^24, d_s(τ)=-2τfracK'(τ)K(τ) =frac8τe^2τ+1. No numerical differentiation is needed for this finite example. vfigspectral_dimension headBoth ends reflect the fin…
Exact mathematical passage
K(\tau)=\frac{1+2e^{-2\tau}+e^{-4\tau}}4 =\frac{(1+e^{-2\tau})^2}4, \qquad d_s(\tau)=-2\tau\frac{K'(\tau)}{K(\tau)} =\frac{8\tau}{e^{2\tau}+1}.
e^{-8t}Use in source · 1 passage
Search aliases: e^-8t
Use in source · atlas-pages.tex, line 95
For later discrete calculations define K=I+Q/8. This matrix is stochastic, but it is not e^Q/8. It is the step kernel in the uniformization identity e^tQ=e^-8t∑_n≥0(8t)^nK^n/n!. takeawayChanging packet coordinates does not change the evolution. Replacing continuous time by one discrete step does.
Exact mathematical passage
e^{tQ}=e^{-8t}\sum_{n\ge0}(8t)^nK^n/n!
\e^{-a(E_j-E_0)}Use in source · 1 passage
Search aliases: e^-a(E_j-E_0) e mathrm
Use in source · 02_strengthened.tex, line 1852
Diagonalize H. Each eigenvalue of T is e^-a(E_j-E_0) and therefore lies in (0,1]. The power law follows from the functional calculus.
Exact mathematical passage
\e^{-a(E_j-E_0)}
\e^{-a(H-E_0I)}Use in source · 1 passage
Search aliases: e^-a(H-E_0I) e mathrm
Use in source · 02_strengthened.tex, line 1838
Let H=H^* on a nonzero finite dimensional Hilbert space and let E_0=minSpec H. For a lattice time step a>0, define T= e^-a(H-E_0I).
Exact mathematical passage
T=\e^{-a(H-E_0I)}.
e^{-a_nt}Use in source · 1 passage
Search aliases: e^-a_nt
Use in source · atlas-pages.tex, line 712
vpage31A tail bound states what refinement buysSUMMABLE DEFECTS / A SCALAR COMPLETIONChapter VI, Part II: Theorem E4.5 and Corollary E4.7. This example has identity connecting maps. Take X_n= R, C_n=R_n=1, and a_n=1-2^-n, A_n=-a_n, T_n(t)=e^-a_nt (n≥0). Each semigroup is contractive for t≥0. The exact adjacent generator-defect magnitude is |A_n+1-A_n|=2^-(n+1), ∑_j=n^∞|A_j+1-A_j|=2^-n. vfigext_tails platecaptionThe bars show the certified tail 2^-n at horizon T=1. They are bounds on a whole time…
Exact mathematical passage
a_n=1-2^{-n},\qquad A_n=-a_n,\qquad T_n(t)=e^{-a_nt}\quad(n\ge0).
\e^{-\abs{n}^2/(4t)}Use in source · 2 passages
Search aliases: e^-absn^2/(4t) e abs mathrm abs
Use in source · 02_strengthened.tex, line 2107
[Theta transformation] The continuum flat torus heat trace satisfies Z_d(t) = ∑_k∈ Z^d e^-4π^2tabsk^2 = (4π t)^-d/2 ∑_n∈ Z^d e^-absn^2/(4t). This is the Poisson summation formula for a Gaussian.
Exact mathematical passage
Z_d(t) = \sum_{k\in\ZZ^d}\e^{-4\pi^2t\abs{k}^2} = (4\pi t)^{-d/2} \sum_{n\in\ZZ^d}\e^{-\abs{n}^2/(4t)}.
\e^{-aH_T}Use in source · 1 passage
Search aliases: e^-aH_T e mathrm
Use in source · 02_strengthened.tex, line 1867
[Hamiltonian reconstruction from transfer] Let T be positive definite, self adjoint, and contractive. Then H_T=-a^-1log T is a nonnegative self adjoint operator and T^n= e^-naH_T. It is the unique nonnegative self adjoint operator with T= e^-aH_T.
Exact mathematical passage
T=\e^{-aH_T}
\e^{-at}Use in source · 1 passage
Search aliases: e^-at e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 2838
Choose a≥max_i(-Q_ii). The matrix Q+aI is nonnegative, so its exponential is nonnegative. Hence e^tQ= e^-at e^t(Q+aI) is nonnegative. Since 1^TQ=0, differentiation gives 1^T e^tQ= 1^T.
Exact mathematical passage
\e^{tQ}=\e^{-at}\e^{t(Q+aI)}
\e^{-bt}Use in source · 1 passage
Search aliases: e^-bt e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 5870
d · -b , a,b>0, and retain the first coordinate. The memory kernel and self-energy are K(t)=cd e^-bt, Σ(z)=fraccdz+b. The retained coordinate has an exact time local closure for every retained initial value precisely when cd=0. When cd≠0, one auxiliary scalar state realizes the memory exactly.
Exact mathematical passage
\Kcal(t)=cd\e^{-bt}, \qquad \Sigma(z)=\frac{cd}{z+b}.
e^{-c_N|\rho|}Use in source · 1 passage
Search aliases: e^-c_N|ρ| rho
Use in source · Paper_IV_Traditional_Final.tex, line 2942
For N≥3, [source reference] gives | D_N'(ρ)|^-1 ≤ C e^-c_N|ρ| for every sufficiently large simple zero. Let L be a compact subset of the s plane. Because the chosen arguments of the zeros lie in a fixed bounded interval, there are constants C_L and M_L such that sup_s∈ L|e^-sLogρ| ≤ C_L|ρ|^M_L for every sufficiently large zero. Hence the absolute value of the corresponding summand is at most C_L'|ρ|^M_Le^-q c_N|ρ|. The linear zero counting estimate makes this majorant summable. The Weierstrass t…
Exact mathematical passage
|\Dfun_N'(\rho)|^{-1} \leq C e^{-c_N|\rho|}
e^{-e^{-ia_N}t_m}Use in source · 1 passage
Search aliases: e^-e^-ia_Nt_m
Use in source · Paper_IV_Traditional_Final.tex, line 2128
At the real point t_m, the normalized value N e^-e^-ia_Nt_m D_N(t_m) is purely imaginary. The two dominant terms cancel there, so R_N(t_m) is purely imaginary and the displayed correction is real.
Exact mathematical passage
N e^{-e^{-ia_N}t_m}\Dfun_N(t_m)
e^{-e^{-ia_N}z}Use in source · 2 passages
Search aliases: e^-e^-ia_Nz
Use in source · Paper_IV_Traditional_Final.tex, line 1882
η≠ e^ia_N,e^-ia_N exp((η-e^-ia_N)z). Then N e^-e^-ia_Nz D_N(z) = 1+e^2is_Nz+R_N(z), and, for q=0,1,2, |R_N^(q)(z)| ≤ C e^-γ_Nx.
Exact mathematical passage
N e^{-e^{-ia_N}z}\Dfun_N(z) = 1+e^{2is_Nz}+R_N(z),
e^{-H_n}Use in source · 1 passage
Search aliases: e^-H_n
Use in source · monograph_atlas.tex, line 797
For transfers on two states T_n=diag(1,q_n)=e^-H_n, with H_n=diag(0,-log q_n) at unit time, the error in trace norm is 2^-(n+2) and the heat traces converge to 3/2. The criterion for convergence of spectral dimension additionally assumes convergence of the first heat moments in trace norm and a nonzero limiting heat trace. Convergence of the heat trace alone does not determine that quotient. headThe retained evidence and the remaining open targets The extension includes retained exact checks ove…
Exact mathematical passage
T_n=\diag(1,q_n)=e^{-H_n}
e^{-i(t-s)H_{QQ}}Use in source · 1 passage
Search aliases: e^-i(t-s)H_QQ
Use in source · 01_finite_core.tex, line 973
idot y · =H_QPx+H_QQy. Solving the second equation gives y(t) =e^-itH_QQy(0) -i∫_0^t e^-i(t-s)H_QQH_QPx(s) ds. Substitution yields the exact Volterra equation idot x(t) = · H_PPx(t) +H_PQe^-itH_QQy(0)
Exact mathematical passage
y(t) =e^{-itH_{QQ}}y(0) -i\int_0^t e^{-i(t-s)H_{QQ}}H_{QP}x(s)\,ds.
e^{-ia_N}Use in source · 11 passages
Search aliases: e^-ia_N
Use in source · Paper_IV_Traditional_Final.tex, line 1874
On the strip x≥ R and |y|≤ Y, define R_N(z) = ∑_substackη^N=-1 η≠ e^ia_N,e^-ia_N exp((η-e^-ia_N)z). Then N e^-e^-ia_Nz D_N(z) = 1+e^2is_Nz+R_N(z), and, for q=0,1,2, |R_N^(q)(z)| ≤ C e^-γ_Nx.
Exact mathematical passage
R_N(z) = \sum_{\substack{\eta^N=-1\\ \eta\ne e^{ia_N},e^{-ia_N}}} \exp\bigl((\eta-e^{-ia_N})z\bigr).
\e^{-\ii sH}Use in source · 1 passage
Search aliases: e^-ii sH e^- i sH e ii mathrm mathrm
Use in source · 02_strengthened.tex, line 2445
The summability condition makes (H_n) Cauchy in operator norm. The limit is self adjoint. Duhamel's identity gives e^- i tH_n- e^- i tH =- i∫_0^t e^- i(t-s)H_n(H_n-H) e^- i sH d s. Both unitary factors have norm one, which gives the first estimate. The same argument gives the Euclidean estimate because the semigroups generated by nonnegative Hamiltonians are contractions.
Exact mathematical passage
\e^{-\ii tH_n}-\e^{-\ii tH} =-\ii\int_0^t \e^{-\ii(t-s)H_n}(H_n-H)\e^{-\ii sH}\dd s.
\e^{-\ii sH_{\rm d}}Use in source · 1 passage
Search aliases: e^-ii sH_rm d e^- i sH_rm d e ii rm mathrm mathrm rm
Use in source · 02_strengthened.tex, line 582
Let H_rm d=PHP+QHQ, H_rm o=PHQ+QHP. The propagator generated by H_rm d preserves P H and Q H. Duhamel's formula relative to H_rm d gives Q e^- i tHP = - i ∫_0^t Q e^- i(t-s)HH_rm o e^- i sH_rm dP d s. Since H_rm oP=QHP and both propagators are unitary, the norm is at most abstnormQHP. Use [source reference] for the last equality.
Exact mathematical passage
Q\e^{-\ii tH}P = -\ii \int_0^t Q\e^{-\ii(t-s)H}H_{\rm o}\e^{-\ii sH_{\rm d}}P \dd s.
e^{-\ii t}Use in source · 1 passage
Search aliases: e^-ii t e^- i t ii mathrm
Use in source · monograph_atlas.tex, line 680
4/5 · 0 · 2. Its memory kernel is K(t)=frac925e^- i t+frac1625e^-2 i t, K(0)=1. Each oscillatory coefficient is positive although K(t) need not be real or pointwise nonnegative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structural identity is ∑_j,koverline v_jK(t_j-t_k)v_k = ∑_k e^ i t_kDB^*v_k ^2≥0. The exact worked check specializes to t=(0,π/2,π) and v=(1,2,-1). Its phase…
Exact mathematical passage
K(t)=\frac9{25}e^{-\ii t}+\frac{16}{25}e^{-2\ii t}, \qquad K(0)=1.
\e^{-\ii tA}Use in source · 1 passage
Search aliases: e^-ii tA e^- i tA e ii mathrm mathrm
Use in source · 02_strengthened.tex, line 1770
If B=0, the retained and hidden equations decouple. Conversely, suppose that every solution with y(0)=0 satisfies x(t)= e^- i tAx(0). The full block equation gives ddot x(0) = -(A^2+BB^*)x(0), while the declared local equation gives ddot x(0)=-A^2x(0). Since this holds for every retained initial vector, BB^*=0. Hence B=0.
Exact mathematical passage
x(t)=\e^{-\ii tA}x(0)
\e^{-\ii tD}Use in source · 3 passages
Search aliases: e^-ii tD e^- i tD e ii mathrm mathrm
Use in source · 02_strengthened.tex, line 1680
[Exact retained memory equation] The retained coordinate satisfies idot x(t) = · Ax(t)+B e^- i tDy(0) · - i∫_0^tK(t-s)x(s) d s, where K(t)=B e^- i tDB^*. For Res>0, ∫_0^∞ e^-stK(t) d t = B(s+ i D)^-1B^*.
Exact mathematical passage
\begin{aligned} \ii\dot x(t) ={}&Ax(t)+B\e^{-\ii tD}y(0)\\ &-\ii\int_0^tK(t-s)x(s)\dd s, \end{aligned}
e^{-\ii tD}Use in source · 1 passage
Search aliases: e^-ii tD e^- i tD ii mathrm
Use in source · monograph_atlas.tex, line 696
A nonzero initial hidden vector also supplies the forcing term Be^- i tDy(0). It is additional data, not part of the convolution kernel. The retained checks cover the displayed kernel values, exact Gram factorization, and acceleration defect. The integral equation follows from the written elimination argument. takeawayMemory is positive definite through a Gram factorization, not through pointwise signs. Exact bare local closure requires the coupling to vanish.
Exact mathematical passage
Be^{-\ii tD}y(0)
\e^{-\ii tE_r}Use in source · 1 passage
Search aliases: e^-ii tE_r e^- i tE_r e ii mathrm mathrm
Use in source · 02_strengthened.tex, line 1744
[Finite oscillatory memory expansion] If D=∑_rE_rQ_r is the spectral decomposition of the hidden Hamiltonian, then K(t)=∑_r e^- i tE_rBQ_rB^*. Every coefficient BQ_rB^* is positive semidefinite.
Exact mathematical passage
K(t)=\sum_r\e^{-\ii tE_r}BQ_rB^*.
\e^{-\ii tH}Use in source · 11 passages
Search aliases: e^-ii tH e^- i tH e ii mathrm mathrm
Use in source · 02_strengthened.tex, line 525
[Exact invariant packets] The following conditions are equivalent: item The Hamiltonian commutes with each projector: [H,P_a]=0 for every a. item Transfer blocks between distinct packets vanish: P_bHP_a=0 whenever a≠ b. item Every packet P_a H is invariant under e^- i tH for every real t. When they hold, H=∑_aP_aHP_a and e^- i tH=∑_aP_a e^- i tHP_a.
Exact mathematical passage
\e^{-\ii tH}
e^{-\ii tH}Use in source · 3 passages
Search aliases: e^-ii tH e^- i tH ii mathrm
Use in source · monograph_atlas.tex, line 31
H · Self-adjoint Hamiltonian · It generates evolution in real time through e^- i tH.
Exact mathematical passage
e^{-\ii tH}
\e^{-\ii tH_n}Use in source · 4 passages
Search aliases: e^-ii tH_n e^- i tH_n e ii mathrm mathrm
Use in source · 02_strengthened.tex, line 2428
[Norm summable Hamiltonian tower] Let (H_n) be bounded self adjoint operators on one Hilbert space and assume ∑_n=0^∞normH_n+1-H_n<∞. Then H_n converges in norm to a bounded self adjoint operator H. For every T>0, sup_abs t≤ T norm e^- i tH_n- e^- i tH ≤ TnormH_n-H. If every H_n≥0, then sup_0≤ t≤ T norm e^-tH_n- e^-tH ≤ TnormH_n-H.
Exact mathematical passage
\sup_{\abs t\leq T} \norm{\e^{-\ii tH_n}-\e^{-\ii tH}} \leq T\norm{H_n-H}.
e^{-\ii tH_n}Use in source · 1 passage
Search aliases: e^-ii tH_n e^- i tH_n ii mathrm
Use in source · monograph_atlas.tex, line 779
For the norm route, work instead on the single fixed space C^2 and set H_n=diag(0,1+2^-n), H=diag(0,1). Here normH_n+1-H_n=2^-(n+1) and normH_n-H=2^-n. The summable tail supplies an error bound for every n. For |t|≤ T, the paper's Duhamel estimate gives norme^- i tH_n-e^- i tH≤ T2^-n. The Euclidean counterpart holds on 0≤ t≤ T because both Hamiltonians are nonnegative. If packet projectors also converge in norm, leakage operators converge uniformly on these time intervals. headExecution scope Ex…
Exact mathematical passage
\norm{e^{-\ii tH_n}-e^{-\ii tH}}\leq T2^{-n}.
\e^{-\ii tH_\ell}Use in source · 1 passage
Search aliases: e^-ii tH_ℓ e^- i tH_ℓ e ii ell mathrm mathrm ell
Use in source · 02_strengthened.tex, line 2380
[Bounded Hamiltonian direct limit] Assume H_ℓ+1R_ℓ=R_ℓ H_ℓ and sup_ℓnormH_ℓ<∞. Then there is a unique bounded self adjoint operator H_∞ on H_∞ such that H_∞ι_ℓ=ι_ℓ H_ℓ for every ℓ. It satisfies e^- i tH_∞ι_ℓ = ι_ℓ e^- i tH_ℓ and e^-tH_∞ι_ℓ = ι_ℓ e^-tH_ℓ when the Hamiltonians are nonnegative.
Exact mathematical passage
\e^{-\ii tH_\infty}\iota_\ell = \iota_\ell\e^{-\ii tH_\ell}
\e^{-\ii tH_\infty}Use in source · 1 passage
Search aliases: e^-ii tH_∞ e^- i tH_∞ e ii infty mathrm mathrm infty
Use in source · 02_strengthened.tex, line 2380
[Bounded Hamiltonian direct limit] Assume H_ℓ+1R_ℓ=R_ℓ H_ℓ and sup_ℓnormH_ℓ<∞. Then there is a unique bounded self adjoint operator H_∞ on H_∞ such that H_∞ι_ℓ=ι_ℓ H_ℓ for every ℓ. It satisfies e^- i tH_∞ι_ℓ = ι_ℓ e^- i tH_ℓ and e^-tH_∞ι_ℓ = ι_ℓ e^-tH_ℓ when the Hamiltonians are nonnegative.
Exact mathematical passage
\e^{-\ii tH_\infty}\iota_\ell = \iota_\ell\e^{-\ii tH_\ell}
\e^{-\ii(t-s)D}Use in source · 1 passage
Search aliases: e^-ii(t-s)D e^- i(t-s)D e ii mathrm mathrm
Use in source · 02_strengthened.tex, line 1705
The hidden equation is idot y=B^*x+Dy. Variation of constants gives y(t)= e^- i tDy(0) - i∫_0^t e^- i(t-s)DB^*x(s) d s. Substitute into idot x=Ax+By. The Laplace formula follows by diagonalizing D or integrating the norm convergent finite dimensional exponential.
Exact mathematical passage
y(t)=\e^{-\ii tD}y(0) -\ii\int_0^t\e^{-\ii(t-s)D}B^*x(s)\dd s.
\e^{-\ii(t-s)H}Use in source · 1 passage
Search aliases: e^-ii(t-s)H e^- i(t-s)H e ii mathrm mathrm
Use in source · 02_strengthened.tex, line 582
Let H_rm d=PHP+QHQ, H_rm o=PHQ+QHP. The propagator generated by H_rm d preserves P H and Q H. Duhamel's formula relative to H_rm d gives Q e^- i tHP = - i ∫_0^t Q e^- i(t-s)HH_rm o e^- i sH_rm dP d s. Since H_rm oP=QHP and both propagators are unitary, the norm is at most abstnormQHP. Use [source reference] for the last equality.
Exact mathematical passage
Q\e^{-\ii tH}P = -\ii \int_0^t Q\e^{-\ii(t-s)H}H_{\rm o}\e^{-\ii sH_{\rm d}}P \dd s.
\e^{-\ii(t-s)H_n}Use in source · 1 passage
Search aliases: e^-ii(t-s)H_n e^- i(t-s)H_n e ii mathrm mathrm
Use in source · 02_strengthened.tex, line 2445
The summability condition makes (H_n) Cauchy in operator norm. The limit is self adjoint. Duhamel's identity gives e^- i tH_n- e^- i tH =- i∫_0^t e^- i(t-s)H_n(H_n-H) e^- i sH d s. Both unitary factors have norm one, which gives the first estimate. The same argument gives the Euclidean estimate because the semigroups generated by nonnegative Hamiltonians are contractions.
Exact mathematical passage
\e^{-\ii tH_n}-\e^{-\ii tH} =-\ii\int_0^t \e^{-\ii(t-s)H_n}(H_n-H)\e^{-\ii sH}\dd s.
\e^{-\ii(t_j-t_k)D}Use in source · 1 passage
Search aliases: e^-ii(t_j-t_k)D e^- i(t_j-t_k)D e ii mathrm mathrm
Use in source · 02_strengthened.tex, line 1727
Set w=∑_k=1^m e^ i t_kDB^*v_k. Then norm w^2 = ∑_j,k ipB^*v_j e^- i(t_j-t_k)DB^*v_k = ∑_j,k ipv_jK(t_j-t_k)v_k.
Exact mathematical passage
\norm w^2 = \sum_{j,k} \ip{B^*v_j}{\e^{-\ii(t_j-t_k)D}B^*v_k} = \sum_{j,k} \ip{v_j}{K(t_j-t_k)v_k}.
e^{-\ii\pi L/2}Use in source · 1 passage
Search aliases: e^-iiπ L/2 e^- iπ L/2 ii pi mathrm pi
Use in source · monograph_atlas.tex, line 495
One calculation of transfer to the opposite siteSEEDS, CHARACTERS, INTERFERENCE, AND DEPHASINGChapter VII: Corollary II.3.3, Theorem II.4.4, Theorem II.7.4. Return to the shift on four sites S and Laplacian L=2I-S-S^*. The same character projectors have energies (0,2,4,2). At time t=π/2, their evolution phases are (1,-1,1,-1), exactly the character eigenvalues of S^2. Hence e^- iπ L/2=S^2, ket0⟼ket2. vfigperfect_transfer At halfway time π/4, the actual amplitude vector is (- i,1, i,1)^T/2, givin…
Exact mathematical passage
e^{-\ii\pi L/2}=S^2,\qquad \ket0\longmapsto\ket2.
e^{-is_N\tau_m}Use in source · 1 passage
Search aliases: e^-is_Nτ_m tau
Use in source · Paper_IV_Traditional_Final.tex, line 2134
For the derivative, use the formula in the proof of [source reference] and e^-is_Nτ_m = (-1)^m(-i) (1+O(e^-γ_Nτ_m)). The stated real sign follows.
Exact mathematical passage
e^{-is_N\tau_m} = (-1)^m(-i) \left(1+O(e^{-\gamma_N\tau_m})\right).
e^{-itH_{QQ}}Use in source · 3 passages
Search aliases: e^-itH_QQ
Use in source · 01_finite_core.tex, line 973
idot y · =H_QPx+H_QQy. Solving the second equation gives y(t) =e^-itH_QQy(0) -i∫_0^t e^-i(t-s)H_QQH_QPx(s) ds. Substitution yields the exact Volterra equation idot x(t) = · H_PPx(t) +H_PQe^-itH_QQy(0)
Exact mathematical passage
y(t) =e^{-itH_{QQ}}y(0) -i\int_0^t e^{-i(t-s)H_{QQ}}H_{QP}x(s)\,ds.
\e^{-na(E_j-E_0)}Use in source · 1 passage
Search aliases: e^-na(E_j-E_0) e mathrm
Use in source · 02_strengthened.tex, line 1934
In the spectral decomposition, T^n-P_0 has eigenvalues zero on the ground space and e^-na(E_j-E_0) on the excited eigenspaces. The largest is e^-naΔ.
Exact mathematical passage
\e^{-na(E_j-E_0)}
\e^{-na(H-E_0I)}Use in source · 1 passage
Search aliases: e^-na(H-E_0I) e mathrm
Use in source · 02_strengthened.tex, line 1844
[Finite Euclidean transfer] The operator T is positive definite, self adjoint, and contractive. Its spectrum lies in (0,1], and T^n= e^-na(H-E_0I) for every n∈ N.
Exact mathematical passage
T^n=\e^{-na(H-E_0I)}
\e^{-naH_T}Use in source · 1 passage
Search aliases: e^-naH_T e mathrm
Use in source · 02_strengthened.tex, line 1864
[Hamiltonian reconstruction from transfer] Let T be positive definite, self adjoint, and contractive. Then H_T=-a^-1log T is a nonnegative self adjoint operator and T^n= e^-naH_T. It is the unique nonnegative self adjoint operator with T= e^-aH_T.
Exact mathematical passage
T^n=\e^{-naH_T}.
\e^{-na\Delta}Use in source · 2 passages
Search aliases: e^-naΔ e Delta mathrm Delta
Use in source · 02_strengthened.tex, line 1928
[Ground packet convergence] Let P_0 be the spectral projector of H at E_0, and let Δ=min{E_j-E_0:E_j>E_0}. If an excited eigenvalue exists, then normT^n-P_0= e^-naΔ.
Exact mathematical passage
\norm{T^n-P_0}=\e^{-na\Delta}.
\e^{-nt}Use in source · 4 passages
Search aliases: e^-nt e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 3140
[Ornstein--Uhlenbeck semigroup] The operators ( O_t)_t≥0 form a positive Markov contraction semigroup on L^p(γ) for every 1≤ p≤∞. On L^2(γ), the semigroup is self-adjoint and O_t h_n= e^-nth_n. The family (h_n)_n≥0 is an orthonormal basis of L^2(γ).
Exact mathematical passage
\Ocal_t h_n=\e^{-nt}h_n.
e^{-nt}Use in source · 1 passage
Search aliases: e^-nt
Use in source · Paper_VI_Traditional_Final.tex, line 717
[Gaussian chaos realization] For the standard Gaussian measure, the normalized probabilists' Hermite polynomials form orthonormal chaos packets. The Ornstein--Uhlenbeck semigroup acts diagonally: T_tH_n=e^-ntH_n. Thus conditional expectation, martingale differences, and Gaussian chaos are all projector realizations of stochastic refinement.
Exact mathematical passage
T_tH_n=e^{-nt}H_n.
e^{-q c_N|\rho|}Use in source · 1 passage
Search aliases: e^-q c_N|ρ| rho
Use in source · Paper_IV_Traditional_Final.tex, line 2956
-1 ≤ C e^-c_N|ρ| for every sufficiently large simple zero. Let L be a compact subset of the s plane. Because the chosen arguments of the zeros lie in a fixed bounded interval, there are constants C_L and M_L such that sup_s∈ L|e^-sLogρ| ≤ C_L|ρ|^M_L for every sufficiently large zero. Hence the absolute value of the corresponding summand is at most C_L'|ρ|^M_Le^-q c_N|ρ|. The linear zero counting estimate makes this majorant summable. The Weierstrass test gives absolute local uniform convergence …
Exact mathematical passage
C_L'|\rho|^{M_L}e^{-q c_N|\rho|}.
e^{-r}Use in source · 1 passage
Search aliases: e^-r
Use in source · atlas-pages.tex, line 142
The continuous counterpart is equally explicit. With fine generator -1, coarse generator -2, and C=1, the generator defect is 1 and e^-t-e^-2t=∫_0^t e^-2(t-r)e^-r dr. Its resolvent version is 1/(z+1)-1/(z+2)=1/((z+1)(z+2)) for z>0. takeawayA small defect does not disappear. The identity specifies how it accumulates, and the norm assumptions turn that identity into a finite-horizon certificate.
Exact mathematical passage
e^{-t}-e^{-2t}=\int_0^t e^{-2(t-r)}e^{-r}\,dr.
e^{-rA^\alpha}Use in source · 2 passages
Search aliases: e^-rA^α alpha
Use in source · Paper_VI_Traditional_Final.tex, line 1297
[Fractional semigroup refinement] For s≥0, Ce^-sA^α-e^-sB^αC = -∫_0^s e^-(s-r)B^αD_α e^-rA^α dr. Hence normCe^-sA^α-e^-sB^αC ≤ snormD_α. Under the common gap A,B≥ mI, normCe^-sA^α-e^-sB^αC ≤ s e^-sm^αα m^α-1norm D.
Exact mathematical passage
\boxed{ Ce^{-sA^\alpha}-e^{-sB^\alpha}C = -\int_0^s e^{-(s-r)B^\alpha}D_\alpha e^{-rA^\alpha}\,dr. }
\e^{-s}Use in source · 1 passage
Search aliases: e^-s e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 3148
Positivity and preservation of constants follow from the integral formula. If X and Y are independent standard Gaussian variables, then e^-s ( e^-tx+sqrt1- e^-2tX ) + sqrt1- e^-2sY has the same distribution as e^-(s+t)x + sqrt1- e^-2(s+t)Z with Z standard Gaussian. This proves the semigroup law. Jensen's inequality and invariance of γ give the L^p contraction. The joint Gaussian pair ( X, e^-tX+sqrt1- e^-2tY ) is symmetric under interchange, which proves self-adjointness.
Exact mathematical passage
\e^{-s} \left( \e^{-t}x+\sqrt{1-\e^{-2t}}X \right) + \sqrt{1-\e^{-2s}}Y
e^{-s}Use in source · 1 passage
Search aliases: e^-s
Use in source · atlas-pages.tex, line 931
/2 and f_2(t)=e^-2t. vfigext_risks platecaptionAt t=log2, surviving mass is 3/8. Cause 1 has accumulated probability 1/4 and cause 2 has accumulated probability 3/8. The three pieces sum to one. headCondition on survival before restarting the clock At this time the transient masses are (1/4,1/8), so the conditional distribution of survivors is (2/3,1/3). Consequently fracS(t+s)S(t)=frac23e^-s+frac13e^-2s, E[τ-tmidτ>t]=frac23+frac16=frac56. The original mean lifetime was tfrac12·1+tfrac12·tfrac12…
Exact mathematical passage
\frac{S(t+s)}{S(t)}=\frac23e^{-s}+\frac13e^{-2s}, \qquad \EE[\tau-t\mid\tau>t]=\frac23+\frac16=\frac56.
e^{-sA^\alpha}Use in source · 3 passages
Search aliases: e^-sA^α alpha
Use in source · Paper_VI_Traditional_Final.tex, line 1297
[Fractional semigroup refinement] For s≥0, Ce^-sA^α-e^-sB^αC = -∫_0^s e^-(s-r)B^αD_α e^-rA^α dr. Hence normCe^-sA^α-e^-sB^αC ≤ snormD_α. Under the common gap A,B≥ mI, normCe^-sA^α-e^-sB^αC ≤ s e^-sm^αα m^α-1norm D.
Exact mathematical passage
\boxed{ Ce^{-sA^\alpha}-e^{-sB^\alpha}C = -\int_0^s e^{-(s-r)B^\alpha}D_\alpha e^{-rA^\alpha}\,dr. }
e^{-sB^\alpha}Use in source · 3 passages
Search aliases: e^-sB^α alpha
Use in source · Paper_VI_Traditional_Final.tex, line 1297
[Fractional semigroup refinement] For s≥0, Ce^-sA^α-e^-sB^αC = -∫_0^s e^-(s-r)B^αD_α e^-rA^α dr. Hence normCe^-sA^α-e^-sB^αC ≤ snormD_α. Under the common gap A,B≥ mI, normCe^-sA^α-e^-sB^αC ≤ s e^-sm^αα m^α-1norm D.
Exact mathematical passage
\boxed{ Ce^{-sA^\alpha}-e^{-sB^\alpha}C = -\int_0^s e^{-(s-r)B^\alpha}D_\alpha e^{-rA^\alpha}\,dr. }
\e^{-sL}Use in source · 2 passages
Search aliases: e^-sL e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 4084
nstandardfact[Stable subordination] Let 0<α<1 and t>0. There is a probability measure η_α,t on [0,∞) such that ∫_0^∞ e^-sλη_α,t( d s) = e^-tλ^α for every λ≥0. Consequently, for a finite dimensional positive self-adjoint operator, e^-tL^α = ∫_0^∞ e^-sLη_α,t( d s) in operator norm. The measures satisfy the convolution law η_α,s*η_α,t = η_α,s+t.
Exact mathematical passage
\e^{-tL^\alpha} = \int_0^\infty \e^{-sL}\eta_{\alpha,t}(\dd s)
e^{-sL}Use in source · 2 passages
Search aliases: e^-sL
Use in source · Paper_VI_Traditional_Final.tex, line 1167
[Poisson subordination] For t>0 and L≥0, e^-tsqrt L = fract2sqrtπ ∫_0^∞ s^-3/2e^-t^2/(4s)e^-sL ds. The integral converges in operator norm in finite dimension.
Exact mathematical passage
\boxed{ e^{-t\sqrt L} = \frac{t}{2\sqrt\pi} \int_0^\infty s^{-3/2}e^{-t^2/(4s)}e^{-sL}\,ds. }
e^{-sm^\alpha}Use in source · 2 passages
Search aliases: e^-sm^α alpha
Use in source · Paper_VI_Traditional_Final.tex, line 1311
[Fractional semigroup refinement] For s≥0, Ce^-sA^α-e^-sB^αC = -∫_0^s e^-(s-r)B^αD_α e^-rA^α dr. Hence normCe^-sA^α-e^-sB^αC ≤ snormD_α. Under the common gap A,B≥ mI, normCe^-sA^α-e^-sB^αC ≤ s e^-sm^αα m^α-1norm D.
Exact mathematical passage
\boxed{ \norm{Ce^{-sA^\alpha}-e^{-sB^\alpha}C} \le s e^{-sm^\alpha}\alpha m^{\alpha-1}\norm D. }
e^{-\sqrt{2j}\,t}Use in source · 1 passage
Search aliases: e^-sqrt2j t sqrt
Use in source · atlas-pages.tex, line 330
wers. Return to the four-state generator on pagerefV4 and put L=-Q. It is real symmetric and nonnegative, with orthogonal projectors P_0,P_1,P_2,P_3 and eigenvalues 0,2,4,6. Its square root is L^1/2=sqrt2P_1+2P_2+sqrt6P_3, (L^1/2)^2=L. Both heat evolutions retain these modes. Ordinary diffusion multiplies mode j by e^-2jt. Square-root diffusion multiplies it by e^-sqrt2j t. The constant mode is unchanged in both. vfigfractional platecaptionThe highest mode decays at rates 6 and sqrt6. The graph …
Exact mathematical passage
e^{-\sqrt{2j}\,t}
\e^{-st}Use in source · 2 passages
Search aliases: e^-st e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 4997
The Laplace transform follows by integrating the density in [source reference]. Stability of A gives ∫_0^∞ e^-st e^tA d t = (sI-A)^-1. At s=0, a^T(-A)^-1p_0 = 1^Tp_0=1. For the moments, ∫_0^∞ t^m e^tA d t = m!(-A)^-m-1. Multiplying by a^T= 1^T(-A) gives the result. These are the standard matrix formulas for phase type laws [source reference].
Exact mathematical passage
\int_0^\infty \e^{-st}\e^{tA}\dd t = (sI-A)^{-1}.
e^{-st}Use in source · 1 passage
Search aliases: e^-st
Use in source · Paper_VI_Traditional_Final.tex, line 5771
For the survival consequences, A is a finite Metzler matrix with nonpositive column sums and spectrum in the open left half-plane, p_0 is a probability vector, S(t)= 1^T e^tAp_0, and f(t)=-S'(t). The one-phase model has λ>0. The survival Laplace identity of Part II also holds for every complex s with Res≥0: stability gives absolute convergence of ∫_0^∞ e^-ste^tA dt=(sI-A)^-1, and multiplying by the exit row and p_0 gives the same transform. [Phase type rationality] The Laplace transform of every…
Exact mathematical passage
\int_0^\infty e^{-st}e^{tA}\,dt=(sI-A)^{-1}
\e^{-s\lambda}Use in source · 1 passage
Search aliases: e^-sλ e lambda mathrm lambda
Use in source · Paper_VI_Traditional_Final.tex, line 4078
[Stable subordination] Let 0<α<1 and t>0. There is a probability measure η_α,t on [0,∞) such that ∫_0^∞ e^-sλη_α,t( d s) = e^-tλ^α for every λ≥0. Consequently, for a finite dimensional positive self-adjoint operator, e^-tL^α = ∫_0^∞ e^-sLη_α,t( d s) in operator norm. The measures satisfy the convolution law η_α,s*η_α,t = η_α,s+t.
Exact mathematical passage
\int_0^\infty \e^{-s\lambda}\eta_{\alpha,t}(\dd s) = \e^{-t\lambda^\alpha}
\e^{-s\tau}Use in source · 1 passage
Search aliases: e^-sτ e tau mathrm tau
Use in source · Paper_VI_Traditional_Final.tex, line 4983
[Laplace transform and moments] Assume the spectrum of A lies in the open left half plane. Then absorption occurs with probability one. For s≥0, E[ e^-sτ] = a^T(sI-A)^-1p_0. For every integer m≥1, E[τ^m] = m! 1^T(-A)^-mp_0.
Exact mathematical passage
\Exp[\e^{-s\tau}] = a^T(sI-A)^{-1}p_0.
\e^{-t}Use in source · 7 passages
Search aliases: e^-t e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 3121
Let γ be the standard Gaussian probability measure on R. For t≥0, define the Ornstein--Uhlenbeck transfer ( O_t f)(x) = ∫_ R f( e^-tx+sqrt1- e^-2t y ) γ( d y). Let H_n be the probabilists' Hermite polynomial defined by exp(ux-fracu^22) = ∑_n=0^∞ H_n(x)fracu^nn!, and set h_n=H_n/sqrtn!.
Exact mathematical passage
(\Ocal_t f)(x) = \int_\RR f\left( \e^{-t}x+\sqrt{1-\e^{-2t}}\,y \right) \gamma(\dd y).
e^{-t}Use in source · 15 passages
Search aliases: e^-t
Use in source · atlas-pages.tex, line 323
vpage18Two large terms nearly cancelTHE CUBIC MODELChapter IV: Lemma 11.5, Theorem 11.7, Corollary 11.9. For real t, order three has the exact formula D_3(t)=frace^-t+2e^t/2cos(sqrt3t/2)3. Divide by the nonzero factor 2e^t/2/3. The resulting equation separates the dominant cosine from the small third exponential. vfigtwo_term Ignoring the last term gives model zeros t_m=frac2πsqrt3(m+1/2). At t_m, the full normalized expression equals tfrac12e^-3t_m/2>0, so t_m is not an exact zero. Linearizing …
Exact mathematical passage
D_3(t)=\frac{e^{-t}+2e^{t/2}\cos(\sqrt3t/2)}{3}.
\e^{-t(-\Lcal)^\alpha}Use in source · 1 passage
Search aliases: e^-t(-Lcal)^α e^-t(- L)^α e Lcal alpha mathrm mathcal alpha
Use in source · Paper_VI_Traditional_Final.tex, line 3307
[Fractional Hermite transfer] Let - L be the nonnegative Ornstein--Uhlenbeck number operator with - L h_n=nh_n. For every α>0, e^-t(- L)^αf = ∑_n=0^∞ e^-tn^αP_nf. This is a contraction semigroup on L^2(γ).
Exact mathematical passage
\e^{-t(-\Lcal)^\alpha}f = \sum_{n=0}^\infty \e^{-tn^\alpha}P_nf.
e^{-t(H-E_0I)}Use in source · 1 passage
Search aliases: e^-t(H-E_0I)
Use in source · monograph_atlas.tex, line 714
For a family with nontrivial excited spectra, a uniform exponential rate needs a common positive lower bound on the gap. Ground multiplicity does not cause a problem: P_0 selects the entire eigenspace at the minimum energy. For a Hamiltonian whose minimum is E_0≠0, the normalized transfer is e^-t(H-E_0I) before taking the limit. headPositivity is retained at several Euclidean times For arbitrary vectors u_j and nonnegative integer times n_j, the reflected kernel obeys ∑_j,k⟨ u_j,T^n_j+n_ku_k⟩ = …
Exact mathematical passage
e^{-t(H-E_0I)}
\e^{-t(k\pi)^{2\alpha}}Use in source · 1 passage
Search aliases: e^-t(kπ)^2α e pi alpha mathrm pi alpha
Use in source · Paper_VI_Traditional_Final.tex, line 5594
set λ_k,h=4h^-2sin^2(kπ h/2). The retained explicit coefficient below is valid for every discrete mode. [Fractional low mode dispersion] Let 0<α≤1. Then 0 ≤ (kπ)^2α-λ_k,h^α ≤ fracα(4k^2)^α-1(kπ)^412h^2. For every t≥0, abs e^-tλ_k,h^α - e^-t(kπ)^2α ≤ fractα(4k^2)^α-1(kπ)^412h^2.
Exact mathematical passage
\abs{ \e^{-t\lambda_{k,h}^\alpha} - \e^{-t(k\pi)^{2\alpha}} } \leq \frac{t\alpha(4k^2)^{\alpha-1}(k\pi)^4}{12}h^2.
e^{-t(x-e)}Use in source · 2 passages
Search aliases: e^-t(x-e)
Use in source · 03_supplement_03.tex, line 143
Connecting power series and spectral calculus. The shifted generator is proved Hermitian. The implementation then identifies the actual matrix exponential with continuous functional calculus: T_e(t)=f_t(H), f_t(x)=e^-t(x-e). The formal proof uses composition, scalar multiplication, subtraction, and the established agreement between the exponential in functional calculus and the exponential defined by a power series. Spectral mapping is consequently available for the operator that was actually de…
Exact mathematical passage
\boxed{T_e(t)=f_t(H),\qquad f_t(x)=e^{-t(x-e)}.}
\e^{-t^2/(4s)}Use in source · 1 passage
Search aliases: e^-t^2/(4s) e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 4102
[Poisson subordination] For t>0, e^-tsqrt L = fract2sqrtπ ∫_0^∞ s^-3/2 e^-t^2/(4s) e^-sL d s. The integral converges in operator norm.
Exact mathematical passage
\e^{-t\sqrt L} = \frac{t}{2\sqrt\pi} \int_0^\infty s^{-3/2} \e^{-t^2/(4s)} \e^{-sL} \dd s.
e^{-t^2/(4s)}Use in source · 2 passages
Search aliases: e^-t^2/(4s)
Use in source · Paper_VI_Traditional_Final.tex, line 1167
[Poisson subordination] For t>0 and L≥0, e^-tsqrt L = fract2sqrtπ ∫_0^∞ s^-3/2e^-t^2/(4s)e^-sL ds. The integral converges in operator norm in finite dimension.
Exact mathematical passage
\boxed{ e^{-t\sqrt L} = \frac{t}{2\sqrt\pi} \int_0^\infty s^{-3/2}e^{-t^2/(4s)}e^{-sL}\,ds. }
\e^{-t^2/(4s)-\lambda s}Use in source · 1 passage
Search aliases: e^-t^2/(4s)-λ s e lambda mathrm lambda
Use in source · Paper_VI_Traditional_Final.tex, line 4117
On a spectral packet with eigenvalue λ, the formula reduces to e^-tsqrtλ = fract2sqrtπ ∫_0^∞ s^-3/2 e^-t^2/(4s)-λ s d s. This scalar identity follows from the Gaussian integral. Recombine the finitely many spectral packets.
Exact mathematical passage
\e^{-t\sqrt\lambda} = \frac{t}{2\sqrt\pi} \int_0^\infty s^{-3/2}\e^{-t^2/(4s)-\lambda s}\dd s.
e^{-t^2/(4s)-\lambda s}Use in source · 1 passage
Search aliases: e^-t^2/(4s)-λ s lambda
Use in source · Paper_VI_Traditional_Final.tex, line 1181
By the spectral theorem it suffices to prove the scalar identity e^-tsqrtλ = fract2sqrtπ ∫_0^∞ s^-3/2e^-t^2/(4s)-λ s ds for λ≥0. This is the classical Gaussian integral after the substitution u=t/(2sqrt s). Applying the identity to every eigenvalue and recombining the spectral projectors proves the operator formula.
Exact mathematical passage
e^{-t\sqrt\lambda} = \frac{t}{2\sqrt\pi} \int_0^\infty s^{-3/2}e^{-t^2/(4s)-\lambda s}\,ds
e^{-t_aH}Use in source · 1 passage
Search aliases: e^-t_aH
Use in source · monograph_atlas.tex, line 431
nnegative entries and column sums one. Conjugate by the unitary Z. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become -1/4. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation. headA useful finite positive kernel For this nonnegative H, vectors u_a and times t_a≥0 satisfy ∑_a,b⟨ u_a,e^-(t_a+t_b)Hu_b⟩ = ∑_a e^-t_aHu_a ^2≥0. The factorization makes positivity transparent. It remains a fini…
Exact mathematical passage
\sum_{a,b}\langle u_a,e^{-(t_a+t_b)H}u_b\rangle =\left\|\sum_a e^{-t_aH}u_a\right\|^2\ge0.
\e^{-ta}Use in source · 1 passage
Search aliases: e^-ta e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 5614
proof The chord bound sin x≥2x/π on [0,π/2] gives λ_k,h≥4k^2. Since x↦ x^α is increasing and has decreasing derivative, (kπ)^2α-λ_k,h^α ≤ αλ_k,h^α-1 ((kπ)^2-λ_k,h). Use [source reference] and the lower bound for λ_k,h. The exponential estimate follows from abs e^-ta- e^-tb ≤ tabsa-b for nonnegative a and b.
Exact mathematical passage
\abs{\e^{-ta}-\e^{-tb}} \leq t\abs{a-b}
\e^{-tb}Use in source · 1 passage
Search aliases: e^-tb e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 5614
proof The chord bound sin x≥2x/π on [0,π/2] gives λ_k,h≥4k^2. Since x↦ x^α is increasing and has decreasing derivative, (kπ)^2α-λ_k,h^α ≤ αλ_k,h^α-1 ((kπ)^2-λ_k,h). Use [source reference] and the lower bound for λ_k,h. The exponential estimate follows from abs e^-ta- e^-tb ≤ tabsa-b for nonnegative a and b.
Exact mathematical passage
\abs{\e^{-ta}-\e^{-tb}} \leq t\abs{a-b}
\e^{-tD_V/n}Use in source · 1 passage
Search aliases: e^-tD_V/n e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 4420
[Alternating propagation and killing] For every t≥0, e^t(Q^T-D_V) = lim_n→∞ ( e^-tD_V/n e^tQ^T/n )^n in operator norm. Each finite product alternates one Markov propagation step with one diagonal survival weight.
Exact mathematical passage
\e^{t(Q^T-D_V)} = \lim_{n\to\infty} \left( \e^{-tD_V/n}\e^{tQ^T/n} \right)^n
\e^{-tH}Use in source · 1 passage
Search aliases: e^-tH e mathrm
Use in source · 02_strengthened.tex, line 2435
Let (H_n) be bounded self adjoint operators on one Hilbert space and assume ∑_n=0^∞normH_n+1-H_n<∞. Then H_n converges in norm to a bounded self adjoint operator H. For every T>0, sup_abs t≤ T norm e^- i tH_n- e^- i tH ≤ TnormH_n-H. If every H_n≥0, then sup_0≤ t≤ T norm e^-tH_n- e^-tH ≤ TnormH_n-H.
Exact mathematical passage
\sup_{0\leq t\leq T} \norm{\e^{-tH_n}-\e^{-tH}} \leq T\norm{H_n-H}.
\e^{-tH_n}Use in source · 1 passage
Search aliases: e^-tH_n e mathrm
Use in source · 02_strengthened.tex, line 2435
Let (H_n) be bounded self adjoint operators on one Hilbert space and assume ∑_n=0^∞normH_n+1-H_n<∞. Then H_n converges in norm to a bounded self adjoint operator H. For every T>0, sup_abs t≤ T norm e^- i tH_n- e^- i tH ≤ TnormH_n-H. If every H_n≥0, then sup_0≤ t≤ T norm e^-tH_n- e^-tH ≤ TnormH_n-H.
Exact mathematical passage
\sup_{0\leq t\leq T} \norm{\e^{-tH_n}-\e^{-tH}} \leq T\norm{H_n-H}.
\e^{-tH_\ell}Use in source · 1 passage
Search aliases: e^-tH_ℓ e ell mathrm ell
Use in source · 02_strengthened.tex, line 2386
imit] Assume H_ℓ+1R_ℓ=R_ℓ H_ℓ and sup_ℓnormH_ℓ<∞. Then there is a unique bounded self adjoint operator H_∞ on H_∞ such that H_∞ι_ℓ=ι_ℓ H_ℓ for every ℓ. It satisfies e^- i tH_∞ι_ℓ = ι_ℓ e^- i tH_ℓ and e^-tH_∞ι_ℓ = ι_ℓ e^-tH_ℓ when the Hamiltonians are nonnegative.
Exact mathematical passage
\e^{-tH_\infty}\iota_\ell = \iota_\ell\e^{-tH_\ell}
\e^{-tH_\infty}Use in source · 1 passage
Search aliases: e^-tH_∞ e infty mathrm infty
Use in source · 02_strengthened.tex, line 2386
imit] Assume H_ℓ+1R_ℓ=R_ℓ H_ℓ and sup_ℓnormH_ℓ<∞. Then there is a unique bounded self adjoint operator H_∞ on H_∞ such that H_∞ι_ℓ=ι_ℓ H_ℓ for every ℓ. It satisfies e^- i tH_∞ι_ℓ = ι_ℓ e^- i tH_ℓ and e^-tH_∞ι_ℓ = ι_ℓ e^-tH_ℓ when the Hamiltonians are nonnegative.
Exact mathematical passage
\e^{-tH_\infty}\iota_\ell = \iota_\ell\e^{-tH_\ell}
e^{-tL}Use in source · 3 passages
Search aliases: e^-tL
Use in source · Paper_VI_Traditional_Final.tex, line 304
Let X be a finite-dimensional real or complex vector space, let A∈End(X), and let s∈ X be a primitive seed. The basic continuous transfer is T(t)=e^tA, t≥ 0, while a positive operator L is more naturally written H(t)=e^-tL. Both satisfy the semigroup law. The sign convention changes with the domain: Markov generators and evolution equations are usually written with e^tA. Positive diffusion and elliptic operators are usually written with e^-tL.
Exact mathematical passage
H(t)=e^{-tL}.
\e^{-tL^\alpha}Use in source · 4 passages
Search aliases: e^-tL^α e alpha mathrm alpha
Use in source · Paper_VI_Traditional_Final.tex, line 2003
For a positive operator L, spectral packets define L^α and e^-tL^α. Exact intertwining passes to every fractional power. When both fine and coarse operators have a common positive lower bound m, the Balakrishnan formula gives the quantitative estimate normCL_f^α-L_c^α C ≤ α m^α-1 normCL_f-L_cC, 0<α<1. A Duhamel argument then controls the fractional semigroup defect. This closes a finite version of fractional naturality under explicit spectral hypotheses.
Exact mathematical passage
\e^{-tL^\alpha}
e^{-tL^\alpha}Use in source · 2 passages
Search aliases: e^-tL^α alpha
Use in source · Paper_VI_Traditional_Final.tex, line 1130
Let L=L^*≥0 on a finite-dimensional Hilbert space. Its spectral resolution is L=∑_jλ_jP_j. For α>0, define L^α=∑_jλ_j^α P_j. The associated fractional semigroup is T_α(t)=e^-tL^α =∑_je^-tλ_j^αP_j.
Exact mathematical passage
T_\alpha(t)=e^{-tL^\alpha} =\sum_je^{-t\lambda_j^\alpha}P_j.
\e^{-tL_c^\alpha}Use in source · 2 passages
Search aliases: e^-tL_c^α e alpha mathrm alpha
Use in source · Paper_VI_Traditional_Final.tex, line 3991
[Exact fractional naturality] Let L_f and L_c be finite dimensional positive self-adjoint operators and let C satisfy CL_f=L_cC. Then, for every α>0 and t≥0, CL_f^α=L_c^α C and C e^-tL_f^α = e^-tL_c^αC.
Exact mathematical passage
C\e^{-tL_f^\alpha} = \e^{-tL_c^\alpha}C.
\e^{-tL_f^\alpha}Use in source · 2 passages
Search aliases: e^-tL_f^α e alpha mathrm alpha
Use in source · Paper_VI_Traditional_Final.tex, line 3991
[Exact fractional naturality] Let L_f and L_c be finite dimensional positive self-adjoint operators and let C satisfy CL_f=L_cC. Then, for every α>0 and t≥0, CL_f^α=L_c^α C and C e^-tL_f^α = e^-tL_c^αC.
Exact mathematical passage
C\e^{-tL_f^\alpha} = \e^{-tL_c^\alpha}C.
\e^{-tL_n}Use in source · 3 passages
Search aliases: e^-tL_n e mathrm
Use in source · 02_strengthened.tex, line 2512
[Heat trace convergence from trace norm] Let L_n,L be nonnegative self adjoint operators on one Hilbert space. For a fixed t>0, assume their heat operators are trace class and that e^-tL_n⟶ e^-tL in trace norm. Then Tr( e^-tL_n)⟶Tr( e^-tL). If also the heat moment operators are trace class and L_n e^-tL_n⟶ L e^-tL in trace norm and the limiting heat trace is nonzero, then the spectral dimensions defined by the heat moment ratio converge at that time.
Exact mathematical passage
\e^{-tL_n}\longrightarrow\e^{-tL}
\e^{-tn^\alpha}Use in source · 1 passage
Search aliases: e^-tn^α e alpha mathrm alpha
Use in source · Paper_VI_Traditional_Final.tex, line 3307
[Fractional Hermite transfer] Let - L be the nonnegative Ornstein--Uhlenbeck number operator with - L h_n=nh_n. For every α>0, e^-t(- L)^αf = ∑_n=0^∞ e^-tn^αP_nf. This is a contraction semigroup on L^2(γ).
Exact mathematical passage
\e^{-t(-\Lcal)^\alpha}f = \sum_{n=0}^\infty \e^{-tn^\alpha}P_nf.
\e^{-t\sqrt L}Use in source · 1 passage
Search aliases: e^-tsqrt L e sqrt mathrm sqrt
Use in source · Paper_VI_Traditional_Final.tex, line 4102
[Poisson subordination] For t>0, e^-tsqrt L = fract2sqrtπ ∫_0^∞ s^-3/2 e^-t^2/(4s) e^-sL d s. The integral converges in operator norm.
Exact mathematical passage
\e^{-t\sqrt L} = \frac{t}{2\sqrt\pi} \int_0^\infty s^{-3/2} \e^{-t^2/(4s)} \e^{-sL} \dd s.
e^{-t\sqrt L}Use in source · 2 passages
Search aliases: e^-tsqrt L sqrt
Use in source · Paper_VI_Traditional_Final.tex, line 1167
[Poisson subordination] For t>0 and L≥0, e^-tsqrt L = fract2sqrtπ ∫_0^∞ s^-3/2e^-t^2/(4s)e^-sL ds. The integral converges in operator norm in finite dimension.
Exact mathematical passage
\boxed{ e^{-t\sqrt L} = \frac{t}{2\sqrt\pi} \int_0^\infty s^{-3/2}e^{-t^2/(4s)}e^{-sL}\,ds. }
\e^{-t\sqrt\lambda}Use in source · 1 passage
Search aliases: e^-tsqrtλ e sqrt lambda mathrm sqrt lambda
Use in source · Paper_VI_Traditional_Final.tex, line 4117
On a spectral packet with eigenvalue λ, the formula reduces to e^-tsqrtλ = fract2sqrtπ ∫_0^∞ s^-3/2 e^-t^2/(4s)-λ s d s. This scalar identity follows from the Gaussian integral. Recombine the finitely many spectral packets.
Exact mathematical passage
\e^{-t\sqrt\lambda} = \frac{t}{2\sqrt\pi} \int_0^\infty s^{-3/2}\e^{-t^2/(4s)-\lambda s}\dd s.
e^{-t\sqrt\lambda}Use in source · 2 passages
Search aliases: e^-tsqrtλ sqrt lambda
Use in source · Paper_VI_Traditional_Final.tex, line 1181
By the spectral theorem it suffices to prove the scalar identity e^-tsqrtλ = fract2sqrtπ ∫_0^∞ s^-3/2e^-t^2/(4s)-λ s ds for λ≥0. This is the classical Gaussian integral after the substitution u=t/(2sqrt s). Applying the identity to every eigenvalue and recombining the spectral projectors proves the operator formula.
Exact mathematical passage
e^{-t\sqrt\lambda} = \frac{t}{2\sqrt\pi} \int_0^\infty s^{-3/2}e^{-t^2/(4s)-\lambda s}\,ds
\e^{-tx^\alpha}Use in source · 1 passage
Search aliases: e^-tx^α e alpha mathrm alpha
Use in source · Paper_VI_Traditional_Final.tex, line 3999
The union of the two finite spectra is a finite set. There is a polynomial that agrees with x^α on that set, and another polynomial that agrees with e^-tx^α. Exact intertwining passes to every polynomial by induction.
Exact mathematical passage
\e^{-tx^\alpha}
e^{-t\lambda}Use in source · 1 passage
Search aliases: e^-tλ lambda
Use in source · monograph_atlas.tex, line 728
At the single time t=log2/16, two small grids admit exact rational heat weights: c|c|c|c|c N · spectrum · Z_N · ∑λ e^-tλ · d_N/log2 2 · (0,16) · 3/2 · 8 · 2/3 4 · (0,32,32,64) · 25/16 · 20 · 8/5 These are unnormalized traces. Normalizing by N changes Z_N but leaves the logarithmic derivative at that fixed grid unchanged. xfigheat_limits headThe theorem uses two separate choices For each ε>0, first choose a fixed positive time t_ε so that the continuum torus dimension differs from one by less tha…
Exact mathematical passage
\begin{array}{c|c|c|c|c} N&\text{spectrum}&Z_N&\sum\lambda e^{-t\lambda}&d_N/\log2\\\hline 2&(0,16)&3/2&8&2/3\\ 4&(0,32,32,64)&25/16&20&8/5 \end{array}
\e^{-t\lambda^\alpha}Use in source · 2 passages
Search aliases: e^-tλ^α e lambda alpha mathrm lambda alpha
Use in source · Paper_VI_Traditional_Final.tex, line 4078
[Stable subordination] Let 0<α<1 and t>0. There is a probability measure η_α,t on [0,∞) such that ∫_0^∞ e^-sλη_α,t( d s) = e^-tλ^α for every λ≥0. Consequently, for a finite dimensional positive self-adjoint operator, e^-tL^α = ∫_0^∞ e^-sLη_α,t( d s) in operator norm. The measures satisfy the convolution law η_α,s*η_α,t = η_α,s+t.
Exact mathematical passage
\int_0^\infty \e^{-s\lambda}\eta_{\alpha,t}(\dd s) = \e^{-t\lambda^\alpha}
\e^{-t\lambda_j^\alpha}Use in source · 1 passage
Search aliases: e^-tλ_j^α e lambda alpha mathrm lambda alpha
Use in source · Paper_VI_Traditional_Final.tex, line 3930
Let H be a finite dimensional Hilbert space and let L=L^*≥0. Write its spectral resolution as L=∑_jλ_jP_j. For α>0, define L^α = ∑_jλ_j^α P_j and T_α(t) = e^-tL^α = ∑_j e^-tλ_j^αP_j.
Exact mathematical passage
T_\alpha(t) = \e^{-tL^\alpha} = \sum_j\e^{-t\lambda_j^\alpha}P_j.
e^{-t\lambda_j^\alpha}Use in source · 1 passage
Search aliases: e^-tλ_j^α lambda alpha
Use in source · Paper_VI_Traditional_Final.tex, line 1130
Let L=L^*≥0 on a finite-dimensional Hilbert space. Its spectral resolution is L=∑_jλ_jP_j. For α>0, define L^α=∑_jλ_j^α P_j. The associated fractional semigroup is T_α(t)=e^-tL^α =∑_je^-tλ_j^αP_j.
Exact mathematical passage
T_\alpha(t)=e^{-tL^\alpha} =\sum_je^{-t\lambda_j^\alpha}P_j.
\e^{-t\lambda_k}Use in source · 1 passage
Search aliases: e^-tλ_k e lambda mathrm lambda
Use in source · Paper_VI_Traditional_Final.tex, line 4313
[Heat and fractional mode transfer] Let λ_k=(kπ)^2. For t≥0, 0≤ e^-tλ_k,h- e^-tλ_k ≤ fract(kπ)^412h^2 e^-tλ_k,h. For 0<α≤1 and fixed k, λ_k,h^α = λ_k^α+O(h^2) and e^-tλ_k,h^α = e^-tλ_k^α+O(th^2) uniformly for t in bounded intervals.
Exact mathematical passage
0\leq \e^{-t\lambda_{k,h}}-\e^{-t\lambda_k} \leq \frac{t(k\pi)^4}{12}h^2\e^{-t\lambda_{k,h}}.
\e^{-t\lambda_{k,h}}Use in source · 1 passage
Search aliases: e^-tλ_k,h e lambda mathrm lambda
Use in source · Paper_VI_Traditional_Final.tex, line 4313
[Heat and fractional mode transfer] Let λ_k=(kπ)^2. For t≥0, 0≤ e^-tλ_k,h- e^-tλ_k ≤ fract(kπ)^412h^2 e^-tλ_k,h. For 0<α≤1 and fixed k, λ_k,h^α = λ_k^α+O(h^2) and e^-tλ_k,h^α = e^-tλ_k^α+O(th^2) uniformly for t in bounded intervals.
Exact mathematical passage
0\leq \e^{-t\lambda_{k,h}}-\e^{-t\lambda_k} \leq \frac{t(k\pi)^4}{12}h^2\e^{-t\lambda_{k,h}}.
\e^{-t\lambda_{k,h}^\alpha}Use in source · 2 passages
Search aliases: e^-tλ_k,h^α e lambda alpha mathrm lambda alpha
Use in source · Paper_VI_Traditional_Final.tex, line 4326
[Heat and fractional mode transfer] Let λ_k=(kπ)^2. For t≥0, 0≤ e^-tλ_k,h- e^-tλ_k ≤ fract(kπ)^412h^2 e^-tλ_k,h. For 0<α≤1 and fixed k, λ_k,h^α = λ_k^α+O(h^2) and e^-tλ_k,h^α = e^-tλ_k^α+O(th^2) uniformly for t in bounded intervals.
Exact mathematical passage
\e^{-t\lambda_{k,h}^\alpha} = \e^{-t\lambda_k^\alpha}+O(th^2)
\e^{-t\lambda_k^\alpha}Use in source · 1 passage
Search aliases: e^-tλ_k^α e lambda alpha mathrm lambda alpha
Use in source · Paper_VI_Traditional_Final.tex, line 4326
[Heat and fractional mode transfer] Let λ_k=(kπ)^2. For t≥0, 0≤ e^-tλ_k,h- e^-tλ_k ≤ fract(kπ)^412h^2 e^-tλ_k,h. For 0<α≤1 and fixed k, λ_k,h^α = λ_k^α+O(h^2) and e^-tλ_k,h^α = e^-tλ_k^α+O(th^2) uniformly for t in bounded intervals.
Exact mathematical passage
\e^{-t\lambda_{k,h}^\alpha} = \e^{-t\lambda_k^\alpha}+O(th^2)
\e^{-t\lambda_N(k)}Use in source · 2 passages
Search aliases: e^-tλ_N(k) e lambda mathrm lambda
Use in source · 02_strengthened.tex, line 2061
[Fixed time heat trace convergence] For every t>0, ∑_k∈Λ_N^* e^-tλ_N(k) ⟶ ∑_k∈ Z^d e^-4π^2tabsk^2 as N→∞. The first spectral moments also converge: ∑_k∈Λ_N^* λ_N(k) e^-tλ_N(k) ⟶ ∑_k∈ Z^d 4π^2absk^2 e^-4π^2tabsk^2. Consequently, the discrete spectral dimensions converge to the continuum torus spectral dimension at every fixed positive time.
Exact mathematical passage
\sum_{k\in\Lambda_N^*} \e^{-t\lambda_N(k)} \longrightarrow \sum_{k\in\ZZ^d} \e^{-4\pi^2t\abs{k}^2}
e^{-t\lambda_N(k)}Use in source · 2 passages
Search aliases: e^-tλ_N(k) lambda
Use in source · monograph_atlas.tex, line 725
vpage37The dimension window has an order of limitsFINITE HEAT DATA AND THE QUANTIFIERSChapter VII: Theorem II.11.5, Corollary II.11.8. For the unit circle grid with N sites, the scaled eigenvalues are λ_N(k)=4N^2sin^2(π k/N). Put Z_N(t)=∑_ke^-tλ_N(k) and d_N(t)=2t∑_kλ_N(k)e^-tλ_N(k)/Z_N(t).
Exact mathematical passage
Z_N(t)=\sum_ke^{-t\lambda_N(k)}
e^{-u^2}Use in source · 1 passage
Search aliases: e^-u^2
Use in source · atlas-pages.tex, line 385
ime, and s is the ordinary diffusion time being mixed. They are not interchangeable variables. vfigrandom_clock platecaptionThe density η_1 peaks at s=1/6 but has a long right tail. The drawing ends at s=2. It does not claim that the missing tail has zero mass. headCheck the normalizer before interpreting a density Under s=t^2/(4u^2), reversing the integration limits gives η_t(s) |ds|=frac2sqrtπe^-u^2 du. Its integral over u>0 is one, so η_t is a probability density. On each eigenmode of L, the …
Exact mathematical passage
\eta_t(s)\,|ds|=\frac2{\sqrt\pi}e^{-u^2}\,du.
\e^{-V}Use in source · 1 passage
Search aliases: e^-V e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 4354
[Discrete Feynman--Kac formula] Define M_V=diag( e^-V)K^T. For terminal data f, (M_V^nf)_i = E_i[ exp( -∑_r=0^n-1V(X_r) ) f(X_n) ].
Exact mathematical passage
M_V=\diag(\e^{-V})K^T.
e^{-V}Use in source · 2 passages
Search aliases: e^-V
Use in source · Paper_VI_Traditional_Final.tex, line 1443
Let K be a column-stochastic matrix, let V be a potential on the states, and let f be terminal data. Define M_V=diag(e^-V)K^T. The backward Feynman--Kac value after n steps is u_n=M_V^nf.
Exact mathematical passage
M_V=\diag(e^{-V})K^T.
\e^{-V(i)}Use in source · 1 passage
Search aliases: e^-V(i) e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 4372
For one step, (M_Vf)_i = e^-V(i)∑_jK_jif_j. Iterating expands the matrix product over all paths. The product of transition entries is the path probability, and the diagonal factors give the exponential potential weight.
Exact mathematical passage
(M_Vf)_i = \e^{-V(i)}\sum_jK_{ji}f_j.
e^{-V(i)}Use in source · 2 passages
Search aliases: e^-V(i)
Use in source · Paper_VI_Traditional_Final.tex, line 1466
For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).
Exact mathematical passage
(M_Vf)_i=e^{-V(i)}\sum_jK_{j i}f_j,
e^{-V(i_r)}Use in source · 1 passage
Search aliases: e^-V(i_r)
Use in source · Paper_VI_Traditional_Final.tex, line 1469
For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).
Exact mathematical passage
\prod_re^{-V(i_r)}
\e^{-v_0t}Use in source · 2 passages
Search aliases: e^-v_0t e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 4395
[Continuous time finite state formula] Let Q be a finite Markov generator and let (X_t)_t≥0 be the corresponding chain. Then ( e^t(Q^T-D_V)f )_i = E_i[ exp( -∫_0^tV(X_s) d s ) f(X_t) ]. If V≥ v_0, then norm e^t(Q^T-D_V)_∞→∞ ≤ e^-v_0t.
Exact mathematical passage
\norm{\e^{t(Q^T-D_V)}}_{\infty\to\infty} \leq \e^{-v_0t}.
e^{-w}Use in source · 1 passage
Search aliases: e^-w
Use in source · atlas-pages.tex, line 398
and integral comparison. Thus truncation has an explicit error mechanism. takeawayThe infinite product is justified by the analytic theorem. Finite determinants provide controlled approximations, not substitutes for that theorem. platecaptionThe exact finite determinant test includes a repeated eigenvalue and rejects deduplication. The curves use known cosine zeros. Order one is excluded: Δ_1(w)=e^-w has no zeros, so its empty product would be 1, not the function.
Exact mathematical passage
\Delta_1(w)=e^{-w}
e^{-z}Use in source · 2 passages
Search aliases: e^-z
Use in source · Paper_IV_Traditional_Final.tex, line 752
The first denominators are D_1(z)=e^-z, D_2(z)=cos z. For N=3, D_3(z) = frac13( e^-z +2e^z/2cos(fracsqrt3z2) ). For N=4, D_4(z) = cos(frac zsqrt2) cosh(frac zsqrt2).
Exact mathematical passage
\Dfun_1(z)=e^{-z}, \qquad \Dfun_2(z)=\cos z.
\e^{-zt}Use in source · 10 passages
Search aliases: e^-zt e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 2118
[Laplace resolvent and packet transform] Let A∈ B(X), let T(t)= e^tA, and let Rez>norm A. Then the Bochner integral converges in operator norm and (zI-A)^-1 = ∫_0^∞ e^-ztT(t) d t. For every pair of packets, P_β(zI-A)^-1P_α = ∫_0^∞ e^-ztT_βα(t) d t.
Exact mathematical passage
(zI-A)^{-1} = \int_0^\infty \e^{-zt}T(t)\dd t.
e^{-zt}Use in source · 2 passages
Search aliases: e^-zt
Use in source · Paper_VI_Traditional_Final.tex, line 1547
[Memory/self-energy correspondence] For finite complex block matrices, assume Rez>Reλ for every λ∈σ(A_QQ). Then the following improper integral is absolutely convergent and ∫_0^∞ e^-zt K(t) dt =A_PQ(zI-A_QQ)^-1A_QP =Σ_P(z).
Exact mathematical passage
\boxed{ \int_0^\infty e^{-zt}\cK(t)\,dt =A_{PQ}(zI-A_{QQ})^{-1}A_{QP} =\Sigma_P(z). }
\e^{-\gamma t}Use in source · 2 passages
Search aliases: e^-γ t e gamma mathrm gamma
Use in source · Paper_VI_Traditional_Final.tex, line 4558
[Memory and self-energy] If z lies to the right of the spectrum of A_QQ, then ∫_0^∞ e^-zt K(t) d t = Σ_P(z). If, for constants M≥0 and γ>0, norm e^tA_QQ ≤ M e^-γ t, then norm K(t) ≤ MnormA_PQnormA_QP e^-γ t and norm K_L^1(0,∞) ≤ fracMγ normA_PQnormA_QP.
Exact mathematical passage
\norm{\e^{tA_{QQ}}} \leq M\e^{-\gamma t},
e^{-\gamma t_m}Use in source · 3 passages
Search aliases: e^-γ t_m gamma
Use in source · Paper_IV_Traditional_Final.tex, line 1971
[Large zero strings] Assume N≥3. There are constants m_0, C, and γ>0 and a sequence of positive real simple zeros (τ_m)_m≥ m_0 of D_N such that τ_m = t_m+O(e^-γ t_m). Every zero of sufficiently large modulus is of the form ω^kτ_m for a unique 0≤ k<N and a unique m≥ m_0.
Exact mathematical passage
\tau_m = t_m+O(e^{-\gamma t_m}).
e^{-\gamma_N\Re z}Use in source · 2 passages
Search aliases: e^-γ_NRe z gamma Re
Use in source · Paper_IV_Traditional_Final.tex, line 2009
If G(z)=0, then |1+e^2is_Nz| ≤ C e^-γ_NRe z. The real part of 2is_Nz is bounded in the strip. Apply [source reference] with w=2is_Nz. There is a unique integer m such that |z-t_m|≤ C_1e^-γ_NRe z, where t_m=frac(2m+1)π2s_N. In particular, |Re z-t_m|=o(1). For all sufficiently large zeros we have |Re z-t_m|≤1, and hence |z-t_m|≤ C_2e^-γ_Nt_m with a constant independent of m. Because Re z→+∞ in this argument, the corresponding integers m are eventually nonnegative and tend to infinity.
Exact mathematical passage
|1+e^{2is_Nz}| \leq C e^{-\gamma_N\Re z}.
e^{-\gamma_Nt_m}Use in source · 4 passages
Search aliases: e^-γ_Nt_m gamma
Use in source · Paper_IV_Traditional_Final.tex, line 2026
G(z)=0, then |1+e^2is_Nz| ≤ C e^-γ_NRe z. The real part of 2is_Nz is bounded in the strip. Apply [source reference] with w=2is_Nz. There is a unique integer m such that |z-t_m|≤ C_1e^-γ_NRe z, where t_m=frac(2m+1)π2s_N. In particular, |Re z-t_m|=o(1). For all sufficiently large zeros we have |Re z-t_m|≤1, and hence |z-t_m|≤ C_2e^-γ_Nt_m with a constant independent of m. Because Re z→+∞ in this argument, the corresponding integers m are eventually nonnegative and tend to infinity.
Exact mathematical passage
|z-t_m|\leq C_2e^{-\gamma_Nt_m}
e^{-\gamma_Nx}Use in source · 2 passages
Search aliases: e^-γ_Nx gamma
Use in source · Paper_IV_Traditional_Final.tex, line 1888
η≠ e^ia_N,e^-ia_N exp((η-e^-ia_N)z). Then N e^-e^-ia_Nz D_N(z) = 1+e^2is_Nz+R_N(z), and, for q=0,1,2, |R_N^(q)(z)| ≤ C e^-γ_Nx.
Exact mathematical passage
|R_N^{(q)}(z)| \leq C e^{-\gamma_Nx}.
e^{-\gamma_N\tau_m}Use in source · 3 passages
Search aliases: e^-γ_Nτ_m gamma tau
Use in source · Paper_IV_Traditional_Final.tex, line 2075
Differentiate the normalized expression. At z=τ_m, G'(τ_m) = 2is_Ne^2is_Nτ_m+R_N'(τ_m) = -2is_N+O(e^-γ_Nτ_m). Since G(τ_m)=0, D_N'(τ_m) = frac1N e^e^-ia_Nτ_mG'(τ_m). Taking absolute values gives the derivative asymptotic. Differentiating D_N(ω z)= D_N(z) gives the rotation formula.
Exact mathematical passage
G'(\tau_m) = 2is_Ne^{2is_N\tau_m}+R_N'(\tau_m) = -2is_N+O(e^{-\gamma_N\tau_m}).
e^{-\gamma\tau_m}Use in source · 1 passage
Search aliases: e^-γτ_m gamma tau
Use in source · Paper_IV_Traditional_Final.tex, line 1989
The derivatives satisfy D_N'(ω^kτ_m) = ω^-k D_N'(τ_m) and | D_N'(τ_m)| = frac2s_NNe^c_Nτ_m (1+O(e^-γτ_m)).
Exact mathematical passage
|\Dfun_N'(\tau_m)| = \frac{2s_N}{N}e^{c_N\tau_m} \left(1+O(e^{-\gamma\tau_m})\right).
e^{-\delta r}Use in source · 1 passage
Search aliases: e^-δ r delta
Use in source · Paper_IV_Traditional_Final.tex, line 1818
The indicator of D_N equals h_N, with the limsup taken over radii where D_N(re^iθ)≠0: limsup_r→∞ frac1rlog| D_N(re^iθ)| = h_N(θ). If K is a compact set of directions disjoint from T_N, then there are δ>0 and R>0 such that, for each θ∈ K, one root η(θ) of -1 satisfies D_N(re^iθ) = frac1N exp(η(θ)re^iθ) (1+O(e^-δ r)) uniformly for r≥ R. In particular, no sufficiently large zero can lie in those directions.
Exact mathematical passage
\Dfun_N(re^{i\theta}) = \frac1N \exp\bigl(\eta(\theta)re^{i\theta}\bigr) \left(1+O(e^{-\delta r})\right)
\e^{-\delta t}Use in source · 2 passages
Search aliases: e^-δ t e delta mathrm delta
Use in source · Paper_VI_Traditional_Final.tex, line 3839
[Concentration on the maximal score packet] Let G=max{g_i:p_i(0)>0} and let M be the set of supported indices with score G. If M is the full support, then the conclusion below is immediate. Otherwise, suppose δ = min{G-g_i:p_i(0)>0, i∉ M}>0. Then 1-∑_i∈ Mp_i(t) ≤ frac1-p_M(0)p_M(0) e^-δ t, where p_M(0)=∑_i∈ Mp_i(0).
Exact mathematical passage
1-\sum_{i\in M}p_i(t) \leq \frac{1-p_M(0)}{p_M(0)}\e^{-\delta t},
e^{-\lambda s}Use in source · 1 passage
Search aliases: e^-λ s lambda
Use in source · atlas-pages.tex, line 389
ends at s=2. It does not claim that the missing tail has zero mass. headCheck the normalizer before interpreting a density Under s=t^2/(4u^2), reversing the integration limits gives η_t(s) |ds|=frac2sqrtπe^-u^2 du. Its integral over u>0 is one, so η_t is a probability density. On each eigenmode of L, the operator formula reduces to the scalar Laplace identity ∫_0^∞η_t(s)e^-λ s ds=e^-tsqrtλ. The manuscript supplies that scalar identity and its spectral assembly. The appendix separately checks the…
Exact mathematical passage
\int_0^\infty\eta_t(s)e^{-\lambda s}\,ds=e^{-t\sqrt\lambda}.
\e^{-\lambda t}Use in source · 3 passages
Search aliases: e^-λ t e lambda mathrm lambda
Use in source · Paper_VI_Traditional_Final.tex, line 5782
[One phase model] If A=(-λ) and p_0=1, then S(t)= e^-λ t, f(t)=λ e^-λ t, h(t)=λ.
Exact mathematical passage
S(t)=\e^{-\lambda t}, \qquad f(t)=\lambda\e^{-\lambda t}, \qquad h(t)=\lambda.
e^{-\lambda t}Use in source · 1 passage
Search aliases: e^-λ t lambda
Use in source · Paper_VI_Traditional_Final.tex, line 552
Choose λ≥max_i(-Q_ii). Then B=Q+λ I has nonnegative entries, so e^tB is entrywise nonnegative by its power series. Hence e^tQ=e^-λ te^tB is nonnegative. Since 1^TQ=0, differentiation gives fracddt 1^Te^tQ= 1^TQe^tQ=0, so 1^Te^tQ= 1^T.
Exact mathematical passage
e^{tQ}=e^{-\lambda t}e^{tB}
\e^{-\mu t}Use in source · 2 passages
Search aliases: e^-μ t e mu mathrm mu
Use in source · Paper_VI_Traditional_Final.tex, line 5891
0, where λ≠μ are positive. The first phase moves to the second at rate λ, and the second is absorbed at rate μ. Direct calculation gives S(t)= fracμ e^-λ t-λ e^-μ tμ-λ, f(t)= fracλμμ-λ ( e^-λ t- e^-μ t). The ratio f(t)/S(t) is generally not constant. A time local Markov model on two transient phases can therefore produce a nonconstant hazard after the phase coordinate is hidden.
Exact mathematical passage
S(t)= \frac{\mu\e^{-\lambda t}-\lambda\e^{-\mu t}}{\mu-\lambda},
e^{-\tau H}Use in source · 1 passage
Search aliases: e^-τ H tau
Use in source · monograph_atlas.tex, line 422
vpage23Positive transfer is not a stochastic matrixSPECTRAL POSITIVITY IS BASIS INDEPENDENTChapter VII: Theorem II.10.1, Theorem II.10.3. For H=I-X, the two eigenvalues are zero and two. Its Euclidean transfer is T(τ)=e^-τ H =frac12mat1+q · 1-q 1-q · 1+q, q=e^-2τ. For τ≥0, the eigenvalues 1,q lie in (0,1], and multiplying two such matrices replaces q,r by qr. This is the semigroup law in spectral coordinates. vfigeuclidean At q=1/2, the displayed T happens also to have nonnegative entries and co…
Exact mathematical passage
T(\tau)=e^{-\tau H} =\frac12\mat{1+q&1-q\\1-q&1+q},\qquad q=e^{-2\tau}.
e^{-\tau(E_j-E_0)}Use in source · 1 passage
Search aliases: e^-τ(E_j-E_0) tau
Use in source · 01_finite_core.tex, line 1072
Diagonalize H=∑_jE_jP_j. Then T(τ)=∑_j e^-τ(E_j-E_0)P_j. Every eigenvalue is strictly positive and at most one. Multiplication adds the exponents, proving the semigroup law and T(0)=I. Positivity here is operator positivity, not an assertion that every matrix entry is nonnegative.
Exact mathematical passage
T(\tau)=\sum_j e^{-\tau(E_j-E_0)}P_j.
e^{-\tau(H-E_0I)}Use in source · 1 passage
Search aliases: e^-τ(H-E_0I) tau
Use in source · 01_finite_core.tex, line 1056
[Finite Euclidean transfer] Let H be Hermitian on a nonzero finite-dimensional Hilbert space. For all real τ,σ≥0, T(τ)=e^-τ(H-E_0I), E_0=minSpec H, is positive definite and satisfies T(τ+σ)=T(τ)T(σ). Its spectrum is contained in (0,1].
Exact mathematical passage
T(\tau)=e^{-\tau(H-E_0I)}, \qquad E_0=\min\Spec H,
e^{-\tau\lambda_j}Use in source · 2 passages
Search aliases: e^-τλ_j tau lambda
Use in source · 01_finite_core.tex, line 1116
Let L≥0 be a finite Laplacian on N≥1 states, with eigenvalues λ_j and at least one zero eigenvalue. For diffusion time τ>0, define the normalized heat trace K(τ)=frac1N∑_j e^-τλ_j and the scale-dependent spectral dimension d_s(τ) =-2fracdlog K(τ)dlogτ. For a finite system, d_s(τ) is a diagnostic, not a topological invariant. Indeed, finite differentiation gives d_s(τ)=2τ frac∑_jλ_j e^-τλ_j∑_j e^-τλ_j. It tends to zero as τdownarrow0 and, under the stated assumption of a zero mode, as τ→∞. If a g…
Exact mathematical passage
K(\tau)=\frac1N\sum_j e^{-\tau\lambda_j}
e^{1/4}Use in source · 1 passage
Search aliases: e^1/4
Use in source · atlas-pages.tex, line 888
und is deliberately shown separately from the trajectory. headTrack every constant before applying the estimate Here M_T=1, B=sup widetilde x =1, and L= K _L^1(0,T)=T, ε= K-widetilde K _L^1(0,T)=T. The theorem yields sup_0≤ t≤ T x(t)-widetilde x(t) =fracT^22≤ T^2e^T^2. At T=1/2, the exact error is 1/8 and the certificate is e^1/4/4. Nilpotence makes this example polynomial. It does not remove the time horizon from the general theorem.
Exact mathematical passage
e^{1/4}/4
e^{10}Use in source · 1 passage
Search aliases: e^10
Use in source · atlas-pages.tex, line 691
1/2, σ=1/50, δ=1/20, and a_min=2/25. Since log(10240)<10<(16/5)^2, choosing R=16 makes the right-hand side strictly less than frac1256+frac150frac165 =frac217332000<frac225. Thus sixteen independent observations per sample site suffice for this bound. It is a sufficient budget, not a claim of optimal sample complexity. The exact checker certifies e^10>10240 using a rational Taylor partial sum, so the displayed logarithm bound is not based on rounded numerical logarithms. headDependence between a…
Exact mathematical passage
e^{10}>10240
E^2Use in source · 3 passages
Search aliases: E^2
Use in source · 04_arithmetic.tex, line 156
The B ezout identity gives N=rI-UF. The two stated commutation relations and FG=0 give N(UF)=VGUF=VUGF=VUFG=0. Therefore N^2=N(rI-UF)=rN. Since r≠0, E^2=r^-2N^2=E. If Fx=0, the evaluated B ezout identity gives VGx=rx, hence Ex=x. If Gx=0, then Ex=0. The lattice assertion follows directly from rE=VG.
Exact mathematical passage
E^2=r^{-2}N^2=E
\mathcal E^2Use in source · 2 passages
Search aliases: E^2 mathcal
Use in source · strengthening.tex, line 305
dependent variables mean that moments of polynomials in separate variables factor. Odd moments vanish. Frobenius's congruence is v_2(∑_r=0^m(-1)^m-rbinom mr Euler_2n+2r) =m+v_2(m!) (n,m≥0). For m=0, the assertion is that every even Euler number is odd. With P=( E^2-1)/2 and γ_m(P)=P^m/m!, the case n=0 says L(γ_m(P))∈ Z_2^× (m≥0). Introduce an independent Euler variable F and Q=( F^2-1)/2.
Exact mathematical passage
P=(\mathcal E^2-1)/2
e^2Use in source · 2 passages
Search aliases: e^2
Use in source · Paper_XII_Traditional_Final.tex, line 332
Let e^2=e. Since e+(1-e)=1, at least one of e and 1-e is a unit. If e is a unit, then e=1. If 1-e is a unit, then e=0.
Exact mathematical passage
e^2=e
e^{2is_N(t_m+\delta_m)}Use in source · 1 passage
Search aliases: e^2is_N(t_m+δ_m) delta
Use in source · Paper_IV_Traditional_Final.tex, line 2114
Put δ_m=τ_m-t_m. The large zero theorem gives δ_m=O(e^-γ_Nt_m). Since e^2is_Nt_m=-1, 1+e^2is_N(t_m+δ_m) = -2is_Nδ_m+O(δ_m^2). The remainder estimate through the first derivative gives R_N(t_m+δ_m) = R_N(t_m)+O(e^-γ_Nt_m|δ_m|). Substitution in the zero equation gives the displacement formula.
Exact mathematical passage
1+e^{2is_N(t_m+\delta_m)} = -2is_N\delta_m+O(\delta_m^2).
e^{2is_Nt_m}Use in source · 1 passage
Search aliases: e^2is_Nt_m
Use in source · Paper_IV_Traditional_Final.tex, line 2113
Put δ_m=τ_m-t_m. The large zero theorem gives δ_m=O(e^-γ_Nt_m). Since e^2is_Nt_m=-1, 1+e^2is_N(t_m+δ_m) = -2is_Nδ_m+O(δ_m^2). The remainder estimate through the first derivative gives R_N(t_m+δ_m) = R_N(t_m)+O(e^-γ_Nt_m|δ_m|). Substitution in the zero equation gives the displacement formula.
Exact mathematical passage
e^{2is_Nt_m}=-1
e^{2is_Nz}Use in source · 3 passages
Search aliases: e^2is_Nz
Use in source · Paper_IV_Traditional_Final.tex, line 1882
η≠ e^ia_N,e^-ia_N exp((η-e^-ia_N)z). Then N e^-e^-ia_Nz D_N(z) = 1+e^2is_Nz+R_N(z), and, for q=0,1,2, |R_N^(q)(z)| ≤ C e^-γ_Nx.
Exact mathematical passage
N e^{-e^{-ia_N}z}\Dfun_N(z) = 1+e^{2is_Nz}+R_N(z),
e^{2is_N\tau_m}Use in source · 1 passage
Search aliases: e^2is_Nτ_m tau
Use in source · Paper_IV_Traditional_Final.tex, line 2075
Differentiate the normalized expression. At z=τ_m, G'(τ_m) = 2is_Ne^2is_Nτ_m+R_N'(τ_m) = -2is_N+O(e^-γ_Nτ_m). Since G(τ_m)=0, D_N'(τ_m) = frac1N e^e^-ia_Nτ_mG'(τ_m). Taking absolute values gives the derivative asymptotic. Differentiating D_N(ω z)= D_N(z) gives the rotation formula.
Exact mathematical passage
G'(\tau_m) = 2is_Ne^{2is_N\tau_m}+R_N'(\tau_m) = -2is_N+O(e^{-\gamma_N\tau_m}).
e^{2pt}Use in source · 4 passages
Search aliases: e^2pt
Use in source · 20_arithmetic.tex, line 818
The exponential generating series of the source is the finite sum S_p(t)=∑_a=0^p-1(-1)^a e^(2a+1)t. Multiplication by cosh t telescopes. Since p is odd, S_p(t)=frac1+e^2pt2secht. Taking its even part gives ∑_n≥0S_2n(p)fract^2n(2n)! =cosh^2(pt)secht. Multiplication by the reciprocal sech^2(pt) yields secht. The EGF product rule now gives the reconstruction coefficient by coefficient. Replacing m by n-q produces the displayed lower triangular row. Its diagonal and weights follow from γ_0=1 and the…
Exact mathematical passage
\mathcal S_p(t)=\frac{1+e^{2pt}}{2}\operatorname{sech}t.
\e^{2qt}Use in source · 1 passage
Search aliases: e^2qt e mathrm
Use in source · 72_recovered.tex, line 937
The finite geometric sum gives ∑_a=0^q-1(-1)^a e^(2a+1)t = e^tfrac1+ e^2qt1+ e^2t = e^qtcosh(qt)secht. Averaging at t and -t gives the even part.
Exact mathematical passage
\sum_{a=0}^{q-1}(-1)^a\e^{(2a+1)t} = \e^t\frac{1+\e^{2qt}}{1+\e^{2t}} = \e^{qt}\cosh(qt)\operatorname{sech}t.
\e^{2t}Use in source · 1 passage
Search aliases: e^2t e mathrm
Use in source · 72_recovered.tex, line 937
The finite geometric sum gives ∑_a=0^q-1(-1)^a e^(2a+1)t = e^tfrac1+ e^2qt1+ e^2t = e^qtcosh(qt)secht. Averaging at t and -t gives the even part.
Exact mathematical passage
\sum_{a=0}^{q-1}(-1)^a\e^{(2a+1)t} = \e^t\frac{1+\e^{2qt}}{1+\e^{2t}} = \e^{qt}\cosh(qt)\operatorname{sech}t.
e^{2t}Use in source · 1 passage
Search aliases: e^2t
Use in source · strengthening.tex, line 620
The variable B=2Z+1 has transform e^tfrac2te^2t-1=fractsinh t. Its odd moments vanish, and P=( B^2-1)/4=Z(Z+1) has the divided-power moments of Lemma~[source reference]. Introduce an independent copy D and Q=( D^2-1)/4. The exact factorization F_2(t)=fract^2cosh t-cos t =fracatsinh(at)fracbtsinh(bt) shows that U=a B+b D satisfies L(U^4n)=C_2(n). Direct expansion gives H_2:=fracU^4-14 =P+Q-P^2-Q^2+6PQ+i B D(P-Q).
Exact mathematical passage
e^t\frac{2t}{e^{2t}-1}=\frac{t}{\sinh t}.
e^{2Y}Use in source · 2 passages
Search aliases: e^2Y
Use in source · Paper_IV_Traditional_Final.tex, line 1929
ng strip, the roots e^ia_N and e^-ia_N have real part c_N, while each remaining root η=u+iv satisfies u≤cos(3a_N)=c_N-γ_N. Hence Re((η-e^-ia_N)z) ≤-γ_Nx+2Y. Division by e^e^-ia_Nz gives the displayed normalization. Each differentiation multiplies a remainder term by η-e^-ia_N, whose modulus is at most two. Thus, for q=0,1,2, |R_N^(q)(z)|≤(N-2)2^qe^2Ye^-γ_Nx. One may take R=1 and C=4(N-2)e^2Y.
Exact mathematical passage
|R_N^{(q)}(z)|\leq(N-2)2^qe^{2Y}e^{-\gamma_Nx}.
\e^{2\pi i/M}Use in source · 1 passage
Search aliases: e^2π i/M e pi mathrm pi
Use in source · Paper_XI_Traditional_Final.tex, line 313
Assume that P_K acts trivially. After passing to a finite quotient, tame inertia is cyclic. Let t generate a quotient of order M, and put ζ_M= e^2π i/M. For j∈ Z/M Z, define P_j = frac1M ∑_a=0^M-1 ζ_M^-jar(t)^a.
Exact mathematical passage
\zeta_M=\e^{2\pi i/M}
e^{2\pi i/N}Use in source · 2 passages
Search aliases: e^2π i/N pi
Use in source · Paper_IV_Traditional_Final.tex, line 359
Fix N≥1 and write ω=e^2π i/N. Let Ω be a disk centered at the origin, with the whole plane allowed, and assume that Ω is invariant under multiplication by ω. For a holomorphic function F on Ω, define ( B_N,jF)(z) = frac1N∑_ℓ=0^N-1ω^-jℓF(ω^ℓ z), 0≤ j<N. The same formula is valid for formal power series.
Exact mathematical passage
\omega=e^{2\pi i/N}.
\e^{2\pi iz/p}Use in source · 1 passage
Search aliases: e^2π iz/p e pi mathrm pi
Use in source · Paper_XI_Traditional_Final.tex, line 806
Let p be odd. Let X be an r dimensional vector space over F_p, and put V=X⊕ X^*. Define the alternating form b((x,ξ),(y,η)) = η(x)-ξ(y). Let Z= F_p, and define the Heisenberg group H(V)=V× Z with multiplication (v,z)(w,t) = (v+w,z+t+frac12 b(v,w)). Fix the additive character ψ(z)= e^2π iz/p.
Exact mathematical passage
\psi(z)=\e^{2\pi iz/p}.
\e^{2\pi\ii k/N}Use in source · 1 passage
Search aliases: e^2πii k/N e^2π i k/N e pi ii mathrm pi mathrm
Use in source · Paper_V_Traditional_Final.tex, line 3692
+1) into C^-1 and simplify. Preservation of H follows because C maps it to D, rotation preserves D, and C^-1 maps back. The group law follows by conjugating the rotation group law. When θ=2π/N, the Nth power is the identity. Conversely, if its kth power is the identity, conjugation by C and evaluation at w=1/2 give e^2π i k/N=1, so N divides k. Hence the order is exactly N.
Exact mathematical passage
\e^{2\pi\ii k/N}=1
\e^{2\pi\ii/N}Use in source · 1 passage
Search aliases: e^2πii/N e^2π i/N e pi ii mathrm pi mathrm
Use in source · 02_strengthened.tex, line 270
Let N≥1 be an integer, S unitary with S^N=I, and ω= e^2π i/N.
Exact mathematical passage
\omega=\e^{2\pi\ii/N}
e^{2\pi\ii/N}Use in source · 1 passage
Search aliases: e^2πii/N e^2π i/N pi ii pi mathrm
Use in source · Paper_V_Traditional_Final.tex, line 1134
Fix ω_N=e^2π i/N and rotate the reflection coefficient by R_N,k(Γ)=ω_N^kΓ.
Exact mathematical passage
\omega_N=e^{2\pi\ii/N}
e^{2\tau}Use in source · 1 passage
Search aliases: e^2τ tau
Use in source · monograph_atlas.tex, line 442
vpage24A dimension curve is a scale diagnosticTHE EXACT HEAT TRACE ON FOUR SITESChapter VII: Definition II.11.1, Proposition II.11.2. For the Laplacian on a cycle with four sites L=2I-S-S^*, the eigenvalues are 0,2,2,4. The normalized heat trace and its diagnostic of spectral dimension are therefore K(τ)=frac1+2e^-2τ+e^-4τ4 =frac(1+e^-2τ)^24, d_s(τ)=-2τfracK'(τ)K(τ) =frac8τe^2τ+1. No numerical differentiation is needed for this finite example. vfigspectral_dimension headBoth ends reflect the fin…
Exact mathematical passage
K(\tau)=\frac{1+2e^{-2\tau}+e^{-4\tau}}4 =\frac{(1+e^{-2\tau})^2}4, \qquad d_s(\tau)=-2\tau\frac{K'(\tau)}{K(\tau)} =\frac{8\tau}{e^{2\tau}+1}.
e^{a_\ell z}Use in source · 1 passage
Search aliases: e^a_ℓ z ell
Use in source · Paper_IV_Traditional_Final.tex, line 520
At the origin, the derivative matrix is diagonal with entries 1,λ,…,λ^N-1. For an explicit determinant proof, write each basis function as a finite sum of exponentials with exponents a_ℓ=λω^ℓ. Its derivative matrix factors as W(z)=V diag(e^a_ℓ z) C, V_kℓ=a_ℓ^k, where C is the constant matrix of Fourier weights. Thus det W(z)=e^z∑_ℓ a_ℓdet W(0)=det W(0), because N≥2 implies ∑_ℓω^ℓ=0. The product of the diagonal initial values is the stated power of λ.
Exact mathematical passage
W(z)=V\,\operatorname{diag}(e^{a_\ell z})\,C, \qquad V_{k\ell}=a_\ell^k,
\e^{at}Use in source · 2 passages
Search aliases: e^at e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 5809
c · b and let the two coordinate axes be the packet spaces. If a≠ b, then e^tA = e^at · 0 displaystyle cfrac e^at- e^bta-b · e^bt . The lower left block is produced by one transfer from the first packet to the second. Its expansion begins with ct, in agreement with [source reference]. When a=b, the same block is ct e^at.
Exact mathematical passage
\e^{tA} = \begin{pmatrix} \e^{at}&0\\ \displaystyle c\frac{\e^{at}-\e^{bt}}{a-b}&\e^{bt} \end{pmatrix}.
\e^{bt}Use in source · 1 passage
Search aliases: e^bt e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 5809
c · b and let the two coordinate axes be the packet spaces. If a≠ b, then e^tA = e^at · 0 displaystyle cfrac e^at- e^bta-b · e^bt . The lower left block is produced by one transfer from the first packet to the second. Its expansion begins with ct, in agreement with [source reference]. When a=b, the same block is ct e^at.
Exact mathematical passage
\e^{tA} = \begin{pmatrix} \e^{at}&0\\ \displaystyle c\frac{\e^{at}-\e^{bt}}{a-b}&\e^{bt} \end{pmatrix}.
e^{c_Nr}Use in source · 1 passage
Search aliases: e^c_Nr
Use in source · Paper_IV_Traditional_Final.tex, line 1843
At a tie direction exactly two adjacent vertices attain the maximum. After rotation it is enough to take θ=0. With c_N=cos(π/N) and s_N=sin(π/N), the two dominant terms give frac2N e^c_Nrcos(s_Nr) and every other term is exponentially smaller. Along a sequence on which |cos(s_Nr)|≥1/2, the lower indicator bound is c_N=h_N(0). Rotational symmetry treats every tie direction.
Exact mathematical passage
\frac2N e^{c_Nr}\cos(s_Nr)
e^{c_N\tau_m}Use in source · 2 passages
Search aliases: e^c_Nτ_m tau
Use in source · Paper_IV_Traditional_Final.tex, line 1989
The derivatives satisfy D_N'(ω^kτ_m) = ω^-k D_N'(τ_m) and | D_N'(τ_m)| = frac2s_NNe^c_Nτ_m (1+O(e^-γτ_m)).
Exact mathematical passage
|\Dfun_N'(\tau_m)| = \frac{2s_N}{N}e^{c_N\tau_m} \left(1+O(e^{-\gamma\tau_m})\right).
\mathcal E^dUse in source · 1 passage
Search aliases: E^d mathcal
Use in source · strengthening.tex, line 290
Let L be the ordinary Euler evaluation functional on a variable E, defined by L( E^d)=Euler_d, ∑_d≥0Euler_dfract^dd! =secht. The functional is linear. It is not a multiplicative evaluation. Independent variables mean that moments of polynomials in separate variables factor. Odd moments vanish. Frobenius's congruence is v_2(∑_r=0^m(-1)^m-rbinom mr Euler_2n+2r) =m+v_2(m!) (n,m≥0). For m=0, the assertion is that every even Euler number is odd. With P=( E^2-1)/2 and γ_m(P)=P^m/m!, the case n=0 says …
Exact mathematical passage
\mathcal L(\mathcal E^d)=\operatorname{Euler}_d, \qquad \sum_{d\geq0}\operatorname{Euler}_d\frac{t^d}{d!} =\operatorname{sech}t.
e^{e^{-ia_N}z}Use in source · 1 passage
Search aliases: e^e^-ia_Nz
Use in source · Paper_IV_Traditional_Final.tex, line 1926
On the resulting strip, the roots e^ia_N and e^-ia_N have real part c_N, while each remaining root η=u+iv satisfies u≤cos(3a_N)=c_N-γ_N. Hence Re((η-e^-ia_N)z) ≤-γ_Nx+2Y. Division by e^e^-ia_Nz gives the displayed normalization. Each differentiation multiplies a remainder term by η-e^-ia_N, whose modulus is at most two. Thus, for q=0,1,2, |R_N^(q)(z)|≤(N-2)2^qe^2Ye^-γ_Nx. One may take R=1 and C=4(N-2)e^2Y.
Exact mathematical passage
e^{e^{-ia_N}z}
e^{e^{-ia_N}\tau_m}Use in source · 1 passage
Search aliases: e^e^-ia_Nτ_m tau
Use in source · Paper_IV_Traditional_Final.tex, line 2083
Differentiate the normalized expression. At z=τ_m, G'(τ_m) = 2is_Ne^2is_Nτ_m+R_N'(τ_m) = -2is_N+O(e^-γ_Nτ_m). Since G(τ_m)=0, D_N'(τ_m) = frac1N e^e^-ia_Nτ_mG'(τ_m). Taking absolute values gives the derivative asymptotic. Differentiating D_N(ω z)= D_N(z) gives the rotation formula.
Exact mathematical passage
\Dfun_N'(\tau_m) = \frac1N e^{e^{-ia_N}\tau_m}G'(\tau_m).
e^{g(z)}Use in source · 1 passage
Search aliases: e^g(z)
Use in source · Paper_IV_Traditional_Final.tex, line 1343
For a function g, define the complete differential Bell polynomial Y_N by fracd^Ndz^Ne^g(z) = e^g(z)Y_N(g',g'',…,g^(N)).
Exact mathematical passage
\frac{d^N}{dz^N}e^{g(z)} = e^{g(z)}Y_N\bigl(g',g'',\ldots,g^{(N)}\bigr).
e^{ia_N}Use in source · 2 passages
Search aliases: e^ia_N
Use in source · Paper_IV_Traditional_Final.tex, line 1874
On the strip x≥ R and |y|≤ Y, define R_N(z) = ∑_substackη^N=-1 η≠ e^ia_N,e^-ia_N exp((η-e^-ia_N)z). Then N e^-e^-ia_Nz D_N(z) = 1+e^2is_Nz+R_N(z), and, for q=0,1,2, |R_N^(q)(z)| ≤ C e^-γ_Nx.
Exact mathematical passage
R_N(z) = \sum_{\substack{\eta^N=-1\\ \eta\ne e^{ia_N},e^{-ia_N}}} \exp\bigl((\eta-e^{-ia_N})z\bigr).
\e^{\ii d\omega}Use in source · 1 passage
Search aliases: e^ii dω e^ i dω e ii omega mathrm mathrm omega
Use in source · Paper_V_Traditional_Final.tex, line 2053
[Quantitative imperfect reconstruction] Define the delay compensated defect D( e^ iω) = e^ i dωR( e^ iω)E( e^ iω)-I_N. If ε_rec = *ess sup_ω D( e^ iω) _op<∞, then the delay compensated reconstruction widetilde x satisfies widetilde x-x _2 ≤ ε_rec x _2. The constant is sharp as an operator norm.
Exact mathematical passage
D(\e^{\ii\omega}) = \e^{\ii d\omega}R(\e^{\ii\omega})E(\e^{\ii\omega})-I_N.
\e^{\ii t_kD}Use in source · 1 passage
Search aliases: e^ii t_kD e^ i t_kD e ii mathrm mathrm
Use in source · 02_strengthened.tex, line 1723
Set w=∑_k=1^m e^ i t_kDB^*v_k. Then norm w^2 = ∑_j,k ipB^*v_j e^- i(t_j-t_k)DB^*v_k = ∑_j,k ipv_jK(t_j-t_k)v_k.
Exact mathematical passage
w=\sum_{k=1}^{m}\e^{\ii t_kD}B^*v_k.
e^{\ii t_kD}Use in source · 1 passage
Search aliases: e^ii t_kD e^ i t_kD ii mathrm
Use in source · monograph_atlas.tex, line 688
ts memory kernel is K(t)=frac925e^- i t+frac1625e^-2 i t, K(0)=1. Each oscillatory coefficient is positive although K(t) need not be real or pointwise nonnegative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structural identity is ∑_j,koverline v_jK(t_j-t_k)v_k = ∑_k e^ i t_kDB^*v_k ^2≥0. The exact worked check specializes to t=(0,π/2,π) and v=(1,2,-1). Its phases lie in QQ( i…
Exact mathematical passage
\sum_{j,k}\overline v_jK(t_j-t_k)v_k =\left\|\sum_k e^{\ii t_kD}B^*v_k\right\|^2\geq0.
\e^{\ii(\varphi+\theta)}Use in source · 1 passage
Search aliases: e^ii(φ+θ) e^ i(φ+θ) e ii varphi theta mathrm mathrm varphi theta
Use in source · Paper_V_Traditional_Final.tex, line 3715
[Radial and phase coordinates] Write C(z)=ρ e^ iφ, 0≤ρ<1. Then C(Φ_θ(z)) = ρ e^ i(φ+θ). The radial mismatch ρ is invariant and the phase coordinate advances by θ.
Exact mathematical passage
C(\Phi_\theta(z)) = \rho\e^{\ii(\varphi+\theta)}.
\e^{\ii\theta}Use in source · 3 passages
Search aliases: e^iiθ e^ iθ e ii theta mathrm mathrm theta
Use in source · Paper_V_Traditional_Final.tex, line 3663
Let H = {z∈ C:Re z>0} and define C(z)=fracz-1z+1. It maps H conformally onto the unit disk D, with inverse C^-1(w)=frac1+w1-w. For θ∈ R, let R_θ(w)= e^ iθw.
Exact mathematical passage
R_\theta(w)=\e^{\ii\theta}w.
e^{\ii\theta}Use in source · 2 passages
Search aliases: e^iiθ e^ iθ ii theta mathrm theta
Use in source · Paper_V_Traditional_Final.tex, line 1191
For 0≤ρ<1 and angle θ, Γ=ρ e^ iθ gives z(ρ,θ) =frac1+ρ e^ iθ1-ρ e^ iθ, with r(ρ,θ)=frac1-ρ^21-2ρcosθ+ρ^2, x(ρ,θ)=frac2ρsinθ1-2ρcosθ+ρ^2. A fixed ρ is a constant-mismatch circle. Refining N increases angular resolution without changing ρ.
Exact mathematical passage
\Gam=\rho e^{\ii\theta}
\e^{\ii\pi}Use in source · 1 passage
Search aliases: e^iiπ e^ iπ e ii pi mathrm mathrm pi
Use in source · Paper_V_Traditional_Final.tex, line 3704
Insert e^ iπ=-1 in the formula of [source reference].
Exact mathematical passage
\e^{\ii\pi}=-1
\e^{\ii\varphi}Use in source · 1 passage
Search aliases: e^iiφ e^ iφ e ii varphi mathrm mathrm varphi
Use in source · Paper_V_Traditional_Final.tex, line 3709
[Radial and phase coordinates] Write C(z)=ρ e^ iφ, 0≤ρ<1. Then C(Φ_θ(z)) = ρ e^ i(φ+θ). The radial mismatch ρ is invariant and the phase coordinate advances by θ.
Exact mathematical passage
C(z)=\rho\e^{\ii\varphi}, \qquad 0\leq\rho<1.
e^{\ii\phi}Use in source · 1 passage
Search aliases: e^iiϕ e^ iϕ ii phi mathrm phi
Use in source · monograph_atlas.tex, line 283
n·σ). This is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form P( d Pwedge d P)P. With the paper's scalar convention, its component is F_θϕ=- iTr (P[∂_θ P,∂_ϕ P]) =frac12sinθ. vfigbloch_curvature headA local frame gives the same answer Away from the south pole, use u_N=(cos(θ/2),e^ iϕsin(θ/2))^T. Then P=u_Nu_N^* and the scalar one-form is A_N=- i u_N^* d u_N=frac1-cosθ2 dϕ, d A_N=frac12sinθ dθwedge dϕ. Thus the curvature flux through the cap 0≤θ…
Exact mathematical passage
u_N=(\cos(\theta/2),e^{\ii\phi}\sin(\theta/2))^T
\e^{\ii\omega}Use in source · 8 passages
Search aliases: e^iiω e^ iω e ii omega mathrm mathrm omega
Use in source · Paper_V_Traditional_Final.tex, line 2053
[Quantitative imperfect reconstruction] Define the delay compensated defect D( e^ iω) = e^ i dωR( e^ iω)E( e^ iω)-I_N. If ε_rec = *ess sup_ω D( e^ iω) _op<∞, then the delay compensated reconstruction widetilde x satisfies widetilde x-x _2 ≤ ε_rec x _2. The constant is sharp as an operator norm.
Exact mathematical passage
D(\e^{\ii\omega}) = \e^{\ii d\omega}R(\e^{\ii\omega})E(\e^{\ii\omega})-I_N.
e^{ik\pi/4}Use in source · 1 passage
Search aliases: e^ikπ/4 pi
Use in source · visual_atlas.tex, line 510
A NONLINEAR PHASE ORBITChapter II, mathematical text: Section 12.3, Smith-phase identity (p. 43) Smith normal form concerns linear congruences. The Smith impedance chart uses the nonlinear coordinate change Γ=fracz-1z+1, z=frac1+Γ1-Γ. They share a name, not a mathematical definition. Sample the reflection circle at the exact algebraic points Γ_k=tfrac12 e^ikπ/4.
Exact mathematical passage
\Gamma_k=\tfrac12 e^{ik\pi/4}
e^{isH}Use in source · 2 passages
Search aliases: e^isH
Use in source · 01_finite_core.tex, line 442
Set F(s)=e^isHPe^-isH. Differentiating the exponential gives F'(s)=i e^isH[H,P]e^-isH, F(0)=P. Unitary conjugation preserves the operator norm, so F'(s) = [H,P] . The mean-value estimate on the interval between 0 and t yields F(t)-P ≤ |t| [H,P] . Finally, e^itHQe^-itHP=(P-F(t))P and P ≤1. Left multiplication by e^itH preserves the norm and gives the desired inequality. This argument includes positive, negative and zero time.
Exact mathematical passage
F(s)=e^{isH}Pe^{-isH}
e^{itH}Use in source · 2 passages
Search aliases: e^itH
Use in source · 01_finite_core.tex, line 450
Set F(s)=e^isHPe^-isH. Differentiating the exponential gives F'(s)=i e^isH[H,P]e^-isH, F(0)=P. Unitary conjugation preserves the operator norm, so F'(s) = [H,P] . The mean-value estimate on the interval between 0 and t yields F(t)-P ≤ |t| [H,P] . Finally, e^itHQe^-itHP=(P-F(t))P and P ≤1. Left multiplication by e^itH preserves the norm and gives the desired inequality. This argument includes positive, negative and zero time.
Exact mathematical passage
e^{itH}Qe^{-itH}P=(P-F(t))P
e^{i\theta}Use in source · 8 passages
Search aliases: e^iθ theta
Use in source · Paper_IV_Traditional_Final.tex, line 1810
[Indicator polygon and dominant sectors] The indicator of D_N equals h_N, with the limsup taken over radii where D_N(re^iθ)≠0: limsup_r→∞ frac1rlog| D_N(re^iθ)| = h_N(θ). If K is a compact set of directions disjoint from T_N, then there are δ>0 and R>0 such that, for each θ∈ K, one root η(θ) of -1 satisfies D_N(re^iθ) = frac1N exp(η(θ)re^iθ) (1+O(e^-δ r)) uniformly for r≥ R. In particular, no sufficiently large zero can lie in those directions.
Exact mathematical passage
\Dfun_N(re^{i\theta})\ne0
e^{L(z)}Use in source · 1 passage
Search aliases: e^L(z)
Use in source · atlas-pages.tex, line 231
vpage13A complex power needs a domainBRANCH BOOKKEEPINGChapter IV: Section 2, Definition 7.1, Corollary 8.4. A formal unit series has a formal logarithm at the origin. To interpret that germ analytically, choose a simply connected zero-free region containing 0 and normalize the logarithm to vanish there. D_N(z)^-r=exp(-rL(z)), e^L(z)=D_N(z), L(0)=0. vfigbranch_domain For D_2=cos z, the open disk |z|<π/2 is an admissible domain. A small loop around the simple zero ρ=π/2 is not contained in that d…
Exact mathematical passage
D_N(z)^{-r}=\exp(-rL(z)),\qquad e^{L(z)}=D_N(z),\quad L(0)=0.
e^{L+\pi i}Use in source · 1 passage
Search aliases: e^L+π i pi
Use in source · Paper_IV_Traditional_Final.tex, line 1957
In fact, the constants ε_M=1/2 and C_M=3/2 work for every complex w, without the strip restriction. Put u=-(1+e^w) and L=log(1+u)=log(-e^w), using the principal logarithm. The elementary logarithm estimate on |u|≤1/2 gives |L|≤(3/2)|u|. Since e^L+π i=e^w, there is an integer m such that w=L+(2m+1)π i. This gives the required distance bound. Distinct centers are separated by at least 2π, whereas two centers satisfying the bound would be separated by at most 3|1+e^w|≤3/2. Thus the integer is uniqu…
Exact mathematical passage
e^{L+\pi i}=e^w
E^{n-1}Use in source · 1 passage
Search aliases: E^n-1
Use in source · 03_nonlinear_coordinates.tex, line 1150
The class of (e,0) vanishes exactly when there are e'∈ E^n-1 and c∈ C^n with (e,0)=d_Cone(e',c). This means e=d_Ee'+Dc, d_Cc=0. Equivalently, [e]=H^n(D)[c]. The degree zero statement follows directly from the definition of the cone differential.
Exact mathematical passage
e'\in E^{n-1}
e^{Q/8}Use in source · 1 passage
Search aliases: e^Q/8
Use in source · atlas-pages.tex, line 95
For later discrete calculations define K=I+Q/8. This matrix is stochastic, but it is not e^Q/8. It is the step kernel in the uniformization identity e^tQ=e^-8t∑_n≥0(8t)^nK^n/n!. takeawayChanging packet coordinates does not change the evolution. Replacing continuous time by one discrete step does.
Exact mathematical passage
e^{Q/8}
\e^{qt}Use in source · 2 passages
Search aliases: e^qt e mathrm
Use in source · 72_recovered.tex, line 918
[Odd source generating function] For every positive odd q, ∑_n≥0S_n(q)fract^nn! = e^qtcosh(qt)secht. Its even part is ∑_n≥0S_2n(q)fract^2n(2n)! = cosh^2(qt)secht.
Exact mathematical passage
\boxed{ \sum_{n\geq0}S_n(q)\frac{t^n}{n!} = \e^{qt}\cosh(qt)\operatorname{sech}t. }
e^{rA_f}Use in source · 3 passages
Search aliases: e^rA_f
Use in source · Paper_VI_Traditional_Final.tex, line 463
[Exact Duhamel refinement identity] For every t≥0, Ce^tA_f-e^tA_cC = ∫_0^t e^(t-r)A_cΔ e^rA_f dr. In particular, Δ=0 implies exact semigroup intertwining at every time.
Exact mathematical passage
\boxed{ Ce^{tA_f}-e^{tA_c}C = \int_0^t e^{(t-r)A_c}\Delta e^{rA_f}\,dr. }
\e^{\operatorname{Re}\mu_jt}Use in source · 1 passage
Search aliases: e^Reμ_jt e operatorname mu mathrm operatorname mu
Use in source · Paper_VI_Traditional_Final.tex, line 4715
esolvent expansion is (zI-A_QQ)^-1Q_j = ∑_k=0^ν_j-1 fracN_j^kQ_j(z-μ_j)^k+1. Multiplication by the coupling blocks gives the self-energy formula for every z outside the hidden spectrum. Equivalently, on a right half plane it is the Laplace transform identity ∫_0^∞ e^-zt e^μ_jt fract^kk! d t = frac1(z-μ_j)^k+1. Under stability, ∫_0^∞ e^Reμ_jt fract^kk! d t = frac1(-Reμ_j)^k+1. Sum the omitted terms.
Exact mathematical passage
\int_0^\infty \e^{\operatorname{Re}\mu_jt} \frac{t^k}{k!} \dd t = \frac{1}{(-\operatorname{Re}\mu_j)^{k+1}}.
e^{r\Re(\eta e^{i\theta})}Use in source · 1 passage
Search aliases: e^rRe(η e^iθ) Re eta theta
Use in source · atlas-pages.tex, line 308
vpage17Growth comes from a polygonDOMINANT EXPONENTSChapter IV: Definition 11.3, Theorem 11.4. Write the signed denominator as a finite exponential sum: D_N(z)=frac1N∑_η^N=-1e^η z. For z=re^iθ, the size of one exponential is e^rRe(η e^iθ). The maximal projection determines the directional exponential growth. vfigindicator The indicator is h_N(θ)=max_0≤ k<Ncos(θ+frac(2k+1)πN). Away from a tie direction, one exponential dominates all the others by a uniform positive projection gap on compact angul…
Exact mathematical passage
e^{r\Re(\eta e^{i\theta})}
e^{rz}Use in source · 1 passage
Search aliases: e^rz
Use in source · Paper_IV_Traditional_Final.tex, line 1228
For N=1, one has D_1(z)^-r=e^rz and P_1,n(r)=r^n. For N=2, P_2,1(r)=r, P_2,2(r)=r(3r+2), and P_2,3(r)=r(15r^2+30r+16). At r=1, these are the positive secant coefficients.
Exact mathematical passage
\Dfun_1(z)^{-r}=e^{rz}
\e^{s_1D}Use in source · 1 passage
Search aliases: e^s_1D e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 2236
[Dyson expansion over packet paths] For t≥0, e^tA = · e^tD + ∑_m=1^∞ ∫_0<s_1<⋯<s_m<t e^(t-s_m)DN e^(s_m-s_m-1)D ⋯ N e^s_1D d s_1⋯ d s_m. The series converges absolutely in operator norm. After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.
Exact mathematical passage
\begin{aligned} \e^{tA} ={}&\e^{tD} + \sum_{m=1}^{\infty} \int_{0<s_1<\cdots<s_m<t} \e^{(t-s_m)D}N\e^{(s_m-s_{m-1})D} \cdots N\e^{s_1D} \dd s_1\cdots\dd s_m. \end{aligned}
\e^{sA}Use in source · 3 passages
Search aliases: e^sA e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 2254
Write the variation of constants identity e^tA = e^tD + ∫_0^t e^(t-s)DN e^sA d s. Iterating the identity gives the displayed series. The norm of the term of order m is at most e^tnorm Dfrac(tnorm N)^mm!. The majorant is summable. Inserting the packet resolution around every factor N leaves only products whose consecutive packet labels are joined by nonzero transfer blocks.
Exact mathematical passage
\e^{tA} = \e^{tD} + \int_0^t\e^{(t-s)D}N\e^{sA}\dd s.
e^{sA}Use in source · 3 passages
Search aliases: e^sA
Use in source · Paper_VI_Traditional_Final.tex, line 335
[Finite semigroup law] For every matrix A and all s,t∈ R, e^(s+t)A=e^sAe^tA. Moreover, T(0)=I and T'(t)=AT(t)=T(t)A.
Exact mathematical passage
\boxed{e^{(s+t)A}=e^{sA}e^{tA}.}
\e^{sA_f}Use in source · 3 passages
Search aliases: e^sA_f e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 2290
[Duhamel defect identity] For every t≥0, D(t) = ∫_0^t e^(t-s)A_cΔ e^sA_f d s. If norm e^tA_f≤ M_f e^ω_f t, norm e^tA_c≤ M_c e^ω_c t, then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-s)+ω_fs d s. For contraction semigroups, norm D(t)≤ tnormΔ.
Exact mathematical passage
\mathfrak D(t) = \int_0^t \e^{(t-s)A_c}\Delta\e^{sA_f} \dd s.
\e^{sg}Use in source · 1 passage
Search aliases: e^sg e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 3796
The semigroup law follows from multiplication of the likelihoods e^sg and e^tg. Differentiating the normalized formula gives the replicator equation. A coordinate that starts at zero remains zero, while a positive coordinate remains positive. Dividing two explicit coordinate formulas gives the odds identity.
Exact mathematical passage
\e^{sg}
e^{sg}Use in source · 1 passage
Search aliases: e^sg
Use in source · Paper_VI_Traditional_Final.tex, line 1057
The semigroup law is the likelihood-composition theorem with e^sge^tg=e^(s+t)g. Differentiating the normalized exponential formula gives dot p_i=g_ip_i-p_ifrac∑_jg_je^tg_jp_j(0)∑_je^tg_jp_j(0). The quotient is ⟨ g,p(t)⟩.
Exact mathematical passage
e^{sg}e^{tg}=e^{(s+t)g}
e^{sQ}Use in source · 1 passage
Search aliases: e^sQ
Use in source · atlas-pages.tex, line 90
Any other complete packet family still has exact block convolution. For the coordinate groups A and B, P_b e^(s+t)QP_a=∑_c∈{ A, B}(P_b e^sQP_c)(P_c e^tQP_a). The intermediate packet cannot generally be omitted. Only a reducing family eliminates all cross terms.
Exact mathematical passage
P_b e^{(s+t)Q}P_a=\sum_{c\in\{\mathrm A,\mathrm B\}}(P_b e^{sQ}P_c)(P_c e^{tQ}P_a).
e^{sT_f}Use in source · 4 passages
Search aliases: e^sT_f
Use in source · 05_defects.tex, line 182
[Exponential defect identity] Let C:X_f→ X_c, T_f, and T_c be bounded linear maps between Banach spaces over a common field. For real t, use real or complex spaces. For complex t, use complex spaces and complex-linear maps. Then Ce^tT_f-e^tT_cC = ∫_0^t e^(t-s)T_cΔ e^sT_f ds, where the integral is taken along the straight segment from 0 to t when t is complex. For real t≥0, Ce^tT_f-e^tT_cC ≤ Δ ∫_0^t e^(t-s)T_c e^sT_f ds.
Exact mathematical passage
Ce^{tT_f}-e^{tT_c}C = \int_0^t e^{(t-s)T_c}\Delta e^{sT_f}\,ds,
\e^tUse in source · 1 passage
Search aliases: e^t e mathrm
Use in source · 72_recovered.tex, line 937
The finite geometric sum gives ∑_a=0^q-1(-1)^a e^(2a+1)t = e^tfrac1+ e^2qt1+ e^2t = e^qtcosh(qt)secht. Averaging at t and -t gives the even part.
Exact mathematical passage
\sum_{a=0}^{q-1}(-1)^a\e^{(2a+1)t} = \e^t\frac{1+\e^{2qt}}{1+\e^{2t}} = \e^{qt}\cosh(qt)\operatorname{sech}t.
\e^{t(B+C)}Use in source · 1 passage
Search aliases: e^t(B+C) e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 4442
Set B=-D_V and C=Q^T. For bounded matrices, e^tB/n e^tC/n = I+fractn(B+C)+O(n^-2) uniformly in operator norm. The standard telescoping comparison with (I+t(B+C)/n)^n shows that the product converges to e^t(B+C). This is the displayed factor order, so the limit follows.
Exact mathematical passage
\e^{t(B+C)}
e^{t(B-zI)}Use in source · 1 passage
Search aliases: e^t(B-zI)
Use in source · Paper_VI_Traditional_Final.tex, line 1557
Put B=A_QQ and T_z(t)=e^-zte^tB=e^t(B-zI). The generalized-eigenspace decomposition in finite dimensions writes T_z as a finite sum of constant matrix coefficients times t^k e^(λ-z)t. This follows by expanding the nilpotent part on each generalized eigenspace. Equivalently, apply the decomposition to left multiplication by B on the matrix algebra and evaluate the exponential at I. The strict spectral inequality makes every such term absolutely integrable on (0,∞) and forces it to tend to zero. H…
Exact mathematical passage
T_z(t)=e^{-zt}e^{tB}=e^{t(B-zI)}
\e^{t(g_i-g_j)}Use in source · 1 passage
Search aliases: e^t(g_i-g_j) e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 3787
[Bayesian flow and replicator equation] The maps Φ_t form a nonlinear semigroup: Φ_s+t=Φ_s∘Φ_t. Every coordinate satisfies frac d d tp_i(t) = (g_i-ipgp(t))p_i(t). The support of p(t) is independent of t, and for any two coordinates in the support, fracp_i(t)p_j(t) = e^t(g_i-g_j) fracp_i(0)p_j(0).
Exact mathematical passage
\frac{p_i(t)}{p_j(t)} = \e^{t(g_i-g_j)} \frac{p_i(0)}{p_j(0)}.
\e^{t(\overline Q^T-\overline V)}Use in source · 1 passage
Search aliases: e^t(overline Q^T-overline V) e overline overline mathrm overline overline
Use in source · Paper_VI_Traditional_Final.tex, line 5649
[Coarse Feynman--Kac transfer] Assume CQ=overline QC, and let J=C^T lift a coarse function to the corresponding block constant fine function. Suppose the potential is block constant, so VJ=Joverline V. Then e^t(Q^T-V)J = J e^t(overline Q^T-overline V) for every t≥0.
Exact mathematical passage
\boxed{ \e^{t(Q^T-V)}J = J\e^{t(\overline Q^T-\overline V)} }
\e^{t(Q+aI)}Use in source · 1 passage
Search aliases: e^t(Q+aI) e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 2838
Choose a≥max_i(-Q_ii). The matrix Q+aI is nonnegative, so its exponential is nonnegative. Hence e^tQ= e^-at e^t(Q+aI) is nonnegative. Since 1^TQ=0, differentiation gives 1^T e^tQ= 1^T.
Exact mathematical passage
\e^{tQ}=\e^{-at}\e^{t(Q+aI)}
\e^{t(Q-D_V)}Use in source · 1 passage
Search aliases: e^t(Q-D_V) e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 4943
Average [source reference] with terminal function f= 1 against the initial distribution. Transposition gives p_0^T e^t(Q^T-D_V) 1 = 1^T e^t(Q-D_V)p_0. Since 1^TQ=0, differentiation gives S_V'(t) = - 1^TD_V e^tAp_0.
Exact mathematical passage
p_0^T\e^{t(Q^T-D_V)}\one = \one^T\e^{t(Q-D_V)}p_0.
\e^{t(Q^T-D_V)}Use in source · 5 passages
Search aliases: e^t(Q^T-D_V) e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 4382
[Continuous time finite state formula] Let Q be a finite Markov generator and let (X_t)_t≥0 be the corresponding chain. Then ( e^t(Q^T-D_V)f )_i = E_i[ exp( -∫_0^tV(X_s) d s ) f(X_t) ]. If V≥ v_0, then norm e^t(Q^T-D_V)_∞→∞ ≤ e^-v_0t.
Exact mathematical passage
\left( \e^{t(Q^T-D_V)}f \right)_i = \Exp_i\left[ \exp\left( -\int_0^tV(X_s)\dd s \right) f(X_t) \right].
\e^{t(Q^T-V)}Use in source · 2 passages
Search aliases: e^t(Q^T-V) e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 5626
Let Q and overline Q be finite Markov generators in the column convention. All times in this section satisfy t≥0. [Potential perturbation bound] Let V and W be nonnegative diagonal matrices. In the supremum norm, norm e^t(Q^T-V) - e^t(Q^T-W) _∞→∞ ≤ tnormV-W_∞→∞.
Exact mathematical passage
\boxed{ \norm{ \e^{t(Q^T-V)} - \e^{t(Q^T-W)} }_{\infty\to\infty} \leq t\norm{V-W}_{\infty\to\infty}. }
\e^{t(Q^T-W)}Use in source · 1 passage
Search aliases: e^t(Q^T-W) e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 5626
Let Q and overline Q be finite Markov generators in the column convention. All times in this section satisfy t≥0. [Potential perturbation bound] Let V and W be nonnegative diagonal matrices. In the supremum norm, norm e^t(Q^T-V) - e^t(Q^T-W) _∞→∞ ≤ tnormV-W_∞→∞.
Exact mathematical passage
\boxed{ \norm{ \e^{t(Q^T-V)} - \e^{t(Q^T-W)} }_{\infty\to\infty} \leq t\norm{V-W}_{\infty\to\infty}. }
\e^{t(QAQ)|_{QX}}Use in source · 1 passage
Search aliases: e^t(QAQ)|_QX e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 2750
[Generator control of range leakage] Let T(t)= e^tA on a Banach space X, let P be a bounded projection, and set Q=I-P. Assume normT(t)≤ M e^ω t and norm e^t(QAQ)|_QX ≤ M_Q e^ω_Qt. Then normQT(t)P ≤ M M_Qnorm P^2normQAP ∫_0^t e^ω_Q(t-s)+ω s d s. In particular, for orthogonal projections and contraction bounds, sup_0≤ t≤ T normQT(t)P ≤ TnormQAP.
Exact mathematical passage
\norm{\e^{t(QAQ)|_{QX}}} \leq M_Q\e^{\omega_Qt}.
e^{t/2}Use in source · 2 passages
Search aliases: e^t/2
Use in source · atlas-pages.tex, line 323
vpage18Two large terms nearly cancelTHE CUBIC MODELChapter IV: Lemma 11.5, Theorem 11.7, Corollary 11.9. For real t, order three has the exact formula D_3(t)=frace^-t+2e^t/2cos(sqrt3t/2)3. Divide by the nonzero factor 2e^t/2/3. The resulting equation separates the dominant cosine from the small third exponential. vfigtwo_term Ignoring the last term gives model zeros t_m=frac2πsqrt3(m+1/2). At t_m, the full normalized expression equals tfrac12e^-3t_m/2>0, so t_m is not an exact zero. Linearizing …
Exact mathematical passage
D_3(t)=\frac{e^{-t}+2e^{t/2}\cos(\sqrt3t/2)}{3}.
e^{T^2}Use in source · 1 passage
Search aliases: e^T^2
Use in source · atlas-pages.tex, line 884
trajectory platecaptionThe first-coordinate error is exactly t^2/2. The certificate bounds its maximum on the entire interval. The bound is deliberately shown separately from the trajectory. headTrack every constant before applying the estimate Here M_T=1, B=sup widetilde x =1, and L= K _L^1(0,T)=T, ε= K-widetilde K _L^1(0,T)=T. The theorem yields sup_0≤ t≤ T x(t)-widetilde x(t) =fracT^22≤ T^2e^T^2. At T=1/2, the exact error is 1/8 and the certificate is e^1/4/4. Nilpotence makes this example po…
Exact mathematical passage
\sup_{0\le t\le T}\|x(t)-\widetilde x(t)\| =\frac{T^2}{2}\le T^2e^{T^2}.
e^{tA}Use in source · 17 passages
Search aliases: e^tA
Use in source · Paper_VI_Traditional_Final.tex, line 300
Let X be a finite-dimensional real or complex vector space, let A∈End(X), and let s∈ X be a primitive seed. The basic continuous transfer is T(t)=e^tA, t≥ 0, while a positive operator L is more naturally written H(t)=e^-tL. Both satisfy the semigroup law. The sign convention changes with the domain: Markov generators and evolution equations are usually written with e^tA. Positive diffusion and elliptic operators are usually written with e^-tL.
Exact mathematical passage
T(t)=e^{tA},\qquad t\ge 0,
\e^{tA_c}Use in source · 4 passages
Search aliases: e^tA_c e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 2284
Let X_f and X_c be complex Banach spaces. Let A_f∈ B(X_f), A_c∈ B(X_c), and C∈ B(X_f,X_c). Define Δ=CA_f-A_cC and D(t)=C e^tA_f- e^tA_cC.
Exact mathematical passage
\mathfrak D(t)=C\e^{tA_f}-\e^{tA_c}C.
e^{tA_c}Use in source · 4 passages
Search aliases: e^tA_c
Use in source · Paper_VI_Traditional_Final.tex, line 457
Let X_f and X_c be fine and coarse spaces, with generators A_f and A_c, and let C:X_f→ X_c be a coarsening map. Define the generator-level refinement defect Δ=CA_f-A_cC. The corresponding semigroup defect is D(t)=Ce^tA_f-e^tA_cC.
Exact mathematical passage
\mathfrak D(t)=Ce^{tA_f}-e^{tA_c}C.
\e^{tA_f}Use in source · 4 passages
Search aliases: e^tA_f e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 2284
Let X_f and X_c be complex Banach spaces. Let A_f∈ B(X_f), A_c∈ B(X_c), and C∈ B(X_f,X_c). Define Δ=CA_f-A_cC and D(t)=C e^tA_f- e^tA_cC.
Exact mathematical passage
\mathfrak D(t)=C\e^{tA_f}-\e^{tA_c}C.
e^{tA_f}Use in source · 4 passages
Search aliases: e^tA_f
Use in source · Paper_VI_Traditional_Final.tex, line 457
Let X_f and X_c be fine and coarse spaces, with generators A_f and A_c, and let C:X_f→ X_c be a coarsening map. Define the generator-level refinement defect Δ=CA_f-A_cC. The corresponding semigroup defect is D(t)=Ce^tA_f-e^tA_cC.
Exact mathematical passage
\mathfrak D(t)=Ce^{tA_f}-e^{tA_c}C.
\e^{tA_n}Use in source · 1 passage
Search aliases: e^tA_n e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 2801
[A usable closure condition for refinement towers] Assume the contractive summability hypothesis of [source reference] holds on every bounded time interval. Suppose every P_m,n is an orthogonal projection on a Hilbert space and T_n(t)= e^tA_n is contractive. If, for every fixed m, norm(I-P_m,n)A_nP_m,n ⟶0, then the limiting family U_m is a semigroup.
Exact mathematical passage
T_n(t)=\e^{tA_n}
\e^{tA_{PP}}Use in source · 1 passage
Search aliases: e^tA_PP e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 4590
[Criterion for exact local closure] Assume x_Q(0)=0 and d=dim X_Q<∞. The following conditions are equivalent. item For every retained initial state, P e^tAP = e^tA_PPP for all t≥0. item Σ_P(z)=0 for every sufficiently large z. item K(t)=0 for every t≥0. item If d=dim X_Q, then A_PQA_QQ^kA_QP=0 for 0≤ k<d.
Exact mathematical passage
P\e^{tA}P = \e^{tA_{PP}}P
\e^{tA_{QQ}}Use in source · 5 passages
Search aliases: e^tA_QQ e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 4519
For the homogeneous equation dot x=Ax, variation of constants gives x_Q(t) = e^tA_QQx_Q(0) + ∫_0^t e^(t-s)A_QQA_QPx_P(s) d s. Substitution into the retained equation gives dot x_P(t) = · A_PPx_P(t) + A_PQ e^tA_QQx_Q(0)
Exact mathematical passage
x_Q(t) = \e^{tA_{QQ}}x_Q(0) + \int_0^t \e^{(t-s)A_{QQ}}A_{QP}x_P(s) \dd s.
e^{tA_{QQ}}Use in source · 3 passages
Search aliases: e^tA_QQ
Use in source · Paper_VI_Traditional_Final.tex, line 1528
For the homogeneous system dot x=Ax, variation of constants gives x_Q(t)=e^tA_QQx_Q(0) +∫_0^te^(t-r)A_QQA_QPx_P(r) dr. Substitution yields dot x_P(t) =A_PPx_P(t)+A_PQe^tA_QQx_Q(0) +∫_0^t K(t-r)x_P(r) dr, where K(t)=A_PQe^tA_QQA_QP.
Exact mathematical passage
x_Q(t)=e^{tA_{QQ}}x_Q(0) +\int_0^te^{(t-r)A_{QQ}}A_{QP}x_P(r)\,dr.
\e^{tA_{\alpha\alpha}}Use in source · 1 passage
Search aliases: e^tA_αα e alpha alpha mathrm alpha alpha
Use in source · Paper_VI_Traditional_Final.tex, line 2190
thm:reducing_packets The following conditions are equivalent. item AP_α=P_α A for every α. item A_βα=0 whenever β≠α. item T_βα(t)=0 whenever β≠α and t∈ R. item Every packet space P_α X is invariant under T(t) for every t. When these conditions hold, T(t)|_P_α X = e^tA_αα.
Exact mathematical passage
T(t)|_{P_\alpha X} = \e^{tA_{\alpha\alpha}}.
\e^{tA_\ell}Use in source · 3 passages
Search aliases: e^tA_ℓ e ell mathrm ell
Use in source · Paper_VI_Traditional_Final.tex, line 2484
If T_ℓ(t)= e^tA_ℓ, the generators are bounded, and sup_ℓnormA_ℓ<∞, C_ℓ A_ℓ+1=A_ℓ C_ℓ, then the continuity condition holds and A_∞(x_ℓ)_ℓ=(A_ℓ x_ℓ)_ℓ is the bounded generator of T_∞.
Exact mathematical passage
T_\ell(t)=\e^{tA_\ell}
e^{tA_\ell}Use in source · 1 passage
Search aliases: e^tA_ℓ ell
Use in source · Paper_VI_Traditional_Final.tex, line 1767
[Uniform refinement control] For a nested family of physically or analytically meaningful discretizations, there exist norms and constants independent of resolution such that normC_ℓ e^tA_ℓ+1-e^tA_ℓC_ℓ ≤ M t normC_ℓ A_ℓ+1-A_ℓ C_ℓ on every bounded time interval. Summable generator defects would then produce a coherent inverse-limit evolution.
Exact mathematical passage
\norm{C_\ell e^{tA_{\ell+1}}-e^{tA_\ell}C_\ell} \le M\,t\,\norm{C_\ell A_{\ell+1}-A_\ell C_\ell}
e^{tA_{\ell+1}}Use in source · 1 passage
Search aliases: e^tA_ℓ+1 ell
Use in source · Paper_VI_Traditional_Final.tex, line 1767
[Uniform refinement control] For a nested family of physically or analytically meaningful discretizations, there exist norms and constants independent of resolution such that normC_ℓ e^tA_ℓ+1-e^tA_ℓC_ℓ ≤ M t normC_ℓ A_ℓ+1-A_ℓ C_ℓ on every bounded time interval. Summable generator defects would then produce a coherent inverse-limit evolution.
Exact mathematical passage
\norm{C_\ell e^{tA_{\ell+1}}-e^{tA_\ell}C_\ell} \le M\,t\,\norm{C_\ell A_{\ell+1}-A_\ell C_\ell}
\e^{tB}Use in source · 3 passages
Search aliases: e^tB e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 4580
pace, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every integer n≥0, after increasing M_β to cover the finitely many initial values. Write t=n+r, where 0≤ r<1. Then norm e^tB≤ M_βexp(norm B+max(0,-β)) e^β t. Thus e^-zt e^tB is Bochner integrable and tends to zero. Integrating its derivative gives both inverse identities for R=∫_0^∞ e^-zt e^tB d t, so R=(zI-B)^-1. This is the…
Exact mathematical passage
\norm{\e^{tB}}\leq M_\beta\exp(\norm B+\max(0,-\beta))\e^{\beta t}.
e^{tB}Use in source · 4 passages
Search aliases: e^tB
Use in source · Paper_VI_Traditional_Final.tex, line 551
Choose λ≥max_i(-Q_ii). Then B=Q+λ I has nonnegative entries, so e^tB is entrywise nonnegative by its power series. Hence e^tQ=e^-λ te^tB is nonnegative. Since 1^TQ=0, differentiation gives fracddt 1^Te^tQ= 1^TQe^tQ=0, so 1^Te^tQ= 1^T.
Exact mathematical passage
e^{tB}
\e^{tB/n}Use in source · 1 passage
Search aliases: e^tB/n e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 4433
Set B=-D_V and C=Q^T. For bounded matrices, e^tB/n e^tC/n = I+fractn(B+C)+O(n^-2) uniformly in operator norm. The standard telescoping comparison with (I+t(B+C)/n)^n shows that the product converges to e^t(B+C). This is the displayed factor order, so the limit follows.
Exact mathematical passage
\e^{tB/n}\e^{tC/n} = I+\frac{t}{n}(B+C)+O(n^{-2})
e^{t\bar Q}Use in source · 1 passage
Search aliases: e^tbar Q bar
Use in source · atlas-pages.tex, line 119
2 · -2. The Duhamel identity therefore gives Ce^tQ=e^tbar QC. This is a second consequence of the corresponding generator identity, not an inference that K is a fixed-time sample of Q.
Exact mathematical passage
Ce^{tQ}=e^{t\bar Q}C
\e^{tC/n}Use in source · 1 passage
Search aliases: e^tC/n e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 4433
Set B=-D_V and C=Q^T. For bounded matrices, e^tB/n e^tC/n = I+fractn(B+C)+O(n^-2) uniformly in operator norm. The standard telescoping comparison with (I+t(B+C)/n)^n shows that the product converges to e^t(B+C). This is the displayed factor order, so the limit follows.
Exact mathematical passage
\e^{tB/n}\e^{tC/n} = I+\frac{t}{n}(B+C)+O(n^{-2})
\e^{tD}Use in source · 5 passages
Search aliases: e^tD e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 2236
[Dyson expansion over packet paths] For t≥0, e^tA = · e^tD + ∑_m=1^∞ ∫_0<s_1<⋯<s_m<t e^(t-s_m)DN e^(s_m-s_m-1)D ⋯ N e^s_1D d s_1⋯ d s_m. The series converges absolutely in operator norm. After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.
Exact mathematical passage
\begin{aligned} \e^{tA} ={}&\e^{tD} + \sum_{m=1}^{\infty} \int_{0<s_1<\cdots<s_m<t} \e^{(t-s_m)D}N\e^{(s_m-s_{m-1})D} \cdots N\e^{s_1D} \dd s_1\cdots\dd s_m. \end{aligned}
e^{tD}Use in source · 5 passages
Search aliases: e^tD
Use in source · Paper_VI_Traditional_Final.tex, line 5732
A finite auxiliary realization of a matrix kernel means K(t)=B e^tDC, with finite matrices of compatible sizes. Laplace transforms are taken in a right half-plane of absolute convergence. The rational characterization concerns continuous kernels of exponential order, so uniqueness of the ordinary Laplace transform applies. [Rational memory realization] A matrix valued memory transform has a finite auxiliary realization if and only if it is a strictly proper rational matrix function. More precise…
Exact mathematical passage
\Kcal(t)=B e^{tD}C
\e^{tG}Use in source · 1 passage
Search aliases: e^tG e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 3848
Factor e^tG from the normalizing constant. The total weight outside M is at most (1-p_M(0)) e^-δ t, while the weight on M is p_M(0).
Exact mathematical passage
\e^{tG}
\e^{tg}Use in source · 1 passage
Search aliases: e^tg e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 3796
The semigroup law follows from multiplication of the likelihoods e^sg and e^tg. Differentiating the normalized formula gives the replicator equation. A coordinate that starts at zero remains zero, while a positive coordinate remains positive. Dividing two explicit coordinate formulas gives the odds identity.
Exact mathematical passage
\e^{tg}
e^{tG}Use in source · 1 passage
Search aliases: e^tG
Use in source · atlas-pages.tex, line 59
1 · -1. The total-mass direction (1,1) has eigenvalue zero. The contrast (1,-1) has eigenvalue -2. Resolving those two directions gives the full transfer, not only its first-order approximation: T(t)=e^tG=frac121+e^-2t · 1-e^-2t 1-e^-2t · 1+e^-2t. vfigclock platecaptionA point seed at state zero approaches the uniform distribution. The displayed times were selected so that every bar label is rational. This is continuous-time evolution, not a simulated trajectory. headTwo checks explain the mecha…
Exact mathematical passage
T(t)=e^{tG}=\frac12\begin{pmatrix}1+e^{-2t}&1-e^{-2t}\\1-e^{-2t}&1+e^{-2t}\end{pmatrix}.
e^{tg}Use in source · 1 passage
Search aliases: e^tg
Use in source · Paper_VI_Traditional_Final.tex, line 1057
The semigroup law is the likelihood-composition theorem with e^sge^tg=e^(s+t)g. Differentiating the normalized exponential formula gives dot p_i=g_ip_i-p_ifrac∑_jg_je^tg_jp_j(0)∑_je^tg_jp_j(0). The quotient is ⟨ g,p(t)⟩.
Exact mathematical passage
e^{sg}e^{tg}=e^{(s+t)g}
\e^{tg_i}Use in source · 2 passages
Search aliases: e^tg_i e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 3768
Fix a score vector g∈ R^n. For t≥0, define Φ_t(p)_i = frac e^tg_ip_i ∑_j e^tg_jp_j.
Exact mathematical passage
\Phi_t(p)_i = \frac{\e^{tg_i}p_i} {\sum_j\e^{tg_j}p_j}.
e^{tg_i}Use in source · 3 passages
Search aliases: e^tg_i
Use in source · Paper_VI_Traditional_Final.tex, line 1030
Fix a score vector g. The likelihood flow is λ_i(t)=e^tg_i, and the Bayesian flow is Φ_t(p)_i = frace^tg_ip_i∑_je^tg_jp_j.
Exact mathematical passage
\lambda_i(t)=e^{tg_i},
\e^{tg_j}Use in source · 1 passage
Search aliases: e^tg_j e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 3768
Fix a score vector g∈ R^n. For t≥0, define Φ_t(p)_i = frac e^tg_ip_i ∑_j e^tg_jp_j.
Exact mathematical passage
\Phi_t(p)_i = \frac{\e^{tg_i}p_i} {\sum_j\e^{tg_j}p_j}.
e^{tg_j}Use in source · 3 passages
Search aliases: e^tg_j
Use in source · Paper_VI_Traditional_Final.tex, line 1034
Fix a score vector g. The likelihood flow is λ_i(t)=e^tg_i, and the Bayesian flow is Φ_t(p)_i = frace^tg_ip_i∑_je^tg_jp_j.
Exact mathematical passage
\Phi_t(p)_i = \frac{e^{tg_i}p_i}{\sum_je^{tg_j}p_j}.
\e^{t\Gcal}Use in source · 2 passages
Search aliases: e^tGcal e^t G e Gcal mathrm mathcal
Use in source · 02_strengthened.tex, line 1244
a Lindblad generator] Let S be unitary and N≥1 an integer with S^N=I. Assume SHS^*=H and, for every jump operator, there is a charge q_r such that SL_rS^*=ω^q_rL_r. Then the Lindblad generator G preserves every coherence order: G L_p= L_p G. The quantum dynamical semigroup e^t G has the same property.
Exact mathematical passage
\e^{t\Gcal}
e^{tJ}Use in source · 1 passage
Search aliases: e^tJ
Use in source · atlas-pages.tex, line 520
0 · 0, N^2=0. Choose the entrance column V=(0,1)^ T and exit row U=(1,0). The hidden exponential is not e^-2tI: e^tJ=e^-2t(I+tN), K_rm mem(t)=Ue^tJV=t e^-2t. The nilpotent link carries an input from the second hidden coordinate to the first. vfigjordan platecaptionThe memory kernel starts at zero, becomes positive, and peaks at t=1/2. A rule that retained the eigenvalue -2 but discarded the Jordan chain would predict the zero kernel for these entrance and exit maps. headThe transfer function see…
Exact mathematical passage
e^{tJ}=e^{-2t}(I+tN), \qquad K_{\rm mem}(t)=Ue^{tJ}V=t e^{-2t}.
e^{tL}Use in source · 1 passage
Search aliases: e^tL
Use in source · atlas-pages.tex, line 36
L · A nonnegative self-adjoint operator. · Its evolution is e^-tL, not e^tL.
Exact mathematical passage
e^{tL}
\e^{tM_A}Use in source · 1 passage
Search aliases: e^tM_A e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 2501
For bounded generators with common bound M_A, sup_ℓ norm( e^tA_ℓ-I)x_ℓ ≤ ( e^tM_A-1)sup_ℓnormx_ℓ. This tends to zero with t. Exact generator intertwining makes (A_ℓ x_ℓ)_ℓ compatible, and coordinatewise differentiation identifies the generator.
Exact mathematical passage
\sup_\ell \norm{(\e^{tA_\ell}-I)x_\ell} \leq (\e^{tM_A}-1)\sup_\ell\norm{x_\ell}.
\e^{t\norm A}Use in source · 1 passage
Search aliases: e^tnorm A e norm mathrm norm
Use in source · Paper_VI_Traditional_Final.tex, line 2133
The estimate normT(t)≤ e^tnorm A gives absolute convergence of the operator valued integral. Let R(z)=∫_0^∞ e^-ztT(t) d t. Since T'(t)=AT(t)=T(t)A, integration by parts gives (zI-A)R(z) = -∫_0^∞ frac d d t ( e^-ztT(t)) d t =I. The same calculation on the other side gives R(z)(zI-A)=I . Multiplication by P_β and P_α may be passed under the norm convergent integral, which gives the packet formula.
Exact mathematical passage
\norm{T(t)}\leq\e^{t\norm A}
\e^{t\norm D}Use in source · 1 passage
Search aliases: e^tnorm D e norm mathrm norm
Use in source · Paper_VI_Traditional_Final.tex, line 2262
Write the variation of constants identity e^tA = e^tD + ∫_0^t e^(t-s)DN e^sA d s. Iterating the identity gives the displayed series. The norm of the term of order m is at most e^tnorm Dfrac(tnorm N)^mm!. The majorant is summable. Inserting the packet resolution around every factor N leaves only products whose consecutive packet labels are joined by nonzero transfer blocks.
Exact mathematical passage
\e^{t\norm D}\frac{(t\norm N)^m}{m!}.
\e^{t\overline Q}Use in source · 1 passage
Search aliases: e^toverline Q e overline mathrm overline
Use in source · Paper_VI_Traditional_Final.tex, line 2945
[Continuous time lumping] A coarse Markov generator overline Q satisfying CQ=overline QC exists if and only if the aggregated rates ∑_r∈ B_αQ_ri depend only on the source block of i. In that case C e^tQ= e^toverline QC for every t≥0.
Exact mathematical passage
C\e^{tQ}=\e^{t\overline Q}C
\e^{tQ}Use in source · 5 passages
Search aliases: e^tQ e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 2027
Markov chain · K^n or e^tQ · equality of aggregated transition masses
Exact mathematical passage
\e^{tQ}
e^{tQ}Use in source · 10 passages
Search aliases: e^tQ
Use in source · Paper_VI_Traditional_Final.tex, line 542
Use column probability vectors. A finite continuous-time Markov generator Q has nonnegative off-diagonal entries and column sums zero. The stochastic semigroup is K(t)=e^tQ.
Exact mathematical passage
K(t)=e^{tQ}.
\e^{tQ^T/n}Use in source · 1 passage
Search aliases: e^tQ^T/n e mathrm
Use in source · Paper_VI_Traditional_Final.tex, line 4420
[Alternating propagation and killing] For every t≥0, e^t(Q^T-D_V) = lim_n→∞ ( e^-tD_V/n e^tQ^T/n )^n in operator norm. Each finite product alternates one Markov propagation step with one diagonal survival weight.
Exact mathematical passage
\e^{t(Q^T-D_V)} = \lim_{n\to\infty} \left( \e^{-tD_V/n}\e^{tQ^T/n} \right)^n
e^{tT_c}Use in source · 2 passages
Search aliases: e^tT_c
Use in source · 05_defects.tex, line 182
[Exponential defect identity] Let C:X_f→ X_c, T_f, and T_c be bounded linear maps between Banach spaces over a common field. For real t, use real or complex spaces. For complex t, use complex spaces and complex-linear maps. Then Ce^tT_f-e^tT_cC = ∫_0^t e^(t-s)T_cΔ e^sT_f ds, where the integral is taken along the straight segment from 0 to t when t is complex. For real t≥0, Ce^tT_f-e^tT_cC ≤ Δ ∫_0^t e^(t-s)T_c e^sT_f ds.
Exact mathematical passage
Ce^{tT_f}-e^{tT_c}C = \int_0^t e^{(t-s)T_c}\Delta e^{sT_f}\,ds,
e^{tT_f}Use in source · 2 passages
Search aliases: e^tT_f
Use in source · 05_defects.tex, line 182
[Exponential defect identity] Let C:X_f→ X_c, T_f, and T_c be bounded linear maps between Banach spaces over a common field. For real t, use real or complex spaces. For complex t, use complex spaces and complex-linear maps. Then Ce^tT_f-e^tT_cC = ∫_0^t e^(t-s)T_cΔ e^sT_f ds, where the integral is taken along the straight segment from 0 to t when t is complex. For real t≥0, Ce^tT_f-e^tT_cC ≤ Δ ∫_0^t e^(t-s)T_c e^sT_f ds.
Exact mathematical passage
Ce^{tT_f}-e^{tT_c}C = \int_0^t e^{(t-s)T_c}\Delta e^{sT_f}\,ds,
e^{tu}Use in source · 1 passage
Search aliases: e^tu
Use in source · atlas-pages.tex, line 822
vpage36The score rises because it has varianceBAYESIAN FLOW / EVIDENCE CUMULANTSChapter VI, Part II: Theorems E10.1 and E10.2. Exponential tilting of a fixed finite prior. Start with equal prior masses on scores u=(0,1,2) and likelihood e^tu. Put q=e^t. The evidence and tilted distribution are Z(t)=frac1+q+q^23, p(t)=frac(1,q,q^2)1+q+q^2. At t=log2, they are Z=7/3 and p=(1,2,4)/7. vfigext_cumulants platecaptionThe score values stay fixed. Their probabilities change. The rising mean score is a pr…
Exact mathematical passage
e^{tu}
e^wUse in source · 6 passages
Search aliases: e^w
Use in source · Paper_IV_Traditional_Final.tex, line 1941
[Near cancellation of two exponentials] For every M>0, there are constants ε_M>0 and C_M>0 such that the following holds. If |Re w|≤ M and |1+e^w|≤ε_M, then there is a unique integer m with |w-(2m+1)π i| ≤ C_M|1+e^w|.
Exact mathematical passage
|1+e^w|\leq\varepsilon_M,
e^{w\xi}Use in source · 2 passages
Search aliases: e^wξ xi
Use in source · Paper_IV_Traditional_Final.tex, line 817
Differentiation multiplies by λξ in the first algebra and by ξ in the second. Reduction of the product by the defining relation gives the derivative cycles. The identity e^(z+w)ξ=e^zξe^wξ gives the addition laws after comparison of the coefficients of 1,ξ,…,ξ^N-1.
Exact mathematical passage
e^{(z+w)\xi}=e^{z\xi}e^{w\xi}
e^{yz}Use in source · 2 passages
Search aliases: e^yz
Use in source · Paper_IV_Traditional_Final.tex, line 1509
Differentiate the generating function with respect to x for the first identity. Multiply it by e^yz and compare coefficients for the second.
Exact mathematical passage
e^{yz}
e^zUse in source · 1 passage
Search aliases: e^z
Use in source · 20_arithmetic.tex, line 80
[The Bernoulli boundary address] At N=2, the address j=1 gives a second classical family. Define the Bernoulli numbers by B(z)=fracze^z-1=∑_r≥0B_rfracz^rr!, B_0=1, B_1=-frac12. All quotients here are regular formal series. The identity 2B(z)-B(2z)=fraczsinh z =∑_n≥0C_2,1(n)fracz^2n(2n)! follows by putting the two fractions on a common denominator and using [source reference]. Comparing even coefficients yields C_2,1(n)=(2-2^2n)B_2n (n∈ N). This classical derivation restores the boundary example …
Exact mathematical passage
B(z)=\frac{z}{e^z-1}=\sum_{r\ge0}B_r\frac{z^r}{r!}, \qquad B_0=1,\quad B_1=-\frac12.
e^{z/2}Use in source · 1 passage
Search aliases: e^z/2
Use in source · Paper_IV_Traditional_Final.tex, line 758
The first denominators are D_1(z)=e^-z, D_2(z)=cos z. For N=3, D_3(z) = frac13( e^-z +2e^z/2cos(fracsqrt3z2) ). For N=4, D_4(z) = cos(frac zsqrt2) cosh(frac zsqrt2).
Exact mathematical passage
\Dfun_3(z) = \frac13\left( e^{-z} +2e^{z/2}\cos\left(\frac{\sqrt3z}{2}\right) \right).
e^{zH}Use in source · 2 passages
Search aliases: e^zH
Use in source · 03_supplement_01.tex, line 59
Symmetry and measurement. Exact symmetry: source Theorem 5.1. If HP_a=P_aH for every packet, then P_bHP_a=0 (b≠ a), H=∑_aP_aHP_a. In a complex Banach algebra, commutation also propagates to the actual exponential: e^zHP_a=P_ae^zH (z∈ C). Consequently, e^zH=∑_aP_ae^zHP_a, P_be^zHP_a=0 (b≠ a). Taking z=-it gives evolution in real time. The Lean proof applies the exponential commutation theorem directly, so invariance of the packets is a proved consequence of commutation. Self-adjointness of H is n…
Exact mathematical passage
e^{zH}P_a=P_ae^{zH}\qquad(z\in\C).
e^{zJ_3}Use in source · 2 passages
Search aliases: e^zJ_3
Use in source · atlas-pages.tex, line 85
vpage5One signed edge determines the cycleDIFFERENTIAL SYSTEMChapter IV: Theorem 5.3, Theorem 6.4, Corollary 6.5, Corollary 6.6. Let J_3e_0=e_1, J_3e_1=e_2, and J_3e_2=-e_0. A complete turn changes sign, so J_3^3=-I. The signed sections are the coordinates of e^zJ_3e_0. vfigsigned_shift Here D_j inside the matrix means D_3,j(z). For example, D_3,0=1-fracz^33!+fracz^66!-⋯, D_3,1=z-fracz^44!+⋯. Rows are output coordinates and columns are input coordinates. The upper-triangular minus signs record c…
Exact mathematical passage
e^{zJ_3}e_0
e^{zJ_N}Use in source · 9 passages
Search aliases: e^zJ_N
Use in source · Paper_IV_Traditional_Final.tex, line 847
D_N,N-1(z) . Then D_N(z)=e^zJ_Ne_0 and D_N(z)=frac1Ntr(e^zJ_N). The eigenvalues of J_N are the N roots of -1, each with multiplicity one.
Exact mathematical passage
\mathbf D_N(z)=e^{zJ_N}e_0
e^{z\tr J_N}Use in source · 1 passage
Search aliases: e^ztr J_N tr operatorname
Use in source · Paper_IV_Traditional_Final.tex, line 936
The first two identities are the exponential group law. Since tr J_N=0, det(e^zJ_N)=e^ztr J_N=1. The first column statement is [source reference].
Exact mathematical passage
\det(e^{zJ_N})=e^{z\tr J_N}=1.
e^{z\xi}Use in source · 4 passages
Search aliases: e^zξ xi
Use in source · Paper_IV_Traditional_Final.tex, line 805
[Finite algebra exponential identities] In A_N^+ one has e^λ zξ = ∑_j=0^N-1 E_N,j(λ z)ξ^j. In A_N^- one has e^zξ = ∑_j=0^N-1 D_N,j(z)ξ^j. Differentiation and multiplication of these identities recover the cyclic derivative systems and the two addition laws.
Exact mathematical passage
e^{z\xi} = \sum_{j=0}^{N-1}\Dfun_{N,j}(z)\xi^j.
e^{z\sum_\ell a_\ell}Use in source · 1 passage
Search aliases: e^z∑_ℓ a_ℓ sum ell ell
Use in source · Paper_IV_Traditional_Final.tex, line 525
igin, the derivative matrix is diagonal with entries 1,λ,…,λ^N-1. For an explicit determinant proof, write each basis function as a finite sum of exponentials with exponents a_ℓ=λω^ℓ. Its derivative matrix factors as W(z)=V diag(e^a_ℓ z) C, V_kℓ=a_ℓ^k, where C is the constant matrix of Fourier weights. Thus det W(z)=e^z∑_ℓ a_ℓdet W(0)=det W(0), because N≥2 implies ∑_ℓω^ℓ=0. The product of the diagonal initial values is the stated power of λ.
Exact mathematical passage
\det W(z)=e^{z\sum_\ell a_\ell}\det W(0)=\det W(0),
e^{|z|}Use in source · 1 passage
Search aliases: e^|z|
Use in source · Paper_IV_Traditional_Final.tex, line 1779
The finite exponential representation gives | D_N(z)|≤ e^|z|, so the type is at most one. Choose a root η_0 and put z=toverlineη_0 with t>0. The term e^η_0z=e^t has strictly larger real exponent than every other term. Hence D_N(toverlineη_0) = frac1Ne^t(1+o(1)). The type is therefore one.
Exact mathematical passage
|\Dfun_N(z)|\leq e^{|z|},
\e^{\beta t}Use in source · 1 passage
Search aliases: e^β t e beta mathrm beta
Use in source · Paper_VI_Traditional_Final.tex, line 4580
pace, and put s(B)=maxReSpec(B). Choose a real β with s(B)<β<Rez. Spectral mapping and the spectral-radius formula give r( e^B)= e^s(B)< e^β, hence norm e^nB≤ M_β e^β n for every integer n≥0, after increasing M_β to cover the finitely many initial values. Write t=n+r, where 0≤ r<1. Then norm e^tB≤ M_βexp(norm B+max(0,-β)) e^β t. Thus e^-zt e^tB is Bochner integrable and tends to zero. Integrating its derivative gives both inverse identities for R=∫_0^∞ e^-zt e^tB d t, so R=(zI-B)^-1. This is the…
Exact mathematical passage
\norm{\e^{tB}}\leq M_\beta\exp(\norm B+\max(0,-\beta))\e^{\beta t}.
e^{\eta z}Use in source · 4 passages
Search aliases: e^η z eta
Use in source · Paper_IV_Traditional_Final.tex, line 269
The second half of this chapter concerns the zeros of D_N. The finite exponential sum D_N(z)=frac1N∑_η^N=-1e^η z has a regular polygon of exponents. The general relation between an exponent polygon and the zero strips of an exponential sum goes back to P olya and Langer. Later developments include the work of Moreno and the treatment in Levin's monograph [source reference]. Bleher and Mallison give a particularly close general result: for convex-polygon exponential sums they locate all sufficien…
Exact mathematical passage
\Dfun_N(z)=\frac1N\sum_{\eta^N=-1}e^{\eta z}
e^{\eta_0z}Use in source · 1 passage
Search aliases: e^η_0z eta
Use in source · Paper_IV_Traditional_Final.tex, line 1782
The finite exponential representation gives | D_N(z)|≤ e^|z|, so the type is at most one. Choose a root η_0 and put z=toverlineη_0 with t>0. The term e^η_0z=e^t has strictly larger real exponent than every other term. Hence D_N(toverlineη_0) = frac1Ne^t(1+o(1)). The type is therefore one.
Exact mathematical passage
e^{\eta_0z}=e^t
e^{\eta_kz}Use in source · 1 passage
Search aliases: e^η_kz eta
Use in source · Paper_IV_Traditional_Final.tex, line 1748
We now assume N≥2. Write η_k=e^(2k+1)π i/N, 0≤ k<N. These are the vertices of a regular polygon and D_N(z)=frac1N∑_k=0^N-1e^η_kz.
Exact mathematical passage
\Dfun_N(z)=\frac1N\sum_{k=0}^{N-1}e^{\eta_kz}.
e^{\lambda w}Use in source · 1 passage
Search aliases: e^λ w lambda
Use in source · Paper_IV_Traditional_Final.tex, line 543
Expand e^λ(z+w)=e^λ ze^λ w as a bivariate power series. A term z^a w^b contributes to the kth section exactly when a+b≡ kbmod N. Grouping first by i≡ abmod N and j≡ bbmod N gives the displayed sum.
Exact mathematical passage
e^{\lambda(z+w)}=e^{\lambda z}e^{\lambda w}
e^{\lambda z\xi}Use in source · 1 passage
Search aliases: e^λ zξ lambda xi
Use in source · Paper_IV_Traditional_Final.tex, line 799
[Finite algebra exponential identities] In A_N^+ one has e^λ zξ = ∑_j=0^N-1 E_N,j(λ z)ξ^j. In A_N^- one has e^zξ = ∑_j=0^N-1 D_N,j(z)ξ^j. Differentiation and multiplication of these identities recover the cyclic derivative systems and the two addition laws.
Exact mathematical passage
e^{\lambda z\xi} = \sum_{j=0}^{N-1}\Efun_{N,j}(\lambda z)\xi^j.
e^{\lambda(z+w)}Use in source · 1 passage
Search aliases: e^λ(z+w) lambda
Use in source · Paper_IV_Traditional_Final.tex, line 543
Expand e^λ(z+w)=e^λ ze^λ w as a bivariate power series. A term z^a w^b contributes to the kth section exactly when a+b≡ kbmod N. Grouping first by i≡ abmod N and j≡ bbmod N gives the displayed sum.
Exact mathematical passage
e^{\lambda(z+w)}=e^{\lambda z}e^{\lambda w}
\e^{\lambda_jt}Use in source · 1 passage
Search aliases: e^λ_jt e lambda mathrm lambda
Use in source · Paper_VI_Traditional_Final.tex, line 5760
[Exponential sum memory] If K(t)=∑_j=1^mB_j e^λ_jtC_j, then K has a finite auxiliary realization with one auxiliary block for each exponential. Repeated poles produce polynomial factors in t and are realized by Jordan blocks.
Exact mathematical passage
\Kcal(t)=\sum_{j=1}^{m}B_j\e^{\lambda_jt}C_j,
e^{\lambda\eta}Use in source · 1 passage
Search aliases: e^λη lambda eta
Use in source · atlas-pages.tex, line 679
vpage36Repeated observations shrink the noise side of the gapSUBGAUSSIAN RECOVERY / A CONSERVATIVE SAMPLE BUDGETChapter V: Theorem E9.8, Corollary E9.9, Corollary E9.11. Suppose all noise observations are independent and centered, with moment-generating bound E e^λη≤ e^σ^2λ^2/2. Averaging R observations at each of n+1 sample sites replaces σ by σ/sqrt R. A sufficient recovery condition is a_min>2Mh+frac4σsqrt Rsqrtlog(2n/δ). The probability of simultaneous recovery is then at least 1-δ, subject …
Exact mathematical passage
\mathbb E e^{\lambda\eta}\le e^{\sigma^2\lambda^2/2}
\e^{\lambda\eta_k}Use in source · 1 passage
Search aliases: e^λη_k e lambda eta mathrm lambda eta
Use in source · Paper_V_Traditional_Final.tex, line 3091
Assume the noise variables are independent, centered, and σ sub-Gaussian: E e^λη_k ≤ e^σ^2λ^2/2.
Exact mathematical passage
\mathbb E\e^{\lambda\eta_k} \leq \e^{\sigma^2\lambda^2/2}.
\e^{\mu_jt}Use in source · 4 passages
Search aliases: e^μ_jt e mu mathrm mu
Use in source · Paper_VI_Traditional_Final.tex, line 4658
xtended:thm:modal_memory Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt ∑_k=0^ν_j-1 fract^kk!B_j,k and Σ_P(z) = ∑_j=1^r ∑_k=0^ν_j-1 fracB_j,k(z-μ_j)^k+1. Suppose Reμ_j<0 for every j. If J⊆{1,…,r} and K_J retains the complete pri…
Exact mathematical passage
\Kcal(t) = \sum_{j=1}^r \e^{\mu_jt} \sum_{k=0}^{\nu_j-1} \frac{t^k}{k!}B_{j,k}
e^{\rho\omega^\ell z}Use in source · 1 passage
Search aliases: e^ρω^ℓ z rho omega ell
Use in source · Paper_IV_Traditional_Final.tex, line 708
[Finite exponential representation] Let R_N={η∈ Cmathrel:η^N=-1}. Then D_N(z) = frac1N∑_η∈ R_Ne^η z. More generally, D_N,j(z) = fracρ^-jN ∑_ℓ=0^N-1 ω^-jℓe^ρω^ℓ z.
Exact mathematical passage
\Dfun_{N,j}(z) = \frac{\rho^{-j}}N \sum_{\ell=0}^{N-1} \omega^{-j\ell}e^{\rho\omega^\ell z}.
\e^{\sigma^2\lambda^2/2}Use in source · 1 passage
Search aliases: e^σ^2λ^2/2 e sigma lambda mathrm sigma lambda
Use in source · Paper_V_Traditional_Final.tex, line 3091
Assume the noise variables are independent, centered, and σ sub-Gaussian: E e^λη_k ≤ e^σ^2λ^2/2.
Exact mathematical passage
\mathbb E\e^{\lambda\eta_k} \leq \e^{\sigma^2\lambda^2/2}.
e^{\sigma^2\lambda^2/2}Use in source · 1 passage
Search aliases: e^σ^2λ^2/2 sigma lambda
Use in source · atlas-pages.tex, line 679
vpage36Repeated observations shrink the noise side of the gapSUBGAUSSIAN RECOVERY / A CONSERVATIVE SAMPLE BUDGETChapter V: Theorem E9.8, Corollary E9.9, Corollary E9.11. Suppose all noise observations are independent and centered, with moment-generating bound E e^λη≤ e^σ^2λ^2/2. Averaging R observations at each of n+1 sample sites replaces σ by σ/sqrt R. A sufficient recovery condition is a_min>2Mh+frac4σsqrt Rsqrtlog(2n/δ). The probability of simultaneous recovery is then at least 1-δ, subject …
Exact mathematical passage
\mathbb E e^{\lambda\eta}\le e^{\sigma^2\lambda^2/2}
\e^{\omega t}Use in source · 2 passages
Search aliases: e^ω t e omega mathrm omega
Use in source · Paper_VI_Traditional_Final.tex, line 2467
[Exact projective limit semigroup] For each ℓ, let T_ℓ(t) be a strongly continuous semigroup on X_ℓ. Assume C_ℓ T_ℓ+1(t)=T_ℓ(t)C_ℓ for every ℓ and t≥0, and assume the uniform bound sup_ℓnormT_ℓ(t)≤ M e^ω t. Define T_∞(t)(x_ℓ)_ℓ = (T_ℓ(t)x_ℓ)_ℓ. Then T_∞(t) is a bounded semigroup on X_∞. It is strongly continuous if lim_tdownarrow0 sup_ℓ normT_ℓ(t)x_ℓ-x_ℓ=0 for every (x_ℓ)_ℓ∈ X_∞.
Exact mathematical passage
\sup_\ell\norm{T_\ell(t)}\leq M\e^{\omega t}.
e^{\omega t}Use in source · 1 passage
Search aliases: e^ω t omega
Use in source · Paper_VI_Traditional_Final.tex, line 5180
section renewcommandthefigureRarabicfigure renewcommandtheHfigureR.arabicfigure renewcommandthetableRarabictable renewcommandtheHtableR.arabictable FloatBarrier Transfer paths and product-topology limits. Let T(t) be an arbitrary strongly continuous semigroup on a complex Banach space X, with generator A, and suppose normT(t)≤ M e^ω t for M≥1 and ω∈ R. Let (P_α)_α∈ I_0 be a finite family of bounded pairwise orthogonal idempotents summing to I, and put T_βα(t)=P_β T(t)P_α. The generator need not …
Exact mathematical passage
\norm{T(t)}\leq M e^{\omega t}
\e^{\omega_c t}Use in source · 1 passage
Search aliases: e^ω_c t e omega mathrm omega
Use in source · Paper_VI_Traditional_Final.tex, line 2298
[Duhamel defect identity] For every t≥0, D(t) = ∫_0^t e^(t-s)A_cΔ e^sA_f d s. If norm e^tA_f≤ M_f e^ω_f t, norm e^tA_c≤ M_c e^ω_c t, then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-s)+ω_fs d s. For contraction semigroups, norm D(t)≤ tnormΔ.
Exact mathematical passage
\norm{\e^{tA_f}}\leq M_f\e^{\omega_f t}, \qquad \norm{\e^{tA_c}}\leq M_c\e^{\omega_c t},
e^{\omega_c t}Use in source · 1 passage
Search aliases: e^ω_c t omega
Use in source · Paper_VI_Traditional_Final.tex, line 487
[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.
Exact mathematical passage
\norm{e^{tA_f}}\le M_f e^{\omega_f t}, \qquad \norm{e^{tA_c}}\le M_c e^{\omega_c t}.
e^{\omega_c(t-r)}Use in source · 1 passage
Search aliases: e^ω_c(t-r) omega
Use in source · Paper_VI_Traditional_Final.tex, line 493
[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.
Exact mathematical passage
\norm{\mathfrak D(t)} \le M_fM_c\norm{\Delta} \int_0^t e^{\omega_c(t-r)}e^{\omega_fr}\,dr.
\e^{\omega_c(t-s)+\omega_fs}Use in source · 1 passage
Search aliases: e^ω_c(t-s)+ω_fs e omega omega mathrm omega omega
Use in source · Paper_VI_Traditional_Final.tex, line 2304
[Duhamel defect identity] For every t≥0, D(t) = ∫_0^t e^(t-s)A_cΔ e^sA_f d s. If norm e^tA_f≤ M_f e^ω_f t, norm e^tA_c≤ M_c e^ω_c t, then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-s)+ω_fs d s. For contraction semigroups, norm D(t)≤ tnormΔ.
Exact mathematical passage
\norm{\mathfrak D(t)} \leq M_fM_c\norm\Delta \int_0^t \e^{\omega_c(t-s)+\omega_fs} \dd s.
\e^{\omega_f t}Use in source · 1 passage
Search aliases: e^ω_f t e omega mathrm omega
Use in source · Paper_VI_Traditional_Final.tex, line 2298
[Duhamel defect identity] For every t≥0, D(t) = ∫_0^t e^(t-s)A_cΔ e^sA_f d s. If norm e^tA_f≤ M_f e^ω_f t, norm e^tA_c≤ M_c e^ω_c t, then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-s)+ω_fs d s. For contraction semigroups, norm D(t)≤ tnormΔ.
Exact mathematical passage
\norm{\e^{tA_f}}\leq M_f\e^{\omega_f t}, \qquad \norm{\e^{tA_c}}\leq M_c\e^{\omega_c t},
e^{\omega_f t}Use in source · 1 passage
Search aliases: e^ω_f t omega
Use in source · Paper_VI_Traditional_Final.tex, line 487
[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.
Exact mathematical passage
\norm{e^{tA_f}}\le M_f e^{\omega_f t}, \qquad \norm{e^{tA_c}}\le M_c e^{\omega_c t}.
e^{\omega_fr}Use in source · 1 passage
Search aliases: e^ω_fr omega
Use in source · Paper_VI_Traditional_Final.tex, line 493
[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.
Exact mathematical passage
\norm{\mathfrak D(t)} \le M_fM_c\norm{\Delta} \int_0^t e^{\omega_c(t-r)}e^{\omega_fr}\,dr.
\e^{\omega_Q(t-s)+\omega s}Use in source · 1 passage
Search aliases: e^ω_Q(t-s)+ω s e omega omega mathrm omega omega
Use in source · Paper_VI_Traditional_Final.tex, line 2755
[Generator control of range leakage] Let T(t)= e^tA on a Banach space X, let P be a bounded projection, and set Q=I-P. Assume normT(t)≤ M e^ω t and norm e^t(QAQ)|_QX ≤ M_Q e^ω_Qt. Then normQT(t)P ≤ M M_Qnorm P^2normQAP ∫_0^t e^ω_Q(t-s)+ω s d s. In particular, for orthogonal projections and contraction bounds, sup_0≤ t≤ T normQT(t)P ≤ TnormQAP.
Exact mathematical passage
\norm{QT(t)P} \leq M M_Q\norm P^2\norm{QAP} \int_0^t \e^{\omega_Q(t-s)+\omega s} \dd s.
\e^{\omega_Qt}Use in source · 1 passage
Search aliases: e^ω_Qt e omega mathrm omega
Use in source · Paper_VI_Traditional_Final.tex, line 2750
[Generator control of range leakage] Let T(t)= e^tA on a Banach space X, let P be a bounded projection, and set Q=I-P. Assume normT(t)≤ M e^ω t and norm e^t(QAQ)|_QX ≤ M_Q e^ω_Qt. Then normQT(t)P ≤ M M_Qnorm P^2normQAP ∫_0^t e^ω_Q(t-s)+ω s d s. In particular, for orthogonal projections and contraction bounds, sup_0≤ t≤ T normQT(t)P ≤ TnormQAP.
Exact mathematical passage
\norm{\e^{t(QAQ)|_{QX}}} \leq M_Q\e^{\omega_Qt}.
e_{\mathfrak n}Use in source · 1 passage
Search aliases: e_ n mathfrak
Use in source · Paper_XII_Traditional_Final.tex, line 1033
[Artin decomposition] Let E' be a field and let T be a finite dimensional commutative E' algebra. Then T ≅ ∏_ m∈Max T T_ m. Each factor is an Artin local algebra. Equivalently, there is a unique complete family of central idempotents (e_ m) such that e_ me_ n=0 ( m≠ n), ∑_ me_ m=1, and e_ m T≅ T_ m [source reference].
Exact mathematical passage
e_{\frakm}e_{\mathfrak n}=0 \quad(\frakm\neq\mathfrak n), \qquad \sum_{\frakm}e_{\frakm}=1,
E_{(p-1)x+j}Use in source · 1 passage
Search aliases: E_(p-1)x+j
Use in source · 72_recovered.tex, line 1858
[Odd prime analytic Euler branch] The series defining mathscr E_p,j^× converges in the Tate algebra on Z_p. For every nonnegative integer x, mathscr E_p,j^×(x) = (1-σ_pp^(p-1)x+j) E_(p-1)x+j.
Exact mathematical passage
\boxed{ \mathscr E_{p,j}^\times(x) = \left(1-\sigma_pp^{(p-1)x+j}\right) E_{(p-1)x+j}. }
E_+Use in source · 15 passages
Search aliases: E_+
Use in source · 04_arithmetic.tex, line 563
0 · -1 act on Z_p^2. Over Q_p, the sign projectors are E_+=fracI+A2 = 1 · 1/2 0 · 0 , E_-=fracI-A2 = 0 · -1/2
Exact mathematical passage
E_+=\frac{I+A}{2} = \begin{pmatrix} 1&1/2\\ 0&0 \end{pmatrix},
e_+Use in source · 1 passage
Search aliases: e_+
Use in source · Paper_XII_Traditional_Final.tex, line 1581
[Failure of integral sign projectors] Let C_2=⟨ g⟩. Over Q_2, the two sign projectors are e_+=frac1+g2, e_-=frac1-g2. Neither belongs to Z_2[C_2]. After reduction modulo two, F_2[C_2] ≅ F_2[X]/(X^2), X=g-1. This ring is local and has no nontrivial idempotents.
Exact mathematical passage
\boxed{ e_+=\frac{1+g}{2}, \qquad e_-=\frac{1-g}{2}. }
E_-Use in source · 7 passages
Search aliases: E_-
Use in source · 04_arithmetic.tex, line 571
0 · 0 , E_-=fracI-A2 = 0 · -1/2 0 · 1 . For every odd prime p, both projectors preserve Z_p^2. At p=2, s_2(E_+)=s_2(E_-)=1. The factors x-1 and x+1 have resultant 2.
Exact mathematical passage
E_-=\frac{I-A}{2} = \begin{pmatrix} 0&-1/2\\ 0&1 \end{pmatrix}.
e_-Use in source · 1 passage
Search aliases: e_-
Use in source · Paper_XII_Traditional_Final.tex, line 1581
[Failure of integral sign projectors] Let C_2=⟨ g⟩. Over Q_2, the two sign projectors are e_+=frac1+g2, e_-=frac1-g2. Neither belongs to Z_2[C_2]. After reduction modulo two, F_2[C_2] ≅ F_2[X]/(X^2), X=g-1. This ring is local and has no nontrivial idempotents.
Exact mathematical passage
\boxed{ e_+=\frac{1+g}{2}, \qquad e_-=\frac{1-g}{2}. }
E_0Use in source · 29 passages
Search aliases: E_0
Use in source · 20_arithmetic.tex, line 869
Return now to the classical signed Euler numbers secht=frac1cosh t =∑_n≥0E_2nfract^2n(2n)!. Thus E_2n=E_2n^(2,0) in the convention [source reference]. In particular E_0=1, E_2=-1, and E_4=5. The following construction concerns these N=2 coefficients, not C_4,0.
Exact mathematical passage
E_0=1
E_{00}Use in source · 1 passage
Search aliases: E_00
Use in source · visual_atlas.tex, line 210
The words are written matrix products: in ab, the map b acts first. Therefore ab=8E_00, ba=8E_11, aba=16E_01. The two products ab and ba cannot be identified merely because they have equal weight. Both their order and their source/target information matter.
Exact mathematical passage
ab=8E_{00},\qquad ba=8E_{11},\qquad aba=16E_{01}.
E_{01}Use in source · 11 passages
Search aliases: E_01
Use in source · visual_atlas.tex, line 201
vpage9Which words can survive precision five?MODEL B / ORDERED TRANSFERS IN A FILTERED RINGChapter II, mathematical text: Theorem 5.10, Corollary 5.11 Let E_ij send the jth basis vector to the ith. In integral 2×2 matrices set a=2E_01, b=4E_10, A=a+b. Their certified dyadic weights are 1 and 2. At precision L=5, every word of weight at least five vanishes modulo 32.
Exact mathematical passage
a=2E_{01},\qquad b=4E_{10},\qquad A=a+b.
e_{01}Use in source · 1 passage
Search aliases: e_01
Use in source · Paper_X_Block27_Amended.tex, line 2216
For the continuous Z_2 carrier ρ(s)=diag(1,-1,1), conjugate the three coordinate projectors by 1+2e_01 and reduce modulo 4. They form a complete equivariant family. The general theorem constructs three nonzero integral summands retaining this family modulo 2. No exact equivariant family retains the whole prescribed family modulo 4. Separate checked examples show that independent equivariant idempotent lifts can lose orthogonality, and that distinct complete families with the same reduction can b…
Exact mathematical passage
1+2e_{01}
e_0^{(m,p)}Use in source · 1 passage
Search aliases: e_0^(m,p)
Use in source · 72_recovered.tex, line 374
[Compressed inverse recurrence] The compressed coefficients satisfy ∑_q=0^n binommnmq e_q^(m,p) = δ_n0. In particular, e_0^(m,p)=1, and the positive coefficients are determined recursively because the diagonal is one.
Exact mathematical passage
e_0^{(m,p)}=1
E_0^*Use in source · 1 passage
Search aliases: E_0^*
Use in source · 03_supplement_05.tex, line 211
A genuine channel can still be uncorrectable. For full qubit dephasing, E_0=diag(1,0) and E_1=diag(0,1), the Kraus normalization is checked. But E_0^*E_0 cannot be a scalar multiple of I, as its two diagonal entries differ. Necessity therefore proves that no finite recovery channel corrects the full code space. The case of the zero code is also checked independently.
Exact mathematical passage
E_0^*E_0
E_0^{p,q}Use in source · 1 passage
Search aliases: E_0^p,q
Use in source · 04_correction_transport.tex, line 435
[Spectral sequence of a finite filtered complex] Let C^bullet be a cochain complex with a finite decreasing filtration by subcomplexes C^bullet=F^0C^bullet⊇ F^1C^bullet⊇⋯⊇ F^N+1C^bullet=0. There is a natural spectral sequence with E_0^p,q=gr^pC^p+q, E_1^p,q=H^p+q(gr^pC^bullet), which converges to the associated graded object of H^p+q(C^bullet) with its induced filtration. A filtered cochain map induces a morphism of spectral sequences.
Exact mathematical passage
E_0^{p,q}=\gr^pC^{p+q}, \qquad E_1^{p,q}=H^{p+q}(\gr^pC^\bullet),
E_1Use in source · 8 passages
Search aliases: E_1
Use in source · atlas-pages.tex, line 171
vpage8Where the variance goesMARTINGALE PACKETS / THREE EXACT PIECESChapter VI, Part I: Theorem 6.2. Nested weighted projections and their differences. Keep the weights and observable from pagerefV7. Let E_0f be the constant global mean, let E_1=E average within A and B, and let E_2=I retain every value. Define D_0=E_0, D_1=E_1-E_0, D_2=I-E_1. Then D_0f is the mean, D_1f describes the contrast between block means, and D_2f is the fluctuation left inside each block. vfigvariance platecaptionThe s…
Exact mathematical passage
E_1=E
E_{10}Use in source · 3 passages
Search aliases: E_10
Use in source · visual_atlas.tex, line 201
vpage9Which words can survive precision five?MODEL B / ORDERED TRANSFERS IN A FILTERED RINGChapter II, mathematical text: Theorem 5.10, Corollary 5.11 Let E_ij send the jth basis vector to the ith. In integral 2×2 matrices set a=2E_01, b=4E_10, A=a+b. Their certified dyadic weights are 1 and 2. At precision L=5, every word of weight at least five vanishes modulo 32.
Exact mathematical passage
a=2E_{01},\qquad b=4E_{10},\qquad A=a+b.
E_{11}Use in source · 1 passage
Search aliases: E_11
Use in source · visual_atlas.tex, line 210
The words are written matrix products: in ab, the map b acts first. Therefore ab=8E_00, ba=8E_11, aba=16E_01. The two products ab and ba cannot be identified merely because they have equal weight. Both their order and their source/target information matter.
Exact mathematical passage
ab=8E_{00},\qquad ba=8E_{11},\qquad aba=16E_{01}.
E_{12}Use in source · 3 passages
Search aliases: E_12
Use in source · visual_atlas.tex, line 352
vpage16Resolve the entire obstructionMODEL F / PACKET BLOCKS FROM INPUT TO OUTPUTChapter III, mathematical text: Theorem 7.5, Corollary 7.6 Use three coordinate packets with projectors P_0,P_1,P_2, and set H=I+2E_01+3E_12, Ob=H-I. Here H is invertible, so it can be the loop product of a reversible connection on three charts. Its entries outside the diagonal encode two effects that mix packets.
Exact mathematical passage
H=I+2E_{01}+3E_{12},\qquad \Ob=H-I.
E_1^{p,n-p}Use in source · 2 passages
Search aliases: E_1^p,n-p
Use in source · 04_correction_transport.tex, line 450
[Fine obstruction survival] A class on the first page contributes to the total obstruction only if it is represented by a permanent cycle, so it is neither killed nor identified as a boundary on any later page. If the spectral sequence degenerates at the first page, then H^n(C^bullet)=0 exactly when E_1^p,n-p=0 for every p.
Exact mathematical passage
E_1^{p,n-p}=0
E_1^{p,q}Use in source · 1 passage
Search aliases: E_1^p,q
Use in source · 04_correction_transport.tex, line 435
[Spectral sequence of a finite filtered complex] Let C^bullet be a cochain complex with a finite decreasing filtration by subcomplexes C^bullet=F^0C^bullet⊇ F^1C^bullet⊇⋯⊇ F^N+1C^bullet=0. There is a natural spectral sequence with E_0^p,q=gr^pC^p+q, E_1^p,q=H^p+q(gr^pC^bullet), which converges to the associated graded object of H^p+q(C^bullet) with its induced filtration. A filtered cochain map induces a morphism of spectral sequences.
Exact mathematical passage
E_0^{p,q}=\gr^pC^{p+q}, \qquad E_1^{p,q}=H^{p+q}(\gr^pC^\bullet),
E_2Use in source · 12 passages
Search aliases: E_2
Use in source · 04_correction_transport.tex, line 982
Taylor's theorem in the Banach space of bounded operators gives P_α(t+h) = P_α(t)+hdot P_α(t)+E_α(t,h), E_α(t,h) ≤ C_1h^2, uniformly in t and α. Hence S( P(t+h), P(t)) = I+hK(t)+E(t,h), E(t,h) ≤ C_2h^2. Since K(t)^*=-K(t), S^*S=I+E_2(t,h), E_2(t,h) ≤ C_3h^2. For small h, the spectrum of S^*S lies in a fixed compact subinterval of (0,∞). The functional calculus for z^-1/2 gives (S^*S)^-1/2=I+E_3(t,h), E_3(t,h) ≤ C_4h^2. Multiplying the last two expansions proves the estimate.
Exact mathematical passage
S^*S=I+E_2(t,h), \qquad \|E_2(t,h)\|\le C_3h^2.
e_2Use in source · 28 passages
Search aliases: e_2
Use in source · visual_atlas.tex, line 19
[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…
Exact mathematical passage
4pt] \textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt] \textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology? \end{tabularx} A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. \takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''} \vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}} Most of the chapter can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix. {\small \begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule \textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule $X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\ $S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\ $\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\ $\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\ $P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\ $X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\ $T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\ $F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\ $U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\ $\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\ $\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule \end{tabularx}} \head{Read products from right to left} In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column. \head{Where to begin} For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}. \vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$} Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define \[ Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0, \qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.
E_{20}Use in source · 1 passage
Search aliases: E_20
Use in source · visual_atlas.tex, line 55
On a vector (x_0,x_1,x_2)^ T, the first step produces (0,2x_0,0)^ T, and the second produces (0,0,6x_0)^ T. Thus L_fL_e=6E_20, L_eL_f=0. For general edges, a summand in the entrywise product can survive only when the same intermediate packet is both tgt(e) and src(f): (L_fL_e)_yx=∑_z(L_f)_yz(L_e)_zx. This is why a broken interface annihilates the entire product, even if both edge coefficients are nonzero.
Exact mathematical passage
L_fL_e=6E_{20},\qquad L_eL_f=0.
E_{21}Use in source · 4 passages
Search aliases: E_21
Use in source · visual_atlas.tex, line 255
The first pattern contains the diagonal and permits 1→2 and 2→3, but forbids 1→3. Yet E_32E_21=E_31≠0. Both factors obey the pattern and their product does not. Thus the whole supported matrix space is not an algebra.
Exact mathematical passage
E_{32}E_{21}=E_{31}\ne0.
E_{2M+2}Use in source · 1 passage
Search aliases: E_2M+2
Use in source · 20_arithmetic.tex, line 988
[Optimal uniform dyadic tower] For all M,n∈ N, E_2n-T_M(n)∈2^4M+3 Z_2. At the first omitted row n=M+1, the error is nonzero and v_2(E_2M+2-T_M(M+1))=4M+3. Thus the exponent 4M+3 cannot be increased uniformly in the Euler row.
Exact mathematical passage
v_2\bigl(E_{2M+2}-T_M(M+1)\bigr)=4M+3.
E_{2n}Use in source · 15 passages
Search aliases: E_2n
Use in source · 20_arithmetic.tex, line 749
The classical signed Euler numbers have generating function secht=∑_n≥0E_2nt^2n/(2n)!. They are the N=2, j=0 member of [source reference]. The coordinate theorem reconstructs them from finite alternating sums of odd powers.
Exact mathematical passage
\operatorname{sech}t=\sum_{n\ge0}E_{2n}t^{2n}/(2n)!
\mathcal E_{2N}Use in source · 2 passages
Search aliases: E_2N mathcal
Use in source · root_scope.tex, line 56
The classical dyadic Bernoulli denominator law v_2(B_2m)=-1 and Legendre's formula yield v_2(β_m)=s_2(m)-1, v_2(κ_m)=4m-1. The source identity ∑_N≥0A(N)fract^2N(2N)! =fracsinh(2t)sinh t2t^2 implies, after multiplication by 4t^2/sinh^2(2t), E_2N=∑_m=0^N κ_mbinom2N2mA(N-m). For a retained depth M≥0, put T_M(N)= E_2N -∑_m=0^min(M,N)κ_mbinom2N2mA(N-m). An empty tail, N≤ M, is exactly zero.
Exact mathematical passage
\mathcal E_{2N}=\sum_{m=0}^N \kappa_m\binom{2N}{2m}A(N-m). \label{eq:root:packet}
E_{2n}^{(2,0)}Use in source · 1 passage
Search aliases: E_2n^(2,0)
Use in source · 20_arithmetic.tex, line 868
Return now to the classical signed Euler numbers secht=frac1cosh t =∑_n≥0E_2nfract^2n(2n)!. Thus E_2n=E_2n^(2,0) in the convention [source reference]. In particular E_0=1, E_2=-1, and E_4=5. The following construction concerns these N=2 coefficients, not C_4,0.
Exact mathematical passage
E_{2n}=E_{2n}^{(2,0)}
E_{2r}Use in source · 8 passages
Search aliases: E_2r
Use in source · supplement_02.tex, line 78
The source defines E_2r as the integer numerator of that rational coefficient and proves that its rational cast equals C_2,0(r). Set E_2r+1=0. The actual exponential-section identity from the first block proves ∑_n≥0E_nfract^nn!=sech t. Thus the integer sequence is connected to the intended signed Euler family, rather than merely assigned the name Euler.
Exact mathematical passage
E_{2r}
E_{2r+1}Use in source · 1 passage
Search aliases: E_2r+1
Use in source · supplement_02.tex, line 78
The source defines E_2r as the integer numerator of that rational coefficient and proves that its rational cast equals C_2,0(r). Set E_2r+1=0. The actual exponential-section identity from the first block proves ∑_n≥0E_nfract^nn!=sech t. Thus the integer sequence is connected to the intended signed Euler family, rather than merely assigned the name Euler.
Exact mathematical passage
E_{2r+1}=0
E_3Use in source · 2 passages
Search aliases: E_3
Use in source · 04_correction_transport.tex, line 988
P_α(t)+E_α(t,h), E_α(t,h) ≤ C_1h^2, uniformly in t and α. Hence S( P(t+h), P(t)) = I+hK(t)+E(t,h), E(t,h) ≤ C_2h^2. Since K(t)^*=-K(t), S^*S=I+E_2(t,h), E_2(t,h) ≤ C_3h^2. For small h, the spectrum of S^*S lies in a fixed compact subinterval of (0,∞). The functional calculus for z^-1/2 gives (S^*S)^-1/2=I+E_3(t,h), E_3(t,h) ≤ C_4h^2. Multiplying the last two expansions proves the estimate.
Exact mathematical passage
(S^*S)^{-1/2}=I+E_3(t,h), \qquad \|E_3(t,h)\|\le C_4h^2.
e_3Use in source · 14 passages
Search aliases: e_3
Use in source · 05_worked_models.tex, line 43
[Typed reachability certificate] In the preceding quiver realization, no seed supported at U can generate the full representation, whereas the two seeds e_1∈ U, e_3∈ V do generate it.
Exact mathematical passage
e_1\in U, \qquad e_3\in V
E_{3,1}Use in source · 1 passage
Search aliases: E_3,1
Use in source · atlas-pages.tex, line 130
because the hypergeometric convention already supplies (1)_n=n!. vfighypergeometric headCheck the displayed case For N=3,j=1,n=2, the six factors after the initial 1! are 2,3,4,5,6,7. Dividing by three and grouping them gives the three rows of the plate. The middle row supplies 2!. The others supply (2/3)_2 and (4/3)_2. The right-hand side is exactly 5040. Consequently, E_3,1(z)=z(1+fracz^34!+fracz^67!+⋯). The prefactor z retains the forced zero at the origin. The hypergeometric argument z^3/27 …
Exact mathematical passage
E_{3,1}(z)=z\left(1+\frac{z^3}{4!}+\frac{z^6}{7!}+\cdots\right).
E_{3,j}Use in source · 1 passage
Search aliases: E_3,j
Use in source · atlas-pages.tex, line 111
Multiply the coordinate forms of e^zξ and e^wξ. Their product is e^(z+w)ξ. The coefficient of 1 receives one contribution without a wrap and two contributions with a wrap. Reading that coefficient gives the displayed addition law. headCompare the unsigned model In C[ξ]/(ξ^3-1), the same table has no minus signs. Its exponential coordinates are the unsigned sections E_3,j. Thus the difference between the two differential systems is a single algebraic relation, not a change in how coefficients are…
Exact mathematical passage
E_{3,j}
E_{31}Use in source · 1 passage
Search aliases: E_31
Use in source · visual_atlas.tex, line 255
The first pattern contains the diagonal and permits 1→2 and 2→3, but forbids 1→3. Yet E_32E_21=E_31≠0. Both factors obey the pattern and their product does not. Thus the whole supported matrix space is not an algebra.
Exact mathematical passage
E_{32}E_{21}=E_{31}\ne0.
E_{32}Use in source · 1 passage
Search aliases: E_32
Use in source · visual_atlas.tex, line 255
The first pattern contains the diagonal and permits 1→2 and 2→3, but forbids 1→3. Yet E_32E_21=E_31≠0. Both factors obey the pattern and their product does not. Thus the whole supported matrix space is not an algebra.
Exact mathematical passage
E_{32}E_{21}=E_{31}\ne0.
E_4Use in source · 3 passages
Search aliases: E_4
Use in source · 20_arithmetic.tex, line 869
Return now to the classical signed Euler numbers secht=frac1cosh t =∑_n≥0E_2nfract^2n(2n)!. Thus E_2n=E_2n^(2,0) in the convention [source reference]. In particular E_0=1, E_2=-1, and E_4=5. The following construction concerns these N=2 coefficients, not C_4,0.
Exact mathematical passage
E_4=5
e_4Use in source · 5 passages
Search aliases: e_4
Use in source · visual_atlas.tex, line 221
vpage9Two visible routes can cancel exactlyREACHABILITY / NECESSARY IS NOT SUFFICIENTMain text: [source reference] Let a single operator D on Q^4 act by De_1=e_2+e_3, De_2=e_4, De_3=-e_4, De_4=0. Every vertex of its transfer graph for the coordinate packets is reachable from vertex 1. Nevertheless, D^2e_1=e_4-e_4=0, D^2=0.
Exact mathematical passage
De_1=e_2+e_3,\qquad De_2=e_4,\qquad De_3=-e_4,\qquad De_4=0.
E_{4(n+H)}^{(4,j)}Use in source · 1 passage
Search aliases: E_4(n+H)^(4,j)
Use in source · strengthening.tex, line 1013
[Exact four-address first differences] For every n≥0, H>0, and j∈{0,1,2,3}, v_2(C_j(n+H)-C_j(n)) · =v_2(H)+b_j, · (b_0,b_1,b_2,b_3) · =(1,1,4,2), v_2(E_4(n+H)^(4,j)-E_4n^(4,j)) · =v_2(H)+B_j, · (B_0,B_1,B_2,B_3) · =(1,1,5,3). Consequently, the least positive row period modulo 2^r is exactly P_j(r)=2^max(0,r-B_j) (r≥1). The corresponding displacement in the original Euler subscript is 4P_j(r).
Exact mathematical passage
v_2(C_j(n+H)-C_j(n))&=v_2(H)+b_j, &(b_0,b_1,b_2,b_3)&=(1,1,4,2),\label{eq:rank-four-normalized-isometry}\\ v_2(E_{4(n+H)}^{(4,j)}-E_{4n}^{(4,j)})&=v_2(H)+B_j, &(B_0,B_1,B_2,B_3)&=(1,1,5,3). \label{eq:rank-four-original-isometry}
E_{4(n+iH)}^{(4,j)}Use in source · 2 passages
Search aliases: E_4(n+iH)^(4,j)
Use in source · edition_summary.tex, line 11
The strongest common-step conclusion in width four is compact. For n≥0 and k,H≥1, ∑_i=0^k(-1)^k-ibinom ki E_4(n+iH)^(4,j) ≡0pmod2^ v_2(k!)+kv_2(H)+A_j(k), where A_0(k)=A_1(k)=k, A_2(k)=max{k+4, 2k+lfloor k/2rfloor+1}, A_3(k)=k+2. There is no starting-row threshold. The exponent is exact in addresses zero and one for every order and positive step. It is exact in address two at every power-of-two order. The proof chapters also give mixed-step forms, the cancellation criterion in addresses zero and…
Exact mathematical passage
\sum_{i=0}^k(-1)^{k-i}\binom ki E_{4(n+iH)}^{(4,j)} \equiv0\pmod{2^{\,v_2(k!)+kv_2(H)+A_j(k)}},
E_{4n}^{(4,j)}Use in source · 1 passage
Search aliases: E_4n^(4,j)
Use in source · strengthening.tex, line 1013
[Exact four-address first differences] For every n≥0, H>0, and j∈{0,1,2,3}, v_2(C_j(n+H)-C_j(n)) · =v_2(H)+b_j, · (b_0,b_1,b_2,b_3) · =(1,1,4,2), v_2(E_4(n+H)^(4,j)-E_4n^(4,j)) · =v_2(H)+B_j, · (B_0,B_1,B_2,B_3) · =(1,1,5,3). Consequently, the least positive row period modulo 2^r is exactly P_j(r)=2^max(0,r-B_j) (r≥1). The corresponding displacement in the original Euler subscript is 4P_j(r).
Exact mathematical passage
v_2(C_j(n+H)-C_j(n))&=v_2(H)+b_j, &(b_0,b_1,b_2,b_3)&=(1,1,4,2),\label{eq:rank-four-normalized-isometry}\\ v_2(E_{4(n+H)}^{(4,j)}-E_{4n}^{(4,j)})&=v_2(H)+B_j, &(B_0,B_1,B_2,B_3)&=(1,1,5,3). \label{eq:rank-four-original-isometry}
E_6Use in source · 1 passage
Search aliases: E_6
Use in source · supplement_02.tex, line 174
The displayed manuscript row is kernel-checked. The scalar anchors are derived from the same general coefficient recurrence: (E_0,E_2,E_4,E_6,E_8)=(1,-1,5,-61,1385), (b_0,b_1,b_2,b_3,b_4)=(1,-2,16,-272,7936). The literal source sums at p=5 give (S_0,S_2,S_4,S_6,S_8)=(1,49,4705,428689,37665985). Consequently, T_2(5,8)=2 731 001 385, E_8-T_2(5,8)=-2 731 000 000. The source proves both 5^6mid E_8-T_2(5,8), 5^7nmid E_8-T_2(5,8). Thus the guaranteed power is attained in this example. No stronger unif…
Exact mathematical passage
(E_0,E_2,E_4,E_6,E_8)=(1,-1,5,-61,1385),
E_8Use in source · 7 passages
Search aliases: E_8
Use in source · supplement_02.tex, line 174
The displayed manuscript row is kernel-checked. The scalar anchors are derived from the same general coefficient recurrence: (E_0,E_2,E_4,E_6,E_8)=(1,-1,5,-61,1385), (b_0,b_1,b_2,b_3,b_4)=(1,-2,16,-272,7936). The literal source sums at p=5 give (S_0,S_2,S_4,S_6,S_8)=(1,49,4705,428689,37665985). Consequently, T_2(5,8)=2 731 001 385, E_8-T_2(5,8)=-2 731 000 000. The source proves both 5^6mid E_8-T_2(5,8), 5^7nmid E_8-T_2(5,8). Thus the guaranteed power is attained in this example. No stronger unif…
Exact mathematical passage
(E_0,E_2,E_4,E_6,E_8)=(1,-1,5,-61,1385),
E_AUse in source · 3 passages
Search aliases: E_A
Use in source · 02_supplement_01.tex, line 10
0 · b≠ a, ∑_aP_a=1. Write A_ba=P_bAP_a and let E_A={(b,a):A_ba≠0}. These are exact algebraic zeros, not values discarded by a floating-point threshold.
Exact mathematical passage
E_A=\{(b,a):A_{ba}\ne0\}
E_{ab}Use in source · 1 passage
Search aliases: E_ab
Use in source · monograph_atlas.tex, line 70
For the exact calculation, apply the four matrix units E_ab to each seed matrix, flatten the outputs, and compute their column rank. The observed dimensions are two and four, respectively. The calculation directly exhibits the independent rows counted by the theorem. takeawaySchmidt rank measures the remote multiplicity directions accessible to the local operator span. It is not permission for unrestricted deterministic local state preparation.
Exact mathematical passage
E_{ab}
E_b^*Use in source · 1 passage
Search aliases: E_b^*
Use in source · 03_supplement_05.tex, line 169
Cross terms for the original errors. For arbitrary positive Gram data, the same recovery satisfies R(E_aρ E_b^*)=C_baρ (Pρ P=ρ). The index order ba is intentional: the defining Gram relation is PE_a^*E_bP=C_abP. The proof first derives scalar corrected Kraus composites for the constructed recovery and then uses its trace-preserving completeness to identify the original Gram entries.
Exact mathematical passage
\boxed{\mathcal R(E_a\rho E_b^*)=C_{ba}\rho\qquad(P\rho P=\rho).}
E_cUse in source · 5 passages
Search aliases: E_c
Use in source · 20_arithmetic.tex, line 305
pnmid c, differentiation of the section identities gives tR_c'=R_c(1+R_c-1-R_c). Writing q=R_c-1-(c-1) and y=R_c-c, this becomes (tD+c)y=cq+yq-y^2. On degrees divisible by p, the inverse of tD+c divides the degree-d coefficient by the unit d+c. Starting with y_0=0, form y_s+1=(tD+c)^-1(cq+y_sq-y_s^2). Flatness and support are preserved. Indeed, put E_c=tD+c and C_p=pD^p^2-D^p. The commutator is [B_p,E_c]=C_p, and iteration gives B_p^rE_cy=E_cB_p^ry+rC_pB_p^r-1y. For the inverse equation E_cy=f, …
Exact mathematical passage
E_c=tD+c
E_eUse in source · 6 passages
Search aliases: E_e
Use in source · pages.tex, line 333
For orthogonal coordinate blocks, the squared Frobenius norms add exactly. Operator norms instead require a suitable bound or further spectral information. The maximum omitted block norm is generally insufficient. headInput sensitivity is additional information The packet perturbation estimate is (widehat A-A)x ≤∑_e E_e P_α(e)x . It can be much tighter for an input that scarcely occupies an affected source packet. To obtain a uniform operator bound, the input dependence must be bounded uniformly…
Exact mathematical passage
\|(\widehat A-A)x\|\le\sum_e\|E_e\|\,\|P_{\alpha(e)}x\|.
E_FUse in source · 4 passages
Search aliases: E_F
Use in source · 30_lifting.tex, line 798
For the error estimate, let δ=b-a and D=M-s. Taylor's theorem for the preconditioned polynomial gives E_F=(AF)(b)-(AF)(a)-p^sδ∈ p^2DR^d. Polynomial congruence gives E_H=(AH)(b)-(AH)(a)∈ p^DR^d. The equations F(a)=0 and G(b)=0 imply p^sδ+p^MAH(a)=-E_F-p^ME_H∈ p^2DR^d, because M+D≥2D. Cancel p^s in the integral domain R to obtain δ+p^M-sAH(a)∈ p^2M-3sR^d. This integral vector is ε. After inclusion into Q_p, p^M-sA=p^MJ^-1 gives [source reference].
Exact mathematical passage
E_F=(AF)(b)-(AF)(a)-p^s\delta\in p^{2D}R^d.
E_fUse in source · 7 passages
Search aliases: E_f
Use in source · 04_arithmetic.tex, line 19
[Bézout spectral projector] Assume that χ_A(x)=f(x)g(x) with monic polynomials f,g∈ O[x] that are coprime in K[x]. Choose u,v∈ O[x] and a nonzero r∈ O such that u(x)f(x)+v(x)g(x)=r. Then E_f=r^-1v(A)g(A) is the projector of V onto Ker f(A) along Ker g(A). In particular, E_f^2=E_f, rE_f∈End_ O(Λ).
Exact mathematical passage
E_f=r^{-1}v(A)g(A)
E_f^2Use in source · 1 passage
Search aliases: E_f^2
Use in source · 04_arithmetic.tex, line 23
[Bézout spectral projector] Assume that χ_A(x)=f(x)g(x) with monic polynomials f,g∈ O[x] that are coprime in K[x]. Choose u,v∈ O[x] and a nonzero r∈ O such that u(x)f(x)+v(x)g(x)=r. Then E_f=r^-1v(A)g(A) is the projector of V onto Ker f(A) along Ker g(A). In particular, E_f^2=E_f, rE_f∈End_ O(Λ).
Exact mathematical passage
E_f^2=E_f, \qquad rE_f\in\End_{\OO}(\Lambda).
e_{\frakm}Use in source · 10 passages
Search aliases: e_frakm e_ m frakm mathfrak
Use in source · Paper_XII_Traditional_Final.tex, line 1032
[Artin decomposition] Let E' be a field and let T be a finite dimensional commutative E' algebra. Then T ≅ ∏_ m∈Max T T_ m. Each factor is an Artin local algebra. Equivalently, there is a unique complete family of central idempotents (e_ m) such that e_ me_ n=0 ( m≠ n), ∑_ me_ m=1, and e_ m T≅ T_ m [source reference].
Exact mathematical passage
(e_{\frakm})
E_HUse in source · 6 passages
Search aliases: E_H
Use in source · 30_lifting.tex, line 802
For the error estimate, let δ=b-a and D=M-s. Taylor's theorem for the preconditioned polynomial gives E_F=(AF)(b)-(AF)(a)-p^sδ∈ p^2DR^d. Polynomial congruence gives E_H=(AH)(b)-(AH)(a)∈ p^DR^d. The equations F(a)=0 and G(b)=0 imply p^sδ+p^MAH(a)=-E_F-p^ME_H∈ p^2DR^d, because M+D≥2D. Cancel p^s in the integral domain R to obtain δ+p^M-sAH(a)∈ p^2M-3sR^d. This integral vector is ε. After inclusion into Q_p, p^M-sA=p^MJ^-1 gives [source reference].
Exact mathematical passage
E_H=(AH)(b)-(AH)(a)\in p^DR^d
\bm E_{\mathrm{H}}Use in source · 2 passages
Search aliases: E_H bm mathrm
Use in source · Paper_V_Traditional_Final.tex, line 525
The Haar analysis matrix is E_H = frac1sqrt2 1 · 1 1 · -1 . It is orthogonal, so R_H= E_H^ T and R_H E_H=I_2. The channels are a[n]=fracx[2n]+x[2n+1]sqrt2, d[n]=fracx[2n]-x[2n+1]sqrt2.
Exact mathematical passage
\bm E_{\mathrm{H}} = \frac1{\sqrt2} \begin{pmatrix} 1&1\\ 1&-1 \end{pmatrix}.
\bm E_{\mathrm{H}}^{\mathsf T}Use in source · 1 passage
Search aliases: E_H^ T bm mathrm mathsf
Use in source · Paper_V_Traditional_Final.tex, line 534
1 · -1 . It is orthogonal, so R_H= E_H^ T and R_H E_H=I_2. The channels are a[n]=fracx[2n]+x[2n+1]sqrt2, d[n]=fracx[2n]-x[2n+1]sqrt2.
Exact mathematical passage
\bm R_{\mathrm{H}}=\bm E_{\mathrm{H}}^{\mathsf T}
E_iUse in source · 20 passages
Search aliases: E_i
Use in source · 04_arithmetic.tex, line 63
[Several coprime spectral packets] Assume χ_A(x)=∏_i=1^rf_i(x), where the monic factors f_i∈ O[x] are pairwise coprime in K[x]. Put g_i=∏_j≠ if_j. There are complementary projectors E_i with im E_i=Ker f_i(A), ∑_iE_i=I, E_iE_j=0 (i≠ j). One may choose E_i so that Res(f_i,g_i)E_i∈End_ O(Λ) up to multiplication of the resultant by a unit.
Exact mathematical passage
E_i
E_i^2Use in source · 1 passage
Search aliases: E_i^2
Use in source · Paper_X_Block27_Amended.tex, line 2168
Keep the complete domain, supplied finite-group carrier and group-order hypotheses of the hyperref[book:x-construction-18]supplied-carrier summand-lifting construction. Let J be any finite label set and k>0. Suppose (bar E_i)_i∈ J is a complete orthogonal family of equivariant idempotents modulo (π)^e+k. There exists an integral equivariant family (E_i)_i∈ J satisfying E_i^2=E_i, E_iE_j=0 (i≠ j), ∑_i∈ JE_i=1, E_ibmod(π)^k=bar E_ibmod(π)^k. The loss is at most e, independent of |J| and the ambien…
Exact mathematical passage
E_i^2=E_i,\qquad E_iE_j=0\ (i\ne j),\qquad \sum_{i\in J}E_i=1,\qquad E_i\bmod(\pi)^k=\bar E_i\bmod(\pi)^k.
E_{ii}^{\lambda}Use in source · 1 passage
Search aliases: E_ii^λ lambda
Use in source · 02_strengthened.tex, line 259
x an irreducible unitary representation π_λ:G→ U(d_λ) and an orthonormal basis of its representation space. Define E_ij^λ = fracd_λabsG ∑_g∈ G [π_λ(g^-1)]_jiU(g). Then E_ij^λE_kℓ^μ = δ_λμδ_jkE_iℓ^λ, (E_ij^λ)^*=E_ji^λ, and P_λ=∑_i=1^d_λE_ii^λ.
Exact mathematical passage
P_\lambda=\sum_{i=1}^{d_\lambda}E_{ii}^{\lambda}.
E_{ij}^{\lambda}Use in source · 2 passages
Search aliases: E_ij^λ lambda
Use in source · 02_strengthened.tex, line 243
[Representation matrix units] Fix an irreducible unitary representation π_λ:G→ U(d_λ) and an orthonormal basis of its representation space. Define E_ij^λ = fracd_λabsG ∑_g∈ G [π_λ(g^-1)]_jiU(g). Then E_ij^λE_kℓ^μ = δ_λμδ_jkE_iℓ^λ, (E_ij^λ)^*=E_ji^λ, and P_λ=∑_i=1^d_λE_ii^λ.
Exact mathematical passage
E_{ij}^{\lambda} = \frac{d_\lambda}{\abs{G}} \sum_{g\in G} \bigl[\pi_\lambda(g^{-1})\bigr]_{ji}U(g).
E_{i\ell}^{\lambda}Use in source · 1 passage
Search aliases: E_iℓ^λ ell lambda
Use in source · 02_strengthened.tex, line 251
[Representation matrix units] Fix an irreducible unitary representation π_λ:G→ U(d_λ) and an orthonormal basis of its representation space. Define E_ij^λ = fracd_λabsG ∑_g∈ G [π_λ(g^-1)]_jiU(g). Then E_ij^λE_kℓ^μ = δ_λμδ_jkE_iℓ^λ, (E_ij^λ)^*=E_ji^λ, and P_λ=∑_i=1^d_λE_ii^λ.
Exact mathematical passage
E_{ij}^{\lambda}E_{k\ell}^{\mu} = \delta_{\lambda\mu}\delta_{jk}E_{i\ell}^{\lambda}, \qquad \bigl(E_{ij}^{\lambda}\bigr)^*=E_{ji}^{\lambda},
e_{j-1}Use in source · 3 passages
Search aliases: e_j-1
Use in source · 05_worked_models.tex, line 381
C, a Fourier quotient built from a primitive root of unity supplies a matching scalar obstruction of dimension two. Hence, for every n≥3, two is the least finite number of adjacency seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any field. For the lower bound over C, let ζ be a primitive nth root of unity and define q(…
Exact mathematical passage
Ae_j=e_{j-1}+e_{j+1}
E_j^*Use in source · 1 passage
Search aliases: E_j^*
Use in source · monograph_examples.tex, line 17
resolvent elimination give different results. Normalization changes the map. Let P=|0⟩⟨0| on C^2 and Z=diag(1,-1). The normalized Kraus family sqrt3/4 I, sqrt1/4 Z satisfies ∑_j K_j^*K_j=I and maps P to P. The unnormalized family sqrt3 I, Z satisfies ∑_j E_j^*E_j=4I and maps P to 4P. The two families have the same relative weights, but their total normalization differs. Both outputs are proportional to the same projector, but their traces differ. [H]centering includegraphics[width=.94textwidth]m…
Exact mathematical passage
\sum_j E_j^*E_j=4I
E_{ji}Use in source · 2 passages
Search aliases: E_ji
Use in source · visual_atlas.tex, line 18
vpage2An atlas of the objectsORIENTATION / KEEP THE TYPES VISIBLEChapter III, mathematical text: Definition 2.1, Definition 5.5, Definition 7.4 Rows are targets and columns are sources throughout. If E_ji is a matrix unit, then E_jie_i=e_j. The notation e_i means a basis vector. A named graph edge is written a,b,c,d or e,f when its role is explicit.
Exact mathematical passage
E_{ji}
E_{ji}^{\lambda}Use in source · 1 passage
Search aliases: E_ji^λ lambda
Use in source · 02_strengthened.tex, line 251
[Representation matrix units] Fix an irreducible unitary representation π_λ:G→ U(d_λ) and an orthonormal basis of its representation space. Define E_ij^λ = fracd_λabsG ∑_g∈ G [π_λ(g^-1)]_jiU(g). Then E_ij^λE_kℓ^μ = δ_λμδ_jkE_iℓ^λ, (E_ij^λ)^*=E_ji^λ, and P_λ=∑_i=1^d_λE_ii^λ.
Exact mathematical passage
E_{ij}^{\lambda}E_{k\ell}^{\mu} = \delta_{\lambda\mu}\delta_{jk}E_{i\ell}^{\lambda}, \qquad \bigl(E_{ij}^{\lambda}\bigr)^*=E_{ji}^{\lambda},
e_{jj}Use in source · 3 passages
Search aliases: e_jj
Use in source · 02_foundations.tex, line 752
For one factor M_d(D), let e_jj be the diagonal matrix units. The left regular module decomposes as M_d(D)=⊕_j=1^dM_d(D)e_jj. Each minimal left ideal M_d(D)e_jj is isomorphic to the natural column module D^d. Taking the direct product of factors gives the stated decomposition.
Exact mathematical passage
e_{jj}
E_kUse in source · 13 passages
Search aliases: E_k
Use in source · monograph_atlas.tex, line 29
E_k · Positive measurement effect · It gives the outcome probability Tr(E_kρ).
Exact mathematical passage
E_k
E_{k\ell}^{\mu}Use in source · 1 passage
Search aliases: E_kℓ^μ ell mu
Use in source · 02_strengthened.tex, line 251
[Representation matrix units] Fix an irreducible unitary representation π_λ:G→ U(d_λ) and an orthonormal basis of its representation space. Define E_ij^λ = fracd_λabsG ∑_g∈ G [π_λ(g^-1)]_jiU(g). Then E_ij^λE_kℓ^μ = δ_λμδ_jkE_iℓ^λ, (E_ij^λ)^*=E_ji^λ, and P_λ=∑_i=1^d_λE_ii^λ.
Exact mathematical passage
E_{ij}^{\lambda}E_{k\ell}^{\mu} = \delta_{\lambda\mu}\delta_{jk}E_{i\ell}^{\lambda}, \qquad \bigl(E_{ij}^{\lambda}\bigr)^*=E_{ji}^{\lambda},
e_mUse in source · 10 passages
Search aliases: e_m
Use in source · visual_atlas.tex, line 481
vpage20Filter first or split first: same outputMODEL F / A FINITE POLYPHASE REALIZATIONChapter II, mathematical text: Section B.1 Use the finitely supported signal x=(1,2,3,4,5,6) and the filter h=(1,2,1). Outside the displayed support, the input is zero. Let e_m=x_2m and o_m=x_2m+1.
Exact mathematical passage
e_m=x_{2m}
E_m'Use in source · 1 passage
Search aliases: E_m'
Use in source · atlas-pages.tex, line 274
The four real roots of this polynomial are ±1,±sqrt5. They are not the zeros of the original analytic denominator. The plot distinguishes the input polynomial and its transformed output. headAn inverse operation, not a truncation error Applying cosh(∂_x) returns x^4 exactly. The general inverse uses the reciprocal formal symbol. The lowering and translation identities are E_m'=mE_m-1, E_m(x+y)=∑_q=0^mbinom mqE_q(x)y^m-q. Both follow from the generating function. No operator-series convergence as…
Exact mathematical passage
E_m'=mE_{m-1},\qquad E_m(x+y)=\sum_{q=0}^m\binom mqE_q(x)y^{m-q}.
e_{m+1}Use in source · 3 passages
Search aliases: e_m+1
Use in source · Paper_XII_Traditional_Final.tex, line 1151
[Nested packets in one finite carrier] Let V be a finite dimensional vector space. Let e_1,e_2,… be idempotents such that e_m+1e_m=e_me_m+1=e_m+1 for every m. If e_mV≠0 for every m, then the descending chain stabilizes and ⋂_m≥1e_mV ≠0.
Exact mathematical passage
e_{m+1}e_m=e_me_{m+1}=e_{m+1}
e_{m+2}Use in source · 1 passage
Search aliases: e_m+2
Use in source · Paper_XII_Traditional_Final.tex, line 1257
[Tail escape in an infinite carrier] Let V=ℓ^2( N), and let P_m be the orthogonal projection onto the closed span of e_m,e_m+1,e_m+2,…. Every range P_mV is nonzero and the ranges are nested, but ⋂_m≥1P_mV=0. Thus the finite carrier hypothesis in [source reference] cannot be removed.
Exact mathematical passage
e_m,e_{m+1},e_{m+2},\ldots.
E_{m-1}Use in source · 1 passage
Search aliases: E_m-1
Use in source · atlas-pages.tex, line 274
The four real roots of this polynomial are ±1,±sqrt5. They are not the zeros of the original analytic denominator. The plot distinguishes the input polynomial and its transformed output. headAn inverse operation, not a truncation error Applying cosh(∂_x) returns x^4 exactly. The general inverse uses the reciprocal formal symbol. The lowering and translation identities are E_m'=mE_m-1, E_m(x+y)=∑_q=0^mbinom mqE_q(x)y^m-q. Both follow from the generating function. No operator-series convergence as…
Exact mathematical passage
E_m'=mE_{m-1},\qquad E_m(x+y)=\sum_{q=0}^m\binom mqE_q(x)y^{m-q}.
E_{m-1}^{(N,j,r)}Use in source · 1 passage
Search aliases: E_m-1^(N,j,r)
Use in source · Paper_IV_Traditional_Final.tex, line 1493
[Appell laws] For m≥1, the polynomials satisfy fracddxE_m^(N,j,r)(x) = mE_m-1^(N,j,r)(x) and, for m≥0, E_m^(N,j,r)(x+y) = ∑_q=0^m binom mq E_q^(N,j,r)(x)y^m-q.
Exact mathematical passage
\frac{d}{dx}E_m^{(N,j,r)}(x) = mE_{m-1}^{(N,j,r)}(x)
E_{m-q}^{(N,j,s)}Use in source · 1 passage
Search aliases: E_m-q^(N,j,s)
Use in source · Paper_IV_Traditional_Final.tex, line 1530
[Joint Appell convolution in the space and order parameters] For all r,s,x,y∈ C and m≥0, E_m^(N,j,r+s)(x+y) = ∑_q=0^m binom mq E_q^(N,j,r)(x) E_m-q^(N,j,s)(y).
Exact mathematical passage
E_m^{(N,j,r+s)}(x+y) = \sum_{q=0}^{m} \binom mq E_q^{(N,j,r)}(x) E_{m-q}^{(N,j,s)}(y).
E_m^{(N,j,r+s)}Use in source · 1 passage
Search aliases: E_m^(N,j,r+s)
Use in source · Paper_IV_Traditional_Final.tex, line 1530
[Joint Appell convolution in the space and order parameters] For all r,s,x,y∈ C and m≥0, E_m^(N,j,r+s)(x+y) = ∑_q=0^m binom mq E_q^(N,j,r)(x) E_m-q^(N,j,s)(y).
Exact mathematical passage
E_m^{(N,j,r+s)}(x+y) = \sum_{q=0}^{m} \binom mq E_q^{(N,j,r)}(x) E_{m-q}^{(N,j,s)}(y).
E_{mpn+qp^r}^{(mp,j)}Use in source · 1 passage
Search aliases: E_mpn+qp^r^(mp,j)
Use in source · edition_summary.tex, line 4
Principal results. addcontentslinetocsectionPrincipal results The original external problem is Nishibuchi's Conjecture 3.9 for generalized Euler numbers. Its full parameter range is resolved. If p is prime, m is positive with mmid p-1 or m=2, 0≤ j<mp, and q=lcm(2,p-1), then the original Euler coefficients are p-adically integral and E_mpn+qp^r^(mp,j)≡ E_mpn^(mp,j)pmodp^r (r≥1, n≥ r). The chapter states the generating function and the conversion between ordinary degree and packet row before using…
Exact mathematical passage
E_{mpn+qp^r}^{(mp,j)}\equiv E_{mpn}^{(mp,j)}\pmod{p^r} \qquad (r\ge1,\ n\ge r).
E_{mpn}^{(mp,j)}Use in source · 1 passage
Search aliases: E_mpn^(mp,j)
Use in source · edition_summary.tex, line 4
Principal results. addcontentslinetocsectionPrincipal results The original external problem is Nishibuchi's Conjecture 3.9 for generalized Euler numbers. Its full parameter range is resolved. If p is prime, m is positive with mmid p-1 or m=2, 0≤ j<mp, and q=lcm(2,p-1), then the original Euler coefficients are p-adically integral and E_mpn+qp^r^(mp,j)≡ E_mpn^(mp,j)pmodp^r (r≥1, n≥ r). The chapter states the generating function and the conversion between ordinary degree and packet row before using…
Exact mathematical passage
E_{mpn+qp^r}^{(mp,j)}\equiv E_{mpn}^{(mp,j)}\pmod{p^r} \qquad (r\ge1,\ n\ge r).
E_nUse in source · 16 passages
Search aliases: E_n
Use in source · 72_recovered.tex, line 904
Let q be a positive odd integer. Define S_n(q) = ∑_a=0^q-1 (-1)^a(2a+1)^n. Let the signed Euler numbers and Euler zigzag numbers be defined by secht = ∑_n≥0E_nfract^nn! and sec t+tan t = ∑_n≥0Z_nfract^nn!.
Exact mathematical passage
\operatorname{sech}t = \sum_{n\geq0}E_n\frac{t^n}{n!}
\mathcal E_nUse in source · 1 passage
Search aliases: E_n mathcal
Use in source · root_scope.tex, line 24
To distinguish ordinary signed Euler numbers from the generalized numbers of the main text, write secht=∑_n≥0 E_nfract^nn!, fracte^t-1=∑_n≥0B_nfract^nn!, B_1=-frac12. Throughout this appendix, v_2(2)=1 and v_2(0)=∞. Let s_2(n) be the number of ones in the binary expansion of the nonnegative integer n. Define A(N)=frac3^2N+2-14(2N+1)(2N+2), (frac zsinh z)^2=∑_m≥0β_m z^2m, κ_m=2^2m(2m)!β_m. The initial values are β_0=κ_0=1. For m≥1, differentiation of zcoth z gives β_m=-frac2^2m(2m-1)B_2m(2m)!, κ_…
Exact mathematical passage
\operatorname{sech}t=\sum_{n\ge0}\mathcal E_n\frac{t^n}{n!}, \qquad \frac{t}{e^t-1}=\sum_{n\ge0}B_n\frac{t^n}{n!}, \qquad B_1=-\frac12.
e_{n+1}Use in source · 1 passage
Search aliases: e_n+1
Use in source · 03_nonlinear_coordinates.tex, line 815
Let e_n= CF_f^n(x)-F_c^n(Cx) . The definition of the defect and the Lipschitz property give e_n+1≤ε_n+qe_n. Iterating this recurrence proves the first estimate. The second is the geometric sum.
Exact mathematical passage
e_{n+1}\le\varepsilon_n+qe_n.
E_{N,0}Use in source · 1 passage
Search aliases: E_N,0
Use in source · atlas-pages.tex, line 282
vpage16Scale and order enter differentlyTHETA COUPLINGChapter IV: Theorem 10.2, Proposition 10.3, Proposition 10.4. Let H_N(z)=E_N,0(z). The theta family uses the same parameter u twice, but in two different roles: H_N(uz)^-1H_N(z)^-u =∑_n≥0Θ_N,n(u)fracz^Nn(Nn)!. Argument scaling contributes u^Nm at logarithmic block m. Reciprocal order contributes u. vfigtheta Their combined logarithm has coefficient (-1)^mκ_N,mfracu^Nm+um. At the first block this gives Θ_N,1(u)=-(u^N+u). Every positive-index t…
Exact mathematical passage
H_N(z)=E_{N,0}(z)
E_{N,j}Use in source · 7 passages
Search aliases: E_N,j
Use in source · atlas-pages.tex, line 29
midrule E_N,j(z) · displaystyle∑_n≥0fracz^Nn+j(Nn+j)! selects one residue class of the exponential.
Exact mathematical passage
E_{N,j}(z)
e_{n-1}Use in source · 5 passages
Search aliases: e_n-1
Use in source · 01_finite_coordinates.tex, line 249
For a nonempty composable word w=(e_1,…,e_n), define its reverse coefficient product c^←(w)=c(e_n)c(e_n-1)⋯ c(e_1). The reversal is essential over a noncommutative semiring.
Exact mathematical passage
c^\leftarrow(w)=c(e_n)c(e_{n-1})\cdots c(e_1).
E_{n-2r}Use in source · 3 passages
Search aliases: E_n-2r
Use in source · 72_recovered.tex, line 976
[Odd modulus Euler coordinate pair] For every positive odd q and every even n≥0, E_n = ∑_r=0^n/2 (-1)^rZ_2r+1 binom n2r q^2rS_n-2r(q). The inverse transform is S_n(q) = E_n + ∑_r=1^n/2 2^2r-1 binom n2r q^2rE_n-2r. Both transforms are lower triangular with diagonal one.
Exact mathematical passage
\boxed{ S_n(q) = E_n + \sum_{r=1}^{n/2} 2^{2r-1} \binom n{2r} q^{2r}E_{n-2r}. }
E_{n-r}Use in source · 1 passage
Search aliases: E_n-r
Use in source · supplement_02.tex, line 86
The inverse coefficients are integral by construction. Define the integer binomial convolution B_n=∑_r=0^nbinom nr E_rE_n-r, b_m=B_2m. The product formula for actual EGFs gives ∑_n≥0B_nfract^nn!=sech^2 t. Odd coefficients vanish by parity. In particular, every b_m is an integer. This is the integrality required by the later congruence.
Exact mathematical passage
B_n=\sum_{r=0}^{n}\binom nr E_rE_{n-r},\qquad b_m=B_{2m}.
e_n^{(2,7)}Use in source · 1 passage
Search aliases: e_n^(2,7)
Use in source · 72_recovered.tex, line 449
ction. The positive-index standard-window tail printed below is erroneous. The defining recurrence gives e_1=-1≡6, e_2=5 and e_3=-61≡2pmod7, so the correct positive tail is 6,5,2,6,5,2,…. The period-three conclusion remains correct. The original line is retained only as a corrected historical record. Take p=7 and m=2, so h=3. The recurrence gives e_n^(2,7) = 1,6,4,1,6,4,… (n≥1). The positive tail has exact period three.
Exact mathematical passage
e_n^{(2,7)} = 1,6,4,1,6,4,\ldots \qquad(n\geq1).
E_{Nn+qp^r}^{(N,j)}Use in source · 1 passage
Search aliases: E_Nn+qp^r^(N,j)
Use in source · 20_arithmetic.tex, line 170
[Nishibuchi congruence] Let p be prime, let m>0, and suppose that mmid p-1 or m=2. Put N=mp and q=lcm(2,p-1). For every 0≤ j<N, all U_N,j(n) belong to Z_p. For r≥1 and n≥ r, E_Nn+qp^r^(N,j)≡ E_Nn^(N,j)pmodp^r. Thus the positive sufficient starting index n_0=r works uniformly. If mmid p-1, the stronger sufficient condition r≤ mn is valid.
Exact mathematical passage
\label{eq:arith-nishibuchi} E_{Nn+qp^r}^{(N,j)}\equiv E_{Nn}^{(N,j)}\pmod{p^r}.
E_pUse in source · 3 passages
Search aliases: E_p
Use in source · Paper_X_Block27_Amended.tex, line 1794
For each supplied required packet p, let A_p be its finite-dimensional carrier over a characteristic-zero field E_p, and let e_p be the actual joint spectral projector for its finite list of split commuting conditions. No order on E_p is assumed. The source notation Tr(e_p)>0 means that the unique natural number n_p with Tr(e_p)=(n_p:E_p) is positive. The checked projector formula identifies n_p with the dimension of the actual joint eigenspace.
Exact mathematical passage
E_p
e_pUse in source · 7 passages
Search aliases: e_p
Use in source · 02_foundations.tex, line 916
Let the minimal polynomial be m_T(t)=∏_pp(t)^e_p. The acting algebra is 𝕜[T]≅𝕜[t]/(m_T) ≅∏_pR_p, R_p:=𝕜[t]/(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with maximal ideal generated by the class of p and residue division field K_p=𝕜[t]/(p). By [source reference], srank_R_p(X_p) = dim_K_pX_p/p(T)X_p = μ_p(T). …
Exact mathematical passage
m_T(t)=\prod_{p}p(t)^{e_p}.
E_{p,j}^\timesUse in source · 4 passages
Search aliases: E_p,j^× times
Use in source · 72_recovered.tex, line 1841
Fix an even address j∈{0,2,…,p-3}. For x∈ Z_p, define mathscr E_p,j^×(x) = ∑_r≥0 · (-1)^rZ_2r+1p^2r binom(p-1)x+j2r · × ∑_substack0≤ a<p 2a+1≠ p (-1)^a (2a+1)^j-2r ((2a+1)^p-1)^x.
Exact mathematical passage
\begin{aligned} \mathscr E_{p,j}^\times(x) = \sum_{r\geq0} &(-1)^rZ_{2r+1}p^{2r} \binom{(p-1)x+j}{2r}\\ &\times \sum_{\substack{0\leq a<p\\2a+1\neq p}} (-1)^a (2a+1)^{j-2r} \left((2a+1)^{p-1}\right)^x. \end{aligned}
E_qUse in source · 6 passages
Search aliases: E_q
Use in source · atlas-pages.tex, line 274
The four real roots of this polynomial are ±1,±sqrt5. They are not the zeros of the original analytic denominator. The plot distinguishes the input polynomial and its transformed output. headAn inverse operation, not a truncation error Applying cosh(∂_x) returns x^4 exactly. The general inverse uses the reciprocal formal symbol. The lowering and translation identities are E_m'=mE_m-1, E_m(x+y)=∑_q=0^mbinom mqE_q(x)y^m-q. Both follow from the generating function. No operator-series convergence as…
Exact mathematical passage
E_m'=mE_{m-1},\qquad E_m(x+y)=\sum_{q=0}^m\binom mqE_q(x)y^{m-q}.
e_qUse in source · 1 passage
Search aliases: e_q
Use in source · Paper_IV_Traditional_Final.tex, line 902
For every nonnegative integer n, J_N^Nn+d_pqe_q = (-1)^nε_pqe_p. Only powers whose exponent is congruent to d_pq modulo N contribute to the (p,q) entry of the exponential. Therefore (e^zJ_N)_pq = ε_pq ∑_n≥0(-1)^n fracz^Nn+d_pq(Nn+d_pq)!. This is the stated formula.
Exact mathematical passage
J_N^{Nn+d_{pq}}e_q = (-1)^n\varepsilon_{pq}e_p.
E_q^{(N,j,r)}Use in source · 2 passages
Search aliases: E_q^(N,j,r)
Use in source · Paper_IV_Traditional_Final.tex, line 1499
[Appell laws] For m≥1, the polynomials satisfy fracddxE_m^(N,j,r)(x) = mE_m-1^(N,j,r)(x) and, for m≥0, E_m^(N,j,r)(x+y) = ∑_q=0^m binom mq E_q^(N,j,r)(x)y^m-q.
Exact mathematical passage
E_m^{(N,j,r)}(x+y) = \sum_{q=0}^{m} \binom mq E_q^{(N,j,r)}(x)y^{m-q}.
E_rUse in source · 5 passages
Search aliases: E_r
Use in source · supplement_02.tex, line 86
The inverse coefficients are integral by construction. Define the integer binomial convolution B_n=∑_r=0^nbinom nr E_rE_n-r, b_m=B_2m. The product formula for actual EGFs gives ∑_n≥0B_nfract^nn!=sech^2 t. Odd coefficients vanish by parity. In particular, every b_m is an integer. This is the integrality required by the later congruence.
Exact mathematical passage
B_n=\sum_{r=0}^{n}\binom nr E_rE_{n-r},\qquad b_m=B_{2m}.
e_{r,0}Use in source · 1 passage
Search aliases: e_r,0
Use in source · root_scope.tex, line 428
The latest continuation retains the exact Bernoulli undepletion factor e_r,q=frac1-2^2r-11-2^2r-14^q, q≥0. It is a dyadic unit, including e_r,0=1. Multiplication by this factor preserves the all-q error of the shifted Jacobi moment model. Consequently the actual model critical values satisfy, for every m≥25, v_2(Φ_r,m^act-Φ_r^act) ≥ L_m=8m-4s_2(m)-178, L_m⟶∞. The estimate includes every packet and controls motion of the critical point. There is no ceiling at the undepletion exponent 2r-25. For e…
Exact mathematical passage
e_{r,0}=1
e_{r,q}Use in source · 1 passage
Search aliases: e_r,q
Use in source · root_scope.tex, line 425
The latest continuation retains the exact Bernoulli undepletion factor e_r,q=frac1-2^2r-11-2^2r-14^q, q≥0. It is a dyadic unit, including e_r,0=1. Multiplication by this factor preserves the all-q error of the shifted Jacobi moment model. Consequently the actual model critical values satisfy, for every m≥25, v_2(Φ_r,m^act-Φ_r^act) ≥ L_m=8m-4s_2(m)-178, L_m⟶∞. The estimate includes every packet and controls motion of the critical point. There is no ceiling at the undepletion exponent 2r-25. For e…
Exact mathematical passage
e_{r,q}=\frac{1-2^{2r-1}}{1-2^{2r-1}4^q},\qquad q\ge0.
e_r^{(m,p)}Use in source · 1 passage
Search aliases: e_r^(m,p)
Use in source · 72_recovered.tex, line 407
[Rational ordinary generating function] In F_p[[z]], ∑_n≥0e_n^(m,p)z^n = 1+ frac ∑_r=1^he_r^(m,p)z^r 1-z^h.
Exact mathematical passage
\boxed{ \sum_{n\geq0}e_n^{(m,p)}z^n = 1+ \frac{ \sum_{r=1}^{h}e_r^{(m,p)}z^r }{1-z^h}. }
E_{rs}Use in source · 2 passages
Search aliases: E_rs
Use in source · 03_nonlinear_coordinates.tex, line 704
[Matrix difference grading] Let A=M_N(𝕜) and assign the matrix unit E_rs the label r-s modulo N. Since E_rsE_uv=δ_suE_rv, the output label is (r-s)+(u-v)=r-v whenever the product is nonzero. Matrix multiplication therefore obeys the cyclic convolution law exactly. The same grading resolves commutators and polynomial matrix equations.
Exact mathematical passage
E_{rs}
E_{rv}Use in source · 1 passage
Search aliases: E_rv
Use in source · 03_nonlinear_coordinates.tex, line 705
[Matrix difference grading] Let A=M_N(𝕜) and assign the matrix unit E_rs the label r-s modulo N. Since E_rsE_uv=δ_suE_rv, the output label is (r-s)+(u-v)=r-v whenever the product is nonzero. Matrix multiplication therefore obeys the cyclic convolution law exactly. The same grading resolves commutators and polynomial matrix equations.
Exact mathematical passage
E_{rs}E_{uv}=\delta_{su}E_{rv},
e_TUse in source · 6 passages
Search aliases: e_T
Use in source · Paper_X_Block27_Amended.tex, line 1620
Let E be a field of characteristic zero and let T_1,…,T_N be commuting operators on a finite-dimensional E-vector space A, each diagonalizable over E. Thus the selected spectra split over the coefficient field. Let a_i∈ E be a selected target scalar for T_i, possibly absent from its spectrum. Define e_i = ∏_μ∈Spec(T_i)∖{a_i} fracT_i-μ Ia_i-μ, e_T=∏_i=1^Ne_i, with e_i=0 if a_i is absent.
Exact mathematical passage
e_i = \prod_{\mu\in\Spec(T_i)\setminus\{a_i\}} \frac{T_i-\mu I}{a_i-\mu}, \qquad e_T=\prod_{i=1}^{N}e_i,
e_{T,a}Use in source · 1 passage
Search aliases: e_T,a
Use in source · Paper_X_Block27_Amended.tex, line 1678
Naturality and finite hypotheses. If F T_i=U_i F and the source operators split, then F e_T,a=e_U,aF. This follows first on source eigenvectors and then on their span. In particular, a change of basis conjugates the constructed projector and preserves its trace. Adding conditions decreases the actual joint eigenspace and cannot increase its dimension. These statements do not assume an orthonormal basis or a positive definite inner product.
Exact mathematical passage
F e_{T,a}=e_{U,a}F
E_{\tgt(e),\src(e)}Use in source · 1 passage
Search aliases: E_tgt(e),src(e) tgt src operatorname operatorname
Use in source · visual_atlas.tex, line 24
L_e=c(e)E_tgt(e),src(e) · One operator supported on a packet. · pagerefV3
Exact mathematical passage
L_e=c(e)E_{\tgt(e),\src(e)}
e_UUse in source · 5 passages
Search aliases: e_U
Use in source · 05_worked_models.tex, line 14
Let the quiver have vertices U,V and arrows R:U→ U, A:U→ V, B:V→ U, with the relation in the path algebra BA=e_U, whose realization is the operator identity BA=I_U. Realize U=𝕜^2, V=𝕜^3, by R= 0 · -1
Exact mathematical passage
BA=e_U,
e_{U,a}Use in source · 1 passage
Search aliases: e_U,a
Use in source · Paper_X_Block27_Amended.tex, line 1678
Naturality and finite hypotheses. If F T_i=U_i F and the source operators split, then F e_T,a=e_U,aF. This follows first on source eigenvectors and then on their span. In particular, a change of basis conjugates the constructed projector and preserves its trace. Adding conditions decreases the actual joint eigenspace and cannot increase its dimension. These statements do not assume an orthonormal basis or a positive definite inner product.
Exact mathematical passage
F e_{T,a}=e_{U,a}F
E_{uv}Use in source · 1 passage
Search aliases: E_uv
Use in source · 03_nonlinear_coordinates.tex, line 705
[Matrix difference grading] Let A=M_N(𝕜) and assign the matrix unit E_rs the label r-s modulo N. Since E_rsE_uv=δ_suE_rv, the output label is (r-s)+(u-v)=r-v whenever the product is nonzero. Matrix multiplication therefore obeys the cyclic convolution law exactly. The same grading resolves commutators and polynomial matrix equations.
Exact mathematical passage
E_{rs}E_{uv}=\delta_{su}E_{rv},
e_vUse in source · 8 passages
Search aliases: e_v
Use in source · 02_foundations.tex, line 73
[Typed realization and module equivalence] The category of realizations of G is equivalent to the category of unital left modules over Λ_ G. Explicitly, a realization X determines the collapsed space |X| of [source reference] with the vertex idempotent e_v acting as the coordinate projector onto X_v and each arrow acting in the block between its declared source and target. Conversely, a unital module M over Λ_ G determines the realization M_v=e_vM, M_a(x)=ax. The two constructions are inverse up…
Exact mathematical passage
e_v
e_wUse in source · 2 passages
Search aliases: e_w
Use in source · 02_foundations.tex, line 93
n a realization, define the action of a path by composition of its arrow maps and extend linearly. Every element of I acts as zero, so the action factors through Λ_ G. The vertex idempotents give |X|=⊕_v e_v|X|=⊕_vX_v. Conversely, if M is unital, then M=1M=∑_ve_vM. Orthogonality makes the sum direct. For an arrow a:v→ w, the identities a=e_wae_v imply a(e_vM)⊆ e_wM. The relations in I act as zero because M is a module over Λ_ G. Applying both constructions returns the original vertex spaces and …
Exact mathematical passage
a=e_wae_v
E_xUse in source · 1 passage
Search aliases: E_x
Use in source · monograph_atlas.tex, line 253
vpage14A chart failure is not a singular fiberTHE RANGE BUNDLE IS CONSTRUCTED LOCALLYChapter VII: Theorem II.6.7. Details of the range atlas: Chapter VII, Part III, Sections III.24--III.26 (range-bundle construction). The fiber of a projector field is the actual fixed subspace E_x={v:P(x)v=v}. To compare it with a nearby anchor fiber E_a, the completion proof constructs T_a(x)=P(x)P(a)+(I-P(x))(I-P(a)). It equals the identity at the anchor and intertwines the two projectors. Where it is invertib…
Exact mathematical passage
E_x=\{v:P(x)v=v\}
E_zUse in source · 2 passages
Search aliases: E_z
Use in source · 03_supplement_04.tex, line 130
very operator supported on the code, Pρ P=ρ, the central identity is R(V_aρ V_b^*)=δ_abρ. Consequently normalized mixtures satisfy R (∑_ap_aV_aρ V_a^*)=ρ when ∑_ap_a=1. The identity is algebraic and is proved for complex weights. Physical mixtures use the nonnegative real case. For a coherent error E_z=∑_a z_aV_a, the calculation of the cross terms yields R(E_zρ E_z^*)=(∑_a z_aoverlinez_a)ρ. Thus the same map also corrects normalized coherent superpositions. This construction supplies the recove…
Exact mathematical passage
E_z=\sum_a z_aV_a
E_z^*Use in source · 1 passage
Search aliases: E_z^*
Use in source · 03_supplement_04.tex, line 131
identity is R(V_aρ V_b^*)=δ_abρ. Consequently normalized mixtures satisfy R (∑_ap_aV_aρ V_a^*)=ρ when ∑_ap_a=1. The identity is algebraic and is proved for complex weights. Physical mixtures use the nonnegative real case. For a coherent error E_z=∑_a z_aV_a, the calculation of the cross terms yields R(E_zρ E_z^*)=(∑_a z_aoverlinez_a)ρ. Thus the same map also corrects normalized coherent superpositions. This construction supplies the recovery stage used in the full Knill--Laflamme proof that foll…
Exact mathematical passage
\mathcal R(E_z\rho E_z^*)=\left(\sum_a z_a\overline{z_a}\right)\rho.
E_\pmUse in source · 1 passage
Search aliases: E_± pm
Use in source · visual_atlas.tex, line 366
headThe special fiber does not retain the rational splitting Modulo two, x-1=x+1 and A becomes a nontrivial unipotent matrix. The entries 1/2 in E_± cannot be reduced modulo two. By contrast, the coefficient selectors on Plate II.6 use only zero and one and remain available.
Exact mathematical passage
E_\pm
E_{\alpha}Use in source · 1 passage
Search aliases: E_α alpha
Use in source · 04_correction_transport.tex, line 966
Taylor's theorem in the Banach space of bounded operators gives P_α(t+h) = P_α(t)+hdot P_α(t)+E_α(t,h), E_α(t,h) ≤ C_1h^2, uniformly in t and α. Hence S( P(t+h), P(t)) = I+hK(t)+E(t,h), E(t,h) ≤ C_2h^2. Since K(t)^*=-K(t), S^*S=I+E_2(t,h), E_2(t,h) ≤ C_3h^2. For small h, the spectrum of S^*S lies in a fixed compact subinterval of (0,∞). The functional calculus for z^-1/2 gives (S^*S)^-1/2=I+E_3(t,h), E_3(t,h) ≤ C_4h^2. Multiplying the last two expansions proves the estimate.
Exact mathematical passage
P_\alpha(t+h) = P_\alpha(t)+h\dot P_\alpha(t)+E_{\alpha}(t,h), \qquad \|E_{\alpha}(t,h)\|\le C_1h^2,
e_\thetaUse in source · 3 passages
Search aliases: e_θ theta
Use in source · Paper_XI_Traditional_Final.tex, line 466
[Primitive cuspidal idempotent] Let χ_θ be the irreducible character of κ_θ, and let d_θ=χ_θ(1). Then e_θ = fracd_θabsG_n ∑_g∈ G_n χ_θ(g^-1)g is the primitive central idempotent selecting the κ_θ block in C[G_n].
Exact mathematical passage
e_\theta = \frac{d_\theta}{\abs{G_n}} \sum_{g\in G_n} \chi_\theta(g^{-1})g
E_\lambdaUse in source · 1 passage
Search aliases: E_λ lambda
Use in source · Paper_X_Block27_Amended.tex, line 1056
ion] Let mathscr W_1,…,mathscr W_r be pairwise nonisomorphic strictly compatible systems whose members are absolutely irreducible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficient-field embedding σ_λ:E↪ E_λ therefore sends C to an invertible matrix. Its entries agree with the ordinary representation traces Tr(W_i,λ(Frob_v_j)) when the selected places …
Exact mathematical passage
\sigma_\lambda:E\hookrightarrow E_\lambda
e_\lambdaUse in source · 12 passages
Search aliases: e_λ lambda
Use in source · 02_foundations.tex, line 352
X, continuity of a:X→ X supplies continuous seminorms controlling p(aj(e)), so the induced action on E is continuous and extends uniquely to widehat E. The subspace is dense by hypothesis. A Cauchy net in E is mapped to a Cauchy net in X, which converges because X is complete. This defines a continuous extension widehat j:widehat E→ X. For any x∈ X, choose a net (e_λ) in E with j(e_λ)→ x. The transported topology makes (e_λ) Cauchy, so it converges to some ξ∈widehat E, and continuity gives wideh…
Exact mathematical passage
(e_\lambda)
E_\muUse in source · 2 passages
Search aliases: E_μ mu
Use in source · Paper_VI_Traditional_Final.tex, line 1656
For clarity, the finite-dimensional holomorphic calculus used below has the following canonical definition. For A acting on C^d, its generalized eigenspaces E_μ=ker(A-μ I)^d form a direct-sum decomposition over the eigenvalues. Define the unique linear operator F(A) by .F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent of a basis and of increasing the truncation index: the extra nilpotent powers vanish. It depends only on the finite Taylor jets of F at the spectrum. I…
Exact mathematical passage
E_\mu=\ker(A-\mu I)^d
E_\piUse in source · 1 passage
Search aliases: E_π pi
Use in source · atlas-pages.tex, line 34
E_π · Conditional expectation of observables. · A weighted average, self-adjoint for the π inner product.
Exact mathematical passage
E_\pi
E_\ellUse in source · 8 passages
Search aliases: E_ℓ ell
Use in source · Paper_VI_Traditional_Final.tex, line 2600
[Contractive summable defect completion] Assume normC_ℓ≤1, normR_ℓ≤1, normT_ℓ(t)≤1 for every ℓ and t≥0. If ∑_ℓ=0^∞ sup_0≤ t≤ TnormE_ℓ(t)<∞, then the conclusion of [source reference] holds, and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_ℓ=n^∞ sup_0≤ t≤ TnormE_ℓ(t).
Exact mathematical passage
\sum_{\ell=0}^{\infty} \sup_{0\leq t\leq T}\norm{E_\ell(t)}<\infty,
E_\ell^*Use in source · 1 passage
Search aliases: E_ℓ^* ell
Use in source · 01_finite_core.tex, line 1019
order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_ℓ: H_ℓ→ H_ℓ+1 and C_ℓ: H_ℓ+1→ H_ℓ. The left inverse relation C_ℓ E_ℓ=I alone does not imply C_ℓ=E_ℓ^* or make the embedding isometric.
Exact mathematical passage
C_\ell=E_\ell^*
\EcalUse in source · 3 passages
Search aliases: Ecal E mathcal
Use in source · 72_recovered.tex, line 576
For integers N≥1 and 0≤ j<N, define E_N,j(t) = ∑_n≥0 fract^Nn+j(Nn+j)!. The forced factor t^j may be removed without changing the residue address.
Exact mathematical passage
\Ecal_{N,j}(t) = \sum_{n\geq0} \frac{t^{Nn+j}}{(Nn+j)!}.
\Ecal_{N,j}Use in source · 3 passages
Search aliases: Ecal_N,j E_N,j Ecal mathcal
Use in source · 72_recovered.tex, line 576
For integers N≥1 and 0≤ j<N, define E_N,j(t) = ∑_n≥0 fract^Nn+j(Nn+j)!. The forced factor t^j may be removed without changing the residue address.
Exact mathematical passage
\Ecal_{N,j}(t) = \sum_{n\geq0} \frac{t^{Nn+j}}{(Nn+j)!}.
\EEUse in source · 12 passages
Search aliases: EE E mathbb EE E mathcal
Use in source · supplement_11.tex, line 5
Nishibuchi conjecture completion Checkpoint scope. This complete source body retains the status of its dated original supplement. Earlier pending or partial-coverage labels are historical. The final core records their resolution. 1. The complete external statement. For N>0 and 0≤ j<N, use the original unnormalized convention ∑_d≥0 E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The existing packet convention satisfies E_Nn^(N,j)=j!C_N,j(n). The formalization defines the degree-indexed coefficient…
Exact mathematical passage
\sum_{d\ge0}\EE_d^{(N,j)}\frac{t^d}{d!} =\left(\sum_{n\ge0}\frac{t^{Nn}}{(Nn+j)!}\right)^{-1}.
\EE_0Use in source · 1 passage
Search aliases: EE_0 E_0 EE mathbb
Use in source · atlas-pages.tex, line 448
1 · 3, start at state 0, use weights w=(1,1/2), and terminal observable f=(2,1). For three transitions, there are eight possible state sequences after the fixed initial state. We evaluate E_0[∏_r=0^2w(X_r) f(X_3)]. Each path contributes its transition probability times its three departure weights times its terminal value. vfigpaths platecaptionEvery possible path is included. The probability column sums to one. The contribution column sums to 311/256, agreeing exactly with the transfer calculati…
Exact mathematical passage
\EE_0\left[\prod_{r=0}^{2}w(X_r)\,f(X_3)\right].
\EE_d^{(N,j)}Use in source · 1 passage
Search aliases: EE_d^(N,j) E_d^(N,j) EE mathcal
Use in source · supplement_11.tex, line 5
Nishibuchi conjecture completion Checkpoint scope. This complete source body retains the status of its dated original supplement. Earlier pending or partial-coverage labels are historical. The final core records their resolution. 1. The complete external statement. For N>0 and 0≤ j<N, use the original unnormalized convention ∑_d≥0 E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The existing packet convention satisfies E_Nn^(N,j)=j!C_N,j(n). The formalization defines the degree-indexed coefficient…
Exact mathematical passage
\sum_{d\ge0}\EE_d^{(N,j)}\frac{t^d}{d!} =\left(\sum_{n\ge0}\frac{t^{Nn}}{(Nn+j)!}\right)^{-1}.
\EE_{Nn+qp^r}^{(N,j)}Use in source · 1 passage
Search aliases: EE_Nn+qp^r^(N,j) E_Nn+qp^r^(N,j) EE mathcal
Use in source · supplement_11.tex, line 14
Let p be prime, m>0, and assume mmid p-1 or m=2. Put N=mp and q=lcm(2,p-1). Every E_Nn^(N,j) is p-adically integral. For every r≥1, E_Nn+qp^r^(N,j)≡ E_Nn^(N,j)pmodp^r (0≤ j<N, n≥ r). In particular, the positive sufficient threshold n_0=r works uniformly in all allowed parameters.
Exact mathematical passage
\boxed{\EE_{Nn+qp^r}^{(N,j)}\equiv\EE_{Nn}^{(N,j)}\pmod{p^r} \qquad(0\le j<N,\ n\ge r).}
\EE_{Nn}^{(N,j)}Use in source · 3 passages
Search aliases: EE_Nn^(N,j) E_Nn^(N,j) EE mathcal
Use in source · supplement_11.tex, line 10
ed original supplement. Earlier pending or partial-coverage labels are historical. The final core records their resolution. 1. The complete external statement. For N>0 and 0≤ j<N, use the original unnormalized convention ∑_d≥0 E_d^(N,j)fract^dd! =(∑_n≥0fract^Nn(Nn+j)!)^-1. The existing packet convention satisfies E_Nn^(N,j)=j!C_N,j(n). The formalization defines the degree-indexed coefficients through the actual factorial-scaled series coefficients, proves their packet identification, and checks …
Exact mathematical passage
\EE_{Nn}^{(N,j)}=j!C_{N,j}(n)
\EE_{p(t)}Use in source · 3 passages
Search aliases: EE_p(t) E_p(t) EE mathbb
Use in source · Paper_VI_Traditional_Final.tex, line 5517
Along the Bayesian flow p_i(t)=e^tg_ip_i(0)/∑_j e^tg_jp_j(0), the following statement holds for every real observable h. [Covariance identity] For any observable h∈ R^n, frac d d t E_p(t)[h] = Cov_p(t)(h,g). In particular, frac d d t E_p(t)[g] = Var_p(t)(g)≥0.
Exact mathematical passage
\frac{\dd}{\dd t}\EE_{p(t)}[h] = \Cov_{p(t)}(h,g).
\mathrm{eff}Use in source · 7 passages
Search aliases: eff mathrm
Use in source · 01_finite_core.tex, line 907
[Exact Feshbach resolvent] Assume that both zI_Q-H_QQ and zI-H are invertible. Define H_eff(z) = H_PP +H_PQ(z-H_QQ)^-1H_QP. Then P(z-H)^-1P = (z-H_eff(z))^-1 where the left side is restricted to P H, the right side is an inverse on that retained space, and both are written in the same orthonormal basis. Here z-H_QQ and z-H_eff abbreviate the appropriately sized scalar matrices minus the indicated blocks.
Exact mathematical passage
\boxed{ H_{\mathrm{eff}}(z) = H_{PP} +H_{PQ}(z-H_{QQ})^{-1}H_{QP}. }
\Efun_{2,0}Use in source · 1 passage
Search aliases: Efun_2,0 E_2,0 Efun mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 594
For N=1, the only section is e^λ z. For N=2, E_2,0(z)=cosh z, E_2,1(z)=sinh z. These are classical instances of the same cyclic basis.
Exact mathematical passage
\Efun_{2,0}(z)=\cosh z, \qquad \Efun_{2,1}(z)=\sinh z.
\Efun_{2,1}Use in source · 1 passage
Search aliases: Efun_2,1 E_2,1 Efun mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 594
For N=1, the only section is e^λ z. For N=2, E_2,0(z)=cosh z, E_2,1(z)=sinh z. These are classical instances of the same cyclic basis.
Exact mathematical passage
\Efun_{2,0}(z)=\cosh z, \qquad \Efun_{2,1}(z)=\sinh z.
\Efun_{N,i}Use in source · 1 passage
Search aliases: Efun_N,i E_N,i Efun mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 534
[Addition law] For 0≤ k<N, E_N,k(λ(z+w)) = ∑_substack0≤ i,j<N i+j≡ kbmod N E_N,i(λ z) E_N,j(λ w).
Exact mathematical passage
\Efun_{N,k}(\lambda(z+w)) = \sum_{\substack{0\leq i,j<N\\ i+j\equiv k\bmod N}} \Efun_{N,i}(\lambda z)\Efun_{N,j}(\lambda w).
\Efun_{N,j-1}Use in source · 1 passage
Search aliases: Efun_N,j-1 E_N,j-1 Efun mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 470
[Differential cycle and initial data] For 1≤ j<N, fracddz E_N,j(λ z) = λ E_N,j-1(λ z), while fracddz E_N,0(λ z) = λ E_N,N-1(λ z). Thus fracd^Ndz^N E_N,j(λ z) = λ^N E_N,j(λ z). For 0≤ k<N, .fracd^kdz^k E_N,j(λ z)|_z=0 = δ_kjλ^j. If λ≠0, the N functions E_N,j(λ z) form a basis of the solution space of y^(N)=λ^Ny.
Exact mathematical passage
\frac{d}{dz}\Efun_{N,j}(\lambda z) = \lambda\Efun_{N,j-1}(\lambda z),
\Efun_{N,k}Use in source · 1 passage
Search aliases: Efun_N,k E_N,k Efun mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 534
[Addition law] For 0≤ k<N, E_N,k(λ(z+w)) = ∑_substack0≤ i,j<N i+j≡ kbmod N E_N,i(λ z) E_N,j(λ w).
Exact mathematical passage
\Efun_{N,k}(\lambda(z+w)) = \sum_{\substack{0\leq i,j<N\\ i+j\equiv k\bmod N}} \Efun_{N,i}(\lambda z)\Efun_{N,j}(\lambda w).
\Efun_{N,N-1}Use in source · 2 passages
Search aliases: Efun_N,N-1 E_N,N-1 Efun mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 476
[Differential cycle and initial data] For 1≤ j<N, fracddz E_N,j(λ z) = λ E_N,j-1(λ z), while fracddz E_N,0(λ z) = λ E_N,N-1(λ z). Thus fracd^Ndz^N E_N,j(λ z) = λ^N E_N,j(λ z). For 0≤ k<N, .fracd^kdz^k E_N,j(λ z)|_z=0 = δ_kjλ^j. If λ≠0, the N functions E_N,j(λ z) form a basis of the solution space of y^(N)=λ^Ny.
Exact mathematical passage
\frac{d}{dz}\Efun_{N,0}(\lambda z) = \lambda\Efun_{N,N-1}(\lambda z).
\operatorname{End}Use in source · 4 passages
Search aliases: End operatorname
Use in source · Paper_X_Block27_Amended.tex, line 1386
Explicit summands inside a supplied decomposition. Let K be a field, let G be a monoid, and let ρ_i:G→End_K(W_i) be actual representations on finite-dimensional spaces, indexed by a finite set I. For natural numbers m_i≤ c_i, set U_m=∏_i∈ I∏_k<m_iW_i, U_c=∏_i∈ I∏_k<c_iW_i. These finite products realize the corresponding finite direct sums. The representation on U_m acts componentwise. Define j:U_m→ U_c by retaining the first m_i copies and filling every remaining coordinate with zero. Define r:U…
Exact mathematical passage
\rho_i:G\to\operatorname{End}_K(W_i)
\operatorname{End}_{\bbR}Use in source · 1 passage
Search aliases: End_bbR End_ R operatorname bbR operatorname mathbb
Use in source · 01_finite_core.tex, line 573
[Projected connection] Let M be a smooth finite-dimensional real manifold without boundary, let V be a finite-dimensional real normed vector space, and let P:M→End_ R(V) be smooth with P(x)^2=P(x). In the quantum application, V is the underlying real space of a finite complex Hilbert space and P is an orthogonal complex-linear projector.
Exact mathematical passage
P:M\to\operatorname{End}_{\bbR}(V)
\operatorname{End}_{E_0}Use in source · 1 passage
Search aliases: End_E_0 operatorname
Use in source · dyadic_synthesis.tex, line 310
Use the actual rank-four outer approximation field E_0/F of degree two and its centralizer B=End_E_0(F^4). The odd lower critical level is r=3h_1=2m+1. The half leading level is a=3h_2>r. Define P=J U_B^m, where J=J^1 is the actual compact type group. The filtered centralizer estimate for the actual final field, without assuming E_0⊂ K, proves that P is a compact group with J normal of index two. For a concrete new coset representative u, λ=θ^u/θ on the actual H=H^1 is nontrivial with values {±1…
Exact mathematical passage
B=\operatorname{End}_{E_0}(F^4)
\End_KUse in source · 1 passage
Search aliases: End_K End operatorname
Use in source · 04_arithmetic.tex, line 121
[Evaluated B ezout spectral projector] Let F,G,U,V∈End_K(Λ⊗_ O K) and let r∈ K^×. Suppose UF+VG=rI, FG=0, FG=GF, GU=UG. Put N=VG. Then N^2=rN. Consequently the operator E=r^-1VG is idempotent, acts as the identity on ker F, and acts as zero on ker G. If V and G preserve Λ, then rE=VG∈End_ O(Λ).
Exact mathematical passage
F,G,U,V\in\End_K(\Lambda\otimes_\OO K)
\operatorname{End}_KUse in source · 1 passage
Search aliases: End_K operatorname
Use in source · Paper_X_Block27_Amended.tex, line 1386
Explicit summands inside a supplied decomposition. Let K be a field, let G be a monoid, and let ρ_i:G→End_K(W_i) be actual representations on finite-dimensional spaces, indexed by a finite set I. For natural numbers m_i≤ c_i, set U_m=∏_i∈ I∏_k<m_iW_i, U_c=∏_i∈ I∏_k<c_iW_i. These finite products realize the corresponding finite direct sums. The representation on U_m acts componentwise. Define j:U_m→ U_c by retaining the first m_i copies and filling every remaining coordinate with zero. Define r:U…
Exact mathematical passage
\rho_i:G\to\operatorname{End}_K(W_i)
\End_\OOUse in source · 2 passages
Search aliases: End_OO End_ O End OO operatorname mathcal
Use in source · 04_arithmetic.tex, line 114
Let O be a DVR, K=Frac( O), and A∈End_ O(Λ) for a finite free lattice Λ. Suppose the characteristic polynomial factors over O as χ_A(x)=f(x)g(x), where f and g are coprime over K[x].
Exact mathematical passage
A\in\End_\OO(\Lambda)
\End_{\OO}Use in source · 3 passages
Search aliases: End_OO End_ O End OO operatorname mathcal
Use in source · 04_arithmetic.tex, line 7
Let O be a discrete valuation ring with fraction field K and uniformizer π. Let Λ be a finite free O module and put V=K⊗_ OΛ. Suppose A∈End_ O(Λ).
Exact mathematical passage
A\in\End_{\OO}(\Lambda)
\epsUse in source · 10 passages
Search aliases: eps ε varepsilon
Use in source · 01_manuscript.tex, line 1281
[Packet perturbation certificate] Let A= ∑_e∈ EA_e, widehat A = ∑_e∈ E(A_e+E_e), where edge e starts at packet α(e), and assume A_e=A_eP_α(e), E_e=E_eP_α(e). Then norm(widehat A-A)x ≤ ∑_e∈ E normE_e normP_α(e)x. If normE_e ≤ ε_enormA_e, then norm(widehat A-A)x ≤ ∑_e∈ E ε_enormA_e normP_α(e)x.
Exact mathematical passage
\norm{E_e} \leq \eps_e\norm{A_e},
\eps_eUse in source · 10 passages
Search aliases: eps_e ε_e eps varepsilon
Use in source · 01_manuscript.tex, line 1281
[Packet perturbation certificate] Let A= ∑_e∈ EA_e, widehat A = ∑_e∈ E(A_e+E_e), where edge e starts at packet α(e), and assume A_e=A_eP_α(e), E_e=E_eP_α(e). Then norm(widehat A-A)x ≤ ∑_e∈ E normE_e normP_α(e)x. If normE_e ≤ ε_enormA_e, then norm(widehat A-A)x ≤ ∑_e∈ E ε_enormA_e normP_α(e)x.
Exact mathematical passage
\norm{E_e} \leq \eps_e\norm{A_e},
\errUse in source · 13 passages
Search aliases: err operatorname
Use in source · pages.tex, line 387
ne. Keeping either a or b costs three with error two. Splitting a costs four with zero error, and splitting b costs five with zero error. vfigtree_frontier The five complete prunings have (cost,error) pairs (1,6), (7,4), (9,2), (8,2), (10,0). At budget two the unique cheapest pruning is {a_0,a_1,b}, costing eight. It is Pareto optimal, but never minimizes cost+λerr for any λ≥0. headSee the incompatible inequalities To beat the root, that pruning would require 8+2λ≤1+6λ, hence λ≥7/4. To beat the …
Exact mathematical passage
\cost+\lambda\err
\esssupUse in source · 4 passages
Search aliases: esssup *ess sup operatorname
Use in source · Paper_V_Traditional_Final.tex, line 2059
[Quantitative imperfect reconstruction] Define the delay compensated defect D( e^ iω) = e^ i dωR( e^ iω)E( e^ iω)-I_N. If ε_rec = *ess sup_ω D( e^ iω) _op<∞, then the delay compensated reconstruction widetilde x satisfies widetilde x-x _2 ≤ ε_rec x _2. The constant is sharp as an operator norm.
Exact mathematical passage
\varepsilon_{\mathrm{rec}} = \esssup_{\omega}\|D(\e^{\ii\omega})\|_{\mathrm{op}}<\infty,
\esssup_\omegaUse in source · 2 passages
Search aliases: esssup_ω *ess sup_ω esssup omega operatorname omega
Use in source · Paper_V_Traditional_Final.tex, line 2088
0 · 0.02sinω . Then *ess sup_ω D( e^ iω) _op =0.02. Every input therefore has relative reconstruction error at most two percent in ℓ^2. The estimate is independent of the input length.
Exact mathematical passage
\esssup_\omega\|D(\e^{\ii\omega})\|_{\mathrm{op}} =0.02.
\esssup_{\omega}Use in source · 1 passage
Search aliases: esssup_ω *ess sup_ω esssup omega operatorname omega
Use in source · Paper_V_Traditional_Final.tex, line 2059
[Quantitative imperfect reconstruction] Define the delay compensated defect D( e^ iω) = e^ i dωR( e^ iω)E( e^ iω)-I_N. If ε_rec = *ess sup_ω D( e^ iω) _op<∞, then the delay compensated reconstruction widetilde x satisfies widetilde x-x _2 ≤ ε_rec x _2. The constant is sharp as an operator norm.
Exact mathematical passage
\varepsilon_{\mathrm{rec}} = \esssup_{\omega}\|D(\e^{\ii\omega})\|_{\mathrm{op}}<\infty,
\operatorname{Euler}Use in source · 2 passages
Search aliases: Euler operatorname
Use in source · strengthening.tex, line 290
Let L be the ordinary Euler evaluation functional on a variable E, defined by L( E^d)=Euler_d, ∑_d≥0Euler_dfract^dd! =secht. The functional is linear. It is not a multiplicative evaluation. Independent variables mean that moments of polynomials in separate variables factor. Odd moments vanish. Frobenius's congruence is v_2(∑_r=0^m(-1)^m-rbinom mr Euler_2n+2r) =m+v_2(m!) (n,m≥0). For m=0, the assertion is that every even Euler number is odd. With P=( E^2-1)/2 and γ_m(P)=P^m/m!, the case n=0 says …
Exact mathematical passage
\mathcal L(\mathcal E^d)=\operatorname{Euler}_d, \qquad \sum_{d\geq0}\operatorname{Euler}_d\frac{t^d}{d!} =\operatorname{sech}t.
\operatorname{Euler}_{2n+2r}Use in source · 1 passage
Search aliases: Euler_2n+2r operatorname
Use in source · strengthening.tex, line 298
ariable E, defined by L( E^d)=Euler_d, ∑_d≥0Euler_dfract^dd! =secht. The functional is linear. It is not a multiplicative evaluation. Independent variables mean that moments of polynomials in separate variables factor. Odd moments vanish. Frobenius's congruence is v_2(∑_r=0^m(-1)^m-rbinom mr Euler_2n+2r) =m+v_2(m!) (n,m≥0). For m=0, the assertion is that every even Euler number is odd. With P=( E^2-1)/2 and γ_m(P)=P^m/m!, the case n=0 says L(γ_m(P))∈ Z_2^× (m≥0). Introduce an independent Euler v…
Exact mathematical passage
v_2\left(\sum_{r=0}^m(-1)^{m-r}\binom mr \operatorname{Euler}_{2n+2r}\right) =m+v_2(m!)\qquad(n,m\geq0). \label{eq:classical-frobenius-exact}
\operatorname{Euler}_dUse in source · 1 passage
Search aliases: Euler_d operatorname
Use in source · strengthening.tex, line 290
Let L be the ordinary Euler evaluation functional on a variable E, defined by L( E^d)=Euler_d, ∑_d≥0Euler_dfract^dd! =secht. The functional is linear. It is not a multiplicative evaluation. Independent variables mean that moments of polynomials in separate variables factor. Odd moments vanish. Frobenius's congruence is v_2(∑_r=0^m(-1)^m-rbinom mr Euler_2n+2r) =m+v_2(m!) (n,m≥0). For m=0, the assertion is that every even Euler number is odd. With P=( E^2-1)/2 and γ_m(P)=P^m/m!, the case n=0 says …
Exact mathematical passage
\mathcal L(\mathcal E^d)=\operatorname{Euler}_d, \qquad \sum_{d\geq0}\operatorname{Euler}_d\frac{t^d}{d!} =\operatorname{sech}t.
\mathrm{even}Use in source · 3 passages
Search aliases: even mathrm
Use in source · 01_finite_core.tex, line 228
For parity, S=Z^⊗ n, P_even=fracI+S2, P_odd=fracI-S2.
Exact mathematical passage
S=Z^{\otimes n}, \qquad P_{\mathrm{even}}=\frac{I+S}{2}, \qquad P_{\mathrm{odd}}=\frac{I-S}{2}.
\operatorname{Even}Use in source · 1 passage
Search aliases: Even operatorname
Use in source · supplement_03.tex, line 129
Both row indexing by r and the literal even-degree version with hypothesis Even(n) are checked. All rows include degree zero. The exact identity is a rational equality with the Euler coefficient's integer-to-rational cast made explicit.
Exact mathematical passage
\operatorname{Even}(n)
\operatorname{EvenPart}Use in source · 1 passage
Search aliases: EvenPart operatorname
Use in source · supplement_02.tex, line 59
Even projection is identified coefficientwise. For any formal series f, let EvenPart(f)=(f(t)+f(-t))/2. Its nth coefficient is proved to be the coefficient of f when n is even and zero otherwise. Rescaling the inverse law proves sech(-t)=sech(t). The exponential multiplication identity then gives cosh^2(pt)=frac2+e^2pt+e^-2pt4, and hence ∑_r≥0S_2r(p)fract^2r(2r)! =cosh^2(pt) sech t. Both the full and even EGF formulas are derived from the original integer sum. Neither is supplied as a hypothesis…
Exact mathematical passage
\operatorname{EvenPart}(f)=(f(t)+f(-t))/2
\ExpUse in source · 9 passages
Search aliases: Exp E mathbb
Use in source · Paper_VI_Traditional_Final.tex, line 580
[Symmetric random walk] Starting from δ_0 and applying the transfer T=frac12(S+S^-1) for n steps gives P(X_n=k)= 2^-nbinomn(n+k)/2 when k≡ npmod2, and zero otherwise. Hence E[X_n]=0, Var(X_n)=n. The parity packet is exact: all mass remains in the one residue class permitted by n.
Exact mathematical passage
\Exp[X_n]=0, \qquad \Var(X_n)=n.
\Exp_iUse in source · 3 passages
Search aliases: Exp_i E_i Exp mathbb
Use in source · Paper_VI_Traditional_Final.tex, line 1453
[Discrete Feynman--Kac path formula] For an initial state i, (u_n)_i = E_i[ exp(-∑_r=0^n-1V(X_r))f(X_n) ].
Exact mathematical passage
\boxed{ (u_n)_i = \Exp_i\left[ \exp\left(-\sum_{r=0}^{n-1}V(X_r)\right)f(X_n) \right]. }
\Exp_{p_0}Use in source · 1 passage
Search aliases: Exp_p_0 E_p_0 Exp mathbb
Use in source · Paper_VI_Traditional_Final.tex, line 4922
[State dependent killing as survival] Let Q be a finite Markov generator, let V≥0, and set A=Q-D_V. For an initial distribution p_0, define S_V(t)= 1^T e^tAp_0. Then S_V(t) = E_p_0 [ exp( -∫_0^tV(X_s) d s ) ]. Also, -S_V'(t) = 1^TD_V e^tAp_0. The matrix A=Q-D_V has nonnegative off-diagonal entries and nonpositive column sums, so it is the subgenerator of the killed, possibly nontransient chain. Thus the potential in the finite Feynman--Kac formula is a state dependent killing rate in the forward…
Exact mathematical passage
S_V(t) = \Exp_{p_0} \left[ \exp\left( -\int_0^tV(X_s)\dd s \right) \right].
\Exp_\gammaUse in source · 1 passage
Search aliases: Exp_γ E_γ Exp gamma mathbb gamma
Use in source · Paper_VI_Traditional_Final.tex, line 3287
A mean zero function has no h_0 component, so every surviving Hermite multiplier is at most e^-t. For a polynomial f- E_γ f=∑_n≥1c_nh_n. Then Var_γ(f)=∑_n≥1absc_n^2 ≤ ∑_n≥1nabsc_n^2 = ip- L ff. Apply [source reference].
Exact mathematical passage
f-\Exp_\gamma f=\sum_{n\geq1}c_nh_n.
\Exp_\piUse in source · 1 passage
Search aliases: Exp_π E_π Exp pi mathbb pi
Use in source · Paper_VI_Traditional_Final.tex, line 3092
[Law of total variance] For a real valued f, let P_0f= E_π[f] 1. Then Var_π(f) = Var_π(P_ Qf) + normf-P_ Qf_π^2.
Exact mathematical passage
P_0f=\Exp_\pi[f]\one
\operatorname{Ext}Use in source · 2 passages
Search aliases: Ext operatorname
Use in source · 50_open_questions.tex, line 158
The accompanying nolinkurlnative/Native_Conjectures_Progress.md records further results in restricted settings: an intrinsic finite Ext^1 obstruction for finitely presented commutative integer algebras, finite CRT reconstruction for polynomial solution sets, and a characterization of rectangular finite relations by equality of common labels. The obstruction result is a stated specialization of a classical external theorem. The CRT and rectangularity results have prose proofs in the note. These a…
Exact mathematical passage
\operatorname{Ext}^1
\operatorname{Ext}^1Use in source · 1 passage
Search aliases: Ext^1 operatorname
Use in source · 50_open_questions.tex, line 158
The accompanying nolinkurlnative/Native_Conjectures_Progress.md records further results in restricted settings: an intrinsic finite Ext^1 obstruction for finitely presented commutative integer algebras, finite CRT reconstruction for polynomial solution sets, and a characterization of rectangular finite relations by equality of common labels. The obstruction result is a stated specialization of a classical external theorem. The CRT and rectangularity results have prose proofs in the note. These a…
Exact mathematical passage
\operatorname{Ext}^1
\operatorname{Ext}^1_BUse in source · 1 passage
Search aliases: Ext^1_B operatorname
Use in source · 71_native.tex, line 110
o(f)∈Ext^1_B(NL_B/ Z, F_p)
Exact mathematical passage
o(f)\in\operatorname{Ext}^1_B(\mathrm{NL}_{B/\mathbb Z},\mathbb F_p)
\mathrm{extra}Use in source · 2 passages
Search aliases: extra mathrm
Use in source · 03_supplement_04.tex, line 109
The recovery is not left undefined outside the syndrome subspaces. Its explicit Kraus operators are R_a=V_a^* together with R_extra=Q, giving R(ρ)=∑_aV_a^*ρ V_a+Qρ Q. They satisfy exact completeness: ∑_aR_a^*R_a+R_extra^*R_extra =∑_aV_aV_a^*+Q=S+Q=I. For finite complex matrices, trace preservation and complete positivity at every finite auxiliary dimension are proved for this actual map. The amplified map is identified with the Kraus family I_J⊗ R_a, including the complement term. Positivity is …
Exact mathematical passage
R_{\mathrm{extra}}=Q
\FUse in source · 26 passages
Search aliases: F mathbb F mathbf
Use in source · 04_arithmetic.tex, line 668
The two characteristic factors x-1 and x+1 have the same reduction in F_2[x]. The rational matrices E_+ and E_- themselves cannot be reduced modulo 2, because their entries contain 1/2. Consequently the integral special fiber does not inherit their separate idempotent splitting, even though the two projectors remain distinct over Q_2. This is the simplest model of delayed dyadic separation.
Exact mathematical passage
\F_2[x]
\mathbb FUse in source · 50 passages
Search aliases: F mathbb
Use in source · 05_worked_models.tex, line 196
Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. Therefore sequences with finite support lie in its algebraic orbit hull and are dense in ℓ^2( Z).
Exact mathematical passage
X=\ell^2(\Z;\mathbb F)
\mathbf FUse in source · 2 passages
Search aliases: F mathbf
Use in source · Paper_X_Block27_Amended.tex, line 448
An explicit realization of the finite pattern. A separate finite-complex implementation uses F_5^2xrightarrowd_0 F_5^4xrightarrowd_1 F_5^2, d_0(a,b)=(a,0,0,0), d_1(a,b,c,d)=(b,0). The first packet is the exact complex with dimensions (1,2,1), and the second packet has zero differentials and dimensions (1,2,1). The actual middle cohomology is linearly isomorphic to F_5^2, which is checked in Lean. Independent exhaustive coset tests check all three cohomology degrees after changes of basis. This r…
Exact mathematical passage
\mathbf F_5^2\xrightarrow{d_0}\mathbf F_5^4\xrightarrow{d_1}\mathbf F_5^2, \quad d_0(a,b)=(a,0,0,0),\quad d_1(a,b,c,d)=(b,0).
\mathcal FUse in source · 17 passages
Search aliases: F mathcal
Use in source · 30_lifting.tex, line 848
Use the augmented polynomial systems F(Y,T)=(F(Y,T),T-t_0), H(Y,T)=(H(Y,T),-v). Then F(a)=0, and its derivative is B=D F(a)=J · K
Exact mathematical passage
\mathcal F(Y,T)=(F(Y,T),T-t_0),\qquad \mathcal H(Y,T)=(H(Y,T),-v).
\mathrm FUse in source · 5 passages
Search aliases: F mathrm
Use in source · 02_supplement_06.tex, line 54
ubsection:002textitModule SmallCoreNorms; arbitrary ambient competitors For rectangular isometries Q,V define L(G)=QGV^*, C(B)=Q^*BV, Q^*Q=I, V^*V=I. The following statements are proved for the actual matrix rank and actual matrix norms: C( L(G))=G, rank L(G)=rank G, rank C(B)≤rank B, norm L(G)_ν=normG_ν, norm C(B)_ν≤norm B_ν, ν∈{2, F}. For the spectral norm, the proof uses the Gram matrix identity (QG)^*(QG)=G^*G and the C^*-norm square identity, together with right-side adjoint symmetry. Compr…
Exact mathematical passage
\boxed{\norm{\mathcal L(G)}_\nu=\norm{G}_\nu, \qquad \norm{\mathcal C(B)}_\nu\leq\norm B_\nu, \quad \nu\in\{2,\mathrm F\}.}
F'Use in source · 11 passages
Search aliases: F'
Use in source · 05_defects.tex, line 204
For real t, differentiate F(s)=e^(t-s)T_cCe^sT_f. We obtain F'(s)=e^(t-s)T_cΔ e^sT_f. Integration from 0 to t gives the identity. The complex statement follows by the same calculation along the straight segment. The norm bound follows from the integral formula.
Exact mathematical passage
F'(s)=e^{(t-s)T_c}\Delta e^{sT_f}.
f'Use in source · 9 passages
Search aliases: f'
Use in source · odd_synthesis.tex, line 61
Let P_0 extract the constant coefficient in the basis 1,z,…,z^p-1. The normalized corestriction s=κpr, where κ=-f'(z)/z^p-1, satisfies u_0=s(P_0)=a_0/z^p∈ O_E^×, Lift_E(C)=u_0^-1CP_0, s(Lift_E(C))=C. The lift preserves the exact filtration valuation. Its crucial additional identity is tr_L(Lift_E(C)) =Tr_E/L (κ^-1tr_E(C)). Define B(L,t_L;h_1,…,h_d) =α I_p^d-1+ Lift_E(B(E,z;h_1,…,h_d-1)), with the degree-p multiplication matrix as the base case. Induction proves trace zero at every native rank. T…
Exact mathematical passage
\kappa=-f'(z)/z^{p-1}
F''Use in source · 2 passages
Search aliases: F''
Use in source · 20_arithmetic.tex, line 517
For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [source reference] proves that multiplication adds charges. If FG=1, the same formula yields A_-cG=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G'). Induction, starting…
Exact mathematical passage
\label{eq:arith-charged-product} A_{c+d}(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'.
f''Use in source · 4 passages
Search aliases: f''
Use in source · Paper_VI_Traditional_Final.tex, line 3219
[Generator and Dirichlet form] On the polynomial subspace, the infinitesimal generator of ( O_t) is the differential operator L f=f''-xf'. It satisfies L h_n=-nh_n. For every polynomial f, ip- L ff_L^2(γ) = ∫_ Rabsf'(x)^2γ( d x).
Exact mathematical passage
\Lcal f=f''-xf'.
F'''Use in source · 2 passages
Search aliases: F'''
Use in source · 20_arithmetic.tex, line 517
For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [source reference] proves that multiplication adds charges. If FG=1, the same formula yields A_-cG=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G'). Induction, starting…
Exact mathematical passage
\label{eq:arith-charged-product} A_{c+d}(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'.
f^{\mathsf T}Use in source · 2 passages
Search aliases: f^ T mathsf
Use in source · atlas-pages.tex, line 24
vpage2A dictionary with orientationsREAD BEFORE MULTIPLYING / CONVENTIONSChapter VI, Part I: Sections 2, 4, 6, 7, and 12. Column probabilities and backward observables. An apparent disagreement between two transfer formulas often comes from acting on different objects. Throughout this guide, probabilities are columns and observables are functions written as columns. Their pairing is f^ Tp. The transpose distinguishes the forward action on distributions from the backward action on observations. s…
Exact mathematical passage
f^{\mathsf T}p
F^{(k)}Use in source · 2 passages
Search aliases: F^(k)
Use in source · Paper_VI_Traditional_Final.tex, line 1658
For clarity, the finite-dimensional holomorphic calculus used below has the following canonical definition. For A acting on C^d, its generalized eigenspaces E_μ=ker(A-μ I)^d form a direct-sum decomposition over the eigenvalues. Define the unique linear operator F(A) by .F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent of a basis and of increasing the truncation index: the extra nilpotent powers vanish. It depends only on the finite Taylor jets of F at the spectrum. I…
Exact mathematical passage
\left.F(A)\right|_{E_\mu} =\left.\sum_{k=0}^{d-1}\frac{F^{(k)}(\mu)}{k!}(A-\mu I)^k\right|_{E_\mu}.
f^{(m)}Use in source · 3 passages
Search aliases: f^(m)
Use in source · Paper_V_Traditional_Final.tex, line 1673
The singular packet results begin with trace data. If a piecewise smooth function has a jump in its derivative of order m at τ_q, then the packet jumpf^(m)_τ_q frac(t-τ_q)_+^mm! removes exactly that trace jump. Subtracting all packets through order r-1 leaves a global W^r,p remainder. The packet coefficients are uniquely determined by the one-sided trace jumps. This gives a finite and explicit singular part for the class covered by [source reference].
Exact mathematical passage
\jump{f^{(m)}}_{\tau_q} \frac{(t-\tau_q)_+^m}{m!}
f^{(r)}Use in source · 1 passage
Search aliases: f^(r)
Use in source · Paper_V_Traditional_Final.tex, line 2555
sided traces at the endpoints. Define a_q,m = jumpf^(m)_τ_q = f^(m)(τ_q+)-f^(m)(τ_q-) and j_f(t) = ∑_q=1^Q∑_m=0^r-1a_q,mϕ_m,τ_q(t). Then s_f=f-j_f has matching one-sided derivatives through order r-1 at every breakpoint and belongs to W^r,p(0,1). Its weak derivative of order r equals the piecewise classical derivative f^(r) almost everywhere.
Exact mathematical passage
f^{(r)}
F^*Use in source · 8 passages
Search aliases: F^*
Use in source · 02_strengthened.tex, line 1295
Exact correction of an error span means that one completely positive trace preserving recovery R satisfies R(Fρ F^*)=q_Fρ for every F in that span and every code-supported matrix ρ, with a nonnegative scalar q_F depending on F but not on the logical input. For every normalized channel with Kraus operators in the same span, the same recovery restores the code input with total scalar one. Individual unnormalized errors are not required to have unit recovery weight.
Exact mathematical passage
\Rcal(F\rho F^*)=q_F\rho
F^{-1}Use in source · 1 passage
Search aliases: F^-1
Use in source · Paper_XI_Traditional_Final.tex, line 277
[Scalar covariance model] Let F=diag(1,q,q^2,…,q^m-1) and let N_j-1,j=1. Then FNF^-1=q^-1N.
Exact mathematical passage
FNF^{-1}=q^{-1}N.
f^{-1}Use in source · 1 passage
Search aliases: f^-1
Use in source · supplement_08.tex, line 79
The reciprocal is integral. For f∈ I_p with constant coefficient one, the usual power-series reciprocal is used. Its inverse equation yields the exact coefficient recurrence a_f^-1(d)=δ_d0-∑_r=0^d-1 binom dr a_f^-1(r)a_f(d-r). Strong induction proves that all of these coefficients are integral. Combining this construction with the preceding all-order theorem proves flatness of the actual reciprocal. No flatness or integrality of that reciprocal is assumed in the normalized-series specialization.
Exact mathematical passage
a_{f^{-1}}(d)=\delta_{d0}-\sum_{r=0}^{d-1} \binom dr a_{f^{-1}}(r)a_f(d-r).
\mathcal F^2Use in source · 2 passages
Search aliases: F^2 mathcal
Use in source · strengthening.tex, line 313
0). For m=0, the assertion is that every even Euler number is odd. With P=( E^2-1)/2 and γ_m(P)=P^m/m!, the case n=0 says L(γ_m(P))∈ Z_2^× (m≥0). Introduce an independent Euler variable F and Q=( F^2-1)/2.
Exact mathematical passage
Q=(\mathcal F^2-1)/2
F^{2m}Use in source · 1 passage
Search aliases: F^2m
Use in source · 04_correction_transport.tex, line 420
Direct expansion gives Ω(η+h) = Ω(η)+dh+[η,h]+frac12[h,h]. The filtration rule places [h,h] in F^2m L^2, which is contained in F^m+1 L^2 because m≥1. The congruence follows. The last statement is [source reference] applied to the induced graded correction map.
Exact mathematical passage
F^{2m}\Lcal^2
F^4Use in source · 1 passage
Search aliases: F^4
Use in source · dyadic_synthesis.tex, line 310
Use the actual rank-four outer approximation field E_0/F of degree two and its centralizer B=End_E_0(F^4). The odd lower critical level is r=3h_1=2m+1. The half leading level is a=3h_2>r. Define P=J U_B^m, where J=J^1 is the actual compact type group. The filtered centralizer estimate for the actual final field, without assuming E_0⊂ K, proves that P is a compact group with J normal of index two. For a concrete new coset representative u, λ=θ^u/θ on the actual H=H^1 is nontrivial with values {±1…
Exact mathematical passage
B=\operatorname{End}_{E_0}(F^4)
F^\bulletUse in source · 1 passage
Search aliases: F^bullet bullet
Use in source · 03_weighted.tex, line 310
[Finite precision] Let F^bullet R be a multiplicative decreasing filtration and let q_L:R→ R_L be a ring homomorphism that kills F^L R. Suppose H=∑_e∈ EH_e, D^-1H_e∈ F^w(e)R, w(e)≥1, with E finite. Then q_L(D^-1H)^L=0, the finite sum in [source reference](a) is a two-sided inverse after applying q_L, and its word expansion may discard every word of weight at least L. More explicitly, for a length cutoff N, define S^pr_L,N = ∑_n=0^N-1(-1)^n ∑_substackγ∈ E^n
Exact mathematical passage
F^\bullet R
F^{d(\alpha,\beta)}Use in source · 1 passage
Search aliases: F^d(α,β) alpha beta
Use in source · 03_weighted.tex, line 130
[Minimum path weight] For packets α and β, let d(α,β) be the least total weight of a directed path from α to β, with d=∞ when no path exists. Use the convention F^∞ B=⋂_L≥0F^L B=0. If α≠β, then P_β(D+N)^-1P_α ∈ F^d(α,β) B. If there is no path from α to β, the block is zero.
Exact mathematical passage
P_\beta(D+N)^{-1}P_\alpha \in F^{d(\alpha,\beta)}\Bcal.
f^hUse in source · 3 passages
Search aliases: f^h
Use in source · Paper_VI_Traditional_Final.tex, line 4201
[Discrete Green quadrature] If u^h solves L_hu^h=f^h, then every nodal value has the exact representation u_i^h = h∑_j=1^nG(x_i,x_j)f_j^h. Thus the discrete Green operator is the nodal rectangle quadrature of the continuum Green kernel.
Exact mathematical passage
L_hu^h=f^h,
f^h_iUse in source · 1 passage
Search aliases: f^h_i
Use in source · Paper_VI_Traditional_Final.tex, line 4243
[Second order Poisson solution error] Suppose u∈ C^4[0,1] solves the Dirichlet problem above. Let f^h_i=f(x_i) and let u^h solve L_hu^h=f^h. Then max_iabsu^h_i-u(x_i) ≤ frach^296 normu^(4)_∞.
Exact mathematical passage
f^h_i=f(x_i)
F^iUse in source · 1 passage
Search aliases: F^i
Use in source · 04_correction_transport.tex, line 386
[Maurer Cartan lifting obstruction] Let L be a complete filtered differential graded Lie algebra over a ring in which two is invertible. Assume dF^i L⊆ F^i L, [F^i L,F^j L]⊆ F^i+j L. For a degree one element η, put Ω(η)=dη+frac12[η,η]. Suppose Ω(η)∈ F^m L^2 with m≥1. For every h∈ F^m L^1, Ω(η+h) ≡ Ω(η)+d_η h pmodF^m+1 L^2, d_η=d+[η,mathord·]. The obstruction to improving η by one depth is therefore the class of Ω(η) in the cokernel of gr^m(d_η): gr^m L^1 ⟶ gr^m L^2.
Exact mathematical passage
dF^i\Lcal\subseteq F^i\Lcal, \qquad [F^i\Lcal,F^j\Lcal]\subseteq F^{i+j}\Lcal.
F^{i+j}Use in source · 1 passage
Search aliases: F^i+j
Use in source · 04_correction_transport.tex, line 386
[Maurer Cartan lifting obstruction] Let L be a complete filtered differential graded Lie algebra over a ring in which two is invertible. Assume dF^i L⊆ F^i L, [F^i L,F^j L]⊆ F^i+j L. For a degree one element η, put Ω(η)=dη+frac12[η,η]. Suppose Ω(η)∈ F^m L^2 with m≥1. For every h∈ F^m L^1, Ω(η+h) ≡ Ω(η)+d_η h pmodF^m+1 L^2, d_η=d+[η,mathord·]. The obstruction to improving η by one depth is therefore the class of Ω(η) in the cokernel of gr^m(d_η): gr^m L^1 ⟶ gr^m L^2.
Exact mathematical passage
dF^i\Lcal\subseteq F^i\Lcal, \qquad [F^i\Lcal,F^j\Lcal]\subseteq F^{i+j}\Lcal.
F^jUse in source · 1 passage
Search aliases: F^j
Use in source · 04_correction_transport.tex, line 386
[Maurer Cartan lifting obstruction] Let L be a complete filtered differential graded Lie algebra over a ring in which two is invertible. Assume dF^i L⊆ F^i L, [F^i L,F^j L]⊆ F^i+j L. For a degree one element η, put Ω(η)=dη+frac12[η,η]. Suppose Ω(η)∈ F^m L^2 with m≥1. For every h∈ F^m L^1, Ω(η+h) ≡ Ω(η)+d_η h pmodF^m+1 L^2, d_η=d+[η,mathord·]. The obstruction to improving η by one depth is therefore the class of Ω(η) in the cokernel of gr^m(d_η): gr^m L^1 ⟶ gr^m L^2.
Exact mathematical passage
dF^i\Lcal\subseteq F^i\Lcal, \qquad [F^i\Lcal,F^j\Lcal]\subseteq F^{i+j}\Lcal.
F^{N+1}Use in source · 1 passage
Search aliases: F^N+1
Use in source · 04_correction_transport.tex, line 431
[Spectral sequence of a finite filtered complex] Let C^bullet be a cochain complex with a finite decreasing filtration by subcomplexes C^bullet=F^0C^bullet⊇ F^1C^bullet⊇⋯⊇ F^N+1C^bullet=0. There is a natural spectral sequence with E_0^p,q=gr^pC^p+q, E_1^p,q=H^p+q(gr^pC^bullet), which converges to the associated graded object of H^p+q(C^bullet) with its induced filtration. A filtered cochain map induces a morphism of spectral sequences.
Exact mathematical passage
C^\bullet=F^0C^\bullet\supseteq F^1C^\bullet\supseteq\cdots\supseteq F^{N+1}C^\bullet=0.
F^{W(\gamma)}Use in source · 3 passages
Search aliases: F^W(γ) gamma
Use in source · 03_weighted.tex, line 63
l, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 = ∑_γ(-1)^|γ|A_γ D^-1. The series converges in the filtration, and the term indexed by γ belongs to F^W(γ) B.
Exact mathematical passage
F^{W(\gamma)}\Bcal
F^\timesUse in source · 2 passages
Search aliases: F^× times
Use in source · dyadic_synthesis.tex, line 359
There is also a recovery from compact data explicitly recording that root. With A_0=Λ(ϖ_K) and X=c - Ind_J^Gη, define (Tf)(g)=A_0^-1f(ϖ_Kg). Then X/(T-1)X≃π. After imposing the fixed central character, N/(F^× J) is cyclic of order N, and the finite average N^-1∑_i=0^N-1T^i selects this same extension. These operators act on complex representation spaces, so their scalar division by N is valid. This is a different recovery from the Block159 projector onto a sector of η. The exact kernel and finit…
Exact mathematical passage
\mathcal N/(F^\times J)
F^\inftyUse in source · 1 passage
Search aliases: F^∞ infty
Use in source · 03_weighted.tex, line 129
[Minimum path weight] For packets α and β, let d(α,β) be the least total weight of a directed path from α to β, with d=∞ when no path exists. Use the convention F^∞ B=⋂_L≥0F^L B=0. If α≠β, then P_β(D+N)^-1P_α ∈ F^d(α,β) B. If there is no path from α to β, the block is zero.
Exact mathematical passage
F^\infty\Bcal=\bigcap_{L\geq0}F^L\Bcal=0
f_{(k-d)\bmod N}Use in source · 1 passage
Search aliases: f_(k-d)bmod N bmod
Use in source · atlas-pages.tex, line 105
inal sample δ_1. Under DI_2 it stays even but moves to channel address one: this is δ_2. Thus J(D)^2=DI_2, H(D)=I_2+2J(D)-J(D)^2+J(D)^3 recovers the previous page's four-tap transfer matrix. The same delay algebra explains both the off-diagonal phase swap and the channel powers. headSigned delays need the same care For an original-sample delay d at modulus N, y_k[n]=f_(k-d)bmod N [n+lfloorfrack-dNrfloor]. This includes advances when d<0. The finite verification checks the formula on signed sampl…
Exact mathematical passage
y_k[n]=f_{(k-d)\bmod N} \!\left[n+\left\lfloor\frac{k-d}{N}\right\rfloor\right].
F_*Use in source · 4 passages
Search aliases: F_*
Use in source · Paper_V_Traditional_Final.tex, line 2467
[Stability of the optimal cost] Let c and widetilde c be two node cost functions. For an admissible pruning L, set ε( L) = ∑_v∈ L|c(v)-widetilde c(v)| and let ε_* = max_ Lε( L). If F_* and widetilde F_* are the two optimal root costs, then |F_*-widetilde F_*| ≤ ε_*.
Exact mathematical passage
F_*
F_{1,0}Use in source · 1 passage
Search aliases: F_1,0
Use in source · strengthening.tex, line 200
j(n)∈ p^k Z_p. Moreover, Δ_1^k A_N,j(n)=(-1)^n+k(S+1)^k C_N,j(n), so A_N,j has the ordinary p^k difference property. At p=2, L_1=L_-1-1. Its integral binomial expansion on F_N,j gives Δ_1^k C_N,j(n)∈2^k Z_2 directly. For N=1 the interior sum in the product formula is empty, so the same argument applies without change. Equivalently, F_1,0(t)=e^-t.
Exact mathematical passage
F_{1,0}(t)=e^{-t}
F_{11}Use in source · 1 passage
Search aliases: F_11
Use in source · supplement_06.tex, line 159
ndary checksExample 9.3 and supporting results The independently defined Lucas recurrence gives (L_0,L_1,L_2,L_3,L_4,L_5)=(2,1,3,4,7,11). For N=3, the general theorem therefore specializes to u_k+2=4u_k+1+u_k. The implementation proves this for every offset j and every k, not only the displayed sample. At j=2, direct evaluation of the actual Fibonacci sequence gives (F_2,F_5,F_8,F_11,F_14,F_17)=(1,5,21,89,377,1597). This is the exact manuscript example.
Exact mathematical passage
(F_2,F_5,F_8,F_{11},F_{14},F_{17})=(1,5,21,89,377,1597).
F_{14}Use in source · 1 passage
Search aliases: F_14
Use in source · supplement_06.tex, line 159
ndary checksExample 9.3 and supporting results The independently defined Lucas recurrence gives (L_0,L_1,L_2,L_3,L_4,L_5)=(2,1,3,4,7,11). For N=3, the general theorem therefore specializes to u_k+2=4u_k+1+u_k. The implementation proves this for every offset j and every k, not only the displayed sample. At j=2, direct evaluation of the actual Fibonacci sequence gives (F_2,F_5,F_8,F_11,F_14,F_17)=(1,5,21,89,377,1597). This is the exact manuscript example.
Exact mathematical passage
(F_2,F_5,F_8,F_{11},F_{14},F_{17})=(1,5,21,89,377,1597).
F_{17}Use in source · 1 passage
Search aliases: F_17
Use in source · supplement_06.tex, line 159
ndary checksExample 9.3 and supporting results The independently defined Lucas recurrence gives (L_0,L_1,L_2,L_3,L_4,L_5)=(2,1,3,4,7,11). For N=3, the general theorem therefore specializes to u_k+2=4u_k+1+u_k. The implementation proves this for every offset j and every k, not only the displayed sample. At j=2, direct evaluation of the actual Fibonacci sequence gives (F_2,F_5,F_8,F_11,F_14,F_17)=(1,5,21,89,377,1597). This is the exact manuscript example.
Exact mathematical passage
(F_2,F_5,F_8,F_{11},F_{14},F_{17})=(1,5,21,89,377,1597).
F_1^{-1}Use in source · 1 passage
Search aliases: F_1^-1
Use in source · visual_atlas.tex, line 195
Transport from chart a to chart b is forced: J_b← a=F_b^-1F_a. The route 0→1→2 agrees with the direct route because the middle frame cancels: (F_2^-1F_1)(F_1^-1F_0)=F_2^-1F_0. Returning to chart 0 cancels every remaining frame. The final coordinate vector is (3,2)^ T, not a sheared or rotated variant.
Exact mathematical passage
(F_2^{-1}F_1)(F_1^{-1}F_0)=F_2^{-1}F_0.
F_2Use in source · 8 passages
Search aliases: F_2
Use in source · visual_atlas.tex, line 181
vpage9One vector, three exact framesMODEL C / CANONICAL TRANSPORT HAS NO HOLONOMYChapter III, mathematical text: Theorem 4.1, Theorem 4.2 Let F_a synthesize chart coordinates into one ambient QQ^2. Choose F_0=I, F_1=mat1 · 1 0 · 1, F_2=mat2 · 0 1 · 1. The same ambient vector x=(3,2)^ T has chart coordinates u_a=F_a^-1x.
Exact mathematical passage
F_0=I,\qquad F_1=\mat{1&1\\0&1},\qquad F_2=\mat{2&0\\1&1}.
\F_2Use in source · 1 passage
Search aliases: F_2 F mathbb
Use in source · 04_arithmetic.tex, line 668
The two characteristic factors x-1 and x+1 have the same reduction in F_2[x]. The rational matrices E_+ and E_- themselves cannot be reduced modulo 2, because their entries contain 1/2. Consequently the integral special fiber does not inherit their separate idempotent splitting, even though the two projectors remain distinct over Q_2. This is the simplest model of delayed dyadic separation.
Exact mathematical passage
\F_2[x]
\mathbb F_2Use in source · 9 passages
Search aliases: F_2 mathbb
Use in source · visual_atlas.tex, line 133
vpage6Selectors survive without divisionMODEL A / RESIDUE MASKS IN CHARACTERISTIC TWOChapter II, mathematical text: Proposition 4.2 Work first in F_2, where 1+1=0. The eight coefficients below are just a finite display of a coefficient sequence. The even and odd projectors are diagonal masks with entries zero and one.
Exact mathematical passage
\mathbb F_2
f_2Use in source · 8 passages
Search aliases: f_2
Use in source · 02_foundations.tex, line 954
Invariant factors, Jordan blocks, and cyclicity. [Invariant factor count] If X≠0 and the rational canonical decomposition is written with nonconstant invariant factors X≅⊕_j=1^r𝕜[t]/(f_j), f_1mid f_2mid⋯mid f_r, then srank_𝕜[T](X)=r. [source reference]
Exact mathematical passage
X\cong\bigoplus_{j=1}^{r}\bbk[t]/(f_j), \qquad f_1\mid f_2\mid\cdots\mid f_r,
F_2^{-1}Use in source · 1 passage
Search aliases: F_2^-1
Use in source · visual_atlas.tex, line 195
Transport from chart a to chart b is forced: J_b← a=F_b^-1F_a. The route 0→1→2 agrees with the direct route because the middle frame cancels: (F_2^-1F_1)(F_1^-1F_0)=F_2^-1F_0. Returning to chart 0 cancels every remaining frame. The final coordinate vector is (3,2)^ T, not a sheared or rotated variant.
Exact mathematical passage
(F_2^{-1}F_1)(F_1^{-1}F_0)=F_2^{-1}F_0.
\mathbb F_2^2Use in source · 1 passage
Search aliases: F_2^2 mathbb
Use in source · dyadic_synthesis.tex, line 292
e P=U_K^1U_A^m. This is an actual compact open group, with Jtriangleleft P, [P:J]=2 and P∩(K^× J)=J. For g∈ P∖ J, the actual intertwining theorem makes λ=θ^g/θ nontrivial. The commutator [P,J]⊂ U_A^m+1 lies in the forced domain of the additive character, so λ has values {±1}. Put H_0=kerλ. Then P/H_0≃ F_2^2, E=Ind_J^Pθ is irreducible of dimension 2. The actual represented generators are a diagonal sign and an off-diagonal matrix that anticommute. Thus their commutator form is nondegenerate, and …
Exact mathematical passage
P/H_0\simeq\mathbb F_2^2,\qquad E=\operatorname{Ind}_J^P\theta\text{ is irreducible of dimension }2.
\mathbb F_2^4Use in source · 1 passage
Search aliases: F_2^4 mathbb
Use in source · dyadic_synthesis.tex, line 320
) is the actual compact type group. The filtered centralizer estimate for the actual final field, without assuming E_0⊂ K, proves that P is a compact group with J normal of index two. For a concrete new coset representative u, λ=θ^u/θ on the actual H=H^1 is nontrivial with values {±1}. Put H_0=kerλ⊂ H. The calculation proves P/H_0≃ F_2^4, E=Ind_J^Pη is irreducible of dimension 4. The new represented commutators are scalar, its quotient has exponent two, and nondegeneracy follows from its exact s…
Exact mathematical passage
P/H_0\simeq\mathbb F_2^4,\qquad E=\operatorname{Ind}_J^P\eta\text{ is irreducible of dimension }4.
\mathbb F_2^\timesUse in source · 1 passage
Search aliases: F_2^× mathbb times
Use in source · dyadic_synthesis.tex, line 120
For the final totally ramified field, J/J^1= F_2^×=1, so J=J^1 and the canonical beta extension is η. Extend to K^× J, and choose the noncompact scalar to match ω(t). The needed N-th root exists in C^×. The compact type stays fixed. All choices precede H5. See pathcentral/DYADIC_FULL_CHARACTER167.md for the complete proof.
Exact mathematical passage
J/J^1=\mathbb F_2^\times=1
F_3Use in source · 2 passages
Search aliases: F_3
Use in source · strengthening.tex, line 739
e a^-2+b^-2=0. Its constant is one, only degrees divisible by four survive, and the Bernoulli expansion gives D_3(t)=∑_i≥0d_ifract^4i(4i)!, d_i=3(-frac14)^i frac2^4i+2B_4i+2(4i+2)(4i+1). Again v_2(d_i)≥2i, and the inverse coefficients h_i have the same bound. Since F_3=F_2D_3^-1, C_3(n)-C_2(n)=∑_i≥1h_ibinom4n4iC_2(n-i). The scaled-binomial argument [source reference] applies unchanged. The basic bound for C_2 includes order zero. Its continuous extension is available from the threshold-free theo…
Exact mathematical passage
F_3=F_2D_3^{-1}
\F_3Use in source · 1 passage
Search aliases: F_3 F mathbb
Use in source · supplement_07.tex, line 133
Three checked boundaries. The initial row. For (p,m)=(3,2), the same reduced recurrence gives e_0=1 and e_1=-1=2 in F_3. Its proposed period is one, so the positive-tail result cannot in general include row zero.
Exact mathematical passage
\F_3
\mathbb F_3Use in source · 3 passages
Search aliases: F_3 mathbb
Use in source · 02_coordinates.tex, line 443
The field or orthogonality condition is essential. A primitive element of order N in a ring with zero divisors does not by itself force the geometric sums away from the diagonal to vanish. For example, in F_3× F_3 the element (-1,1) has order two, but the second component of its nontrivial character sum is nonzero. The coefficient selector uses no division and remains valid without any Fourier certificate. This distinction becomes decisive at the dyadic prime.
Exact mathematical passage
\mathbb F_3\times\mathbb F_3
F_{3k+j}Use in source · 1 passage
Search aliases: F_3k+j
Use in source · 72_recovered.tex, line 2810
[t] centering includegraphics[width=.95linewidth,height=3.15in,keepaspectratio]../../figures/reconstructed/recurrence_packets.pdf captionNew reconstruction. Fibonacci residues F_3k+j, with F_0=0 and F_1=1. Each packet evolves by the same jump matrix C^3 and has a different initial address.
Exact mathematical passage
F_{3k+j}
F_{4,j}Use in source · 1 passage
Search aliases: F_4,j
Use in source · strengthening.tex, line 252
Write C_j=C_4,j and F_j=F_4,j. The all-row theorem removes the threshold, but at N=4 it leaves a further factorial gain unused. The exact algebra below proves that gain for all four addresses and identifies several infinite families where it is sharp.
Exact mathematical passage
F_j=F_{4,j}
F_5Use in source · 1 passage
Search aliases: F_5
Use in source · supplement_06.tex, line 159
ndary checksExample 9.3 and supporting results The independently defined Lucas recurrence gives (L_0,L_1,L_2,L_3,L_4,L_5)=(2,1,3,4,7,11). For N=3, the general theorem therefore specializes to u_k+2=4u_k+1+u_k. The implementation proves this for every offset j and every k, not only the displayed sample. At j=2, direct evaluation of the actual Fibonacci sequence gives (F_2,F_5,F_8,F_11,F_14,F_17)=(1,5,21,89,377,1597). This is the exact manuscript example.
Exact mathematical passage
(F_2,F_5,F_8,F_{11},F_{14},F_{17})=(1,5,21,89,377,1597).
\F_5Use in source · 1 passage
Search aliases: F_5 F mathbf
Use in source · Paper_X_Block27_Amended.tex, line 433
[Finite packet complex] The exact modular test uses two invariant subcomplexes over F_5. Their cohomology dimensions are (0,0,0), (1,2,1), and the reconstructed total is dim(H^0,H^1,H^2)=(1,2,1). The packet-sum residual is zero.
Exact mathematical passage
\F_5
\mathbf F_5^2Use in source · 2 passages
Search aliases: F_5^2 mathbf
Use in source · Paper_X_Block27_Amended.tex, line 448
An explicit realization of the finite pattern. A separate finite-complex implementation uses F_5^2xrightarrowd_0 F_5^4xrightarrowd_1 F_5^2, d_0(a,b)=(a,0,0,0), d_1(a,b,c,d)=(b,0). The first packet is the exact complex with dimensions (1,2,1), and the second packet has zero differentials and dimensions (1,2,1). The actual middle cohomology is linearly isomorphic to F_5^2, which is checked in Lean. Independent exhaustive coset tests check all three cohomology degrees after changes of basis. This r…
Exact mathematical passage
\mathbf F_5^2\xrightarrow{d_0}\mathbf F_5^4\xrightarrow{d_1}\mathbf F_5^2, \quad d_0(a,b)=(a,0,0,0),\quad d_1(a,b,c,d)=(b,0).
\mathbf F_5^4Use in source · 1 passage
Search aliases: F_5^4 mathbf
Use in source · Paper_X_Block27_Amended.tex, line 448
An explicit realization of the finite pattern. A separate finite-complex implementation uses F_5^2xrightarrowd_0 F_5^4xrightarrowd_1 F_5^2, d_0(a,b)=(a,0,0,0), d_1(a,b,c,d)=(b,0). The first packet is the exact complex with dimensions (1,2,1), and the second packet has zero differentials and dimensions (1,2,1). The actual middle cohomology is linearly isomorphic to F_5^2, which is checked in Lean. Independent exhaustive coset tests check all three cohomology degrees after changes of basis. This r…
Exact mathematical passage
\mathbf F_5^2\xrightarrow{d_0}\mathbf F_5^4\xrightarrow{d_1}\mathbf F_5^2, \quad d_0(a,b)=(a,0,0,0),\quad d_1(a,b,c,d)=(b,0).
\mathbb F_7Use in source · 1 passage
Search aliases: F_7 mathbb
Use in source · visual_atlas.tex, line 156
vpage7A Fourier certificate you can inspectMODEL A / AN EXACT NINE-ENTRY TABLEChapter II, mathematical text: Theorem 4.3 Take N=3 and work in F_7. Set ω=2, ω^-1=4, η=3^-1=5. Here 2^3=1 and neither 2 nor 2^2 is one. The unnormalized geometric sums are the entries in the first matrix.
Exact mathematical passage
\mathbb F_7
F_8Use in source · 1 passage
Search aliases: F_8
Use in source · supplement_06.tex, line 159
ndary checksExample 9.3 and supporting results The independently defined Lucas recurrence gives (L_0,L_1,L_2,L_3,L_4,L_5)=(2,1,3,4,7,11). For N=3, the general theorem therefore specializes to u_k+2=4u_k+1+u_k. The implementation proves this for every offset j and every k, not only the displayed sample. At j=2, direct evaluation of the actual Fibonacci sequence gives (F_2,F_5,F_8,F_11,F_14,F_17)=(1,5,21,89,377,1597). This is the exact manuscript example.
Exact mathematical passage
(F_2,F_5,F_8,F_{11},F_{14},F_{17})=(1,5,21,89,377,1597).
F_aUse in source · 9 passages
Search aliases: F_a
Use in source · 01_finite_coordinates.tex, line 378
Let every chart a carry a linear frame equivalence F_a:Mxrightarrow∼M. The formal Chapter III module defines transport from a to b by J_b← a=F_b^-1F_a. This frame formula means synthesize in chart a, then analyze in chart b.
Exact mathematical passage
F_a:M\xrightarrow{\sim}M
f_{a+b,s+p^at}Use in source · 1 passage
Search aliases: f_a+b,s+p^at
Use in source · 72_recovered.tex, line 2115
[Exact local branch reconstruction] The family {f_a,s}_0≤ s<p^a determines f uniquely on Z_p. More precisely, if y∈ s+p^a Z_p, then f(y)=f_a,s(fracy-sp^a). For a,b≥1, (f_a,s)_b,t = f_a+b,s+p^at. Thus local refinement is exactly the p adic digit tree.
Exact mathematical passage
\boxed{ (f_{a,s})_{b,t} = f_{a+b,s+p^at}. }
f_{a,s}Use in source · 4 passages
Search aliases: f_a,s
Use in source · 72_recovered.tex, line 2103
For a≥1 and 0≤ s<p^a, define f_a,s(x)=f(p^ax+s).
Exact mathematical passage
f_{a,s}(x)=f(p^ax+s).
F_a^*Use in source · 1 passage
Search aliases: F_a^*
Use in source · 03_supplement_05.tex, line 53
Define the transformed errors by an actual finite sum, B_r=∑_a U_arF_a. The formal module proves the identities B_rP=B_r, B_r^*B_s=δ_rsλ_rP, ∑_rB_rρ B_r^*=∑_aF_aρ F_a^*. For ρ supported on the code, the restricted and unrestricted error sums are equal. Thus the unitary change of error coordinates preserves the actual channel on the code.
Exact mathematical passage
B_rP=B_r,\qquad B_r^*B_s=\delta_{rs}\lambda_rP, \qquad \sum_rB_r\rho B_r^*=\sum_aF_a\rho F_a^*.
F_a^{-1}Use in source · 1 passage
Search aliases: F_a^-1
Use in source · visual_atlas.tex, line 185
1 · 1. The same ambient vector x=(3,2)^ T has chart coordinates u_a=F_a^-1x.
Exact mathematical passage
u_a=F_a^{-1}x
f_{AB}^{\ \ C}Use in source · 2 passages
Search aliases: f_AB^ C
Use in source · 01_finite_core.tex, line 1198
At one finite level, let widehat C_A be a finite family of matrix constraints, with fixed scalar structure coefficients, and let Π^phys be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients f_AB^ C, define the anomaly operator A_AB = [widehat C_A,widehat C_B] -i∑_C f_AB^ Cwidehat C_C. Its physical packet is Π^phys A_ABΠ^phys. In this exact finite setting, widehat C_CΠ^phys=0 for every C. Associativity gives A_ABΠ^phys=0, Π^phys…
Exact mathematical passage
f_{AB}^{\ \ C}
F_b^{-1}Use in source · 2 passages
Search aliases: F_b^-1
Use in source · 01_finite_coordinates.tex, line 380
Let every chart a carry a linear frame equivalence F_a:Mxrightarrow∼M. The formal Chapter III module defines transport from a to b by J_b← a=F_b^-1F_a. This frame formula means synthesize in chart a, then analyze in chart b.
Exact mathematical passage
J_{b\leftarrow a}=F_b^{-1}F_a.
f_cUse in source · 3 passages
Search aliases: f_c
Use in source · 03_supplement_01.tex, line 188
A constraint diagnostic that vanishes automatically. Section 12 defines the physical projector using the exact common kernel of the constraints. In finite dimensions this gives C_aP=0 for every a. Absorb the factor i and any structure coefficient into f_c, and write A_ab=C_aC_b-C_bC_a-∑_cf_cC_c. [Exact common-kernel compression] If C_cP=0 for every c, then A_abP=0, P A_abP=0. No idempotence or adjoint property of P is needed for this implication. Associativity gives A_abP=C_a(C_bP)-C_b(C_aP)-∑_c…
Exact mathematical passage
f_c
F_c^nUse in source · 3 passages
Search aliases: F_c^n
Use in source · 03_nonlinear_coordinates.tex, line 796
[Iterated orbit defect] Let F_f:X_f→ X_f and F_c:X_c→ X_c, with coarsening C:X_f→ X_c. Suppose F_c is q-Lipschitz. For a fine orbit starting at x, assume Δ_F(F_f^j x) ≤ε_j. Then CF_f^n(x)-F_c^n(Cx) ≤ ∑_j=0^n-1q^n-1-jε_j. If 0≤ q<1 and every ε_j≤ε, then CF_f^n(x)-F_c^n(Cx) ≤ εfrac1-q^n1-q.
Exact mathematical passage
\left\|CF_f^n(x)-F_c^n(Cx)\right\| \le \sum_{j=0}^{n-1}q^{n-1-j}\varepsilon_j.
F_dUse in source · 1 passage
Search aliases: F_d
Use in source · continuation_06.tex, line 10
Let p be prime, R= Z_p, K= Q_p, and d,s,r∈ N. Let F=(F_1,…,F_d)∈ R[X_1,…,X_d]^d, a∈ R^d, and let J=DF(a)=(∂_jF_i(a))_i,j. Assume J∈GL_d(K), p^sJ^-1∈ M_d(R), F(a)∈ p^rR^d, r>2s. Then ∃! b∈ a+p^s+1R^d: F(b)=0, b-a∈ p^r-sR^d. All vector divisibility assertions are coordinatewise. The uniqueness predicate is the entire ball of depth s+1: competitors need not be assumed to have depth r-s. The theorem permits arbitrary integral p-adic coefficients and centers, every prime including two, and dimension …
Exact mathematical passage
F=(F_1,\ldots,F_d)\in R[X_1,\ldots,X_d]^d
F_f^jUse in source · 1 passage
Search aliases: F_f^j
Use in source · 03_nonlinear_coordinates.tex, line 792
[Iterated orbit defect] Let F_f:X_f→ X_f and F_c:X_c→ X_c, with coarsening C:X_f→ X_c. Suppose F_c is q-Lipschitz. For a fine orbit starting at x, assume Δ_F(F_f^j x) ≤ε_j. Then CF_f^n(x)-F_c^n(Cx) ≤ ∑_j=0^n-1q^n-1-jε_j. If 0≤ q<1 and every ε_j≤ε, then CF_f^n(x)-F_c^n(Cx) ≤ εfrac1-q^n1-q.
Exact mathematical passage
\|\Delta_F(F_f^j x)\|\le\varepsilon_j.
F_f^nUse in source · 3 passages
Search aliases: F_f^n
Use in source · 03_nonlinear_coordinates.tex, line 796
[Iterated orbit defect] Let F_f:X_f→ X_f and F_c:X_c→ X_c, with coarsening C:X_f→ X_c. Suppose F_c is q-Lipschitz. For a fine orbit starting at x, assume Δ_F(F_f^j x) ≤ε_j. Then CF_f^n(x)-F_c^n(Cx) ≤ ∑_j=0^n-1q^n-1-jε_j. If 0≤ q<1 and every ε_j≤ε, then CF_f^n(x)-F_c^n(Cx) ≤ εfrac1-q^n1-q.
Exact mathematical passage
\left\|CF_f^n(x)-F_c^n(Cx)\right\| \le \sum_{j=0}^{n-1}q^{n-1-j}\varepsilon_j.
f_hUse in source · 3 passages
Search aliases: f_h
Use in source · 72_recovered.tex, line 842
Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding term vanishes for positive n. Otherwise ((1+ζ y)^m)^h = (1+ζ y)^p-1 =1. Every summand is therefore h periodic for positive n. Thus T_m,p preserves V_h.…
Exact mathematical passage
d_0=f_h
F_i^{-1}Use in source · 1 passage
Search aliases: F_i^-1
Use in source · Paper_X_Block27_Amended.tex, line 1445
Algebraic monodromy data on the selected copies. Fix a field K and a nonzero scalar q∈ K^×. On each constituent W_i, supply an automorphism F_i and a nilpotent endomorphism N_i with F_iN_iF_i^-1=qN_i. The scalar q is common to this finite family. The componentwise operators F_m,N_m on U_m satisfy the same relation. F_m is invertible and N_m is nilpotent. The split maps constructed above intertwine both operators. The proof first identifies the independently defined kernel through an equivariant …
Exact mathematical passage
F_iN_iF_i^{-1}=qN_i
F_jUse in source · 21 passages
Search aliases: F_j
Use in source · atlas-pages.tex, line 58
F(z)=∑_m=0^12(m+1)z^m. Then B_4,2F=3z^2+7z^6+11z^10. Applying the same selector again changes nothing. Applying a different selector gives zero. Summing the four selected polynomials returns the original F. headSelection is also a symmetry condition For the selected polynomial F_2, rotation gives F_2(iz)=-F_2(z). More generally, a residue-j section satisfies F_j(ω z)=ω^jF_j(z). The phase label therefore describes both the coefficient addresses and the response to rotation. takeawayThis is not an…
Exact mathematical passage
F_j(\omega z)=\omega^jF_j(z)
F_{j-1}Use in source · 2 passages
Search aliases: F_j-1
Use in source · 01_manuscript.tex, line 1437
[Exact discrete precision program] Let the total integer error budget be B. Define F_j(b) as the minimum cost for assigning precisions to the first j edges with total weight exactly b. Then F_0(0)=0, F_0(b)=+∞ for b≠0. Set F_j-1(u)=+∞ whenever u<0 or the state is unreachable. Then F_j(b) = min_k { F_j-1(b-w_j,k)+c_j,k }. The minimum of F_m(b) over 0≤ b≤ B is the exact minimum cost for the quantized model.
Exact mathematical passage
F_{j-1}(u)=+\infty
F_j^*Use in source · 4 passages
Search aliases: F_j^*
Use in source · 02_strengthened.tex, line 1343
For z∈ C^r, let F_z=∑_az_aE_a. Then PF_z^*F_zP=(z^*cz)P≥0, so c≥0. The unitary change of error coordinates gives PF_j^*F_ℓ P = λ_jδ_jℓP. Thus V_j^*V_ℓ=δ_jℓP. Each V_j is an isometry on P H, and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.
Exact mathematical passage
PF_j^*F_\ell P = \lambda_j\delta_{j\ell}P.
f_j^hUse in source · 1 passage
Search aliases: f_j^h
Use in source · Paper_VI_Traditional_Final.tex, line 4205
[Discrete Green quadrature] If u^h solves L_hu^h=f^h, then every nodal value has the exact representation u_i^h = h∑_j=1^nG(x_i,x_j)f_j^h. Thus the discrete Green operator is the nodal rectangle quadrature of the continuum Green kernel.
Exact mathematical passage
u_i^h = h\sum_{j=1}^nG(x_i,x_j)f_j^h.
F_{jk}Use in source · 3 passages
Search aliases: F_jk
Use in source · pages.tex, line 73
vpage5Four coefficients, four exact frequenciesCIRCULANT MULTIPLICATIONChapter VIII: Theorem 4.3. Take c=(2,1,0,-1)^T and define C(c)_jk=c_j-kbmod4. The unitary Fourier matrix is F_jk= i^-jk/2. Its normalization is visible in the exact diagonalization: C(c)=F^*diag(2Fc)F, 2Fc=(2,2-2 i,2,2+2 i)^T. vfigcirculant headFollow an actual input For x=(1,2,-1,0)^T, dense multiplication gives (0,6,0,-2)^T. Transforming x, multiplying by the four displayed eigenvalues, and applying F^* gives exactly the sa…
Exact mathematical passage
F_{jk}=\mathrm i^{-jk}/2
F_kUse in source · 1 passage
Search aliases: F_k
Use in source · Paper_V_Traditional_Final.tex, line 2804
[Background and packet error] Suppose the true and estimated packets use the same amplitude list (a_i)_i=0^I, but possibly different regions. Then j-widehat j _p^p = ∑_i,k|a_i-a_k|^pμ(E_i∩ F_k). Only off-diagonal overlaps contribute.
Exact mathematical passage
\|j-\widehat j\|_p^p = \sum_{i,k}|a_i-a_k|^p\mu(E_i\cap F_k).
F_mUse in source · 4 passages
Search aliases: F_m
Use in source · Paper_X_Block27_Amended.tex, line 1446
Algebraic monodromy data on the selected copies. Fix a field K and a nonzero scalar q∈ K^×. On each constituent W_i, supply an automorphism F_i and a nilpotent endomorphism N_i with F_iN_iF_i^-1=qN_i. The scalar q is common to this finite family. The componentwise operators F_m,N_m on U_m satisfy the same relation. F_m is invertible and N_m is nilpotent. The split maps constructed above intertwine both operators. The proof first identifies the independently defined kernel through an equivariant …
Exact mathematical passage
F_m,N_m
f_{m+1,m}Use in source · 1 passage
Search aliases: f_m+1,m
Use in source · Paper_XII_Traditional_Final.tex, line 1215
Fix y∈ Y_m. For n≥ m+1, let A_n = { z∈im(f_n,m+1): f_m+1,m(z)=y }. Each A_n is a nonempty affine subspace of V_m+1, and the sequence (A_n) is descending. A descending sequence of affine subspaces in a finite dimensional vector space stabilizes whenever every term is nonempty. Choose an element in its stable value. It lies in Y_m+1 and maps to y.
Exact mathematical passage
A_n = \left\{ z\in\im(f_{n,m+1}): f_{m+1,m}(z)=y \right\}.
F_NUse in source · 1 passage
Search aliases: F_N
Use in source · supplement_06.tex, line 126
n independent Lucas sequence Define L_0=2, L_1=1, L_n+2=L_n+L_n+1. Direct finite matrix arithmetic proves C^2=C+I. Multiplication by C^n gives C^n+2=C^n+C^n+1. Taking traces shows that tr(C^n) satisfies the same recurrence and initial values as L_n. Induction yields tr(C^N)=L_N. The implementation separately proves det C=-1, det(C^N)=(-1)^N, L_N=2F_N+1-F_N. No Binet roots or diagonalizability argument is needed.
Exact mathematical passage
\det C=-1,\qquad \det(C^N)=(-1)^N, \qquad L_N=2F_{N+1}-F_N.
f_nUse in source · 3 passages
Search aliases: f_n
Use in source · 72_recovered.tex, line 830
[Fourier invariance of the standard window operator] Let p be odd, let mmid p-1, and let h=(p-1)/m. Define (T_m,pf)_n = ∑_q=0^nbinommnmqf_q in F_p. Let V_h be the vector space of sequences satisfying f_n+h=f_n (n≥1). Then T_m,p(V_h)=V_h. The restriction of T_m,p to V_h is an automorphism.
Exact mathematical passage
f_{n+h}=f_n \qquad(n\geq1).
F_{N+1}Use in source · 1 passage
Search aliases: F_N+1
Use in source · supplement_06.tex, line 126
n independent Lucas sequence Define L_0=2, L_1=1, L_n+2=L_n+L_n+1. Direct finite matrix arithmetic proves C^2=C+I. Multiplication by C^n gives C^n+2=C^n+C^n+1. Taking traces shows that tr(C^n) satisfies the same recurrence and initial values as L_n. Induction yields tr(C^N)=L_N. The implementation separately proves det C=-1, det(C^N)=(-1)^N, L_N=2F_N+1-F_N. No Binet roots or diagonalizability argument is needed.
Exact mathematical passage
\det C=-1,\qquad \det(C^N)=(-1)^N, \qquad L_N=2F_{N+1}-F_N.
f_{n+h}Use in source · 1 passage
Search aliases: f_n+h
Use in source · 72_recovered.tex, line 830
[Fourier invariance of the standard window operator] Let p be odd, let mmid p-1, and let h=(p-1)/m. Define (T_m,pf)_n = ∑_q=0^nbinommnmqf_q in F_p. Let V_h be the vector space of sequences satisfying f_n+h=f_n (n≥1). Then T_m,p(V_h)=V_h. The restriction of T_m,p to V_h is an automorphism.
Exact mathematical passage
f_{n+h}=f_n \qquad(n\geq1).
F_{N+j}Use in source · 1 passage
Search aliases: F_N+j
Use in source · supplement_06.tex, line 174
Uniqueness from the first two packet values. For a fixed N,j, any integer sequence satisfying the proved sampled recurrence and starting at F_j,F_N+j agrees with F_Nk+j for all k. This is proved by two-step induction. It is a consequence of the established coefficients and recurrence, not a premise used to identify Fibonacci values initially.
Exact mathematical passage
F_j,F_{N+j}
F_{N,0}Use in source · 2 passages
Search aliases: F_N,0
Use in source · supplement_10.tex, line 171
Modules. mcodeEulerSeriesInverse, mcodeRiccatiFlatPropagation, mcodeRatioRiccati. 4. All-residue assembly and original coefficients. The standard-window source and inverse results supply R_1 and F_N,0. Induction over 1≤ c<N now has two proved cases. At pmid c, the singular weighted estimate and the new substitution theorem give flatness. At pnmid c, the preceding Riccati propagation applies. Thus every actual ratio R_c is flat.
Exact mathematical passage
F_{N,0}
F_{N,j}Use in source · 10 passages
Search aliases: F_N,j
Use in source · supplement_01.tex, line 104
The formalization does not stop at a recurrence. In the variable z=t^N, construct K_N,j(z)=∑_n≥0fracj!(Nn+j)!z^n, F_N,j(z)=∑_n≥0fracC_N,j(n)(Nn)!z^n. It proves the factorial coefficient identity, then F_N,jK_N,j=1 and therefore F_N,j=K_N,j^-1.
Exact mathematical passage
K_{N,j}(z)=\sum_{n\ge0}\frac{j!}{(Nn+j)!}z^n,\qquad F_{N,j}(z)=\sum_{n\ge0}\frac{C_{N,j}(n)}{(Nn)!}z^n.
F_{N,j+1}Use in source · 1 passage
Search aliases: F_N,j+1
Use in source · supplement_10.tex, line 178
Let K_j(t) be the normalized section kernel with constant coefficient one. The constructed unnormalized Euler series is F_N,j(t)=j! K_j(t)^-1. The source/inverse identities and the factorial successor identity prove F_N,j+1=F_N,jR_j+1, F_N,j=F_N,0∏_c=1^jR_c. The product formula is not an assembly hypothesis. Proved flatness of all factors and the original product-closure theorem establish flatness of the full family.
Exact mathematical passage
F_{N,j+1}=F_{N,j}R_{j+1},\qquad F_{N,j}=F_{N,0}\prod_{c=1}^{j}R_c.
f_{n,m+1}Use in source · 1 passage
Search aliases: f_n,m+1
Use in source · Paper_XII_Traditional_Final.tex, line 1215
Fix y∈ Y_m. For n≥ m+1, let A_n = { z∈im(f_n,m+1): f_m+1,m(z)=y }. Each A_n is a nonempty affine subspace of V_m+1, and the sequence (A_n) is descending. A descending sequence of affine subspaces in a finite dimensional vector space stabilizes whenever every term is nonempty. Choose an element in its stable value. It lies in Y_m+1 and maps to y.
Exact mathematical passage
A_n = \left\{ z\in\im(f_{n,m+1}): f_{m+1,m}(z)=y \right\}.
F_{N-1}Use in source · 1 passage
Search aliases: F_N-1
Use in source · Paper_IV_Traditional_Final.tex, line 565
[Hypergeometric form] Let A_N,j = { fracj+mN mathrel: 1≤ m≤ N and j+m≠ N }. Then E_N,j(λ z) = frac(λ z)^jj! _0F_N-1 ( - A_N,j mathrel, frac(λ z)^NN^N ).
Exact mathematical passage
\Efun_{N,j}(\lambda z) = \frac{(\lambda z)^j}{j!} {}_0F_{N-1} \left( \begin{matrix}-\\ A_{N,j}\end{matrix} \mathrel{,} \frac{(\lambda z)^N}{N^N} \right).
\F_pUse in source · 18 passages
Search aliases: F_p F mathbb
Use in source · 10_foundations.tex, line 173
begingroupsmallraggedright Formal correspondence. leanHurwitz.shift_convolution, leanHurwitz.prime_shift_convolution, leanHurwitz.convolution_left_eq_zero, leanHurwitzInverse.inverse_convolution, and leanHurwitzInverse.inverse_prime_difference. Working directly with coefficient sequences avoids the undefined expression 1/n! in F_p when pmid n!. endgroup
Exact mathematical passage
\F_p
\mathbb F_pUse in source · 16 passages
Search aliases: F_p mathbb
Use in source · 71_native.tex, line 95
odulo 2 has zero obstruction and lifts to 1,3 modulo 4. Both have nonzero obstruction at the next step. There is no root modulo 8 and no compatible infinite lift. item The raw Jacobian cokernel need not be presentation independent: the presentations F=(X) and G=(X,0) define the same solution system. Their raw obstruction groups at zero are respectively zero and F_p. Both actual distinguished obstructions are zero. Thus one cannot claim a canonical isomorphism of these two raw groups merely becau…
Exact mathematical passage
\mathbb F_p
\F_p^2Use in source · 1 passage
Search aliases: F_p^2 F mathbb
Use in source · 50_open_questions.tex, line 64
The raw cokernel already reveals the difficulty. The presentations F=(X) and G=(X,0) describe the same zero locus over every coefficient ring. At zero their Jacobians are, respectively, F_pxrightarrow 1 F_p, F_pxrightarrow (1,0) F_p^2. Their raw cokernels are 0 and F_p. Both distinguished obstruction classes vanish, but the ambient obstruction groups are not isomorphic. The example rules out the raw cokernel as a presentation-independent object while leaving open the possibility of an intrinsic …
Exact mathematical passage
\F_p\xrightarrow{\,1\,}\F_p, \qquad \F_p\xrightarrow{\,(1,0)\,}\F_p^2.
\F_p^dUse in source · 1 passage
Search aliases: F_p^d F mathbb
Use in source · 50_open_questions.tex, line 53
The original conjecture asks for a canonical finite obstruction packet whose vanishing is necessary and sufficient for a one-level lift in a finitely presented arithmetic seed system. For a fixed finite polynomial presentation, the chapter supplies the precise obstruction o_F,bar x=[r]∈ coker (J_F,bar x: F_p^d⟶ F_p^e), where r is the residual digit. We have proved independence from the chosen integer representative. Invariance under a change of equations or presentation requires more.
Exact mathematical passage
o_{F,\bar x}=[r]\in \coker\!\left(J_{F,\bar x}:\F_p^d\longrightarrow\F_p^e\right),
\mathbb F_p^dUse in source · 7 passages
Search aliases: F_p^d mathbb
Use in source · 71_native.tex, line 23
The divisions are exact integer divisions. Regard J as a linear map F_p^d→ F_p^r. Define
Exact mathematical passage
\mathbb F_p^d\to\mathbb F_p^r
\F_p^eUse in source · 3 passages
Search aliases: F_p^e F mathbb
Use in source · 30_lifting.tex, line 314
For prime p, rank-nullity gives dimker J_x=d-ρ_x and dim( F_p^e/im J_x)=e-ρ_x. The cardinality of an n-dimensional F_p-space is p^n. Combine these identities with the obstruction criterion and the residue equivalence.
Exact mathematical passage
\dim(\F_p^e/\im J_x)=e-\rho_x
\mathbb F_p^eUse in source · 8 passages
Search aliases: F_p^e mathbb
Use in source · continuation_02.tex, line 16
Let p be prime, L≥1, and F∈ Z[X_1,…,X_d]^e, x∈ Z^d, F(x)≡0pmodp^L. Write r_x=F(x)/p^L, b_x=r_xbmod p, A_x=DF(x)bmod p: F_p^d⟶ F_p^e, ρ_x=rank_ F_pA_x. The actual lift set is L_x,L= {y∈( Z/p^L+1 Z)^d: y≡ xpmodp^L, F(y)=0}, where polynomial evaluation in the displayed set is modulo p^L+1. Define O_x= F_p^e/imA_x, o_x=[b_x]. Lean proves # L_x,L= p^d-ρ_x, · o_x=0,
Exact mathematical passage
r_x=F(x)/p^L,\qquad b_x=r_x\bmod p, \qquad A_x=DF(x)\bmod p:\mathbb F_p^d\longrightarrow\mathbb F_p^e, \qquad \rho_x=\operatorname{rank}_{\mathbb F_p}A_x.
\mathbb F_p^{e+q}Use in source · 1 passage
Search aliases: F_p^e+q mathbb
Use in source · continuation_05.tex, line 11
Let p be prime, L≥1, e,q≥0, and F∈ Z[X_1,…,X_e+q]^e, x∈ Z^e+q, F(x)≡0pmodp^L. Assume the actual reduced Jacobian A_x=DF(x)bmod p: F_p^e+q⟶ F_p^e is surjective. All vector congruences are coordinatewise.
Exact mathematical passage
A_x=DF(x)\bmod p:\mathbb F_p^{e+q}\longrightarrow\mathbb F_p^e
\F_p^kUse in source · 1 passage
Search aliases: F_p^k F mathbb
Use in source · 30_lifting.tex, line 635
ints] Let q=p be prime in Theorem~[source reference]. For every τ∈ t_0+p^L Z_p^k, there are an integer tower z_n and b∈ Z_p^m+k such that F(b)=0, free(b)=τ, z_n⟶ b, b-z_n∈ p^L+n Z_p^m+k. The initial value is z_0=x. At every stage and for every proposed parameter digit v∈ F_p^k, the corresponding next residue fiber has exactly p^m-e elements. If m=e, then for each τ there is exactly one such root in B_L(x), independently of all choices of integer approximants and correcting digits.
Exact mathematical passage
v\in\F_p^k
\mathbb F_p^kUse in source · 1 passage
Search aliases: F_p^k mathbb
Use in source · continuation_09.tex, line 56
Exact branching. At any stage n, put N=L+n. For every parameter digit v∈ F_p^k, let L_n,v={bar w∈( Z/p^N+1 Z)^m+k: l bar w≡ z(n)pmodp^N,
Exact mathematical passage
v\in\mathbb F_p^k
\mathbb F_p^rUse in source · 5 passages
Search aliases: F_p^r mathbb
Use in source · 71_native.tex, line 23
The divisions are exact integer divisions. Regard J as a linear map F_p^d→ F_p^r. Define
Exact mathematical passage
\mathbb F_p^d\to\mathbb F_p^r
\F_p^\timesUse in source · 1 passage
Search aliases: F_p^× F times mathbb times
Use in source · 10_foundations.tex, line 346
Induction gives the geometric identity. Reduction of a unit to F_p^× gives u_i^p-1=1 modulo p. Taking s powers supplies p^s in each summand. The difference operator is linear over finite sums.
Exact mathematical passage
\F_p^\times
\mathbb F_p^\timesUse in source · 1 passage
Search aliases: F_p^× mathbb times
Use in source · odd_synthesis.tex, line 106
This procedure never substitutes an unjustified extension through a nonmaximal centralizer. It proves full-character compatibility, including shallow units, rather than merely cancellation of a leading trace. The residue factor fixes ω on F_p^×, and an N-th-root choice for the noncompact generator fixes ω(t). Every choice is recorded before H5. See pathcentral/TRACE_ZERO_CHARACTER_INDUCTION166.md for the complete induction.
Exact mathematical passage
\mathbb F_p^\times
F_{\Pcal\leftarrow\Pcal}Use in source · 1 passage
Search aliases: F_Pcal←Pcal F_ P← P Pcal leftarrow Pcal mathcal leftarrow mathcal
Use in source · 03_nonlinear_coordinates.tex, line 157
[Dynamical equivalence] Suppose X=Y and the same packet system is used on both sides. Packet coordinates preserve fixed points, periodic points, injectivity, surjectivity, and invertibility. In particular, F(x)=x Longleftrightarrow F_ P← P( A_ Px)= A_ Px.
Exact mathematical passage
F(x)=x \quad\Longleftrightarrow\quad F_{\Pcal\leftarrow\Pcal}(\Acal_{\Pcal}x)=\Acal_{\Pcal}x.
F_{P\circ f}Use in source · 1 passage
Search aliases: F_P∘ f circ
Use in source · 03_supplement_04.tex, line 86
Tensor properties and the remaining global boundary. The coefficient is proved bilinear, alternating, and supported on the range on both sides. The chain rule proves the pullback identity F_P∘ f(x)(v,w)=F_P(f(x))(Df(x)v,Df(x)w). The main theorem has global regularity hypotheses on normed parameter spaces. The pullback identity alone does not supply localization, smooth local trivializations, or gluing. The final supplement constructs the global range bundle and proves that the local connections …
Exact mathematical passage
F_{P\circ f}(x)(v,w)=F_P(f(x))(Df(x)v,Df(x)w).
F_QUse in source · 13 passages
Search aliases: F_Q
Use in source · 04_correction_transport.tex, line 498
Let U,V,Y_P,Y_Q be Banach spaces. On an open neighborhood of (u_0,v_0)∈ U× V, split the variables and equations as x=(u,v), F(u,v)= F_P(u,v) F_Q(u,v) . Assume F takes values in Y_P× Y_Q, is twice continuously Fr echet differentiable near (u_0,v_0), satisfies F_Q(u_0,v_0)=0, and A=D_vF_Q(u_0,v_0) is a Banach space isomorphism from V to Y_Q. The implicit function theorem gives a local map v=φ(u), with φ(u_0)=v_0, satisfying F_Q(u,φ(u))=0. Define the reduced equation R(u)=F_P(u,φ(u)).
Exact mathematical passage
x=(u,v), \qquad F(u,v)= \begin{pmatrix} F_P(u,v)\\ F_Q(u,v) \end{pmatrix}.
f_qUse in source · 1 passage
Search aliases: f_q
Use in source · 72_recovered.tex, line 823
[Fourier invariance of the standard window operator] Let p be odd, let mmid p-1, and let h=(p-1)/m. Define (T_m,pf)_n = ∑_q=0^nbinommnmqf_q in F_p. Let V_h be the vector space of sequences satisfying f_n+h=f_n (n≥1). Then T_m,p(V_h)=V_h. The restriction of T_m,p to V_h is an automorphism.
Exact mathematical passage
(T_{m,p}f)_n = \sum_{q=0}^{n}\binom{mn}{mq}f_q \quad\text{in }\FF_p.
F_{\Qcal'\leftarrow\Pcal'}Use in source · 2 passages
Search aliases: F_Qcal'←Pcal' F_ Q'← P' Qcal leftarrow Pcal mathcal leftarrow mathcal
Use in source · 03_nonlinear_coordinates.tex, line 176
[Change of packet coordinates] Let P and P' be packet systems on X. Let Q and Q' be packet systems on Y. For any two packet systems U and V on the same space, write J_ U← V = A_ U S_ V. Then F_ Q'← P' = J_ Q'← Q F_ Q← P J_ P← P'.
Exact mathematical passage
F_{\Qcal'\leftarrow\Pcal'} = J_{\Qcal'\leftarrow\Qcal} F_{\Qcal\leftarrow\Pcal} J_{\Pcal\leftarrow\Pcal'}.
f_RUse in source · 2 passages
Search aliases: f_R
Use in source · 71_native.tex, line 251
The component maps f_R:A→ C_R and g_R:B→ C_R have the following universal property. For every set Y and maps h:A→ Y, k:B→ Y satisfying h(a)=k(b) whenever R(a,b), there is a unique map u:C_R→ Y with u∘ f_R=h and u∘ g_R=k.
Exact mathematical passage
f_R:A\to C_R
F_r'Use in source · 1 passage
Search aliases: F_r'
Use in source · supplement_09.tex, line 88
ip 2. Connection to the actual section. An integral coordinate estimate becomes relevant only after connecting it to the manuscript's generating functions. For a residue r∈ Z/N Z, define H_r(t)=∑_d≥01_d≡ r (N)fract^dd!, F_r(T)=H_r(log(1+T)). The implementation uses the actual formal logarithm and substitution in Q[[T]]. It proves H_r'=H_r-1, (1+T)F_r'=F_r-1, and hence the coefficient recurrence (d+1)coefTd+1F_r =coefTdF_r-1-dcoefTdF_r. The initial values are F_r(0)=1_r=0.
Exact mathematical passage
H_r'=H_{r-1},\qquad (1+T)F_r'=F_{r-1},
F_{r-1}Use in source · 2 passages
Search aliases: F_r-1
Use in source · supplement_09.tex, line 88
ip 2. Connection to the actual section. An integral coordinate estimate becomes relevant only after connecting it to the manuscript's generating functions. For a residue r∈ Z/N Z, define H_r(t)=∑_d≥01_d≡ r (N)fract^dd!, F_r(T)=H_r(log(1+T)). The implementation uses the actual formal logarithm and substitution in Q[[T]]. It proves H_r'=H_r-1, (1+T)F_r'=F_r-1, and hence the coefficient recurrence (d+1)coefTd+1F_r =coefTdF_r-1-dcoefTdF_r. The initial values are F_r(0)=1_r=0.
Exact mathematical passage
H_r'=H_{r-1},\qquad (1+T)F_r'=F_{r-1},
F_r^*Use in source · 2 passages
Search aliases: F_r^*
Use in source · 01_finite_core.tex, line 846
textitSufficiency. Diagonalize C=Udiag(λ_r)U^* with U unitary and λ_r≥0. Define compressed transformed errors F_r=∑_aU_arE_aP. Then F_rP=F_r and F_r^*F_s=λ_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal projector. Use the recovery branches K_r=V_r^*, K_unused=I-S. They are complete on the whole ambient space because ∑_rK_r^*K_r+K_unused^*K_unused =S+(…
Exact mathematical passage
F_r^*F_s=\lambda_r\delta_{rs}P
F_r^{-1}Use in source · 1 passage
Search aliases: F_r^-1
Use in source · 01_finite_coordinates.tex, line 402
Every identity is cancellation of adjacent F_rF_r^-1 factors. The Lean definitions use leannameLinearEquiv.trans. The displayed formulas use the equivalent right-to-left notation for function composition fixed in this chapter.
Exact mathematical passage
F_rF_r^{-1}
F_{\rm root}Use in source · 1 passage
Search aliases: F_rm root rm
Use in source · 01_manuscript.tex, line 1663
Assume every keep error is a nonnegative integer w_v. Let B∈ N. For each node v, define F_v(b) as the minimum cost of a pruning of the subtree at v with total error exactly b. The keep option contributes F_v(w_v)≤κ_v. The split option is obtained by min plus convolution of the child tables, followed by addition of σ_v. The smallest value among F_rm root(b), 0≤ b≤ B, is the exact minimum cost among prunings whose quantized error is at most B.
Exact mathematical passage
F_{\rm root}(b), \qquad 0\leq b\leq B,
F_{\mathrm{root}}Use in source · 1 passage
Search aliases: F_root mathrm
Use in source · 02_supplement_03.tex, line 181
The root scan min_0≤ b≤ BF_root(b) is therefore the exact budget-constrained optimum. Increasing the budget cannot increase a finite optimum. The theorem retains feasibility hypotheses where necessary rather than treating codenone as an ordinary real cost.
Exact mathematical passage
\boxed{\min_{0\leq b\leq B}F_{\mathrm{root}}(b)}
F_sUse in source · 1 passage
Search aliases: F_s
Use in source · 01_finite_core.tex, line 846
textitSufficiency. Diagonalize C=Udiag(λ_r)U^* with U unitary and λ_r≥0. Define compressed transformed errors F_r=∑_aU_arE_aP. Then F_rP=F_r and F_r^*F_s=λ_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal projector. Use the recovery branches K_r=V_r^*, K_unused=I-S. They are complete on the whole ambient space because ∑_rK_r^*K_r+K_unused^*K_unused =S+(…
Exact mathematical passage
F_r^*F_s=\lambda_r\delta_{rs}P
F_tUse in source · 1 passage
Search aliases: F_t
Use in source · 02_supplement_03.tex, line 202
Entrywise implementation theorem. For every b≤ B, Lean proves lookup(storedTree(B,t),b)=F_t(b). There is a parallel theorem for stored discrete precision. The proof establishes first that an output entry at b depends only on input entries at most b, then proceeds through cached convolution and the tree recursion. Consequently the optimality, attainment, and infeasibility specification applies to the actual stored result.
Exact mathematical passage
\operatorname{lookup}(\operatorname{storedTree}(B,t),b)=F_t(b).
f_tUse in source · 1 passage
Search aliases: f_t
Use in source · 03_supplement_03.tex, line 143
Connecting power series and spectral calculus. The shifted generator is proved Hermitian. The implementation then identifies the actual matrix exponential with continuous functional calculus: T_e(t)=f_t(H), f_t(x)=e^-t(x-e). The formal proof uses composition, scalar multiplication, subtraction, and the established agreement between the exponential in functional calculus and the exponential defined by a power series. Spectral mapping is consequently available for the operator that was actually de…
Exact mathematical passage
\boxed{T_e(t)=f_t(H),\qquad f_t(x)=e^{-t(x-e)}.}
F_VUse in source · 2 passages
Search aliases: F_V
Use in source · Paper_VI_Traditional_Final.tex, line 4447
[Positivity and composition] The operators F_V(t)= e^t(Q^T-D_V) form a positive semigroup. If f≥0, then F_V(t)f≥0.
Exact mathematical passage
F_V(t)=\e^{t(Q^T-D_V)}
F_vUse in source · 5 passages
Search aliases: F_v
Use in source · pages.tex, line 27
F_v(b) · A budget table · Does b mean exact weight, or an upper bound?
Exact mathematical passage
F_v(b)
F_{v_i}Use in source · 1 passage
Search aliases: F_v_i
Use in source · 02_supplement_03.tex, line 168
Keeping node v costs κ_v and incurs integer error w_v. Splitting costs overhead σ_v in addition to the selected child cuts. The exact-weight recurrence is F_v(b)=min{ l κ_v if b=w_v, [2pt] displaystyleσ_v+ min_∑_i b_i=b∑_iF_v_i(b_i) if v is internal. . Nonexistent alternatives are infeasible. The child sum is implemented by the proved finite min-plus convolution, not by assuming an optimal allocation.
Exact mathematical passage
F_v(b)=\min\left\{ \begin{array}{l} \kappa_v\quad\text{if }b=w_v,\\[2pt] \displaystyle\sigma_v+ \min_{\sum_i b_i=b}\sum_iF_{v_i}(b_i)\quad\text{if }v\text{ is internal}. \end{array}\right.
F_wUse in source · 3 passages
Search aliases: F_w
Use in source · residue_continuation_body.tex, line 164
An actual Mellin endpoint. The counting functions are constant on the integer norm steps. Their measurability, and hence that of S_w, is proved from this fact. For zero-sum w, define F_w(h)=(1+h)∫_1^∞x^-2-hS_w(x) dx (h≥0). Since 1-1/d<1, the logarithmic first moment M_w=∫_1^∞(1+log x) absnx^-2S_w(x) dx is finite. Lean proves the explicit estimate absnF_w(h)-F_w(0)≤ hM_w for every h≥0, and therefore F_w(h)=F_w(0)+O(h) as hdownarrow0. The function here is the displayed integral of the actual ideal…
Exact mathematical passage
F_w(h)=(1+h)\int_1^{\infty}x^{-2-h}S_w(x)\,dx \qquad(h\geq0).
F_zUse in source · 2 passages
Search aliases: F_z
Use in source · 02_strengthened.tex, line 1338
For z∈ C^r, let F_z=∑_az_aE_a. Then PF_z^*F_zP=(z^*cz)P≥0, so c≥0. The unitary change of error coordinates gives PF_j^*F_ℓ P = λ_jδ_jℓP. Thus V_j^*V_ℓ=δ_jℓP. Each V_j is an isometry on P H, and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.
Exact mathematical passage
F_z=\sum_az_aE_a
F_z^*Use in source · 1 passage
Search aliases: F_z^*
Use in source · 02_strengthened.tex, line 1339
For z∈ C^r, let F_z=∑_az_aE_a. Then PF_z^*F_zP=(z^*cz)P≥0, so c≥0. The unitary change of error coordinates gives PF_j^*F_ℓ P = λ_jδ_jℓP. Thus V_j^*V_ℓ=δ_jℓP. Each V_j is an isometry on P H, and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.
Exact mathematical passage
PF_z^*F_zP=(z^*cz)P\geq0,
f_\alphaUse in source · 5 passages
Search aliases: f_α alpha
Use in source · 03_packets_and_graphs.tex, line 150
[Universal mapping property of finite packet coordinates] Let P=(P_α)_α∈ I be a finite packet system on X, and let Y be a vector space over 𝕜. item Every family of linear maps f_α:X_α→ Y determines a unique linear map f:X→ Y with f|_X_α=f_α, namely f=∑_α f_α P_α. item Every family of linear maps g_α:Y→ X_α determines a unique linear map g:Y→ X with P_α g=g_α, namely g=∑_α g_α. Equivalently, Hom(X,Y)≅∏_αHom(X_α,Y), Hom(Y,X)≅∏_αHom(Y,X_α), where the products are finite. [source reference] ExportsA…
Exact mathematical passage
f_\alpha:X_\alpha\to Y
F_{\theta\phi}Use in source · 1 passage
Search aliases: F_θϕ theta phi
Use in source · monograph_atlas.tex, line 277
EXACT BLOCH PROJECTOR CALCULATIONChapter VII: Theorem II.6.7, Corollary II.6.8. On the unit sphere, set n=(sinθcosϕ,sinθsinϕ,cosθ) and P(θ,ϕ)=frac12(I+n·σ). This is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form P( d Pwedge d P)P. With the paper's scalar convention, its component is F_θϕ=- iTr (P[∂_θ P,∂_ϕ P]) =frac12sinθ. vfigbloch_curvature headA local frame gives the same answer Away from the south pole, use u_N=(cos(θ/2),e^ iϕsin(θ/2))^T. …
Exact mathematical passage
F_{\theta\phi}=-\ii\Tr\!\left(P[\partial_\theta P,\partial_\phi P]\right) =\frac12\sin\theta.
F_\lambdaUse in source · 10 passages
Search aliases: F_λ lambda
Use in source · monograph_examples.tex, line 79
The dynamic recurrence gives the same Lagrangian choices without listing prunings. At a, it compares 2+4λ with 4+λ. At r, it compares 1+6λ with 1+F_λ(a). Its exactness follows from the finite tree and additive model in [source reference]. Overlapping decompositions or nonlocal execution costs need another argument.
Exact mathematical passage
1+F_\lambda(a)
f_\pi^2Use in source · 1 passage
Search aliases: f_π^2 pi
Use in source · Paper_VI_Traditional_Final.tex, line 3078
P_0,P_1,…,P_r come from increasingly fine partitions, so that Ran P_0⊆Ran P_1⊆⋯⊆Ran P_r. Define D_0=P_0, D_j=P_j-P_j-1 (1≤ j≤ r). Then P_iP_j=P_min(i,j), D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j. Every D_j is an orthogonal projector, and normP_rf_π^2 = ∑_j=0^rnormD_jf_π^2 and norm f_π^2 = ∑_j=0^rnormD_jf_π^2 + normf-P_rf_π^2.
Exact mathematical passage
\norm f_\pi^2 = \sum_{j=0}^r\norm{D_jf}_\pi^2 + \norm{f-P_rf}_\pi^2.
F_{\chi_i}Use in source · 2 passages
Search aliases: F_χ_i chi
Use in source · residue_continuation_body.tex, line 181
For a finite family of nontrivial ordinary class characters, suppose functions p,Z satisfy h p(h)→ P≠0, h Z(h)→ Q≠0, and, for all sufficiently small h>0, Z(h)=(p(h)∏_i F_χ_i(h))^e, e∈ N. The checked real-axis criterion then gives e=1 and F_χ_i(0)≠0 for every i. The pole limits and product identity remain hypotheses of this application.
Exact mathematical passage
Z(h)=\left(p(h)\prod_i F_{\chi_i}(h)\right)^e, \qquad e\in\NN.
F_\ellUse in source · 1 passage
Search aliases: F_ℓ ell
Use in source · 02_strengthened.tex, line 1343
For z∈ C^r, let F_z=∑_az_aE_a. Then PF_z^*F_zP=(z^*cz)P≥0, so c≥0. The unitary change of error coordinates gives PF_j^*F_ℓ P = λ_jδ_jℓP. Thus V_j^*V_ℓ=δ_jℓP. Each V_j is an isometry on P H, and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.
Exact mathematical passage
PF_j^*F_\ell P = \lambda_j\delta_{j\ell}P.
\fallUse in source · 5 passages
Search aliases: fall
Use in source · 72_recovered.tex, line 1113
[Dyadic Euler coordinate pair] For every even n≥0, E_n = ∑_r=0^n/2 β_r2^2r fall n2r R_n-2r. The inverse transform is R_n = ∑_r=0^n/2 frac2^4r+1(2r+2)! fall n2r E_n-2r.
Exact mathematical passage
\boxed{ E_n = \sum_{r=0}^{n/2} \beta_r2^{2r} \fall n{2r} R_{n-2r}. }
\Fcal_{\theta\phi}Use in source · 3 passages
Search aliases: Fcal_θϕ F_θϕ Fcal theta phi mathcal theta phi
Use in source · 02_strengthened.tex, line 901
[Bloch sphere curvature] Let P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). With the convention F_θϕ =- iTr(P[∂_θ P,∂_ϕ P]), one has F_θϕ=frac12sinθ and frac12π ∫_0^2π∫_0^π F_θϕ dθ dϕ =1.
Exact mathematical passage
\Fcal_{\theta\phi} =-\ii\Tr\bigl(P[\partial_\theta P,\partial_\phi P]\bigr),
\FdetUse in source · 2 passages
Search aliases: Fdet det_ F operatorname
Use in source · Paper_IV_Traditional_Final.tex, line 2335
[Diagonal Fredholm determinant representation] Let I_N be a multiplicity-sensitive countable index set for Λ_N and put H_N=ℓ^2(I_N). On its standard orthonormal basis, define the diagonal operator K_Ne_λ=λ^-1e_λ. Then K_N is trace class and Δ_N(w)=det_ F(I-wK_N). For every m≥1, κ_N,m=tr(K_N^m).
Exact mathematical passage
\Delta_N(w)=\Fdet(I-wK_N).
\FF_2^2Use in source · 1 passage
Search aliases: FF_2^2 F_2^2 FF mathbb
Use in source · Paper_XI_Traditional_Final.tex, line 2122
At rank two, with q=2m+1, enlarge J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=2, and P/H_0≃ F_2^4, with carrier Ind_J^Pη of dimension four ([source reference]). The finite scalar center can be larger than μ_2. No literal extrasp…
Exact mathematical passage
P/H_0\simeq\FF_2^2
\FF_2^4Use in source · 1 passage
Search aliases: FF_2^4 F_2^4 FF mathbb
Use in source · Paper_XI_Traditional_Final.tex, line 2126
(J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=2, and P/H_0≃ F_2^4, with carrier Ind_J^Pη of dimension four ([source reference]). The finite scalar center can be larger than μ_2. No literal extraspecial quotient is claimed.
Exact mathematical passage
P/H_0\simeq\FF_2^4
\FF_2^\timesUse in source · 1 passage
Search aliases: FF_2^× F_2^× FF times mathbb times
Use in source · dyadic_character.tex, line 34
The construction retains the actual canonical beta extension and can be completed with any prescribed central quasicharacter omega_F trivial on U_F 1. Because k_F^×= F_2^× is trivial, such an omega_F is unramified. Thus this covers every tamely ramified central quasicharacter for this particular residue field.
Exact mathematical passage
k_F^\times=\FF_2^\times
\FF_3Use in source · 1 passage
Search aliases: FF_3 F_3 FF mathbb
Use in source · 72_recovered.tex, line 2724
[A singular polynomial obstruction] Let p=3 and consider the class x_1=0 modulo three. For F_0(x)=x^2, the residual and the reduced derivative both vanish. Every correction y∈ F_3 gives F_0(3y)≡0pmod9, so the class has three children. For F_1(x)=x^2+3, the reduced derivative is still zero, but the reduced residual is one. Its class in the cokernel is nonzero, so no lift modulo nine exists.
Exact mathematical passage
y\in\FF_3
\FF_3^2Use in source · 1 passage
Search aliases: FF_3^2 F_3^2 FF mathbb
Use in source · Paper_XI_Traditional_Final.tex, line 1954
[Rank one Heisenberg group] Take V= F_3^2. The Heisenberg group has order 3^3=27, and its irreducible representation with a fixed nontrivial central character has dimension three. The trace vanishes on every noncentral element. Averaging the translation subgroup gives a rank one projector.
Exact mathematical passage
V=\FF_3^2
\FF_7Use in source · 7 passages
Search aliases: FF_7 F_7 FF mathbb
Use in source · Paper_XII_Traditional_Final.tex, line 356
Let G=S_3, let k= F_7, and let V be the standard two dimensional representation. In the basis v_1=(1,-1,0), v_2=(0,1,-1), a transposition and a three cycle act by s= 6 · 1
Exact mathematical passage
k=\FF_7
\FF_7^3Use in source · 1 passage
Search aliases: FF_7^3 F_7^3 FF mathbb
Use in source · Paper_XII_Traditional_Final.tex, line 398
The coordinate map (a,b)⟼(a,-a+b,-b) identifies the displayed matrices with the permutation action on the sum zero subspace of F_7^3. This action is faithful. The four matrices 1,s,t,st span M_2 over every field of characteristic seven: the determinant of their coordinate matrix is 4≠0. A subspace stable under the group is therefore stable under every endomorphism, so the representation is absolutely irreducible. The characteristic polynomials of the identity, transpositions, and three cycles ar…
Exact mathematical passage
\FF_7^3
\FF_p^2Use in source · 1 passage
Search aliases: FF_p^2 F_p^2 FF mathbb
Use in source · Paper_XI_Traditional_Final.tex, line 1958
[Nondegenerate twisted algebra] Let A= F_p^2 with its standard alternating form. A cocycle with that commutator form gives C^c[A]≅ M_p( C). The unique irreducible projective representation has dimension p.
Exact mathematical passage
A=\FF_p^2
\FF_p^dUse in source · 4 passages
Search aliases: FF_p^d F_p^d FF mathbb
Use in source · 72_recovered.tex, line 2551
[One level linear lifting] A class represented by x_L lifts to a solution modulo p^L+1 if and only if Ob^lin_L(x_L)=0. More precisely, a correction y∈ F_p^d gives a lift x_L+1=x_L+p^Ly pmod p^L+1 if and only if overline My=-ρ_L(x_L). When one correction exists, the set of all correction classes is a coset of keroverline M.
Exact mathematical passage
y\in\FF_p^d
\FF_p^\timesUse in source · 3 passages
Search aliases: FF_p^× F_p^× FF times mathbb times
Use in source · 72_recovered.tex, line 251
Fix a divisor m of p-1. Let H_m = {ζ∈ F_p^×:ζ^m=1}. This group has order m.
Exact mathematical passage
H_m = \{\zeta\in\FF_p^\times:\zeta^m=1\}.
\FF_qUse in source · 2 passages
Search aliases: FF_q F_q FF mathbb
Use in source · Paper_XI_Traditional_Final.tex, line 451
Let G_n=GL_n( F_q), T_n≅ F_q^n^× with T_n an elliptic maximal torus.
Exact mathematical passage
G_n=\GL_n(\FF_q), \qquad T_n\cong\FF_{q^n}^{\times}
\FF_{q^L}^{\times}Use in source · 2 passages
Search aliases: FF_q^L^× F_q^L^× FF times mathbb times
Use in source · Paper_XI_Traditional_Final.tex, line 615
[Refined exponent formula] The ith interaction packet on F_q^L^× has exponent c_i = afracq^L-1q^n-1 +bq^ifracq^L-1q^m-1 pmodq^L-1. It is trivial on tame inertia exactly when c_i=0.
Exact mathematical passage
\FF_{q^L}^{\times}
\FF_{q^n}^{\times}Use in source · 3 passages
Search aliases: FF_q^n^× F_q^n^× FF times mathbb times
Use in source · Paper_XI_Traditional_Final.tex, line 406
Characters of F_q^n^× are indexed by exponents modulo q^n-1, and Frobenius acts by multiplication by q.
Exact mathematical passage
\FF_{q^n}^{\times}
\FF_{q^r}^{\times}Use in source · 1 passage
Search aliases: FF_q^r^× F_q^r^× FF times mathbb times
Use in source · Paper_XI_Traditional_Final.tex, line 423
For every divisor rmid n, the characters fixed by Frob^r are exactly those descending to F_q^r^×, and there are q^r-1 of them. Möbius inversion gives the number of characters of exact orbit length n. Division by n gives the number of orbits.
Exact mathematical passage
\FF_{q^r}^{\times}
\operatorname{fill}Use in source · 1 passage
Search aliases: fill operatorname
Use in source · 02_supplement_02.tex, line 101
Correctness of the actual update. For any edited subtree T' and valid context C, Lean proves updateValue(C,F(T'))=F(fill(C,T')). The proof inducts through the context. At each frame, the updated child contributes the new value, and each other child contributes its unchanged cached optimum. The frame recurrence is therefore exactly the full-tree recurrence.
Exact mathematical passage
\operatorname{updateValue}\bigl(C,F(T')\bigr)=F\bigl(\operatorname{fill}(C,T')\bigr).
\mathrm{fine}Use in source · 1 passage
Search aliases: fine mathrm
Use in source · 01_finite_core.tex, line 239
Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.
Exact mathematical passage
P_{\mathbf j}^{\mathrm{fine}} = P^{(1)}_{j_1}\otimes\cdots\otimes P^{(m)}_{j_m}.
\FittUse in source · 8 passages
Search aliases: Fitt operatorname
Use in source · Paper_XII_Traditional_Final.tex, line 609
[Fitting support criterion] Let R be Noetherian and let M be a finite R module. Then Supp_R(M) = V(Fitt_0^R(M)).
Exact mathematical passage
\boxed{ \Supp_R(M) = V\bigl(\Fitt_0^R(M)\bigr). }
\Fitt_0Use in source · 1 passage
Search aliases: Fitt_0 Fitt operatorname
Use in source · Paper_XII_Traditional_Final.tex, line 619
If M=0, the statement follows from the usual convention Fitt_0(0)=R. Assume M≠0, and choose a finite presentation R^mxrightarrowAR^n⟶ M⟶0 with n≥1. The zeroth Fitting ideal is generated by the n by n minors of A. After localization at a prime p, the module M_ p vanishes exactly when A_ p is surjective. A map onto the free module R_ p^n is surjective exactly when one maximal minor is a unit. Thus M_ p=0 exactly when Fitt_0^R(M)not⊆ p.
Exact mathematical passage
\Fitt_0(0)=R
\Fitt_0^RUse in source · 2 passages
Search aliases: Fitt_0^R Fitt operatorname
Use in source · Paper_XII_Traditional_Final.tex, line 609
[Fitting support criterion] Let R be Noetherian and let M be a finite R module. Then Supp_R(M) = V(Fitt_0^R(M)).
Exact mathematical passage
\boxed{ \Supp_R(M) = V\bigl(\Fitt_0^R(M)\bigr). }
\Fitt_j^RUse in source · 2 passages
Search aliases: Fitt_j^R Fitt operatorname
Use in source · Paper_XII_Traditional_Final.tex, line 658
[Fitting ideals and base change] Let R→ S be a ring map and let M be a finitely presented R module. Then for every j≥0, Fitt_j^S(M⊗_RS) = Fitt_j^R(M)S.
Exact mathematical passage
\boxed{ \Fitt_j^S(M\otimes_RS) = \Fitt_j^R(M)S. }
\Fitt_j^SUse in source · 1 passage
Search aliases: Fitt_j^S Fitt operatorname
Use in source · Paper_XII_Traditional_Final.tex, line 658
[Fitting ideals and base change] Let R→ S be a ring map and let M be a finitely presented R module. Then for every j≥0, Fitt_j^S(M⊗_RS) = Fitt_j^R(M)S.
Exact mathematical passage
\boxed{ \Fitt_j^S(M\otimes_RS) = \Fitt_j^R(M)S. }
\Fitt_j^{\TT}Use in source · 1 passage
Search aliases: Fitt_j^TT Fitt_j^ T Fitt TT operatorname mathbb
Use in source · Paper_XII_Traditional_Final.tex, line 935
[Support and Fitting transport] Let φ:R→ T be an isomorphism of Noetherian rings. Let M be a finite T module, viewed as an R module through φ. Then the homeomorphism φ^*:Spec T⟶Spec R satisfies φ^*(Supp_ T(M)) = Supp_R(M). For every j≥0, φ ( Fitt_j^R(M) ) = Fitt_j^ T(M).
Exact mathematical passage
\boxed{ \varphi \left( \Fitt_j^R(M) \right) = \Fitt_j^{\TT}(M). }
\Fitt_{r-1}Use in source · 1 passage
Search aliases: Fitt_r-1 Fitt operatorname
Use in source · Paper_XII_Traditional_Final.tex, line 649
Use a finite presentation with target rank n. If r>n, every fiber has dimension at most n, so the left-hand locus is empty. The convention Fitt_r-1(M)=R for r-1≥ n makes the right-hand locus empty as well. It remains to consider 1≤ r≤ n. At p, the fiber dimension is n-rank A( p). It is at least r exactly when the rank is at most n-r, which is equivalent to the vanishing of every minor of size n-r+1. Those minors generate Fitt_r-1^R(M).
Exact mathematical passage
\Fitt_{r-1}(M)=R
\Fitt_{r-1}^RUse in source · 2 passages
Search aliases: Fitt_r-1^R Fitt operatorname
Use in source · Paper_XII_Traditional_Final.tex, line 636
[Higher Fitting strata] Let R be Noetherian and let M be finite. For r≥1, put μ_ p(M) = dim_κ( p) (M⊗_Rκ( p)). Then { p∈Spec R: μ_ p(M)≥ r } = V(Fitt_r-1^R(M)).
Exact mathematical passage
\boxed{ \left\{ \frakp\in\Spec R: \mu_{\frakp}(M)\geq r \right\} = V\bigl(\Fitt_{r-1}^R(M)\bigr). }
\operatorname{Flat}Use in source · 2 passages
Search aliases: Flat operatorname
Use in source · supplement_08.tex, line 46
All orders, without a circular closure assumption. Let I⊂ A be a subring stable under U. Define Bounded_r(f)iff B^s f∈ I (0≤ s≤ r), Flat(f)iff B^s f∈ I (s≥0). The definitions contain no product or inverse closure requirement. Since B commutes with U, derivatives of a bounded element remain bounded. Induction on r in the displayed product rule then proves closure under multiplication. The finite middle sums use the lower-order induction hypothesis, not the desired next-order conclusion.
Exact mathematical passage
\operatorname{Bounded}_r(f)\iff B^s f\in I\ (0\le s\le r), \qquad \operatorname{Flat}(f)\iff B^s f\in I\ (s\ge0).
\flop_{\rm block}Use in source · 1 passage
Search aliases: flop_rm block flop rm operatorname rm
Use in source · 01_manuscript.tex, line 284
[Equal diagonal packets] Let p≥1 and b≥1 be integers and n=pb. Suppose n is divided into p equal diagonal packets of width b. Then flop_rm block=2qnb, M_rm block=nb. The following factor describes the dense-to-block storage reduction and, when q≥1, the arithmetic reduction: fracnb.
Exact mathematical passage
\flop_{\rm block}=2qnb, \qquad M_{\rm block}=nb.
\flop_{\rm dense}Use in source · 1 passage
Search aliases: flop_rm dense flop rm operatorname rm
Use in source · 01_manuscript.tex, line 251
[Dense and packet application counts] A dense n by n matrix applied to q right hand sides has modeled cost flop_rm dense=2n^2q. A packet representation with active edge set E has modeled cost flop_rm packet = 2q ∑_(β,α)∈ E n_β n_α. It stores M_rm packet = ∑_(β,α)∈ E n_β n_α scalar entries.
Exact mathematical passage
\boxed{ \flop_{\rm dense}=2n^2q. }
\flop_{\rm packet}Use in source · 1 passage
Search aliases: flop_rm packet flop rm operatorname rm
Use in source · 01_manuscript.tex, line 257
[Dense and packet application counts] A dense n by n matrix applied to q right hand sides has modeled cost flop_rm dense=2n^2q. A packet representation with active edge set E has modeled cost flop_rm packet = 2q ∑_(β,α)∈ E n_β n_α. It stores M_rm packet = ∑_(β,α)∈ E n_β n_α scalar entries.
Exact mathematical passage
\boxed{ \flop_{\rm packet} = 2q \sum_{(\beta,\alpha)\in E} n_\beta n_\alpha. }
\FracUse in source · 1 passage
Search aliases: Frac operatorname
Use in source · 04_arithmetic.tex, line 114
Let O be a DVR, K=Frac( O), and A∈End_ O(Λ) for a finite free lattice Λ. Suppose the characteristic polynomial factors over O as χ_A(x)=f(x)g(x), where f and g are coprime over K[x].
Exact mathematical passage
K=\Frac(\OO)
\frakHUse in source · 3 passages
Search aliases: frakH H mathfrak
Use in source · Paper_XI_Traditional_Final.tex, line 796
Let p be odd. Let X be an r dimensional vector space over F_p, and put V=X⊕ X^*. Define the alternating form b((x,ξ),(y,η)) = η(x)-ξ(y). Let Z= F_p, and define the Heisenberg group H(V)=V× Z with multiplication (v,z)(w,t) = (v+w,z+t+frac12 b(v,w)). Fix the additive character ψ(z)= e^2π iz/p.
Exact mathematical passage
\frakH(V)=V\times Z
\frakm_AUse in source · 1 passage
Search aliases: frakm_A m_A frakm mathfrak
Use in source · Paper_XII_Traditional_Final.tex, line 316
and let J⊆ A be the closed local condition ideal. By [source reference], the independently constructed ring R^loc,τ is canonically A/J. The ordinary tensor identity B⊗_A(A/J)≅ B/JB carries the defining ideal of the tensor to the image of m_B. Here the structural map is local by the stated global deformation datum, so it carries m_A into m_B. Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism R^τ≅ B/overlineJB. The constructi…
Exact mathematical passage
\frakm_A
\frakm_BUse in source · 2 passages
Search aliases: frakm_B m_B frakm mathfrak
Use in source · Paper_XII_Traditional_Final.tex, line 316
Put A=R^loc, B=R^square, and let J⊆ A be the closed local condition ideal. By [source reference], the independently constructed ring R^loc,τ is canonically A/J. The ordinary tensor identity B⊗_A(A/J)≅ B/JB carries the defining ideal of the tensor to the image of m_B. Here the structural map is local by the stated global deformation datum, so it carries m_A into m_B. Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism R^τ≅ B/o…
Exact mathematical passage
\frakm_B
\operatorname{free}Use in source · 11 passages
Search aliases: free operatorname
Use in source · 30_lifting.tex, line 461
∈ p^L Z^e. Suppose the full reduced Jacobian is surjective. There is a permutation σ of the coordinates such that, writing σ x=(y_0,t_0) with e solved coordinates and k parameter coordinates, the parameter ball T=t_0+p^L Z_p^k is homeomorphic to the entire root locus R={z∈ B_L(x):F(z)=0}. The inverse homeomorphism is the actual free-coordinate projection z↦free(σ z). If h:T→ R is the forward map, then for every N≥0, h(t)≡ h(u)pmodp^N Longleftrightarrow t≡ upmodp^N. Equivalently, σ h(t)=(ϕ(t),t) …
Exact mathematical passage
z\mapsto\operatorname{free}(\sigma z)
\FrobUse in source · 2 passages
Search aliases: Frob operatorname
Use in source · Paper_XI_Traditional_Final.tex, line 423
For every divisor rmid n, the characters fixed by Frob^r are exactly those descending to F_q^r^×, and there are q^r-1 of them. Möbius inversion gives the number of characters of exact orbit length n. Division by n gives the number of orbits.
Exact mathematical passage
\Frob^r
\operatorname{Frob}Use in source · 1 passage
Search aliases: Frob operatorname
Use in source · Paper_X_Block27_Amended.tex, line 1059
cible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficient-field embedding σ_λ:E↪ E_λ therefore sends C to an invertible matrix. Its entries agree with the ordinary representation traces Tr(W_i,λ(Frob_v_j)) when the selected places satisfy the unramified compatibility convention for that coefficient prime. In particular, places above the residue characte…
Exact mathematical passage
\operatorname{Tr}(W_{i,\lambda}(\operatorname{Frob}_{v_j}))
\Frob^rUse in source · 1 passage
Search aliases: Frob^r Frob operatorname
Use in source · Paper_XI_Traditional_Final.tex, line 423
For every divisor rmid n, the characters fixed by Frob^r are exactly those descending to F_q^r^×, and there are q^r-1 of them. Möbius inversion gives the number of characters of exact orbit length n. Division by n gives the number of orbits.
Exact mathematical passage
\Frob^r
\Frob_vUse in source · 1 passage
Search aliases: Frob_v Frob operatorname
Use in source · Paper_XII_Traditional_Final.tex, line 1356
g T. item An automorphic Hecke datum ( T,H,(P_v)). item A characteristic zero point x:R→ E' lying in Supp_R(H). item A classical automorphic representation π_x occurring in the fiber H_x, together with a semisimple attached representation ρ_π_x:G_F,S⟶GL_n(E') whose unramified characteristic polynomial is x(P_v(X)). item Equality det(X-ρ_x(Frob_v))=x(P_v(X)) for ρ_x=x∘ρ_R at every unramified place in a Chebotarev dense set. Then ρ_x^ss ≅ ρ_π_x. If ρ_x is irreducible, then ρ_x≅ρ_π_x.
Exact mathematical passage
\det\left(X-\rho_x(\Frob_v)\right)=x(P_v(X))
\operatorname{Frob}_{v_j}Use in source · 1 passage
Search aliases: Frob_v_j operatorname
Use in source · Paper_X_Block27_Amended.tex, line 1059
cible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficient-field embedding σ_λ:E↪ E_λ therefore sends C to an invertible matrix. Its entries agree with the ordinary representation traces Tr(W_i,λ(Frob_v_j)) when the selected places satisfy the unramified compatibility convention for that coefficient prime. In particular, places above the residue characte…
Exact mathematical passage
\operatorname{Tr}(W_{i,\lambda}(\operatorname{Frob}_{v_j}))
G'Use in source · 5 passages
Search aliases: G'
Use in source · Paper_IV_Traditional_Final.tex, line 2051
/2)|u| shows that Φ_m maps B_m into itself. The derivative bound |Φ_m'|≤1/2 makes it a contraction. Banach's fixed-point theorem gives a unique fixed point. On B_m, |a(z-t_m)|≤1<π, so the principal logarithm inverts the exponential and the fixed-point equation is equivalent to G(z)=0. Moreover, |z-t_m|≤frac34s_N|R_N(z)|=O(e^-γ_Nt_m). At this zero, |G'(z)|≥ |a| |1+R_N(z)|-|R_N'(z)|≥ s_N/2>0, so the zero is simple. The disks are mutually disjoint because their diameter 1/s_N is smaller than the ce…
Exact mathematical passage
|G'(z)|\geq |a|\,|1+R_N(z)|-|R_N'(z)|\geq s_N/2>0,
G''Use in source · 2 passages
Search aliases: G''
Use in source · 20_arithmetic.tex, line 517
For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [source reference] proves that multiplication adds charges. If FG=1, the same formula yields A_-cG=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G'). Induction, starting…
Exact mathematical passage
\label{eq:arith-charged-product} A_{c+d}(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'.
g''Use in source · 1 passage
Search aliases: g''
Use in source · Paper_IV_Traditional_Final.tex, line 1343
For a function g, define the complete differential Bell polynomial Y_N by fracd^Ndz^Ne^g(z) = e^g(z)Y_N(g',g'',…,g^(N)).
Exact mathematical passage
\frac{d^N}{dz^N}e^{g(z)} = e^{g(z)}Y_N\bigl(g',g'',\ldots,g^{(N)}\bigr).
G'''Use in source · 2 passages
Search aliases: G'''
Use in source · 20_arithmetic.tex, line 517
For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [source reference] proves that multiplication adds charges. If FG=1, the same formula yields A_-cG=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G'). Induction, starting…
Exact mathematical passage
\label{eq:arith-charged-product} A_{c+d}(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'.
g^{(j)}Use in source · 1 passage
Search aliases: g^(j)
Use in source · 03_nonlinear_coordinates.tex, line 317
t↦ x+th. It contains [0,1], and repeated differentiation gives g^(r)(t)=D^rF(x+th)[h,…,h]. The derivatives are continuous on the compact interval, so their Bochner integrals exist in the Banach codomain. Put T_j=frac1j!∫_0^1(1-t)^j g^(j+1)(t) dt (0≤ j≤ m). The fundamental theorem of calculus gives T_0=g(1)-g(0). For j≥1, integration by parts gives T_j=T_j-1-g^(j)(0)/j!. Iterating this relation proves the Taylor identity with remainder T_m=R_m+1(x,h) and its exact factorial normalization.
Exact mathematical passage
T_j=T_{j-1}-g^{(j)}(0)/j!
g^{(j+1)}Use in source · 1 passage
Search aliases: g^(j+1)
Use in source · 03_nonlinear_coordinates.tex, line 312
Set g(t)=F(x+th) on the open preimage of the domain under t↦ x+th. It contains [0,1], and repeated differentiation gives g^(r)(t)=D^rF(x+th)[h,…,h]. The derivatives are continuous on the compact interval, so their Bochner integrals exist in the Banach codomain. Put T_j=frac1j!∫_0^1(1-t)^j g^(j+1)(t) dt (0≤ j≤ m). The fundamental theorem of calculus gives T_0=g(1)-g(0). For j≥1, integration by parts gives T_j=T_j-1-g^(j)(0)/j!. Iterating this relation proves the Taylor identity with remainder T_m…
Exact mathematical passage
T_j=\frac1{j!}\int_0^1(1-t)^j g^{(j+1)}(t)\,dt \qquad(0\le j\le m).
g^{(N)}Use in source · 1 passage
Search aliases: g^(N)
Use in source · Paper_IV_Traditional_Final.tex, line 1343
For a function g, define the complete differential Bell polynomial Y_N by fracd^Ndz^Ne^g(z) = e^g(z)Y_N(g',g'',…,g^(N)).
Exact mathematical passage
\frac{d^N}{dz^N}e^{g(z)} = e^{g(z)}Y_N\bigl(g',g'',\ldots,g^{(N)}\bigr).
g^{(r)}Use in source · 1 passage
Search aliases: g^(r)
Use in source · 03_nonlinear_coordinates.tex, line 311
Set g(t)=F(x+th) on the open preimage of the domain under t↦ x+th. It contains [0,1], and repeated differentiation gives g^(r)(t)=D^rF(x+th)[h,…,h]. The derivatives are continuous on the compact interval, so their Bochner integrals exist in the Banach codomain. Put T_j=frac1j!∫_0^1(1-t)^j g^(j+1)(t) dt (0≤ j≤ m). The fundamental theorem of calculus gives T_0=g(1)-g(0). For j≥1, integration by parts gives T_j=T_j-1-g^(j)(0)/j!. Iterating this relation proves the Taylor identity with remainder T_m…
Exact mathematical passage
g^{(r)}(t)=D^rF(x+th)[h,\ldots,h]
G^*Use in source · 3 passages
Search aliases: G^*
Use in source · 02_strengthened.tex, line 863
[Nonabelian packet connection] The one form with matrix values A=V^* d V is skew adjoint. Under a frame change V'=VG, where G(x)∈ U(r), A' = G^* A G+G^* d G. Its curvature F= d A+ Awedge A satisfies F=V^*( d Pwedge d P)V.
Exact mathematical passage
\Acal' = G^*\Acal G+G^*\dd G.
G^{-1}Use in source · 3 passages
Search aliases: G^-1
Use in source · visual_atlas.tex, line 408
Direct multiplication gives GOb G^-1=mat-1 · 1 -1 · 1. For example, the original (0,0) block vanishes, while the new (0,0) block relative to the old coordinate projectors does not. This is why the theorem about preservation of each block's zero status requires G P_i=P_iG.
Exact mathematical passage
G\Ob G^{-1}=\mat{-1&1\\-1&1}.
g^{-1}Use in source · 8 passages
Search aliases: g^-1
Use in source · Paper_X_Block27_Amended.tex, line 998
Derived comparison bounds. For a finite group Γ whose order is a unit in a commutative local coefficient ring R, any prescribed equivariant residual isomorphism lifts to an integral isomorphism. Lift its matrix to A and define P=|Γ|^-1∑_g∈Γρ(g)Aτ(g^-1). Reindexing proves equivariance, and reduction recovers the prescribed matrix. Its determinant is a unit. Complete local coefficients with finite quotients, occupied compatible finite conditions and this residual type therefore give actual integra…
Exact mathematical passage
P=|\Gamma|^{-1}\sum_{g\in\Gamma}\rho(g)A\tau(g^{-1}).
G^2Use in source · 1 passage
Search aliases: G^2
Use in source · 20_arithmetic.tex, line 525
d product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [source reference] proves that multiplication adds charges. If FG=1, the same formula yields A_-cG=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G'). Induction, starting from the independently established integrality of the reciprocal G, proves that inversion negates the charge. Finally A_c+2=A_c-1, so increasing …
Exact mathematical passage
A_{-c}G=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G').
g^2Use in source · 7 passages
Search aliases: g^2
Use in source · supplement_08.tex, line 53
For an actual inverse fg=1, applying the product rule and multiplying by g gives B(g)=-g^2B(f)-gM_p^2(f,g)+gM_p(f,g). If g∈ I, induction on r proves boundedness of g from boundedness of f. At each successor stage the preceding stage supplies boundedness of g needed on the right. Passing to all finite orders proves reciprocal closure for an integral inverse.
Exact mathematical passage
B(g)=-g^2B(f)-gM_{p^2}(f,g)+gM_p(f,g).
g^{b-a}Use in source · 1 passage
Search aliases: g^b-a
Use in source · 20_arithmetic.tex, line 1074
To prove surjectivity, use compactness. For an isometry g of a compact metric space into itself, the sequence of iterates g^n(x) has a convergent subsequence. Two sufficiently close iterates, with exponents a<b, yield d(x,g^b-a(x)) equally small after removing a isometric iterations. Since b-a>0, the latter point is in the image of g. Thus the image is dense. It is also compact and hence closed, so it is the whole space. Apply this to Φ_j on Z_2.
Exact mathematical passage
d(x,g^{b-a}(x))
g^nUse in source · 1 passage
Search aliases: g^n
Use in source · 20_arithmetic.tex, line 1072
To prove surjectivity, use compactness. For an isometry g of a compact metric space into itself, the sequence of iterates g^n(x) has a convergent subsequence. Two sufficiently close iterates, with exponents a<b, yield d(x,g^b-a(x)) equally small after removing a isometric iterations. Since b-a>0, the latter point is in the image of g. Thus the image is dense. It is also compact and hence closed, so it is the whole space. Apply this to Φ_j on Z_2.
Exact mathematical passage
g^n(x)
G_{\mathbb Q}Use in source · 3 passages
Search aliases: G_ Q mathbb
Use in source · Paper_X_Block27_Amended.tex, line 965
The formalization specializes to the actual maps G_ Q_p→ G_ Q induced by the field embeddings and choices of algebraic closures. A nontrivial continuous quadratic character of G_ Q, constructed from its action on i^2=-1, gives the integral 2-adic actions τ(σ)=diag(1,-1), ρ(σ)=1 · -1
Exact mathematical passage
G_{\mathbb Q_p}\to G_{\mathbb Q}
G_{\mathbb Q_p}Use in source · 2 passages
Search aliases: G_ Q_p mathbb
Use in source · Paper_X_Block27_Amended.tex, line 965
The formalization specializes to the actual maps G_ Q_p→ G_ Q induced by the field embeddings and choices of algebraic closures. A nontrivial continuous quadratic character of G_ Q, constructed from its action on i^2=-1, gives the integral 2-adic actions τ(σ)=diag(1,-1), ρ(σ)=1 · -1
Exact mathematical passage
G_{\mathbb Q_p}\to G_{\mathbb Q}
g_0Use in source · 8 passages
Search aliases: g_0
Use in source · visual_atlas.tex, line 324
vpage15Flat is not the same as calibratedMODEL D / A FLAT CONNECTION WITH NONIDENTITY EDGESChapter III, mathematical text: Theorem 6.3 Make all three canonical charts literally identical, so J_b← a=1 over QQ. Choose scalar gauges g_0=1, g_1=2, and g_2=3, and define Γ_b← a=g_b g_a^-1. The edge 0→1 is multiplication by two, although its canonical counterpart is the identity.
Exact mathematical passage
g_0=1
G_1Use in source · 1 passage
Search aliases: G_1
Use in source · 03_packets_and_graphs.tex, line 957
It is enough to treat a word in the generators and packet projectors. Insert I=∑_γ∈ IP_γ between consecutive nonprojector generators. Every resulting (β,α) block is a finite sum of products P_β G_mP_γ_m-1 G_m-1⋯ P_γ_1G_1P_α. A nonzero product requires every consecutive block to be nonzero, hence gives a directed path α→γ_1→⋯→γ_m-1→β. Packet projectors only insert stationary steps. Thus a word has no block from α to an unreachable β. Linear combinations preserve the same zero pattern.
Exact mathematical passage
P_\beta G_mP_{\gamma_{m-1}} G_{m-1}\cdots P_{\gamma_1}G_1P_\alpha.
g_2Use in source · 3 passages
Search aliases: g_2
Use in source · visual_atlas.tex, line 324
vpage15Flat is not the same as calibratedMODEL D / A FLAT CONNECTION WITH NONIDENTITY EDGESChapter III, mathematical text: Theorem 6.3 Make all three canonical charts literally identical, so J_b← a=1 over QQ. Choose scalar gauges g_0=1, g_1=2, and g_2=3, and define Γ_b← a=g_b g_a^-1. The edge 0→1 is multiplication by two, although its canonical counterpart is the identity.
Exact mathematical passage
g_2=3
g_3Use in source · 2 passages
Search aliases: g_3
Use in source · 20_arithmetic.tex, line 488
The Hurwitz coefficients of the normalized source K_4,j at degrees 4n are g_j(n)=binom4n+j4n^-1. Their explicit expressions are g_0(n) · =1, · g_1(n) · =frac14n+1, g_2(n) · =frac1(4n+1)(2n+1), · g_3(n) · =frac3(4n+1)(2n+1)(4n+3). Every denominator is odd. An affine sequence with even slope satisfies 2^kmidΔ_1^k f(n) for every k,n. The exact product identity Δ_1^s(fg)(n)=∑_i=0^sbinom si Δ_1^if(n)Δ_1^s-ig(n+i) proves closure of this property under products. Applying the formula with the factors in…
Exact mathematical passage
\begin{aligned} g_0(n)&=1,& g_1(n)&=\frac1{4n+1},\\ g_2(n)&=\frac1{(4n+1)(2n+1)},& g_3(n)&=\frac3{(4n+1)(2n+1)(4n+3)}. \end{aligned}
G_{4,1}Use in source · 2 passages
Search aliases: G_4,1
Use in source · strengthening.tex, line 437
The address-one source is G_4,1(t)=(sinh t+sin t)/(2t). The identities a+b=1, a-b=i, and ab=1/2 give D_1(t):=G_4,1(t)F_0(t) =frac12(fractanh(at)at +fractanh(bt)bt). For example, its numerator follows from sinh t+sin t =2bsinh(at)cosh(bt)+2acosh(at)sinh(bt), while cosh t+cos t=2cosh(at)cosh(bt). Only degrees divisible by four survive. Its actual Hurwitz packets are D_1(t)=∑_i≥0d_ifract^4i(4i)!, d_i=(-frac14)^i frac2^4i+2(2^4i+2-1)B_4i+2(4i+2)(4i+1). The constant is one. Since v_2(4i+2)=1
Exact mathematical passage
G_{4,1}(t)=(\sinh t+\sin t)/(2t)
G_{4,3}Use in source · 2 passages
Search aliases: G_4,3
Use in source · strengthening.tex, line 716
The address-three source is G_4,3(t)=3(sinh t-sin t)/t^3. Addition identities give sinh t-sin t =2asinh(at)cosh(bt)+2bcosh(at)sinh(bt), cosh t-cos t=2sinh(at)sinh(bt). Consequently D_3(t):=G_4,3(t)F_2(t) =frac32(fraccoth(at)at +fraccoth(bt)bt). This expression is a power series: its two polar terms cancel because a^-2+b^-2=0. Its constant is one, only degrees divisible by four survive, and the Bernoulli expansion gives D_3(t)=∑_i≥0d_ifract^4i(4i)!, d_i=3(-frac14)^i frac2^4i+2B_4i+2(4i+2)(4i+1). …
Exact mathematical passage
G_{4,3}(t)=3(\sinh t-\sin t)/t^3
g_aUse in source · 2 passages
Search aliases: g_a
Use in source · visual_atlas.tex, line 331
vfig[.97]flat_not_calibrated platecaptionThe rulers display the chosen gauge scales: the marks are g_a n for integer n. They visualize the supplied scalar factors, not an intrinsic metric. Relative transport is the ratio of the target and source scales.
Exact mathematical passage
g_a n
g_a^{-1}Use in source · 2 passages
Search aliases: g_a^-1
Use in source · visual_atlas.tex, line 325
vpage15Flat is not the same as calibratedMODEL D / A FLAT CONNECTION WITH NONIDENTITY EDGESChapter III, mathematical text: Theorem 6.3 Make all three canonical charts literally identical, so J_b← a=1 over QQ. Choose scalar gauges g_0=1, g_1=2, and g_2=3, and define Γ_b← a=g_b g_a^-1. The edge 0→1 is multiplication by two, although its canonical counterpart is the identity.
Exact mathematical passage
\Gamma_{b\leftarrow a}=g_b g_a^{-1}.
g_{aa}Use in source · 1 passage
Search aliases: g_aa
Use in source · 03_supplement_06.tex, line 50
Smooth overlap maps and rank. The coordinate transition is g_ab(x)=A_b(x)B_a(x). It is smooth on U_a∩ U_b, agrees with the actual change between local homeomorphisms, and satisfies g_aa=I, g_bcg_ab=g_ac. A continuous linear equivalence between E_a and E_x also gives local constancy of fiber dimension. No single constant rank across disconnected components is assumed.
Exact mathematical passage
g_{aa}=I,\qquad g_{bc}g_{ab}=g_{ac}.
g_{ab}Use in source · 3 passages
Search aliases: g_ab
Use in source · 03_supplement_06.tex, line 49
Smooth overlap maps and rank. The coordinate transition is g_ab(x)=A_b(x)B_a(x). It is smooth on U_a∩ U_b, agrees with the actual change between local homeomorphisms, and satisfies g_aa=I, g_bcg_ab=g_ac. A continuous linear equivalence between E_a and E_x also gives local constancy of fiber dimension. No single constant rank across disconnected components is assumed.
Exact mathematical passage
g_{ab}(x)=A_b(x)B_a(x)
g_{ac}Use in source · 1 passage
Search aliases: g_ac
Use in source · 03_supplement_06.tex, line 50
Smooth overlap maps and rank. The coordinate transition is g_ab(x)=A_b(x)B_a(x). It is smooth on U_a∩ U_b, agrees with the actual change between local homeomorphisms, and satisfies g_aa=I, g_bcg_ab=g_ac. A continuous linear equivalence between E_a and E_x also gives local constancy of fiber dimension. No single constant rank across disconnected components is assumed.
Exact mathematical passage
g_{aa}=I,\qquad g_{bc}g_{ab}=g_{ac}.
g_bUse in source · 2 passages
Search aliases: g_b
Use in source · visual_atlas.tex, line 325
vpage15Flat is not the same as calibratedMODEL D / A FLAT CONNECTION WITH NONIDENTITY EDGESChapter III, mathematical text: Theorem 6.3 Make all three canonical charts literally identical, so J_b← a=1 over QQ. Choose scalar gauges g_0=1, g_1=2, and g_2=3, and define Γ_b← a=g_b g_a^-1. The edge 0→1 is multiplication by two, although its canonical counterpart is the identity.
Exact mathematical passage
\Gamma_{b\leftarrow a}=g_b g_a^{-1}.
G_b^{-1}Use in source · 2 passages
Search aliases: G_b^-1
Use in source · 02_connections_obstructions.tex, line 341
Substitute the definition of Γ^G. Adjacent factors G_b^-1G_b and G_c^-1G_c cancel. In the defect, both the composite and direct terms acquire the same outer factors G_c and G_a^-1. In the loop, all inner factors cancel and only conjugation at the base remains.
Exact mathematical passage
G_b^{-1}G_b
g_b^{-1}Use in source · 1 passage
Search aliases: g_b^-1
Use in source · visual_atlas.tex, line 334
The route through two steps 0→1→2 has multiplier (3/2)2=3, equal to the direct route 0→2. More generally, (g_cg_b^-1)(g_bg_a^-1)=g_cg_a^-1 for every triple. Thus this connection is flat and reversible, but it is not calibrated to the original J, whose values are identities.
Exact mathematical passage
(g_cg_b^{-1})(g_bg_a^{-1})=g_cg_a^{-1}
g_{bc}Use in source · 1 passage
Search aliases: g_bc
Use in source · 03_supplement_06.tex, line 50
Smooth overlap maps and rank. The coordinate transition is g_ab(x)=A_b(x)B_a(x). It is smooth on U_a∩ U_b, agrees with the actual change between local homeomorphisms, and satisfies g_aa=I, g_bcg_ab=g_ac. A continuous linear equivalence between E_a and E_x also gives local constancy of fiber dimension. No single constant rank across disconnected components is assumed.
Exact mathematical passage
g_{aa}=I,\qquad g_{bc}g_{ab}=g_{ac}.
G_cUse in source · 10 passages
Search aliases: G_c
Use in source · 02_connections_obstructions.tex, line 328
[Gauge covariance] The defect and triangle holonomy transform as Δ^G_c,b,a =G_cΔ_c,b,aG_a^-1, H^G_c,b,a =G_aH_c,b,aG_a^-1. Thus flatness and triviality of triangle holonomy are gauge invariant. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.gauge_cocycleDefect; Komorebi.PaperIII.Transport.ReversibleConnection.gauge_triangleHolonomy; Komorebi.PaperIII.Transport.ReversibleConnection.gauge_flat_iff; Komorebi.PaperIII.Transport.ReversibleConnection.gauge_triangleHolonomy_eq_refl_iff
Exact mathematical passage
\Delta^G_{c,b,a} =G_c\Delta_{c,b,a}G_a^{-1}, \qquad H^G_{c,b,a} =G_aH_{c,b,a}G_a^{-1}.
g_cUse in source · 1 passage
Search aliases: g_c
Use in source · visual_atlas.tex, line 334
The route through two steps 0→1→2 has multiplier (3/2)2=3, equal to the direct route 0→2. More generally, (g_cg_b^-1)(g_bg_a^-1)=g_cg_a^-1 for every triple. Thus this connection is flat and reversible, but it is not calibrated to the original J, whose values are identities.
Exact mathematical passage
(g_cg_b^{-1})(g_bg_a^{-1})=g_cg_a^{-1}
G_c^{-1}Use in source · 2 passages
Search aliases: G_c^-1
Use in source · 02_connections_obstructions.tex, line 341
Substitute the definition of Γ^G. Adjacent factors G_b^-1G_b and G_c^-1G_c cancel. In the defect, both the composite and direct terms acquire the same outer factors G_c and G_a^-1. In the loop, all inner factors cancel and only conjugation at the base remains.
Exact mathematical passage
G_c^{-1}G_c
G_fUse in source · 1 passage
Search aliases: G_f
Use in source · 03_nonlinear_coordinates.tex, line 749
[Exact defect under composition] Let G_f:Y_f→ Z_f and G_c:Y_c→ Z_c, with coarsening C_Z:Z_f→ Z_c. Then Δ_G∘ F(x) = Δ_G(F_f(x)) + G_c(F_c(C_Xx)+Δ_F(x)) - G_c(F_c(C_Xx)). If G_c is L-Lipschitz on the relevant set, then Δ_G∘ F(x) ≤ Δ_G(F_f(x)) + L Δ_F(x) .
Exact mathematical passage
G_f:Y_f\to Z_f
G_{F,S}Use in source · 7 passages
Search aliases: G_F,S
Use in source · Paper_X_Block27_Amended.tex, line 172
For a Galois problem the principal example is barρ:G_F,S⟶ GL_n(k), with a deformation functor Def_barρ cut out by a determinant and selected local conditions.
Exact mathematical passage
\bar\rho:G_{F,S}\longrightarrow \GL_n(k),
G_iUse in source · 1 passage
Search aliases: G_i
Use in source · continuation_04.tex, line 59
If p^Lmid c, the actual identities give G(x)=F(x)+cH(x), DG(x)≡ DF(x)pmod p. Thus if F(x)≡0pmodp^L and A_x is surjective, then ∃ b∈ B_L(x):G(b)=0. This theorem allows rectangular systems. The evaluation, derivative and Jacobian identities are proved from the polynomial definition G_i=F_i+C(c)H_i, rather than supplied as hypotheses. Algebraic congruence helpers also permit composite moduli. Exact p-adic root existence uses prime p.
Exact mathematical passage
G_i=F_i+C(c)H_i
g_iUse in source · 16 passages
Search aliases: g_i
Use in source · 04_arithmetic.tex, line 60
[Several coprime spectral packets] Assume χ_A(x)=∏_i=1^rf_i(x), where the monic factors f_i∈ O[x] are pairwise coprime in K[x]. Put g_i=∏_j≠ if_j. There are complementary projectors E_i with im E_i=Ker f_i(A), ∑_iE_i=I, E_iE_j=0 (i≠ j). One may choose E_i so that Res(f_i,g_i)E_i∈End_ O(Λ) up to multiplication of the resultant by a unit.
Exact mathematical passage
g_i=\prod_{j\ne i}f_j.
G_jUse in source · 6 passages
Search aliases: G_j
Use in source · Paper_IV_Traditional_Final.tex, line 442
Write F=∑_iF_i and G=∑_jG_j, where F_i= B_N,iF and G_j= B_N,jG. Their covariance gives (F_iG_j)(ω z)=ω^i+jF_i(z)G_j(z), so F_iG_j belongs to the packet labeled by i+j modulo N. Expanding the finite product and applying B_N,k gives the formula.
Exact mathematical passage
G=\sum_jG_j
g_jUse in source · 15 passages
Search aliases: g_j
Use in source · 20_arithmetic.tex, line 484
The Hurwitz coefficients of the normalized source K_4,j at degrees 4n are g_j(n)=binom4n+j4n^-1. Their explicit expressions are g_0(n) · =1, · g_1(n) · =frac14n+1,
Exact mathematical passage
g_j(n)=\binom{4n+j}{4n}^{-1}.
g_{j+1}Use in source · 1 passage
Search aliases: g_j+1
Use in source · supplement_08.tex, line 137
For j=0, g_0 is the literal N-divisibility indicator. Because Nmid p(p-1), the two shifts in δ g_0 agree and δ g_0=0. It is flat at every order. The actual binomial identity gives Λ_j+1g_j+1=(j+1)g_j. As 1≤ j+1<p, the nonsingular reflection theorem inductively proves flatness of every g_j in the standard window. There is no uniform-convergence assumption in this argument.
Exact mathematical passage
\Lambda_{j+1}g_{j+1}=(j+1)g_j.
G_j^{-1}Use in source · 1 passage
Search aliases: G_j^-1
Use in source · supplement_11.tex, line 85
Pack g_j into degrees divisible by four and form its actual EGF. The formal coefficient identity is G_j(t)=∑_n≥0g_j(n)fract^4n(4n)! =j!∑_n≥0fract^4n(4n+j)!. Its inverse is therefore the original normalized packet EGF. The source Kummer property gives flatness of G_j. The proved reciprocal closure gives flatness of G_j^-1.
Exact mathematical passage
G_j^{-1}
G_{jk}Use in source · 1 passage
Search aliases: G_jk
Use in source · 02_strengthened.tex, line 1876
[Finite reflection positivity] For vectors v_1,…,v_m∈ H and nonnegative integers n_1,…,n_m, the matrix G_jk=ipv_jT^n_j+n_kv_k is positive semidefinite. Equivalently, ∑_j,koverlinec_jc_k ipv_jT^n_j+n_kv_k ≥0 for every c∈ C^m.
Exact mathematical passage
G_{jk}=\ip{v_j}{T^{n_j+n_k}v_k}
G_KUse in source · 1 passage
Search aliases: G_K
Use in source · Paper_X_Block27_Amended.tex, line 2411
Actual quadratic local conditions and completion. Continue with the fixed packet P_q above. For a characteristic-zero field K, let j_K:G_K⟶ G_ Q be the actual field-induced homomorphism between absolute Galois groups. It includes a chosen embedding of algebraic closures. Let i_K be the image of the chosen square root i under that embedding. Lean proves compatibility with the Galois actions and hence (q∘ j_K)(σ)=1 Longleftrightarrow σ(i_K)=i_K. The fixed-field theorem for the algebraic closure th…
Exact mathematical passage
j_K:G_K\longrightarrow G_{\Q}
G_kUse in source · 2 passages
Search aliases: G_k
Use in source · 02_supplement_06.tex, line 72
Suppose G_k is a best rank-at-most-k approximation to G in either norm. Take an arbitrary ambient matrix B with rank B≤ k. Compression is an admissible small-core competitor and norm L(G)- L(G_k)_ν · =normG-G_k_ν
Exact mathematical passage
G_k
G_mUse in source · 6 passages
Search aliases: G_m
Use in source · 03_packets_and_graphs.tex, line 957
It is enough to treat a word in the generators and packet projectors. Insert I=∑_γ∈ IP_γ between consecutive nonprojector generators. Every resulting (β,α) block is a finite sum of products P_β G_mP_γ_m-1 G_m-1⋯ P_γ_1G_1P_α. A nonzero product requires every consecutive block to be nonzero, hence gives a directed path α→γ_1→⋯→γ_m-1→β. Packet projectors only insert stationary steps. Thus a word has no block from α to an unreachable β. Linear combinations preserve the same zero pattern.
Exact mathematical passage
P_\beta G_mP_{\gamma_{m-1}} G_{m-1}\cdots P_{\gamma_1}G_1P_\alpha.
g_mUse in source · 4 passages
Search aliases: g_m
Use in source · supplement_07.tex, line 78
pageheading2. Identification with the reduced Euler sequenceCorollary 8.5; manuscript page 21 Work over F_p= Z/p Z. For positive mmid p-1, take the literal section g_m(d)=1, · mmid d, 0, · mnmid d. It satisfies g_m(0)=1. Since p≡1pmod m, the conditions mmid d+p and mmid d+1 are equivalent. The source therefore proves U^pg_m=Ug_m.
Exact mathematical passage
g_m(d)=\begin{cases}1,&m\mid d,\\0,&m\nmid d.\end{cases}
G_{m-1}Use in source · 1 passage
Search aliases: G_m-1
Use in source · 03_packets_and_graphs.tex, line 957
It is enough to treat a word in the generators and packet projectors. Insert I=∑_γ∈ IP_γ between consecutive nonprojector generators. Every resulting (β,α) block is a finite sum of products P_β G_mP_γ_m-1 G_m-1⋯ P_γ_1G_1P_α. A nonzero product requires every consecutive block to be nonzero, hence gives a directed path α→γ_1→⋯→γ_m-1→β. Packet projectors only insert stationary steps. Thus a word has no block from α to an unreachable β. Linear combinations preserve the same zero pattern.
Exact mathematical passage
P_\beta G_mP_{\gamma_{m-1}} G_{m-1}\cdots P_{\gamma_1}G_1P_\alpha.
G_nUse in source · 7 passages
Search aliases: G_n
Use in source · Paper_XI_Traditional_Final.tex, line 451
Let G_n=GL_n( F_q), T_n≅ F_q^n^× with T_n an elliptic maximal torus.
Exact mathematical passage
G_n=\GL_n(\FF_q), \qquad T_n\cong\FF_{q^n}^{\times}
g_nUse in source · 1 passage
Search aliases: g_n
Use in source · Paper_V_Traditional_Final.tex, line 3493
[Residue law for Cauchy products] Let F(z)=∑_n≥0f_nz^n, G(z)=∑_n≥0g_nz^n be formal power series. The same statement holds for power series that converge on a common disk. Decompose both series into residue packets modulo N: F=∑_r=0^N-1F_r, G=∑_s=0^N-1G_s. If H=FG, then its residue packet H_t is H_t = ∑_r+s≡ t pmod NF_rG_s.
Exact mathematical passage
F(z)=\sum_{n\geq0}f_nz^n, \qquad G(z)=\sum_{n\geq0}g_nz^n
G_{N,j}Use in source · 4 passages
Search aliases: G_N,j
Use in source · strengthening.tex, line 14
Throughout these sections, the normalized source and its reciprocal are G_N,j(t)=j!∑_n≥0fract^Nn(Nn+j)!, F_N,j(t)=G_N,j(t)^-1 =∑_n≥0C_N,j(n)fract^Nn(Nn)!, where N≥1 and 0≤ j<N. In particular, C_N,j(0)=1. The coefficients in the original convention satisfy E_Nn^(N,j)=j!C_N,j(n), E_d^(N,j)=0 (Nnmid d). Thus every valuation of a nonzero difference gains v_p(j!) on returning to the original coefficients. This factor must be retained when stating optimal periods or the final congruence.
Exact mathematical passage
G_{N,j}(t)=j!\sum_{n\geq0}\frac{t^{Nn}}{(Nn+j)!},\qquad F_{N,j}(t)=G_{N,j}(t)^{-1} =\sum_{n\geq0}C_{N,j}(n)\frac{t^{Nn}}{(Nn)!}, \label{eq:final-normalization}
g_{N,j}Use in source · 4 passages
Search aliases: g_N,j
Use in source · strengthening.tex, line 140
The packed factorial coefficient of G_N,j is g_N,j(n)=fracj!(Nn)!(Nn+j)! =∏_i=1^j(1+frac Ni n)^-1. The empty product is one. If N=p^a and 1≤ i<N, then v_p(i)<a, so N/i∈ p Z_p. Every affine denominator in [source reference] is therefore a unit on Z_p.
Exact mathematical passage
g_{N,j}(n)=\frac{j!(Nn)!}{(Nn+j)!} =\prod_{i=1}^j\left(1+\frac Ni n\right)^{-1}. \label{eq:actual-prime-power-kernel}
G_{\Q_p}Use in source · 1 passage
Search aliases: G_Q_p G_ Q_p Q mathbf
Use in source · Paper_X_Block27_Amended.tex, line 2496
This inertia is the actual residue-action kernel inside D_A. No equality D_A=G_ Q_p is established in this block. Nor is ramification at two proved: the nontrivial full local character there is a separate result. At three the Z_2-valued sign character is nontrivial on the full local Galois group but trivial on every inertia subgroup covered by the theorem. Thus local triviality and the stated inertia triviality are distinguished by checked arithmetic examples.
Exact mathematical passage
D_A=G_{\Q_p}
G_rUse in source · 12 passages
Search aliases: G_r
Use in source · root_scope.tex, line 128
The scalar packet weights have exact valuations λ_r(q)=3q+s_2(r+q)-s_2(r). Put μ_r=min_q≥0λ_r(q), J_r={q:λ_r(q)=μ_r}, G_r=2^-μ_rH_r. Since λ_r(0)=0 and λ_r(q)≥3q+1-s_2(r), every minimizing address satisfies 0≤ q≤lfloorfracs_2(r)-13rfloor. In particular μ_r≤0, and the minimum is found by a finite carry calculation. Reduction of the whole series gives overline G_r(X)=∑_q∈ J_rX^underline q in F_2[X]. Its highest degree is max J_r, with leading coefficient one. Thus the normalization is the actual G…
Exact mathematical passage
\mu_r=\min_{q\ge0}\lambda_r(q),\qquad J_r=\{q:\lambda_r(q)=\mu_r\},\qquad G_r=2^{-\mu_r}H_r.
g_RUse in source · 2 passages
Search aliases: g_R
Use in source · 71_native.tex, line 251
The component maps f_R:A→ C_R and g_R:B→ C_R have the following universal property. For every set Y and maps h:A→ Y, k:B→ Y satisfying h(a)=k(b) whenever R(a,b), there is a unique map u:C_R→ Y with u∘ f_R=h and u∘ g_R=k.
Exact mathematical passage
g_R:B\to C_R
G_r'Use in source · 1 passage
Search aliases: G_r'
Use in source · root_scope.tex, line 294
For a general depth r, call j_r=max J_r its true terminal. This definition depends on the actual carry minimum. Rewriting the same integer as another wall expression does not change j_r. The universal simplicity target is for every integer r≥1 and x∈ Z_2, G_r(x)=0 Longrightarrow G_r'(x)≠0. The following are completed restricted results of the later continuations. Their family statements include a proof that the indicated label is the true terminal throughout the displayed ranges.
Exact mathematical passage
\text{for every integer }r\ge1\text{ and }x\in\mathbb Z_2, \qquad G_r(x)=0\ \Longrightarrow\ G_r'(x)\ne0. \label{eq:root:simplicitytarget}
G_{\Rcal\leftarrow\Qcal}Use in source · 2 passages
Search aliases: G_Rcal←Qcal G_ R← Q Rcal leftarrow Qcal mathcal leftarrow mathcal
Use in source · 03_nonlinear_coordinates.tex, line 127
[Exact nonlinear coordinate theorem] For every map F:X→ Y, S_ QF_ Q← P A_ P=F. If G:Y→ Z and R is a packet system on Z, then (G∘ F)_ R← P = G_ R← Q ∘ F_ Q← P. No linearity, continuity, or differentiability assumption is needed.
Exact mathematical passage
(G\circ F)_{\Rcal\leftarrow\Pcal} = G_{\Rcal\leftarrow\Qcal} \circ F_{\Qcal\leftarrow\Pcal}.
G_{rs}Use in source · 2 passages
Search aliases: G_rs
Use in source · Paper_V_Traditional_Final.tex, line 1966
[Composition of transfer matrices] If G and H are convolution operators with packet matrices (G_rs) and (H_rs), then the packet matrix of GH is (GH)_rt = ∑_s∈ RG_rsH_st. In one-dimensional transform notation, the corresponding matrix symbols multiply pointwise.
Exact mathematical passage
(G_{rs})
G_sUse in source · 3 passages
Search aliases: G_s
Use in source · Paper_V_Traditional_Final.tex, line 3499
[Residue law for Cauchy products] Let F(z)=∑_n≥0f_nz^n, G(z)=∑_n≥0g_nz^n be formal power series. The same statement holds for power series that converge on a common disk. Decompose both series into residue packets modulo N: F=∑_r=0^N-1F_r, G=∑_s=0^N-1G_s. If H=FG, then its residue packet H_t is H_t = ∑_r+s≡ t pmod NF_rG_s.
Exact mathematical passage
F=\sum_{r=0}^{N-1}F_r, \qquad G=\sum_{s=0}^{N-1}G_s.
g_sUse in source · 6 passages
Search aliases: g_s
Use in source · 72_recovered.tex, line 2208
[Mahler polyphase transfer] Let f(x)=∑_r≥0c_rbinom xr be a convergent Mahler expansion. For g_s(q)=f(Mq+s), one has g_s(q)=∑_k≥0d_s,kbinom qk, where d_s,k = ∑_r=k^Mk+s c_rΘ_M,s(r,k). Every refined coefficient uses a finite window of the original Mahler coefficients.
Exact mathematical passage
g_s(q)=f(Mq+s),
G_{v_0}Use in source · 1 passage
Search aliases: G_v_0
Use in source · 04_correction_transport.tex, line 660
[Gauge covariance] For a loop γ based at v_0, Hol'(γ) = G_v_0Hol(γ)G_v_0^-1. Thus the conjugacy class of each loop holonomy is gauge invariant.
Exact mathematical passage
\Hol'(\gamma) = G_{v_0}\Hol(\gamma)G_{v_0}^{-1}.
G_{v_0}^{-1}Use in source · 1 passage
Search aliases: G_v_0^-1
Use in source · 04_correction_transport.tex, line 660
[Gauge covariance] For a loop γ based at v_0, Hol'(γ) = G_v_0Hol(γ)G_v_0^-1. Thus the conjugacy class of each loop holonomy is gauge invariant.
Exact mathematical passage
\Hol'(\gamma) = G_{v_0}\Hol(\gamma)G_{v_0}^{-1}.
g_\alphaUse in source · 6 passages
Search aliases: g_α alpha
Use in source · 03_packets_and_graphs.tex, line 154
] Let P=(P_α)_α∈ I be a finite packet system on X, and let Y be a vector space over 𝕜. item Every family of linear maps f_α:X_α→ Y determines a unique linear map f:X→ Y with f|_X_α=f_α, namely f=∑_α f_α P_α. item Every family of linear maps g_α:Y→ X_α determines a unique linear map g:Y→ X with P_α g=g_α, namely g=∑_α g_α. Equivalently, Hom(X,Y)≅∏_αHom(X_α,Y), Hom(Y,X)≅∏_αHom(Y,X_α), where the products are finite. [source reference] ExportsAssembly of packetwise operators, channels, corrections, …
Exact mathematical passage
g_\alpha:Y\to X_\alpha
g_\betaUse in source · 1 passage
Search aliases: g_β beta
Use in source · 03_packets_and_graphs.tex, line 177
For x=∑_α P_α x, any extension of the f_α must satisfy f(x)=∑_α f_α(P_α x), which proves both existence and uniqueness of the first formula. For the second, the sum g=∑_α g_α satisfies P_β g =∑_α P_β g_α =g_β because g_α(Y)⊆ X_α. Uniqueness follows from ∑_α P_α=I.
Exact mathematical passage
P_\beta g =\sum_\alpha P_\beta g_\alpha =g_\beta
G_{\gamma,B}Use in source · 1 passage
Search aliases: G_γ,B gamma
Use in source · 03_nonlinear_coordinates.tex, line 340
For a block B={i_1<⋯<i_s} and an intermediate packet γ, write G_γ,B(xmid h) = J^s_γmidα_i_1,…,α_i_sG(x) [h_i_1,…,h_i_s].
Exact mathematical passage
G_{\gamma,B}(x\mid h) = J^s_{\gamma\mid\alpha_{i_1},\ldots,\alpha_{i_s}}G(x) [h_{i_1},\ldots,h_{i_s}].
G_{\gamma_1,B_1}Use in source · 1 passage
Search aliases: G_γ_1,B_1 gamma
Use in source · 03_nonlinear_coordinates.tex, line 353
di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n] · = ∑_π={B_1,…,B_k} ∑_γ_1,…,γ_k J^k_δmidγ_1,…,γ_kF(Gx) [ G_γ_1,B_1(xmid h),…, G_γ_k,B_k(xmid h) ]. The first sum runs over all set partitions of {1,…,n}.
Exact mathematical passage
\begin{aligned} &J^{n}_{\delta\mid\alpha_1,\ldots,\alpha_n}(F\circ G)(x) [h_1,\ldots,h_n]\\ &= \sum_{\pi=\{B_1,\ldots,B_k\}} \sum_{\gamma_1,\ldots,\gamma_k} J^{k}_{\delta\mid\gamma_1,\ldots,\gamma_k}F(Gx) \big[ G_{\gamma_1,B_1}(x\mid h),\ldots, G_{\gamma_k,B_k}(x\mid h) \big]. \end{aligned}
G_{\gamma_k,B_k}Use in source · 1 passage
Search aliases: G_γ_k,B_k gamma
Use in source · 03_nonlinear_coordinates.tex, line 353
di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n] · = ∑_π={B_1,…,B_k} ∑_γ_1,…,γ_k J^k_δmidγ_1,…,γ_kF(Gx) [ G_γ_1,B_1(xmid h),…, G_γ_k,B_k(xmid h) ]. The first sum runs over all set partitions of {1,…,n}.
Exact mathematical passage
\begin{aligned} &J^{n}_{\delta\mid\alpha_1,\ldots,\alpha_n}(F\circ G)(x) [h_1,\ldots,h_n]\\ &= \sum_{\pi=\{B_1,\ldots,B_k\}} \sum_{\gamma_1,\ldots,\gamma_k} J^{k}_{\delta\mid\gamma_1,\ldots,\gamma_k}F(Gx) \big[ G_{\gamma_1,B_1}(x\mid h),\ldots, G_{\gamma_k,B_k}(x\mid h) \big]. \end{aligned}
\operatorname{Gal}Use in source · 3 passages
Search aliases: Gal operatorname
Use in source · Paper_X_Block27_Amended.tex, line 1131
Arithmetic coverage for a fixed cyclotomic extension. Let n>0 and let L= Q(ζ_n) be a cyclotomic number field. The canonical isomorphism Gal(L/ Q)≃( Z/n Z)^× defines σ_p by σ_p(ζ_n)=ζ_n^p when pnmid n. For every automorphism σ, finite set B⊂ N and bound b, Lean proves that there is a rational prime p>b, with p∉ B and pnmid n, such that σ_p=σ. The prime-existence step uses the proved Dirichlet theorem in the pinned Mathlib dependency.
Exact mathematical passage
\operatorname{Gal}(L/\Q)\simeq(\mathbb Z/n\mathbb Z)^\times
\Gam_LUse in source · 1 passage
Search aliases: Gam_L Γ_L Gam Gamma
Use in source · Paper_V_Traditional_Final.tex, line 1215
For a lossless transmission line, Γ_rm in=Γ_L e^-2 iβℓ. One N-phase step corresponds to a line-length increment λ/(2N), up to the chosen orientation convention. Hence the phase address has a direct electrical interpretation.
Exact mathematical passage
\Gam_{\rm in}=\Gam_L e^{-2\ii\beta\ell}.
\Gam_{\rm in}Use in source · 1 passage
Search aliases: Gam_rm in Γ_rm in Gam rm Gamma rm
Use in source · Paper_V_Traditional_Final.tex, line 1215
For a lossless transmission line, Γ_rm in=Γ_L e^-2 iβℓ. One N-phase step corresponds to a line-length increment λ/(2N), up to the chosen orientation convention. Hence the phase address has a direct electrical interpretation.
Exact mathematical passage
\Gam_{\rm in}=\Gam_L e^{-2\ii\beta\ell}.
\GcalUse in source · 6 passages
Search aliases: Gcal G mathcal
Use in source · 02_strengthened.tex, line 1214
[Finite Lindblad generator] A norm continuous semigroup of completely positive trace preserving maps on a finite matrix algebra, in the Schr odinger picture, has a generator of the form G(X) = - i[H,X] + ∑_r ( L_rXL_r^* - frac12L_r^*L_rX - frac12XL_r^*L_r ), where H=H^*. Conversely, every operator of this form generates a completely positive trace preserving semigroup [source reference].
Exact mathematical passage
\Gcal(X) = -\ii[H,X] + \sum_r \left( L_rXL_r^* - \frac12L_r^*L_rX - \frac12XL_r^*L_r \right),
\gcdUse in source · 5 passages
Search aliases: gcd
Use in source · supplement_05.tex, line 179
Principal targets. targetUnitRefinement.omega_compatible, .principal_compatible, .principal_power targetDyadicUnits.dyadic_unit_decomposition, .decomposition_unique pageheading4. CRT addresses and the actual packet selectorsTheorem 10.1; manuscript page 22 Let a≥0, let M be odd, and put N=2^aM. Coprimality is derived from oddness: gcd(2^a,M)=1. The construction uses the actual Chinese-remainder ring equivalence Z/N Z ≃ Z/2^a Z× Z/M Z. Its bijectivity and component formula are checked. On a natur…
Exact mathematical passage
\gcd(2^a,M)=1.
\mathrm{GL}Use in source · 4 passages
Search aliases: GL mathrm
Use in source · Paper_X_Block27_Amended.tex, line 2091
Lifting a summand inside a supplied finite-group carrier. Let R be a commutative integral domain complete and separated for the (π)-adic filtration. Let Γ be a finite group, and let ρ:Γ→GL_n(R) be an actual integral representation. Assume that a=|Γ|≠0 in R and amidπ^e. For k>0, suppose bar E∈ M_n(R/(π)^e+k), bar E^2=bar E, barρ(g)bar E=bar Ebarρ(g) (g∈Γ). There is an actual integral matrix E satisfying E^2=E, ρ(g)E=Eρ(g), Ebmod(π)^k=bar Ebmod(π)^k. No integral summand or coherent family of quoti…
Exact mathematical passage
\rho:\Gamma\to\mathrm{GL}_n(R)
\mathrm{GL}_2Use in source · 2 passages
Search aliases: GL_2 mathrm
Use in source · references.tex, line 65
bibitemKisin2010 Mark Kisin. newblock Deformations of G_ Q_p and GL_2( Q_p) representations. newblock In Laurent Berger, Christophe Breuil, and Pierre Colmez, editors, em Repr esentations p-adiques de Groupes p-adiques II: Repr esentations de GL_2( Q_p) et (φ,Γ)-modules, volume 330 of em Ast erisque, pages 511--528. Soci et e Math ematique de France, Paris, 2010.
Exact mathematical passage
\mathrm{GL}_2(\mathbb Q_p)
\GL_DUse in source · 3 passages
Search aliases: GL_D GL operatorname
Use in source · Paper_XII_Traditional_Final.tex, line 1474
For every positive integer n, let X_λ,n be a finite nonempty set of conjugacy classes of actual continuous representations Γ⟶GL_D( O_λ/I_λ^n). Conjugacy means change of basis by an invertible matrix over that quotient. Assume reduction preserves these sets and that X_λ,1 consists of the declared residual class. Each admissible class must have a continuous representative. The auxiliary level zero is the unique class over the trivial quotient and imposes no extra occupancy hypothesis.
Exact mathematical passage
\Gamma\longrightarrow\GL_D(\OO_\lambda/I_\lambda^n).
\operatorname{GL}_dUse in source · 6 passages
Search aliases: GL_d operatorname
Use in source · 30_lifting.tex, line 686
[Weighted nonlinear lifting] Let p be prime, d,s,r∈ N, F∈ R[X_1,…,X_d]^d, and a∈ R^d. Set J=DF(a). Suppose J∈GL_d( Q_p), p^sJ^-1∈ M_d(R), F(a)∈ p^rR^d, r>2s. Then there is exactly one root b of F in a+p^s+1R^d, and it satisfies b-a∈ p^r-sR^d. Uniqueness holds throughout the full ball of depth s+1.
Exact mathematical passage
\label{eq:weighted-hypotheses} J\in\operatorname{GL}_d(\Qp),\qquad p^sJ^{-1}\in M_d(R),\qquad F(a)\in p^rR^d,\qquad r>2s.
\GL_{\dim\rho}Use in source · 2 passages
Search aliases: GL_dimρ GL dim rho operatorname dim rho
Use in source · Paper_XI_Traditional_Final.tex, line 1871
begingroup renewcommand6666 [Arithmetic packet realization] For every ρ∈ A with its supplied H1--H4 assembled finite datum d_ρ, there exists an actual admissible supercuspidal representation π of GL_dimρ(K) and an arithmetic realization witness R(d_ρ,π). endgroup
Exact mathematical passage
\GL_{\dim\rho}(K)
\operatorname{GL}_eUse in source · 9 passages
Search aliases: GL_e operatorname
Use in source · 30_lifting.tex, line 828
[Joint coefficient and parameter deformation] leavevmode Let F,H∈ R[Y_1,…,Y_e,T_1,…,T_k]^e and a=(y_0,t_0)∈ R^e+k satisfy F(a)=0. Define the actual blocks J=D_YF(a)∈ M_e(R), K=D_TF(a)∈ M_e× k(R). Suppose J∈GL_e( Q_p), p^sJ^-1∈ M_e(R), and M>2s. For every v∈ R^k, set τ=t_0+p^Mv, G=F+p^MH. There is exactly one y∈ y_0+p^s+1R^e with G(y,τ)=0. It satisfies y-y_0 · ∈ p^M-sR^e,
Exact mathematical passage
J\in\operatorname{GL}_e(\Qp)
\operatorname{GL}_mUse in source · 1 passage
Search aliases: GL_m operatorname
Use in source · 04_arithmetic.tex, line 705
[Smith normal form over a discrete valuation ring] Let M∈ M_m,n( O). There are invertible matrices U∈GL_m( O), V∈GL_n( O) such that UMV = diag(π^s_1,…,π^s_r,0,…,0), where 0≤ s_1≤⋯≤ s_r and r is the rank over K. The exponents are uniquely determined. See [source reference].
Exact mathematical passage
U\in\operatorname{GL}_m(\OO), \qquad V\in\operatorname{GL}_n(\OO)
\GL_NUse in source · 1 passage
Search aliases: GL_N GL operatorname
Use in source · Paper_XI_Traditional_Final.tex, line 2133
At both ranks K^× normalizes P. Thus N=K^× J is normal of index two in M=K^× P, and Σ=Ind_ N^ MΛ, π≃cInd_ M^GL_N(F)Σ. Induction in stages preserves the full original Λ, its central character, and its noncompact root. See [source reference]. The full marked scalar-pushout comparison applies to these actual enlarged pairs as well as to the original higher-rank types ([source reference]). The full synthesis and scope are in [source reference].
Exact mathematical passage
\Sigma=\Ind_{\mathcal N}^{\mathcal M}\Lambda, \qquad \pi\simeq\cInd_{\mathcal M}^{\GL_N(F)}\Sigma.
\GL_nUse in source · 11 passages
Search aliases: GL_n GL operatorname
Use in source · Paper_X_Block27_Amended.tex, line 172
For a Galois problem the principal example is barρ:G_F,S⟶ GL_n(k), with a deformation functor Def_barρ cut out by a determinant and selected local conditions.
Exact mathematical passage
\bar\rho:G_{F,S}\longrightarrow \GL_n(k),
\mathrm{GL}_NUse in source · 1 passage
Search aliases: GL_N mathrm
Use in source · dyadic_synthesis.tex, line 25
The actual stratum admits a full simple character θ trivial on U_F^1. Its canonical beta extension can be completed to an actual irreducible admissible supercuspidal π of GL_N(F) with any prescribed smooth central quasicharacter ω trivial on U_F^1, provided the noncompact completion is free or already compatible. Over this residue field, such ω is unramified.
Exact mathematical passage
\mathrm{GL}_N(F)
\mathrm{GL}_nUse in source · 1 passage
Search aliases: GL_n mathrm
Use in source · Paper_X_Block27_Amended.tex, line 2091
Lifting a summand inside a supplied finite-group carrier. Let R be a commutative integral domain complete and separated for the (π)-adic filtration. Let Γ be a finite group, and let ρ:Γ→GL_n(R) be an actual integral representation. Assume that a=|Γ|≠0 in R and amidπ^e. For k>0, suppose bar E∈ M_n(R/(π)^e+k), bar E^2=bar E, barρ(g)bar E=bar Ebarρ(g) (g∈Γ). There is an actual integral matrix E satisfying E^2=E, ρ(g)E=Eρ(g), Ebmod(π)^k=bar Ebmod(π)^k. No integral summand or coherent family of quoti…
Exact mathematical passage
\rho:\Gamma\to\mathrm{GL}_n(R)
\operatorname{GL}_nUse in source · 1 passage
Search aliases: GL_n operatorname
Use in source · 04_arithmetic.tex, line 705
[Smith normal form over a discrete valuation ring] Let M∈ M_m,n( O). There are invertible matrices U∈GL_m( O), V∈GL_n( O) such that UMV = diag(π^s_1,…,π^s_r,0,…,0), where 0≤ s_1≤⋯≤ s_r and r is the rank over K. The exponents are uniquely determined. See [source reference].
Exact mathematical passage
U\in\operatorname{GL}_m(\OO), \qquad V\in\operatorname{GL}_n(\OO)
\grUse in source · 2 passages
Search aliases: gr operatorname
Use in source · 04_correction_transport.tex, line 405
i+j L. For a degree one element η, put Ω(η)=dη+frac12[η,η]. Suppose Ω(η)∈ F^m L^2 with m≥1. For every h∈ F^m L^1, Ω(η+h) ≡ Ω(η)+d_η h pmodF^m+1 L^2, d_η=d+[η,mathord·]. The obstruction to improving η by one depth is therefore the class of Ω(η) in the cokernel of gr^m(d_η): gr^m L^1 ⟶ gr^m L^2.
Exact mathematical passage
\gr^m(d_\eta): \gr^m\Lcal^1 \longrightarrow \gr^m\Lcal^2.
\gr^mUse in source · 1 passage
Search aliases: gr^m gr operatorname
Use in source · 04_correction_transport.tex, line 405
i+j L. For a degree one element η, put Ω(η)=dη+frac12[η,η]. Suppose Ω(η)∈ F^m L^2 with m≥1. For every h∈ F^m L^1, Ω(η+h) ≡ Ω(η)+d_η h pmodF^m+1 L^2, d_η=d+[η,mathord·]. The obstruction to improving η by one depth is therefore the class of Ω(η) in the cokernel of gr^m(d_η): gr^m L^1 ⟶ gr^m L^2.
Exact mathematical passage
\gr^m(d_\eta): \gr^m\Lcal^1 \longrightarrow \gr^m\Lcal^2.
\gr^pUse in source · 1 passage
Search aliases: gr^p gr operatorname
Use in source · 04_correction_transport.tex, line 435
[Spectral sequence of a finite filtered complex] Let C^bullet be a cochain complex with a finite decreasing filtration by subcomplexes C^bullet=F^0C^bullet⊇ F^1C^bullet⊇⋯⊇ F^N+1C^bullet=0. There is a natural spectral sequence with E_0^p,q=gr^pC^p+q, E_1^p,q=H^p+q(gr^pC^bullet), which converges to the associated graded object of H^p+q(C^bullet) with its induced filtration. A filtered cochain map induces a morphism of spectral sequences.
Exact mathematical passage
E_0^{p,q}=\gr^pC^{p+q}, \qquad E_1^{p,q}=H^{p+q}(\gr^pC^\bullet),
\bm HUse in source · 3 passages
Search aliases: H bm
Use in source · Paper_V_Traditional_Final.tex, line 439
[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.
Exact mathematical passage
\bm Y(z)=\bm H^{(N)}(z)\bm X(z), \qquad H^{(N)}_{ks}(z)=\sum_m h[Nm+k-s]z^{-m}.
\mathbb HUse in source · 4 passages
Search aliases: H mathbb
Use in source · Paper_V_Traditional_Final.tex, line 3649
Let H = {z∈ C:Re z>0} and define C(z)=fracz-1z+1. It maps H conformally onto the unit disk D, with inverse C^-1(w)=frac1+w1-w. For θ∈ R, let R_θ(w)= e^ iθw.
Exact mathematical passage
\mathbb H = \{z\in\CC:\Re z>0\}
\mathrm{H}Use in source · 3 passages
Search aliases: H mathrm
Use in source · Paper_V_Traditional_Final.tex, line 525
The Haar analysis matrix is E_H = frac1sqrt2 1 · 1 1 · -1 . It is orthogonal, so R_H= E_H^ T and R_H E_H=I_2. The channels are a[n]=fracx[2n]+x[2n+1]sqrt2, d[n]=fracx[2n]-x[2n+1]sqrt2.
Exact mathematical passage
\bm E_{\mathrm{H}} = \frac1{\sqrt2} \begin{pmatrix} 1&1\\ 1&-1 \end{pmatrix}.
h'Use in source · 2 passages
Search aliases: h'
Use in source · 04_correction_transport.tex, line 305
If x'-x∈ F^mX, then filtration congruence preservation gives Φ(x')-Φ(x)∈ F^mY. Thus one defect lies in F^mY exactly when the other does. If h-h'∈ F^m+1X, then x_m+h and x_m+h' define the same depth m+1 class. Conversely, two depth m+1 classes reducing to [x_m]_m differ from x_m by elements of F^mX, and they are equal precisely when those elements agree modulo F^m+1X. The first-order congruence shows that such a child has defect in F^m+1Y exactly when its correction class solves the associated gr…
Exact mathematical passage
h-h'\in F^{m+1}X
\bm H^{(N)}Use in source · 1 passage
Search aliases: H^(N) bm
Use in source · Paper_V_Traditional_Final.tex, line 439
[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.
Exact mathematical passage
\bm Y(z)=\bm H^{(N)}(z)\bm X(z), \qquad H^{(N)}_{ks}(z)=\sum_m h[Nm+k-s]z^{-m}.
H^{(N)}_{ks}Use in source · 1 passage
Search aliases: H^(N)_ks
Use in source · Paper_V_Traditional_Final.tex, line 439
[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.
Exact mathematical passage
\bm Y(z)=\bm H^{(N)}(z)\bm X(z), \qquad H^{(N)}_{ks}(z)=\sum_m h[Nm+k-s]z^{-m}.
H^{-1}Use in source · 6 passages
Search aliases: H^-1
Use in source · 10_foundations.tex, line 89
[Constructed unit-triangular coordinates] Let h: N→ K satisfy h_0=1, write H= E(h) and H^-1= E(k), and define U_Ha= D(H E(a)). Then U_H is an equivalence, with inverse U_H^-1a= D(H^-1 E(a)). Its coefficient formulas are (U_Ha)_n=∑_r=0^nbinom nr h_r a_n-r, (U_H^-1a)_n=∑_r=0^nbinom nr k_r a_n-r. The resulting triangular matrix has diagonal entries 1. The transported selectors widehat P_N,j=U_HP_N,jU_H^-1 are idempotent and pairwise orthogonal, and their sum is the identity.
Exact mathematical passage
H^{-1}=\mathcal E(k)
h^{-1}Use in source · 1 passage
Search aliases: h^-1
Use in source · continuation_05.tex, line 22
There is a coordinate permutation σ, with (σ z)_i=z_σ(i). Write σ z=(y,t), using e solved coordinates and q free coordinates. Define (y_0,t_0)=σ x, B=x+p^L Z_p^e+q, T=t_0+p^L Z_p^q, R={z∈ B:F(z)=0}. There is a homeomorphism h:Txrightarrow ∼ R, h^-1(z)=free(σ z). Equivalently, a continuous function ϕ:T→ y_0+p^L Z_p^e gives the entire root locus through σ h(t)=(ϕ(t),t). Every parameter has exactly one corresponding root in B. For every N≥0, h(t)≡ h(u)pmodp^Niff t≡ upmodp^N.
Exact mathematical passage
h:T\xrightarrow{\ \sim\ }\mathcal R,\qquad h^{-1}(z)=\operatorname{free}(\sigma z).
h^{-2}Use in source · 3 passages
Search aliases: h^-2
Use in source · Paper_VI_Traditional_Final.tex, line 5583
FloatBarrier Fractional modes and killed transfer. For the one-dimensional Dirichlet difference operator, let h=1/M with an integer M≥2, let 1≤ k≤ M-1, and set λ_k,h=4h^-2sin^2(kπ h/2). The retained explicit coefficient below is valid for every discrete mode. [Fractional low mode dispersion] Let 0<α≤1. Then 0 ≤ (kπ)^2α-λ_k,h^α ≤ fracα(4k^2)^α-1(kπ)^412h^2. For every t≥0, abs e^-tλ_k,h^α - e^-t(kπ)^2α ≤ fractα(4k^2)^α-1(kπ)^412h^2.
Exact mathematical passage
\lambda_{k,h}=4h^{-2}\sin^2(k\pi h/2)
H^0Use in source · 4 passages
Search aliases: H^0
Use in source · 03_nonlinear_coordinates.tex, line 1144
[Cohomological obstruction through the cone] Let e∈ E^n satisfy d_Ee=0. The class of (e,0) in H^n(Cone(D)) vanishes if and only if the class [e]∈ H^n(E) lies in the image of H^n(D):H^n(C)⟶ H^n(E). When C and E are concentrated in degree zero, H^0(Cone(D))≅coker D.
Exact mathematical passage
H^0(\Cone(D))\cong\coker D.
H^1_jUse in source · 1 passage
Search aliases: H^1_j
Use in source · Paper_X_Block27_Amended.tex, line 365
[Cohomology packet decomposition] Under these hypotheses, each packet is a subcomplex and H^r(C^bullet) ≅ ⊕_jH^r(P_jC^bullet). In particular, H^1=⊕_jH^1_j, H^2=⊕_jH^2_j, so tangent and obstruction spaces reconstruct exactly from their packet components.
Exact mathematical passage
H^1=\bigoplus_jH^1_j, \qquad H^2=\bigoplus_jH^2_j,
h^2Use in source · 47 passages
Search aliases: h^2
Use in source · 04_correction_transport.tex, line 950
[One-step direct rotation estimate] Assume P(t) is twice continuously differentiable in operator norm on [0,T]. There are constants h_0>0 and C>0 such that, whenever t,t+h∈[0,T] and |h|≤ h_0, U( P(t+h), P(t)) - I - hK(t) ≤ Ch^2. The constants can be chosen uniformly in t.
Exact mathematical passage
\left\| \Ucal(\Pcal(t+h),\Pcal(t)) - I - hK(t) \right\| \le Ch^2.
H^2_jUse in source · 1 passage
Search aliases: H^2_j
Use in source · Paper_X_Block27_Amended.tex, line 365
[Cohomology packet decomposition] Under these hypotheses, each packet is a subcomplex and H^r(C^bullet) ≅ ⊕_jH^r(P_jC^bullet). In particular, H^1=⊕_jH^1_j, H^2=⊕_jH^2_j, so tangent and obstruction spaces reconstruct exactly from their packet components.
Exact mathematical passage
H^1=\bigoplus_jH^1_j, \qquad H^2=\bigoplus_jH^2_j,
H^aUse in source · 1 passage
Search aliases: H^a
Use in source · odd_capacity.tex, line 132
For H, BK 3.1.9(ii) states that H^a(β_i)=H^a(β_i+1) as soon as
Exact mathematical passage
H^a(\beta_i)=H^a(\beta_{i+1})
H^DUse in source · 1 passage
Search aliases: H^D
Use in source · visual_atlas.tex, line 297
vpage14Gauge changes the picture, not flatnessMODEL E / CONJUGATION AND ITS LIMITSChapter III, mathematical text: Theorem 6.2 At the base chart, change coordinates by D=diag(2,1). Transform every connection edge by the prescribed endpoint gauges. The loop transforms by conjugation, giving H^D=DHD^-1=mat1 · 2 0 · 1. The shear looks stronger in these fixed numerical axes, but the verdict on flatness is unchanged by gauge.
Exact mathematical passage
H^D=DHD^{-1}=\mat{1&2\\0&1}.
H^G_{c,b,a}Use in source · 1 passage
Search aliases: H^G_c,b,a
Use in source · 02_connections_obstructions.tex, line 328
[Gauge covariance] The defect and triangle holonomy transform as Δ^G_c,b,a =G_cΔ_c,b,aG_a^-1, H^G_c,b,a =G_aH_c,b,aG_a^-1. Thus flatness and triviality of triangle holonomy are gauge invariant. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.gauge_cocycleDefect; Komorebi.PaperIII.Transport.ReversibleConnection.gauge_triangleHolonomy; Komorebi.PaperIII.Transport.ReversibleConnection.gauge_flat_iff; Komorebi.PaperIII.Transport.ReversibleConnection.gauge_triangleHolonomy_eq_refl_iff
Exact mathematical passage
\Delta^G_{c,b,a} =G_c\Delta_{c,b,a}G_a^{-1}, \qquad H^G_{c,b,a} =G_aH_{c,b,a}G_a^{-1}.
H^iUse in source · 4 passages
Search aliases: H^i
Use in source · Paper_X_Block27_Amended.tex, line 2002
Modular Gaussian elimination computes the ranks of the differentials. For C^i-1xrightarrowd_i-1C^ixrightarrowd_iC^i+1, the finite dimension is evaluated as dim H^i = dim C^i-rank d_i-rank d_i-1. Packet subcomplexes are computed separately and checked against the total.
Exact mathematical passage
\dim H^i = \dim C^i-\rank d_i-\rank d_{i-1}.
H^mUse in source · 1 passage
Search aliases: H^m
Use in source · odd_capacity.tex, line 189
The right side is the Heisenberg commutator pairing for the minimal simple stratum of gamma. Its quotient J^1(γ)/H^1(γ) is precisely U^m(A)/H^m(γ), because J^1(γ)=U^1(B_order)U^m(A), with U^1(B_order)⊂ H^1(γ). BK's nondegeneracy theorem for simple-character commutators therefore says that the form on S is nondegenerate.
Exact mathematical passage
U^m(A)/H^m(\gamma)
H^{m+1}Use in source · 1 passage
Search aliases: H^m+1
Use in source · dyadic_synthesis.tex, line 108
For a later native step, first transfer the full beta-determinant unit character to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar units, Θ_j(uI)=θ_j(u^2)=1. For odd critical level r, use m=lfloor r/2rfloor and D=H^m+1(β)=H^m+1(γ). The endpoint is valid because 2(m+1)-r=1: the cross term for the additive character lies in the radical on which the matrix additive character is trivial. BK 3.3.18 supplies ϑ=Θ_j|_D ψ_β-γ. Integral trace makes it trivial on D∩ U_F^1.
Exact mathematical passage
D=H^{m+1}(\beta)=H^{m+1}(\gamma)
H^nUse in source · 8 passages
Search aliases: H^n
Use in source · 03_nonlinear_coordinates.tex, line 1136
[Cohomological obstruction through the cone] Let e∈ E^n satisfy d_Ee=0. The class of (e,0) in H^n(Cone(D)) vanishes if and only if the class [e]∈ H^n(E) lies in the image of H^n(D):H^n(C)⟶ H^n(E). When C and E are concentrated in degree zero, H^0(Cone(D))≅coker D.
Exact mathematical passage
H^n(\Cone(D))
H^{p+q}Use in source · 2 passages
Search aliases: H^p+q
Use in source · 04_correction_transport.tex, line 435
[Spectral sequence of a finite filtered complex] Let C^bullet be a cochain complex with a finite decreasing filtration by subcomplexes C^bullet=F^0C^bullet⊇ F^1C^bullet⊇⋯⊇ F^N+1C^bullet=0. There is a natural spectral sequence with E_0^p,q=gr^pC^p+q, E_1^p,q=H^p+q(gr^pC^bullet), which converges to the associated graded object of H^p+q(C^bullet) with its induced filtration. A filtered cochain map induces a morphism of spectral sequences.
Exact mathematical passage
E_0^{p,q}=\gr^pC^{p+q}, \qquad E_1^{p,q}=H^{p+q}(\gr^pC^\bullet),
H^rUse in source · 5 passages
Search aliases: H^r
Use in source · Paper_X_Block27_Amended.tex, line 357
[Cohomology packet decomposition] Under these hypotheses, each packet is a subcomplex and H^r(C^bullet) ≅ ⊕_jH^r(P_jC^bullet). In particular, H^1=⊕_jH^1_j, H^2=⊕_jH^2_j, so tangent and obstruction spaces reconstruct exactly from their packet components.
Exact mathematical passage
\boxed{ H^r(C^\bullet) \cong \bigoplus_jH^r(P_jC^\bullet). }
h_0Use in source · 8 passages
Search aliases: h_0
Use in source · 04_correction_transport.tex, line 949
[One-step direct rotation estimate] Assume P(t) is twice continuously differentiable in operator norm on [0,T]. There are constants h_0>0 and C>0 such that, whenever t,t+h∈[0,T] and |h|≤ h_0, U( P(t+h), P(t)) - I - hK(t) ≤ Ch^2. The constants can be chosen uniformly in t.
Exact mathematical passage
h_0>0
h_1Use in source · 16 passages
Search aliases: h_1
Use in source · 03_nonlinear_coordinates.tex, line 235
[Exact reconstruction of derivatives] For h_1,…,h_r∈ X, D^rF(x)[h_1,…,h_r] = ∑_γ ∑_α_1,…,α_r J^r_γmidα_1,…,α_rF(x) [P_α_1h_1,…,P_α_rh_r].
Exact mathematical passage
h_1,\ldots,h_r\in X
H_{1/3}Use in source · 1 passage
Search aliases: H_1/3
Use in source · atlas-pages.tex, line 579
vpage31Derivative jumps have their own packetsTRACE DATA / A STEP AND A RAMPChapter V: Theorem E7.2, Corollary E7.3. On [0,1], let H_τ=1_[τ,1] and take f(t)=t^2+2H_1/3(t)-3(t-2/3)_+. The function has a value jump 2 at 1/3 and a first-derivative jump -3 at 2/3. Its packet and remainder are j_f(t)=2ϕ_0,1/3(t)-3ϕ_1,2/3(t), s_f(t)=t^2. vfig[.92]ext_trace platecaptionThe displayed curves are exact polynomial pieces. The solid jump is not connected across its discontinuity. The dashed curve is the glo…
Exact mathematical passage
f(t)=t^2+2H_{1/3}(t)-3(t-2/3)_+.
\mathcal H_2Use in source · 2 passages
Search aliases: H_2 mathcal
Use in source · 20_arithmetic.tex, line 516
For an integer charge c, put A_c=fracD^4-c2. A series is flat of charge c when applying any power A_c^s to it gives an element of H_2. The required product formula is A_c+d(FG)=(A_cF)G+F(A_dG) +2F'G'''+3F''G''+2F'''G'. All middle coefficients are integral. Derivatives preserve H_2 and commute with A_c. Induction on the number of iterations in [source reference] proves that multiplication adds charges. If FG=1, the same formula yields A_-cG=-G^2A_cF-G(2F'G'''+3F''G''+2F'''G'). Induction, starting…
Exact mathematical passage
\mathcal H_2
h_2Use in source · 7 passages
Search aliases: h_2
Use in source · 03_nonlinear_coordinates.tex, line 400
[The second-order composition law] At order two, the partition formula has two shapes. For packet directions h_1 and h_2, R_δ D^2(F∘ G)(x)[h_1,h_2] = · ∑_γ_1,γ_2 J^2_δmidγ_1,γ_2F(Gx) [J^1_γ_1midα_1G(x)h_1,J^1_γ_2midα_2G(x)h_2]
Exact mathematical passage
h_2
H_{2,1,0}Use in source · 5 passages
Search aliases: H_2,1,0
Use in source · 02_connections_obstructions.tex, line 72
[The minimal API admits nontrivial holonomy] There is a connection on three points over Q for which transport 0→1 is multiplication by 2, every other transport between distinct endpoints is the identity, and H_2,1,0(1)=2. In particular H_2,1,0≠id. statusleanthreeKomorebi.PaperIII.Transport.twistedThreePointConnection; Komorebi.PaperIII.Transport.twisted_triangleHolonomy_apply_one; Komorebi.PaperIII.Transport.twisted_triangleHolonomy_ne_refl
Exact mathematical passage
H_{2,1,0}(1)=2.
H_{2n}Use in source · 1 passage
Search aliases: H_2n
Use in source · pages.tex, line 126
vpage8See every butterfly valueWALSH RECURSION IN EIGHT COORDINATESChapter VIII: Theorem 4.7. The unnormalized Walsh matrix satisfies H_1=[1] and H_2n=matH_n · H_n H_n · -H_n, H_nH_n^T=nI. Apply the three add/subtract stages to (1,2,3,4,5,6,7,8)^T. The full intermediate ledger is displayed below. vfigwalsh headArithmetic can be counted exactly Each stage visits eight coordinates, producing one sum or difference per coordinate. Three stages therefore use exactly 24 additions or subtractions. No m…
Exact mathematical passage
H_{2n}=\mat{H_n&H_n\\H_n&-H_n},\qquad H_nH_n^T=nI.
H_{3,0}Use in source · 1 passage
Search aliases: H_3,0
Use in source · supplement_09.tex, line 202
It also derives coefT3H_3,0(log(1+T))=1/6 from the actual differential recurrence and checks its valuation v_3(1/6)=-1. Thus weighted membership does not mean ordinary coefficient integrality. This value attains the allowed floor weight at degree three.
Exact mathematical passage
\coef{T}{3}H_{3,0}(\log(1+T))=1/6
H_4Use in source · 2 passages
Search aliases: H_4
Use in source · atlas-pages.tex, line 753
vpage33Gaussian packets have separate clocksORNSTEIN--UHLENBECK / HERMITE ENERGYChapter VI, Part II: Theorem E7.1 and Corollaries E7.2, E7.4, and E7.5. Standard Gaussian measure. In L^2(γ) for the standard Gaussian law on R, use the probabilists' Hermite polynomials H_1(x)=x, H_2(x)=x^2-1, H_4(x)=x^4-6x^2+3. The Ornstein--Uhlenbeck generator is Lf=f''-xf'. Direct polynomial differentiation gives LH_n=-nH_n for these three polynomials. Their squared norms are n! and different Hermite degrees are …
Exact mathematical passage
H_1(x)=x,\qquad H_2(x)=x^2-1,\qquad H_4(x)=x^4-6x^2+3.
\bm H_4Use in source · 1 passage
Search aliases: H_4 bm
Use in source · Paper_V_Traditional_Final.tex, line 2015
[A four phase finite impulse response operator] Let h(0)=1, h(1)=2, h(2)=3, h(3)=4, h(4)=5, with all other taps zero. For four residue channels, the transfer symbol is H_4(z) = 1+5z^-1 · 4z^-1 · 3z^-1 · 2z^-1 2 · 1+5z^-1 · 4z^-1 · 3z^-1 3 · 2 · 1+5z^-1 · 4z^-1 4 · 3 · 2 · 1+5z^-1 . The delay term records the tap that crosses a coarse cell boundary. No approximation is introduced by the matrix representation.
Exact mathematical passage
\bm H_4(z) = \begin{pmatrix} 1+5z^{-1}&4z^{-1}&3z^{-1}&2z^{-1}\\ 2&1+5z^{-1}&4z^{-1}&3z^{-1}\\ 3&2&1+5z^{-1}&4z^{-1}\\ 4&3&2&1+5z^{-1} \end{pmatrix}.
H_8Use in source · 1 passage
Search aliases: H_8
Use in source · pages.tex, line 134
The result (36,-4,-8,0,-16,0,0,0)^T still reconstructs the input: the inverse is H_8^T/8. Because the displayed transform is unnormalized, its squared output norm is eight times the squared input norm. The normalized transform H_8/sqrt8 would instead be orthogonal. headWhat the zeros mean The zero coefficients arise from exact cancellation among the input values. They do not imply that the transform's defining matrix is sparse. The butterfly structure, not the final coefficient support, explains…
Exact mathematical passage
H_8/\sqrt8
H_8^TUse in source · 1 passage
Search aliases: H_8^T
Use in source · pages.tex, line 134
The result (36,-4,-8,0,-16,0,0,0)^T still reconstructs the input: the inverse is H_8^T/8. Because the displayed transform is unnormalized, its squared output norm is eight times the squared input norm. The normalized transform H_8/sqrt8 would instead be orthogonal. headWhat the zeros mean The zero coefficients arise from exact cancellation among the input values. They do not imply that the transform's defining matrix is sparse. The butterfly structure, not the final coefficient support, explains…
Exact mathematical passage
H_8^T/8
H_aUse in source · 10 passages
Search aliases: H_a
Use in source · 72_recovered.tex, line 2330
[Scalar polyphase precision amplification] Let u∈ Z_p^×, let a,r≥0, and put H_a= (p-1)p^a, · p odd, 2^a+1, · p=2. Then Δ_H_a^ ru^n = u^n(u^H_a-1)^r. For odd p, v_p(Δ_H_a^ ru^n) ≥(a+1)r. For p=2 and odd u, v_2(Δ_H_a^ ru^n) ≥(a+3)r. Equivalently, every additional prime power digit increases the certified Mahler slope by one.
Exact mathematical passage
H_a= \begin{cases} (p-1)p^a,&p\text{ odd},\\ 2^{a+1},&p=2. \end{cases}
\mathcal H_AUse in source · 1 passage
Search aliases: H_A mathcal
Use in source · 03_supplement_03.tex, line 81
Full support is expressed as surjectivity of the reduced matrix's linear map. Assuming d_A>0, the checked equivalence is O_A(ψ)= H_A⊗ H_B Longleftrightarrow Ranρ_B= H_B. Normalization of ψ is unnecessary for these dimension and support statements. The zero state has rank 0 because its decomposition needs no product terms.
Exact mathematical passage
\boxed{\mathcal O_A(\psi)=\mathcal H_A\otimes\mathcal H_B \quad\Longleftrightarrow\quad \Ran\rho_B=\mathcal H_B.}
\mathcal H_BUse in source · 1 passage
Search aliases: H_B mathcal
Use in source · 03_supplement_03.tex, line 81
Full support is expressed as surjectivity of the reduced matrix's linear map. Assuming d_A>0, the checked equivalence is O_A(ψ)= H_A⊗ H_B Longleftrightarrow Ranρ_B= H_B. Normalization of ψ is unnecessary for these dimension and support statements. The zero state has rank 0 because its decomposition needs no product terms.
Exact mathematical passage
\boxed{\mathcal O_A(\psi)=\mathcal H_A\otimes\mathcal H_B \quad\Longleftrightarrow\quad \Ran\rho_B=\mathcal H_B.}
H_cUse in source · 3 passages
Search aliases: H_c
Use in source · 20_arithmetic.tex, line 216
Set H_c(t)=∑_n≥0fract^Nn+c(Nn+c)!, A_c(t)=∑_n≥0fract^Nn(Nn+c)!, R_c(t)=fracA_c-1(t)A_c(t) (1≤ c<N). The denominator of this last quotient has nonzero constant coefficient. The section identities give R_cH_c=tH_c-1, R_c(0)=c, j! P_N,j= P_N,0∏_c=1^jR_c. Cancellation of consecutive source sections proves the product identity and ties the ratios directly to the family under study.
Exact mathematical passage
H_c(t)=\sum_{n\ge0}\frac{t^{Nn+c}}{(Nn+c)!},\qquad A_c(t)=\sum_{n\ge0}\frac{t^{Nn}}{(Nn+c)!},\qquad R_c(t)=\frac{A_{c-1}(t)}{A_c(t)}\quad(1\le c<N).
H_{c-1}Use in source · 1 passage
Search aliases: H_c-1
Use in source · 20_arithmetic.tex, line 223
Set H_c(t)=∑_n≥0fract^Nn+c(Nn+c)!, A_c(t)=∑_n≥0fract^Nn(Nn+c)!, R_c(t)=fracA_c-1(t)A_c(t) (1≤ c<N). The denominator of this last quotient has nonzero constant coefficient. The section identities give R_cH_c=tH_c-1, R_c(0)=c, j! P_N,j= P_N,0∏_c=1^jR_c. Cancellation of consecutive source sections proves the product identity and ties the ratios directly to the family under study.
Exact mathematical passage
R_cH_c=tH_{c-1},\qquad R_c(0)=c,\qquad j!\mathcal P_{N,j}=\mathcal P_{N,0}\prod_{c=1}^{j}R_c.
h_{d-1}Use in source · 2 passages
Search aliases: h_d-1
Use in source · dyadic_synthesis.tex, line 62
erse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact filtration valuation, and give the identity tr_LLift_E(C) =Tr_E/L (αtr_E C). The actual recursive matrix is B(L,t_L;h_1,…,h_d) =α I_2^d-1+ Lift_E(B(E,z;h_1,…,h_d-1)), with multiplication by α as the rank-two base case.
Exact mathematical passage
B(L,t_L;h_1,\ldots,h_d) =\alpha I_{2^{d-1}}+ \operatorname{Lift}_E\bigl(B(E,z;h_1,\ldots,h_{d-1})\bigr),
H_eUse in source · 4 passages
Search aliases: H_e
Use in source · 03_weighted.tex, line 247
A packet decomposition resolves H=∑_e∈ EH_e and assigns a certified filtration gain D^-1H_e∈ F^w(e) B, w(e)∈ Z_>0. Each e is a directed transfer edge. For an ordered transfer word γ=(e_1,…,e_r), define W(γ)=∑_j=1^rw(e_j), A_γ=A_e_1⋯ A_e_r, A_e=D^-1H_e.
Exact mathematical passage
H=\sum_{e\in E}H_e
H_{e_1}Use in source · 1 passage
Search aliases: H_e_1
Use in source · 03_weighted.tex, line 349
Here A_γ=(D^-1H_e_1)⋯(D^-1H_e_r) in the written order. Multiplicativity places it in F^W(γ)R, so the quotient kills it when W(γ)≥ L. Positive weights also imply |γ|≤ W(γ), making the retained search finite.
Exact mathematical passage
A_\gamma=(D^{-1}H_{e_1})\cdots(D^{-1}H_{e_r})
H_{e_r}Use in source · 1 passage
Search aliases: H_e_r
Use in source · 03_weighted.tex, line 349
Here A_γ=(D^-1H_e_1)⋯(D^-1H_e_r) in the written order. Multiplicativity places it in F^W(γ)R, so the quotient kills it when W(γ)≥ L. Positive weights also imply |γ|≤ W(γ), making the retained search finite.
Exact mathematical passage
A_\gamma=(D^{-1}H_{e_1})\cdots(D^{-1}H_{e_r})
H_{\mathrm{eff}}Use in source · 7 passages
Search aliases: H_eff mathrm
Use in source · 01_finite_core.tex, line 907
[Exact Feshbach resolvent] Assume that both zI_Q-H_QQ and zI-H are invertible. Define H_eff(z) = H_PP +H_PQ(z-H_QQ)^-1H_QP. Then P(z-H)^-1P = (z-H_eff(z))^-1 where the left side is restricted to P H, the right side is an inverse on that retained space, and both are written in the same orthonormal basis. Here z-H_QQ and z-H_eff abbreviate the appropriately sized scalar matrices minus the indicated blocks.
Exact mathematical passage
\boxed{ H_{\mathrm{eff}}(z) = H_{PP} +H_{PQ}(z-H_{QQ})^{-1}H_{QP}. }
H_{\mathrm{eff}}^*Use in source · 1 passage
Search aliases: H_eff^* mathrm
Use in source · 03_supplement_02.tex, line 110
The explicitly defined coordinate curve y(s)=U(s)^*ψ(s) is differentiable at t, with dot y=dot U^*ψ(t)+U(t)^*dotψ. It satisfies idot y=H_effy(t), H_eff=U(t)^*HU(t)-iU(t)^*dot U. The connection term -iU(t)^*dot U is Hermitian. If H^*=H, then H_eff^*=H_eff.
Exact mathematical passage
H_{\mathrm{eff}}^*=H_{\mathrm{eff}}
H_{\mathrm{even}}Use in source · 2 passages
Search aliases: H_even mathrm
Use in source · 01_finite_core.tex, line 395
For two qubits and real parameters J,h, consider H = JX⊗ X +h(Z⊗ I+I⊗ Z). In the ordered even basis {ket00,ket11} and odd basis {ket01,ket10}, H_even = 2h · J J · -2h , H_odd = 0 · J J · 0 . Therefore Spec H_even ={±sqrt4h^2+J^2}, Spec H_odd={± J}.
Exact mathematical passage
H_{\mathrm{even}} = \begin{pmatrix} 2h&J\\ J&-2h \end{pmatrix}, \qquad H_{\mathrm{odd}} = \begin{pmatrix} 0&J\\ J&0 \end{pmatrix}.
H_iUse in source · 1 passage
Search aliases: H_i
Use in source · continuation_04.tex, line 59
If p^Lmid c, the actual identities give G(x)=F(x)+cH(x), DG(x)≡ DF(x)pmod p. Thus if F(x)≡0pmodp^L and A_x is surjective, then ∃ b∈ B_L(x):G(b)=0. This theorem allows rectangular systems. The evaluation, derivative and Jacobian identities are proved from the polynomial definition G_i=F_i+C(c)H_i, rather than supplied as hypotheses. Algebraic congruence helpers also permit composite moduli. Exact p-adic root existence uses prime p.
Exact mathematical passage
G_i=F_i+C(c)H_i
h_{i-r}Use in source · 1 passage
Search aliases: h_i-r
Use in source · strengthening.tex, line 465
(2^4i+2-1)B_4i+2(4i+2)(4i+1). The constant is one. Since v_2(4i+2)=1, the elementary Bernoulli bound shows v_2(d_i)≥2i. Hurwitz inversion therefore gives D_1(t)^-1=∑_i≥0h_ifract^4i(4i)!, h_0=1, v_2(h_i)≥2i. This follows by induction from the inverse recurrence: for packet index i, every product d_rh_i-r has valuation at least 2i, and its binomial multiplier is integral. Because F_1=F_0D_1^-1, C_1(n)-C_0(n) =∑_i≥1h_ibinom4n4iC_0(n-i). At a natural n this is a finite sum, with i≤ n.
Exact mathematical passage
d_rh_{i-r}
H_i^1Use in source · 1 passage
Search aliases: H_i^1
Use in source · odd_synthesis.tex, line 141
Choose a labeled symplectic embedding V_src→ S. For each original full character form P_i=( C^×× J_i^1)/ {(θ_i(h)^-1,h):h∈ H_i^1}. The balanced product and the corresponding host restriction have the same alternating commutator. The center-fixing extension isomorphism, obtained with matched Darboux lifts and complex roots, therefore preserves every supplied value θ_i(h), including the shallow values.
Exact mathematical passage
\mathcal P_i=(\mathbb C^\times\times J_i^1)/ \{(\theta_i(h)^{-1},h):h\in H_i^1\}.
h_{i_1}Use in source · 1 passage
Search aliases: h_i_1
Use in source · 03_nonlinear_coordinates.tex, line 340
For a block B={i_1<⋯<i_s} and an intermediate packet γ, write G_γ,B(xmid h) = J^s_γmidα_i_1,…,α_i_sG(x) [h_i_1,…,h_i_s].
Exact mathematical passage
G_{\gamma,B}(x\mid h) = J^s_{\gamma\mid\alpha_{i_1},\ldots,\alpha_{i_s}}G(x) [h_{i_1},\ldots,h_{i_s}].
h_{i_s}Use in source · 1 passage
Search aliases: h_i_s
Use in source · 03_nonlinear_coordinates.tex, line 340
For a block B={i_1<⋯<i_s} and an intermediate packet γ, write G_γ,B(xmid h) = J^s_γmidα_i_1,…,α_i_sG(x) [h_i_1,…,h_i_s].
Exact mathematical passage
G_{\gamma,B}(x\mid h) = J^s_{\gamma\mid\alpha_{i_1},\ldots,\alpha_{i_s}}G(x) [h_{i_1},\ldots,h_{i_s}].
h_jUse in source · 5 passages
Search aliases: h_j
Use in source · 03_nonlinear_coordinates.tex, line 277
For h_j∈ X_α_j, the inherited norm gives Q_γ D^rF(x)[h_1,…,h_r] ≤ Q_γ D^rF(x) ∏_j h_j . Taking the multilinear operator norm proves the sharper restricted estimate. For ambient inputs, replace h_j by P_α_jh_j and apply the operator norm bound to each projector. Every nonzero idempotent has norm at least one, since it fixes a nonzero vector in its range. The displayed product bound therefore also follows for the restricted jet. If a zero packet is retained as a redundant index, multilinearity mak…
Exact mathematical passage
h_j\in X_{\alpha_j}
h_j^2Use in source · 2 passages
Search aliases: h_j^2
Use in source · 04_correction_transport.tex, line 1024
Let U(t,s) be the exact propagator generated by K. Since K is continuously differentiable, the integral equation for U(t+h,t) gives a uniform estimate U(t+h,t)-I-hK(t) ≤ C_1h^2. By [source reference], the direct rotation V_j from t_j to t_j+1 therefore satisfies V_j-U(t_j+1,t_j) ≤ C_2h_j^2, h_j=t_j+1-t_j. Both V_j and the exact propagator are unitary. A telescoping product comparison gives V_π-U(T,0) ≤ ∑_j V_j-U(t_j+1,t_j) ≤ C_2∑_jh_j^2 ≤ C_2T|π|. This proves norm convergence and the stated firs…
Exact mathematical passage
\|V_j-U(t_{j+1},t_j)\| \le C_2h_j^2, \qquad h_j=t_{j+1}-t_j.
H_j^rUse in source · 1 passage
Search aliases: H_j^r
Use in source · Paper_X_Block27_Amended.tex, line 381
For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet are cochain maps, so they induce maps on these genuine quotients.
Exact mathematical passage
Z_j^r,B_j^r,H_j^r
H_{ji}Use in source · 1 passage
Search aliases: H_ji
Use in source · 02_connections_obstructions.tex, line 653
For a loop represented on one common finite scalar packet coordinate type, the proposition resolves its successive transport matrices into packet walks. If R is a ring, as in the reversible connection API, then (H-I)_ji=H_ji-δ_ji. In particular, off-diagonal entries are unchanged. Thus a nonzero off-diagonal holonomy obstruction in this model with a common index type requires a transport word along the closed chart loop whose packet path runs from its input packet to its output packet. This conc…
Exact mathematical passage
(H-I)_{ji}=H_{ji}-\delta_{ji}
H_kUse in source · 9 passages
Search aliases: H_k
Use in source · strengthening.tex, line 32
Write S f(n)=f(n+1) and Δ_H=S^H-1. A mixed difference Δ_H_1⋯Δ_H_k acts on the row variable n, rather than on the full Euler subscript. We use v_p(0)=+∞. Orders and starting rows are nonnegative integers. Steps are positive integers unless stated otherwise. An empty mixed difference is the original value, and a zero ordinary step makes every nonempty product containing it zero.
Exact mathematical passage
\Delta_{H_1}\cdots\Delta_{H_k}
H_{ks}Use in source · 1 passage
Search aliases: H_ks
Use in source · Paper_V_Traditional_Final.tex, line 1988
For L=N Z with N≥1, first take |z|=1. The assumption h∈ℓ^1( Z) makes the following symbol series absolutely convergent. Define H_ks(z)=∑_m∈ Zh(Nm+k-s)z^-m. For x∈ℓ^1( Z), the channel transforms converge absolutely on the unit circle and satisfy Y(z)= H_N(z) X(z). For x∈ℓ^2( Z), the same identity holds almost everywhere in the L^2 Fourier sense. For finite impulse-response kernels it also holds at nonzero z wherever the input channel transforms are weighted absolutely summable. This is the classi…
Exact mathematical passage
H_{ks}(z)=\sum_{m\in\ZZ}h(Nm+k-s)z^{-m}.
h_{ks}Use in source · 6 passages
Search aliases: h_ks
Use in source · Paper_V_Traditional_Final.tex, line 417
Let h be a finite impulse response and y=h*x, y[n]=∑_q∈ Zh[q]x[n-q]. Resolve both input and output into N phases. For output phase k and input phase s, define the transfer kernel h_ks[m]=h[Nm+k-s]. Out-of-range taps are understood to be zero.
Exact mathematical passage
\boxed{ h_{ks}[m]=h[Nm+k-s]. }
H_mUse in source · 7 passages
Search aliases: H_m
Use in source · 72_recovered.tex, line 251
Fix a divisor m of p-1. Let H_m = {ζ∈ F_p^×:ζ^m=1}. This group has order m.
Exact mathematical passage
H_m = \{\zeta\in\FF_p^\times:\zeta^m=1\}.
h_mUse in source · 4 passages
Search aliases: h_m
Use in source · 04_correction_transport.tex, line 251
[One-step lifting criterion] The approximation x_m has a correction h_m∈ F^mX such that Φ(x_m+h_m)∈ F^m+1Y if and only if Ob_m(x_m)=0. When a correction exists, the set of its classes in F^mX/F^m+1X is a coset of Keroverline L_m. If the graded pieces are vector spaces, this coset is an affine space.
Exact mathematical passage
h_m\in F^mX
h_m'Use in source · 1 passage
Search aliases: h_m'
Use in source · 04_correction_transport.tex, line 272
The next depth condition is equivalent to overline L_m[h_m] =-[Φ(x_m)] in F^mY/F^m+1Y. This is solvable exactly when the displayed defect has zero cokernel class. If h_m and h_m' are two solutions, their difference lies in Keroverline L_m. Adding an element of the kernel to one solution gives another, so the solution set is a coset of the kernel.
Exact mathematical passage
h_m'
h_{m-3}Use in source · 1 passage
Search aliases: h_m-3
Use in source · strengthening.tex, line 606
If m is not a power of two, s_2(m)≥2 makes μ_m even. If m is a positive power of two, the only central indices which can contribute an odd residue are r=1 and r=3, since r-2s_2(r)=-1 only at these two positive integers. The r=1 term is odd: T(m,1)=2^m-1 and v_2(m!)=m-1. For m≥4 a power of two, the r=3 term is even. More explicitly, fracT(m,3)2^m-3=h_m-3(1,3,6), ∑_d≥0h_d(1,3,6)t^d =frac1(1-t)(1-3t)(1-6t). Reduction modulo two gives (1-t)^-2, whose degree m-3 coefficient has parity m-2=0. At m=1,2…
Exact mathematical passage
\frac{T(m,3)}{2^{m-3}}=h_{m-3}(1,3,6), \qquad \sum_{d\geq0}h_d(1,3,6)t^d =\frac1{(1-t)(1-3t)(1-6t)}.
H_nUse in source · 29 passages
Search aliases: H_n
Use in source · Paper_VI_Traditional_Final.tex, line 717
[Gaussian chaos realization] For the standard Gaussian measure, the normalized probabilists' Hermite polynomials form orthonormal chaos packets. The Ornstein--Uhlenbeck semigroup acts diagonally: T_tH_n=e^-ntH_n. Thus conditional expectation, martingale differences, and Gaussian chaos are all projector realizations of stochastic refinement.
Exact mathematical passage
T_tH_n=e^{-nt}H_n.
\bm H_NUse in source · 1 passage
Search aliases: H_N bm
Use in source · Paper_V_Traditional_Final.tex, line 1992
For L=N Z with N≥1, first take |z|=1. The assumption h∈ℓ^1( Z) makes the following symbol series absolutely convergent. Define H_ks(z)=∑_m∈ Zh(Nm+k-s)z^-m. For x∈ℓ^1( Z), the channel transforms converge absolutely on the unit circle and satisfy Y(z)= H_N(z) X(z). For x∈ℓ^2( Z), the same identity holds almost everywhere in the L^2 Fourier sense. For finite impulse-response kernels it also holds at nonzero z wherever the input channel transforms are weighted absolutely summable. This is the classi…
Exact mathematical passage
\bm Y(z)=\bm H_N(z)\bm X(z).
\mathcal H_NUse in source · 1 passage
Search aliases: H_N mathcal
Use in source · Paper_IV_Traditional_Final.tex, line 2329
[Diagonal Fredholm determinant representation] Let I_N be a multiplicity-sensitive countable index set for Λ_N and put H_N=ℓ^2(I_N). On its standard orthonormal basis, define the diagonal operator K_Ne_λ=λ^-1e_λ. Then K_N is trace class and Δ_N(w)=det_ F(I-wK_N). For every m≥1, κ_N,m=tr(K_N^m).
Exact mathematical passage
\mathcal H_N=\ell^2(I_N)
H_{n+1}Use in source · 3 passages
Search aliases: H_n+1
Use in source · monograph_atlas.tex, line 769
vpage39Two routes to a bounded Hamiltonian limitEXACT EMBEDDINGS AND A SUMMABLE TAILChapter VII: Theorem II.13.2, Theorem II.13.4, Corollary II.13.5. For the exact route, let HH_n= C^n with coordinate inclusions and H_n=diag(0,1,1/2,…,1/(n-1)), H_1=(0). The old energies are retained at every level, H_n+1R_n=R_nH_n, and normH_n≤1. On the Hilbert space of the direct limit ℓ^2( N_0), the diagonal sequence defines a bounded self-adjoint H_∞. Projectors onto even addresses also intertwine and give a …
Exact mathematical passage
H_{n+1}R_n=R_nH_n
h_{n+1}Use in source · 1 passage
Search aliases: h_n+1
Use in source · 03_nonlinear_coordinates.tex, line 371
First derive the unprojected partition formula. At order one it is the ordinary chain rule. Suppose it holds at order n and the maps are C^n+1. Differentiating a summand in direction h_n+1 has two kinds of contribution. A derivative of the outer map creates the new singleton block {n+1}. A derivative of an inner block jet adjoins n+1 to that block. Every partition of {1,…,n+1} arises exactly once: if {n+1} is a block, remove that block. Otherwise remove n+1 from the unique block that contains it…
Exact mathematical passage
h_{n+1}
H_{n+1,n}Use in source · 1 passage
Search aliases: H_n+1,n
Use in source · 01_finite_core.tex, line 1079
The paper tests a controlled volume-state model H_nn=2κ+V(n), H_n,n+1=H_n+1,n=-κ, with κ>0 and a real confining potential. This is a finite transfer-matrix test only. Operator positivity alone is not reflection positivity of a Euclidean field measure. It is not derived from a gravitational path integral and is not evidence for a unique microscopic spacetime dynamics.
Exact mathematical passage
H_{nn}=2\kappa+V(n), \qquad H_{n,n+1}=H_{n+1,n}=-\kappa,
H_{N,c}Use in source · 9 passages
Search aliases: H_N,c
Use in source · supplement_09.tex, line 5
Checkpoint scope. This complete source body retains the status of its dated original supplement. Earlier pending or partial-coverage labels are historical. The final core records their resolution. Scope and result. Let p be prime, let mmid p-1, and put N=mp. For 1≤ a<m, put c=ap and define the original exponential section and ratio by H_N,c(t)=∑_n≥0fract^Nn+c(Nn+c)!, R_c(t)=fractH_N,c-1(t)H_N,c(t). The quotient is interpreted after removing its forced zero. This block proves, for the actual rati…
Exact mathematical passage
H_{N,c}(t)=\sum_{n\ge0}\frac{t^{Nn+c}}{(Nn+c)!},\qquad R_c(t)=\frac{tH_{N,c-1}(t)}{H_{N,c}(t)}.
H_{N,c}'Use in source · 1 passage
Search aliases: H_N,c'
Use in source · supplement_10.tex, line 130
Modules. mcodeWeightedDworkTransfer, mcodeSeriesDifferenceBridge, mcodeRatioSeriesSupport. 3. Nonsingular propagation without an assumed limit. For consecutive original sections, the already defined ratios satisfy R_cH_N,c=tH_N,c-1, H_N,c'=H_N,c-1. Differentiating the first relation, using the relation for R_c-1, and cancelling a section whose leading coefficient is proved nonzero gives the actual Riccati equation tR_c'=R_c(1+R_c-1-R_c), 2≤ c<N. It is not an equation imposed on a replacement rat…
Exact mathematical passage
R_cH_{N,c}=tH_{N,c-1},\qquad H_{N,c}'=H_{N,c-1}.
H_{N,c-1}Use in source · 3 passages
Search aliases: H_N,c-1
Use in source · supplement_09.tex, line 5
Checkpoint scope. This complete source body retains the status of its dated original supplement. Earlier pending or partial-coverage labels are historical. The final core records their resolution. Scope and result. Let p be prime, let mmid p-1, and put N=mp. For 1≤ a<m, put c=ap and define the original exponential section and ratio by H_N,c(t)=∑_n≥0fract^Nn+c(Nn+c)!, R_c(t)=fractH_N,c-1(t)H_N,c(t). The quotient is interpreted after removing its forced zero. This block proves, for the actual rati…
Exact mathematical passage
H_{N,c}(t)=\sum_{n\ge0}\frac{t^{Nn+c}}{(Nn+c)!},\qquad R_c(t)=\frac{tH_{N,c-1}(t)}{H_{N,c}(t)}.
H_{n,n+1}Use in source · 1 passage
Search aliases: H_n,n+1
Use in source · 01_finite_core.tex, line 1079
The paper tests a controlled volume-state model H_nn=2κ+V(n), H_n,n+1=H_n+1,n=-κ, with κ>0 and a real confining potential. This is a finite transfer-matrix test only. Operator positivity alone is not reflection positivity of a Euclidean field measure. It is not derived from a gravitational path integral and is not evidence for a unique microscopic spacetime dynamics.
Exact mathematical passage
H_{nn}=2\kappa+V(n), \qquad H_{n,n+1}=H_{n+1,n}=-\kappa,
H_N^TUse in source · 3 passages
Search aliases: H_N^T
Use in source · 01_manuscript.tex, line 640
H_N · -H_N . Then H_NH_N^T=NI. The product H_Nx can be evaluated in exactly Nlog_2N additions or subtractions through the recursion.
Exact mathematical passage
H_NH_N^T=NI.
H_n^TUse in source · 1 passage
Search aliases: H_n^T
Use in source · pages.tex, line 126
vpage8See every butterfly valueWALSH RECURSION IN EIGHT COORDINATESChapter VIII: Theorem 4.7. The unnormalized Walsh matrix satisfies H_1=[1] and H_2n=matH_n · H_n H_n · -H_n, H_nH_n^T=nI. Apply the three add/subtract stages to (1,2,3,4,5,6,7,8)^T. The full intermediate ledger is displayed below. vfigwalsh headArithmetic can be counted exactly Each stage visits eight coordinates, producing one sum or difference per coordinate. Three stages therefore use exactly 24 additions or subtractions. No m…
Exact mathematical passage
H_{2n}=\mat{H_n&H_n\\H_n&-H_n},\qquad H_nH_n^T=nI.
H_{nn}Use in source · 1 passage
Search aliases: H_nn
Use in source · 01_finite_core.tex, line 1079
The paper tests a controlled volume-state model H_nn=2κ+V(n), H_n,n+1=H_n+1,n=-κ, with κ>0 and a real confining potential. This is a finite transfer-matrix test only. Operator positivity alone is not reflection positivity of a Euclidean field measure. It is not derived from a gravitational path integral and is not evidence for a unique microscopic spacetime dynamics.
Exact mathematical passage
H_{nn}=2\kappa+V(n), \qquad H_{n,n+1}=H_{n+1,n}=-\kappa,
H_{\mathrm{odd}}Use in source · 2 passages
Search aliases: H_odd mathrm
Use in source · 01_finite_core.tex, line 395
For two qubits and real parameters J,h, consider H = JX⊗ X +h(Z⊗ I+I⊗ Z). In the ordered even basis {ket00,ket11} and odd basis {ket01,ket10}, H_even = 2h · J J · -2h , H_odd = 0 · J J · 0 . Therefore Spec H_even ={±sqrt4h^2+J^2}, Spec H_odd={± J}.
Exact mathematical passage
H_{\mathrm{even}} = \begin{pmatrix} 2h&J\\ J&-2h \end{pmatrix}, \qquad H_{\mathrm{odd}} = \begin{pmatrix} 0&J\\ J&0 \end{pmatrix}.
H_{PP}Use in source · 6 passages
Search aliases: H_PP
Use in source · monograph_atlas.tex, line 389
1 · 2 and retain its first coordinate. The eliminated block is H_QQ=2. Whenever z-2 and the full zI-H are invertible, H_rm eff(z)=frac1z-2, [(zI-H)^-1]_00 =frac1z-1/(z-2)=fracz-2z^2-2z-1. These are the hypotheses used in the formal companion. vfigfeshbach At z=3, the exact retained resolvent is 1/2. Bare truncation uses only H_PP=0 and gives 1/3, an error of 1/6. The omitted coupling has changed the response even though the retained diagonal entry did not change. headA pole can belong to the eli…
Exact mathematical passage
H_{PP}=0
H_{PQ}Use in source · 7 passages
Search aliases: H_PQ
Use in source · 01_finite_core.tex, line 897
Let P be a retained packet projector and Q=I-P. In packet coordinates, H= H_PP · H_PQ H_QP · H_QQ .
Exact mathematical passage
H= \begin{pmatrix} H_{PP}&H_{PQ}\\ H_{QP}&H_{QQ} \end{pmatrix}.
H_{QP}Use in source · 7 passages
Search aliases: H_QP
Use in source · 01_finite_core.tex, line 897
Let P be a retained packet projector and Q=I-P. In packet coordinates, H= H_PP · H_PQ H_QP · H_QQ .
Exact mathematical passage
H= \begin{pmatrix} H_{PP}&H_{PQ}\\ H_{QP}&H_{QQ} \end{pmatrix}.
H_{QQ}Use in source · 12 passages
Search aliases: H_QQ
Use in source · monograph_atlas.tex, line 381
1 · 2 and retain its first coordinate. The eliminated block is H_QQ=2. Whenever z-2 and the full zI-H are invertible, H_rm eff(z)=frac1z-2, [(zI-H)^-1]_00 =frac1z-1/(z-2)=fracz-2z^2-2z-1. These are the hypotheses used in the formal companion. vfigfeshbach At z=3, the exact retained resolvent is 1/2. Bare truncation uses only H_PP=0 and gives 1/3, an error of 1/6. The omitted coupling has changed the response even though the retained diagonal entry did not change. headA pole can belong to the eli…
Exact mathematical passage
H_{QQ}=2
H_rUse in source · 6 passages
Search aliases: H_r
Use in source · supplement_09.tex, line 83
Modules. mcodeCyclotomicFiltration, mcodePrimeBlockPolynomial, and mcodeCyclicAmiceCoefficients. 2. Connection to the actual section. An integral coordinate estimate becomes relevant only after connecting it to the manuscript's generating functions. For a residue r∈ Z/N Z, define H_r(t)=∑_d≥01_d≡ r (N)fract^dd!, F_r(T)=H_r(log(1+T)). The implementation uses the actual formal logarithm and substitution in Q[[T]]. It proves H_r'=H_r-1, (1+T)F_r'=F_r-1, and hence the coefficient recurrence (d+1)coe…
Exact mathematical passage
H_r(t)=\sum_{d\ge0}\mathbf1_{d\equiv r\ (N)}\frac{t^d}{d!}, \qquad F_r(T)=H_r(\log(1+T)).
h_rUse in source · 10 passages
Search aliases: h_r
Use in source · 03_nonlinear_coordinates.tex, line 235
[Exact reconstruction of derivatives] For h_1,…,h_r∈ X, D^rF(x)[h_1,…,h_r] = ∑_γ ∑_α_1,…,α_r J^r_γmidα_1,…,α_rF(x) [P_α_1h_1,…,P_α_rh_r].
Exact mathematical passage
h_1,\ldots,h_r\in X
H_r'Use in source · 1 passage
Search aliases: H_r'
Use in source · supplement_09.tex, line 88
ip 2. Connection to the actual section. An integral coordinate estimate becomes relevant only after connecting it to the manuscript's generating functions. For a residue r∈ Z/N Z, define H_r(t)=∑_d≥01_d≡ r (N)fract^dd!, F_r(T)=H_r(log(1+T)). The implementation uses the actual formal logarithm and substitution in Q[[T]]. It proves H_r'=H_r-1, (1+T)F_r'=F_r-1, and hence the coefficient recurrence (d+1)coefTd+1F_r =coefTdF_r-1-dcoefTdF_r. The initial values are F_r(0)=1_r=0.
Exact mathematical passage
H_r'=H_{r-1},\qquad (1+T)F_r'=F_{r-1},
H_{r-1}Use in source · 1 passage
Search aliases: H_r-1
Use in source · supplement_09.tex, line 88
ip 2. Connection to the actual section. An integral coordinate estimate becomes relevant only after connecting it to the manuscript's generating functions. For a residue r∈ Z/N Z, define H_r(t)=∑_d≥01_d≡ r (N)fract^dd!, F_r(T)=H_r(log(1+T)). The implementation uses the actual formal logarithm and substitution in Q[[T]]. It proves H_r'=H_r-1, (1+T)F_r'=F_r-1, and hence the coefficient recurrence (d+1)coefTd+1F_r =coefTdF_r-1-dcoefTdF_r. The initial values are F_r(0)=1_r=0.
Exact mathematical passage
H_r'=H_{r-1},\qquad (1+T)F_r'=F_{r-1},
H_{\rm d}Use in source · 4 passages
Search aliases: H_rm d rm
Use in source · 02_strengthened.tex, line 576
Let H_rm d=PHP+QHQ, H_rm o=PHQ+QHP. The propagator generated by H_rm d preserves P H and Q H. Duhamel's formula relative to H_rm d gives Q e^- i tHP = - i ∫_0^t Q e^- i(t-s)HH_rm o e^- i sH_rm dP d s. Since H_rm oP=QHP and both propagators are unitary, the norm is at most abstnormQHP. Use [source reference] for the last equality.
Exact mathematical passage
H_{\rm d}=PHP+QHQ, \qquad H_{\rm o}=PHQ+QHP.
H_{\rm eff}Use in source · 1 passage
Search aliases: H_rm eff rm
Use in source · monograph_atlas.tex, line 382
1 · 2 and retain its first coordinate. The eliminated block is H_QQ=2. Whenever z-2 and the full zI-H are invertible, H_rm eff(z)=frac1z-2, [(zI-H)^-1]_00 =frac1z-1/(z-2)=fracz-2z^2-2z-1. These are the hypotheses used in the formal companion. vfigfeshbach At z=3, the exact retained resolvent is 1/2. Bare truncation uses only H_PP=0 and gives 1/3, an error of 1/6. The omitted coupling has changed the response even though the retained diagonal entry did not change. headA pole can belong to the eli…
Exact mathematical passage
H_{\rm eff}(z)=\frac1{z-2},\qquad \bigl[(zI-H)^{-1}\bigr]_{00} =\frac1{z-1/(z-2)}=\frac{z-2}{z^2-2z-1}.
H_{\rm even}Use in source · 1 passage
Search aliases: H_rm even rm
Use in source · monograph_atlas.tex, line 175
A symmetry reveals blocks, not approximationsONE HAMILTONIAN IN TWO ORDERSChapter VII: Theorem II.5.1. Take the exact rational Hamiltonian H=X⊗ X+frac38(Z⊗ I+I⊗ Z). It commutes with total parity Z⊗ Z. Reordering the computational basis into even addresses (00,11) and odd addresses (01,10) displays the invariant subspaces explicitly. vfighamiltonian H_rm even=mat3/4 · 1 1 · -3/4, H_rm odd=mat0 · 1 1 · 0. The even block squares to (25/16)I, so its eigenvalues are ±5/4. The odd block squares to I, …
Exact mathematical passage
H_{\rm even}=\mat{3/4&1\\1&-3/4},\qquad H_{\rm odd}=\mat{0&1\\1&0}.
H_{\rm frame}Use in source · 1 passage
Search aliases: H_rm frame rm
Use in source · monograph_atlas.tex, line 218
1 · 0. Since U^*dot U=J, the coordinate Hamiltonian is H_rm frame=- i J=mat0 · i - i · 0, idot y=H_rm framey. This matrix is Hermitian. Omitting it would predict a constant y, contradicting the displayed coordinates. In this example the norm of the omitted time derivative is exactly one, so the failure is not a small numerical residual.
Exact mathematical passage
H_{\rm frame}=-\ii J=\mat{0&\ii\\-\ii&0}, \qquad \ii\dot y=H_{\rm frame}y.
H_{\rm Had}Use in source · 4 passages
Search aliases: H_rm Had rm
Use in source · monograph_atlas.tex, line 40
A Hamiltonian and a Hadamard gate are both traditionally denoted H. Where both appear, the gate is H_rm Had and the Hamiltonian keeps the letter H. Numerical labels on a drawing are coordinates or exact example values, not observations of a physical device. takeawayCheck the type before interpreting the picture. An idempotent projector, a density matrix, and a channel matrix have different defining equations.
Exact mathematical passage
H_{\rm Had}
H_{\rm mov}Use in source · 5 passages
Search aliases: H_rm mov rm
Use in source · 02_strengthened.tex, line 704
[Moving frame Hamiltonian] Let U(t) solve idot U(t)=H(t)U(t), U(0)=I, where H(t)=H(t)^* is continuous on the compact time interval under consideration. Set V(t)=W(t)^*U(t). Then idot V(t)=H_rm mov(t)V(t), where H_rm mov(t) = W(t)^*H(t)W(t)- i W(t)^*dot W(t) = W(t)^*(H(t)- i K(t))W(t). The moving frame Hamiltonian is self adjoint.
Exact mathematical passage
\ii\dot V(t)=H_{\rm mov}(t)V(t),
H_{\rm o}Use in source · 3 passages
Search aliases: H_rm o rm
Use in source · 02_strengthened.tex, line 576
Let H_rm d=PHP+QHQ, H_rm o=PHQ+QHP. The propagator generated by H_rm d preserves P H and Q H. Duhamel's formula relative to H_rm d gives Q e^- i tHP = - i ∫_0^t Q e^- i(t-s)HH_rm o e^- i sH_rm dP d s. Since H_rm oP=QHP and both propagators are unitary, the norm is at most abstnormQHP. Use [source reference] for the last equality.
Exact mathematical passage
H_{\rm d}=PHP+QHQ, \qquad H_{\rm o}=PHQ+QHP.
H_{\rm odd}Use in source · 1 passage
Search aliases: H_rm odd rm
Use in source · monograph_atlas.tex, line 175
A symmetry reveals blocks, not approximationsONE HAMILTONIAN IN TWO ORDERSChapter VII: Theorem II.5.1. Take the exact rational Hamiltonian H=X⊗ X+frac38(Z⊗ I+I⊗ Z). It commutes with total parity Z⊗ Z. Reordering the computational basis into even addresses (00,11) and odd addresses (01,10) displays the invariant subspaces explicitly. vfighamiltonian H_rm even=mat3/4 · 1 1 · -3/4, H_rm odd=mat0 · 1 1 · 0. The even block squares to (25/16)I, so its eigenvalues are ±5/4. The odd block squares to I, …
Exact mathematical passage
H_{\rm even}=\mat{3/4&1\\1&-3/4},\qquad H_{\rm odd}=\mat{0&1\\1&0}.
H_{rs}Use in source · 3 passages
Search aliases: H_rs
Use in source · Paper_V_Traditional_Final.tex, line 1934
[Exact convolution transfer law] Let y=h*x. Then y_r(ℓ) = ∑_s∈ R∑_m∈ Lh_rs(m)x_s(ℓ-m). Equivalently, A_LH A_L^-1 = (H_rs)_r,s∈ R, where H_rs is convolution on L by h_rs.
Exact mathematical passage
\Acal_LH\Acal_L^{-1} = (H_{rs})_{r,s\in R},
H_sUse in source · 1 passage
Search aliases: H_s
Use in source · Paper_X_Block27_Amended.tex, line 945
Actual local comparisons from fixed-depth certificates. Let R be an I-adically complete separated commutative ring with finite coefficient quotients and an injective map into a field K. Fix a finite matrix rank. For finitely many local group maps ϕ_s:H_s→ G and integral template representations τ_s, fix detecting depths d_s. At precision m retain matrices P_s,m as part of the admissible object and require ρ_m(ϕ_s(h))P_s,m=P_s,mτ_s,m(h). For m≥ d_s require their determinants to remain nonzero aft…
Exact mathematical passage
\phi_s:H_s\to G
H_{st}Use in source · 1 passage
Search aliases: H_st
Use in source · Paper_V_Traditional_Final.tex, line 1967
[Composition of transfer matrices] If G and H are convolution operators with packet matrices (G_rs) and (H_rs), then the packet matrix of GH is (GH)_rt = ∑_s∈ RG_rsH_st. In one-dimensional transform notation, the corresponding matrix symbols multiply pointwise.
Exact mathematical passage
(GH)_{rt} = \sum_{s\in R}G_{rs}H_{st}.
H_TUse in source · 3 passages
Search aliases: H_T
Use in source · 02_strengthened.tex, line 1860
[Hamiltonian reconstruction from transfer] Let T be positive definite, self adjoint, and contractive. Then H_T=-a^-1log T is a nonnegative self adjoint operator and T^n= e^-naH_T. It is the unique nonnegative self adjoint operator with T= e^-aH_T.
Exact mathematical passage
H_T=-a^{-1}\log T
H_tUse in source · 2 passages
Search aliases: H_t
Use in source · Paper_V_Traditional_Final.tex, line 3504
[Residue law for Cauchy products] Let F(z)=∑_n≥0f_nz^n, G(z)=∑_n≥0g_nz^n be formal power series. The same statement holds for power series that converge on a common disk. Decompose both series into residue packets modulo N: F=∑_r=0^N-1F_r, G=∑_s=0^N-1G_s. If H=FG, then its residue packet H_t is H_t = ∑_r+s≡ t pmod NF_rG_s.
Exact mathematical passage
H_t
H_{\widehat\tau}Use in source · 2 passages
Search aliases: H_widehatτ widehat tau
Use in source · Paper_V_Traditional_Final.tex, line 2725
[Location error for one step] Let 0≤τ,widehatτ≤1 and H_τ= 1_[τ,1]. For 1≤ p<∞, aH_τ-aH_widehatτ _p = |a||τ-widehatτ|^1/p. In the supremum norm, the error equals |a| whenever τ≠widehatτ.
Exact mathematical passage
\|aH_\tau-aH_{\widehat\tau}\|_p = |a||\tau-\widehat\tau|^{1/p}.
H_{\widehat\tau_q}Use in source · 2 passages
Search aliases: H_widehatτ_q widehat tau
Use in source · Paper_V_Traditional_Final.tex, line 3007
[Packet reconstruction after detection] Let j=∑_q=1^Qa_qH_τ_q, widehat j=∑_q=1^Qwidehat a_qH_widehatτ_q. Under the hypotheses of [source reference], for 1≤ p<∞, j-widehat j _p ≤ Qb_h + ( frac h2∑_q=1^Q|a_q|^p )^1/p. If Q_m is bounded and linear and widehat f_m=Q_m(f-widehat j)+widehat j, then f-widehat f_m _p ≤ · s-Q_ms _p
Exact mathematical passage
j=\sum_{q=1}^{Q}a_qH_{\tau_q}, \qquad \widehat j=\sum_{q=1}^{Q}\widehat a_qH_{\widehat\tau_q}.
H_xUse in source · 3 passages
Search aliases: H_x
Use in source · Paper_XII_Traditional_Final.tex, line 1324
[Support gives a Hecke eigensystem] Let R≅ T, let H be a finite T module, and let x:R→ E' be a field valued point. If ker x∈Supp_R(H), then H_x:=H⊗_ T,xE'≠0. Every element of T acts on H_x through the scalar prescribed by x.
Exact mathematical passage
\boxed{ H_x:=H\otimes_{\TT,x}E'\neq0. }
H_\theta^1Use in source · 4 passages
Search aliases: H_θ^1 theta
Use in source · Paper_XI_Traditional_Final.tex, line 1150
Let θ be a maximal simple character for GL_n(K), with compact open subgroups H_θ^1⊆ J_θ^1⊆ J_θ. The quotient V_θ=J_θ^1/H_θ^1 is regarded as a finite vector space over F_p, where p is the residue characteristic. Fix a nontrivial additive character ψ_0: F_p→μ_p⊂ C^×, where μ_p is the group of p-th roots of unity. The simple-character commutator takes values in μ_p. Since ψ_0 is injective, the equation ψ_0(h_θ(bar x,bar y)) = θ([x,y]) uniquely defines the F_p-valued commutator pairing h_θ. We use t…
Exact mathematical passage
H_\theta^1\subseteq J_\theta^1\subseteq J_\theta.
H_\lambdaUse in source · 4 passages
Search aliases: H_λ lambda
Use in source · Paper_XII_Traditional_Final.tex, line 1483
y an integral polynomial P_λ,v∈ O_λ[X] such that i_λ(P_λ,v)=e_λ(P_v) in K_λ[X]. Here and below a ring homomorphism applied to a polynomial acts on its coefficients. Assume every representative of every class in X_λ,n has characteristic polynomial P_λ,vbmod I_λ^n at γ_v. Also supply a subgroup H_λ≤Γ on which all these finite representations are trivial. This is an inertia condition when H_λ is the specified inertia subgroup.
Exact mathematical passage
H_\lambda\leq\Gamma
H_\nuUse in source · 10 passages
Search aliases: H_ν nu
Use in source · strengthening.tex, line 176
[All-row prime-power transfer] Let p be prime, a≥1, N=p^a, and 0≤ j<N. The coefficients C_N,j(n) belong to Z_p for all n≥0. Set A_N,j(n)=(-1)^n C_N,j(n) and T_H=S^H-(-1)^H. For k≥1 and positive H_1,…,H_k, v_p(Δ_H_1⋯Δ_H_kA_N,j(n)) · ≥ k+∑_ν=1^k v_p(H_ν), v_p(T_H_1⋯ T_H_kE_Nn^(N,j)) · ≥ v_p(j!)+k+∑_ν=1^k v_p(H_ν). At p=2, the ordinary differences of C_N,j and of the original coefficients satisfy the same bounds. These dyadic statements also hold for N=1.
Exact mathematical passage
v_p(\Delta_{H_1}\cdots\Delta_{H_k}A_{N,j}(n)) &\geq k+\sum_{\nu=1}^k v_p(H_\nu), \label{eq:prime-power-signed-bound}\\ v_p(T_{H_1}\cdots T_{H_k}E_{Nn}^{(N,j)}) &\geq v_p(j!)+k+\sum_{\nu=1}^k v_p(H_\nu). \label{eq:prime-power-original-twist}
H_{\tau_q}Use in source · 3 passages
Search aliases: H_τ_q tau
Use in source · Paper_V_Traditional_Final.tex, line 2909
Let t_k=kh, k=0,…,n, h=frac1n. Assume f(t) = s(t)+∑_q=1^Qa_qH_τ_q(t), H_τ(t)= 1_[τ,1](t), where s is Lipschitz with constant M, and Q≥1 is finite with nonzero jump amplitudes. The zero-jump case requires no minimum-amplitude condition. Suppose no jump lies at a sample point and no grid cell contains more than one jump. Observe y_k=f(t_k)+η_k. Define the adjacent contrast D_k=y_k+1-y_k.
Exact mathematical passage
f(t) = s(t)+\sum_{q=1}^{Q}a_qH_{\tau_q}(t), \qquad H_\tau(t)=\one_{[\tau,1]}(t),
H_{\ell+1}Use in source · 1 passage
Search aliases: H_ℓ+1 ell
Use in source · 02_strengthened.tex, line 2368
[Bounded Hamiltonian direct limit] Assume H_ℓ+1R_ℓ=R_ℓ H_ℓ and sup_ℓnormH_ℓ<∞. Then there is a unique bounded self adjoint operator H_∞ on H_∞ such that H_∞ι_ℓ=ι_ℓ H_ℓ for every ℓ. It satisfies e^- i tH_∞ι_ℓ = ι_ℓ e^- i tH_ℓ and e^-tH_∞ι_ℓ = ι_ℓ e^-tH_ℓ when the Hamiltonians are nonnegative.
Exact mathematical passage
H_{\ell+1}R_\ell=R_\ell H_\ell
H_\inftyUse in source · 7 passages
Search aliases: H_∞ infty
Use in source · monograph_atlas.tex, line 769
AILChapter VII: Theorem II.13.2, Theorem II.13.4, Corollary II.13.5. For the exact route, let HH_n= C^n with coordinate inclusions and H_n=diag(0,1,1/2,…,1/(n-1)), H_1=(0). The old energies are retained at every level, H_n+1R_n=R_nH_n, and normH_n≤1. On the Hilbert space of the direct limit ℓ^2( N_0), the diagonal sequence defines a bounded self-adjoint H_∞. Projectors onto even addresses also intertwine and give a fixed complete system of two packets in the limit. xfigtower_tail headCompatibili…
Exact mathematical passage
H_\infty
\Hcal_AUse in source · 3 passages
Search aliases: Hcal_A H_A Hcal mathcal
Use in source · 02_strengthened.tex, line 154
[Local orbit and Schmidt rank] Let ψ∈ H_A⊗ H_B, let d_A=dim H_A≥1, and let r be the Schmidt rank of ψ. Then dimspan{(A⊗ I_B)ψ:A∈End( H_A)} = d_A r. Local operators on A generate the whole bipartite Hilbert space exactly when the reduced state on B has full support.
Exact mathematical passage
\psi\in\Hcal_A\otimes\Hcal_B
\Hcal_BUse in source · 1 passage
Search aliases: Hcal_B H_B Hcal mathcal
Use in source · 02_strengthened.tex, line 154
[Local orbit and Schmidt rank] Let ψ∈ H_A⊗ H_B, let d_A=dim H_A≥1, and let r be the Schmidt rank of ψ. Then dimspan{(A⊗ I_B)ψ:A∈End( H_A)} = d_A r. Local operators on A generate the whole bipartite Hilbert space exactly when the reduced state on B has full support.
Exact mathematical passage
\psi\in\Hcal_A\otimes\Hcal_B
\Hcal_{\phys}Use in source · 4 passages
Search aliases: Hcal_phys H_phys Hcal phys mathcal mathrm
Use in source · 02_strengthened.tex, line 2216
[Physical compression is not a closure test] Let H_phys=⋂_aKer C_a and let P_phys be its orthogonal projector. Then P_phys[C_a,C_b]P_phys=0 for every pair a,b, whether or not the constraint span closes.
Exact mathematical passage
\Hcal_{\phys}=\bigcap_a\Ker C_a
\Hcal_{\rm root}Use in source · 1 passage
Search aliases: Hcal_rm root H_rm root Hcal rm mathcal rm
Use in source · Paper_V_Traditional_Final.tex, line 2343
[Reconstruction on a pruned packet tree] Every admissible pruning L defines a unitary analysis map A_ L: H_rm root ⟶ ⊕_v∈ L H_v. Its inverse is obtained by applying the adjoints of the local splits from the leaves toward the root. In particular, x ^2 = ∑_v∈ L x_v ^2.
Exact mathematical passage
\Acal_{\Lcal}:\Hcal_{\rm root} \longrightarrow \bigoplus_{v\in\Lcal}\Hcal_v.
\Hcal_XUse in source · 2 passages
Search aliases: Hcal_X H_X Hcal mathcal
Use in source · 02_strengthened.tex, line 1042
Let Φ:End( H_X)→End( H_Y) be linear. Choose packet systems (P_a) on H_X and (Q_c) on H_Y. Define Φ_cd,ab = Q_cdΦ P_ab.
Exact mathematical passage
\Phi:\End(\Hcal_X)\to\End(\Hcal_Y)
\Hcal_YUse in source · 3 passages
Search aliases: Hcal_Y H_Y Hcal mathcal
Use in source · 02_strengthened.tex, line 1042
Let Φ:End( H_X)→End( H_Y) be linear. Choose packet systems (P_a) on H_X and (Q_c) on H_Y. Define Φ_cd,ab = Q_cdΦ P_ab.
Exact mathematical passage
\Phi:\End(\Hcal_X)\to\End(\Hcal_Y)
\Hcal_ZUse in source · 1 passage
Search aliases: Hcal_Z H_Z Hcal mathcal
Use in source · 02_strengthened.tex, line 1054
[Exact channel block calculus] Every linear map has the exact reconstruction Φ=∑_a,b,c,dΦ_cd,ab. If Ψ:End( H_Y)→End( H_Z), then (ΨΦ)_ef,ab = ∑_c,d Ψ_ef,cdΦ_cd,ab.
Exact mathematical passage
\Psi:\End(\Hcal_Y)\to\End(\Hcal_Z)
\Hcal_\inftyUse in source · 4 passages
Search aliases: Hcal_∞ H_∞ Hcal infty mathcal infty
Use in source · 02_strengthened.tex, line 2360
Let H_∞ be the Hilbert direct limit of the isometric system, and let ι_ℓ: H_ℓ⟶ H_∞ be the canonical isometries.
Exact mathematical passage
\Hcal_\infty
\headUse in source · 1 passage
Search aliases: head
Use in source · visual_atlas.tex, line 19
[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…
Exact mathematical passage
4pt] \textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt] \textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology? \end{tabularx} A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. \takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''} \vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}} Most of the chapter can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix. {\small \begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule \textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule $X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\ $S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\ $\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\ $\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\ $P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\ $X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\ $T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\ $F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\ $U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\ $\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\ $\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule \end{tabularx}} \head{Read products from right to left} In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column. \head{Where to begin} For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}. \vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$} Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define \[ Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0, \qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.
\HHUse in source · 7 passages
Search aliases: HH
Use in source · monograph_atlas.tex, line 25
midrule HH,ψ · Hilbert space and state vector · They specify the space of addresses and its coherent amplitudes.
Exact mathematical passage
\HH,\psi
\HH_nUse in source · 1 passage
Search aliases: HH_n HH
Use in source · monograph_atlas.tex, line 765
vpage39Two routes to a bounded Hamiltonian limitEXACT EMBEDDINGS AND A SUMMABLE TAILChapter VII: Theorem II.13.2, Theorem II.13.4, Corollary II.13.5. For the exact route, let HH_n= C^n with coordinate inclusions and H_n=diag(0,1,1/2,…,1/(n-1)), H_1=(0). The old energies are retained at every level, H_n+1R_n=R_nH_n, and normH_n≤1. On the Hilbert space of the direct limit ℓ^2( N_0), the diagonal sequence defines a bounded self-adjoint H_∞. Projectors onto even addresses also intertwine and give a …
Exact mathematical passage
\HH_n=\CC^n
\HolUse in source · 2 passages
Search aliases: Hol operatorname
Use in source · 04_correction_transport.tex, line 660
[Gauge covariance] For a loop γ based at v_0, Hol'(γ) = G_v_0Hol(γ)G_v_0^-1. Thus the conjugacy class of each loop holonomy is gauge invariant.
Exact mathematical passage
\Hol'(\gamma) = G_{v_0}\Hol(\gamma)G_{v_0}^{-1}.
\operatorname{Hol}Use in source · 1 passage
Search aliases: Hol operatorname
Use in source · Paper_V_Traditional_Final.tex, line 1474
Hol(γ) · closed phase transfer · cycle consistency in a network
Exact mathematical passage
\operatorname{Hol}(\gamma)
\Hol'Use in source · 1 passage
Search aliases: Hol' Hol operatorname
Use in source · 04_correction_transport.tex, line 660
[Gauge covariance] For a loop γ based at v_0, Hol'(γ) = G_v_0Hol(γ)G_v_0^-1. Thus the conjugacy class of each loop holonomy is gauge invariant.
Exact mathematical passage
\Hol'(\gamma) = G_{v_0}\Hol(\gamma)G_{v_0}^{-1}.
\Hol_AUse in source · 1 passage
Search aliases: Hol_A Hol operatorname
Use in source · 04_correction_transport.tex, line 717
[Fundamental holonomy intertwining criterion] The following conditions are equivalent. item There is a family of isomorphisms U_v:V_0→ W_0 with the prescribed root value U_r and satisfying U_vA_vu=B_vuU_u for every oriented edge u→ v. item For every fundamental loop γ based at r, U_rHol_A(γ)=Hol_B(γ)U_r. item The same identity holds for every loop based at r. When these conditions hold, the family (U_v) is uniquely determined by U_r.
Exact mathematical passage
U_r\Hol_A(\gamma)=\Hol_B(\gamma)U_r.
\Hol_BUse in source · 1 passage
Search aliases: Hol_B Hol operatorname
Use in source · 04_correction_transport.tex, line 717
[Fundamental holonomy intertwining criterion] The following conditions are equivalent. item There is a family of isomorphisms U_v:V_0→ W_0 with the prescribed root value U_r and satisfying U_vA_vu=B_vuU_u for every oriented edge u→ v. item For every fundamental loop γ based at r, U_rHol_A(γ)=Hol_B(γ)U_r. item The same identity holds for every loop based at r. When these conditions hold, the family (U_v) is uniquely determined by U_r.
Exact mathematical passage
U_r\Hol_A(\gamma)=\Hol_B(\gamma)U_r.
\operatorname{Hom}Use in source · 1 passage
Search aliases: Hom operatorname
Use in source · 71_native.tex, line 134
coker(Hom_B(Ω_P/ Z⊗_P B, F_p)⟶Hom_B(I/I^2, F_p)).
Exact mathematical passage
\operatorname{coker}\left(\operatorname{Hom}_B(\Omega_{P/\mathbb Z}\otimes_P B,\mathbb F_p)\longrightarrow\operatorname{Hom}_B(I/I^2,\mathbb F_p)\right).
\operatorname{Hom}_BUse in source · 1 passage
Search aliases: Hom_B operatorname
Use in source · 71_native.tex, line 134
coker(Hom_B(Ω_P/ Z⊗_P B, F_p)⟶Hom_B(I/I^2, F_p)).
Exact mathematical passage
\operatorname{coker}\left(\operatorname{Hom}_B(\Omega_{P/\mathbb Z}\otimes_P B,\mathbb F_p)\longrightarrow\operatorname{Hom}_B(I/I^2,\mathbb F_p)\right).
\Hom_{R\text{-alg}}Use in source · 1 passage
Search aliases: Hom_R-alg Hom text operatorname text
Use in source · Paper_X_Block27_Amended.tex, line 2337
Let R be any commutative ring and B any commutative R-algebra. The coordinate ring A_R=R[X]/(X^2-1) represents precisely this specified packet: Hom_R-alg(A_R,B) ≃{x∈ B:x^2=1} ≃Hom(C_2,B^×) ≃ P_q(B). The first map evaluates the coordinate. Its inverse is the quotient universal property. A root is a unit with itself as inverse and determines a character by its value on the generator of C_2. Finally pull back along q. All inverse identities and naturality under coefficient homomorphisms are proved.…
Exact mathematical passage
\Hom_{R\text{-alg}}(A_R,B) \simeq\{x\in B:x^2=1\} \simeq\Hom(C_2,B^\times) \simeq\mathcal P_q(B).
\Hom_{W_K}Use in source · 1 passage
Search aliases: Hom_W_K Hom operatorname
Use in source · Paper_XI_Traditional_Final.tex, line 679
[Tame Hom test] For irreducible tame atoms, dimHom_W_K(τ_ξ^vee,τ_η) is zero or one, and it is one exactly when the refined tensor packet test in [source reference] finds one trivial packet.
Exact mathematical passage
\dim\Hom_{W_K}(\tau_\xi^\vee,\tau_\eta)
\HpUse in source · 3 passages
Search aliases: Hp H( Z_p) mathcal mathbb
Use in source · supplement_10.tex, line 97
The relevant integral subring is the Hurwitz subring H( Z_p)={f∈ Q_p[[t]]: n![t^n]f∈ Z_p for every n}. It is not coefficientwise integrality of ordinary power-series coefficients. The precise connection is n![t^n] E(F)=[T^0]Θ^nF. Since every iterate of Θ preserves W_p, the right side is integral. Therefore E( W_p)⊆ H( Z_p). Applying the same statement to every B_Θ^sF proves f(log(1+T))∈ W_p Longrightarrow B_D^sf∈ H( Z_p) for every s. This is the proved transfer that was missing in the earlier at…
Exact mathematical passage
\Hp=\{f\in\Q_p[[t]]:\ n![t^n]f\in\Z_p\ \text{for every }n\}.
\HSUse in source · 4 passages
Search aliases: HS mathrm
Use in source · 02_strengthened.tex, line 939
Density matrices and observables carry both an input packet and an output packet. Equip End( H) with the Hilbert--Schmidt inner product ipXY_HS=Tr(X^*Y).
Exact mathematical passage
\ip{X}{Y}_{\HS}=\Tr(X^*Y).
\Hull_{(O_i)}Use in source · 1 passage
Search aliases: Hull_(O_i) Hull operatorname
Use in source · 05_worked_models.tex, line 533
[Closure specification for the orbit hull] Let V have finite dimension, let (O_i)_i∈ I be a finite operator family, and let S⊆ V. Define W_0=span_𝕜S, W_n+1=W_n+∑_i∈ IO_i(W_n). Then there is an n∈ N such that W_n+1=W_n and W_n=Hull_(O_i)(S), where the right side is the least subspace containing S and invariant under every O_i. Thus the ascending subspace iteration terminates and its stable value is exactly the finite operator hull.
Exact mathematical passage
W_{n+1}=W_n \qquad\text{and}\qquad W_n=\Hull_{(O_i)}(S),
\Hull_{\AA}Use in source · 2 passages
Search aliases: Hull_AA Hull_ A Hull AA operatorname mathcal
Use in source · visual_atlas.tex, line 19
[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…
Exact mathematical passage
4pt] \textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt] \textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology? \end{tabularx} A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. \takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''} \vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}} Most of the chapter can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix. {\small \begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule \textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule $X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\ $S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\ $\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\ $\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\ $P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\ $X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\ $T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\ $F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\ $U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\ $\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\ $\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule \end{tabularx}} \head{Read products from right to left} In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column. \head{Where to begin} For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}. \vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$} Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define \[ Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0, \qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.
\Hull_{\cA}Use in source · 1 passage
Search aliases: Hull_cA Hull_ A Hull cA operatorname mathcal
Use in source · 05_worked_models.tex, line 439
For a finite ordered seed family S=(s_1,…,s_r), let ε_S: A^⊕ r→ X be the seed evaluation map of [source reference]. Its kernel is the relation module. The free module modulo that kernel presents the generated hull by [source reference]. The invariant content is: · generated hull: Hull_ A(S), · seed complexity: srank_ A(X), · packet coordinates: x↔(P_αx)_α, · operator coordinates: T↔(P_βTP_α)_β,α, · relation module: N_S:=Kerε_S≤ A^⊕ r, · free presentation: A^⊕ r/N_S≅Hull_ A(S), · coordinate invar…
Exact mathematical passage
{ \begin{aligned} &\text{generated hull: }\Hull_{\cA}(S),\\ &\text{seed complexity: }\srank_{\cA}(X),\\ &\text{packet coordinates: }x\leftrightarrow(P_{\alpha}x)_{\alpha},\\ &\text{operator coordinates: }T\leftrightarrow(P_{\beta}TP_{\alpha})_{\beta,\alpha},\\ &\text{relation module: }N_S:=\Ker\varepsilon_S\leq\cA^{\oplus r},\\ &\text{free presentation: }\cA^{\oplus r}/N_S\cong\Hull_{\cA}(S),\\ &\text{coordinate invariance: }P_{\alpha}\mapsto UP_{\alpha}U^{-1},\\ &\text{realization: }\Phi A=\varphi(A)\Phi. \end{aligned} }
\Hull_{\Ccal}Use in source · 2 passages
Search aliases: Hull_Ccal Hull_ C Hull Ccal operatorname mathcal
Use in source · 04_transport.tex, line 164
· =ΨΦ ∑_τ(σ(α))=kP_α. Thus the composite packet map is τ∘σ. If the morphisms are seeded, then Φ(S)⊆Hull_ B(S'), Ψ(S')⊆Hull_ C(S''). By [source reference], ΨΦ(S) ⊆ Ψ(Hull_ B(S')) ⊆ Hull_ C(Ψ(S')) ⊆ Hull_ C(S''). Identity maps satisfy the definitions with identity algebra and packet maps.
Exact mathematical passage
\Phi(S)\subseteq\Hull_{\B}(S'), \qquad \Psi(S')\subseteq\Hull_{\Ccal}(S'').
\Hull_FUse in source · 1 passage
Search aliases: Hull_F Hull operatorname
Use in source · visual_atlas.tex, line 118
IDEAL APPEARSMain text: [source reference] Let F=𝕜⟨ z_1,…,z_r⟩ be the algebra of noncommuting operator words. Evaluation at a fixed seed is Φ_s:F→ X, f↦ f(T_1,…,T_r)s. Two words represent the same generated vector exactly when their difference lies in I_s=KerΦ_s. Consequently F/I_s≅Hull_F(s) as left modules. The relation f(T)s=0 implies g(T)f(T)s=0 for every word g, which explains closure under multiplication on the left.
Exact mathematical passage
F/I_s\cong\Hull_F(s)
\Hull_{\QQ[D]}Use in source · 1 passage
Search aliases: Hull_QQ[D] Hull_ Q[D] Hull QQ operatorname mathbb
Use in source · visual_atlas.tex, line 234
headWhat the seed generates With the fixed algebra Q[D] and seed e_1, Hull_ Q[D](e_1)=span{e_1,e_2+e_3}. It has dimension two, not four. Both the independent difference e_2-e_3 and the last coordinate e_4 are missing.
Exact mathematical passage
\Hull_{\QQ[D]}(e_1)=\Span\{e_1,e_2+e_3\}.
\Hull_{\varphi(\A)}Use in source · 3 passages
Search aliases: Hull_φ(A) Hull_φ( A) Hull varphi A operatorname varphi mathcal
Use in source · 04_transport.tex, line 90
[Naturality of generated hulls and transfer blocks] Let (Φ,φ,σ) be a packet compatible operator morphism. For every subset S⊆ X, Φ(Hull_ A(S)) = Hull_φ( A)(Φ(S)) ⊆ Hull_ B(Φ(S)). For T∈ A and i,j∈ J, Q_jφ(T)Q_iΦ = ∑_substackσ(β)=j
Exact mathematical passage
\boxed{ \Phi(\Hull_{\A}(S)) = \Hull_{\varphi(\A)}(\Phi(S)) \subseteq \Hull_{\B}(\Phi(S)). }
\Hull_{\varphi(\AA)}Use in source · 1 passage
Search aliases: Hull_φ(AA) Hull_φ( A) Hull varphi AA operatorname varphi mathcal
Use in source · visual_atlas.tex, line 290
headWhen the map is not invertible For an intertwiner Φ A=φ(A)Φ, the hull identity uses the image algebra: Φ(Hull_ A(S))=Hull_φ( A)(Φ S). Replacing φ( A) with a larger target algebra can enlarge the target hull. Seed rank need not be equal relative to that larger algebra. A surjective intertwiner gives the appropriate monotonicity inequality, not automatic equality.
Exact mathematical passage
\Phi\bigl(\Hull_{\AA}(S)\bigr)=\Hull_{\varphi(\AA)}(\Phi S).
\mathrm iUse in source · 8 passages
Search aliases: i mathrm
Use in source · extended_pages.tex, line 3
vpage29A complex range needs an adjointTHE INNER PRODUCT IS PART OF THE CONTRACTChapter VIII: Theorem 5.6, Section F.1, Section F.2. The earlier compact example is real. Over C, an isometry means Q^*Q=I, with conjugation. Take the exact Gaussian-rational factors Q_L=mat3/5 · 0 4 i/5 · 0 0 · 1, Q_R=mat1 · 0 0 · i, G=diag(5,2). Their product is a genuinely complex rectangular operator: A=Q_LGQ_R^*=mat3 · 0
Exact mathematical passage
Q_L=\mat{3/5&0\\4\mathrm i/5&0\\0&1},\qquad Q_R=\mat{1&0\\0&\mathrm i},\qquad G=\diag(5,2).
i'Use in source · 4 passages
Search aliases: i'
Use in source · Paper_VI_Traditional_Final.tex, line 899
Conversely, fix two fine states i,i'∈ I_a and one state j∈ I_b in another coarse packet. Vary the distribution of a fixed coarse mass inside I_a between i and i' while leaving every coarse packet mass unchanged. The right side depends only on those coarse masses and is therefore unchanged. The left-side posterior mass of packet a depends on the conditional average of λ_i and λ_i'. It can be independent of that variation only if λ_i=λ_i'. Hence λ is constant on every packet. Write the value on I_…
Exact mathematical passage
i,i'\in I_a
i^{(m-j)\ell}Use in source · 1 passage
Search aliases: i^(m-j)ℓ ell
Use in source · atlas-pages.tex, line 49
vpage3A root filter is an exact voteCYCLIC PROJECTIONChapter IV: Theorem 3.1, Proposition 3.2. For N=4, write ω=i. To extract residue j, evaluate the function at four rotated arguments and attach the phase i^-jℓ: ( B_4,jF)(z)=frac14∑_ℓ=0^3i^-jℓF(i^ℓ z). A monomial a_mz^m is multiplied by the finite average of i^(m-j)ℓ. Matching phases all vote 1. Nonmatching phases cancel exactly. vfigroot_filter headAn example that can be checked coefficient by coefficient Take F(z)=∑_m=0^12(m+1)z^m. Then B_4,2…
Exact mathematical passage
i^{(m-j)\ell}
I^*Use in source · 1 passage
Search aliases: I^*
Use in source · monograph_atlas.tex, line 370
main inside the original code, so the previous syndrome partition reports no bit flip. frac3ket000+4ket1115 ⟼ frac3ket000-4ket1115. The populations are unchanged, but the relative phase is not. vfiglogical_error Let V be the code isometry with columns ket000,ket111. For the proposed error family {I,Z_1}, its off-diagonal error compression is V^*I^*Z_1V=mat1 · 0 0 · -1. This matrix is not cI for any scalar c. Thus the Knill--Laflamme condition fails for that family. Its trace is zero even though …
Exact mathematical passage
V^*I^*Z_1V=\mat{1&0\\0&-1}.
\mathrm i^{-jk}Use in source · 1 passage
Search aliases: i^-jk mathrm
Use in source · pages.tex, line 73
vpage5Four coefficients, four exact frequenciesCIRCULANT MULTIPLICATIONChapter VIII: Theorem 4.3. Take c=(2,1,0,-1)^T and define C(c)_jk=c_j-kbmod4. The unitary Fourier matrix is F_jk= i^-jk/2. Its normalization is visible in the exact diagonalization: C(c)=F^*diag(2Fc)F, 2Fc=(2,2-2 i,2,2+2 i)^T. vfigcirculant headFollow an actual input For x=(1,2,-1,0)^T, dense multiplication gives (0,6,0,-2)^T. Transforming x, multiplying by the four displayed eigenvalues, and applying F^* gives exactly the sa…
Exact mathematical passage
F_{jk}=\mathrm i^{-jk}/2
i^{-j\ell}Use in source · 2 passages
Search aliases: i^-jℓ ell
Use in source · atlas-pages.tex, line 45
vpage3A root filter is an exact voteCYCLIC PROJECTIONChapter IV: Theorem 3.1, Proposition 3.2. For N=4, write ω=i. To extract residue j, evaluate the function at four rotated arguments and attach the phase i^-jℓ: ( B_4,jF)(z)=frac14∑_ℓ=0^3i^-jℓF(i^ℓ z). A monomial a_mz^m is multiplied by the finite average of i^(m-j)ℓ. Matching phases all vote 1. Nonmatching phases cancel exactly. vfigroot_filter headAn example that can be checked coefficient by coefficient Take F(z)=∑_m=0^12(m+1)z^m. Then B_4,2…
Exact mathematical passage
i^{-j\ell}
I^2Use in source · 2 passages
Search aliases: I^2
Use in source · 71_native.tex, line 134
coker(Hom_B(Ω_P/ Z⊗_P B, F_p)⟶Hom_B(I/I^2, F_p)).
Exact mathematical passage
\operatorname{coker}\left(\operatorname{Hom}_B(\Omega_{P/\mathbb Z}\otimes_P B,\mathbb F_p)\longrightarrow\operatorname{Hom}_B(I/I^2,\mathbb F_p)\right).
i^2Use in source · 1 passage
Search aliases: i^2
Use in source · Paper_X_Block27_Amended.tex, line 967
The formalization specializes to the actual maps G_ Q_p→ G_ Q induced by the field embeddings and choices of algebraic closures. A nontrivial continuous quadratic character of G_ Q, constructed from its action on i^2=-1, gives the integral 2-adic actions τ(σ)=diag(1,-1), ρ(σ)=1 · -1
Exact mathematical passage
i^2=-1
I^bUse in source · 1 passage
Search aliases: I^b
Use in source · supplement_09.tex, line 48
ficient in prime row p gives (z-1)^p∈ PI. Every other degree-p product of the two generators of I also belongs to PI. The formal ideal proof therefore establishes I^p⊆ PI. For q=plfloor q/prfloor+(qbmod p), strong induction and the factorial valuation identity give I^q⊆ P^v_p(q!). In detail, write b=lfloor q/prfloor. Then I^q⊆ P^b I^b⊆ P^b+v_p(b!)=P^v_p(q!). No integral-domain hypothesis or division in a field is used here.
Exact mathematical passage
I^q\subseteq P^b I^b\subseteq P^{b+v_p(b!)}=P^{v_p(q!)}.
I^{d_s}Use in source · 1 passage
Search aliases: I^d_s
Use in source · Paper_X_Block27_Amended.tex, line 952
e map into a field K. Fix a finite matrix rank. For finitely many local group maps ϕ_s:H_s→ G and integral template representations τ_s, fix detecting depths d_s. At precision m retain matrices P_s,m as part of the admissible object and require ρ_m(ϕ_s(h))P_s,m=P_s,mτ_s,m(h). For m≥ d_s require their determinants to remain nonzero after reduction to R/I^d_s. Earlier levels omit that test. Finite occupied augmented levels give a compatible branch of both representations and comparisons. The integ…
Exact mathematical passage
R/I^{d_s}
I^kUse in source · 2 passages
Search aliases: I^k
Use in source · Paper_X_Block27_Amended.tex, line 905
Completion from conjugacy classes. Over an I-adically complete separated commutative coefficient ring R, for a fixed finite rank, the formalization now takes finite-level objects to be actual matrix representations modulo conjugation by an invertible matrix. Assume the coefficient quotients R/I^k are finite. Conjugation commutes with coefficient reduction, and each conjugacy class has finitely many frames over a finite coefficient ring. Thus finite allowed class levels give finite full preimages…
Exact mathematical passage
R/I^k
i^kUse in source · 2 passages
Search aliases: i^k
Use in source · atlas-pages.tex, line 142
vpage8Four zeros become one orbit zeroTHE REDUCED VARIABLEChapter IV: Definition 5.7, Example 11.12, Corollary 11.13. For N=4 the factorization is exact: D_4(z)=cos(z/sqrt2)cosh(z/sqrt2), D_4(z)=Δ_4(z^4). Set A_4=sqrt2π. All the zeros are A_4(m+1/2)i^k, with m≥0 and 0≤ k<4. Each zero is simple. vfigorbit_reduction The drawing uses u=z/A_4 and v=w/A_4^4, so its marked coordinates are rational. The first four-point orbit has fourth power 1/16. The second has fourth power 81/16. headMultiplicity is…
Exact mathematical passage
A_4(m+1/2)i^k
I^{\lfloor d/p\rfloor}Use in source · 1 passage
Search aliases: I^lfloor d/prfloor lfloor rfloor
Use in source · supplement_09.tex, line 68
For the falling factorial (x)_d=∏_i=0^d-1(x-i), grouping into consecutive blocks gives (x)_d∈ I^lfloor d/prfloor ⊆(p^v_p(lfloor d/prfloor!)). Every integral residue coordinate therefore satisfies p^v_p(lfloor d/prfloor!)mid [(r)](x)_d in Z. Dividing a coordinate by d! is done only afterward, using v_p(d!)=lfloor d/prfloor+v_p(lfloor d/prfloor!). This establishes the weighted denominator estimate before any field-valued divisibility could make it vacuous.
Exact mathematical passage
(x)_d\in I^{\lfloor d/p\rfloor} \subseteq\bigl(p^{v_p(\lfloor d/p\rfloor!)}\bigr).
I^mUse in source · 9 passages
Search aliases: I^m
Use in source · Paper_X_Block27_Amended.tex, line 865
Checked representation-level completion and its arithmetic boundary. The representation-level completion construction works with actual matrix representations of a fixed group. For a complete separated ring R and ideal I, compatible maps ρ_m:G→GL_d(R/I^m) reconstruct a unique integral representation for that fixed compatible family. Multiplication and inverses are verified after every reduction. If all finite representations are continuous for the discrete quotient topology, the integral represe…
Exact mathematical passage
\rho_m:G\to\operatorname{GL}_d(R/I^m)
I^pUse in source · 2 passages
Search aliases: I^p
Use in source · 20_arithmetic.tex, line 238
At the singular addresses c=ap, 1≤ a<m, use the weighted ring W_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping the falling factorial (x)_d=∏_i=0^d-1(x-i) into consecutive blocks of length p then gives an integral residue-coordinate factor p^v_p(lfloor d/prfloor!)…
Exact mathematical passage
I^p\subseteq(p)I
I^qUse in source · 3 passages
Search aliases: I^q
Use in source · 20_arithmetic.tex, line 239
sses c=ap, 1≤ a<m, use the weighted ring W_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping the falling factorial (x)_d=∏_i=0^d-1(x-i) into consecutive blocks of length p then gives an integral residue-coordinate factor p^v_p(lfloor d/prfloor!). The identity v_p(d!…
Exact mathematical passage
I^q\subseteq(p^{v_p(q!)})
i^\ellUse in source · 1 passage
Search aliases: i^ℓ ell
Use in source · atlas-pages.tex, line 46
vpage3A root filter is an exact voteCYCLIC PROJECTIONChapter IV: Theorem 3.1, Proposition 3.2. For N=4, write ω=i. To extract residue j, evaluate the function at four rotated arguments and attach the phase i^-jℓ: ( B_4,jF)(z)=frac14∑_ℓ=0^3i^-jℓF(i^ℓ z). A monomial a_mz^m is multiplied by the finite average of i^(m-j)ℓ. Matching phases all vote 1. Nonmatching phases cancel exactly. vfigroot_filter headAn example that can be checked coefficient by coefficient Take F(z)=∑_m=0^12(m+1)z^m. Then B_4,2…
Exact mathematical passage
(\mathcal B_{4,j}F)(z)=\frac14\sum_{\ell=0}^{3}i^{-j\ell}F(i^\ell z).
I_{\mathfrak Q_v}Use in source · 2 passages
Search aliases: I_ Q_v mathfrak
Use in source · Paper_X_Block27_Amended.tex, line 1284
The choice of absolute prime and Frobenius lift is also controlled. Let ( Q_u,γ_u) and ( Q_v,γ_v) be any two actual arithmetic witnesses at the same rational prime. Transitivity on primes of A above (p) gives a∈ G_ Q with Q_v=a Q_u, γ_v^-1aγ_u a^-1∈ I_ Q_v. Consequently, any representation trivial on I_ Q_v has the same trace on the two witnesses. For representations factoring through L, triviality on this actual absolute inertia is derived whenever p∉ S_L. The restrictions of all absolute witne…
Exact mathematical passage
\mathfrak Q_v=a\mathfrak Q_u,\qquad \gamma_v^{-1}a\gamma_u a^{-1}\in I_{\mathfrak Q_v}.
i_0Use in source · 3 passages
Search aliases: i_0
Use in source · 02_connections_obstructions.tex, line 609
[Packet path expansion for different matrices] For i,j∈ V and m≥1, set i_0=i and i_m=j. Then (B_m⋯ B_1)_ji = ∑_(i_1,…,i_m-1)∈ V^m-1 B_m[i_m,i_m-1]⋯ B_1[i_1,i_0]. When m=1, the sum is over the unique empty tuple and its summand is B_1[j,i]. For m=0, the product is the identity and the entry is the Kronecker delta δ_ji. If every B_r[q,p] vanishes outside a declared relation R_r(p,q), then the endpoint entry is zero whenever there is no packet path compatible with the successive relations. statusle…
Exact mathematical passage
i_0=i
I_1Use in source · 2 passages
Search aliases: I_1
Use in source · Paper_XII_Traditional_Final.tex, line 263
gintheorem[Completed tensor quotient] Let R_1,…,R_s be complete Noetherian local O algebras with common residue field k. Let I_i⊆ R_i be closed ideals. Put R^loc=R_1mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ OR_s and let J be the closed ideal generated by the images of the I_i. Then there is a canonical isomorphism (R_1/I_1)mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ O(R_s/I_s) ≅ R^loc/J.
Exact mathematical passage
\boxed{ (R_1/I_1)\wideotimes_{\OO}\cdots\wideotimes_{\OO}(R_s/I_s) \cong R^{\mathrm{loc}}/J. }
i_1Use in source · 5 passages
Search aliases: i_1
Use in source · 02_connections_obstructions.tex, line 610
[Packet path expansion for different matrices] For i,j∈ V and m≥1, set i_0=i and i_m=j. Then (B_m⋯ B_1)_ji = ∑_(i_1,…,i_m-1)∈ V^m-1 B_m[i_m,i_m-1]⋯ B_1[i_1,i_0]. When m=1, the sum is over the unique empty tuple and its summand is B_1[j,i]. For m=0, the product is the identity and the entry is the Kronecker delta δ_ji. If every B_r[q,p] vanishes outside a declared relation R_r(p,q), then the endpoint entry is zero whenever there is no packet path compatible with the successive relations. statusle…
Exact mathematical passage
\boxed{ (B_m\cdots B_1)_{ji} = \sum_{(i_1,\ldots,i_{m-1})\in V^{m-1}} B_m[i_m,i_{m-1}]\cdots B_1[i_1,i_0]. }
I_2Use in source · 12 passages
Search aliases: I_2
Use in source · supplement_04.tex, line 165
ar targetUnitModes.unit_mode_valuation_gain targetUnitModes.finite_unit_modes_valuation_gain pageheading4. Exact examples and adversarial checksThe examples use the same definitions as the general results A polynomial anchor. The actual difference table for f(n)=n^2 yields (c_0,c_1,c_2,c_3)=(0,1,2,0). The finite interpolant therefore reconstructs I_2(c)(7)=49. This checks the sign and indexing conventions, not just the names of the defined objects.
Exact mathematical passage
I_2(c)(7)=49
I_{2^{d-1}}Use in source · 1 passage
Search aliases: I_2^d-1
Use in source · dyadic_synthesis.tex, line 62
erse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact filtration valuation, and give the identity tr_LLift_E(C) =Tr_E/L (αtr_E C). The actual recursive matrix is B(L,t_L;h_1,…,h_d) =α I_2^d-1+ Lift_E(B(E,z;h_1,…,h_d-1)), with multiplication by α as the rank-two base case.
Exact mathematical passage
B(L,t_L;h_1,\ldots,h_d) =\alpha I_{2^{d-1}}+ \operatorname{Lift}_E\bigl(B(E,z;h_1,\ldots,h_{d-1})\bigr),
I_3Use in source · 3 passages
Search aliases: I_3
Use in source · monograph_atlas.tex, line 74
vpage5Why one seed is not always enoughAN EXACT MULTIPLICITY LOWER BOUNDChapter VII: Theorem II.2.3. Consider M_2( C)⊗ I_3 acting on C^2⊗ C^3. A seed is a 2×3 coefficient matrix. Its two rows span at most two directions in the three-dimensional multiplicity factor. Therefore no single seed is cyclic for this algebra. C_1=mat1 · 0 · 0
Exact mathematical passage
M_2(\CC)\otimes I_3
I_6Use in source · 1 passage
Search aliases: I_6
Use in source · monograph_atlas.tex, line 544
standard · 2 · 0 · -1 · 2 P_λ=fracd_λ6∑_g∈ S_3χ_λ(g^-1)U(g). All matrices in this example are rational. Multiplying them gives self-adjoint idempotents of ranks 1,1,4, with pairwise zero products and sum I_6. xfigs3 headThe standard packet has dimension four The standard representation occurs twice in the regular representation. Its isotypic component therefore has dimension 2·2=4: one factor carries the two-dimensional irreducible action and the other records multiplicity. A character projector…
Exact mathematical passage
I_6
I_aUse in source · 19 passages
Search aliases: I_a
Use in source · 01_finite_coordinates.tex, line 434
The graph model above resolves a fixed scalar matrix. Coordinate transport is a separate construction. Let X be an R-module, and let P^a=(P^a_i)_i∈ I_a be a finite complete family of pairwise orthogonal linear idempotents: P^a_iP^a_j=δ_ijP^a_i, ∑_i∈ I_aP^a_i=I_X. Define the exact packet coordinate module Coord( P^a) = {(x_i)_i∈ I_a:P^a_ix_i=x_i}. Its analysis and synthesis maps are A_a(x)=(P^a_ix)_i, S_a((x_i)_i)=∑_ix_i. Completeness and orthogonality give S_a A_a=id_X, A_a S_a=id_Coord( P^a).
Exact mathematical passage
\cP^a=(P^a_i)_{i\in I_a}
I_BUse in source · 4 passages
Search aliases: I_B
Use in source · 01_finite_core.tex, line 138
[Local orbit and Schmidt rank] Let ψ∈ H_A⊗ H_B, with dim H_A=d_A≥1, and let r be its Schmidt rank. The reduced positive operator on B is ρ_B=Tr_A(ketψbraψ). Normalization of ψ does not affect its support. Then dimspan{(A⊗ I_B)ψ:A∈ M_d_A( C)} =d_A r. In particular, local operators on A generate the complete bipartite Hilbert space exactly when the reduced state on B has full support.
Exact mathematical passage
\boxed{ \dim\operatorname{span}\{(A\otimes I_B)\psi:A\in M_{d_A}(\bbC)\} =d_A r. }
I_bUse in source · 1 passage
Search aliases: I_b
Use in source · Paper_VI_Traditional_Final.tex, line 899
Conversely, fix two fine states i,i'∈ I_a and one state j∈ I_b in another coarse packet. Vary the distribution of a fixed coarse mass inside I_a between i and i' while leaving every coarse packet mass unchanged. The right side depends only on those coarse masses and is therefore unchanged. The left-side posterior mass of packet a depends on the conditional average of λ_i and λ_i'. It can be independent of that variation only if λ_i=λ_i'. Hence λ is constant on every packet. Write the value on I_…
Exact mathematical passage
j\in I_b
I_dUse in source · 2 passages
Search aliases: I_d
Use in source · 30_lifting.tex, line 701
Let A=p^sJ^-1∈ M_d(R). Inclusion into Q_p and its injectivity give the two integral identities AJ=JA=p^sI_d. These identities are also an equivalent integral certificate for the first two hypotheses of [source reference]. In particular, A is injective: Aw=0 implies p^sw=JAw=0, hence w=0.
Exact mathematical passage
AJ=JA=p^sI_d.
I_{d_j}Use in source · 1 passage
Search aliases: I_d_j
Use in source · Paper_VI_Traditional_Final.tex, line 5767
If B_j and C_j have an intermediate space of dimension d_j, use a block diagonal matrix D with blocks λ_j I_d_j, or the required Jordan blocks when poles repeat. Assemble the matrices B_j and C_j into the output and input maps.
Exact mathematical passage
\lambda_j I_{d_j}
I_eUse in source · 3 passages
Search aliases: I_e
Use in source · 30_lifting.tex, line 860
0 · cI_k. The identities AJ=JA=cI_e give CB=BC=cI_e+k. In the second product, the upper right block is -JAK+cK=0. In the first, it is AK-AK=0. Thus augmentation preserves the original inverse bound s.
Exact mathematical passage
AJ=JA=cI_e
I_{e+k}Use in source · 3 passages
Search aliases: I_e+k
Use in source · 30_lifting.tex, line 860
0 · cI_k. The identities AJ=JA=cI_e give CB=BC=cI_e+k. In the second product, the upper right block is -JAK+cK=0. In the first, it is AK-AK=0. Thus augmentation preserves the original inverse bound s.
Exact mathematical passage
CB=BC=cI_{e+k}
I_{e_1}Use in source · 2 passages
Search aliases: I_e_1
Use in source · visual_atlas.tex, line 125
headThe distinction is visible in two dimensions Use X= Q^2, seed e_1=(1,0)^ T, and the full matrix algebra. Its seed annihilator is the set of matrices with zero first column: I_e_1={mat0 · a 0 · b:a,b∈ Q}.
Exact mathematical passage
I_{e_1}=\left\{\mat{0&a\\0&b}:a,b\in\QQ\right\}.
I_{\frakp}Use in source · 3 passages
Search aliases: I_frakp I_ p frakp mathfrak
Use in source · Paper_XII_Traditional_Final.tex, line 718
Let p⊇ I. Localizing gives (M/IM)_ p ≅ M_ p/I_ pM_ p. If M_ p=0, the quotient is zero. If M_ p≠0, then I_ p lies in the maximal ideal of the local ring R_ p. Nakayama's lemma shows that M_ p/I_ pM_ p≠0. The stated equality follows.
Exact mathematical passage
(M/IM)_{\frakp} \cong M_{\frakp}/I_{\frakp}M_{\frakp}.
I_{H_2}Use in source · 1 passage
Search aliases: I_H_2
Use in source · 04_correction_transport.tex, line 7
[Orthogonal obstruction projector] Let D:H_1→ H_2 be a bounded operator between Hilbert spaces with closed range. Let D^† be its Moore--Penrose inverse. Then P_ob=I_H_2-DD^† is the orthogonal projector onto (im D)^⊥=Ker D^*. For r∈ H_2, the following are equivalent: item Du=-r is solvable item [r]=0 in coker D item P_obr=0. When these conditions hold, -D^† r is the least norm correction. For every r, dist(r,im D)= P_obr .
Exact mathematical passage
P_{\mathrm{ob}}=I_{H_2}-DD^\dagger
I_{\Hcal}Use in source · 1 passage
Search aliases: I_Hcal I_ H Hcal mathcal
Use in source · 02_strengthened.tex, line 215
[Finite group isotypic projectors] The operators P_λ = fracd_λabsG ∑_g∈ G overlineχ_λ(g)U(g) are orthogonal projectors. They satisfy P_λ P_μ=δ_λμP_λ, ∑_λ∈widehat GP_λ=I_ H. The range of P_λ is the λ isotypic component of H.
Exact mathematical passage
P_\lambda P_\mu=\delta_{\lambda\mu}P_\lambda, \qquad \sum_{\lambda\in\widehat G}P_\lambda=I_{\Hcal}.
I_JUse in source · 2 passages
Search aliases: I_J
Use in source · 03_supplement_02.tex, line 211
To prove complete positivity rather than positivity alone, let J be an arbitrary finite auxiliary index set. For a matrix X indexed by J× n, define its blockwise amplification by [Φ^(J)(X)]_(i,u),(j,v) =Φ(X_(i,·),(j,·))_u,v. The explicit index calculation proves the identity Φ^(J)(X)=∑_a (I_J⊗ K_a)X(I_J⊗ K_a)^*. The displayed identity identifies the defined amplification with tensoring by the identity, which is the operation needed for complete positivity. Every summand on the right is positive …
Exact mathematical passage
\boxed{\quad \Phi^{(J)}(X)=\sum_a (I_J\otimes K_a)X(I_J\otimes K_a)^*. \quad}
i_jUse in source · 5 passages
Search aliases: i_j
Use in source · Paper_X_Block27_Amended.tex, line 382
For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet are cochain maps, so they induce maps on these genuine quotients.
Exact mathematical passage
i_j:C_j^\bullet\to C^\bullet
I_{j,k}Use in source · 4 passages
Search aliases: I_j,k
Use in source · Paper_V_Traditional_Final.tex, line 3191
Let I_j,k = [k2^-j,(k+1)2^-j) and let ψ_j,k be the normalized Haar wavelet supported on I_j,k, positive on the left half and negative on the right half.
Exact mathematical passage
I_{j,k} = [k2^{-j},(k+1)2^{-j})
I_KUse in source · 7 passages
Search aliases: I_K
Use in source · Paper_XI_Traditional_Final.tex, line 230
Let W_K be the Weil group of K, let I_K be inertia, let P_K be wild inertia, and choose a geometric Frobenius lift Φ. A Weil Deligne representation is a pair (r,N) with r(w)Nr(w)^-1=absw_K N.
Exact mathematical passage
I_K
I_kUse in source · 11 passages
Search aliases: I_k
Use in source · 30_lifting.tex, line 506
Now suppose a chosen set of e columns of DF(x)bmod p is independent, and put those coordinates first. Apply the square result to the augmented polynomial map Ψ(Y,T)=(F(Y,T),T). Its reduced Jacobian has upper blocks (D_YF,D_TF) and lower blocks (0,I_k), so it is invertible. For t∈ T, the target (0,t) belongs to Ψ(x)+p^L Z_p^e+k because F(x)≡0pmodp^L. The inverse of Ψ at that target gives the unique root with parameters t. The inverse congruence property gives one direction of [source reference]. …
Exact mathematical passage
(0,I_k)
i_KUse in source · 2 passages
Search aliases: i_K
Use in source · Paper_X_Block27_Amended.tex, line 2415
Actual quadratic local conditions and completion. Continue with the fixed packet P_q above. For a characteristic-zero field K, let j_K:G_K⟶ G_ Q be the actual field-induced homomorphism between absolute Galois groups. It includes a chosen embedding of algebraic closures. Let i_K be the image of the chosen square root i under that embedding. Lean proves compatibility with the Galois actions and hence (q∘ j_K)(σ)=1 Longleftrightarrow σ(i_K)=i_K. The fixed-field theorem for the algebraic closure th…
Exact mathematical passage
i_K
I_MUse in source · 1 passage
Search aliases: I_M
Use in source · 72_recovered.tex, line 2083
E_q · 0 · 0 · ⋯ · 0 . It also satisfies S_M(E_q)^M=E_qI_M.
Exact mathematical passage
\boxed{\mathsf S_M(E_q)^M=E_qI_M.}
I_mUse in source · 9 passages
Search aliases: I_m
Use in source · 02_coordinates.tex, line 69
A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.
Exact mathematical passage
\rho_{m,\ell}:I_m\longrightarrow I_\ell \qquad(m\geq\ell)
i_mUse in source · 2 passages
Search aliases: i_m
Use in source · 02_connections_obstructions.tex, line 609
[Packet path expansion for different matrices] For i,j∈ V and m≥1, set i_0=i and i_m=j. Then (B_m⋯ B_1)_ji = ∑_(i_1,…,i_m-1)∈ V^m-1 B_m[i_m,i_m-1]⋯ B_1[i_1,i_0]. When m=1, the sum is over the unique empty tuple and its summand is B_1[j,i]. For m=0, the product is the identity and the entry is the Kronecker delta δ_ji. If every B_r[q,p] vanishes outside a declared relation R_r(p,q), then the endpoint entry is zero whenever there is no packet path compatible with the successive relations. statusle…
Exact mathematical passage
i_m=j
i_{m-1}Use in source · 1 passage
Search aliases: i_m-1
Use in source · 02_connections_obstructions.tex, line 610
[Packet path expansion for different matrices] For i,j∈ V and m≥1, set i_0=i and i_m=j. Then (B_m⋯ B_1)_ji = ∑_(i_1,…,i_m-1)∈ V^m-1 B_m[i_m,i_m-1]⋯ B_1[i_1,i_0]. When m=1, the sum is over the unique empty tuple and its summand is B_1[j,i]. For m=0, the product is the identity and the entry is the Kronecker delta δ_ji. If every B_r[q,p] vanishes outside a declared relation R_r(p,q), then the endpoint entry is zero whenever there is no packet path compatible with the successive relations. statusle…
Exact mathematical passage
\boxed{ (B_m\cdots B_1)_{ji} = \sum_{(i_1,\ldots,i_{m-1})\in V^{m-1}} B_m[i_m,i_{m-1}]\cdots B_1[i_1,i_0]. }
I_{m_r}Use in source · 1 passage
Search aliases: I_m_r
Use in source · 02_strengthened.tex, line 104
Let A ≅ ⊕_r=1^s (M_d_r( C)⊗ I_m_r) act on H ≅ ⊕_r=1^s ( C^d_r⊗ C^m_r). Assume d_r,m_r≥1 for each displayed block. For an empty block family, the seed rank and the displayed finite supremum are defined as zero. A vector in one block is identified with a d_r by m_r coefficient matrix.
Exact mathematical passage
\Acal \cong \bigoplus_{r=1}^{s} \bigl(M_{d_r}(\CC)\otimes I_{m_r}\bigr)
I_{M_\mu}Use in source · 1 passage
Search aliases: I_M_μ mu
Use in source · 02_strengthened.tex, line 225
Complete reducibility gives a unitary decomposition H ≅ ⊕_μ∈widehat G V_μ⊗ M_μ, U(g) ≅ ⊕_μ π_μ(g)⊗ I_M_μ. On the μ summand, the displayed average is central in the irreducible representation π_μ, so it is a scalar by Schur's lemma. Taking its trace and applying character orthogonality shows that the scalar is one when μ=λ and zero otherwise. The asserted identities follow on the direct sum.
Exact mathematical passage
\Hcal \cong \bigoplus_{\mu\in\widehat G} V_\mu\otimes M_\mu, \qquad U(g) \cong \bigoplus_\mu \pi_\mu(g)\otimes I_{M_\mu}.
i_nUse in source · 1 passage
Search aliases: i_n
Use in source · Paper_VI_Traditional_Final.tex, line 1469
For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).
Exact mathematical passage
(i=i_0,i_1,\ldots,i_n)
I_PUse in source · 3 passages
Search aliases: I_P
Use in source · 03_supplement_01.tex, line 101
Corrected Feshbach resolvent. Write a finite complex matrix in retained and eliminated coordinates: H=A · B{ C · D, zI-H=zI_P-A · -B -C · zI_Q-D. Set D_z=zI_Q-D, S_z=zI_P-A-BD_z^-1C. The effective Hamiltonian is H_eff(z)=A+BD_z^-1C, so S_z=zI_P-H_eff(z).
Exact mathematical passage
H=\begin{pmatrix}A&B\\C&D\end{pmatrix},\qquad zI-H=\begin{pmatrix}zI_P-A&-B\\-C&zI_Q-D\end{pmatrix}.
I_pUse in source · 3 passages
Search aliases: I_p
Use in source · supplement_08.tex, line 66
pageheading2. The actual integral Hurwitz latticeConnecting differential expressions to coefficient divisibility The specialization works inside Q_p[[t]] and uses factorial-scaled coefficients a_f(d)=d![t^d]f, I_p={f∈ Q_p[[t]]:a_f(d)∈ Z_p for every d}. The source constructs I_p as a subring. Actual exponential-series multiplication gives a_fg(d)=∑_r=0^dbinom dr a_f(r)a_g(d-r), a_Uf(d)=a_f(d+1). These identities prove multiplication and derivative stability of the lattice. Membership does not req…
Exact mathematical passage
a_f(d)=d![t^d]f,\qquad I_p=\{f\in\Q_p[[t]]:a_f(d)\in\Z_p\text{ for every }d\}.
I_{p^{d-1}}Use in source · 1 passage
Search aliases: I_p^d-1
Use in source · odd_synthesis.tex, line 72
, where κ=-f'(z)/z^p-1, satisfies u_0=s(P_0)=a_0/z^p∈ O_E^×, Lift_E(C)=u_0^-1CP_0, s(Lift_E(C))=C. The lift preserves the exact filtration valuation. Its crucial additional identity is tr_L(Lift_E(C)) =Tr_E/L (κ^-1tr_E(C)). Define B(L,t_L;h_1,…,h_d) =α I_p^d-1+ Lift_E(B(E,z;h_1,…,h_d-1)), with the degree-p multiplication matrix as the base case. Induction proves trace zero at every native rank. This is stronger than the automatic zero trace obtained by multiplying any matrix multiplicity by p in…
Exact mathematical passage
B(L,t_L;h_1,\ldots,h_d) =\alpha I_{p^{d-1}}+ \operatorname{Lift}_E\bigl(B(E,z;h_1,\ldots,h_{d-1})\bigr),
I_QUse in source · 3 passages
Search aliases: I_Q
Use in source · 01_finite_core.tex, line 906
[Exact Feshbach resolvent] Assume that both zI_Q-H_QQ and zI-H are invertible. Define H_eff(z) = H_PP +H_PQ(z-H_QQ)^-1H_QP. Then P(z-H)^-1P = (z-H_eff(z))^-1 where the left side is restricted to P H, the right side is an inverse on that retained space, and both are written in the same orthonormal basis. Here z-H_QQ and z-H_eff abbreviate the appropriately sized scalar matrices minus the indicated blocks.
Exact mathematical passage
zI_Q-H_{QQ}
i_qUse in source · 1 passage
Search aliases: i_q
Use in source · Paper_V_Traditional_Final.tex, line 1076
[Cyclic product law] P_N,k(AB) = ∑_i+j≡ k (mod N)A_iB_j. For q∈ N, the qth-power identity is P_N,k(A^q) = ∑_i_1+⋯+i_q≡ k A_i_1⋯ A_i_q.
Exact mathematical passage
P_{N,k}(A^q) = \sum_{i_1+\cdots+i_q\equiv k} A_{i_1}\cdots A_{i_q}.
i_rUse in source · 2 passages
Search aliases: i_r
Use in source · Paper_VI_Traditional_Final.tex, line 1469
For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).
Exact mathematical passage
\prod_rK_{i_{r+1},i_r}
i_{r+1}Use in source · 1 passage
Search aliases: i_r+1
Use in source · Paper_VI_Traditional_Final.tex, line 1469
For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).
Exact mathematical passage
\prod_rK_{i_{r+1},i_r}
I_{\rm in}Use in source · 1 passage
Search aliases: I_rm in rm
Use in source · 02_strengthened.tex, line 1083
labelstrong:standardfact:001 A linear map Φ:M_n( C)→ M_m( C) is completely positive exactly when it has a Kraus representation Φ(X)=∑_rK_rXK_r^*. It is trace preserving exactly when ∑_rK_r^*K_r=I. Equivalently, its Choi matrix J(Φ) = ∑_i,jΦ(ketibraj)⊗ketibraj is positive, and trace preservation is Tr_rm outJ(Φ)=I_rm in.
Exact mathematical passage
\Tr_{\rm out}J(\Phi)=I_{\rm in}.
I_sUse in source · 4 passages
Search aliases: I_s
Use in source · visual_atlas.tex, line 19
[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…
Exact mathematical passage
4pt] \textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt] \textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology? \end{tabularx} A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. \takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''} \vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}} Most of the chapter can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix. {\small \begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule \textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule $X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\ $S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\ $\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\ $\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\ $P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\ $X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\ $T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\ $F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\ $U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\ $\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\ $\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule \end{tabularx}} \head{Read products from right to left} In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column. \head{Where to begin} For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}. \vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$} Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define \[ Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0, \qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.
i_sUse in source · 2 passages
Search aliases: i_s
Use in source · 03_nonlinear_coordinates.tex, line 339
For a block B={i_1<⋯<i_s} and an intermediate packet γ, write G_γ,B(xmid h) = J^s_γmidα_i_1,…,α_i_sG(x) [h_i_1,…,h_i_s].
Exact mathematical passage
B=\{i_1<\cdots<i_s\}
I_UUse in source · 2 passages
Search aliases: I_U
Use in source · 05_worked_models.tex, line 17
Let the quiver have vertices U,V and arrows R:U→ U, A:U→ V, B:V→ U, with the relation in the path algebra BA=e_U, whose realization is the operator identity BA=I_U. Realize U=𝕜^2, V=𝕜^3, by R= 0 · -1
Exact mathematical passage
BA=I_U
i_UUse in source · 6 passages
Search aliases: i_U
Use in source · 03_nonlinear_coordinates.tex, line 1044
[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is a natural exact sequence 0 · ⟶Ker D_0 ⟶Ker D_f ⟶Ker D_c xrightarrowδcoker D_0
Exact mathematical passage
\begin{array}{ccccccccc} 0&\longrightarrow&U_0&\xrightarrow{i_U}&U_f&\xrightarrow{p_U}&U_c&\longrightarrow&0\\[4pt] &&\mathllap{D_0}\downarrow&&\mathllap{D_f}\downarrow&&\mathllap{D_c}\downarrow&&\\[4pt] 0&\longrightarrow&V_0&\xrightarrow{i_V}&V_f&\xrightarrow{p_V}&V_c&\longrightarrow&0. \end{array}
I_VUse in source · 1 passage
Search aliases: I_V
Use in source · visual_atlas.tex, line 147
0 · 1 · 0. The typed relation is BA=I_U. In the opposite order, AB=diag(1,1,0)≠ I_V. The missing direction is Ker B= Q v_3, not an entirely missing vertex.
Exact mathematical passage
AB=\diag(1,1,0)\ne I_V.
I_vUse in source · 2 passages
Search aliases: I_v
Use in source · 02_connections_obstructions.tex, line 859
Suppose every X_v has a finite complete packet family (P^v_i)_i∈ I_v. A packet labeling assigns an admissible packet index to each edge or node of a typed tree. For multilinear Φ_σ, inserting id_X_v=∑_i∈ I_vP^v_i at every input and output yields a finite sum over trees with packet labels. This is the branching analogue of inserting intermediate projectors in [source reference].
Exact mathematical passage
(P^v_i)_{i\in I_v}
i_VUse in source · 8 passages
Search aliases: i_V
Use in source · 03_nonlinear_coordinates.tex, line 1044
[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is a natural exact sequence 0 · ⟶Ker D_0 ⟶Ker D_f ⟶Ker D_c xrightarrowδcoker D_0
Exact mathematical passage
\begin{array}{ccccccccc} 0&\longrightarrow&U_0&\xrightarrow{i_U}&U_f&\xrightarrow{p_U}&U_c&\longrightarrow&0\\[4pt] &&\mathllap{D_0}\downarrow&&\mathllap{D_f}\downarrow&&\mathllap{D_c}\downarrow&&\\[4pt] 0&\longrightarrow&V_0&\xrightarrow{i_V}&V_f&\xrightarrow{p_V}&V_c&\longrightarrow&0. \end{array}
I_{X/K}Use in source · 1 passage
Search aliases: I_X/K
Use in source · 04_transport.tex, line 238
[Packet descent through an invariant quotient] Under the hypotheses of [source reference], the operators overline P_α(x+K)=P_α x+K are well defined pairwise annihilating idempotents with sum I_X/K, and the quotient map satisfies π P_α=overline P_απ. After zero packets are removed, they form the canonical quotient packet system. [source reference]
Exact mathematical passage
I_{X/K}
I_{X^*}Use in source · 1 passage
Search aliases: I_X^*
Use in source · 03_packets_and_graphs.tex, line 257
Because dualization reverses products, P_α^*P_β^*=(P_β P_α)^* =δ_αβP_α^*. Also ∑_α P_α^*=I_X^*. Thus the dual maps form a packet system. Restriction gives a map from P_α^*X^* to X_α^*. It is injective because a covector in the range of P_α^* vanishes on every other packet, and it is surjective because any covector on X_α extends to X through the direct sum decomposition.
Exact mathematical passage
\sum_\alpha P_\alpha^*=I_{X^*}
I_{X^{\Pcal}}Use in source · 1 passage
Search aliases: I_X^Pcal I_X^ P Pcal mathcal
Use in source · 02_coordinates.tex, line 104
[Exact coordinate change and coherence] For any two packet systems on X, C_ P← QJ_ Q← P=I_X^ P, J_ Q← PC_ P← Q=I_X^ Q. For three packet systems P, Q, and R on X, J_ R← QJ_ Q← P = J_ R← P, C_ P← QC_ Q← R = C_ P← R. If Q refines P through ρ, then (J_ Q← Px)_β = Q_β x_ρ(β), (C_ P← Qy)_α = ∑_ρ(β)=αy_β.
Exact mathematical passage
C_{\Pcal\leftarrow\Qcal}J_{\Qcal\leftarrow\Pcal}=I_{X^{\Pcal}}, \qquad J_{\Qcal\leftarrow\Pcal}C_{\Pcal\leftarrow\Qcal}=I_{X^{\Qcal}}.
I_{X^{\Qcal}}Use in source · 1 passage
Search aliases: I_X^Qcal I_X^ Q Qcal mathcal
Use in source · 02_coordinates.tex, line 104
[Exact coordinate change and coherence] For any two packet systems on X, C_ P← QJ_ Q← P=I_X^ P, J_ Q← PC_ P← Q=I_X^ Q. For three packet systems P, Q, and R on X, J_ R← QJ_ Q← P = J_ R← P, C_ P← QC_ Q← R = C_ P← R. If Q refines P through ρ, then (J_ Q← Px)_β = Q_β x_ρ(β), (C_ P← Qy)_α = ∑_ρ(β)=αy_β.
Exact mathematical passage
C_{\Pcal\leftarrow\Qcal}J_{\Qcal\leftarrow\Pcal}=I_{X^{\Pcal}}, \qquad J_{\Qcal\leftarrow\Pcal}C_{\Pcal\leftarrow\Qcal}=I_{X^{\Qcal}}.
I_{X_m}Use in source · 1 passage
Search aliases: I_X_m
Use in source · Paper_VI_Traditional_Final.tex, line 2529
Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.
Exact mathematical passage
C_{m,m}=I_{X_m}, \qquad R_{m,m}=I_{X_m}.
I_{X_\alpha}Use in source · 1 passage
Search aliases: I_X_α alpha
Use in source · 03_packets_and_graphs.tex, line 813
The identity has diagonal blocks I_X_α and no blocks outside the diagonal, so it belongs to M_R exactly when (α,α)∈ R for every α.
Exact mathematical passage
I_{X_\alpha}
I_{X_\ell}Use in source · 1 passage
Search aliases: I_X_ℓ ell
Use in source · Paper_VI_Traditional_Final.tex, line 2517
Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.
Exact mathematical passage
R_\ell:X_\ell\to X_{\ell+1}, \qquad C_\ell R_\ell=I_{X_\ell}.
I_YUse in source · 7 passages
Search aliases: I_Y
Use in source · 03_packets_and_graphs.tex, line 223
For every pair of packet indices, (P_α⊗ Q_γ)^2 =P_α^2⊗ Q_γ^2 =P_α⊗ Q_γ. If (α,γ)≠(β,δ), then either P_α P_β=0 or Q_γ Q_δ=0, so the corresponding tensor idempotents annihilate each other. Finally, ∑_α,γP_α⊗ Q_γ = (∑_α P_α) ⊗ (∑_γ Q_γ) =I_X⊗ I_Y. The range is generated by pure tensors P_α x⊗ Q_γ y, hence equals X_α⊗ Y_γ. Exact reconstruction follows from [source reference].
Exact mathematical passage
\sum_{\alpha,\gamma}P_\alpha\otimes Q_\gamma = \left(\sum_\alpha P_\alpha\right) \otimes \left(\sum_\gamma Q_\gamma\right) =I_X\otimes I_Y.
I_ZUse in source · 1 passage
Search aliases: I_Z
Use in source · 04_correction_transport.tex, line 38
[Right inverse correction theorem] Let D:X→ Y be a bounded linear map between Banach spaces. Assume Z=im D is closed and there is a bounded linear right inverse R:Z⟶ X, DR=I_Z. Put X_R=R(Z). Let N:overline B_ρ(0)∩ X_R⟶ Z be L-Lipschitz. Let y∈ Z and put q= R L. Assume q<1 and R(y-N(0)) ≤ (1-q)ρ. Then the equation Dx+N(x)=y has a unique solution in overline B_ρ(0)∩ X_R.
Exact mathematical passage
R:Z\longrightarrow X, \qquad DR=I_Z.
I_\lambdaUse in source · 4 passages
Search aliases: I_λ lambda
Use in source · Paper_XII_Traditional_Final.tex, line 1466
Fix a topological group Γ, a finite matrix index set D, and a set Λ of coefficient indices. For each λ, let O_λ be a commutative ring, complete and separated for a finitely generated ideal I_λ, with finite residue ring O_λ/I_λ. Give O_λ its ideal adic topology and every quotient its discrete topology. Let i_λ: O_λ⟶ K_λ, e_λ:L⟶ K_λ be supplied ring homomorphisms, where L and K_λ are fields, K_λ has a declared topology, and i_λ is continuous.
Exact mathematical passage
I_\lambda
i_\lambdaUse in source · 5 passages
Search aliases: i_λ lambda
Use in source · Paper_XII_Traditional_Final.tex, line 1467
Fix a topological group Γ, a finite matrix index set D, and a set Λ of coefficient indices. For each λ, let O_λ be a commutative ring, complete and separated for a finitely generated ideal I_λ, with finite residue ring O_λ/I_λ. Give O_λ its ideal adic topology and every quotient its discrete topology. Let i_λ: O_λ⟶ K_λ, e_λ:L⟶ K_λ be supplied ring homomorphisms, where L and K_λ are fields, K_λ has a declared topology, and i_λ is continuous.
Exact mathematical passage
i_\lambda:\OO_\lambda\longrightarrow K_\lambda, \qquad e_\lambda:L\longrightarrow K_\lambda
I_\lambda^nUse in source · 3 passages
Search aliases: I_λ^n lambda
Use in source · Paper_XII_Traditional_Final.tex, line 1474
For every positive integer n, let X_λ,n be a finite nonempty set of conjugacy classes of actual continuous representations Γ⟶GL_D( O_λ/I_λ^n). Conjugacy means change of basis by an invertible matrix over that quotient. Assume reduction preserves these sets and that X_λ,1 consists of the declared residual class. Each admissible class must have a continuous representative. The auxiliary level zero is the unique class over the trivial quotient and imposes no extra occupancy hypothesis.
Exact mathematical passage
\Gamma\longrightarrow\GL_D(\OO_\lambda/I_\lambda^n).
I_{\oplus X_\alpha}Use in source · 1 passage
Search aliases: I_⊕ X_α oplus alpha
Use in source · 03_packets_and_graphs.tex, line 89
[Exact packet coordinate isomorphism] For every finite packet system, S_ P A_ P=I_X, A_ P S_ P=I_⊕ X_α. Hence X≅⊕_α∈ IX_α. [source reference] ExportsEvery exact packet reconstruction in the series.
Exact mathematical passage
\boxed{ \mathcal S_{\Pcal}\mathcal A_{\Pcal}=I_X, \qquad \mathcal A_{\Pcal}\mathcal S_{\Pcal}=I_{\oplus X_\alpha}. }
\id_{\Coord(\cP^a)}Use in source · 1 passage
Search aliases: id_Coord(cP^a) id_Coord( P^a) id Coord cP mathrm operatorname mathcal
Use in source · 01_finite_coordinates.tex, line 456
i)_i∈ I_a be a finite complete family of pairwise orthogonal linear idempotents: P^a_iP^a_j=δ_ijP^a_i, ∑_i∈ I_aP^a_i=I_X. Define the exact packet coordinate module Coord( P^a) = {(x_i)_i∈ I_a:P^a_ix_i=x_i}. Its analysis and synthesis maps are A_a(x)=(P^a_ix)_i, S_a((x_i)_i)=∑_ix_i. Completeness and orthogonality give S_a A_a=id_X, A_a S_a=id_Coord( P^a).
Exact mathematical passage
\cS_a\cA_a=\id_X, \qquad \cA_a\cS_a=\id_{\Coord(\cP^a)}.
\id_{I_\ell}Use in source · 1 passage
Search aliases: id_I_ℓ id ell mathrm ell
Use in source · 02_coordinates.tex, line 74
A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.
Exact mathematical passage
\rho_{\ell,\ell}=\id_{I_\ell}, \qquad \rho_{n,\ell}=\rho_{m,\ell}\circ\rho_{n,m} \qquad(n\geq m\geq\ell).
\id_XUse in source · 2 passages
Search aliases: id_X id mathrm
Use in source · 01_finite_coordinates.tex, line 456
i)_i∈ I_a be a finite complete family of pairwise orthogonal linear idempotents: P^a_iP^a_j=δ_ijP^a_i, ∑_i∈ I_aP^a_i=I_X. Define the exact packet coordinate module Coord( P^a) = {(x_i)_i∈ I_a:P^a_ix_i=x_i}. Its analysis and synthesis maps are A_a(x)=(P^a_ix)_i, S_a((x_i)_i)=∑_ix_i. Completeness and orthogonality give S_a A_a=id_X, A_a S_a=id_Coord( P^a).
Exact mathematical passage
\cS_a\cA_a=\id_X, \qquad \cA_a\cS_a=\id_{\Coord(\cP^a)}.
\id_{X_v}Use in source · 1 passage
Search aliases: id_X_v id mathrm
Use in source · 02_connections_obstructions.tex, line 861
Suppose every X_v has a finite complete packet family (P^v_i)_i∈ I_v. A packet labeling assigns an admissible packet index to each edge or node of a typed tree. For multilinear Φ_σ, inserting id_X_v=∑_i∈ I_vP^v_i at every input and output yields a finite sum over trees with packet labels. This is the branching analogue of inserting intermediate projectors in [source reference].
Exact mathematical passage
\id_{X_v}=\sum_{i\in I_v}P^v_i
\iffUse in source · 10 passages
Search aliases: iff
Use in source · visual_atlas.tex, line 412
For a specific attainable target (2,4), 2x≡2pmod8iff x∈{1,5}, 4y≡4pmod8iff y∈{1,3,5,7}. The coordinate choices are independent, giving 2·4=8 solutions. The entire carrier is partitioned consistently: 8 attainable targets × 8 inputs per target=64. For an inadmissible target, such as (1,4) or (2,2), the fiber is empty. The same nonzero cardinality formula must not be applied to it.
Exact mathematical passage
2x\equiv2\pmod8\iff x\in\{1,5\},\qquad 4y\equiv4\pmod8\iff y\in\{1,3,5,7\}.
\ii^{-jk}Use in source · 1 passage
Search aliases: ii^-jk i^-jk ii mathrm
Use in source · monograph_atlas.tex, line 93
vpage6Four phases resolve a cyclic shiftCHARACTER PROJECTORS YOU CAN MULTIPLYChapter VII: Corollary II.3.3. On four sites, let Sket k=ketk+1bmod4 and let ω= i. The character projectors are the exact matrices P_j=frac14∑_r=0^3 i^-jrS^r, v_j(k)=frac12 i^-jk, P_j=v_jv_j^*. All entries belong to the Gaussian rationals. Thus no approximation to a root of unity is needed in this model. vfigcyclic_four headReading the columns Each column v_j is a normalized wave around the four sites. Advancing the add…
Exact mathematical passage
P_j=\frac14\sum_{r=0}^3\ii^{-jr}S^r, \qquad v_j(k)=\frac12\ii^{-jk},\qquad P_j=v_jv_j^*.
\ii^{-jr}Use in source · 1 passage
Search aliases: ii^-jr i^-jr ii mathrm
Use in source · monograph_atlas.tex, line 93
vpage6Four phases resolve a cyclic shiftCHARACTER PROJECTORS YOU CAN MULTIPLYChapter VII: Corollary II.3.3. On four sites, let Sket k=ketk+1bmod4 and let ω= i. The character projectors are the exact matrices P_j=frac14∑_r=0^3 i^-jrS^r, v_j(k)=frac12 i^-jk, P_j=v_jv_j^*. All entries belong to the Gaussian rationals. Thus no approximation to a root of unity is needed in this model. vfigcyclic_four headReading the columns Each column v_j is a normalized wave around the four sites. Advancing the add…
Exact mathematical passage
P_j=\frac14\sum_{r=0}^3\ii^{-jr}S^r, \qquad v_j(k)=\frac12\ii^{-jk},\qquad P_j=v_jv_j^*.
\ii^jUse in source · 1 passage
Search aliases: ii^j i^j ii mathrm
Use in source · monograph_atlas.tex, line 100
character projectors are the exact matrices P_j=frac14∑_r=0^3 i^-jrS^r, v_j(k)=frac12 i^-jk, P_j=v_jv_j^*. All entries belong to the Gaussian rationals. Thus no approximation to a root of unity is needed in this model. vfigcyclic_four headReading the columns Each column v_j is a normalized wave around the four sites. Advancing the address multiplies it by i^j. The entries show phase differences rather than four different probability distributions: every column has uniform site probability 1/4.
Exact mathematical passage
\ii^j
\ImUse in source · 1 passage
Search aliases: Im
Use in source · monograph_atlas.tex, line 653
1 · 2, so A=0, B=1, and D=2. In its elimination domain, Σ(z)=1/(z-2). At z=3+ i, Σ(3+ i)=frac1- i2, ImΣ(z)=-(Im z)|(z-2)^-1|^2≤0. The sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix. xfigstability headA stability estimate with a common spectral gap Compare (B,D)=(1,2) with (widetilde B,widetilde D)=(9/8,17/8) at z=3. Both hidden spectral distances are at least δ=3/4, and both couplings are bounded by g=9/8. …
Exact mathematical passage
\Sigma(3+\ii)=\frac{1-\ii}{2},\qquad \Im\Sigma(z)=-(\Im z)\left|(z-2)^{-1}\right|^2\leq0.
\operatorname{Im}Use in source · 3 passages
Search aliases: Im operatorname
Use in source · atlas-pages.tex, line 415
headThe coordinates preserve structure, not Euclidean distances A quarter turn is a rotation in the reflection disk. After pulling back, it is a M obius map, not a Euclidean rotation of the impedance plane. Equal reflection magnitudes need not give equal impedance magnitudes. The opposite points 1/2 and -1/2 become 3 and 1/3. The continuous impedance locus is the circle (Rez-5/3)^2+(Imz)^2=16/9, also drawn in the plate. Finite passive impedances with Rez≥0 map to the closed disk except Γ=1. That…
Exact mathematical passage
(\operatorname{Re}z-5/3)^2+(\operatorname{Im}z)^2=16/9
\operatorname{im}Use in source · 14 passages
Search aliases: im operatorname
Use in source · 71_native.tex, line 25
O_F,bar x= F_p^r/imJ, o_F,bar x=[e].
Exact mathematical passage
O_{F,\bar x}=\mathbb F_p^r/\operatorname{im}J,\qquad o_{F,\bar x}=[e].
\IndUse in source · 9 passages
Search aliases: Ind operatorname
Use in source · Paper_XI_Traditional_Final.tex, line 335
[Tame orbit induction] Let τ be an irreducible complex representation of W_K with trivial wild inertia. Then the nonzero tame character packets form one transitive Frobenius orbit of length f. If K_f/K is the unramified extension of degree f, there is a character ξ of W_K_f such that τ≅Ind_W_K_f^W_Kξ.
Exact mathematical passage
\boxed{ \tau\cong\Ind_{W_{K_f}}^{W_K}\xi. }
\operatorname{Ind}Use in source · 3 passages
Search aliases: Ind operatorname
Use in source · dyadic_synthesis.tex, line 292
e P=U_K^1U_A^m. This is an actual compact open group, with Jtriangleleft P, [P:J]=2 and P∩(K^× J)=J. For g∈ P∖ J, the actual intertwining theorem makes λ=θ^g/θ nontrivial. The commutator [P,J]⊂ U_A^m+1 lies in the forced domain of the additive character, so λ has values {±1}. Put H_0=kerλ. Then P/H_0≃ F_2^2, E=Ind_J^Pθ is irreducible of dimension 2. The actual represented generators are a diagonal sign and an off-diagonal matrix that anticommute. Thus their commutator form is nondegenerate, and …
Exact mathematical passage
P/H_0\simeq\mathbb F_2^2,\qquad E=\operatorname{Ind}_J^P\theta\text{ is irreducible of dimension }2.
\Ind_{\mathcal N}^{\mathcal M}Use in source · 1 passage
Search aliases: Ind_ N^ M Ind mathcal mathcal operatorname mathcal mathcal
Use in source · Paper_XI_Traditional_Final.tex, line 2133
At both ranks K^× normalizes P. Thus N=K^× J is normal of index two in M=K^× P, and Σ=Ind_ N^ MΛ, π≃cInd_ M^GL_N(F)Σ. Induction in stages preserves the full original Λ, its central character, and its noncompact root. See [source reference]. The full marked scalar-pushout comparison applies to these actual enlarged pairs as well as to the original higher-rank types ([source reference]). The full synthesis and scope are in [source reference].
Exact mathematical passage
\Sigma=\Ind_{\mathcal N}^{\mathcal M}\Lambda, \qquad \pi\simeq\cInd_{\mathcal M}^{\GL_N(F)}\Sigma.
\operatorname{Ind}_{\mathcal N}^{\mathcal M}Use in source · 1 passage
Search aliases: Ind_ N^ M operatorname mathcal mathcal
Use in source · dyadic_synthesis.tex, line 344
Consequently M=K^× P is an actual open group compact modulo the center, N=K^× J is normal of index two in M, and Σ=Ind_ N^ MΛ, Σ|_P≃ E, π≃c - Ind_ M^GΣ. Restriction to P is irreducible, so Σ is irreducible. Induction in stages proves the last equality for the original Λ, including its fixed central character and noncompact root. This is an actual enlarged inducing representation. It changes neither the candidate nor its original canonical beta-extension data.
Exact mathematical passage
\Sigma=\operatorname{Ind}_{\mathcal N}^{\mathcal M}\Lambda, \qquad \Sigma|_P\simeq E, \qquad \boxed{\pi\simeq\operatorname{c\!-\!Ind}_{\mathcal M}^G\Sigma.}
\Ind_J^PUse in source · 2 passages
Search aliases: Ind_J^P Ind operatorname
Use in source · Paper_XI_Traditional_Final.tex, line 2122
At rank two, with q=2m+1, enlarge J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=2, and P/H_0≃ F_2^4, with carrier Ind_J^Pη of dimension four ([source reference]). The finite scalar center can be larger than μ_2. No literal extrasp…
Exact mathematical passage
\Ind_J^P\theta
\operatorname{Ind}_J^PUse in source · 2 passages
Search aliases: Ind_J^P operatorname
Use in source · dyadic_synthesis.tex, line 292
e P=U_K^1U_A^m. This is an actual compact open group, with Jtriangleleft P, [P:J]=2 and P∩(K^× J)=J. For g∈ P∖ J, the actual intertwining theorem makes λ=θ^g/θ nontrivial. The commutator [P,J]⊂ U_A^m+1 lies in the forced domain of the additive character, so λ has values {±1}. Put H_0=kerλ. Then P/H_0≃ F_2^2, E=Ind_J^Pθ is irreducible of dimension 2. The actual represented generators are a diagonal sign and an off-diagonal matrix that anticommute. Thus their commutator form is nondegenerate, and …
Exact mathematical passage
P/H_0\simeq\mathbb F_2^2,\qquad E=\operatorname{Ind}_J^P\theta\text{ is irreducible of dimension }2.
\Ind_{W_{K_f}}^{W_K}Use in source · 4 passages
Search aliases: Ind_W_K_f^W_K Ind operatorname
Use in source · Paper_XI_Traditional_Final.tex, line 335
[Tame orbit induction] Let τ be an irreducible complex representation of W_K with trivial wild inertia. Then the nonzero tame character packets form one transitive Frobenius orbit of length f. If K_f/K is the unramified extension of degree f, there is a character ξ of W_K_f such that τ≅Ind_W_K_f^W_Kξ.
Exact mathematical passage
\boxed{ \tau\cong\Ind_{W_{K_f}}^{W_K}\xi. }
\Ind_{W_{K_L}}^{W_K}Use in source · 1 passage
Search aliases: Ind_W_K_L^W_K Ind operatorname
Use in source · Paper_XI_Traditional_Final.tex, line 565
[Greatest common divisor and least common multiple packets] There is a decomposition τ_ξ⊗τ_η ≅ ⊕_i=0^d-1 Ind_W_K_L^W_K ( ξ_L,nη_L,m^Φ^i ), where ξ_L,n and η_L,m are the norm lifts to K_L.
Exact mathematical passage
\boxed{ \tau_\xi\otimes\tau_\eta \cong \bigoplus_{i=0}^{d-1} \Ind_{W_{K_L}}^{W_K} \left( \xi_{L,n}\eta_{L,m}^{\Phi^i} \right), }
\Ind_{W_{K_m}}^{W_K}Use in source · 1 passage
Search aliases: Ind_W_K_m^W_K Ind operatorname
Use in source · Paper_XI_Traditional_Final.tex, line 551
Let τ_ξ=Ind_W_K_n^W_Kξ, τ_η=Ind_W_K_m^W_Kη, where K_n/K and K_m/K are unramified. Put d=gcd(n,m), L=lcm(n,m).
Exact mathematical passage
\tau_\xi=\Ind_{W_{K_n}}^{W_K}\xi, \qquad \tau_\eta=\Ind_{W_{K_m}}^{W_K}\eta,
\Ind_{W_{K_n}}^{W_K}Use in source · 1 passage
Search aliases: Ind_W_K_n^W_K Ind operatorname
Use in source · Paper_XI_Traditional_Final.tex, line 551
Let τ_ξ=Ind_W_K_n^W_Kξ, τ_η=Ind_W_K_m^W_Kη, where K_n/K and K_m/K are unramified. Put d=gcd(n,m), L=lcm(n,m).
Exact mathematical passage
\tau_\xi=\Ind_{W_{K_n}}^{W_K}\xi, \qquad \tau_\eta=\Ind_{W_{K_m}}^{W_K}\eta,
\inf_{j\geq0}Use in source · 1 passage
Search aliases: inf_j≥0 inf geq
Use in source · supplement_04.tex, line 104
For a uniform coefficient hypothesis, the simpler conclusion follows: inf_j≥0 w(s+j)≤ord_p(Δ_1^s f(n)). For each finite L≤ B_p(w;n,s), a separate theorem yields p^LmidΔ_1^s f(n) inside Z_p. Another specialization handles arbitrary finite interpolants with infinite weights beyond their degree.
Exact mathematical passage
\boxed{\inf_{j\geq0} w(s+j)\leq\ord_p(\Delta_1^s f(n)).}
\bm jUse in source · 1 passage
Search aliases: j bm
Use in source · 02_strengthened.tex, line 343
Suppose the ath tensor factor carries a cyclic packet family (P_j^(a))_j∈ Z/N Z. The fine tensor packets are P_ j^rm fine = P_j_1^(1)⊗⋯⊗ P_j_m^(m).
Exact mathematical passage
P_{\bm j}^{\rm fine} = P_{j_1}^{(1)}\otimes\cdots\otimes P_{j_m}^{(m)}.
\mathbf jUse in source · 1 passage
Search aliases: j mathbf
Use in source · 01_finite_core.tex, line 239
Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.
Exact mathematical passage
P_{\mathbf j}^{\mathrm{fine}} = P^{(1)}_{j_1}\otimes\cdots\otimes P^{(m)}_{j_m}.
J'Use in source · 1 passage
Search aliases: J'
Use in source · atlas-pages.tex, line 234
e every node cost changes by at most ε_v=1/32. A pruning may select as many as eight leaves, so its total cost can change by at most 8/32=1/4. The recursive envelope proves this without guessing which pruning will be selected: E(v)=max{ε_v, E(v0)+E(v1)}, E(∅)=tfrac14. For any competitor Λ, the changed cost difference is bounded below by J'(Λ)-J'(Λ_*)≥ Γ-2E(∅)=tfrac12>0. Thus the optimum is unchanged for every permitted simultaneous perturbation, not just a sampled noise realization. The envelope…
Exact mathematical passage
J'(\Lambda)-J'(\Lambda_*)\ge \Gamma-2E(\varnothing)=\tfrac12>0.
j^{-1}Use in source · 1 passage
Search aliases: j^-1
Use in source · rank2.tex, line 139
Finally, if kj∈P with k∈K× and j∈J, then k∈ Pj^-1⊂ U_A^1. Hence k∈U_K 1 and kj∈J. Therefore
Exact mathematical passage
k\in Pj^{-1}\subset U_A^1
J^1_{\gamma_1\mid\alpha_1}Use in source · 1 passage
Search aliases: J^1_γ_1midα_1 gamma mid alpha
Use in source · 03_nonlinear_coordinates.tex, line 401
[The second-order composition law] At order two, the partition formula has two shapes. For packet directions h_1 and h_2, R_δ D^2(F∘ G)(x)[h_1,h_2] = · ∑_γ_1,γ_2 J^2_δmidγ_1,γ_2F(Gx) [J^1_γ_1midα_1G(x)h_1,J^1_γ_2midα_2G(x)h_2] · + ∑_γ J^1_δmidγF(Gx) [J^2_γmidα_1,α_2G(x)[h_1,h_2]]. The first line records two first-order changes entering a quadratic interaction of F. The second records a genuine second-order interaction created inside G before F is applied.
Exact mathematical passage
\begin{aligned} R_\delta D^2(F\circ G)(x)[h_1,h_2] ={}& \sum_{\gamma_1,\gamma_2} J^2_{\delta\mid\gamma_1,\gamma_2}F(Gx) [J^1_{\gamma_1\mid\alpha_1}G(x)h_1,J^1_{\gamma_2\mid\alpha_2}G(x)h_2]\\ &+ \sum_\gamma J^1_{\delta\mid\gamma}F(Gx) [J^2_{\gamma\mid\alpha_1,\alpha_2}G(x)[h_1,h_2]]. \end{aligned}
J^1_{\gamma_2\mid\alpha_2}Use in source · 1 passage
Search aliases: J^1_γ_2midα_2 gamma mid alpha
Use in source · 03_nonlinear_coordinates.tex, line 401
[The second-order composition law] At order two, the partition formula has two shapes. For packet directions h_1 and h_2, R_δ D^2(F∘ G)(x)[h_1,h_2] = · ∑_γ_1,γ_2 J^2_δmidγ_1,γ_2F(Gx) [J^1_γ_1midα_1G(x)h_1,J^1_γ_2midα_2G(x)h_2] · + ∑_γ J^1_δmidγF(Gx) [J^2_γmidα_1,α_2G(x)[h_1,h_2]]. The first line records two first-order changes entering a quadratic interaction of F. The second records a genuine second-order interaction created inside G before F is applied.
Exact mathematical passage
\begin{aligned} R_\delta D^2(F\circ G)(x)[h_1,h_2] ={}& \sum_{\gamma_1,\gamma_2} J^2_{\delta\mid\gamma_1,\gamma_2}F(Gx) [J^1_{\gamma_1\mid\alpha_1}G(x)h_1,J^1_{\gamma_2\mid\alpha_2}G(x)h_2]\\ &+ \sum_\gamma J^1_{\delta\mid\gamma}F(Gx) [J^2_{\gamma\mid\alpha_1,\alpha_2}G(x)[h_1,h_2]]. \end{aligned}
J^1_{\delta\mid\gamma}Use in source · 1 passage
Search aliases: J^1_δmidγ delta mid gamma
Use in source · 03_nonlinear_coordinates.tex, line 401
[The second-order composition law] At order two, the partition formula has two shapes. For packet directions h_1 and h_2, R_δ D^2(F∘ G)(x)[h_1,h_2] = · ∑_γ_1,γ_2 J^2_δmidγ_1,γ_2F(Gx) [J^1_γ_1midα_1G(x)h_1,J^1_γ_2midα_2G(x)h_2] · + ∑_γ J^1_δmidγF(Gx) [J^2_γmidα_1,α_2G(x)[h_1,h_2]]. The first line records two first-order changes entering a quadratic interaction of F. The second records a genuine second-order interaction created inside G before F is applied.
Exact mathematical passage
\begin{aligned} R_\delta D^2(F\circ G)(x)[h_1,h_2] ={}& \sum_{\gamma_1,\gamma_2} J^2_{\delta\mid\gamma_1,\gamma_2}F(Gx) [J^1_{\gamma_1\mid\alpha_1}G(x)h_1,J^1_{\gamma_2\mid\alpha_2}G(x)h_2]\\ &+ \sum_\gamma J^1_{\delta\mid\gamma}F(Gx) [J^2_{\gamma\mid\alpha_1,\alpha_2}G(x)[h_1,h_2]]. \end{aligned}
J^2Use in source · 1 passage
Search aliases: J^2
Use in source · 01_finite_core.tex, line 411
J · 0 . Therefore Spec H_even ={±sqrt4h^2+J^2}, Spec H_odd={± J}.
Exact mathematical passage
\Spec H_{\mathrm{even}} =\{\pm\sqrt{4h^2+J^2}\}, \qquad \Spec H_{\mathrm{odd}}=\{\pm J\}.
J^2_{\gamma\mid\alpha_1,\alpha_2}Use in source · 1 passage
Search aliases: J^2_γmidα_1,α_2 gamma mid alpha alpha
Use in source · 03_nonlinear_coordinates.tex, line 401
[The second-order composition law] At order two, the partition formula has two shapes. For packet directions h_1 and h_2, R_δ D^2(F∘ G)(x)[h_1,h_2] = · ∑_γ_1,γ_2 J^2_δmidγ_1,γ_2F(Gx) [J^1_γ_1midα_1G(x)h_1,J^1_γ_2midα_2G(x)h_2] · + ∑_γ J^1_δmidγF(Gx) [J^2_γmidα_1,α_2G(x)[h_1,h_2]]. The first line records two first-order changes entering a quadratic interaction of F. The second records a genuine second-order interaction created inside G before F is applied.
Exact mathematical passage
\begin{aligned} R_\delta D^2(F\circ G)(x)[h_1,h_2] ={}& \sum_{\gamma_1,\gamma_2} J^2_{\delta\mid\gamma_1,\gamma_2}F(Gx) [J^1_{\gamma_1\mid\alpha_1}G(x)h_1,J^1_{\gamma_2\mid\alpha_2}G(x)h_2]\\ &+ \sum_\gamma J^1_{\delta\mid\gamma}F(Gx) [J^2_{\gamma\mid\alpha_1,\alpha_2}G(x)[h_1,h_2]]. \end{aligned}
J^2_{\delta\mid\gamma_1,\gamma_2}Use in source · 1 passage
Search aliases: J^2_δmidγ_1,γ_2 delta mid gamma gamma
Use in source · 03_nonlinear_coordinates.tex, line 401
[The second-order composition law] At order two, the partition formula has two shapes. For packet directions h_1 and h_2, R_δ D^2(F∘ G)(x)[h_1,h_2] = · ∑_γ_1,γ_2 J^2_δmidγ_1,γ_2F(Gx) [J^1_γ_1midα_1G(x)h_1,J^1_γ_2midα_2G(x)h_2] · + ∑_γ J^1_δmidγF(Gx) [J^2_γmidα_1,α_2G(x)[h_1,h_2]]. The first line records two first-order changes entering a quadratic interaction of F. The second records a genuine second-order interaction created inside G before F is applied.
Exact mathematical passage
\begin{aligned} R_\delta D^2(F\circ G)(x)[h_1,h_2] ={}& \sum_{\gamma_1,\gamma_2} J^2_{\delta\mid\gamma_1,\gamma_2}F(Gx) [J^1_{\gamma_1\mid\alpha_1}G(x)h_1,J^1_{\gamma_2\mid\alpha_2}G(x)h_2]\\ &+ \sum_\gamma J^1_{\delta\mid\gamma}F(Gx) [J^2_{\gamma\mid\alpha_1,\alpha_2}G(x)[h_1,h_2]]. \end{aligned}
J^{k}_{\delta\mid\gamma_1,\ldots,\gamma_k}Use in source · 1 passage
Search aliases: J^k_δmidγ_1,…,γ_k delta mid gamma ldots gamma
Use in source · 03_nonlinear_coordinates.tex, line 353
di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n] · = ∑_π={B_1,…,B_k} ∑_γ_1,…,γ_k J^k_δmidγ_1,…,γ_kF(Gx) [ G_γ_1,B_1(xmid h),…, G_γ_k,B_k(xmid h) ]. The first sum runs over all set partitions of {1,…,n}.
Exact mathematical passage
\begin{aligned} &J^{n}_{\delta\mid\alpha_1,\ldots,\alpha_n}(F\circ G)(x) [h_1,\ldots,h_n]\\ &= \sum_{\pi=\{B_1,\ldots,B_k\}} \sum_{\gamma_1,\ldots,\gamma_k} J^{k}_{\delta\mid\gamma_1,\ldots,\gamma_k}F(Gx) \big[ G_{\gamma_1,B_1}(x\mid h),\ldots, G_{\gamma_k,B_k}(x\mid h) \big]. \end{aligned}
J^{n}_{\delta\mid\alpha_1,\ldots,\alpha_n}Use in source · 1 passage
Search aliases: J^n_δmidα_1,…,α_n delta mid alpha ldots alpha
Use in source · 03_nonlinear_coordinates.tex, line 353
di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n] · = ∑_π={B_1,…,B_k} ∑_γ_1,…,γ_k J^k_δmidγ_1,…,γ_kF(Gx) [ G_γ_1,B_1(xmid h),…, G_γ_k,B_k(xmid h) ]. The first sum runs over all set partitions of {1,…,n}.
Exact mathematical passage
\begin{aligned} &J^{n}_{\delta\mid\alpha_1,\ldots,\alpha_n}(F\circ G)(x) [h_1,\ldots,h_n]\\ &= \sum_{\pi=\{B_1,\ldots,B_k\}} \sum_{\gamma_1,\ldots,\gamma_k} J^{k}_{\delta\mid\gamma_1,\ldots,\gamma_k}F(Gx) \big[ G_{\gamma_1,B_1}(x\mid h),\ldots, G_{\gamma_k,B_k}(x\mid h) \big]. \end{aligned}
J^pUse in source · 1 passage
Search aliases: J^p
Use in source · 72_recovered.tex, line 216
[Augmentation criterion] Let x=∑_u∈ F_pc_u[u] ∈ F_p[ F_p^+]. Then x is a unit if and only if aug(x) = ∑_uc_u is nonzero. If J is the augmentation ideal, then J^p=0.
Exact mathematical passage
J^p=0.
J^r_{\gamma\mid\alpha_1,\ldots,\alpha_r}Use in source · 1 passage
Search aliases: J^r_γmidα_1,…,α_r gamma mid alpha ldots alpha
Use in source · 03_nonlinear_coordinates.tex, line 260
[Jet norm bound] Give every packet range its inherited subspace norm. The restricted jet satisfies the sharper estimate J^r_γmidα_1,…,α_rF(x) ≤ Q_γ D^rF(x) . Its ambient extension inserts P_α_j in the jth slot and has norm at most Q_γ D^rF(x) ∏_j P_α_j . Consequently the operator norm of the restricted packet jet also satisfies J^r_γmidα_1,…,α_rF(x) ≤ Q_γ D^rF(x) ∏_j=1^r P_α_j . When all packets are orthogonal projections on Hilbert spaces, every nonzero projector has norm one and the bound redu…
Exact mathematical passage
\left\|J^r_{\gamma\mid\alpha_1,\ldots,\alpha_r}F(x)\right\| \le \|Q_\gamma\|\,\|D^rF(x)\|.
J^r_{\delta\mid\alpha_1,\ldots,\alpha_r}Use in source · 1 passage
Search aliases: J^r_δmidα_1,…,α_r delta mid alpha ldots alpha
Use in source · 03_nonlinear_coordinates.tex, line 391
[Structural pruning] For a map H with fixed input and output packet systems, let supp_r H(x) be the set of external profiles (δ;α_1,…,α_r) for which the restricted jet J^r_δmidα_1,…,α_rH(x) is nonzero. A profile belongs to the admissible composition set if it admits a partition π={B_1,…,B_k} and intermediate labels γ_1,…,γ_k such that the outer jet of F and every inner block jet of G in [source reference] are nonzero. Then supp_n(F∘ G)(x) is contained in this admissible set. In particular, a sum…
Exact mathematical passage
J^r_{\delta\mid\alpha_1,\ldots,\alpha_r}H(x)
J^s_{\gamma\mid\alpha_{i_1},\ldots,\alpha_{i_s}}Use in source · 1 passage
Search aliases: J^s_γmidα_i_1,…,α_i_s gamma mid alpha ldots alpha
Use in source · 03_nonlinear_coordinates.tex, line 340
For a block B={i_1<⋯<i_s} and an intermediate packet γ, write G_γ,B(xmid h) = J^s_γmidα_i_1,…,α_i_sG(x) [h_i_1,…,h_i_s].
Exact mathematical passage
G_{\gamma,B}(x\mid h) = J^s_{\gamma\mid\alpha_{i_1},\ldots,\alpha_{i_s}}G(x) [h_{i_1},\ldots,h_{i_s}].
J^X_{a\leftarrow a}Use in source · 1 passage
Search aliases: J^X_a← a leftarrow
Use in source · 01_finite_coordinates.tex, line 486
[Identity, inverse, and cocycle for canonical transport] For any exact packet systems a,b,c on X, J^X_a← a=id, (J^X_b← a)^-1=J^X_a← b, J^X_c← bJ^X_b← a=J^X_c← a. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates_self; Komorebi.PaperII.NestedRefinement.coarsenCoordinates_refineCoordinates; Komorebi.PaperII.NestedRefinement.refineCoordinates_coarsenCoordinates; Komorebi.PaperII.NestedRefinement.refineCoordinates_comp; Komorebi.PaperII.NestedRefinement.coordinateRefinementEquiv_tra…
Exact mathematical passage
J^X_{a\leftarrow a}=\id, \qquad \left(J^X_{b\leftarrow a}\right)^{-1}=J^X_{a\leftarrow b}, \qquad J^X_{c\leftarrow b}J^X_{b\leftarrow a}=J^X_{c\leftarrow a}.
J^X_{a\leftarrow b}Use in source · 2 passages
Search aliases: J^X_a← b leftarrow
Use in source · 01_finite_coordinates.tex, line 486
[Identity, inverse, and cocycle for canonical transport] For any exact packet systems a,b,c on X, J^X_a← a=id, (J^X_b← a)^-1=J^X_a← b, J^X_c← bJ^X_b← a=J^X_c← a. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates_self; Komorebi.PaperII.NestedRefinement.coarsenCoordinates_refineCoordinates; Komorebi.PaperII.NestedRefinement.refineCoordinates_coarsenCoordinates; Komorebi.PaperII.NestedRefinement.refineCoordinates_comp; Komorebi.PaperII.NestedRefinement.coordinateRefinementEquiv_tra…
Exact mathematical passage
J^X_{a\leftarrow a}=\id, \qquad \left(J^X_{b\leftarrow a}\right)^{-1}=J^X_{a\leftarrow b}, \qquad J^X_{c\leftarrow b}J^X_{b\leftarrow a}=J^X_{c\leftarrow a}.
J^X_{a\leftarrow c}Use in source · 1 passage
Search aliases: J^X_a← c leftarrow
Use in source · 01_finite_coordinates.tex, line 537
[Canonical triangle holonomy is trivial] For every triple a,b,c, J^X_a← cJ^X_c← bJ^X_b← a=id. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates_comp; Komorebi.PaperII.NestedRefinement.refineCoordinates_self
Exact mathematical passage
J^X_{a\leftarrow c}J^X_{c\leftarrow b}J^X_{b\leftarrow a}=\id.
J^X_{c\leftarrow a}Use in source · 2 passages
Search aliases: J^X_c← a leftarrow
Use in source · 01_finite_coordinates.tex, line 486
[Identity, inverse, and cocycle for canonical transport] For any exact packet systems a,b,c on X, J^X_a← a=id, (J^X_b← a)^-1=J^X_a← b, J^X_c← bJ^X_b← a=J^X_c← a. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates_self; Komorebi.PaperII.NestedRefinement.coarsenCoordinates_refineCoordinates; Komorebi.PaperII.NestedRefinement.refineCoordinates_coarsenCoordinates; Komorebi.PaperII.NestedRefinement.refineCoordinates_comp; Komorebi.PaperII.NestedRefinement.coordinateRefinementEquiv_tra…
Exact mathematical passage
J^X_{a\leftarrow a}=\id, \qquad \left(J^X_{b\leftarrow a}\right)^{-1}=J^X_{a\leftarrow b}, \qquad J^X_{c\leftarrow b}J^X_{b\leftarrow a}=J^X_{c\leftarrow a}.
J^X_{c\leftarrow b}Use in source · 3 passages
Search aliases: J^X_c← b leftarrow
Use in source · 01_finite_coordinates.tex, line 486
[Identity, inverse, and cocycle for canonical transport] For any exact packet systems a,b,c on X, J^X_a← a=id, (J^X_b← a)^-1=J^X_a← b, J^X_c← bJ^X_b← a=J^X_c← a. statusleanpriorKomorebi.PaperII.NestedRefinement.refineCoordinates_self; Komorebi.PaperII.NestedRefinement.coarsenCoordinates_refineCoordinates; Komorebi.PaperII.NestedRefinement.refineCoordinates_coarsenCoordinates; Komorebi.PaperII.NestedRefinement.refineCoordinates_comp; Komorebi.PaperII.NestedRefinement.coordinateRefinementEquiv_tra…
Exact mathematical passage
J^X_{a\leftarrow a}=\id, \qquad \left(J^X_{b\leftarrow a}\right)^{-1}=J^X_{a\leftarrow b}, \qquad J^X_{c\leftarrow b}J^X_{b\leftarrow a}=J^X_{c\leftarrow a}.
J^X_{P_b,P_a}Use in source · 1 passage
Search aliases: J^X_P_b,P_a
Use in source · 02_coordinates.tex, line 193
The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_Q_b,Q_a widehat T_Q_a,P_a = widehat T_Q_b,P_b J^X_P_b,P_a. The identity, cocycle, composition, inverse, and naturality laws are all proved in Lean. Th…
Exact mathematical passage
\boxed{ J^Y_{Q_b,Q_a}\,\widehat T_{Q_a,P_a} = \widehat T_{Q_b,P_b}\,J^X_{P_b,P_a}. }
J^Y_{Q_b,Q_a}Use in source · 1 passage
Search aliases: J^Y_Q_b,Q_a
Use in source · 02_coordinates.tex, line 193
The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_Q_b,Q_a widehat T_Q_a,P_a = widehat T_Q_b,P_b J^X_P_b,P_a. The identity, cocycle, composition, inverse, and naturality laws are all proved in Lean. Th…
Exact mathematical passage
\boxed{ J^Y_{Q_b,Q_a}\,\widehat T_{Q_a,P_a} = \widehat T_{Q_b,P_b}\,J^X_{P_b,P_a}. }
J^Y_{s\leftarrow r}Use in source · 3 passages
Search aliases: J^Y_s← r leftarrow
Use in source · 01_finite_coordinates.tex, line 520
[Canonical coordinate naturality] The exact coordinate representation preserves typed composition and satisfies J^Y_s← rwidehat F_r← a = widehat F_s← bJ^X_b← a. statusleanpriorKomorebi.PaperII.NestedRefinement.coordinateOperator_comp; Komorebi.PaperII.NestedRefinement.coordinateOperator_naturality
Exact mathematical passage
J^Y_{s\leftarrow r}\widehat F_{r\leftarrow a} = \widehat F_{s\leftarrow b}J^X_{b\leftarrow a}.
J_{(a_0)}Use in source · 1 passage
Search aliases: J_(a_0)
Use in source · 01_finite_coordinates.tex, line 410
[Every finite canonical path telescopes] Let p=(a_0,a_1,…,a_n) be any finite vertex list and put J_p =J_a_n← a_n-1⋯ J_a_1← a_0, J_(a_0)=id. Then J_p=J_a_n← a_0. In particular, if a_n=a_0, then J_p=id. statusleanthreeKomorebi.PaperIII.Transport.pathEndpoint; Komorebi.PaperIII.Transport.exactPathTransport; Komorebi.PaperIII.Transport.exactPathTransport_eq_endpoint; Komorebi.PaperIII.Transport.exactPathTransport_loop
Exact mathematical passage
J_p =J_{a_n\leftarrow a_{n-1}}\cdots J_{a_1\leftarrow a_0}, \qquad J_{(a_0)}=\id.
j_1Use in source · 4 passages
Search aliases: j_1
Use in source · 01_finite_core.tex, line 239
Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.
Exact mathematical passage
P_{\mathbf j}^{\mathrm{fine}} = P^{(1)}_{j_1}\otimes\cdots\otimes P^{(m)}_{j_m}.
j_2Use in source · 1 passage
Search aliases: j_2
Use in source · Paper_V_Traditional_Final.tex, line 1236
[Dyadic--odd packet factorization] After identifying the residue labels by the Chinese remainder map, the phase projector factors as P_N,j=P_2^s,j_2P_M,j_o.
Exact mathematical passage
\boxed{ P_{N,j}=P_{2^s,j_2}P_{M,j_o}. }
J_3Use in source · 8 passages
Search aliases: J_3
Use in source · visual_atlas.tex, line 8
Use the same space of dimension six throughout the first comparison: X= Q^6, N=J_3(0)⊕ J_3(0), s=e_0+f_0. Here each J_3(0) moves a basis vector forward along a chain of three basis vectors. The projectors P_e,P_f select the two chains. Their formulas and the proofs of the minimum seed counts appear on pagerefV3 and pagerefV4.
Exact mathematical passage
X=\QQ^6,\qquad N=J_3(0)\oplus J_3(0),\qquad s=e_0+f_0.
J_3^3Use in source · 1 passage
Search aliases: J_3^3
Use in source · atlas-pages.tex, line 85
vpage5One signed edge determines the cycleDIFFERENTIAL SYSTEMChapter IV: Theorem 5.3, Theorem 6.4, Corollary 6.5, Corollary 6.6. Let J_3e_0=e_1, J_3e_1=e_2, and J_3e_2=-e_0. A complete turn changes sign, so J_3^3=-I. The signed sections are the coordinates of e^zJ_3e_0. vfigsigned_shift Here D_j inside the matrix means D_3,j(z). For example, D_3,0=1-fracz^33!+fracz^66!-⋯, D_3,1=z-fracz^44!+⋯. Rows are output coordinates and columns are input coordinates. The upper-triangular minus signs record c…
Exact mathematical passage
J_3^3=-I
J_4Use in source · 1 passage
Search aliases: J_4
Use in source · Paper_IV_Traditional_Final.tex, line 780
The signed shift is a finite carrier for the differential cycle. At order four, three basis steps have coefficient one and the final step has coefficient minus one. Four steps return the vector with reversed sign. The matrix exponential packages every finite path length in one analytic expression. modelfiguresigned_cycleThe exact signed shift J_4. Source columns and target rows describe the same action as the cycle on the left. Direct integer multiplication verifies J_4^4=-I.fig:signed-cycle
Exact mathematical passage
J_4
J_{4,2}Use in source · 2 passages
Search aliases: J_4,2
Use in source · visual_atlas.tex, line 113
The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not merely reuse the same numerical matrix under a new ordering.
Exact mathematical passage
\widehat T_N=R_NTR_N^{-1},\qquad J_{4,2}=R_4R_2^{-1}.
J_4^4Use in source · 1 passage
Search aliases: J_4^4
Use in source · Paper_IV_Traditional_Final.tex, line 780
steps have coefficient one and the final step has coefficient minus one. Four steps return the vector with reversed sign. The matrix exponential packages every finite path length in one analytic expression. modelfiguresigned_cycleThe exact signed shift J_4. Source columns and target rows describe the same action as the cycle on the left. Direct integer multiplication verifies J_4^4=-I.fig:signed-cycle
Exact mathematical passage
J_4^4=-I
J_{8,2}Use in source · 1 passage
Search aliases: J_8,2
Use in source · visual_atlas.tex, line 117
The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not merely reuse the same numerical matrix under a new ordering.
Exact mathematical passage
J_{4,2}\widehat T_2=\widehat T_4J_{4,2},\qquad J_{8,4}J_{4,2}=J_{8,2}.
J_{8,4}Use in source · 1 passage
Search aliases: J_8,4
Use in source · visual_atlas.tex, line 117
The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not merely reuse the same numerical matrix under a new ordering.
Exact mathematical passage
J_{4,2}\widehat T_2=\widehat T_4J_{4,2},\qquad J_{8,4}J_{4,2}=J_{8,2}.
J_aUse in source · 3 passages
Search aliases: J_a
Use in source · continuation_09.tex, line 86
At depth N≥1, suppose F(a)=q^Nr and a desired next integer parameter target satisfies t'-free(a)=q^Nw, with w∈ Z^k. Surjectivity supplies a solved digit u∈ S_q^m satisfying J_a u=-bar r-K_abar w. Choose an integer representative widetilde u, and set a'=a+q^N(widetilde u,w). The parameter quotient is the supplied integer w. Therefore free(a')=t' exactly. The checked polynomial Taylor identity gives F(a')≡ F(a)+q^NDF(a)(widetilde u,w) ≡0pmodq^N+1. Its quadratic remainder is divisible by q^2N, and …
Exact mathematical passage
J_a u=-\bar r-K_a\bar w.
j_AUse in source · 3 passages
Search aliases: j_A
Use in source · 02_coordinates.tex, line 500
[Coprime factorization] Let A and B be coprime positive integers. For each j modulo AB, write j_A=jpmod A, j_B=jpmod B. Then P_AB,j=P_A,j_AP_B,j_B. Every pair (j_A,j_B) occurs for one residue j modulo AB.
Exact mathematical passage
j_A=j\pmod A, \qquad j_B=j\pmod B.
J_{a,a}Use in source · 1 passage
Search aliases: J_a,a
Use in source · 02_coordinates.tex, line 177
Typed coordinate naturality. These coordinate changes are coherent through every finite tower. If three complete packet systems are denoted by a,b,c, then J_c,bJ_b,a=J_c,a, J_a,a=I, J_b,a^-1=J_a,b. These laws hold for arbitrary exact packet systems representing the same ambient module. The parent maps supply the component formulas linking children to their parents. Coordinate coherence itself does not require them.
Exact mathematical passage
J_{c,b}J_{b,a}=J_{c,a}, \qquad J_{a,a}=I, \qquad J_{b,a}^{-1}=J_{a,b}.
J_{a,b}Use in source · 1 passage
Search aliases: J_a,b
Use in source · 02_coordinates.tex, line 177
Typed coordinate naturality. These coordinate changes are coherent through every finite tower. If three complete packet systems are denoted by a,b,c, then J_c,bJ_b,a=J_c,a, J_a,a=I, J_b,a^-1=J_a,b. These laws hold for arbitrary exact packet systems representing the same ambient module. The parent maps supply the component formulas linking children to their parents. Coordinate coherence itself does not require them.
Exact mathematical passage
J_{c,b}J_{b,a}=J_{c,a}, \qquad J_{a,a}=I, \qquad J_{b,a}^{-1}=J_{a,b}.
J_{a_1\leftarrow a_0}Use in source · 1 passage
Search aliases: J_a_1← a_0 leftarrow
Use in source · 01_finite_coordinates.tex, line 410
[Every finite canonical path telescopes] Let p=(a_0,a_1,…,a_n) be any finite vertex list and put J_p =J_a_n← a_n-1⋯ J_a_1← a_0, J_(a_0)=id. Then J_p=J_a_n← a_0. In particular, if a_n=a_0, then J_p=id. statusleanthreeKomorebi.PaperIII.Transport.pathEndpoint; Komorebi.PaperIII.Transport.exactPathTransport; Komorebi.PaperIII.Transport.exactPathTransport_eq_endpoint; Komorebi.PaperIII.Transport.exactPathTransport_loop
Exact mathematical passage
J_p =J_{a_n\leftarrow a_{n-1}}\cdots J_{a_1\leftarrow a_0}, \qquad J_{(a_0)}=\id.
J_{a_n\leftarrow a_0}Use in source · 1 passage
Search aliases: J_a_n← a_0 leftarrow
Use in source · 01_finite_coordinates.tex, line 417
[Every finite canonical path telescopes] Let p=(a_0,a_1,…,a_n) be any finite vertex list and put J_p =J_a_n← a_n-1⋯ J_a_1← a_0, J_(a_0)=id. Then J_p=J_a_n← a_0. In particular, if a_n=a_0, then J_p=id. statusleanthreeKomorebi.PaperIII.Transport.pathEndpoint; Komorebi.PaperIII.Transport.exactPathTransport; Komorebi.PaperIII.Transport.exactPathTransport_eq_endpoint; Komorebi.PaperIII.Transport.exactPathTransport_loop
Exact mathematical passage
J_p=J_{a_n\leftarrow a_0}.
J_{a_n\leftarrow a_{n-1}}Use in source · 1 passage
Search aliases: J_a_n← a_n-1 leftarrow
Use in source · 01_finite_coordinates.tex, line 410
[Every finite canonical path telescopes] Let p=(a_0,a_1,…,a_n) be any finite vertex list and put J_p =J_a_n← a_n-1⋯ J_a_1← a_0, J_(a_0)=id. Then J_p=J_a_n← a_0. In particular, if a_n=a_0, then J_p=id. statusleanthreeKomorebi.PaperIII.Transport.pathEndpoint; Komorebi.PaperIII.Transport.exactPathTransport; Komorebi.PaperIII.Transport.exactPathTransport_eq_endpoint; Komorebi.PaperIII.Transport.exactPathTransport_loop
Exact mathematical passage
J_p =J_{a_n\leftarrow a_{n-1}}\cdots J_{a_1\leftarrow a_0}, \qquad J_{(a_0)}=\id.
J_{a\leftarrow a}Use in source · 1 passage
Search aliases: J_a← a leftarrow
Use in source · 01_finite_coordinates.tex, line 389
[Exact frame transport is coherent] For every a,b,c, J_a← a=id, J_a← bJ_b← a=id, J_c← bJ_b← a=J_c← a, and therefore J_a← cJ_c← bJ_b← a=id. statusleanthreeKomorebi.PaperIII.Transport.exactTransport_self; Komorebi.PaperIII.Transport.exactTransport_roundTrip; Komorebi.PaperIII.Transport.exactTransport_cocycle; Komorebi.PaperIII.Transport.exactTransport_triangleLoop
Exact mathematical passage
J_{a\leftarrow a}=\id,\qquad J_{a\leftarrow b}J_{b\leftarrow a}=\id,\qquad J_{c\leftarrow b}J_{b\leftarrow a}=J_{c\leftarrow a},
J_{a\leftarrow b}Use in source · 1 passage
Search aliases: J_a← b leftarrow
Use in source · 01_finite_coordinates.tex, line 389
[Exact frame transport is coherent] For every a,b,c, J_a← a=id, J_a← bJ_b← a=id, J_c← bJ_b← a=J_c← a, and therefore J_a← cJ_c← bJ_b← a=id. statusleanthreeKomorebi.PaperIII.Transport.exactTransport_self; Komorebi.PaperIII.Transport.exactTransport_roundTrip; Komorebi.PaperIII.Transport.exactTransport_cocycle; Komorebi.PaperIII.Transport.exactTransport_triangleLoop
Exact mathematical passage
J_{a\leftarrow a}=\id,\qquad J_{a\leftarrow b}J_{b\leftarrow a}=\id,\qquad J_{c\leftarrow b}J_{b\leftarrow a}=J_{c\leftarrow a},
J_{a\leftarrow c}Use in source · 1 passage
Search aliases: J_a← c leftarrow
Use in source · 01_finite_coordinates.tex, line 395
[Exact frame transport is coherent] For every a,b,c, J_a← a=id, J_a← bJ_b← a=id, J_c← bJ_b← a=J_c← a, and therefore J_a← cJ_c← bJ_b← a=id. statusleanthreeKomorebi.PaperIII.Transport.exactTransport_self; Komorebi.PaperIII.Transport.exactTransport_roundTrip; Komorebi.PaperIII.Transport.exactTransport_cocycle; Komorebi.PaperIII.Transport.exactTransport_triangleLoop
Exact mathematical passage
J_{a\leftarrow c}J_{c\leftarrow b}J_{b\leftarrow a}=\id.
J_{a\leftarrow o}Use in source · 2 passages
Search aliases: J_a← o leftarrow
Use in source · 02_connections_obstructions.tex, line 366
Reversibility gives G_a^-1=J_a← oΓ_o← a. Therefore G_bJ_b← aG_a^-1 · = Γ_b← o J_o← bJ_b← a J_a← o Γ_o← a
Exact mathematical passage
G_a^{-1}=J_{a\leftarrow o}\Gamma_{o\leftarrow a}
j_BUse in source · 3 passages
Search aliases: j_B
Use in source · 02_coordinates.tex, line 500
[Coprime factorization] Let A and B be coprime positive integers. For each j modulo AB, write j_A=jpmod A, j_B=jpmod B. Then P_AB,j=P_A,j_AP_B,j_B. Every pair (j_A,j_B) occurs for one residue j modulo AB.
Exact mathematical passage
j_A=j\pmod A, \qquad j_B=j\pmod B.
J_{b,a}Use in source · 1 passage
Search aliases: J_b,a
Use in source · 02_coordinates.tex, line 177
Typed coordinate naturality. These coordinate changes are coherent through every finite tower. If three complete packet systems are denoted by a,b,c, then J_c,bJ_b,a=J_c,a, J_a,a=I, J_b,a^-1=J_a,b. These laws hold for arbitrary exact packet systems representing the same ambient module. The parent maps supply the component formulas linking children to their parents. Coordinate coherence itself does not require them.
Exact mathematical passage
J_{c,b}J_{b,a}=J_{c,a}, \qquad J_{a,a}=I, \qquad J_{b,a}^{-1}=J_{a,b}.
J_{b,a}^{-1}Use in source · 1 passage
Search aliases: J_b,a^-1
Use in source · 02_coordinates.tex, line 177
Typed coordinate naturality. These coordinate changes are coherent through every finite tower. If three complete packet systems are denoted by a,b,c, then J_c,bJ_b,a=J_c,a, J_a,a=I, J_b,a^-1=J_a,b. These laws hold for arbitrary exact packet systems representing the same ambient module. The parent maps supply the component formulas linking children to their parents. Coordinate coherence itself does not require them.
Exact mathematical passage
J_{c,b}J_{b,a}=J_{c,a}, \qquad J_{a,a}=I, \qquad J_{b,a}^{-1}=J_{a,b}.
J_{c,a}Use in source · 1 passage
Search aliases: J_c,a
Use in source · 02_coordinates.tex, line 177
Typed coordinate naturality. These coordinate changes are coherent through every finite tower. If three complete packet systems are denoted by a,b,c, then J_c,bJ_b,a=J_c,a, J_a,a=I, J_b,a^-1=J_a,b. These laws hold for arbitrary exact packet systems representing the same ambient module. The parent maps supply the component formulas linking children to their parents. Coordinate coherence itself does not require them.
Exact mathematical passage
J_{c,b}J_{b,a}=J_{c,a}, \qquad J_{a,a}=I, \qquad J_{b,a}^{-1}=J_{a,b}.
J_{c,b}Use in source · 1 passage
Search aliases: J_c,b
Use in source · 02_coordinates.tex, line 177
Typed coordinate naturality. These coordinate changes are coherent through every finite tower. If three complete packet systems are denoted by a,b,c, then J_c,bJ_b,a=J_c,a, J_a,a=I, J_b,a^-1=J_a,b. These laws hold for arbitrary exact packet systems representing the same ambient module. The parent maps supply the component formulas linking children to their parents. Coordinate coherence itself does not require them.
Exact mathematical passage
J_{c,b}J_{b,a}=J_{c,a}, \qquad J_{a,a}=I, \qquad J_{b,a}^{-1}=J_{a,b}.
J_{c\leftarrow a}Use in source · 1 passage
Search aliases: J_c← a leftarrow
Use in source · 01_finite_coordinates.tex, line 389
[Exact frame transport is coherent] For every a,b,c, J_a← a=id, J_a← bJ_b← a=id, J_c← bJ_b← a=J_c← a, and therefore J_a← cJ_c← bJ_b← a=id. statusleanthreeKomorebi.PaperIII.Transport.exactTransport_self; Komorebi.PaperIII.Transport.exactTransport_roundTrip; Komorebi.PaperIII.Transport.exactTransport_cocycle; Komorebi.PaperIII.Transport.exactTransport_triangleLoop
Exact mathematical passage
J_{a\leftarrow a}=\id,\qquad J_{a\leftarrow b}J_{b\leftarrow a}=\id,\qquad J_{c\leftarrow b}J_{b\leftarrow a}=J_{c\leftarrow a},
J_{c\leftarrow b}Use in source · 2 passages
Search aliases: J_c← b leftarrow
Use in source · 01_finite_coordinates.tex, line 389
[Exact frame transport is coherent] For every a,b,c, J_a← a=id, J_a← bJ_b← a=id, J_c← bJ_b← a=J_c← a, and therefore J_a← cJ_c← bJ_b← a=id. statusleanthreeKomorebi.PaperIII.Transport.exactTransport_self; Komorebi.PaperIII.Transport.exactTransport_roundTrip; Komorebi.PaperIII.Transport.exactTransport_cocycle; Komorebi.PaperIII.Transport.exactTransport_triangleLoop
Exact mathematical passage
J_{a\leftarrow a}=\id,\qquad J_{a\leftarrow b}J_{b\leftarrow a}=\id,\qquad J_{c\leftarrow b}J_{b\leftarrow a}=J_{c\leftarrow a},
j_fUse in source · 4 passages
Search aliases: j_f
Use in source · Paper_V_Traditional_Final.tex, line 2546
_packet_decomposition Let 0=τ_0<τ_1<⋯<τ_Q<τ_Q+1=1. Suppose the restriction of f to every interval (τ_q,τ_q+1) belongs to W^r,p, and choose the one-dimensional Sobolev representatives whose derivatives through order r-1 have one-sided traces at the endpoints. Define a_q,m = jumpf^(m)_τ_q = f^(m)(τ_q+)-f^(m)(τ_q-) and j_f(t) = ∑_q=1^Q∑_m=0^r-1a_q,mϕ_m,τ_q(t). Then s_f=f-j_f has matching one-sided derivatives through order r-1 at every breakpoint and belongs to W^r,p(0,1). Its weak derivative of or…
Exact mathematical passage
j_f(t) = \sum_{q=1}^{Q}\sum_{m=0}^{r-1}a_{q,m}\phi_{m,\tau_q}(t).
J_{F,\bar x}Use in source · 1 passage
Search aliases: J_F,bar x bar
Use in source · 50_open_questions.tex, line 53
The original conjecture asks for a canonical finite obstruction packet whose vanishing is necessary and sufficient for a one-level lift in a finitely presented arithmetic seed system. For a fixed finite polynomial presentation, the chapter supplies the precise obstruction o_F,bar x=[r]∈ coker (J_F,bar x: F_p^d⟶ F_p^e), where r is the residual digit. We have proved independence from the chosen integer representative. Invariance under a change of equations or presentation requires more.
Exact mathematical passage
o_{F,\bar x}=[r]\in \coker\!\left(J_{F,\bar x}:\F_p^d\longrightarrow\F_p^e\right),
J_i^1Use in source · 1 passage
Search aliases: J_i^1
Use in source · odd_synthesis.tex, line 141
Choose a labeled symplectic embedding V_src→ S. For each original full character form P_i=( C^×× J_i^1)/ {(θ_i(h)^-1,h):h∈ H_i^1}. The balanced product and the corresponding host restriction have the same alternating commutator. The center-fixing extension isomorphism, obtained with matched Darboux lifts and complex roots, therefore preserves every supplied value θ_i(h), including the shallow values.
Exact mathematical passage
\mathcal P_i=(\mathbb C^\times\times J_i^1)/ \{(\theta_i(h)^{-1},h):h\in H_i^1\}.
J_{ij}Use in source · 1 passage
Search aliases: J_ij
Use in source · 71_native.tex, line 20
e_i=F_i(x)/p^Lpmod p, J_ij=frac∂ F_i∂ X_j(x)pmod p.
Exact mathematical passage
e_i=F_i(x)/p^L\pmod p,\qquad J_{ij}=\frac{\partial F_i}{\partial X_j}(x)\pmod p.
j_KUse in source · 4 passages
Search aliases: j_K
Use in source · Paper_X_Block27_Amended.tex, line 2411
Actual quadratic local conditions and completion. Continue with the fixed packet P_q above. For a characteristic-zero field K, let j_K:G_K⟶ G_ Q be the actual field-induced homomorphism between absolute Galois groups. It includes a chosen embedding of algebraic closures. Let i_K be the image of the chosen square root i under that embedding. Lean proves compatibility with the Galois actions and hence (q∘ j_K)(σ)=1 Longleftrightarrow σ(i_K)=i_K. The fixed-field theorem for the algebraic closure th…
Exact mathematical passage
j_K:G_K\longrightarrow G_{\Q}
j_mUse in source · 4 passages
Search aliases: j_m
Use in source · 01_finite_core.tex, line 239
Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.
Exact mathematical passage
P_{\mathbf j}^{\mathrm{fine}} = P^{(1)}_{j_1}\otimes\cdots\otimes P^{(m)}_{j_m}.
J_N^{N-1}Use in source · 1 passage
Search aliases: J_N^N-1
Use in source · Paper_IV_Traditional_Final.tex, line 861
The vectors e_0,J_Ne_0,…,J_N^N-1e_0 are the standard basis. Thus e_0 is cyclic for J_N, so the minimal polynomial of J_N has degree N. Since J_N^N=-I, that minimal polynomial divides x^N+1 and therefore equals x^N+1. The characteristic polynomial has the same degree and is divisible by the minimal polynomial, so it too equals x^N+1. This polynomial has distinct roots. Consequently J_N is diagonalizable and its spectrum consists of all N roots of -1, each once. Hence tr(e^zJ_N) = ∑_η^N=-1e^η z = …
Exact mathematical passage
e_0,J_Ne_0,\ldots,J_N^{N-1}e_0
J_N^{Nn+d_{pq}}Use in source · 1 passage
Search aliases: J_N^Nn+d_pq
Use in source · Paper_IV_Traditional_Final.tex, line 902
For every nonnegative integer n, J_N^Nn+d_pqe_q = (-1)^nε_pqe_p. Only powers whose exponent is congruent to d_pq modulo N contribute to the (p,q) entry of the exponential. Therefore (e^zJ_N)_pq = ε_pq ∑_n≥0(-1)^n fracz^Nn+d_pq(Nn+d_pq)!. This is the stated formula.
Exact mathematical passage
J_N^{Nn+d_{pq}}e_q = (-1)^n\varepsilon_{pq}e_p.
j_oUse in source · 1 passage
Search aliases: j_o
Use in source · Paper_V_Traditional_Final.tex, line 1236
[Dyadic--odd packet factorization] After identifying the residue labels by the Chinese remainder map, the phase projector factors as P_N,j=P_2^s,j_2P_M,j_o.
Exact mathematical passage
\boxed{ P_{N,j}=P_{2^s,j_2}P_{M,j_o}. }
J_{o\leftarrow a}Use in source · 2 passages
Search aliases: J_o← a leftarrow
Use in source · 02_connections_obstructions.tex, line 351
[Flat connections are gauges of the canonical pair transport] Suppose the C_a are exact packet coordinate modules for one ambient module, so the canonical J_b← a is available. If Γ is a flat normalized reversible pair connection, choose o∈ B and set G_a=Γ_a← oJ_o← a. Then every pair transport satisfies Γ_b← a =G_bJ_b← aG_a^-1. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.flatGauge; Komorebi.PaperIII.Transport.ReversibleConnection.flat_isGaugeOfCanonical
Exact mathematical passage
G_a=\Gamma_{a\leftarrow o}J_{o\leftarrow a}.
J_{o\leftarrow b}Use in source · 1 passage
Search aliases: J_o← b leftarrow
Use in source · 02_connections_obstructions.tex, line 367
Reversibility gives G_a^-1=J_a← oΓ_o← a. Therefore G_bJ_b← aG_a^-1 · = Γ_b← o J_o← bJ_b← a J_a← o Γ_o← a · = Γ_b← oΓ_o← a · = Γ_b← a. The first reduction uses the canonical cocycle twice. The second uses flatness of Γ.
Exact mathematical passage
\begin{aligned} G_bJ_{b\leftarrow a}G_a^{-1} &= \Gamma_{b\leftarrow o} J_{o\leftarrow b}J_{b\leftarrow a} J_{a\leftarrow o} \Gamma_{o\leftarrow a}\\ &= \Gamma_{b\leftarrow o}\Gamma_{o\leftarrow a}\\ &= \Gamma_{b\leftarrow a}. \end{aligned}
J_pUse in source · 3 passages
Search aliases: J_p
Use in source · 01_finite_coordinates.tex, line 410
[Every finite canonical path telescopes] Let p=(a_0,a_1,…,a_n) be any finite vertex list and put J_p =J_a_n← a_n-1⋯ J_a_1← a_0, J_(a_0)=id. Then J_p=J_a_n← a_0. In particular, if a_n=a_0, then J_p=id. statusleanthreeKomorebi.PaperIII.Transport.pathEndpoint; Komorebi.PaperIII.Transport.exactPathTransport; Komorebi.PaperIII.Transport.exactPathTransport_eq_endpoint; Komorebi.PaperIII.Transport.exactPathTransport_loop
Exact mathematical passage
J_p =J_{a_n\leftarrow a_{n-1}}\cdots J_{a_1\leftarrow a_0}, \qquad J_{(a_0)}=\id.
J_{\Pcal\leftarrow\Pcal'}Use in source · 1 passage
Search aliases: J_Pcal←Pcal' J_ P← P' Pcal leftarrow Pcal mathcal leftarrow mathcal
Use in source · 03_nonlinear_coordinates.tex, line 176
[Change of packet coordinates] Let P and P' be packet systems on X. Let Q and Q' be packet systems on Y. For any two packet systems U and V on the same space, write J_ U← V = A_ U S_ V. Then F_ Q'← P' = J_ Q'← Q F_ Q← P J_ P← P'.
Exact mathematical passage
F_{\Qcal'\leftarrow\Pcal'} = J_{\Qcal'\leftarrow\Qcal} F_{\Qcal\leftarrow\Pcal} J_{\Pcal\leftarrow\Pcal'}.
j_qUse in source · 4 passages
Search aliases: j_q
Use in source · Paper_V_Traditional_Final.tex, line 682
[Several exact edge packets] If f=s+∑_q=1^Mj_q and every j_q is retained exactly, then f-(Q_ms+∑_qj_q) = s-Q_ms . Curved edges, corners, and different jump amplitudes do not alter the identity. The finite sum j=∑_q=1^M j_q is a single retained packet. Applying the preceding cancellation identity gives the result. The empty family M=0 is included with sum zero.
Exact mathematical passage
f=s+\sum_{q=1}^{M}j_q
J_{\Qcal'\leftarrow\Qcal}Use in source · 1 passage
Search aliases: J_Qcal'←Qcal J_ Q'← Q Qcal leftarrow Qcal mathcal leftarrow mathcal
Use in source · 03_nonlinear_coordinates.tex, line 176
[Change of packet coordinates] Let P and P' be packet systems on X. Let Q and Q' be packet systems on Y. For any two packet systems U and V on the same space, write J_ U← V = A_ U S_ V. Then F_ Q'← P' = J_ Q'← Q F_ Q← P J_ P← P'.
Exact mathematical passage
F_{\Qcal'\leftarrow\Pcal'} = J_{\Qcal'\leftarrow\Qcal} F_{\Qcal\leftarrow\Pcal} J_{\Pcal\leftarrow\Pcal'}.
J_rUse in source · 6 passages
Search aliases: J_r
Use in source · root_scope.tex, line 128
The scalar packet weights have exact valuations λ_r(q)=3q+s_2(r+q)-s_2(r). Put μ_r=min_q≥0λ_r(q), J_r={q:λ_r(q)=μ_r}, G_r=2^-μ_rH_r. Since λ_r(0)=0 and λ_r(q)≥3q+1-s_2(r), every minimizing address satisfies 0≤ q≤lfloorfracs_2(r)-13rfloor. In particular μ_r≤0, and the minimum is found by a finite carry calculation. Reduction of the whole series gives overline G_r(X)=∑_q∈ J_rX^underline q in F_2[X]. Its highest degree is max J_r, with leading coefficient one. Thus the normalization is the actual G…
Exact mathematical passage
\mu_r=\min_{q\ge0}\lambda_r(q),\qquad J_r=\{q:\lambda_r(q)=\mu_r\},\qquad G_r=2^{-\mu_r}H_r.
j_rUse in source · 5 passages
Search aliases: j_r
Use in source · root_scope.tex, line 290
For a general depth r, call j_r=max J_r its true terminal. This definition depends on the actual carry minimum. Rewriting the same integer as another wall expression does not change j_r. The universal simplicity target is for every integer r≥1 and x∈ Z_2, G_r(x)=0 Longrightarrow G_r'(x)≠0. The following are completed restricted results of the later continuations. Their family statements include a proof that the indicated label is the true terminal throughout the displayed ranges.
Exact mathematical passage
j_r=\max J_r
J_{\Rcal\leftarrow\Pcal}Use in source · 1 passage
Search aliases: J_Rcal←Pcal J_ R← P Rcal leftarrow Pcal mathcal leftarrow mathcal
Use in source · 02_coordinates.tex, line 110
[Exact coordinate change and coherence] For any two packet systems on X, C_ P← QJ_ Q← P=I_X^ P, J_ Q← PC_ P← Q=I_X^ Q. For three packet systems P, Q, and R on X, J_ R← QJ_ Q← P = J_ R← P, C_ P← QC_ Q← R = C_ P← R. If Q refines P through ρ, then (J_ Q← Px)_β = Q_β x_ρ(β), (C_ P← Qy)_α = ∑_ρ(β)=αy_β.
Exact mathematical passage
J_{\Rcal\leftarrow\Qcal}J_{\Qcal\leftarrow\Pcal} = J_{\Rcal\leftarrow\Pcal},
J_{\Rcal\leftarrow\Qcal}Use in source · 1 passage
Search aliases: J_Rcal←Qcal J_ R← Q Rcal leftarrow Qcal mathcal leftarrow mathcal
Use in source · 02_coordinates.tex, line 110
[Exact coordinate change and coherence] For any two packet systems on X, C_ P← QJ_ Q← P=I_X^ P, J_ Q← PC_ P← Q=I_X^ Q. For three packet systems P, Q, and R on X, J_ R← QJ_ Q← P = J_ R← P, C_ P← QC_ Q← R = C_ P← R. If Q refines P through ρ, then (J_ Q← Px)_β = Q_β x_ρ(β), (C_ P← Qy)_α = ∑_ρ(β)=αy_β.
Exact mathematical passage
J_{\Rcal\leftarrow\Qcal}J_{\Qcal\leftarrow\Pcal} = J_{\Rcal\leftarrow\Pcal},
J_{\rm keep}Use in source · 1 passage
Search aliases: J_rm keep rm
Use in source · extended_pages.tex, line 53
bottomrule vfig[.45]shared_family With reuse weights (r_1,r_2)=(1,3), the aggregated objectives are J_rm keep=4+2+3·7=27, J_rm split=2+8+3·2=16. The exact shared optimum is split. Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the declared one-pruning problem. headA hand-computable decision threshold Fix r_1=1 and vary r_2=r≥0. Keeping costs 6+7r and splitting costs 10+2r. Splitting …
Exact mathematical passage
J_{\rm keep}=4+2+3\cdot7=27,\qquad J_{\rm split}=2+8+3\cdot2=16.
J_{\rm split}Use in source · 1 passage
Search aliases: J_rm split rm
Use in source · extended_pages.tex, line 53
bottomrule vfig[.45]shared_family With reuse weights (r_1,r_2)=(1,3), the aggregated objectives are J_rm keep=4+2+3·7=27, J_rm split=2+8+3·2=16. The exact shared optimum is split. Operator 1 alone prefers keep on its per-use costs, while operator 2 prefers split. Choosing independently and trying to combine those decisions would not solve the declared one-pruning problem. headA hand-computable decision threshold Fix r_1=1 and vary r_2=r≥0. Keeping costs 6+7r and splitting costs 10+2r. Splitting …
Exact mathematical passage
J_{\rm keep}=4+2+3\cdot7=27,\qquad J_{\rm split}=2+8+3\cdot2=16.
j_sUse in source · 1 passage
Search aliases: j_s
Use in source · 02_strengthened.tex, line 2476
[Reflection positivity is closed under limits] Suppose C_n(j,k) are reflection positive kernels on N, meaning every finite matrix (C_n(j_r,j_s))_r,s is positive semidefinite. If C_n(j,k)⟶ C(j,k) for every pair, then C is reflection positive.
Exact mathematical passage
\bigl(C_n(j_r,j_s)\bigr)_{r,s}
J_tUse in source · 2 passages
Search aliases: J_t
Use in source · 40_regularity.tex, line 174
[Regularity throughout an open parameter set] Let U⊆ K^k be open, and let h be continuous on U. Suppose that for every t∈ U, F(h(t),t)=0, J_t=D_YF(h(t),t)∈GL_e( K). Then h is analytic at every point of U and its strict derivative is Dh(t)=-J_t^-1D_TF(h(t),t) (t∈ U). At each t∈ U, openness turns continuity on U into continuity at t and makes the root equation hold on a neighborhood of t. Apply Theorem~[source reference] at (h(t),t). This is leanKomorebi.PolynomialAnalyticSelection.continuous_sele…
Exact mathematical passage
F(h(t),t)=0, \qquad J_t=D_YF(h(t),t)\in\operatorname{GL}_e(\mathbb K).
J_t^{-1}Use in source · 2 passages
Search aliases: J_t^-1
Use in source · 40_regularity.tex, line 179
[Regularity throughout an open parameter set] Let U⊆ K^k be open, and let h be continuous on U. Suppose that for every t∈ U, F(h(t),t)=0, J_t=D_YF(h(t),t)∈GL_e( K). Then h is analytic at every point of U and its strict derivative is Dh(t)=-J_t^-1D_TF(h(t),t) (t∈ U). At each t∈ U, openness turns continuity on U into continuity at t and makes the root equation hold on a neighborhood of t. Apply Theorem~[source reference] at (h(t),t). This is leanKomorebi.PolynomialAnalyticSelection.continuous_sele…
Exact mathematical passage
Dh(t)=-J_t^{-1}D_TF(h(t),t)\qquad(t\in U).
J_{\Ucal\leftarrow\Vcal}Use in source · 1 passage
Search aliases: J_Ucal←Vcal J_ U← V Ucal leftarrow Vcal mathcal leftarrow mathcal
Use in source · 03_nonlinear_coordinates.tex, line 170
[Change of packet coordinates] Let P and P' be packet systems on X. Let Q and Q' be packet systems on Y. For any two packet systems U and V on the same space, write J_ U← V = A_ U S_ V. Then F_ Q'← P' = J_ Q'← Q F_ Q← P J_ P← P'.
Exact mathematical passage
J_{\Ucal\leftarrow\Vcal} = \Acal_{\Ucal}\Scal_{\Vcal}.
J_{x'}Use in source · 3 passages
Search aliases: J_x'
Use in source · 30_lifting.tex, line 134
[Independence of the integer representative] Under the hypotheses of Theorem~[source reference], let x'=x+q^Ls and write F(x')=q^Lr_x'. Then J_x'=J_x, overliner_x'=overliner_x+J_xoverline s. Consequently the two residual classes coincide in the common quotient S_q^e/im J_x.
Exact mathematical passage
J_{x'}=J_x,\qquad \overline{r_{x'}}=\overline{r_x}+J_x\overline s.
J_{y'}Use in source · 2 passages
Search aliases: J_y'
Use in source · Paper_V_Traditional_Final.tex, line 950
imizer is unique and at least one competing pruning exists, its gap Γ_y = min_Λ≠Λ_y^* (J_y(Λ)-J_y(Λ_y^*)) is attained and strictly positive. If the pruning set is a singleton, its only element minimizes every criterion. No real-valued minimum over an empty competitor set is introduced. item In the case with competitors, if new data y' satisfy sup_Λ∈ P( T) |J_y'(Λ)-J_y(Λ)| <fracΓ_y2, then Λ_y'^*=Λ_y^*. item Every selected minimizer preserves the exact synthesis identity of the preceding theorem.
Exact mathematical passage
\sup_{\Lambda\in\mathfrak P(\mathcal T)} |J_{y'}(\Lambda)-J_y(\Lambda)| <\frac{\Gamma_y}{2},
J_{z(n)}Use in source · 1 passage
Search aliases: J_z(n)
Use in source · continuation_09.tex, line 37
_0, t(n+1)-t(n)∈ q^L+n Z^k, there is a single tower z: N→ Z^m+k with z(0)=x, free(z(n))=t(n), F(z(n))∈ q^L+n Z^e, z(n+1)-z(n)∈ q^L+n Z^m+k. The parameter equality is exact in integers. For a≤ b, z(b)-z(a)∈ q^L+a Z^m+k, DF(z(n))bmod q=DF(x)bmod q. Consequently J_z(n)=J_x at every level. No later surjectivity or per-step solvability is assumed. This theorem includes positive composite bases and q=1.
Exact mathematical passage
J_{z(n)}=J_x
J_{z_n}Use in source · 4 passages
Search aliases: J_z_n
Use in source · 30_lifting.tex, line 338
[Compatible polynomial tower] Let q≥1, L≥1, F∈ Z[X_1,…,X_d]^e, and x∈ Z^d satisfy F(x)∈ q^L Z^e. Suppose J_x:S_q^d→ S_q^e is surjective. There is a single sequence z_n∈ Z^d such that z_0 · =x, · F(z_n) · ∈ q^L+n Z^e, z_n+1-z_n · ∈ q^L+n Z^d, · J_z_n · =J_x. For a≤ b, it also satisfies z_b-z_a∈ q^L+a Z^d.
Exact mathematical passage
z_0&=x,& F(z_n)&\in q^{L+n}\Z^e,\\ z_{n+1}-z_n&\in q^{L+n}\Z^d,& J_{z_n}&=J_x.
J_{z_{n+1}}Use in source · 1 passage
Search aliases: J_z_n+1
Use in source · 30_lifting.tex, line 350
At level N=L+n, divide the residual by q^N. Surjectivity supplies t∈ S_q^d with J_z_nt=-overlineF(z_n)/q^N. Choose an integer representative and apply [source reference] to obtain z_n+1. The new point is congruent to z_n modulo q. Evaluation of each formal partial derivative therefore gives J_z_n+1=J_z_n.
Exact mathematical passage
J_{z_{n+1}}=J_{z_n}
J_\thetaUse in source · 4 passages
Search aliases: J_θ theta
Use in source · Paper_XI_Traditional_Final.tex, line 1150
Let θ be a maximal simple character for GL_n(K), with compact open subgroups H_θ^1⊆ J_θ^1⊆ J_θ. The quotient V_θ=J_θ^1/H_θ^1 is regarded as a finite vector space over F_p, where p is the residue characteristic. Fix a nontrivial additive character ψ_0: F_p→μ_p⊂ C^×, where μ_p is the group of p-th roots of unity. The simple-character commutator takes values in μ_p. Since ψ_0 is injective, the equation ψ_0(h_θ(bar x,bar y)) = θ([x,y]) uniquely defines the F_p-valued commutator pairing h_θ. We use t…
Exact mathematical passage
H_\theta^1\subseteq J_\theta^1\subseteq J_\theta.
J_\theta^1Use in source · 7 passages
Search aliases: J_θ^1 theta
Use in source · Paper_XI_Traditional_Final.tex, line 1150
Let θ be a maximal simple character for GL_n(K), with compact open subgroups H_θ^1⊆ J_θ^1⊆ J_θ. The quotient V_θ=J_θ^1/H_θ^1 is regarded as a finite vector space over F_p, where p is the residue characteristic. Fix a nontrivial additive character ψ_0: F_p→μ_p⊂ C^×, where μ_p is the group of p-th roots of unity. The simple-character commutator takes values in μ_p. Since ψ_0 is injective, the equation ψ_0(h_θ(bar x,bar y)) = θ([x,y]) uniquely defines the F_p-valued commutator pairing h_θ. We use t…
Exact mathematical passage
H_\theta^1\subseteq J_\theta^1\subseteq J_\theta.
J_\lambdaUse in source · 3 passages
Search aliases: J_λ lambda
Use in source · 01_manuscript.tex, line 1529
[Exact Lagrangian packet compiler] Fix λ≥0 and define J_λ( L) = cost( L) + λerr( L). For every node v, set K_λ(v) = κ_v+λϵ_v and S_λ(v) = σ_v + ∑_w∈child(v)F_λ(w). For a terminal node, set F_λ(v)=K_λ(v). For an internal node, the recurrence F_λ(v) = min{K_λ(v),S_λ(v)} computes the exact minimum Lagrangian objective over every admissible pruning of the subtree rooted at v.
Exact mathematical passage
J_\lambda(\Lcal) = \cost(\Lcal) + \lambda\err(\Lcal).
J_{\lambda,u}Use in source · 5 passages
Search aliases: J_λ,u lambda
Use in source · Paper_XII_Traditional_Final.tex, line 1489
Finally, put B_λ= O_λ[x_g,a,b:g∈Γ, a,b∈ D]. For each declared local condition index u, supply an ideal J_λ,u⊆ B_λ. Evaluation at a representation sends x_g,a,b to its (a,b) matrix entry at g. Assume that evaluation at every admissible finite representative kills J_λ,u, with polynomial coefficients reduced modulo I_λ^n. For this class based theorem, the equations must hold in every admissible frame. A condition imposed only in one chosen frame needs a separate compatible frame argument. No arbitr…
Exact mathematical passage
J_{\lambda,u}\subseteq B_\lambda
\JacUse in source · 8 passages
Search aliases: Jac operatorname
Use in source · 01_orientation.tex, line 47
How many seeds are required? · srank_A(M) · For a finitely generated module, pass to M/Jac(A)M. When the quotient algebra is semisimple Artinian, compute its simple multiplicities.
Exact mathematical passage
M/\Jac(A)M
\jumpUse in source · 3 passages
Search aliases: jump
Use in source · Paper_V_Traditional_Final.tex, line 1673
The singular packet results begin with trace data. If a piecewise smooth function has a jump in its derivative of order m at τ_q, then the packet jumpf^(m)_τ_q frac(t-τ_q)_+^mm! removes exactly that trace jump. Subtracting all packets through order r-1 leaves a global W^r,p remainder. The packet coefficients are uniquely determined by the one-sided trace jumps. This gives a finite and explicit singular part for the class covered by [source reference].
Exact mathematical passage
\jump{f^{(m)}}_{\tau_q} \frac{(t-\tau_q)_+^m}{m!}
\mathcal KUse in source · 6 passages
Search aliases: K mathcal
Use in source · atlas-pages.tex, line 38
K(t) · The memory kernel after hidden-state elimination. · It is not the transition matrix K.
Exact mathematical passage
\mathcal K(t)
K'Use in source · 3 passages
Search aliases: K'
Use in source · monograph_atlas.tex, line 442
vpage24A dimension curve is a scale diagnosticTHE EXACT HEAT TRACE ON FOUR SITESChapter VII: Definition II.11.1, Proposition II.11.2. For the Laplacian on a cycle with four sites L=2I-S-S^*, the eigenvalues are 0,2,2,4. The normalized heat trace and its diagnostic of spectral dimension are therefore K(τ)=frac1+2e^-2τ+e^-4τ4 =frac(1+e^-2τ)^24, d_s(τ)=-2τfracK'(τ)K(τ) =frac8τe^2τ+1. No numerical differentiation is needed for this finite example. vfigspectral_dimension headBoth ends reflect the fin…
Exact mathematical passage
K(\tau)=\frac{1+2e^{-2\tau}+e^{-4\tau}}4 =\frac{(1+e^{-2\tau})^2}4, \qquad d_s(\tau)=-2\tau\frac{K'(\tau)}{K(\tau)} =\frac{8\tau}{e^{2\tau}+1}.
K^{\mathsf T}Use in source · 1 passage
Search aliases: K^ T mathsf
Use in source · atlas-pages.tex, line 31
K^ Tf · Expected next observable given the current state. · The same dynamics, with the opposite action on the dual object.
Exact mathematical passage
K^{\mathsf T}f
K^*Use in source · 2 passages
Search aliases: K^*
Use in source · 04_correction_transport.tex, line 837
Differentiating P_α^2=P_α gives P_αdot P_α P_α=0. Differentiating P_β P_α=0 for α≠β gives P_βdot P_α=-dot P_β P_α. Hence K P_β=dot P_β P_β and P_β K=-dot P_β(I-P_β). Their difference is dot P_β. Also, K+K^* = ∑_α(dot P_α P_α+P_αdot P_α) = ∑_αdot P_α =0. endgroup
Exact mathematical passage
K+K^* = \sum_\alpha(\dot P_\alpha P_\alpha+P_\alpha\dot P_\alpha) = \sum_\alpha\dot P_\alpha =0.
K^2Use in source · 4 passages
Search aliases: K^2
Use in source · monograph_atlas.tex, line 454
For the Cartesian product of two identical cycles with four sites, the Laplacian eigenvalues add and the heat traces multiply: K_2=K^2. Consequently d_s,2=2d_s exactly. The product identity determines the second curve exactly from the first.
Exact mathematical passage
K_2=K^2
k^2Use in source · 12 passages
Search aliases: k^2
Use in source · Paper_VI_Traditional_Final.tex, line 5586
:04 For the one-dimensional Dirichlet difference operator, let h=1/M with an integer M≥2, let 1≤ k≤ M-1, and set λ_k,h=4h^-2sin^2(kπ h/2). The retained explicit coefficient below is valid for every discrete mode. [Fractional low mode dispersion] Let 0<α≤1. Then 0 ≤ (kπ)^2α-λ_k,h^α ≤ fracα(4k^2)^α-1(kπ)^412h^2. For every t≥0, abs e^-tλ_k,h^α - e^-t(kπ)^2α ≤ fractα(4k^2)^α-1(kπ)^412h^2.
Exact mathematical passage
0 \leq (k\pi)^{2\alpha}-\lambda_{k,h}^\alpha \leq \frac{\alpha(4k^2)^{\alpha-1}(k\pi)^4}{12}h^2.
k^4Use in source · 6 passages
Search aliases: k^4
Use in source · monograph_atlas.tex, line 464
vpage25Refine the grid while holding the mode fixedAN EXACT ERROR ENCLOSURE, NOT A FITTED SLOPEChapter VII: Theorem II.11.4. For the circle symbol, the continuum value is k^2 and the grid value is λ_k,h=frac4h^2sin^2 (frackh2), h≠0. For a fixed low mode, the paper bounds the error by 0≤ k^2-λ_k,h≤ k^4h^2/12. The spacing condition matters because the displayed expression divides by h^2. vfigdispersion headWhat is certified in the displayed table? Take k=1 and h=1/4,1/8,1/16. For x=h/2∈[0,1], alte…
Exact mathematical passage
0\le k^2-\lambda_{k,h}\le k^4h^2/12
K^dUse in source · 1 passage
Search aliases: K^d
Use in source · continuation_07.tex, line 17
mathbb Q_p, and d,s,M∈ N. Let F,H∈ R[X_1,…,X_d]^d, let a∈ R^d satisfy F(a)=0, and set J=DF(a), G=F+p^M H. Assume J∈GL_d(K), p^sJ^-1∈ M_d(R), M>2s. Then there is exactly one root b of G in a+p^s+1R^d. It satisfies b-a∈ p^M-sR^d, and there is ε∈ p^2M-3sR^d such that, in K^d, b-a=-p^M J^-1H(a)+ε. The inverse in this formula belongs to the original Jacobian DF(a). The theorem constructs the perturbed root from these hypotheses and proves uniqueness throughout the full ball of depth s+1. It does not …
Exact mathematical passage
K^d
K^hUse in source · 1 passage
Search aliases: K^h
Use in source · pages.tex, line 239
dicts possible support but not actual support. Replacing the inclusion supp(AB)⊆supp(A)∘supp(B) by equality would be wrong even in this four-dimensional model. headWhat an acyclic graph does guarantee If the longest strict directed path has h edges, then no term of K^h+1 can survive. Thus K^h+1=0. This is an upper bound on the nilpotence index, not an assertion that K^h≠0. Here the graph has paths of length two, while coefficient cancellation already gives K^2=0.
Exact mathematical passage
K^h\ne0
K^{h+1}Use in source · 4 passages
Search aliases: K^h+1
Use in source · pages.tex, line 239
, K^2=0. The matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the inclusion supp(AB)⊆supp(A)∘supp(B) by equality would be wrong even in this four-dimensional model. headWhat an acyclic graph does guarantee If the longest strict directed path has h edges, then no term of K^h+1 can survive. Thus K^h+1=0. This is an upper bound on the nilpotence index, not an assertion that K^h≠0. Here the graph has path…
Exact mathematical passage
K^{h+1}
K^jUse in source · 4 passages
Search aliases: K^j
Use in source · Paper_VI_Traditional_Final.tex, line 613
[Finite-horizon defect formula] For arbitrary Δ=CK-bar KC, CK^n-bar K^nC = ∑_j=0^n-1bar K^n-1-jΔ K^j. If K and bar K are contractions in the chosen norm, then normCK^n-bar K^nC≤ nnormΔ.
Exact mathematical passage
\boxed{ CK^n-\bar K^nC = \sum_{j=0}^{n-1}\bar K^{n-1-j}\Delta K^j. }
K^{j+1}Use in source · 2 passages
Search aliases: K^j+1
Use in source · Paper_VI_Traditional_Final.tex, line 628
The identity follows by telescoping: bar K^n-1-jCK^j+1-bar K^n-jCK^j =bar K^n-1-j(CK-bar KC)K^j. Summing from j=0 to n-1 leaves CK^n-bar K^nC. The norm bound follows termwise.
Exact mathematical passage
\bar K^{n-1-j}CK^{j+1}-\bar K^{n-j}CK^j =\bar K^{n-1-j}(CK-\bar KC)K^j.
K^nUse in source · 12 passages
Search aliases: K^n
Use in source · Paper_VI_Traditional_Final.tex, line 598
[Exact lumpability] If Δ=0, then CK^n=bar K^nC for every n≥0. Thus fine evolution followed by coarse reconstruction equals coarse evolution from the beginning.
Exact mathematical passage
\boxed{CK^n=\bar K^nC}
\mathbb K^nUse in source · 1 passage
Search aliases: K^n mathbb
Use in source · 40_regularity.tex, line 28
[Derivative of polynomial evaluation] For P∈ K[X_1,…,X_n], polynomial evaluation is analytic at every point. At a∈ K^n its strict Fr echet derivative is v⟼∑_j=1^n (∂_jP)(a)v_j. For a finite system, its strict derivative is the continuous linear map represented by the matrix of evaluated formal partial derivatives. For the derivative statement, use polynomial induction on constants, addition, and multiplication by a variable. Constants have zero derivative. The sum rule agrees with the additivity…
Exact mathematical passage
a\in\mathbb K^n
K^{n+1}Use in source · 1 passage
Search aliases: K^n+1
Use in source · Paper_VI_Traditional_Final.tex, line 606
The result is immediate for n=0. If CK^n=bar K^nC, then CK^n+1=bar K^nCK=bar K^n+1C.
Exact mathematical passage
CK^{n+1}=\bar K^nCK=\bar K^{n+1}C.
K^TUse in source · 4 passages
Search aliases: K^T
Use in source · Paper_VI_Traditional_Final.tex, line 1443
Let K be a column-stochastic matrix, let V be a potential on the states, and let f be terminal data. Define M_V=diag(e^-V)K^T. The backward Feynman--Kac value after n steps is u_n=M_V^nf.
Exact mathematical passage
M_V=\diag(e^{-V})K^T.
K^\timesUse in source · 15 passages
Search aliases: K^× times
Use in source · 04_arithmetic.tex, line 121
[Evaluated B ezout spectral projector] Let F,G,U,V∈End_K(Λ⊗_ O K) and let r∈ K^×. Suppose UF+VG=rI, FG=0, FG=GF, GU=UG. Put N=VG. Then N^2=rN. Consequently the operator E=r^-1VG is idempotent, acts as the identity on ker F, and acts as zero on ker G. If V and G preserve Λ, then rE=VG∈End_ O(Λ).
Exact mathematical passage
r\in K^\times
K^{\times}Use in source · 1 passage
Search aliases: K^× times
Use in source · Paper_XI_Traditional_Final.tex, line 1812
Local class field theory identifies one dimensional Weil characters with characters of K^× [source reference]. The tame induction theorem extends this explicit picture to higher dimensional tame atoms. The local polynomial and conductor calculations in [source reference] therefore give explicit finite invariants for the declared tame packets. Finite Gauss sums give the residue field phase. The full local factor, however, also includes ramification depth, Haar measure normalization, the additive …
Exact mathematical passage
K^{\times}
k^\timesUse in source · 1 passage
Search aliases: k^× times
Use in source · odd_synthesis.tex, line 190
Indeed, for residue cardinality q, let B=ω(Teich(k^×)). The actual inertia hypothesis gives χ(C)^N⊂ B. Since N is a power of p and χ(C) has order prime to p, this implies χ(C)⊂ B. Choose k_0 inverse to N modulo q-1, and set δ(u)=ω(Teich(u))^k_0, widetilde C={(c,u):χ(c)=δ(u)}. The matrix Teich(u^k_0)I_N acts by χ(c). The kernel acts trivially on the entire sector induction and ambient compact induction, because these matrices are central. This preserves the fixed Λ, its canonical beta-extension f…
Exact mathematical passage
B=\omega(\operatorname{Teich}(k^\times))
k^{\times}Use in source · 4 passages
Search aliases: k^× times
Use in source · Paper_XI_Traditional_Final.tex, line 711
Let k= F_q, let ψ:k→ C^× be a nontrivial additive character, and let χ:k^×→ C^× be a multiplicative character. Define G(χ,ψ) = ∑_x∈ k^×χ(x)ψ(x). The standard theory of Gauss and Jacobi sums will be used in the form recorded in [source reference].
Exact mathematical passage
\chi:k^{\times}\to\CC^{\times}
K_0Use in source · 5 passages
Search aliases: K_0
Use in source · monograph_atlas.tex, line 312
vpage17A channel acts on four matrix entriesPOPULATIONS AND COHERENCES TRANSFORM DIFFERENTLYChapter VII: Theorem II.7.6, Example II.7.12. Choose amplitude damping with γ=9/25: K_0=mat1 · 0 0 · 4/5, K_1=mat0 · 3/5 0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter keeps every displayed coefficient rational. vfigdamping The superoperator uses the entry order (ρ_00,ρ_01,ρ_10,ρ_11). Population leaves the excited state with fa…
Exact mathematical passage
K_0=\mat{1&0\\0&4/5},\qquad K_1=\mat{0&3/5\\0&0},\qquad K_0^*K_0+K_1^*K_1=I.
k_0Use in source · 9 passages
Search aliases: k_0
Use in source · 03_packets_and_graphs.tex, line 383
Let x^(k)=(x_n^(k)) be Cauchy in · _u. For every fixed n, x_n^(k)-x_n^(ℓ) ≤ x^(k)-x^(ℓ) _u, so x_n^(k) converges to some x_n∈ X_n because X_n is closed. Given ε>0, choose k_0 such that x^(k)-x^(ℓ) _u<ε (k,ℓ≥ k_0). For finite F and |a_n|≤1, pass ℓ→∞ in the finite sum to obtain ∑_n∈ Fa_n(x_n^(k)-x_n) ≤ε (k≥ k_0). Hence x^(k)-x _u≤ε once x is known to be unconditionally summable.
Exact mathematical passage
k_0
K_0^*Use in source · 2 passages
Search aliases: K_0^*
Use in source · monograph_atlas.tex, line 312
vpage17A channel acts on four matrix entriesPOPULATIONS AND COHERENCES TRANSFORM DIFFERENTLYChapter VII: Theorem II.7.6, Example II.7.12. Choose amplitude damping with γ=9/25: K_0=mat1 · 0 0 · 4/5, K_1=mat0 · 3/5 0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter keeps every displayed coefficient rational. vfigdamping The superoperator uses the entry order (ρ_00,ρ_01,ρ_10,ρ_11). Population leaves the excited state with fa…
Exact mathematical passage
K_0=\mat{1&0\\0&4/5},\qquad K_1=\mat{0&3/5\\0&0},\qquad K_0^*K_0+K_1^*K_1=I.
K_1Use in source · 6 passages
Search aliases: K_1
Use in source · monograph_atlas.tex, line 312
vpage17A channel acts on four matrix entriesPOPULATIONS AND COHERENCES TRANSFORM DIFFERENTLYChapter VII: Theorem II.7.6, Example II.7.12. Choose amplitude damping with γ=9/25: K_0=mat1 · 0 0 · 4/5, K_1=mat0 · 3/5 0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter keeps every displayed coefficient rational. vfigdamping The superoperator uses the entry order (ρ_00,ρ_01,ρ_10,ρ_11). Population leaves the excited state with fa…
Exact mathematical passage
K_0=\mat{1&0\\0&4/5},\qquad K_1=\mat{0&3/5\\0&0},\qquad K_0^*K_0+K_1^*K_1=I.
k_1Use in source · 2 passages
Search aliases: k_1
Use in source · 04_correction_transport.tex, line 164
[Quadratic reduced obstruction] Assume the hypotheses of [source reference] with r≥2. For k_1,k_2∈ K, D^2κ(0)[k_1,k_2] = π_O D^2F(0)[k_1,k_2].
Exact mathematical passage
k_1,k_2\in K
K_{10}Use in source · 1 passage
Search aliases: K_10
Use in source · pages.tex, line 233
vpage14Two routes can cancel exactlySUPPORT IS AN INCLUSIONChapter VIII: Theorem 2.5, Theorem 6.1, Corollary 6.3. Let K have transfers 0→1, 0→2, 1→3, and 2→3, with coefficients 1,1,1,-1. There are two length-two routes from packet zero to packet three, but their contributions sum to zero. vfigcancellation (K^2)_30=K_31K_10+K_32K_20=1-1=0, K^2=0. The matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the…
Exact mathematical passage
(K^2)_{30}=K_{31}K_{10}+K_{32}K_{20}=1-1=0, \qquad K^2=0.
K_1^*Use in source · 2 passages
Search aliases: K_1^*
Use in source · monograph_atlas.tex, line 312
vpage17A channel acts on four matrix entriesPOPULATIONS AND COHERENCES TRANSFORM DIFFERENTLYChapter VII: Theorem II.7.6, Example II.7.12. Choose amplitude damping with γ=9/25: K_0=mat1 · 0 0 · 4/5, K_1=mat0 · 3/5 0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter keeps every displayed coefficient rational. vfigdamping The superoperator uses the entry order (ρ_00,ρ_01,ρ_10,ρ_11). Population leaves the excited state with fa…
Exact mathematical passage
K_0=\mat{1&0\\0&4/5},\qquad K_1=\mat{0&3/5\\0&0},\qquad K_0^*K_0+K_1^*K_1=I.
k_2Use in source · 2 passages
Search aliases: k_2
Use in source · 04_correction_transport.tex, line 164
[Quadratic reduced obstruction] Assume the hypotheses of [source reference] with r≥2. For k_1,k_2∈ K, D^2κ(0)[k_1,k_2] = π_O D^2F(0)[k_1,k_2].
Exact mathematical passage
k_1,k_2\in K
K_{20}Use in source · 1 passage
Search aliases: K_20
Use in source · pages.tex, line 233
vpage14Two routes can cancel exactlySUPPORT IS AN INCLUSIONChapter VIII: Theorem 2.5, Theorem 6.1, Corollary 6.3. Let K have transfers 0→1, 0→2, 1→3, and 2→3, with coefficients 1,1,1,-1. There are two length-two routes from packet zero to packet three, but their contributions sum to zero. vfigcancellation (K^2)_30=K_31K_10+K_32K_20=1-1=0, K^2=0. The matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the…
Exact mathematical passage
(K^2)_{30}=K_{31}K_{10}+K_{32}K_{20}=1-1=0, \qquad K^2=0.
K_2^{\mathsf T}Use in source · 4 passages
Search aliases: K_2^ T mathsf
Use in source · atlas-pages.tex, line 47
1/4 · 3/4, p=e_0, f=(2,1)^ T, we obtain K_2p=(3/4,1/4)^ T and f^ TK_2p=7/4=(K_2^ Tf)_0. With departure weights w_i=e^-V(i), the backward Feynman--Kac operator is diag(w)K_2^ T. Its forward dual is K_2diag(w). Interchanging these factors changes when the weight is applied. takeawayAlways name the object being transported. The transpose and the position of the likelihood or potential are mathematical choices, not typographic details.
Exact mathematical passage
f^{\mathsf T}K_2p=7/4=(K_2^{\mathsf T}f)_0.
K_{31}Use in source · 1 passage
Search aliases: K_31
Use in source · pages.tex, line 233
vpage14Two routes can cancel exactlySUPPORT IS AN INCLUSIONChapter VIII: Theorem 2.5, Theorem 6.1, Corollary 6.3. Let K have transfers 0→1, 0→2, 1→3, and 2→3, with coefficients 1,1,1,-1. There are two length-two routes from packet zero to packet three, but their contributions sum to zero. vfigcancellation (K^2)_30=K_31K_10+K_32K_20=1-1=0, K^2=0. The matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the…
Exact mathematical passage
(K^2)_{30}=K_{31}K_{10}+K_{32}K_{20}=1-1=0, \qquad K^2=0.
K_{32}Use in source · 1 passage
Search aliases: K_32
Use in source · pages.tex, line 233
vpage14Two routes can cancel exactlySUPPORT IS AN INCLUSIONChapter VIII: Theorem 2.5, Theorem 6.1, Corollary 6.3. Let K have transfers 0→1, 0→2, 1→3, and 2→3, with coefficients 1,1,1,-1. There are two length-two routes from packet zero to packet three, but their contributions sum to zero. vfigcancellation (K^2)_30=K_31K_10+K_32K_20=1-1=0, K^2=0. The matrix of path counts sees the two possibilities and forgets the signs. It therefore predicts possible support but not actual support. Replacing the…
Exact mathematical passage
(K^2)_{30}=K_{31}K_{10}+K_{32}K_{20}=1-1=0, \qquad K^2=0.
K_{4,j}Use in source · 2 passages
Search aliases: K_4,j
Use in source · 20_arithmetic.tex, line 482
The Hurwitz coefficients of the normalized source K_4,j at degrees 4n are g_j(n)=binom4n+j4n^-1. Their explicit expressions are g_0(n) · =1, · g_1(n) · =frac14n+1,
Exact mathematical passage
K_{4,j}
K_aUse in source · 15 passages
Search aliases: K_a
Use in source · continuation_09.tex, line 86
At depth N≥1, suppose F(a)=q^Nr and a desired next integer parameter target satisfies t'-free(a)=q^Nw, with w∈ Z^k. Surjectivity supplies a solved digit u∈ S_q^m satisfying J_a u=-bar r-K_abar w. Choose an integer representative widetilde u, and set a'=a+q^N(widetilde u,w). The parameter quotient is the supplied integer w. Therefore free(a')=t' exactly. The checked polynomial Taylor identity gives F(a')≡ F(a)+q^NDF(a)(widetilde u,w) ≡0pmodq^N+1. Its quadratic remainder is divisible by q^2N, and …
Exact mathematical passage
J_a u=-\bar r-K_a\bar w.
k_AUse in source · 2 passages
Search aliases: k_A
Use in source · Paper_X_Block27_Amended.tex, line 2477
The inertia theorem uses actual residue actions. For a valuation subring A of an extension field L/K, let D_A be its decomposition subgroup in Aut_K(L) and put I_A=ker(D_A⟶Aut(k_A)), where k_A is the residue field. If 2 is a unit in A, every member of I_A fixes every y∈ L with y^2=-1. Indeed y^-1=-y, so y belongs to A and is a unit there. An automorphism sends y to y or -y. Their residues are distinct because 2 and y are units, whereas inertia fixes residues. The second possibility is excluded.
Exact mathematical passage
I_A=\ker\bigl(D_A\longrightarrow\operatorname{Aut}(k_A)\bigr),
K_a^*Use in source · 8 passages
Search aliases: K_a^*
Use in source · monograph_atlas.tex, line 316
0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter keeps every displayed coefficient rational. vfigdamping The superoperator uses the entry order (ρ_00,ρ_01,ρ_10,ρ_11). Population leaves the excited state with factor 9/25, while coherences contract by 4/5. These are not the same rate or the same coordinate.
Exact mathematical passage
\Phi(\rho)=\sum_aK_a\rho K_a^*
K_cUse in source · 4 passages
Search aliases: K_c
Use in source · supplement_09.tex, line 151
Removing a singular forced zero. Put c=ap<N and define the actual normalized kernel K_c(t)=c!∑_n≥0fract^Nn(Nn+c)!, K_c(0)=1. Let ℓ(T)=log(1+T)/T, defined by shifting logarithm coefficients, not by inverting T. Set U(T)=ℓ(T)^cK_c(log(1+T)). The formal proof establishes U(0)=1 and the exact factorization H_N,c(log(1+T))=fracT^cc!U(T). It follows coefficientwise that coefTdU=c!coefTd+cH_N,c(log(1+T)). Because p^amid(ap)! and lfloor(d+ap)/prfloor=a+lfloor d/prfloor, the already proved section estima…
Exact mathematical passage
K_c(t)=c!\sum_{n\ge0}\frac{t^{Nn}}{(Nn+c)!},\qquad K_c(0)=1.
K_{c-1}Use in source · 1 passage
Search aliases: K_c-1
Use in source · supplement_09.tex, line 172
Boundary. The original positive-address section has zero constant coefficient. It is not the series to which unit inverse closure is applied. 4. The singular ratio estimate. For a positive address c, the rational ratio is independently constructed from the original kernels: R_c(t)=c K_c-1(t)K_c(t)^-1. The scalar factor c is necessary. The source proves the cleared identity R_c(t)H_N,c(t)=tH_N,c-1(t). This connects the constructed object to the original unnormalized ratio A_c-1/A_c. It does not a…
Exact mathematical passage
R_c(t)=c\,K_{c-1}(t)K_c(t)^{-1}.
K_fUse in source · 7 passages
Search aliases: K_f
Use in source · Paper_XI_Traditional_Final.tex, line 334
[Tame orbit induction] Let τ be an irreducible complex representation of W_K with trivial wild inertia. Then the nonzero tame character packets form one transitive Frobenius orbit of length f. If K_f/K is the unramified extension of degree f, there is a character ξ of W_K_f such that τ≅Ind_W_K_f^W_Kξ.
Exact mathematical passage
K_f/K
k_F^\timesUse in source · 1 passage
Search aliases: k_F^× times
Use in source · dyadic_character.tex, line 34
The construction retains the actual canonical beta extension and can be completed with any prescribed central quasicharacter omega_F trivial on U_F 1. Because k_F^×= F_2^× is trivial, such an omega_F is unramified. Thus this covers every tamely ramified central quasicharacter for this particular residue field.
Exact mathematical passage
k_F^\times=\FF_2^\times
k_iUse in source · 3 passages
Search aliases: k_i
Use in source · root_scope.tex, line 219
For 0≤ i<j, let d_i=v_2(i!)+v_2((j-1-i)!), k_i=1+lfloorlog_2max(i,j-1-i)rfloor, B_i=i+2^k_i Z_2, t_i=d_i+k_i. At j=1 use d_0=k_0=t_0=0. The balls B_i partition Z_2. Indeed a residue represented twice among 0,…,j-1 splits into its two children. Every other represented residue remains one ball. On each such ball, P_j(i+2^k_iy)=2^t_iyQ_i(y), Q_i∈ Z_2⟨ y⟩^×, Q_i≡1pmod2.
Exact mathematical passage
d_i=v_2(i!)+v_2((j-1-i)!),\qquad k_i=1+\left\lfloor\log_2\max(i,j-1-i)\right\rfloor,
K_{i_{r+1},i_r}Use in source · 1 passage
Search aliases: K_i_r+1,i_r
Use in source · Paper_VI_Traditional_Final.tex, line 1469
For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).
Exact mathematical passage
\prod_rK_{i_{r+1},i_r}
K_{ij}Use in source · 1 passage
Search aliases: K_ij
Use in source · atlas-pages.tex, line 30
K · One discrete stochastic step. · ones^ TK=ones^ T and K_ij≥0.
Exact mathematical passage
K_{ij}\ge0
K_jUse in source · 8 passages
Search aliases: K_j
Use in source · supplement_10.tex, line 173
Let K_j(t) be the normalized section kernel with constant coefficient one. The constructed unnormalized Euler series is F_N,j(t)=j! K_j(t)^-1. The source/inverse identities and the factorial successor identity prove F_N,j+1=F_N,jR_j+1, F_N,j=F_N,0∏_c=1^jR_c. The product formula is not an assembly hypothesis. Proved flatness of all factors and the original product-closure theorem establish flatness of the full family.
Exact mathematical passage
K_j(t)
k_jUse in source · 2 passages
Search aliases: k_j
Use in source · 02_strengthened.tex, line 2022
Let N≥2 and let Λ_N=( Z/N Z)^d be the unit flat torus grid with spacing 1/N. The standard periodic difference Laplacian has eigenvalues λ_N(k) = 4N^2 ∑_j=1^d sin^2(fracπ k_jN), where k runs through centered residue representatives. This is the standard periodic finite difference Laplacian [source reference].
Exact mathematical passage
\lambda_N(k) = 4N^2 \sum_{j=1}^{d} \sin^2\left(\frac{\pi k_j}{N}\right),
K_{j i}Use in source · 1 passage
Search aliases: K_j i
Use in source · Paper_VI_Traditional_Final.tex, line 1466
For n=1, (M_Vf)_i=e^-V(i)∑_jK_j if_j, which is the stated expectation. Iterating and expanding the matrix product sums over every path (i=i_0,i_1,…,i_n) with probability ∏_rK_i_r+1,i_r and weight ∏_re^-V(i_r).
Exact mathematical passage
(M_Vf)_i=e^{-V(i)}\sum_jK_{j i}f_j,
\mathcal K_{j+1}Use in source · 1 passage
Search aliases: K_j+1 mathcal
Use in source · 02_supplement_03.tex, line 120
Define the comparison space L_j+1=⊕_i K_j+1(P_iDP_i,P_iv). Packet diagonality is first shown to imply commutation. The exact theorem then proves D^jv∈ L_j+1. Using that vector as a witness in the infimum distance gives dist(A^jv, L_j+1) ≤ jM^j-1normKnormv. This statement uses Lean's actual metric infimum distance to the subspace. It does not assume that the comparison vector lies there. The block-Jacobi condition bound and the preconditioned conjugate-gradient corollary are not completed by thes…
Exact mathematical passage
\mathcal L_{j+1}=\bigoplus_i\mathcal K_{j+1}(P_iDP_i,P_iv).
K_j^*Use in source · 2 passages
Search aliases: K_j^*
Use in source · monograph_examples.tex, line 17
e, and the plotted values are exact fractions. The example shows why direct compression and resolvent elimination give different results. Normalization changes the map. Let P=|0⟩⟨0| on C^2 and Z=diag(1,-1). The normalized Kraus family sqrt3/4 I, sqrt1/4 Z satisfies ∑_j K_j^*K_j=I and maps P to P. The unnormalized family sqrt3 I, Z satisfies ∑_j E_j^*E_j=4I and maps P to 4P. The two families have the same relative weights, but their total normalization differs. Both outputs are proportional to th…
Exact mathematical passage
\sum_j K_j^*K_j=I
k_j^4Use in source · 1 passage
Search aliases: k_j^4
Use in source · 02_strengthened.tex, line 2033
[Low mode dispersion in dimension d] For every centered mode k, 0 ≤ 4π^2absk^2-λ_N(k) ≤ frac4π^43N^2 ∑_j=1^dk_j^4.
Exact mathematical passage
0 \leq 4\pi^2\abs{k}^2-\lambda_N(k) \leq \frac{4\pi^4}{3N^2} \sum_{j=1}^{d}k_j^4.
K_{ji}Use in source · 1 passage
Search aliases: K_ji
Use in source · Paper_VI_Traditional_Final.tex, line 4372
For one step, (M_Vf)_i = e^-V(i)∑_jK_jif_j. Iterating expands the matrix product over all paths. The product of transition entries is the path probability, and the diagonal factors give the exponential potential weight.
Exact mathematical passage
(M_Vf)_i = \e^{-V(i)}\sum_jK_{ji}f_j.
K_LUse in source · 4 passages
Search aliases: K_L
Use in source · Paper_XI_Traditional_Final.tex, line 565
[Greatest common divisor and least common multiple packets] There is a decomposition τ_ξ⊗τ_η ≅ ⊕_i=0^d-1 Ind_W_K_L^W_K ( ξ_L,nη_L,m^Φ^i ), where ξ_L,n and η_L,m are the norm lifts to K_L.
Exact mathematical passage
\boxed{ \tau_\xi\otimes\tau_\eta \cong \bigoplus_{i=0}^{d-1} \Ind_{W_{K_L}}^{W_K} \left( \xi_{L,n}\eta_{L,m}^{\Phi^i} \right), }
K_{L^1(0,T)}Use in source · 1 passage
Search aliases: K_L^1(0,T)
Use in source · Paper_VI_Traditional_Final.tex, line 4841
· + ∫_0^tS(t-s) ∫_0^sK(s-r)x(r) d r d s. Assume sup_0≤ t≤ Tnormwidetilde x(t)≤ B. Set L=norm K_L^1(0,T), ε=normK-widetilde K_L^1(0,T). Then sup_0≤ t≤ T normx(t)-widetilde x(t) ≤ M_TBTε exp(M_TLT).
Exact mathematical passage
L=\norm K_{L^1(0,T)}, \qquad \varepsilon=\norm{K-\widetilde K}_{L^1(0,T)}.
K_mUse in source · 3 passages
Search aliases: K_m
Use in source · Paper_XI_Traditional_Final.tex, line 551
Let τ_ξ=Ind_W_K_n^W_Kξ, τ_η=Ind_W_K_m^W_Kη, where K_n/K and K_m/K are unramified. Put d=gcd(n,m), L=lcm(n,m).
Exact mathematical passage
\tau_\xi=\Ind_{W_{K_n}}^{W_K}\xi, \qquad \tau_\eta=\Ind_{W_{K_m}}^{W_K}\eta,
\mathcal K_mUse in source · 3 passages
Search aliases: K_m mathcal
Use in source · pages.tex, line 283
vpage17Invariant does not mean independently addressableA STRICT KRYLOV INCLUSIONChapter VIII: Proposition 7.2, Theorem 7.3. Take A=I_2, split into the two coordinate packets, and start with v=(1,1)^T. Every packet is invariant. Nevertheless every polynomial applies the same scalar to both coordinates: p(A)v=p(1)(1,1)^T, dim K_m(A,v)=1 (m≥1). vfigkrylov The direct sum of the two local Krylov spans is all of QQ^2. It permits independent coefficients in the two packets. The global Krylov space doe…
Exact mathematical passage
p(A)v=p(1)(1,1)^T,\qquad \dim\mathcal K_m(A,v)=1\quad(m\ge1).
k_mUse in source · 3 passages
Search aliases: k_m
Use in source · supplement_03.tex, line 37
The unit kernel and its reciprocal. Use a squared formal variable u and define k_m=frac2^2m+1(2m+2)!, K(u)=∑_m≥0k_mu^m, β_m=[u^m]K(u)^-1. The constant term is proved to be one. The factorial valuation bound gives v_2(k_m)=2m+1-v_2((2m+2)!)≥0. Thus every k_m belongs to O_2. A general coefficientwise subring theorem is proved by strong induction using the actual unit-inverse coefficient formula. Applied to K, it establishes β_m∈ O_2 for every m≥0. The reciprocal's coefficients are therefore proved…
Exact mathematical passage
k_m=\frac{2^{2m+1}}{(2m+2)!},\qquad K(u)=\sum_{m\geq0}k_mu^m,\qquad \beta_m=[u^m]K(u)^{-1}.
K_nUse in source · 6 passages
Search aliases: K_n
Use in source · 05_worked_models.tex, line 349
…,s_r, 𝕜^n=span_𝕜{p(A_P_n)s_j:p∈𝕜[t], 1≤ j≤ r} }. Over C, the familiar spectral description has the distinct eigenvalues λ_k=2cosfrackπn+1, 1≤ k≤ n, and the normalized eigenvectors have endpoint coordinate proportional to sinfrackπn+1≠0. There is also a combinatorial proof valid over every field. Let K_n=[e_1,Ae_1,…,A^n-1e_1]. Its (i,j) entry counts walks of length j-1 from the first vertex to vertex i. This entry is zero when i>j, while the diagonal entry is one because there is a unique shorte…
Exact mathematical passage
K_n=[e_1,Ae_1,\ldots,A^{n-1}e_1].
K_{N,j}Use in source · 4 passages
Search aliases: K_N,j
Use in source · 20_arithmetic.tex, line 29
The normalized source and its inverse are K_N,j(t)=j!∑_n≥0fract^Nn(Nn+j)!, P_N,j(t)=∑_n≥0C_N,j(n)fract^Nn(Nn)!. Their product is one. Clearing factorials in its coefficient of t^Nn gives the triangular system ∑_q=0^nbinomNn+jNqC_N,j(q)=δ_n0. In particular, for n>0, C_N,j(n)=-binomNn+jNn^-1 ∑_q=0^n-1binomNn+jNqC_N,j(q). The diagonal binomial coefficient is positive, so this recurrence determines the sequence uniquely. The Lean construction begins with this recurrence and derives the reciprocal id…
Exact mathematical passage
\label{eq:arith-kernel} K_{N,j}(t)=j!\sum_{n\ge0}\frac{t^{Nn}}{(Nn+j)!},\qquad \mathcal P_{N,j}(t)=\sum_{n\ge0}C_{N,j}(n)\frac{t^{Nn}}{(Nn)!}.
K_{N,j}^{-1}Use in source · 1 passage
Search aliases: K_N,j^-1
Use in source · supplement_01.tex, line 108
The formalization does not stop at a recurrence. In the variable z=t^N, construct K_N,j(z)=∑_n≥0fracj!(Nn+j)!z^n, F_N,j(z)=∑_n≥0fracC_N,j(n)(Nn)!z^n. It proves the factorial coefficient identity, then F_N,jK_N,j=1 and therefore F_N,j=K_N,j^-1.
Exact mathematical passage
F_{N,j}=K_{N,j}^{-1}
K_N^mUse in source · 2 passages
Search aliases: K_N^m
Use in source · Paper_IV_Traditional_Final.tex, line 2339
[Diagonal Fredholm determinant representation] Let I_N be a multiplicity-sensitive countable index set for Λ_N and put H_N=ℓ^2(I_N). On its standard orthonormal basis, define the diagonal operator K_Ne_λ=λ^-1e_λ. Then K_N is trace class and Δ_N(w)=det_ F(I-wK_N). For every m≥1, κ_N,m=tr(K_N^m).
Exact mathematical passage
\kappa_{N,m}=\tr(K_N^m).
K_N^qUse in source · 1 passage
Search aliases: K_N^q
Use in source · atlas-pages.tex, line 402
vpage22Finite coefficients give infinite momentsZERO POWER SUMSChapter IV: Theorem 12.6, Corollary 12.7, Corollary 12.9, Corollary 12.10. Taking the logarithm of the canonical product identifies κ_N,q=σ_N,q:=∑_λ∈Λ_Nλ^-q =Tr(K_N^q), q≥1. The same exact numbers can be computed from finitely many denominator coefficients by Newton's recurrence. No numerical zero list is needed for that calculation. vfigmoments For order two, σ_2,1=frac12, σ_2,2=frac16, σ_2,3=frac115, σ_2,4=frac17630. The plate show…
Exact mathematical passage
\kappa_{N,q}=\sigma_{N,q}:=\sum_{\lambda\in\Lambda_N}\lambda^{-q} =\Tr(K_N^q),\qquad q\ge1.
K_rUse in source · 9 passages
Search aliases: K_r
Use in source · root_scope.tex, line 361
The current focused obstruction is the admitted ordinary-index family r=(17659+2^18T)2^640-161, T∈ Z_≥0, j_r=161. On its surviving chart define the actual, undepleted function K_r(Z)=2^-178G_r(10202+2^15Z). The complete coefficient theorem gives K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z), a_r,b_r∈ Z_2, u_r∈ Z_2^×, R_r∈ Z_2⟨ Z⟩. Its half derivative is a dyadic isometry. It therefore has a unique critical point c_r∈ Z_2. Set Φ_r^act=K_r(c_r). The earlier independent-undepletion notation for this same sc…
Exact mathematical passage
K_r(Z)=2^{-178}G_r(10202+2^{15}Z).
k_rUse in source · 4 passages
Search aliases: k_r
Use in source · 10_foundations.tex, line 95
[Constructed unit-triangular coordinates] Let h: N→ K satisfy h_0=1, write H= E(h) and H^-1= E(k), and define U_Ha= D(H E(a)). Then U_H is an equivalence, with inverse U_H^-1a= D(H^-1 E(a)). Its coefficient formulas are (U_Ha)_n=∑_r=0^nbinom nr h_r a_n-r, (U_H^-1a)_n=∑_r=0^nbinom nr k_r a_n-r. The resulting triangular matrix has diagonal entries 1. The transported selectors widehat P_N,j=U_HP_N,jU_H^-1 are idempotent and pairwise orthogonal, and their sum is the identity.
Exact mathematical passage
(U_Ha)_n=\sum_{r=0}^{n}\binom nr h_r a_{n-r},\qquad (U_H^{-1}a)_n=\sum_{r=0}^{n}\binom nr k_r a_{n-r}.
K_r^*Use in source · 3 passages
Search aliases: K_r^*
Use in source · 01_finite_core.tex, line 855
_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal projector. Use the recovery branches K_r=V_r^*, K_unused=I-S. They are complete on the whole ambient space because ∑_rK_r^*K_r+K_unused^*K_unused =S+(I-S)^2=I. The inverse unitary change of coordinates reconstructs each E_aP from the F_r. Every recovery branch of a coherent combination is therefore scala…
Exact mathematical passage
\sum_rK_r^*K_r+K_{\mathrm{unused}}^*K_{\mathrm{unused}} =S+(I-S)^2=I.
K_{ri}Use in source · 3 passages
Search aliases: K_ri
Use in source · Paper_VI_Traditional_Final.tex, line 2898
[Coordinate form of the lumping obstruction] For the partition aggregation above, Ker C is spanned by differences e_i-e_j with i and j in one source block. Also, Ob_C(K)(e_i-e_j) = ( ∑_r∈ B_α(K_ri-K_rj) )_α=1^m.
Exact mathematical passage
\Ob_C(K)(e_i-e_j) = \left( \sum_{r\in B_\alpha}(K_{ri}-K_{rj}) \right)_{\alpha=1}^m.
K_{rj}Use in source · 2 passages
Search aliases: K_rj
Use in source · Paper_VI_Traditional_Final.tex, line 2898
[Coordinate form of the lumping obstruction] For the partition aggregation above, Ker C is spanned by differences e_i-e_j with i and j in one source block. Also, Ob_C(K)(e_i-e_j) = ( ∑_r∈ B_α(K_ri-K_rj) )_α=1^m.
Exact mathematical passage
\Ob_C(K)(e_i-e_j) = \left( \sum_{r\in B_\alpha}(K_{ri}-K_{rj}) \right)_{\alpha=1}^m.
K_{\rm mem}Use in source · 2 passages
Search aliases: K_rm mem rm
Use in source · atlas-pages.tex, line 520
0 · 0, N^2=0. Choose the entrance column V=(0,1)^ T and exit row U=(1,0). The hidden exponential is not e^-2tI: e^tJ=e^-2t(I+tN), K_rm mem(t)=Ue^tJV=t e^-2t. The nilpotent link carries an input from the second hidden coordinate to the first. vfigjordan platecaptionThe memory kernel starts at zero, becomes positive, and peaks at t=1/2. A rule that retained the eigenvalue -2 but discarded the Jordan chain would predict the zero kernel for these entrance and exit maps. headThe transfer function see…
Exact mathematical passage
e^{tJ}=e^{-2t}(I+tN), \qquad K_{\rm mem}(t)=Ue^{tJ}V=t e^{-2t}.
K_tUse in source · 2 passages
Search aliases: K_t
Use in source · continuation_11.tex, line 95
There is also an explicit open-set version. Let U be open, let g be continuous on U, and suppose for every t∈ U that F(g(t),t)=0, J_t:=D_YF(g(t),t)∈GL_e( K). Writing K_t=D_TF(g(t),t), the theorem mcodecontinuous_selection_analyticOnNhd proves g is analytic at every t∈ U, Dg(t)=-J_t^-1K_t (t∈ U), with a strict Fréchet derivative at each point. The invertibility hypothesis at each point is part of this theorem. This is a transfer statement for continuous local root selections satisfying its hypoth…
Exact mathematical passage
K_t=D_TF(g(t),t)
K_{\mathrm{unused}}Use in source · 2 passages
Search aliases: K_unused mathrm
Use in source · 01_finite_core.tex, line 851
efine compressed transformed errors F_r=∑_aU_arE_aP. Then F_rP=F_r and F_r^*F_s=λ_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal projector. Use the recovery branches K_r=V_r^*, K_unused=I-S. They are complete on the whole ambient space because ∑_rK_r^*K_r+K_unused^*K_unused =S+(I-S)^2=I. The inverse unitary change of coordinates reconstructs each E_aP…
Exact mathematical passage
K_r=V_r^*,\qquad K_{\mathrm{unused}}=I-S.
K_{\mathrm{unused}}^*Use in source · 1 passage
Search aliases: K_unused^* mathrm
Use in source · 01_finite_core.tex, line 855
_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal projector. Use the recovery branches K_r=V_r^*, K_unused=I-S. They are complete on the whole ambient space because ∑_rK_r^*K_r+K_unused^*K_unused =S+(I-S)^2=I. The inverse unitary change of coordinates reconstructs each E_aP from the F_r. Every recovery branch of a coherent combination is therefore scala…
Exact mathematical passage
\sum_rK_r^*K_r+K_{\mathrm{unused}}^*K_{\mathrm{unused}} =S+(I-S)^2=I.
K_xUse in source · 8 passages
Search aliases: K_x
Use in source · 30_lifting.tex, line 539
Suppose now that the split into solved variables and parameter variables is fixed in advance. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0), and assume F(x)=q^Lr_x with q≥1 and L≥1. Write the reduced Jacobian as DF(x)bmod q=(J_x K_x), J_x:S_q^m→ S_q^e, K_x:S_q^k→ S_q^e. For a prescribed parameter digit v∈ S_q^k, let L_x,L,v be the set of next residue roots over x whose parameter coordinates are t_0+q^Lwidetilde v modulo q^L+1.
Exact mathematical passage
DF(x)\bmod q=(J_x\ K_x),\qquad J_x:S_q^m\to S_q^e,\quad K_x:S_q^k\to S_q^e.
k_xUse in source · 1 passage
Search aliases: k_x
Use in source · Paper_V_Traditional_Final.tex, line 821
[Two-dimensional errors] For the 192×192 disk experiment, a square Fourier mask retains indices |k_x|,|k_y|≤16, or 33^2=1089 bins. The reported errors are lrr method · full-image RMSE · boundary-band RMSE
Exact mathematical passage
|k_x|,|k_y|\le16
k_yUse in source · 1 passage
Search aliases: k_y
Use in source · Paper_V_Traditional_Final.tex, line 821
[Two-dimensional errors] For the 192×192 disk experiment, a square Fourier mask retains indices |k_x|,|k_y|≤16, or 33^2=1089 bins. The reported errors are lrr method · full-image RMSE · boundary-band RMSE
Exact mathematical passage
|k_x|,|k_y|\le16
K_{z_n}Use in source · 2 passages
Search aliases: K_z_n
Use in source · 30_lifting.tex, line 610
At level N=L+n, write the supplied parameter increment exactly as t_n+1-t_n=q^Nw with w∈ Z^k. Surjectivity of the solved block gives a digit u satisfying J_z_nu=-overlineF(z_n)/q^N-K_z_noverline w. Choose an integer representative widetilde u and set z_n+1=z_n+q^N(widetilde u,w). Its parameter coordinates equal the supplied target exactly. Taylor expansion proves the next root condition, while congruence modulo q preserves the full reduced Jacobian. Construct the sequence recursively, preserving…
Exact mathematical passage
J_{z_n}u=-\overline{F(z_n)/q^N}-K_{z_n}\overline w
K_\alphaUse in source · 7 passages
Search aliases: K_α alpha
Use in source · 04_transport.tex, line 188
[Packet exact sequence for an invariant subspace] Let K≤ X satisfy P_α K⊆ K for every α. Define K_α:=P_α K=K∩ X_α and overline P_α(x+K):=P_α x+K. Then: [label=(roman*)] item The nonzero restrictions P_α|_K form a packet system on K. item The nonzero overline P_α form a packet system on X/K. item For every α there is a short exact sequence 0⟶ K_α ⟶ X_α ⟶ overline P_α(X/K) ⟶0, and overline P_α(X/K) ≅ X_α/(X_α∩ K) ≅ (X_α+K)/K. item The full exact sequence decomposes packetwise as the direct sum of …
Exact mathematical passage
K_\alpha:=P_\alpha K=K\cap X_\alpha
K_\betaUse in source · 2 passages
Search aliases: K_β beta
Use in source · 04_transport.tex, line 256
[Restriction and quotient of transfer blocks] Under the hypotheses of [source reference], let T∈End(X) preserve K. Then every block T_βα maps K_α into K_β, and the induced quotient operator overline T satisfies overline T_βα π = π T_βα, where π:X↠ X/K. [source reference]
Exact mathematical passage
K_\beta
K_{\beta\alpha}Use in source · 1 passage
Search aliases: K_βα beta alpha
Use in source · 01_manuscript.tex, line 1026
The equation in packet β is D_β x_β = b_β - ∑_α→βK_βαx_α. Every predecessor of β lies in an earlier level. Once those values are available, x_β is obtained from the local solve with D_β. Vertices in the same level have no directed edge between them, so their solves are independent.
Exact mathematical passage
D_\beta x_\beta = b_\beta - \sum_{\alpha\to\beta}K_{\beta\alpha}x_\alpha.
K_\varepsilonUse in source · 6 passages
Search aliases: K_ε varepsilon
Use in source · atlas-pages.tex, line 125
vpage6A broken lump and its error budgetDEFECTS / A GUARANTEE WITHOUT A FITChapter VI, Part I: Theorems 3.1, 3.3, and 5.2. Continuous and discrete defect identities. Modify only column zero of K: move 1/16 of its mass from target state zero to target state two. The perturbed matrix K_ε remains stochastic, but its aggregate column now differs from column one. Relative to the unchanged coarse kernel, Δ=CK_ε-bar KC =-1/16 · 0 · 0 · 0
Exact mathematical passage
K_\varepsilon
K_\varepsilon^jUse in source · 1 passage
Search aliases: K_ε^j varepsilon
Use in source · atlas-pages.tex, line 135
1/16 · 0 · 0 · 0, Δ _1=1/8. vfigdefect_budget platecaptionThe solid values are computed with rational matrix powers. The dashed certificate uses only stochasticity and the exact one-step defect. It need not be sharp. headTelescoping tells us where every error originates For n≥0, CK_ε^n-bar K^nC =∑_j=0^n-1bar K^n-1-jΔ K_ε^j. Each summand propagates one defect between an earlier fine evolution and a later coarse evolution. Column-stochastic matrices have induced ℓ^1 norm one, so the norm is at mos…
Exact mathematical passage
CK_\varepsilon^n-\bar K^nC =\sum_{j=0}^{n-1}\bar K^{n-1-j}\Delta K_\varepsilon^j.
K_\varepsilon^nUse in source · 1 passage
Search aliases: K_ε^n varepsilon
Use in source · atlas-pages.tex, line 135
1/16 · 0 · 0 · 0, Δ _1=1/8. vfigdefect_budget platecaptionThe solid values are computed with rational matrix powers. The dashed certificate uses only stochasticity and the exact one-step defect. It need not be sharp. headTelescoping tells us where every error originates For n≥0, CK_ε^n-bar K^nC =∑_j=0^n-1bar K^n-1-jΔ K_ε^j. Each summand propagates one defect between an earlier fine evolution and a later coarse evolution. Column-stochastic matrices have induced ℓ^1 norm one, so the norm is at mos…
Exact mathematical passage
CK_\varepsilon^n-\bar K^nC =\sum_{j=0}^{n-1}\bar K^{n-1-j}\Delta K_\varepsilon^j.
K_\ellUse in source · 3 passages
Search aliases: K_ℓ ell
Use in source · 72_recovered.tex, line 1277
the even source subsequences, equivalently the evenized generating functions H_p. It does not identify their full odd-degree source values. For p=3, the original odd source has A_3,1=S_1(3)=3, whereas every odd dyadic source coefficient is zero. The even kernel L_3K_2 cannot change that odd coefficient. Read the coefficient consequence below only at even indices. The displayed identity H_p=L_pK_ℓ H_ℓ and its proof remain valid. For any two primes p and ℓ, the source sequences A_p,n and A_ℓ,n are…
Exact mathematical passage
H_p=L_pK_\ell H_\ell
\Kcal_JUse in source · 4 passages
Search aliases: Kcal_J K_J Kcal mathcal
Use in source · Paper_VI_Traditional_Final.tex, line 4674
he least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt ∑_k=0^ν_j-1 fract^kk!B_j,k and Σ_P(z) = ∑_j=1^r ∑_k=0^ν_j-1 fracB_j,k(z-μ_j)^k+1. Suppose Reμ_j<0 for every j. If J⊆{1,…,r} and K_J retains the complete primary blocks indexed by J, then norm K- K_J_L^1(0,∞) ≤ ∑_j∉ J ∑_k=0^ν_j-1 fracnormB_j,k (-Reμ_j)^k+1.
Exact mathematical passage
\Kcal_J
\Kcal_{j+1}Use in source · 1 passage
Search aliases: Kcal_j+1 K_j+1 Kcal mathcal
Use in source · 01_manuscript.tex, line 1160
[Approximate packet Krylov bound] Write A=D+K, where D is packet diagonal. For every positive integer j, normA^j-D^j ≤ jM^j-1normK, where M=max{normA,normD}. Consequently, dist(A^jv,⊕_α K_j+1(D_α,v_α)) ≤ jM^j-1normKnormv.
Exact mathematical passage
\dist(A^jv,\bigoplus_\alpha\Kcal_{j+1}(D_\alpha,v_\alpha)) \leq jM^{j-1}\norm{K}\norm{v}.
\kkUse in source · 4 passages
Search aliases: kk 𝕜 Bbbk
Use in source · visual_atlas.tex, line 113
vpage5Relations identify words, not just outputsPRESENTATIONS / WHY A LEFT IDEAL APPEARSMain text: [source reference] Let F=𝕜⟨ z_1,…,z_r⟩ be the algebra of noncommuting operator words. Evaluation at a fixed seed is Φ_s:F→ X, f↦ f(T_1,…,T_r)s. Two words represent the same generated vector exactly when their difference lies in I_s=KerΦ_s. Consequently F/I_s≅Hull_F(s) as left modules. The relation f(T)s=0 implies g(T)f(T)s=0 for every word g, which explains closure under multiplication on the left.
Exact mathematical passage
F=\kk\langle z_1,\ldots,z_r\rangle
\kk^2Use in source · 1 passage
Search aliases: kk^2 𝕜^2 kk Bbbk
Use in source · visual_atlas.tex, line 304
The corresponding minimum seed numbers are srank_𝕜 I(𝕜^2)=2, srank_𝕜×𝕜(𝕜⊕𝕜)=1. In the independent case, the algebra contains (1,0) and (0,1), which recover the two coordinates of s separately.
Exact mathematical passage
\srank_{\kk I}(\kk^2)=2, \qquad \srank_{\kk\times\kk}(\kk\oplus\kk)=1.
L'Use in source · 7 passages
Search aliases: L'
Use in source · Paper_V_Traditional_Final.tex, line 1649
a finite index sublattice of Z^d, and let R contain one representative from every coset. The analysis map x⟼ (x_r)_r∈ R, x_r(ℓ)=x(r+ℓ), is an isometric isomorphism on every ℓ^p space. This statement includes the usual one-dimensional polyphase decomposition, but it also applies to arbitrary finite index lattice packets in several dimensions. If L'≤ L, the coarse packets split exactly into their fine descendants. A convolution operator becomes a matrix of convolution operators on L, with entries …
Exact mathematical passage
L'\leq L
L^*Use in source · 2 passages
Search aliases: L^*
Use in source · Paper_VI_Traditional_Final.tex, line 1121
Let L=L^*≥0 on a finite-dimensional Hilbert space. Its spectral resolution is L=∑_jλ_jP_j. For α>0, define L^α=∑_jλ_j^α P_j. The associated fractional semigroup is T_α(t)=e^-tL^α =∑_je^-tλ_j^αP_j.
Exact mathematical passage
L=L^*\ge0
L^{-1/2}Use in source · 1 passage
Search aliases: L^-1/2
Use in source · Paper_V_Traditional_Final.tex, line 3209
Let L=2^-j and let a be the left endpoint of the support. Pair every point t∈[a,a+L/2] with t+L/2. Then ⟨ s,ψ_j,k⟩ = L^-1/2 ∫_a^a+L/2 overlines(t)-s(t+L/2) d t. The integrand has magnitude at most ML/2, and the integration interval has length L/2. The result is ML^3/2/4.
Exact mathematical passage
\langle s,\psi_{j,k}\rangle = L^{-1/2} \int_a^{a+L/2} \overline{s(t)-s(t+L/2)}\,\dd t.
L^1Use in source · 18 passages
Search aliases: L^1
Use in source · atlas-pages.tex, line 298
ehatΩ=[tfrac516,tfrac1316], a=widehat a=3. The two intervals have the same length. Their disagreement consists of one missed strip and one extra strip, each of width 1/16. vfigmask_translation |ΩmathbintrianglewidehatΩ|=tfrac18, j-widehat j=3(1_Ω-1_widehatΩ). c|ccc error measure · L^1 norm · squared L^2 norm · L^∞ norm
Exact mathematical passage
L^1
L^{1/2}Use in source · 11 passages
Search aliases: L^1/2
Use in source · Paper_V_Traditional_Final.tex, line 3225
[Exact Haar coefficient of a step] Let τ∈ I_j,k, let L=2^-j, and write u=fracτ-kLL∈[0,1]. Then | ⟨ aH_τ,ψ_j,k ⟩ | = |a|L^1/2min{u,1-u}. Every Haar coefficient whose support does not cross τ is zero.
Exact mathematical passage
\left| \left\langle aH_\tau,\psi_{j,k} \right\rangle \right| = |a|L^{1/2}\min\{u,1-u\}.
L^2Use in source · 20 passages
Search aliases: L^2
Use in source · Paper_V_Traditional_Final.tex, line 1995
ZZ with N≥1, first take |z|=1. The assumption h∈ℓ^1( Z) makes the following symbol series absolutely convergent. Define H_ks(z)=∑_m∈ Zh(Nm+k-s)z^-m. For x∈ℓ^1( Z), the channel transforms converge absolutely on the unit circle and satisfy Y(z)= H_N(z) X(z). For x∈ℓ^2( Z), the same identity holds almost everywhere in the L^2 Fourier sense. For finite impulse-response kernels it also holds at nonzero z wherever the input channel transforms are weighted absolutely summable. This is the classical pol…
Exact mathematical passage
L^2
L^{3/2}Use in source · 3 passages
Search aliases: L^3/2
Use in source · Paper_V_Traditional_Final.tex, line 3216
Let L=2^-j and let a be the left endpoint of the support. Pair every point t∈[a,a+L/2] with t+L/2. Then ⟨ s,ψ_j,k⟩ = L^-1/2 ∫_a^a+L/2 overlines(t)-s(t+L/2) d t. The integrand has magnitude at most ML/2, and the integration interval has length L/2. The result is ML^3/2/4.
Exact mathematical passage
ML^{3/2}/4
L^pUse in source · 21 passages
Search aliases: L^p
Use in source · Paper_V_Traditional_Final.tex, line 712
[Packet perturbation bound] For 1≤ p<∞, interpreted first as an inequality of extended L^p seminorms, j-widehat j _L^p ≤ |a-widehat a| μ(Ω)^1/p + |widehat a| μ(ΩtrianglewidehatΩ)^1/p. If μ(Ω) and μ(ΩtrianglewidehatΩ) are finite, all terms in this displayed jump bound are finite and it is an ordinary real-valued L^p inequality. For the pointwise supremum, and consequently also the essential L^∞ seminorm, j-widehat j _∞ ≤ |a-widehat a|+|widehat a| 1_ΩtrianglewidehatΩ≠∅.
Exact mathematical passage
L^p
L^\alphaUse in source · 15 passages
Search aliases: L^α alpha
Use in source · Paper_VI_Traditional_Final.tex, line 1126
Let L=L^*≥0 on a finite-dimensional Hilbert space. Its spectral resolution is L=∑_jλ_jP_j. For α>0, define L^α=∑_jλ_j^α P_j. The associated fractional semigroup is T_α(t)=e^-tL^α =∑_je^-tλ_j^αP_j.
Exact mathematical passage
\boxed{L^\alpha=\sum_j\lambda_j^\alpha P_j.}
L^{\alpha+\beta}Use in source · 2 passages
Search aliases: L^α+β alpha beta
Use in source · Paper_VI_Traditional_Final.tex, line 1137
[Fractional packet calculus] For α,β>0, L^α L^β=L^α+β, and for s,t≥0, T_α(s)T_α(t)=T_α(s+t). Every spectral packet P_j reduces L^α and T_α(t).
Exact mathematical passage
L^\alpha L^\beta=L^{\alpha+\beta},
L^\betaUse in source · 3 passages
Search aliases: L^β beta
Use in source · Paper_VI_Traditional_Final.tex, line 1137
[Fractional packet calculus] For α,β>0, L^α L^β=L^α+β, and for s,t≥0, T_α(s)T_α(t)=T_α(s+t). Every spectral packet P_j reduces L^α and T_α(t).
Exact mathematical passage
L^\alpha L^\beta=L^{\alpha+\beta},
L^\inftyUse in source · 2 passages
Search aliases: L^∞ infty
Use in source · Paper_V_Traditional_Final.tex, line 722
f extended L^p seminorms, j-widehat j _L^p ≤ |a-widehat a| μ(Ω)^1/p + |widehat a| μ(ΩtrianglewidehatΩ)^1/p. If μ(Ω) and μ(ΩtrianglewidehatΩ) are finite, all terms in this displayed jump bound are finite and it is an ordinary real-valued L^p inequality. For the pointwise supremum, and consequently also the essential L^∞ seminorm, j-widehat j _∞ ≤ |a-widehat a|+|widehat a| 1_ΩtrianglewidehatΩ≠∅.
Exact mathematical passage
L^\infty
L_{-1}Use in source · 1 passage
Search aliases: L_-1
Use in source · strengthening.tex, line 197
Reciprocal transfer gives L_-1^kF_N,j∈ H_p. Extraction of the packed coefficient gives (S+1)^k C_N,j(n)∈ p^k Z_p. Moreover, Δ_1^k A_N,j(n)=(-1)^n+k(S+1)^k C_N,j(n), so A_N,j has the ordinary p^k difference property. At p=2, L_1=L_-1-1. Its integral binomial expansion on F_N,j gives Δ_1^k C_N,j(n)∈2^k Z_2 directly. For N=1 the interior sum in the product formula is empty, so the same argument applies without change. Equivalently, F_1,0(t)=e^-t.
Exact mathematical passage
L_1=L_{-1}-1
L_{-1}^kUse in source · 1 passage
Search aliases: L_-1^k
Use in source · strengthening.tex, line 190
Reciprocal transfer gives L_-1^kF_N,j∈ H_p. Extraction of the packed coefficient gives (S+1)^k C_N,j(n)∈ p^k Z_p. Moreover, Δ_1^k A_N,j(n)=(-1)^n+k(S+1)^k C_N,j(n), so A_N,j has the ordinary p^k difference property. At p=2, L_1=L_-1-1. Its integral binomial expansion on F_N,j gives Δ_1^k C_N,j(n)∈2^k Z_2 directly. For N=1 the interior sum in the product formula is empty, so the same argument applies without change. Equivalently, F_1,0(t)=e^-t.
Exact mathematical passage
L_{-1}^kF_{N,j}\in\mathcal H_p
L_{-P}Use in source · 1 passage
Search aliases: L_-P
Use in source · Paper_V_Traditional_Final.tex, line 2281
[Exact lifting update] The lifting matrices are invertible with L_P^-1=L_-P, U_Q^-1=U_-Q. If R_0E_0=I, then every finite product V of lifting matrices gives another exact bank by E=VE_0, R=R_0V^-1. Every lifting step has determinant one.
Exact mathematical passage
L_P^{-1}=L_{-P}, \qquad U_Q^{-1}=U_{-Q}.
L_{-\lambda}^bUse in source · 1 passage
Search aliases: L_-λ^b lambda
Use in source · strengthening.tex, line 122
Set B=A^-1. Its order-zero integrality was established in [source reference]. Assume by induction that L_-λ^bB∈ H_p for b<k. Since AB=1, L_0^k(AB)=(fracD^Np)^k1=0 (k>0). Use [source reference] with μ=-λ. Exactly one term has inverse order b=k: it has a=0 and all e_i=0, and is A L_-λ^k B. In every other term b<k, so its second factor is integral by induction and derivative-shift integrality. Its first factor is integral by the flatness of A, and its scalar is integral. Hence A L_-λ^k B∈ H_p. Mult…
Exact mathematical passage
L_{-\lambda}^bB\in\mathcal H_p
L_{-\lambda}^kUse in source · 3 passages
Search aliases: L_-λ^k lambda
Use in source · strengthening.tex, line 128
beginproof Set B=A^-1. Its order-zero integrality was established in [source reference]. Assume by induction that L_-λ^bB∈ H_p for b<k. Since AB=1, L_0^k(AB)=(fracD^Np)^k1=0 (k>0). Use [source reference] with μ=-λ. Exactly one term has inverse order b=k: it has a=0 and all e_i=0, and is A L_-λ^k B. In every other term b<k, so its second factor is integral by induction and derivative-shift integrality. Its first factor is integral by the flatness of A, and its scalar is integral. Hence A L_-λ^k B…
Exact mathematical passage
A L_{-\lambda}^k B
L_0Use in source · 3 passages
Search aliases: L_0
Use in source · supplement_06.tex, line 118
An independent Lucas sequence. Define L_0=2, L_1=1, L_n+2=L_n+L_n+1. Direct finite matrix arithmetic proves C^2=C+I. Multiplication by C^n gives C^n+2=C^n+C^n+1. Taking traces shows that tr(C^n) satisfies the same recurrence and initial values as L_n. Induction yields tr(C^N)=L_N. The implementation separately proves det C=-1, det(C^N)=(-1)^N, L_N=2F_N+1-F_N. No Binet roots or diagonalizability argument is needed.
Exact mathematical passage
L_0=2,\quad L_1=1,\quad L_{n+2}=L_n+L_{n+1}.
L_{03}Use in source · 1 passage
Search aliases: L_03
Use in source · atlas-pages.tex, line 334
by e^-2jt. Square-root diffusion multiplies it by e^-sqrt2j t. The constant mode is unchanged in both. vfigfractional platecaptionThe highest mode decays at rates 6 and sqrt6. The graph beside the curves shows the additional couplings in the fractional operator. Curves display the explicit formulas, not an empirical fit. headFractional does not mean entrywise Although L_03=0, its spectral square root satisfies (L^1/2)_03=fracsqrt6-sqrt2-24<0. Thus -L^1/2 permits a direct transition between state…
Exact mathematical passage
L_{03}=0
L_0^kUse in source · 1 passage
Search aliases: L_0^k
Use in source · strengthening.tex, line 123
Set B=A^-1. Its order-zero integrality was established in [source reference]. Assume by induction that L_-λ^bB∈ H_p for b<k. Since AB=1, L_0^k(AB)=(fracD^Np)^k1=0 (k>0). Use [source reference] with μ=-λ. Exactly one term has inverse order b=k: it has a=0 and all e_i=0, and is A L_-λ^k B. In every other term b<k, so its second factor is integral by induction and derivative-shift integrality. Its first factor is integral by the flatness of A, and its scalar is integral. Hence A L_-λ^k B∈ H_p. Mult…
Exact mathematical passage
L_0^k(AB)=\left(\frac{D^N}{p}\right)^k1=0\qquad(k>0).
L_1Use in source · 9 passages
Search aliases: L_1
Use in source · 04_correction_transport.tex, line 122
[Local Lyapunov--Schmidt reduction] Let U⊆ X be an open neighborhood of zero and let F: U→ Y be a C^r map between Banach spaces with r≥1, F(0)=0, and L=DF(0). Assume there are closed decompositions X=K⊕ X_1, Y=R⊕ O, where K=Ker L, R=im L, and L_1=L|_X_1:X_1⟶ R is a Banach space isomorphism. Let π_R and π_O be the two projections on Y. Then there are neighborhoods U_K⊆ K and U_1⊆ X_1 of zero and a unique C^r map ψ:U_K⟶ U_1 with ψ(0)=0 such that π_RF(k+ψ(k))=0 for every k∈ U_K. After the neighborh…
Exact mathematical passage
L_1=L|_{X_1}:X_1\longrightarrow R
L_1^kUse in source · 1 passage
Search aliases: L_1^k
Use in source · strengthening.tex, line 166
n adds the factor -(k+1)s in the numerator. Its denominator is a unit, so it belongs to p^k Z_p. The discrete Leibniz formula Δ_1^k(fg)(n) =∑_r=0^kbinom kr (Δ_1^r f)(n)(Δ_1^k-rg)(n+r) then proves Δ_1^k g_N,j(n)∈ p^k Z_p for every k,n. The coefficient at degree Nn of L_1^kG_N,j is precisely p^-kΔ_1^k g_N,j(n), and the other residue classes vanish. Thus the actual source is 1-flat.
Exact mathematical passage
L_1^kG_{N,j}
L_2Use in source · 5 passages
Search aliases: L_2
Use in source · 72_recovered.tex, line 1256
dfrac2^4r+1(2r+1)(2r+2), · p=2. Then for every prime p and every even n≥0, A_p,n = ∑_r=0^n/2 binom n2r λ_p,r E_n-2r. Equivalently, if L_p(t)=K_p(t)^-1, then H_p(t)=L_p(t)secht, where L_p(t)=cosh^2(pt) for odd p, and L_2(t)=(fracsinh(2t)2t)^2 for p=2.
Exact mathematical passage
L_2(t)=\left(\frac{\sinh(2t)}{2t}\right)^2
L_3Use in source · 2 passages
Search aliases: L_3
Use in source · 72_recovered.tex, line 1277
l source charts] Editorial range correction. This comparison holds for the even source subsequences, equivalently the evenized generating functions H_p. It does not identify their full odd-degree source values. For p=3, the original odd source has A_3,1=S_1(3)=3, whereas every odd dyadic source coefficient is zero. The even kernel L_3K_2 cannot change that odd coefficient. Read the coefficient consequence below only at even indices. The displayed identity H_p=L_pK_ℓ H_ℓ and its proof remain vali…
Exact mathematical passage
L_3K_2
L_4Use in source · 1 passage
Search aliases: L_4
Use in source · supplement_06.tex, line 151
s.state_is_orbit, FibonacciPackets.packet_is_fibonacci targetFibonacciPackets.trace_power_is_lucas, FibonacciPackets.det_power targetFibonacciPackets.jump_charpoly targetFibonacciPackets.fibonacci_packet_recurrence pageheading4. Exact examples and meaningful boundary checksExample 9.3 and supporting results The independently defined Lucas recurrence gives (L_0,L_1,L_2,L_3,L_4,L_5)=(2,1,3,4,7,11). For N=3, the general theorem therefore specializes to u_k+2=4u_k+1+u_k. The implementation proves th…
Exact mathematical passage
(L_0,L_1,L_2,L_3,L_4,L_5)=(2,1,3,4,7,11).
L_5Use in source · 1 passage
Search aliases: L_5
Use in source · supplement_06.tex, line 151
s.state_is_orbit, FibonacciPackets.packet_is_fibonacci targetFibonacciPackets.trace_power_is_lucas, FibonacciPackets.det_power targetFibonacciPackets.jump_charpoly targetFibonacciPackets.fibonacci_packet_recurrence pageheading4. Exact examples and meaningful boundary checksExample 9.3 and supporting results The independently defined Lucas recurrence gives (L_0,L_1,L_2,L_3,L_4,L_5)=(2,1,3,4,7,11). For N=3, the general theorem therefore specializes to u_k+2=4u_k+1+u_k. The implementation proves th…
Exact mathematical passage
(L_0,L_1,L_2,L_3,L_4,L_5)=(2,1,3,4,7,11).
L_cUse in source · 7 passages
Search aliases: L_c
Use in source · Paper_VI_Traditional_Final.tex, line 2004
For a positive operator L, spectral packets define L^α and e^-tL^α. Exact intertwining passes to every fractional power. When both fine and coarse operators have a common positive lower bound m, the Balakrishnan formula gives the quantitative estimate normCL_f^α-L_c^α C ≤ α m^α-1 normCL_f-L_cC, 0<α<1. A Duhamel argument then controls the fractional semigroup defect. This closes a finite version of fractional naturality under explicit spectral hypotheses.
Exact mathematical passage
\norm{CL_f^\alpha-L_c^\alpha C} \leq \alpha m^{\alpha-1} \norm{CL_f-L_cC}, \qquad 0<\alpha<1.
L_c^\alphaUse in source · 7 passages
Search aliases: L_c^α alpha
Use in source · Paper_VI_Traditional_Final.tex, line 2004
For a positive operator L, spectral packets define L^α and e^-tL^α. Exact intertwining passes to every fractional power. When both fine and coarse operators have a common positive lower bound m, the Balakrishnan formula gives the quantitative estimate normCL_f^α-L_c^α C ≤ α m^α-1 normCL_f-L_cC, 0<α<1. A Duhamel argument then controls the fractional semigroup defect. This closes a finite version of fractional naturality under explicit spectral hypotheses.
Exact mathematical passage
\norm{CL_f^\alpha-L_c^\alpha C} \leq \alpha m^{\alpha-1} \norm{CL_f-L_cC}, \qquad 0<\alpha<1.
L_dUse in source · 1 passage
Search aliases: L_d
Use in source · Paper_X_Block27_Amended.tex, line 2234
General finite logarithmic recovery. Let A be a rational algebra, without any commutativity assumption on A. For a unipotent element U and a positive integer d satisfying (U-1)^d=0, put L_d(U)=∑_j=1^d-1frac(-1)^j+1j(U-1)^j. The value is independent of the chosen annihilating exponent. The finite logarithm and the nilpotent exponential satisfy L(exp N)=N, exp(L(U))=U. Both maps preserve every annihilating exponent, and hence preserve the exact nilpotency class of N and exp(N)-1, or of U-1 and L(U…
Exact mathematical passage
L_d(U)=\sum_{j=1}^{d-1}\frac{(-1)^{j+1}}{j}(U-1)^j.
L_fUse in source · 21 passages
Search aliases: L_f
Use in source · 01_finite_coordinates.tex, line 85
Words are written in execution order. In L_fL_e, the rightmost operator L_e acts first. For a written word (e_1,…,e_n), e_1 acts first and the operator product is L_e_n⋯ L_e_1.
Exact mathematical passage
L_fL_e
\mathcal L_{F,L}Use in source · 1 passage
Search aliases: L_F,L mathcal
Use in source · 50_open_questions.tex, line 21
For a fixed polynomial system F∈ Z[X_1,…,X_d]^e, a precision L≥1, and a root bar x modulo p^L, let L_F,L(bar x) ={x'∈( Z/p^L+1 Z)^d: F(x')=0, x'≡bar xpmodp^L}. The finite obstruction theorem decides whether this set is empty from the residual digit and reduced Jacobian. When it is nonempty, the theorem also determines its cardinality and affine kernel structure. The conclusion concerns one level: vanishing of the obstruction permits the next step, but does not guarantee that subsequent steps are…
Exact mathematical passage
\mathcal L_{F,L}(\bar x) =\{x'\in(\Z/p^{L+1}\Z)^d: F(x')=0,\ x'\equiv\bar x\pmod{p^L}\}.
L_f^\alphaUse in source · 7 passages
Search aliases: L_f^α alpha
Use in source · Paper_VI_Traditional_Final.tex, line 2004
For a positive operator L, spectral packets define L^α and e^-tL^α. Exact intertwining passes to every fractional power. When both fine and coarse operators have a common positive lower bound m, the Balakrishnan formula gives the quantitative estimate normCL_f^α-L_c^α C ≤ α m^α-1 normCL_f-L_cC, 0<α<1. A Duhamel argument then controls the fractional semigroup defect. This closes a finite version of fractional naturality under explicit spectral hypotheses.
Exact mathematical passage
\norm{CL_f^\alpha-L_c^\alpha C} \leq \alpha m^{\alpha-1} \norm{CL_f-L_cC}, \qquad 0<\alpha<1.
L_hUse in source · 9 passages
Search aliases: L_h
Use in source · Paper_VI_Traditional_Final.tex, line 1368
For n interior points with spacing h=1/(n+1), define L_h=frac1h^2 2 · -1 · · -1 · 2 · ddots · · ddots · ddots · -1 · · -1 · 2 .
Exact mathematical passage
L_h=\frac1{h^2} \begin{pmatrix} 2&-1&&\\ -1&2&\ddots&\\ &\ddots&\ddots&-1\\ &&-1&2 \end{pmatrix}.
L_h^{-1}Use in source · 9 passages
Search aliases: L_h^-1
Use in source · Paper_VI_Traditional_Final.tex, line 1380
[Exact discrete Green reconstruction at grid nodes] Let x_i=ih. Then frac1h(L_h^-1)_ij =G(x_i,x_j). Thus the scaled discrete inverse reproduces the continuum Dirichlet Green kernel exactly at the grid nodes.
Exact mathematical passage
\boxed{ \frac1h(L_h^{-1})_{ij} =G(x_i,x_j). }
L_iUse in source · 4 passages
Search aliases: L_i
Use in source · Paper_VI_Traditional_Final.tex, line 5405
tly positive finite likelihood and let B_L(p)=Lodot p/ipLp for a probability vector p. Fix probability vectors r^(a) supported on B_a, and restrict the priors to p_i=q_a r_i^(a) for i∈ B_a, with q a coarse probability vector. [Sufficiency relative to a prior disintegration] Define overline L_a = ∑_i∈ B_aL_ir_i^(a). For every prior in the declared family, C B_L(p) = B_overline L(q).
Exact mathematical passage
\overline L_a = \sum_{i\in B_a}L_ir_i^{(a)}.
L_jUse in source · 6 passages
Search aliases: L_j
Use in source · root_scope.tex, line 204
For a wall label j≥1 and 0≤ q<j, define e_j(q)=3q+s_2(j-1)-s_2(j-q-1), E_j=min_q<je_j(q), Θ(j)=3j+1+s_2(j-1), L_j=Θ(j)-E_j. Here E_j is a carry minimum, not an Euler number. A wall label is regular when e_j(q)>0 for 1≤ q<j. For r=2^L-j with L>L_j, put δ=L-L_j. The carry identities give μ_r=E_j-δ, J_r={j}, G_r(x)=P_j(x)U(x-j)+2^δ V(x), where U≡1pmod2. For a regular label, also E_j=0 and V≡1pmod2.
Exact mathematical passage
\Theta(j)=3j+1+s_2(j-1),\qquad L_j=\Theta(j)-E_j.
\mathcal L_{j+1}Use in source · 3 passages
Search aliases: L_j+1 mathcal
Use in source · 02_supplement_03.tex, line 120
Define the comparison space L_j+1=⊕_i K_j+1(P_iDP_i,P_iv). Packet diagonality is first shown to imply commutation. The exact theorem then proves D^jv∈ L_j+1. Using that vector as a witness in the infimum distance gives dist(A^jv, L_j+1) ≤ jM^j-1normKnormv. This statement uses Lean's actual metric infimum distance to the subspace. It does not assume that the comparison vector lies there. The block-Jacobi condition bound and the preconditioned conjugate-gradient corollary are not completed by thes…
Exact mathematical passage
\mathcal L_{j+1}=\bigoplus_i\mathcal K_{j+1}(P_iDP_i,P_iv).
L_kUse in source · 5 passages
Search aliases: L_k
Use in source · 01_manuscript.tex, line 825
R=Q_RR_R, where the columns of Q_L and Q_R are orthonormal. Form the small core G=R_LR_R^* and let G=WΣ Z^* be an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer k≥0, retain min{k,d} columns, where d is the number of supplied thin singular directions. Tail singular values beyond d are zero. Define L_k=Q_LW_kΣ_k^1/2, R_k=Q_RZ_kΣ_k^1/2. Then L_kR_k^* is a best rank at most k approximation to LR^*. Its spectral and Frobenius errors are the s…
Exact mathematical passage
L_k=Q_LW_k\Sigma_k^{1/2}, \qquad R_k=Q_RZ_k\Sigma_k^{1/2}.
L_mUse in source · 10 passages
Search aliases: L_m
Use in source · 04_correction_transport.tex, line 217
Suppose x_m satisfies Φ(x_m)∈ F^mY. Assume that corrections h∈ F^mX obey a first-order congruence Φ(x_m+h) ≡ Φ(x_m)+L_mh pmodF^m+1Y, where L_m:F^mX→ F^mY is a module homomorphism satisfying L_m(F^m+1X)⊆ F^m+1Y and therefore induces overline L_m: F^mX/F^m+1X ⟶ F^mY/F^m+1Y.
Exact mathematical passage
\Phi(x_m+h) \equiv \Phi(x_m)+L_mh \pmod{F^{m+1}Y},
L_m'Use in source · 1 passage
Search aliases: L_m'
Use in source · 04_correction_transport.tex, line 364
[Naturality of filtered obstructions] Suppose a morphism of filtered lifting problems sends x_m to x_m', intertwines L_m with L_m', and sends Φ(x_m) to Φ'(x_m') on the associated graded pieces. Then the induced cokernel map sends Ob_m(x_m) to Ob_m'(x_m').
Exact mathematical passage
L_m'
L_{m-1}Use in source · 2 passages
Search aliases: L_m-1
Use in source · 02_supplement_03.tex, line 19
Construction. Suppose P_m-1=D^(m-1)+L_m-1R_m-1^*. The new factors are L_m · =[D_mL_m-1+U_m(V_m^*L_m-1) U_m],
Exact mathematical passage
P_{m-1}=D^{(m-1)}+L_{m-1}R_{m-1}^*
L_nUse in source · 6 passages
Search aliases: L_n
Use in source · supplement_06.tex, line 118
An independent Lucas sequence. Define L_0=2, L_1=1, L_n+2=L_n+L_n+1. Direct finite matrix arithmetic proves C^2=C+I. Multiplication by C^n gives C^n+2=C^n+C^n+1. Taking traces shows that tr(C^n) satisfies the same recurrence and initial values as L_n. Induction yields tr(C^N)=L_N. The implementation separately proves det C=-1, det(C^N)=(-1)^N, L_N=2F_N+1-F_N. No Binet roots or diagonalizability argument is needed.
Exact mathematical passage
L_0=2,\quad L_1=1,\quad L_{n+2}=L_n+L_{n+1}.
L_{n+1}Use in source · 1 passage
Search aliases: L_n+1
Use in source · supplement_06.tex, line 118
An independent Lucas sequence. Define L_0=2, L_1=1, L_n+2=L_n+L_n+1. Direct finite matrix arithmetic proves C^2=C+I. Multiplication by C^n gives C^n+2=C^n+C^n+1. Taking traces shows that tr(C^n) satisfies the same recurrence and initial values as L_n. Induction yields tr(C^N)=L_N. The implementation separately proves det C=-1, det(C^N)=(-1)^N, L_N=2F_N+1-F_N. No Binet roots or diagonalizability argument is needed.
Exact mathematical passage
L_0=2,\quad L_1=1,\quad L_{n+2}=L_n+L_{n+1}.
L_{n+2}Use in source · 1 passage
Search aliases: L_n+2
Use in source · supplement_06.tex, line 118
An independent Lucas sequence. Define L_0=2, L_1=1, L_n+2=L_n+L_n+1. Direct finite matrix arithmetic proves C^2=C+I. Multiplication by C^n gives C^n+2=C^n+C^n+1. Taking traces shows that tr(C^n) satisfies the same recurrence and initial values as L_n. Induction yields tr(C^N)=L_N. The implementation separately proves det C=-1, det(C^N)=(-1)^N, L_N=2F_N+1-F_N. No Binet roots or diagonalizability argument is needed.
Exact mathematical passage
L_0=2,\quad L_1=1,\quad L_{n+2}=L_n+L_{n+1}.
\mathcal L_{n,v}Use in source · 2 passages
Search aliases: L_n,v mathcal
Use in source · continuation_09.tex, line 56
Exact branching. At any stage n, put N=L+n. For every parameter digit v∈ F_p^k, let L_n,v={bar w∈( Z/p^N+1 Z)^m+k: l bar w≡ z(n)pmodp^N, F(bar w)=0, free(bar w)=t(n)+p^Nvpmodp^N+1 }. Here p^Nv means any integer representative of v, multiplied by p^N. The residue class does not depend on the representative. Then # L_n,v=p^m-e for every n and every v. Surjectivity implies e≤ m. This is a count of distinct actual next-level residue classes. It does not count integer representatives or entire infini…
Exact mathematical passage
\mathcal L_{n,v}=\left\{\bar w\in(\mathbb Z/p^{N+1}\mathbb Z)^{m+k}: \begin{array}{l} \bar w\equiv z(n)\pmod{p^N},\\ F(\bar w)=0,\\ \operatorname{free}(\bar w)=t(n)+p^Nv\pmod{p^{N+1}} \end{array}\right\}.
L_PUse in source · 1 passage
Search aliases: L_P
Use in source · Paper_V_Traditional_Final.tex, line 2263
Let P(z) and Q(z) be Laurent polynomials. Define L_P(z) = 1 · 0 P(z) · 1 , U_Q(z) = 1 · Q(z) 0 · 1 .
Exact mathematical passage
L_P(z) = \begin{pmatrix} 1&0\\ P(z)&1 \end{pmatrix}, \qquad U_Q(z) = \begin{pmatrix} 1&Q(z)\\ 0&1 \end{pmatrix}.
L_pUse in source · 11 passages
Search aliases: L_p
Use in source · 71_native.tex, line 207
For a finite set S of primes, positive precisions L_p, and a fixed integer polynomial system F, put
Exact mathematical passage
L_p
L_P^{-1}Use in source · 1 passage
Search aliases: L_P^-1
Use in source · Paper_V_Traditional_Final.tex, line 2281
[Exact lifting update] The lifting matrices are invertible with L_P^-1=L_-P, U_Q^-1=U_-Q. If R_0E_0=I, then every finite product V of lifting matrices gives another exact bank by E=VE_0, R=R_0V^-1. Every lifting step has determinant one.
Exact mathematical passage
L_P^{-1}=L_{-P}, \qquad U_Q^{-1}=U_{-Q}.
L_rUse in source · 4 passages
Search aliases: L_r
Use in source · 02_strengthened.tex, line 1214
[Finite Lindblad generator] A norm continuous semigroup of completely positive trace preserving maps on a finite matrix algebra, in the Schr odinger picture, has a generator of the form G(X) = - i[H,X] + ∑_r ( L_rXL_r^* - frac12L_r^*L_rX - frac12XL_r^*L_r ), where H=H^*. Conversely, every operator of this form generates a completely positive trace preserving semigroup [source reference].
Exact mathematical passage
\Gcal(X) = -\ii[H,X] + \sum_r \left( L_rXL_r^* - \frac12L_r^*L_rX - \frac12XL_r^*L_r \right),
L_r^*Use in source · 3 passages
Search aliases: L_r^*
Use in source · 02_strengthened.tex, line 1214
[Finite Lindblad generator] A norm continuous semigroup of completely positive trace preserving maps on a finite matrix algebra, in the Schr odinger picture, has a generator of the form G(X) = - i[H,X] + ∑_r ( L_rXL_r^* - frac12L_r^*L_rX - frac12XL_r^*L_r ), where H=H^*. Conversely, every operator of this form generates a completely positive trace preserving semigroup [source reference].
Exact mathematical passage
\Gcal(X) = -\ii[H,X] + \sum_r \left( L_rXL_r^* - \frac12L_r^*L_rX - \frac12XL_r^*L_r \right),
L_TUse in source · 5 passages
Search aliases: L_T
Use in source · 20_arithmetic.tex, line 258
To pass from the section to the singular ratio, factor out its initial zero: for c=ap, the section has the form H_c(log(1+T))=fracT^cc! V(T), V(0)=1, V∈ W_p. The weight of c! compensates for the coefficient shift by c. The weighted ring is closed under inversion of series with constant coefficient one and under L_T=log(1+T)(1+T)d/dT. With ℓ(T)=log(1+T)/T, interpreted by coefficient shifting, differentiation of the displayed factorization gives R_c(log(1+T))=c(1+T)ℓ(T)+(L_TV)V^-1. Since pℓ(T)∈ W_…
Exact mathematical passage
L_T=\log(1+T)(1+T)d/dT
\mathcal L_{x,L}Use in source · 6 passages
Search aliases: L_x,L mathcal
Use in source · 30_lifting.tex, line 281
At positive polynomial depth, restrict it to the correction fiber T_x={t∈ S_q^d:overliner_x+J_xt=0} and the lift set L_x,L= {y∈( Z/q^L+1 Z)^d:y≡ xpmodq^L, F(y)=0}. The polynomial predicate in the second display is evaluation in the residue ring. Evaluation on canonical integer representatives gives the same predicate, because integer polynomial evaluation respects congruence.
Exact mathematical passage
\mathcal L_{x,L}= \{y\in(\Z/q^{L+1}\Z)^d:y\equiv x\pmod{q^L},\ F(y)=0\}.
\mathcal L_{x,L,v}Use in source · 3 passages
Search aliases: L_x,L,v mathcal
Use in source · 30_lifting.tex, line 543
Suppose now that the split into solved variables and parameter variables is fixed in advance. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0), and assume F(x)=q^Lr_x with q≥1 and L≥1. Write the reduced Jacobian as DF(x)bmod q=(J_x K_x), J_x:S_q^m→ S_q^e, K_x:S_q^k→ S_q^e. For a prescribed parameter digit v∈ S_q^k, let L_x,L,v be the set of next residue roots over x whose parameter coordinates are t_0+q^Lwidetilde v modulo q^L+1.
Exact mathematical passage
\mathcal L_{x,L,v}
L_zUse in source · 2 passages
Search aliases: L_z
Use in source · Paper_VI_Traditional_Final.tex, line 4498
Set L_z= I · -A_PQ(zI-A_QQ)^-1 0 · I and S_z= F_P(z) · 0
Exact mathematical passage
L_z= \begin{pmatrix} I&-A_{PQ}(zI-A_{QQ})^{-1}\\ 0&I \end{pmatrix}
\mathcal L_{z_n,L+n}Use in source · 1 passage
Search aliases: L_z_n,L+n mathcal
Use in source · continuation_03.tex, line 98
Surjectivity implies rank_ F_pA_x=e, e≤ d. Combining this with the preceding exact residue-lift count gives # L_z_n,L+n=p^d-e for every n. These are distinct solutions modulo p^L+n+1 lying above the selected point modulo p^L+n. The original finite obstruction group at each smooth point has cardinality one.
Exact mathematical passage
\boxed{\#\mathcal L_{z_n,L+n}=p^{d-e}\quad\text{for every }n.}
L_{\lambda+\mu}Use in source · 1 passage
Search aliases: L_λ+μ lambda mu
Use in source · strengthening.tex, line 75
[Translated product formula] For N=p^a, a≥1, and λ,μ∈ Z_p, L_λ+μ(AB) =(L_λ A)B+A(L_μ B) +∑_i=1^N-1c_i(D^iA)(D^N-iB), c_i=frac1pbinom Ni∈ Z. Flatness types add under multiplication.
Exact mathematical passage
L_{\lambda+\mu}(AB) =(L_\lambda A)B+A(L_\mu B) +\sum_{i=1}^{N-1}c_i(D^iA)(D^{N-i}B), \qquad c_i=\frac1p\binom Ni\in\mathbb Z. \label{eq:translated-product}
L_{\lambda+\mu}^kUse in source · 1 passage
Search aliases: L_λ+μ^k lambda mu
Use in source · strengthening.tex, line 99
For completeness, the iterated formula is an integral multinomial formula, rather than a product rule with unspecified remainders. On the tensor of the two factors use the mutually commuting operators L_λ⊗1, 1⊗ L_μ, c_iD^i⊗ D^N-i (1≤ i<N). Applying multiplication after the kth power of their sum gives exactly L_λ+μ^k(AB). Expanding it yields terms indexed by a,b,e_1,…,e_N-1≥0, with a+b+∑_i e_i=k: frack!a!b!∏_i e_i!∏_i c_i^e_i (D^α L_λ^a A)(D^β L_μ^b B), α=∑_i i e_i, β=∑_i(N-i)e_i. Every scalar i…
Exact mathematical passage
L_{\lambda+\mu}^k(AB)
L_\lambda^aUse in source · 1 passage
Search aliases: L_λ^a lambda
Use in source · strengthening.tex, line 101
e with unspecified remainders. On the tensor of the two factors use the mutually commuting operators L_λ⊗1, 1⊗ L_μ, c_iD^i⊗ D^N-i (1≤ i<N). Applying multiplication after the kth power of their sum gives exactly L_λ+μ^k(AB). Expanding it yields terms indexed by a,b,e_1,…,e_N-1≥0, with a+b+∑_i e_i=k: frack!a!b!∏_i e_i!∏_i c_i^e_i (D^α L_λ^a A)(D^β L_μ^b B), α=∑_i i e_i, β=∑_i(N-i)e_i. Every scalar is integral. If A and B have the stated flatness types, both factors in every term belong to H_p, as …
Exact mathematical passage
\frac{k!}{a!b!\prod_i e_i!}\prod_i c_i^{e_i} (D^\alpha L_\lambda^a A)(D^\beta L_\mu^b B), \quad \alpha=\sum_i i e_i,\quad \beta=\sum_i(N-i)e_i. \label{eq:translated-multinomial}
L_\muUse in source · 2 passages
Search aliases: L_μ mu
Use in source · strengthening.tex, line 75
[Translated product formula] For N=p^a, a≥1, and λ,μ∈ Z_p, L_λ+μ(AB) =(L_λ A)B+A(L_μ B) +∑_i=1^N-1c_i(D^iA)(D^N-iB), c_i=frac1pbinom Ni∈ Z. Flatness types add under multiplication.
Exact mathematical passage
L_{\lambda+\mu}(AB) =(L_\lambda A)B+A(L_\mu B) +\sum_{i=1}^{N-1}c_i(D^iA)(D^{N-i}B), \qquad c_i=\frac1p\binom Ni\in\mathbb Z. \label{eq:translated-product}
L_\mu^bUse in source · 1 passage
Search aliases: L_μ^b mu
Use in source · strengthening.tex, line 101
e with unspecified remainders. On the tensor of the two factors use the mutually commuting operators L_λ⊗1, 1⊗ L_μ, c_iD^i⊗ D^N-i (1≤ i<N). Applying multiplication after the kth power of their sum gives exactly L_λ+μ^k(AB). Expanding it yields terms indexed by a,b,e_1,…,e_N-1≥0, with a+b+∑_i e_i=k: frack!a!b!∏_i e_i!∏_i c_i^e_i (D^α L_λ^a A)(D^β L_μ^b B), α=∑_i i e_i, β=∑_i(N-i)e_i. Every scalar is integral. If A and B have the stated flatness types, both factors in every term belong to H_p, as …
Exact mathematical passage
\frac{k!}{a!b!\prod_i e_i!}\prod_i c_i^{e_i} (D^\alpha L_\lambda^a A)(D^\beta L_\mu^b B), \quad \alpha=\sum_i i e_i,\quad \beta=\sum_i(N-i)e_i. \label{eq:translated-multinomial}
L_\inftyUse in source · 1 passage
Search aliases: L_∞ infty
Use in source · 03_nonlinear_coordinates.tex, line 645
[Packet equations for L_∞ operations] Let L be complete and carry operations ℓ_r that respect a finite abelian group grading. Assume the packet projectors are continuous for the topology in which the series below converges. Assume the Maurer Cartan series ∑_r≥1frac1r!ℓ_r(η,…,η) converges and the required factorials are invertible. Then its packet of label k is ∑_r≥1frac1r! ∑_g_1+⋯+g_r=k ℓ_r(η_g_1,…,η_g_r). The full Maurer Cartan equation holds if and only if every packet equation vanishes.
Exact mathematical passage
L_\infty
\landUse in source · 1 passage
Search aliases: land
Use in source · Paper_X_Block27_Amended.tex, line 1803
An admissible realization here includes continuous coefficient-field members and membership of every finite reduction in its designated admissible class. Under the completion hypotheses and these supplied carrier hypotheses, Lean proves · ∃ ρ, (x_p)_p: ρ is an admissible realization of the full chosen spine, 0≠ x_p∈im(e_p) for every p · Longleftrightarrow (∀λ, X_λ,1( C)≠∅) land (∀λ, ∀ m≥2, X_λ,m( C)≠∅) land (∀ p, n_p>0). The packet index may be infinite. Each x_p belongs to its own supplied carr…
Exact mathematical passage
\begin{aligned} &\exists\,\rho,\ (x_p)_p:\quad \rho\text{ is an admissible realization of the full chosen spine},\quad 0\ne x_p\in\operatorname{im}(e_p)\text{ for every }p\\ &\quad\Longleftrightarrow\quad \bigl(\forall\lambda,\ X_{\lambda,1}(\Compat)\ne\varnothing\bigr) \ \land\ \bigl(\forall\lambda,\ \forall m\ge2, X_{\lambda,m}(\Compat)\ne\varnothing\bigr) \ \land\ \bigl(\forall p,\ n_p>0\bigr). \end{aligned}
\LcalUse in source · 54 passages
Search aliases: Lcal L mathcal
Use in source · 03_nonlinear_coordinates.tex, line 615
[Packet Maurer Cartan equations] Let L=⊕_g∈ G L_g be a differential graded Lie algebra over a ring in which two is invertible. Assume d L_g⊆ L_g, [ L_g, L_h]⊆ L_g+h. For a degree one element η=∑_gη_g, the Maurer Cartan equation dη+frac12[η,η]=0 is equivalent to dη_k + frac12 ∑_g+h=k[η_g,η_h] =0 for every k∈ G.
Exact mathematical passage
\Lcal=\bigoplus_{g\in G}\Lcal_g
\Lcal^1Use in source · 2 passages
Search aliases: Lcal^1 L^1 Lcal mathcal
Use in source · 04_correction_transport.tex, line 395
lifting obstruction] Let L be a complete filtered differential graded Lie algebra over a ring in which two is invertible. Assume dF^i L⊆ F^i L, [F^i L,F^j L]⊆ F^i+j L. For a degree one element η, put Ω(η)=dη+frac12[η,η]. Suppose Ω(η)∈ F^m L^2 with m≥1. For every h∈ F^m L^1, Ω(η+h) ≡ Ω(η)+d_η h pmodF^m+1 L^2, d_η=d+[η,mathord·]. The obstruction to improving η by one depth is therefore the class of Ω(η) in the cokernel of gr^m(d_η): gr^m L^1 ⟶ gr^m L^2.
Exact mathematical passage
h\in F^m\Lcal^1
\Lcal^2Use in source · 5 passages
Search aliases: Lcal^2 L^2 Lcal mathcal
Use in source · 04_correction_transport.tex, line 395
[Maurer Cartan lifting obstruction] Let L be a complete filtered differential graded Lie algebra over a ring in which two is invertible. Assume dF^i L⊆ F^i L, [F^i L,F^j L]⊆ F^i+j L. For a degree one element η, put Ω(η)=dη+frac12[η,η]. Suppose Ω(η)∈ F^m L^2 with m≥1. For every h∈ F^m L^1, Ω(η+h) ≡ Ω(η)+d_η h pmodF^m+1 L^2, d_η=d+[η,mathord·]. The obstruction to improving η by one depth is therefore the class of Ω(η) in the cokernel of gr^m(d_η): gr^m L^1 ⟶ gr^m L^2.
Exact mathematical passage
\Omega(\eta)\in F^m\Lcal^2
\Lcal_*Use in source · 2 passages
Search aliases: Lcal_* L_* Lcal mathcal
Use in source · Paper_V_Traditional_Final.tex, line 2486
[Stability of the selected basis] Assume the optimal pruning L_* for c is unique and at least one competing pruning exists. Define g=min_ L≠ L_*(C( L)-C( L_*))>0. If 2ε_*<g, then the optimal pruning for widetilde c is the same. If only one admissible pruning exists, it remains selected under every cost change without a gap condition.
Exact mathematical passage
\Lcal_*
\Lcal_0Use in source · 1 passage
Search aliases: Lcal_0 L_0 Lcal mathcal
Use in source · monograph_atlas.tex, line 580
vpage31Coherence order survives a covariant channelCHANNEL, CHOI, AND GENERATOR CHECKSChapter VII: Theorem II.7.8, Theorem II.7.10, Theorem II.7.14. Take S=Z=diag(1,-1) on C^2. Under conjugation X↦ ZXZ, diagonal matrices have coherence order zero and off-diagonal matrices have coherence order one modulo two. Its projectors on Hilbert--Schmidt operator space are L_0(X)=(X+ZXZ)/2 and L_1(X)=(X-ZXZ)/2.
Exact mathematical passage
\Lcal_0(X)=(X+ZXZ)/2
\Lcal_1Use in source · 7 passages
Search aliases: Lcal_1 L_1 Lcal mathcal
Use in source · monograph_atlas.tex, line 580
ence order survives a covariant channelCHANNEL, CHOI, AND GENERATOR CHECKSChapter VII: Theorem II.7.8, Theorem II.7.10, Theorem II.7.14. Take S=Z=diag(1,-1) on C^2. Under conjugation X↦ ZXZ, diagonal matrices have coherence order zero and off-diagonal matrices have coherence order one modulo two. Its projectors on Hilbert--Schmidt operator space are L_0(X)=(X+ZXZ)/2 and L_1(X)=(X-ZXZ)/2.
Exact mathematical passage
\Lcal_1(X)=(X-ZXZ)/2
\Lcal_2Use in source · 6 passages
Search aliases: Lcal_2 L_2 Lcal mathcal
Use in source · 01_manuscript.tex, line 1601
[Monotone Lagrangian frontier] Let 0≤λ_1<λ_2. Choose optimal prunings L_1 and L_2 for the two Lagrangian parameters. Then err( L_2) ≤ err( L_1) and cost( L_2) ≥ cost( L_1).
Exact mathematical passage
\Lcal_2
\Lcal_{2g}Use in source · 1 passage
Search aliases: Lcal_2g L_2g Lcal mathcal
Use in source · 03_nonlinear_coordinates.tex, line 674
[Divided square packet equations] Assume q( L_g)⊆ L_2g. Choose an ordering of the finite label set. The equation dη+q(η)=0 is equivalent to dη_k + ∑_2g=kq(η_g) + ∑_substackg<h
Exact mathematical passage
q(\Lcal_g)\subseteq\Lcal_{2g}
\Lcal_gUse in source · 3 passages
Search aliases: Lcal_g L_g Lcal mathcal
Use in source · 03_nonlinear_coordinates.tex, line 615
[Packet Maurer Cartan equations] Let L=⊕_g∈ G L_g be a differential graded Lie algebra over a ring in which two is invertible. Assume d L_g⊆ L_g, [ L_g, L_h]⊆ L_g+h. For a degree one element η=∑_gη_g, the Maurer Cartan equation dη+frac12[η,η]=0 is equivalent to dη_k + frac12 ∑_g+h=k[η_g,η_h] =0 for every k∈ G.
Exact mathematical passage
\Lcal=\bigoplus_{g\in G}\Lcal_g
\Lcal_{g+h}Use in source · 1 passage
Search aliases: Lcal_g+h L_g+h Lcal mathcal
Use in source · 03_nonlinear_coordinates.tex, line 619
[Packet Maurer Cartan equations] Let L=⊕_g∈ G L_g be a differential graded Lie algebra over a ring in which two is invertible. Assume d L_g⊆ L_g, [ L_g, L_h]⊆ L_g+h. For a degree one element η=∑_gη_g, the Maurer Cartan equation dη+frac12[η,η]=0 is equivalent to dη_k + frac12 ∑_g+h=k[η_g,η_h] =0 for every k∈ G.
Exact mathematical passage
d\Lcal_g\subseteq\Lcal_g, \qquad [\Lcal_g,\Lcal_h]\subseteq\Lcal_{g+h}.
\Lcal_hUse in source · 1 passage
Search aliases: Lcal_h L_h Lcal mathcal
Use in source · 03_nonlinear_coordinates.tex, line 619
[Packet Maurer Cartan equations] Let L=⊕_g∈ G L_g be a differential graded Lie algebra over a ring in which two is invertible. Assume d L_g⊆ L_g, [ L_g, L_h]⊆ L_g+h. For a degree one element η=∑_gη_g, the Maurer Cartan equation dη+frac12[η,η]=0 is equivalent to dη_k + frac12 ∑_g+h=k[η_g,η_h] =0 for every k∈ G.
Exact mathematical passage
d\Lcal_g\subseteq\Lcal_g, \qquad [\Lcal_g,\Lcal_h]\subseteq\Lcal_{g+h}.
\Lcal_kUse in source · 1 passage
Search aliases: Lcal_k L_k Lcal mathcal
Use in source · 03_nonlinear_coordinates.tex, line 640
The differential preserves the group label and the bracket adds labels. Apply the projector onto L_k and use exact reconstruction.
Exact mathematical passage
\Lcal_k
\Lcal_pUse in source · 2 passages
Search aliases: Lcal_p L_p Lcal mathcal
Use in source · 02_strengthened.tex, line 1241
[Charge preservation by a Lindblad generator] Let S be unitary and N≥1 an integer with S^N=I. Assume SHS^*=H and, for every jump operator, there is a charge q_r such that SL_rS^*=ω^q_rL_r. Then the Lindblad generator G preserves every coherence order: G L_p= L_p G. The quantum dynamical semigroup e^t G has the same property.
Exact mathematical passage
\Gcal\Lcal_p=\Lcal_p\Gcal.
\Lcal_qUse in source · 5 passages
Search aliases: Lcal_q L_q Lcal mathcal
Use in source · 02_strengthened.tex, line 1099
[Coherence order projectors] For q∈ Z/N Z, define L_q(X) = frac1N ∑_r=0^N-1 ω^-qrS^rXS^-r. Then the L_q form a complete orthogonal packet system on Hilbert--Schmidt operator space. Their ranges are the operators satisfying SXS^*=ω^qX. If P_j are the state charge projectors, then Ran L_q = ⊕_jP_j+qEnd( H)P_j.
Exact mathematical passage
\Lcal_q(X) = \frac1N \sum_{r=0}^{N-1} \omega^{-qr}S^rXS^{-r}.
\Lcal_wUse in source · 1 passage
Search aliases: Lcal_w L_w Lcal mathcal
Use in source · Paper_V_Traditional_Final.tex, line 2407
Assume the child subtree bounds have been proved. Then ∑_w∑_ℓ∈ L_w x_ℓ ^2 ≥ min_w a(w) ∑_w (U_vx)_w ^2 ≥ A_vmin_w a(w) x ^2. The upper bound follows in the same way with the maximum and B_v. Induction completes the proof.
Exact mathematical passage
\sum_{w}\sum_{\ell\in\Lcal_w}\|x_\ell\|^2 \geq \min_w a(w) \sum_w\|(U_vx)_w\|^2 \geq A_v\min_w a(w)\|x\|^2.
\lcmUse in source · 2 passages
Search aliases: lcm operatorname
Use in source · Paper_XI_Traditional_Final.tex, line 557
Let τ_ξ=Ind_W_K_n^W_Kξ, τ_η=Ind_W_K_m^W_Kη, where K_n/K and K_m/K are unramified. Put d=gcd(n,m), L=lcm(n,m).
Exact mathematical passage
d=\gcd(n,m), \qquad L=\lcm(n,m).
\operatorname{lcm}Use in source · 4 passages
Search aliases: lcm operatorname
Use in source · edition_summary.tex, line 3
Principal results. addcontentslinetocsectionPrincipal results The original external problem is Nishibuchi's Conjecture 3.9 for generalized Euler numbers. Its full parameter range is resolved. If p is prime, m is positive with mmid p-1 or m=2, 0≤ j<mp, and q=lcm(2,p-1), then the original Euler coefficients are p-adically integral and E_mpn+qp^r^(mp,j)≡ E_mpn^(mp,j)pmodp^r (r≥1, n≥ r). The chapter states the generating function and the conversion between ordinary degree and packet row before using…
Exact mathematical passage
q=\operatorname{lcm}(2,p-1)
\leadsto_{H}Use in source · 1 passage
Search aliases: leadsto_H leadsto
Use in source · 02_supplement_05.tex, line 101
Hidden elimination. For the actual Schur complement S=A_RR-A_RHA_HH^-1A_HR, the hidden inverse is expanded through the hidden projectors. A nonzero retained correction block requires indices j,k with nonzero entrance and exit blocks and a hidden path jleadsto k. Therefore a nonzero block S_ba requires either a direct retained block, or a⟶ j leadsto_H k⟶ b. The entrance and exit matrices may be rectangular. Retained and hidden coordinate sizes and packet systems are independently typed. The hidde…
Exact mathematical passage
a\longrightarrow j\ \leadsto_{H}\ k\longrightarrow b.
\lengthUse in source · 1 passage
Search aliases: length operatorname
Use in source · 04_arithmetic.tex, line 329
[Total gluing and worst precision] If G_Λ, P ≅ ⊕_j=1^r O/π^a_j O, then length_ O(G_Λ, P) = ∑_j=1^ra_j, s_Λ( P)=max({0}∪{a_j:1≤ j≤ r}). Thus the lattice index measures the total amount of gluing, while the separation weight measures the worst precision loss in one direction.
Exact mathematical passage
\length_{\OO}(G_{\Lambda,\Pcal}) = \sum_{j=1}^ra_j, \qquad s_\Lambda(\Pcal)=\max\bigl(\{0\}\cup\{a_j:1\leq j\leq r\}\bigr).
\operatorname{length}Use in source · 2 passages
Search aliases: length operatorname
Use in source · 05_defects.tex, line 158
olomorphic on an open set U containing both spectra. Let Γ be an admissible finite rectifiable Cauchy cycle in U and in both resolvent sets, with winding number one on the union of the spectra and zero outside U. The length of a cycle counts multiplicities. Then Ch(T_f)-h(T_c)C = frac12π i ∫_Γ h(z)(z-T_c)^-1Δ(z-T_f)^-1 dz. Consequently, Ch(T_f)-h(T_c)C · ≤ fraclength(Γ)2π sup_z∈Γ|h(z)| Δ · × sup_z∈Γ (z-T_c)^-1 sup_z∈Γ (z-T_f)^-1 .
Exact mathematical passage
\|Ch(T_f)-h(T_c)C\| &\leq \frac{\operatorname{length}(\Gamma)}{2\pi} \sup_{z\in\Gamma}|h(z)|\,\|\Delta\|\\ &\quad\times \sup_{z\in\Gamma}\|(z-T_c)^{-1}\| \sup_{z\in\Gamma}\|(z-T_f)^{-1}\|.
\length_{\OO}Use in source · 1 passage
Search aliases: length_OO length_ O length OO operatorname mathcal
Use in source · 04_arithmetic.tex, line 329
[Total gluing and worst precision] If G_Λ, P ≅ ⊕_j=1^r O/π^a_j O, then length_ O(G_Λ, P) = ∑_j=1^ra_j, s_Λ( P)=max({0}∪{a_j:1≤ j≤ r}). Thus the lattice index measures the total amount of gluing, while the separation weight measures the worst precision loss in one direction.
Exact mathematical passage
\length_{\OO}(G_{\Lambda,\Pcal}) = \sum_{j=1}^ra_j, \qquad s_\Lambda(\Pcal)=\max\bigl(\{0\}\cup\{a_j:1\leq j\leq r\}\bigr).
\lfloorUse in source · 49 passages
Search aliases: lfloor
Use in source · Paper_IV_Traditional_Final.tex, line 679
[Twisted addition law] For 0≤ k<N, D_N,k(z+w) = ∑_substack0≤ i,j<N i+j≡ kbmod N (-1)^lfloor(i+j)/Nrfloor D_N,i(z) D_N,j(w).
Exact mathematical passage
\Dfun_{N,k}(z+w) = \sum_{\substack{0\leq i,j<N\\ i+j\equiv k\bmod N}} (-1)^{\lfloor(i+j)/N\rfloor} \Dfun_{N,i}(z)\Dfun_{N,j}(w).
\operatorname{Lift}Use in source · 7 passages
Search aliases: Lift operatorname
Use in source · dyadic_synthesis.tex, line 53
(t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two, and z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact filtration valuation, and give the identity tr_LLift_E(C) =Tr_E/L (αtr_E C). The actual recu…
Exact mathematical passage
\kappa=\alpha^{-1},\qquad s=\kappa\operatorname{pr},\qquad s(P_0)=t_L/z^2,\qquad \operatorname{Lift}_E(C)=(z^2/t_L)CP_0.
\operatorname{Lift}_EUse in source · 7 passages
Search aliases: Lift_E operatorname
Use in source · dyadic_synthesis.tex, line 53
(t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two, and z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact filtration valuation, and give the identity tr_LLift_E(C) =Tr_E/L (αtr_E C). The actual recu…
Exact mathematical passage
\kappa=\alpha^{-1},\qquad s=\kappa\operatorname{pr},\qquad s(P_0)=t_L/z^2,\qquad \operatorname{Lift}_E(C)=(z^2/t_L)CP_0.
\lim_{n\to\infty}Use in source · 1 passage
Search aliases: lim_n→∞ lim to infty
Use in source · Paper_VI_Traditional_Final.tex, line 4420
[Alternating propagation and killing] For every t≥0, e^t(Q^T-D_V) = lim_n→∞ ( e^-tD_V/n e^tQ^T/n )^n in operator norm. Each finite product alternates one Markov propagation step with one diagonal survival weight.
Exact mathematical passage
\e^{t(Q^T-D_V)} = \lim_{n\to\infty} \left( \e^{-tD_V/n}\e^{tQ^T/n} \right)^n
\lim_{t\downarrow0}Use in source · 2 passages
Search aliases: lim_tdownarrow0 lim downarrow
Use in source · Paper_VI_Traditional_Final.tex, line 2477
ℓ, let T_ℓ(t) be a strongly continuous semigroup on X_ℓ. Assume C_ℓ T_ℓ+1(t)=T_ℓ(t)C_ℓ for every ℓ and t≥0, and assume the uniform bound sup_ℓnormT_ℓ(t)≤ M e^ω t. Define T_∞(t)(x_ℓ)_ℓ = (T_ℓ(t)x_ℓ)_ℓ. Then T_∞(t) is a bounded semigroup on X_∞. It is strongly continuous if lim_tdownarrow0 sup_ℓ normT_ℓ(t)x_ℓ-x_ℓ=0 for every (x_ℓ)_ℓ∈ X_∞.
Exact mathematical passage
\lim_{t\downarrow0} \sup_\ell \norm{T_\ell(t)x_\ell-x_\ell}=0
\lim_{t\to0}Use in source · 1 passage
Search aliases: lim_t→0 lim to
Use in source · Paper_VI_Traditional_Final.tex, line 2332
[Infinitesimal recovery of the defect] The generator defect is the first order semigroup defect: lim_t→0 frac D(t)t = Δ in operator norm. Consequently, C e^tA_f= e^tA_cC hboxfor every t if and only if CA_f=A_cC.
Exact mathematical passage
\lim_{t\to0} \frac{\mathfrak D(t)}{t} = \Delta
\lim_{t\to\infty}Use in source · 2 passages
Search aliases: lim_t→∞ lim to infty
Use in source · Paper_VI_Traditional_Final.tex, line 1610
[Semigroup survival identities] For t≥0, the finite-time exit density and, wherever S(t)>0, the hazard are f(t)=a^Te^tAp_0=-S'(t), h(t)=fracf(t)S(t). Consequently, S is nonincreasing and S(t)=exp(-∫_0^th(r) dr) whenever S(t)>0. The density may be defective: its total mass is 1-lim_t→∞S(t). It is a probability density when absorption is eventual, so that this limit is zero.
Exact mathematical passage
1-\lim_{t\to\infty}S(t)
\limsupUse in source · 2 passages
Search aliases: limsup
Use in source · Paper_IV_Traditional_Final.tex, line 1754
[Order from Taylor coefficients] If f(w)=∑_n≥0a_nw^n is an entire function with infinitely many nonzero coefficients, then ord f = limsup_substackn→∞ a_n≠0 fracnlog nlog(1/|a_n|). Canonical products for entire functions of order below one have genus zero. We use these standard facts in the form given in [source reference].
Exact mathematical passage
\ord f = \limsup_{\substack{n\to\infty\\a_n\ne0}} \frac{n\log n}{\log(1/|a_n|)}.
\limsup_{r\to\infty}Use in source · 1 passage
Search aliases: limsup_r→∞ limsup to infty
Use in source · Paper_IV_Traditional_Final.tex, line 1811
[Indicator polygon and dominant sectors] The indicator of D_N equals h_N, with the limsup taken over radii where D_N(re^iθ)≠0: limsup_r→∞ frac1rlog| D_N(re^iθ)| = h_N(θ). If K is a compact set of directions disjoint from T_N, then there are δ>0 and R>0 such that, for each θ∈ K, one root η(θ) of -1 satisfies D_N(re^iθ) = frac1N exp(η(θ)re^iθ) (1+O(e^-δ r)) uniformly for r≥ R. In particular, no sufficiently large zero can lie in those directions.
Exact mathematical passage
\limsup_{r\to\infty} \frac1r\log|\Dfun_N(re^{i\theta})| = h_N(\theta).
\limsup_{\substack{n\to\infty\\a_n\ne0}}Use in source · 1 passage
Search aliases: limsup_substackn→∞ a_n≠0 limsup substack to infty a ne
Use in source · Paper_IV_Traditional_Final.tex, line 1754
[Order from Taylor coefficients] If f(w)=∑_n≥0a_nw^n is an entire function with infinitely many nonzero coefficients, then ord f = limsup_substackn→∞ a_n≠0 fracnlog nlog(1/|a_n|). Canonical products for entire functions of order below one have genus zero. We use these standard facts in the form given in [source reference].
Exact mathematical passage
\ord f = \limsup_{\substack{n\to\infty\\a_n\ne0}} \frac{n\log n}{\log(1/|a_n|)}.
\mathrm{lin}Use in source · 3 passages
Search aliases: lin mathrm
Use in source · 72_recovered.tex, line 2533
[Linear lift obstruction] The linear obstruction at depth L is Ob^lin_L(x_L) = [ρ_L(x_L)] ∈ coker(overline M).
Exact mathematical passage
\boxed{ \operatorname{Ob}^{\mathrm{lin}}_L(x_L) = [\rho_L(x_L)] \in \coker(\overline M). }
\linewidthUse in source · 1 passage
Search aliases: linewidth
Use in source · visual_atlas.tex, line 19
[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…
Exact mathematical passage
4pt] \textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt] \textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology? \end{tabularx} A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. \takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''} \vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}} Most of the chapter can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix. {\small \begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule \textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule $X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\ $S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\ $\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\ $\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\ $P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\ $X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\ $T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\ $F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\ $U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\ $\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\ $\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule \end{tabularx}} \head{Read products from right to left} In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column. \head{Where to begin} For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}. \vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$} Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define \[ Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0, \qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.
\operatorname{LocallyTrivial}Use in source · 4 passages
Search aliases: LocallyTrivial operatorname
Use in source · Paper_X_Block27_Amended.tex, line 2436
For every commutative coefficient ring B and χ∈ P_q(B), define χ_K=χ∘ j_K. The local character is continuous and commutes with coefficient change. The precise local admissibility condition is LocallyTrivial_K(χ) Longleftrightarrow χ_K=1. Descent through the fixed quadratic quotient proves LocallyTrivial_K(χ) Longleftrightarrow (∃ x∈ K, x^2=-1) lor χ=1. No reducedness, characteristic-zero or domain hypothesis is imposed on B. In particular, the formula applies to the finite coefficient quotients …
Exact mathematical passage
\operatorname{LocallyTrivial}_K(\chi) \quad\Longleftrightarrow\quad \chi_K=1.
\operatorname{LocallyTrivial}_KUse in source · 3 passages
Search aliases: LocallyTrivial_K operatorname
Use in source · Paper_X_Block27_Amended.tex, line 2436
For every commutative coefficient ring B and χ∈ P_q(B), define χ_K=χ∘ j_K. The local character is continuous and commutes with coefficient change. The precise local admissibility condition is LocallyTrivial_K(χ) Longleftrightarrow χ_K=1. Descent through the fixed quadratic quotient proves LocallyTrivial_K(χ) Longleftrightarrow (∃ x∈ K, x^2=-1) lor χ=1. No reducedness, characteristic-zero or domain hypothesis is imposed on B. In particular, the formula applies to the finite coefficient quotients …
Exact mathematical passage
\operatorname{LocallyTrivial}_K(\chi) \quad\Longleftrightarrow\quad \chi_K=1.
\operatorname{LocallyTrivial}_{\Q_p}Use in source · 1 passage
Search aliases: LocallyTrivial_Q_p LocallyTrivial_ Q_p operatorname Q operatorname mathbf
Use in source · Paper_X_Block27_Amended.tex, line 2462
l. Reduction modulo three rules out a root in Q_3, and reduction modulo four rules out a root in Q_2. At an odd prime, any residue root of X^2+1 has nonzero derivative and lifts by the pinned Mathlib Hensel theorem. The finite-field square criterion then gives the stated congruence. An explicit Hensel application at the residue root 2 modulo 5 supplies a root in Z_5. Consequently LocallyTrivial_ Q_p(χ) Longleftrightarrow p≡1pmod4 lor χ=1. The arithmetic prime p is independent of the coefficient …
Exact mathematical passage
\boxed{ \operatorname{LocallyTrivial}_{\Q_p}(\chi) \quad\Longleftrightarrow\quad p\equiv1\pmod4\ \lor\ \chi=1. }
\log_2Use in source · 15 passages
Search aliases: log_2 log
Use in source · 20_arithmetic.tex, line 616
[Finite Newton dominance] Let f: N→ Q and write a_k=Δ_1^kf(0). Suppose that c∈ N, a_1≠0, v_2(a_1)=c, and v_2(a_k)≥ c+1+lfloorlog_2krfloor (k≥2). Then for every n≥0 and T>0, f(n+T)-f(n)≠0, v_2(f(n+T)-f(n))=v_2(T)+c.
Exact mathematical passage
v_2(a_k)\ge c+1+\lfloor\log_2k\rfloor\qquad(k\ge2).
\operatorname{lookup}Use in source · 1 passage
Search aliases: lookup operatorname
Use in source · 02_supplement_03.tex, line 202
Entrywise implementation theorem. For every b≤ B, Lean proves lookup(storedTree(B,t),b)=F_t(b). There is a parallel theorem for stored discrete precision. The proof establishes first that an output entry at b depends only on input entries at most b, then proceeds through cached convolution and the tree recursion. Consequently the optimality, attainment, and infeasibility specification applies to the actual stored result.
Exact mathematical passage
\operatorname{lookup}(\operatorname{storedTree}(B,t),b)=F_t(b).
\lorUse in source · 3 passages
Search aliases: lor
Use in source · Paper_X_Block27_Amended.tex, line 2441
For every commutative coefficient ring B and χ∈ P_q(B), define χ_K=χ∘ j_K. The local character is continuous and commutes with coefficient change. The precise local admissibility condition is LocallyTrivial_K(χ) Longleftrightarrow χ_K=1. Descent through the fixed quadratic quotient proves LocallyTrivial_K(χ) Longleftrightarrow (∃ x∈ K, x^2=-1) lor χ=1. No reducedness, characteristic-zero or domain hypothesis is imposed on B. In particular, the formula applies to the finite coefficient quotients …
Exact mathematical passage
\operatorname{LocallyTrivial}_K(\chi) \quad\Longleftrightarrow\quad \bigl(\exists x\in K,\ x^2=-1\bigr)\ \lor\ \chi=1.
\lVertUse in source · 1 passage
Search aliases: lVert
Use in source · 03_weighted.tex, line 281
lineskip [Certified weighted inverse] Let R be a possibly noncommutative ring, let D^-1 be a supplied two-sided inverse of D, and put A=D^-1H. [label=(alph*)] item If A^r=0, then D+H has the two-sided inverse (D+H)^-1=(∑_k=0^r-1(-A)^k)D^-1. item If R is a complete normed ring and lVert ArVert<1, then ∑_k≥0(-A)^k converges and (D+H)^-1=(∑_k≥0(-A)^k)D^-1.
Exact mathematical passage
\lVert A\rVert<1
\lvertUse in source · 1 passage
Search aliases: lvert
Use in source · Paper_X_Block27_Amended.tex, line 2455
The p-adic square-root criterion is proved for every prime: (∃ x∈ Q_p, x^2=-1) Longleftrightarrow p≡1pmod4. A root satisfies lvert xrvert_p^2=1 and is therefore integral. Reduction modulo three rules out a root in Q_3, and reduction modulo four rules out a root in Q_2. At an odd prime, any residue root of X^2+1 has nonzero derivative and lifts by the pinned Mathlib Hensel theorem. The finite-field square criterion then gives the stated congruence. An explicit Hensel application at the residue ro…
Exact mathematical passage
\lvert x\rvert_p^2=1
\bm mUse in source · 2 passages
Search aliases: m bm
Use in source · Paper_X_Block27_Amended.tex, line 1362
Use the same multiplicities in every member of the compatible carrier. Direct sums preserve strict compatibility, and m_i≤ c_i makes V_λ a summand of W_λ. At the separating places, the target trace vector is C m. Full column rank makes m the unique solution.
Exact mathematical passage
C\bm m
M'Use in source · 3 passages
Search aliases: M'
Use in source · Paper_V_Traditional_Final.tex, line 1896
[Rectangular image phases] For L=M Z× N Z⊂ Z^2, one may take R={0,…,M-1}×{0,…,N-1}. The analysis map separates an image into MN exact phases. Refining (M,N) to (M',N'), where Mmid M' and Nmid N', splits every coarse phase into fracM'MfracN'N fine phases.
Exact mathematical passage
(M',N')
M^*Use in source · 2 passages
Search aliases: M^*
Use in source · 02_supplement_05.tex, line 171
Lean identifies the actual Frobenius norm with its trace expression, normM_rm F^2=Retr(M^*M), then expands the finite sum and removes each cross term. The result is normA-Π(A)_rm F^2 =∑_b≠ anormP_bAP_a_rm F^2. No Pythagorean equality for the target residual is assumed.
Exact mathematical passage
\norm{M}_{\rm F}^2=\operatorname{Re}\operatorname{tr}(M^*M),
M^{-1/2}Use in source · 1 passage
Search aliases: M^-1/2
Use in source · 01_manuscript.tex, line 1211
[Preconditioned conjugate gradient convergence] Let A and M be Hermitian positive definite on a nonempty finite coordinate space, and set κ = cond_2(M^-1/2AM^-1/2). In exact arithmetic, the energy norm error of preconditioned conjugate gradient satisfies normx_m-x_*_A ≤ 2 ( fracsqrtκ-1sqrtκ+1 )^m normx_0-x_*_A.
Exact mathematical passage
\kappa = \cond_2(M^{-1/2}AM^{-1/2}).
m^2Use in source · 8 passages
Search aliases: m^2
Use in source · pages.tex, line 119
9+12 i · 6. Replacing A^T with A^* changes the signs of the imaginary terms. A real-only test would conceal that error, which is why the worked model uses an imaginary coefficient. headWhy modewise contraction matters For A of order m and B of order n, the explicit tensor matrix stores m^2n^2 entries. Its factors store m^2+n^2. Applying through BXA^T costs 2n^2m+2nm^2 modeled operations, rather than forming the full tensor matrix first.
Exact mathematical passage
m^2n^2
m^dUse in source · 1 passage
Search aliases: m^d
Use in source · residue_continuation_body.tex, line 73
The residue norm and its sign. For m∈ N, reducing the multiplication matrices of O modulo m and taking their determinants constructs a monoid homomorphism overline n: O/m O⟶ Z/m Z, overline n([x])=n(x)bmod m. The construction is proved independent of the integral basis. For m≠0, its source is finite of cardinality m^d. Consequently x-y∈ m O implies n(x)≡ n(y)pmod m. No surjectivity of overline n is asserted.
Exact mathematical passage
m^d
M^{j-1}Use in source · 6 passages
Search aliases: M^j-1
Use in source · pages.tex, line 292
With A=diag(1,2), the vectors v and Av=(1,2)^T are independent. Two polynomial samples now distinguish the packet labels. Alternatively, allowing the packet projectors as extra address operators separates the coordinates even when A=I_2. Both changes enlarge the addressing power, but in different ways. headWhen invariance is only approximate If A=D+K with D packet diagonal, then A^j-D^j ≤ jM^j-1 K , M=max{ A , D }. This controls departure from the decoupled iterate. It is not a claim that the di…
Exact mathematical passage
\|A^j-D^j\|\le jM^{j-1}\|K\|, \qquad M=\max\{\|A\|,\|D\|\}.
M^kUse in source · 3 passages
Search aliases: M^k
Use in source · 03_nonlinear_coordinates.tex, line 378
[Number of labeled composition terms] For n≥1, if Y has M packets, then the number of packet labeled terms at order n before zero jets are removed is N_n,M = ∑_k=1^nS(n,k)M^k, where S(n,k) is a Stirling number of the second kind. Thus N_n,M is the nth Touchard polynomial evaluated at M.
Exact mathematical passage
N_{n,M} = \sum_{k=1}^{n}S(n,k)M^k,
m^NUse in source · 1 passage
Search aliases: m^N
Use in source · atlas-pages.tex, line 438
vpage24Convergence is not continuationZERO ZETA FUNCTIONSChapter IV: Theorem 14.2, Theorem 14.3, Corollary 14.4. Reduced zeros grow like m^N. Their absolute zero-zeta series therefore has half-plane of absolute convergence Re s>1/N. The absolute zero zeta in the original variable instead requires Re s>1. vfigzeta_plane For N=2 and N=4, the lattices are exact: Z_2^rm red(s)=π^-2sζ(2s,1/2), Z_4^rm red(s)=(sqrt2π)^-4sζ(4s,1/2). Each identity first holds where its defining series converges. The righ…
Exact mathematical passage
m^N
m^{r-2}Use in source · 3 passages
Search aliases: m^r-2
Use in source · Paper_IV_Traditional_Final.tex, line 2676
[Leading dominant zero term] Under the hypotheses of [source reference], P_N,n(r) = · m!binomm+r-1r-1 ∑_ρ∈ Z_R fracρ^-m(-ρ D_N'(ρ))^r · +O(m!m^r-2R^-m) +O_R_*(m!R_*^-m). For r=1, the first error term is absent.
Exact mathematical passage
\begin{aligned} P_{N,n}(r) ={}&m!\binom{m+r-1}{r-1} \sum_{\rho\in\Zcal_R} \frac{\rho^{-m}}{(-\rho\Dfun_N'(\rho))^r} \\ &+O\left(m!m^{r-2}R^{-m}\right) +O_{R_*}\left(m!R_*^{-m}\right). \end{aligned}
m^\alphaUse in source · 2 passages
Search aliases: m^α alpha
Use in source · Paper_VI_Traditional_Final.tex, line 1311
[Fractional semigroup refinement] For s≥0, Ce^-sA^α-e^-sB^αC = -∫_0^s e^-(s-r)B^αD_α e^-rA^α dr. Hence normCe^-sA^α-e^-sB^αC ≤ snormD_α. Under the common gap A,B≥ mI, normCe^-sA^α-e^-sB^αC ≤ s e^-sm^αα m^α-1norm D.
Exact mathematical passage
\boxed{ \norm{Ce^{-sA^\alpha}-e^{-sB^\alpha}C} \le s e^{-sm^\alpha}\alpha m^{\alpha-1}\norm D. }
m^{\alpha-1}Use in source · 9 passages
Search aliases: m^α-1 alpha
Use in source · Paper_VI_Traditional_Final.tex, line 1225
D_α = fracsin(πα)π ∫_0^∞ t^α(tI+B)^-1D(tI+A)^-1 dt. The improper integral converges in operator norm. Moreover, normD_α ≤ K_αnormC^1-αnormD^α, K_α= frac2^1-αsin(πα) πα(1-α). If A,B≥ mI for some m>0, then the stronger linear estimate holds: normD_α ≤ α m^α-1normD. In particular, exact generator intertwining CA=BC implies exact fractional intertwining CA^α=B^α C.
Exact mathematical passage
\boxed{ \norm{D_\alpha} \le \alpha m^{\alpha-1}\norm{D}. }
M^\perpUse in source · 1 passage
Search aliases: M^⊥ perp
Use in source · Paper_XI_Traditional_Final.tex, line 1792
The invariant orthogonal decomposition gives M=mathop⊥limits_s∈ SM_s. Symplectic invariance gives M^⊥=ker(g-I). Fixed point freeness on M therefore makes the restricted form nondegenerate on M and on each M_s. Each restriction of g_s-I to M_s is invertible, so [source reference] applies to the moving spaces. A zero moving space contributes one. In characteristic two, apply [source reference].
Exact mathematical passage
M^\perp=\ker(g-I)
m_{\mathrm A}Use in source · 2 passages
Search aliases: m_ A mathrm
Use in source · atlas-pages.tex, line 205
vfigevidence platecaptionThe A packet becomes heavier and its internal shape changes. The B packet retains an equal internal split but loses total mass. These are different effects of the same likelihood. headReconstruct from packet mass times packet shape The unnormalized packet evidences are Z_ A=5/4 and Z_ B=1/2. The resulting masses and conditional shapes are m_ A=5/7, q_ A=(4/5,1/5,0,0)^ T, m_ B=2/7, q_ B=(0,0,1/2,1/2)^ T. Thus m_ Aq_ A+m_ Bq_ B is exactly the displayed posterior. Keeping o…
Exact mathematical passage
m_{\mathrm A}=5/7,\quad q_{\mathrm A}=(4/5,1/5,0,0)^{\mathsf T}, \qquad m_{\mathrm B}=2/7,\quad q_{\mathrm B}=(0,0,1/2,1/2)^{\mathsf T}.
m_{\mathrm B}Use in source · 2 passages
Search aliases: m_ B mathrm
Use in source · atlas-pages.tex, line 205
vfigevidence platecaptionThe A packet becomes heavier and its internal shape changes. The B packet retains an equal internal split but loses total mass. These are different effects of the same likelihood. headReconstruct from packet mass times packet shape The unnormalized packet evidences are Z_ A=5/4 and Z_ B=1/2. The resulting masses and conditional shapes are m_ A=5/7, q_ A=(4/5,1/5,0,0)^ T, m_ B=2/7, q_ B=(0,0,1/2,1/2)^ T. Thus m_ Aq_ A+m_ Bq_ B is exactly the displayed posterior. Keeping o…
Exact mathematical passage
m_{\mathrm A}=5/7,\quad q_{\mathrm A}=(4/5,1/5,0,0)^{\mathsf T}, \qquad m_{\mathrm B}=2/7,\quad q_{\mathrm B}=(0,0,1/2,1/2)^{\mathsf T}.
m_0Use in source · 7 passages
Search aliases: m_0
Use in source · Paper_IV_Traditional_Final.tex, line 1970
[Large zero strings] Assume N≥3. There are constants m_0, C, and γ>0 and a sequence of positive real simple zeros (τ_m)_m≥ m_0 of D_N such that τ_m = t_m+O(e^-γ t_m). Every zero of sufficiently large modulus is of the form ω^kτ_m for a unique 0≤ k<N and a unique m≥ m_0.
Exact mathematical passage
m_0
M_1Use in source · 2 passages
Search aliases: M_1
Use in source · 72_recovered.tex, line 2017
[Nested discrete refinement] For positive integers M,N, (a_r^[M])_s^[N] = a_r+Ms^[MN]. For a mixed radix path (M_1,…,M_ℓ) with digits 0≤ s_i<M_i, the final address is r( s) = s_1+M_1s_2+⋯+ (∏_i=1^ℓ-1M_i)s_ℓ. Only the product radix and accumulated address determine the final branch.
Exact mathematical passage
(M_1,\ldots,M_\ell)
m_1Use in source · 2 passages
Search aliases: m_1
Use in source · strengthening.tex, line 394
The conversion from unit steps to arbitrary mixed steps is used several times, so it is useful to make it explicit. Write Δ=Δ_1. For H>0, Δ_H=∑_m=1^Hbinom HmΔ^m. A mixed term with local orders m_1,…,m_k≥1 and M=∑_ν m_ν therefore has valuation at least M+v_2(M!)+β +∑_ν(v_2(H_ν)-v_2(m_ν)) if the unit-order estimate has offset β for every positive order. The inequalities m-1≥ v_2(m), M≥ k, v_2(M!)≥ v_2(k!) make this at least k+v_2(k!)+β+∑_ν v_2(H_ν). Every expansion is finite. This proves the desir…
Exact mathematical passage
m_1,\ldots,m_k\geq1
M_2Use in source · 12 passages
Search aliases: M_2
Use in source · 01_orientation.tex, line 87
[title=R2: a matrix algebra with multiplicity] Let A=M_2(𝕜) act diagonally on M=(𝕜^2)^⊕3. One seed can contribute at most two independent multiplicity directions, because the regular module contains two copies of the simple module. Hence srank_A(M)=lceilfrac32rceil=2. This is the simplest nontrivial instance of the semisimple multiplicity formula.
Exact mathematical passage
A=M_2(\bbk)
M_{2\times5}Use in source · 2 passages
Search aliases: M_2×5 times
Use in source · visual_atlas.tex, line 325
vpage13How much row space can one seed supply?SPLIT SEMISIMPLE MODEL / A CONSTRUCTIVE FORMULAMain text: [source reference] Let M_2( Q) act on X=M_2×5( Q) by left multiplication. Think of the five columns as five copies of the same natural module of dimension two. One seed is a 2×5 matrix V.
Exact mathematical passage
X=M_{2\times5}(\QQ)
M_3Use in source · 1 passage
Search aliases: M_3
Use in source · 05_worked_models.tex, line 499
M_3( C) acting on seven copies, seed rank · 3
Exact mathematical passage
M_3(\C)
M_4Use in source · 1 passage
Search aliases: M_4
Use in source · 05_worked_models.tex, line 502
Full M_4( R) orbit dimensions for two sampled nonzero seeds · 4,4
Exact mathematical passage
M_4(\R)
M_AUse in source · 2 passages
Search aliases: M_A
Use in source · Paper_VI_Traditional_Final.tex, line 2500
For bounded generators with common bound M_A, sup_ℓ norm( e^tA_ℓ-I)x_ℓ ≤ ( e^tM_A-1)sup_ℓnormx_ℓ. This tends to zero with t. Exact generator intertwining makes (A_ℓ x_ℓ)_ℓ compatible, and coordinatewise differentiation identifies the generator.
Exact mathematical passage
M_A
M_aUse in source · 1 passage
Search aliases: M_a
Use in source · 02_foundations.tex, line 74
unital left modules over Λ_ G. Explicitly, a realization X determines the collapsed space |X| of [source reference] with the vertex idempotent e_v acting as the coordinate projector onto X_v and each arrow acting in the block between its declared source and target. Conversely, a unital module M over Λ_ G determines the realization M_v=e_vM, M_a(x)=ax. The two constructions are inverse up to canonical natural isomorphism. [source reference]
Exact mathematical passage
M_v=e_vM, \qquad M_a(x)=ax.
M_cUse in source · 4 passages
Search aliases: M_c
Use in source · Paper_VI_Traditional_Final.tex, line 487
[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.
Exact mathematical passage
\norm{e^{tA_f}}\le M_f e^{\omega_f t}, \qquad \norm{e^{tA_c}}\le M_c e^{\omega_c t}.
M_dUse in source · 32 passages
Search aliases: M_d
Use in source · 02_foundations.tex, line 752
For one factor M_d(D), let e_jj be the diagonal matrix units. The left regular module decomposes as M_d(D)=⊕_j=1^dM_d(D)e_jj. Each minimal left ideal M_d(D)e_jj is isomorphic to the natural column module D^d. Taking the direct product of factors gives the stated decomposition.
Exact mathematical passage
M_d(D)
m_dUse in source · 1 passage
Search aliases: m_d
Use in source · Paper_XII_Traditional_Final.tex, line 601
Conversely, suppose M_ p=0. Choose generators m_1,…,m_d of M. For each i, there is an element s_i∉ p such that s_im_i=0. The product s=s_1⋯ s_d lies outside p and annihilates every generator. Thus s∈Ann_R(M)∖ p. This proves the equality.
Exact mathematical passage
m_1,\ldots,m_d
M_{d_A}Use in source · 2 passages
Search aliases: M_d_A
Use in source · 01_finite_core.tex, line 138
[Local orbit and Schmidt rank] Let ψ∈ H_A⊗ H_B, with dim H_A=d_A≥1, and let r be its Schmidt rank. The reduced positive operator on B is ρ_B=Tr_A(ketψbraψ). Normalization of ψ does not affect its support. Then dimspan{(A⊗ I_B)ψ:A∈ M_d_A( C)} =d_A r. In particular, local operators on A generate the complete bipartite Hilbert space exactly when the reduced state on B has full support.
Exact mathematical passage
\boxed{ \dim\operatorname{span}\{(A\otimes I_B)\psi:A\in M_{d_A}(\bbC)\} =d_A r. }
M_{d_r}Use in source · 1 passage
Search aliases: M_d_r
Use in source · 02_strengthened.tex, line 104
Let A ≅ ⊕_r=1^s (M_d_r( C)⊗ I_m_r) act on H ≅ ⊕_r=1^s ( C^d_r⊗ C^m_r). Assume d_r,m_r≥1 for each displayed block. For an empty block family, the seed rank and the displayed finite supremum are defined as zero. A vector in one block is identified with a d_r by m_r coefficient matrix.
Exact mathematical passage
\Acal \cong \bigoplus_{r=1}^{s} \bigl(M_{d_r}(\CC)\otimes I_{m_r}\bigr)
M_{d\times m}Use in source · 2 passages
Search aliases: M_d× m times
Use in source · 03_supplement_03.tex, line 2
Actual orbits and sharp seed counts. Let C∈ M_d× m( C) be the coefficient matrix of one seed. Define its orbit by actual matrix multiplication: O(C)={AC:A∈ M_d( C)}. This set is already a linear subspace. The source proves that it equals the linear span of the orbit before computing its dimension.
Exact mathematical passage
C\in M_{d\times m}(\C)
m_eUse in source · 23 passages
Search aliases: m_e
Use in source · pages.tex, line 352
Now impose upper tolerances m=(1/10,1/2,1/2). The first edge must stop at its tighter cap. With multiplier λ=10/3, the clipped solution is ϵ_e=min{m_e,c_e/(λ s_e)}, ϵ=(1/10,3/20,3/20). Its weighted contributions are (1/10,3/10,3/5) and still sum to one. The checker verifies the active cap, the multiplier equations, and the budget exactly. headFrom tolerance to a real data type The continuous objective is a model. Actual kernels offer discrete precisions and may use different accumulation types. …
Exact mathematical passage
\epsilon_e=\min\{m_e,c_e/(\lambda s_e)\},\qquad \epsilon=(1/10,3/20,3/20).
M_fUse in source · 5 passages
Search aliases: M_f
Use in source · Paper_IV_Traditional_Final.tex, line 1774
The coefficient of w^n in Δ_N has modulus 1/(Nn)!. Stirling's formula and [source reference] give order 1/N. Since M_f( · ^N)(r)=M_f(r^N) for the maximum modulus, composition with z^N multiplies the order by N and gives order one for D_N.
Exact mathematical passage
M_{f(\,\cdot\,^N)}(r)=M_f(r^N)
M_{f(\,\cdot\,^N)}Use in source · 1 passage
Search aliases: M_f( · ^N) cdot
Use in source · Paper_IV_Traditional_Final.tex, line 1774
The coefficient of w^n in Δ_N has modulus 1/(Nn)!. Stirling's formula and [source reference] give order 1/N. Since M_f( · ^N)(r)=M_f(r^N) for the maximum modulus, composition with z^N multiplies the order by N and gives order one for D_N.
Exact mathematical passage
M_{f(\,\cdot\,^N)}(r)=M_f(r^N)
M_IUse in source · 1 passage
Search aliases: M_I
Use in source · supplement_06.tex, line 30
pageheading1. Actual matrix orbits and arithmetic jumpsTheorem 9.1; manuscript page 22 Let R be a commutative ring, I a finite coordinate type and d=|I|. For C∈ M_I(R) and v_0∈ R^I, define orbit(C,v_0,n)=C^nv_0, u_k=packet(C,v_0,N,j,k)=C^Nk+jv_0. The implementation uses actual matrix powers and matrix--vector multiplication. A separately supplied sequence v with v_n+1=Cv_n is proved by induction to equal this power orbit. Thus the source-sequence formulation is connected to the construction.
Exact mathematical passage
C\in M_I(R)
M_iUse in source · 2 passages
Search aliases: M_i
Use in source · 72_recovered.tex, line 2017
[Nested discrete refinement] For positive integers M,N, (a_r^[M])_s^[N] = a_r+Ms^[MN]. For a mixed radix path (M_1,…,M_ℓ) with digits 0≤ s_i<M_i, the final address is r( s) = s_1+M_1s_2+⋯+ (∏_i=1^ℓ-1M_i)s_ℓ. Only the product radix and accumulated address determine the final branch.
Exact mathematical passage
0\leq s_i<M_i
M_{ij}Use in source · 5 passages
Search aliases: M_ij
Use in source · Paper_V_Traditional_Final.tex, line 2782
[Overlap matrix formula] Define M_ij=μ(E_i∩ F_j). For 1≤ p<∞, ∑_i=0^Ia_i 1_E_i - ∑_j=0^Jb_j 1_F_j _p^p = ∑_i=0^I∑_j=0^J|a_i-b_j|^pM_ij. The row sums of M are μ(E_i), and the column sums are μ(F_j).
Exact mathematical passage
M_{ij}=\mu(E_i\cap F_j).
m_jUse in source · 2 passages
Search aliases: m_j
Use in source · 02_foundations.tex, line 493
The lower inequality is quotient monotonicity. For the upper inequality, choose generating families (ℓ_i)_i∈ I of L and (n_j)_j∈ J of N with the least possible cardinalities. Choose a lift m_j∈ M of each n_j. For any m∈ M, a finite subset F⊆ J and coefficients a_j∈ A give π(m)=∑_j∈ Fa_jn_j. The vector m-∑_j∈ Fa_jm_j belongs to ι(L), so it is a finite linear combination of the ι(ℓ_i). The union of the lifted family and the image of the generating family of L therefore generates M. Its cardinality…
Exact mathematical passage
m_j\in M
M_KUse in source · 6 passages
Search aliases: M_K
Use in source · 02_foundations.tex, line 595
[Monotonicity under extension of scalars] Let K/𝕜 be a field extension, let A be a unital algebra over 𝕜, and let M be a finitely generated left module over A. Set A_K:=K⊗_𝕜A, M_K:=K⊗_𝕜M. Then srank_A_K(M_K)≤srank_A(M). Every generating family S⊆ M extends to the generating family 1⊗ S⊆ M_K. [source reference] ExportsExtension of the coefficient field in realizations from arithmetic, quantum theory, and representation theory.
Exact mathematical passage
A_K:=K\otimes_{\bbk}A, \qquad M_K:=K\otimes_{\bbk}M.
m_kUse in source · 1 passage
Search aliases: m_k
Use in source · strengthening.tex, line 394
The conversion from unit steps to arbitrary mixed steps is used several times, so it is useful to make it explicit. Write Δ=Δ_1. For H>0, Δ_H=∑_m=1^Hbinom HmΔ^m. A mixed term with local orders m_1,…,m_k≥1 and M=∑_ν m_ν therefore has valuation at least M+v_2(M!)+β +∑_ν(v_2(H_ν)-v_2(m_ν)) if the unit-order estimate has offset β for every positive order. The inequalities m-1≥ v_2(m), M≥ k, v_2(M!)≥ v_2(k!) make this at least k+v_2(k!)+β+∑_ν v_2(H_ν). Every expansion is finite. This proves the desir…
Exact mathematical passage
m_1,\ldots,m_k\geq1
M_LUse in source · 3 passages
Search aliases: M_L
Use in source · Paper_IV_Traditional_Final.tex, line 2949
For N≥3, [source reference] gives | D_N'(ρ)|^-1 ≤ C e^-c_N|ρ| for every sufficiently large simple zero. Let L be a compact subset of the s plane. Because the chosen arguments of the zeros lie in a fixed bounded interval, there are constants C_L and M_L such that sup_s∈ L|e^-sLogρ| ≤ C_L|ρ|^M_L for every sufficiently large zero. Hence the absolute value of the corresponding summand is at most C_L'|ρ|^M_Le^-q c_N|ρ|. The linear zero counting estimate makes this majorant summable. The Weierstrass t…
Exact mathematical passage
M_L
M_mUse in source · 1 passage
Search aliases: M_m
Use in source · 02_strengthened.tex, line 1068
[Finite Kraus and Choi criteria] A linear map Φ:M_n( C)→ M_m( C) is completely positive exactly when it has a Kraus representation Φ(X)=∑_rK_rXK_r^*. It is trace preserving exactly when ∑_rK_r^*K_r=I. Equivalently, its Choi matrix J(Φ) = ∑_i,jΦ(ketibraj)⊗ketibraj is positive, and trace preservation is Tr_rm outJ(Φ)=I_rm in.
Exact mathematical passage
\Phi:M_n(\CC)\to M_m(\CC)
M_{m,n}Use in source · 2 passages
Search aliases: M_m,n
Use in source · 04_arithmetic.tex, line 704
[Smith normal form over a discrete valuation ring] Let M∈ M_m,n( O). There are invertible matrices U∈GL_m( O), V∈GL_n( O) such that UMV = diag(π^s_1,…,π^s_r,0,…,0), where 0≤ s_1≤⋯≤ s_r and r is the rank over K. The exponents are uniquely determined. See [source reference].
Exact mathematical passage
M\in M_{m,n}(\OO)
M_NUse in source · 4 passages
Search aliases: M_N
Use in source · 03_nonlinear_coordinates.tex, line 704
[Matrix difference grading] Let A=M_N(𝕜) and assign the matrix unit E_rs the label r-s modulo N. Since E_rsE_uv=δ_suE_rv, the output label is (r-s)+(u-v)=r-v whenever the product is nonzero. Matrix multiplication therefore obeys the cyclic convolution law exactly. The same grading resolves commutators and polynomial matrix equations.
Exact mathematical passage
A=M_N(\bbk)
m_nUse in source · 3 passages
Search aliases: m_n
Use in source · 03_packets_and_graphs.tex, line 346
For x∈ X, choose m_n∈ M with m_n→ x. Continuity gives P_α m_n→ P_α x, so P_α X⊆clP_α M. The reverse inclusion follows because P_α X is closed and contains P_α M. The bounded extensions remain pairwise annihilating idempotents whose finite sum is the identity.
Exact mathematical passage
m_n\in M
M_pUse in source · 3 passages
Search aliases: M_p
Use in source · supplement_08.tex, line 39
∑_a=1^q-1frac1pbinom qa (U^a f)(U^q-ag), q=p,p^2. The coefficients of each M_q are proved integral: the middle entries of a prime-power binomial row are divisible by p. In Lean they are implemented as natural quotients, with exact division proved before the product identity is used. Induction on derivative order establishes the full Leibniz formula and hence B(fg)=B(f)g+fB(g)+M_p^2(f,g)-M_p(f,g). The middle terms are essential. At p=3, f=t and g=t^2, the constant coefficient of B(fg) is -2, wher…
Exact mathematical passage
\boxed{B(fg)=B(f)g+fB(g)+M_{p^2}(f,g)-M_p(f,g).}
M_{p^2}Use in source · 2 passages
Search aliases: M_p^2
Use in source · supplement_08.tex, line 39
∑_a=1^q-1frac1pbinom qa (U^a f)(U^q-ag), q=p,p^2. The coefficients of each M_q are proved integral: the middle entries of a prime-power binomial row are divisible by p. In Lean they are implemented as natural quotients, with exact division proved before the product identity is used. Induction on derivative order establishes the full Leibniz formula and hence B(fg)=B(f)g+fB(g)+M_p^2(f,g)-M_p(f,g). The middle terms are essential. At p=3, f=t and g=t^2, the constant coefficient of B(fg) is -2, wher…
Exact mathematical passage
\boxed{B(fg)=B(f)g+fB(g)+M_{p^2}(f,g)-M_p(f,g).}
M_QUse in source · 2 passages
Search aliases: M_Q
Use in source · Paper_VI_Traditional_Final.tex, line 2750
[Generator control of range leakage] Let T(t)= e^tA on a Banach space X, let P be a bounded projection, and set Q=I-P. Assume normT(t)≤ M e^ω t and norm e^t(QAQ)|_QX ≤ M_Q e^ω_Qt. Then normQT(t)P ≤ M M_Qnorm P^2normQAP ∫_0^t e^ω_Q(t-s)+ω s d s. In particular, for orthogonal projections and contraction bounds, sup_0≤ t≤ T normQT(t)P ≤ TnormQAP.
Exact mathematical passage
\norm{\e^{t(QAQ)|_{QX}}} \leq M_Q\e^{\omega_Qt}.
M_qUse in source · 2 passages
Search aliases: M_q
Use in source · supplement_08.tex, line 33
pageheading1. The characteristic-zero product ruleThe normalized defect is not an ordinary derivation Let A be a commutative algebra over a characteristic-zero field K, and let U be an actual K-derivation. The source defines B=fracU^p^2-U^pp, M_q(f,g)=∑_a=1^q-1frac1pbinom qa (U^a f)(U^q-ag), q=p,p^2. The coefficients of each M_q are proved integral: the middle entries of a prime-power binomial row are divisible by p. In Lean they are implemented as natural quotients, with exact division proved b…
Exact mathematical passage
B=\frac{U^{p^2}-U^p}{p},\qquad M_q(f,g)=\sum_{a=1}^{q-1}\frac{1}{p}\binom qa (U^a f)(U^{q-a}g),\quad q=p,p^2.
M_rUse in source · 1 passage
Search aliases: M_r
Use in source · Paper_X_Block27_Amended.tex, line 1053
[Finite Frobenius separation] Let mathscr W_1,…,mathscr W_r be pairwise nonisomorphic strictly compatible systems whose members are absolutely irreducible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficient-field embedding σ_λ:E↪ E_λ therefore sends C to an invertible matrix. Its entries agree with the ordinary representation traces Tr(W_i,λ(Frob_v_j)) …
Exact mathematical passage
C=(t_i(v_j))_{j,i}\in M_r(E),\qquad \det C\ne0.
m_rUse in source · 7 passages
Search aliases: m_r
Use in source · 02_strengthened.tex, line 104
Let A ≅ ⊕_r=1^s (M_d_r( C)⊗ I_m_r) act on H ≅ ⊕_r=1^s ( C^d_r⊗ C^m_r). Assume d_r,m_r≥1 for each displayed block. For an empty block family, the seed rank and the displayed finite supremum are defined as zero. A vector in one block is identified with a d_r by m_r coefficient matrix.
Exact mathematical passage
\Acal \cong \bigoplus_{r=1}^{s} \bigl(M_{d_r}(\CC)\otimes I_{m_r}\bigr)
M_{r,s}Use in source · 1 passage
Search aliases: M_r,s
Use in source · 02_coordinates.tex, line 279
[htbp] centering includegraphics[width=0.96textwidth]03_refinement_incidence.pdf captionThe incidence matrix M_r,s=1 when the fine residue s belongs to the coarse residue r. Its rows implement exact aggregation from fine packets to coarse packets. Common refinements.
Exact mathematical passage
M_{r,s}=1
\mathfrak M_R^{\mathrm b}Use in source · 3 passages
Search aliases: M_R^ b mathfrak mathrm
Use in source · 03_packets_and_graphs.tex, line 835
[Bounded support patterns closed under *] Let X be Hilbert, let the packet projectors be orthogonal, assume X_α≠0 for every α∈ I, and put M_R^ b:= M_R∩ B(X). Then M_R^ b is closed under adjoints if and only if R is symmetric. Hence M_R^ b is a unital * subalgebra of B(X) if and only if R is an equivalence relation. [source reference]
Exact mathematical passage
\mathfrak M_R^{\mathrm b}:=\mathfrak M_R\cap\mathcal B(X).
\mathfrak M_{R_\Gamma}Use in source · 1 passage
Search aliases: M_R_Γ mathfrak Gamma
Use in source · 03_packets_and_graphs.tex, line 942
[Reachability support of the generated algebra] For every A∈ A, P_β AP_α=0 whenever β∉Reach(α). Equivalently, A⊆ M_R_Γ, R_Γ:={(β,α):β∈Reach(α)}. [source reference]
Exact mathematical passage
\A\subseteq\mathfrak M_{R_\Gamma}, \qquad R_\Gamma:=\{(\beta,\alpha):\beta\in\Reach(\alpha)\}.
M_{\rm block}Use in source · 1 passage
Search aliases: M_rm block rm
Use in source · 01_manuscript.tex, line 284
[Equal diagonal packets] Let p≥1 and b≥1 be integers and n=pb. Suppose n is divided into p equal diagonal packets of width b. Then flop_rm block=2qnb, M_rm block=nb. The following factor describes the dense-to-block storage reduction and, when q≥1, the arithmetic reduction: fracnb.
Exact mathematical passage
\flop_{\rm block}=2qnb, \qquad M_{\rm block}=nb.
M_{\rm packet}Use in source · 1 passage
Search aliases: M_rm packet rm
Use in source · 01_manuscript.tex, line 267
[Dense and packet application counts] A dense n by n matrix applied to q right hand sides has modeled cost flop_rm dense=2n^2q. A packet representation with active edge set E has modeled cost flop_rm packet = 2q ∑_(β,α)∈ E n_β n_α. It stores M_rm packet = ∑_(β,α)∈ E n_β n_α scalar entries.
Exact mathematical passage
\boxed{ M_{\rm packet} = \sum_{(\beta,\alpha)\in E} n_\beta n_\alpha }
M_sUse in source · 5 passages
Search aliases: M_s
Use in source · Paper_XI_Traditional_Final.tex, line 1780
[Breakwise phase product] In odd residue characteristic, let g∈Sp(V) preserve every break packet and suppose its restriction to M=im(g-I) is fixed point free. Put g_s=g|_V_s and M_s=im(g_s-I). The moving restrictions are symplectic, and Φ_p(g|_M,M) = ∏_s∈ SΦ_p(g_s|_M_s,M_s). In residue characteristic two, the analogous normalized Gauss sign is the product of the signs determined by the breakwise quadratic refinements.
Exact mathematical passage
M_s=\im(g_s-I)
M_{\sqrt{\abs A}}Use in source · 1 passage
Search aliases: M_sqrtabs A sqrt abs
Use in source · Paper_XI_Traditional_Final.tex, line 955
[Nondegenerate twisted group algebra] Assume that b_c is nondegenerate. Then abs A is a square, and C^c[A] ≅ M_sqrtabs A( C). Consequently, C^c[A] has one irreducible module up to isomorphism, of dimension sqrtabs A.
Exact mathematical passage
\boxed{ \CC^c[A] \cong M_{\sqrt{\abs A}}(\CC). }
M_TUse in source · 5 passages
Search aliases: M_T
Use in source · Paper_VI_Traditional_Final.tex, line 4821
[Volterra stability under kernel compression] Let S(t) be a strongly continuous semigroup on a Banach space X, fix T>0, and assume normS(t)≤ M_T (0≤ t≤ T). Let K,widetilde K∈ L^1((0,T), B(X)), let g∈ L^1((0,T),X), and let x,widetilde x∈ C([0,T],X). Suppose these are mild solutions with the same initial state and forcing, except that K is replaced by widetilde K: x(t) = · S(t)x_0+ ∫_0^tS(t-s)g(s) d s
Exact mathematical passage
\norm{S(t)}\leq M_T \quad(0\leq t\leq T).
m_TUse in source · 2 passages
Search aliases: m_T
Use in source · 02_foundations.tex, line 916
Let the minimal polynomial be m_T(t)=∏_pp(t)^e_p. The acting algebra is 𝕜[T]≅𝕜[t]/(m_T) ≅∏_pR_p, R_p:=𝕜[t]/(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with maximal ideal generated by the class of p and residue division field K_p=𝕜[t]/(p). By [source reference], srank_R_p(X_p) = dim_K_pX_p/p(T)X_p = μ_p(T). …
Exact mathematical passage
m_T(t)=\prod_{p}p(t)^{e_p}.
M_VUse in source · 4 passages
Search aliases: M_V
Use in source · Paper_VI_Traditional_Final.tex, line 1443
Let K be a column-stochastic matrix, let V be a potential on the states, and let f be terminal data. Define M_V=diag(e^-V)K^T. The backward Feynman--Kac value after n steps is u_n=M_V^nf.
Exact mathematical passage
M_V=\diag(e^{-V})K^T.
M_vUse in source · 1 passage
Search aliases: M_v
Use in source · 02_foundations.tex, line 74
unital left modules over Λ_ G. Explicitly, a realization X determines the collapsed space |X| of [source reference] with the vertex idempotent e_v acting as the coordinate projector onto X_v and each arrow acting in the block between its declared source and target. Conversely, a unital module M over Λ_ G determines the realization M_v=e_vM, M_a(x)=ax. The two constructions are inverse up to canonical natural isomorphism. [source reference]
Exact mathematical passage
M_v=e_vM, \qquad M_a(x)=ax.
M_V^nUse in source · 2 passages
Search aliases: M_V^n
Use in source · Paper_VI_Traditional_Final.tex, line 1447
Let K be a column-stochastic matrix, let V be a potential on the states, and let f be terminal data. Define M_V=diag(e^-V)K^T. The backward Feynman--Kac value after n steps is u_n=M_V^nf.
Exact mathematical passage
u_n=M_V^nf.
M_wUse in source · 2 passages
Search aliases: M_w
Use in source · residue_continuation_body.tex, line 169
An actual Mellin endpoint. The counting functions are constant on the integer norm steps. Their measurability, and hence that of S_w, is proved from this fact. For zero-sum w, define F_w(h)=(1+h)∫_1^∞x^-2-hS_w(x) dx (h≥0). Since 1-1/d<1, the logarithmic first moment M_w=∫_1^∞(1+log x) absnx^-2S_w(x) dx is finite. Lean proves the explicit estimate absnF_w(h)-F_w(0)≤ hM_w for every h≥0, and therefore F_w(h)=F_w(0)+O(h) as hdownarrow0. The function here is the displayed integral of the actual ideal…
Exact mathematical passage
M_w=\int_1^{\infty}(1+\log x)\,\absn{x^{-2}S_w(x)}\,dx
M_xUse in source · 7 passages
Search aliases: M_x
Use in source · 05_worked_models.tex, line 81
Take the seed s=1 and let M_x,M_y be multiplication by x and y. Then M_x^iM_y^js=x^iy^j, so the algebra, regarded as a module over its multiplication algebra, is generated by one seed.
Exact mathematical passage
M_x,M_y
M_x^3Use in source · 1 passage
Search aliases: M_x^3
Use in source · 05_worked_models.tex, line 126
The relation is reconstructed as an operator identity: M_y^2=M_x^3-M_x.
Exact mathematical passage
M_y^2=M_x^3-M_x.
M_{x^3-x}Use in source · 2 passages
Search aliases: M_x^3-x
Use in source · 05_worked_models.tex, line 107
Identify the packet coefficients by A_0⊕ A_1⟶𝕜[x]⊕𝕜[x], (a,yb)⟼(a,b). In these ordered coefficient coordinates, with M_x and M_x^3-x denoting multiplication on 𝕜[x], M_x= M_x · 0
Exact mathematical passage
M_{x^3-x}
M_x^iUse in source · 1 passage
Search aliases: M_x^i
Use in source · 05_worked_models.tex, line 82
Take the seed s=1 and let M_x,M_y be multiplication by x and y. Then M_x^iM_y^js=x^iy^j, so the algebra, regarded as a module over its multiplication algebra, is generated by one seed.
Exact mathematical passage
M_x^iM_y^js=x^iy^j,
M_y^jUse in source · 1 passage
Search aliases: M_y^j
Use in source · 05_worked_models.tex, line 82
Take the seed s=1 and let M_x,M_y be multiplication by x and y. Then M_x^iM_y^js=x^iy^j, so the algebra, regarded as a module over its multiplication algebra, is generated by one seed.
Exact mathematical passage
M_x^iM_y^js=x^iy^j,
m_\alphaUse in source · 8 passages
Search aliases: m_α alpha
Use in source · Paper_VI_Traditional_Final.tex, line 763
Let the state indices be partitioned into packets I_α. Define Z_α=∑_i∈ I_αλ_ip_i, m_α=fracZ_α∑_β Z_β, q_α,i= λ_ip_i/Z_α, · Z_α>0, i∈ I_α, 0, · otherwise.
Exact mathematical passage
Z_\alpha=\sum_{i\in I_\alpha}\lambda_ip_i, \qquad m_\alpha=\frac{Z_\alpha}{\sum_\beta Z_\beta}, \qquad q_{\alpha,i}=\begin{cases} \lambda_ip_i/Z_\alpha,&Z_\alpha>0,\ i\in I_\alpha,\\ 0,&\text{otherwise}. \end{cases}
m_\lambdaUse in source · 3 passages
Search aliases: m_λ lambda
Use in source · 02_foundations.tex, line 452
For the topological statement, let S generate M densely. Given n=Φ(m), choose a net m_λ∈ AS converging to m. Continuity gives Φ(m_λ)⟶ n, and every Φ(m_λ) lies in BΦ(S). Thus BΦ(S) is dense in N.
Exact mathematical passage
m_\lambda\in AS
m_{\lambda}Use in source · 6 passages
Search aliases: m_λ lambda
Use in source · 05_worked_models.tex, line 283
[Certified graph seed number] Let 𝕜 be a field and let Λ be a nonempty finite set. Suppose there are pairwise distinct scalars a_λ∈𝕜, multiplicities m_λ∈ N, and an eigenbasis {b_λ,j:λ∈Λ, 1≤ j≤ m_λ} of 𝕜^V(G) satisfying A_Gb_λ,j=a_λb_λ,j. If m_λ≤ R for all λ and m_λ_*=R for some λ_*, then R is the least natural number of adjacency seeds: R=min{r∈ N: l there exist s_1,…,s_r with
Exact mathematical passage
m_{\lambda}\in\N
m_{\lambda_*}Use in source · 1 passage
Search aliases: m_λ_* lambda
Use in source · 05_worked_models.tex, line 292
r] Let 𝕜 be a field and let Λ be a nonempty finite set. Suppose there are pairwise distinct scalars a_λ∈𝕜, multiplicities m_λ∈ N, and an eigenbasis {b_λ,j:λ∈Λ, 1≤ j≤ m_λ} of 𝕜^V(G) satisfying A_Gb_λ,j=a_λb_λ,j. If m_λ≤ R for all λ and m_λ_*=R for some λ_*, then R is the least natural number of adjacency seeds: R=min{r∈ N: l there exist s_1,…,s_r with
Exact mathematical passage
m_{\lambda_*}=R
M_\muUse in source · 1 passage
Search aliases: M_μ mu
Use in source · 02_strengthened.tex, line 225
Complete reducibility gives a unitary decomposition H ≅ ⊕_μ∈widehat G V_μ⊗ M_μ, U(g) ≅ ⊕_μ π_μ(g)⊗ I_M_μ. On the μ summand, the displayed average is central in the irreducible representation π_μ, so it is a scalar by Schur's lemma. Taking its trace and applying character orthogonality shows that the scalar is one when μ=λ and zero otherwise. The asserted identities follow on the direct sum.
Exact mathematical passage
\Hcal \cong \bigoplus_{\mu\in\widehat G} V_\mu\otimes M_\mu, \qquad U(g) \cong \bigoplus_\mu \pi_\mu(g)\otimes I_{M_\mu}.
m_\nuUse in source · 8 passages
Search aliases: m_ν nu
Use in source · strengthening.tex, line 395
The conversion from unit steps to arbitrary mixed steps is used several times, so it is useful to make it explicit. Write Δ=Δ_1. For H>0, Δ_H=∑_m=1^Hbinom HmΔ^m. A mixed term with local orders m_1,…,m_k≥1 and M=∑_ν m_ν therefore has valuation at least M+v_2(M!)+β +∑_ν(v_2(H_ν)-v_2(m_ν)) if the unit-order estimate has offset β for every positive order. The inequalities m-1≥ v_2(m), M≥ k, v_2(M!)≥ v_2(k!) make this at least k+v_2(k!)+β+∑_ν v_2(H_ν). Every expansion is finite. This proves the desir…
Exact mathematical passage
M=\sum_\nu m_\nu
M_\ellUse in source · 1 passage
Search aliases: M_ℓ ell
Use in source · 72_recovered.tex, line 2017
[Nested discrete refinement] For positive integers M,N, (a_r^[M])_s^[N] = a_r+Ms^[MN]. For a mixed radix path (M_1,…,M_ℓ) with digits 0≤ s_i<M_i, the final address is r( s) = s_1+M_1s_2+⋯+ (∏_i=1^ℓ-1M_i)s_ℓ. Only the product radix and accumulated address determine the final branch.
Exact mathematical passage
(M_1,\ldots,M_\ell)
\operatorname{Mat}Use in source · 2 passages
Search aliases: Mat operatorname
Use in source · Paper_V_Traditional_Final.tex, line 491
Work with finite impulse-response banks. Let L= C[z,z^-1] be the Laurent-polynomial algebra, with the monomial z^-m acting on a bilateral sequence by x[n]↦ x[n-m]. Let E∈Mat_N× N( L) and R∈Mat_N× N( L) be analysis and synthesis matrices. Their composition acts on all bilateral channel sequences by finite convolutions. At a nonzero z where the input transforms are absolutely convergent, this gives widehat X(z)= R(z) E(z) X(z).
Exact mathematical passage
\bm E\in\operatorname{Mat}_{N\times N}(\mathcal L)
\operatorname{Mat}_{N\times N}Use in source · 2 passages
Search aliases: Mat_N× N operatorname times
Use in source · Paper_V_Traditional_Final.tex, line 491
Work with finite impulse-response banks. Let L= C[z,z^-1] be the Laurent-polynomial algebra, with the monomial z^-m acting on a bilateral sequence by x[n]↦ x[n-m]. Let E∈Mat_N× N( L) and R∈Mat_N× N( L) be analysis and synthesis matrices. Their composition acts on all bilateral channel sequences by finite convolutions. At a nonzero z where the input transforms are absolutely convergent, this gives widehat X(z)= R(z) E(z) X(z).
Exact mathematical passage
\bm E\in\operatorname{Mat}_{N\times N}(\mathcal L)
\mathllapUse in source · 2 passages
Search aliases: mathllap
Use in source · 03_nonlinear_coordinates.tex, line 953
[Naturality of obstruction classes] Suppose ccc U · xrightarrowD · V mathllapadownarrow · · downarrowmathrlapb U' · xrightarrowD' · V' is a commuting square. Then there is a unique induced map overline b:coker D⟶coker D' satisfying overline b Ob_D(r)=Ob_D'(br).
Exact mathematical passage
\begin{array}{ccc} U & \xrightarrow{D} & V\\ \mathllap{a}\downarrow & & \downarrow\mathrlap{b}\\ U' & \xrightarrow{D'} & V' \end{array}
\mathordUse in source · 1 passage
Search aliases: mathord
Use in source · 04_correction_transport.tex, line 396
Let L be a complete filtered differential graded Lie algebra over a ring in which two is invertible. Assume dF^i L⊆ F^i L, [F^i L,F^j L]⊆ F^i+j L. For a degree one element η, put Ω(η)=dη+frac12[η,η]. Suppose Ω(η)∈ F^m L^2 with m≥1. For every h∈ F^m L^1, Ω(η+h) ≡ Ω(η)+d_η h pmodF^m+1 L^2, d_η=d+[η,mathord·]. The obstruction to improving η by one depth is therefore the class of Ω(η) in the cokernel of gr^m(d_η): gr^m L^1 ⟶ gr^m L^2.
Exact mathematical passage
\Omega(\eta+h) \equiv \Omega(\eta)+d_\eta h \pmod{F^{m+1}\Lcal^2}, \qquad d_\eta=d+[\eta,\mathord\cdot].
\mathrelUse in source · 3 passages
Search aliases: mathrel
Use in source · Paper_IV_Traditional_Final.tex, line 553
[Hypergeometric form] Let A_N,j = { fracj+mN mathrel: 1≤ m≤ N and j+m≠ N }. Then E_N,j(λ z) = frac(λ z)^jj! _0F_N-1 ( -
Exact mathematical passage
A_{N,j} = \left\{ \frac{j+m}{N} \mathrel{:} 1\leq m\leq N \text{ and } j+m\ne N \right\}.
\mathrlapUse in source · 1 passage
Search aliases: mathrlap
Use in source · 03_nonlinear_coordinates.tex, line 953
[Naturality of obstruction classes] Suppose ccc U · xrightarrowD · V mathllapadownarrow · · downarrowmathrlapb U' · xrightarrowD' · V' is a commuting square. Then there is a unique induced map overline b:coker D⟶coker D' satisfying overline b Ob_D(r)=Ob_D'(br).
Exact mathematical passage
\begin{array}{ccc} U & \xrightarrow{D} & V\\ \mathllap{a}\downarrow & & \downarrow\mathrlap{b}\\ U' & \xrightarrow{D'} & V' \end{array}
\MaxUse in source · 2 passages
Search aliases: Max operatorname
Use in source · Paper_XII_Traditional_Final.tex, line 1025
[Artin decomposition] Let E' be a field and let T be a finite dimensional commutative E' algebra. Then T ≅ ∏_ m∈Max T T_ m. Each factor is an Artin local algebra. Equivalently, there is a unique complete family of central idempotents (e_ m) such that e_ me_ n=0 ( m≠ n), ∑_ me_ m=1, and e_ m T≅ T_ m [source reference].
Exact mathematical passage
\boxed{ \TT \cong \prod_{\frakm\in\Max\TT}\TT_{\frakm}. }
\max_{0\le k<N}Use in source · 1 passage
Search aliases: max_0≤ k<N max le
Use in source · atlas-pages.tex, line 311
vpage17Growth comes from a polygonDOMINANT EXPONENTSChapter IV: Definition 11.3, Theorem 11.4. Write the signed denominator as a finite exponential sum: D_N(z)=frac1N∑_η^N=-1e^η z. For z=re^iθ, the size of one exponential is e^rRe(η e^iθ). The maximal projection determines the directional exponential growth. vfigindicator The indicator is h_N(θ)=max_0≤ k<Ncos(θ+frac(2k+1)πN). Away from a tie direction, one exponential dominates all the others by a uniform positive projection gap on compact angul…
Exact mathematical passage
h_N(\theta)=\max_{0\le k<N}\cos\left(\theta+\frac{(2k+1)\pi}{N}\right).
\max_{1\leq i\leq n}Use in source · 1 passage
Search aliases: max_1≤ i≤ n max leq leq
Use in source · Paper_VI_Traditional_Final.tex, line 4223
[Discrete maximum bound] The inverse L_h^-1 has nonnegative entries and normL_h^-1_∞→∞ = max_1≤ i≤ n fracx_i(1-x_i)2 ≤ frac18.
Exact mathematical passage
\norm{L_h^{-1}}_{\infty\to\infty} = \max_{1\leq i\leq n} \frac{x_i(1-x_i)}{2} \leq \frac18.
\max_{1\leq i\leq q}Use in source · 2 passages
Search aliases: max_1≤ i≤ q max leq leq
Use in source · 02_foundations.tex, line 770
[Semisimple seed rank formula over arbitrary division rings] Let A≅∏_i=1^qM_d_i(D_i) be semisimple Artinian and let M≅⊕_i=1^qS_i^⊕ m_i be a finitely generated left module over A. Then srank_A(M) = max_1≤ i≤ q lceilfracm_id_irceil. If A is any algebra of finite dimension over a field, the same formula applied to the decomposition of M/Jac(A)M over A/Jac(A) computes srank_A(M). [source reference] ExportsEvery calculation of seed rank for semisimple algebras of finite dimension in the series.
Exact mathematical passage
\boxed{ \srank_A(M) = \max_{1\leq i\leq q} \left\lceil\frac{m_i}{d_i}\right\rceil. }
\max_{1\le r\le s}Use in source · 1 passage
Search aliases: max_1≤ r≤ s max le le
Use in source · 02_strengthened.tex, line 135
[Quantum seed rank] For the displayed multiplicity decomposition, srank_ A( H) = max_1≤ r≤ s lceilfracm_rd_rrceil.
Exact mathematical passage
\operatorname{srank}_{\Acal}(\Hcal) = \max_{1\le r\le s} \left\lceil\frac{m_r}{d_r}\right\rceil.
\max_{i\in B_a}Use in source · 1 passage
Search aliases: max_i∈ B_a max in
Use in source · Paper_VI_Traditional_Final.tex, line 5491
[Oscillation bound] Suppose overline L_a is any representative likelihood for block B_a, and set ω_a = max_i∈ B_aabsL_i-overline L_a. Then d(pmid L,overline L) ≤ ∑_aq_aω_a. Thus small within block likelihood oscillation gives a small coarse posterior defect whenever the evidence is bounded away from zero.
Exact mathematical passage
\omega_a = \max_{i\in B_a}\abs{L_i-\overline L_a}.
\max_{i\in I_\alpha}Use in source · 1 passage
Search aliases: max_i∈ I_α max in alpha
Use in source · Paper_VI_Traditional_Final.tex, line 3720
[Information created by a Bayesian update] Let q= B_λ(p) with positive likelihood and positive prior. Then D(qVert p) = ∑_iq_ilogλ_i - log Z_λ(p). Inside a packet, D(q_αVert r_α) = ∑_i∈ I_α q_α,i logfracλ_ioverlineλ_α(p). Hence D(q_αVert r_α) ≤ log fracmax_i∈ I_αλ_i min_i∈ I_αλ_i.
Exact mathematical passage
D(q_\alpha\Vert r_\alpha) \leq \log \frac{\max_{i\in I_\alpha}\lambda_i} {\min_{i\in I_\alpha}\lambda_i}.
\max_kUse in source · 1 passage
Search aliases: max_k max
Use in source · Paper_V_Traditional_Final.tex, line 3126
inproof If σ=0, every centered zero-sub-Gaussian noise variable vanishes almost surely, and the noise-free deterministic conclusion applies. For σ>0, the difference η_k+1-η_k is sub-Gaussian with variance proxy 2σ^2, so P(|η_k+1-η_k|>t) ≤ 2exp(-fract^24σ^2). A union bound over the n cells gives P( max_k|η_k+1-η_k|>t_n,δ ) ≤ δ. On the complementary event, repeat the deterministic proof.
Exact mathematical passage
\mathbb P\left( \max_k|\eta_{k+1}-\eta_k|>t_{n,\delta} \right) \leq \delta.
\max_{\Lcal}Use in source · 1 passage
Search aliases: max_Lcal max_ L max Lcal max mathcal
Use in source · Paper_V_Traditional_Final.tex, line 2462
[Stability of the optimal cost] Let c and widetilde c be two node cost functions. For an admissible pruning L, set ε( L) = ∑_v∈ L|c(v)-widetilde c(v)| and let ε_* = max_ Lε( L). If F_* and widetilde F_* are the two optimal root costs, then |F_*-widetilde F_*| ≤ ε_*.
Exact mathematical passage
\varepsilon_* = \max_{\Lcal}\varepsilon(\Lcal).
\max_pUse in source · 3 passages
Search aliases: max_p max
Use in source · 02_foundations.tex, line 901
[Seed rank for a single operator] Let X have finite dimension over 𝕜 and let T∈End_𝕜(X). Regard X as a module over 𝕜[t] by t· x=Tx. For every monic irreducible p, let X_p be the primary component for p and set μ_p(T):= dim_𝕜[t]/(p)(X_p/p(T)X_p). Then srank_𝕜[T](X)=max_pμ_p(T). Moreover, μ_p(T) is the number of cyclic primary summands for p in the decomposition into elementary divisors of X_p. DependsnolinkurlPSC-I-CI-PID-PRIMARY-DECOMPOSITION, nolinkurlPSC-I-C-LOCAL-ALGEBRA-FORMULA, and nolinkur…
Exact mathematical passage
\boxed{ \srank_{\bbk[T]}(X)=\max_p\mu_p(T). }
\max_rUse in source · 1 passage
Search aliases: max_r max
Use in source · 02_strengthened.tex, line 147
Set k=max_rlceilfracm_rd_rrceil. For each r, partition a basis of C^m_r into at most k groups of size at most d_r. Choose the coefficient matrix of the jth seed so that its row space spans the jth group. The combined row spaces span every multiplicity factor. The preceding theorem then shows that the k seeds generate every summand.
Exact mathematical passage
k=\max_r\left\lceil\frac{m_r}{d_r}\right\rceil.
\max_{v\in V}Use in source · 1 passage
Search aliases: max_v∈ V max in
Use in source · 02_supplement_05.tex, line 31
A valid schedule s:V→ N assigns positive stages and strictly increases on each edge. Induction proves the stronger pointwise statement ℓ(v)≤ s(v) for every v. Consequently the constructed schedule achieves the least possible makespan, d=max_v∈ Vℓ(v). For a nonempty graph, d equals the number of vertices on an attained longest path. Vertices of the same level have no dependency edge in either direction.
Exact mathematical passage
\boxed{d=\max_{v\in V}\ell(v).}
\max_wUse in source · 1 passage
Search aliases: max_w max
Use in source · Paper_V_Traditional_Final.tex, line 2395
A_vmin_w a(w), · v split, and b(v)= 1, · v retained, B_vmax_w b(w), · v split. Then the leaf analysis of the pruned tree has bounds a(rm root) and b(rm root).
Exact mathematical passage
b(v)= \begin{cases} 1,&v\text{ retained},\\ B_v\max_w b(w),&v\text{ split}. \end{cases}
\max_\alphaUse in source · 1 passage
Search aliases: max_α max alpha
Use in source · 03_nonlinear_coordinates.tex, line 215
It is therefore Banach with its inherited norm. The same holds for every Y_γ. Give the finite product of ranges the maximum norm. Packet analysis and synthesis are bounded, since they are formed from finitely many bounded projectors and a finite sum. Their algebraic inverse identities from [source reference] make them bounded linear isomorphisms. For a nonempty input family, A_ P ≤max_α P_α and S_ P ≤ |I|. An empty complete family can occur only on the zero space. Its coordinate product is the z…
Exact mathematical passage
\|\Acal_{\Pcal}\|\le\max_\alpha\|P_\alpha\|
\max_{\Lambda\in\mathfrak P_v}Use in source · 1 passage
Search aliases: max_Λ∈ P_v max Lambda in mathfrak
Use in source · Paper_V_Traditional_Final.tex, line 1016
[1mm] displaystyle max{ε_v, ∑_u∈ch(v)E(u)}, · v internal. For the complete prunings P_v below v, the envelope is exactly E(v)=max_Λ∈ P_v∑_a∈Λε_a. The maximum is attained. Every complete-pruning cost changes by at most E(∅), and the two globally optimal values differ by at most this quantity. Consequently, for a unique optimum with competitors, Γ_y>2E(∅) is a deterministic certificate that the globally optimal pruning is unchanged. A singleton pruning set is stable without a gap condition. Assume…
Exact mathematical passage
E(v)=\max_{\Lambda\in\mathfrak P_v}\sum_{a\in\Lambda}\varepsilon_a.
\max_{\lambda\in\operatorname{Spec}(T)}Use in source · 1 passage
Search aliases: max_λ∈Spec(T) max lambda in operatorname
Use in source · 02_foundations.tex, line 974
[Formula for geometric multiplicity] If 𝕜 is algebraically closed, then srank_𝕜[T](X) = max_λ∈Spec(T) dim_𝕜Ker(T-λ I). [source reference]
Exact mathematical passage
\boxed{ \srank_{\bbk[T]}(X) = \max_{\lambda\in\operatorname{Spec}(T)} \dim_{\bbk}\Ker(T-\lambda I). }
\max_{\mu\in\Spec_K(A)}Use in source · 1 passage
Search aliases: max_μ∈Spec_K(A) max mu in Spec max mu in operatorname
Use in source · 05_worked_models.tex, line 565
[Exact graph seed rank from a certificate] For a finite square matrix A over a field K, define the noncomputable exact spectral quantity r_spec(A) = max_μ∈Spec_K(A) dim_Kker(A-μ I), with value zero for an empty spectrum. Suppose an exact certificate supplies a finite eigenbasis indexed by labels and multiplicities, the eigenvalue equations, a multiplicity bound attained by one label, projector polynomials that extract the padded eigenbasis generators, coverage of every spectral value by a label,…
Exact mathematical passage
r_{\mathrm{spec}}(A) = \max_{\mu\in\Spec_K(A)} \dim_K\ker(A-\mu I),
\mcodeUse in source · 1 passage
Search aliases: mcode
Use in source · supplement_05.tex, line 221
_bijective, .address_components, .residue_equivalence targetCRTPackets.packet_factorization, .series_packet_factorization targetFiniteArithmeticExamples.outside_window_not_integral pageheading5. Verification, regressions and remaining workCumulative release evidence The fresh build started with an empty output directory under Lean 4.34.0-rc2 and Mathlib revision mcodeaeb72f1e9981b6d35b426b16dca7759be3e7ba06. All nine dependency revisions matched the runtime manifest and repository heads.
Exact mathematical passage
\mcode{aeb72f1e9981b6d35b426b16dca7759be3e7ba06}.
\midruleUse in source · 4 passages
Search aliases: midrule
Use in source · visual_atlas.tex, line 19
[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…
Exact mathematical passage
4pt] \textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt] \textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology? \end{tabularx} A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. \takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''} \vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}} Most of the chapter can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix. {\small \begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule \textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule $X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\ $S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\ $\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\ $\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\ $P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\ $X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\ $T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\ $F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\ $U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\ $\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\ $\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule \end{tabularx}} \head{Read products from right to left} In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column. \head{Where to begin} For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}. \vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$} Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define \[ Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0, \qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.
\min_{(w,c)\text{ allowed},\;w\leq b}Use in source · 1 passage
Search aliases: min_(w,c) allowed, w≤ b min text leq
Use in source · 02_supplement_03.tex, line 153
If D is the already solved tail table, the next edge has exact-weight recurrence F(b)=min_(w,c) allowed, w≤ b{D(b-w)+c}. The final answer is min_0≤ b≤ BF(b). Lean proves that this is the exact minimum over legal assignments with total error at most B, or codenone exactly when no such assignment exists.
Exact mathematical passage
F(b)=\min_{(w,c)\text{ allowed},\;w\leq b}\{D(b-w)+c\}.
\min_{0\leq b\leq B}Use in source · 2 passages
Search aliases: min_0≤ b≤ B min leq leq
Use in source · 02_supplement_03.tex, line 156
If D is the already solved tail table, the next edge has exact-weight recurrence F(b)=min_(w,c) allowed, w≤ b{D(b-w)+c}. The final answer is min_0≤ b≤ BF(b). Lean proves that this is the exact minimum over legal assignments with total error at most B, or codenone exactly when no such assignment exists.
Exact mathematical passage
\min_{0\leq b\leq B}F(b)
\min_{0\leq j\leq n}Use in source · 2 passages
Search aliases: min_0≤ j≤ n min leq leq
Use in source · 10_foundations.tex, line 306
[Weighted difference bound] Let f: N→ Z_p and let w: N→ N∪{∞} satisfy w(s+j)≤ord_p(c_s+j(f)) for 0≤ j≤ n. Then ord_p(Δ_1^s f(n))≥ min_0≤ j≤ n {w(s+j)+v_pbinom nj}.
Exact mathematical passage
\operatorname{ord}_p(\Delta_1^s f(n))\geq \min_{0\leq j\leq n} \left\{w(s+j)+v_p\binom nj\right\}.
\min_{0\leq k\leq b}Use in source · 1 passage
Search aliases: min_0≤ k≤ b min leq leq
Use in source · 02_supplement_03.tex, line 145
For exact-weight tables F,G, min-plus convolution is (F⋆ G)(b)=min_0≤ k≤ b{F(k)+G(b-k)}, with an infeasible sum whenever either operand is infeasible. Its correctness theorem identifies every finite answer with an actual feasible pair of component weights and costs. Repeated convolution handles arbitrary finite lists of components.
Exact mathematical passage
(F\star G)(b)=\min_{0\leq k\leq b}\{F(k)+G(b-k)\},
\min_eUse in source · 1 passage
Search aliases: min_e min
Use in source · 01_manuscript.tex, line 1409
upper bound. The weighted sum ϕ(λ) = ∑_es_e min{ m_e, fracc_eλ s_e } is continuous and nonincreasing. It decreases from the maximal feasible value to zero. For a strictly submaximal budget, once any coordinate is unsaturated, ϕ is strictly decreasing, which gives a unique multiplier. At maximal budget the only feasible vector is (m_e)_e and every 0<λ≤min_e c_e/(s_em_e) produces it. Thus the minimizing vector is unique at this endpoint while its multiplier need not be. Strict convexity gives uniq…
Exact mathematical passage
0<\lambda\le\min_e c_e/(s_em_e)
\min_{\eps_e>0}Use in source · 1 passage
Search aliases: min_eps_e>0 min_ε_e>0 min eps min varepsilon
Use in source · 01_manuscript.tex, line 1337
[Continuous precision allocation] Let E be a finite nonempty edge set, let s_e>0 be sensitivities and c_e>0 be costs, and let τ>0. Consider min_ε_e>0 ∑_e c_elogfrac1ε_e subject to ∑_es_eε_e=τ. The unique minimizer is ε_e = fracτ c_es_e∑_fc_f.
Exact mathematical passage
\min_{\eps_e>0} \sum_e c_e\log\frac1{\eps_e}
\min_{I_\alpha}Use in source · 1 passage
Search aliases: min_I_α min alpha
Use in source · Paper_VI_Traditional_Final.tex, line 3604
Subtract r_α,i from the local posterior formula in [source reference]. The norm identity follows because the factor multiplying η_i is nonnegative. The last estimate uses ∑_i r_α,iabsη_i ≤ osc_I_α(λ) and overlineλ_α(p)≥min_I_αλ.
Exact mathematical passage
\overline\lambda_\alpha(p)\geq\min_{I_\alpha}\lambda
\min_{i\in I_\alpha}Use in source · 2 passages
Search aliases: min_i∈ I_α min in alpha
Use in source · Paper_VI_Traditional_Final.tex, line 3589
] For every packet with positive mass and evidence, q_α,i-r_α,i = fracr_α,iη_i(p) overlineλ_α(p). Consequently, normq_α-r_α_1 = frac ∑_i∈ I_αr_α,iabsη_i(p) overlineλ_α(p). If λ is positive on the packet, then normq_α-r_α_1 ≤ fracosc_I_α(λ) min_i∈ I_αλ_i.
Exact mathematical passage
\norm{q_\alpha-r_\alpha}_1 \leq \frac{\osc_{I_\alpha}(\lambda)} {\min_{i\in I_\alpha}\lambda_i}.
\min_kUse in source · 1 passage
Search aliases: min_k min
Use in source · 01_manuscript.tex, line 1438
ct discrete precision program] Let the total integer error budget be B. Define F_j(b) as the minimum cost for assigning precisions to the first j edges with total weight exactly b. Then F_0(0)=0, F_0(b)=+∞ for b≠0. Set F_j-1(u)=+∞ whenever u<0 or the state is unreachable. Then F_j(b) = min_k { F_j-1(b-w_j,k)+c_j,k }. The minimum of F_m(b) over 0≤ b≤ B is the exact minimum cost for the quantized model.
Exact mathematical passage
\boxed{ F_j(b) = \min_k \left\{ F_{j-1}(b-w_{j,k})+c_{j,k} \right\}. }
\min_{k\leq r\leq Mk+s}Use in source · 1 passage
Search aliases: min_k≤ r≤ Mk+s min leq leq
Use in source · 72_recovered.tex, line 2262
[Polyphase valuation ledger] For every refined coefficient, v_p(d_s,k) ≥ min_k≤ r≤ Mk+s ( v_p(c_r)+v_p(Θ_M,s(r,k)) ). If v_p(c_r)≥ w(r) for a nondecreasing function w, then v_p(d_s,k)≥ w(k). Thus every monotone Mahler precision ladder survives arbitrary polyphase refinement.
Exact mathematical passage
\boxed{ \vp(d_{s,k}) \geq \min_{k\leq r\leq Mk+s} \left( \vp(c_r)+\vp(\Theta_{M,s}(r,k)) \right). }
\min_{\Lcal\ne\Lcal_*}Use in source · 1 passage
Search aliases: min_Lcal≠Lcal_* min_ L≠ L_* min Lcal ne Lcal min mathcal ne mathcal
Use in source · Paper_V_Traditional_Final.tex, line 2486
[Stability of the selected basis] Assume the optimal pruning L_* for c is unique and at least one competing pruning exists. Define g=min_ L≠ L_*(C( L)-C( L_*))>0. If 2ε_*<g, then the optimal pruning for widetilde c is the same. If only one admissible pruning exists, it remains selected under every cost change without a gap condition.
Exact mathematical passage
g=\min_{\Lcal\ne\Lcal_*}(C(\Lcal)-C(\Lcal_*))>0
\min_{M+1\leq r\leq n/2}Use in source · 1 passage
Search aliases: min_M+1≤ r≤ n/2 min leq leq
Use in source · 72_recovered.tex, line 1409
[Optimal dyadic truncation] Let T_2,M(n) = ∑_r=0^min(M,n/2) κ_2,rbinom n2rR_n-2r. Then E_n≡ T_2,M(n)pmod 2^4M+3. More precisely, v_2(E_n-T_2,M(n)) ≥ min_M+1≤ r≤ n/2 ( 4r-1+v_2binom n2r ). The uniform exponent 4M+3 is best possible.
Exact mathematical passage
\boxed{ \vTwo\bigl(E_n-T_{2,M}(n)\bigr) \geq \min_{M+1\leq r\leq n/2} \left( 4r-1+\vTwo\binom n{2r} \right). }
\min_qUse in source · 1 passage
Search aliases: min_q min
Use in source · Paper_V_Traditional_Final.tex, line 2954
[Deterministic jump cell recovery] Let a_min=min_q|a_q| and assume a_min>2b_h. Choose a threshold θ satisfying b_h<θ<a_min-b_h. Then widehat K = {k:|D_k|>θ} is exactly the set of grid cells containing jumps. If widehatτ_q is the midpoint of the detected cell and widehat a_q=D_k, then after matching by order, |widehatτ_q-τ_q| ≤ frac h2 and |widehat a_q-a_q| ≤ b_h.
Exact mathematical passage
a_{\min}=\min_q|a_q|
\min_{q<j}Use in source · 1 passage
Search aliases: min_q<j min
Use in source · root_scope.tex, line 200
For a wall label j≥1 and 0≤ q<j, define e_j(q)=3q+s_2(j-1)-s_2(j-q-1), E_j=min_q<je_j(q), Θ(j)=3j+1+s_2(j-1), L_j=Θ(j)-E_j. Here E_j is a carry minimum, not an Euler number. A wall label is regular when e_j(q)>0 for 1≤ q<j. For r=2^L-j with L>L_j, put δ=L-L_j. The carry identities give μ_r=E_j-δ, J_r={j}, G_r(x)=P_j(x)U(x-j)+2^δ V(x), where U≡1pmod2. For a regular label, also E_j=0 and V≡1pmod2.
Exact mathematical passage
e_j(q)=3q+s_2(j-1)-s_2(j-q-1),\qquad E_j=\min_{q<j}e_j(q),
\min_{q\ge0}Use in source · 1 passage
Search aliases: min_q≥0 min ge
Use in source · root_scope.tex, line 128
The scalar packet weights have exact valuations λ_r(q)=3q+s_2(r+q)-s_2(r). Put μ_r=min_q≥0λ_r(q), J_r={q:λ_r(q)=μ_r}, G_r=2^-μ_rH_r. Since λ_r(0)=0 and λ_r(q)≥3q+1-s_2(r), every minimizing address satisfies 0≤ q≤lfloorfracs_2(r)-13rfloor. In particular μ_r≤0, and the minimum is found by a finite carry calculation. Reduction of the whole series gives overline G_r(X)=∑_q∈ J_rX^underline q in F_2[X]. Its highest degree is max J_r, with leading coefficient one. Thus the normalization is the actual G…
Exact mathematical passage
\mu_r=\min_{q\ge0}\lambda_r(q),\qquad J_r=\{q:\lambda_r(q)=\mu_r\},\qquad G_r=2^{-\mu_r}H_r.
\min_{r\geq s}Use in source · 1 passage
Search aliases: min_r≥ s min geq
Use in source · 72_recovered.tex, line 1497
[Mahler valuation ledger] If c_r∈ Z_p and v_p(c_r)≥ w(r), then v_p(Δ^sf(n)) ≥ min_r≥ s ( w(r)+v_pbinom nr-s ).
Exact mathematical passage
\boxed{ \vp(\Delta^sf(n)) \geq \min_{r\geq s} \left( w(r)+\vp\binom n{r-s} \right). }
\min_{\substack{0\leq r\leq n\\w(r)\geq L}}Use in source · 1 passage
Search aliases: min_substack0≤ r≤ n w(r)≥ L min substack leq leq w geq
Use in source · 72_recovered.tex, line 553
w(r)<L binom nr c_rp^w(r)S_n-r pmod p^L. When the omitted set is nonempty, the tail has valuation at least min_substack0≤ r≤ n w(r)≥ L ( w(r)+v_pbinom nr+v_p(S_n-r) ).
Exact mathematical passage
\boxed{ \min_{\substack{0\leq r\leq n\\w(r)\geq L}} \left( w(r)+\vp\binom nr+\vp(S_{n-r}) \right). }
\min_wUse in source · 2 passages
Search aliases: min_w min
Use in source · Paper_V_Traditional_Final.tex, line 2387
[Bounds for nonunitary local splits] Suppose every internal node has an analysis map U_v and constants 0≤ A_v≤ B_v<∞ satisfying A_v x ^2 ≤ ∑_w∈child(v) (U_vx)_w ^2 ≤ B_v x ^2. Define recursively a(v)= 1, · v retained, A_vmin_w a(w), · v split, and b(v)= 1, · v retained,
Exact mathematical passage
a(v)= \begin{cases} 1,&v\text{ retained},\\ A_v\min_w a(w),&v\text{ split}, \end{cases}
\min_{\varepsilon_e>0}Use in source · 1 passage
Search aliases: min_ε_e>0 min varepsilon
Use in source · 02_supplement_02.tex, line 120
The continuous allocation theorem. textitSource item 8.4, with an explicit positive budget Let E be a finite nonempty edge set. Assume s_e>0, c_e>0, τ>0, and define C=∑_ec_e. The optimization problem is min_ε_e>0∑_ec_elogfrac1ε_e subject to ∑_es_eε_e=τ. Lean constructs the displayed candidate ε_e^*=fracτ c_es_eC and proves strict positivity and the exact budget equality. It then proves global optimality and uniqueness for the actual logarithmic objective.
Exact mathematical passage
\min_{\varepsilon_e>0}\sum_ec_e\log\frac1{\varepsilon_e} \qquad\text{subject to}\qquad \sum_es_e\varepsilon_e=\tau.
\min_{\Lambda\ne\Lambda_y^*}Use in source · 1 passage
Search aliases: min_Λ≠Λ_y^* min Lambda ne Lambda
Use in source · Paper_V_Traditional_Final.tex, line 942
[label=(roman*)] item A global minimizer Λ_y^*∈*argmin_Λ∈ P( T)J_y(Λ) always exists. item If the minimizer is unique and at least one competing pruning exists, its gap Γ_y = min_Λ≠Λ_y^* (J_y(Λ)-J_y(Λ_y^*)) is attained and strictly positive. If the pruning set is a singleton, its only element minimizes every criterion. No real-valued minimum over an empty competitor set is introduced. item In the case with competitors, if new data y' satisfy sup_Λ∈ P( T) |J_y'(Λ)-J_y(Λ)| <fracΓ_y2, then Λ_y'^*=Λ_…
Exact mathematical passage
\Gamma_y = \min_{\Lambda\ne\Lambda_y^*} \bigl(J_y(\Lambda)-J_y(\Lambda_y^*)\bigr)
\min_{\sum_i b_i=b}Use in source · 1 passage
Search aliases: min_∑_i b_i=b min sum
Use in source · 02_supplement_03.tex, line 168
Keeping node v costs κ_v and incurs integer error w_v. Splitting costs overhead σ_v in addition to the selected child cuts. The exact-weight recurrence is F_v(b)=min{ l κ_v if b=w_v, [2pt] displaystyleσ_v+ min_∑_i b_i=b∑_iF_v_i(b_i) if v is internal. . Nonexistent alternatives are infeasible. The child sum is implemented by the proved finite min-plus convolution, not by assuming an optimal allocation.
Exact mathematical passage
F_v(b)=\min\left\{ \begin{array}{l} \kappa_v\quad\text{if }b=w_v,\\[2pt] \displaystyle\sigma_v+ \min_{\sum_i b_i=b}\sum_iF_{v_i}(b_i)\quad\text{if }v\text{ is internal}. \end{array}\right.
\mathrm{mod}Use in source · 5 passages
Search aliases: mod mathrm
Use in source · Paper_V_Traditional_Final.tex, line 316
Fix an integer N≥1. For 0≤ r<N, define (P_N,rx)[n] = 1_n≡ r (mod N)x[n]. The corresponding decimated channel is x_r[q]=x[Nq+r].
Exact mathematical passage
(P_{N,r}x)[n] = \mathbf 1_{n\equiv r\, (\mathrm{mod}\,N)}x[n].
\bm nUse in source · 3 passages
Search aliases: n bm
Use in source · 02_strengthened.tex, line 891
[Bloch sphere curvature] Let P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). With the convention F_θϕ =- iTr(P[∂_θ P,∂_ϕ P]), one has F_θϕ=frac12sinθ and frac12π ∫_0^2π∫_0^π F_θϕ dθ dϕ =1.
Exact mathematical passage
P(\theta,\phi) = \frac12\left(I+\bm n(\theta,\phi)\cdot\bm\sigma\right),
\mathbf NUse in source · 1 passage
Search aliases: N mathbf
Use in source · Paper_X_Block27_Amended.tex, line 711
Why the finite hypothesis cannot be dropped. Take vertices {*}sqcup{v_n:n∈ N}, with * isolated and edges v_n+1→ v_n. Every strongly connected component is a singleton, and {*} is the unique source component. No finite seed set covers all vertices: it can reach only indices no larger than the largest chosen index. Both the unique-source statement and the failure of every finite cover are checked in Lean. Hence one source component in an infinite condensation does not imply that one seed reaches a…
Exact mathematical passage
\{*\}\sqcup\{v_n:n\in\mathbf N\}
\mathbf nUse in source · 5 passages
Search aliases: n mathbf
Use in source · 01_finite_core.tex, line 637
For an orthogonal rank-one complex projector, use the scalar connection convention A=-i⟨ u,du⟩ in a local unit frame. The associated scalar curvature density is F_μν =-iTr(P[∂_μ P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)·σ. Using Tr(P a·σ)= n· a therefore yields F_θϕ =frac12 n·(∂_θ nmathbin×∂_ϕ n) =frac12sinθ. The scalar sign changes if the opposite convention for A i…
Exact mathematical passage
P(\theta,\phi) = \frac12\left(I+\mathbf n(\theta,\phi)\cdot\boldsymbol\sigma\right),
\mathcal NUse in source · 10 passages
Search aliases: N mathcal
Use in source · Paper_XI_Traditional_Final.tex, line 2131
At both ranks K^× normalizes P. Thus N=K^× J is normal of index two in M=K^× P, and Σ=Ind_ N^ MΛ, π≃cInd_ M^GL_N(F)Σ. Induction in stages preserves the full original Λ, its central character, and its noncompact root. See [source reference]. The full marked scalar-pushout comparison applies to these actual enlarged pairs as well as to the original higher-rank types ([source reference]). The full synthesis and scope are in [source reference].
Exact mathematical passage
\mathcal N=K^\times J
\mathfrak NUse in source · 8 passages
Search aliases: N mathfrak
Use in source · 03_packets_and_graphs.tex, line 859
[Preorder support algebra decomposition] Let R be a preorder on the finite set I. Define α∼β Longleftrightarrow (β,α)∈ R and (α,β)∈ R. Let overline I=I/∼, which carries the induced partial order, and put X_C:=⊕_α∈ CX_α. Let N_R⊆ M_R consist of operators whose blocks vanish whenever source and target belong to the same equivalence class. Then: [label=(roman*)] item N_R is a nilpotent ideal on both sides. item If h is the maximal number of strict steps in a chain of overline I, then N_R^ h+1=0. it…
Exact mathematical passage
\mathfrak N_R\subseteq\mathfrak M_R
\mathfrak nUse in source · 1 passage
Search aliases: n mathfrak
Use in source · Paper_XII_Traditional_Final.tex, line 1033
[Artin decomposition] Let E' be a field and let T be a finite dimensional commutative E' algebra. Then T ≅ ∏_ m∈Max T T_ m. Each factor is an Artin local algebra. Equivalently, there is a unique complete family of central idempotents (e_ m) such that e_ me_ n=0 ( m≠ n), ∑_ me_ m=1, and e_ m T≅ T_ m [source reference].
Exact mathematical passage
e_{\frakm}e_{\mathfrak n}=0 \quad(\frakm\neq\mathfrak n), \qquad \sum_{\frakm}e_{\frakm}=1,
N'Use in source · 3 passages
Search aliases: N'
Use in source · Paper_V_Traditional_Final.tex, line 1896
[Rectangular image phases] For L=M Z× N Z⊂ Z^2, one may take R={0,…,M-1}×{0,…,N-1}. The analysis map separates an image into MN exact phases. Refining (M,N) to (M',N'), where Mmid M' and Nmid N', splits every coarse phase into fracM'MfracN'N fine phases.
Exact mathematical passage
(M',N')
N^{-1}Use in source · 2 passages
Search aliases: N^-1
Use in source · 02_coordinates.tex, line 427
0, · r≠ j, 0≤ j,r<N. Then P_N,j = η∑_k=0^N-1(ω^vee)^jkU^k. In particular, if R is a field, N>0, and ω is a primitive Nth root of unity, the certificate holds with ω^vee=ω^-1 and η=N^-1.
Exact mathematical passage
\eta=N^{-1}
n^{-1/2}Use in source · 1 passage
Search aliases: n^-1/2
Use in source · 02_supplement_04.tex, line 22
Normalization and inversion. The proof identifies matrix application with the ordinary counting-measure discrete Fourier transform: Fx=n^-1/2DFT(x). It separately identifies F^*x with the appropriately scaled inverse transform. Mathlib's checked Fourier inversion theorem and the proved scalar normalization yield both identities F^*F=I, FF^*=I. No unitarity assumption is inserted.
Exact mathematical passage
Fx=n^{-1/2}\operatorname{DFT}(x).
n^{-2}Use in source · 1 passage
Search aliases: n^-2
Use in source · Paper_VI_Traditional_Final.tex, line 4433
Set B=-D_V and C=Q^T. For bounded matrices, e^tB/n e^tC/n = I+fractn(B+C)+O(n^-2) uniformly in operator norm. The standard telescoping comparison with (I+t(B+C)/n)^n shows that the product converges to e^t(B+C). This is the displayed factor order, so the limit follows.
Exact mathematical passage
\e^{tB/n}\e^{tC/n} = I+\frac{t}{n}(B+C)+O(n^{-2})
n^2Use in source · 26 passages
Search aliases: n^2
Use in source · supplement_04.tex, line 161
rincipal targets. targetUnitModes.geometric_difference, UnitModes.unit_fermat_divisibility targetUnitModes.unit_mode_valuation_gain targetUnitModes.finite_unit_modes_valuation_gain pageheading4. Exact examples and adversarial checksThe examples use the same definitions as the general results A polynomial anchor. The actual difference table for f(n)=n^2 yields (c_0,c_1,c_2,c_3)=(0,1,2,0). The finite interpolant therefore reconstructs I_2(c)(7)=49. This checks the sign and indexing conventions, no…
Exact mathematical passage
f(n)=n^2
N^3Use in source · 3 passages
Search aliases: N^3
Use in source · visual_atlas.tex, line 60
vpage3Two chains can stay linkedMODEL A / AN OPERATOR IN SIX DIMENSIONSMain text: [source reference]. Matrices over Q Use the ordered basis (e_0,e_1,e_2,f_0,f_1,f_2). Define Ne_0=e_1, Ne_1=e_2, Ne_2=0, Nf_0=f_1, Nf_1=f_2, Nf_2=0. Starting with s=e_0+f_0, the two chains advance together: s=e_0+f_0, Ns=e_1+f_1, N^2s=e_2+f_2, N^3s=0.
Exact mathematical passage
s=e_0+f_0,\qquad Ns=e_1+f_1,\qquad N^2s=e_2+f_2,\qquad N^3s=0.
N^5Use in source · 1 passage
Search aliases: N^5
Use in source · Paper_X_Block27_Amended.tex, line 2294
Actual affine representations with monodromy chains of every finite length invoke these theorems. In particular a length-six example has N^5≠0 and N^6=0 and is recovered from its actual reconstructed action. The length-four example still defeats the old quadratic logarithm, while the full finite logarithm recovers it.
Exact mathematical passage
N^5\ne0
N^6Use in source · 1 passage
Search aliases: N^6
Use in source · Paper_X_Block27_Amended.tex, line 2294
Actual affine representations with monodromy chains of every finite length invoke these theorems. In particular a length-six example has N^5≠0 and N^6=0 and is recovered from its actual reconstructed action. The length-four example still defeats the old quadratic logarithm, while the full finite logarithm recovers it.
Exact mathematical passage
N^6=0
N^dUse in source · 2 passages
Search aliases: N^d
Use in source · Paper_X_Block27_Amended.tex, line 1575
Finite-exponential reconstruction and a cubic inverse. Let K be a characteristic-zero field and let (r,N) be supplied group data with nilpotent N and r(g)N=ν(g)Nr(g). For a supplied scalar map satisfying t(gh)=t(g)+ν(g)t(h), the formalization constructs ρ(g)=exp(t(g)N)r(g), exp(aN)=∑_j=0^d-1fraca^jj!N^j (N^d=0). Lean proves the representation laws, actual finite-copy and split-map compatibility, and joint continuity from continuity of N, r and t with the stated continuous vector-space operations…
Exact mathematical passage
\rho(g)=\exp(t(g)N)r(g),\qquad \exp(aN)=\sum_{j=0}^{d-1}\frac{a^j}{j!}N^j\quad(N^d=0).
N^jUse in source · 1 passage
Search aliases: N^j
Use in source · Paper_X_Block27_Amended.tex, line 1575
Finite-exponential reconstruction and a cubic inverse. Let K be a characteristic-zero field and let (r,N) be supplied group data with nilpotent N and r(g)N=ν(g)Nr(g). For a supplied scalar map satisfying t(gh)=t(g)+ν(g)t(h), the formalization constructs ρ(g)=exp(t(g)N)r(g), exp(aN)=∑_j=0^d-1fraca^jj!N^j (N^d=0). Lean proves the representation laws, actual finite-copy and split-map compatibility, and joint continuity from continuity of N, r and t with the stated continuous vector-space operations…
Exact mathematical passage
\rho(g)=\exp(t(g)N)r(g),\qquad \exp(aN)=\sum_{j=0}^{d-1}\frac{a^j}{j!}N^j\quad(N^d=0).
N^mUse in source · 2 passages
Search aliases: N^m
Use in source · Paper_XI_Traditional_Final.tex, line 259
[Exact block reconstruction] Let P_j be the projector onto V_j. Then N=∑_j=1^m-1P_j-1NP_j, N^m=0. Every power N^r is the sum of the packet paths of length r along the chain.
Exact mathematical passage
N=\sum_{j=1}^{m-1}P_{j-1}NP_j, \qquad N^m=0.
N^NUse in source · 2 passages
Search aliases: N^N
Use in source · Paper_IV_Traditional_Final.tex, line 565
[Hypergeometric form] Let A_N,j = { fracj+mN mathrel: 1≤ m≤ N and j+m≠ N }. Then E_N,j(λ z) = frac(λ z)^jj! _0F_N-1 ( - A_N,j mathrel, frac(λ z)^NN^N ).
Exact mathematical passage
\Efun_{N,j}(\lambda z) = \frac{(\lambda z)^j}{j!} {}_0F_{N-1} \left( \begin{matrix}-\\ A_{N,j}\end{matrix} \mathrel{,} \frac{(\lambda z)^N}{N^N} \right).
N^nUse in source · 1 passage
Search aliases: N^n
Use in source · Paper_X_Block27_Amended.tex, line 1469
The exact special chain has basis e_0,…,e_n-1 and operators Ne_0=0, Ne_j=e_j-1 (j>0), Fe_j=aq^-je_j, a,q∈ K^×. Lean proves N^n=0, and for n>0 also N^n-1≠0 and ker N=Ke_0. Consequently its kernel polynomial is X-a, whereas its whole-space polynomial is ∏_j=0^n-1(X-aq^-j). The zero-length chain is included as a zero-dimensional space. The assertion ker N=Ke_0 is made only for positive length.
Exact mathematical passage
N^n=0
N^{n-1}Use in source · 1 passage
Search aliases: N^n-1
Use in source · Paper_X_Block27_Amended.tex, line 1469
The exact special chain has basis e_0,…,e_n-1 and operators Ne_0=0, Ne_j=e_j-1 (j>0), Fe_j=aq^-je_j, a,q∈ K^×. Lean proves N^n=0, and for n>0 also N^n-1≠0 and ker N=Ke_0. Consequently its kernel polynomial is X-a, whereas its whole-space polynomial is ∏_j=0^n-1(X-aq^-j). The zero-length chain is included as a zero-dimensional space. The assertion ker N=Ke_0 is made only for positive length.
Exact mathematical passage
N^{n-1}\ne0
N^{Nn}Use in source · 2 passages
Search aliases: N^Nn
Use in source · Paper_IV_Traditional_Final.tex, line 580
The multiplication formula for the gamma function gives (Nn+j)! = j!N^Nnn! ∏_a∈ A_N,j(a)_n. Indeed, the N Pochhammer factors supplied by the multiplication formula are ((j+m)/N)_n for 1≤ m≤ N. The excluded factor is ((j+N-j)/N)_n=(1)_n=n!. Insertion into the defining series gives the stated hypergeometric expansion.
Exact mathematical passage
(Nn+j)! = j!N^{Nn}n! \prod_{a\in A_{N,j}}(a)_n.
n^pUse in source · 1 passage
Search aliases: n^p
Use in source · 72_recovered.tex, line 245
By [source reference], n^p=0. The finite geometric sum gives the inverse.
Exact mathematical passage
n^p=0
N^rUse in source · 1 passage
Search aliases: N^r
Use in source · Paper_XI_Traditional_Final.tex, line 264
[Exact block reconstruction] Let P_j be the projector onto V_j. Then N=∑_j=1^m-1P_j-1NP_j, N^m=0. Every power N^r is the sum of the packet paths of length r along the chain.
Exact mathematical passage
N^r
\mathbb N^rUse in source · 1 passage
Search aliases: N^r mathbb
Use in source · 03_weighted.tex, line 445
[Several independent weights] Suppose every edge has a weight vector w(e)=(w_1(e),…,w_r(e))∈ N^r∖{0}. Replace t^w(e) by t_1^w_1(e)⋯ t_r^w_r(e). The resulting multigraded transfer series is rational in t_1,…,t_r.
Exact mathematical passage
w(e)=(w_1(e),\ldots,w_r(e))\in\mathbb N^r\setminus\{0\}.
n^\alphaUse in source · 1 passage
Search aliases: n^α alpha
Use in source · Paper_VI_Traditional_Final.tex, line 3307
[Fractional Hermite transfer] Let - L be the nonnegative Ornstein--Uhlenbeck number operator with - L h_n=nh_n. For every α>0, e^-t(- L)^αf = ∑_n=0^∞ e^-tn^αP_nf. This is a contraction semigroup on L^2(γ).
Exact mathematical passage
\e^{-t(-\Lcal)^\alpha}f = \sum_{n=0}^\infty \e^{-tn^\alpha}P_nf.
N_{\bm w}Use in source · 1 passage
Search aliases: N_ w bm
Use in source · Paper_X_Block27_Amended.tex, line 504
[Weighted smooth-chart count] For weights w=(w_1,…,w_d), N_ w(m) = q^∑_i=1^dmax(m-w_i,0).
Exact mathematical passage
\boxed{ N_{\bm w}(m) = q^{\sum_{i=1}^d\max(m-w_i,0)}. }
n_0Use in source · 3 passages
Search aliases: n_0
Use in source · 20_arithmetic.tex, line 173
chi congruence] Let p be prime, let m>0, and suppose that mmid p-1 or m=2. Put N=mp and q=lcm(2,p-1). For every 0≤ j<N, all U_N,j(n) belong to Z_p. For r≥1 and n≥ r, E_Nn+qp^r^(N,j)≡ E_Nn^(N,j)pmodp^r. Thus the positive sufficient starting index n_0=r works uniformly. If mmid p-1, the stronger sufficient condition r≤ mn is valid.
Exact mathematical passage
n_0=r
N_{01}Use in source · 2 passages
Search aliases: N_01
Use in source · 01_orientation.tex, line 56
A simple two packet model already contains all four notions. Let X=X_0⊕ X_1, D=D_0 · 0 0 · D_1, N=0 · N_01{ N_10 · 0. A finer family may split X_0 and X_1 into smaller summands. The blocks N_01 and N_10 may enter at different filtration weights. Over a fraction field, the rational packet spaces may fail to split the chosen lattice. A coarse operator on one selected summand may also fail to intertwine with the full matrix. The later sections isolate these four effects one at a time.
Exact mathematical passage
X=X_0\oplus X_1, \qquad D=\begin{pmatrix}D_0&0\\0&D_1\end{pmatrix}, \qquad N=\begin{pmatrix}0&N_{01}\\N_{10}&0\end{pmatrix}.
N_1Use in source · 1 passage
Search aliases: N_1
Use in source · 72_recovered.tex, line 3026
[Several coprime moduli] Let N_1,…,N_s be pairwise coprime and put N=∏_j=1^sN_j. For compatible addresses r_j, P_N,r = ∏_j=1^sP_N_j,r_j, where r is the unique combined address modulo N.
Exact mathematical passage
N_1,\ldots,N_s
n_1Use in source · 4 passages
Search aliases: n_1
Use in source · dyadic_synthesis.tex, line 19
((t)), fix an additive character ψ_F of conductor t O_F, and take increasing odd positive integers h_1<⋯<h_d. Put q_i=3h_i, N=2^d, n_j=2^d-jq_d-j+1. There is an explicit β∈ M_N(F) generating a separable totally ramified degree-N field that normalizes the complete lattice chain. It gives a maximal simple stratum with leading level n_1, final critical exponent -q_1, and successive approximation-field degrees 2,4,…,2^d, with critical levels n_2,…,n_d.
Exact mathematical passage
n_1
N_{10}Use in source · 2 passages
Search aliases: N_10
Use in source · 01_orientation.tex, line 56
A simple two packet model already contains all four notions. Let X=X_0⊕ X_1, D=D_0 · 0 0 · D_1, N=0 · N_01{ N_10 · 0. A finer family may split X_0 and X_1 into smaller summands. The blocks N_01 and N_10 may enter at different filtration weights. Over a fraction field, the rational packet spaces may fail to split the chosen lattice. A coarse operator on one selected summand may also fail to intertwine with the full matrix. The later sections isolate these four effects one at a time.
Exact mathematical passage
X=X_0\oplus X_1, \qquad D=\begin{pmatrix}D_0&0\\0&D_1\end{pmatrix}, \qquad N=\begin{pmatrix}0&N_{01}\\N_{10}&0\end{pmatrix}.
n_2Use in source · 2 passages
Search aliases: n_2
Use in source · dyadic_synthesis.tex, line 21
cdots<h_d. Put q_i=3h_i, N=2^d, n_j=2^d-jq_d-j+1. There is an explicit β∈ M_N(F) generating a separable totally ramified degree-N field that normalizes the complete lattice chain. It gives a maximal simple stratum with leading level n_1, final critical exponent -q_1, and successive approximation-field degrees 2,4,…,2^d, with critical levels n_2,…,n_d.
Exact mathematical passage
n_2,\ldots,n_d
N_DUse in source · 1 passage
Search aliases: N_D
Use in source · Paper_X_Block27_Amended.tex, line 2271
For supplied covariance data and the twisted cocycle of the hyperref[book:x-construction-13]finite-exponential reconstruction, let ρ_D(g)=exp(t(g)N_D)r_D(g). If a common element u satisfies r_D(u)=r_E(u)=1 and t(u)≠0, every intertwiner of ρ_D and ρ_E recovers both the original r-intertwiner and the N-intertwiner. There is no cubic nilpotence restriction. This recovers equivariant projectors for both original components. Reconstructed-action isomorphisms induce actual isomorphisms of the inertia-…
Exact mathematical passage
\rho_D(g)=\exp(t(g)N_D)r_D(g)
n_dUse in source · 3 passages
Search aliases: n_d
Use in source · dyadic_synthesis.tex, line 21
cdots<h_d. Put q_i=3h_i, N=2^d, n_j=2^d-jq_d-j+1. There is an explicit β∈ M_N(F) generating a separable totally ramified degree-N field that normalizes the complete lattice chain. It gives a maximal simple stratum with leading level n_1, final critical exponent -q_1, and successive approximation-field degrees 2,4,…,2^d, with critical levels n_2,…,n_d.
Exact mathematical passage
n_2,\ldots,n_d
n_{D_N}Use in source · 2 passages
Search aliases: n_D_N
Use in source · atlas-pages.tex, line 369
ions are v=(m+1/2)^4. vfigcounting The two rows describe the same zero orbits. The upper row counts four zeros each time a new radius is reached. The lower row counts one reduced zero each time its fourth-power location is reached. Both formulas include zeros on the boundary of the disk. headSeparate radius conversion from orbit multiplicity In unscaled variables the general relation is n_D_N(R^1/N)=Nn_Δ_N(R). There are two operations here: replace the radius by its Nth root and divide the count…
Exact mathematical passage
n_{D_N}(R^{1/N})=Nn_{\Delta_N}(R).
n_{\Dfun_N}Use in source · 3 passages
Search aliases: n_Dfun_N n_ D_N Dfun mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 2244
[Zero counting asymptotics] Let n_ D_N(R) count the zeros of D_N in |z|≤ R, with multiplicity. Let n_Δ_N(R) count the zeros of Δ_N in |w|≤ R. Then, as R→∞, n_ D_N(R) = fracNsin(π/N)πR+O(1) and n_Δ_N(R) = fracsin(π/N)πR^1/N+O(1).
Exact mathematical passage
n_{\Dfun_N}(R)
\mathcal N_EUse in source · 5 passages
Search aliases: N_E mathcal
Use in source · 03_supplement_05.tex, line 3
The precise finite channel statement. Let H= C^n be a finite coordinate space and let P=P^*=P^2. A matrix is code supported when Pρ P=ρ. For a finite family of square matrices E_a, put N_E(ρ)=∑_a E_aρ E_a^*. The input channel normalization is the actual matrix identity ∑_a E_a^*E_a=I. A finite Kraus recovery has the form R(σ)=∑_r R_rσ R_r^*, with ∑_rR_r^*R_r=I. This representation, together with the checked amplification results, provides complete positivity and trace preservation on the entire …
Exact mathematical passage
\mathcal N_E(\rho)=\sum_a E_a\rho E_a^*.
n_eUse in source · 4 passages
Search aliases: n_e
Use in source · 02_supplement_02.tex, line 161
Explicit storage and operation models. textitSource items 3.1, 3.2, 3.4, 3.6, 3.8--3.10, and 4.2 For a finite collection of rectangular blocks, the scalar addresses are actual elements of coprod_e∈ E(Fin(m_e)×Fin(n_e)). The corresponding operation addresses include a right-hand-side index and one of two scalar-operation tags. Lean proves their cardinalities rather than merely defining a formula named ``cost.''
Exact mathematical passage
\coprod_{e\in E}\bigl(\operatorname{Fin}(m_e)\times\operatorname{Fin}(n_e)\bigr).
N_iUse in source · 6 passages
Search aliases: N_i
Use in source · Paper_X_Block27_Amended.tex, line 1444
Algebraic monodromy data on the selected copies. Fix a field K and a nonzero scalar q∈ K^×. On each constituent W_i, supply an automorphism F_i and a nilpotent endomorphism N_i with F_iN_iF_i^-1=qN_i. The scalar q is common to this finite family. The componentwise operators F_m,N_m on U_m satisfy the same relation. F_m is invertible and N_m is nilpotent. The split maps constructed above intertwine both operators. The proof first identifies the independently defined kernel through an equivariant …
Exact mathematical passage
N_i
n_i^2Use in source · 2 passages
Search aliases: n_i^2
Use in source · 02_supplement_02.tex, line 175
BDLR · ∑_i n_i^2+2nr · 2q(∑_i n_i^2+2nr)
Exact mathematical passage
\sum_i n_i^2+2nr
N_jUse in source · 4 passages
Search aliases: N_j
Use in source · 72_recovered.tex, line 3027
[Several coprime moduli] Let N_1,…,N_s be pairwise coprime and put N=∏_j=1^sN_j. For compatible addresses r_j, P_N,r = ∏_j=1^sP_N_j,r_j, where r is the unique combined address modulo N.
Exact mathematical passage
N=\prod_{j=1}^{s}N_j.
n_jUse in source · 19 passages
Search aliases: n_j
Use in source · 02_foundations.tex, line 493
The lower inequality is quotient monotonicity. For the upper inequality, choose generating families (ℓ_i)_i∈ I of L and (n_j)_j∈ J of N with the least possible cardinalities. Choose a lift m_j∈ M of each n_j. For any m∈ M, a finite subset F⊆ J and coefficients a_j∈ A give π(m)=∑_j∈ Fa_jn_j. The vector m-∑_j∈ Fa_jm_j belongs to ι(L), so it is a finite linear combination of the ι(ℓ_i). The union of the lifted family and the image of the generating family of L therefore generates M. Its cardinality…
Exact mathematical passage
(n_j)_{j\in J}
N_{j-1,j}Use in source · 1 passage
Search aliases: N_j-1,j
Use in source · Paper_XI_Traditional_Final.tex, line 276
[Scalar covariance model] Let F=diag(1,q,q^2,…,q^m-1) and let N_j-1,j=1. Then FNF^-1=q^-1N.
Exact mathematical passage
N_{j-1,j}=1
N_j^kUse in source · 3 passages
Search aliases: N_j^k
Use in source · Paper_VI_Traditional_Final.tex, line 4650
mory of a finite hidden system] Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt ∑_k=0^ν_j-1 fract^kk!B_j,k and Σ_P(z) = ∑_j=1^r ∑_k=0^ν_j-1 fracB_j,k(z-μ_j)^k+1. Suppose Reμ_j<0 for every j. If J⊆{1,…,r} and K_J retains the compl…
Exact mathematical passage
B_{j,k} = A_{PQ}N_j^kQ_jA_{QP} \qquad (0\leq k<\nu_j).
N_j^{\nu_j}Use in source · 1 passage
Search aliases: N_j^ν_j nu
Use in source · Paper_VI_Traditional_Final.tex, line 4649
rem[Polynomial exponential memory of a finite hidden system] Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt ∑_k=0^ν_j-1 fract^kk!B_j,k and Σ_P(z) = ∑_j=1^r ∑_k=0^ν_j-1 fracB_j,k(z-μ_j)^k+1. Suppose Reμ_j<0 for every j. If J⊆{1,……
Exact mathematical passage
N_j^{\nu_j}=0
n_kUse in source · 6 passages
Search aliases: n_k
Use in source · monograph_atlas.tex, line 717
tive lower bound on the gap. Ground multiplicity does not cause a problem: P_0 selects the entire eigenspace at the minimum energy. For a Hamiltonian whose minimum is E_0≠0, the normalized transfer is e^-t(H-E_0I) before taking the limit. headPositivity is retained at several Euclidean times For arbitrary vectors u_j and nonnegative integer times n_j, the reflected kernel obeys ∑_j,k⟨ u_j,T^n_j+n_ku_k⟩ = ∑_jT^n_ju_j ^2. This is a finite Gram factorization. The retained run checks the integer sem…
Exact mathematical passage
\sum_{j,k}\langle u_j,T^{n_j+n_k}u_k\rangle =\left\|\sum_jT^{n_j}u_j\right\|^2.
N_mUse in source · 4 passages
Search aliases: N_m
Use in source · Paper_X_Block27_Amended.tex, line 1446
Algebraic monodromy data on the selected copies. Fix a field K and a nonzero scalar q∈ K^×. On each constituent W_i, supply an automorphism F_i and a nilpotent endomorphism N_i with F_iN_iF_i^-1=qN_i. The scalar q is common to this finite family. The componentwise operators F_m,N_m on U_m satisfy the same relation. F_m is invertible and N_m is nilpotent. The split maps constructed above intertwine both operators. The proof first identifies the independently defined kernel through an equivariant …
Exact mathematical passage
F_m,N_m
n_mUse in source · 1 passage
Search aliases: n_m
Use in source · 02_strengthened.tex, line 1875
[Finite reflection positivity] For vectors v_1,…,v_m∈ H and nonnegative integers n_1,…,n_m, the matrix G_jk=ipv_jT^n_j+n_kv_k is positive semidefinite. Equivalently, ∑_j,koverlinec_jc_k ipv_jT^n_j+n_kv_k ≥0 for every c∈ C^m.
Exact mathematical passage
n_1,\ldots,n_m
N_{n,M}Use in source · 2 passages
Search aliases: N_n,M
Use in source · 03_nonlinear_coordinates.tex, line 378
[Number of labeled composition terms] For n≥1, if Y has M packets, then the number of packet labeled terms at order n before zero jets are removed is N_n,M = ∑_k=1^nS(n,k)M^k, where S(n,k) is a Stirling number of the second kind. Thus N_n,M is the nth Touchard polynomial evaluated at M.
Exact mathematical passage
N_{n,M} = \sum_{k=1}^{n}S(n,k)M^k,
n_pUse in source · 4 passages
Search aliases: n_p
Use in source · Paper_X_Block27_Amended.tex, line 1797
For each supplied required packet p, let A_p be its finite-dimensional carrier over a characteristic-zero field E_p, and let e_p be the actual joint spectral projector for its finite list of split commuting conditions. No order on E_p is assumed. The source notation Tr(e_p)>0 means that the unique natural number n_p with Tr(e_p)=(n_p:E_p) is positive. The checked projector formula identifies n_p with the dimension of the actual joint eigenspace.
Exact mathematical passage
n_p
N_{p,w}Use in source · 5 passages
Search aliases: N_p,w
Use in source · 01_manuscript.tex, line 465
[Equal block banded count] Let p,b≥1 be integers, n=pb, and let p=n/b equal packets have width b. Suppose a block is active precisely when absi-j≤ w, 0≤ w≤ p-1. The number of active blocks is N_p,w = p(2w+1)-w(w+1). For q right hand sides, the modeled application cost and scalar storage are 2qb^2N_p,w and b^2N_p,w.
Exact mathematical passage
\boxed{ N_{p,w} = p(2w+1)-w(w+1). }
\mathfrak N_RUse in source · 7 passages
Search aliases: N_R mathfrak
Use in source · 03_packets_and_graphs.tex, line 859
[Preorder support algebra decomposition] Let R be a preorder on the finite set I. Define α∼β Longleftrightarrow (β,α)∈ R and (α,β)∈ R. Let overline I=I/∼, which carries the induced partial order, and put X_C:=⊕_α∈ CX_α. Let N_R⊆ M_R consist of operators whose blocks vanish whenever source and target belong to the same equivalence class. Then: [label=(roman*)] item N_R is a nilpotent ideal on both sides. item If h is the maximal number of strict steps in a chain of overline I, then N_R^ h+1=0. it…
Exact mathematical passage
\mathfrak N_R\subseteq\mathfrak M_R
n_rUse in source · 1 passage
Search aliases: n_r
Use in source · Paper_XII_Traditional_Final.tex, line 495
[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.
Exact mathematical passage
\boxed{ k[\Delta] \cong k[X_1,\ldots,X_r] \big/ (X_1^{p^{n_1}},\ldots,X_r^{p^{n_r}}). }
\mathfrak N_R^{\,h+1}Use in source · 1 passage
Search aliases: N_R^ h+1 mathfrak
Use in source · 03_packets_and_graphs.tex, line 863
rtial order, and put X_C:=⊕_α∈ CX_α. Let N_R⊆ M_R consist of operators whose blocks vanish whenever source and target belong to the same equivalence class. Then: [label=(roman*)] item N_R is a nilpotent ideal on both sides. item If h is the maximal number of strict steps in a chain of overline I, then N_R^ h+1=0. item The map to the diagonal classes induces an algebra isomorphism M_R/ N_R ≅ ∏_C∈overline IEnd_𝕜(X_C). [source reference] ExportsFiltered transfer algebras and triangular packet inver…
Exact mathematical passage
\mathfrak N_R^{\,h+1}=0.
N_SUse in source · 5 passages
Search aliases: N_S
Use in source · 02_foundations.tex, line 229
[Finite free presentation] For the map in [source reference], imε_S=Hull_A(S), N_S:=Kerε_S≤ A^⊕ r, and the induced map is an isomorphism of left modules over A A^⊕ r/N_S≅Hull_A(S). Consequently, S generates M if and only if ε_S is surjective. [source reference] ExportsAll later free seed presentations.
Exact mathematical passage
\im\varepsilon_S=\Hull_A(S), \qquad N_S:=\Ker\varepsilon_S\leq A^{\oplus r},
N_sUse in source · 1 passage
Search aliases: N_s
Use in source · 72_recovered.tex, line 3026
[Several coprime moduli] Let N_1,…,N_s be pairwise coprime and put N=∏_j=1^sN_j. For compatible addresses r_j, P_N,r = ∏_j=1^sP_N_j,r_j, where r is the unique combined address modulo N.
Exact mathematical passage
N_1,\ldots,N_s
n_\alpha^2Use in source · 4 passages
Search aliases: n_α^2 alpha
Use in source · 01_manuscript.tex, line 321
[BDLR application and storage] For q right hand sides, AZ=DZ+U(V^*Z) has modeled cost 2q ( ∑_α n_α^2+2nr ). The representation stores ∑_α n_α^2+2nr scalars. For n=pb, with integers p,b≥1, the following factor describes the ideal dense storage reduction and, when q≥1, the arithmetic reduction: fracnb+2r.
Exact mathematical passage
\boxed{ 2q \left( \sum_\alpha n_\alpha^2+2nr \right). }
n_\betaUse in source · 6 passages
Search aliases: n_β beta
Use in source · 01_manuscript.tex, line 257
[Dense and packet application counts] A dense n by n matrix applied to q right hand sides has modeled cost flop_rm dense=2n^2q. A packet representation with active edge set E has modeled cost flop_rm packet = 2q ∑_(β,α)∈ E n_β n_α. It stores M_rm packet = ∑_(β,α)∈ E n_β n_α scalar entries.
Exact mathematical passage
\boxed{ \flop_{\rm packet} = 2q \sum_{(\beta,\alpha)\in E} n_\beta n_\alpha. }
n_{\Delta_N}Use in source · 5 passages
Search aliases: n_Δ_N Delta
Use in source · Paper_IV_Traditional_Final.tex, line 2244
[Zero counting asymptotics] Let n_ D_N(R) count the zeros of D_N in |z|≤ R, with multiplicity. Let n_Δ_N(R) count the zeros of Δ_N in |w|≤ R. Then, as R→∞, n_ D_N(R) = fracNsin(π/N)πR+O(1) and n_Δ_N(R) = fracsin(π/N)πR^1/N+O(1).
Exact mathematical passage
n_{\Delta_N}(R)
N_\varepsilonUse in source · 5 passages
Search aliases: N_ε varepsilon
Use in source · monograph_atlas.tex, line 738
· 25/16 · 20 · 8/5 These are unnormalized traces. Normalizing by N changes Z_N but leaves the logarithmic derivative at that fixed grid unchanged. xfigheat_limits headThe theorem uses two separate choices For each ε>0, first choose a fixed positive time t_ε so that the continuum torus dimension differs from one by less than ε/2. Then choose N_ε so that every N≥ N_ε differs from that continuum value by less than ε/2 at the same time. This proves one controlled intermediate time for all sufficient…
Exact mathematical passage
N_\varepsilon
\negUse in source · 2 passages
Search aliases: neg
Use in source · 02_connections_obstructions.tex, line 422
[Exact logical force of a local obstruction] Every obstruction packet for x proves that x is not a solution, and negIsSolution_r(x) Longleftrightarrow there exists an obstruction packet for x. The reverse implication from failure to a packet witness is classical and does not require the packet type to be finite. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.ObstructionPacket.not_isSolution; Komorebi.PaperIII.NonlinearObstructions.obstructionPacket_of_not_isSolution; Komorebi.PaperIII.No…
Exact mathematical passage
\neg\operatorname{IsSolution}_r(x) \quad\Longleftrightarrow\quad \text{there exists an obstruction packet for }x.
\nilUse in source · 1 passage
Search aliases: nil Nil operatorname
Use in source · Paper_XII_Traditional_Final.tex, line 680
[Near faithfulness is full support] Let R be Noetherian and let M be a finite R module. The following conditions are equivalent. item M is nearly faithful. item Supp_R(M)=Spec R. item Ann_R(M)⊆Nil(R). If R is reduced, these conditions are equivalent to faithfulness.
Exact mathematical passage
\Ann_R(M)\subseteq\nil(R)
\mathrm{NL}Use in source · 1 passage
Search aliases: NL mathrm
Use in source · 71_native.tex, line 110
o(f)∈Ext^1_B(NL_B/ Z, F_p)
Exact mathematical passage
o(f)\in\operatorname{Ext}^1_B(\mathrm{NL}_{B/\mathbb Z},\mathbb F_p)
\mathrm{NL}_{B/\mathbb Z}Use in source · 1 passage
Search aliases: NL_B/ Z mathrm mathbb
Use in source · 71_native.tex, line 110
o(f)∈Ext^1_B(NL_B/ Z, F_p)
Exact mathematical passage
o(f)\in\operatorname{Ext}^1_B(\mathrm{NL}_{B/\mathbb Z},\mathbb F_p)
\nmidUse in source · 29 passages
Search aliases: nmid
Use in source · 20_arithmetic.tex, line 291
At a nonsingular address 2≤ c<N with pnmid c, differentiation of the section identities gives tR_c'=R_c(1+R_c-1-R_c). Writing q=R_c-1-(c-1) and y=R_c-c, this becomes (tD+c)y=cq+yq-y^2. On degrees divisible by p, the inverse of tD+c divides the degree-d coefficient by the unit d+c. Starting with y_0=0, form y_s+1=(tD+c)^-1(cq+y_sq-y_s^2). Flatness and support are preserved. Indeed, put E_c=tD+c and C_p=pD^p^2-D^p. The commutator is [B_p,E_c]=C_p, and iteration gives B_p^rE_cy=E_cB_p^ry+rC_pB_p^r-…
Exact mathematical passage
p\nmid c
\NNUse in source · 11 passages
Search aliases: NN N mathbb
Use in source · 72_recovered.tex, line 124
[Hurwitz series ring] Let R be a commutative ring with identity. The Hurwitz series ring H(R) is the set R^ N with componentwise addition and product (amathbin⋆ b)_n = ∑_r=0^n binom nr a_r b_n-r. Its identity is δ=(1,0,0,…).
Exact mathematical passage
R^{\NN}
\NN_0Use in source · 1 passage
Search aliases: NN_0 N_0 NN mathbb
Use in source · monograph_atlas.tex, line 769
utes to a bounded Hamiltonian limitEXACT EMBEDDINGS AND A SUMMABLE TAILChapter VII: Theorem II.13.2, Theorem II.13.4, Corollary II.13.5. For the exact route, let HH_n= C^n with coordinate inclusions and H_n=diag(0,1,1/2,…,1/(n-1)), H_1=(0). The old energies are retained at every level, H_n+1R_n=R_nH_n, and normH_n≤1. On the Hilbert space of the direct limit ℓ^2( N_0), the diagonal sequence defines a bounded self-adjoint H_∞. Projectors onto even addresses also intertwine and give a fixed complet…
Exact mathematical passage
\ell^2(\NN_0)
OUse in source · 71 passages
Search aliases: O
Use in source · 05_worked_models.tex, line 525
[Closure specification for the orbit hull] Let V have finite dimension, let (O_i)_i∈ I be a finite operator family, and let S⊆ V. Define W_0=span_𝕜S, W_n+1=W_n+∑_i∈ IO_i(W_n). Then there is an n∈ N such that W_n+1=W_n and W_n=Hull_(O_i)(S), where the right side is the least subspace containing S and invariant under every O_i. Thus the ascending subspace iteration terminates and its stable value is exactly the finite operator hull.
Exact mathematical passage
(O_i)_{i\in I}
O_1Use in source · 1 passage
Search aliases: O_1
Use in source · 01_finite_core.tex, line 58
The generated state space is H(ψ_0; O) = span { O_r⋯ O_1ψ_0: O_j∈ O, r≥0 }. The quantum seed is cyclic for the grammar when H(ψ_0; O)= H. This is a statement about the pair (ψ_0, O), not about the vector alone. A state may be cyclic for the full matrix algebra and noncyclic for a restricted physical gate set.
Exact mathematical passage
\cH(\psi_0;\cO) = \operatorname{span} \left\{ O_r\cdots O_1\psi_0: O_j\in\cO, \ r\ge0 \right\}.
\mathcal O_2Use in source · 5 passages
Search aliases: O_2 mathcal
Use in source · supplement_03.tex, line 30
cients is not universally formalized here. Formal-series identities do not establish analytic convergence. The corrected global odd-prime finite-difference theorem remains pending. Historical numerical evidence is not presented as a fresh reproduction. pageheading1. Integral coefficients and an exact unit sourceLemmas 5.1 and 5.2; manuscript page 14 Define the rational subring O_2={q∈ Q:|q|_2≤1}. Its ring laws follow from the actual nonarchimedean norm. For every q∈ O_2, the code constructs its …
Exact mathematical passage
\mathcal O_2=\{q\in\Q:|q|_2\leq1\}.
\mathcal O_AUse in source · 1 passage
Search aliases: O_A mathcal
Use in source · 03_supplement_03.tex, line 81
Full support is expressed as surjectivity of the reduced matrix's linear map. Assuming d_A>0, the checked equivalence is O_A(ψ)= H_A⊗ H_B Longleftrightarrow Ranρ_B= H_B. Normalization of ψ is unnecessary for these dimension and support statements. The zero state has rank 0 because its decomposition needs no product terms.
Exact mathematical passage
\boxed{\mathcal O_A(\psi)=\mathcal H_A\otimes\mathcal H_B \quad\Longleftrightarrow\quad \Ran\rho_B=\mathcal H_B.}
\mathcal O_E^\timesUse in source · 1 passage
Search aliases: O_E^× mathcal times
Use in source · odd_synthesis.tex, line 62
Let P_0 extract the constant coefficient in the basis 1,z,…,z^p-1. The normalized corestriction s=κpr, where κ=-f'(z)/z^p-1, satisfies u_0=s(P_0)=a_0/z^p∈ O_E^×, Lift_E(C)=u_0^-1CP_0, s(Lift_E(C))=C. The lift preserves the exact filtration valuation. Its crucial additional identity is tr_L(Lift_E(C)) =Tr_E/L (κ^-1tr_E(C)). Define B(L,t_L;h_1,…,h_d) =α I_p^d-1+ Lift_E(B(E,z;h_1,…,h_d-1)), with the degree-p multiplication matrix as the base case. Induction proves trace zero at every native rank. T…
Exact mathematical passage
u_0=s(P_0)=a_0/z^p\in\mathcal O_E^\times, \qquad \operatorname{Lift}_E(C)=u_0^{-1}CP_0, \qquad s(\operatorname{Lift}_E(C))=C.
\mathcal O_FUse in source · 2 passages
Search aliases: O_F mathcal
Use in source · dyadic_synthesis.tex, line 14
Let F= F_2((t)), fix an additive character ψ_F of conductor t O_F, and take increasing odd positive integers h_1<⋯<h_d. Put q_i=3h_i, N=2^d, n_j=2^d-jq_d-j+1. There is an explicit β∈ M_N(F) generating a separable totally ramified degree-N field that normalizes the complete lattice chain. It gives a maximal simple stratum with leading level n_1, final critical exponent -q_1, and successive approximation-field degrees 2,4,…,2^d, with critical levels n_2,…,n_d.
Exact mathematical passage
t\mathcal O_F
O_{F,\bar x}Use in source · 1 passage
Search aliases: O_F,bar x bar
Use in source · 71_native.tex, line 25
O_F,bar x= F_p^r/imJ, o_F,bar x=[e].
Exact mathematical passage
O_{F,\bar x}=\mathbb F_p^r/\operatorname{im}J,\qquad o_{F,\bar x}=[e].
o_{F,\bar x}Use in source · 2 passages
Search aliases: o_F,bar x bar
Use in source · 50_open_questions.tex, line 53
The original conjecture asks for a canonical finite obstruction packet whose vanishing is necessary and sufficient for a one-level lift in a finitely presented arithmetic seed system. For a fixed finite polynomial presentation, the chapter supplies the precise obstruction o_F,bar x=[r]∈ coker (J_F,bar x: F_p^d⟶ F_p^e), where r is the residual digit. We have proved independence from the chosen integer representative. Invariance under a change of equations or presentation requires more.
Exact mathematical passage
o_{F,\bar x}=[r]\in \coker\!\left(J_{F,\bar x}:\F_p^d\longrightarrow\F_p^e\right),
O_iUse in source · 6 passages
Search aliases: O_i
Use in source · 05_worked_models.tex, line 525
[Closure specification for the orbit hull] Let V have finite dimension, let (O_i)_i∈ I be a finite operator family, and let S⊆ V. Define W_0=span_𝕜S, W_n+1=W_n+∑_i∈ IO_i(W_n). Then there is an n∈ N such that W_n+1=W_n and W_n=Hull_(O_i)(S), where the right side is the least subspace containing S and invariant under every O_i. Thus the ascending subspace iteration terminates and its stable value is exactly the finite operator hull.
Exact mathematical passage
(O_i)_{i\in I}
O_jUse in source · 1 passage
Search aliases: O_j
Use in source · 01_finite_core.tex, line 58
The generated state space is H(ψ_0; O) = span { O_r⋯ O_1ψ_0: O_j∈ O, r≥0 }. The quantum seed is cyclic for the grammar when H(ψ_0; O)= H. This is a statement about the pair (ψ_0, O), not about the vector alone. A state may be cyclic for the full matrix algebra and noncyclic for a restricted physical gate set.
Exact mathematical passage
\cH(\psi_0;\cO) = \operatorname{span} \left\{ O_r\cdots O_1\psi_0: O_j\in\cO, \ r\ge0 \right\}.
O_{K,C}Use in source · 1 passage
Search aliases: O_K,C
Use in source · residue_continuation_body.tex, line 138
Ordinary class characters: an unconditional power saving. The preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl( O)→ C, define the actual finite ideal sum S_w(X)=∑_substackI≠0
Exact mathematical passage
A_C(X)=c_KX+O_{K,C}(X^{1-1/d}),\qquad c_K=\frac{2^{r_1}(2\pi)^{r_2}R_K}{\#\mu(K)\sqrt{\absn{\Delta_K}}}.
O_{K,J,m,\chi}Use in source · 1 passage
Search aliases: O_K,J,m,χ chi
Use in source · residue_continuation_body.tex, line 225
The assembled ideal norm-character estimate. Let C∈Cl( O), let J≠0 be an integral ideal with [J]=C^-1, and let m>0. Let χ: Z/m Z→ C preserve multiplication, 0, and 1. Assume a residue-ring unit satisfying [source reference], and assume χ( N Jbmod m)≠0. Then T_C,χ(X) =∑_substackI≠0, [I]=C N I≤ X χ( N Ibmod m) =O_K,J,m,χ(X^1-1/d). More precisely, some B≥0 bounds its absolute value by BX^1-1/d for every real X≥1.
Exact mathematical passage
T_{C,\chi}(X) =\sum_{\substack{I\ne0,\ [I]=C\\\Nm I\leq X}} \chi(\Nm I\bmod m) =O_{K,J,m,\chi}(X^{1-1/d}).
o_mUse in source · 1 passage
Search aliases: o_m
Use in source · visual_atlas.tex, line 481
vpage20Filter first or split first: same outputMODEL F / A FINITE POLYPHASE REALIZATIONChapter II, mathematical text: Section B.1 Use the finitely supported signal x=(1,2,3,4,5,6) and the filter h=(1,2,1). Outside the displayed support, the input is zero. Let e_m=x_2m and o_m=x_2m+1.
Exact mathematical passage
o_m=x_{2m+1}
O_rUse in source · 1 passage
Search aliases: O_r
Use in source · 01_finite_core.tex, line 58
The generated state space is H(ψ_0; O) = span { O_r⋯ O_1ψ_0: O_j∈ O, r≥0 }. The quantum seed is cyclic for the grammar when H(ψ_0; O)= H. This is a statement about the pair (ψ_0, O), not about the vector alone. A state may be cyclic for the full matrix algebra and noncyclic for a restricted physical gate set.
Exact mathematical passage
\cH(\psi_0;\cO) = \operatorname{span} \left\{ O_r\cdots O_1\psi_0: O_j\in\cO, \ r\ge0 \right\}.
O_{R_*}Use in source · 4 passages
Search aliases: O_R_*
Use in source · Paper_IV_Traditional_Final.tex, line 2632
rcle_expansion Let r∈ Z_>0. Suppose the zeros of D_N of least modulus R are simple and form the finite set Z_R. Suppose the next zero modulus is R_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_ρ,ℓ binomm+ℓ-1ℓ-1 ρ^-m + O_R_*(m!R_*^-m). The leading Laurent coefficient is a_ρ,r = (-ρ D_N'(ρ))^-r.
Exact mathematical passage
P_{N,n}(r) = m! \sum_{\rho\in\Zcal_R} \sum_{\ell=1}^{r} a_{\rho,\ell} \binom{m+\ell-1}{\ell-1} \rho^{-m} + O_{R_*}(m!R_*^{-m}).
\mathcal O_xUse in source · 7 passages
Search aliases: O_x mathcal
Use in source · 30_lifting.tex, line 72
[Finite polynomial obstruction] Let q≥1, L≥1, x∈ Z^d, and suppose F(x)∈ q^L Z^e. Define the exact integer residual r_x by F(x)=q^Lr_x, and put O_x=S_q^e/im J_x, o_x=[overliner_x]. Then O_x is finite and o_x=0 Longleftrightarrow ∃ y∈ Z^d: y≡ xpmodq^L, F(y)≡0pmodq^L+1. No surjectivity or nonsingularity assumption is required.
Exact mathematical passage
\mathcal O_x=S_q^e/\im J_x,\qquad o_x=[\overline{r_x}].
\mathrm{ob}Use in source · 3 passages
Search aliases: ob mathrm
Use in source · 04_correction_transport.tex, line 7
[Orthogonal obstruction projector] Let D:H_1→ H_2 be a bounded operator between Hilbert spaces with closed range. Let D^† be its Moore--Penrose inverse. Then P_ob=I_H_2-DD^† is the orthogonal projector onto (im D)^⊥=Ker D^*. For r∈ H_2, the following are equivalent: item Du=-r is solvable item [r]=0 in coker D item P_obr=0. When these conditions hold, -D^† r is the least norm correction. For every r, dist(r,im D)= P_obr .
Exact mathematical passage
P_{\mathrm{ob}}=I_{H_2}-DD^\dagger
\operatorname{Ob}Use in source · 6 passages
Search aliases: Ob operatorname
Use in source · 72_recovered.tex, line 2533
[Linear lift obstruction] The linear obstruction at depth L is Ob^lin_L(x_L) = [ρ_L(x_L)] ∈ coker(overline M).
Exact mathematical passage
\boxed{ \operatorname{Ob}^{\mathrm{lin}}_L(x_L) = [\rho_L(x_L)] \in \coker(\overline M). }
\operatorname{Ob}^{F}_LUse in source · 3 passages
Search aliases: Ob^F_L operatorname
Use in source · 72_recovered.tex, line 2656
[Polynomial lift obstruction] The polynomial obstruction at depth L is Ob^F_L(x_L) = [ρ_L^F(x_L)] ∈ coker(overline J_F(x_L)).
Exact mathematical passage
\boxed{ \operatorname{Ob}^{F}_L(x_L) = [\rho_L^F(x_L)] \in \coker\bigl(\overline J_F(x_L)\bigr). }
\Ob^GUse in source · 2 passages
Search aliases: Ob^G Ob operatorname
Use in source · 02_connections_obstructions.tex, line 576
[Gauge covariance when packet projectors are preserved] If G_aΠ^a_μ=Π^a_μ G_a for every μ, then (Ob^G)^μν = G_aOb^μνG_a^-1. Consequently the zero/nonzero status of each obstruction block is preserved by gauges that preserve the packet projectors. statusleanthreeKomorebi.PaperIII.HolonomyBlocks.obstructionBlock_gauge_of_commutes; Komorebi.PaperIII.HolonomyBlocks.obstructionBlock_gauge_eq_zero_iff
Exact mathematical passage
\left(\Ob^G\right)^{\mu\nu} = G_a\Ob^{\mu\nu}G_a^{-1}.
\operatorname{Ob}^{\mathrm{lin}}_LUse in source · 3 passages
Search aliases: Ob^lin_L operatorname mathrm
Use in source · 72_recovered.tex, line 2533
[Linear lift obstruction] The linear obstruction at depth L is Ob^lin_L(x_L) = [ρ_L(x_L)] ∈ coker(overline M).
Exact mathematical passage
\boxed{ \operatorname{Ob}^{\mathrm{lin}}_L(x_L) = [\rho_L(x_L)] \in \coker(\overline M). }
\Ob^{\mu\nu}Use in source · 1 passage
Search aliases: Ob^μν Ob mu nu operatorname mu nu
Use in source · 02_connections_obstructions.tex, line 576
[Gauge covariance when packet projectors are preserved] If G_aΠ^a_μ=Π^a_μ G_a for every μ, then (Ob^G)^μν = G_aOb^μνG_a^-1. Consequently the zero/nonzero status of each obstruction block is preserved by gauges that preserve the packet projectors. statusleanthreeKomorebi.PaperIII.HolonomyBlocks.obstructionBlock_gauge_of_commutes; Komorebi.PaperIII.HolonomyBlocks.obstructionBlock_gauge_eq_zero_iff
Exact mathematical passage
\left(\Ob^G\right)^{\mu\nu} = G_a\Ob^{\mu\nu}G_a^{-1}.
\Ob_{D'}Use in source · 1 passage
Search aliases: Ob_D' Ob operatorname
Use in source · 03_nonlinear_coordinates.tex, line 965
U' · xrightarrowD' · V' is a commuting square. Then there is a unique induced map overline b:coker D⟶coker D' satisfying overline b Ob_D(r)=Ob_D'(br).
Exact mathematical passage
\overline b\,\Ob_D(r)=\Ob_{D'}(br).
\Ob_m'Use in source · 1 passage
Search aliases: Ob_m' Ob operatorname
Use in source · 04_correction_transport.tex, line 365
[Naturality of filtered obstructions] Suppose a morphism of filtered lifting problems sends x_m to x_m', intertwines L_m with L_m', and sends Φ(x_m) to Φ'(x_m') on the associated graded pieces. Then the induced cokernel map sends Ob_m(x_m) to Ob_m'(x_m').
Exact mathematical passage
\Ob_m(x_m) \quad\text{to}\quad \Ob_m'(x_m').
\Ocal_tUse in source · 12 passages
Search aliases: Ocal_t O_t Ocal mathcal
Use in source · Paper_VI_Traditional_Final.tex, line 3121
Let γ be the standard Gaussian probability measure on R. For t≥0, define the Ornstein--Uhlenbeck transfer ( O_t f)(x) = ∫_ R f( e^-tx+sqrt1- e^-2t y ) γ( d y). Let H_n be the probabilists' Hermite polynomial defined by exp(ux-fracu^22) = ∑_n=0^∞ H_n(x)fracu^nn!, and set h_n=H_n/sqrtn!.
Exact mathematical passage
(\Ocal_t f)(x) = \int_\RR f\left( \e^{-t}x+\sqrt{1-\e^{-2t}}\,y \right) \gamma(\dd y).
\Ocal_\lambdaUse in source · 2 passages
Search aliases: Ocal_λ O_λ Ocal lambda mathcal lambda
Use in source · Paper_X_Block27_Amended.tex, line 882
All coefficients of the characteristic polynomial commute with reduction. Consequently, prescribed integral characteristic polynomials at every precision hold exactly in the reconstructed representation. For a common coefficient field E, local integral polynomials lie in O_λ[X] and are compared with P_v∈ E[X] only after mapping both into the local coefficient field. No field embedding E→ O_λ is assumed. The allowed-place predicate is explicit and may depend on λ. Trivial action on a supplied sub…
Exact mathematical passage
\Ocal_\lambda[X]
\odotUse in source · 12 passages
Search aliases: odot
Use in source · Paper_VI_Traditional_Final.tex, line 731
Let p=(p_i) be a probability vector and let λ=(λ_i) be a nonnegative likelihood with positive evidence Z_λ(p)=∑_iλ_ip_i. Define B_λ(p) = fracλodot pZ_λ(p). This map is nonlinear solely because of its scalar reconstruction factor.
Exact mathematical passage
\boxed{ \cB_\lambda(p) = \frac{\lambda\odot p}{Z_\lambda(p)}. }
\ointUse in source · 1 passage
Search aliases: oint
Use in source · Paper_VI_Traditional_Final.tex, line 1667
(μ)k!(A-μ I)^k|_E_μ. The formula is independent of a basis and of increasing the truncation index: the extra nilpotent powers vanish. It depends only on the finite Taylor jets of F at the spectrum. If F is holomorphic on an open neighborhood U of the spectrum, each eigenvalue admits a radius r_μ>0 with overlineD(μ,r_μ)⊂ U, and Cauchy's formula gives fracF^(k)(μ)k! =frac12π ioint_|z-μ|=r_μ fracF(z)(z-μ)^k+1 dz. Consequently the same canonical operator is obtained from any such local Cauchy circle…
Exact mathematical passage
\frac{F^{(k)}(\mu)}{k!} =\frac{1}{2\pi i}\oint_{|z-\mu|=r_\mu} \frac{F(z)}{(z-\mu)^{k+1}}\,dz.
\oint_{|z-\mu|=r_\mu}Use in source · 1 passage
Search aliases: oint_|z-μ|=r_μ oint mu mu
Use in source · Paper_VI_Traditional_Final.tex, line 1667
(μ)k!(A-μ I)^k|_E_μ. The formula is independent of a basis and of increasing the truncation index: the extra nilpotent powers vanish. It depends only on the finite Taylor jets of F at the spectrum. If F is holomorphic on an open neighborhood U of the spectrum, each eigenvalue admits a radius r_μ>0 with overlineD(μ,r_μ)⊂ U, and Cauchy's formula gives fracF^(k)(μ)k! =frac12π ioint_|z-μ|=r_μ fracF(z)(z-μ)^k+1 dz. Consequently the same canonical operator is obtained from any such local Cauchy circle…
Exact mathematical passage
\frac{F^{(k)}(\mu)}{k!} =\frac{1}{2\pi i}\oint_{|z-\mu|=r_\mu} \frac{F(z)}{(z-\mu)^{k+1}}\,dz.
\one^TUse in source · 15 passages
Search aliases: one^T 1^T one bm
Use in source · Paper_VI_Traditional_Final.tex, line 2841
Choose a≥max_i(-Q_ii). The matrix Q+aI is nonnegative, so its exponential is nonnegative. Hence e^tQ= e^-at e^t(Q+aI) is nonnegative. Since 1^TQ=0, differentiation gives 1^T e^tQ= 1^T.
Exact mathematical passage
\one^TQ=0
\one_{[-a,a]}Use in source · 1 passage
Search aliases: one_[-a,a] 1_[-a,a] one mathbf
Use in source · Paper_V_Traditional_Final.tex, line 3469
[Square wave Fourier tail] Let j(t)= 1_[-a,a](t) on the circle and let s be smooth. If Q_m is a Fourier partial sum, then Q_m(s+j) has the ordinary oscillatory boundary layer. In contrast, Q_ms+j has error exactly s-Q_ms. The visual comparison in [source reference] illustrates this change in approximation target. It is not a new Fourier summability theorem.
Exact mathematical passage
j(t)=\one_{[-a,a]}(t)
\one_EUse in source · 4 passages
Search aliases: one_E 1_E one mathbf
Use in source · Paper_V_Traditional_Final.tex, line 2702
[Exact scalar region formula] Let E,F be measurable subsets of a measure space, let a,b∈ C, and let 1≤ p<∞. Then a 1_E-b 1_F _p^p = · |a-b|^pμ(E∩ F) +|a|^pμ(E∖ F) · +|b|^pμ(F∖ E). For equal amplitudes, a 1_E-a 1_F _p = |a|μ(Emathbintriangle F)^1/p.
Exact mathematical passage
\begin{aligned} \|a\one_E-b\one_F\|_p^p ={}& |a-b|^p\mu(E\cap F) +|a|^p\mu(E\setminus F)\\ &+|b|^p\mu(F\setminus E). \end{aligned}
\one_{E_i}Use in source · 1 passage
Search aliases: one_E_i 1_E_i one mathbf
Use in source · Paper_V_Traditional_Final.tex, line 2786
[Overlap matrix formula] Define M_ij=μ(E_i∩ F_j). For 1≤ p<∞, ∑_i=0^Ia_i 1_E_i - ∑_j=0^Jb_j 1_F_j _p^p = ∑_i=0^I∑_j=0^J|a_i-b_j|^pM_ij. The row sums of M are μ(E_i), and the column sums are μ(F_j).
Exact mathematical passage
\left\| \sum_{i=0}^{I}a_i\one_{E_i} - \sum_{j=0}^{J}b_j\one_{F_j} \right\|_p^p = \sum_{i=0}^{I}\sum_{j=0}^{J}|a_i-b_j|^pM_{ij}.
\one_FUse in source · 4 passages
Search aliases: one_F 1_F one mathbf
Use in source · Paper_V_Traditional_Final.tex, line 2702
[Exact scalar region formula] Let E,F be measurable subsets of a measure space, let a,b∈ C, and let 1≤ p<∞. Then a 1_E-b 1_F _p^p = · |a-b|^pμ(E∩ F) +|a|^pμ(E∖ F) · +|b|^pμ(F∖ E). For equal amplitudes, a 1_E-a 1_F _p = |a|μ(Emathbintriangle F)^1/p.
Exact mathematical passage
\begin{aligned} \|a\one_E-b\one_F\|_p^p ={}& |a-b|^p\mu(E\cap F) +|a|^p\mu(E\setminus F)\\ &+|b|^p\mu(F\setminus E). \end{aligned}
\one_{F_j}Use in source · 1 passage
Search aliases: one_F_j 1_F_j one mathbf
Use in source · Paper_V_Traditional_Final.tex, line 2786
[Overlap matrix formula] Define M_ij=μ(E_i∩ F_j). For 1≤ p<∞, ∑_i=0^Ia_i 1_E_i - ∑_j=0^Jb_j 1_F_j _p^p = ∑_i=0^I∑_j=0^J|a_i-b_j|^pM_ij. The row sums of M are μ(E_i), and the column sums are μ(F_j).
Exact mathematical passage
\left\| \sum_{i=0}^{I}a_i\one_{E_i} - \sum_{j=0}^{J}b_j\one_{F_j} \right\|_p^p = \sum_{i=0}^{I}\sum_{j=0}^{J}|a_i-b_j|^pM_{ij}.
\one_{i\in B_a}Use in source · 1 passage
Search aliases: one_i∈ B_a 1_i∈ B_a one in bm in
Use in source · Paper_VI_Traditional_Final.tex, line 5349
The product topology used here differs from the bounded inverse-limit norm of Part II. No uniform boundedness of coordinate generators is required by the coordinatewise statement. FloatBarrier Reversible coarse dynamics. Let Q be a finite Markov generator acting on column probability vectors, let (B_a) be a partition into nonempty blocks, and let C_ai= 1_i∈ B_a. Strong lumpability means CQ=overline Q C for a coarse Markov generator. For observables, the backward generator is Q^T. Weighted L^2 in…
Exact mathematical passage
C_{ai}=\one_{i\in B_a}
\one_{\{i\in B_\alpha\}}Use in source · 1 passage
Search aliases: one_{i∈ B_α} 1_{i∈ B_α} one in alpha bm in alpha
Use in source · Paper_VI_Traditional_Final.tex, line 2848
Let {B_1,…,B_m} be a partition of the fine state set. Its aggregation matrix is C_α i= 1_{i∈ B_α}.
Exact mathematical passage
C_{\alpha i}=\one_{\{i\in B_\alpha\}}.
\onesUse in source · 8 passages
Search aliases: ones
Use in source · atlas-pages.tex, line 28
midrule p · A nonnegative column with ones^ Tp=1. · It is not a vector of function values.
Exact mathematical passage
\ones^{\mathsf T}p=1
\ones^{\mathsf T}Use in source · 4 passages
Search aliases: ones^ T ones mathsf
Use in source · atlas-pages.tex, line 28
midrule p · A nonnegative column with ones^ Tp=1. · It is not a vector of function values.
Exact mathematical passage
\ones^{\mathsf T}p=1
\ones^TUse in source · 1 passage
Search aliases: ones^T ones
Use in source · pages.tex, line 322
vpage19Twelve small blocks, one coherent errorDISCOVERY AND PERTURBATION CERTIFICATESChapter VIII: Theorem 8.1, Proposition 9.2. Split QQ^4 into scalar packets and let A=I+E, E=frac1100(onesones^T-I). Dropping every off-diagonal block leaves I. Each discarded entry is only 1/100, but the total error is not bounded by that single-entry size. vfigblock_error headTwo norms answer different questions There are twelve omitted entries, so E _F^2=12/10^4. The all-one vector is an eigenvector with eigen…
Exact mathematical passage
A=I+E,\qquad E=\frac1{100}(\ones\ones^T-I).
\OO^2Use in source · 1 passage
Search aliases: OO^2 O^2 OO mathcal
Use in source · 04_arithmetic.tex, line 103
[A resultant bound that is not exact] Let A=diag(0,π^s) on the standard lattice Λ= O^2. The factors x and x-π^s have resultant -π^s, up to the harmless choice of resultant order, but the two spectral projectors are the integral coordinate projectors. Their exact separation weight is zero. The resultant sees the closeness of the two eigenvalues. It does not see that the chosen lattice already splits along their eigenspaces.
Exact mathematical passage
\Lambda=\OO^2
\OO^dUse in source · 2 passages
Search aliases: OO^d O^d OO mathcal
Use in source · Paper_XII_Traditional_Final.tex, line 529
[Coinvariant recovery] Let M be a free Λ module of rank d. Then M/ a M ≅ M⊗_Λ,ϵ O ≅ O^d. The statement is valid for every coefficient prime, including two.
Exact mathematical passage
\boxed{ M/\fraka M \cong M\otimes_{\Lambda,\epsilon}\OO \cong \OO^d. }
\OO_FUse in source · 1 passage
Search aliases: OO_F O_F OO mathcal
Use in source · Paper_XI_Traditional_Final.tex, line 2078
[Restricted dyadic marked assembly realization] Let F= F_2((t)), fix an additive character of conductor t O_F, fix d≥1, and let h_1<⋯<h_d be positive odd integers. Put q_i=3h_i, N=2^d, and n_j=2^d-jq_d-j+1. The explicit recursive matrix generates a separable totally ramified degree-N maximal parameter field, normalizes the complete lattice chain, and has the literal defining sequence of degrees 2,4,…,2^d. It admits a full simple character trivial on U_F^1 and an actual supercuspidal completion w…
Exact mathematical passage
t\OO_F
\OO_K^\timesUse in source · 1 passage
Search aliases: OO_K^× O_K^× OO times mathcal times
Use in source · dyadic_character.tex, line 358
Here O_K^×=U_K^1 because the residue field is F_2. Since U_K 1 is contained in J 1, we have
Exact mathematical passage
\OO_K^\times=U_K^1
\OO_\lambdaUse in source · 9 passages
Search aliases: OO_λ O_λ OO lambda mathcal lambda
Use in source · Paper_XII_Traditional_Final.tex, line 1466
Fix a topological group Γ, a finite matrix index set D, and a set Λ of coefficient indices. For each λ, let O_λ be a commutative ring, complete and separated for a finitely generated ideal I_λ, with finite residue ring O_λ/I_λ. Give O_λ its ideal adic topology and every quotient its discrete topology. Let i_λ: O_λ⟶ K_λ, e_λ:L⟶ K_λ be supplied ring homomorphisms, where L and K_λ are fields, K_λ has a declared topology, and i_λ is continuous.
Exact mathematical passage
\OO_\lambda
\mathrm{op}Use in source · 4 passages
Search aliases: op mathrm
Use in source · Paper_V_Traditional_Final.tex, line 2059
[Quantitative imperfect reconstruction] Define the delay compensated defect D( e^ iω) = e^ i dωR( e^ iω)E( e^ iω)-I_N. If ε_rec = *ess sup_ω D( e^ iω) _op<∞, then the delay compensated reconstruction widetilde x satisfies widetilde x-x _2 ≤ ε_rec x _2. The constant is sharp as an operator norm.
Exact mathematical passage
\varepsilon_{\mathrm{rec}} = \esssup_{\omega}\|D(\e^{\ii\omega})\|_{\mathrm{op}}<\infty,
\operatorname{orbit}Use in source · 1 passage
Search aliases: orbit operatorname
Use in source · supplement_06.tex, line 31
pageheading1. Actual matrix orbits and arithmetic jumpsTheorem 9.1; manuscript page 22 Let R be a commutative ring, I a finite coordinate type and d=|I|. For C∈ M_I(R) and v_0∈ R^I, define orbit(C,v_0,n)=C^nv_0, u_k=packet(C,v_0,N,j,k)=C^Nk+jv_0. The implementation uses actual matrix powers and matrix--vector multiplication. A separately supplied sequence v with v_n+1=Cv_n is proved by induction to equal this power orbit. Thus the source-sequence formulation is connected to the construction.
Exact mathematical passage
\operatorname{orbit}(C,v_0,n)=C^nv_0, \qquad u_k=\operatorname{packet}(C,v_0,N,j,k)=C^{Nk+j}v_0.
\operatorname{ord}Use in source · 4 passages
Search aliases: ord operatorname
Use in source · 10_foundations.tex, line 295
For x∈ Z_p, extend the valuation to zero by setting ord_p(x)= v_p(x), · x≠0, ∞, · x=0. Thus L≤ord_p(x) is equivalent to x∈ p^L Z_p.
Exact mathematical passage
\operatorname{ord}_p(x)= \begin{cases}v_p(x),&x\ne0,\\\infty,&x=0.\end{cases}
\ordUse in source · 10 passages
Search aliases: ord operatorname
Use in source · Paper_IV_Traditional_Final.tex, line 1754
[Order from Taylor coefficients] If f(w)=∑_n≥0a_nw^n is an entire function with infinitely many nonzero coefficients, then ord f = limsup_substackn→∞ a_n≠0 fracnlog nlog(1/|a_n|). Canonical products for entire functions of order below one have genus zero. We use these standard facts in the form given in [source reference].
Exact mathematical passage
\ord f = \limsup_{\substack{n\to\infty\\a_n\ne0}} \frac{n\log n}{\log(1/|a_n|)}.
\ord_3Use in source · 1 passage
Search aliases: ord_3 ord operatorname
Use in source · supplement_04.tex, line 177
ubsection*A sharp binomial-weight improvement Let F(n)=81binom n1+3binom n2+3binom n3+81binom n4. At n=3 and s=1, the four coefficient valuations are (4,1,1,4), while the binomial valuations are (0,1,1,0). Thus the refined bound is B_3(w;3,1)=min(4,2,2,4)=2, stronger than the coefficient-only lower bound one. The source checks both the minimum and the actual result: Δ F(3)=180, ord_3(180)=2. The valuation equality is proved in the actual completed ring Z_3, with an independent exact integer divi…
Exact mathematical passage
\boxed{\Delta F(3)=180,\qquad\ord_3(180)=2.}
\operatorname{ord}_pUse in source · 4 passages
Search aliases: ord_p operatorname
Use in source · 10_foundations.tex, line 295
For x∈ Z_p, extend the valuation to zero by setting ord_p(x)= v_p(x), · x≠0, ∞, · x=0. Thus L≤ord_p(x) is equivalent to x∈ p^L Z_p.
Exact mathematical passage
\operatorname{ord}_p(x)= \begin{cases}v_p(x),&x\ne0,\\\infty,&x=0.\end{cases}
\ord_pUse in source · 7 passages
Search aliases: ord_p ord operatorname
Use in source · supplement_04.tex, line 24
The additive valuation is extended to ord_p(0)=∞. This avoids incorrectly applying Mathlib's finite-valued default at zero to an inequality that must also cover cancellation and identically zero differences.
Exact mathematical passage
\ord_p(0)=\infty
\mathrm{order}Use in source · 2 passages
Search aliases: order mathrm
Use in source · odd_capacity.tex, line 190
The right side is the Heisenberg commutator pairing for the minimal simple stratum of gamma. Its quotient J^1(γ)/H^1(γ) is precisely U^m(A)/H^m(γ), because J^1(γ)=U^1(B_order)U^m(A), with U^1(B_order)⊂ H^1(γ). BK's nondegeneracy theorem for simple-character commutators therefore says that the form on S is nondegenerate.
Exact mathematical passage
J^1(\gamma)=U^1(B_{\mathrm{order}})U^m(A)
\oscUse in source · 2 passages
Search aliases: osc operatorname
Use in source · Paper_VI_Traditional_Final.tex, line 3589
] For every packet with positive mass and evidence, q_α,i-r_α,i = fracr_α,iη_i(p) overlineλ_α(p). Consequently, normq_α-r_α_1 = frac ∑_i∈ I_αr_α,iabsη_i(p) overlineλ_α(p). If λ is positive on the packet, then normq_α-r_α_1 ≤ fracosc_I_α(λ) min_i∈ I_αλ_i.
Exact mathematical passage
\norm{q_\alpha-r_\alpha}_1 \leq \frac{\osc_{I_\alpha}(\lambda)} {\min_{i\in I_\alpha}\lambda_i}.
\osc_{I_\alpha}Use in source · 2 passages
Search aliases: osc_I_α osc alpha operatorname alpha
Use in source · Paper_VI_Traditional_Final.tex, line 3589
] For every packet with positive mass and evidence, q_α,i-r_α,i = fracr_α,iη_i(p) overlineλ_α(p). Consequently, normq_α-r_α_1 = frac ∑_i∈ I_αr_α,iabsη_i(p) overlineλ_α(p). If λ is positive on the packet, then normq_α-r_α_1 ≤ fracosc_I_α(λ) min_i∈ I_αλ_i.
Exact mathematical passage
\norm{q_\alpha-r_\alpha}_1 \leq \frac{\osc_{I_\alpha}(\lambda)} {\min_{i\in I_\alpha}\lambda_i}.
\overline aUse in source · 1 passage
Search aliases: overline a overline
Use in source · Paper_X_Block27_Amended.tex, line 1188
The arithmetic witnesses use the integral closure A of Z in overline Q. For each rational prime p, Lean constructs a maximal ideal Q of A above (p) and an actual element γ_p∈ G_ Q such that γ_p∈Stab_G_ Q( Q), overlineγ_p(a)=overline a^ p (a∈ A). The existence argument applies Mathlib's proved profinite stabilizer-surjectivity theorem to this actual invariant ring. The residue p-power map is proved bijective by taking p-th roots in overline Q and showing that they remain integral when their p-th …
Exact mathematical passage
\gamma_p\in\operatorname{Stab}_{G_{\Q}}(\mathfrak Q),\qquad \overline{\gamma_p(a)}=\overline a^{\,p} \quad(a\in\mathcal A).
\overline a^{\,p}Use in source · 1 passage
Search aliases: overline a^ p overline
Use in source · Paper_X_Block27_Amended.tex, line 1188
The arithmetic witnesses use the integral closure A of Z in overline Q. For each rational prime p, Lean constructs a maximal ideal Q of A above (p) and an actual element γ_p∈ G_ Q such that γ_p∈Stab_G_ Q( Q), overlineγ_p(a)=overline a^ p (a∈ A). The existence argument applies Mathlib's proved profinite stabilizer-surjectivity theorem to this actual invariant ring. The residue p-power map is proved bijective by taking p-th roots in overline Q and showing that they remain integral when their p-th …
Exact mathematical passage
\gamma_p\in\operatorname{Stab}_{G_{\Q}}(\mathfrak Q),\qquad \overline{\gamma_p(a)}=\overline a^{\,p} \quad(a\in\mathcal A).
\overline BUse in source · 5 passages
Search aliases: overline B overline
Use in source · 03_nonlinear_coordinates.tex, line 455
[Polynomial tree expansion] Suppose there is R>0 such that ϕ(R)≤ R and q= ∑_r=2^dr b_rR^r-1<1. Then the map Φ(x)=s+∑_r=2^dB_r(x,…,x) has a unique fixed point in the closed ball overline B_R(0). The sum of the values of all finite ordered rooted trees converges absolutely to that fixed point.
Exact mathematical passage
\overline B_R(0)
\overline bUse in source · 2 passages
Search aliases: overline b overline
Use in source · 03_nonlinear_coordinates.tex, line 961
U' · xrightarrowD' · V' is a commuting square. Then there is a unique induced map overline b:coker D⟶coker D' satisfying overline b Ob_D(r)=Ob_D'(br).
Exact mathematical passage
\overline b:\coker D\longrightarrow\coker D'
\overline B_RUse in source · 2 passages
Search aliases: overline B_R overline
Use in source · 03_nonlinear_coordinates.tex, line 455
[Polynomial tree expansion] Suppose there is R>0 such that ϕ(R)≤ R and q= ∑_r=2^dr b_rR^r-1<1. Then the map Φ(x)=s+∑_r=2^dB_r(x,…,x) has a unique fixed point in the closed ball overline B_R(0). The sum of the values of all finite ordered rooted trees converges absolutely to that fixed point.
Exact mathematical passage
\overline B_R(0)
\overline B_\rhoUse in source · 3 passages
Search aliases: overline B_ρ overline rho
Use in source · 04_correction_transport.tex, line 44
[Right inverse correction theorem] Let D:X→ Y be a bounded linear map between Banach spaces. Assume Z=im D is closed and there is a bounded linear right inverse R:Z⟶ X, DR=I_Z. Put X_R=R(Z). Let N:overline B_ρ(0)∩ X_R⟶ Z be L-Lipschitz. Let y∈ Z and put q= R L. Assume q<1 and R(y-N(0)) ≤ (1-q)ρ. Then the equation Dx+N(x)=y has a unique solution in overline B_ρ(0)∩ X_R.
Exact mathematical passage
N:\overline B_\rho(0)\cap X_R\longrightarrow Z
\overline CUse in source · 4 passages
Search aliases: overline C overline
Use in source · 20_arithmetic.tex, line 135
[Lucas reduction and address independence] Let p be prime, m>0, and N=mp. For 0≤ j<p and n,q∈ N, binommpn+jmpq≡binommnmqpmod p. Every C_mp,j(n) belongs to Z_p. Its reduction is the unique sequence overline C_m: N→ F_p satisfying ∑_q=0^nbinommnmqoverline C_m(q)=δ_n0 in F_p. Consequently, for any two addresses 0≤ j,k<p, C_mp,j(n)≡ C_mp,k(n)pmod p (n∈ N).
Exact mathematical passage
\overline C_m:\N\to\F_p
\overline C_mUse in source · 4 passages
Search aliases: overline C_m overline
Use in source · 20_arithmetic.tex, line 135
[Lucas reduction and address independence] Let p be prime, m>0, and N=mp. For 0≤ j<p and n,q∈ N, binommpn+jmpq≡binommnmqpmod p. Every C_mp,j(n) belongs to Z_p. Its reduction is the unique sequence overline C_m: N→ F_p satisfying ∑_q=0^nbinommnmqoverline C_m(q)=δ_n0 in F_p. Consequently, for any two addresses 0≤ j,k<p, C_mp,j(n)≡ C_mp,k(n)pmod p (n∈ N).
Exact mathematical passage
\overline C_m:\N\to\F_p
\overline DUse in source · 1 passage
Search aliases: overline D overline
Use in source · atlas-pages.tex, line 614
Two distinct equalities justify the coarse computation. First, CK=K_cC transports prediction. Second, the likelihood is constant on each packet, so CD_r=overline D_rC transports observation weighting. Their composition preserves both the posterior packet totals and the evidence factors.
Exact mathematical passage
CD_r=\overline D_rC
\overline D_rUse in source · 1 passage
Search aliases: overline D_r overline
Use in source · atlas-pages.tex, line 614
Two distinct equalities justify the coarse computation. First, CK=K_cC transports prediction. Second, the likelihood is constant on each packet, so CD_r=overline D_rC transports observation weighting. Their composition preserves both the posterior packet totals and the evidence factors.
Exact mathematical passage
CD_r=\overline D_rC
\overline GUse in source · 2 passages
Search aliases: overline G overline
Use in source · root_scope.tex, line 140
ad J_r={q:λ_r(q)=μ_r}, G_r=2^-μ_rH_r. Since λ_r(0)=0 and λ_r(q)≥3q+1-s_2(r), every minimizing address satisfies 0≤ q≤lfloorfracs_2(r)-13rfloor. In particular μ_r≤0, and the minimum is found by a finite carry calculation. Reduction of the whole series gives overline G_r(X)=∑_q∈ J_rX^underline q in F_2[X]. Its highest degree is max J_r, with leading coefficient one. Thus the normalization is the actual Gauss normalization.
Exact mathematical passage
\overline G_r(X)=\sum_{q\in J_r}X^{\underline q} \quad\text{in }\mathbb F_2[X]. \label{eq:root:reduction}
\overline G_rUse in source · 2 passages
Search aliases: overline G_r overline
Use in source · root_scope.tex, line 140
ad J_r={q:λ_r(q)=μ_r}, G_r=2^-μ_rH_r. Since λ_r(0)=0 and λ_r(q)≥3q+1-s_2(r), every minimizing address satisfies 0≤ q≤lfloorfracs_2(r)-13rfloor. In particular μ_r≤0, and the minimum is found by a finite carry calculation. Reduction of the whole series gives overline G_r(X)=∑_q∈ J_rX^underline q in F_2[X]. Its highest degree is max J_r, with leading coefficient one. Thus the normalization is the actual Gauss normalization.
Exact mathematical passage
\overline G_r(X)=\sum_{q\in J_r}X^{\underline q} \quad\text{in }\mathbb F_2[X]. \label{eq:root:reduction}
\overline IUse in source · 4 passages
Search aliases: overline I overline
Use in source · 03_packets_and_graphs.tex, line 855
[Preorder support algebra decomposition] Let R be a preorder on the finite set I. Define α∼β Longleftrightarrow (β,α)∈ R and (α,β)∈ R. Let overline I=I/∼, which carries the induced partial order, and put X_C:=⊕_α∈ CX_α. Let N_R⊆ M_R consist of operators whose blocks vanish whenever source and target belong to the same equivalence class. Then: [label=(roman*)] item N_R is a nilpotent ideal on both sides. item If h is the maximal number of strict steps in a chain of overline I, then N_R^ h+1=0. it…
Exact mathematical passage
\overline I=I/{\sim}
\overline KUse in source · 27 passages
Search aliases: overline K overline
Use in source · Paper_VI_Traditional_Final.tex, line 2864
[Descent through an aggregation map] Let C:X_f→ X_c be a surjective linear map between finite dimensional spaces. A unique operator overline K:X_c→ X_c satisfying CK=overline KC exists if and only if Ob_C(K)=0. For any right inverse R of C, the only possible coarse operator is overline K=CKR, and CK-overline KC = CK(I-RC).
Exact mathematical passage
\overline K:X_c\to X_c
\overline K^nUse in source · 2 passages
Search aliases: overline K^n overline
Use in source · Paper_VI_Traditional_Final.tex, line 2961
[Finite horizon lumping defect] For arbitrary fine and coarse operators K and overline K, with the fixed aggregation map C, set Δ_K=CK-overline KC. Then CK^n-overline K^nC = ∑_j=0^n-1 overline K^n-1-jΔ_KK^j. If K and overline K are column stochastic, then normCK^n-overline K^nC_1→1 ≤ nnormΔ_K_1→1. For every probability vector p, the total variation discrepancy is at most frac n2normΔ_K_1→1.
Exact mathematical passage
CK^n-\overline K^nC = \sum_{j=0}^{n-1} \overline K^{n-1-j}\Delta_KK^j.
\overline K^{n-1-j}Use in source · 2 passages
Search aliases: overline K^n-1-j overline
Use in source · Paper_VI_Traditional_Final.tex, line 2961
[Finite horizon lumping defect] For arbitrary fine and coarse operators K and overline K, with the fixed aggregation map C, set Δ_K=CK-overline KC. Then CK^n-overline K^nC = ∑_j=0^n-1 overline K^n-1-jΔ_KK^j. If K and overline K are column stochastic, then normCK^n-overline K^nC_1→1 ≤ nnormΔ_K_1→1. For every probability vector p, the total variation discrepancy is at most frac n2normΔ_K_1→1.
Exact mathematical passage
CK^n-\overline K^nC = \sum_{j=0}^{n-1} \overline K^{n-1-j}\Delta_KK^j.
\overline K^{n-j}Use in source · 1 passage
Search aliases: overline K^n-j overline
Use in source · Paper_VI_Traditional_Final.tex, line 2981
Sum the telescoping identities overline K^n-1-jCK^j+1 - overline K^n-jCK^j = overline K^n-1-jΔ_KK^j. Stochastic matrices are contractions in the ℓ^1 norm. The last statement uses total variation equal to one half of the ℓ^1 distance.
Exact mathematical passage
\overline K^{n-1-j}CK^{j+1} - \overline K^{n-j}CK^j = \overline K^{n-1-j}\Delta_KK^j.
\overline K_{\alpha\beta}Use in source · 1 passage
Search aliases: overline K_αβ overline alpha beta
Use in source · Paper_VI_Traditional_Final.tex, line 2923
Let K be column stochastic. A coarse column stochastic matrix overline K satisfying CK=overline KC exists if and only if, for every source block B_β, every pair i,j∈ B_β, and every target block B_α, ∑_r∈ B_αK_ri = ∑_r∈ B_αK_rj. When it exists, overline K is unique and overline K_αβ = ∑_r∈ B_αK_ri for any i∈ B_β.
Exact mathematical passage
\overline K_{\alpha\beta} = \sum_{r\in B_\alpha}K_{ri}
\overline L_aUse in source · 6 passages
Search aliases: overline L_a overline
Use in source · Paper_VI_Traditional_Final.tex, line 5405
tly positive finite likelihood and let B_L(p)=Lodot p/ipLp for a probability vector p. Fix probability vectors r^(a) supported on B_a, and restrict the priors to p_i=q_a r_i^(a) for i∈ B_a, with q a coarse probability vector. [Sufficiency relative to a prior disintegration] Define overline L_a = ∑_i∈ B_aL_ir_i^(a). For every prior in the declared family, C B_L(p) = B_overline L(q).
Exact mathematical passage
\overline L_a = \sum_{i\in B_a}L_ir_i^{(a)}.
\overline L_yUse in source · 6 passages
Search aliases: overline L_y overline
Use in source · Paper_VI_Traditional_Final.tex, line 5535
For stochastic matrices K,overline K and positive likelihoods L_y,overline L_y, put Φ_y= B_L_y∘ K and overlineΦ_y= B_overline L_y∘overline K. [Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯∘Φ_y_1 = overlineΦ_y_r∘⋯∘overlineΦ_y_1C.
Exact mathematical passage
L_y,\overline L_y
\overline mUse in source · 2 passages
Search aliases: overline m overline
Use in source · Paper_VI_Traditional_Final.tex, line 5559
[One step filter defect] Let r=Kp, q=Cr, and let ℓ_y(r) be the exact coarse likelihood obtained by averaging L_y inside each block with respect to r. Assume 0<overline m≤(overline L_y)_a≤overline M. Then normCΦ_y(p)-overlineΦ_y(Cp)_1 ≤ · frac2d(rmid L_y,overline L_y) max{ipL_yr,ipoverline L_yq}
Exact mathematical passage
0<\overline m\leq(\overline L_y)_a\leq\overline M.
\overline nUse in source · 8 passages
Search aliases: overline n overline
Use in source · residue_continuation_body.tex, line 68
The residue norm and its sign. For m∈ N, reducing the multiplication matrices of O modulo m and taking their determinants constructs a monoid homomorphism overline n: O/m O⟶ Z/m Z, overline n([x])=n(x)bmod m. The construction is proved independent of the integral basis. For m≠0, its source is finite of cardinality m^d. Consequently x-y∈ m O implies n(x)≡ n(y)pmod m. No surjectivity of overline n is asserted.
Exact mathematical passage
\overline n:\OO/m\OO\longrightarrow\ZZ/m\ZZ, \qquad \overline n([x])=n(x)\bmod m.
\overline n_JUse in source · 2 passages
Search aliases: overline n_J overline
Use in source · residue_continuation_body.tex, line 202
An actual unit action on ideal residues. Fix m>0 and a nonzero integral ideal J. Multiplication by u∈( O/m O)^× gives a proved permutation of J/mJ. Its inverse follows from (ab-1)x∈ mJ when ab-1∈ m O and x∈ J. No surjectivity of J/mJ→ O/m O or coprimality condition on J is used. The well-defined residue norm on this quotient satisfies overline n_J(u· a)=overline n(u) overline n_J(a). For a multiplicative χ: Z/m Z→ C preserving 1, an exhibited unit with u∈( O/m O)^×, χ(overline n(u))≠1 therefore …
Exact mathematical passage
\overline n_J(u\cdot a)=\overline n(u)\,\overline n_J(a).
\overline n_{\varepsilon}Use in source · 3 passages
Search aliases: overline n_ε overline varepsilon
Use in source · residue_continuation_body.tex, line 89
For ε∈{1,-1} define overline n_ε(a)=εoverline n(a). On the corresponding sign portion, N(x)bmod m=overline n_ε([x]). The sign cannot be dropped: -1≡2pmod3, whereas N(-1)=1 and N(2)=2 for principal ideals of Z.
Exact mathematical passage
\overline n_{\varepsilon}(a)=\varepsilon\overline n(a)
\overline PUse in source · 11 passages
Search aliases: overline P overline
Use in source · 04_transport.tex, line 192
[Packet exact sequence for an invariant subspace] Let K≤ X satisfy P_α K⊆ K for every α. Define K_α:=P_α K=K∩ X_α and overline P_α(x+K):=P_α x+K. Then: [label=(roman*)] item The nonzero restrictions P_α|_K form a packet system on K. item The nonzero overline P_α form a packet system on X/K. item For every α there is a short exact sequence 0⟶ K_α ⟶ X_α ⟶ overline P_α(X/K) ⟶0, and overline P_α(X/K) ≅ X_α/(X_α∩ K) ≅ (X_α+K)/K. item The full exact sequence decomposes packetwise as the direct sum of …
Exact mathematical passage
\overline P_\alpha(x+K):=P_\alpha x+K.
\overline pUse in source · 1 passage
Search aliases: overline p overline
Use in source · 03_supplement_05.tex, line 98
Rank-one rigidity of the corrected channel. The module tests the map equality on actual matrix units and flattens each A_ra into a row. If X is the matrix of those rows and p is the vector of entries of P, the resulting Gram identity is X^*X=overline p p^ T. The checked rank-one Gram lemma proves that every row of X is proportional to p. For nonzero p, a nonzero coordinate is selected and a test vector supported on two coordinates is placed in the null space of the Gram matrix. The implication X…
Exact mathematical passage
X^*X=\overline p\,p^{\mathsf T}.
\overline P_\alphaUse in source · 9 passages
Search aliases: overline P_α overline alpha
Use in source · 04_transport.tex, line 192
[Packet exact sequence for an invariant subspace] Let K≤ X satisfy P_α K⊆ K for every α. Define K_α:=P_α K=K∩ X_α and overline P_α(x+K):=P_α x+K. Then: [label=(roman*)] item The nonzero restrictions P_α|_K form a packet system on K. item The nonzero overline P_α form a packet system on X/K. item For every α there is a short exact sequence 0⟶ K_α ⟶ X_α ⟶ overline P_α(X/K) ⟶0, and overline P_α(X/K) ≅ X_α/(X_α∩ K) ≅ (X_α+K)/K. item The full exact sequence decomposes packetwise as the direct sum of …
Exact mathematical passage
\overline P_\alpha(x+K):=P_\alpha x+K.
\overline P_\betaUse in source · 1 passage
Search aliases: overline P_β overline beta
Use in source · 04_transport.tex, line 275
If x∈ K_α, then Tx∈ K, so T_βαx=P_β Tx∈ K∩ X_β=K_β. On the quotient, overline T_βαπ = overline P_βoverline Toverline P_απ = π P_β TP_α.
Exact mathematical passage
\overline T_{\beta\alpha}\pi = \overline P_\beta\overline T\overline P_\alpha\pi = \pi P_\beta TP_\alpha.
\overline QUse in source · 20 passages
Search aliases: overline Q overline
Use in source · 03_nonlinear_coordinates.tex, line 1015
[Projectors on the obstruction quotient] Let (Q_α)_α∈ I be a complete family of pairwise annihilating idempotents on V. Assume Q_α(im D)⊆im D for every α. Then each Q_α induces an idempotent overline Q_α on coker D, and coker D = ⊕_α∈ Ioverline Q_α(coker D). For every defect r∈ V, overline Q_αOb_D(r) = [Q_α r]. No packet decomposition of the correction space U is required.
Exact mathematical passage
\overline Q_\alpha
\overline Q^TUse in source · 4 passages
Search aliases: overline Q^T overline
Use in source · Paper_VI_Traditional_Final.tex, line 5388
[Coarse Dirichlet form] Under the hypotheses of [source reference], let Jg be the fine function that is constant with value g_a on B_a. Then -ipJgQ^TJg_L^2(π) = -ipgoverline Q^Tg_L^2(overlineπ).
Exact mathematical passage
-\ip{Jg}{Q^TJg}_{L^2(\pi)} = -\ip{g}{\overline Q^Tg}_{L^2(\overline\pi)}.
\overline Q_{ab}Use in source · 2 passages
Search aliases: overline Q_ab overline
Use in source · Paper_VI_Traditional_Final.tex, line 5360
the stated stationary probability. [Reversibility descends] Assume Q is strongly lumpable and reversible with respect to a positive stationary distribution π, so Q_ijπ_j=Q_jiπ_i. Define overlineπ_a=∑_i∈ B_aπ_i. Then overlineπ is stationary for overline Q, and overline Q_aboverlineπ_b = overline Q_baoverlineπ_a. Thus the coarse chain is reversible.
Exact mathematical passage
\overline Q_{ab}\overline\pi_b = \overline Q_{ba}\overline\pi_a.
\overline Q_{ba}Use in source · 2 passages
Search aliases: overline Q_ba overline
Use in source · Paper_VI_Traditional_Final.tex, line 5360
the stated stationary probability. [Reversibility descends] Assume Q is strongly lumpable and reversible with respect to a positive stationary distribution π, so Q_ijπ_j=Q_jiπ_i. Define overlineπ_a=∑_i∈ B_aπ_i. Then overlineπ is stationary for overline Q, and overline Q_aboverlineπ_b = overline Q_baoverlineπ_a. Thus the coarse chain is reversible.
Exact mathematical passage
\overline Q_{ab}\overline\pi_b = \overline Q_{ba}\overline\pi_a.
\overline Q_\alphaUse in source · 3 passages
Search aliases: overline Q_α overline alpha
Use in source · 03_nonlinear_coordinates.tex, line 1015
[Projectors on the obstruction quotient] Let (Q_α)_α∈ I be a complete family of pairwise annihilating idempotents on V. Assume Q_α(im D)⊆im D for every α. Then each Q_α induces an idempotent overline Q_α on coker D, and coker D = ⊕_α∈ Ioverline Q_α(coker D). For every defect r∈ V, overline Q_αOb_D(r) = [Q_α r]. No packet decomposition of the correction space U is required.
Exact mathematical passage
\overline Q_\alpha
\overline rUse in source · 2 passages
Search aliases: overline r overline
Use in source · 30_lifting.tex, line 657
The solved Jacobian remains surjective, so every shifted obstruction [overline r+K_z_nv] vanishes. Formula~[source reference] therefore gives p^m-e choices for every parameter digit at every stage. When m=e, the fixed-coordinate chart in the proof of Theorem~[source reference] contains every root in the initial ball with the specified parameters. Its uniqueness proves the endpoint assertion. This is uniqueness of the actual endpoint, not of the integer representatives used to approach it.
Exact mathematical passage
[\overline r+K_{z_n}v]
\overline sUse in source · 1 passage
Search aliases: overline s overline
Use in source · 30_lifting.tex, line 134
[Independence of the integer representative] Under the hypotheses of Theorem~[source reference], let x'=x+q^Ls and write F(x')=q^Lr_x'. Then J_x'=J_x, overliner_x'=overliner_x+J_xoverline s. Consequently the two residual classes coincide in the common quotient S_q^e/im J_x.
Exact mathematical passage
J_{x'}=J_x,\qquad \overline{r_{x'}}=\overline{r_x}+J_x\overline s.
\overline TUse in source · 3 passages
Search aliases: overline T overline
Use in source · 04_transport.tex, line 256
[Restriction and quotient of transfer blocks] Under the hypotheses of [source reference], let T∈End(X) preserve K. Then every block T_βα maps K_α into K_β, and the induced quotient operator overline T satisfies overline T_βα π = π T_βα, where π:X↠ X/K. [source reference]
Exact mathematical passage
\overline T
\overline tUse in source · 2 passages
Search aliases: overline t overline
Use in source · 30_lifting.tex, line 92
Every point in the lower residue class has the form y=x+q^Lt with t∈ Z^d. By Lemma~[source reference] and 2L≥ L+1, F(x+q^Lt)≡ q^L(r_x+DF(x)t)pmodq^L+1. The integer q^L is nonzero. Cancelling it in the corresponding divisibility condition proves F(x+q^Lt)≡0pmodq^L+1 Longleftrightarrow overliner_x+J_xoverline t=0. Integer vectors reduce onto S_q^d. The equation on the right is solvable exactly when overliner_x belongs to im J_x, since that image is closed under negation. This is precisely o_x=0. F…
Exact mathematical passage
\label{eq:digit-correction} F(x+q^Lt)\equiv0\pmod{q^{L+1}} \quad\Longleftrightarrow\quad \overline{r_x}+J_x\overline t=0.
\overline T_{\beta\alpha}Use in source · 2 passages
Search aliases: overline T_βα overline beta alpha
Use in source · 04_transport.tex, line 257
[Restriction and quotient of transfer blocks] Under the hypotheses of [source reference], let T∈End(X) preserve K. Then every block T_βα maps K_α into K_β, and the induced quotient operator overline T satisfies overline T_βα π = π T_βα, where π:X↠ X/K. [source reference]
Exact mathematical passage
\boxed{ \overline T_{\beta\alpha}\,\pi = \pi T_{\beta\alpha}, }
\overline UUse in source · 4 passages
Search aliases: overline U overline
Use in source · 02_strengthened.tex, line 37
orm a complete orthogonal packet system on Hilbert--Schmidt operator space. Dephasing is the diagonal projection. Every channel has input and output superblocks, with the same composition law as ordinary matrix blocks. A covariant channel preserves the isotypic packets of the conjugation action. In the cyclic case these are coherence orders. The Choi matrix carries the equivalent symmetry under U⊗overline U.
Exact mathematical passage
U\otimes\overline U
\overline VUse in source · 4 passages
Search aliases: overline V overline
Use in source · Paper_VI_Traditional_Final.tex, line 5645
[Coarse Feynman--Kac transfer] Assume CQ=overline QC, and let J=C^T lift a coarse function to the corresponding block constant fine function. Suppose the potential is block constant, so VJ=Joverline V. Then e^t(Q^T-V)J = J e^t(overline Q^T-overline V) for every t≥0.
Exact mathematical passage
VJ=J\overline V.
\overline vUse in source · 1 passage
Search aliases: overline v overline
Use in source · monograph_atlas.tex, line 688
ts memory kernel is K(t)=frac925e^- i t+frac1625e^-2 i t, K(0)=1. Each oscillatory coefficient is positive although K(t) need not be real or pointwise nonnegative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structural identity is ∑_j,koverline v_jK(t_j-t_k)v_k = ∑_k e^ i t_kDB^*v_k ^2≥0. The exact worked check specializes to t=(0,π/2,π) and v=(1,2,-1). Its phases lie in QQ( i…
Exact mathematical passage
\sum_{j,k}\overline v_jK(t_j-t_k)v_k =\left\|\sum_k e^{\ii t_kD}B^*v_k\right\|^2\geq0.
\overline v_jUse in source · 1 passage
Search aliases: overline v_j overline
Use in source · monograph_atlas.tex, line 688
ts memory kernel is K(t)=frac925e^- i t+frac1625e^-2 i t, K(0)=1. Each oscillatory coefficient is positive although K(t) need not be real or pointwise nonnegative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structural identity is ∑_j,koverline v_jK(t_j-t_k)v_k = ∑_k e^ i t_kDB^*v_k ^2≥0. The exact worked check specializes to t=(0,π/2,π) and v=(1,2,-1). Its phases lie in QQ( i…
Exact mathematical passage
\sum_{j,k}\overline v_jK(t_j-t_k)v_k =\left\|\sum_k e^{\ii t_kD}B^*v_k\right\|^2\geq0.
\overline wUse in source · 1 passage
Search aliases: overline w overline
Use in source · 30_lifting.tex, line 610
At level N=L+n, write the supplied parameter increment exactly as t_n+1-t_n=q^Nw with w∈ Z^k. Surjectivity of the solved block gives a digit u satisfying J_z_nu=-overlineF(z_n)/q^N-K_z_noverline w. Choose an integer representative widetilde u and set z_n+1=z_n+q^N(widetilde u,w). Its parameter coordinates equal the supplied target exactly. Taylor expansion proves the next root condition, while congruence modulo q preserves the full reduced Jacobian. Construct the sequence recursively, preserving…
Exact mathematical passage
J_{z_n}u=-\overline{F(z_n)/q^N}-K_{z_n}\overline w
\overline yUse in source · 1 passage
Search aliases: overline y overline
Use in source · Paper_X_Block27_Amended.tex, line 1210
When pnmid n, the comparison with the finite cyclotomic sample is proved: q_L(γ_p)=σ_p, σ_p(ζ_n)=ζ_n^p. Reduction distinguishes n-th roots of unity at such a prime. Indeed, if x^n=y^n=1 and their reductions agree, the factorization of x^n-y^n gives (x-y)∑_i=0^n-1x^i y^n-1-i=0. The second factor reduces to noverline y^ n-1≠0, so the domain property gives x=y. This proves both Frobenius recognition on the cyclotomic generator and triviality of its inertia action. Consequently, every representation…
Exact mathematical passage
n\overline y^{\,n-1}\ne0
\overline y^{\,n-1}Use in source · 1 passage
Search aliases: overline y^ n-1 overline
Use in source · Paper_X_Block27_Amended.tex, line 1210
When pnmid n, the comparison with the finite cyclotomic sample is proved: q_L(γ_p)=σ_p, σ_p(ζ_n)=ζ_n^p. Reduction distinguishes n-th roots of unity at such a prime. Indeed, if x^n=y^n=1 and their reductions agree, the factorization of x^n-y^n gives (x-y)∑_i=0^n-1x^i y^n-1-i=0. The second factor reduces to noverline y^ n-1≠0, so the domain property gives x=y. This proves both Frobenius recognition on the cyclotomic generator and triviality of its inertia action. Consequently, every representation…
Exact mathematical passage
n\overline y^{\,n-1}\ne0
\overline zUse in source · 3 passages
Search aliases: overline z overline
Use in source · Paper_IV_Traditional_Final.tex, line 740
[Symmetries] The denominator satisfies D_N(ω z)= D_N(z) and D_N(overline z)=overline D_N(z). Its nonzero zeros occur in orbits under multiplication by the Nth roots of unity and under complex conjugation.
Exact mathematical passage
\Dfun_N(\overline z)=\overline{\Dfun_N(z)}.
\overline{(G_r-W_kU_k)/2^k}Use in source · 1 passage
Search aliases: overline(G_r-W_kU_k)/2^k overline
Use in source · root_scope.tex, line 179
For preparation, start from the monic polynomial g=overline G_r and its zero-one lift W_1, with U_1=1. Suppose G_r-W_kU_k is coefficientwise divisible by 2^k. Its divided reduction is a finite polynomial because G_r is restricted. Divide it by g in F_2[X]: overline(G_r-W_kU_k)/2^k=gq+s, deg s<d_r. With zero-one coefficient lifts, set W_k+1=W_k+2^ks and U_k+1=U_k+2^kq. The error is now divisible by 2^k+1, the polynomial remains monic of degree d_r, and the unit reduces to one. The sequences conve…
Exact mathematical passage
\overline{(G_r-W_kU_k)/2^k}=gq+s,\qquad\deg s<d_r.
\overline{\left(\sum_{j\in B}\pi_j g_j\right)}Use in source · 1 passage
Search aliases: overline(∑_j∈ Bπ_j g_j) overline left sum in pi right
Use in source · Paper_VI_Traditional_Final.tex, line 675
After one application, the result is constant on every block, so applying the same block average again changes nothing. Weighted self-adjointness follows by expanding both inner products block by block. Both equal ∑_B∈ Q frac1π(B) (∑_i∈ Bπ_i f_i) overline(∑_j∈ Bπ_j g_j).
Exact mathematical passage
\sum_{B\in\mathcal Q} \frac{1}{\pi(B)} \left(\sum_{i\in B}\pi_i f_i\right) \overline{\left(\sum_{j\in B}\pi_j g_j\right)}.
\overline{\left(\sum_{j\in B}\pi_jg_j\right)}Use in source · 1 passage
Search aliases: overline(∑_j∈ Bπ_jg_j) overline left sum in pi right
Use in source · Paper_VI_Traditional_Final.tex, line 3037
One application produces a block constant function, so a second application changes nothing. Expanding both weighted inner products block by block gives ipP_ Qfg_π = ∑_B∈ Q frac (∑_i∈ Bπ_if_i) overline(∑_j∈ Bπ_jg_j) π(B) = ipfP_ Qg_π. The range is exactly the block constant subspace.
Exact mathematical passage
\ip{P_\mathcal Qf}{g}_\pi = \sum_{B\in\mathcal Q} \frac{ \left(\sum_{i\in B}\pi_if_i\right) \overline{\left(\sum_{j\in B}\pi_jg_j\right)} }{\pi(B)} = \ip{f}{P_\mathcal Qg}_\pi.
\overline{c_b}Use in source · 1 passage
Search aliases: overlinec_b overline
Use in source · 03_supplement_05.tex, line 175
For E(c)=∑_a c_aE_a, finite-sum expansion gives R(E(c)ρ E(c)^*) =(∑_a,bc_aoverlinec_bC_ba)ρ =(c^*Cc)ρ. The formula with a double sum is the literal checked conclusion. The final notation as a quadratic form follows by renaming finite indices. It is not restricted to orthogonal errors or real coefficients.
Exact mathematical passage
\mathcal R\bigl(E(c)\rho E(c)^*\bigr) =\left(\sum_{a,b}c_a\overline{c_b}C_{ba}\right)\rho =(c^*Cc)\rho.
\overline{c_j}Use in source · 1 passage
Search aliases: overlinec_j overline
Use in source · 02_strengthened.tex, line 1880
[Finite reflection positivity] For vectors v_1,…,v_m∈ H and nonnegative integers n_1,…,n_m, the matrix G_jk=ipv_jT^n_j+n_kv_k is positive semidefinite. Equivalently, ∑_j,koverlinec_jc_k ipv_jT^n_j+n_kv_k ≥0 for every c∈ C^m.
Exact mathematical passage
\sum_{j,k}\overline{c_j}c_k \ip{v_j}{T^{n_j+n_k}v_k} \geq0
\overline{D(\mu,r_\mu)}Use in source · 1 passage
Search aliases: overlineD(μ,r_μ) overline mu mu
Use in source · Paper_VI_Traditional_Final.tex, line 1666
.F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent of a basis and of increasing the truncation index: the extra nilpotent powers vanish. It depends only on the finite Taylor jets of F at the spectrum. If F is holomorphic on an open neighborhood U of the spectrum, each eigenvalue admits a radius r_μ>0 with overlineD(μ,r_μ)⊂ U, and Cauchy's formula gives fracF^(k)(μ)k! =frac12π ioint_|z-μ|=r_μ fracF(z)(z-μ)^k+1 dz. Consequently the same canonical operator is obtained fr…
Exact mathematical passage
\overline{D(\mu,r_\mu)}\subset U
\overline{d_{ja}}Use in source · 1 passage
Search aliases: overlined_ja overline
Use in source · 01_finite_core.tex, line 833
he left is a sum of nonnegative squared absolute values and equals zero. Each L_jψ therefore lies in the span of ψ. A linear operator that preserves every line in C is scalar there: apply it to two independent vectors and their sum. The one-dimensional case is immediate, and the claim is vacuous for the zero code. Consequently K_jE_aP=d_jaP. Recovery completeness now gives PE_a^*E_bP · =∑_j (K_jE_aP)^*(K_jE_bP) · =(∑_joverlined_jad_jb)P. The coefficients form a Gram matrix, hence are positive se…
Exact mathematical passage
\begin{aligned} PE_a^*E_bP &=\sum_j (K_jE_aP)^*(K_jE_bP)\\ &=\left(\sum_j\overline{d_{ja}}d_{jb}\right)P. \end{aligned}
\overline{\Dfun_N(z)}Use in source · 1 passage
Search aliases: overlineDfun_N(z) overline D_N(z) overline Dfun overline mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 740
[Symmetries] The denominator satisfies D_N(ω z)= D_N(z) and D_N(overline z)=overline D_N(z). Its nonzero zeros occur in orbits under multiplication by the Nth roots of unity and under complex conjugation.
Exact mathematical passage
\Dfun_N(\overline z)=\overline{\Dfun_N(z)}.
\overline{F(z_n)/q^N}Use in source · 2 passages
Search aliases: overlineF(z_n)/q^N overline
Use in source · 30_lifting.tex, line 347
At level N=L+n, divide the residual by q^N. Surjectivity supplies t∈ S_q^d with J_z_nt=-overlineF(z_n)/q^N. Choose an integer representative and apply [source reference] to obtain z_n+1. The new point is congruent to z_n modulo q. Evaluation of each formal partial derivative therefore gives J_z_n+1=J_z_n.
Exact mathematical passage
J_{z_n}t=-\overline{F(z_n)/q^N}
\overline{g(x)}Use in source · 1 passage
Search aliases: overlineg(x) overline
Use in source · Paper_VI_Traditional_Final.tex, line 3262
-xf'(x))+o(t). Every omitted term is a polynomial in the Gaussian variable multiplied by a quantity that is o(t), so the expansion may be integrated term by term. Therefore .frac d d t O_tf|_t=0 =f''-xf'. The Hermite eigenvalue identity also follows by differentiating O_th_n= e^-nth_n at zero. Gaussian integration by parts gives ∫_ R f'(x)overlineg(x)γ( d x) = ∫_ R f(x)overlinexg(x)-g'(x)γ( d x). Set g=f' and rearrange.
Exact mathematical passage
\int_\RR f'(x)\overline{g(x)}\gamma(\dd x) = \int_\RR f(x)\overline{xg(x)-g'(x)}\gamma(\dd x).
\overline{JB}Use in source · 1 passage
Search aliases: overlineJB overline
Use in source · Paper_XII_Traditional_Final.tex, line 317
is canonically A/J. The ordinary tensor identity B⊗_A(A/J)≅ B/JB carries the defining ideal of the tensor to the image of m_B. Here the structural map is local by the stated global deformation datum, so it carries m_A into m_B. Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism R^τ≅ B/overlineJB. The construction commutes with the canonical map from B, which is therefore surjective. It is also unchanged by transporting the t…
Exact mathematical passage
R^\tau\cong B/\overline{JB}.
\overline{L\widehat S}Use in source · 2 passages
Search aliases: overlineLwidehat S overline widehat
Use in source · Paper_XII_Traditional_Final.tex, line 285
Here completion means the actual Hausdorff completion for the additive uniformity of the indicated adic topology. For every adic ring S and ideal L, there is a canonical bicontinuous ring isomorphism widehat S/overlineLwidehat S≅widehatS/L. Indeed, the completed quotient map kills the closed extended ideal. In the other direction, the quotient map from S/L to widehat S/overlineLwidehat S extends to its completion. The latter quotient is complete: the adic uniformity has a countable base, and the…
Exact mathematical passage
\widehat S\big/\overline{L\widehat S}\cong\widehat{S/L}.
\overline{\Q_p}Use in source · 1 passage
Search aliases: overlineQ_p overline Q_p overline Q overline mathbf
Use in source · Paper_X_Block27_Amended.tex, line 2487
For every prime p, Lean constructs a valuation subring A⊆overline Q_p containing the integral image of Z_p such that the induced map Z_p→ A is local. For odd p, two is a unit. The preceding argument applies to every valuation subring containing that integral image, and hence every member of P_q(B) is trivial on its inertia subgroup after actual local restriction. The existence theorem supplies a locally dominating choice, so this assertion is not vacuous.
Exact mathematical passage
A\subseteq\overline{\Q_p}
\overline{r_x}Use in source · 7 passages
Search aliases: overliner_x overline
Use in source · 30_lifting.tex, line 72
[Finite polynomial obstruction] Let q≥1, L≥1, x∈ Z^d, and suppose F(x)∈ q^L Z^e. Define the exact integer residual r_x by F(x)=q^Lr_x, and put O_x=S_q^e/im J_x, o_x=[overliner_x]. Then O_x is finite and o_x=0 Longleftrightarrow ∃ y∈ Z^d: y≡ xpmodq^L, F(y)≡0pmodq^L+1. No surjectivity or nonsingularity assumption is required.
Exact mathematical passage
\mathcal O_x=S_q^e/\im J_x,\qquad o_x=[\overline{r_x}].
\overline{r_{x'}}Use in source · 1 passage
Search aliases: overliner_x' overline
Use in source · 30_lifting.tex, line 134
[Independence of the integer representative] Under the hypotheses of Theorem~[source reference], let x'=x+q^Ls and write F(x')=q^Lr_x'. Then J_x'=J_x, overliner_x'=overliner_x+J_xoverline s. Consequently the two residual classes coincide in the common quotient S_q^e/im J_x.
Exact mathematical passage
J_{x'}=J_x,\qquad \overline{r_{x'}}=\overline{r_x}+J_x\overline s.
\overline{s(t)-s(t+L/2)}Use in source · 1 passage
Search aliases: overlines(t)-s(t+L/2) overline
Use in source · Paper_V_Traditional_Final.tex, line 3209
Let L=2^-j and let a be the left endpoint of the support. Pair every point t∈[a,a+L/2] with t+L/2. Then ⟨ s,ψ_j,k⟩ = L^-1/2 ∫_a^a+L/2 overlines(t)-s(t+L/2) d t. The integrand has magnitude at most ML/2, and the integration interval has length L/2. The result is ML^3/2/4.
Exact mathematical passage
\langle s,\psi_{j,k}\rangle = L^{-1/2} \int_a^{a+L/2} \overline{s(t)-s(t+L/2)}\,\dd t.
\overline{\operatorname{span}\{S^n\delta_0:n\in\bbZ\}}Use in source · 1 passage
Search aliases: overlinespan{S^nδ_0:n∈bbZ} overlinespan{S^nδ_0:n∈ Z} overline operatorname delta in bbZ overline operatorname delta in mathbb
Use in source · Paper_V_Traditional_Final.tex, line 277
[Impulse seed theorem] The closed shift-generated hull of δ_0 is all of ℓ^2( Z): overlinespan{S^nδ_0:n∈ Z}^ ℓ^2 =ℓ^2( Z). Every finitely supported signal has the exact finite expression x=∑_n∈ Zx[n]S^nδ_0.
Exact mathematical passage
\boxed{ \overline{\operatorname{span}\{S^n\delta_0:n\in\bbZ\}}^{\,\ell^2} =\ell^2(\bbZ). }
\overline{\operatorname{span}\{S^n\delta_0:n\in\bbZ\}}^{\,\ell^2}Use in source · 1 passage
Search aliases: overlinespan{S^nδ_0:n∈bbZ}^ ℓ^2 overlinespan{S^nδ_0:n∈ Z}^ ℓ^2 overline operatorname delta in bbZ ell overline operatorname delta in mathbb ell
Use in source · Paper_V_Traditional_Final.tex, line 277
[Impulse seed theorem] The closed shift-generated hull of δ_0 is all of ℓ^2( Z): overlinespan{S^nδ_0:n∈ Z}^ ℓ^2 =ℓ^2( Z). Every finitely supported signal has the exact finite expression x=∑_n∈ Zx[n]S^nδ_0.
Exact mathematical passage
\boxed{ \overline{\operatorname{span}\{S^n\delta_0:n\in\bbZ\}}^{\,\ell^2} =\ell^2(\bbZ). }
\overline{\Span\{T_a\delta_0:a\in\ZZ^d\}}Use in source · 1 passage
Search aliases: overlineSpan{T_aδ_0:a∈ZZ^d} overlinespan{T_aδ_0:a∈ Z^d} overline Span delta in ZZ overline operatorname delta in mathbb
Use in source · Paper_V_Traditional_Final.tex, line 1733
[Impulse generation] For 1≤ p<∞, overlinespan{T_aδ_0:a∈ Z^d}^ℓ^p = ℓ^p( Z^d). The same statement holds in c_0( Z^d) with the supremum norm. In ℓ^∞( Z^d), the closed span of the translation orbit is exactly c_0( Z^d).
Exact mathematical passage
\overline{\Span\{T_a\delta_0:a\in\ZZ^d\}}^{\ell^p} = \ell^p(\ZZ^d).
\overline{\Span\{T_a\delta_0:a\in\ZZ^d\}}^{\ell^p}Use in source · 1 passage
Search aliases: overlineSpan{T_aδ_0:a∈ZZ^d}^ℓ^p overlinespan{T_aδ_0:a∈ Z^d}^ℓ^p overline Span delta in ZZ ell overline operatorname delta in mathbb ell
Use in source · Paper_V_Traditional_Final.tex, line 1733
[Impulse generation] For 1≤ p<∞, overlinespan{T_aδ_0:a∈ Z^d}^ℓ^p = ℓ^p( Z^d). The same statement holds in c_0( Z^d) with the supremum norm. In ℓ^∞( Z^d), the closed span of the translation orbit is exactly c_0( Z^d).
Exact mathematical passage
\overline{\Span\{T_a\delta_0:a\in\ZZ^d\}}^{\ell^p} = \ell^p(\ZZ^d).
\overline{U(g)}Use in source · 1 passage
Search aliases: overlineU(g) overline
Use in source · 02_strengthened.tex, line 1146
[Choi symmetry criterion] Let U:G→ U( H) be a finite group representation. A channel Φ is covariant, Φ(U(g)XU(g)^*) = U(g)Φ(X)U(g)^*, if and only if (U(g)⊗overlineU(g)) J(Φ) (U(g)^*⊗ U(g)^ T) = J(Φ) for every g∈ G.
Exact mathematical passage
\bigl(U(g)\otimes\overline{U(g)}\bigr) J(\Phi) \bigl(U(g)^*\otimes U(g)^{\mathsf T}\bigr) = J(\Phi)
\overline{W_{ja}}Use in source · 1 passage
Search aliases: overlineW_ja overline
Use in source · 02_strengthened.tex, line 1323
[Orthogonal syndrome construction] Assume the Knill--Laflamme identities. Then the coefficient matrix c is positive semidefinite. Choose a unitary matrix W such that WcW^*=diag(λ_1,…,λ_r), λ_j≥0, and define F_j=∑_aoverlineW_jaE_a. For every j with λ_j>0, the operator V_j=λ_j^-1/2F_jP is an isometry from the code into H. Its range projector Q_j=V_jV_j^* is a syndrome packet, and the nonzero Q_j are pairwise orthogonal.
Exact mathematical passage
F_j=\sum_a\overline{W_{ja}}E_a.
\overline{x}Use in source · 1 passage
Search aliases: overlinex overline
Use in source · Paper_X_Block27_Amended.tex, line 1144
The arithmetic attachment is checked in this finite cyclotomic case. There is an actual maximal prime P of O_L above (p), its ramification index is one, and σ_p belongs to its decomposition group. For every integral x∈ O_L, overlineσ_p(x)=overlinex^ p in O_L/ P. The residue identity is proved on all integral elements using O_L= Z[ζ_n]. Existence of P is obtained by lying over, rather than assumed. This specifies arithmetic Frobenius, with p-power residue action. The action and unramifiedness sta…
Exact mathematical passage
\overline{\sigma_p(x)}=\overline{x}^{\,p} \quad\text{in }\mathcal O_L/\mathfrak P.
\overline{x}^{\,p}Use in source · 1 passage
Search aliases: overlinex^ p overline
Use in source · Paper_X_Block27_Amended.tex, line 1144
The arithmetic attachment is checked in this finite cyclotomic case. There is an actual maximal prime P of O_L above (p), its ramification index is one, and σ_p belongs to its decomposition group. For every integral x∈ O_L, overlineσ_p(x)=overlinex^ p in O_L/ P. The residue identity is proved on all integral elements using O_L= Z[ζ_n]. Existence of P is obtained by lying over, rather than assumed. This specifies arithmetic Frobenius, with p-power residue action. The action and unramifiedness sta…
Exact mathematical passage
\overline{\sigma_p(x)}=\overline{x}^{\,p} \quad\text{in }\mathcal O_L/\mathfrak P.
\overline{xg(x)-g'(x)}Use in source · 1 passage
Search aliases: overlinexg(x)-g'(x) overline
Use in source · Paper_VI_Traditional_Final.tex, line 3262
-xf'(x))+o(t). Every omitted term is a polynomial in the Gaussian variable multiplied by a quantity that is o(t), so the expansion may be integrated term by term. Therefore .frac d d t O_tf|_t=0 =f''-xf'. The Hermite eigenvalue identity also follows by differentiating O_th_n= e^-nth_n at zero. Gaussian integration by parts gives ∫_ R f'(x)overlineg(x)γ( d x) = ∫_ R f(x)overlinexg(x)-g'(x)γ( d x). Set g=f' and rearrange.
Exact mathematical passage
\int_\RR f'(x)\overline{g(x)}\gamma(\dd x) = \int_\RR f(x)\overline{xg(x)-g'(x)}\gamma(\dd x).
\overline{z_a}Use in source · 1 passage
Search aliases: overlinez_a overline
Use in source · 03_supplement_04.tex, line 131
identity is R(V_aρ V_b^*)=δ_abρ. Consequently normalized mixtures satisfy R (∑_ap_aV_aρ V_a^*)=ρ when ∑_ap_a=1. The identity is algebraic and is proved for complex weights. Physical mixtures use the nonnegative real case. For a coherent error E_z=∑_a z_aV_a, the calculation of the cross terms yields R(E_zρ E_z^*)=(∑_a z_aoverlinez_a)ρ. Thus the same map also corrects normalized coherent superpositions. This construction supplies the recovery stage used in the full Knill--Laflamme proof that foll…
Exact mathematical passage
\mathcal R(E_z\rho E_z^*)=\left(\sum_a z_a\overline{z_a}\right)\rho.
\overline{z_{ra}}Use in source · 1 passage
Search aliases: overlinez_ra overline
Use in source · 03_supplement_05.tex, line 111
Trace-preserving recovery supplies the original Gram data. Let Z_ra=z_ra. The recovery normalization gives PE_a^*E_bP · =PE_a^*(∑_rR_r^*R_r)E_bP · =∑_r(R_rE_aP)^*(R_rE_bP) · =(∑_roverlinez_raz_rb)P =(Z^*Z)_abP. Thus C=Z^*Z is positive semidefinite and satisfies the required compressed identities. The theorem pathgram_of_recovery implements this calculation with the original error and recovery matrices.
Exact mathematical passage
PE_a^*E_bP &=PE_a^*\Bigl(\sum_rR_r^*R_r\Bigr)E_bP\\ &=\sum_r(R_rE_aP)^*(R_rE_bP)\\ &=\Bigl(\sum_r\overline{z_{ra}}z_{rb}\Bigr)P =(Z^*Z)_{ab}P.
\overline{\gamma_p(a)}Use in source · 1 passage
Search aliases: overlineγ_p(a) overline gamma
Use in source · Paper_X_Block27_Amended.tex, line 1188
The arithmetic witnesses use the integral closure A of Z in overline Q. For each rational prime p, Lean constructs a maximal ideal Q of A above (p) and an actual element γ_p∈ G_ Q such that γ_p∈Stab_G_ Q( Q), overlineγ_p(a)=overline a^ p (a∈ A). The existence argument applies Mathlib's proved profinite stabilizer-surjectivity theorem to this actual invariant ring. The residue p-power map is proved bijective by taking p-th roots in overline Q and showing that they remain integral when their p-th …
Exact mathematical passage
\gamma_p\in\operatorname{Stab}_{G_{\Q}}(\mathfrak Q),\qquad \overline{\gamma_p(a)}=\overline a^{\,p} \quad(a\in\mathcal A).
\overline{\eta_0}Use in source · 2 passages
Search aliases: overlineη_0 overline eta
Use in source · Paper_IV_Traditional_Final.tex, line 1782
The finite exponential representation gives | D_N(z)|≤ e^|z|, so the type is at most one. Choose a root η_0 and put z=toverlineη_0 with t>0. The term e^η_0z=e^t has strictly larger real exponent than every other term. Hence D_N(toverlineη_0) = frac1Ne^t(1+o(1)). The type is therefore one.
Exact mathematical passage
z=t\overline{\eta_0}
\overline\lambda_\betaUse in source · 1 passage
Search aliases: overlineλ_β overline lambda beta
Use in source · Paper_VI_Traditional_Final.tex, line 3501
The posterior mass of packet I_α is frac∑_i∈ I_αλ_ip_i ∑_jλ_jp_j = frac(Cp)_αoverlineλ_α(p) ∑_β(Cp)_βoverlineλ_β(p). This is the coarse Bayesian formula. Dividing the fine posterior coordinates by the packet mass gives the local formula.
Exact mathematical passage
\frac{\sum_{i\in I_\alpha}\lambda_ip_i} {\sum_j\lambda_jp_j} = \frac{(Cp)_\alpha\overline\lambda_\alpha(p)} {\sum_\beta(Cp)_\beta\overline\lambda_\beta(p)}.
\overline\piUse in source · 6 passages
Search aliases: overlineπ overline pi
Use in source · Paper_VI_Traditional_Final.tex, line 5356
se Markov generator. For observables, the backward generator is Q^T. Weighted L^2 inner products use the stated stationary probability. [Reversibility descends] Assume Q is strongly lumpable and reversible with respect to a positive stationary distribution π, so Q_ijπ_j=Q_jiπ_i. Define overlineπ_a=∑_i∈ B_aπ_i. Then overlineπ is stationary for overline Q, and overline Q_aboverlineπ_b = overline Q_baoverlineπ_a. Thus the coarse chain is reversible.
Exact mathematical passage
\overline\pi_a=\sum_{i\in B_a}\pi_i.
\overline\pi_aUse in source · 3 passages
Search aliases: overlineπ_a overline pi
Use in source · Paper_VI_Traditional_Final.tex, line 5356
se Markov generator. For observables, the backward generator is Q^T. Weighted L^2 inner products use the stated stationary probability. [Reversibility descends] Assume Q is strongly lumpable and reversible with respect to a positive stationary distribution π, so Q_ijπ_j=Q_jiπ_i. Define overlineπ_a=∑_i∈ B_aπ_i. Then overlineπ is stationary for overline Q, and overline Q_aboverlineπ_b = overline Q_baoverlineπ_a. Thus the coarse chain is reversible.
Exact mathematical passage
\overline\pi_a=\sum_{i\in B_a}\pi_i.
\overline\pi_bUse in source · 2 passages
Search aliases: overlineπ_b overline pi
Use in source · Paper_VI_Traditional_Final.tex, line 5360
the stated stationary probability. [Reversibility descends] Assume Q is strongly lumpable and reversible with respect to a positive stationary distribution π, so Q_ijπ_j=Q_jiπ_i. Define overlineπ_a=∑_i∈ B_aπ_i. Then overlineπ is stationary for overline Q, and overline Q_aboverlineπ_b = overline Q_baoverlineπ_a. Thus the coarse chain is reversible.
Exact mathematical passage
\overline Q_{ab}\overline\pi_b = \overline Q_{ba}\overline\pi_a.
\overline{\sigma_p(x)}Use in source · 1 passage
Search aliases: overlineσ_p(x) overline sigma
Use in source · Paper_X_Block27_Amended.tex, line 1144
The arithmetic attachment is checked in this finite cyclotomic case. There is an actual maximal prime P of O_L above (p), its ramification index is one, and σ_p belongs to its decomposition group. For every integral x∈ O_L, overlineσ_p(x)=overlinex^ p in O_L/ P. The residue identity is proved on all integral elements using O_L= Z[ζ_n]. Existence of P is obtained by lying over, rather than assumed. This specifies arithmetic Frobenius, with p-power residue action. The action and unramifiedness sta…
Exact mathematical passage
\overline{\sigma_p(x)}=\overline{x}^{\,p} \quad\text{in }\mathcal O_L/\mathfrak P.
\overline\PhiUse in source · 4 passages
Search aliases: overlineΦ overline Phi
Use in source · Paper_VI_Traditional_Final.tex, line 5535
For stochastic matrices K,overline K and positive likelihoods L_y,overline L_y, put Φ_y= B_L_y∘ K and overlineΦ_y= B_overline L_y∘overline K. [Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯∘Φ_y_1 = overlineΦ_y_r∘⋯∘overlineΦ_y_1C.
Exact mathematical passage
\overline\Phi_y=\Bcal_{\overline L_y}\circ\overline K
\overline\varphiUse in source · 2 passages
Search aliases: overlineφ overline varphi
Use in source · 03_nonlinear_coordinates.tex, line 927
[Cokernel obstruction criterion] A linear correction exists if and only if Ob_D(r)=0. The quotient map q_D:V⟶coker D has the following universal property. If φ:V→ W is linear and φ D=0, then there is a unique linear map overlineφ:coker D⟶ W with φ=overlineφ q_D.
Exact mathematical passage
\overline\varphi:\coker D\longrightarrow W
\overline\Phi_yUse in source · 3 passages
Search aliases: overlineΦ_y overline Phi
Use in source · Paper_VI_Traditional_Final.tex, line 5535
For stochastic matrices K,overline K and positive likelihoods L_y,overline L_y, put Φ_y= B_L_y∘ K and overlineΦ_y= B_overline L_y∘overline K. [Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯∘Φ_y_1 = overlineΦ_y_r∘⋯∘overlineΦ_y_1C.
Exact mathematical passage
\overline\Phi_y=\Bcal_{\overline L_y}\circ\overline K
\overline\Phi_{y_1}Use in source · 1 passage
Search aliases: overlineΦ_y_1 overline Phi
Use in source · Paper_VI_Traditional_Final.tex, line 5546
[Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯∘Φ_y_1 = overlineΦ_y_r∘⋯∘overlineΦ_y_1C.
Exact mathematical passage
C\Phi_{y_r}\circ\cdots\circ\Phi_{y_1} = \overline\Phi_{y_r}\circ\cdots\circ\overline\Phi_{y_1}C.
\overline\Phi_{y_r}Use in source · 1 passage
Search aliases: overlineΦ_y_r overline Phi
Use in source · Paper_VI_Traditional_Final.tex, line 5546
[Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯∘Φ_y_1 = overlineΦ_y_r∘⋯∘overlineΦ_y_1C.
Exact mathematical passage
C\Phi_{y_r}\circ\cdots\circ\Phi_{y_1} = \overline\Phi_{y_r}\circ\cdots\circ\overline\Phi_{y_1}C.
\overline{\chi(r)}Use in source · 1 passage
Search aliases: overlineχ(r) overline chi
Use in source · 03_supplement_03.tex, line 126
Orthogonality of the projectors. Unitarity supplies ρ(r)^*=ρ(-r), while the character obeys overlineχ(r)=χ(-r). Reindexing r↦-r proves P_j^*=P_j. The final concrete matrix theorem includes Hermiticity together with multiplication, completeness, and eigenaction. Together these identities establish orthogonal spectral packets.
Exact mathematical passage
\overline{\chi(r)}=\chi(-r)
\overline{\chi_\lambda(g)}Use in source · 1 passage
Search aliases: overlineχ_λ(g) overline chi lambda
Use in source · 02_strengthened.tex, line 207
[Finite group isotypic projectors] The operators P_λ = fracd_λabsG ∑_g∈ G overlineχ_λ(g)U(g) are orthogonal projectors. They satisfy P_λ P_μ=δ_λμP_λ, ∑_λ∈widehat GP_λ=I_ H. The range of P_λ is the λ isotypic component of H.
Exact mathematical passage
P_\lambda = \frac{d_\lambda}{\abs{G}} \sum_{g\in G} \overline{\chi_\lambda(g)}U(g)
\overline{\omega_\ell(a)}Use in source · 1 passage
Search aliases: overlineω_ℓ(a) overline omega ell
Use in source · supplement_05.tex, line 94
The root equation and residue are derived. Euler's unit-group exponent gives ω_ℓ(a)^p-1 =a^p^ℓ-1(p-1)=a^φ(p^ℓ)=1. Reduction to the actual field F_p commutes with powers. Its Frobenius identity gives overlineω_ℓ(a) =bar a^p^ℓ-1=bar a. Consequently, a=ω_ℓ(a)⟨ a⟩_ℓ, ω_ℓ(a)^p-1=1, overline⟨ a⟩_ℓ=1. The manuscript's finite existence conclusion is obtained directly. The proof does not assume a Hensel lift or an already supplied root representative.
Exact mathematical passage
\overline{\omega_\ell(a)} =\bar a^{p^{\ell-1}}=\bar a.
\overline{\langle a\rangle_\ell}Use in source · 1 passage
Search aliases: overline⟨ a⟩_ℓ overline langle rangle ell
Use in source · supplement_05.tex, line 99
The root equation and residue are derived. Euler's unit-group exponent gives ω_ℓ(a)^p-1 =a^p^ℓ-1(p-1)=a^φ(p^ℓ)=1. Reduction to the actual field F_p commutes with powers. Its Frobenius identity gives overlineω_ℓ(a) =bar a^p^ℓ-1=bar a. Consequently, a=ω_ℓ(a)⟨ a⟩_ℓ, ω_ℓ(a)^p-1=1, overline⟨ a⟩_ℓ=1. The manuscript's finite existence conclusion is obtained directly. The proof does not assume a Hensel lift or an already supplied root representative.
Exact mathematical passage
\boxed{a=\omega_\ell(a)\langle a\rangle_\ell,\quad \omega_\ell(a)^{p-1}=1,\quad \overline{\langle a\rangle_\ell}=1.}
\mathbb PUse in source · 4 passages
Search aliases: P mathbb
Use in source · Paper_V_Traditional_Final.tex, line 1143
[Polyphase M obius action] Regard normalized impedance as the projective line P^1( C)= C∪{∞}. For N≥1, the pullback is a globally defined projective action. In the finite chart, wherever the denominator is nonzero, it is T_N,k(z) = frac(1+ω_N^k)z+(1-ω_N^k) (1-ω_N^k)z+(1+ω_N^k). As maps on P^1( C) it satisfies T_N,k∘ T_N,ℓ=T_N,k+ℓ (mod N).
Exact mathematical passage
\mathbb P^1(\bbC)=\bbC\cup\{\infty\}
\mathcal PUse in source · 22 passages
Search aliases: P mathcal
Use in source · 20_arithmetic.tex, line 29
The normalized source and its inverse are K_N,j(t)=j!∑_n≥0fract^Nn(Nn+j)!, P_N,j(t)=∑_n≥0C_N,j(n)fract^Nn(Nn)!. Their product is one. Clearing factorials in its coefficient of t^Nn gives the triangular system ∑_q=0^nbinomNn+jNqC_N,j(q)=δ_n0. In particular, for n>0, C_N,j(n)=-binomNn+jNn^-1 ∑_q=0^n-1binomNn+jNqC_N,j(q). The diagonal binomial coefficient is positive, so this recurrence determines the sequence uniquely. The Lean construction begins with this recurrence and derives the reciprocal id…
Exact mathematical passage
\label{eq:arith-kernel} K_{N,j}(t)=j!\sum_{n\ge0}\frac{t^{Nn}}{(Nn+j)!},\qquad \mathcal P_{N,j}(t)=\sum_{n\ge0}C_{N,j}(n)\frac{t^{Nn}}{(Nn)!}.
P'Use in source · 1 passage
Search aliases: P'
Use in source · 04_correction_transport.tex, line 1051
[htbp] centering includegraphics[width=.96linewidth]rotating_projector_transport.pdf captionA rotating rank-one packet and its orthogonal complement. With u(θ)=(cosθ,sinθ) and P(θ)=u(θ)u(θ)^ T, the canonical generator is [P',P]=(0 · -1 1 · 0). For a positive step δ<π/2, the polar unitary is the usual rotation R(δ). On each displayed mesh, the direct rotation product equals R(π/3) exactly. This special planar example illustrates the construction. The general theorem supplies convergence for famil…
Exact mathematical passage
[P',P]=\bigl(\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\bigr)
P^{\mathsf T}Use in source · 5 passages
Search aliases: P^ T mathsf
Use in source · Paper_VI_Traditional_Final.tex, line 927
[Bayesian information-loss packet] Assume nonnegative fine and declared coarse likelihoods, with both Z_r>0 and Z_barλ>0. The coarsening defect is C_π B_λ(p) - B_barλ(P) = frac1Z_r (diag(P)-bar bP^ T) δ(p). The factor matrix annihilates barλ. This records the common-scale null direction at the declared coarse likelihood. The full exact defect remains homogeneous under simultaneous likelihood rescaling because the numerator and evidence denominator scale together. Moreover, C_π B_λ(p)- B_barλ(P) …
Exact mathematical passage
\boxed{ C_\pi\cB_\lambda(p) - \cB_{\bar\lambda}(P) = \frac{1}{Z_r} \left(\diag(P)-\bar bP^{\mathsf T}\right) \delta(p).}
p^{\mathsf T}Use in source · 1 passage
Search aliases: p^ T mathsf
Use in source · 03_supplement_05.tex, line 98
Rank-one rigidity of the corrected channel. The module tests the map equality on actual matrix units and flattens each A_ra into a row. If X is the matrix of those rows and p is the vector of entries of P, the resulting Gram identity is X^*X=overline p p^ T. The checked rank-one Gram lemma proves that every row of X is proportional to p. For nonzero p, a nonzero coordinate is selected and a test vector supported on two coordinates is placed in the null space of the Gram matrix. The implication X…
Exact mathematical passage
X^*X=\overline p\,p^{\mathsf T}.
P^{(1)}_{j_1}Use in source · 2 passages
Search aliases: P^(1)_j_1
Use in source · 01_finite_core.tex, line 239
Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.
Exact mathematical passage
P_{\mathbf j}^{\mathrm{fine}} = P^{(1)}_{j_1}\otimes\cdots\otimes P^{(m)}_{j_m}.
p^{(a)}Use in source · 2 passages
Search aliases: p^(a)
Use in source · atlas-pages.tex, line 216
vpage10Two priors that fool the coarse modelSUFFICIENCY / A COMPUTABLE FAILURE WITNESSChapter VI, Part I: Theorem 8.2 and the one-packet exception. Keep the same likelihood λ=(4,1,1,1)^ T and partition A={0,1}, B={2,3}. Compare p^(a)=(1/2,0,1/2,0)^ T, p^(b)=(0,1/2,1/2,0)^ T. Both have coarse prior (1/2,1/2)^ T. A procedure given only that two-entry vector cannot distinguish them. vfiginsufficiency platecaptionAfter the identical fine likelihood is applied, the coarse posteriors are (4/5,1/5) and…
Exact mathematical passage
p^{(a)}=(1/2,0,1/2,0)^{\mathsf T},\qquad p^{(b)}=(0,1/2,1/2,0)^{\mathsf T}.
P^{(a)}_0Use in source · 1 passage
Search aliases: P^(a)_0
Use in source · 01_finite_core.tex, line 238
Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.
Exact mathematical passage
P^{(a)}_0,\ldots,P^{(a)}_{N-1}
P^{(a)}_{N-1}Use in source · 1 passage
Search aliases: P^(a)_N-1
Use in source · 01_finite_core.tex, line 238
Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.
Exact mathematical passage
P^{(a)}_0,\ldots,P^{(a)}_{N-1}
p^{(b)}Use in source · 2 passages
Search aliases: p^(b)
Use in source · atlas-pages.tex, line 216
vpage10Two priors that fool the coarse modelSUFFICIENCY / A COMPUTABLE FAILURE WITNESSChapter VI, Part I: Theorem 8.2 and the one-packet exception. Keep the same likelihood λ=(4,1,1,1)^ T and partition A={0,1}, B={2,3}. Compare p^(a)=(1/2,0,1/2,0)^ T, p^(b)=(0,1/2,1/2,0)^ T. Both have coarse prior (1/2,1/2)^ T. A procedure given only that two-entry vector cannot distinguish them. vfiginsufficiency platecaptionAfter the identical fine likelihood is applied, the coarse posteriors are (4/5,1/5) and…
Exact mathematical passage
p^{(a)}=(1/2,0,1/2,0)^{\mathsf T},\qquad p^{(b)}=(0,1/2,1/2,0)^{\mathsf T}.
P^{(i)}_{a_i}Use in source · 1 passage
Search aliases: P^(i)_a_i
Use in source · 03_supplement_02.tex, line 54
Suppose N>0 and each factor has a complete orthogonal self-adjoint projector family (P^(i)_j)_j∈ Z/N Z. For a fine address a:F→ Z/N Z, set Q_a= T(i↦ P^(i)_a_i), C_r=∑_∑_i a_i=rQ_a.
Exact mathematical passage
Q_a=\mathsf T\bigl(i\mapsto P^{(i)}_{a_i}\bigr), \qquad C_r=\sum_{\sum_i a_i=r}Q_a.
P^{(i)}_jUse in source · 2 passages
Search aliases: P^(i)_j
Use in source · 03_supplement_02.tex, line 53
Suppose N>0 and each factor has a complete orthogonal self-adjoint projector family (P^(i)_j)_j∈ Z/N Z. For a fine address a:F→ Z/N Z, set Q_a= T(i↦ P^(i)_a_i), C_r=∑_∑_i a_i=rQ_a.
Exact mathematical passage
(P^{(i)}_j)_{j\in\mathbb Z/N\mathbb Z}
p^{(L+a)r}Use in source · 1 passage
Search aliases: p^(L+a)r
Use in source · 72_recovered.tex, line 2401
by the retained proof. Without integral observation vectors or positive L, the preceding counterexamples invalidate the claimed precision. Assume C^H=I+p^LB. Put H_a=Hp^a. Then C^H_a=I+p^L+aB_a for an integral matrix B_a. Every refined packet b_a,j(x)=a_H_ax+j has Mahler coefficients satisfying v_p(c_a,j,r)≥(L+a)r. Equivalently, Δ_H_a^ ra_n ≡0 pmod p^(L+a)r.
Exact mathematical passage
\boxed{ \Delta_{H_a}^{\,r}a_n \equiv0 \pmod {p^{(L+a)r}}. }
P^{(m)}_{j_m}Use in source · 2 passages
Search aliases: P^(m)_j_m
Use in source · 01_finite_core.tex, line 239
Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.
Exact mathematical passage
P_{\mathbf j}^{\mathrm{fine}} = P^{(1)}_{j_1}\otimes\cdots\otimes P^{(m)}_{j_m}.
p^{(N-1)/2}Use in source · 1 passage
Search aliases: p^(N-1)/2
Use in source · odd_synthesis.tex, line 36
The same candidate has rank(π) · =N, · t(π) · =1, Sw(π) · =p^d-1(p+1)h_d, · dim_ F_p(J^1/H^1) · =N-1, dimη · =p^(N-1)/2, a(π^vee×π) · =p^2d-1+∑_i=1^d(p^2i-p^2i-2)h_i. These numerical identities do not identify a Weil representation, an endo-class, or the pair epsilon factors required by H5.
Exact mathematical passage
\begin{aligned} \operatorname{rank}(\pi)&=N, &t(\pi)&=1,\\ \operatorname{Sw}(\pi)&=p^{d-1}(p+1)h_d, & \dim_{\mathbb F_p}(J^1/H^1)&=N-1,\\ \dim\eta&=p^{(N-1)/2},\\ a(\pi^\vee\times\pi) &=p^{2d}-1+\sum_{i=1}^d(p^{2i}-p^{2i-2})h_i. \end{aligned}
P^*Use in source · 4 passages
Search aliases: P^*
Use in source · 01_finite_core.tex, line 792
Let P=P^*=P^2 project onto a code space C⊂ H, and let E_1,…,E_r be a finite family of complex error matrices. Neither a nonzero code nor channel normalization is assumed unless stated. The operators P E_a^*E_bP are the obstruction packets on the code. If they retain information about the logical state beyond one scalar multiple of P, the environment can distinguish logical states and exact correction is impossible.
Exact mathematical passage
P=P^*=P^2
p^{-(L+n)}Use in source · 2 passages
Search aliases: p^-(L+n)
Use in source · 30_lifting.tex, line 378
Cross-level compatibility bounds the supremum distance between later terms by p^-(L+n). Completeness gives a limit, and closed norm bounds retain the congruence to every z_n. View the coefficients of the original integer polynomials in Z_p. Their evaluations satisfy both F_i(z_n)→ F_i(a), by continuity, and F_i(z_n)→0, by their growing divisibility. Uniqueness of limits gives exact vanishing.
Exact mathematical passage
p^{-(L+n)}
p^{-1}Use in source · 1 passage
Search aliases: p^-1
Use in source · 30_lifting.tex, line 15
Unless a different coefficient ring is specified, a polynomial system is a finite list of integer polynomials. All divisibility conditions on vectors are coordinatewise. For a positive integer q, write S_q= Z/q Z and use an overline for reduction modulo q. For a prime p, write F_p=S_p, normalize |p|_p=p^-1, and put B_L(x)=x+p^L Z_p^d. An integer center is included in this notation by its natural image in Z_p^d. All dimensions below are finite and may be zero. A statement over S_q allows composit…
Exact mathematical passage
|p|_p=p^{-1}
p^{-2r+\vp((2r)!)}Use in source · 1 passage
Search aliases: p^-2r+vp((2r)!) p^-2r+v_p((2r)!) vp
Use in source · 72_recovered.tex, line 1874
Every retained base is a unit, and its (p-1)st power lies in 1+p Z_p. The polynomial p^2rbinom(p-1)x+j2r has Gauss norm at most p^-2r+v_p((2r)!). This tends to zero because p is odd. The remaining factors have bounded Gauss norm. Thus the series converges in the Tate algebra. At a nonnegative integer x, the binomial factor vanishes for large r, and [source reference] gives the value.
Exact mathematical passage
p^{-2r+\vp((2r)!)}.
p^{-k}Use in source · 3 passages
Search aliases: p^-k
Use in source · 20_arithmetic.tex, line 333
Here is the coefficient extraction explicitly. If a_d=d![t^d]F, then d![t^d]B_p^kF=p^-kΔ_p(p-1)^ka_d+pk. For F=j! P_N,j and k≤ mn, take d=mpn-pk. Support in multiples of mp turns this into p^-kΔ_h^kU_N,j(n), so flatness gives divisibility by p^k. For k>mn, expand Δ_h^k-mn applied to Δ_h^mnU_N,j(n). Every shifted row is at least n, so each summand is divisible by p^mn. Finally Δ_uh=((I+Δ_h)^u-I), whose kth power for u>0 is an integral combination of powers of Δ_h of order at least k. The case u=0…
Exact mathematical passage
d![t^d]B_p^kF=p^{-k}\Delta_{p(p-1)}^ka_{d+pk}.
p^{-s}Use in source · 2 passages
Search aliases: p^-s
Use in source · supplement_08.tex, line 91
The earlier coefficient criterion is recovered exactly. Define the literal coefficient defect (δ a)(d)=a(d+p^2)-a(d+p). The new source proves, for every s,d, a_B^s f(d)=p^-s(δ^s a_f)(d). For an integral coefficient x, it proves the bridge p^-sx∈ Z_p Longleftrightarrow p^smid x in Z_p. Thus the differential definition is equivalent to the existing coefficient-level Dwork criterion. Division occurs in Q_p, while the assertion after division is membership in Z_p. It is not divisibility in the field…
Exact mathematical passage
a_{B^s f}(d)=p^{-s}(\delta^s a_f)(d).
\mathbb P^1Use in source · 2 passages
Search aliases: P^1 mathbb
Use in source · Paper_V_Traditional_Final.tex, line 1143
[Polyphase M obius action] Regard normalized impedance as the projective line P^1( C)= C∪{∞}. For N≥1, the pullback is a globally defined projective action. In the finite chart, wherever the denominator is nonzero, it is T_N,k(z) = frac(1+ω_N^k)z+(1-ω_N^k) (1-ω_N^k)z+(1+ω_N^k). As maps on P^1( C) it satisfies T_N,k∘ T_N,ℓ=T_N,k+ℓ (mod N).
Exact mathematical passage
\mathbb P^1(\bbC)=\bbC\cup\{\infty\}
p^2Use in source · 19 passages
Search aliases: p^2
Use in source · 20_arithmetic.tex, line 209
Work in the Hurwitz integral ring of formal series H_p={F∈ Q_p[[t]]: d![t^d]F∈ Z_p for every d}. For D=d/dt, let B_p=(D^p^2-D^p)/p. Call F flat if B_p^sF∈ H_p for every s≥0. We will prove that j! P_N,j is flat at every address. The main difficulty is to include the addresses at which the usual unit-pivot argument fails.
Exact mathematical passage
B_p=(D^{p^2}-D^p)/p
p^{2(n-q)}Use in source · 3 passages
Search aliases: p^2(n-q)
Use in source · 20_arithmetic.tex, line 805
[Classical odd-length reconstruction] For every positive odd integer p and every n∈ N, E_2n=∑_m=0^nbinom2n2mγ_m p^2mS_2(n-m)(p). Equivalently, E_2n=∑_q=0^n V_nqS_2q(p) for the lower triangular matrix V_nq=binom2n2qγ_n-qp^2(n-q) (q≤ n), V_nq=0 (q>n). It has V_nn=1 and p^2(n-q)mid V_nq when q≤ n.
Exact mathematical passage
V_{nq}=\binom{2n}{2q}\gamma_{n-q}p^{2(n-q)}\quad(q\le n), \qquad V_{nq}=0\quad(q>n).
p^{2(r-q)}Use in source · 2 passages
Search aliases: p^2(r-q)
Use in source · supplement_02.tex, line 142
The row action is connected to the matrix entries. Reversing the finite sum and using binomial symmetry gives the actual source-index row action E_2r=∑_q=0^rU_r,q(p)S_2q(p), where U_r,q(p)= displaystylebinom2r2qb_r-qp^2(r-q), · q≤ r, 0, · q>r. The row-action equality, the diagonal identity U_r,r=1, and vanishing above the diagonal are all checked. The arithmetic weight is also explicit: p^2(r-q)mid U_r,q(p) in Z, 0≤ q≤ r. No inverse-coordinate identity, unit diagonal or weight bound is assumed f…
Exact mathematical passage
U_{r,q}(p)= \begin{cases} \displaystyle\binom{2r}{2q}b_{r-q}p^{2(r-q)},&q\le r,\\ 0,&q>r. \end{cases}
p^{2a(M+1)}Use in source · 1 passage
Search aliases: p^2a(M+1)
Use in source · 72_recovered.tex, line 1020
[Odd modulus truncation] For every M≥0, every positive odd q, and every even n, E_n ≡ ∑_r=0^min(M,n/2) (-1)^rZ_2r+1 binom n2rq^2rS_n-2r(q) pmod q^2M+2. If pmid q and a=v_p(q), the same truncation is valid modulo p^2a(M+1).
Exact mathematical passage
p^{2a(M+1)}
p^{2D}Use in source · 2 passages
Search aliases: p^2D
Use in source · 30_lifting.tex, line 798
For the error estimate, let δ=b-a and D=M-s. Taylor's theorem for the preconditioned polynomial gives E_F=(AF)(b)-(AF)(a)-p^sδ∈ p^2DR^d. Polynomial congruence gives E_H=(AH)(b)-(AH)(a)∈ p^DR^d. The equations F(a)=0 and G(b)=0 imply p^sδ+p^MAH(a)=-E_F-p^ME_H∈ p^2DR^d, because M+D≥2D. Cancel p^s in the integral domain R to obtain δ+p^M-sAH(a)∈ p^2M-3sR^d. This integral vector is ε. After inclusion into Q_p, p^M-sA=p^MJ^-1 gives [source reference].
Exact mathematical passage
E_F=(AF)(b)-(AF)(a)-p^s\delta\in p^{2D}R^d.
p^{2L}Use in source · 4 passages
Search aliases: p^2L
Use in source · 71_native.tex, line 41
Indeed, each term involving at least two correction factors is divisible by p^2L, and 2L≥ L+1. Constant and linear correction terms give the two displayed summands. Adding the monomial expansions proves (1) for every polynomial, with no denominator or analytic convergence issue.
Exact mathematical passage
p^{2L}
p^{2M+2}Use in source · 6 passages
Search aliases: p^2M+2
Use in source · 10_foundations.tex, line 237
begingroupsmallraggedright Formal correspondence. leanFiltered.filtered_triangular_reconstruction and leanFiltered.padic_filtered_reconstruction. For weights 2r, retaining r≤ M gives the bound p^2M+2 through leanFiltered.even_weight_tail. A sharper kernel weight can improve this bound, as in the classical dyadic reconstruction later in the chapter. Each coefficient row is finite, so no hypothesis w(r)→∞ is needed here. endgroup
Exact mathematical passage
p^{2M+2}
p^{2M-3s}Use in source · 9 passages
Search aliases: p^2M-3s
Use in source · 30_lifting.tex, line 767
[Weighted coefficient deformation] Let F,H∈ R[X_1,…,X_d]^d, F(a)=0, and J=DF(a). Suppose J∈GL_d( Q_p), p^sJ^-1∈ M_d(R), and M>2s. For G=F+p^MH, there is exactly one root b∈ a+p^s+1R^d. It satisfies b-a∈ p^M-sR^d and, in Q_p^d, b-a=-p^MJ^-1H(a)+ε, ε∈ p^2M-3sR^d. Only the original Jacobian DF(a) is subject to an inverse hypothesis.
Exact mathematical passage
\label{eq:coefficient-first-order} b-a=-p^MJ^{-1}H(a)+\varepsilon, \qquad\varepsilon\in p^{2M-3s}R^d.
p^{2r}Use in source · 5 passages
Search aliases: p^2r
Use in source · 72_recovered.tex, line 1166
0, · p=2 and n odd, and κ_p,r = (-1)^rZ_2r+1p^2r, · p odd, β_r2^2r(2r)!, · p=2.
Exact mathematical passage
\kappa_{p,r} = \begin{cases} (-1)^rZ_{2r+1}p^{2r},&p\text{ odd},\\ \beta_r2^{2r}(2r)!,&p=2. \end{cases}
p^{2s+1}Use in source · 11 passages
Search aliases: p^2s+1
Use in source · 30_lifting.tex, line 711
Precondition the polynomial system by A and set c=p^s+1. Its derivative at a is AJ=p^sI_d. Lemma~[source reference] gives a polynomial vector Q with Q(0)=0 and (AF)(a+cu)=(AF)(a)+p^2s+1u+p^2s+2Q(u). Since A is integral, (AF)(a)∈ p^rR^d. The strict integer inequality r>2s permits k=r-(2s+1)≥0 and a vector C∈ p^kR^d such that (AF)(a)=p^2s+1C. The normalized root equation is therefore u=-C-pQ(u).
Exact mathematical passage
(AF)(a+cu)=(AF)(a)+p^{2s+1}u+p^{2s+2}Q(u).
p^{2s+2}Use in source · 4 passages
Search aliases: p^2s+2
Use in source · 30_lifting.tex, line 711
Precondition the polynomial system by A and set c=p^s+1. Its derivative at a is AJ=p^sI_d. Lemma~[source reference] gives a polynomial vector Q with Q(0)=0 and (AF)(a+cu)=(AF)(a)+p^2s+1u+p^2s+2Q(u). Since A is integral, (AF)(a)∈ p^rR^d. The strict integer inequality r>2s permits k=r-(2s+1)≥0 and a vector C∈ p^kR^d such that (AF)(a)=p^2s+1C. The normalized root equation is therefore u=-C-pQ(u).
Exact mathematical passage
(AF)(a+cu)=(AF)(a)+p^{2s+1}u+p^{2s+2}Q(u).
p^{[r]}Use in source · 1 passage
Search aliases: p^[r]
Use in source · 10_foundations.tex, line 209
To isolate the filtration argument, let p be any element of a commutative ring and let w: N→ N∪{∞}. Set p^[r]=p^r for finite r and p^[∞]=0. The latter notation means the coefficient vanishes.
Exact mathematical passage
p^{[r]}=p^r
p^{[w(r)]}Use in source · 2 passages
Search aliases: p^[w(r)]
Use in source · 10_foundations.tex, line 215
[Filtered finite reconstruction] For every n,L∈ N and sequences c,S in that ring, · ∑_r=0^nbinom nr c_rp^[w(r)]S_n-r · ≡ ∑_substack0≤ r≤ n w(r)<L binom nr c_rp^[w(r)]S_n-rpmodp^L. Here congruence means divisibility of the difference by p^L in the coefficient ring. In Z_p, it means membership in p^L Z_p.
Exact mathematical passage
\begin{split} &\sum_{r=0}^{n}\binom nr c_rp^{[w(r)]}S_{n-r}\\ &\hspace{8mm}\equiv \sum_{\substack{0\leq r\leq n\\w(r)<L}} \binom nr c_rp^{[w(r)]}S_{n-r}\pmod{p^L}. \end{split}
p^{[w]}Use in source · 1 passage
Search aliases: p^[w]
Use in source · supplement_01.tex, line 131
ach earlier clearing denominator to the common product, then cancels the new nonzero diagonal. This is a denominator bound, not a claim that every normalized coefficient is integral. pageheading3. Filtered tails and the standard windowSource items 2.2 and 8.2 Filtration at a finite coefficient row. Let R be any commutative ring. For w∈ N∪{∞} define p^[w]=p^w, · w<∞, 0, · w=∞. The infinity convention expresses a vanishing coefficient, not an infinite exponent operation. For any finite index set, …
Exact mathematical passage
p^{[w]}=\begin{cases}p^w,&w<\infty,\\0,&w=\infty.\end{cases}
p^{[\infty]}Use in source · 1 passage
Search aliases: p^[∞] infty
Use in source · 10_foundations.tex, line 210
To isolate the filtration argument, let p be any element of a commutative ring and let w: N→ N∪{∞}. Set p^[r]=p^r for finite r and p^[∞]=0. The latter notation means the coefficient vanishes.
Exact mathematical passage
p^{[\infty]}=0
P^aUse in source · 1 passage
Search aliases: P^a
Use in source · odd_capacity.tex, line 72
Put P=rad(A), U^a(A)=1+P^a for atextgreater=1, and m=N/2.
Exact mathematical passage
U^a(A)=1+P^a
p^{a+b}Use in source · 1 passage
Search aliases: p^a+b
Use in source · 72_recovered.tex, line 2127
The cosets s+p^a Z_p form a disjoint partition of Z_p. The first formula is the inverse affine change of variables on one coset. The second follows from p^a(p^bx+t)+s = p^a+bx+p^at+s.
Exact mathematical passage
p^a(p^bx+t)+s = p^{a+b}x+p^at+s.
P^a_iUse in source · 4 passages
Search aliases: P^a_i
Use in source · 01_finite_coordinates.tex, line 434
The graph model above resolves a fixed scalar matrix. Coordinate transport is a separate construction. Let X be an R-module, and let P^a=(P^a_i)_i∈ I_a be a finite complete family of pairwise orthogonal linear idempotents: P^a_iP^a_j=δ_ijP^a_i, ∑_i∈ I_aP^a_i=I_X. Define the exact packet coordinate module Coord( P^a) = {(x_i)_i∈ I_a:P^a_ix_i=x_i}. Its analysis and synthesis maps are A_a(x)=(P^a_ix)_i, S_a((x_i)_i)=∑_ix_i. Completeness and orthogonality give S_a A_a=id_X, A_a S_a=id_Coord( P^a).
Exact mathematical passage
\cP^a=(P^a_i)_{i\in I_a}
P^a_jUse in source · 1 passage
Search aliases: P^a_j
Use in source · 01_finite_coordinates.tex, line 438
The graph model above resolves a fixed scalar matrix. Coordinate transport is a separate construction. Let X be an R-module, and let P^a=(P^a_i)_i∈ I_a be a finite complete family of pairwise orthogonal linear idempotents: P^a_iP^a_j=δ_ijP^a_i, ∑_i∈ I_aP^a_i=I_X. Define the exact packet coordinate module Coord( P^a) = {(x_i)_i∈ I_a:P^a_ix_i=x_i}. Its analysis and synthesis maps are A_a(x)=(P^a_ix)_i, S_a((x_i)_i)=∑_ix_i. Completeness and orthogonality give S_a A_a=id_X, A_a S_a=id_Coord( P^a).
Exact mathematical passage
P^a_iP^a_j=\delta_{ij}P^a_i, \qquad \sum_{i\in I_a}P^a_i=I_X.
P^bUse in source · 1 passage
Search aliases: P^b
Use in source · supplement_09.tex, line 48
ficient in prime row p gives (z-1)^p∈ PI. Every other degree-p product of the two generators of I also belongs to PI. The formal ideal proof therefore establishes I^p⊆ PI. For q=plfloor q/prfloor+(qbmod p), strong induction and the factorial valuation identity give I^q⊆ P^v_p(q!). In detail, write b=lfloor q/prfloor. Then I^q⊆ P^b I^b⊆ P^b+v_p(b!)=P^v_p(q!). No integral-domain hypothesis or division in a field is used here.
Exact mathematical passage
I^q\subseteq P^b I^b\subseteq P^{b+v_p(b!)}=P^{v_p(q!)}.
p^bUse in source · 1 passage
Search aliases: p^b
Use in source · 72_recovered.tex, line 2127
The cosets s+p^a Z_p form a disjoint partition of Z_p. The first formula is the inverse affine change of variables on one coset. The second follows from p^a(p^bx+t)+s = p^a+bx+p^at+s.
Exact mathematical passage
p^a(p^bx+t)+s = p^{a+b}x+p^at+s.
P^{b+v_p(b!)}Use in source · 1 passage
Search aliases: P^b+v_p(b!)
Use in source · supplement_09.tex, line 48
ficient in prime row p gives (z-1)^p∈ PI. Every other degree-p product of the two generators of I also belongs to PI. The formal ideal proof therefore establishes I^p⊆ PI. For q=plfloor q/prfloor+(qbmod p), strong induction and the factorial valuation identity give I^q⊆ P^v_p(q!). In detail, write b=lfloor q/prfloor. Then I^q⊆ P^b I^b⊆ P^b+v_p(b!)=P^v_p(q!). No integral-domain hypothesis or division in a field is used here.
Exact mathematical passage
I^q\subseteq P^b I^b\subseteq P^{b+v_p(b!)}=P^{v_p(q!)}.
p^DUse in source · 1 passage
Search aliases: p^D
Use in source · 30_lifting.tex, line 802
For the error estimate, let δ=b-a and D=M-s. Taylor's theorem for the preconditioned polynomial gives E_F=(AF)(b)-(AF)(a)-p^sδ∈ p^2DR^d. Polynomial congruence gives E_H=(AH)(b)-(AH)(a)∈ p^DR^d. The equations F(a)=0 and G(b)=0 imply p^sδ+p^MAH(a)=-E_F-p^ME_H∈ p^2DR^d, because M+D≥2D. Cancel p^s in the integral domain R to obtain δ+p^M-sAH(a)∈ p^2M-3sR^d. This integral vector is ε. After inclusion into Q_p, p^M-sA=p^MJ^-1 gives [source reference].
Exact mathematical passage
E_H=(AH)(b)-(AH)(a)\in p^DR^d
p^{d-2}Use in source · 1 passage
Search aliases: p^d-2
Use in source · odd_synthesis.tex, line 82
For the inductive field step d≥2, put n=p^d-1q_d and R=p^d-2q_d-1. The derived matrix is the recursively constructed maximal-field parameter, with k_0=-q_1. The inequalities n>pR>2R imply -2R>-n-q_1. Bushnell--Kutzko Remark 2.2.7 therefore applies to the actual displayed matrix. It supplies its maximal field, normalization, and critical exponent. Valuation preservation and literal truncation of the recursion supply the whole defining sequence. The full details, quotient count, and conductor depe…
Exact mathematical passage
R=p^{d-2}q_{d-1}
p^{d-d}Use in source · 2 passages
Search aliases: p^d-d
Use in source · 30_lifting.tex, line 392
For square uniqueness, let a,b∈ B_L(x) be arbitrary exact roots. Their canonical integer truncations are genuine roots in the respective finite residue rings, since evaluation commutes with reduction. They agree at depth L. If they agree at depth N≥ L, both next residues belong to the same lift set, which has cardinality p^d-d=1. Induction gives agreement at every depth. The residue maps separate p-adic integers, so a=b. Existence was already supplied by the tower.
Exact mathematical passage
p^{d-d}=1
p^{d-e}Use in source · 3 passages
Search aliases: p^d-e
Use in source · 30_lifting.tex, line 372
[Completion and smooth branching] In Theorem~[source reference], let q=p be prime. The tower can be chosen with a limit a∈ Z_p^d such that F(a)=0, z_n⟶ a, a-z_n∈ p^L+n Z_p^d (n≥0). At each stage, the number of distinct next-level residue roots over z_n is exactly p^d-e, and the finite obstruction module has one element. If d=e, there is exactly one root of F in B_L(x).
Exact mathematical passage
p^{d-e}
p^{d-j}Use in source · 2 passages
Search aliases: p^d-j
Use in source · Paper_XI_Traditional_Final.tex, line 2024
[Restricted odd-prime arithmetic realization] Let p be an odd prime, F= F_p((t)), fix d≥1, and let 0<h_1<⋯<h_d, pnmid h_i, q_i=(p+1)h_i, N=p^d. There is an explicit maximal simple stratum in M_N(F) with totally ramified full-degree parameter field, a literal defining sequence of degrees p,p^2,…,p^d, and levels n_j=p^d-jq_d-j+1. Every native parameter and consecutive correction has trace zero over its recursive base field. The stratum admits a full simple character trivial on U_F^1. Its fixed can…
Exact mathematical passage
n_j=p^{d-j}q_{d-j+1}
p^{d-\operatorname{rank}J}Use in source · 1 passage
Search aliases: p^d-rankJ operatorname
Use in source · 71_native.tex, line 33
Theorem. For this fixed polynomial presentation, the group and distinguished class are independent of the chosen integer representative. The class vanishes if and only if bar x lifts to a solution modulo p^L+1. The obstruction group has p^r-rankJ elements. When lifts exist, their set is a torsor for ker J and contains exactly p^d-rankJ elements.
Exact mathematical passage
p^{d-\operatorname{rank}J}
p^{d-\rho}Use in source · 1 passage
Search aliases: p^d-ρ rho
Use in source · 72_recovered.tex, line 2589
[Exact number of linear children] If ρ=rank_ F_poverline M, then a solvable depth L class has exactly p^d-ρ children modulo p^L+1.
Exact mathematical passage
\boxed{ p^{d-\rho} }
p^{d-\rho_x}Use in source · 2 passages
Search aliases: p^d-ρ_x rho
Use in source · 30_lifting.tex, line 295
[Torsor and lift cardinalities] Under the hypotheses of Theorem~[source reference], there is an equivalence T_x≃ L_x,L. Every nonempty fiber is a torsor for ker J_x under addition of correction digits. If q=p is prime and ρ_x=rank_ F_pJ_x, then # L_x,L= p^d-ρ_x, · o_x=0, 0, · o_x≠0, # O_x=p^e-ρ_x.
Exact mathematical passage
\label{eq:lift-count} \#\mathcal L_{x,L}= \begin{cases}p^{d-\rho_x},&o_x=0,\\0,&o_x\ne0,\end{cases} \qquad \#\mathcal O_x=p^{e-\rho_x}.
p^{e-\rho}Use in source · 1 passage
Search aliases: p^e-ρ rho
Use in source · continuation_08.tex, line 62
When q=p is prime, put ρ=rank_ F_pJ. Then # L_v= p^m-ρ, · o_x(v)=0, 0, · o_x(v)≠0, # C_x=p^e-ρ. The count is of distinct actual residue classes at level L+1. The solved dimension m, rather than the total number m+k of variables, appears in the exponent because the parameter digit is fixed. These rank formulas are asserted only over prime fields.
Exact mathematical passage
\boxed{\#\mathcal L_v= \begin{cases} p^{m-\rho},&o_x(v)=0,\\ 0,&o_x(v)\ne0, \end{cases}} \qquad \boxed{\#\mathcal C_x=p^{e-\rho}.}
p^{e-\rho_x}Use in source · 3 passages
Search aliases: p^e-ρ_x rho
Use in source · 30_lifting.tex, line 295
[Torsor and lift cardinalities] Under the hypotheses of Theorem~[source reference], there is an equivalence T_x≃ L_x,L. Every nonempty fiber is a torsor for ker J_x under addition of correction digits. If q=p is prime and ρ_x=rank_ F_pJ_x, then # L_x,L= p^d-ρ_x, · o_x=0, 0, · o_x≠0, # O_x=p^e-ρ_x.
Exact mathematical passage
\label{eq:lift-count} \#\mathcal L_{x,L}= \begin{cases}p^{d-\rho_x},&o_x=0,\\0,&o_x\ne0,\end{cases} \qquad \#\mathcal O_x=p^{e-\rho_x}.
p^{e_p}Use in source · 1 passage
Search aliases: p^e_p
Use in source · 02_foundations.tex, line 920
Let the minimal polynomial be m_T(t)=∏_pp(t)^e_p. The acting algebra is 𝕜[T]≅𝕜[t]/(m_T) ≅∏_pR_p, R_p:=𝕜[t]/(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with maximal ideal generated by the class of p and residue division field K_p=𝕜[t]/(p). By [source reference], srank_R_p(X_p) = dim_K_pX_p/p(T)X_p = μ_p(T). …
Exact mathematical passage
\bbk[T]\cong\bbk[t]/(m_T) \cong\prod_pR_p, \qquad R_p:=\bbk[t]/(p^{e_p}),
p^iUse in source · 1 passage
Search aliases: p^i
Use in source · strengthening.tex, line 209
To retain the valuation of a step, let B be any sequence satisfying Δ_1^m B∈ p^m Z_p at every order and row. Use the finite operator identity Δ_H=∑_i=1^Hbinom HiΔ_1^i. The relation ibinom Hi=HbinomH-1i-1 implies v_p(p^ibinom Hi) ≥ i+v_p(H)-v_p(i)≥1+v_p(H). Multiplying these identities for the k steps, every term of total order M has its scalar multiplied by p^M, and is divisible by p^k+∑_ν v_p(H_ν). This proves the mixed-step bound. Finally, Δ_H A_N,j(n)=(-1)^n+HT_H C_N,j(n). Iteration contribut…
Exact mathematical passage
v_p\left(p^i\binom Hi\right) \geq i+v_p(H)-v_p(i)\geq1+v_p(H).
p^jUse in source · 1 passage
Search aliases: p^j
Use in source · odd_synthesis.tex, line 93
The full-character argument uses a specific order of operations. First construct a full character at native rank p^j, trivial on U_F^1. Transfer its full beta-determinant unit character to multiplicity p, using BK 3.6.13(ii) and 3.6.14. On a scalar u∈ U_F^1, this gives θ_j(u^p)=1. Next add the actual trace-zero correction, using BK 3.3.18. It remains trivial on the scalar intersection of its domain. Finally extend at the next maximal native parameter field: the forced deep character and the triv…
Exact mathematical passage
p^j
p^kUse in source · 15 passages
Search aliases: p^k
Use in source · 20_arithmetic.tex, line 338
Here is the coefficient extraction explicitly. If a_d=d![t^d]F, then d![t^d]B_p^kF=p^-kΔ_p(p-1)^ka_d+pk. For F=j! P_N,j and k≤ mn, take d=mpn-pk. Support in multiples of mp turns this into p^-kΔ_h^kU_N,j(n), so flatness gives divisibility by p^k. For k>mn, expand Δ_h^k-mn applied to Δ_h^mnU_N,j(n). Every shifted row is at least n, so each summand is divisible by p^mn. Finally Δ_uh=((I+Δ_h)^u-I), whose kth power for u>0 is an integral combination of powers of Δ_h of order at least k. The case u=0…
Exact mathematical passage
p^k
p^{k(1+v_p(u))}Use in source · 1 passage
Search aliases: p^k(1+v_p(u))
Use in source · 20_arithmetic.tex, line 557
To incorporate v_2(u), expand a displacement using finite Newton coefficients. For any prime p, u>0, and 1≤ a≤ u, p^v_p(u)+1midbinom ua p^a, as follows from abinom ua=ubinomu-1a-1 and v_p(a)<a. Repeated displacement, retaining the remaining base-difference order at each induction step, gives a factor p^k(1+v_p(u)) from k displacements. This induction is formalized in leanKomorebiPaperIX.StepAmplification.repeated_precision. Applying it with p=2 proves [source reference]. Multiplication by j! pro…
Exact mathematical passage
p^{k(1+v_p(u))}
p^{k+\sum_\nu v_p(H_\nu)}Use in source · 1 passage
Search aliases: p^k+∑_ν v_p(H_ν) sum nu nu
Use in source · strengthening.tex, line 215
ng Δ_1^m B∈ p^m Z_p at every order and row. Use the finite operator identity Δ_H=∑_i=1^Hbinom HiΔ_1^i. The relation ibinom Hi=HbinomH-1i-1 implies v_p(p^ibinom Hi) ≥ i+v_p(H)-v_p(i)≥1+v_p(H). Multiplying these identities for the k steps, every term of total order M has its scalar multiplied by p^M, and is divisible by p^k+∑_ν v_p(H_ν). This proves the mixed-step bound. Finally, Δ_H A_N,j(n)=(-1)^n+HT_H C_N,j(n). Iteration contributes only a global sign. Multiplying by j! proves [source reference…
Exact mathematical passage
p^{k+\sum_\nu v_p(H_\nu)}
p^{k-1}Use in source · 1 passage
Search aliases: p^k-1
Use in source · supplement_05.tex, line 145
c branch For 1≤ k≤ℓ, let ρ_ℓ,k: Z/p^ℓ Z⟶ Z/p^k Z be the actual reduction homomorphism. It induces a map on units. The explicit construction satisfies ρ_ℓ,k(ω_ℓ(a)) =ω_k(ρ_ℓ,k(a)), ρ_ℓ,k(⟨ a⟩_ℓ) =⟨ρ_ℓ,k(a)⟩_k. The proof uses the exponent factorization p^ℓ-1=p^k-1p^ℓ-k and the already proved Frobenius invariance of a lifted root. Principal factors also satisfy ⟨ a⟩_ℓ^ p^ℓ-1=1. At precision one the principal factor is exactly one. These are coherent finite certificates. No infinite inverse limit or…
Exact mathematical passage
p^{\ell-1}=p^{k-1}p^{\ell-k}
p^{ks}Use in source · 1 passage
Search aliases: p^ks
Use in source · supplement_04.tex, line 120
-field reductionTheorem 7.4; manuscript page 18 For u in any commutative ring, and every natural step h, iteration proves Δ_h^s(u^n)=u^n(u^h-1)^s. The proof unfolds the actual forward-difference operator and uses the power multiplication law. No valuation premise or geometric-difference identity is assumed. More generally, a proved divisibility p^kmid u^h-1 implies p^ksmidΔ_h^s(u^n).
Exact mathematical passage
p^{ks}\mid\Delta_h^s(u^n).
p^{L(R+1-d)}Use in source · 1 passage
Search aliases: p^L(R+1-d)
Use in source · 72_recovered.tex, line 1657
f degree d. Let H=Λ_p,L, fix an address j, and write Q_j(x)=P(Hx+j), y=u^H, q=y-1. Then f(Hx+j)=∑_r≥0c_j,rbinom xr, where c_j,r = u^j ∑_k=0^min(d,r) binom rk Δ^kQ_j(0)y^kq^r-k. Consequently, v_p(c_j,r)≥ Lmax(r-d,0). For R≥ d, truncation at depth R is valid modulo p^L(R+1-d) uniformly on Z_p.
Exact mathematical passage
p^{L(R+1-d)}
p^{L+1}Use in source · 19 passages
Search aliases: p^L+1
Use in source · 10_foundations.tex, line 363
[Finite unit coordinates] Let p be prime, L≥0, and a∈( Z/p^L+1 Z)^×. Define ω_L(a)=a^p^L, ⟨ a⟩_L=aω_L(a)^-1. Then a=ω_L(a)⟨ a⟩_L, ω_L(a)^p-1=1, ⟨ a⟩_L≡1pmod p. The two factors are uniquely determined by these properties and are compatible with reduction to lower precision. For a∈( Z/2^L+2 Z)^×, there is instead a unique decomposition a=ϵ(a)⟨ a⟩_4, ϵ(a)∈{1,-1}, ⟨ a⟩_4≡1pmod4.
Exact mathematical passage
a\in(\Z/p^{L+1}\Z)^\times
p^{L+a}Use in source · 3 passages
Search aliases: p^L+a
Use in source · 72_recovered.tex, line 2373
r Editorial hypothesis correction. Require L≥1. At L=0 the standalone claim is false: for p=3, d=1 and X=2=1+B, one has X^3-1=7∉3 Z_3. With L≥1, every higher binomial term in the retained proof has the required valuation, including p=2. If X=I+p^LB with B∈ M_d( Z_p), then for every a≥0 there is B_a∈ M_d( Z_p) such that X^p^a=I+p^L+aB_a.
Exact mathematical passage
\boxed{X^{p^a}=I+p^{L+a}B_a.}
p^{L+n}Use in source · 13 passages
Search aliases: p^L+n
Use in source · 30_lifting.tex, line 367
[Completion and smooth branching] In Theorem~[source reference], let q=p be prime. The tower can be chosen with a limit a∈ Z_p^d such that F(a)=0, z_n⟶ a, a-z_n∈ p^L+n Z_p^d (n≥0). At each stage, the number of distinct next-level residue roots over z_n is exactly p^d-e, and the finite obstruction module has one element. If d=e, there is exactly one root of F in B_L(x).
Exact mathematical passage
F(a)=0,\qquad z_n\longrightarrow a,\qquad a-z_n\in p^{L+n}\Zp^d\quad(n\ge0).
p^{L+n+1}Use in source · 1 passage
Search aliases: p^L+n+1
Use in source · continuation_03.tex, line 100
Surjectivity implies rank_ F_pA_x=e, e≤ d. Combining this with the preceding exact residue-lift count gives # L_z_n,L+n=p^d-e for every n. These are distinct solutions modulo p^L+n+1 lying above the selected point modulo p^L+n. The original finite obstruction group at each smooth point has cardinality one.
Exact mathematical passage
p^{L+n+1}
p^{L-1}Use in source · 1 passage
Search aliases: p^L-1
Use in source · 72_recovered.tex, line 1538
For odd p and L≥1, put Λ_p,L=(p-1)p^L-1. For p=2 and L≥3, put Λ_2,L=2^L-2. These are exponents of the unit groups modulo the declared prime powers.
Exact mathematical passage
\Lambda_{p,L}=(p-1)p^{L-1}.
p^{L_p}Use in source · 3 passages
Search aliases: p^L_p
Use in source · 71_native.tex, line 210
N=∏_p∈ Sp^L_p, Sol_F(R)={x∈ R^d:F(x)=0}.
Exact mathematical passage
N=\prod_{p\in S}p^{L_p},\qquad\operatorname{Sol}_F(R)=\{x\in R^d:F(x)=0\}.
p^{\lfloor d/p\rfloor}Use in source · 5 passages
Search aliases: p^lfloor d/prfloor lfloor rfloor
Use in source · 20_arithmetic.tex, line 231
At the singular addresses c=ap, 1≤ a<m, use the weighted ring W_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping the falling factorial (x)_d=∏_i=0^d-1(x-i) into consecutive blocks of length p then gives an integral residue-coordinate factor p^v_p(lfloor d/prfloor!)…
Exact mathematical passage
\mathcal W_p=\left\{\sum_{d\ge0}a_dT^d: p^{\lfloor d/p\rfloor}a_d\in\Z_p\right\}.
p^{Lr}Use in source · 2 passages
Search aliases: p^Lr
Use in source · 72_recovered.tex, line 1681
means the matrix-binomial interpolation. With p=3, C=4, L=1, w=1 and v_0=1/3, one has Δ_1a_0=1, which is not in 3 Z_3. The printed claim without these restrictions is therefore not a certificate as written. The full original proof follows under the corrected hypotheses. Assume C^H=I+p^LB with B∈ M_d( Z_p). For every address j and every x∈ Z_p, a_Hx+j = ∑_r≥0 p^Lrbinom xr w^TC^jB^rv_0. The series converges on Z_p, and Δ_H^ra_n ≡0 pmod p^Lr.
Exact mathematical passage
\boxed{ a_{Hx+j} = \sum_{r\geq0} p^{Lr}\binom xr w^TC^jB^rv_0. }
P^mUse in source · 3 passages
Search aliases: P^m
Use in source · strengthening.tex, line 306
at moments of polynomials in separate variables factor. Odd moments vanish. Frobenius's congruence is v_2(∑_r=0^m(-1)^m-rbinom mr Euler_2n+2r) =m+v_2(m!) (n,m≥0). For m=0, the assertion is that every even Euler number is odd. With P=( E^2-1)/2 and γ_m(P)=P^m/m!, the case n=0 says L(γ_m(P))∈ Z_2^× (m≥0). Introduce an independent Euler variable F and Q=( F^2-1)/2.
Exact mathematical passage
\gamma_m(P)=P^m/m!
p^MUse in source · 31 passages
Search aliases: p^M
Use in source · 30_lifting.tex, line 418
[Smooth coefficient stability] Let p be prime, M≥1, F,H∈ Z[X_1,…,X_d]^e, and c∈ p^M Z. Suppose DF(x)bmod p is surjective for an integer x∈ Z^d. If a∈ Z_p^d satisfies F(a)=0 and a≡ xpmod p, then there is a root b of G=F+cH with b-a∈ p^M Z_p^d.
Exact mathematical passage
c\in p^M\Z
p^mUse in source · 6 passages
Search aliases: p^m
Use in source · strengthening.tex, line 203
To retain the valuation of a step, let B be any sequence satisfying Δ_1^m B∈ p^m Z_p at every order and row. Use the finite operator identity Δ_H=∑_i=1^Hbinom HiΔ_1^i. The relation ibinom Hi=HbinomH-1i-1 implies v_p(p^ibinom Hi) ≥ i+v_p(H)-v_p(i)≥1+v_p(H). Multiplying these identities for the k steps, every term of total order M has its scalar multiplied by p^M, and is divisible by p^k+∑_ν v_p(H_ν). This proves the mixed-step bound. Finally, Δ_H A_N,j(n)=(-1)^n+HT_H C_N,j(n). Iteration contribut…
Exact mathematical passage
\Delta_1^m B\in p^m\mathbb Z_p
p^{M-(2s+1)}Use in source · 1 passage
Search aliases: p^M-(2s+1)
Use in source · 30_lifting.tex, line 787
Retain the original certificate A=p^sJ^-1 and put c=p^s+1. Taylor expansion and a polynomial difference identity yield polynomial vectors Q_F,Q_H, both zero at the origin, such that (AF)(a+cu)=p^2s+1u+p^2s+2Q_F(u), (AH)(a+cu)=AH(a)+cQ_H(u). Consequently (AG)(a+cu)=p^2s+1(C+u+pQ(u)), where C=p^M-(2s+1)AH(a), Q=Q_F+p^M-s-1Q_H, Q(0)=0. All exponents are nonnegative. The same contraction and divisibility argument as in Theorem~[source reference] gives the root, the displacement p^M-s, and full-ball …
Exact mathematical passage
C=p^{M-(2s+1)}AH(a),\qquad Q=Q_F+p^{M-s-1}Q_H,\qquad Q(0)=0.
p^{m-1}Use in source · 2 passages
Search aliases: p^m-1
Use in source · Paper_X_Block27_Amended.tex, line 1973
The residual module partitions coordinates modulo p, verifies compatibility x_m+1≡ x_mpmodp^m, and reconstructs the digit a_m-1 by a_m-1=lfloorfracx_mp^m-1rfloorpmod p.
Exact mathematical passage
a_{m-1}=\left\lfloor\frac{x_m}{p^{m-1}}\right\rfloor\pmod p.
p^{m-e}Use in source · 4 passages
Search aliases: p^m-e
Use in source · 30_lifting.tex, line 636
Theorem~[source reference]. For every τ∈ t_0+p^L Z_p^k, there are an integer tower z_n and b∈ Z_p^m+k such that F(b)=0, free(b)=τ, z_n⟶ b, b-z_n∈ p^L+n Z_p^m+k. The initial value is z_0=x. At every stage and for every proposed parameter digit v∈ F_p^k, the corresponding next residue fiber has exactly p^m-e elements. If m=e, then for each τ there is exactly one such root in B_L(x), independently of all choices of integer approximants and correcting digits.
Exact mathematical passage
p^{m-e}
p^{M-s}Use in source · 15 passages
Search aliases: p^M-s
Use in source · 30_lifting.tex, line 766
[Weighted coefficient deformation] Let F,H∈ R[X_1,…,X_d]^d, F(a)=0, and J=DF(a). Suppose J∈GL_d( Q_p), p^sJ^-1∈ M_d(R), and M>2s. For G=F+p^MH, there is exactly one root b∈ a+p^s+1R^d. It satisfies b-a∈ p^M-sR^d and, in Q_p^d, b-a=-p^MJ^-1H(a)+ε, ε∈ p^2M-3sR^d. Only the original Jacobian DF(a) is subject to an inverse hypothesis.
Exact mathematical passage
b-a\in p^{M-s}R^d
p^{M-s-1}Use in source · 2 passages
Search aliases: p^M-s-1
Use in source · 30_lifting.tex, line 787
Retain the original certificate A=p^sJ^-1 and put c=p^s+1. Taylor expansion and a polynomial difference identity yield polynomial vectors Q_F,Q_H, both zero at the origin, such that (AF)(a+cu)=p^2s+1u+p^2s+2Q_F(u), (AH)(a+cu)=AH(a)+cQ_H(u). Consequently (AG)(a+cu)=p^2s+1(C+u+pQ(u)), where C=p^M-(2s+1)AH(a), Q=Q_F+p^M-s-1Q_H, Q(0)=0. All exponents are nonnegative. The same contraction and divisibility argument as in Theorem~[source reference] gives the root, the displacement p^M-s, and full-ball …
Exact mathematical passage
C=p^{M-(2s+1)}AH(a),\qquad Q=Q_F+p^{M-s-1}Q_H,\qquad Q(0)=0.
p^{m-\rho}Use in source · 1 passage
Search aliases: p^m-ρ rho
Use in source · continuation_08.tex, line 62
When q=p is prime, put ρ=rank_ F_pJ. Then # L_v= p^m-ρ, · o_x(v)=0, 0, · o_x(v)≠0, # C_x=p^e-ρ. The count is of distinct actual residue classes at level L+1. The solved dimension m, rather than the total number m+k of variables, appears in the exponent because the parameter digit is fixed. These rank formulas are asserted only over prime fields.
Exact mathematical passage
\boxed{\#\mathcal L_v= \begin{cases} p^{m-\rho},&o_x(v)=0,\\ 0,&o_x(v)\ne0, \end{cases}} \qquad \boxed{\#\mathcal C_x=p^{e-\rho}.}
p^{m-\rho_x}Use in source · 1 passage
Search aliases: p^m-ρ_x rho
Use in source · 30_lifting.tex, line 556
[Prescribed-parameter obstruction and count] Define o_x(v)=[overliner_x+K_xv]∈ S_q^e/im J_x. Then L_x,L,v is nonempty if and only if o_x(v)=0. When nonempty, it is a torsor for ker J_x. If q=p is prime and ρ_x=rank_ F_pJ_x, then # L_x,L,v =p^m-ρ_x, · o_x(v)=0, 0, · o_x(v)≠0, #( F_p^e/im J_x)=p^e-ρ_x. The criterion holds uniformly for every prescribed v without a surjectivity hypothesis.
Exact mathematical passage
\label{eq:prescribed-count} \#\mathcal L_{x,L,v} =\begin{cases}p^{m-\rho_x},&o_x(v)=0,\\0,&o_x(v)\ne0,\end{cases} \qquad \#(\F_p^e/\im J_x)=p^{e-\rho_x}.
p^{\min(k(1+\nu_p(u)),mn)}Use in source · 1 passage
Search aliases: p^min(k(1+ν_p(u)),mn) min nu
Use in source · 20_arithmetic.tex, line 571
[Step-sensitive divisor-branch bound] Under the hypotheses of Theorem~[source reference], for all u,k,n∈ N, Δ_u(p-1)/m^kU_mp,j(n) ∈ p^min(k(1+ν_p(u)),mn) Z_p. For C_mp,j the corresponding valuation lower bound is min(k(1+ν_p(u)),mn)-v_p(j!).
Exact mathematical passage
\Delta_{u(p-1)/m}^{k}U_{mp,j}(n) \in p^{\min(k(1+\nu_p(u)),mn)}\Z_p.
p^{\min(k,r)}Use in source · 2 passages
Search aliases: p^min(k,r) min
Use in source · 20_arithmetic.tex, line 416
[Prime-power period from finite differences] Let R be a commutative ring, p a prime, f: N→ R, h,n∈ N, and r≥1. If p^min(k,r)midΔ_h^k f(n) for every k∈ N, then p^rmid f(n+p^r-1h)-f(n).
Exact mathematical passage
p^{\min(k,r)}\mid\Delta_h^k f(n)\quad\text{for every }k\in\N,
p^{\min(s,L)}Use in source · 3 passages
Search aliases: p^min(s,L) min
Use in source · 04_arithmetic.tex, line 856
Lean proves the one-coordinate arithmetic directly over Z/p^L Z for every p>1: (∃ x∈ Z/p^L Z: p^s x=bpmodp^L) Longleftrightarrow p^min(s,L)mid b_val, where b_val is the canonical natural representative, and every admissible fiber has exactly p^min(s,L) elements. The coordinates are independent, so the product equivalence gives the displayed global count. If one coordinate fiber is empty, the global fiber is empty. Lean proves both the global solvability equivalence and the conditional count, inc…
Exact mathematical passage
\bigl(\exists x\in\mathbb Z/p^L\mathbb Z:\quad p^s x=b\pmod{p^L}\bigr) \quad\Longleftrightarrow\quad p^{\min(s,L)}\mid b_{\mathrm{val}},
p^{\min(s_i,L)}Use in source · 1 passage
Search aliases: p^min(s_i,L) min
Use in source · 04_arithmetic.tex, line 838
[Concrete Smith-weighted congruence solve] Let p>1 and consider the diagonal system over ( Z/(p^L) Z)^d diag(p^s_1,…,p^s_d)x=b pmodp^L. The system is solvable if and only if p^min(s_i,L)mid (b_i)_val for every i, where (b_i)_val is the canonical natural representative. Conditional on this admissibility condition, the number of solutions is p^∑_i=1^dmin(s_i,L). If any coordinate is inadmissible, the solution set is empty and its cardinality is zero.
Exact mathematical passage
p^{\min(s_i,L)}\mid (b_i)_{\mathrm{val}}
p^{mn}Use in source · 1 passage
Search aliases: p^mn
Use in source · 20_arithmetic.tex, line 341
ly. If a_d=d![t^d]F, then d![t^d]B_p^kF=p^-kΔ_p(p-1)^ka_d+pk. For F=j! P_N,j and k≤ mn, take d=mpn-pk. Support in multiples of mp turns this into p^-kΔ_h^kU_N,j(n), so flatness gives divisibility by p^k. For k>mn, expand Δ_h^k-mn applied to Δ_h^mnU_N,j(n). Every shifted row is at least n, so each summand is divisible by p^mn. Finally Δ_uh=((I+Δ_h)^u-I), whose kth power for u>0 is an integral combination of powers of Δ_h of order at least k. The case u=0 gives zero for k>0, and order zero is the …
Exact mathematical passage
p^{mn}
p^NUse in source · 8 passages
Search aliases: p^N
Use in source · 30_lifting.tex, line 463
t, writing σ x=(y_0,t_0) with e solved coordinates and k parameter coordinates, the parameter ball T=t_0+p^L Z_p^k is homeomorphic to the entire root locus R={z∈ B_L(x):F(z)=0}. The inverse homeomorphism is the actual free-coordinate projection z↦free(σ z). If h:T→ R is the forward map, then for every N≥0, h(t)≡ h(u)pmodp^N Longleftrightarrow t≡ upmodp^N. Equivalently, σ h(t)=(ϕ(t),t) for a continuous ϕ:T→ y_0+p^L Z_p^e whose graph is the entire reordered root locus.
Exact mathematical passage
\label{eq:chart-congruence} h(t)\equiv h(u)\pmod{p^N} \quad\Longleftrightarrow\quad t\equiv u\pmod{p^N}.
p^nUse in source · 9 passages
Search aliases: p^n
Use in source · 30_lifting.tex, line 315
For prime p, rank-nullity gives dimker J_x=d-ρ_x and dim( F_p^e/im J_x)=e-ρ_x. The cardinality of an n-dimensional F_p-space is p^n. Combine these identities with the obstruction criterion and the residue equivalence.
Exact mathematical passage
p^n
p^{N+1}Use in source · 3 passages
Search aliases: p^N+1
Use in source · 30_lifting.tex, line 475
First consider a square integer polynomial map P whose reduced Jacobian at x is surjective. Its Jacobian is invertible over F_p. A digit comparison using Taylor expansion shows that equal outputs modulo p^N+1 force equal inputs modulo p^N+1 whenever the inputs already agree modulo p^N and belong to B_1(x). Induction and integer truncation give, for arbitrary a,b∈ B_1(x), P(a)≡ P(b)pmodp^N Longleftrightarrow a≡ bpmodp^N.
Exact mathematical passage
p^{N+1}
p^{n+1}Use in source · 2 passages
Search aliases: p^n+1
Use in source · 30_lifting.tex, line 730
To obtain the sharper displacement, begin with u∈ R^d. If u∈ p^nR^d, then Q(0)=0 and polynomial congruence give Q(u)∈ p^nR^d. For n<k, both C and pQ(u) are divisible by p^n+1. Equation~[source reference] therefore improves the divisibility of u by one digit. Induction gives u∈ p^kR^d. Thus b=a+p^s+1u satisfies [source reference], because s+1+k=r-s. Its normalized preconditioned value vanishes, and injectivity of A gives F(b)=0.
Exact mathematical passage
p^{n+1}
p^{n_1}Use in source · 1 passage
Search aliases: p^n_1
Use in source · Paper_XII_Traditional_Final.tex, line 495
[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.
Exact mathematical passage
\boxed{ k[\Delta] \cong k[X_1,\ldots,X_r] \big/ (X_1^{p^{n_1}},\ldots,X_r^{p^{n_r}}). }
p^{n_i}Use in source · 3 passages
Search aliases: p^n_i
Use in source · Paper_XII_Traditional_Final.tex, line 490
[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.
Exact mathematical passage
\Delta\cong \prod_{i=1}^{r}C_{p^{n_i}}.
p^{n_r}Use in source · 1 passage
Search aliases: p^n_r
Use in source · Paper_XII_Traditional_Final.tex, line 495
[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.
Exact mathematical passage
\boxed{ k[\Delta] \cong k[X_1,\ldots,X_r] \big/ (X_1^{p^{n_1}},\ldots,X_r^{p^{n_r}}). }
p^pUse in source · 6 passages
Search aliases: p^p
Use in source · Paper_V_Traditional_Final.tex, line 1777
For finite p, equip ⊕_r∈ Rℓ^p(L) with (u_r) ^p=∑_r∈ R u_r _p^p. For p=∞, use the maximum norm.
Exact mathematical passage
\|(u_r)\|^p=\sum_{r\in R}\|u_r\|_p^p.
p^qUse in source · 2 passages
Search aliases: p^q
Use in source · supplement_10.tex, line 69
The scalar input is proved for every natural b, including nonunits: p^2mid b^p^2-b^p in Z. Because Θ(1+T)^b=b(1+T)^b, the binomial expansion of T^p=((1+T)-1)^p then proves stability of the divided generator T^p/p. For d=pq+r, 0≤ r<p, every weighted monomial has the exact factorization a_dT^d=(p^qa_d)(fracT^pp)^qT^r. The scalar p^qa_d is integral by the actual weight condition. Thus all weighted monomials, and all finite weighted polynomials, satisfy the defect condition.
Exact mathematical passage
a_dT^d=(p^qa_d)\left(\frac{T^p}{p}\right)^qT^r.
P^rUse in source · 3 passages
Search aliases: P^r
Use in source · dyadic_tower.tex, line 120
B-bimodularity gives s(Lift_E(C))=C. The lift preserves exact valuation: u_0 (-1) is an E-unit, and expansion in 1,z gives coefficient valuations distinct modulo 2. Each coefficient entry has the same grade in the interleaved L-chain and the E-chain. In particular s(P^r)=Q^r.
Exact mathematical passage
s(P^r)=Q^r
p^rUse in source · 25 passages
Search aliases: p^r
Use in source · edition_summary.tex, line 4
Principal results. addcontentslinetocsectionPrincipal results The original external problem is Nishibuchi's Conjecture 3.9 for generalized Euler numbers. Its full parameter range is resolved. If p is prime, m is positive with mmid p-1 or m=2, 0≤ j<mp, and q=lcm(2,p-1), then the original Euler coefficients are p-adically integral and E_mpn+qp^r^(mp,j)≡ E_mpn^(mp,j)pmodp^r (r≥1, n≥ r). The chapter states the generating function and the conversion between ordinary degree and packet row before using…
Exact mathematical passage
E_{mpn+qp^r}^{(mp,j)}\equiv E_{mpn}^{(mp,j)}\pmod{p^r} \qquad (r\ge1,\ n\ge r).
p^{r-1}Use in source · 10 passages
Search aliases: p^r-1
Use in source · 20_arithmetic.tex, line 419
[Prime-power period from finite differences] Let R be a commutative ring, p a prime, f: N→ R, h,n∈ N, and r≥1. If p^min(k,r)midΔ_h^k f(n) for every k∈ N, then p^rmid f(n+p^r-1h)-f(n).
Exact mathematical passage
p^r\mid f(n+p^{r-1}h)-f(n)
p^{r-\operatorname{rank}J}Use in source · 1 passage
Search aliases: p^r-rankJ operatorname
Use in source · 71_native.tex, line 31
Theorem. For this fixed polynomial presentation, the group and distinguished class are independent of the chosen integer representative. The class vanishes if and only if bar x lifts to a solution modulo p^L+1. The obstruction group has p^r-rankJ elements. When lifts exist, their set is a torsor for ker J and contains exactly p^d-rankJ elements.
Exact mathematical passage
p^{r-\operatorname{rank}J}
p^{r-s}Use in source · 5 passages
Search aliases: p^r-s
Use in source · 30_lifting.tex, line 692
hted nonlinear lifting] Let p be prime, d,s,r∈ N, F∈ R[X_1,…,X_d]^d, and a∈ R^d. Set J=DF(a). Suppose J∈GL_d( Q_p), p^sJ^-1∈ M_d(R), F(a)∈ p^rR^d, r>2s. Then there is exactly one root b of F in a+p^s+1R^d, and it satisfies b-a∈ p^r-sR^d. Uniqueness holds throughout the full ball of depth s+1.
Exact mathematical passage
\label{eq:weighted-displacement} b-a\in p^{r-s}R^d.
p^{r-s+1}Use in source · 2 passages
Search aliases: p^r-s+1
Use in source · 30_lifting.tex, line 757
and displacement boundary] For p=2, F(X)=X^2+3, and a=1, one has F(a)=4 and DF(a)=2. Thus s=1 and r=2 satisfy every displayed hypothesis except the strict inequality. No root exists in Z_2, since none exists modulo eight. Hence r≥2s cannot replace r>2s uniformly. Also, F(X)=p^sX+p^r has root -p^r-s, whose displacement is not divisible by p^r-s+1. The guaranteed displacement cannot uniformly gain a digit.
Exact mathematical passage
p^{r-s+1}
p^sUse in source · 43 passages
Search aliases: p^s
Use in source · 04_arithmetic.tex, line 856
Lean proves the one-coordinate arithmetic directly over Z/p^L Z for every p>1: (∃ x∈ Z/p^L Z: p^s x=bpmodp^L) Longleftrightarrow p^min(s,L)mid b_val, where b_val is the canonical natural representative, and every admissible fiber has exactly p^min(s,L) elements. The coordinates are independent, so the product equivalence gives the displayed global count. If one coordinate fiber is empty, the global fiber is empty. Lean proves both the global solvability equivalence and the conditional count, inc…
Exact mathematical passage
\bigl(\exists x\in\mathbb Z/p^L\mathbb Z:\quad p^s x=b\pmod{p^L}\bigr) \quad\Longleftrightarrow\quad p^{\min(s,L)}\mid b_{\mathrm{val}},
p^{s+1}Use in source · 20 passages
Search aliases: p^s+1
Use in source · 30_lifting.tex, line 690
[Weighted nonlinear lifting] Let p be prime, d,s,r∈ N, F∈ R[X_1,…,X_d]^d, and a∈ R^d. Set J=DF(a). Suppose J∈GL_d( Q_p), p^sJ^-1∈ M_d(R), F(a)∈ p^rR^d, r>2s. Then there is exactly one root b of F in a+p^s+1R^d, and it satisfies b-a∈ p^r-sR^d. Uniqueness holds throughout the full ball of depth s+1.
Exact mathematical passage
a+p^{s+1}R^d
p^{s+1+k}Use in source · 1 passage
Search aliases: p^s+1+k
Use in source · continuation_06.tex, line 59
For completeness, the sharp precision is additional to this fixed-point existence. Inductively, if u∈ p^nR^d, then polynomial congruence and Q(0)=0 imply Q(u)∈ p^nR^d. For n<k, the equation u=-C-pQ(u) therefore gives u∈ p^n+1R^d. Starting at n=0, this proves u∈ p^kR^d. Hence b=a+tu satisfies b-a∈ p^s+1+kR^d=p^r-sR^d. The normalization and injectivity of A give F(b)=0.
Exact mathematical passage
b-a\in p^{s+1+k}R^d=p^{r-s}R^d.
p^{s_1}Use in source · 1 passage
Search aliases: p^s_1
Use in source · 04_arithmetic.tex, line 833
[Concrete Smith-weighted congruence solve] Let p>1 and consider the diagonal system over ( Z/(p^L) Z)^d diag(p^s_1,…,p^s_d)x=b pmodp^L. The system is solvable if and only if p^min(s_i,L)mid (b_i)_val for every i, where (b_i)_val is the canonical natural representative. Conditional on this admissibility condition, the number of solutions is p^∑_i=1^dmin(s_i,L). If any coordinate is inadmissible, the solution set is empty and its cardinality is zero.
Exact mathematical passage
\diag(p^{s_1},\ldots,p^{s_d})x=b \pmod{p^L}.
p^{s_d}Use in source · 1 passage
Search aliases: p^s_d
Use in source · 04_arithmetic.tex, line 833
[Concrete Smith-weighted congruence solve] Let p>1 and consider the diagonal system over ( Z/(p^L) Z)^d diag(p^s_1,…,p^s_d)x=b pmodp^L. The system is solvable if and only if p^min(s_i,L)mid (b_i)_val for every i, where (b_i)_val is the canonical natural representative. Conditional on this admissibility condition, the number of solutions is p^∑_i=1^dmin(s_i,L). If any coordinate is inadmissible, the solution set is empty and its cardinality is zero.
Exact mathematical passage
\diag(p^{s_1},\ldots,p^{s_d})x=b \pmod{p^L}.
p^{s_i}Use in source · 1 passage
Search aliases: p^s_i
Use in source · 04_arithmetic.tex, line 886
If a Smith form is rank deficient, a literal zero diagonal factor requires the corresponding target coordinate to vanish modulo p^L and then contributes all p^L possible coordinate values. In the concrete quotient it can equivalently be represented by any weight s_i≥ L, since p^s_i=0 in Z/(p^L) Z. The theorem then gives exactly that condition and factor. The numbered theorem takes the diagonal data as its input.
Exact mathematical passage
p^{s_i}=0
P^v_iUse in source · 2 passages
Search aliases: P^v_i
Use in source · 02_connections_obstructions.tex, line 859
Suppose every X_v has a finite complete packet family (P^v_i)_i∈ I_v. A packet labeling assigns an admissible packet index to each edge or node of a typed tree. For multilinear Φ_σ, inserting id_X_v=∑_i∈ I_vP^v_i at every input and output yields a finite sum over trees with packet labels. This is the branching analogue of inserting intermediate projectors in [source reference].
Exact mathematical passage
(P^v_i)_{i\in I_v}
p^{v_p(\lfloor d/p\rfloor!)}Use in source · 3 passages
Search aliases: p^v_p(lfloor d/prfloor!) lfloor rfloor
Use in source · 20_arithmetic.tex, line 242
1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping the falling factorial (x)_d=∏_i=0^d-1(x-i) into consecutive blocks of length p then gives an integral residue-coordinate factor p^v_p(lfloor d/prfloor!). The identity v_p(d!)=lfloor d/prfloor+v_p(lfloor d/prfloor!) accounts for precisely the weight in W_p after dividing by d!. Comparison o…
Exact mathematical passage
p^{v_p(\lfloor d/p\rfloor!)}
P^{v_p(q!)}Use in source · 2 passages
Search aliases: P^v_p(q!)
Use in source · supplement_09.tex, line 44
p). Expanding ((z-1)+1)^p=1 and using divisibility of every interior binomial coefficient in prime row p gives (z-1)^p∈ PI. Every other degree-p product of the two generators of I also belongs to PI. The formal ideal proof therefore establishes I^p⊆ PI. For q=plfloor q/prfloor+(qbmod p), strong induction and the factorial valuation identity give I^q⊆ P^v_p(q!). In detail, write b=lfloor q/prfloor. Then I^q⊆ P^b I^b⊆ P^b+v_p(b!)=P^v_p(q!). No integral-domain hypothesis or division in a field is u…
Exact mathematical passage
\boxed{I^q\subseteq P^{v_p(q!)}}.
p^{v_p(q!)}Use in source · 1 passage
Search aliases: p^v_p(q!)
Use in source · 20_arithmetic.tex, line 239
sses c=ap, 1≤ a<m, use the weighted ring W_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping the falling factorial (x)_d=∏_i=0^d-1(x-i) into consecutive blocks of length p then gives an integral residue-coordinate factor p^v_p(lfloor d/prfloor!). The identity v_p(d!…
Exact mathematical passage
I^q\subseteq(p^{v_p(q!)})
p^{v_p(u)+1}Use in source · 1 passage
Search aliases: p^v_p(u)+1
Use in source · 20_arithmetic.tex, line 552
To incorporate v_2(u), expand a displacement using finite Newton coefficients. For any prime p, u>0, and 1≤ a≤ u, p^v_p(u)+1midbinom ua p^a, as follows from abinom ua=ubinomu-1a-1 and v_p(a)<a. Repeated displacement, retaining the remaining base-difference order at each induction step, gives a factor p^k(1+v_p(u)) from k displacements. This induction is formalized in leanKomorebiPaperIX.StepAmplification.repeated_precision. Applying it with p=2 proves [source reference]. Multiplication by j! pro…
Exact mathematical passage
p^{v_p(u)+1}\mid\binom ua p^a,
p^wUse in source · 1 passage
Search aliases: p^w
Use in source · supplement_01.tex, line 131
ach earlier clearing denominator to the common product, then cancels the new nonzero diagonal. This is a denominator bound, not a claim that every normalized coefficient is integral. pageheading3. Filtered tails and the standard windowSource items 2.2 and 8.2 Filtration at a finite coefficient row. Let R be any commutative ring. For w∈ N∪{∞} define p^[w]=p^w, · w<∞, 0, · w=∞. The infinity convention expresses a vanishing coefficient, not an infinite exponent operation. For any finite index set, …
Exact mathematical passage
p^{[w]}=\begin{cases}p^w,&w<\infty,\\0,&w=\infty.\end{cases}
p^{w(r)}Use in source · 2 passages
Search aliases: p^w(r)
Use in source · 72_recovered.tex, line 530
[Filtered triangular reconstruction] Let p be prime. Suppose A_n = ∑_r=0^n binom nr c_rp^w(r)S_n-r, where c_r∈ Z_p, S_n∈ Z_p, and w(r)∈ N. For every precision L≥1, A_n ≡ ∑_substack0≤ r≤ n
Exact mathematical passage
A_n = \sum_{r=0}^{n} \binom nr c_rp^{w(r)}S_{n-r},
p^\ellUse in source · 5 passages
Search aliases: p^ℓ ell
Use in source · supplement_05.tex, line 25
The specification matters. Rational coefficients are not reduced through an imaginary map Q→ F_p. The proof first constructs their integral lifts in Z_p and establishes equality in Q_p. Root representatives are actual units at the requested precision, not unchanged residues that need not satisfy the root equation modulo p^ℓ.
Exact mathematical passage
p^\ell
p^{\ell-1}Use in source · 6 passages
Search aliases: p^ℓ-1 ell
Use in source · supplement_05.tex, line 81
ffPadic_coe, .standard_window_integral targetWindowIndependence.residue_independence, .reduced_unique targetWindowIndependence.two_window_addresses_congruent pageheading2. Explicit finite prime-power decompositionProposition 6.2; odd-prime branch Let p be prime, ℓ≥1, and a∈( Z/p^ℓ Z)^×. Define actual units in the same ring by ω_ℓ(a)=a^p^ℓ-1, ⟨ a⟩_ℓ=a ω_ℓ(a)^-1. The Lean source writes ℓ=L+1, so zero precision is excluded without truncated natural-number subtraction.
Exact mathematical passage
\boxed{\omega_\ell(a)=a^{p^{\ell-1}},\qquad \langle a\rangle_\ell=a\,\omega_\ell(a)^{-1}.}
p^{\ell-k}Use in source · 1 passage
Search aliases: p^ℓ-k ell
Use in source · supplement_05.tex, line 145
c branch For 1≤ k≤ℓ, let ρ_ℓ,k: Z/p^ℓ Z⟶ Z/p^k Z be the actual reduction homomorphism. It induces a map on units. The explicit construction satisfies ρ_ℓ,k(ω_ℓ(a)) =ω_k(ρ_ℓ,k(a)), ρ_ℓ,k(⟨ a⟩_ℓ) =⟨ρ_ℓ,k(a)⟩_k. The proof uses the exponent factorization p^ℓ-1=p^k-1p^ℓ-k and the already proved Frobenius invariance of a lifted root. Principal factors also satisfy ⟨ a⟩_ℓ^ p^ℓ-1=1. At precision one the principal factor is exactly one. These are coherent finite certificates. No infinite inverse limit or…
Exact mathematical passage
p^{\ell-1}=p^{k-1}p^{\ell-k}
p^{\sum_{i=1}^d\min(s_i,L)}Use in source · 1 passage
Search aliases: p^∑_i=1^dmin(s_i,L) sum min
Use in source · 04_arithmetic.tex, line 844
ongruence solve] Let p>1 and consider the diagonal system over ( Z/(p^L) Z)^d diag(p^s_1,…,p^s_d)x=b pmodp^L. The system is solvable if and only if p^min(s_i,L)mid (b_i)_val for every i, where (b_i)_val is the canonical natural representative. Conditional on this admissibility condition, the number of solutions is p^∑_i=1^dmin(s_i,L). If any coordinate is inadmissible, the solution set is empty and its cardinality is zero.
Exact mathematical passage
\boxed{ p^{\sum_{i=1}^d\min(s_i,L)}. }
P_{\mathbf j}^{\mathrm{fine}}Use in source · 1 passage
Search aliases: P_ j^fine mathbf mathrm
Use in source · 01_finite_core.tex, line 239
Fix N≥1. Suppose every factor a has a complete orthogonal family P^(a)_0,…,P^(a)_N-1. The fine tensor packets are P_ j^fine = P^(1)_j_1⊗⋯⊗ P^(m)_j_m.
Exact mathematical passage
P_{\mathbf j}^{\mathrm{fine}} = P^{(1)}_{j_1}\otimes\cdots\otimes P^{(m)}_{j_m}.
P_{\bm j}^{\rm fine}Use in source · 1 passage
Search aliases: P_ j^rm fine bm rm
Use in source · 02_strengthened.tex, line 343
Suppose the ath tensor factor carries a cyclic packet family (P_j^(a))_j∈ Z/N Z. The fine tensor packets are P_ j^rm fine = P_j_1^(1)⊗⋯⊗ P_j_m^(m).
Exact mathematical passage
P_{\bm j}^{\rm fine} = P_{j_1}^{(1)}\otimes\cdots\otimes P_{j_m}^{(m)}.
P_{\mathcal Q}Use in source · 5 passages
Search aliases: P_ Q mathcal
Use in source · Paper_VI_Traditional_Final.tex, line 650
Let Ω={1,…,n} carry strictly positive probabilities π_i, and let Q={B_1,…,B_m} be a partition. Define (P_ Qf)_i = frac∑_j∈ B(i)π_jf_j∑_j∈ B(i)π_j, where B(i) is the block containing i.
Exact mathematical passage
(P_{\mathcal Q}f)_i = \frac{\sum_{j\in B(i)}\pi_jf_j}{\sum_{j\in B(i)}\pi_j},
P_{\mathcal Q}^2Use in source · 1 passage
Search aliases: P_ Q^2 mathcal
Use in source · Paper_VI_Traditional_Final.tex, line 663
[Finite conditional-expectation projector] With the weighted inner product ⟨ f,g⟩_π=∑_iπ_if_ioverlineg_i, the operator P_ Q satisfies P_ Q^2=P_ Q, ⟨ P_ Qf,g⟩_π = ⟨ f,P_ Qg⟩_π. Its range is the space of functions constant on each partition block.
Exact mathematical passage
P_{\mathcal Q}^2=P_{\mathcal Q}, \qquad \langle P_{\mathcal Q}f,g\rangle_\pi = \langle f,P_{\mathcal Q}g\rangle_\pi.
p_0Use in source · 37 passages
Search aliases: p_0
Use in source · visual_atlas.tex, line 363
headThe missing separating operation is now a polynomial Define p_2(t)=frac3t^5-15t^4+20t^316, p_0(t)=1-p_2(t). These polynomials satisfy p_2≡0pmodt^3, p_2≡1pmod(t-2)^3. Consequently p_0(T)=P_e and p_2(T)=P_f. Applying them to s extracts both chain heads, after which polynomial powers generate all six directions. Unlike the model on pagerefV3, the separating maps already belong to the polynomial grammar.
Exact mathematical passage
p_2(t)=\frac{3t^5-15t^4+20t^3}{16},\qquad p_0(t)=1-p_2(t).
P_{000}Use in source · 1 passage
Search aliases: P_000
Use in source · monograph_atlas.tex, line 109
vpage7A coarse packet can still contain coherenceTENSOR REFINEMENT BY TOTAL PARITYChapter VII: Theorem II.3.7, Corollary II.3.8. For three qubits, the eight fine projectors are P_a=ket abra a. Grouping addresses according to the parity of their Hamming weight gives P_rm even=P_000+P_011+P_101+P_110 =fracI+Z⊗ Z⊗ Z2, P_rm odd=I-P_rm even. This is a sum of subspaces, not an averaging of their coordinates. The rank-four even packet retains all complex amplitudes on its four children. vfigparity head…
Exact mathematical passage
P_{\rm even}=P_{000}+P_{011}+P_{101}+P_{110} =\frac{I+Z\otimes Z\otimes Z}{2}, \qquad P_{\rm odd}=I-P_{\rm even}.
P_{011}Use in source · 1 passage
Search aliases: P_011
Use in source · monograph_atlas.tex, line 109
vpage7A coarse packet can still contain coherenceTENSOR REFINEMENT BY TOTAL PARITYChapter VII: Theorem II.3.7, Corollary II.3.8. For three qubits, the eight fine projectors are P_a=ket abra a. Grouping addresses according to the parity of their Hamming weight gives P_rm even=P_000+P_011+P_101+P_110 =fracI+Z⊗ Z⊗ Z2, P_rm odd=I-P_rm even. This is a sum of subspaces, not an averaging of their coordinates. The rank-four even packet retains all complex amplitudes on its four children. vfigparity head…
Exact mathematical passage
P_{\rm even}=P_{000}+P_{011}+P_{101}+P_{110} =\frac{I+Z\otimes Z\otimes Z}{2}, \qquad P_{\rm odd}=I-P_{\rm even}.
p_0^TUse in source · 1 passage
Search aliases: p_0^T
Use in source · Paper_VI_Traditional_Final.tex, line 4943
Average [source reference] with terminal function f= 1 against the initial distribution. Transposition gives p_0^T e^t(Q^T-D_V) 1 = 1^T e^t(Q-D_V)p_0. Since 1^TQ=0, differentiation gives S_V'(t) = - 1^TD_V e^tAp_0.
Exact mathematical passage
p_0^T\e^{t(Q^T-D_V)}\one = \one^T\e^{t(Q-D_V)}p_0.
P_1Use in source · 23 passages
Search aliases: P_1
Use in source · 01_orientation.tex, line 96
[title=R3: an exact but oblique packet split] On X=𝕜^2, set P_1=1 · -1 0 · 0, P_2=0 · 1 0 · 1. Then P_i^2=P_i, P_1P_2=P_2P_1=0, and P_1+P_2=I. The ranges form a direct sum, but over R the projectors are not orthogonal in the Euclidean inner product. Exact reconstruction and numerical conditioning are therefore separate questions.
Exact mathematical passage
P_1=\begin{pmatrix}1&-1\\0&0\end{pmatrix}, \qquad P_2=\begin{pmatrix}0&1\\0&1\end{pmatrix}.
p_1Use in source · 6 passages
Search aliases: p_1
Use in source · Paper_IV_Traditional_Final.tex, line 1131
[Logarithmic Newton recurrence] Set b_N,n(r)=fracP_N,n(r)(Nn)!. Then b_N,0(r)=1 and for every n≥1, nb_N,n(r) = r∑_k=1^nκ_N,kb_N,n-k(r). Equivalently, P_N,n(r) = frac r n ∑_k=1^n frac(Nn)!(N(n-k))! κ_N,kP_N,n-k(r). If p_k=rκ_N,k, then, for n≥1, b_N,n(r) = frac1n! det p_1 · -1 · 0 · ⋯ · 0 p_2 · p_1 · -2 · ⋯ · 0 p_3 · p_2 · p_1 · ddots · vdots vdots · vdots · ddots · ddots · -(n-1) p_n · p_n-1 · ⋯ · p_2 · p_1 .
Exact mathematical passage
b_{N,n}(r) = \frac1{n!} \det \begin{pmatrix} p_1&-1&0&\cdots&0\\ p_2&p_1&-2&\cdots&0\\ p_3&p_2&p_1&\ddots&\vdots\\ \vdots&\vdots&\ddots&\ddots&-(n-1)\\ p_n&p_{n-1}&\cdots&p_2&p_1 \end{pmatrix}.
P_{1,n}Use in source · 1 passage
Search aliases: P_1,n
Use in source · Paper_IV_Traditional_Final.tex, line 1228
For N=1, one has D_1(z)^-r=e^rz and P_1,n(r)=r^n. For N=2, P_2,1(r)=r, P_2,2(r)=r(3r+2), and P_2,3(r)=r(15r^2+30r+16). At r=1, these are the positive secant coefficients.
Exact mathematical passage
P_{1,n}(r)=r^n
P_{101}Use in source · 1 passage
Search aliases: P_101
Use in source · monograph_atlas.tex, line 109
vpage7A coarse packet can still contain coherenceTENSOR REFINEMENT BY TOTAL PARITYChapter VII: Theorem II.3.7, Corollary II.3.8. For three qubits, the eight fine projectors are P_a=ket abra a. Grouping addresses according to the parity of their Hamming weight gives P_rm even=P_000+P_011+P_101+P_110 =fracI+Z⊗ Z⊗ Z2, P_rm odd=I-P_rm even. This is a sum of subspaces, not an averaging of their coordinates. The rank-four even packet retains all complex amplitudes on its four children. vfigparity head…
Exact mathematical passage
P_{\rm even}=P_{000}+P_{011}+P_{101}+P_{110} =\frac{I+Z\otimes Z\otimes Z}{2}, \qquad P_{\rm odd}=I-P_{\rm even}.
P_{110}Use in source · 1 passage
Search aliases: P_110
Use in source · monograph_atlas.tex, line 109
vpage7A coarse packet can still contain coherenceTENSOR REFINEMENT BY TOTAL PARITYChapter VII: Theorem II.3.7, Corollary II.3.8. For three qubits, the eight fine projectors are P_a=ket abra a. Grouping addresses according to the parity of their Hamming weight gives P_rm even=P_000+P_011+P_101+P_110 =fracI+Z⊗ Z⊗ Z2, P_rm odd=I-P_rm even. This is a sum of subspaces, not an averaging of their coordinates. The rank-four even packet retains all complex amplitudes on its four children. vfigparity head…
Exact mathematical passage
P_{\rm even}=P_{000}+P_{011}+P_{101}+P_{110} =\frac{I+Z\otimes Z\otimes Z}{2}, \qquad P_{\rm odd}=I-P_{\rm even}.
P_{12,7}Use in source · 2 passages
Search aliases: P_12,7
Use in source · 02_coordinates.tex, line 530
[Residues modulo twelve] The pair (jpmod4,jpmod3)=(3,1) corresponds to j=7 modulo 12. Hence P_12,7=P_4,3P_3,1. This factorization concerns addresses. It does not say that the two factors have the same integral spectral behavior.
Exact mathematical passage
P_{12,7}=P_{4,3}P_{3,1}.
P_{12,k}Use in source · 1 passage
Search aliases: P_12,k
Use in source · atlas-pages.tex, line 449
FACTORIZATION / TWELVE ADDRESSESChapter V: Proposition 14.1. At phase order 12=4·3, the map k⟼(kbmod4, kbmod3) is a bijection from twelve residue addresses to the 4×3 grid. Every cell below contains its unique original address. vfigcrt_refinement headThe factorization is an intersection of conditions For integer sample addresses, including negative ones, P_12,k=P_4,kbmod4 P_3,kbmod3. The product on the right is composition of two residue masks: both congruences must hold. Coprimality makes their…
Exact mathematical passage
P_{12,k}=P_{4,k\bmod4}\,P_{3,k\bmod3}.
P_2Use in source · 14 passages
Search aliases: P_2
Use in source · 01_orientation.tex, line 96
[title=R3: an exact but oblique packet split] On X=𝕜^2, set P_1=1 · -1 0 · 0, P_2=0 · 1 0 · 1. Then P_i^2=P_i, P_1P_2=P_2P_1=0, and P_1+P_2=I. The ranges form a direct sum, but over R the projectors are not orthogonal in the Euclidean inner product. Exact reconstruction and numerical conditioning are therefore separate questions.
Exact mathematical passage
P_1=\begin{pmatrix}1&-1\\0&0\end{pmatrix}, \qquad P_2=\begin{pmatrix}0&1\\0&1\end{pmatrix}.
p_2Use in source · 8 passages
Search aliases: p_2
Use in source · visual_atlas.tex, line 363
headThe missing separating operation is now a polynomial Define p_2(t)=frac3t^5-15t^4+20t^316, p_0(t)=1-p_2(t). These polynomials satisfy p_2≡0pmodt^3, p_2≡1pmod(t-2)^3. Consequently p_0(T)=P_e and p_2(T)=P_f. Applying them to s extracts both chain heads, after which polynomial powers generate all six directions. Unlike the model on pagerefV3, the separating maps already belong to the polynomial grammar.
Exact mathematical passage
p_2(t)=\frac{3t^5-15t^4+20t^3}{16},\qquad p_0(t)=1-p_2(t).
P_{2,1}Use in source · 2 passages
Search aliases: P_2,1
Use in source · Paper_IV_Traditional_Final.tex, line 1229
For N=1, one has D_1(z)^-r=e^rz and P_1,n(r)=r^n. For N=2, P_2,1(r)=r, P_2,2(r)=r(3r+2), and P_2,3(r)=r(15r^2+30r+16). At r=1, these are the positive secant coefficients.
Exact mathematical passage
P_{2,1}(r)=r,
P_{2,2}Use in source · 2 passages
Search aliases: P_2,2
Use in source · Paper_IV_Traditional_Final.tex, line 1232
For N=1, one has D_1(z)^-r=e^rz and P_1,n(r)=r^n. For N=2, P_2,1(r)=r, P_2,2(r)=r(3r+2), and P_2,3(r)=r(15r^2+30r+16). At r=1, these are the positive secant coefficients.
Exact mathematical passage
P_{2,2}(r)=r(3r+2),
P_{2,3}Use in source · 2 passages
Search aliases: P_2,3
Use in source · Paper_IV_Traditional_Final.tex, line 1236
For N=1, one has D_1(z)^-r=e^rz and P_1,n(r)=r^n. For N=2, P_2,1(r)=r, P_2,2(r)=r(3r+2), and P_2,3(r)=r(15r^2+30r+16). At r=1, these are the positive secant coefficients.
Exact mathematical passage
P_{2,3}(r)=r(15r^2+30r+16).
P_{2,n}Use in source · 2 passages
Search aliases: P_2,n
Use in source · Paper_IV_Traditional_Final.tex, line 2726
[Secant asymptotic] For N=2 and r=1, P_2,n(1) ∼ 2(2n)!(frac2π)^2n+1.
Exact mathematical passage
P_{2,n}(1) \sim 2(2n)!\left(\frac2\pi\right)^{2n+1}.
P_{22}^2Use in source · 1 passage
Search aliases: P_22^2
Use in source · Paper_X_Block27_Amended.tex, line 981
0 · 1, every integral intertwiner with ρ_1 has determinant aP_22^2. Taking a=π^n for a nonzero π∈ I rules out every fixed detecting depth, even though all the field-valued approximants are isomorphic and their integral limit is trivial. This does not refute Claim 8.2, which already assumes inverse-limit closure of its chosen local conditions.
Exact mathematical passage
aP_{22}^2
P_{2^a,\,r\bmod2^a}Use in source · 1 passage
Search aliases: P_2^a, rbmod2^a bmod
Use in source · supplement_05.tex, line 201
0, · otherwise, take the canonical label 0≤ r<N. The logical equivalence yields the actual equality of sequence operators P_N,rf= P_2^a, rbmod2^a(P_M, rbmod Mf). This statement holds for coefficients in any type with zero. A separate theorem lifts the equality coefficientwise to the existing formal power-series selectors over any semiring.
Exact mathematical passage
\boxed{P_{N,r}f= P_{2^a,\,r\bmod2^a}\bigl(P_{M,\,r\bmod M}f\bigr).}
P_{2^a,r\bmod 2^a}Use in source · 1 passage
Search aliases: P_2^a,rbmod 2^a bmod
Use in source · 10_foundations.tex, line 456
[Dyadic--odd packet factorization] Let a≥0 and let M be a positive odd integer. The Chinese remainder map is a ring isomorphism Z/(2^aM) Z≃ Z/2^a Z× Z/M Z. For 0≤ r<2^aM and any coefficient sequence f, P_2^aM,rf =P_2^a,rbmod 2^a P_M,rbmod Mf. The same identity holds for formal-series coefficient selectors.
Exact mathematical passage
P_{2^aM,r}f =P_{2^a,r\bmod 2^a}\,P_{M,r\bmod M}f.
P_{2^aM,r}Use in source · 1 passage
Search aliases: P_2^aM,r
Use in source · 10_foundations.tex, line 456
[Dyadic--odd packet factorization] Let a≥0 and let M be a positive odd integer. The Chinese remainder map is a ring isomorphism Z/(2^aM) Z≃ Z/2^a Z× Z/M Z. For 0≤ r<2^aM and any coefficient sequence f, P_2^aM,rf =P_2^a,rbmod 2^a P_M,rbmod Mf. The same identity holds for formal-series coefficient selectors.
Exact mathematical passage
P_{2^aM,r}f =P_{2^a,r\bmod 2^a}\,P_{M,r\bmod M}f.
P_{2^s,j_2}Use in source · 1 passage
Search aliases: P_2^s,j_2
Use in source · Paper_V_Traditional_Final.tex, line 1236
[Dyadic--odd packet factorization] After identifying the residue labels by the Chinese remainder map, the phase projector factors as P_N,j=P_2^s,j_2P_M,j_o.
Exact mathematical passage
\boxed{ P_{N,j}=P_{2^s,j_2}P_{M,j_o}. }
P_3Use in source · 4 passages
Search aliases: P_3
Use in source · visual_atlas.tex, line 199
headReconstruction retains every block The identity P_1+P_2+P_3=I gives T=∑_i,j=1^3 P_iTP_j. Selecting only the diagonal would change the operator. The entries 2,3,4 carry transfers between different packets.
Exact mathematical passage
P_1+P_2+P_3=I
p_3Use in source · 4 passages
Search aliases: p_3
Use in source · Paper_IV_Traditional_Final.tex, line 1131
[Logarithmic Newton recurrence] Set b_N,n(r)=fracP_N,n(r)(Nn)!. Then b_N,0(r)=1 and for every n≥1, nb_N,n(r) = r∑_k=1^nκ_N,kb_N,n-k(r). Equivalently, P_N,n(r) = frac r n ∑_k=1^n frac(Nn)!(N(n-k))! κ_N,kP_N,n-k(r). If p_k=rκ_N,k, then, for n≥1, b_N,n(r) = frac1n! det p_1 · -1 · 0 · ⋯ · 0 p_2 · p_1 · -2 · ⋯ · 0 p_3 · p_2 · p_1 · ddots · vdots vdots · vdots · ddots · ddots · -(n-1) p_n · p_n-1 · ⋯ · p_2 · p_1 .
Exact mathematical passage
b_{N,n}(r) = \frac1{n!} \det \begin{pmatrix} p_1&-1&0&\cdots&0\\ p_2&p_1&-2&\cdots&0\\ p_3&p_2&p_1&\ddots&\vdots\\ \vdots&\vdots&\ddots&\ddots&-(n-1)\\ p_n&p_{n-1}&\cdots&p_2&p_1 \end{pmatrix}.
P_{3,0}Use in source · 1 passage
Search aliases: P_3,0
Use in source · atlas-pages.tex, line 369
)·(P_N,ky). Here the dot means multiplication at the same sample, not convolution. headCauchy multiplication adds degrees For formal series A(t),B(t), the coefficient of t^n is a sum over a+b=n. Consequently the output residue is i+jpmod N: P_N,k(AB)=∑_i+j≡ k (N)(P_N,iA)(P_N,jB). For N=3, take A=1+2t+3t^2 and B=2+t^2. Direct multiplication gives AB=2+4t+7t^2+2t^3+3t^4, P_3,0(AB)=2+2t^3. The 2t^3 term comes from residues 1+2, not 0+0. At power zero, the empty product contributes 1 only in residue…
Exact mathematical passage
AB=2+4t+7t^2+2t^3+3t^4, \qquad P_{3,0}(AB)=2+2t^3.
P_{3,1}Use in source · 2 passages
Search aliases: P_3,1
Use in source · 02_coordinates.tex, line 530
[Residues modulo twelve] The pair (jpmod4,jpmod3)=(3,1) corresponds to j=7 modulo 12. Hence P_12,7=P_4,3P_3,1. This factorization concerns addresses. It does not say that the two factors have the same integral spectral behavior.
Exact mathematical passage
P_{12,7}=P_{4,3}P_{3,1}.
P_{3,k\bmod3}Use in source · 1 passage
Search aliases: P_3,kbmod3 bmod
Use in source · atlas-pages.tex, line 449
FACTORIZATION / TWELVE ADDRESSESChapter V: Proposition 14.1. At phase order 12=4·3, the map k⟼(kbmod4, kbmod3) is a bijection from twelve residue addresses to the 4×3 grid. Every cell below contains its unique original address. vfigcrt_refinement headThe factorization is an intersection of conditions For integer sample addresses, including negative ones, P_12,k=P_4,kbmod4 P_3,kbmod3. The product on the right is composition of two residue masks: both congruences must hold. Coprimality makes their…
Exact mathematical passage
P_{12,k}=P_{4,k\bmod4}\,P_{3,k\bmod3}.
P_{4,3}Use in source · 2 passages
Search aliases: P_4,3
Use in source · 02_coordinates.tex, line 530
[Residues modulo twelve] The pair (jpmod4,jpmod3)=(3,1) corresponds to j=7 modulo 12. Hence P_12,7=P_4,3P_3,1. This factorization concerns addresses. It does not say that the two factors have the same integral spectral behavior.
Exact mathematical passage
P_{12,7}=P_{4,3}P_{3,1}.
\mathcal P_{4,j}Use in source · 1 passage
Search aliases: P_4,j mathcal
Use in source · 20_arithmetic.tex, line 539
Packing g_j into degrees divisible by four gives K_4,j. Its source property is precisely charge-one flatness. Inversion gives charge -1, and the charge shift restores charge one for the reciprocal P_4,j. If dec(F)(d)=d![t^d]F, the exact coefficient identity is dec(A_1^sF)(d) =2^-sΔ_4^sdec(F)(d). At d=4n, packing turns the right side into 2^-sΔ_1^sC_j(n). This extraction requires no negative degree and therefore holds at every n, including zero. It proves the unit-step bound leanKomorebiPaperIX.D…
Exact mathematical passage
\mathcal P_{4,j}
P_{4,k\bmod4}Use in source · 1 passage
Search aliases: P_4,kbmod4 bmod
Use in source · atlas-pages.tex, line 449
FACTORIZATION / TWELVE ADDRESSESChapter V: Proposition 14.1. At phase order 12=4·3, the map k⟼(kbmod4, kbmod3) is a bijection from twelve residue addresses to the 4×3 grid. Every cell below contains its unique original address. vfigcrt_refinement headThe factorization is an intersection of conditions For integer sample addresses, including negative ones, P_12,k=P_4,kbmod4 P_3,kbmod3. The product on the right is composition of two residue masks: both congruences must hold. Coprimality makes their…
Exact mathematical passage
P_{12,k}=P_{4,k\bmod4}\,P_{3,k\bmod3}.
P_{4,r}Use in source · 1 passage
Search aliases: P_4,r
Use in source · atlas-pages.tex, line 42
y cell contains an original address and its rational sample value. Shading marks the known jump packet, not an estimated feature or a frequency band. headWhat has changed? Only where the values are stored The complete tuple of channels contains sixteen values, just as the original vector does. A single decimated channel is shorter, but keeping all four loses no samples. The residue projector P_4,rf is different: it keeps the original lattice and writes zeros outside phase r. S_4 A_4f=f, f _2^2=∑…
Exact mathematical passage
P_{4,r}f
P_{8,2}Use in source · 1 passage
Search aliases: P_8,2
Use in source · visual_atlas.tex, line 59
The address 6 is even, is 2 modulo 4, and is 6 modulo 8. Its coefficient therefore lies in the successive ranges P_8,6X⊆ P_4,2X⊆ P_2,0X. The corresponding identities hold as operators: P_2,0=P_4,0+P_4,2, P_4,2=P_8,2+P_8,6. Applying either composite parent selector to P_8,6x leaves it unchanged. Applying the odd selector kills it. This is absorption between a parent and a child, rather than an approximate resemblance between nearby scales.
Exact mathematical passage
P_{2,0}=P_{4,0}+P_{4,2},\qquad P_{4,2}=P_{8,2}+P_{8,6}.
P_{8,6}Use in source · 3 passages
Search aliases: P_8,6
Use in source · visual_atlas.tex, line 55
The address 6 is even, is 2 modulo 4, and is 6 modulo 8. Its coefficient therefore lies in the successive ranges P_8,6X⊆ P_4,2X⊆ P_2,0X. The corresponding identities hold as operators: P_2,0=P_4,0+P_4,2, P_4,2=P_8,2+P_8,6. Applying either composite parent selector to P_8,6x leaves it unchanged. Applying the odd selector kills it. This is absorption between a parent and a child, rather than an approximate resemblance between nearby scales.
Exact mathematical passage
P_{8,6}X\subseteq P_{4,2}X\subseteq P_{2,0}X.
p_aUse in source · 1 passage
Search aliases: p_a
Use in source · 03_supplement_04.tex, line 126
Exact correction, including coherent errors. For every operator supported on the code, Pρ P=ρ, the central identity is R(V_aρ V_b^*)=δ_abρ. Consequently normalized mixtures satisfy R (∑_ap_aV_aρ V_a^*)=ρ when ∑_ap_a=1. The identity is algebraic and is proved for complex weights. Physical mixtures use the nonnegative real case. For a coherent error E_z=∑_a z_aV_a, the calculation of the cross terms yields R(E_zρ E_z^*)=(∑_a z_aoverlinez_a)ρ. Thus the same map also corrects normalized coherent sup…
Exact mathematical passage
\mathcal R\!\left(\sum_ap_aV_a\rho V_a^*\right)=\rho \quad\text{when}\quad\sum_ap_a=1.
P_{a(e)}Use in source · 5 passages
Search aliases: P_a(e)
Use in source · 02_supplement_01.tex, line 127
Packet perturbation certificates. textitSource item 8.1 and two uniform consequences Let X be a normed space over a nontrivially normed field. The formal objects are actual continuous linear maps X→ X, with their operator norm. Let a(e) identify the source packet of edge e, and suppose Δ_e=Δ_e P_a(e). Set A=∑_e A_e, widehat A=∑_e(A_e+Δ_e). The exact operator and vector identities are proved first: widehat A-A=∑_eΔ_e P_a(e), (widehat A-A)x=∑_eΔ_e(P_a(e)x). Applying the finite triangle inequality …
Exact mathematical passage
\Delta_e=\Delta_e P_{a(e)}.
P_{A,j_A}Use in source · 1 passage
Search aliases: P_A,j_A
Use in source · 02_coordinates.tex, line 506
[Coprime factorization] Let A and B be coprime positive integers. For each j modulo AB, write j_A=jpmod A, j_B=jpmod B. Then P_AB,j=P_A,j_AP_B,j_B. Every pair (j_A,j_B) occurs for one residue j modulo AB.
Exact mathematical passage
P_{AB,j}=P_{A,j_A}P_{B,j_B}.
P_{A,r_A}Use in source · 1 passage
Search aliases: P_A,r_A
Use in source · 72_recovered.tex, line 3010
[Coprime packet factorization] Let A and B be coprime positive integers. Let r_A be an address modulo A and r_B an address modulo B. Let r be the unique address modulo AB satisfying r≡ r_Apmod A, r≡ r_Bpmod B. Then P_AB,r = P_A,r_AP_B,r_B = P_B,r_BP_A,r_A.
Exact mathematical passage
\boxed{ P_{AB,r} = P_{A,r_A}P_{B,r_B} = P_{B,r_B}P_{A,r_A}. }
P_a^*Use in source · 3 passages
Search aliases: P_a^*
Use in source · 02_strengthened.tex, line 1013
The projectors P_a form a Kraus family because ∑_aP_a^*P_a=I. Thus the map is completely positive and trace preserving. It is unital for the same reason. Orthogonality and idempotence follow from [source reference]. Its range is the sum of the diagonal operator packets. This is exactly the commutant of the projector family.
Exact mathematical passage
\sum_aP_a^*P_a=I
P_a^2Use in source · 2 passages
Search aliases: P_a^2
Use in source · 01_finite_core.tex, line 541
Differentiating P_a^2=P_a gives dot P_aP_a+P_adot P_a=dot P_a. Summing over a and using ∑_adot P_a=0 yields ∑_aP_adot P_a=-∑_adot P_aP_a, which proves both displayed formulas for C and its skew-Hermiticity.
Exact mathematical passage
P_a^2=P_a
P_A^jUse in source · 1 passage
Search aliases: P_A^j
Use in source · rank4.tex, line 56
Use U_A^j=1+P_A^j and U_B^j=1+Q^j. BK 3.1.7--9 give the actual groups
Exact mathematical passage
U_A^j=1+P_A^j
P_A^{n-r}Use in source · 1 passage
Search aliases: P_A^n-r
Use in source · dyadic_synthesis.tex, line 337
At both low ranks, K^× normalizes the new P. In rank two this follows directly from normalization of A and its radical. In rank four, a K-uniformizer is approximated by the derived chain shift R times a unit of B, with relative error in P_A^n-r. Its conjugation error on U_B^m lies in U_A^n-r+m⊂ J, which proves normalization. The exact argument is pathlowrank/RANK4_INDUCTION_BY_STAGES167.md.
Exact mathematical passage
P_A^{n-r}
P_{a_0}Use in source · 4 passages
Search aliases: P_a_0
Use in source · 02_strengthened.tex, line 390
[Exact circuit path expansion] For source packet a_0 and target packet a_m, P_a_mUP_a_0 = ∑_a_1,…,a_m-1 P_a_mU_mP_a_m-1 ⋯ P_a_1U_1P_a_0. Summing over a_0 and a_m reconstructs U.
Exact mathematical passage
P_{a_m}UP_{a_0} = \sum_{a_1,\ldots,a_{m-1}} P_{a_m}U_mP_{a_{m-1}} \cdots P_{a_1}U_1P_{a_0}.
P_{a_1}Use in source · 3 passages
Search aliases: P_a_1
Use in source · 02_strengthened.tex, line 390
[Exact circuit path expansion] For source packet a_0 and target packet a_m, P_a_mUP_a_0 = ∑_a_1,…,a_m-1 P_a_mU_mP_a_m-1 ⋯ P_a_1U_1P_a_0. Summing over a_0 and a_m reconstructs U.
Exact mathematical passage
P_{a_m}UP_{a_0} = \sum_{a_1,\ldots,a_{m-1}} P_{a_m}U_mP_{a_{m-1}} \cdots P_{a_1}U_1P_{a_0}.
P_{a_m}Use in source · 3 passages
Search aliases: P_a_m
Use in source · 02_strengthened.tex, line 390
[Exact circuit path expansion] For source packet a_0 and target packet a_m, P_a_mUP_a_0 = ∑_a_1,…,a_m-1 P_a_mU_mP_a_m-1 ⋯ P_a_1U_1P_a_0. Summing over a_0 and a_m reconstructs U.
Exact mathematical passage
P_{a_m}UP_{a_0} = \sum_{a_1,\ldots,a_{m-1}} P_{a_m}U_mP_{a_{m-1}} \cdots P_{a_1}U_1P_{a_0}.
P_{a_{m-1}}Use in source · 3 passages
Search aliases: P_a_m-1
Use in source · 02_strengthened.tex, line 390
[Exact circuit path expansion] For source packet a_0 and target packet a_m, P_a_mUP_a_0 = ∑_a_1,…,a_m-1 P_a_mU_mP_a_m-1 ⋯ P_a_1U_1P_a_0. Summing over a_0 and a_m reconstructs U.
Exact mathematical passage
P_{a_m}UP_{a_0} = \sum_{a_1,\ldots,a_{m-1}} P_{a_m}U_mP_{a_{m-1}} \cdots P_{a_1}U_1P_{a_0}.
P_{AB,j}Use in source · 1 passage
Search aliases: P_AB,j
Use in source · 02_coordinates.tex, line 506
[Coprime factorization] Let A and B be coprime positive integers. For each j modulo AB, write j_A=jpmod A, j_B=jpmod B. Then P_AB,j=P_A,j_AP_B,j_B. Every pair (j_A,j_B) occurs for one residue j modulo AB.
Exact mathematical passage
P_{AB,j}=P_{A,j_A}P_{B,j_B}.
P_{AB,r}Use in source · 1 passage
Search aliases: P_AB,r
Use in source · 72_recovered.tex, line 3010
[Coprime packet factorization] Let A and B be coprime positive integers. Let r_A be an address modulo A and r_B an address modulo B. Let r be the unique address modulo AB satisfying r≡ r_Apmod A, r≡ r_Bpmod B. Then P_AB,r = P_A,r_AP_B,r_B = P_B,r_BP_A,r_A.
Exact mathematical passage
\boxed{ P_{AB,r} = P_{A,r_A}P_{B,r_B} = P_{B,r_B}P_{A,r_A}. }
P_{B,j_B}Use in source · 1 passage
Search aliases: P_B,j_B
Use in source · 02_coordinates.tex, line 506
[Coprime factorization] Let A and B be coprime positive integers. For each j modulo AB, write j_A=jpmod A, j_B=jpmod B. Then P_AB,j=P_A,j_AP_B,j_B. Every pair (j_A,j_B) occurs for one residue j modulo AB.
Exact mathematical passage
P_{AB,j}=P_{A,j_A}P_{B,j_B}.
P_{B,r_B}Use in source · 1 passage
Search aliases: P_B,r_B
Use in source · 72_recovered.tex, line 3010
[Coprime packet factorization] Let A and B be coprime positive integers. Let r_A be an address modulo A and r_B an address modulo B. Let r be the unique address modulo AB satisfying r≡ r_Apmod A, r≡ r_Bpmod B. Then P_AB,r = P_A,r_AP_B,r_B = P_B,r_BP_A,r_A.
Exact mathematical passage
\boxed{ P_{AB,r} = P_{A,r_A}P_{B,r_B} = P_{B,r_B}P_{A,r_A}. }
P_b^2Use in source · 1 passage
Search aliases: P_b^2
Use in source · 03_supplement_01.tex, line 41
UU^*=1 · Longrightarrow ∑_aU_ba(U_da)^*=δ_bdP_b. For the first law, expand each corner, collapse P_b^2 and sum: ∑_bP_aU^*P_bUP_c =P_aU^*(∑_bP_b)UP_c=P_aP_c. The second law follows by the analogous calculation and has its own formal proof. For complex matrices, the adjoint is the conjugate transpose. The order of the operator products is preserved throughout.
Exact mathematical passage
P_b^2
P_cUse in source · 9 passages
Search aliases: P_c
Use in source · Paper_VI_Traditional_Final.tex, line 685
[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.
Exact mathematical passage
\boxed{ P_cP_f=P_fP_c=P_c. }
P_c^2Use in source · 1 passage
Search aliases: P_c^2
Use in source · 03_supplement_01.tex, line 16
0, · a≠ b, ∑_a∈ IP_a=1. Adjoint conditions are added when needed. For an operator U, define the transfer corner U_ba=P_bUP_a. The target index comes first. Completeness and distributivity give ∑_b,aU_ba=U, (UV)_ba=∑_c U_bcV_ca. The second identity uses P_c^2=P_c when adjacent transfer corners are multiplied.
Exact mathematical passage
P_c^2=P_c
P_dUse in source · 2 passages
Search aliases: P_d
Use in source · 02_strengthened.tex, line 970
Idempotence and completeness follow from the projector identities. For Hilbert--Schmidt orthogonality, ipP_aXP_bP_cYP_d_HS · = Tr(P_bX^*P_aP_cYP_d) · = δ_acδ_bd Tr(P_bX^*P_aYP_b). Since the packets are orthogonal and sum to the original operator, their squared norms add to its squared norm.
Exact mathematical passage
\begin{aligned} \ip{P_aXP_b}{P_cYP_d}_{\HS} &= \Tr(P_bX^*P_aP_cYP_d)\\ &= \delta_{ac}\delta_{bd} \Tr(P_bX^*P_aYP_b). \end{aligned}
P_{D,\alpha}Use in source · 1 passage
Search aliases: P_D,α alpha
Use in source · Paper_XII_Traditional_Final.tex, line 434
0 · 1, Tr(P_s,1W_std)=6≠0. For an operator D with (D-α I)(D-β I)=0 and α≠β, the canonical projector is P_D,α=(α-β)^-1(D-β I). Its image equals ker((D-α I)^n) for every n≥1, and it kills ker((D-β I)^n). This identifies the displayed matrices with the generalized eigenspace projectors required in the adequacy definition.
Exact mathematical passage
P_{D,\alpha}=(\alpha-\beta)^{-1}(D-\beta I).
P_eUse in source · 6 passages
Search aliases: P_e
Use in source · visual_atlas.tex, line 12
Use the same space of dimension six throughout the first comparison: X= Q^6, N=J_3(0)⊕ J_3(0), s=e_0+f_0. Here each J_3(0) moves a basis vector forward along a chain of three basis vectors. The projectors P_e,P_f select the two chains. Their formulas and the proofs of the minimum seed counts appear on pagerefV3 and pagerefV4.
Exact mathematical passage
P_e,P_f
P_{\mathrm{even}}Use in source · 1 passage
Search aliases: P_even mathrm
Use in source · 01_finite_core.tex, line 228
For parity, S=Z^⊗ n, P_even=fracI+S2, P_odd=fracI-S2.
Exact mathematical passage
S=Z^{\otimes n}, \qquad P_{\mathrm{even}}=\frac{I+S}{2}, \qquad P_{\mathrm{odd}}=\frac{I-S}{2}.
P_fUse in source · 7 passages
Search aliases: P_f
Use in source · visual_atlas.tex, line 12
Use the same space of dimension six throughout the first comparison: X= Q^6, N=J_3(0)⊕ J_3(0), s=e_0+f_0. Here each J_3(0) moves a basis vector forward along a chain of three basis vectors. The projectors P_e,P_f select the two chains. Their formulas and the proofs of the minimum seed counts appear on pagerefV3 and pagerefV4.
Exact mathematical passage
P_e,P_f
P_gUse in source · 3 passages
Search aliases: P_g
Use in source · Paper_XI_Traditional_Final.tex, line 1359
[Characteristic polynomial certificate] Let P_g(X)=det(XI-gmid V). Then Φ_p(g,V) = (frac-1p)^r (fracP_g(1)p).
Exact mathematical passage
P_g(X)=\det(XI-g\mid V).
P_iUse in source · 31 passages
Search aliases: P_i
Use in source · 01_orientation.tex, line 101
0 · 1. Then P_i^2=P_i, P_1P_2=P_2P_1=0, and P_1+P_2=I. The ranges form a direct sum, but over R the projectors are not orthogonal in the Euclidean inner product. Exact reconstruction and numerical conditioning are therefore separate questions.
Exact mathematical passage
P_i^2=P_i
\mathcal P_iUse in source · 1 passage
Search aliases: P_i mathcal
Use in source · odd_synthesis.tex, line 141
Choose a labeled symplectic embedding V_src→ S. For each original full character form P_i=( C^×× J_i^1)/ {(θ_i(h)^-1,h):h∈ H_i^1}. The balanced product and the corresponding host restriction have the same alternating commutator. The center-fixing extension isomorphism, obtained with matched Darboux lifts and complex roots, therefore preserves every supplied value θ_i(h), including the shallow values.
Exact mathematical passage
\mathcal P_i=(\mathbb C^\times\times J_i^1)/ \{(\theta_i(h)^{-1},h):h\in H_i^1\}.
p_i'Use in source · 1 passage
Search aliases: p_i'
Use in source · atlas-pages.tex, line 843
tatistical meanings The first derivative of log evidence is the current mean score, and the second is its variance: fracddtlog Z(t)= E_p(t)u, fracd^2dt^2log Z(t)=Var_p(t)(u)≥0. At q=2 the exact values are E u=frac107, E u^2=frac187, Var(u)=frac2649. Thus the derivative of the mean is 26/49 at this time. The flow equation p_i'=(u_i- E u)p_i gives derivatives (-10,-6,16)/49, which sum to zero.
Exact mathematical passage
p_i'=(u_i-\EE u)p_i
p_{i,a}Use in source · 2 passages
Search aliases: p_i,a
Use in source · Paper_X_Block27_Amended.tex, line 1649
For each operator, its finite set of actual eigenvalues is used without multiplicity. The polynomial p_i,a(X)=∏_μ∈Spec(T_i)∖{a} (a-μ)^-1(X-μ) is used when a occurs, and the zero polynomial otherwise. For an eigenvector T_i v=bv, evaluation gives p_i,a(T_i)v=v · b=a,
Exact mathematical passage
p_{i,a}(X)=\prod_{\mu\in\Spec(T_i)\setminus\{a\}} (a-\mu)^{-1}(X-\mu)
p_{i,a_i}Use in source · 1 passage
Search aliases: p_i,a_i
Use in source · Paper_X_Block27_Amended.tex, line 1658
0 · b≠ a. Because the eigenspaces span A, this proves that e_i=p_i,a_i(T_i) maps all of A into the selected eigenspace and fixes that eigenspace pointwise. It therefore proves the individual projector and image assertions, including an absent target.
Exact mathematical passage
e_i=p_{i,a_i}(T_i)
P_{i,g}Use in source · 1 passage
Search aliases: P_i,g
Use in source · Paper_X_Block27_Amended.tex, line 1419
(XI-ρ_m(g)) =∏_i∈ Idet(XI-ρ_i(g))^m_i, Tr(ρ_m(g)) =∑_i∈ I(m_i:K)Tr(ρ_i(g)). Constituents may have different dimensions. Zero multiplicities, zero-dimensional constituents, and the empty index set are included. If at a permitted sample g the constituent polynomials are the images of P_i,g∈ E[X] under a specified map E→ K, the summand polynomial is the image of ∏_iP_i,g^m_i. Applying this independently at each coefficient field preserves a supplied common polynomial family. This does not establish…
Exact mathematical passage
P_{i,g}\in E[X]
P_{i,g}^{m_i}Use in source · 1 passage
Search aliases: P_i,g^m_i
Use in source · Paper_X_Block27_Amended.tex, line 1421
eratornameTr(ρ_m(g)) =∑_i∈ I(m_i:K)Tr(ρ_i(g)). Constituents may have different dimensions. Zero multiplicities, zero-dimensional constituents, and the empty index set are included. If at a permitted sample g the constituent polynomials are the images of P_i,g∈ E[X] under a specified map E→ K, the summand polynomial is the image of ∏_iP_i,g^m_i. Applying this independently at each coefficient field preserves a supplied common polynomial family. This does not establish the Hodge--Tate or Weil--Del…
Exact mathematical passage
\prod_iP_{i,g}^{m_i}
P_i^2Use in source · 1 passage
Search aliases: P_i^2
Use in source · 01_orientation.tex, line 101
0 · 1. Then P_i^2=P_i, P_1P_2=P_2P_1=0, and P_1+P_2=I. The ranges form a direct sum, but over R the projectors are not orthogonal in the Euclidean inner product. Exact reconstruction and numerical conditioning are therefore separate questions.
Exact mathematical passage
P_i^2=P_i
P_i^rUse in source · 1 passage
Search aliases: P_i^r
Use in source · Paper_X_Block27_Amended.tex, line 347
Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.
Exact mathematical passage
\sum_jP_j^r=I, \qquad P_i^rP_j^r=\delta_{ij}P_j^r, \qquad d_rP_j^r=P_j^{r+1}d_r.
P_jUse in source · 62 passages
Search aliases: P_j
Use in source · visual_atlas.tex, line 193
vpage8A transfer block has an input and an outputOPERATOR COORDINATES / MULTIPLICATION AS A SUMMain text: [source reference] Use the three coordinate packets of dimension one in Q^3. The block T_ij=P_iTP_j sends input packet j into output packet i. For this example, its nonzero scalar is the (i,j) matrix entry.
Exact mathematical passage
T_{ij}=P_iTP_j
p_jUse in source · 13 passages
Search aliases: p_j
Use in source · Paper_X_Block27_Amended.tex, line 383
h packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet are cochain maps, so they induce maps on these genuine quotients.
Exact mathematical passage
p_j:C^\bullet\to C_j^\bullet
P_{j+1}Use in source · 2 passages
Search aliases: P_j+1
Use in source · 02_supplement_03.tex, line 35
Actual noncommutative products. The error module defines the left-accumulated product recursively, with P_0=I and P_j+1=A_j+1P_j. For a perturbed sequence widehat A_j, it derives the discrepancy recursion widehat P_j+1-P_j+1 =widehat A_j+1(widehat P_j-P_j) +(widehat A_j+1-A_j+1)P_j. The identity follows from the actual products. It is not supplied as an assumption to a norm estimate, and it does not commute the operators. Lean uses zero-based indices internally, so index j in source code corresp…
Exact mathematical passage
P_{j+1}=A_{j+1}P_j
P_{j+q}Use in source · 3 passages
Search aliases: P_j+q
Use in source · 02_strengthened.tex, line 306
[Charge selection rule] If A has charge q, then P_kAP_j=0 unless k=j+q in Z/N Z. In fact, AP_j=P_j+qAP_j.
Exact mathematical passage
AP_j=P_{j+q}AP_j.
P_{j,a(e)}Use in source · 2 passages
Search aliases: P_j,a(e)
Use in source · 02_supplement_03.tex, line 75
Packetwise mixed precision. At each stage let A_j=∑_e A_j,e, widehat A_j=∑_e(A_j,e+Δ_j,e), Δ_j,e=Δ_j,eP_j,a(e). The previous packet certificate supplies the uniform operator bound normwidehat A_j-A_j≤ δ_j:=∑_enormΔ_j,enormP_j,a(e). The new theorem substitutes these δ_j into the actual product-chain estimate. The resulting vector error is at most the operator certificate times normx. A vector-specific packet bound is not silently reused as an operator norm.
Exact mathematical passage
A_j=\sum_e A_{j,e},\qquad \widehat A_j=\sum_e(A_{j,e}+\Delta_{j,e}),\qquad \Delta_{j,e}=\Delta_{j,e}P_{j,a(e)}.
P_{j-1}Use in source · 5 passages
Search aliases: P_j-1
Use in source · Paper_XI_Traditional_Final.tex, line 259
[Exact block reconstruction] Let P_j be the projector onto V_j. Then N=∑_j=1^m-1P_j-1NP_j, N^m=0. Every power N^r is the sum of the packet paths of length r along the chain.
Exact mathematical passage
N=\sum_{j=1}^{m-1}P_{j-1}NP_j, \qquad N^m=0.
P_j^{(a)}Use in source · 1 passage
Search aliases: P_j^(a)
Use in source · 02_strengthened.tex, line 339
Suppose the ath tensor factor carries a cyclic packet family (P_j^(a))_j∈ Z/N Z. The fine tensor packets are P_ j^rm fine = P_j_1^(1)⊗⋯⊗ P_j_m^(m).
Exact mathematical passage
\bigl(P_j^{(a)}\bigr)_{j\in\ZZ/N\ZZ}.
P_j^*Use in source · 2 passages
Search aliases: P_j^*
Use in source · 01_finite_core.tex, line 221
Using S^r+s=S^r+sbmod N and the finite character orthogonality relation, frac1N∑_r=0^N-1ω^(k-j)r=δ_jk, we obtain P_jP_k=δ_jkP_j. Summing over j cancels every term indexed by r≠0 and retains the r=0 term, which proves completeness. Unitarity and the change of residue r↦-r give P_j^*=frac1N∑_r=0^N-1ω^jrS^-r=P_j. Finally, multiplication by S shifts the summation index and contributes the factor ω^j.
Exact mathematical passage
P_j^*=\frac1N\sum_{r=0}^{N-1}\omega^{jr}S^{-r}=P_j.
P_j^2Use in source · 1 passage
Search aliases: P_j^2
Use in source · monograph_atlas.tex, line 102
Multiplying the actual projectors gives P_jP_k=0 for j≠ k, P_j^2=P_j, and ∑_jP_j=I. In particular, the complete packet tuple (P_jψ)_j retains the original vector exactly. Recording only the four packet norms is a different operation and can discard phase information between packets.
Exact mathematical passage
P_j^2=P_j
p_j^{a_j}Use in source · 2 passages
Search aliases: p_j^a_j
Use in source · 72_recovered.tex, line 3054
[Finite local reconstruction] Let N=∏_j=1^sp_j^a_j be the prime power factorization of a positive integer. The map Z/N Z ⟶ ∏_j=1^s Z/p_j^a_j Z is a ring isomorphism. Thus a compatible collection of prime power residues reconstructs one residue modulo N [source reference].
Exact mathematical passage
N=\prod_{j=1}^{s}p_j^{a_j}
P_j^rUse in source · 4 passages
Search aliases: P_j^r
Use in source · Paper_X_Block27_Amended.tex, line 346
Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.
Exact mathematical passage
P_j^r:C^r\to C^r
P_j^{r+1}Use in source · 1 passage
Search aliases: P_j^r+1
Use in source · Paper_X_Block27_Amended.tex, line 347
Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.
Exact mathematical passage
\sum_jP_j^r=I, \qquad P_i^rP_j^r=\delta_{ij}P_j^r, \qquad d_rP_j^r=P_j^{r+1}d_r.
P_{j_1}^{(1)}Use in source · 2 passages
Search aliases: P_j_1^(1)
Use in source · 02_strengthened.tex, line 343
Suppose the ath tensor factor carries a cyclic packet family (P_j^(a))_j∈ Z/N Z. The fine tensor packets are P_ j^rm fine = P_j_1^(1)⊗⋯⊗ P_j_m^(m).
Exact mathematical passage
P_{\bm j}^{\rm fine} = P_{j_1}^{(1)}\otimes\cdots\otimes P_{j_m}^{(m)}.
P_{j_m}^{(m)}Use in source · 2 passages
Search aliases: P_j_m^(m)
Use in source · 02_strengthened.tex, line 343
Suppose the ath tensor factor carries a cyclic packet family (P_j^(a))_j∈ Z/N Z. The fine tensor packets are P_ j^rm fine = P_j_1^(1)⊗⋯⊗ P_j_m^(m).
Exact mathematical passage
P_{\bm j}^{\rm fine} = P_{j_1}^{(1)}\otimes\cdots\otimes P_{j_m}^{(m)}.
P_KUse in source · 2 passages
Search aliases: P_K
Use in source · Paper_XI_Traditional_Final.tex, line 230
Let W_K be the Weil group of K, let I_K be inertia, let P_K be wild inertia, and choose a geometric Frobenius lift Φ. A Weil Deligne representation is a pair (r,N) with r(w)Nr(w)^-1=absw_K N.
Exact mathematical passage
P_K
P_kUse in source · 8 passages
Search aliases: P_k
Use in source · 03_nonlinear_coordinates.tex, line 583
[Graded packet convolution] If a_j=∑_g∈ Ga_j,g and μ respects the grading, then P_kμ(a_1,…,a_r) = ∑_g_1+⋯+g_r=k μ(a_1,g_1,…,a_r,g_r).
Exact mathematical passage
P_k\mu(a_1,\ldots,a_r) = \sum_{g_1+\cdots+g_r=k} \mu(a_{1,g_1},\ldots,a_{r,g_r}).
\mathfrak p_K^mUse in source · 1 passage
Search aliases: p_K^m mathfrak
Use in source · rank2.tex, line 132
For if ku∈U_A m with k∈U_K 1 and u∈U_A (m+1), then k∈ K^×∩ U_A^m=U_K^m. The reverse inclusion is immediate. The complete period-two A-grade R^m/R^m+1 has F2-dimension 2, and its field grade p_K^m/ p_K^m+1 has dimension 1. Thus
Exact mathematical passage
\mathfrak p_K^m/\mathfrak p_K^{m+1}
\mathfrak p_K^{m+1}Use in source · 1 passage
Search aliases: p_K^m+1 mathfrak
Use in source · rank2.tex, line 132
For if ku∈U_A m with k∈U_K 1 and u∈U_A (m+1), then k∈ K^×∩ U_A^m=U_K^m. The reverse inclusion is immediate. The complete period-two A-grade R^m/R^m+1 has F2-dimension 2, and its field grade p_K^m/ p_K^m+1 has dimension 1. Thus
Exact mathematical passage
\mathfrak p_K^m/\mathfrak p_K^{m+1}
P_LUse in source · 5 passages
Search aliases: P_L
Use in source · atlas-pages.tex, line 390
ltiplicity-sensitive genus-zero product is Δ_N(w)=∏_λ∈Λ_N(1-w/λ). Putting K_Ne_λ=λ^-1e_λ on ℓ^2(Λ_N) gives the corresponding diagonal Fredholm determinant. Repeated zeros require repeated diagonal entries. vfigproduct headFinite compression is an approximation For N=2, λ_m=π^2(m+1/2)^2. The finite determinant P_L is exactly the displayed finite product, but it is not exactly Δ_2. For 0≤ w≤1 and L≥1, an elementary tail estimate gives 0≤1-fracΔ_2(w)P_L(w) ≤ w∑_m≥ Lλ_m^-1 ≤frac19(L-1/2). The last b…
Exact mathematical passage
P_L
P_mUse in source · 12 passages
Search aliases: P_m
Use in source · 03_packets_and_graphs.tex, line 410
[Countable unconditional packet isomorphism] Let X be a real or complex Banach space and let (P_n)_n≥1⊆ B(X) satisfy P_nP_m=δ_nmP_n and x=∑_n≥1P_nx unconditionally for every x∈ X. Assume K:= sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nP_n <∞. Then analysis and synthesis define inverse Banach space isomorphisms X≅⊕_n≥1^uP_nX. More precisely, A_ P ≤ K, S_ P ≤1. [source reference] ExportsCountable analytic packet decompositions under explicit convergence control.
Exact mathematical passage
P_nP_m=\delta_{nm}P_n
\mathcal P_MUse in source · 3 passages
Search aliases: P_M mathcal
Use in source · 72_recovered.tex, line 1977
For a sequence a=(a_n)_n≥0 and M≥2, define a_r^[M](q)=a_Mq+r (0≤ r<M). The analysis map is P_Ma = (a_0^[M],…,a_M-1^[M]). Synthesis interleaves the branches by the rule a_Mq+r=a_r^[M](q).
Exact mathematical passage
\mathcal P_Ma = \left(a_0^{[M]},\ldots,a_{M-1}^{[M]}\right).
p_MUse in source · 4 passages
Search aliases: p_M
Use in source · Paper_VI_Traditional_Final.tex, line 3839
[Concentration on the maximal score packet] Let G=max{g_i:p_i(0)>0} and let M be the set of supported indices with score G. If M is the full support, then the conclusion below is immediate. Otherwise, suppose δ = min{G-g_i:p_i(0)>0, i∉ M}>0. Then 1-∑_i∈ Mp_i(t) ≤ frac1-p_M(0)p_M(0) e^-δ t, where p_M(0)=∑_i∈ Mp_i(0).
Exact mathematical passage
1-\sum_{i\in M}p_i(t) \leq \frac{1-p_M(0)}{p_M(0)}\e^{-\delta t},
P_{M,\,r\bmod M}Use in source · 1 passage
Search aliases: P_M, rbmod M bmod
Use in source · supplement_05.tex, line 201
0, · otherwise, take the canonical label 0≤ r<N. The logical equivalence yields the actual equality of sequence operators P_N,rf= P_2^a, rbmod2^a(P_M, rbmod Mf). This statement holds for coefficients in any type with zero. A separate theorem lifts the equality coefficientwise to the existing formal power-series selectors over any semiring.
Exact mathematical passage
\boxed{P_{N,r}f= P_{2^a,\,r\bmod2^a}\bigl(P_{M,\,r\bmod M}f\bigr).}
P_{M,a}Use in source · 1 passage
Search aliases: P_M,a
Use in source · atlas-pages.tex, line 66
leaved to recover the coarse channel. Adding them entrywise instead produces sums of different original samples. For example, at the first descendant address the values f[0]=0 and f[2]=1/2 are two separate values. Their sum 1/2 cannot tell us both of them. Exact refinement is a disjoint partition, not an averaging rule or a new measurement. headThe general formula When N divides M, P_N,r=∑_substack0≤ a<M a≡ rpmod NP_M,a. The divisibility hypothesis ensures that every fine residue class lies enti…
Exact mathematical passage
P_{N,r}=\sum_{\substack{0\le a<M\\a\equiv r\pmod N}}P_{M,a}.
P_{M,j_o}Use in source · 1 passage
Search aliases: P_M,j_o
Use in source · Paper_V_Traditional_Final.tex, line 1236
[Dyadic--odd packet factorization] After identifying the residue labels by the Chinese remainder map, the phase projector factors as P_N,j=P_2^s,j_2P_M,j_o.
Exact mathematical passage
\boxed{ P_{N,j}=P_{2^s,j_2}P_{M,j_o}. }
P_{m,n}Use in source · 11 passages
Search aliases: P_m,n
Use in source · Paper_VI_Traditional_Final.tex, line 2650
[Asymptotic closure criterion] Assume the weighted convergence in [source reference] holds for every bounded time interval, and define P_m,n=R_n,mC_m,n on X_n. Suppose that for every T>0, ρ_m,n(T) = sup_0≤ s,t≤ T norm C_m,nT_n(s)(I-P_m,n)T_n(t)R_n,m ⟶0. Then U_m(s+t)=U_m(s)U_m(t) for all s,t≥0. If the convergence in [source reference] is locally uniform and every T_n is norm continuous, then U_m is a norm continuous semigroup.
Exact mathematical passage
P_{m,n}=R_{n,m}C_{m,n}
P_{M,r}Use in source · 2 passages
Search aliases: P_M,r
Use in source · 10_foundations.tex, line 50
[Residue reconstruction] The selectors satisfy P_N,j^2=P_N,j, P_N,iP_N,j=0 (i≠ j), ∑_j=0^N-1P_N,j=I. If Nmid M and M>0, then P_N,ja=∑_substack0≤ r<M r≡ jpmod NP_M,ra. The same identities hold coefficientwise for formal power series.
Exact mathematical passage
P_{N,j}a=\sum_{\substack{0\leq r<M\\r\equiv j\pmod N}}P_{M,r}a.
P_{M,r\bmod M}Use in source · 1 passage
Search aliases: P_M,rbmod M bmod
Use in source · 10_foundations.tex, line 456
[Dyadic--odd packet factorization] Let a≥0 and let M be a positive odd integer. The Chinese remainder map is a ring isomorphism Z/(2^aM) Z≃ Z/2^a Z× Z/M Z. For 0≤ r<2^aM and any coefficient sequence f, P_2^aM,rf =P_2^a,rbmod 2^a P_M,rbmod Mf. The same identity holds for formal-series coefficient selectors.
Exact mathematical passage
P_{2^aM,r}f =P_{2^a,r\bmod 2^a}\,P_{M,r\bmod M}f.
P_{M,s}Use in source · 3 passages
Search aliases: P_M,s
Use in source · 02_coordinates.tex, line 478
[Nested residue refinement] Let N,M be positive integers with N dividing M, and let 0≤ j<N. Then P_N,j = ∑_substack0≤ s<M s≡ jpmod NP_M,s. When R is nonzero, the packet system modulo M refines the packet system modulo N.
Exact mathematical passage
P_{N,j} = \sum_{\substack{0\leq s<M\\s\equiv j\pmod N}}P_{M,s}.
P_{m-1}Use in source · 1 passage
Search aliases: P_m-1
Use in source · 02_supplement_03.tex, line 19
Construction. Suppose P_m-1=D^(m-1)+L_m-1R_m-1^*. The new factors are L_m · =[D_mL_m-1+U_m(V_m^*L_m-1) U_m],
Exact mathematical passage
P_{m-1}=D^{(m-1)}+L_{m-1}R_{m-1}^*
P_{\min(i,j)}Use in source · 3 passages
Search aliases: P_min(i,j) min
Use in source · Paper_VI_Traditional_Final.tex, line 705
A coarse-block average of a fine-block average is the coarse-block average, and a fine-block average of a coarse-constant function leaves it unchanged. The difference identities follow from P_iP_j=P_min(i,j).
Exact mathematical passage
P_iP_j=P_{\min(i,j)}
P_nUse in source · 22 passages
Search aliases: P_n
Use in source · 03_packets_and_graphs.tex, line 409
[Countable unconditional packet isomorphism] Let X be a real or complex Banach space and let (P_n)_n≥1⊆ B(X) satisfy P_nP_m=δ_nmP_n and x=∑_n≥1P_nx unconditionally for every x∈ X. Assume K:= sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nP_n <∞. Then analysis and synthesis define inverse Banach space isomorphisms X≅⊕_n≥1^uP_nX. More precisely, A_ P ≤ K, S_ P ≤1. [source reference] ExportsCountable analytic packet decompositions under explicit convergence control.
Exact mathematical passage
(P_n)_{n\geq1}\subseteq\mathcal B(X)
p_nUse in source · 1 passage
Search aliases: p_n
Use in source · Paper_IV_Traditional_Final.tex, line 1131
[Logarithmic Newton recurrence] Set b_N,n(r)=fracP_N,n(r)(Nn)!. Then b_N,0(r)=1 and for every n≥1, nb_N,n(r) = r∑_k=1^nκ_N,kb_N,n-k(r). Equivalently, P_N,n(r) = frac r n ∑_k=1^n frac(Nn)!(N(n-k))! κ_N,kP_N,n-k(r). If p_k=rκ_N,k, then, for n≥1, b_N,n(r) = frac1n! det p_1 · -1 · 0 · ⋯ · 0 p_2 · p_1 · -2 · ⋯ · 0 p_3 · p_2 · p_1 · ddots · vdots vdots · vdots · ddots · ddots · -(n-1) p_n · p_n-1 · ⋯ · p_2 · p_1 .
Exact mathematical passage
b_{N,n}(r) = \frac1{n!} \det \begin{pmatrix} p_1&-1&0&\cdots&0\\ p_2&p_1&-2&\cdots&0\\ p_3&p_2&p_1&\ddots&\vdots\\ \vdots&\vdots&\ddots&\ddots&-(n-1)\\ p_n&p_{n-1}&\cdots&p_2&p_1 \end{pmatrix}.
P_{N,0}Use in source · 1 passage
Search aliases: P_N,0
Use in source · Paper_IV_Traditional_Final.tex, line 1028
[Parameter recurrence] We have P_N,0(r)=1. For n≥1, P_N,n(r) = · - ∑_k=1^n-1 (-1)^k binomNn-1Nk P_N,n-k(r)
Exact mathematical passage
P_{N,0}(r)=1
\mathcal P_{N,0}Use in source · 1 passage
Search aliases: P_N,0 mathcal
Use in source · 20_arithmetic.tex, line 223
Set H_c(t)=∑_n≥0fract^Nn+c(Nn+c)!, A_c(t)=∑_n≥0fract^Nn(Nn+c)!, R_c(t)=fracA_c-1(t)A_c(t) (1≤ c<N). The denominator of this last quotient has nonzero constant coefficient. The section identities give R_cH_c=tH_c-1, R_c(0)=c, j! P_N,j= P_N,0∏_c=1^jR_c. Cancellation of consecutive source sections proves the product identity and ties the ratios directly to the family under study.
Exact mathematical passage
R_cH_c=tH_{c-1},\qquad R_c(0)=c,\qquad j!\mathcal P_{N,j}=\mathcal P_{N,0}\prod_{c=1}^{j}R_c.
P_{N,1}Use in source · 1 passage
Search aliases: P_N,1
Use in source · Paper_IV_Traditional_Final.tex, line 1175
[The first three reciprocal polynomials] For every N≥1, P_N,1(r)=r, P_N,2(r) = r[ frac12binom2NN(r+1)-1 ], and P_N,3(r) = r[ frac(3N)!6(N!)^3(r+1)(r+2) - binom3NN(r+1) +1 ].
Exact mathematical passage
P_{N,1}(r)=r,
P_{N,2}Use in source · 1 passage
Search aliases: P_N,2
Use in source · Paper_IV_Traditional_Final.tex, line 1178
[The first three reciprocal polynomials] For every N≥1, P_N,1(r)=r, P_N,2(r) = r[ frac12binom2NN(r+1)-1 ], and P_N,3(r) = r[ frac(3N)!6(N!)^3(r+1)(r+2) - binom3NN(r+1) +1 ].
Exact mathematical passage
P_{N,2}(r) = r\left[ \frac12\binom{2N}{N}(r+1)-1 \right],
P_{N,3}Use in source · 1 passage
Search aliases: P_N,3
Use in source · Paper_IV_Traditional_Final.tex, line 1186
[The first three reciprocal polynomials] For every N≥1, P_N,1(r)=r, P_N,2(r) = r[ frac12binom2NN(r+1)-1 ], and P_N,3(r) = r[ frac(3N)!6(N!)^3(r+1)(r+2) - binom3NN(r+1) +1 ].
Exact mathematical passage
P_{N,3}(r) = r\left[ \frac{(3N)!}{6(N!)^3}(r+1)(r+2) - \binom{3N}{N}(r+1) +1 \right].
P_{N,i}Use in source · 7 passages
Search aliases: P_N,i
Use in source · 02_coordinates.tex, line 383
[Residue packet algebra] For every commutative ring R, the operators (P_N,j)_0≤ j<N satisfy the identities below. If R is nonzero, they form a packet system on R[[z]]: P_N,iP_N,j=δ_ijP_N,j, ∑_j=0^N-1P_N,j=I.
Exact mathematical passage
P_{N,i}P_{N,j}=\delta_{ij}P_{N,j}, \qquad \sum_{j=0}^{N-1}P_{N,j}=I.
\mathcal P_{N,j}Use in source · 5 passages
Search aliases: P_N,j mathcal
Use in source · 20_arithmetic.tex, line 29
The normalized source and its inverse are K_N,j(t)=j!∑_n≥0fract^Nn(Nn+j)!, P_N,j(t)=∑_n≥0C_N,j(n)fract^Nn(Nn)!. Their product is one. Clearing factorials in its coefficient of t^Nn gives the triangular system ∑_q=0^nbinomNn+jNqC_N,j(q)=δ_n0. In particular, for n>0, C_N,j(n)=-binomNn+jNn^-1 ∑_q=0^n-1binomNn+jNqC_N,j(q). The diagonal binomial coefficient is positive, so this recurrence determines the sequence uniquely. The Lean construction begins with this recurrence and derives the reciprocal id…
Exact mathematical passage
\label{eq:arith-kernel} K_{N,j}(t)=j!\sum_{n\ge0}\frac{t^{Nn}}{(Nn+j)!},\qquad \mathcal P_{N,j}(t)=\sum_{n\ge0}C_{N,j}(n)\frac{t^{Nn}}{(Nn)!}.
P_{N,j+1}Use in source · 2 passages
Search aliases: P_N,j+1
Use in source · 05_worked_models.tex, line 234
0, · otherwise. They form an orthogonal packet system and satisfy S_1P_N,j=P_N,j+1S_1, with indices modulo N. The shift is therefore a cyclic transfer between residue packets.
Exact mathematical passage
S_1P_{N,j}=P_{N,j+1}S_1,
P_{N,j+r}Use in source · 1 passage
Search aliases: P_N,j+r
Use in source · 06_models.tex, line 24
A banded operator need not remain banded after residue reordering, but its nonzero couplings become a structured packet graph. With the convention (S_rx)_n=x_n-r, the only active transfer is P_N,j+rS_rP_N,j, with indices taken modulo N. More general finite stencils produce a finite set of active phase differences.
Exact mathematical passage
P_{N,j+r}S_rP_{N,j},
P_{N,j}^2Use in source · 1 passage
Search aliases: P_N,j^2
Use in source · 10_foundations.tex, line 45
[Residue reconstruction] The selectors satisfy P_N,j^2=P_N,j, P_N,iP_N,j=0 (i≠ j), ∑_j=0^N-1P_N,j=I. If Nmid M and M>0, then P_N,ja=∑_substack0≤ r<M
Exact mathematical passage
P_{N,j}^2=P_{N,j},\qquad P_{N,i}P_{N,j}=0\quad(i\ne j), \qquad \sum_{j=0}^{N-1}P_{N,j}=I.
P_{N,n-k}Use in source · 3 passages
Search aliases: P_N,n-k
Use in source · Paper_IV_Traditional_Final.tex, line 1029
[Parameter recurrence] We have P_N,0(r)=1. For n≥1, P_N,n(r) = · - ∑_k=1^n-1 (-1)^k binomNn-1Nk P_N,n-k(r) · -r ∑_k=1^n (-1)^k binomNn-1Nk-1 P_N,n-k(r).
Exact mathematical passage
\begin{aligned} P_{N,n}(r) ={}&- \sum_{k=1}^{n-1} (-1)^k \binom{Nn-1}{Nk} P_{N,n-k}(r) \\ &-r \sum_{k=1}^{n} (-1)^k \binom{Nn-1}{Nk-1} P_{N,n-k}(r). \end{aligned}
P_{N,n-q}Use in source · 3 passages
Search aliases: P_N,n-q
Use in source · Paper_IV_Traditional_Final.tex, line 1071
[Additive parameter law for reciprocal powers] For all r,s∈ C, S_N^(r)(z) S_N^(s)(z) = S_N^(r+s)(z) as a formal identity and on every common zero-free region with a fixed logarithm. Consequently, P_N,n(r+s) = ∑_q=0^n binomNnNq P_N,q(r)P_N,n-q(s).
Exact mathematical passage
P_{N,n}(r+s) = \sum_{q=0}^{n} \binom{Nn}{Nq} P_{N,q}(r)P_{N,n-q}(s).
P_{N,q}Use in source · 3 passages
Search aliases: P_N,q
Use in source · Paper_IV_Traditional_Final.tex, line 1071
[Additive parameter law for reciprocal powers] For all r,s∈ C, S_N^(r)(z) S_N^(s)(z) = S_N^(r+s)(z) as a formal identity and on every common zero-free region with a fixed logarithm. Consequently, P_N,n(r+s) = ∑_q=0^n binomNnNq P_N,q(r)P_N,n-q(s).
Exact mathematical passage
P_{N,n}(r+s) = \sum_{q=0}^{n} \binom{Nn}{Nq} P_{N,q}(r)P_{N,n-q}(s).
p_{n-1}Use in source · 1 passage
Search aliases: p_n-1
Use in source · Paper_IV_Traditional_Final.tex, line 1131
[Logarithmic Newton recurrence] Set b_N,n(r)=fracP_N,n(r)(Nn)!. Then b_N,0(r)=1 and for every n≥1, nb_N,n(r) = r∑_k=1^nκ_N,kb_N,n-k(r). Equivalently, P_N,n(r) = frac r n ∑_k=1^n frac(Nn)!(N(n-k))! κ_N,kP_N,n-k(r). If p_k=rκ_N,k, then, for n≥1, b_N,n(r) = frac1n! det p_1 · -1 · 0 · ⋯ · 0 p_2 · p_1 · -2 · ⋯ · 0 p_3 · p_2 · p_1 · ddots · vdots vdots · vdots · ddots · ddots · -(n-1) p_n · p_n-1 · ⋯ · p_2 · p_1 .
Exact mathematical passage
b_{N,n}(r) = \frac1{n!} \det \begin{pmatrix} p_1&-1&0&\cdots&0\\ p_2&p_1&-2&\cdots&0\\ p_3&p_2&p_1&\ddots&\vdots\\ \vdots&\vdots&\ddots&\ddots&-(n-1)\\ p_n&p_{n-1}&\cdots&p_2&p_1 \end{pmatrix}.
P_{N_j,r_j}Use in source · 1 passage
Search aliases: P_N_j,r_j
Use in source · 72_recovered.tex, line 3031
[Several coprime moduli] Let N_1,…,N_s be pairwise coprime and put N=∏_j=1^sN_j. For compatible addresses r_j, P_N,r = ∏_j=1^sP_N_j,r_j, where r is the unique combined address modulo N.
Exact mathematical passage
\boxed{ P_{N,r} = \prod_{j=1}^{s}P_{N_j,r_j}, }
P_{\mathrm{ob}}Use in source · 3 passages
Search aliases: P_ob mathrm
Use in source · 04_correction_transport.tex, line 7
[Orthogonal obstruction projector] Let D:H_1→ H_2 be a bounded operator between Hilbert spaces with closed range. Let D^† be its Moore--Penrose inverse. Then P_ob=I_H_2-DD^† is the orthogonal projector onto (im D)^⊥=Ker D^*. For r∈ H_2, the following are equivalent: item Du=-r is solvable item [r]=0 in coker D item P_obr=0. When these conditions hold, -D^† r is the least norm correction. For every r, dist(r,im D)= P_obr .
Exact mathematical passage
P_{\mathrm{ob}}=I_{H_2}-DD^\dagger
P_{\mathrm{odd}}Use in source · 1 passage
Search aliases: P_odd mathrm
Use in source · 01_finite_core.tex, line 228
For parity, S=Z^⊗ n, P_even=fracI+S2, P_odd=fracI-S2.
Exact mathematical passage
S=Z^{\otimes n}, \qquad P_{\mathrm{even}}=\frac{I+S}{2}, \qquad P_{\mathrm{odd}}=\frac{I-S}{2}.
P_{\phys}Use in source · 6 passages
Search aliases: P_phys phys mathrm
Use in source · 02_strengthened.tex, line 45
Constraint closure needs a different obstruction. If C_a are self adjoint constraints and P_phys projects onto their common kernel, then P_phys[C_a,C_b]P_phys=0 holds automatically. This compression cannot test closure. The correct finite obstruction is the class of [C_a,C_b] in the quotient of End( H) by the span of the constraints. Vanishing means that the constraint span is closed under commutators.
Exact mathematical passage
P_{\phys}
P_qUse in source · 2 passages
Search aliases: P_q
Use in source · root_scope.tex, line 107
Set r=M+1≥1 and define P_q(x)=∏_i=0^q-1(x-i)(2x-(2i+1)), P_0=1, H_r(x)=∑_q≥08^qfracβ_r+qβ_r P_q(x)mathscr A(x-q). The coefficient valuations tend to infinity. At any nonnegative integer x, terms with q>x vanish. Cancelling the factorials in [source reference] therefore gives the exact identity T_r-1(N)=κ_rbinom2N2rH_r(N-r) (N≥ r). This is a finite rational identity at every original Euler row.
Exact mathematical passage
P_q(x)=\prod_{i=0}^{q-1}(x-i)(2x-(2i+1)),\qquad P_0=1,
\mathcal P_qUse in source · 12 passages
Search aliases: P_q mathcal
Use in source · Paper_X_Block27_Amended.tex, line 2325
Fix the continuous quotient q:G_ Q⟶ C_2 obtained by the action on a chosen square root i of -1 in the algebraic closure. Its surjectivity and the equivalence q(σ)=1 Longleftrightarrow σ(i)=i are proved using the existing actual quadratic Galois construction. For a commutative ring B, define the arithmetic packet P_q(B)= {χ:G_ Q⟶ B^×midker q⊆kerχ}, where the maps are group homomorphisms. Unique descent through the actual surjective map q is derived, rather than supplied. Every member has finite i…
Exact mathematical passage
\mathcal P_q(B)= \{\chi:G_{\Q}\longrightarrow B^\times\mid\ker q\subseteq\ker\chi\},
P_{qj}Use in source · 2 passages
Search aliases: P_qj
Use in source · Paper_XI_Traditional_Final.tex, line 324
[Tame character projectors] The operators P_j are pairwise orthogonal idempotents whose sum is the identity. Frobenius permutes them by Φ P_jΦ^-1=P_qj.
Exact mathematical passage
\Phi P_j\Phi^{-1}=P_{qj}.
p_rUse in source · 8 passages
Search aliases: p_r
Use in source · Paper_VI_Traditional_Final.tex, line 1093
Let K be column stochastic and let λ^(r) be the likelihood associated with the rth observation. The filter recursion is widehat p_r=Kp_r-1, p_r= B_λ^(r)(widehat p_r).
Exact mathematical passage
\widehat p_r=Kp_{r-1}, \qquad p_r=\cB_{\lambda^{(r)}}(\widehat p_r).
P_{r+1}Use in source · 1 passage
Search aliases: P_r+1
Use in source · strengthening.tex, line 581
Expand X^m=∑_r=0^mT(m,r)P_r(X). Since XP_r(X)=P_r+1(X)+r(r+1)P_r(X), these integer coefficients satisfy T(m+1,r)=T(m,r-1)+r(r+1)T(m,r), T(0,0)=1. Every r(r+1) is even. Induction, including the boundary indices, proves 2^m-rmid T(m,r) for 0≤ r≤ m. For m>0 the r=0 term vanishes. Each remaining summand of L(P^m/m!) has valuation at least m-r+2v_2(r!)-v_2(m!) =r+s_2(m)-2s_2(r)≥ s_2(m)-1≥0. Here r≥2s_2(r)-1 follows from r≥2^s_2(r)-1 and 2^u≥2u for u≥1. This proves integrality and the stronger bound.
Exact mathematical passage
XP_r(X)=P_{r+1}(X)+r(r+1)P_r(X)
p_{r-1}Use in source · 4 passages
Search aliases: p_r-1
Use in source · Paper_VI_Traditional_Final.tex, line 1093
Let K be column stochastic and let λ^(r) be the likelihood associated with the rth observation. The filter recursion is widehat p_r=Kp_r-1, p_r= B_λ^(r)(widehat p_r).
Exact mathematical passage
\widehat p_r=Kp_{r-1}, \qquad p_r=\cB_{\lambda^{(r)}}(\widehat p_r).
P_r^{\mathrm{coarse}}Use in source · 2 passages
Search aliases: P_r^coarse mathrm
Use in source · 01_finite_core.tex, line 247
[Tensor/polyphase refinement] For each residue r∈ Z/N Z, define P_r^coarse = ∑_j_1+⋯+j_m≡ rpmod N P^(1)_j_1⊗⋯⊗ P^(m)_j_m. Then the P_r^coarse are complete orthogonal projectors, and every coarse packet is the exact sum of its fine descendants.
Exact mathematical passage
\boxed{ P_r^{\mathrm{coarse}} = \sum_{j_1+\cdots+j_m\equiv r\pmod N} P^{(1)}_{j_1}\otimes\cdots\otimes P^{(m)}_{j_m}. }
P_r^{L}Use in source · 1 passage
Search aliases: P_r^L
Use in source · Paper_V_Traditional_Final.tex, line 1875
[Exact lattice refinement] For every coarse packet, P_r^L = ∑_s∈Child(r)P_s^L'. The fine packets in the sum are pairwise disjoint. Refinement through several nested sublattices is transitive.
Exact mathematical passage
P_r^{L} = \sum_{s\in\operatorname{Child}(r)}P_s^{L'}.
P_r^{\rm total}Use in source · 2 passages
Search aliases: P_r^rm total rm
Use in source · 02_strengthened.tex, line 351
[Tensor charge refinement] For r∈ Z/N Z, define P_r^rm total = ∑_j_1+⋯+j_m=r P_j_1^(1)⊗⋯⊗ P_j_m^(m). Then the P_r^rm total form a complete orthogonal packet system. Every total charge packet is the exact sum of its fine tensor descendants.
Exact mathematical passage
P_r^{\rm total} = \sum_{j_1+\cdots+j_m=r} P_{j_1}^{(1)}\otimes\cdots\otimes P_{j_m}^{(m)}.
P_{\rm even}Use in source · 2 passages
Search aliases: P_rm even rm
Use in source · monograph_atlas.tex, line 109
vpage7A coarse packet can still contain coherenceTENSOR REFINEMENT BY TOTAL PARITYChapter VII: Theorem II.3.7, Corollary II.3.8. For three qubits, the eight fine projectors are P_a=ket abra a. Grouping addresses according to the parity of their Hamming weight gives P_rm even=P_000+P_011+P_101+P_110 =fracI+Z⊗ Z⊗ Z2, P_rm odd=I-P_rm even. This is a sum of subspaces, not an averaging of their coordinates. The rank-four even packet retains all complex amplitudes on its four children. vfigparity head…
Exact mathematical passage
P_{\rm even}=P_{000}+P_{011}+P_{101}+P_{110} =\frac{I+Z\otimes Z\otimes Z}{2}, \qquad P_{\rm odd}=I-P_{\rm even}.
P_{\rm odd}Use in source · 2 passages
Search aliases: P_rm odd rm
Use in source · monograph_atlas.tex, line 109
vpage7A coarse packet can still contain coherenceTENSOR REFINEMENT BY TOTAL PARITYChapter VII: Theorem II.3.7, Corollary II.3.8. For three qubits, the eight fine projectors are P_a=ket abra a. Grouping addresses according to the parity of their Hamming weight gives P_rm even=P_000+P_011+P_101+P_110 =fracI+Z⊗ Z⊗ Z2, P_rm odd=I-P_rm even. This is a sum of subspaces, not an averaging of their coordinates. The rank-four even packet retains all complex amplitudes on its four children. vfigparity head…
Exact mathematical passage
P_{\rm even}=P_{000}+P_{011}+P_{101}+P_{110} =\frac{I+Z\otimes Z\otimes Z}{2}, \qquad P_{\rm odd}=I-P_{\rm even}.
P_sUse in source · 1 passage
Search aliases: P_s
Use in source · Paper_V_Traditional_Final.tex, line 1794
[Exact lattice polyphase decomposition] For every 1≤ p≤∞, the analysis map A_L:ℓ^p( Z^d)⟶⊕_r∈ Rℓ^p(L), A_Lx=(x_r)_r∈ R, is an isometric isomorphism. Its inverse is the interleaving map S_L(u)(r+ℓ)=u_r(ℓ). The projectors satisfy P_rP_s=δ_rsP_r, ∑_r∈ RP_r=I.
Exact mathematical passage
P_rP_s=\delta_{rs}P_r, \qquad \sum_{r\in R}P_r=I.
p_sUse in source · 2 passages
Search aliases: p_s
Use in source · Paper_X_Block27_Amended.tex, line 2519
Suppose now that R is I-adically complete and that every positive quotient R/I^m is finite. Let C_m⊆ P_q(R/I^m) be allowed subsets preserved by reduction, and let (p_s)_s∈ S be any indexed family of arithmetic primes. The index set S need not be finite. Lean proves the simultaneous arithmetic completion criterion · ∃χ∈ P_q(R), χ continuous, χbmod I^m∈ C_m (m≥1), χ_ Q_p_s=1 (s∈ S)
Exact mathematical passage
(p_s)_{s\in S}
P_{s,-1}Use in source · 1 passage
Search aliases: P_s,-1
Use in source · Paper_XII_Traditional_Final.tex, line 440
For detection on every irreducible submodule, one can check the stronger assertion that canonical projectors detect every nonzero matrix. In addition to the two projectors above, use P_s,-1=1 · 3 0 · 0, P_tst^-1,1=1 · 0 4 · 0. The four trace pairings with these matrices have trivial common kernel in M_2( F_7), by row reduction. Every irreducible adjoint submodule contains a nonzero matrix, so some canonical projector has nonzero trace pairing on that submodule. This verifies the last adequacy cl…
Exact mathematical passage
P_{s,-1}=\begin{pmatrix}1&3\\0&0\end{pmatrix}, \qquad P_{tst^{-1},1}=\begin{pmatrix}1&0\\4&0\end{pmatrix}.
P_{s,1}Use in source · 1 passage
Search aliases: P_s,1
Use in source · Paper_XII_Traditional_Final.tex, line 429
3 · 6, Tr(P_t,2W_sgn)=3≠0, P_s,1=0 · 4 0 · 1, Tr(P_s,1W_std)=6≠0. For an operator D with (D-α I)(D-β I)=0 and α≠β, the canonical projector is P_D,α=(α-β)^-1(D-β I). Its image equals ker((D-α I)^n) for every n≥1, and it kills ker((D-β I)^n). This identifies the displayed matrices with the generalized eigenspace projectors required in the adequacy definition.
Exact mathematical passage
P_{s,1}=\begin{pmatrix}0&4\\0&1\end{pmatrix}, \qquad \Tr(P_{s,1}W_{\mathrm{std}})=6\ne0.
P_{s,m}Use in source · 2 passages
Search aliases: P_s,m
Use in source · Paper_X_Block27_Amended.tex, line 946
om fixed-depth certificates. Let R be an I-adically complete separated commutative ring with finite coefficient quotients and an injective map into a field K. Fix a finite matrix rank. For finitely many local group maps ϕ_s:H_s→ G and integral template representations τ_s, fix detecting depths d_s. At precision m retain matrices P_s,m as part of the admissible object and require ρ_m(ϕ_s(h))P_s,m=P_s,mτ_s,m(h). For m≥ d_s require their determinants to remain nonzero after reduction to R/I^d_s. Ea…
Exact mathematical passage
P_{s,m}
P_{s,t}Use in source · 4 passages
Search aliases: P_s,t
Use in source · 03_weighted.tex, line 507
w(e)≤ n a_n-w(e)(target(e),t). Let T(t)=I-A(t), Q(t)=det T(t), P_s,t(t)=adj(T(t))_s,t. Then Q(0)=1, hence Q≠0 and Q is a unit in Z[[t]], and the independently enumerated walk series satisfies Z_s,t(t)=Q(t)^-1P_s,t(t), Q(t)Z_s,t(t)=P_s,t(t) in Z[[t]].
Exact mathematical passage
T(t)=I-A(t),\qquad Q(t)=\det T(t),\qquad P_{s,t}(t)=\operatorname{adj}(T(t))_{s,t}.
P_s^{L'}Use in source · 1 passage
Search aliases: P_s^L'
Use in source · Paper_V_Traditional_Final.tex, line 1875
[Exact lattice refinement] For every coarse packet, P_r^L = ∑_s∈Child(r)P_s^L'. The fine packets in the sum are pairwise disjoint. Refinement through several nested sublattices is transitive.
Exact mathematical passage
P_r^{L} = \sum_{s\in\operatorname{Child}(r)}P_s^{L'}.
p_tUse in source · 2 passages
Search aliases: p_t
Use in source · Paper_VI_Traditional_Final.tex, line 5020
[Residual life as a restarted semigroup] For S(t)>0, define the conditional transient distribution p_t = frac e^tAp_0S(t). Then fracS(t+s)S(t) = 1^T e^sAp_t.
Exact mathematical passage
p_t = \frac{\e^{tA}p_0}{S(t)}.
P_{t,2}Use in source · 1 passage
Search aliases: P_t,2
Use in source · Paper_XII_Traditional_Final.tex, line 425
The sign witness is W_sgn=Z, while W_std=W(6,4,3)=3A+3B belongs to the standard summand. The corresponding projectors and trace pairings are P_t,2=2 · 4 3 · 6, Tr(P_t,2W_sgn)=3≠0, P_s,1=0 · 4
Exact mathematical passage
P_{t,2}=\begin{pmatrix}2&4\\3&6\end{pmatrix}, \qquad \Tr(P_{t,2}W_{\mathrm{sgn}})=3\ne0,
P_{tst^{-1},1}Use in source · 1 passage
Search aliases: P_tst^-1,1
Use in source · Paper_XII_Traditional_Final.tex, line 440
For detection on every irreducible submodule, one can check the stronger assertion that canonical projectors detect every nonzero matrix. In addition to the two projectors above, use P_s,-1=1 · 3 0 · 0, P_tst^-1,1=1 · 0 4 · 0. The four trace pairings with these matrices have trivial common kernel in M_2( F_7), by row reduction. Every irreducible adjoint submodule contains a nonzero matrix, so some canonical projector has nonzero trace pairing on that submodule. This verifies the last adequacy cl…
Exact mathematical passage
P_{s,-1}=\begin{pmatrix}1&3\\0&0\end{pmatrix}, \qquad P_{tst^{-1},1}=\begin{pmatrix}1&0\\4&0\end{pmatrix}.
P_UUse in source · 2 passages
Search aliases: P_U
Use in source · visual_atlas.tex, line 159
headWhy this does not prove an unrestricted minimum of two The bound path algebra includes the vertex idempotents P_U,P_V. A mixed seed s=u_1+v_3∈ U⊕ V satisfies P_Us=u_1 and P_Vs=v_3. It therefore generates the entire representation by itself. The chapter proves that no seed supported at U is enough and that two displayed pure seeds suffice. It does not claim that two arbitrary mixed seeds are necessary.
Exact mathematical passage
P_U,P_V
p_UUse in source · 4 passages
Search aliases: p_U
Use in source · 03_nonlinear_coordinates.tex, line 1044
[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is a natural exact sequence 0 · ⟶Ker D_0 ⟶Ker D_f ⟶Ker D_c xrightarrowδcoker D_0
Exact mathematical passage
\begin{array}{ccccccccc} 0&\longrightarrow&U_0&\xrightarrow{i_U}&U_f&\xrightarrow{p_U}&U_c&\longrightarrow&0\\[4pt] &&\mathllap{D_0}\downarrow&&\mathllap{D_f}\downarrow&&\mathllap{D_c}\downarrow&&\\[4pt] 0&\longrightarrow&V_0&\xrightarrow{i_V}&V_f&\xrightarrow{p_V}&V_c&\longrightarrow&0. \end{array}
p_uUse in source · 6 passages
Search aliases: p_u
Use in source · Paper_XI_Traditional_Final.tex, line 1693
[Exact edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD_eA_p_u^-1.
Exact mathematical passage
p_u
P_VUse in source · 2 passages
Search aliases: P_V
Use in source · visual_atlas.tex, line 159
headWhy this does not prove an unrestricted minimum of two The bound path algebra includes the vertex idempotents P_U,P_V. A mixed seed s=u_1+v_3∈ U⊕ V satisfies P_Us=u_1 and P_Vs=v_3. It therefore generates the entire representation by itself. The chapter proves that no seed supported at U is enough and that two displayed pure seeds suffice. It does not claim that two arbitrary mixed seeds are necessary.
Exact mathematical passage
P_U,P_V
\mathfrak P_vUse in source · 2 passages
Search aliases: P_v mathfrak
Use in source · Paper_V_Traditional_Final.tex, line 1015
[1mm] displaystyle max{ε_v, ∑_u∈ch(v)E(u)}, · v internal. For the complete prunings P_v below v, the envelope is exactly E(v)=max_Λ∈ P_v∑_a∈Λε_a. The maximum is attained. Every complete-pruning cost changes by at most E(∅), and the two globally optimal values differ by at most this quantity. Consequently, for a unique optimum with competitors, Γ_y>2E(∅) is a deterministic certificate that the globally optimal pruning is unchanged. A singleton pruning set is stable without a gap condition. Assume…
Exact mathematical passage
\mathfrak P_v
p_VUse in source · 7 passages
Search aliases: p_V
Use in source · 03_nonlinear_coordinates.tex, line 1044
[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is a natural exact sequence 0 · ⟶Ker D_0 ⟶Ker D_f ⟶Ker D_c xrightarrowδcoker D_0
Exact mathematical passage
\begin{array}{ccccccccc} 0&\longrightarrow&U_0&\xrightarrow{i_U}&U_f&\xrightarrow{p_U}&U_c&\longrightarrow&0\\[4pt] &&\mathllap{D_0}\downarrow&&\mathllap{D_f}\downarrow&&\mathllap{D_c}\downarrow&&\\[4pt] 0&\longrightarrow&V_0&\xrightarrow{i_V}&V_f&\xrightarrow{p_V}&V_c&\longrightarrow&0. \end{array}
p_vUse in source · 6 passages
Search aliases: p_v
Use in source · Paper_XI_Traditional_Final.tex, line 1693
[Exact edge residual from cycle defect] Fix a spanning tree and construct the vertex maps by tree transport from U_r. Let e:u→ v be a non tree edge. Let p_u and p_v be the tree paths from the root to u and v. Define the fundamental cycle defect D_e = U_rA_p_v^-1A_eA_p_u - B_p_v^-1B_eB_p_uU_r. Then the edge residual is exactly U_vA_e-B_eU_u = B_p_vD_eA_p_u^-1.
Exact mathematical passage
p_v
P_{v0}Use in source · 1 passage
Search aliases: P_v0
Use in source · atlas-pages.tex, line 200
es, not time intervals in the original signal. vfigcomplete_cuts headThree candidate selections The complete selection {0,10,11} covers every terminal branch once. The selection {0,00,1} covers the 00 branch twice: it is not an antichain. The selection {00,01,10} misses the 11 branch entirely. For actual Haar packets, set P_v=W_v^*W_v. Parent-to-child conservation gives P_v=P_v0+P_v1. Replacing unselected parents by their children until the complete pruning is reached proves ∑_v∈ΛP_v=I. This is …
Exact mathematical passage
P_v=P_{v0}+P_{v1}.
P_{v1}Use in source · 1 passage
Search aliases: P_v1
Use in source · atlas-pages.tex, line 200
es, not time intervals in the original signal. vfigcomplete_cuts headThree candidate selections The complete selection {0,10,11} covers every terminal branch once. The selection {0,00,1} covers the 00 branch twice: it is not an antichain. The selection {00,01,10} misses the 11 branch entirely. For actual Haar packets, set P_v=W_v^*W_v. Parent-to-child conservation gives P_v=P_v0+P_v1. Replacing unselected parents by their children until the complete pruning is reached proves ∑_v∈ΛP_v=I. This is …
Exact mathematical passage
P_v=P_{v0}+P_{v1}.
P_{\alpha}Use in source · 2 passages
Search aliases: P_α alpha
Use in source · 05_worked_models.tex, line 439
For a finite ordered seed family S=(s_1,…,s_r), let ε_S: A^⊕ r→ X be the seed evaluation map of [source reference]. Its kernel is the relation module. The free module modulo that kernel presents the generated hull by [source reference]. The invariant content is: · generated hull: Hull_ A(S), · seed complexity: srank_ A(X), · packet coordinates: x↔(P_αx)_α, · operator coordinates: T↔(P_βTP_α)_β,α, · relation module: N_S:=Kerε_S≤ A^⊕ r, · free presentation: A^⊕ r/N_S≅Hull_ A(S), · coordinate invar…
Exact mathematical passage
{ \begin{aligned} &\text{generated hull: }\Hull_{\cA}(S),\\ &\text{seed complexity: }\srank_{\cA}(X),\\ &\text{packet coordinates: }x\leftrightarrow(P_{\alpha}x)_{\alpha},\\ &\text{operator coordinates: }T\leftrightarrow(P_{\beta}TP_{\alpha})_{\beta,\alpha},\\ &\text{relation module: }N_S:=\Ker\varepsilon_S\leq\cA^{\oplus r},\\ &\text{free presentation: }\cA^{\oplus r}/N_S\cong\Hull_{\cA}(S),\\ &\text{coordinate invariance: }P_{\alpha}\mapsto UP_{\alpha}U^{-1},\\ &\text{realization: }\Phi A=\varphi(A)\Phi. \end{aligned} }
p_\alphaUse in source · 3 passages
Search aliases: p_α alpha
Use in source · Paper_VI_Traditional_Final.tex, line 3654
[Exact information decomposition] Let p and q be probability vectors on a finite partition (I_α)_α, and assume q_i=0 whenever p_i=0. Let m=Cp, n=Cq, and let p_α and q_α be the conditional distributions inside packet I_α whenever the corresponding mass is positive. Then D(qVert p) = D(nVert m) + ∑_α:n_α>0 n_α D(q_αVert p_α). The first term is the information retained by the coarse masses. The second term is the exact information carried by changes inside the packets.
Exact mathematical passage
p_\alpha
P_{\alpha'}Use in source · 1 passage
Search aliases: P_α' alpha
Use in source · 02_coordinates.tex, line 303
Commutativity gives R_αβ^2=P_α^2Q_β^2=R_αβ. If (α,β)≠(α',β'), then either P_α P_α'=0 or Q_β Q_β'=0, so the corresponding products annihilate one another. Finally, ∑_α,βR_αβ = (∑_α P_α) (∑_β Q_β) =I_X. For a fixed α, P_α =P_α∑_β Q_β = ∑_β R_αβ, and the same argument works for a fixed β. Since each P_α and Q_β is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite f…
Exact mathematical passage
P_\alpha P_{\alpha'}=0
P_{\alpha(e)}Use in source · 5 passages
Search aliases: P_α(e) alpha
Use in source · pages.tex, line 333
For orthogonal coordinate blocks, the squared Frobenius norms add exactly. Operator norms instead require a suitable bound or further spectral information. The maximum omitted block norm is generally insufficient. headInput sensitivity is additional information The packet perturbation estimate is (widehat A-A)x ≤∑_e E_e P_α(e)x . It can be much tighter for an input that scarcely occupies an affected source packet. To obtain a uniform operator bound, the input dependence must be bounded uniformly…
Exact mathematical passage
\|(\widehat A-A)x\|\le\sum_e\|E_e\|\,\|P_{\alpha(e)}x\|.
P_{\alpha(j,e)}Use in source · 2 passages
Search aliases: P_α(j,e) alpha
Use in source · 01_manuscript.tex, line 1322
[Mixed precision product chain] If each fixed linear factor A_j is replaced by a fixed packetwise linear approximation widehat A_j, write E_j=widehat A_j-A_j=∑_eΔ_j,eP_α(j,e). A uniform certificate is E_j ≤δ_j:=∑_e Δ_j,e P_α(j,e) . For orthogonal packet projectors, δ_j≤∑_e Δ_j,e . The final operator error is bounded by [source reference] with these uniform δ_j. The corresponding vector error is this operator bound multiplied by the input norm.
Exact mathematical passage
E_j=\widehat A_j-A_j=\sum_e\Delta_{j,e}P_{\alpha(j,e)}
p_{\alpha,i}Use in source · 2 passages
Search aliases: p_α,i alpha
Use in source · Paper_VI_Traditional_Final.tex, line 3668
For i∈ I_α, write p_i=m_α p_α,i, q_i=n_α q_α,i. Then logfracq_ip_i = logfracn_αm_α + logfracq_α,ip_α,i. Multiply by q_i, sum first inside each packet, and then sum over the packets. This is the finite relative entropy chain rule. Standard information theoretic treatments include [source reference].
Exact mathematical passage
p_i=m_\alpha p_{\alpha,i}, \qquad q_i=n_\alpha q_{\alpha,i}.
P_{\alpha_0}Use in source · 2 passages
Search aliases: P_α_0 alpha
Use in source · Paper_VI_Traditional_Final.tex, line 432
[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.
Exact mathematical passage
P_{\alpha_m}A^mP_{\alpha_0} = \sum_{\alpha_1,\ldots,\alpha_{m-1}} P_{\alpha_m}AP_{\alpha_{m-1}}\cdots P_{\alpha_1}AP_{\alpha_0}.
P_{\alpha_1}Use in source · 4 passages
Search aliases: P_α_1 alpha
Use in source · 03_nonlinear_coordinates.tex, line 236
[Exact reconstruction of derivatives] For h_1,…,h_r∈ X, D^rF(x)[h_1,…,h_r] = ∑_γ ∑_α_1,…,α_r J^r_γmidα_1,…,α_rF(x) [P_α_1h_1,…,P_α_rh_r].
Exact mathematical passage
D^rF(x)[h_1,\ldots,h_r] = \sum_{\gamma} \sum_{\alpha_1,\ldots,\alpha_r} J^{r}_{\gamma\mid\alpha_1,\ldots,\alpha_r}F(x) [P_{\alpha_1}h_1,\ldots,P_{\alpha_r}h_r].
P_{\alpha_j}Use in source · 4 passages
Search aliases: P_α_j alpha
Use in source · 03_nonlinear_coordinates.tex, line 264
[Jet norm bound] Give every packet range its inherited subspace norm. The restricted jet satisfies the sharper estimate J^r_γmidα_1,…,α_rF(x) ≤ Q_γ D^rF(x) . Its ambient extension inserts P_α_j in the jth slot and has norm at most Q_γ D^rF(x) ∏_j P_α_j . Consequently the operator norm of the restricted packet jet also satisfies J^r_γmidα_1,…,α_rF(x) ≤ Q_γ D^rF(x) ∏_j=1^r P_α_j . When all packets are orthogonal projections on Hilbert spaces, every nonzero projector has norm one and the bound redu…
Exact mathematical passage
P_{\alpha_j}
P_{\alpha_m}Use in source · 2 passages
Search aliases: P_α_m alpha
Use in source · Paper_VI_Traditional_Final.tex, line 432
[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.
Exact mathematical passage
P_{\alpha_m}A^mP_{\alpha_0} = \sum_{\alpha_1,\ldots,\alpha_{m-1}} P_{\alpha_m}AP_{\alpha_{m-1}}\cdots P_{\alpha_1}AP_{\alpha_0}.
P_{\alpha_{m-1}}Use in source · 2 passages
Search aliases: P_α_m-1 alpha
Use in source · Paper_VI_Traditional_Final.tex, line 432
[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.
Exact mathematical passage
P_{\alpha_m}A^mP_{\alpha_0} = \sum_{\alpha_1,\ldots,\alpha_{m-1}} P_{\alpha_m}AP_{\alpha_{m-1}}\cdots P_{\alpha_1}AP_{\alpha_0}.
P_{\alpha_r}Use in source · 2 passages
Search aliases: P_α_r alpha
Use in source · 03_nonlinear_coordinates.tex, line 236
[Exact reconstruction of derivatives] For h_1,…,h_r∈ X, D^rF(x)[h_1,…,h_r] = ∑_γ ∑_α_1,…,α_r J^r_γmidα_1,…,α_rF(x) [P_α_1h_1,…,P_α_rh_r].
Exact mathematical passage
D^rF(x)[h_1,\ldots,h_r] = \sum_{\gamma} \sum_{\alpha_1,\ldots,\alpha_r} J^{r}_{\gamma\mid\alpha_1,\ldots,\alpha_r}F(x) [P_{\alpha_1}h_1,\ldots,P_{\alpha_r}h_r].
P_{\beta}Use in source · 1 passage
Search aliases: P_β beta
Use in source · 05_worked_models.tex, line 439
For a finite ordered seed family S=(s_1,…,s_r), let ε_S: A^⊕ r→ X be the seed evaluation map of [source reference]. Its kernel is the relation module. The free module modulo that kernel presents the generated hull by [source reference]. The invariant content is: · generated hull: Hull_ A(S), · seed complexity: srank_ A(X), · packet coordinates: x↔(P_αx)_α, · operator coordinates: T↔(P_βTP_α)_β,α, · relation module: N_S:=Kerε_S≤ A^⊕ r, · free presentation: A^⊕ r/N_S≅Hull_ A(S), · coordinate invar…
Exact mathematical passage
{ \begin{aligned} &\text{generated hull: }\Hull_{\cA}(S),\\ &\text{seed complexity: }\srank_{\cA}(X),\\ &\text{packet coordinates: }x\leftrightarrow(P_{\alpha}x)_{\alpha},\\ &\text{operator coordinates: }T\leftrightarrow(P_{\beta}TP_{\alpha})_{\beta,\alpha},\\ &\text{relation module: }N_S:=\Ker\varepsilon_S\leq\cA^{\oplus r},\\ &\text{free presentation: }\cA^{\oplus r}/N_S\cong\Hull_{\cA}(S),\\ &\text{coordinate invariance: }P_{\alpha}\mapsto UP_{\alpha}U^{-1},\\ &\text{realization: }\Phi A=\varphi(A)\Phi. \end{aligned} }
P_\beta^*Use in source · 1 passage
Search aliases: P_β^* beta
Use in source · 03_packets_and_graphs.tex, line 253
Because dualization reverses products, P_α^*P_β^*=(P_β P_α)^* =δ_αβP_α^*. Also ∑_α P_α^*=I_X^*. Thus the dual maps form a packet system. Restriction gives a map from P_α^*X^* to X_α^*. It is injective because a covector in the range of P_α^* vanishes on every other packet, and it is surjective because any covector on X_α extends to X through the direct sum decomposition.
Exact mathematical passage
P_\alpha^*P_\beta^*=(P_\beta P_\alpha)^* =\delta_{\alpha\beta}P_\alpha^*.
P_\gammaUse in source · 10 passages
Search aliases: P_γ gamma
Use in source · 03_packets_and_graphs.tex, line 546
Insert I=∑_β P_β between S and T: P_γ STP_α = ∑_β P_γ SP_β TP_α.
Exact mathematical passage
P_\gamma STP_\alpha = \sum_\beta P_\gamma SP_\beta TP_\alpha.
P_{\gamma_1}Use in source · 1 passage
Search aliases: P_γ_1 gamma
Use in source · 03_packets_and_graphs.tex, line 957
It is enough to treat a word in the generators and packet projectors. Insert I=∑_γ∈ IP_γ between consecutive nonprojector generators. Every resulting (β,α) block is a finite sum of products P_β G_mP_γ_m-1 G_m-1⋯ P_γ_1G_1P_α. A nonzero product requires every consecutive block to be nonzero, hence gives a directed path α→γ_1→⋯→γ_m-1→β. Packet projectors only insert stationary steps. Thus a word has no block from α to an unreachable β. Linear combinations preserve the same zero pattern.
Exact mathematical passage
P_\beta G_mP_{\gamma_{m-1}} G_{m-1}\cdots P_{\gamma_1}G_1P_\alpha.
P_{\gamma_{m-1}}Use in source · 1 passage
Search aliases: P_γ_m-1 gamma
Use in source · 03_packets_and_graphs.tex, line 957
It is enough to treat a word in the generators and packet projectors. Insert I=∑_γ∈ IP_γ between consecutive nonprojector generators. Every resulting (β,α) block is a finite sum of products P_β G_mP_γ_m-1 G_m-1⋯ P_γ_1G_1P_α. A nonzero product requires every consecutive block to be nonzero, hence gives a directed path α→γ_1→⋯→γ_m-1→β. Packet projectors only insert stationary steps. Thus a word has no block from α to an unreachable β. Linear combinations preserve the same zero pattern.
Exact mathematical passage
P_\beta G_mP_{\gamma_{m-1}} G_{m-1}\cdots P_{\gamma_1}G_1P_\alpha.
P_\deltaUse in source · 1 passage
Search aliases: P_δ delta
Use in source · 01_finite_core.tex, line 751
Packet superoperator blocks are defined by Φ_αβ;γδ(X) = P_α Φ(P_γ XP_δ) P_β. Summing all four packet indices reconstructs Φ(X) exactly.
Exact mathematical passage
\Phi_{\alpha\beta;\gamma\delta}(X) = P_\alpha \Phi(P_\gamma XP_\delta) P_\beta.
P_\lambdaUse in source · 5 passages
Search aliases: P_λ lambda
Use in source · monograph_atlas.tex, line 534
cibility, not unproved continuum claims or a new theorem count. vpage29Nonabelian packets carry multiplicityA SIX-DIMENSIONAL REGULAR REPRESENTATIONChapter VII: Theorem II.3.1, Theorem II.3.2. Let HH= C[S_3] have the six orthonormal basis vectors ket g, with left action U(h)ket g=kethg. The irreducible characters on the identity, transposition, and three-cycle classes are c|rrr|r · e · (12) · (123) · d_λ trivial · 1 · 1 · 1 · 1 sign · 1 · -1 · 1 · 1 standard · 2 · 0 · -1 · 2 P_λ=fracd_λ6∑_g∈ S_3…
Exact mathematical passage
\begin{array}{c|rrr|r} & e & (12)&(123)&d_\lambda\\\hline \text{trivial}&1&1&1&1\\ \text{sign}&1&-1&1&1\\ \text{standard}&2&0&-1&2 \end{array} \qquad P_\lambda=\frac{d_\lambda}{6}\sum_{g\in S_3}\chi_\lambda(g^{-1})U(g).
P_{\lambda,v}Use in source · 4 passages
Search aliases: P_λ,v lambda
Use in source · Paper_XII_Traditional_Final.tex, line 1479
Choose elements γ_v∈Γ on a sample set Ω, with declared good subsets Ω_λ⊆Ω. In the arithmetic application these are the chosen Frobenius elements at the permitted places. For each v∈Ω_λ, supply an integral polynomial P_λ,v∈ O_λ[X] such that i_λ(P_λ,v)=e_λ(P_v) in K_λ[X]. Here and below a ring homomorphism applied to a polynomial acts on its coefficients. Assume every representative of every class in X_λ,n has characteristic polynomial P_λ,vbmod I_λ^n at γ_v. Also supply a subgroup H_λ≤Γ on which …
Exact mathematical passage
P_{\lambda,v}\in\OO_\lambda[X]
P_\muUse in source · 3 passages
Search aliases: P_μ mu
Use in source · visual_atlas.tex, line 358
vfig[.95]all_obstruction_blocks platecaptionThe board on the right contains all nine actual embedded blocks P_μOb P_ν. Row labels are output packets and column labels are input packets. Only the addresses (0,1) and (1,2) carry nonzero maps.
Exact mathematical passage
P_\mu\Ob P_\nu
P_\nuUse in source · 2 passages
Search aliases: P_ν nu
Use in source · visual_atlas.tex, line 358
vfig[.95]all_obstruction_blocks platecaptionThe board on the right contains all nine actual embedded blocks P_μOb P_ν. Row labels are output packets and column labels are input packets. Only the addresses (0,1) and (1,2) carry nonzero maps.
Exact mathematical passage
P_\mu\Ob P_\nu
P_{\rho(\beta)}Use in source · 1 passage
Search aliases: P_ρ(β) rho beta
Use in source · 02_coordinates.tex, line 57
0 · ρ(β)≠α. Hence Q_β X⊆ P_ρ(β)X. If a third packet system R refines Q, then R refines P through the composite parent map.
Exact mathematical passage
Q_\beta X\subseteq P_{\rho(\beta)}X
P_{\sigma^{-1}(i)}Use in source · 1 passage
Search aliases: P_σ^-1(i) sigma
Use in source · 04_transport.tex, line 122
For the block formula, use intertwining and packet compatibility twice: Q_jφ(T)Q_iΦ · =Q_jφ(T)Φ P_σ^-1(i) · =Q_jΦ TP_σ^-1(i) · =Φ P_σ^-1(j)TP_σ^-1(i) · =∑_substackσ(β)=j σ(α)=i Φ P_β TP_α. If Φ is surjective and Hull_ A(S)=X, the hull equality gives Y=Hull_φ( A)(Φ(S)). If Φ(S)⊆Hull_ B(S'), then Hull_ B(Φ(S))⊆Hull_ B(S'), so S' generates Y.
Exact mathematical passage
Q_j\varphi(T)Q_i\Phi &=Q_j\varphi(T)\Phi P_{\sigma^{-1}(i)}\\ &=Q_j\Phi TP_{\sigma^{-1}(i)}\\ &=\Phi P_{\sigma^{-1}(j)}TP_{\sigma^{-1}(i)}\\ &=\sum_{\substack{\sigma(\beta)=j\\\sigma(\alpha)=i}} \Phi P_\beta TP_\alpha.
P_{\ell+1,a}Use in source · 1 passage
Search aliases: P_ℓ+1,a ell
Use in source · 02_strengthened.tex, line 2404
[Exact packet direct limit] Assume the packet families have one fixed finite address set at every level and each packet address a is compatible in the sense P_ℓ+1,aR_ℓ=R_ℓ P_ℓ,a. Then there is an orthogonal projector P_∞,a on H_∞ satisfying P_∞,aι_ℓ=ι_ℓ P_ℓ,a. Finite completeness and orthogonality pass to the direct limit.
Exact mathematical passage
P_{\ell+1,a}R_\ell=R_\ell P_{\ell,a}.
P_{\ell+1,\beta}Use in source · 1 passage
Search aliases: P_ℓ+1,β ell beta
Use in source · 01_finite_core.tex, line 1026
A coherent refinement should satisfy C_ℓ E_ℓ=I and P_ℓ,α = C_ℓ ( ∑_βsuccαP_ℓ+1,β ) E_ℓ. For an operator D, define the refinement defect Δ_ℓ(D) =C_ℓ D_ℓ+1E_ℓ-D_ℓ. Exact resolution independence is Δ_ℓ(D)=0. When exact equality is too strong, the correct continuum problem is to prove convergence of the resolvents and uniform control of the effective self-energies.
Exact mathematical passage
\boxed{ P_{\ell,\alpha} = C_\ell \left( \sum_{\beta\succ\alpha}P_{\ell+1,\beta} \right) E_\ell. }
P_{\infty,a}Use in source · 2 passages
Search aliases: P_∞,a infty
Use in source · 02_strengthened.tex, line 2407
[Exact packet direct limit] Assume the packet families have one fixed finite address set at every level and each packet address a is compatible in the sense P_ℓ+1,aR_ℓ=R_ℓ P_ℓ,a. Then there is an orthogonal projector P_∞,a on H_∞ satisfying P_∞,aι_ℓ=ι_ℓ P_ℓ,a. Finite completeness and orthogonality pass to the direct limit.
Exact mathematical passage
P_{\infty,a}
\operatorname{pack}Use in source · 1 passage
Search aliases: pack operatorname
Use in source · supplement_11.tex, line 88
For a packed coefficient sequence e, the exact extraction formula is (U^8-U^4)^kpack_4(e)(4n-4k) =Δ_1^k e(n), k≤ n. Here U is the coefficient shift. The nonnegative extraction index is why the first estimate covers k≤ n. The existing saturation lemma extends it to all k with exponent min(k,n), and the step-multiple lemma extends step one to every natural u.
Exact mathematical passage
\bigl(U^8-U^4\bigr)^k\operatorname{pack}_4(e)(4n-4k) =\Delta_1^k e(n),\qquad k\le n.
\operatorname{pack}_4Use in source · 1 passage
Search aliases: pack_4 operatorname
Use in source · supplement_11.tex, line 88
For a packed coefficient sequence e, the exact extraction formula is (U^8-U^4)^kpack_4(e)(4n-4k) =Δ_1^k e(n), k≤ n. Here U is the coefficient shift. The nonnegative extraction index is why the first estimate covers k≤ n. The existing saturation lemma extends it to all k with exponent min(k,n), and the step-multiple lemma extends step one to every natural u.
Exact mathematical passage
\bigl(U^8-U^4\bigr)^k\operatorname{pack}_4(e)(4n-4k) =\Delta_1^k e(n),\qquad k\le n.
\mathsf{Packet}Use in source · 2 passages
Search aliases: Packet mathsf
Use in source · 02_connections_obstructions.tex, line 397
Let Candidate, Packet, and Residual be arbitrary types, with a distinguished 0∈Residual, and let r:Candidate⟶ Packet⟶Residual be a residual function.
Exact mathematical passage
\mathsf{Packet}
\operatorname{packet}Use in source · 1 passage
Search aliases: packet operatorname
Use in source · supplement_06.tex, line 31
pageheading1. Actual matrix orbits and arithmetic jumpsTheorem 9.1; manuscript page 22 Let R be a commutative ring, I a finite coordinate type and d=|I|. For C∈ M_I(R) and v_0∈ R^I, define orbit(C,v_0,n)=C^nv_0, u_k=packet(C,v_0,N,j,k)=C^Nk+jv_0. The implementation uses actual matrix powers and matrix--vector multiplication. A separately supplied sequence v with v_n+1=Cv_n is proved by induction to equal this power orbit. Thus the source-sequence formulation is connected to the construction.
Exact mathematical passage
\operatorname{orbit}(C,v_0,n)=C^nv_0, \qquad u_k=\operatorname{packet}(C,v_0,N,j,k)=C^{Nk+j}v_0.
\mathrm{packets}Use in source · 1 passage
Search aliases: packets mathrm
Use in source · visual_atlas.tex, line 481
headWhat one shift does in these coordinates Let S=S_1 and z=Sx. Substituting 3k+j into the shift definition gives z_1[k]=x_0[k], z_2[k]=x_1[k], z_0[k]=x_2[k-1]. The passage from phase 2 to phase 0 carries a shift of the coarse address. If Lh[k]=h[k-1], the exact packet matrix is S_packets=mat0 · 0 · L I · 0 · 0 0 · I · 0. Replacing L by a scalar 1 would preserve the phase graph but would lose the operator's action on the infinite sequences.
Exact mathematical passage
S_{\mathrm{packets}}=\mat{0&0&L\\I&0&0\\0&I&0}.
\pagerefVUse in source · 1 passage
Search aliases: pagerefV
Use in source · visual_atlas.tex, line 19
[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…
Exact mathematical passage
4pt] \textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt] \textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology? \end{tabularx} A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. \takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''} \vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}} Most of the chapter can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix. {\small \begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule \textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule $X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\ $S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\ $\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\ $\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\ $P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\ $X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\ $T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\ $F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\ $U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\ $\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\ $\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule \end{tabularx}} \head{Read products from right to left} In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column. \head{Where to begin} For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}. \vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$} Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define \[ Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0, \qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.
\Pcal'Use in source · 3 passages
Search aliases: Pcal' P' Pcal mathcal
Use in source · 03_nonlinear_coordinates.tex, line 169
[Change of packet coordinates] Let P and P' be packet systems on X. Let Q and Q' be packet systems on Y. For any two packet systems U and V on the same space, write J_ U← V = A_ U S_ V. Then F_ Q'← P' = J_ Q'← Q F_ Q← P J_ P← P'.
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\Pcal'
\Pcal_mUse in source · 1 passage
Search aliases: Pcal_m P_m Pcal mathcal
Use in source · 02_coordinates.tex, line 73
A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.
Exact mathematical passage
\Pcal_m
\Pcal_\ellUse in source · 2 passages
Search aliases: Pcal_ℓ P_ℓ Pcal ell mathcal ell
Use in source · 02_coordinates.tex, line 68
A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.
Exact mathematical passage
(\Pcal_\ell)_{\ell\geq0}
\mathrm{phys}Use in source · 5 passages
Search aliases: phys mathrm
Use in source · 01_finite_core.tex, line 1003
At resolution ℓ, define Q_ℓ = ( A_ℓ, H_ℓ, D_ℓ, {P_ℓ,α}, E_ℓ, C_ℓ, Π_ℓ^phys, T_ℓ ). In order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_ℓ: H_ℓ→ H_ℓ+1 and C_ℓ: H_ℓ+1→ H_ℓ. The left inverse relation C_ℓ E_ℓ=I alone does not imply C_ℓ=E_ℓ^* or make the embedding isometric.
Exact mathematical passage
\boxed{ \mathfrak Q_\ell = \left( \cA_\ell, \cH_\ell, D_\ell, \{P_{\ell,\alpha}\}, E_\ell, C_\ell, \Pi_\ell^{\mathrm{phys}}, T_\ell \right). }
\physUse in source · 10 passages
Search aliases: phys mathrm
Use in source · 02_strengthened.tex, line 45
Constraint closure needs a different obstruction. If C_a are self adjoint constraints and P_phys projects onto their common kernel, then P_phys[C_a,C_b]P_phys=0 holds automatically. This compression cannot test closure. The correct finite obstruction is the class of [C_a,C_b] in the quotient of End( H) by the span of the constraints. Vanishing means that the constraint span is closed under commutators.
Exact mathematical passage
P_{\phys}
\mathbf{PktMod}Use in source · 1 passage
Search aliases: PktMod mathbf
Use in source · 04_transport.tex, line 140
[Category of seeded packet systems] Packet compatible operator morphisms compose, seeded morphisms compose, and identity morphisms satisfy both definitions. Thus packet resolved operator modules form a category PktMod and seeded packet systems form a category SeedSys. DependsnolinkurlPSC-I-D-PACKET-MORPHISM, nolinkurlPSC-I-D-SEED-MORPHISM, and nolinkurlPSC-I-T-NATURALITY.
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\mathbf{PktMod}
\PPUse in source · 3 passages
Search aliases: PP P mathcal
Use in source · supplement_11.tex, line 54
Define a Kummer sequence over a commutative ring by P_a(f) Longleftrightarrow a^kmidΔ_1^k f(n) for all k,n. The exact discrete product rule Δ_1^s(fg)(n)=∑_i=0^sbinom si Δ_1^i f(n) Δ_1^s-ig(n+i) is proved by induction. It proves closure under products. Applied to fg=1, the same formula isolates Δ_1^s g(n)f(n+s). Induction and multiplication by g(n+s) prove closure under actual pointwise inverses.
Exact mathematical passage
\PP_a(f)\quad\Longleftrightarrow\quad a^k\mid\Delta_1^k f(n)\ \text{for all }k,n.
\PP_2Use in source · 1 passage
Search aliases: PP_2 P_2 PP mathcal
Use in source · supplement_11.tex, line 66
An affine sequence whose slope is divisible by a satisfies P_a. Consequently the four displayed source sequences satisfy P_2. No numerical interpolation or finite set of samples is used in that deduction.
Exact mathematical passage
\PP_2
\PP_aUse in source · 2 passages
Search aliases: PP_a P_a PP mathcal
Use in source · supplement_11.tex, line 54
Define a Kummer sequence over a commutative ring by P_a(f) Longleftrightarrow a^kmidΔ_1^k f(n) for all k,n. The exact discrete product rule Δ_1^s(fg)(n)=∑_i=0^sbinom si Δ_1^i f(n) Δ_1^s-ig(n+i) is proved by induction. It proves closure under products. Applied to fg=1, the same formula isolates Δ_1^s g(n)f(n+s). Induction and multiplication by g(n+s) prove closure under actual pointwise inverses.
Exact mathematical passage
\PP_a(f)\quad\Longleftrightarrow\quad a^k\mid\Delta_1^k f(n)\ \text{for all }k,n.
\PrUse in source · 1 passage
Search aliases: Pr
Use in source · monograph_atlas.tex, line 198
in(ε t)X, normQe^- i tHP=|sin(ε t)|, norm[H,P]=|ε|. The general commutator estimate becomes the elementary bound |sin u|≤ |u|. Unitarity also bounds the amplitude by one. vfigleakage headFrom a norm to a probability For this initial state, the escape probability is sin^2(ε t), not |sin(ε t)|. Consequently Pr(hidden packet at t)≤min{1,ε^2t^2}. At ε t=π/6, the amplitude is 1/2 and the probability is 1/4. Near zero, amplitudes are first order while escape probabilities are second order. Mixing thos…
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\Pr(\text{hidden packet at }t)\le\min\{1,\varepsilon^2t^2\}.
\mathrm{pr}Use in source · 3 passages
Search aliases: pr mathrm
Use in source · 03_weighted.tex, line 321
et q_L:R→ R_L be a ring homomorphism that kills F^L R. Suppose H=∑_e∈ EH_e, D^-1H_e∈ F^w(e)R, w(e)≥1, with E finite. Then q_L(D^-1H)^L=0, the finite sum in [source reference](a) is a two-sided inverse after applying q_L, and its word expansion may discard every word of weight at least L. More explicitly, for a length cutoff N, define S^pr_L,N = ∑_n=0^N-1(-1)^n ∑_substackγ∈ E^n W(γ)<LA_γ. Lean proves directly that q_L(S^pr_L,N) equals the mapped geometric inverse whenever q_L(A^N)=0, and hence is…
Exact mathematical passage
S^{\mathrm{pr}}_{L,N} = \sum_{n=0}^{N-1}(-1)^n \sum_{\substack{\gamma\in E^n\\W(\gamma)<L}}A_\gamma.
\operatorname{pr}Use in source · 2 passages
Search aliases: pr operatorname
Use in source · dyadic_synthesis.tex, line 53
(t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two, and z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact filtration valuation, and give the identity tr_LLift_E(C) =Tr_E/L (αtr_E C). The actual recu…
Exact mathematical passage
\kappa=\alpha^{-1},\qquad s=\kappa\operatorname{pr},\qquad s(P_0)=t_L/z^2,\qquad \operatorname{Lift}_E(C)=(z^2/t_L)CP_0.
\primeUse in source · 2 passages
Search aliases: prime
Use in source · 04_correction_transport.tex, line 863
The continuous bounded generator has an evolution inverse. Indeed, solve dot V=-VK with V(0)=I. Differentiation gives (VU)^prime=0, hence VU=I. The product W=UV satisfies dot W=KW-WK and W(0)=I. Uniqueness of this bounded linear equation gives UV=I. Thus U is invertible.
Exact mathematical passage
(VU)^{\prime}=0
\psrank_{A,\Pcal}Use in source · 1 passage
Search aliases: psrank_A,Pcal psrank_A, P psrank Pcal operatorname mathcal
Use in source · 03_packets_and_graphs.tex, line 1037
[Basic packet pure rank bounds] For every A module resolved into packets with P_α∈ A, srank_A(X) ≤ psrank_A, P(X) ≤ ∑_α∈ I srank_P_α A P_α(X_α). The corner algebra on the right is unital on X_α with identity P_α. [source reference]
Exact mathematical passage
\boxed{ \srank_A(X) \leq \psrank_{A,\Pcal}(X) \leq \sum_{\alpha\in I} \srank_{P_\alpha A P_\alpha}(X_\alpha). }
\mathbb
QUse in source · 1 passage
Search aliases: Q mathbb
Use in source · references.tex, line 65
bibitemKisin2010 Mark Kisin. newblock Deformations of G_ Q_p and GL_2( Q_p) representations. newblock In Laurent Berger, Christophe Breuil, and Pierre Colmez, editors, em Repr esentations p-adiques de Groupes p-adiques II: Repr esentations de GL_2( Q_p) et (φ,Γ)-modules, volume 330 of em Ast erisque, pages 511--528. Soci et e Math ematique de France, Paris, 2010.
Exact mathematical passage
\mathrm{GL}_2(\mathbb Q_p)
\mathbf QUse in source · 1 passage
Search aliases: Q mathbf
Use in source · Paper_X_Block27_Amended.tex, line 1436
For the three-character example below, the construction produces an actual three-dimensional summand inside the carrier of dimension seven. At the sample (0,1) its characteristic polynomial is (X-1)^2(X+1). A separate checked counterexample uses the coordinate inclusion Q→ Q^2, its coordinate retraction, and the swap involution: the split law holds, but compression sends the swap to zero and therefore fails to preserve its square. Thus selecting whole constituent copies, with the proved equivari…
Exact mathematical passage
\mathbf Q\to\mathbf Q^2
Q'Use in source · 1 passage
Search aliases: Q'
Use in source · 20_arithmetic.tex, line 962
Let Q(z)=zcoth z, the regular formal series with constant coefficient one. The elementary differential identity (frac zsinh z)^2=Q(z)-zQ'(z) and the Bernoulli expansion Q(z)=1+∑_m≥1frac2^2mB_2m(2m)!z^2m give β_m=-2^2m(2m-1)B_2m/(2m)!. Multiplication by 2^2m(2m)! proves the identity for κ_m. The Bernoulli denominator theorem gives v_2(B_2m)=-1 for m>0: its denominator contains each prime q with (q-1)mid 2m exactly once, including 2[source reference]. Since 2m-1 is odd, the valuation is exactly 4m…
Exact mathematical passage
\left(\frac z{\sinh z}\right)^2=Q(z)-zQ'(z)
q'Use in source · 2 passages
Search aliases: q'
Use in source · 72_recovered.tex, line 2058
Write the input indices as Mq+u and Mq'+v. Since 0≤ u,v<M, their sum has residue u+v when u+v<M, and residue u+v-M with one additional coarse index when u+v≥ M.
Exact mathematical passage
Mq'+v
q^{(n-r)L+\sum_{i=1}^r\min(s_i,L)}Use in source · 1 passage
Search aliases: q^(n-r)L+∑_i=1^rmin(s_i,L) sum min
Use in source · 04_arithmetic.tex, line 770
rdinality q, fix L≥1, and consider Mx≡ bpmodπ^L. Put widetilde b=Ub for a Smith matrix U as in [source reference]. The congruence is solvable if and only if widetilde b_i≡0 pmodπ^min(s_i,L) (1≤ i≤ r) and widetilde b_i≡0 pmodπ^L (r<i≤ m). When it is solvable, the number of solutions in ( O/π^L O)^n is q^(n-r)L+∑_i=1^rmin(s_i,L).
Exact mathematical passage
\boxed{ q^{(n-r)L+\sum_{i=1}^r\min(s_i,L)}. }
q^{-j}Use in source · 3 passages
Search aliases: q^-j
Use in source · Paper_X_Block27_Amended.tex, line 1465
The exact special chain has basis e_0,…,e_n-1 and operators Ne_0=0, Ne_j=e_j-1 (j>0), Fe_j=aq^-je_j, a,q∈ K^×. Lean proves N^n=0, and for n>0 also N^n-1≠0 and ker N=Ke_0. Consequently its kernel polynomial is X-a, whereas its whole-space polynomial is ∏_j=0^n-1(X-aq^-j). The zero-length chain is included as a zero-dimensional space. The assertion ker N=Ke_0 is made only for positive length.
Exact mathematical passage
Ne_0=0,\qquad Ne_j=e_{j-1}\ (j>0),\qquad Fe_j=aq^{-j}e_j, \qquad a,q\in K^\times.
q^{-L}Use in source · 1 passage
Search aliases: q^-L
Use in source · continuation_08.tex, line 14
Let q>0, L≥1, and m,k,e≥0. Write S= Z/q Z. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0)∈ Z^m+k, F(x)≡0pmodq^L. The actual lower root supplies the unique integral residual r=q^-LF(x)∈ Z^e, bar r=rbmod q∈ S^e. Let the actual reduced Jacobian blocks be J=(D_YF)(x)bmod q:S^m→ S^e, K=(D_TF)(x)bmod q:S^k→ S^e. For a prescribed parameter digit v∈ S^k, choose its canonical integral representative widehat v. Define L_v to be the set of actual residue vectors z∈( Z/q^L+1 Z)^m+k satisfying z≡ xpmodq^L, F(z…
Exact mathematical passage
r=q^{-L}F(x)\in\mathbb Z^e,\qquad \bar r=r\bmod q\in S^e.
q^{-W}Use in source · 1 passage
Search aliases: q^-W
Use in source · Paper_X_Block27_Amended.tex, line 560
[Weighted seed zeta function] Let 1≤ w_1≤⋯≤ w_d, put W=∑_iw_i, and let a_m=q^∑_imax(m-w_i,0). Then Z_ w(T) = ∑_m=1^w_da_mT^m + q^-Wfrac(q^dT)^w_d+11-q^dT. In particular, every weighted smooth chart has a rational seed zeta function.
Exact mathematical passage
\boxed{ Z_{\bm w}(T) = \sum_{m=1}^{w_d}a_mT^m + q^{-W}\frac{(q^dT)^{w_d+1}}{1-q^dT}. }
q^{1/2}Use in source · 1 passage
Search aliases: q^1/2
Use in source · Paper_XI_Traditional_Final.tex, line 721
[Gauss magnitude] If χ is nontrivial, then absG(χ,ψ)=q^1/2.
Exact mathematical passage
\boxed{ \abs{G(\chi,\psi)}=q^{1/2}. }
Q^2Use in source · 6 passages
Search aliases: Q^2
Use in source · visual_atlas.tex, line 172
0 · 1. These maps satisfy P^2=P, Q^2=Q, PQ=QP=0 and P+Q=I. Their images are the horizontal line and the diagonal line, respectively. They are not self-adjoint in the usual Euclidean inner product.
Exact mathematical passage
Q^2=Q
\mathbf Q^2Use in source · 1 passage
Search aliases: Q^2 mathbf
Use in source · Paper_X_Block27_Amended.tex, line 1436
For the three-character example below, the construction produces an actual three-dimensional summand inside the carrier of dimension seven. At the sample (0,1) its characteristic polynomial is (X-1)^2(X+1). A separate checked counterexample uses the coordinate inclusion Q→ Q^2, its coordinate retraction, and the swap involution: the split law holds, but compression sends the swap to zero and therefore fails to preserve its square. Thus selecting whole constituent copies, with the proved equivari…
Exact mathematical passage
\mathbf Q\to\mathbf Q^2
q^2Use in source · 3 passages
Search aliases: q^2
Use in source · Paper_XI_Traditional_Final.tex, line 273
[Scalar covariance model] Let F=diag(1,q,q^2,…,q^m-1) and let N_j-1,j=1. Then FNF^-1=q^-1N.
Exact mathematical passage
F=\operatorname{diag}(1,q,q^2,\ldots,q^{m-1})
q^{2M+2}Use in source · 1 passage
Search aliases: q^2M+2
Use in source · 72_recovered.tex, line 1010
[Odd modulus truncation] For every M≥0, every positive odd q, and every even n, E_n ≡ ∑_r=0^min(M,n/2) (-1)^rZ_2r+1 binom n2rq^2rS_n-2r(q) pmod q^2M+2. If pmid q and a=v_p(q), the same truncation is valid modulo p^2a(M+1).
Exact mathematical passage
\boxed{ E_n \equiv \sum_{r=0}^{\min(M,n/2)} (-1)^rZ_{2r+1} \binom n{2r}q^{2r}S_{n-2r}(q) \pmod {q^{2M+2}}. }
q^{2N}Use in source · 1 passage
Search aliases: q^2N
Use in source · continuation_09.tex, line 95
^m satisfying J_a u=-bar r-K_abar w. Choose an integer representative widetilde u, and set a'=a+q^N(widetilde u,w). The parameter quotient is the supplied integer w. Therefore free(a')=t' exactly. The checked polynomial Taylor identity gives F(a')≡ F(a)+q^NDF(a)(widetilde u,w) ≡0pmodq^N+1. Its quadratic remainder is divisible by q^2N, and 2N≥ N+1. Also a'≡ apmod q, so evaluation of every polynomial partial derivative gives the same reduced Jacobian. Induction on the level, carrying the root, exa…
Exact mathematical passage
q^{2N}
q^{2r}Use in source · 5 passages
Search aliases: q^2r
Use in source · 72_recovered.tex, line 965
[Odd modulus Euler coordinate pair] For every positive odd q and every even n≥0, E_n = ∑_r=0^n/2 (-1)^rZ_2r+1 binom n2r q^2rS_n-2r(q). The inverse transform is S_n(q) = E_n + ∑_r=1^n/2 2^2r-1 binom n2r q^2rE_n-2r. Both transforms are lower triangular with diagonal one.
Exact mathematical passage
\boxed{ E_n = \sum_{r=0}^{n/2} (-1)^rZ_{2r+1} \binom n{2r} q^{2r}S_{n-2r}(q). }
q^3Use in source · 1 passage
Search aliases: q^3
Use in source · atlas-pages.tex, line 84
basis u_j. Their projectors P_j=u_ju_j^ T sum to the identity and satisfy P_iP_j=0 for i≠ j. vfiggenerator_modes platecaptionState coordinates have off-diagonal transfers. Spectral packets reduce the generator and evolve independently. The two dashed state groups are the coarse partition used next. headThe same evolution has a closed formula Writing q=e^-2t gives e^tQ=P_0+qP_1+q^2P_2+q^3P_3. At q=1/2, the point seed e_0 becomes (15,5,9,3)^ T/32. The polynomial identity T(q)T(r)=T(qr) checks comp…
Exact mathematical passage
e^{tQ}=P_0+qP_1+q^2P_2+q^3P_3.
\Q^4Use in source · 1 passage
Search aliases: Q^4 Q mathbf
Use in source · Paper_X_Block27_Amended.tex, line 1685
A specified finite realization. On Q^4, take the diagonal operators with diagonal lists (1,1,1,0), (2,2,0,0), (3,0,0,0), (0,4,4,4) and target scalars (1,2,3,4). All four targets occur individually. Imposing these conditions successively gives actual projector ranks 3,2,1,0. The last packet is empty despite individual feasibility. The formal code checks these operators, projector formulas, traces and dimensions. This is a newly specified realization, not a claim to have recovered the original com…
Exact mathematical passage
\Q^4
q^dUse in source · 3 passages
Search aliases: q^d
Use in source · continuation_02.tex, line 54
For every positive integer q, including composite moduli and modulus one, and every L≥0, the checked map Φ_x,L:( Z/q Z)^d⟶ {ybmod q^L+1:y≡ xpmodq^L}, t⟼ x+q^Lwidetilde t is a bijection. Here widetilde t is the canonical integer representative of the digit vector. The entire residue fiber therefore has q^d elements. Injectivity is proved by cancelling q^L in the corresponding integer divisibility statement. Surjectivity comes from the lower-residue congruence.
Exact mathematical passage
q^d
q^{d(m-1)}Use in source · 3 passages
Search aliases: q^d(m-1)
Use in source · Paper_X_Block27_Amended.tex, line 298
[Smooth formal branch count] If R_bar x≅ O[[t_1,…,t_d]] as complete local O-algebras, then #X_m(bar x)=q^d(m-1). Fix a uniformizer and a section s:k→ O of reduction, with s(0)=0. Every O-valued branch is encoded by d positive-depth digit streams t_i=s(a_i,1)ϖ+s(a_i,2)ϖ^2+⋯, a_i,j∈ k,
Exact mathematical passage
\boxed{ \#X_m(\bar x)=q^{d(m-1)}. }
q^{dm-W}Use in source · 1 passage
Search aliases: q^dm-W
Use in source · Paper_X_Block27_Amended.tex, line 574
The finite sum records the depths before all coordinates are active. For m≥ w_d+1, a_m=q^dm-W, so the remaining terms form one geometric series.
Exact mathematical passage
a_m=q^{dm-W},
Q^eUse in source · 1 passage
Search aliases: Q^e
Use in source · continuation_10.tex, line 29
)∈ M_e× k(R). Suppose, for natural numbers s,M, that J∈GL_e(Q), p^sJ^-1∈ M_e(R), M>2s. For every v∈ R^k, set τ=t_0+p^Mv, G=F+p^MH. Then there is exactly one y∈ y_0+p^s+1R^e with G(y,τ)=0. It satisfies y-y_0∈ p^M-sR^e and there is an integral vector ε∈ p^2M-3sR^e such that, in Q^e, displaystyle y-y_0=-p^MJ^-1(H(a)+Kv)+ε. The same original exponent s appears throughout. No inverse or rank assumption on the perturbed Jacobian is imposed. The statement includes arbitrary integral p-adic coefficients…
Exact mathematical passage
Q^e
q^{e-1}Use in source · 1 passage
Search aliases: q^e-1
Use in source · Paper_X_Block27_Amended.tex, line 592
[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.
Exact mathematical passage
\boxed{ Z_e(T) = \frac{\sum_{r=1}^{e}q^{r-1}T^r} {1-q^{e-1}T^e}. }
q^iUse in source · 2 passages
Search aliases: q^i
Use in source · Paper_XI_Traditional_Final.tex, line 616
[Refined exponent formula] The ith interaction packet on F_q^L^× has exponent c_i = afracq^L-1q^n-1 +bq^ifracq^L-1q^m-1 pmodq^L-1. It is trivial on tame inertia exactly when c_i=0.
Exact mathematical passage
\boxed{ c_i = a\frac{q^L-1}{q^n-1} +bq^i\frac{q^L-1}{q^m-1} \pmod{q^L-1}. }
Q^jUse in source · 1 passage
Search aliases: Q^j
Use in source · rank4.tex, line 56
Use U_A^j=1+P_A^j and U_B^j=1+Q^j. BK 3.1.7--9 give the actual groups
Exact mathematical passage
U_B^j=1+Q^j
q^{j-1}Use in source · 1 passage
Search aliases: q^j-1
Use in source · Paper_XI_Traditional_Final.tex, line 283
The only nonzero entry in column j and row j-1 is multiplied by q^j-1q^-j=q^-1 under conjugation.
Exact mathematical passage
q^{j-1}q^{-j}=q^{-1}
q^{L+1}Use in source · 18 passages
Search aliases: q^L+1
Use in source · 30_lifting.tex, line 76
[Finite polynomial obstruction] Let q≥1, L≥1, x∈ Z^d, and suppose F(x)∈ q^L Z^e. Define the exact integer residual r_x by F(x)=q^Lr_x, and put O_x=S_q^e/im J_x, o_x=[overliner_x]. Then O_x is finite and o_x=0 Longleftrightarrow ∃ y∈ Z^d: y≡ xpmodq^L, F(y)≡0pmodq^L+1. No surjectivity or nonsingularity assumption is required.
Exact mathematical passage
\label{eq:finite-obstruction} o_x=0\quad\Longleftrightarrow\quad \exists y\in\Z^d:\quad y\equiv x\pmod{q^L},\qquad F(y)\equiv0\pmod{q^{L+1}}.
q^{L+a}Use in source · 4 passages
Search aliases: q^L+a
Use in source · 30_lifting.tex, line 342
z_n+1-z_n · ∈ q^L+n Z^d, · J_z_n · =J_x. For a≤ b, it also satisfies z_b-z_a∈ q^L+a Z^d.
Exact mathematical passage
z_b-z_a\in q^{L+a}\Z^d
q^{L+n}Use in source · 5 passages
Search aliases: q^L+n
Use in source · 30_lifting.tex, line 338
[Compatible polynomial tower] Let q≥1, L≥1, F∈ Z[X_1,…,X_d]^e, and x∈ Z^d satisfy F(x)∈ q^L Z^e. Suppose J_x:S_q^d→ S_q^e is surjective. There is a single sequence z_n∈ Z^d such that z_0 · =x, · F(z_n) · ∈ q^L+n Z^e, z_n+1-z_n · ∈ q^L+n Z^d, · J_z_n · =J_x. For a≤ b, it also satisfies z_b-z_a∈ q^L+a Z^d.
Exact mathematical passage
z_0&=x,& F(z_n)&\in q^{L+n}\Z^e,\\ z_{n+1}-z_n&\in q^{L+n}\Z^d,& J_{z_n}&=J_x.
q^mUse in source · 2 passages
Search aliases: q^m
Use in source · Paper_XI_Traditional_Final.tex, line 608
Represent the residue characters by exponents apmodq^n-1, bpmodq^m-1.
Exact mathematical passage
a\pmod{q^n-1}, \qquad b\pmod{q^m-1}.
q^{m-1}Use in source · 1 passage
Search aliases: q^m-1
Use in source · Paper_XI_Traditional_Final.tex, line 273
[Scalar covariance model] Let F=diag(1,q,q^2,…,q^m-1) and let N_j-1,j=1. Then FNF^-1=q^-1N.
Exact mathematical passage
F=\operatorname{diag}(1,q,q^2,\ldots,q^{m-1})
q^{m-\lceil m/e\rceil}Use in source · 1 passage
Search aliases: q^m-lceil m/erceil lceil rceil
Use in source · Paper_X_Block27_Amended.tex, line 586
[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.
Exact mathematical passage
\boxed{ q^{m-\lceil m/e\rceil} }
q^{m-w_i}Use in source · 1 passage
Search aliases: q^m-w_i
Use in source · Paper_X_Block27_Amended.tex, line 514
The image of ϖ^w_i O in O/ϖ^m has one element when m≤ w_i, and q^m-w_i elements when m>w_i. Multiplying the independent coordinate counts gives the formula.
Exact mathematical passage
q^{m-w_i}
q^{\min(s_i,L)}Use in source · 1 passage
Search aliases: q^min(s_i,L) min
Use in source · 04_arithmetic.tex, line 784
Set x=Vy. Multiplying the congruence by U reduces it to independent equations π^s_iy_i≡widetilde b_ipmodπ^L (1≤ i≤ r) and zero equations in the remaining rows. Multiplication by π^s_i on O/π^L O has image equal to the elements divisible by π^min(s_i,L) and kernel of size q^min(s_i,L). Each of the n-r unused variables is free and contributes q^L choices. Multiplying the independent counts gives the formula.
Exact mathematical passage
q^{\min(s_i,L)}.
Q^NUse in source · 3 passages
Search aliases: Q^N
Use in source · 72_recovered.tex, line 2869
[Residue subsequences of a second order recurrence] Suppose x_n+2=Px_n+1-Qx_n over a commutative ring. Define V_0=2, V_1=P, and V_n+2=PV_n+1-QV_n. For every positive integer N and every address j, the subsequence u_k=x_Nk+j satisfies u_k+2=V_Nu_k+1-Q^Nu_k.
Exact mathematical passage
\boxed{ u_{k+2}=V_Nu_{k+1}-Q^Nu_k.}
q^NUse in source · 7 passages
Search aliases: q^N
Use in source · 30_lifting.tex, line 346
At level N=L+n, divide the residual by q^N. Surjectivity supplies t∈ S_q^d with J_z_nt=-overlineF(z_n)/q^N. Choose an integer representative and apply [source reference] to obtain z_n+1. The new point is congruent to z_n modulo q. Evaluation of each formal partial derivative therefore gives J_z_n+1=J_z_n.
Exact mathematical passage
q^N
q^nUse in source · 8 passages
Search aliases: q^n
Use in source · 03_nonlinear_coordinates.tex, line 802
rated orbit defect] Let F_f:X_f→ X_f and F_c:X_c→ X_c, with coarsening C:X_f→ X_c. Suppose F_c is q-Lipschitz. For a fine orbit starting at x, assume Δ_F(F_f^j x) ≤ε_j. Then CF_f^n(x)-F_c^n(Cx) ≤ ∑_j=0^n-1q^n-1-jε_j. If 0≤ q<1 and every ε_j≤ε, then CF_f^n(x)-F_c^n(Cx) ≤ εfrac1-q^n1-q.
Exact mathematical passage
\left\|CF_f^n(x)-F_c^n(Cx)\right\| \le \varepsilon\frac{1-q^n}{1-q}.
q^{N+1}Use in source · 1 passage
Search aliases: q^N+1
Use in source · continuation_09.tex, line 92
s t'-free(a)=q^Nw, with w∈ Z^k. Surjectivity supplies a solved digit u∈ S_q^m satisfying J_a u=-bar r-K_abar w. Choose an integer representative widetilde u, and set a'=a+q^N(widetilde u,w). The parameter quotient is the supplied integer w. Therefore free(a')=t' exactly. The checked polynomial Taylor identity gives F(a')≡ F(a)+q^NDF(a)(widetilde u,w) ≡0pmodq^N+1. Its quadratic remainder is divisible by q^2N, and 2N≥ N+1. Also a'≡ apmod q, so evaluation of every polynomial partial derivative give…
Exact mathematical passage
F(a')\equiv F(a)+q^NDF(a)(\widetilde u,w) \equiv0\pmod{q^{N+1}}.
q^{n-1-j}Use in source · 1 passage
Search aliases: q^n-1-j
Use in source · 03_nonlinear_coordinates.tex, line 796
[Iterated orbit defect] Let F_f:X_f→ X_f and F_c:X_c→ X_c, with coarsening C:X_f→ X_c. Suppose F_c is q-Lipschitz. For a fine orbit starting at x, assume Δ_F(F_f^j x) ≤ε_j. Then CF_f^n(x)-F_c^n(Cx) ≤ ∑_j=0^n-1q^n-1-jε_j. If 0≤ q<1 and every ε_j≤ε, then CF_f^n(x)-F_c^n(Cx) ≤ εfrac1-q^n1-q.
Exact mathematical passage
\left\|CF_f^n(x)-F_c^n(Cx)\right\| \le \sum_{j=0}^{n-1}q^{n-1-j}\varepsilon_j.
q^{n/d}Use in source · 1 passage
Search aliases: q^n/d
Use in source · Paper_XI_Traditional_Final.tex, line 410
[Möbius orbit count] The number of regular Frobenius orbits of length n is c_n(q) = frac1n ∑_dmid n μ(d) (q^n/d-1).
Exact mathematical passage
\boxed{ c_n(q) = \frac1n \sum_{d\mid n} \mu(d) \bigl(q^{n/d}-1\bigr). }
Q^rUse in source · 5 passages
Search aliases: Q^r
Use in source · 72_recovered.tex, line 1719
/3 in the one-prime example above loses the advertised digit. Invertibility of C alone does not restore it. The original proof is retained under these hypotheses. Let Q=∏_p∈ Sp^L_p for a finite set of primes. Let C∈ M_d( Z) be invertible modulo Q, and choose H such that C^H=I+QB with B∈ M_d( Z). Then for every nonnegative integer x, a_Hx+j = ∑_r=0^xQ^rbinom xr w^TC^jB^rv_0. For every r, Δ_H^ra_n≡0pmod Q^r.
Exact mathematical passage
\boxed{ a_{Hx+j} = \sum_{r=0}^{x}Q^r\binom xr w^TC^jB^rv_0. }
q^rUse in source · 2 passages
Search aliases: q^r
Use in source · Paper_XI_Traditional_Final.tex, line 423
For every divisor rmid n, the characters fixed by Frob^r are exactly those descending to F_q^r^×, and there are q^r-1 of them. Möbius inversion gives the number of characters of exact orbit length n. Division by n gives the number of orbits.
Exact mathematical passage
\FF_{q^r}^{\times}
q^{r-1}Use in source · 1 passage
Search aliases: q^r-1
Use in source · Paper_X_Block27_Amended.tex, line 592
[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.
Exact mathematical passage
\boxed{ Z_e(T) = \frac{\sum_{r=1}^{e}q^{r-1}T^r} {1-q^{e-1}T^e}. }
q^{r-d}Use in source · 1 passage
Search aliases: q^r-d
Use in source · 72_recovered.tex, line 1661
On the polynomial factor the difference operator acts as q+yΔ. The two terms commute. Expand (q+yΔ)^r and evaluate at zero. Since Δ^kQ_j=0 for k>d, every surviving term contains at least q^r-d.
Exact mathematical passage
q^{r-d}
q^{r-k}Use in source · 1 passage
Search aliases: q^r-k
Use in source · 72_recovered.tex, line 1641
[Polynomial exponential Mahler transfer] Let f(n)=P(n)u^n with P∈ Z_p[n] of degree d. Let H=Λ_p,L, fix an address j, and write Q_j(x)=P(Hx+j), y=u^H, q=y-1. Then f(Hx+j)=∑_r≥0c_j,rbinom xr, where c_j,r = u^j ∑_k=0^min(d,r) binom rk Δ^kQ_j(0)y^kq^r-k. Consequently, v_p(c_j,r)≥ Lmax(r-d,0). For R≥ d, truncation at depth R is valid modulo p^L(R+1-d) uniformly on Z_p.
Exact mathematical passage
\boxed{ c_{j,r} = u^j \sum_{k=0}^{\min(d,r)} \binom rk \Delta^kQ_j(0)y^kq^{r-k}. }
Q^TUse in source · 13 passages
Search aliases: Q^T
Use in source · Paper_VI_Traditional_Final.tex, line 4382
[Continuous time finite state formula] Let Q be a finite Markov generator and let (X_t)_t≥0 be the corresponding chain. Then ( e^t(Q^T-D_V)f )_i = E_i[ exp( -∫_0^tV(X_s) d s ) f(X_t) ]. If V≥ v_0, then norm e^t(Q^T-D_V)_∞→∞ ≤ e^-v_0t.
Exact mathematical passage
\left( \e^{t(Q^T-D_V)}f \right)_i = \Exp_i\left[ \exp\left( -\int_0^tV(X_s)\dd s \right) f(X_t) \right].
q^{\sum_{i=1}^d\max(m-w_i,0)}Use in source · 1 passage
Search aliases: q^∑_i=1^dmax(m-w_i,0) sum max
Use in source · Paper_X_Block27_Amended.tex, line 504
[Weighted smooth-chart count] For weights w=(w_1,…,w_d), N_ w(m) = q^∑_i=1^dmax(m-w_i,0).
Exact mathematical passage
\boxed{ N_{\bm w}(m) = q^{\sum_{i=1}^d\max(m-w_i,0)}. }
q^{\sum_i\max(m-w_i,0)}Use in source · 1 passage
Search aliases: q^∑_imax(m-w_i,0) sum max
Use in source · Paper_X_Block27_Amended.tex, line 556
[Weighted seed zeta function] Let 1≤ w_1≤⋯≤ w_d, put W=∑_iw_i, and let a_m=q^∑_imax(m-w_i,0). Then Z_ w(T) = ∑_m=1^w_da_mT^m + q^-Wfrac(q^dT)^w_d+11-q^dT. In particular, every weighted smooth chart has a rational seed zeta function.
Exact mathematical passage
a_m=q^{\sum_i\max(m-w_i,0)}.
q^{\sum_i\min(s_i,L)}Use in source · 1 passage
Search aliases: q^∑_imin(s_i,L) sum min
Use in source · 04_arithmetic.tex, line 871
For an arbitrary DVR O with residue field cardinality q, the analogous formula q^∑_imin(s_i,L) follows from the solvability and fiber count for each coordinate. The formal correspondence records direct proofs of this DVR coordinate arithmetic and the rectangular transport as separate targets. The inherited abstract product interface remains available when the coordinate facts are supplied as certificates. A Smith form and its changes of basis are inputs to the transport statement. It does not pr…
Exact mathematical passage
q^{\sum_i\min(s_i,L)}
Q^\perpUse in source · 2 passages
Search aliases: Q^⊥ perp
Use in source · 02_strengthened.tex, line 1385
The map R_0 is completely positive. It is trace preserving on operators supported in Q H because Tr(V_j^*Q_jXQ_jV_j) = Tr(Q_jXQ_j). Fix a code state ρ_0 and write Q^⊥=I-Q. Then R(X) = R_0(QXQ) + Tr(Q^⊥ X)ρ_0 is completely positive and trace preserving. It extends R_0 on operators supported in Q H.
Exact mathematical passage
Q^\perp=I-Q
q_{\mathrm A}Use in source · 2 passages
Search aliases: q_ A mathrm
Use in source · atlas-pages.tex, line 205
vfigevidence platecaptionThe A packet becomes heavier and its internal shape changes. The B packet retains an equal internal split but loses total mass. These are different effects of the same likelihood. headReconstruct from packet mass times packet shape The unnormalized packet evidences are Z_ A=5/4 and Z_ B=1/2. The resulting masses and conditional shapes are m_ A=5/7, q_ A=(4/5,1/5,0,0)^ T, m_ B=2/7, q_ B=(0,0,1/2,1/2)^ T. Thus m_ Aq_ A+m_ Bq_ B is exactly the displayed posterior. Keeping o…
Exact mathematical passage
m_{\mathrm A}=5/7,\quad q_{\mathrm A}=(4/5,1/5,0,0)^{\mathsf T}, \qquad m_{\mathrm B}=2/7,\quad q_{\mathrm B}=(0,0,1/2,1/2)^{\mathsf T}.
q_{\mathrm B}Use in source · 2 passages
Search aliases: q_ B mathrm
Use in source · atlas-pages.tex, line 205
vfigevidence platecaptionThe A packet becomes heavier and its internal shape changes. The B packet retains an equal internal split but loses total mass. These are different effects of the same likelihood. headReconstruct from packet mass times packet shape The unnormalized packet evidences are Z_ A=5/4 and Z_ B=1/2. The resulting masses and conditional shapes are m_ A=5/7, q_ A=(4/5,1/5,0,0)^ T, m_ B=2/7, q_ B=(0,0,1/2,1/2)^ T. Thus m_ Aq_ A+m_ Bq_ B is exactly the displayed posterior. Keeping o…
Exact mathematical passage
m_{\mathrm A}=5/7,\quad q_{\mathrm A}=(4/5,1/5,0,0)^{\mathsf T}, \qquad m_{\mathrm B}=2/7,\quad q_{\mathrm B}=(0,0,1/2,1/2)^{\mathsf T}.
\mathcal Q_{\mathcal L}Use in source · 3 passages
Search aliases: Q_ L mathcal mathcal
Use in source · Paper_VI_Traditional_Final.tex, line 3551
[Coarsest fixed sufficient partition] Let L be a nonempty family of strictly positive likelihood vectors. Define i∼_ Lj Longleftrightarrow λ_i=λ_j hboxfor every λ∈ L. Let Q_ L be the partition into equivalence classes. A partition Q gives a fixed coarse sufficient description in the sense of [source reference] for every λ∈ L if and only if every block of Q is contained in a block of Q_ L. Hence Q_ L is the unique coarsest partition with this property.
Exact mathematical passage
\mathcal Q_{\mathcal L}
q_*Use in source · 3 passages
Search aliases: q_*
Use in source · Paper_XII_Traditional_Final.tex, line 475
[Functorial group algebra transfer] A group homomorphism q:Δ'→Δ induces an O algebra homomorphism q_*: O[Δ']⟶ O[Δ]. If q is surjective, then q_* is surjective. Augmentation commutes with the transfer: ϵ_Δ q_*=ϵ_Δ'.
Exact mathematical passage
q_*:\OO[\Delta']\longrightarrow\OO[\Delta].
q_0Use in source · 2 passages
Search aliases: q_0
Use in source · Paper_XI_Traditional_Final.tex, line 1966
[Dyadic phase ambiguity] On a symplectic plane over F_2, choose refinements q_0 and q_1 with Arf invariants zero and one. Their polar forms are the same, while ∑_x(-1)^q_0(x)=2, ∑_x(-1)^q_1(x)=-2. Thus the alternating packet alone cannot select the dyadic sign.
Exact mathematical passage
q_0
Q_{00}Use in source · 1 passage
Search aliases: Q_00
Use in source · 02_coordinates.tex, line 352
[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coarse direct summands. Fine analysis recovers the four entries exactly. Nothing has been approximated.
Exact mathematical passage
P_0=Q_{00}+Q_{01}, \qquad P_1=Q_{10}+Q_{11}
Q_{01}Use in source · 1 passage
Search aliases: Q_01
Use in source · 02_coordinates.tex, line 352
[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coarse direct summands. Fine analysis recovers the four entries exactly. Nothing has been approximated.
Exact mathematical passage
P_0=Q_{00}+Q_{01}, \qquad P_1=Q_{10}+Q_{11}
Q_0^\mathsfUse in source · 1 passage
Search aliases: Q_0^ mathsf
Use in source · visual_atlas.tex, line 389
headExact reconstruction does not require right angles For the displayed Q_0,Q_1, Q_0+Q_1=I, Q_i^2=Q_i, Q_0Q_1=Q_1Q_0=0, but Q_0^ T≠ Q_0. Expanding both copies of I still proves ∑_i,jQ_iOb Q_j=Ob. The block reconstruction identity itself uses the condition that the projectors sum to the identity. Idempotence is additionally used when constructing a witness that remains supported in one packet after projection. The general formal reconstruction statement does not require pairwise annihilation.
Exact mathematical passage
Q_0^\mathsf T\ne Q_0
q_1Use in source · 16 passages
Search aliases: q_1
Use in source · 03_weighted.tex, line 422
[Recurrence for path counts] Write Z_u,v(t)=fracP(t)Q(t), Q(t)=1+q_1t+⋯+q_dt^d. Use the convention deg 0=-∞. Then, for every nonnegative integer n>deg P, a_n+q_1a_n-1+⋯+q_da_n-d=0.
Exact mathematical passage
Z_{u,v}(t)=\frac{P(t)}{Q(t)}, \qquad Q(t)=1+q_1t+\cdots+q_dt^d.
Q_{10}Use in source · 1 passage
Search aliases: Q_10
Use in source · 02_coordinates.tex, line 352
[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coarse direct summands. Fine analysis recovers the four entries exactly. Nothing has been approximated.
Exact mathematical passage
P_0=Q_{00}+Q_{01}, \qquad P_1=Q_{10}+Q_{11}
Q_{11}Use in source · 1 passage
Search aliases: Q_11
Use in source · 02_coordinates.tex, line 352
[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coarse direct summands. Fine analysis recovers the four entries exactly. Nothing has been approximated.
Exact mathematical passage
P_0=Q_{00}+Q_{01}, \qquad P_1=Q_{10}+Q_{11}
Q_2Use in source · 1 passage
Search aliases: Q_2
Use in source · visual_atlas.tex, line 281
Indeed, Q_1T^UQ_2=Q_2T^UQ_1=0, Q_iT^UQ_j=U(P_iTP_j)U^-1. A change in raw matrix sparsity is therefore not necessarily a change in the packet transfer graph. Conjugacy preserves the exact block identities and their zero pattern. A nonunitary shear need not preserve their Euclidean norms.
Exact mathematical passage
Q_1T^UQ_2=Q_2T^UQ_1=0, \qquad Q_iT^UQ_j=U(P_iTP_j)U^{-1}.
\mathbb Q_2Use in source · 6 passages
Search aliases: Q_2 mathbb
Use in source · 04_arithmetic.tex, line 590
Modulo 2, the two eigenvalues become equal. The rational sign packets remain distinct over Q_2, but the standard lattice does not split into their intersections until one more dyadic digit is retained.
Exact mathematical passage
\mathbb Q_2
q_2Use in source · 4 passages
Search aliases: q_2
Use in source · Paper_XI_Traditional_Final.tex, line 1520
[Orthogonal dyadic multiplicativity] For an orthogonal sum (V,q)=(V_1,q_1)mathop⊥limits(V_2,q_2), we have Arf(q)=Arf(q_1)+Arf(q_2) and the normalized Gauss sign is the product of the two normalized signs.
Exact mathematical passage
(V,q)=(V_1,q_1)\perpbig(V_2,q_2),
Q_2^{\mathrm{red}}Use in source · 1 passage
Search aliases: Q_2^red mathrm
Use in source · Paper_IV_Traditional_Final.tex, line 2809
For N=2, [source reference] and D_2(z)=Δ_2(z^2) give λ_m=π^2(m+frac12)^2, m≥0. Thus the first formula holds with Q_2^red=0.
Exact mathematical passage
Q_2^{\mathrm{red}}=0
\Q_3Use in source · 1 passage
Search aliases: Q_3 Q mathbf
Use in source · Paper_X_Block27_Amended.tex, line 2456
The p-adic square-root criterion is proved for every prime: (∃ x∈ Q_p, x^2=-1) Longleftrightarrow p≡1pmod4. A root satisfies lvert xrvert_p^2=1 and is therefore integral. Reduction modulo three rules out a root in Q_3, and reduction modulo four rules out a root in Q_2. At an odd prime, any residue root of X^2+1 has nonzero derivative and lifts by the pinned Mathlib Hensel theorem. The finite-field square criterion then gives the stated congruence. An explicit Hensel application at the residue ro…
Exact mathematical passage
\Q_3
Q_8Use in source · 1 passage
Search aliases: Q_8
Use in source · Paper_XI_Traditional_Final.tex, line 2154
r an independently fixed native quadratic marking, a preassigned embedding must satisfy q_W(f(v))=q_rm src(v). At equal capacity, a free embedding requires equal Arf invariant. Spare capacity for d≥4 admits either source Arf type when the embedding is free. Retaining only the central {±1} would erase the distinction between whole marked finite extensions such as D_8 and Q_8.
Exact mathematical passage
Q_8
Q_aUse in source · 15 passages
Search aliases: Q_a
Use in source · 02_coordinates.tex, line 186
The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_Q_b,Q_a widehat T_Q_a,P_a = widehat T_Q_b,P_b J^X_P_b,P_a. The identity, cocycle, composition, inverse, and naturality laws are all proved in Lean. Th…
Exact mathematical passage
Q_a,Q_b
q_aUse in source · 11 passages
Search aliases: q_a
Use in source · Paper_VI_Traditional_Final.tex, line 5402
FloatBarrier Prior-restricted sufficiency and filtering. Let L be a strictly positive finite likelihood and let B_L(p)=Lodot p/ipLp for a probability vector p. Fix probability vectors r^(a) supported on B_a, and restrict the priors to p_i=q_a r_i^(a) for i∈ B_a, with q a coarse probability vector. [Sufficiency relative to a prior disintegration] Define overline L_a = ∑_i∈ B_aL_ir_i^(a). For every prior in the declared family, C B_L(p) = B_overline L(q).
Exact mathematical passage
p_i=q_a r_i^{(a)}
Q_a^*Use in source · 1 passage
Search aliases: Q_a^*
Use in source · 03_supplement_02.tex, line 74
Expanding the finite product of sums gives T(A) T(B)= T(i↦ A_iB_i). Consequently, if two fine addresses differ in any coordinate, the corresponding product contains a zero factor. If the addresses agree, the product is the same fine projector. The adjoint passes through the finite coordinate product, so Q_a^*=Q_a. Summing over all address functions and using the same expansion yields ∑_a Q_a= T(i↦∑_jP^(i)_j)=I. A separate algebraic regrouping theorem applies to any finite orthogonal family and a…
Exact mathematical passage
Q_a^*=Q_a
Q_bUse in source · 2 passages
Search aliases: Q_b
Use in source · 02_coordinates.tex, line 186
The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_Q_b,Q_a widehat T_Q_a,P_a = widehat T_Q_b,P_b J^X_P_b,P_a. The identity, cocycle, composition, inverse, and naturality laws are all proved in Lean. Th…
Exact mathematical passage
Q_a,Q_b
Q_cUse in source · 3 passages
Search aliases: Q_c
Use in source · atlas-pages.tex, line 344
-1 · 1C. The coarse matrix is the positive square root of -Q_c. This is a concrete naturality identity, checked on the full matrices. It is not a claim that an arbitrary partition commutes with every fractional construction.
Exact mathematical passage
-Q_c
\mathcal Q_cUse in source · 1 passage
Search aliases: Q_c mathcal
Use in source · Paper_VI_Traditional_Final.tex, line 684
[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.
Exact mathematical passage
\mathcal Q_c
Q_DUse in source · 2 passages
Search aliases: Q_D
Use in source · Paper_X_Block27_Amended.tex, line 1508
Writing Q_D(T) for the reversed local characteristic polynomial, Lean proves Q_D(T)=det(1-T r(φ)mid E(D,H)), Q_D(0)=1, Q_⊕_iD_i^⊕ m_i(T)=∏_iQ_D_i(T)^m_i. The abstract formulas do not need characteristic zero. Normality and finite dimension appear explicitly where used. Arithmetic identification of G,H,ν and φ remains separate. In the geometric-Frobenius convention, ν(φ) is the inverse residue cardinality. It is this unit, rather than the residue cardinality, that enters the earlier special-chain…
Exact mathematical passage
Q_D(T)
Q_dUse in source · 1 passage
Search aliases: Q_d
Use in source · 02_supplement_01.tex, line 33
Adjoint and tensor products. With an involution and self-adjoint projectors, the formalization proves (A_ba)^*=(A^*)_ab. For actual finite matrices and actual Kronecker products, it proves (P_b⊗ Q_d)(A⊗ B)(P_a⊗ Q_c) =A_ba⊗ B_dc. The self-adjointness hypothesis is explicit in the adjoint theorem. The tensor theorem is not a statement about an uninterpreted tensor symbol.
Exact mathematical passage
(P_b\otimes Q_d)(A\otimes B)(P_a\otimes Q_c) =A_{ba}\otimes B_{dc}.
q_DUse in source · 2 passages
Search aliases: q_D
Use in source · 03_nonlinear_coordinates.tex, line 923
[Cokernel obstruction criterion] A linear correction exists if and only if Ob_D(r)=0. The quotient map q_D:V⟶coker D has the following universal property. If φ:V→ W is linear and φ D=0, then there is a unique linear map overlineφ:coker D⟶ W with φ=overlineφ q_D.
Exact mathematical passage
q_D:V\longrightarrow\coker D
q_dUse in source · 8 passages
Search aliases: q_d
Use in source · 03_weighted.tex, line 422
[Recurrence for path counts] Write Z_u,v(t)=fracP(t)Q(t), Q(t)=1+q_1t+⋯+q_dt^d. Use the convention deg 0=-∞. Then, for every nonnegative integer n>deg P, a_n+q_1a_n-1+⋯+q_da_n-d=0.
Exact mathematical passage
Z_{u,v}(t)=\frac{P(t)}{Q(t)}, \qquad Q(t)=1+q_1t+\cdots+q_dt^d.
q_{D'}Use in source · 1 passage
Search aliases: q_D'
Use in source · 03_nonlinear_coordinates.tex, line 971
The relation bD=D'a shows that the quotient map q_D'b vanishes on im D. Apply the universal property in [source reference].
Exact mathematical passage
q_{D'}b
q_{d-1}Use in source · 2 passages
Search aliases: q_d-1
Use in source · dyadic_synthesis.tex, line 76
At an inductive field step d≥2, set n=2^d-1q_d, R=2^d-2q_d-1. Then n>2R and -2R>-n-q_1. BK Remark 2.2.7 gives the maximal field generated by the actual displayed matrix. Literal truncation and the filtered lift give the defining sequence. Nonzero native trace also proves separability: the irreducible characteristic polynomial has a nonzero coefficient of X^N-1, so its derivative is nonzero. See pathtower/DYADIC_AS_TOWER167.md for all field and filtration details.
Exact mathematical passage
R=2^{d-2}q_{d-1}
q_{d-j+1}Use in source · 4 passages
Search aliases: q_d-j+1
Use in source · Paper_XI_Traditional_Final.tex, line 2024
[Restricted odd-prime arithmetic realization] Let p be an odd prime, F= F_p((t)), fix d≥1, and let 0<h_1<⋯<h_d, pnmid h_i, q_i=(p+1)h_i, N=p^d. There is an explicit maximal simple stratum in M_N(F) with totally ramified full-degree parameter field, a literal defining sequence of degrees p,p^2,…,p^d, and levels n_j=p^d-jq_d-j+1. Every native parameter and consecutive correction has trace zero over its recursive base field. The stratum admits a full simple character trivial on U_F^1. Its fixed can…
Exact mathematical passage
n_j=p^{d-j}q_{d-j+1}
Q_{D_i}Use in source · 1 passage
Search aliases: Q_D_i
Use in source · Paper_X_Block27_Amended.tex, line 1509
Writing Q_D(T) for the reversed local characteristic polynomial, Lean proves Q_D(T)=det(1-T r(φ)mid E(D,H)), Q_D(0)=1, Q_⊕_iD_i^⊕ m_i(T)=∏_iQ_D_i(T)^m_i. The abstract formulas do not need characteristic zero. Normality and finite dimension appear explicitly where used. Arithmetic identification of G,H,ν and φ remains separate. In the geometric-Frobenius convention, ν(φ) is the inverse residue cardinality. It is this unit, rather than the residue cardinality, that enters the earlier special-chain…
Exact mathematical passage
Q_D(T)=\det(1-T r(\varphi)\mid E(D,H)),\qquad Q_D(0)=1, \qquad Q_{\bigoplus_iD_i^{\oplus m_i}}(T)=\prod_iQ_{D_i}(T)^{m_i}.
Q_FUse in source · 6 passages
Search aliases: Q_F
Use in source · 30_lifting.tex, line 777
Retain the original certificate A=p^sJ^-1 and put c=p^s+1. Taylor expansion and a polynomial difference identity yield polynomial vectors Q_F,Q_H, both zero at the origin, such that (AF)(a+cu)=p^2s+1u+p^2s+2Q_F(u), (AH)(a+cu)=AH(a)+cQ_H(u). Consequently (AG)(a+cu)=p^2s+1(C+u+pQ(u)), where C=p^M-(2s+1)AH(a), Q=Q_F+p^M-s-1Q_H, Q(0)=0. All exponents are nonnegative. The same contraction and divisibility argument as in Theorem~[source reference] gives the root, the displacement p^M-s, and full-ball …
Exact mathematical passage
Q_F,Q_H
\mathcal Q_fUse in source · 1 passage
Search aliases: Q_f mathcal
Use in source · Paper_VI_Traditional_Final.tex, line 684
[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.
Exact mathematical passage
\mathcal Q_f
q_FUse in source · 2 passages
Search aliases: q_F
Use in source · 02_strengthened.tex, line 1295
Exact correction of an error span means that one completely positive trace preserving recovery R satisfies R(Fρ F^*)=q_Fρ for every F in that span and every code-supported matrix ρ, with a nonnegative scalar q_F depending on F but not on the logical input. For every normalized channel with Kraus operators in the same span, the same recovery restores the code input with total scalar one. Individual unnormalized errors are not required to have unit recovery weight.
Exact mathematical passage
\Rcal(F\rho F^*)=q_F\rho
Q_HUse in source · 6 passages
Search aliases: Q_H
Use in source · 30_lifting.tex, line 777
Retain the original certificate A=p^sJ^-1 and put c=p^s+1. Taylor expansion and a polynomial difference identity yield polynomial vectors Q_F,Q_H, both zero at the origin, such that (AF)(a+cu)=p^2s+1u+p^2s+2Q_F(u), (AH)(a+cu)=AH(a)+cQ_H(u). Consequently (AG)(a+cu)=p^2s+1(C+u+pQ(u)), where C=p^M-(2s+1)AH(a), Q=Q_F+p^M-s-1Q_H, Q(0)=0. All exponents are nonnegative. The same contraction and divisibility argument as in Theorem~[source reference] gives the root, the displacement p^M-s, and full-ball …
Exact mathematical passage
Q_F,Q_H
Q_iUse in source · 9 passages
Search aliases: Q_i
Use in source · 04_transport.tex, line 100
[Naturality of generated hulls and transfer blocks] Let (Φ,φ,σ) be a packet compatible operator morphism. For every subset S⊆ X, Φ(Hull_ A(S)) = Hull_φ( A)(Φ(S)) ⊆ Hull_ B(Φ(S)). For T∈ A and i,j∈ J, Q_jφ(T)Q_iΦ = ∑_substackσ(β)=j σ(α)=i Φ P_β TP_α. If Φ is surjective and S generates X, then Φ(S) generates Y under φ( A) and hence under B. If, in addition, the morphism is seeded, then S' generates Y. [source reference] ExportsThe canonical naturality theorem for every later realization.
Exact mathematical passage
\boxed{ Q_j\varphi(T)Q_i\Phi = \sum_{\substack{\sigma(\beta)=j\\\sigma(\alpha)=i}} \Phi P_\beta TP_\alpha. }
Q_i^2Use in source · 1 passage
Search aliases: Q_i^2
Use in source · visual_atlas.tex, line 386
headExact reconstruction does not require right angles For the displayed Q_0,Q_1, Q_0+Q_1=I, Q_i^2=Q_i, Q_0Q_1=Q_1Q_0=0, but Q_0^ T≠ Q_0. Expanding both copies of I still proves ∑_i,jQ_iOb Q_j=Ob. The block reconstruction identity itself uses the condition that the projectors sum to the identity. Idempotence is additionally used when constructing a witness that remains supported in one packet after projection. The general formal reconstruction statement does not require pairwise annihilation.
Exact mathematical passage
Q_0+Q_1=I,\qquad Q_i^2=Q_i,\qquad Q_0Q_1=Q_1Q_0=0,
Q_{ii}Use in source · 2 passages
Search aliases: Q_ii
Use in source · Paper_VI_Traditional_Final.tex, line 551
Choose λ≥max_i(-Q_ii). Then B=Q+λ I has nonnegative entries, so e^tB is entrywise nonnegative by its power series. Hence e^tQ=e^-λ te^tB is nonnegative. Since 1^TQ=0, differentiation gives fracddt 1^Te^tQ= 1^TQe^tQ=0, so 1^Te^tQ= 1^T.
Exact mathematical passage
\lambda\ge\max_i(-Q_{ii})
Q_{ij}Use in source · 3 passages
Search aliases: Q_ij
Use in source · Paper_VI_Traditional_Final.tex, line 5352
ty means CQ=overline Q C for a coarse Markov generator. For observables, the backward generator is Q^T. Weighted L^2 inner products use the stated stationary probability. [Reversibility descends] Assume Q is strongly lumpable and reversible with respect to a positive stationary distribution π, so Q_ijπ_j=Q_jiπ_i. Define overlineπ_a=∑_i∈ B_aπ_i. Then overlineπ is stationary for overline Q, and overline Q_aboverlineπ_b = overline Q_baoverlineπ_a. Thus the coarse chain is reversible.
Exact mathematical passage
Q_{ij}\pi_j=Q_{ji}\pi_i.
\mathfrak Q_jUse in source · 1 passage
Search aliases: Q_j mathfrak
Use in source · Paper_X_Block27_Amended.tex, line 1227
For a compatible family ρ_i,λ of representations of the fixed finite group, set V_i,λ=ρ_i,λ∘ q_L. Suppose the actual absolute-Galois members are irreducible and the systems are pairwise componentwise nonisomorphic. The finite prime separator now yields distinct permitted rational primes p_j, actual prime ideals Q_j and arithmetic Frobenius witnesses γ_j as above, outside any prescribed finite set, such that the common algebraic trace matrix has nonzero determinant. Every coefficient embedding pr…
Exact mathematical passage
\mathfrak Q_j
Q_j^2Use in source · 1 passage
Search aliases: Q_j^2
Use in source · 03_packets_and_graphs.tex, line 61
Pairwise annihilation of the P_α gives Q_j^2=Q_j, Q_jQ_k=0 (j≠ k). Summing over j recovers ∑_α P_α=I. The range statement follows from the directness forced by pairwise annihilation.
Exact mathematical passage
Q_j^2=Q_j, \qquad Q_jQ_k=0\quad(j\neq k).
Q_{ji}Use in source · 2 passages
Search aliases: Q_ji
Use in source · Paper_VI_Traditional_Final.tex, line 5352
ty means CQ=overline Q C for a coarse Markov generator. For observables, the backward generator is Q^T. Weighted L^2 inner products use the stated stationary probability. [Reversibility descends] Assume Q is strongly lumpable and reversible with respect to a positive stationary distribution π, so Q_ijπ_j=Q_jiπ_i. Define overlineπ_a=∑_i∈ B_aπ_i. Then overlineπ is stationary for overline Q, and overline Q_aboverlineπ_b = overline Q_baoverlineπ_a. Thus the coarse chain is reversible.
Exact mathematical passage
Q_{ij}\pi_j=Q_{ji}\pi_i.
Q_kUse in source · 1 passage
Search aliases: Q_k
Use in source · 03_packets_and_graphs.tex, line 61
Pairwise annihilation of the P_α gives Q_j^2=Q_j, Q_jQ_k=0 (j≠ k). Summing over j recovers ∑_α P_α=I. The range statement follows from the directness forced by pairwise annihilation.
Exact mathematical passage
Q_j^2=Q_j, \qquad Q_jQ_k=0\quad(j\neq k).
q_kUse in source · 4 passages
Search aliases: q_k
Use in source · 03_weighted.tex, line 557
[Linear recurrence] With the canonical polynomials from [source reference], write Q(t)=∑_kq_kt^k, Z_s,t(t)=∑_n≥0a_n(s,t)t^n. Then the walk counts satisfy the recurrence with constant coefficients determined by Q after the finite initial range contributed by P_s,t, with deg 0=-∞: ∑_k=0^nq_k a_n-k(s,t)=0 (n>deg P_s,t).
Exact mathematical passage
Q(t)=\sum_kq_kt^k, \qquad Z_{s,t}(t)=\sum_{n\ge0}a_n(s,t)t^n.
Q_LUse in source · 18 passages
Search aliases: Q_L
Use in source · extended_pages.tex, line 3
vpage29A complex range needs an adjointTHE INNER PRODUCT IS PART OF THE CONTRACTChapter VIII: Theorem 5.6, Section F.1, Section F.2. The earlier compact example is real. Over C, an isometry means Q^*Q=I, with conjugation. Take the exact Gaussian-rational factors Q_L=mat3/5 · 0 4 i/5 · 0 0 · 1, Q_R=mat1 · 0 0 · i, G=diag(5,2). Their product is a genuinely complex rectangular operator: A=Q_LGQ_R^*=mat3 · 0
Exact mathematical passage
Q_L=\mat{3/5&0\\4\mathrm i/5&0\\0&1},\qquad Q_R=\mat{1&0\\0&\mathrm i},\qquad G=\diag(5,2).
q_LUse in source · 16 passages
Search aliases: q_L
Use in source · 03_weighted.tex, line 311
[Finite precision] Let F^bullet R be a multiplicative decreasing filtration and let q_L:R→ R_L be a ring homomorphism that kills F^L R. Suppose H=∑_e∈ EH_e, D^-1H_e∈ F^w(e)R, w(e)≥1, with E finite. Then q_L(D^-1H)^L=0, the finite sum in [source reference](a) is a two-sided inverse after applying q_L, and its word expansion may discard every word of weight at least L. More explicitly, for a length cutoff N, define S^pr_L,N = ∑_n=0^N-1(-1)^n ∑_substackγ∈ E^n
Exact mathematical passage
q_L:R\to R_L
Q_L^*Use in source · 3 passages
Search aliases: Q_L^*
Use in source · pages.tex, line 208
0 · 0 · 0. Keeping only its first singular direction gives a rank-one approximation with spectral and Frobenius error exactly two. Keeping none gives errors five and sqrt29, while keeping both gives zero. vfigoptimality headWhy an outside competitor cannot do better For isometric outer factors, compression C(B)=Q_L^*BQ_R cannot increase rank or either matrix norm. Every ambient rank-k competitor therefore supplies an admissible core competitor. The lifted core residual has exactly the core resid…
Exact mathematical passage
C(B)=Q_L^*BQ_R
Q_L^TUse in source · 1 passage
Search aliases: Q_L^T
Use in source · extended_pages.tex, line 14
0 · -2 i, A^*A=diag(25,4). vfigcomplex_core headThe small calculation exposes a wrong convention The first column of Q_L has Hermitian squared length 9/25+16/25=1. Its bilinear square without conjugation is 9/25-16/25=-7/25. Thus Q_L^TQ_L is not the identity. The portable negative control replaces the adjoint by transpose and rejects the resulting isometry claim.
Exact mathematical passage
Q_L^TQ_L
q_mUse in source · 3 passages
Search aliases: q_m
Use in source · 02_strengthened.tex, line 369
[Tensor selection rule] If A_a has local charge q_a, then A_1⊗⋯⊗ A_m has total charge q_1+⋯+q_m.
Exact mathematical passage
q_1+\cdots+q_m
Q_{m,n}Use in source · 3 passages
Search aliases: Q_m,n
Use in source · Paper_VI_Traditional_Final.tex, line 2810
A bounded generator of a contraction semigroup on a Hilbert space is dissipative. If Q_m,n=I-P_m,n and y∈ Q_m,nX_n, then ReipQ_m,nA_nQ_m,nyy = ReipA_nyy ≤0. Hence the compressed hidden semigroup is contractive. Apply [source reference] to P_m,n. Since P_m,nR_n,m=R_n,m, η_m,n(T) ≤ Tnorm(I-P_m,n)A_nP_m,n normR_n,m. The right side tends to zero. Use [source reference].
Exact mathematical passage
Q_{m,n}=I-P_{m,n}
Q_nUse in source · 3 passages
Search aliases: Q_n
Use in source · Paper_V_Traditional_Final.tex, line 3383
[Classical square-wave Gibbs behavior] Let q(x)=tfrac12sign(sin x), whose jump at zero has size one. For n≥1, its partial sum through the first n positive odd frequencies is Q_n(x)=frac2π∑_k=0^n-1fracsin((2k+1)x)2k+1. At x_n=π/(2n), x_n⟶0, Q_n(x_n)⟶fracSi(π)π =frac12+G, G≈0.089489872236, where Si(z)=∫_0^zsin(t)/t dt and G=Si(π)/π-1/2>0. Thus a nonzero overshoot persists along points approaching the jump. Scaling by a real amplitude a gives an overshoot or undershoot of magnitude |a|G.
Exact mathematical passage
Q_n(x)=\frac{2}{\pi}\sum_{k=0}^{n-1}\frac{\sin((2k+1)x)}{2k+1}.
q_nUse in source · 4 passages
Search aliases: q_n
Use in source · monograph_atlas.tex, line 788
vpage40Choose the topology that preserves the quantityREFLECTION GRAMS, TRACES, AND FINAL SCOPEChapter VII: Theorem II.13.6, Theorem II.13.8, Section II.14. Let q_n=1/2+2^-(n+2) and define the scalar reflected kernel C_n(j,k)=1+q_n^j+k on nonnegative integers. Each finite Gram matrix is a sum of two positive rank-one matrices, and pointwise convergence gives C(j,k)=1+(1/2)^j+k. For times (0,1,2) and coefficients (1,-2,1), ∑_j,kv_jC_n(j,k)v_k=(1-q_n)^4≥0. This exact check illustrates how finite q…
Exact mathematical passage
q_n=1/2+2^{-(n+2)}
Q_N^{\mathrm{abs}}Use in source · 3 passages
Search aliases: Q_N^abs mathrm
Use in source · Paper_IV_Traditional_Final.tex, line 2789
There is an entire function Q_N^abs such that, initially on Re s>1, Z_N^abs(s) = N A_N^-sζ(s,frac12)+Q_N^abs(s). Thus Z_N^abs continues meromorphically to C with one simple pole at s=1, of residue fracNsin(π/N)π.
Exact mathematical passage
Q_N^{\mathrm{abs}}
q_n^{j+k}Use in source · 1 passage
Search aliases: q_n^j+k
Use in source · monograph_atlas.tex, line 788
vpage40Choose the topology that preserves the quantityREFLECTION GRAMS, TRACES, AND FINAL SCOPEChapter VII: Theorem II.13.6, Theorem II.13.8, Section II.14. Let q_n=1/2+2^-(n+2) and define the scalar reflected kernel C_n(j,k)=1+q_n^j+k on nonnegative integers. Each finite Gram matrix is a sum of two positive rank-one matrices, and pointwise convergence gives C(j,k)=1+(1/2)^j+k. For times (0,1,2) and coefficients (1,-2,1), ∑_j,kv_jC_n(j,k)v_k=(1-q_n)^4≥0. This exact check illustrates how finite q…
Exact mathematical passage
C_n(j,k)=1+q_n^{j+k}
Q_N^{\mathrm{red}}Use in source · 4 passages
Search aliases: Q_N^red mathrm
Use in source · Paper_IV_Traditional_Final.tex, line 2778
[Meromorphic continuation of the symmetry-orbit zero zeta] Put A_N=fracπsin(π/N). There is an entire function Q_N^red such that Z_N^red(s) = A_N^-Nsζ(Ns,frac12)+Q_N^red(s) on Re s>1/N. The right-hand side gives a meromorphic continuation to C with one simple pole at s=1/N. Its residue is fracsin(π/N)Nπ.
Exact mathematical passage
Q_N^{\mathrm{red}}
\mathbb
Q_pUse in source · 1 passage
Search aliases: Q_p mathbb
Use in source · references.tex, line 65
bibitemKisin2010 Mark Kisin. newblock Deformations of G_ Q_p and GL_2( Q_p) representations. newblock In Laurent Berger, Christophe Breuil, and Pierre Colmez, editors, em Repr esentations p-adiques de Groupes p-adiques II: Repr esentations de GL_2( Q_p) et (φ,Γ)-modules, volume 330 of em Ast erisque, pages 511--528. Soci et e Math ematique de France, Paris, 2010.
Exact mathematical passage
\mathrm{GL}_2(\mathbb Q_p)
\mathbb Q_p^eUse in source · 1 passage
Search aliases: Q_p^e mathbb
Use in source · continuation_11.tex, line 115
hbb Z_p^e+k, F(a)=0. Let ι: Z_p↪ Q_p be the coefficient inclusion, applied entrywise to points and matrices. Assume only that ι(D_YF(a))∈GL_e( Q_p). The theorem mcodePolynomialRegularityIntegral.exists_analytic_integral_implicit constructs the preceding analytic graph in the ambient spaces Q_p^k and Q_p^e. Its root equation evaluates the original integral polynomial coefficients using ι, and its base derivative is -ι(D_YF(a))^-1 ι(D_TF(a)). The source proves that polynomial evaluation, formal pa…
Exact mathematical passage
\mathbb Q_p^e
\mathbb Q_p^kUse in source · 1 passage
Search aliases: Q_p^k mathbb
Use in source · continuation_11.tex, line 115
,T]^e, a∈ Z_p^e+k, F(a)=0. Let ι: Z_p↪ Q_p be the coefficient inclusion, applied entrywise to points and matrices. Assume only that ι(D_YF(a))∈GL_e( Q_p). The theorem mcodePolynomialRegularityIntegral.exists_analytic_integral_implicit constructs the preceding analytic graph in the ambient spaces Q_p^k and Q_p^e. Its root equation evaluates the original integral polynomial coefficients using ι, and its base derivative is -ι(D_YF(a))^-1 ι(D_TF(a)). The source proves that polynomial evaluation, for…
Exact mathematical passage
\mathbb Q_p^k
\Q_{p_s}Use in source · 1 passage
Search aliases: Q_p_s Q mathbf
Use in source · Paper_X_Block27_Amended.tex, line 2522
Suppose now that R is I-adically complete and that every positive quotient R/I^m is finite. Let C_m⊆ P_q(R/I^m) be allowed subsets preserved by reduction, and let (p_s)_s∈ S be any indexed family of arithmetic primes. The index set S need not be finite. Lean proves the simultaneous arithmetic completion criterion · ∃χ∈ P_q(R), χ continuous, χbmod I^m∈ C_m (m≥1), χ_ Q_p_s=1 (s∈ S) · Longleftrightarrow ∀ m≥1, ∃ψ∈ C_m, ∀ s∈ S, p_s≡1pmod4 lor ψ=1. The constructed character also satisfies every odd-p…
Exact mathematical passage
\begin{split} &\exists\chi\in\mathcal P_q(R),\quad \chi\text{ continuous},\quad \chi\bmod I^m\in C_m\ (m\ge1),\quad \chi_{\Q_{p_s}}=1\ (s\in S)\\ &\quad\Longleftrightarrow\quad \forall m\ge1,\ \exists\psi\in C_m,\quad \forall s\in S,\ p_s\equiv1\pmod4\ \lor\ \psi=1. \end{split}
Q_RUse in source · 9 passages
Search aliases: Q_R
Use in source · extended_pages.tex, line 3
vpage29A complex range needs an adjointTHE INNER PRODUCT IS PART OF THE CONTRACTChapter VIII: Theorem 5.6, Section F.1, Section F.2. The earlier compact example is real. Over C, an isometry means Q^*Q=I, with conjugation. Take the exact Gaussian-rational factors Q_L=mat3/5 · 0 4 i/5 · 0 0 · 1, Q_R=mat1 · 0 0 · i, G=diag(5,2). Their product is a genuinely complex rectangular operator: A=Q_LGQ_R^*=mat3 · 0
Exact mathematical passage
Q_L=\mat{3/5&0\\4\mathrm i/5&0\\0&1},\qquad Q_R=\mat{1&0\\0&\mathrm i},\qquad G=\diag(5,2).
Q_rUse in source · 3 passages
Search aliases: Q_r
Use in source · 02_strengthened.tex, line 1740
[Finite oscillatory memory expansion] If D=∑_rE_rQ_r is the spectral decomposition of the hidden Hamiltonian, then K(t)=∑_r e^- i tE_rBQ_rB^*. Every coefficient BQ_rB^* is positive semidefinite.
Exact mathematical passage
D=\sum_{r}E_rQ_r
q_rUse in source · 5 passages
Search aliases: q_r
Use in source · 10_foundations.tex, line 415
[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalization hypothesis is required.
Exact mathematical passage
\chi_{C^N}(X)=X^d+\sum_{r<d}q_rX^r
Q_R^*Use in source · 4 passages
Search aliases: Q_R^*
Use in source · extended_pages.tex, line 8
0 · i, G=diag(5,2). Their product is a genuinely complex rectangular operator: A=Q_LGQ_R^*=mat3 · 0 4 i · 0 0 · -2 i, A^*A=diag(25,4). vfigcomplex_core headThe small calculation exposes a wrong convention The first column of Q_L has Hermitian squared length 9/25+16/25=1. Its bilinear square without conjugation is 9/25-16/25=-7/25. Thus Q_L^TQ_L is not the identity. The portable negative control replaces the adjoint by transpose and rejects the resulting isometry claim.
Exact mathematical passage
A=Q_LGQ_R^*=\mat{3&0\\4\mathrm i&0\\0&-2\mathrm i}, \qquad A^*A=\diag(25,4).
Q_{ri}Use in source · 1 passage
Search aliases: Q_ri
Use in source · Paper_VI_Traditional_Final.tex, line 2941
[Continuous time lumping] A coarse Markov generator overline Q satisfying CQ=overline QC exists if and only if the aggregated rates ∑_r∈ B_αQ_ri depend only on the source block of i. In that case C e^tQ= e^toverline QC for every t≥0.
Exact mathematical passage
\sum_{r\in B_\alpha}Q_{ri}
q_{\rm src}Use in source · 2 passages
Search aliases: q_rm src rm
Use in source · Paper_XI_Traditional_Final.tex, line 2150
The original low-rank maximal-type capacity obstruction remains true. The new pair is a common arithmetic assembly carrier. It is not renamed as the original maximal-simple-character pair. For an independently fixed native quadratic marking, a preassigned embedding must satisfy q_W(f(v))=q_rm src(v). At equal capacity, a free embedding requires equal Arf invariant. Spare capacity for d≥4 admits either source Arf type when the embedding is free. Retaining only the central {±1} would erase the dis…
Exact mathematical passage
q_W(f(v))=q_{\rm src}(v)
\mathfrak Q_uUse in source · 2 passages
Search aliases: Q_u mathfrak
Use in source · Paper_X_Block27_Amended.tex, line 1281
The choice of absolute prime and Frobenius lift is also controlled. Let ( Q_u,γ_u) and ( Q_v,γ_v) be any two actual arithmetic witnesses at the same rational prime. Transitivity on primes of A above (p) gives a∈ G_ Q with Q_v=a Q_u, γ_v^-1aγ_u a^-1∈ I_ Q_v. Consequently, any representation trivial on I_ Q_v has the same trace on the two witnesses. For representations factoring through L, triviality on this actual absolute inertia is derived whenever p∉ S_L. The restrictions of all absolute witne…
Exact mathematical passage
(\mathfrak Q_u,\gamma_u)
q_{u,L}Use in source · 2 passages
Search aliases: q_u,L
Use in source · 72_recovered.tex, line 1549
[One unit mode] Let u∈ Z_p^×, let H=Λ_p,L, and put q_u,L=u^H-1∈ p^L Z_p. For every integer address j and every x∈ Z_p, u^Hx+j = u^j∑_r≥0q_u,L^ rbinom xr. The rth Mahler coefficient has valuation at least Lr.
Exact mathematical passage
q_{u,L}=u^H-1\in p^L\Zp.
q_{u,L}^{\,r}Use in source · 1 passage
Search aliases: q_u,L^ r
Use in source · 72_recovered.tex, line 1553
[One unit mode] Let u∈ Z_p^×, let H=Λ_p,L, and put q_u,L=u^H-1∈ p^L Z_p. For every integer address j and every x∈ Z_p, u^Hx+j = u^j∑_r≥0q_u,L^ rbinom xr. The rth Mahler coefficient has valuation at least Lr.
Exact mathematical passage
\boxed{ u^{Hx+j} = u^j\sum_{r\geq0}q_{u,L}^{\,r}\binom xr. }
\mathfrak Q_vUse in source · 3 passages
Search aliases: Q_v mathfrak
Use in source · Paper_X_Block27_Amended.tex, line 1281
The choice of absolute prime and Frobenius lift is also controlled. Let ( Q_u,γ_u) and ( Q_v,γ_v) be any two actual arithmetic witnesses at the same rational prime. Transitivity on primes of A above (p) gives a∈ G_ Q with Q_v=a Q_u, γ_v^-1aγ_u a^-1∈ I_ Q_v. Consequently, any representation trivial on I_ Q_v has the same trace on the two witnesses. For representations factoring through L, triviality on this actual absolute inertia is derived whenever p∉ S_L. The restrictions of all absolute witne…
Exact mathematical passage
(\mathfrak Q_v,\gamma_v)
q_vUse in source · 5 passages
Search aliases: q_v
Use in source · Paper_XI_Traditional_Final.tex, line 1545
[Translation of quadratic refinements] For v∈ V, put q_v(x)=q(x)+b(v,x). Then Arf(q_v)=Arf(q)+q(v).
Exact mathematical passage
q_v(x)=q(x)+b(v,x).
q_WUse in source · 2 passages
Search aliases: q_W
Use in source · Paper_XI_Traditional_Final.tex, line 2150
The original low-rank maximal-type capacity obstruction remains true. The new pair is a common arithmetic assembly carrier. It is not renamed as the original maximal-simple-character pair. For an independently fixed native quadratic marking, a preassigned embedding must satisfy q_W(f(v))=q_rm src(v). At equal capacity, a free embedding requires equal Arf invariant. Spare capacity for d≥4 admits either source Arf type when the embedding is free. Retaining only the central {±1} would erase the dis…
Exact mathematical passage
q_W(f(v))=q_{\rm src}(v)
q_\alphaUse in source · 12 passages
Search aliases: q_α alpha
Use in source · Paper_VI_Traditional_Final.tex, line 775
[Bayesian packet reconstruction] B_λ(p) = ∑_α m_α q_α. Only packets with positive evidence carry a conditional probability vector. A zero-evidence packet has zero posterior mass and contributes the zero vector. The nonlinear coupling between packets occurs only through the total evidence used to normalize the packet evidences.
Exact mathematical passage
\boxed{ \cB_\lambda(p) = \sum_\alpha m_\alpha q_\alpha. }
q_{\alpha,i}Use in source · 10 passages
Search aliases: q_α,i alpha
Use in source · Paper_VI_Traditional_Final.tex, line 763
Let the state indices be partitioned into packets I_α. Define Z_α=∑_i∈ I_αλ_ip_i, m_α=fracZ_α∑_β Z_β, q_α,i= λ_ip_i/Z_α, · Z_α>0, i∈ I_α, 0, · otherwise.
Exact mathematical passage
Z_\alpha=\sum_{i\in I_\alpha}\lambda_ip_i, \qquad m_\alpha=\frac{Z_\alpha}{\sum_\beta Z_\beta}, \qquad q_{\alpha,i}=\begin{cases} \lambda_ip_i/Z_\alpha,&Z_\alpha>0,\ i\in I_\alpha,\\ 0,&\text{otherwise}. \end{cases}
Q_{\beta'}Use in source · 1 passage
Search aliases: Q_β' beta
Use in source · 02_coordinates.tex, line 303
Commutativity gives R_αβ^2=P_α^2Q_β^2=R_αβ. If (α,β)≠(α',β'), then either P_α P_α'=0 or Q_β Q_β'=0, so the corresponding products annihilate one another. Finally, ∑_α,βR_αβ = (∑_α P_α) (∑_β Q_β) =I_X. For a fixed α, P_α =P_α∑_β Q_β = ∑_β R_αβ, and the same argument works for a fixed β. Since each P_α and Q_β is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite f…
Exact mathematical passage
Q_\beta Q_{\beta'}=0
Q_\beta^2Use in source · 1 passage
Search aliases: Q_β^2 beta
Use in source · 02_coordinates.tex, line 300
Commutativity gives R_αβ^2=P_α^2Q_β^2=R_αβ. If (α,β)≠(α',β'), then either P_α P_α'=0 or Q_β Q_β'=0, so the corresponding products annihilate one another. Finally, ∑_α,βR_αβ = (∑_α P_α) (∑_β Q_β) =I_X. For a fixed α, P_α =P_α∑_β Q_β = ∑_β R_αβ, and the same argument works for a fixed β. Since each P_α and Q_β is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite f…
Exact mathematical passage
R_{\alpha\beta}^2=P_\alpha^2Q_\beta^2=R_{\alpha\beta}.
Q_\gamma^2Use in source · 1 passage
Search aliases: Q_γ^2 gamma
Use in source · 03_packets_and_graphs.tex, line 217
For every pair of packet indices, (P_α⊗ Q_γ)^2 =P_α^2⊗ Q_γ^2 =P_α⊗ Q_γ. If (α,γ)≠(β,δ), then either P_α P_β=0 or Q_γ Q_δ=0, so the corresponding tensor idempotents annihilate each other. Finally, ∑_α,γP_α⊗ Q_γ = (∑_α P_α) ⊗ (∑_γ Q_γ) =I_X⊗ I_Y. The range is generated by pure tensors P_α x⊗ Q_γ y, hence equals X_α⊗ Y_γ. Exact reconstruction follows from [source reference].
Exact mathematical passage
(P_\alpha\otimes Q_\gamma)^2 =P_\alpha^2\otimes Q_\gamma^2 =P_\alpha\otimes Q_\gamma.
Q_\deltaUse in source · 6 passages
Search aliases: Q_δ delta
Use in source · 03_packets_and_graphs.tex, line 222
For every pair of packet indices, (P_α⊗ Q_γ)^2 =P_α^2⊗ Q_γ^2 =P_α⊗ Q_γ. If (α,γ)≠(β,δ), then either P_α P_β=0 or Q_γ Q_δ=0, so the corresponding tensor idempotents annihilate each other. Finally, ∑_α,γP_α⊗ Q_γ = (∑_α P_α) ⊗ (∑_γ Q_γ) =I_X⊗ I_Y. The range is generated by pure tensors P_α x⊗ Q_γ y, hence equals X_α⊗ Y_γ. Exact reconstruction follows from [source reference].
Exact mathematical passage
Q_\gamma Q_\delta=0
Q_{\tau^{-1}(k)}Use in source · 1 passage
Search aliases: Q_τ^-1(k) tau
Use in source · 04_transport.tex, line 157
Let (Φ,φ,σ):(X, A, P)→(Y, B, Q) and (Ψ,ψ,τ):(Y, B, Q)→(Z, C, R) be morphisms. Intertwining is preserved because ΨΦ(Ax) = Ψ(φ(A)Φ x) = (ψ∘φ)(A)ΨΦ x. For k in the target packet set, R_kΨΦ · =Ψ Q_τ^-1(k)Φ · =ΨΦ ∑_τ(σ(α))=kP_α. Thus the composite packet map is τ∘σ. If the morphisms are seeded, then Φ(S)⊆Hull_ B(S'), Ψ(S')⊆Hull_ C(S''). By [source reference], ΨΦ(S) ⊆ Ψ(Hull_ B(S')) ⊆ Hull_ C(Ψ(S')) ⊆ Hull_ C(S''). Identity maps satisfy the definitions with identity algebra and packet maps.
Exact mathematical passage
R_k\Psi\Phi &=\Psi Q_{\tau^{-1}(k)}\Phi\\ &=\Psi\Phi \sum_{\tau(\sigma(\alpha))=k}P_\alpha.
\mathfrak Q_\ellUse in source · 1 passage
Search aliases: Q_ℓ mathfrak ell
Use in source · 01_finite_core.tex, line 1003
At resolution ℓ, define Q_ℓ = ( A_ℓ, H_ℓ, D_ℓ, {P_ℓ,α}, E_ℓ, C_ℓ, Π_ℓ^phys, T_ℓ ). In order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_ℓ: H_ℓ→ H_ℓ+1 and C_ℓ: H_ℓ+1→ H_ℓ. The left inverse relation C_ℓ E_ℓ=I alone does not imply C_ℓ=E_ℓ^* or make the embedding isometric.
Exact mathematical passage
\boxed{ \mathfrak Q_\ell = \left( \cA_\ell, \cH_\ell, D_\ell, \{P_{\ell,\alpha}\}, E_\ell, C_\ell, \Pi_\ell^{\mathrm{phys}}, T_\ell \right). }
q_\ellUse in source · 2 passages
Search aliases: q_ℓ ell
Use in source · 03_nonlinear_coordinates.tex, line 852
[Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.
Exact mathematical passage
q_\ell<1
Q_{\bigoplus_iD_i^{\oplus m_i}}Use in source · 1 passage
Search aliases: Q_⊕_iD_i^⊕ m_i bigoplus oplus
Use in source · Paper_X_Block27_Amended.tex, line 1509
Writing Q_D(T) for the reversed local characteristic polynomial, Lean proves Q_D(T)=det(1-T r(φ)mid E(D,H)), Q_D(0)=1, Q_⊕_iD_i^⊕ m_i(T)=∏_iQ_D_i(T)^m_i. The abstract formulas do not need characteristic zero. Normality and finite dimension appear explicitly where used. Arithmetic identification of G,H,ν and φ remains separate. In the geometric-Frobenius convention, ν(φ) is the inverse residue cardinality. It is this unit, rather than the residue cardinality, that enters the earlier special-chain…
Exact mathematical passage
Q_D(T)=\det(1-T r(\varphi)\mid E(D,H)),\qquad Q_D(0)=1, \qquad Q_{\bigoplus_iD_i^{\oplus m_i}}(T)=\prod_iQ_{D_i}(T)^{m_i}.
\Qcal'Use in source · 3 passages
Search aliases: Qcal' Q' Qcal mathcal
Use in source · 03_nonlinear_coordinates.tex, line 169
[Change of packet coordinates] Let P and P' be packet systems on X. Let Q and Q' be packet systems on Y. For any two packet systems U and V on the same space, write J_ U← V = A_ U S_ V. Then F_ Q'← P' = J_ Q'← Q F_ Q← P J_ P← P'.
Exact mathematical passage
\Qcal'
\Qcal_{cd}Use in source · 1 passage
Search aliases: Qcal_cd Q_cd Qcal mathcal
Use in source · 02_strengthened.tex, line 1043
Let Φ:End( H_X)→End( H_Y) be linear. Choose packet systems (P_a) on H_X and (Q_c) on H_Y. Define Φ_cd,ab = Q_cdΦ P_ab.
Exact mathematical passage
\Phi_{cd,ab} = \Qcal_{cd}\Phi\Pcal_{ab}.
\qedhereUse in source · 1 passage
Search aliases: qedhere
Use in source · 04_correction_transport.tex, line 1132
For a section s satisfying Ps=s, ∇^2s=P d(P ds)=P(dP)wedge ds. Since ds=(dP)s+Pds and P(dP)P=0, the surviving term is P(dPwedge dP)Ps.qedhere
Exact mathematical passage
P(dP\wedge dP)Ps.\qedhere
\Qp^dUse in source · 1 passage
Search aliases: Qp^d Q_p^d Qp mathbb
Use in source · 30_lifting.tex, line 766
[Weighted coefficient deformation] Let F,H∈ R[X_1,…,X_d]^d, F(a)=0, and J=DF(a). Suppose J∈GL_d( Q_p), p^sJ^-1∈ M_d(R), and M>2s. For G=F+p^MH, there is exactly one root b∈ a+p^s+1R^d. It satisfies b-a∈ p^M-sR^d and, in Q_p^d, b-a=-p^MJ^-1H(a)+ε, ε∈ p^2M-3sR^d. Only the original Jacobian DF(a) is subject to an inverse hypothesis.
Exact mathematical passage
\Qp^d
\Qp^eUse in source · 1 passage
Search aliases: Qp^e Q_p^e Qp mathbb
Use in source · 30_lifting.tex, line 842
y-y_0 · =-p^MJ^-1(H(a)+Kv)+ε, ε∈ p^2M-3sR^e. The equality involving J^-1 is in Q_p^e, with an integral error vector. The parameter equality is exact.
Exact mathematical passage
\Qp^e
\QQ^{16}Use in source · 1 passage
Search aliases: QQ^16 QQ
Use in source · atlas-pages.tex, line 5
tion 11, Section 13. A packet calculus can reorganize a signal exactly, choose among finitely many representations, or describe a nonlinear coordinate geometry. These are not the same task. This guide keeps one concrete object in view for each, so that the reader can compute rather than only follow arrows. vfigthree_worlds headA signal that can be checked by hand Our main finite signal is f=s+j∈QQ^16. The background repeats tfrac14(0,1,2,1,0,-1,-2,-1), and j[n]=3 at 6≤ n<12, with j[n]=0 elsewher…
Exact mathematical passage
f=s+j\in\QQ^{16}
\QQ^2Use in source · 12 passages
Search aliases: QQ^2 QQ QQ^2 Q^2 QQ mathbb
Use in source · visual_atlas.tex, line 86
Define a surjective linear map q:X⟶ Q^2, q(∑ a_je_j+∑ b_jf_j)=(a_0,b_0).
Exact mathematical passage
q:X\longrightarrow\QQ^2,\qquad q\Bigl(\sum a_je_j+\sum b_jf_j\Bigr)=(a_0,b_0).
\QQ^3Use in source · 3 passages
Search aliases: QQ^3 QQ QQ^3 Q^3 QQ mathbb
Use in source · visual_atlas.tex, line 139
vpage6A reached type can still hide a directionTYPED MODEL / U= Q^2 AND V= Q^3Main text: [source reference] The chapter's typed model has a rotation R:U→ U, an inclusion A:U→ V, and a retraction B:V→ U: R=mat0 · -1
Exact mathematical passage
V=\QQ^3
\QQ^4Use in source · 2 passages
Search aliases: QQ^4 QQ QQ^4 Q^4 QQ mathbb
Use in source · visual_atlas.tex, line 220
vpage9Two visible routes can cancel exactlyREACHABILITY / NECESSARY IS NOT SUFFICIENTMain text: [source reference] Let a single operator D on Q^4 act by De_1=e_2+e_3, De_2=e_4, De_3=-e_4, De_4=0. Every vertex of its transfer graph for the coordinate packets is reachable from vertex 1. Nevertheless, D^2e_1=e_4-e_4=0, D^2=0.
Exact mathematical passage
\QQ^4
\QQ^5Use in source · 1 passage
Search aliases: QQ^5 QQ
Use in source · visual_atlas.tex, line 136
vpage7Reachability is only an upper boundMODEL A / AN ORBIT THAT STAYS IN TWO DIMENSIONSChapter III, mathematical text: Corollary 3.11 Keep the diamond and add an isolated packet 4. Start with the single seed e_0∈QQ^5. Direct multiplication gives Te_0=e_1+e_2, T^2e_0=0. Every later orbit vector is zero. The orbit hull generated by T is therefore exactly the span of the first two displayed columns.
Exact mathematical passage
e_0\in\QQ^5
\QQ^6Use in source · 4 passages
Search aliases: QQ^6 QQ QQ^6 Q^6 QQ mathbb
Use in source · visual_atlas.tex, line 8
Use the same space of dimension six throughout the first comparison: X= Q^6, N=J_3(0)⊕ J_3(0), s=e_0+f_0. Here each J_3(0) moves a basis vector forward along a chain of three basis vectors. The projectors P_e,P_f select the two chains. Their formulas and the proofs of the minimum seed counts appear on pagerefV3 and pagerefV4.
Exact mathematical passage
X=\QQ^6,\qquad N=J_3(0)\oplus J_3(0),\qquad s=e_0+f_0.
\QQ_2Use in source · 2 passages
Search aliases: QQ_2 QQ QQ_2 Q_2 QQ mathbb
Use in source · visual_atlas.tex, line 353
vfig[.94]dyadic_eigenlattice platecaptionA picture of the ordinary integer lattice, not a drawing of the topology of QQ_2. Filled dots have even second coordinate. Unfilled dots lie in the other coset. The hollow point at (1/2,0) records the image of an integral vector under E_+.
Exact mathematical passage
\QQ_2
\QQ_pUse in source · 1 passage
Search aliases: QQ_p Q_p QQ mathbb
Use in source · Paper_XII_Traditional_Final.tex, line 202
Let F be a number field. Fix a prime p, a finite extension E/ Q_p, its valuation ring O, uniformizer ϖ, and residue field k. Let barρ:G_F,S⟶GL_n(k) be a continuous residual representation, where S is a finite set of places containing every place at which the deformation problem is allowed to ramify.
Exact mathematical passage
E/\QQ_p
\RUse in source · 8 passages
Search aliases: R mathbb
Use in source · 05_worked_models.tex, line 195
Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. Therefore sequences with finite support lie in its algebraic orbit hull and are dense in ℓ^2( Z).
Exact mathematical passage
\mathbb{F}\in\{\R,\C\}
R'Use in source · 3 passages
Search aliases: R'
Use in source · Paper_V_Traditional_Final.tex, line 1866
Let L'≤ L≤ Z^d, with both indices finite. Choose representative systems R for Z^d/L and R' for Z^d/L'. For r∈ R, define Child(r) = {s∈ R':s+L'⊂ r+L}.
Exact mathematical passage
R'
r^{(a)}Use in source · 1 passage
Search aliases: r^(a)
Use in source · Paper_VI_Traditional_Final.tex, line 5402
FloatBarrier Prior-restricted sufficiency and filtering. Let L be a strictly positive finite likelihood and let B_L(p)=Lodot p/ipLp for a probability vector p. Fix probability vectors r^(a) supported on B_a, and restrict the priors to p_i=q_a r_i^(a) for i∈ B_a, with q a coarse probability vector. [Sufficiency relative to a prior disintegration] Define overline L_a = ∑_i∈ B_aL_ir_i^(a). For every prior in the declared family, C B_L(p) = B_overline L(q).
Exact mathematical passage
r^{(a)}
R^*Use in source · 6 passages
Search aliases: R^*
Use in source · monograph_atlas.tex, line 556
vpage30Rotate whole packets by a polar factorAN EXACT LOCAL CHART AND ITS FAILUREChapter VII: Theorem II.6.5, Theorem II.6.6. On HH= C^2, put P_0=diag(1,0), P_1=I-P_0, and R=mat3/5 · -4/5 4/5 · 3/5, Q_a=RP_aR^*, S=∑_a=0^1Q_aP_a=frac35R. The exact Gram matrices give (S-I)^*(S-I)=frac1625I, S^*S=frac925I, normS-I=frac45<1. Consequently the unitary polar factor is S(S^*S)^-1/2=R, and Q_aR=RP_a for both complete packets. xfigpolar headThe norm condition selects a chart If the target rank-one packet …
Exact mathematical passage
R=\mat{3/5&-4/5\\4/5&3/5},\qquad Q_a=RP_aR^*, \qquad S=\sum_{a=0}^1Q_aP_a=\frac35R.
r^{-1}Use in source · 5 passages
Search aliases: r^-1
Use in source · 04_arithmetic.tex, line 19
[Bézout spectral projector] Assume that χ_A(x)=f(x)g(x) with monic polynomials f,g∈ O[x] that are coprime in K[x]. Choose u,v∈ O[x] and a nonzero r∈ O such that u(x)f(x)+v(x)g(x)=r. Then E_f=r^-1v(A)g(A) is the projector of V onto Ker f(A) along Ker g(A). In particular, E_f^2=E_f, rE_f∈End_ O(Λ).
Exact mathematical passage
E_f=r^{-1}v(A)g(A)
r^{-2}Use in source · 1 passage
Search aliases: r^-2
Use in source · 04_arithmetic.tex, line 156
The B ezout identity gives N=rI-UF. The two stated commutation relations and FG=0 give N(UF)=VGUF=VUGF=VUFG=0. Therefore N^2=N(rI-UF)=rN. Since r≠0, E^2=r^-2N^2=E. If Fx=0, the evaluated B ezout identity gives VGx=rx, hence Ex=x. If Gx=0, then Ex=0. The lattice assertion follows directly from rE=VG.
Exact mathematical passage
E^2=r^{-2}N^2=E
R^{-m}Use in source · 3 passages
Search aliases: R^-m
Use in source · Paper_IV_Traditional_Final.tex, line 2676
[Leading dominant zero term] Under the hypotheses of [source reference], P_N,n(r) = · m!binomm+r-1r-1 ∑_ρ∈ Z_R fracρ^-m(-ρ D_N'(ρ))^r · +O(m!m^r-2R^-m) +O_R_*(m!R_*^-m). For r=1, the first error term is absent.
Exact mathematical passage
\begin{aligned} P_{N,n}(r) ={}&m!\binom{m+r-1}{r-1} \sum_{\rho\in\Zcal_R} \frac{\rho^{-m}}{(-\rho\Dfun_N'(\rho))^r} \\ &+O\left(m!m^{r-2}R^{-m}\right) +O_{R_*}\left(m!R_*^{-m}\right). \end{aligned}
R^{1/N}Use in source · 5 passages
Search aliases: R^1/N
Use in source · Paper_IV_Traditional_Final.tex, line 2251
[Zero counting asymptotics] Let n_ D_N(R) count the zeros of D_N in |z|≤ R, with multiplicity. Let n_Δ_N(R) count the zeros of Δ_N in |w|≤ R. Then, as R→∞, n_ D_N(R) = fracNsin(π/N)πR+O(1) and n_Δ_N(R) = fracsin(π/N)πR^1/N+O(1).
Exact mathematical passage
n_{\Delta_N}(R) = \frac{\sin(\pi/N)}{\pi}R^{1/N}+O(1).
R^2Use in source · 1 passage
Search aliases: R^2
Use in source · 03_nonlinear_coordinates.tex, line 514
Assume now that a>0 and b>0. Take R= frac1-sqrt1-4ab2b. Then a+bR^2=R and 2bR<1. Apply [source reference]. The number of full ordered binary trees with n leaves is the Catalan number C_n-1, which gives the displayed majorant.
Exact mathematical passage
a+bR^2=R
\mathbb R^2Use in source · 3 passages
Search aliases: R^2 mathbb
Use in source · 03_nonlinear_coordinates.tex, line 946
[htbp] centering includegraphics[width=.96linewidth]correction_cokernel_plane.pdf captionA specified correction range in R^2. For L(a,b)=(a,0), the image is the horizontal axis. The defect (2,3) splits into (2,0) in that image and the remaining representative (0,3). Subtracting the image component leaves the same cokernel class. The functional (x,y)↦ y vanishes on the correction range and detects the value three. The correction equation Lu=-(2,3) therefore has no solution.
Exact mathematical passage
\mathbb R^2
r^2Use in source · 4 passages
Search aliases: r^2
Use in source · Paper_IV_Traditional_Final.tex, line 1236
For N=1, one has D_1(z)^-r=e^rz and P_1,n(r)=r^n. For N=2, P_2,1(r)=r, P_2,2(r)=r(3r+2), and P_2,3(r)=r(15r^2+30r+16). At r=1, these are the positive secant coefficients.
Exact mathematical passage
P_{2,3}(r)=r(15r^2+30r+16).
r^3Use in source · 1 passage
Search aliases: r^3
Use in source · atlas-pages.tex, line 185
ynomials collect the possible ways logarithmic terms contribute total degree n. vfigbell_partitions For N=2, the first three logarithmic coefficients are κ_1=frac12, κ_2=frac16, κ_3=frac115. The third Bell polynomial is B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3. Substituting x_1=rκ_1, x_2=rκ_2, x_3=2rκ_3 and multiplying by 6!/3! gives P_2,3(r)=15r^3+30r^2+16r. At r=1 this is 61. At r=2 it is 272. At r=0 it vanishes, as every positive-index reciprocal coefficient must. headThe leading term has a simple …
Exact mathematical passage
P_{2,3}(r)=15r^3+30r^2+16r.
R^dUse in source · 44 passages
Search aliases: R^d
Use in source · 30_lifting.tex, line 37
[Polynomial Taylor remainder] Let R be a commutative ring, P∈ R[X_1,…,X_d], a∈ R^d, and c∈ R. There is a polynomial Q∈ R[U_1,…,U_d], with Q(0)=0, such that for every u∈ R^d, P(a+cu)=P(a)+c∑_j=1^d(∂_jP)(a)u_j+c^2Q(u). In particular, when R= Z and a,u,c are integral, subtracting the first two terms on the right from P(a+cu) gives a multiple of c^2.
Exact mathematical passage
a\in R^d
R^eUse in source · 9 passages
Search aliases: R^e
Use in source · 30_lifting.tex, line 833
rmation] leavevmode Let F,H∈ R[Y_1,…,Y_e,T_1,…,T_k]^e and a=(y_0,t_0)∈ R^e+k satisfy F(a)=0. Define the actual blocks J=D_YF(a)∈ M_e(R), K=D_TF(a)∈ M_e× k(R). Suppose J∈GL_e( Q_p), p^sJ^-1∈ M_e(R), and M>2s. For every v∈ R^k, set τ=t_0+p^Mv, G=F+p^MH. There is exactly one y∈ y_0+p^s+1R^e with G(y,τ)=0. It satisfies y-y_0 · ∈ p^M-sR^e,
Exact mathematical passage
y\in y_0+p^{s+1}R^e
R^{e+k}Use in source · 6 passages
Search aliases: R^e+k
Use in source · 30_lifting.tex, line 824
[Joint coefficient and parameter deformation] leavevmode Let F,H∈ R[Y_1,…,Y_e,T_1,…,T_k]^e and a=(y_0,t_0)∈ R^e+k satisfy F(a)=0. Define the actual blocks J=D_YF(a)∈ M_e(R), K=D_TF(a)∈ M_e× k(R). Suppose J∈GL_e( Q_p), p^sJ^-1∈ M_e(R), and M>2s. For every v∈ R^k, set τ=t_0+p^Mv, G=F+p^MH. There is exactly one y∈ y_0+p^s+1R^e with G(y,τ)=0. It satisfies y-y_0 · ∈ p^M-sR^e,
Exact mathematical passage
a=(y_0,t_0)\in R^{e+k}
R^IUse in source · 2 passages
Search aliases: R^I
Use in source · supplement_06.tex, line 30
pageheading1. Actual matrix orbits and arithmetic jumpsTheorem 9.1; manuscript page 22 Let R be a commutative ring, I a finite coordinate type and d=|I|. For C∈ M_I(R) and v_0∈ R^I, define orbit(C,v_0,n)=C^nv_0, u_k=packet(C,v_0,N,j,k)=C^Nk+jv_0. The implementation uses actual matrix powers and matrix--vector multiplication. A separately supplied sequence v with v_n+1=Cv_n is proved by induction to equal this power orbit. Thus the source-sequence formulation is connected to the construction.
Exact mathematical passage
v_0\in R^I
R^kUse in source · 3 passages
Search aliases: R^k
Use in source · 30_lifting.tex, line 829
[Joint coefficient and parameter deformation] leavevmode Let F,H∈ R[Y_1,…,Y_e,T_1,…,T_k]^e and a=(y_0,t_0)∈ R^e+k satisfy F(a)=0. Define the actual blocks J=D_YF(a)∈ M_e(R), K=D_TF(a)∈ M_e× k(R). Suppose J∈GL_e( Q_p), p^sJ^-1∈ M_e(R), and M>2s. For every v∈ R^k, set τ=t_0+p^Mv, G=F+p^MH. There is exactly one y∈ y_0+p^s+1R^e with G(y,τ)=0. It satisfies y-y_0 · ∈ p^M-sR^e,
Exact mathematical passage
v\in R^k
R^mUse in source · 2 passages
Search aliases: R^m
Use in source · rank2.tex, line 131
For if ku∈U_A m with k∈U_K 1 and u∈U_A (m+1), then k∈ K^×∩ U_A^m=U_K^m. The reverse inclusion is immediate. The complete period-two A-grade R^m/R^m+1 has F2-dimension 2, and its field grade p_K^m/ p_K^m+1 has dimension 1. Thus
Exact mathematical passage
R^m/R^{m+1}
R^{m+1}Use in source · 1 passage
Search aliases: R^m+1
Use in source · rank2.tex, line 131
For if ku∈U_A m with k∈U_K 1 and u∈U_A (m+1), then k∈ K^×∩ U_A^m=U_K^m. The reverse inclusion is immediate. The complete period-two A-grade R^m/R^m+1 has F2-dimension 2, and its field grade p_K^m/ p_K^m+1 has dimension 1. Thus
Exact mathematical passage
R^m/R^{m+1}
R^nUse in source · 4 passages
Search aliases: R^n
Use in source · Paper_X_Block27_Amended.tex, line 2129
The actual submodules U=imE and W=ker E are invariant, and addition gives an equivariant isomorphism U⊕ W≃ R^n. The inclusion i:U↪ R^n and retraction r(v)=Ev are constructed as linear maps and satisfy ri=1_U and ir=E. Their equivariance is proved. Coordinatewise reduction maps U onto im(bar Ebmod(π)^k) and maps W onto ker(bar Ebmod(π)^k). Both equalities are surjectivity statements onto the actual submodules over the finite-level quotient, not merely containment. The same projector defines a sum…
Exact mathematical passage
U\oplus W\simeq R^n.
r^NUse in source · 1 passage
Search aliases: r^N
Use in source · Paper_IV_Traditional_Final.tex, line 1774
The coefficient of w^n in Δ_N has modulus 1/(Nn)!. Stirling's formula and [source reference] give order 1/N. Since M_f( · ^N)(r)=M_f(r^N) for the maximum modulus, composition with z^N multiplies the order by N and gives order one for D_N.
Exact mathematical passage
M_{f(\,\cdot\,^N)}(r)=M_f(r^N)
r^nUse in source · 4 passages
Search aliases: r^n
Use in source · Paper_IV_Traditional_Final.tex, line 1228
For N=1, one has D_1(z)^-r=e^rz and P_1,n(r)=r^n. For N=2, P_2,1(r)=r, P_2,2(r)=r(3r+2), and P_2,3(r)=r(15r^2+30r+16). At r=1, these are the positive secant coefficients.
Exact mathematical passage
P_{1,n}(r)=r^n
R^{\NN}Use in source · 1 passage
Search aliases: R^NN R^ N NN mathbb
Use in source · 72_recovered.tex, line 124
[Hurwitz series ring] Let R be a commutative ring with identity. The Hurwitz series ring H(R) is the set R^ N with componentwise addition and product (amathbin⋆ b)_n = ∑_r=0^n binom nr a_r b_n-r. Its identity is δ=(1,0,0,…).
Exact mathematical passage
R^{\NN}
R^{r-1}Use in source · 2 passages
Search aliases: R^r-1
Use in source · 03_nonlinear_coordinates.tex, line 447
[Polynomial tree expansion] Suppose there is R>0 such that ϕ(R)≤ R and q= ∑_r=2^dr b_rR^r-1<1. Then the map Φ(x)=s+∑_r=2^dB_r(x,…,x) has a unique fixed point in the closed ball overline B_R(0). The sum of the values of all finite ordered rooted trees converges absolutely to that fixed point.
Exact mathematical passage
q= \sum_{r=2}^{d}r b_rR^{r-1}<1.
R^TUse in source · 2 passages
Search aliases: R^T
Use in source · pages.tex, line 190
1 · 0 · -1. Set L=Q_LR_L and R=R_R. The two equal columns of L do not create a third range direction. In contrast, R has rank three. vfigcompact_core The actual core is G=R_LR_R^T=mat5 · 0 · 0 0 · 2 · 0, LR^T=Q_LG. Its dimensions are the two actual input ranks, two and three. This is not inferred by merely discarding a column with a small norm. The range factor Q_L has orthonormal columns and reconstructs L exactly. headThe preserved formal boundary The small-core recompression construction in A…
Exact mathematical passage
G=R_LR_R^T=\mat{5&0&0\\0&2&0},\qquad LR^T=Q_LG.
R^VUse in source · 1 passage
Search aliases: R^V
Use in source · 01_finite_coordinates.tex, line 353
[Orbit support obstruction] Let S⊆ V, and let x∈ R^V be supported on S. If j is not reachable by a directed path from any vertex of S, then (T^nx)_j=0 for every n∈ N. Hence the orbit under T of x cannot generate a vector with a nonzero coordinate at j. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Reachable; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp_pow_mulVec_apply_eq_zero_of_unreachable_support statusleanpriorKomorebi.PacketNaturalityCompletion.R…
Exact mathematical passage
x\in R^V
R^{V\times V}Use in source · 1 passage
Search aliases: R^V× V times
Use in source · 02_connections_obstructions.tex, line 604
The passage from words to walks also applies to products of different matrices. Let V be one finite packet type, let R be a semiring, and let B_1,…,B_m∈ R^V× V be a finite sequence of possibly different packet matrices. Rows are targets and columns are sources as before.
Exact mathematical passage
B_1,\ldots,B_m\in R^{V\times V}
R^\tauUse in source · 2 passages
Search aliases: R^τ tau
Use in source · Paper_XII_Traditional_Final.tex, line 317
is canonically A/J. The ordinary tensor identity B⊗_A(A/J)≅ B/JB carries the defining ideal of the tensor to the image of m_B. Here the structural map is local by the stated global deformation datum, so it carries m_A into m_B. Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism R^τ≅ B/overlineJB. The construction commutes with the canonical map from B, which is therefore surjective. It is also unchanged by transporting the t…
Exact mathematical passage
R^\tau\cong B/\overline{JB}.
R^\nablaUse in source · 2 passages
Search aliases: R^∇ nabla
Use in source · 03_supplement_06.tex, line 125
Cancellation of the second derivatives. The product rule and the identity for the ambient Lie bracket reduce curvature to R^∇(X,Y)s=P d P[X] d s[Y]-P d P[Y] d s[X]. Substitute the identity obtained by differentiating the range condition for the section. The terms containing P d P[v]P vanish, leaving R^∇(X,Y)s= P( d P[X] d P[Y]- d P[Y] d P[X])P s. Lean also proves that this depends only on X(x), Y(x) and s(x), not on their extensions. The operator-valued form is alternating, additive and homogene…
Exact mathematical passage
R^\nabla(X,Y)s=P\dd P[X]\dd s[Y]-P\dd P[Y]\dd s[X].
R_*Use in source · 7 passages
Search aliases: R_*
Use in source · Paper_IV_Traditional_Final.tex, line 2631
rest-pole coefficient expansion] Let r∈ Z_>0. Suppose the zeros of D_N of least modulus R are simple and form the finite set Z_R. Suppose the next zero modulus is R_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_ρ,ℓ binomm+ℓ-1ℓ-1 ρ^-m + O_R_*(m!R_*^-m). The leading Laurent coefficient is a_ρ,r = (-ρ D_N'(ρ))^-r.
Exact mathematical passage
R_*
R_*^{-m}Use in source · 4 passages
Search aliases: R_*^-m
Use in source · Paper_IV_Traditional_Final.tex, line 2632
rcle_expansion Let r∈ Z_>0. Suppose the zeros of D_N of least modulus R are simple and form the finite set Z_R. Suppose the next zero modulus is R_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_ρ,ℓ binomm+ℓ-1ℓ-1 ρ^-m + O_R_*(m!R_*^-m). The leading Laurent coefficient is a_ρ,r = (-ρ D_N'(ρ))^-r.
Exact mathematical passage
P_{N,n}(r) = m! \sum_{\rho\in\Zcal_R} \sum_{\ell=1}^{r} a_{\rho,\ell} \binom{m+\ell-1}{\ell-1} \rho^{-m} + O_{R_*}(m!R_*^{-m}).
R_0Use in source · 9 passages
Search aliases: R_0
Use in source · 72_recovered.tex, line 1324
[The first nontrivial dyadic row] For n=4, R_4=frac9115, R_2=frac53, R_0=1. The exact kernel coefficients are κ_2,1=-frac83, κ_2,2=frac1285. Therefore E_4 = frac9115 -frac83binom42frac53 +frac1285 =5. The rational source values are dyadic units, while the transfer depth is carried by the kernel.
Exact mathematical passage
R_4=\frac{91}{15}, \qquad R_2=\frac53, \qquad R_0=1.
r_0Use in source · 3 passages
Search aliases: r_0
Use in source · Paper_V_Traditional_Final.tex, line 2103
[Stability under filter perturbation] Assume RE=z^-dI_N. Let widetilde E=E+Δ E, widetilde R=R+Δ R. Set e_0 · =*ess sup E _op, · r_0 · =*ess sup R _op, δ_E · =*ess sup Δ E _op, · δ_R · =*ess sup Δ R _op. Then the delay compensated reconstruction defect obeys ε_rec ≤ r_0δ_E+e_0δ_R+δ_Rδ_E.
Exact mathematical passage
\begin{aligned} e_0&=\esssup\|E\|_{\mathrm{op}}, & r_0&=\esssup\|R\|_{\mathrm{op}},\\ \delta_E&=\esssup\|\Delta E\|_{\mathrm{op}}, & \delta_R&=\esssup\|\Delta R\|_{\mathrm{op}}. \end{aligned}
R_1Use in source · 8 passages
Search aliases: R_1
Use in source · 20_arithmetic.tex, line 324
The standard window supplies the initial ratio R_1. The weighted estimate handles pmid c, and Riccati propagation handles pnmid c. Flatness is closed under products, so the exact factorization establishes flatness of j! P_N,j. Extracting its supported Hurwitz coefficients and applying the proved saturation and step-multiple lemmas yields [source reference]--[source reference].
Exact mathematical passage
R_1
R_1^*Use in source · 1 passage
Search aliases: R_1^*
Use in source · monograph_atlas.tex, line 613
The syndrome support is smaller than the ambient space. Define a total recovery with Kraus operators R_1=P, R_2=ket0bra2, R_1^*R_1+R_2^*R_2=I_3. It acts as the identity on operators supported on the code and sends the unused complement population into a chosen code state. The declared error map preserves the trace of code states, so recovery returns every code matrix unit, including both off-diagonal units, exactly. It need not preserve states on the unused complement. headExceptional cases and …
Exact mathematical passage
R_1=P,\qquad R_2=\ket0\bra2, \qquad R_1^*R_1+R_2^*R_2=I_3.
R_1^2Use in source · 1 passage
Search aliases: R_1^2
Use in source · monograph_atlas.tex, line 571
3/5 · 4/5 obeys R_1^2=mat7/25 · -24/25 24/25 · 7/25. The multiplication is exact and each step has a valid local overlap. The convergence theorem has a separate analytic contract: a twice continuously differentiable complete projector path on a compact time interval, sufficiently fine partitions, and a constant depending on the path in an O(|π|) estimate in operator norm. headEvidence boundary The retained run checks the Gram identities, polar factor, intertwining, and product of two rational st…
Exact mathematical passage
R_1^2=\mat{7/25&-24/25\\24/25&7/25}.
R_2Use in source · 8 passages
Search aliases: R_2
Use in source · Paper_IV_Traditional_Final.tex, line 2620
[Nearest-pole coefficient expansion] Let r∈ Z_>0. Suppose the zeros of D_N of least modulus R are simple and form the finite set Z_R. Suppose the next zero modulus is R_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_ρ,ℓ binomm+ℓ-1ℓ-1 ρ^-m + O_R_*(m!R_*^-m). The leading Laurent coefficient is a_ρ,r = (-ρ D_N'(ρ))^-r.
Exact mathematical passage
R_2>R
R_{2(r-m)}Use in source · 2 passages
Search aliases: R_2(r-m)
Use in source · supplement_03.tex, line 122
rgetDyadicSeries.dyadic_reconstruction_series pageheading3. Exact reconstruction and genuine dyadic congruenceTheorem 5.3 and Corollary 5.4; manuscript pages 14--15 The reciprocal coefficients β_m use an ordinary power expansion. In contrast, the source and Euler series use exponential coefficients. Extracting degree 2r from B(2t)F(t)=sech t therefore gives E_2r=∑_m=0^rβ_m2^2mfall2r2mR_2(r-m), fall2r2m=frac(2r)!(2r-2m)!. The formal proof uses the actual natural-number descending factorial and pr…
Exact mathematical passage
\boxed{E_{2r}=\sum_{m=0}^{r}\beta_m2^{2m}\fall{2r}{2m}R_{2(r-m)},} \qquad \fall{2r}{2m}=\frac{(2r)!}{(2r-2m)!}.
R_2^*Use in source · 1 passage
Search aliases: R_2^*
Use in source · monograph_atlas.tex, line 613
The syndrome support is smaller than the ambient space. Define a total recovery with Kraus operators R_1=P, R_2=ket0bra2, R_1^*R_1+R_2^*R_2=I_3. It acts as the identity on operators supported on the code and sends the unused complement population into a chosen code state. The declared error map preserves the trace of code states, so recovery returns every code matrix unit, including both off-diagonal units, exactly. It need not preserve states on the unused complement. headExceptional cases and …
Exact mathematical passage
R_1=P,\qquad R_2=\ket0\bra2, \qquad R_1^*R_1+R_2^*R_2=I_3.
R_2^{-1}Use in source · 1 passage
Search aliases: R_2^-1
Use in source · visual_atlas.tex, line 113
The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not merely reuse the same numerical matrix under a new ordering.
Exact mathematical passage
\widehat T_N=R_NTR_N^{-1},\qquad J_{4,2}=R_4R_2^{-1}.
R_{2n}Use in source · 2 passages
Search aliases: R_2n
Use in source · 72_recovered.tex, line 1898
[Analytic dyadic source] The function mathscr R belongs to the dyadic Tate algebra and takes values in Z_2. For every nonnegative integer n, mathscr R(n)=R_2n.
Exact mathematical passage
\boxed{\mathscr R(n)=R_{2n}.}
R_{2r}Use in source · 4 passages
Search aliases: R_2r
Use in source · supplement_03.tex, line 54
The source has valuation exactly zero. At even degree 2r, define the literal rational source R_2r=frac3^2r+2-14(2r+1)(2r+2). Both numerator and denominator are nonzero. The pinned Mathlib lifting-the-exponent theorem gives v_2(3^2r+2-1)=2+v_2(2r+2). Because 2r+1 is odd, the denominator has the same valuation. Consequently, v_2(R_2r)=0, |R_2r|_2=1. The code proves actual mcodeIsUnit status for the constructed Z_2 element. This strengthens the manuscript's integrality conclusion without changing i…
Exact mathematical passage
R_{2r}=\frac{3^{2r+2}-1}{4(2r+1)(2r+2)}.
R_3Use in source · 1 passage
Search aliases: R_3
Use in source · Paper_IV_Traditional_Final.tex, line 2161
The only nondominant exponent is -1. Hence R_3(t) = exp((-frac32+fracsqrt32i)t). At t=t_m, its phase is i(-1)^m. Insert this value into [source reference].
Exact mathematical passage
R_3(t) = \exp\left(\left(-\frac32+\frac{\sqrt3}{2}i\right)t\right).
R_4Use in source · 3 passages
Search aliases: R_4
Use in source · visual_atlas.tex, line 113
The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not merely reuse the same numerical matrix under a new ordering.
Exact mathematical passage
\widehat T_N=R_NTR_N^{-1},\qquad J_{4,2}=R_4R_2^{-1}.
R_6Use in source · 1 passage
Search aliases: R_6
Use in source · supplement_03.tex, line 163
al checksExample 5.5 and controls against incorrect normalizations The actual reciprocal recurrence gives (β_0,β_1,β_2,β_3,β_4) =(1,-frac13,frac115,-frac2189,frac1675). These values are obtained from the general formal-series definition. They are not substituted as independent input data for the reconstruction theorem. The source formula gives (R_0,R_2,R_4,R_6,R_8) =(1,frac53,frac9115,frac2057,frac738145). Using the retained Euler anchor E_8=1385, Lean checks the literal decomposition E_8=frac73…
Exact mathematical passage
(R_0,R_2,R_4,R_6,R_8) =\left(1,\frac53,\frac{91}{15},\frac{205}{7},\frac{7381}{45}\right).
R_8Use in source · 1 passage
Search aliases: R_8
Use in source · supplement_03.tex, line 163
al checksExample 5.5 and controls against incorrect normalizations The actual reciprocal recurrence gives (β_0,β_1,β_2,β_3,β_4) =(1,-frac13,frac115,-frac2189,frac1675). These values are obtained from the general formal-series definition. They are not substituted as independent input data for the reconstruction theorem. The source formula gives (R_0,R_2,R_4,R_6,R_8) =(1,frac53,frac9115,frac2057,frac738145). Using the retained Euler anchor E_8=1385, Lean checks the literal decomposition E_8=frac73…
Exact mathematical passage
(R_0,R_2,R_4,R_6,R_8) =\left(1,\frac53,\frac{91}{15},\frac{205}{7},\frac{7381}{45}\right).
R_aUse in source · 6 passages
Search aliases: R_a
Use in source · monograph_atlas.tex, line 353
rthogonal rank-two packets Q_a=E_aPE_a^* that sum to the full identity. vfigsyndromes Use the logical state ψ=(3ket0_L+4ket1_L)/5 and channel probabilities w=(1,2,3,4)/10. The noisy state is ∑_aw_aE_aψψ^*E_a^*. A syndrome measurement returns label a with probability w_a, independent of the logical amplitudes. headThe recovery is an actual channel The operators R_a=E_a^*Q_a satisfy ∑_aR_a^*R_a=I. Applying the recovery R(X)=∑_aR_aXR_a^* to the noisy state gives exactly ψψ^*. The off-diagonal logic…
Exact mathematical passage
R_a=E_a^*Q_a
r_AUse in source · 7 passages
Search aliases: r_A
Use in source · 72_recovered.tex, line 3003
[Coprime packet factorization] Let A and B be coprime positive integers. Let r_A be an address modulo A and r_B an address modulo B. Let r be the unique address modulo AB satisfying r≡ r_Apmod A, r≡ r_Bpmod B. Then P_AB,r = P_A,r_AP_B,r_B = P_B,r_BP_A,r_A.
Exact mathematical passage
r_A
r_aUse in source · 10 passages
Search aliases: r_a
Use in source · supplement_10.tex, line 120
The actual ratio is also proved to be an expansion in t^N, so its coefficients vanish outside multiples of N. Constructed Z_p lifts and the established coefficient criterion now give, for its literal Hurwitz coefficients r_a(n), Δ_uh^kr_a(n)∈ p^min(k,mn) Z_p. The separate rational precision theorem is obtained through the coefficient-lift equality. Divisibility in the field Q_p is never substituted for this assertion.
Exact mathematical passage
r_a(n)
R_a^*Use in source · 3 passages
Search aliases: R_a^*
Use in source · monograph_atlas.tex, line 353
ets Q_a=E_aPE_a^* that sum to the full identity. vfigsyndromes Use the logical state ψ=(3ket0_L+4ket1_L)/5 and channel probabilities w=(1,2,3,4)/10. The noisy state is ∑_aw_aE_aψψ^*E_a^*. A syndrome measurement returns label a with probability w_a, independent of the logical amplitudes. headThe recovery is an actual channel The operators R_a=E_a^*Q_a satisfy ∑_aR_a^*R_a=I. Applying the recovery R(X)=∑_aR_aXR_a^* to the noisy state gives exactly ψψ^*. The off-diagonal logical coherence 12/25 is r…
Exact mathematical passage
\sum_aR_a^*R_a=I
R_{ap}Use in source · 4 passages
Search aliases: R_ap
Use in source · 20_arithmetic.tex, line 235
At the singular addresses c=ap, 1≤ a<m, use the weighted ring W_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping the falling factorial (x)_d=∏_i=0^d-1(x-i) into consecutive blocks of length p then gives an integral residue-coordinate factor p^v_p(lfloor d/prfloor!)…
Exact mathematical passage
R_{ap}(\log(1+T))\in\mathcal W_p
r_BUse in source · 7 passages
Search aliases: r_B
Use in source · 72_recovered.tex, line 3003
[Coprime packet factorization] Let A and B be coprime positive integers. Let r_A be an address modulo A and r_B an address modulo B. Let r be the unique address modulo AB satisfying r≡ r_Apmod A, r≡ r_Bpmod B. Then P_AB,r = P_A,r_AP_B,r_B = P_B,r_BP_A,r_A.
Exact mathematical passage
r_B
r_bUse in source · 1 passage
Search aliases: r_b
Use in source · Paper_VI_Traditional_Final.tex, line 866
For P=C_π p and Z=∑_iλ_ip_i=∑_aP_ar_a(p), (C_π B_λ(p))_a = frac∑_i∈ I_aλ_ip_iZ = fracP_ar_a(p)∑_bP_br_b(p) = ( B_r(p)(P))_a.
Exact mathematical passage
\bigl(C_\pi\cB_\lambda(p)\bigr)_a = \frac{\sum_{i\in I_a}\lambda_ip_i}{Z} = \frac{P_ar_a(p)}{\sum_bP_br_b(p)} = \bigl(\cB_{r(p)}(P)\bigr)_a.
R_{\bar x}Use in source · 7 passages
Search aliases: R_bar x bar
Use in source · Paper_X_Block27_Amended.tex, line 259
[Represented complete deformation problems] When a deformation functor is represented by a complete local ring R_bar x, its O-valued points are continuous local maps R_bar x→ O. Completeness identifies such maps with compatible maps to every quotient O/ϖ^m.
Exact mathematical passage
R_{\bar x}
R_cUse in source · 31 passages
Search aliases: R_c
Use in source · 05_defects.tex, line 119
Inverse witnesses from one side and norm control. For continuous linear maps, the same identity gives the certified norm bound C R_f-R_c C ≤ R_c Δ R_f , from a left inverse witness for the coarse scalar shift and a right inverse witness for the fine scalar shift. These witnesses suffice for the displayed identity. On their own, they need not define inverses on both sides and hence need not define resolvents. If Δ=0, Lean also proves, for every finite polynomial p, the exact compatibility C p(T_f…
Exact mathematical passage
\boxed{ \|C R_f-R_c C\| \le \|R_c\|\,\|\Delta\|\,\|R_f\|, }
R_c'Use in source · 2 passages
Search aliases: R_c'
Use in source · 20_arithmetic.tex, line 293
At a nonsingular address 2≤ c<N with pnmid c, differentiation of the section identities gives tR_c'=R_c(1+R_c-1-R_c). Writing q=R_c-1-(c-1) and y=R_c-c, this becomes (tD+c)y=cq+yq-y^2. On degrees divisible by p, the inverse of tD+c divides the degree-d coefficient by the unit d+c. Starting with y_0=0, form y_s+1=(tD+c)^-1(cq+y_sq-y_s^2). Flatness and support are preserved. Indeed, put E_c=tD+c and C_p=pD^p^2-D^p. The commutator is [B_p,E_c]=C_p, and iteration gives B_p^rE_cy=E_cB_p^ry+rC_pB_p^r-…
Exact mathematical passage
tR_c'=R_c(1+R_{c-1}-R_c).
R_{c-1}Use in source · 5 passages
Search aliases: R_c-1
Use in source · 20_arithmetic.tex, line 293
At a nonsingular address 2≤ c<N with pnmid c, differentiation of the section identities gives tR_c'=R_c(1+R_c-1-R_c). Writing q=R_c-1-(c-1) and y=R_c-c, this becomes (tD+c)y=cq+yq-y^2. On degrees divisible by p, the inverse of tD+c divides the degree-d coefficient by the unit d+c. Starting with y_0=0, form y_s+1=(tD+c)^-1(cq+y_sq-y_s^2). Flatness and support are preserved. Indeed, put E_c=tD+c and C_p=pD^p^2-D^p. The commutator is [B_p,E_c]=C_p, and iteration gives B_p^rE_cy=E_cB_p^ry+rC_pB_p^r-…
Exact mathematical passage
tR_c'=R_c(1+R_{c-1}-R_c).
R_{\mathrm{can}}Use in source · 2 passages
Search aliases: R_can mathrm
Use in source · Paper_V_Traditional_Final.tex, line 2165
[Coefficient noise under canonical synthesis] Under the hypotheses of [source reference], suppose the observed coefficient vector is c= A_Ex+ν. Canonical synthesis gives widehat x=R_canc and widehat x-x _2 ≤ frac1sqrt A ν _2.
Exact mathematical passage
\widehat x=R_{\mathrm{can}}c
R_DUse in source · 4 passages
Search aliases: R_D
Use in source · Paper_VI_Traditional_Final.tex, line 2156
[Resolvent expansion over packet paths] With A=D+N as in the directed path decomposition, let R_D(z)=(zI-D)^-1. If normR_D(z)N<1, then (zI-A)^-1 = ∑_m=0^∞ (R_D(z)N)^mR_D(z). After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.
Exact mathematical passage
R_D(z)=(zI-D)^{-1}.
r_DUse in source · 4 passages
Search aliases: r_D
Use in source · Paper_X_Block27_Amended.tex, line 1593
The hyperref[book:x-construction-13]finite-exponential reconstruction also established an extracted inverse on the explicit class N^3=0: log_2 U=(U-I)-tfrac12(U-I)^2, a^-1log_2(exp(aN))=N (a≠0). Both exponential/logarithmic inverse identities are checked on the cubic nilpotent and unipotent loci. If a supplied common sample u has r_D(u)=r_E(u)=I and t(u)≠0, an intertwiner of the reconstructed cubic actions recovers both original group equivariance and monodromy equivariance. The general nilpoten…
Exact mathematical passage
r_D(u)=r_E(u)=I
r_EUse in source · 3 passages
Search aliases: r_E
Use in source · Paper_X_Block27_Amended.tex, line 1593
The hyperref[book:x-construction-13]finite-exponential reconstruction also established an extracted inverse on the explicit class N^3=0: log_2 U=(U-I)-tfrac12(U-I)^2, a^-1log_2(exp(aN))=N (a≠0). Both exponential/logarithmic inverse identities are checked on the cubic nilpotent and unipotent loci. If a supplied common sample u has r_D(u)=r_E(u)=I and t(u)≠0, an intertwiner of the reconstructed cubic actions recovers both original group equivariance and monodromy equivariance. The general nilpoten…
Exact mathematical passage
r_D(u)=r_E(u)=I
r_eUse in source · 5 passages
Search aliases: r_e
Use in source · 02_supplement_02.tex, line 132
Global comparison. For any other feasible ε, set r_e=ε_e/ε_e^*>0. The budget identities give ∑_ec_er_e=C. Using the standard analytic inequality log r≤ r-1 yields ∑_ec_elog r_e≤∑_ec_e(r_e-1)=0. Lean separately proves the exact objective-difference identity J(ε)-J(ε^*)=-∑_ec_elog r_e. Therefore J(ε)≥ J(ε^*).
Exact mathematical passage
r_e=\varepsilon_e/\varepsilon_e^*>0
R_{\mathrm{extra}}Use in source · 2 passages
Search aliases: R_extra mathrm
Use in source · 03_supplement_04.tex, line 109
The recovery is not left undefined outside the syndrome subspaces. Its explicit Kraus operators are R_a=V_a^* together with R_extra=Q, giving R(ρ)=∑_aV_a^*ρ V_a+Qρ Q. They satisfy exact completeness: ∑_aR_a^*R_a+R_extra^*R_extra =∑_aV_aV_a^*+Q=S+Q=I. For finite complex matrices, trace preservation and complete positivity at every finite auxiliary dimension are proved for this actual map. The amplified map is identified with the Kraus family I_J⊗ R_a, including the complement term. Positivity is …
Exact mathematical passage
R_{\mathrm{extra}}=Q
R_{\mathrm{extra}}^*Use in source · 1 passage
Search aliases: R_extra^* mathrm
Use in source · 03_supplement_04.tex, line 114
The recovery is not left undefined outside the syndrome subspaces. Its explicit Kraus operators are R_a=V_a^* together with R_extra=Q, giving R(ρ)=∑_aV_a^*ρ V_a+Qρ Q. They satisfy exact completeness: ∑_aR_a^*R_a+R_extra^*R_extra =∑_aV_aV_a^*+Q=S+Q=I. For finite complex matrices, trace preservation and complete positivity at every finite auxiliary dimension are proved for this actual map. The amplified map is identified with the Kraus family I_J⊗ R_a, including the complement term. Positivity is …
Exact mathematical passage
\sum_aR_a^*R_a+R_{\mathrm{extra}}^*R_{\mathrm{extra}} =\sum_aV_aV_a^*+Q=S+Q=I.
R_fUse in source · 14 passages
Search aliases: R_f
Use in source · 05_defects.tex, line 119
Inverse witnesses from one side and norm control. For continuous linear maps, the same identity gives the certified norm bound C R_f-R_c C ≤ R_c Δ R_f , from a left inverse witness for the coarse scalar shift and a right inverse witness for the fine scalar shift. These witnesses suffice for the displayed identity. On their own, they need not define inverses on both sides and hence need not define resolvents. If Δ=0, Lean also proves, for every finite polynomial p, the exact compatibility C p(T_f…
Exact mathematical passage
\boxed{ \|C R_f-R_c C\| \le \|R_c\|\,\|\Delta\|\,\|R_f\|, }
r_fUse in source · 3 passages
Search aliases: r_f
Use in source · 03_nonlinear_coordinates.tex, line 1098
[Coarse vanishing and fine vanishing] Let [r_f]∈coker D_f. If its image in coker D_c is zero, then [r_f] comes from coker D_0. Coarse vanishing therefore means that the fine obstruction is supported in information removed by coarsening. It does not imply [r_f]=0.
Exact mathematical passage
[r_f]\in\coker D_f
R_{\frakp}Use in source · 3 passages
Search aliases: R_frakp R_ p frakp mathfrak
Use in source · Paper_XII_Traditional_Final.tex, line 598
Let p∈Spec R. If Ann_R(M)not⊆ p, choose a∈Ann_R(M)∖ p. The element a becomes a unit in R_ p, so M_ p=0.
Exact mathematical passage
R_{\frakp}
R_{\frakp}^nUse in source · 1 passage
Search aliases: R_frakp^n R_ p^n frakp mathfrak
Use in source · Paper_XII_Traditional_Final.tex, line 624
from the usual convention Fitt_0(0)=R. Assume M≠0, and choose a finite presentation R^mxrightarrowAR^n⟶ M⟶0 with n≥1. The zeroth Fitting ideal is generated by the n by n minors of A. After localization at a prime p, the module M_ p vanishes exactly when A_ p is surjective. A map onto the free module R_ p^n is surjective exactly when one maximal minor is a unit. Thus M_ p=0 exactly when Fitt_0^R(M)not⊆ p.
Exact mathematical passage
R_{\frakp}^n
\bm R_{\mathrm{H}}Use in source · 2 passages
Search aliases: R_H bm mathrm
Use in source · Paper_V_Traditional_Final.tex, line 534
1 · -1 . It is orthogonal, so R_H= E_H^ T and R_H E_H=I_2. The channels are a[n]=fracx[2n]+x[2n+1]sqrt2, d[n]=fracx[2n]-x[2n+1]sqrt2.
Exact mathematical passage
\bm R_{\mathrm{H}}=\bm E_{\mathrm{H}}^{\mathsf T}
r_iUse in source · 3 passages
Search aliases: r_i
Use in source · Paper_XI_Traditional_Final.tex, line 654
[Tame tensor local factor] The local factor of the tame tensor product is the product of (1-a_i u^r_i)^-1 over precisely those refined packets whose inertia exponent vanishes.
Exact mathematical passage
(1-a_i u^{r_i})^{-1}
r_i^{(a)}Use in source · 3 passages
Search aliases: r_i^(a)
Use in source · Paper_VI_Traditional_Final.tex, line 5402
FloatBarrier Prior-restricted sufficiency and filtering. Let L be a strictly positive finite likelihood and let B_L(p)=Lodot p/ipLp for a probability vector p. Fix probability vectors r^(a) supported on B_a, and restrict the priors to p_i=q_a r_i^(a) for i∈ B_a, with q a coarse probability vector. [Sufficiency relative to a prior disintegration] Define overline L_a = ∑_i∈ B_aL_ir_i^(a). For every prior in the declared family, C B_L(p) = B_overline L(q).
Exact mathematical passage
p_i=q_a r_i^{(a)}
r_jUse in source · 14 passages
Search aliases: r_j
Use in source · 72_recovered.tex, line 3030
[Several coprime moduli] Let N_1,…,N_s be pairwise coprime and put N=∏_j=1^sN_j. For compatible addresses r_j, P_N,r = ∏_j=1^sP_N_j,r_j, where r is the unique combined address modulo N.
Exact mathematical passage
r_j
R_{j+1}Use in source · 1 passage
Search aliases: R_j+1
Use in source · supplement_10.tex, line 178
Let K_j(t) be the normalized section kernel with constant coefficient one. The constructed unnormalized Euler series is F_N,j(t)=j! K_j(t)^-1. The source/inverse identities and the factorial successor identity prove F_N,j+1=F_N,jR_j+1, F_N,j=F_N,0∏_c=1^jR_c. The product formula is not an assembly hypothesis. Proved flatness of all factors and the original product-closure theorem establish flatness of the full family.
Exact mathematical passage
F_{N,j+1}=F_{N,j}R_{j+1},\qquad F_{N,j}=F_{N,0}\prod_{c=1}^{j}R_c.
r_{j-i}Use in source · 1 passage
Search aliases: r_j-i
Use in source · 02_supplement_04.tex, line 48
Toeplitz padding and column-major vectorization. textitSource items 4.4 and 4.5 Toeplitz embedding. For n>0, define T_ij= c_i-j, · j≤ i, r_j-i, · i<j. The diagonal is supplied by c_0, and the coordinate r_0 is unused. The constructed generator of length 2n is g=(c_0,…,c_n-1,0,r_n-1,…,r_1), and the padding map is the actual finite-index sum eliminator that appends n zeros.
Exact mathematical passage
T_{ij}= \begin{cases} c_{i-j},&j\le i,\\ r_{j-i},&i<j. \end{cases}
R_KUse in source · 2 passages
Search aliases: R_K
Use in source · residue_continuation_body.tex, line 138
Ordinary class characters: an unconditional power saving. The preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl( O)→ C, define the actual finite ideal sum S_w(X)=∑_substackI≠0
Exact mathematical passage
A_C(X)=c_KX+O_{K,C}(X^{1-1/d}),\qquad c_K=\frac{2^{r_1}(2\pi)^{r_2}R_K}{\#\mu(K)\sqrt{\absn{\Delta_K}}}.
R_kUse in source · 3 passages
Search aliases: R_k
Use in source · 04_transport.tex, line 157
Let (Φ,φ,σ):(X, A, P)→(Y, B, Q) and (Ψ,ψ,τ):(Y, B, Q)→(Z, C, R) be morphisms. Intertwining is preserved because ΨΦ(Ax) = Ψ(φ(A)Φ x) = (ψ∘φ)(A)ΨΦ x. For k in the target packet set, R_kΨΦ · =Ψ Q_τ^-1(k)Φ · =ΨΦ ∑_τ(σ(α))=kP_α. Thus the composite packet map is τ∘σ. If the morphisms are seeded, then Φ(S)⊆Hull_ B(S'), Ψ(S')⊆Hull_ C(S''). By [source reference], ΨΦ(S) ⊆ Ψ(Hull_ B(S')) ⊆ Hull_ C(Ψ(S')) ⊆ Hull_ C(S''). Identity maps satisfy the definitions with identity algebra and packet maps.
Exact mathematical passage
R_k\Psi\Phi &=\Psi Q_{\tau^{-1}(k)}\Phi\\ &=\Psi\Phi \sum_{\tau(\sigma(\alpha))=k}P_\alpha.
r_kUse in source · 2 passages
Search aliases: r_k
Use in source · Paper_V_Traditional_Final.tex, line 512
Original-sample delay. For an arbitrary original-sample delay D∈ Z, the output phase k has y_k[n]=x_r_k[n+q_k], r_k=(k-D)bmod N, q_k=(k-D)mathbindivN. Its channel matrix has the sole nonzero entry z^q_k in row k, column r_k. In particular, a one-sample delay at N=2 has matrix 0 · z^-1
Exact mathematical passage
y_k[n]=x_{r_k}[n+q_k],\qquad r_k=(k-D)\bmod N,\qquad q_k=(k-D)\mathbin{\mathrm{div}}N.
r_{k-j}Use in source · 2 passages
Search aliases: r_k-j
Use in source · pages.tex, line 94
2,-1)^T. The diagonal pattern is Toeplitz, not circulant. To use a circular convolution, embed it into dimension six with first column g=(2,3,4,0,5,-1)^T and pad the input with three zeros. vfigtoeplitz headRead the wrapped indices For output index j<3, an input index k≤ j uses the first-column coefficient c_j-k. An input index k>j wraps to the stored first-row coefficient r_k-j. The padded input positions contribute nothing. Keeping the first three outputs therefore gives Tx=(-5,8,8)^T. The rem…
Exact mathematical passage
r_{k-j}
R_k^*Use in source · 3 passages
Search aliases: R_k^*
Use in source · 01_manuscript.tex, line 831
Form the small core G=R_LR_R^* and let G=WΣ Z^* be an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer k≥0, retain min{k,d} columns, where d is the number of supplied thin singular directions. Tail singular values beyond d are zero. Define L_k=Q_LW_kΣ_k^1/2, R_k=Q_RZ_kΣ_k^1/2. Then L_kR_k^* is a best rank at most k approximation to LR^*. Its spectral and Frobenius errors are the singular value tails in [source reference]. The core dimensio…
Exact mathematical passage
L_kR_k^*
R_LUse in source · 14 passages
Search aliases: R_L
Use in source · 03_weighted.tex, line 187
[Exact finite precision truncation] Under the hypotheses of [source reference], define R_L = ∑_W(γ)<L(-1)^|γ|A_γ D^-1. Then T^-1-R_L∈ F^L B, TR_L-I∈ F^L B, R_LT-I∈ F^L B. Thus R_L is the exact inverse in B/F^L B.
Exact mathematical passage
R_L = \sum_{W(\gamma)<L}(-1)^{|\gamma|}A_\gamma D^{-1}.
r_LUse in source · 5 passages
Search aliases: r_L
Use in source · worked_atlas.tex, line 68
AtlasHeadA compatible root is built one digit at a timeatlas:smooth Take F(x)=x^2-7 over the 3-adic integers and start with x_1=1 modulo three. If F(x_L)=3^Lr_L, the next representative has the form x_L+1=x_L+3^Lt_L. Its correcting digit satisfies r_L+2x_Lt_L≡0pmod3. AtlasImagesmooth_tower For example, F(1)/3=-2 and 2x_1=2, so the next digit is one and x_2=4. Then F(4)/9=1, again giving digit one and x_3=13. At the next stage the residual is divisible by three, so digit zero is correct. A zero d…
Exact mathematical passage
F(x_L)=3^Lr_L
R_mUse in source · 3 passages
Search aliases: R_m
Use in source · Paper_VI_Traditional_Final.tex, line 2523
Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.
Exact mathematical passage
C_{m,n}=C_mC_{m+1}\cdots C_{n-1}, \qquad R_{n,m}=R_{n-1}\cdots R_m.
r_mUse in source · 1 passage
Search aliases: r_m
Use in source · Paper_VI_Traditional_Final.tex, line 5242
Every nonzero term in [source reference] contains at least m_0 time integrations. The simplex 0<r_1<⋯<r_m<t has volume t^m/m!, so the first possible order is t^m_0. If there is no path, every packet term vanishes.
Exact mathematical passage
0<r_1<\cdots<r_m<t
R_{m+1}Use in source · 4 passages
Search aliases: R_m+1
Use in source · 03_nonlinear_coordinates.tex, line 287
[Packet Taylor formula] Assume F is m+1 times continuously Fr echet differentiable on an open set containing the segment x+th for 0≤ t≤1. Then Q_γ F(x+h) = · Q_γ F(x) + ∑_r=1^mfrac1r! ∑_α_1,…,α_r J^r_γmidα_1,…,α_rF(x) [P_α_1h,…,P_α_rh] · +Q_γ R_m+1(x,h), where the Bochner integral remainder is R_m+1(x,h) = frac1m! ∫_0^1(1-t)^m D^m+1F(x+th)[h,…,h] dt.
Exact mathematical passage
\begin{aligned} Q_\gamma F(x+h) ={}&Q_\gamma F(x) + \sum_{r=1}^{m}\frac1{r!} \sum_{\alpha_1,\ldots,\alpha_r} J^{r}_{\gamma\mid\alpha_1,\ldots,\alpha_r}F(x) [P_{\alpha_1}h,\ldots,P_{\alpha_r}h]\\ &+Q_\gamma R_{m+1}(x,h), \end{aligned}
r_{m+1,m}Use in source · 2 passages
Search aliases: r_m+1,m
Use in source · Paper_X_Block27_Amended.tex, line 250
Fix bar x∈ M(k). For m≥1, set X_m(bar x)= M_bar x( O/ϖ^m). Reduction gives transition maps r_m+1,m:X_m+1(bar x)⟶ X_m(bar x). A compatible branch is a sequence (x_m)_m≥1 satisfying r_m+1,m(x_m+1)=x_m.
Exact mathematical passage
r_{m+1,m}:X_{m+1}(\bar x)\longrightarrow X_m(\bar x).
R_{m,m}Use in source · 1 passage
Search aliases: R_m,m
Use in source · Paper_VI_Traditional_Final.tex, line 2529
Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.
Exact mathematical passage
C_{m,m}=I_{X_m}, \qquad R_{m,m}=I_{X_m}.
R_{m-1}Use in source · 1 passage
Search aliases: R_m-1
Use in source · 02_supplement_03.tex, line 20
Construction. Suppose P_m-1=D^(m-1)+L_m-1R_m-1^*. The new factors are L_m · =[D_mL_m-1+U_m(V_m^*L_m-1) U_m], R_m · =[R_m-1 (D^(m-1))^*V_m]. Substitution into the already checked two-factor product identity proves exactness. This is not a rank-only existence argument. The recursively defined factors are themselves audited.
Exact mathematical passage
L_m&=\bigl[D_mL_{m-1}+U_m(V_m^*L_{m-1})\quad U_m\bigr],\\ R_m&=\bigl[R_{m-1}\quad (D^{(m-1)})^*V_m\bigr].
R_{m-1}^*Use in source · 1 passage
Search aliases: R_m-1^*
Use in source · 02_supplement_03.tex, line 19
Construction. Suppose P_m-1=D^(m-1)+L_m-1R_m-1^*. The new factors are L_m · =[D_mL_m-1+U_m(V_m^*L_m-1) U_m],
Exact mathematical passage
P_{m-1}=D^{(m-1)}+L_{m-1}R_{m-1}^*
R_m^*Use in source · 1 passage
Search aliases: R_m^*
Use in source · 02_supplement_03.tex, line 14
Low-rank factors through a product chain. textitSource item 5.2 Let A_j=D_j+U_jV_j^*, U_j,V_j∈ C^n× r_j, with all D_j diagonal in one packet layout. Set P_m=A_m⋯ A_1, D^(m)=D_m⋯ D_1. The formalization constructs thin matrices L_m,R_m such that P_m=D^(m)+L_mR_m^*. Their column index type is a recursively constructed disjoint sum. Its cardinality is proved to equal ∑_j=1^m r_j.
Exact mathematical passage
\boxed{P_m=D^{(m)}+L_mR_m^*.}
R_{\min}Use in source · 2 passages
Search aliases: R_min min
Use in source · 01_manuscript.tex, line 1759
[Amortization threshold] Let s≥0 be a one time setup cost. Let t_b,t_c≥0 be the baseline and candidate times per use and let R∈ N be a reuse count. If t_b>t_c, the candidate is faster after R uses exactly when R> fracst_b-t_c. The smallest integer reuse count is R_min = lfloor fracst_b-t_c rfloor+1. If t_b≤ t_c, no positive reuse count amortizes a positive setup cost.
Exact mathematical passage
\boxed{ R_{\min} = \left\lfloor \frac{s}{t_b-t_c} \right\rfloor+1. }
R_nUse in source · 15 passages
Search aliases: R_n
Use in source · atlas-pages.tex, line 431
t poles are ρ=±π/2. They contribute equally to even-degree coefficients. Including both poles gives P_2,n(1)∼2(2n)!(frac2π)^2n+1. The factor two is orbit multiplicity, not an adjustable fitted constant. vfigpole_asymptotic headA finite quantitative check For n=1,…,8, the exact integer coefficients and a rational enclosure for π bound the ratio R_n strictly between zero and one. In this tested range, 1-R_n<3^-(2n+1). The plot displays the logarithm of the enclosed error, not an independently asse…
Exact mathematical passage
R_n
R_N'Use in source · 3 passages
Search aliases: R_N'
Use in source · Paper_IV_Traditional_Final.tex, line 2036
Fix a large m, put a=2is_N, and consider the closed disk B_m={z:|z-t_m|≤ 1/(2s_N)}. The remainder bounds hold uniformly on these fixed-radius disks. For sufficiently large m, |R_N|≤ 1/2 and |R_N'|≤ s_N/2 there. Define Φ_m(z)=t_m+fraclog(1+R_N(z))2is_N, using the principal logarithm. The estimate |log(1+u)|≤(3/2)|u| shows that Φ_m maps B_m into itself. The derivative bound |Φ_m'|≤1/2 makes it a contraction. Banach's fixed-point theorem gives a unique fixed point. On B_m, |a(z-t_m)|≤1<π, so the pr…
Exact mathematical passage
|R_N'|\leq s_N/2
R_{N,k}Use in source · 2 passages
Search aliases: R_N,k
Use in source · Paper_V_Traditional_Final.tex, line 1138
Fix ω_N=e^2π i/N and rotate the reflection coefficient by R_N,k(Γ)=ω_N^kΓ.
Exact mathematical passage
R_{N,k}(\Gam)=\omega_N^k\Gam.
R_{n-1}Use in source · 1 passage
Search aliases: R_n-1
Use in source · Paper_VI_Traditional_Final.tex, line 2523
Assume now that each C_ℓ:X_ℓ+1→ X_ℓ has a bounded right inverse R_ℓ:X_ℓ→ X_ℓ+1, C_ℓ R_ℓ=I_X_ℓ. For m<n, set C_m,n=C_mC_m+1⋯ C_n-1, R_n,m=R_n-1⋯ R_m. The empty products are C_m,m=I_X_m, R_m,m=I_X_m.
Exact mathematical passage
C_{m,n}=C_mC_{m+1}\cdots C_{n-1}, \qquad R_{n,m}=R_{n-1}\cdots R_m.
r_{n-1}Use in source · 3 passages
Search aliases: r_n-1
Use in source · 01_manuscript.tex, line 569
[Toeplitz circulant embedding] Let n≥1 be an integer and T∈ C^n× n be Toeplitz, with first column (c_0,c_1,…,c_n-1)^T and first row (c_0,r_1,…,r_n-1). Set g = (c_0,c_1,…,c_n-1,0,r_n-1,…,r_1)^T ∈ C^2n and append n zeros to x∈ C^n. The first n entries of the circulant product C(g)(x,0)^T equal Tx.
Exact mathematical passage
(c_0,r_1,\ldots,r_{n-1}).
R_{n-2}Use in source · 1 passage
Search aliases: R_n-2
Use in source · 72_recovered.tex, line 1427
The first three sharp truncations are E_n · ≡ R_n · · pmod 8, E_n · ≡ R_n-frac83binom n2R_n-2 · · pmod 128, E_n · ≡ R_n-frac83binom n2R_n-2 +frac1285binom n4R_n-4 · · pmod 2048.
Exact mathematical passage
\begin{aligned} E_n&\equiv R_n&&\pmod 8,\\ E_n&\equiv R_n-\frac83\binom n2R_{n-2}&&\pmod {128},\\ E_n&\equiv R_n-\frac83\binom n2R_{n-2} +\frac{128}{5}\binom n4R_{n-4}&&\pmod {2048}. \end{aligned}
R_{n-2r}Use in source · 4 passages
Search aliases: R_n-2r
Use in source · 72_recovered.tex, line 1113
[Dyadic Euler coordinate pair] For every even n≥0, E_n = ∑_r=0^n/2 β_r2^2r fall n2r R_n-2r. The inverse transform is R_n = ∑_r=0^n/2 frac2^4r+1(2r+2)! fall n2r E_n-2r.
Exact mathematical passage
\boxed{ E_n = \sum_{r=0}^{n/2} \beta_r2^{2r} \fall n{2r} R_{n-2r}. }
R_{n-4}Use in source · 1 passage
Search aliases: R_n-4
Use in source · 72_recovered.tex, line 1427
The first three sharp truncations are E_n · ≡ R_n · · pmod 8, E_n · ≡ R_n-frac83binom n2R_n-2 · · pmod 128, E_n · ≡ R_n-frac83binom n2R_n-2 +frac1285binom n4R_n-4 · · pmod 2048.
Exact mathematical passage
\begin{aligned} E_n&\equiv R_n&&\pmod 8,\\ E_n&\equiv R_n-\frac83\binom n2R_{n-2}&&\pmod {128},\\ E_n&\equiv R_n-\frac83\binom n2R_{n-2} +\frac{128}{5}\binom n4R_{n-4}&&\pmod {2048}. \end{aligned}
R_N^{(q)}Use in source · 2 passages
Search aliases: R_N^(q)
Use in source · Paper_IV_Traditional_Final.tex, line 1888
η≠ e^ia_N,e^-ia_N exp((η-e^-ia_N)z). Then N e^-e^-ia_Nz D_N(z) = 1+e^2is_Nz+R_N(z), and, for q=0,1,2, |R_N^(q)(z)| ≤ C e^-γ_Nx.
Exact mathematical passage
|R_N^{(q)}(z)| \leq C e^{-\gamma_Nx}.
R_N^{-1}Use in source · 1 passage
Search aliases: R_N^-1
Use in source · visual_atlas.tex, line 113
The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not merely reuse the same numerical matrix under a new ordering.
Exact mathematical passage
\widehat T_N=R_NTR_N^{-1},\qquad J_{4,2}=R_4R_2^{-1}.
R_PUse in source · 3 passages
Search aliases: R_P
Use in source · 03_supplement_04.tex, line 70
Retaining the bracket of vector fields. Define the actual curvature action by R_P(X,Y)s=∇_X∇_Ys-∇_Y∇_Xs-∇_[X,Y]s. The source uses the Lie bracket defined through Fr echet derivatives. Symmetry of second derivatives is supplied by the C^2 hypothesis, not postulated separately. Expanding the connection gives R_P(X,Y)s=P(DP[X]Ds[Y]-DP[Y]Ds[X]). Substituting the section identity and removing the zero sandwich terms proves R_P(X,Y)s=P(DP[X]DP[Y]-DP[Y]DP[X])P s. Thus the coefficient is precisely P( d …
Exact mathematical passage
R_P(X,Y)s=\nabla_X\nabla_Ys-\nabla_Y\nabla_Xs-\nabla_{[X,Y]}s.
R_pUse in source · 5 passages
Search aliases: R_p
Use in source · 02_foundations.tex, line 920
Let the minimal polynomial be m_T(t)=∏_pp(t)^e_p. The acting algebra is 𝕜[T]≅𝕜[t]/(m_T) ≅∏_pR_p, R_p:=𝕜[t]/(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with maximal ideal generated by the class of p and residue division field K_p=𝕜[t]/(p). By [source reference], srank_R_p(X_p) = dim_K_pX_p/p(T)X_p = μ_p(T). …
Exact mathematical passage
\bbk[T]\cong\bbk[t]/(m_T) \cong\prod_pR_p, \qquad R_p:=\bbk[t]/(p^{e_p}),
R_RUse in source · 4 passages
Search aliases: R_R
Use in source · pages.tex, line 182
vpage11Stored width is not intrinsic rankACTUAL RANGE COORDINATESChapter VIII: Theorem 5.6, Section F.1. Choose the isometry Q_L=mat3/5 · 0 4/5 · 0 0 · 1 0 · 0, R_L=mat1 · 0 · 1 0 · 1 · 0, R_R=mat3 · 0 · 2 0 · 2 · 0 1 · 0 · -1. Set L=Q_LR_L and R=R_R. The two equal columns of L do not create a third range direction. In contrast, R has rank three. vfigcompact_core The actual core is G=R_LR_R^T=mat5 · 0 · 0
Exact mathematical passage
Q_L=\mat{3/5&0\\4/5&0\\0&1\\0&0},\quad R_L=\mat{1&0&1\\0&1&0},\quad R_R=\mat{3&0&2\\0&2&0\\1&0&-1}.
R_rUse in source · 6 passages
Search aliases: R_r
Use in source · root_scope.tex, line 365
The current focused obstruction is the admitted ordinary-index family r=(17659+2^18T)2^640-161, T∈ Z_≥0, j_r=161. On its surviving chart define the actual, undepleted function K_r(Z)=2^-178G_r(10202+2^15Z). The complete coefficient theorem gives K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z), a_r,b_r∈ Z_2, u_r∈ Z_2^×, R_r∈ Z_2⟨ Z⟩. Its half derivative is a dyadic isometry. It therefore has a unique critical point c_r∈ Z_2. Set Φ_r^act=K_r(c_r). The earlier independent-undepletion notation for this same sc…
Exact mathematical passage
K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z), \quad a_r,b_r\in\mathbb Z_2,\quad u_r\in\mathbb Z_2^\times,\quad R_r\in\mathbb Z_2\langle Z\rangle. \label{eq:root:quadraticclass}
\mathcal R_rUse in source · 1 passage
Search aliases: R_r mathcal
Use in source · 02_connections_obstructions.tex, line 622
in V and m≥1, set i_0=i and i_m=j. Then (B_m⋯ B_1)_ji = ∑_(i_1,…,i_m-1)∈ V^m-1 B_m[i_m,i_m-1]⋯ B_1[i_1,i_0]. When m=1, the sum is over the unique empty tuple and its summand is B_1[j,i]. For m=0, the product is the identity and the entry is the Kronecker delta δ_ji. If every B_r[q,p] vanishes outside a declared relation R_r(p,q), then the endpoint entry is zero whenever there is no packet path compatible with the successive relations. statusleanthreeKomorebi.PaperIII.EdgeOperators.heterogeneousP…
Exact mathematical passage
\mathcal R_r(p,q)
r_RUse in source · 3 passages
Search aliases: r_R
Use in source · 02_supplement_06.tex, line 10
Compact factors from actual column spaces. textitModules CompactRangeFactors and SmallCoreTransfer Let L∈ C^m× t, R∈ C^n× t, A=LR^*. Use the ordinary Euclidean inner products on the coordinate spaces. Define S_L=range(L), r_L=dim S_L, S_R=range(R), r_R=dim S_R. The implementation constructs an orthonormal basis of each actual range subspace. Let Q_L and Q_R be the matrices of the corresponding isometric inclusions. Express the range-restricted input maps in these bases to obtain R_L,R_R.
Exact mathematical passage
S_L=\operatorname{range}(L),\quad r_L=\dim S_L, \qquad S_R=\operatorname{range}(R),\quad r_R=\dim S_R.
R_R^*Use in source · 3 passages
Search aliases: R_R^*
Use in source · 01_manuscript.tex, line 817
[Small core recompression] Let L∈ C^m× r, R∈ C^n× r. Take compact QR factorizations L=Q_LR_L, R=Q_RR_R, where the columns of Q_L and Q_R are orthonormal. Form the small core G=R_LR_R^* and let G=WΣ Z^* be an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer k≥0, retain min{k,d} columns, where d is the number of supplied thin singular directions. Tail singular values beyond d are zero. Define L_k=Q_LW_kΣ_k^1/2, R_k=Q_RZ_kΣ_k^1/2. Then L_kR_k…
Exact mathematical passage
G=R_LR_R^*
R_r^*Use in source · 3 passages
Search aliases: R_r^*
Use in source · 03_supplement_05.tex, line 10
recise finite channel statement Let H= C^n be a finite coordinate space and let P=P^*=P^2. A matrix is code supported when Pρ P=ρ. For a finite family of square matrices E_a, put N_E(ρ)=∑_a E_aρ E_a^*. The input channel normalization is the actual matrix identity ∑_a E_a^*E_a=I. A finite Kraus recovery has the form R(σ)=∑_r R_rσ R_r^*, with ∑_rR_r^*R_r=I. This representation, together with the checked amplification results, provides complete positivity and trace preservation on the entire ambien…
Exact mathematical passage
\mathcal R(\sigma)=\sum_r R_r\sigma R_r^*
R_R^TUse in source · 1 passage
Search aliases: R_R^T
Use in source · pages.tex, line 190
1 · 0 · -1. Set L=Q_LR_L and R=R_R. The two equal columns of L do not create a third range direction. In contrast, R has rank three. vfigcompact_core The actual core is G=R_LR_R^T=mat5 · 0 · 0 0 · 2 · 0, LR^T=Q_LG. Its dimensions are the two actual input ranks, two and three. This is not inferred by merely discarding a column with a small norm. The range factor Q_L has orthonormal columns and reconstructs L exactly. headThe preserved formal boundary The small-core recompression construction in A…
Exact mathematical passage
G=R_LR_R^T=\mat{5&0&0\\0&2&0},\qquad LR^T=Q_LG.
R_{\rm can}Use in source · 2 passages
Search aliases: R_rm can rm
Use in source · Paper_V_Traditional_Final.tex, line 2147
[Frame bounds from the polyphase symbol] Assume that A I_N ≤ E( e^ iω)^*E( e^ iω) ≤ B I_N for almost every ω, where 0<A≤ B<∞. Then the analysis bank satisfies A x _2^2 ≤ A_E x _2^2 ≤ B x _2^2. The pointwise matrix R_rm can( e^ iω) = (E^*E)^-1E^* is a bounded left inverse and gives the canonical dual synthesis bank.
Exact mathematical passage
R_{\rm can}(\e^{\ii\omega}) = (E^*E)^{-1}E^*
R_sUse in source · 3 passages
Search aliases: R_s
Use in source · Paper_XII_Traditional_Final.tex, line 258
[Completed tensor quotient] Let R_1,…,R_s be complete Noetherian local O algebras with common residue field k. Let I_i⊆ R_i be closed ideals. Put R^loc=R_1mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ OR_s and let J be the closed ideal generated by the images of the I_i. Then there is a canonical isomorphism (R_1/I_1)mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ O(R_s/I_s) ≅ R^loc/J.
Exact mathematical passage
R_1,\ldots,R_s
r_{\mathrm{spec}}Use in source · 2 passages
Search aliases: r_spec mathrm
Use in source · 05_worked_models.tex, line 565
[Exact graph seed rank from a certificate] For a finite square matrix A over a field K, define the noncomputable exact spectral quantity r_spec(A) = max_μ∈Spec_K(A) dim_Kker(A-μ I), with value zero for an empty spectrum. Suppose an exact certificate supplies a finite eigenbasis indexed by labels and multiplicities, the eigenvalue equations, a multiplicity bound attained by one label, projector polynomials that extract the padded eigenbasis generators, coverage of every spectral value by a label,…
Exact mathematical passage
r_{\mathrm{spec}}(A) = \max_{\mu\in\Spec_K(A)} \dim_K\ker(A-\mu I),
r_{\widehat\tau}Use in source · 1 passage
Search aliases: r_widehatτ widehat tau
Use in source · atlas-pages.tex, line 649
y on the interval between its two locations: H_τ-H_widehatτ _2=sqrtδ=tfrac14, H_τ-H_widehatτ _∞=1. vfig[.89]ext_location headFor a ramp, the error height shrinks too Write r_τ(t)=(t-τ)_+. The difference grows linearly from zero to δ on [τ,widehatτ] and stays at δ on [widehatτ,1]. Direct integration gives r_τ-r_widehatτ _2^2 =fracδ^33+(1-widehatτ)δ^2 =frac176144<frac1256=δ^2. Thus the L^2 ramp error is at most δ, and its supremum error is exactly δ. Positive-order truncated powers are Lipschitz i…
Exact mathematical passage
\|r_\tau-r_{\widehat\tau}\|_2^2 =\frac{\delta^3}{3}+(1-\widehat\tau)\delta^2 =\frac{17}{6144}<\frac1{256}=\delta^2.
R_xUse in source · 3 passages
Search aliases: R_x
Use in source · atlas-pages.tex, line 472
1 · -1. Retain the first coordinate and eliminate the second. For z away from the relevant poles, the retained entry of (zI-A)^-1 is R_x(z)=[z+1-frac1z+1]^-1 =fracz+1z(z+2). The term Σ(z)=1/(z+1) is the hidden coordinate's self-energy. vfigresolvent platecaptionThe full inverse gives a rational response with two poles, at 0 and -2. Its partial fractions are R_x(z)=frac12z^-1+frac12(z+2)^-1. The hidden intermediate inverse has its own excluded point z=-1. headAn eliminated variable has not disapp…
Exact mathematical passage
R_x(z)=\left[z+1-\frac1{z+1}\right]^{-1} =\frac{z+1}{z(z+2)}.
r_{x'}Use in source · 4 passages
Search aliases: r_x'
Use in source · 30_lifting.tex, line 133
[Independence of the integer representative] Under the hypotheses of Theorem~[source reference], let x'=x+q^Ls and write F(x')=q^Lr_x'. Then J_x'=J_x, overliner_x'=overliner_x+J_xoverline s. Consequently the two residual classes coincide in the common quotient S_q^e/im J_x.
Exact mathematical passage
F(x')=q^Lr_{x'}
r_\alphaUse in source · 8 passages
Search aliases: r_α alpha
Use in source · 03_nonlinear_coordinates.tex, line 991
of an obstruction] Suppose U=⊕_α∈ IU_α, V=⊕_α∈ IV_α, and D(U_α)⊆ V_α for every α. Let D_α:U_α⟶ V_α be the restriction. Then coker D ≅ ⊕_α∈ Icoker D_α. Under this isomorphism, the obstruction class of r=∑_α r_α is the tuple ([r_α])_α. Hence r is correctable if and only if every packet obstruction vanishes.
Exact mathematical passage
r=\sum_\alpha r_\alpha
R_{\alpha\beta}Use in source · 4 passages
Search aliases: R_αβ alpha beta
Use in source · 02_coordinates.tex, line 292
[Common refinement of commuting packet systems] Let P=(P_α)_α∈ I and Q=(Q_β)_β∈ J be packet systems on X. Assume P_α Q_β=Q_β P_α for all α and β. Remove the zero operators from the family R_αβ=P_α Q_β. The remaining operators form a packet system that refines both P and Q. The same statement holds for any finite collection of pairwise commuting packet systems.
Exact mathematical passage
R_{\alpha\beta}=P_\alpha Q_\beta.
R_{\alpha\beta}^2Use in source · 1 passage
Search aliases: R_αβ^2 alpha beta
Use in source · 02_coordinates.tex, line 300
Commutativity gives R_αβ^2=P_α^2Q_β^2=R_αβ. If (α,β)≠(α',β'), then either P_α P_α'=0 or Q_β Q_β'=0, so the corresponding products annihilate one another. Finally, ∑_α,βR_αβ = (∑_α P_α) (∑_β Q_β) =I_X. For a fixed α, P_α =P_α∑_β Q_β = ∑_β R_αβ, and the same argument works for a fixed β. Since each P_α and Q_β is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite f…
Exact mathematical passage
R_{\alpha\beta}^2=P_\alpha^2Q_\beta^2=R_{\alpha\beta}.
R_\GammaUse in source · 1 passage
Search aliases: R_Γ Gamma
Use in source · 03_packets_and_graphs.tex, line 942
[Reachability support of the generated algebra] For every A∈ A, P_β AP_α=0 whenever β∉Reach(α). Equivalently, A⊆ M_R_Γ, R_Γ:={(β,α):β∈Reach(α)}. [source reference]
Exact mathematical passage
\A\subseteq\mathfrak M_{R_\Gamma}, \qquad R_\Gamma:=\{(\beta,\alpha):\beta\in\Reach(\alpha)\}.
R_\gammaUse in source · 6 passages
Search aliases: R_γ gamma
Use in source · 02_coordinates.tex, line 326
Let R_γ be a packet in R. Because R refines P, there is a unique α with P_α R_γ=R_γ. Likewise there is a unique β with Q_β R_γ=R_γ. Therefore P_α Q_β R_γ=R_γ. Every packet of R lies in one product packet. Multiplying ∑_γ R_γ=I_X by P_α Q_β and using the parent identities gives P_α Q_β =∑_substackγ: P_α R_γ=R_γ
Exact mathematical passage
R_\gamma
R_\deltaUse in source · 2 passages
Search aliases: R_δ delta
Use in source · 03_nonlinear_coordinates.tex, line 373
Insert I_Y=∑_γ Q_γ in each argument of the outer derivative and expand the finite sums by multilinearity. Apply R_δ to the output. The directions already lie in the declared input ranges, and each intermediate projection lands in its literal output range. The resulting summands are exactly the displayed packet jets. List directions within a block in increasing index order and blocks in increasing least-element order. Symmetry of the higher derivative makes the final value independent of the bloc…
Exact mathematical passage
R_\delta
R_\thetaUse in source · 3 passages
Search aliases: R_θ theta
Use in source · Paper_V_Traditional_Final.tex, line 3663
Let H = {z∈ C:Re z>0} and define C(z)=fracz-1z+1. It maps H conformally onto the unit disk D, with inverse C^-1(w)=frac1+w1-w. For θ∈ R, let R_θ(w)= e^ iθw.
Exact mathematical passage
R_\theta(w)=\e^{\ii\theta}w.
r_\muUse in source · 2 passages
Search aliases: r_μ mu
Use in source · Paper_VI_Traditional_Final.tex, line 1665
ator F(A) by .F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent of a basis and of increasing the truncation index: the extra nilpotent powers vanish. It depends only on the finite Taylor jets of F at the spectrum. If F is holomorphic on an open neighborhood U of the spectrum, each eigenvalue admits a radius r_μ>0 with overlineD(μ,r_μ)⊂ U, and Cauchy's formula gives fracF^(k)(μ)k! =frac12π ioint_|z-μ|=r_μ fracF(z)(z-μ)^k+1 dz. Consequently the same canonical operator i…
Exact mathematical passage
r_\mu>0
r_\tauUse in source · 2 passages
Search aliases: r_τ tau
Use in source · atlas-pages.tex, line 648
LITY / A ONE-SIXTEENTH DISPLACEMENTChapter V: Corollary E8.2, Theorem E8.3, Theorem E8.7. Set τ=1/4, widehatτ=5/16, and δ=1/16. A unit step differs only on the interval between its two locations: H_τ-H_widehatτ _2=sqrtδ=tfrac14, H_τ-H_widehatτ _∞=1. vfig[.89]ext_location headFor a ramp, the error height shrinks too Write r_τ(t)=(t-τ)_+. The difference grows linearly from zero to δ on [τ,widehatτ] and stays at δ on [widehatτ,1]. Direct integration gives r_τ-r_widehatτ _2^2 =fracδ^33+(1-widehatτ)δ…
Exact mathematical passage
r_\tau(t)=(t-\tau)_+
\raggedrightUse in source · 1 passage
Search aliases: raggedright
Use in source · visual_atlas.tex, line 19
[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…
Exact mathematical passage
4pt] \textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt] \textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology? \end{tabularx} A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. \takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''} \vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}} Most of the chapter can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix. {\small \begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule \textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule $X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\ $S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\ $\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\ $\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\ $P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\ $X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\ $T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\ $F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\ $U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\ $\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\ $\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule \end{tabularx}} \head{Read products from right to left} In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column. \head{Where to begin} For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}. \vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$} Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define \[ Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0, \qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.
\RanUse in source · 9 passages
Search aliases: Ran operatorname
Use in source · Paper_VI_Traditional_Final.tex, line 1504
[Schur--Feshbach projected resolvent] Assume both zI-A and zI-A_QQ are invertible. Define F_P(z) =zI-A_PP-A_PQ(zI-A_QQ)^-1A_QP. Then .P(zI-A)^-1|_Ran P=F_P(z)^-1.
Exact mathematical passage
\boxed{ \left.P(zI-A)^{-1}\right|_{\Ran P}=F_P(z)^{-1}. }
\operatorname{range}Use in source · 1 passage
Search aliases: range operatorname
Use in source · 02_supplement_06.tex, line 10
Compact factors from actual column spaces. textitModules CompactRangeFactors and SmallCoreTransfer Let L∈ C^m× t, R∈ C^n× t, A=LR^*. Use the ordinary Euclidean inner products on the coordinate spaces. Define S_L=range(L), r_L=dim S_L, S_R=range(R), r_R=dim S_R. The implementation constructs an orthonormal basis of each actual range subspace. Let Q_L and Q_R be the matrices of the corresponding isometric inclusions. Express the range-restricted input maps in these bases to obtain R_L,R_R.
Exact mathematical passage
S_L=\operatorname{range}(L),\quad r_L=\dim S_L, \qquad S_R=\operatorname{range}(R),\quad r_R=\dim S_R.
\operatorname{rank}Use in source · 7 passages
Search aliases: rank operatorname
Use in source · 71_native.tex, line 31
Theorem. For this fixed polynomial presentation, the group and distinguished class are independent of the chosen integer representative. The class vanishes if and only if bar x lifts to a solution modulo p^L+1. The obstruction group has p^r-rankJ elements. When lifts exist, their set is a torsor for ker J and contains exactly p^d-rankJ elements.
Exact mathematical passage
p^{r-\operatorname{rank}J}
\operatorname{rank}_{\mathbb F_p}Use in source · 3 passages
Search aliases: rank_ F_p operatorname mathbb
Use in source · continuation_02.tex, line 16
Let p be prime, L≥1, and F∈ Z[X_1,…,X_d]^e, x∈ Z^d, F(x)≡0pmodp^L. Write r_x=F(x)/p^L, b_x=r_xbmod p, A_x=DF(x)bmod p: F_p^d⟶ F_p^e, ρ_x=rank_ F_pA_x. The actual lift set is L_x,L= {y∈( Z/p^L+1 Z)^d: y≡ xpmodp^L, F(y)=0}, where polynomial evaluation in the displayed set is modulo p^L+1. Define O_x= F_p^e/imA_x, o_x=[b_x]. Lean proves # L_x,L= p^d-ρ_x, · o_x=0,
Exact mathematical passage
r_x=F(x)/p^L,\qquad b_x=r_x\bmod p, \qquad A_x=DF(x)\bmod p:\mathbb F_p^d\longrightarrow\mathbb F_p^e, \qquad \rho_x=\operatorname{rank}_{\mathbb F_p}A_x.
\rank_{E'}Use in source · 1 passage
Search aliases: rank_E' rank operatorname
Use in source · Paper_XII_Traditional_Final.tex, line 1093
[Rank and trace detection in characteristic zero] Assume that E' has characteristic zero. For every maximal Hecke packet, Tr_E'(e_ m|V) = rank_E'(e_ m|V) = dim_E'e_ mV. In particular, the packet is occupied exactly when its projector has positive rank, equivalently nonzero trace.
Exact mathematical passage
\boxed{ \Tr_{E'}(e_{\frakm}|V) = \rank_{E'}(e_{\frakm}|V) = \dim_{E'}e_{\frakm}V. }
\rank_{\F_p}Use in source · 2 passages
Search aliases: rank_F_p rank_ F_p rank F operatorname mathbb
Use in source · 30_lifting.tex, line 294
[Torsor and lift cardinalities] Under the hypotheses of Theorem~[source reference], there is an equivalence T_x≃ L_x,L. Every nonempty fiber is a torsor for ker J_x under addition of correction digits. If q=p is prime and ρ_x=rank_ F_pJ_x, then # L_x,L= p^d-ρ_x, · o_x=0,
Exact mathematical passage
\rho_x=\rank_{\F_p}J_x
\rank_{\FF_p}Use in source · 1 passage
Search aliases: rank_FF_p rank_ F_p rank FF operatorname mathbb
Use in source · 72_recovered.tex, line 2588
[Exact number of linear children] If ρ=rank_ F_poverline M, then a solvable depth L class has exactly p^d-ρ children modulo p^L+1.
Exact mathematical passage
\rho=\rank_{\FF_p}\overline M
\Rcal_0Use in source · 4 passages
Search aliases: Rcal_0 R_0 Rcal mathcal
Use in source · 02_strengthened.tex, line 1361
[Exact recovery on the error span] Under the hypotheses of [source reference], define on the syndrome support Q=∑_λ_j>0Q_j the completely positive map R_0(X) = ∑_λ_j>0 V_j^*Q_jXQ_jV_j. It extends to a quantum channel R on End( H). If Φ(X)=∑_aE_aXE_a^* is trace preserving on code states, then RΦ(ρ)=ρ for every density matrix ρ=Pρ P.
Exact mathematical passage
\Rcal_0(X) = \sum_{\lambda_j>0} V_j^*Q_jXQ_jV_j.
\Rcal_NUse in source · 2 passages
Search aliases: Rcal_N R_N Rcal mathcal
Use in source · Paper_IV_Traditional_Final.tex, line 698
[Finite exponential representation] Let R_N={η∈ Cmathrel:η^N=-1}. Then D_N(z) = frac1N∑_η∈ R_Ne^η z. More generally, D_N,j(z) = fracρ^-jN ∑_ℓ=0^N-1 ω^-jℓe^ρω^ℓ z.
Exact mathematical passage
\Rcal_N=\{\eta\in\CC\mathrel{:}\eta^N=-1\}.
\ReachUse in source · 3 passages
Search aliases: Reach operatorname
Use in source · 03_packets_and_graphs.tex, line 937
[Reachability support of the generated algebra] For every A∈ A, P_β AP_α=0 whenever β∉Reach(α). Equivalently, A⊆ M_R_Γ, R_Γ:={(β,α):β∈Reach(α)}. [source reference]
Exact mathematical passage
P_\beta AP_\alpha=0 \qquad\text{whenever }\beta\notin\Reach(\alpha).
\mathrm{rec}Use in source · 3 passages
Search aliases: rec mathrm
Use in source · Paper_V_Traditional_Final.tex, line 2059
[Quantitative imperfect reconstruction] Define the delay compensated defect D( e^ iω) = e^ i dωR( e^ iω)E( e^ iω)-I_N. If ε_rec = *ess sup_ω D( e^ iω) _op<∞, then the delay compensated reconstruction widetilde x satisfies widetilde x-x _2 ≤ ε_rec x _2. The constant is sharp as an operator norm.
Exact mathematical passage
\varepsilon_{\mathrm{rec}} = \esssup_{\omega}\|D(\e^{\ii\omega})\|_{\mathrm{op}}<\infty,
\ResUse in source · 6 passages
Search aliases: Res operatorname
Use in source · 04_arithmetic.tex, line 73
[Several coprime spectral packets] Assume χ_A(x)=∏_i=1^rf_i(x), where the monic factors f_i∈ O[x] are pairwise coprime in K[x]. Put g_i=∏_j≠ if_j. There are complementary projectors E_i with im E_i=Ker f_i(A), ∑_iE_i=I, E_iE_j=0 (i≠ j). One may choose E_i so that Res(f_i,g_i)E_i∈End_ O(Λ) up to multiplication of the resultant by a unit.
Exact mathematical passage
\Res(f_i,g_i)E_i\in\End_{\OO}(\Lambda)
\operatorname{Res}Use in source · 1 passage
Search aliases: Res operatorname
Use in source · visual_atlas.tex, line 345
For the original sign factors the concrete resultant is Res(x-1,x+1)=2 and the bound is sharp. That equality is special to this displayed calculation, not a universal minimality theorem for every resultant certificate.
Exact mathematical passage
\operatorname{Res}(x-1,x+1)=2
\Res_yUse in source · 2 passages
Search aliases: Res_y Res operatorname
Use in source · 72_recovered.tex, line 2815
We use the resultant convention Res_y(f,g) = ∏_f(α)=0g(α) when f is monic, with roots counted with multiplicity.
Exact mathematical passage
\Res_y(f,g) = \prod_{f(\alpha)=0}g(\alpha)
\mathsf{Residual}Use in source · 3 passages
Search aliases: Residual mathsf
Use in source · 02_connections_obstructions.tex, line 397
Let Candidate, Packet, and Residual be arbitrary types, with a distinguished 0∈Residual, and let r:Candidate⟶ Packet⟶Residual be a residual function.
Exact mathematical passage
\mathsf{Residual}
\rfloorUse in source · 49 passages
Search aliases: rfloor
Use in source · Paper_IV_Traditional_Final.tex, line 679
[Twisted addition law] For 0≤ k<N, D_N,k(z+w) = ∑_substack0≤ i,j<N i+j≡ kbmod N (-1)^lfloor(i+j)/Nrfloor D_N,i(z) D_N,j(w).
Exact mathematical passage
\Dfun_{N,k}(z+w) = \sum_{\substack{0\leq i,j<N\\ i+j\equiv k\bmod N}} (-1)^{\lfloor(i+j)/N\rfloor} \Dfun_{N,i}(z)\Dfun_{N,j}(w).
\RMSEUse in source · 1 passage
Search aliases: RMSE operatorname
Use in source · Paper_V_Traditional_Final.tex, line 781
[One-dimensional errors] For 1024 samples on [-1,1) and Fourier cutoff m=18, the retained frequency indices are -18,…,18: 37 bins including the zero-frequency bin. The measured values were lrr method · RMSE · maximum absolute error global Fourier truncation · 5.4023×10^-2 · 3.4968×10^-1 known edge packet · 1.3692×10^-4 · 1.4503×10^-3 estimated edge packet · 2.4029×10^-2 · 5.9530×10^-2 The simple estimator recovered the binary mask exactly in this deliberately separated two-class example. That re…
Exact mathematical passage
\begin{array}{lrr} \toprule \text{method}&\RMSE&\text{maximum absolute error}\\ \midrule \text{global Fourier truncation}&5.4023\times10^{-2}&3.4968\times10^{-1}\\ \text{known edge packet}&1.3692\times10^{-4}&1.4503\times10^{-3}\\ \text{estimated edge packet}&2.4029\times10^{-2}&5.9530\times10^{-2}\\ \bottomrule \end{array}
\mathrm{root}Use in source · 1 passage
Search aliases: root mathrm
Use in source · 02_supplement_03.tex, line 181
The root scan min_0≤ b≤ BF_root(b) is therefore the exact budget-constrained optimum. Increasing the budget cannot increase a finite optimum. The theorem retains feasibility hypotheses where necessary rather than treating codenone as an ordinary real cost.
Exact mathematical passage
\boxed{\min_{0\leq b\leq B}F_{\mathrm{root}}(b)}
\operatorname{Row}Use in source · 2 passages
Search aliases: Row operatorname
Use in source · 03_supplement_03.tex, line 8
A matrix X belongs to O(C) exactly when each of its rows belongs to Row(C). One direction follows by expanding AC. For the converse, choose the coefficients producing each row separately and place them in the corresponding row of A. An explicit linear equivalence therefore gives O(C)≃ Row(C)^d, dim_ C O(C)=d rank C. If d>0, equality with the full d× m coefficient space is equivalent to rank C=m. Consequently a cyclic seed requires m≤ d. The assumption that there is at least one row is essential …
Exact mathematical passage
\operatorname{Row}(C)
\RR^2Use in source · 5 passages
Search aliases: RR^2 R^2 RR mathbb
Use in source · visual_atlas.tex, line 168
vpage7Exact does not mean orthogonalPACKET COORDINATES / AN OBLIQUE SPLITTINGMain text: [source reference] In R^2, take P=mat1 · -1
Exact mathematical passage
\RR^2
\RR^{m\times p}Use in source · 1 passage
Search aliases: RR^m× p R^m× p RR times mathbb times
Use in source · 01_manuscript.tex, line 1309
[Floating point matrix product model] For real A∈ R^m× p and B∈ R^p× n, consider ordinary dot-product matrix multiplication in uniform precision with unit roundoff u, no overflow or underflow, and pu<1. Its computed output widehat C satisfies the componentwise forward-error bound |widehat C-AB|≤γ_p|A||B|, γ_p=fracpu1-pu. Absolute values and the inequality are entrywise. Complex arithmetic, fused kernels, mixed accumulation, and tensor kernels require their own stated constants. This fact does no…
Exact mathematical passage
A\in\RR^{m\times p}
\RR^nUse in source · 3 passages
Search aliases: RR^n R^n RR mathbb
Use in source · Paper_VI_Traditional_Final.tex, line 3331
Let Δ_n-1 = {p∈ R^n: p_i≥0, ∑_ip_i=1 } be the probability simplex. Let λ∈[0,∞)^n have positive evidence Z_λ(p)=∑_iλ_ip_i. The Bayesian transfer is B_λ(p) = fracλodot pZ_λ(p).
Exact mathematical passage
\Delta_{n-1} = \left\{p\in\RR^n: p_i\geq0, \quad \sum_ip_i=1 \right\}
\RR^{p\times n}Use in source · 1 passage
Search aliases: RR^p× n R^p× n RR times mathbb times
Use in source · 01_manuscript.tex, line 1309
[Floating point matrix product model] For real A∈ R^m× p and B∈ R^p× n, consider ordinary dot-product matrix multiplication in uniform precision with unit roundoff u, no overflow or underflow, and pu<1. Its computed output widehat C satisfies the componentwise forward-error bound |widehat C-AB|≤γ_p|A||B|, γ_p=fracpu1-pu. Absolute values and the inequality are entrywise. Complex arithmetic, fused kernels, mixed accumulation, and tensor kernels require their own stated constants. This fact does no…
Exact mathematical passage
B\in\RR^{p\times n}
\mathrm{run}Use in source · 1 passage
Search aliases: run mathrm
Use in source · 02_supplement_02.tex, line 203
Contracts, grouping, and amortization. textitSource items 2.9, 10.2, 10.4--10.6, and 11.3 Output conversion. A decoded run returns the actual pair (decode(compressed(x)), t_run(x)+t_decode(compressed(x))). Under the declared output equality, Lean proves that this returns the same explicit result and charges the conversion time. It also proves the downstream composition identity when compressed output is consumed directly. Omitting a positive decode cost strictly understates the modeled explicit-…
Exact mathematical passage
\bigl(\operatorname{decode}(\operatorname{compressed}(x)), \;t_{\mathrm{run}}(x)+t_{\mathrm{decode}}(\operatorname{compressed}(x))\bigr).
\rVertUse in source · 1 passage
Search aliases: rVert
Use in source · 03_weighted.tex, line 281
lineskip [Certified weighted inverse] Let R be a possibly noncommutative ring, let D^-1 be a supplied two-sided inverse of D, and put A=D^-1H. [label=(alph*)] item If A^r=0, then D+H has the two-sided inverse (D+H)^-1=(∑_k=0^r-1(-A)^k)D^-1. item If R is a complete normed ring and lVert ArVert<1, then ∑_k≥0(-A)^k converges and (D+H)^-1=(∑_k≥0(-A)^k)D^-1.
Exact mathematical passage
\lVert A\rVert<1
\rvertUse in source · 1 passage
Search aliases: rvert
Use in source · Paper_X_Block27_Amended.tex, line 2455
The p-adic square-root criterion is proved for every prime: (∃ x∈ Q_p, x^2=-1) Longleftrightarrow p≡1pmod4. A root satisfies lvert xrvert_p^2=1 and is therefore integral. Reduction modulo three rules out a root in Q_3, and reduction modulo four rules out a root in Q_2. At an odd prime, any residue root of X^2+1 has nonzero derivative and lifts by the pinned Mathlib Hensel theorem. The finite-field square criterion then gives the stated congruence. An explicit Hensel application at the residue ro…
Exact mathematical passage
\lvert x\rvert_p^2=1
\rvert_p^2Use in source · 1 passage
Search aliases: rvert_p^2 rvert
Use in source · Paper_X_Block27_Amended.tex, line 2455
The p-adic square-root criterion is proved for every prime: (∃ x∈ Q_p, x^2=-1) Longleftrightarrow p≡1pmod4. A root satisfies lvert xrvert_p^2=1 and is therefore integral. Reduction modulo three rules out a root in Q_3, and reduction modulo four rules out a root in Q_2. At an odd prime, any residue root of X^2+1 has nonzero derivative and lifts by the pinned Mathlib Hensel theorem. The finite-field square criterion then gives the stated congruence. An explicit Hensel application at the residue ro…
Exact mathematical passage
\lvert x\rvert_p^2=1
\boldsymbol sUse in source · 1 passage
Search aliases: s boldsymbol
Use in source · 72_recovered.tex, line 2018
[Nested discrete refinement] For positive integers M,N, (a_r^[M])_s^[N] = a_r+Ms^[MN]. For a mixed radix path (M_1,…,M_ℓ) with digits 0≤ s_i<M_i, the final address is r( s) = s_1+M_1s_2+⋯+ (∏_i=1^ℓ-1M_i)s_ℓ. Only the product radix and accumulated address determine the final branch.
Exact mathematical passage
\boxed{ r(\boldsymbol s) = s_1+M_1s_2+\cdots+ \left(\prod_{i=1}^{\ell-1}M_i\right)s_\ell. }
\mathsf SUse in source · 3 passages
Search aliases: S mathsf
Use in source · 72_recovered.tex, line 2063
[Polyphase shift matrix] Let E be the scalar shift (Ea)_n=a_n+1, and let E_q shift the coarse index. Then P_M(Ea) = S_M(E_q) P_Ma, where S_M(E_q) = 0 · 1 · 0 · ⋯ · 0
Exact mathematical passage
\boxed{ \mathcal P_M(Ea) = \mathsf S_M(E_q)\mathcal P_Ma, }
S''Use in source · 2 passages
Search aliases: S''
Use in source · 04_transport.tex, line 164
· =ΨΦ ∑_τ(σ(α))=kP_α. Thus the composite packet map is τ∘σ. If the morphisms are seeded, then Φ(S)⊆Hull_ B(S'), Ψ(S')⊆Hull_ C(S''). By [source reference], ΨΦ(S) ⊆ Ψ(Hull_ B(S')) ⊆ Hull_ C(Ψ(S')) ⊆ Hull_ C(S''). Identity maps satisfy the definitions with identity algebra and packet maps.
Exact mathematical passage
\Phi(S)\subseteq\Hull_{\B}(S'), \qquad \Psi(S')\subseteq\Hull_{\Ccal}(S'').
S^{-1}Use in source · 9 passages
Search aliases: S^-1
Use in source · Paper_VI_Traditional_Final.tex, line 571
[Symmetric random walk] Starting from δ_0 and applying the transfer T=frac12(S+S^-1) for n steps gives P(X_n=k)= 2^-nbinomn(n+k)/2 when k≡ npmod2, and zero otherwise. Hence E[X_n]=0, Var(X_n)=n. The parity packet is exact: all mass remains in the one residue class permitted by n.
Exact mathematical passage
T=\frac12(S+S^{-1})
s^{-1/2}Use in source · 1 passage
Search aliases: s^-1/2
Use in source · atlas-pages.tex, line 394
A normalized density need not have a finite mean. For large s, the exponential factor tends to one and η_t(s) is proportional to s^-3/2. Consequently sη_t(s) has a nonintegrable s^-1/2 tail. The clock has infinite mean even though its total mass is one. takeawaySubordination is a weighted mixture of diffusion times. Replacing the clock distribution by a single average time is not justified, and here that average is infinite.
Exact mathematical passage
s^{-1/2}
s^{-3/2}Use in source · 6 passages
Search aliases: s^-3/2
Use in source · Paper_VI_Traditional_Final.tex, line 1167
[Poisson subordination] For t>0 and L≥0, e^-tsqrt L = fract2sqrtπ ∫_0^∞ s^-3/2e^-t^2/(4s)e^-sL ds. The integral converges in operator norm in finite dimension.
Exact mathematical passage
\boxed{ e^{-t\sqrt L} = \frac{t}{2\sqrt\pi} \int_0^\infty s^{-3/2}e^{-t^2/(4s)}e^{-sL}\,ds. }
S^{-r}Use in source · 2 passages
Search aliases: S^-r
Use in source · 01_finite_core.tex, line 221
Using S^r+s=S^r+sbmod N and the finite character orthogonality relation, frac1N∑_r=0^N-1ω^(k-j)r=δ_jk, we obtain P_jP_k=δ_jkP_j. Summing over j cancels every term indexed by r≠0 and retains the r=0 term, which proves completeness. Unitarity and the change of residue r↦-r give P_j^*=frac1N∑_r=0^N-1ω^jrS^-r=P_j. Finally, multiplication by S shifts the summation index and contributes the factor ω^j.
Exact mathematical passage
P_j^*=\frac1N\sum_{r=0}^{N-1}\omega^{jr}S^{-r}=P_j.
s^{[N]}Use in source · 3 passages
Search aliases: s^[N]
Use in source · 10_foundations.tex, line 488
polyphase vocabulary. We explain it through elementary identities with explicit algebraic derivations. These identities have not been separately formalized in Lean for this consolidation. For a sequence a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient and remainder give A(z)=∑_r=0^M-1z^rA_r(z^M), (a_r^[M])_s^[N]=a_r+Ms^[MN]. For ordinary Cauchy multiplication, the phase-r product has a term without a …
Exact mathematical passage
A(z)=\sum_{r=0}^{M-1}z^rA_r(z^M),\qquad \bigl(a_r^{[M]}\bigr)_s^{[N]}=a_{r+Ms}^{[MN]}.
S^eUse in source · 3 passages
Search aliases: S^e
Use in source · continuation_08.tex, line 14
Let q>0, L≥1, and m,k,e≥0. Write S= Z/q Z. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0)∈ Z^m+k, F(x)≡0pmodq^L. The actual lower root supplies the unique integral residual r=q^-LF(x)∈ Z^e, bar r=rbmod q∈ S^e. Let the actual reduced Jacobian blocks be J=(D_YF)(x)bmod q:S^m→ S^e, K=(D_TF)(x)bmod q:S^k→ S^e. For a prescribed parameter digit v∈ S^k, choose its canonical integral representative widehat v. Define L_v to be the set of actual residue vectors z∈( Z/q^L+1 Z)^m+k satisfying z≡ xpmodq^L, F(z…
Exact mathematical passage
r=q^{-L}F(x)\in\mathbb Z^e,\qquad \bar r=r\bmod q\in S^e.
S^HUse in source · 3 passages
Search aliases: S^H
Use in source · strengthening.tex, line 31
Write S f(n)=f(n+1) and Δ_H=S^H-1. A mixed difference Δ_H_1⋯Δ_H_k acts on the row variable n, rather than on the full Euler subscript. We use v_p(0)=+∞. Orders and starting rows are nonnegative integers. Steps are positive integers unless stated otherwise. An empty mixed difference is the original value, and a zero ordinary step makes every nonempty product containing it zero.
Exact mathematical passage
\Delta_H=S^H-1
S^kUse in source · 2 passages
Search aliases: S^k
Use in source · continuation_08.tex, line 16
Let q>0, L≥1, and m,k,e≥0. Write S= Z/q Z. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0)∈ Z^m+k, F(x)≡0pmodq^L. The actual lower root supplies the unique integral residual r=q^-LF(x)∈ Z^e, bar r=rbmod q∈ S^e. Let the actual reduced Jacobian blocks be J=(D_YF)(x)bmod q:S^m→ S^e, K=(D_TF)(x)bmod q:S^k→ S^e. For a prescribed parameter digit v∈ S^k, choose its canonical integral representative widehat v. Define L_v to be the set of actual residue vectors z∈( Z/q^L+1 Z)^m+k satisfying z≡ xpmodq^L, F(z…
Exact mathematical passage
J=(D_YF)(x)\bmod q:S^m\to S^e, \qquad K=(D_TF)(x)\bmod q:S^k\to S^e.
S^mUse in source · 2 passages
Search aliases: S^m
Use in source · continuation_08.tex, line 16
Let q>0, L≥1, and m,k,e≥0. Write S= Z/q Z. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0)∈ Z^m+k, F(x)≡0pmodq^L. The actual lower root supplies the unique integral residual r=q^-LF(x)∈ Z^e, bar r=rbmod q∈ S^e. Let the actual reduced Jacobian blocks be J=(D_YF)(x)bmod q:S^m→ S^e, K=(D_TF)(x)bmod q:S^k→ S^e. For a prescribed parameter digit v∈ S^k, choose its canonical integral representative widehat v. Define L_v to be the set of actual residue vectors z∈( Z/q^L+1 Z)^m+k satisfying z≡ xpmodq^L, F(z…
Exact mathematical passage
J=(D_YF)(x)\bmod q:S^m\to S^e, \qquad K=(D_TF)(x)\bmod q:S^k\to S^e.
s^mUse in source · 1 passage
Search aliases: s^m
Use in source · Paper_VI_Traditional_Final.tex, line 343
Both exponentials are absolutely convergent matrix power series. Since every power of A commutes with every other power, the Cauchy product gives e^sAe^tA =∑_m,n≥0fracs^mt^nm!n!A^m+n =∑_r≥0frac(s+t)^rr!A^r. Termwise differentiation is valid in finite dimension and gives the derivative identities.
Exact mathematical passage
e^{sA}e^{tA} =\sum_{m,n\ge0}\frac{s^mt^n}{m!n!}A^{m+n} =\sum_{r\ge0}\frac{(s+t)^r}{r!}A^r.
S^NUse in source · 6 passages
Search aliases: S^N
Use in source · 01_finite_core.tex, line 189
Let N≥1 be an integer, and let S be a unitary symmetry whose order divides N: S^N=I. Set ω=e^2π i/N.
Exact mathematical passage
S^N=I.
S^{\mathrm{pr}}_{L,N}Use in source · 3 passages
Search aliases: S^pr_L,N mathrm
Use in source · 03_weighted.tex, line 321
et q_L:R→ R_L be a ring homomorphism that kills F^L R. Suppose H=∑_e∈ EH_e, D^-1H_e∈ F^w(e)R, w(e)≥1, with E finite. Then q_L(D^-1H)^L=0, the finite sum in [source reference](a) is a two-sided inverse after applying q_L, and its word expansion may discard every word of weight at least L. More explicitly, for a length cutoff N, define S^pr_L,N = ∑_n=0^N-1(-1)^n ∑_substackγ∈ E^n W(γ)<LA_γ. Lean proves directly that q_L(S^pr_L,N) equals the mapped geometric inverse whenever q_L(A^N)=0, and hence is…
Exact mathematical passage
S^{\mathrm{pr}}_{L,N} = \sum_{n=0}^{N-1}(-1)^n \sum_{\substack{\gamma\in E^n\\W(\gamma)<L}}A_\gamma.
S^rUse in source · 5 passages
Search aliases: S^r
Use in source · 06_models.tex, line 6
Let x=(x_n)_n∈ Z∈ c_00( Z;R) and let the seed be δ_0 under the bilateral shifts (S^rδ_0)_n=δ_n,r. This makes the algebraic span and every displayed sum finite. The N packets are x_j[m]=x[Nm+j]. Synthesis interleaves these sequences. A linear filter with a finite impulse response that is invariant under shifts becomes an N× N matrix of transfer blocks. Refining from N to M with Nmid M splits each coarse phase into its descendants without changing the reconstructed signal. The same formulas extend…
Exact mathematical passage
(S^r\delta_0)_n=\delta_{n,r}
S^{r+s}Use in source · 1 passage
Search aliases: S^r+s
Use in source · 01_finite_core.tex, line 216
Using S^r+s=S^r+sbmod N and the finite character orthogonality relation, frac1N∑_r=0^N-1ω^(k-j)r=δ_jk, we obtain P_jP_k=δ_jkP_j. Summing over j cancels every term indexed by r≠0 and retains the r=0 term, which proves completeness. Unitarity and the change of residue r↦-r give P_j^*=frac1N∑_r=0^N-1ω^jrS^-r=P_j. Finally, multiplication by S shifts the summation index and contributes the factor ω^j.
Exact mathematical passage
S^{r+s}=S^{r+s\bmod N}
S^{r+s\bmod N}Use in source · 1 passage
Search aliases: S^r+sbmod N bmod
Use in source · 01_finite_core.tex, line 216
Using S^r+s=S^r+sbmod N and the finite character orthogonality relation, frac1N∑_r=0^N-1ω^(k-j)r=δ_jk, we obtain P_jP_k=δ_jkP_j. Summing over j cancels every term indexed by r≠0 and retains the r=0 term, which proves completeness. Unitarity and the change of residue r↦-r give P_j^*=frac1N∑_r=0^N-1ω^jrS^-r=P_j. Finally, multiplication by S shifts the summation index and contributes the factor ω^j.
Exact mathematical passage
S^{r+s}=S^{r+s\bmod N}
S^{\operatorname{val}(r)}Use in source · 1 passage
Search aliases: S^val(r) operatorname
Use in source · 03_supplement_03.tex, line 98
003 Let N>0 and let S be a unitary complex matrix with S^N=I. The implementation uses the standard additive character of Z/N Z and proves its expression through a root of unity: ω=exp(2π i/N), χ(r)=ω^val(r). Here val(r) is the canonical representative in {0,…,N-1}. The representation is constructed as ρ(r)=S^val(r). Reduction of exponents modulo N is proved from S^N=I, giving ρ(r+s)=ρ(r)ρ(s) and ρ(0)=I. The final matrix theorem derives the representation law and character orthogonality from the …
Exact mathematical passage
\rho(r)=S^{\operatorname{val}(r)}.
s^\alphaUse in source · 1 passage
Search aliases: s^α alpha
Use in source · Paper_VI_Traditional_Final.tex, line 4057
Set R_f(s)=(sI+L_f)^-1, R_c(s)=(sI+L_c)^-1. The resolvent identity gives CR_f(s)-R_c(s)C = -R_c(s)Δ_LR_f(s). Since LR=I-sR, CL_fR_f(s)-L_cR_c(s)C = sR_c(s)Δ_LR_f(s). Insert this into [source reference]. The spectral lower bound gives normR_f(s),normR_c(s) ≤ frac1s+m. Therefore normCL_f^α-L_c^α C · ≤ fracsin(πα)π normΔ_L ∫_0^∞ fracs^α(s+m)^2 d s · = α m^α-1normΔ_L. The semigroup estimate follows from [source reference], applied to the contraction generators -L_f^α and -L_c^α.
Exact mathematical passage
\begin{aligned} \norm{CL_f^\alpha-L_c^\alpha C} &\leq \frac{\sin(\pi\alpha)}{\pi} \norm{\Delta_L} \int_0^\infty \frac{s^\alpha}{(s+m)^2}\dd s\\ &= \alpha m^{\alpha-1}\norm{\Delta_L}. \end{aligned}
s^{\alpha-1}Use in source · 2 passages
Search aliases: s^α-1 alpha
Use in source · Paper_VI_Traditional_Final.tex, line 3956
[Balakrishnan formula in finite dimension] For 0<α<1, L^α = fracsin(πα)π ∫_0^∞ s^α-1L(sI+L)^-1 d s. The integral converges in operator norm.
Exact mathematical passage
L^\alpha = \frac{\sin(\pi\alpha)}{\pi} \int_0^\infty s^{\alpha-1}L(sI+L)^{-1} \dd s.
S^{\oplus m}Use in source · 2 passages
Search aliases: S^⊕ m oplus
Use in source · 02_foundations.tex, line 847
[Multiplicity obstruction] For the diagonal action of M_d(D) on S^⊕ m, srank_M_d(D)(S^⊕ m) = lceilfracmdrceil. [source reference]
Exact mathematical passage
S^{\oplus m}
S_0Use in source · 4 passages
Search aliases: S_0
Use in source · 05_worked_models.tex, line 204
Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. Therefore sequences with finite support lie in its algebraic orbit hull and are dense in ℓ^2( Z).
Exact mathematical passage
S_0=I
s_0Use in source · 1 passage
Search aliases: s_0
Use in source · visual_atlas.tex, line 453
vpage19Precision exposes directional freedomMODEL E / A FIBER BUDGET AT EACH DEPTHChapter II, mathematical text: Theorem 10.6 Take p=2 and the supplied diagonal weights (s_0,s_1,s_2)=(0,1,3). At depth L, an admissible target has 2^W(L) solutions, W(L)=min(0,L)+min(1,L)+min(3,L). Each direction contributes only the amount visible at the current depth.
Exact mathematical passage
(s_0,s_1,s_2)=(0,1,3)
S_1Use in source · 4 passages
Search aliases: S_1
Use in source · 05_worked_models.tex, line 204
Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. Therefore sequences with finite support lie in its algebraic orbit hull and are dense in ℓ^2( Z).
Exact mathematical passage
S_1
S_2Use in source · 2 passages
Search aliases: S_2
Use in source · 72_recovered.tex, line 1310
[The first odd reconstruction] For p=3 and n=2, S_2(3)=1-9+25=17. Since Z_1=1, Z_3=2, and S_0(3)=1, the odd coordinate formula gives E_2 =17-2·9 =-1. The large source value and the depth one correction are both necessary.
Exact mathematical passage
S_2(3)=1-9+25=17.
\mathcal S_2Use in source · 1 passage
Search aliases: S_2 mathcal
Use in source · visual_atlas.tex, line 87
Define X_N^rm pkt=⊕_j P_N,jX, A_Nx=(P_N,jx)_j, S_N(x_j)_j=∑_jx_j. Completeness and mutual annihilation give both inverse laws. Consequently, J= A_4 S_2, C= A_2 S_4, CJ=I, JC=I. The larger list of fine packets has smaller individual packet spaces.
Exact mathematical passage
J=\mathcal A_4\mathcal S_2,\qquad C=\mathcal A_2\mathcal S_4, \qquad CJ=I,\quad JC=I.
s_2Use in source · 22 passages
Search aliases: s_2
Use in source · 04_arithmetic.tex, line 580
0 · 1 . For every odd prime p, both projectors preserve Z_p^2. At p=2, s_2(E_+)=s_2(E_-)=1. The factors x-1 and x+1 have resultant 2.
Exact mathematical passage
s_2(E_+)=s_2(E_-)=1.
S_{2(r-m)}Use in source · 2 passages
Search aliases: S_2(r-m)
Use in source · supplement_02.tex, line 125
ing3. Exact reconstruction and the triangular matrixTheorem 4.2; manuscript page 11 For odd p, rescaling the already proved inverse law gives sech(pt)cosh(pt)=1. Multiplying the even-source identity by sech^2(pt) therefore proves sech^2(pt)(∑_r≥0S_2r(p)fract^2r(2r)!)=sech t. Actual EGF multiplication now gives the integer coefficient formula E_2r=∑_m=0^rbinom2r2mb_m p^2mS_2(r-m)(p). The formal proof first extracts the sum over all indices and then proves that the odd summands vanish. A checked e…
Exact mathematical passage
\boxed{E_{2r}=\sum_{m=0}^{r}\binom{2r}{2m}b_m p^{2m}S_{2(r-m)}(p).}
s_{2,\Z_2^2}Use in source · 1 passage
Search aliases: s_2,Z_2^2 s_2, Z_2^2 Z mathbb
Use in source · 04_arithmetic.tex, line 650
[Odd-adic versus dyadic behavior] For every prime p≠2, every entry of E_+ and E_- is p-integral, and each projector has exact local denominator weight zero. At p=2, each projector has exact local denominator weight one. Multiplication by 2 makes it integral, while the unscaled matrix is not integral: s_2, Z_2^2(E_+)=s_2, Z_2^2(E_-)=1. The characteristic factors x-1 and x+1 have resultant 2. Its 2-adic valuation is one, matching both local projector weights above, and its p-adic valuation is zero…
Exact mathematical passage
s_{2,\Z_2^2}(E_+)=s_{2,\Z_2^2}(E_-)=1.
S_{2n}Use in source · 3 passages
Search aliases: S_2n
Use in source · 20_arithmetic.tex, line 822
The exponential generating series of the source is the finite sum S_p(t)=∑_a=0^p-1(-1)^a e^(2a+1)t. Multiplication by cosh t telescopes. Since p is odd, S_p(t)=frac1+e^2pt2secht. Taking its even part gives ∑_n≥0S_2n(p)fract^2n(2n)! =cosh^2(pt)secht. Multiplication by the reciprocal sech^2(pt) yields secht. The EGF product rule now gives the reconstruction coefficient by coefficient. Replacing m by n-q produces the displayed lower triangular row. Its diagonal and weights follow from γ_0=1 and the…
Exact mathematical passage
\sum_{n\ge0}S_{2n}(p)\frac{t^{2n}}{(2n)!} =\cosh^2(pt)\operatorname{sech}t.
S_{2q}Use in source · 2 passages
Search aliases: S_2q
Use in source · 20_arithmetic.tex, line 803
[Classical odd-length reconstruction] For every positive odd integer p and every n∈ N, E_2n=∑_m=0^nbinom2n2mγ_m p^2mS_2(n-m)(p). Equivalently, E_2n=∑_q=0^n V_nqS_2q(p) for the lower triangular matrix V_nq=binom2n2qγ_n-qp^2(n-q) (q≤ n), V_nq=0 (q>n). It has V_nn=1 and p^2(n-q)mid V_nq when q≤ n.
Exact mathematical passage
E_{2n}=\sum_{q=0}^n V_{nq}S_{2q}(p)
S_{2r}Use in source · 2 passages
Search aliases: S_2r
Use in source · supplement_02.tex, line 64
Even projection is identified coefficientwise. For any formal series f, let EvenPart(f)=(f(t)+f(-t))/2. Its nth coefficient is proved to be the coefficient of f when n is even and zero otherwise. Rescaling the inverse law proves sech(-t)=sech(t). The exponential multiplication identity then gives cosh^2(pt)=frac2+e^2pt+e^-2pt4, and hence ∑_r≥0S_2r(p)fract^2r(2r)! =cosh^2(pt) sech t. Both the full and even EGF formulas are derived from the original integer sum. Neither is supplied as a hypothesis…
Exact mathematical passage
\boxed{\sum_{r\ge0}S_{2r}(p)\frac{t^{2r}}{(2r)!} =\cosh^2(pt)\,\sech t.}
S_3Use in source · 10 passages
Search aliases: S_3
Use in source · Paper_XII_Traditional_Final.tex, line 356
Let G=S_3, let k= F_7, and let V be the standard two dimensional representation. In the basis v_1=(1,-1,0), v_2=(0,1,-1), a transposition and a three cycle act by s= 6 · 1
Exact mathematical passage
G=S_3
S_4Use in source · 1 passage
Search aliases: S_4
Use in source · supplement_02.tex, line 181
The displayed manuscript row is kernel-checked. The scalar anchors are derived from the same general coefficient recurrence: (E_0,E_2,E_4,E_6,E_8)=(1,-1,5,-61,1385), (b_0,b_1,b_2,b_3,b_4)=(1,-2,16,-272,7936). The literal source sums at p=5 give (S_0,S_2,S_4,S_6,S_8)=(1,49,4705,428689,37665985). Consequently, T_2(5,8)=2 731 001 385, E_8-T_2(5,8)=-2 731 000 000. The source proves both 5^6mid E_8-T_2(5,8), 5^7nmid E_8-T_2(5,8). Thus the guaranteed power is attained in this example. No stronger unif…
Exact mathematical passage
(S_0,S_2,S_4,S_6,S_8)=(1,49,4705,428689,37665985).
\mathcal S_4Use in source · 2 passages
Search aliases: S_4 mathcal
Use in source · visual_atlas.tex, line 87
Define X_N^rm pkt=⊕_j P_N,jX, A_Nx=(P_N,jx)_j, S_N(x_j)_j=∑_jx_j. Completeness and mutual annihilation give both inverse laws. Consequently, J= A_4 S_2, C= A_2 S_4, CJ=I, JC=I. The larger list of fine packets has smaller individual packet spaces.
Exact mathematical passage
J=\mathcal A_4\mathcal S_2,\qquad C=\mathcal A_2\mathcal S_4, \qquad CJ=I,\quad JC=I.
S_6Use in source · 1 passage
Search aliases: S_6
Use in source · supplement_02.tex, line 181
The displayed manuscript row is kernel-checked. The scalar anchors are derived from the same general coefficient recurrence: (E_0,E_2,E_4,E_6,E_8)=(1,-1,5,-61,1385), (b_0,b_1,b_2,b_3,b_4)=(1,-2,16,-272,7936). The literal source sums at p=5 give (S_0,S_2,S_4,S_6,S_8)=(1,49,4705,428689,37665985). Consequently, T_2(5,8)=2 731 001 385, E_8-T_2(5,8)=-2 731 000 000. The source proves both 5^6mid E_8-T_2(5,8), 5^7nmid E_8-T_2(5,8). Thus the guaranteed power is attained in this example. No stronger unif…
Exact mathematical passage
(S_0,S_2,S_4,S_6,S_8)=(1,49,4705,428689,37665985).
S_8Use in source · 1 passage
Search aliases: S_8
Use in source · supplement_02.tex, line 181
The displayed manuscript row is kernel-checked. The scalar anchors are derived from the same general coefficient recurrence: (E_0,E_2,E_4,E_6,E_8)=(1,-1,5,-61,1385), (b_0,b_1,b_2,b_3,b_4)=(1,-2,16,-272,7936). The literal source sums at p=5 give (S_0,S_2,S_4,S_6,S_8)=(1,49,4705,428689,37665985). Consequently, T_2(5,8)=2 731 001 385, E_8-T_2(5,8)=-2 731 000 000. The source proves both 5^6mid E_8-T_2(5,8), 5^7nmid E_8-T_2(5,8). Thus the guaranteed power is attained in this example. No stronger unif…
Exact mathematical passage
(S_0,S_2,S_4,S_6,S_8)=(1,49,4705,428689,37665985).
S_{90}Use in source · 1 passage
Search aliases: S_90
Use in source · atlas-pages.tex, line 347
V: Theorem 11.7, Theorem 11.8, Corollary 11.9. The root finder is used only to suggest intervals. Each displayed interval is accepted by exact rational arithmetic: opposite endpoint signs prove existence, and an interval bound for D_3' excludes zero throughout the interval, proving uniqueness inside it. vfigcubic_certificates headHow the certificate works Evaluate the Taylor polynomial S_90(t)=∑_n=0^90frac(-1)^nt^3n(3n)! at the two rational endpoints. From the next term onward, magnitudes decrea…
Exact mathematical passage
S_{90}(t)=\sum_{n=0}^{90}\frac{(-1)^nt^{3n}}{(3n)!}
S_aUse in source · 2 passages
Search aliases: S_a
Use in source · 03_supplement_04.tex, line 104
Constructing the recovery on the whole ambient space. Define the syndrome projectors and their complement by S_a=V_aV_a^*, S=∑_aS_a, Q=1-S. The source proves self-adjointness, S_aS_b=δ_abS_a, S^2=S, and Q^2=Q=Q^*. It also proves QV_a=0 and V_a^*Q=0.
Exact mathematical passage
S_a=V_aV_a^*,\qquad S=\sum_aS_a,\qquad Q=1-S.
\mathcal S_aUse in source · 1 passage
Search aliases: S_a mathcal
Use in source · visual_atlas.tex, line 28
A_a, S_a · Inverse analysis and synthesis maps, which may be maps of sets. · pagerefV10
Exact mathematical passage
\mathcal A_a,\mathcal S_a
S_bUse in source · 1 passage
Search aliases: S_b
Use in source · 03_supplement_04.tex, line 107
Constructing the recovery on the whole ambient space. Define the syndrome projectors and their complement by S_a=V_aV_a^*, S=∑_aS_a, Q=1-S. The source proves self-adjointness, S_aS_b=δ_abS_a, S^2=S, and Q^2=Q=Q^*. It also proves QV_a=0 and V_a^*Q=0.
Exact mathematical passage
S_aS_b=\delta_{ab}S_a
S_{ba}Use in source · 1 passage
Search aliases: S_ba
Use in source · 02_supplement_05.tex, line 100
Hidden elimination. For the actual Schur complement S=A_RR-A_RHA_HH^-1A_HR, the hidden inverse is expanded through the hidden projectors. A nonzero retained correction block requires indices j,k with nonzero entrance and exit blocks and a hidden path jleadsto k. Therefore a nonzero block S_ba requires either a direct retained block, or a⟶ j leadsto_H k⟶ b. The entrance and exit matrices may be rectangular. Retained and hidden coordinate sizes and packet systems are independently typed. The hidde…
Exact mathematical passage
S_{ba}
S_CUse in source · 2 passages
Search aliases: S_C
Use in source · 03_packets_and_graphs.tex, line 1093
For the reverse inequality, choose for each source component C a packet pure family S_C of size r_C generating X_C under A_C, and put S=⋃_CS_C. Then every source component lies in A S. Traverse the finite condensation graph in a topological order. Suppose all components preceding a component D that is not a source have already been shown to lie in A S. Choose an incoming condensation edge from such a component to D, represented by a graph edge α→β with α already generated and β∈ D. Surjectivity …
Exact mathematical passage
S_C
s_dUse in source · 2 passages
Search aliases: s_d
Use in source · 04_arithmetic.tex, line 833
[Concrete Smith-weighted congruence solve] Let p>1 and consider the diagonal system over ( Z/(p^L) Z)^d diag(p^s_1,…,p^s_d)x=b pmodp^L. The system is solvable if and only if p^min(s_i,L)mid (b_i)_val for every i, where (b_i)_val is the canonical natural representative. Conditional on this admissibility condition, the number of solutions is p^∑_i=1^dmin(s_i,L). If any coordinate is inadmissible, the solution set is empty and its cardinality is zero.
Exact mathematical passage
\diag(p^{s_1},\ldots,p^{s_d})x=b \pmod{p^L}.
s_eUse in source · 30 passages
Search aliases: s_e
Use in source · pages.tex, line 25
s_e,ϵ_e · Sensitivity and tolerance · Is the budget input-dependent or uniform over unit inputs?
Exact mathematical passage
s_e,\epsilon_e
s_fUse in source · 3 passages
Search aliases: s_f
Use in source · Paper_V_Traditional_Final.tex, line 2552
Suppose the restriction of f to every interval (τ_q,τ_q+1) belongs to W^r,p, and choose the one-dimensional Sobolev representatives whose derivatives through order r-1 have one-sided traces at the endpoints. Define a_q,m = jumpf^(m)_τ_q = f^(m)(τ_q+)-f^(m)(τ_q-) and j_f(t) = ∑_q=1^Q∑_m=0^r-1a_q,mϕ_m,τ_q(t). Then s_f=f-j_f has matching one-sided derivatives through order r-1 at every breakpoint and belongs to W^r,p(0,1). Its weak derivative of order r equals the piecewise classical derivative f^(…
Exact mathematical passage
s_f=f-j_f
s_iUse in source · 22 passages
Search aliases: s_i
Use in source · 02_foundations.tex, line 446
Let S generate M. For every n∈ N, choose m∈ M with Φ(m)=n and write m=∑_ia_is_i. Then n=∑_iφ(a_i)Φ(s_i), so Φ(S) generates N.
Exact mathematical passage
m=\sum_ia_is_i
S_i^{\oplus m_i}Use in source · 1 passage
Search aliases: S_i^⊕ m_i oplus
Use in source · 02_foundations.tex, line 766
[Semisimple seed rank formula over arbitrary division rings] Let A≅∏_i=1^qM_d_i(D_i) be semisimple Artinian and let M≅⊕_i=1^qS_i^⊕ m_i be a finitely generated left module over A. Then srank_A(M) = max_1≤ i≤ q lceilfracm_id_irceil. If A is any algebra of finite dimension over a field, the same formula applied to the decomposition of M/Jac(A)M over A/Jac(A) computes srank_A(M). [source reference] ExportsEvery calculation of seed rank for semisimple algebras of finite dimension in the series.
Exact mathematical passage
M\cong\bigoplus_{i=1}^{q}S_i^{\oplus m_i}
S_i^{\oplus rd_i}Use in source · 1 passage
Search aliases: S_i^⊕ rd_i oplus
Use in source · 02_foundations.tex, line 786
By [source reference], M is generated by r elements exactly when it is a quotient of A^⊕ r. By [source reference], A^⊕ r≅⊕_iS_i^⊕ rd_i. Every short exact sequence of modules over a semisimple ring splits. Therefore a quotient isomorphic to M exists if and only if m_i≤ rd_i for every i. The smallest such r is the displayed maximum. For a nonsemisimple algebra of finite dimension, [source reference] reduces the calculation to the semisimple module M/Jac(A)M over A/Jac(A).
Exact mathematical passage
A^{\oplus r}\cong\bigoplus_iS_i^{\oplus rd_i}.
S_LUse in source · 6 passages
Search aliases: S_L
Use in source · Paper_X_Block27_Amended.tex, line 1270
The exceptional primes are controlled by the actual field discriminant: S_L={p rational prime:pmiddisc(L)}. This set is finite. For p∉ S_L, every prime of O_L above (p) is unramified. Lean proves that the actual inertia cardinality is the ramification index. It follows that inertia is trivial and the arithmetic Frobenius at a fixed finite prime is unique as a field automorphism. Neither trivial inertia nor uniqueness is assumed in place of the arithmetic unramifiedness condition.
Exact mathematical passage
S_L=\{p\text{ rational prime}:p\mid\operatorname{disc}(L)\}.
\mathsf S_MUse in source · 3 passages
Search aliases: S_M mathsf
Use in source · 72_recovered.tex, line 2063
[Polyphase shift matrix] Let E be the scalar shift (Ea)_n=a_n+1, and let E_q shift the coarse index. Then P_M(Ea) = S_M(E_q) P_Ma, where S_M(E_q) = 0 · 1 · 0 · ⋯ · 0
Exact mathematical passage
\boxed{ \mathcal P_M(Ea) = \mathsf S_M(E_q)\mathcal P_Ma, }
s_mUse in source · 2 passages
Search aliases: s_m
Use in source · visual_atlas.tex, line 135
headSeveral seeds give a module of relations For (s_1,…,s_m), evaluation is (f_1,…,f_m)↦∑_j f_j(T)s_j. Its kernel is a submodule of F^m and can include relations between seeds, not only relations annihilating each seed separately.
Exact mathematical passage
(s_1,\ldots,s_m)
S_{m+r}Use in source · 1 passage
Search aliases: S_m+r
Use in source · 05_worked_models.tex, line 204
Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. Therefore sequences with finite support lie in its algebraic orbit hull and are dense in ℓ^2( Z).
Exact mathematical passage
S_{m+r}=S_mS_r
s_{m-1}Use in source · 1 passage
Search aliases: s_m-1
Use in source · Paper_VI_Traditional_Final.tex, line 2236
[Dyson expansion over packet paths] For t≥0, e^tA = · e^tD + ∑_m=1^∞ ∫_0<s_1<⋯<s_m<t e^(t-s_m)DN e^(s_m-s_m-1)D ⋯ N e^s_1D d s_1⋯ d s_m. The series converges absolutely in operator norm. After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.
Exact mathematical passage
\begin{aligned} \e^{tA} ={}&\e^{tD} + \sum_{m=1}^{\infty} \int_{0<s_1<\cdots<s_m<t} \e^{(t-s_m)D}N\e^{(s_m-s_{m-1})D} \cdots N\e^{s_1D} \dd s_1\cdots\dd s_m. \end{aligned}
S_NUse in source · 2 passages
Search aliases: S_N
Use in source · visual_atlas.tex, line 250
vpage11The residual tells you when to stopMODEL B / A CERTIFIED CUTOFF, NOT A HEURISTICChapter II, mathematical text: Theorem 5.10, Corollary 5.11, Section B.2 For S_N=∑_k=0^N-1(-A)^k, telescoping gives (I+A)S_N=S_N(I+A)=I-(-A)^N. Thus the exact residual is known. To certify precision L, it is enough to show that every residual entry is divisible by 2^L.
Exact mathematical passage
S_N=\sum_{k=0}^{N-1}(-A)^k
\mathcal S_NUse in source · 2 passages
Search aliases: S_N mathcal
Use in source · visual_atlas.tex, line 27
A_N, S_N · Split into range-restricted packet vectors, then sum them. · A permutation/interleaving of eight coordinates.
Exact mathematical passage
\mathcal A_N,\mathcal S_N
s_NUse in source · 31 passages
Search aliases: s_N
Use in source · Paper_IV_Traditional_Final.tex, line 1842
At a tie direction exactly two adjacent vertices attain the maximum. After rotation it is enough to take θ=0. With c_N=cos(π/N) and s_N=sin(π/N), the two dominant terms give frac2N e^c_Nrcos(s_Nr) and every other term is exponentially smaller. Along a sequence on which |cos(s_Nr)|≥1/2, the lower indicator bound is c_N=h_N(0). Rotational symmetry treats every tie direction.
Exact mathematical passage
s_N=\sin(\pi/N)
s_nUse in source · 1 passage
Search aliases: s_n
Use in source · supplement_05.tex, line 44
First construct the integral solution. For a ring R, a triangular row system T, right-hand side b, and proved unit pivots d_n=T_nn, define s_n=d_n^-1(b_n-∑_q<nT_nqs_q). The new ring-level solver uses well-founded recursion. Its row identities and unique-solution property are proved. The source also proves compatibility of this solver with ring homomorphisms into fields.
Exact mathematical passage
s_n=d_n^{-1}\left(b_n-\sum_{q<n}T_{nq}s_q\right).
S_{n-2m}Use in source · 1 passage
Search aliases: S_n-2m
Use in source · supplement_02.tex, line 131
Substituting the independently proved signed-zigzag identity recovers the manuscript's notation. For every even degree n, E_n=∑_m=0^n/2(-1)^m Z_2m+1binom n2mp^2mS_n-2m(p). The Lean version uses n=2r to avoid ambiguous natural-number division. Every even natural degree has that form. Primality is not used in this equality.
Exact mathematical passage
\boxed{E_n=\sum_{m=0}^{n/2}(-1)^m Z_{2m+1}\binom n{2m}p^{2m}S_{n-2m}(p).}
S_{n-2r}Use in source · 3 passages
Search aliases: S_n-2r
Use in source · 72_recovered.tex, line 965
[Odd modulus Euler coordinate pair] For every positive odd q and every even n≥0, E_n = ∑_r=0^n/2 (-1)^rZ_2r+1 binom n2r q^2rS_n-2r(q). The inverse transform is S_n(q) = E_n + ∑_r=1^n/2 2^2r-1 binom n2r q^2rE_n-2r. Both transforms are lower triangular with diagonal one.
Exact mathematical passage
\boxed{ E_n = \sum_{r=0}^{n/2} (-1)^rZ_{2r+1} \binom n{2r} q^{2r}S_{n-2r}(q). }
S_{n-2r}^\timesUse in source · 1 passage
Search aliases: S_n-2r^× times
Use in source · 72_recovered.tex, line 1825
[Depleted odd prime reconstruction] For every even n, (1-σ_pp^n)E_n = ∑_r=0^n/2 (-1)^rZ_2r+1 binom n2rp^2rS_n-2r^×(p).
Exact mathematical passage
\boxed{ \left(1-\sigma_pp^n\right)E_n = \sum_{r=0}^{n/2} (-1)^rZ_{2r+1} \binom n{2r}p^{2r}S_{n-2r}^\times(p). }
S_{n-r}Use in source · 6 passages
Search aliases: S_n-r
Use in source · 10_foundations.tex, line 215
[Filtered finite reconstruction] For every n,L∈ N and sequences c,S in that ring, · ∑_r=0^nbinom nr c_rp^[w(r)]S_n-r · ≡ ∑_substack0≤ r≤ n w(r)<L binom nr c_rp^[w(r)]S_n-rpmodp^L. Here congruence means divisibility of the difference by p^L in the coefficient ring. In Z_p, it means membership in p^L Z_p.
Exact mathematical passage
\begin{split} &\sum_{r=0}^{n}\binom nr c_rp^{[w(r)]}S_{n-r}\\ &\hspace{8mm}\equiv \sum_{\substack{0\leq r\leq n\\w(r)<L}} \binom nr c_rp^{[w(r)]}S_{n-r}\pmod{p^L}. \end{split}
S_n^\timesUse in source · 1 passage
Search aliases: S_n^× times
Use in source · 72_recovered.tex, line 1819
For an odd prime p, the source S_n(p) contains one nonunit base. Put σ_p=(-1)^(p-1)/2 and S_n^×(p)=S_n(p)-σ_pp^n.
Exact mathematical passage
S_n^\times(p)=S_n(p)-\sigma_pp^n.
S_pUse in source · 1 passage
Search aliases: S_p
Use in source · 30_lifting.tex, line 15
Unless a different coefficient ring is specified, a polynomial system is a finite list of integer polynomials. All divisibility conditions on vectors are coordinatewise. For a positive integer q, write S_q= Z/q Z and use an overline for reduction modulo q. For a prime p, write F_p=S_p, normalize |p|_p=p^-1, and put B_L(x)=x+p^L Z_p^d. An integer center is included in this notation by its natural image in Z_p^d. All dimensions below are finite and may be zero. A statement over S_q allows composit…
Exact mathematical passage
\F_p=S_p
\mathcal S_pUse in source · 2 passages
Search aliases: S_p mathcal
Use in source · 20_arithmetic.tex, line 814
The exponential generating series of the source is the finite sum S_p(t)=∑_a=0^p-1(-1)^a e^(2a+1)t. Multiplication by cosh t telescopes. Since p is odd, S_p(t)=frac1+e^2pt2secht. Taking its even part gives ∑_n≥0S_2n(p)fract^2n(2n)! =cosh^2(pt)secht. Multiplication by the reciprocal sech^2(pt) yields secht. The EGF product rule now gives the reconstruction coefficient by coefficient. Replacing m by n-q produces the displayed lower triangular row. Its diagonal and weights follow from γ_0=1 and the…
Exact mathematical passage
\mathcal S_p(t)=\sum_{a=0}^{p-1}(-1)^a e^{(2a+1)t}.
s_pUse in source · 2 passages
Search aliases: s_p
Use in source · 72_recovered.tex, line 1039
[Odd row weight] For an odd prime p, the rth nonzero term in the reconstruction has valuation W_p(n,r) = 2r+v_p(Z_2r+1) +v_pbinom n2r +v_p(S_n-2r(p)). The binomial contribution is v_pbinom n2r = fracs_p(2r)+s_p(n-2r)-s_p(n)p-1, where s_p is the base p digit sum.
Exact mathematical passage
\boxed{ \vp\binom n{2r} = \frac{s_p(2r)+s_p(n-2r)-s_p(n)}{p-1}, }
S_p^eUse in source · 1 passage
Search aliases: S_p^e
Use in source · continuation_09.tex, line 120
For every proposed next parameter digit, the previously checked obstruction is [bar r+K_av]∈ S_p^e/imJ_a. The fixed solved Jacobian is surjective, so this class vanishes for every digit at every stage. The actual lift fiber is a torsor for ker J_a. Rank-nullity gives its cardinality p^m-e.
Exact mathematical passage
[\bar r+K_av]\in S_p^e/\operatorname{im}J_a
S_{\mathrm{packets}}Use in source · 1 passage
Search aliases: S_packets mathrm
Use in source · visual_atlas.tex, line 481
headWhat one shift does in these coordinates Let S=S_1 and z=Sx. Substituting 3k+j into the shift definition gives z_1[k]=x_0[k], z_2[k]=x_1[k], z_0[k]=x_2[k-1]. The passage from phase 2 to phase 0 carries a shift of the coarse address. If Lh[k]=h[k-1], the exact packet matrix is S_packets=mat0 · 0 · L I · 0 · 0 0 · I · 0. Replacing L by a scalar 1 would preserve the phase graph but would lose the operator's action on the infinite sequences.
Exact mathematical passage
S_{\mathrm{packets}}=\mat{0&0&L\\I&0&0\\0&I&0}.
S_qUse in source · 4 passages
Search aliases: S_q
Use in source · 30_lifting.tex, line 13
Unless a different coefficient ring is specified, a polynomial system is a finite list of integer polynomials. All divisibility conditions on vectors are coordinatewise. For a positive integer q, write S_q= Z/q Z and use an overline for reduction modulo q. For a prime p, write F_p=S_p, normalize |p|_p=p^-1, and put B_L(x)=x+p^L Z_p^d. An integer center is included in this notation by its natural image in Z_p^d. All dimensions below are finite and may be zero. A statement over S_q allows composit…
Exact mathematical passage
S_q=\Z/q\Z
s_qUse in source · 1 passage
Search aliases: s_q
Use in source · supplement_05.tex, line 44
First construct the integral solution. For a ring R, a triangular row system T, right-hand side b, and proved unit pivots d_n=T_nn, define s_n=d_n^-1(b_n-∑_q<nT_nqs_q). The new ring-level solver uses well-founded recursion. Its row identities and unique-solution property are proved. The source also proves compatibility of this solver with ring homomorphisms into fields.
Exact mathematical passage
s_n=d_n^{-1}\left(b_n-\sum_{q<n}T_{nq}s_q\right).
S_q^dUse in source · 7 passages
Search aliases: S_q^d
Use in source · 30_lifting.tex, line 28
For F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, define its evaluated Jacobian by the formal partial derivatives: DF(x)_ij=(∂_jF_i)(x), J_x=DF(x)bmod q:S_q^d⟶ S_q^e. The obstruction therefore uses the Jacobian of the given polynomials. The following algebraic Taylor identity relates this linear map to polynomial evaluation.
Exact mathematical passage
DF(x)_{ij}=(\partial_jF_i)(x),\qquad J_x=DF(x)\bmod q:S_q^d\longrightarrow S_q^e.
S_q^eUse in source · 10 passages
Search aliases: S_q^e
Use in source · 30_lifting.tex, line 28
For F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, define its evaluated Jacobian by the formal partial derivatives: DF(x)_ij=(∂_jF_i)(x), J_x=DF(x)bmod q:S_q^d⟶ S_q^e. The obstruction therefore uses the Jacobian of the given polynomials. The following algebraic Taylor identity relates this linear map to polynomial evaluation.
Exact mathematical passage
DF(x)_{ij}=(\partial_jF_i)(x),\qquad J_x=DF(x)\bmod q:S_q^d\longrightarrow S_q^e.
S_q^fUse in source · 1 passage
Search aliases: S_q^f
Use in source · 30_lifting.tex, line 216
[Simultaneous polynomial lifting] Let F∈ Z[X_1,…,X_d]^e and G∈ Z[X_1,…,X_d]^f have a common lower root x modulo q^L, where q≥1 and L≥1. Use the integer-domain reduced Jacobians J: Z^d→ S_q^e and K: Z^d→ S_q^f. Suppose u,v∈ Z^d are separately valid corrections for F and G, respectively. Then [u]=[v] in Z^d/(ker J+ker K) if and only if a single y≡ xpmodq^L solves both systems modulo q^L+1. Each separate label is independent of the chosen valid correction. The common quotient is finite.
Exact mathematical passage
K:\Z^d\to S_q^f
S_q^kUse in source · 3 passages
Search aliases: S_q^k
Use in source · 30_lifting.tex, line 539
Suppose now that the split into solved variables and parameter variables is fixed in advance. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0), and assume F(x)=q^Lr_x with q≥1 and L≥1. Write the reduced Jacobian as DF(x)bmod q=(J_x K_x), J_x:S_q^m→ S_q^e, K_x:S_q^k→ S_q^e. For a prescribed parameter digit v∈ S_q^k, let L_x,L,v be the set of next residue roots over x whose parameter coordinates are t_0+q^Lwidetilde v modulo q^L+1.
Exact mathematical passage
DF(x)\bmod q=(J_x\ K_x),\qquad J_x:S_q^m\to S_q^e,\quad K_x:S_q^k\to S_q^e.
S_q^mUse in source · 4 passages
Search aliases: S_q^m
Use in source · 30_lifting.tex, line 539
Suppose now that the split into solved variables and parameter variables is fixed in advance. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0), and assume F(x)=q^Lr_x with q≥1 and L≥1. Write the reduced Jacobian as DF(x)bmod q=(J_x K_x), J_x:S_q^m→ S_q^e, K_x:S_q^k→ S_q^e. For a prescribed parameter digit v∈ S_q^k, let L_x,L,v be the set of next residue roots over x whose parameter coordinates are t_0+q^Lwidetilde v modulo q^L+1.
Exact mathematical passage
DF(x)\bmod q=(J_x\ K_x),\qquad J_x:S_q^m\to S_q^e,\quad K_x:S_q^k\to S_q^e.
S_RUse in source · 1 passage
Search aliases: S_R
Use in source · 02_supplement_06.tex, line 10
Compact factors from actual column spaces. textitModules CompactRangeFactors and SmallCoreTransfer Let L∈ C^m× t, R∈ C^n× t, A=LR^*. Use the ordinary Euclidean inner products on the coordinate spaces. Define S_L=range(L), r_L=dim S_L, S_R=range(R), r_R=dim S_R. The implementation constructs an orthonormal basis of each actual range subspace. Let Q_L and Q_R be the matrices of the corresponding isometric inclusions. Express the range-restricted input maps in these bases to obtain R_L,R_R.
Exact mathematical passage
S_L=\operatorname{range}(L),\quad r_L=\dim S_L, \qquad S_R=\operatorname{range}(R),\quad r_R=\dim S_R.
S_rUse in source · 3 passages
Search aliases: S_r
Use in source · 05_worked_models.tex, line 204
Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. Therefore sequences with finite support lie in its algebraic orbit hull and are dense in ℓ^2( Z).
Exact mathematical passage
S_{m+r}=S_mS_r
S_VUse in source · 2 passages
Search aliases: S_V
Use in source · Paper_VI_Traditional_Final.tex, line 4918
[State dependent killing as survival] Let Q be a finite Markov generator, let V≥0, and set A=Q-D_V. For an initial distribution p_0, define S_V(t)= 1^T e^tAp_0. Then S_V(t) = E_p_0 [ exp( -∫_0^tV(X_s) d s ) ]. Also, -S_V'(t) = 1^TD_V e^tAp_0. The matrix A=Q-D_V has nonnegative off-diagonal entries and nonpositive column sums, so it is the subgenerator of the killed, possibly nontransient chain. Thus the potential in the finite Feynman--Kac formula is a state dependent killing rate in the forward…
Exact mathematical passage
S_V(t)=\one^T\e^{tA}p_0.
S_V'Use in source · 2 passages
Search aliases: S_V'
Use in source · Paper_VI_Traditional_Final.tex, line 4933
[State dependent killing as survival] Let Q be a finite Markov generator, let V≥0, and set A=Q-D_V. For an initial distribution p_0, define S_V(t)= 1^T e^tAp_0. Then S_V(t) = E_p_0 [ exp( -∫_0^tV(X_s) d s ) ]. Also, -S_V'(t) = 1^TD_V e^tAp_0. The matrix A=Q-D_V has nonnegative off-diagonal entries and nonpositive column sums, so it is the subgenerator of the killed, possibly nontransient chain. Thus the potential in the finite Feynman--Kac formula is a state dependent killing rate in the forward…
Exact mathematical passage
-S_V'(t) = \one^TD_V\e^{tA}p_0.
S_wUse in source · 6 passages
Search aliases: S_w
Use in source · residue_continuation_body.tex, line 145
e preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl( O)→ C, define the actual finite ideal sum S_w(X)=∑_substackI≠0 N I≤ Xw([I]). Exact grouping by ideal class gives S_w(X)=∑_C w(C)A_C(X). If ∑_Cw(C)=0, the common leading coefficient cancels, so there is B_w≥0 s…
Exact mathematical passage
S_w(X)=\sum_{\substack{I\ne0\\\Nm I\leq X}}w([I]).
S_zUse in source · 7 passages
Search aliases: S_z
Use in source · Paper_VI_Traditional_Final.tex, line 4506
0 · I and S_z= F_P(z) · 0 -A_QP · zI-A_QQ . Direct multiplication gives zI-A=L_zS_z. The first factor is invertible. The claims follow from triangular inversion and the determinant of the second factor. This is the finite block form of the Feshbach reduction [source reference].
Exact mathematical passage
S_z= \begin{pmatrix} F_P(z)&0\\ -A_{QP}&zI-A_{QQ} \end{pmatrix}.
S_z^{-1}Use in source · 1 passage
Search aliases: S_z^-1
Use in source · 03_supplement_01.tex, line 114
Correct statement: source Theorem 9.1. Assume both D_z and zI-H are invertible. Then S_z is invertible and ((zI-H)^-1)_PP=S_z^-1. The left side is the submatrix of the full inverse on the retained coordinates. This matches the paper's compressed operator when written in an orthonormal basis of the retained subspace. The proof is valid for arbitrary complex blocks of compatible sizes and does not require Hermiticity.
Exact mathematical passage
\boxed{\bigl((zI-H)^{-1}\bigr)_{PP}=S_z^{-1}.}
S_\alphaUse in source · 3 passages
Search aliases: S_α alpha
Use in source · 03_packets_and_graphs.tex, line 1053
Every packet pure generating family is an unrestricted generating family, which gives the first inequality. For each α, choose a minimal generating family S_α⊆ X_α for the corner action P_α A P_α. The union S=⋃_α S_α is packet pure. Since every x∈ X has the unique decomposition x=∑_α x_α and every x_α lies in (P_α A P_α)S_α⊆ AS, the union generates X. Its cardinality is at most the displayed sum.
Exact mathematical passage
S_\alpha\subseteq X_\alpha
S_{\beta\alpha}Use in source · 2 passages
Search aliases: S_βα beta alpha
Use in source · 03_packets_and_graphs.tex, line 601
[Tensor product transfer blocks] Let S∈End(X) and T∈End(Y), with packet systems P and Q. Relative to the tensor product packet system of [source reference], (S⊗ T)_(β,δ),(α,γ) = S_βα⊗ T_δγ. [source reference] ExportsTensor network, quantum circuit, and multivariate packet calculations.
Exact mathematical passage
\boxed{ (S\otimes T)_{(\beta,\delta),(\alpha,\gamma)} = S_{\beta\alpha}\otimes T_{\delta\gamma}. }
S_{\gamma\beta}Use in source · 2 passages
Search aliases: S_γβ gamma beta
Use in source · 03_packets_and_graphs.tex, line 532
[Packet matrix multiplication] For S,T∈End_𝕜(X), (ST)_γα = ∑_β∈ IS_γβT_βα. [source reference] ExportsAll packet path expansions.
Exact mathematical passage
\boxed{ (ST)_{\gamma\alpha} = \sum_{\beta\in I}S_{\gamma\beta}T_{\beta\alpha}. }
S_{\gamma\delta}Use in source · 3 passages
Search aliases: S_γδ gamma delta
Use in source · 03_packets_and_graphs.tex, line 626
[Blockwise intertwining equations] Let T∈End(X), S∈End(Y), let P=(P_α)_α∈ I and Q=(Q_γ)_γ∈ J be finite packet systems, and let U:X→ Y be linear. Define T_βα=P_β TP_α, S_γδ=Q_γ SQ_δ, U_γα=Q_γ UP_α. Then UT=SU if and only if, for every γ∈ J and α∈ I, ∑_β∈ IU_γβT_βα = ∑_δ∈ JS_γδU_δα. [source reference] ExportsPacketwise equivalence, gluing, symmetry, and local representation comparison.
Exact mathematical passage
T_{\beta\alpha}=P_\beta TP_\alpha, \qquad S_{\gamma\delta}=Q_\gamma SQ_\delta, \qquad U_{\gamma\alpha}=Q_\gamma UP_\alpha.
s_{\Lambda_s}Use in source · 1 passage
Search aliases: s_Λ_s Lambda
Use in source · 04_arithmetic.tex, line 522
with two lattices] Let V=Ke_1⊕ Ke_2 and let E project onto Ke_1. On the split lattice Λ_0= O e_1⊕ O e_2, the weight is zero. Fix s≥1 and consider Λ_s = O e_1+ O(π^-se_1+e_2). Then Λ_s∩ Ke_1= O e_1, Λ_s∩ Ke_2=π^s O e_2. The gluing module is isomorphic to O/π^s O, so s_Λ_s(E)=s. The rational decomposition has not changed. Only the lattice has changed.
Exact mathematical passage
s_{\Lambda_s}(E)=s.
s_\ellUse in source · 1 passage
Search aliases: s_ℓ ell
Use in source · 72_recovered.tex, line 2018
[Nested discrete refinement] For positive integers M,N, (a_r^[M])_s^[N] = a_r+Ms^[MN]. For a mixed radix path (M_1,…,M_ℓ) with digits 0≤ s_i<M_i, the final address is r( s) = s_1+M_1s_2+⋯+ (∏_i=1^ℓ-1M_i)s_ℓ. Only the product radix and accumulated address determine the final branch.
Exact mathematical passage
\boxed{ r(\boldsymbol s) = s_1+M_1s_2+\cdots+ \left(\prod_{i=1}^{\ell-1}M_i\right)s_\ell. }
\Scal_{\Vcal}Use in source · 1 passage
Search aliases: Scal_Vcal S_ V Scal Vcal mathcal mathcal
Use in source · 03_nonlinear_coordinates.tex, line 170
[Change of packet coordinates] Let P and P' be packet systems on X. Let Q and Q' be packet systems on Y. For any two packet systems U and V on the same space, write J_ U← V = A_ U S_ V. Then F_ Q'← P' = J_ Q'← Q F_ Q← P J_ P← P'.
Exact mathematical passage
J_{\Ucal\leftarrow\Vcal} = \Acal_{\Ucal}\Scal_{\Vcal}.
\secUse in source · 9 passages
Search aliases: sec
Use in source · Paper_IV_Traditional_Final.tex, line 948
A unit constant term makes the formal reciprocal unambiguous. To compute one coefficient, only finitely many earlier coefficients are needed. The later analytic theory explains how those exact coefficients relate to the full zero set. modelfigurereciprocal_normalizationAt N=2 and r=1, the reciprocal is sec z. The upper row gives block exponential coefficients and the lower row gives ordinary Taylor coefficients. Both rows encode the same five terms.fig:reciprocal-normalization
Exact mathematical passage
\sec z
\sec^2Use in source · 3 passages
Search aliases: sec^2 sec
Use in source · 20_arithmetic.tex, line 790
The same inverse coefficients can be written in the signed zigzag notation of the original manuscript. Define Z_r by sec t+tan t=∑_r≥0Z_rfract^rr!. Then γ_m=(-1)^m Z_2m+1 (m∈ N). Indeed, the odd coefficients come from tan t, whose derivative is sec^2t. Changing the sign of the squared variable in the even series for sech^2t gives sec^2t, so its degree-2m Hurwitz coefficient is (-1)^mγ_m. Differentiation identifies this coefficient with Z_2m+1, proving the formula. The Lean bridge is leanKomorebi…
Exact mathematical passage
\sec^2t
\sech^2Use in source · 3 passages
Search aliases: sech^2 sech operatorname
Use in source · supplement_02.tex, line 90
The inverse coefficients are integral by construction. Define the integer binomial convolution B_n=∑_r=0^nbinom nr E_rE_n-r, b_m=B_2m. The product formula for actual EGFs gives ∑_n≥0B_nfract^nn!=sech^2 t. Odd coefficients vanish by parity. In particular, every b_m is an integer. This is the integrality required by the later congruence.
Exact mathematical passage
\sum_{n\ge0}B_n\frac{t^n}{n!}=\sech^2 t.
\mathrm{setup}Use in source · 1 passage
Search aliases: setup mathrm
Use in source · 02_supplement_02.tex, line 29
Shared operator families. For reuse weights r_j, the local keep model can aggregate setup and per-operator costs: κ_v=κ_v^setup+∑_jr_jκ_v^(j), ϵ_v=∑_jr_jϵ_v^(j). Lean proves the aggregation identity at the cut level, including split overhead, before applying the compiler theorem. The proof is valid over an ordered field, including Q and R. The intended nonnegative model is a specialization.
Exact mathematical passage
\kappa_v=\kappa_v^{\mathrm{setup}}+\sum_jr_j\kappa_v^{(j)}, \qquad \epsilon_v=\sum_jr_j\epsilon_v^{(j)}.
\Sfun_N^{(1)}Use in source · 1 passage
Search aliases: Sfun_N^(1) S_N^(1) Sfun mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 321
acz^Nn+j(Nn+j)! is the residue j part of the exponential. Second, D_N,j(z) = ∑_n≥0(-1)^nfracz^Nn+j(Nn+j)! is the signed basis for y^(N)=-y. The section D_N= D_N,0 is the denominator used in the reciprocal family S_N^(r)= D_N^-r. Set T_N=- D_N^prime/ D_N. Equivalently, T_N is the logarithmic derivative of S_N^(1).
Exact mathematical passage
\Sfun_N^{(1)}
\Sfun_N^{(r+s)}Use in source · 2 passages
Search aliases: Sfun_N^(r+s) S_N^(r+s) Sfun mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 1065
[Additive parameter law for reciprocal powers] For all r,s∈ C, S_N^(r)(z) S_N^(s)(z) = S_N^(r+s)(z) as a formal identity and on every common zero-free region with a fixed logarithm. Consequently, P_N,n(r+s) = ∑_q=0^n binomNnNq P_N,q(r)P_N,n-q(s).
Exact mathematical passage
\Sfun_N^{(r)}(z)\Sfun_N^{(s)}(z) = \Sfun_N^{(r+s)}(z)
\Sfun_N^{(s)}Use in source · 2 passages
Search aliases: Sfun_N^(s) S_N^(s) Sfun mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 1065
[Additive parameter law for reciprocal powers] For all r,s∈ C, S_N^(r)(z) S_N^(s)(z) = S_N^(r+s)(z) as a formal identity and on every common zero-free region with a fixed logarithm. Consequently, P_N,n(r+s) = ∑_q=0^n binomNnNq P_N,q(r)P_N,n-q(s).
Exact mathematical passage
\Sfun_N^{(r)}(z)\Sfun_N^{(s)}(z) = \Sfun_N^{(r+s)}(z)
\mathrm{sgn}Use in source · 2 passages
Search aliases: sgn mathrm
Use in source · Paper_XII_Traditional_Final.tex, line 423
The sign witness is W_sgn=Z, while W_std=W(6,4,3)=3A+3B belongs to the standard summand. The corresponding projectors and trace pairings are P_t,2=2 · 4
Exact mathematical passage
W_{\mathrm{sgn}}=Z
\operatorname{sgn}Use in source · 2 passages
Search aliases: sgn operatorname
Use in source · Paper_XII_Traditional_Final.tex, line 387
[Finite adequacy certificate] The standard representation of S_3 over F_7 is adequate. More precisely, H^0(S_3,ad^0V)=0, H^1(S_3, F_7)=0, H^1(S_3,ad^0V)=0, and ad^0V≅sgn⊕ V. The sign summand is detected by an eigenprojector of a three cycle, and the standard summand is detected by an eigenprojector of a transposition.
Exact mathematical passage
\ad^0V\cong\operatorname{sgn}\oplus V.
\sgnUse in source · 1 passage
Search aliases: sgn operatorname
Use in source · Paper_V_Traditional_Final.tex, line 2991
[Jump count and sign recovery] Under the hypotheses of [source reference], the number of detected cells equals Q, and sgn(widehat a_q)=sgn(a_q) for every real valued jump.
Exact mathematical passage
\sgn(\widehat a_q)=\sgn(a_q)
\operatorname{Si}Use in source · 4 passages
Search aliases: Si operatorname
Use in source · Paper_V_Traditional_Final.tex, line 3387
[Classical square-wave Gibbs behavior] Let q(x)=tfrac12sign(sin x), whose jump at zero has size one. For n≥1, its partial sum through the first n positive odd frequencies is Q_n(x)=frac2π∑_k=0^n-1fracsin((2k+1)x)2k+1. At x_n=π/(2n), x_n⟶0, Q_n(x_n)⟶fracSi(π)π =frac12+G, G≈0.089489872236, where Si(z)=∫_0^zsin(t)/t dt and G=Si(π)/π-1/2>0. Thus a nonzero overshoot persists along points approaching the jump. Scaling by a real amplitude a gives an overshoot or undershoot of magnitude |a|G.
Exact mathematical passage
x_n\longrightarrow0,\qquad Q_n(x_n)\longrightarrow\frac{\operatorname{Si}(\pi)}{\pi} =\frac12+G,\qquad G\approx0.089489872236,
\mathrm{sign}Use in source · 1 passage
Search aliases: sign mathrm
Use in source · Paper_X_Block27_Amended.tex, line 1039
0 · -1) and τ(g)=diag(1,-1). It preserves the residual comparison and has determinant one. Pullback through the quadratic character of the actual G_ Q gives continuous arithmetic examples. In contrast, the identity and diag(1,sign) over Z/4 agree modulo two but are not integrally conjugate. Their group order is not a unit. These restricted results do not supply general arithmetic packet attachment, all finite-level existence statements, or cross-coefficient compatibility.
Exact mathematical passage
\operatorname{diag}(1,\mathrm{sign})
\operatorname{sign}Use in source · 1 passage
Search aliases: sign operatorname
Use in source · Paper_V_Traditional_Final.tex, line 3382
[Classical square-wave Gibbs behavior] Let q(x)=tfrac12sign(sin x), whose jump at zero has size one. For n≥1, its partial sum through the first n positive odd frequencies is Q_n(x)=frac2π∑_k=0^n-1fracsin((2k+1)x)2k+1. At x_n=π/(2n), x_n⟶0, Q_n(x_n)⟶fracSi(π)π =frac12+G, G≈0.089489872236, where Si(z)=∫_0^zsin(t)/t dt and G=Si(π)/π-1/2>0. Thus a nonzero overshoot persists along points approaching the jump. Scaling by a real amplitude a gives an overshoot or undershoot of magnitude |a|G.
Exact mathematical passage
q(x)=\tfrac12\operatorname{sign}(\sin x)
\sinUse in source · 89 passages
Search aliases: sin
Use in source · 05_worked_models.tex, line 345
quently one is the least finite number of adjacency seeds: 1=min{r∈ N: ∃ s_1,…,s_r, 𝕜^n=span_𝕜{p(A_P_n)s_j:p∈𝕜[t], 1≤ j≤ r} }. Over C, the familiar spectral description has the distinct eigenvalues λ_k=2cosfrackπn+1, 1≤ k≤ n, and the normalized eigenvectors have endpoint coordinate proportional to sinfrackπn+1≠0. There is also a combinatorial proof valid over every field. Let K_n=[e_1,Ae_1,…,A^n-1e_1]. Its (i,j) entry counts walks of length j-1 from the first vertex to vertex i. This entry is ze…
Exact mathematical passage
\sin\frac{k\pi}{n+1}\ne0.
\sin^2Use in source · 22 passages
Search aliases: sin^2 sin
Use in source · supplement_02.tex, line 101
Connecting to the chapter's Z_n convention. The chapter defines Z_n through sec t+tan t. The Lean construction uses Mathlib's actual formal sine and cosine series and defines sec t=(cos t)^-1, tan t=(sin t)sec t, Z_n=n. It proves the sine and cosine derivative identities, sin^2t+cos^2t=1, and (tan t)'=sec^2t. Sign substitution in the squared variable carries the hyperbolic cosine kernel to the actual cosine kernel. The same homomorphism carries their inverses and their squares…
Exact mathematical passage
\sin^2t+\cos^2t=1
\sinh_aUse in source · 1 passage
Search aliases: sinh_a sinh
Use in source · supplement_03.tex, line 73
. targetDyadic.kernel_coefficient_integral, Dyadic.beta_integral targetDyadic.source_valuation, Dyadic.sourcePadic_isUnit pageheading2. Connecting the definitions to actual formal seriesGenerating-function foundations for Theorem 5.3 Work in Q[[t]]. The formal exponential and rescaling homomorphism are the same checked objects used in the second block. Define sinh_a(t)=frace^at-e^-at2, cosh_a(t)=frace^at+e^-at2. Let B(t)=K(t^2)^-1=∑_m≥0β_mt^2m and let F(t)=∑_r≥0R_2rfract^2r(2r)!. These series ar…
Exact mathematical passage
\sinh_a(t)=\frac{e^{at}-e^{-at}}2,\qquad \cosh_a(t)=\frac{e^{at}+e^{-at}}2.
\smallUse in source · 1 passage
Search aliases: small
Use in source · visual_atlas.tex, line 19
[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…
Exact mathematical passage
4pt] \textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt] \textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology? \end{tabularx} A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. \takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''} \vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}} Most of the chapter can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix. {\small \begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule \textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule $X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\ $S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\ $\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\ $\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\ $P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\ $X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\ $T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\ $F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\ $U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\ $\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\ $\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule \end{tabularx}} \head{Read products from right to left} In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column. \head{Where to begin} For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}. \vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$} Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define \[ Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0, \qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.
\mathrm{Smith}Use in source · 1 passage
Search aliases: Smith mathrm
Use in source · 06_models.tex, line 79
The symmetric weight w_Smith(Γ) = frac1|1-Γ|^2 + frac1|1+Γ|^2 indicates where small errors in reflection coordinates produce large impedance or admittance errors.
Exact mathematical passage
w_{\mathrm{Smith}}(\Gamma) = \frac1{|1-\Gamma|^2} + \frac1{|1+\Gamma|^2}
\operatorname{Sol}Use in source · 2 passages
Search aliases: Sol operatorname
Use in source · 71_native.tex, line 210
N=∏_p∈ Sp^L_p, Sol_F(R)={x∈ R^d:F(x)=0}.
Exact mathematical passage
N=\prod_{p\in S}p^{L_p},\qquad\operatorname{Sol}_F(R)=\{x\in R^d:F(x)=0\}.
\operatorname{Sol}_FUse in source · 2 passages
Search aliases: Sol_F operatorname
Use in source · 71_native.tex, line 210
N=∏_p∈ Sp^L_p, Sol_F(R)={x∈ R^d:F(x)=0}.
Exact mathematical passage
N=\prod_{p\in S}p^{L_p},\qquad\operatorname{Sol}_F(R)=\{x\in R^d:F(x)=0\}.
\mathrm{source}Use in source · 1 passage
Search aliases: source mathrm
Use in source · visual_atlas.tex, line 289
headThe first-edge decomposition Let a_n(s,t) count walks of total weight n from s to t. Removing the first edge gives a_n(s,t)=1_s=t, n=0 +∑_substacke: source(e)=s w(e)≤ n a_n-w(e)(target(e),t). The empty walk is allowed precisely on the diagonal at weight zero. Positive weights ensure that every contributing walk has length at most n, so enumeration at each fixed weight is finite.
Exact mathematical passage
a_n(s,t)=\mathbf1_{s=t,\,n=0} +\sum_{\substack{e:\,\mathrm{source}(e)=s\\w(e)\le n}} a_{n-w(e)}(\mathrm{target}(e),t).
\operatorname{source}Use in source · 1 passage
Search aliases: source operatorname
Use in source · 03_weighted.tex, line 499
[Automatic finite-graph transfer-zeta rationality] Assume the vertex and edge sets are finite and every edge has positive integer weight. For every n, exact enumeration of edge tuples of lengths at most n, followed by endpoint and total-weight filtering, returns a_n(s,t). These coefficients obey the recurrence obtained by removing the first edge a_n(s,t) = 1_s=t, n=0 +∑_substacke: source(e)=s w(e)≤ n a_n-w(e)(target(e),t). Let T(t)=I-A(t), Q(t)=det T(t), P_s,t(t)=adj(T(t))_s,t. Then Q(0)=1, henc…
Exact mathematical passage
a_n(s,t) = \mathbf 1_{s=t,\,n=0} +\sum_{\substack{e:\,\operatorname{source}(e)=s\\w(e)\le n}} a_{n-w(e)}(\operatorname{target}(e),t).
\SpUse in source · 3 passages
Search aliases: Sp operatorname
Use in source · Paper_XI_Traditional_Final.tex, line 1288
Let p be odd, let (V,b) be a finite symplectic F_p space of dimension 2r, and let g∈Sp(V) have no fixed vector. Fix one of the standard finite Weil normalizations used in the literature [source reference].
Exact mathematical passage
g\in\Sp(V)
\operatorname{Sp}Use in source · 2 passages
Search aliases: Sp operatorname
Use in source · Paper_XI_Traditional_Final.tex, line 237
[Indecomposable monodromy blocks] Every Frobenius semisimple Weil Deligne representation is a direct sum of blocks Sp_m(τ), where τ is an irreducible Weil representation and the nilpotent operator is one Jordan chain of length m.
Exact mathematical passage
\operatorname{Sp}_m(\tau),
\operatorname{Sp}_mUse in source · 2 passages
Search aliases: Sp_m operatorname
Use in source · Paper_XI_Traditional_Final.tex, line 237
[Indecomposable monodromy blocks] Every Frobenius semisimple Weil Deligne representation is a direct sum of blocks Sp_m(τ), where τ is an irreducible Weil representation and the nilpotent operator is one Jordan chain of length m.
Exact mathematical passage
\operatorname{Sp}_m(\tau),
\SpanUse in source · 10 passages
Search aliases: Span Span span operatorname
Use in source · 05_worked_models.tex, line 34
2,0), B(v_1,v_2,v_3)=(v_1,v_2). Starting from the seed e_1∈ U, repeated application of R spans U, and A carries that span onto the first two coordinates of V. Any path from U to V ends with A and therefore lands in im A. The third coordinate direction, which spans ker B, is unreachable from a seed supported only at U. Therefore the generated representation is U⊕span{(1,0,0),(0,1,0)} subsetneq U⊕ V. Adding the seed (0,0,1)∈ V generates the full representation.
Exact mathematical passage
U\oplus\Span\{(1,0,0),(0,1,0)\} \subsetneq U\oplus V.
\Span_{\bbk}Use in source · 3 passages
Search aliases: Span_bbk span_𝕜 Span bbk operatorname Bbbk
Use in source · 05_worked_models.tex, line 294
se there are pairwise distinct scalars a_λ∈𝕜, multiplicities m_λ∈ N, and an eigenbasis {b_λ,j:λ∈Λ, 1≤ j≤ m_λ} of 𝕜^V(G) satisfying A_Gb_λ,j=a_λb_λ,j. If m_λ≤ R for all λ and m_λ_*=R for some λ_*, then R is the least natural number of adjacency seeds: R=min{r∈ N: l there exist s_1,…,s_r with [2pt] 𝕜^V(G)=span_𝕜 {p(A_G)s_j:p∈𝕜[t], 1≤ j≤ r} }.
Exact mathematical passage
{ R=\min\left\{r\in\N: \begin{array}{l} \text{there exist }s_1,\ldots,s_r\text{ with}\\[2pt] \bbk^{V(G)}=\Span_{\bbk} \{p(A_G)s_j:p\in\bbk[t],\ 1\le j\le r\} \end{array} \right\}. }
\Span_{\C}Use in source · 1 passage
Search aliases: Span_C span_ C Span C operatorname mathbb
Use in source · 05_worked_models.tex, line 372
Two adjacent vertex vectors generate the adjacency module of every simple cycle. Over C, a Fourier quotient built from a primitive root of unity supplies a matching scalar obstruction of dimension two. Hence, for every n≥3, two is the least finite number of adjacency seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any f…
Exact mathematical passage
{ 2=\min\left\{r\in\N: \exists s_1,\ldots,s_r,\quad \C^n=\Span_{\C}\{p(A_{C_n})s_j:p\in\C[t],\ 1\le j\le r\} \right\}. }
\operatorname{span}_{\CC}Use in source · 1 passage
Search aliases: span_CC span_ C operatorname CC operatorname mathbb
Use in source · 02_strengthened.tex, line 2175
Let C_1,…,C_m be self adjoint constraint operators on a finite dimensional Hilbert space. The distinction between constraint action and constraint closure is standard in constrained Hamiltonian theory [source reference]. Write c=span_ C{C_1,…,C_m} ⊆End( H).
Exact mathematical passage
\mathfrak c=\operatorname{span}_{\CC}\{C_1,\ldots,C_m\} \subseteq\End(\Hcal).
\operatorname{span}_RUse in source · 1 passage
Search aliases: span_R operatorname
Use in source · Paper_X_Block27_Amended.tex, line 1013
For a second sector let R be a DVR with uniformizer π, fraction field K and matrix rank n. If a fixed template τ has full K-linear action span, finite generation and denominator clearing give a fixed c such that π^c M_n(R)⊆span_R{τ(g):g∈ G}. Normalize a field comparison to an integral P with a unit entry. The displayed span constructs an integral B with BP=π^cI, so det Pmidπ^cn, det Pnot≡0pmodπ^cn+1. One c works for all integral representations in the fixed field-isomorphism class. Jacobson dens…
Exact mathematical passage
\pi^c M_n(R)\subseteq\operatorname{span}_R\{\tau(g):g\in G\}.
\spanofUse in source · 1 passage
Search aliases: spanof span operatorname
Use in source · 03_supplement_03.tex, line 63
The matrix-orbit dimension and the coefficient equivalence now yield dim_ Cspan{(A⊗ I_B)ψ:A∈ M_d_A( C)} =d_A r(ψ).
Exact mathematical passage
\boxed{\dim_\C\spanof\{(A\otimes I_B)\psi:A\in M_{d_A}(\C)\} =d_A r(\psi).}
\mathrm{spec}Use in source · 2 passages
Search aliases: spec mathrm
Use in source · 05_worked_models.tex, line 565
[Exact graph seed rank from a certificate] For a finite square matrix A over a field K, define the noncomputable exact spectral quantity r_spec(A) = max_μ∈Spec_K(A) dim_Kker(A-μ I), with value zero for an empty spectrum. Suppose an exact certificate supplies a finite eigenbasis indexed by labels and multiplicities, the eigenvalue equations, a multiplicity bound attained by one label, projector polynomials that extract the padded eigenbasis generators, coverage of every spectral value by a label,…
Exact mathematical passage
r_{\mathrm{spec}}(A) = \max_{\mu\in\Spec_K(A)} \dim_K\ker(A-\mu I),
\operatorname{Spec}Use in source · 1 passage
Search aliases: Spec operatorname
Use in source · 02_foundations.tex, line 974
[Formula for geometric multiplicity] If 𝕜 is algebraically closed, then srank_𝕜[T](X) = max_λ∈Spec(T) dim_𝕜Ker(T-λ I). [source reference]
Exact mathematical passage
\boxed{ \srank_{\bbk[T]}(X) = \max_{\lambda\in\operatorname{Spec}(T)} \dim_{\bbk}\Ker(T-\lambda I). }
\Spec_KUse in source · 1 passage
Search aliases: Spec_K Spec operatorname
Use in source · 05_worked_models.tex, line 565
[Exact graph seed rank from a certificate] For a finite square matrix A over a field K, define the noncomputable exact spectral quantity r_spec(A) = max_μ∈Spec_K(A) dim_Kker(A-μ I), with value zero for an empty spectrum. Suppose an exact certificate supplies a finite eigenbasis indexed by labels and multiplicities, the eigenvalue equations, a multiplicity bound attained by one label, projector polynomials that extract the padded eigenbasis generators, coverage of every spectral value by a label,…
Exact mathematical passage
r_{\mathrm{spec}}(A) = \max_{\mu\in\Spec_K(A)} \dim_K\ker(A-\mu I),
\Spec_\RUse in source · 2 passages
Search aliases: Spec_R Spec_ R Spec R operatorname mathbb
Use in source · 03_supplement_03.tex, line 147
tual matrix exponential with continuous functional calculus: T_e(t)=f_t(H), f_t(x)=e^-t(x-e). The formal proof uses composition, scalar multiplication, subtraction, and the established agreement between the exponential in functional calculus and the exponential defined by a power series. Spectral mapping is consequently available for the operator that was actually defined: Spec_ R T_e(t)={e^-t(x-e):x∈Spec_ R H}. Hermiticity also identifies the full complex spectrum with the image of these real e…
Exact mathematical passage
\Spec_\R T_e(t)=\{e^{-t(x-e)}:x\in\Spec_\R H\}.
\sqcupUse in source · 6 passages
Search aliases: sqcup
Use in source · 71_native.tex, line 249
for maps from A and B to a common set if and only if R is rectangular. In that case a canonical finite choice of common set is the connected-component set C_R of the bipartite graph with vertex set Asqcup B and edges given by R.
Exact mathematical passage
A\sqcup B
\operatorname{srank}Use in source · 3 passages
Search aliases: srank operatorname
Use in source · monograph_atlas.tex, line 83
dimensional state space. vfigseed_rank headFrom the example to the finite formula For a block M_d( C)⊗ I_m, k seed matrices contribute at most kd independent multiplicity directions. Hence k≥lceil m/drceil. Partitioning a multiplicity basis into groups of size at most d attains this bound. For a direct sum of such blocks, the worst block determines the required count: srank_AA(HH)=max_ilceil m_i/d_irceil. This is an invariant of the declared representation of the algebra, and depends on more tha…
Exact mathematical passage
\operatorname{srank}_{\AA}(\HH)=\max_i\lceil m_i/d_i\rceil.
\srank_{\A}Use in source · 1 passage
Search aliases: srank_A srank_ A srank A operatorname mathcal
Use in source · 03_packets_and_graphs.tex, line 1143
] The source component formulas concern packet pure seeds. They do not give the unrestricted seed rank. For example, let A=𝕜×𝕜, X=𝕜⊕𝕜 with the coordinate action and the two coordinate packet projectors. The transfer graph has two source components and psrank_ A, P(X)=2, but the mixed vector (1,1) is cyclic, so srank_ A(X)=1.
Exact mathematical passage
\srank_{\A}(X)=1.
\srank_{A,\tau_M}Use in source · 1 passage
Search aliases: srank_A,τ_M srank tau operatorname tau
Use in source · 02_foundations.tex, line 437
onotonicity] Let φ:A→ B be a unital ring homomorphism and let Φ:M↠ N be a surjective additive map satisfying Φ(am)=φ(a)Φ(m) (a∈ A, m∈ M). Then srank_B(N)≤srank_A(M). If M and N are topological modules, Φ is continuous and surjective, and the actions are continuous, then srank_B,τ_N(N)≤srank_A,τ_M(M). [source reference] ExportsAll quotient lower bounds and realization inequalities.
Exact mathematical passage
\srank_{B,\tau_N}(N)\leq\srank_{A,\tau_M}(M).
\srank_{A/J}Use in source · 2 passages
Search aliases: srank_A/J srank operatorname
Use in source · 02_foundations.tex, line 678
radical] Let A be a unital ring, let J=Jac(A), and let M be a finitely generated left module over A. A family S⊆ M generates M if and only if its image generates M/JM as a module over A/J. When A/J is semisimple Artinian, this quotient is the usual semisimple top. No semisimplicity of A/J is required for the general equality. Consequently, srank_A(M)=srank_A/J(M/JM). [source reference] ExportsGenerator formulas in finite dimension, quantum multiplicity calculations, and residual deformation modu…
Exact mathematical passage
\boxed{ \srank_A(M)=\srank_{A/J}(M/JM). }
\srank_{A_K}Use in source · 1 passage
Search aliases: srank_A_K srank operatorname
Use in source · 02_foundations.tex, line 601
[Monotonicity under extension of scalars] Let K/𝕜 be a field extension, let A be a unital algebra over 𝕜, and let M be a finitely generated left module over A. Set A_K:=K⊗_𝕜A, M_K:=K⊗_𝕜M. Then srank_A_K(M_K)≤srank_A(M). Every generating family S⊆ M extends to the generating family 1⊗ S⊆ M_K. [source reference] ExportsExtension of the coefficient field in realizations from arithmetic, quantum theory, and representation theory.
Exact mathematical passage
\boxed{ \srank_{A_K}(M_K)\leq\srank_A(M). }
\operatorname{srank}_{\AA}Use in source · 1 passage
Search aliases: srank_AA operatorname AA
Use in source · monograph_atlas.tex, line 83
dimensional state space. vfigseed_rank headFrom the example to the finite formula For a block M_d( C)⊗ I_m, k seed matrices contribute at most kd independent multiplicity directions. Hence k≥lceil m/drceil. Partitioning a multiplicity basis into groups of size at most d attains this bound. For a direct sum of such blocks, the worst block determines the required count: srank_AA(HH)=max_ilceil m_i/d_irceil. This is an invariant of the declared representation of the algebra, and depends on more tha…
Exact mathematical passage
\operatorname{srank}_{\AA}(\HH)=\max_i\lceil m_i/d_i\rceil.
\srank_{\AA}Use in source · 2 passages
Search aliases: srank_AA srank_ A srank AA operatorname mathcal
Use in source · visual_atlas.tex, line 313
headThis is why the independent formula uses a maximum For independent actions of A on X and B on Y, a common list of seed positions can carry the seeds for both spaces. Pad the shorter list with zeros. Independent operators then select the two components, giving srank_ A× B(X⊕ Y) =max{srank_ A(X),srank_ B(Y)}. With a shared action, this selection need not be available. The general bound is the sum.
Exact mathematical passage
\srank_{\AA\times\BB}(X\oplus Y) =\max\{\srank_{\AA}(X),\srank_{\BB}(Y)\}.
\srank_{\AA\times\BB}Use in source · 1 passage
Search aliases: srank_AA×BB srank_ A× B srank AA times BB operatorname mathcal times mathcal
Use in source · visual_atlas.tex, line 313
headThis is why the independent formula uses a maximum For independent actions of A on X and B on Y, a common list of seed positions can carry the seeds for both spaces. Pad the shorter list with zeros. Independent operators then select the two components, giving srank_ A× B(X⊕ Y) =max{srank_ A(X),srank_ B(Y)}. With a shared action, this selection need not be available. The general bound is the sum.
Exact mathematical passage
\srank_{\AA\times\BB}(X\oplus Y) =\max\{\srank_{\AA}(X),\srank_{\BB}(Y)\}.
\srank_{\AA\otimes\BB}Use in source · 1 passage
Search aliases: srank_AA⊗BB srank_ A⊗ B srank AA otimes BB operatorname mathcal otimes mathcal
Use in source · visual_atlas.tex, line 528
headThe resulting upper bound Since simple tensors span the algebraic tensor product, the complete rectangular family proves srank_ A⊗ B(X⊗ Y) ≤srank_ A(X) srank_ B(Y). This is not a universal equality claim. Changing the algebra changes the answer: the full independent matrix action can make a single nonzero pure tensor cyclic. Nor does the algebraic argument itself establish a theorem for completed topological tensor products.
Exact mathematical passage
\boxed{\srank_{\AA\otimes\BB}(X\otimes Y) \le\srank_{\AA}(X)\,\srank_{\BB}(Y).}
\operatorname{srank}_{\Acal}Use in source · 1 passage
Search aliases: srank_Acal srank_ A operatorname Acal operatorname mathcal
Use in source · 02_strengthened.tex, line 135
[Quantum seed rank] For the displayed multiplicity decomposition, srank_ A( H) = max_1≤ r≤ s lceilfracm_rd_rrceil.
Exact mathematical passage
\operatorname{srank}_{\Acal}(\Hcal) = \max_{1\le r\le s} \left\lceil\frac{m_r}{d_r}\right\rceil.
\srank_{A\times B}Use in source · 1 passage
Search aliases: srank_A× B srank times operatorname times
Use in source · 02_foundations.tex, line 531
[Direct sum formula for independent actions] Let M be a left module over A and N a left module over B. Under the action of the product algebra (a,b)(m,n)=(am,bn), one has srank_A× B(M⊕ N) = max{srank_A(M),srank_B(N)}. [source reference]
Exact mathematical passage
\boxed{ \srank_{A\times B}(M\oplus N) = \max\{\srank_A(M),\srank_B(N)\}. }
\srank_{A\otimes B}Use in source · 1 passage
Search aliases: srank_A⊗ B srank otimes operatorname otimes
Use in source · 02_foundations.tex, line 560
Tensor generation and witnesses for lower bounds. [Tensor product bound] Let A and B be unital algebras over 𝕜, let M be a left module over A, and let N be a left module over B. Under the natural A⊗_𝕜B action, srank_A⊗ B(M⊗_𝕜N) ≤ srank_A(M)srank_B(N). [source reference]
Exact mathematical passage
\boxed{ \srank_{A\otimes B}(M\otimes_{\bbk}N) \leq \srank_A(M)\srank_B(N). }
\srank_BUse in source · 4 passages
Search aliases: srank_B srank operatorname
Use in source · 02_foundations.tex, line 403
[Monotonicity under enlargement of the acting algebra] Let B⊆ A⊆End_𝕜(X) be unital operator algebras with the same identity. Then srank_A(X)≤srank_B(X). The same statement holds for topological seed rank when both algebras act continuously. [source reference]
Exact mathematical passage
\boxed{\srank_A(X)\leq\srank_B(X).}
\srank_{B,\tau_N}Use in source · 1 passage
Search aliases: srank_B,τ_N srank tau operatorname tau
Use in source · 02_foundations.tex, line 437
onotonicity] Let φ:A→ B be a unital ring homomorphism and let Φ:M↠ N be a surjective additive map satisfying Φ(am)=φ(a)Φ(m) (a∈ A, m∈ M). Then srank_B(N)≤srank_A(M). If M and N are topological modules, Φ is continuous and surjective, and the actions are continuous, then srank_B,τ_N(N)≤srank_A,τ_M(M). [source reference] ExportsAll quotient lower bounds and realization inequalities.
Exact mathematical passage
\srank_{B,\tau_N}(N)\leq\srank_{A,\tau_M}(M).
\srank_{\BB}Use in source · 2 passages
Search aliases: srank_BB srank_ B srank BB operatorname mathcal
Use in source · visual_atlas.tex, line 313
headThis is why the independent formula uses a maximum For independent actions of A on X and B on Y, a common list of seed positions can carry the seeds for both spaces. Pad the shorter list with zeros. Independent operators then select the two components, giving srank_ A× B(X⊕ Y) =max{srank_ A(X),srank_ B(Y)}. With a shared action, this selection need not be available. The general bound is the sum.
Exact mathematical passage
\srank_{\AA\times\BB}(X\oplus Y) =\max\{\srank_{\AA}(X),\srank_{\BB}(Y)\}.
\srank_{\bbk[T]}Use in source · 5 passages
Search aliases: srank_bbk[T] srank_𝕜[T] srank bbk operatorname Bbbk
Use in source · 02_foundations.tex, line 901
[Seed rank for a single operator] Let X have finite dimension over 𝕜 and let T∈End_𝕜(X). Regard X as a module over 𝕜[t] by t· x=Tx. For every monic irreducible p, let X_p be the primary component for p and set μ_p(T):= dim_𝕜[t]/(p)(X_p/p(T)X_p). Then srank_𝕜[T](X)=max_pμ_p(T). Moreover, μ_p(T) is the number of cyclic primary summands for p in the decomposition into elementary divisors of X_p. DependsnolinkurlPSC-I-CI-PID-PRIMARY-DECOMPOSITION, nolinkurlPSC-I-C-LOCAL-ALGEBRA-FORMULA, and nolinkur…
Exact mathematical passage
\boxed{ \srank_{\bbk[T]}(X)=\max_p\mu_p(T). }
\operatorname{srank}_{\cA}Use in source · 1 passage
Search aliases: srank_cA srank_ A operatorname cA operatorname mathcal
Use in source · 01_finite_core.tex, line 117
[Quantum seed-rank formula] For A=⊕_i(M_d_i( C)⊗ I_m_i) on H=⊕_i( C^d_i⊗ C^m_i), the minimum number of seeds required to generate H is srank_ A( H) = max_ilceilfracm_id_irceil.
Exact mathematical passage
\boxed{ \operatorname{srank}_{\cA}(\cH) = \max_i\left\lceil\frac{m_i}{d_i}\right\rceil. }
\srank_{\cA}Use in source · 1 passage
Search aliases: srank_cA srank_ A srank cA operatorname mathcal
Use in source · 05_worked_models.tex, line 439
For a finite ordered seed family S=(s_1,…,s_r), let ε_S: A^⊕ r→ X be the seed evaluation map of [source reference]. Its kernel is the relation module. The free module modulo that kernel presents the generated hull by [source reference]. The invariant content is: · generated hull: Hull_ A(S), · seed complexity: srank_ A(X), · packet coordinates: x↔(P_αx)_α, · operator coordinates: T↔(P_βTP_α)_β,α, · relation module: N_S:=Kerε_S≤ A^⊕ r, · free presentation: A^⊕ r/N_S≅Hull_ A(S), · coordinate invar…
Exact mathematical passage
{ \begin{aligned} &\text{generated hull: }\Hull_{\cA}(S),\\ &\text{seed complexity: }\srank_{\cA}(X),\\ &\text{packet coordinates: }x\leftrightarrow(P_{\alpha}x)_{\alpha},\\ &\text{operator coordinates: }T\leftrightarrow(P_{\beta}TP_{\alpha})_{\beta,\alpha},\\ &\text{relation module: }N_S:=\Ker\varepsilon_S\leq\cA^{\oplus r},\\ &\text{free presentation: }\cA^{\oplus r}/N_S\cong\Hull_{\cA}(S),\\ &\text{coordinate invariance: }P_{\alpha}\mapsto UP_{\alpha}U^{-1},\\ &\text{realization: }\Phi A=\varphi(A)\Phi. \end{aligned} }
\srank_{\kk I}Use in source · 1 passage
Search aliases: srank_kk I srank_𝕜 I srank kk operatorname Bbbk
Use in source · visual_atlas.tex, line 304
The corresponding minimum seed numbers are srank_𝕜 I(𝕜^2)=2, srank_𝕜×𝕜(𝕜⊕𝕜)=1. In the independent case, the algebra contains (1,0) and (0,1), which recover the two coordinates of s separately.
Exact mathematical passage
\srank_{\kk I}(\kk^2)=2, \qquad \srank_{\kk\times\kk}(\kk\oplus\kk)=1.
\srank_{\kk\times\kk}Use in source · 1 passage
Search aliases: srank_kk×kk srank_𝕜×𝕜 srank kk times kk operatorname Bbbk times Bbbk
Use in source · visual_atlas.tex, line 304
The corresponding minimum seed numbers are srank_𝕜 I(𝕜^2)=2, srank_𝕜×𝕜(𝕜⊕𝕜)=1. In the independent case, the algebra contains (1,0) and (0,1), which recover the two coordinates of s separately.
Exact mathematical passage
\srank_{\kk I}(\kk^2)=2, \qquad \srank_{\kk\times\kk}(\kk\oplus\kk)=1.
\srank_{M_2(\QQ)}Use in source · 1 passage
Search aliases: srank_M_2(QQ) srank_M_2( Q) srank QQ operatorname mathbb
Use in source · visual_atlas.tex, line 342
6 · 7 · 8 · 9 · 10, choose A_1 from columns 1,2 of W, A_2 from columns 3,4, and A_3 with first column equal to column 5 of W and second column zero. Then A_1V_1+A_2V_2+A_3V_3=W. The same construction works for arbitrary entries of W, so the lower bound is attained: srank_M_2( Q)M_2×5( Q)=lceil5/2rceil=3.
Exact mathematical passage
\boxed{\srank_{M_2(\QQ)}M_{2\times5}(\QQ)=\lceil5/2\rceil=3.}
\srank_{M_d(D)}Use in source · 2 passages
Search aliases: srank_M_d(D) srank operatorname
Use in source · 02_foundations.tex, line 833
[Full matrix algebra] For the natural module S=D^d over M_d(D), srank_M_d(D)(S)=1. More strongly, every nonzero vector of S is cyclic. [source reference]
Exact mathematical passage
\srank_{M_d(D)}(S)=1.
\srank_{P_\alpha A P_\alpha}Use in source · 1 passage
Search aliases: srank_P_α A P_α srank alpha alpha operatorname alpha alpha
Use in source · 03_packets_and_graphs.tex, line 1037
[Basic packet pure rank bounds] For every A module resolved into packets with P_α∈ A, srank_A(X) ≤ psrank_A, P(X) ≤ ∑_α∈ I srank_P_α A P_α(X_α). The corner algebra on the right is unital on X_α with identity P_α. [source reference]
Exact mathematical passage
\boxed{ \srank_A(X) \leq \psrank_{A,\Pcal}(X) \leq \sum_{\alpha\in I} \srank_{P_\alpha A P_\alpha}(X_\alpha). }
\srank_{\QQ[N]}Use in source · 1 passage
Search aliases: srank_QQ[N] srank_ Q[N] srank QQ operatorname mathbb
Use in source · visual_atlas.tex, line 102
headLower and upper bounds meet If r seeds generated X, their images would generate the quotient. Since scalar multiplication cannot create a new independent direction, r≥2. Conversely, the two seeds e_0,f_0 generate their respective chains, so srank_ Q[N](X)=2. This proof uses the displayed operator and quotient. It does not ask a computer to search every possible seed, and it does not need an existence theorem for Jordan form.
Exact mathematical passage
\boxed{\srank_{\QQ[N]}(X)=2.}
\srank_{R_p}Use in source · 2 passages
Search aliases: srank_R_p srank operatorname
Use in source · 02_foundations.tex, line 927
Let the minimal polynomial be m_T(t)=∏_pp(t)^e_p. The acting algebra is 𝕜[T]≅𝕜[t]/(m_T) ≅∏_pR_p, R_p:=𝕜[t]/(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with maximal ideal generated by the class of p and residue division field K_p=𝕜[t]/(p). By [source reference], srank_R_p(X_p) = dim_K_pX_p/p(T)X_p = μ_p(T). …
Exact mathematical passage
\srank_{\bbk[T]}(X)=\max_p\srank_{R_p}(X_p).
\mathrm{src}Use in source · 4 passages
Search aliases: src mathrm
Use in source · odd_synthesis.tex, line 120
Assume the one-vertex scalar-pushout contract: there are d labeled, actually supplied full character extensions with whole symplectic quotients V_i of dimension two over F_p. Their balanced C^×-central product has quotient V_src=⊕_iV_i and Heisenberg carrier dimension p^d=N. Initially take the declared stabilizer C={1}.
Exact mathematical passage
V_{\mathrm{src}}=\bigoplus_iV_i
\mathrm{ss}Use in source · 6 passages
Search aliases: ss mathrm
Use in source · Paper_XII_Traditional_Final.tex, line 1362
tomorphic representation π_x occurring in the fiber H_x, together with a semisimple attached representation ρ_π_x:G_F,S⟶GL_n(E') whose unramified characteristic polynomial is x(P_v(X)). item Equality det(X-ρ_x(Frob_v))=x(P_v(X)) for ρ_x=x∘ρ_R at every unramified place in a Chebotarev dense set. Then ρ_x^ss ≅ ρ_π_x. If ρ_x is irreducible, then ρ_x≅ρ_π_x.
Exact mathematical passage
\boxed{ \rho_x^{\mathrm{ss}} \cong \rho_{\pi_x}. }
\operatorname{St}Use in source · 1 passage
Search aliases: St operatorname
Use in source · Paper_XI_Traditional_Final.tex, line 295
[Formal atomic reconstruction] Assume that every irreducible Weil representation τ has been assigned a supercuspidal representation σ_τ, compatibly with isomorphism and unramified twist. Then the rules Sp_m(τ) ⟼ St_m(σ_τ) and normalized parabolic induction determine a candidate correspondence on all Frobenius semisimple Weil Deligne parameters.
Exact mathematical passage
\operatorname{Sp}_m(\tau) \longmapsto \operatorname{St}_m(\sigma_\tau)
\operatorname{St}_mUse in source · 1 passage
Search aliases: St_m operatorname
Use in source · Paper_XI_Traditional_Final.tex, line 295
[Formal atomic reconstruction] Assume that every irreducible Weil representation τ has been assigned a supercuspidal representation σ_τ, compatibly with isomorphism and unramified twist. Then the rules Sp_m(τ) ⟼ St_m(σ_τ) and normalized parabolic induction determine a candidate correspondence on all Frobenius semisimple Weil Deligne parameters.
Exact mathematical passage
\operatorname{Sp}_m(\tau) \longmapsto \operatorname{St}_m(\sigma_\tau)
\operatorname{Stab}Use in source · 1 passage
Search aliases: Stab operatorname
Use in source · Paper_X_Block27_Amended.tex, line 1188
The arithmetic witnesses use the integral closure A of Z in overline Q. For each rational prime p, Lean constructs a maximal ideal Q of A above (p) and an actual element γ_p∈ G_ Q such that γ_p∈Stab_G_ Q( Q), overlineγ_p(a)=overline a^ p (a∈ A). The existence argument applies Mathlib's proved profinite stabilizer-surjectivity theorem to this actual invariant ring. The residue p-power map is proved bijective by taking p-th roots in overline Q and showing that they remain integral when their p-th …
Exact mathematical passage
\gamma_p\in\operatorname{Stab}_{G_{\Q}}(\mathfrak Q),\qquad \overline{\gamma_p(a)}=\overline a^{\,p} \quad(a\in\mathcal A).
\operatorname{Stab}_{G_{\Q}}Use in source · 1 passage
Search aliases: Stab_G_Q Stab_G_ Q operatorname Q operatorname mathbf
Use in source · Paper_X_Block27_Amended.tex, line 1188
The arithmetic witnesses use the integral closure A of Z in overline Q. For each rational prime p, Lean constructs a maximal ideal Q of A above (p) and an actual element γ_p∈ G_ Q such that γ_p∈Stab_G_ Q( Q), overlineγ_p(a)=overline a^ p (a∈ A). The existence argument applies Mathlib's proved profinite stabilizer-surjectivity theorem to this actual invariant ring. The residue p-power map is proved bijective by taking p-th roots in overline Q and showing that they remain integral when their p-th …
Exact mathematical passage
\gamma_p\in\operatorname{Stab}_{G_{\Q}}(\mathfrak Q),\qquad \overline{\gamma_p(a)}=\overline a^{\,p} \quad(a\in\mathcal A).
\starhUse in source · 4 passages
Search aliases: starh mathbin⋆ mathbin star
Use in source · 72_recovered.tex, line 34
This article begins with the Hurwitz product (amathbin⋆ b)_n = ∑_r=0^n binom nr a_rb_n-r. Over a field of characteristic zero it is multiplication of exponential generating functions. Over a finite field it realizes the additive group algebra through the exponential modes ϵ_u=(u^n)_n≥0. This finite algebra gives an exact proof of the standard window period. Hurwitz series rings are treated in [source reference]. The finite difference and umbral background may be found in [source reference].
Exact mathematical passage
(a\starh b)_n = \sum_{r=0}^{n} \binom nr a_rb_{n-r}.
\mathrm{std}Use in source · 2 passages
Search aliases: std mathrm
Use in source · Paper_XII_Traditional_Final.tex, line 424
The sign witness is W_sgn=Z, while W_std=W(6,4,3)=3A+3B belongs to the standard summand. The corresponding projectors and trace pairings are P_t,2=2 · 4
Exact mathematical passage
W_{\mathrm{std}}=W(6,4,3)=3A+3B
\operatorname{step}Use in source · 1 passage
Search aliases: step operatorname
Use in source · 02_supplement_02.tex, line 86
Let v be the new optimum of the edited subtree, and let j be the edited child slot at one frame. The update performed at that frame is step(v)= min{K, σ+ ∑_i v, · i=j, widehat F_i, · i≠ j. }. This result is passed to the next ancestor. The update function does not recursively evaluate sibling subtrees. It reads their stored values widehat F_i.
Exact mathematical passage
\boxed{ \operatorname{step}(v)= \min\left\{K,\;\sigma+ \sum_i\begin{cases} v,&i=j,\\ \widehat F_i,&i\ne j. \end{cases}\right\}. }
\operatorname{storedTree}Use in source · 1 passage
Search aliases: storedTree operatorname
Use in source · 02_supplement_03.tex, line 202
Entrywise implementation theorem. For every b≤ B, Lean proves lookup(storedTree(B,t),b)=F_t(b). There is a parallel theorem for stored discrete precision. The proof establishes first that an output entry at b depends only on input entries at most b, then proceeds through cached convolution and the tree recursion. Consequently the optimality, attainment, and infeasibility specification applies to the actual stored result.
Exact mathematical passage
\operatorname{lookup}(\operatorname{storedTree}(B,t),b)=F_t(b).
\mathrm{stream}Use in source · 1 passage
Search aliases: stream mathrm
Use in source · 02_supplement_02.tex, line 184
Streaming. The explicit event type counts one block read, one source read, and a target read plus write for every edge. With scalar size s, its byte count is B_stream=s∑_e(m_en_e+n_eq+2m_eq). No cross-edge cache reuse is assumed. Sparse index metadata and temporary workspace remain separate.
Exact mathematical passage
B_{\mathrm{stream}}=s\sum_e(m_en_e+n_eq+2m_eq).
\sTwoUse in source · 3 passages
Search aliases: sTwo s_2
Use in source · 72_recovered.tex, line 1091
[Dyadic inverse coefficients] For r≥1, β_r = - frac2^2r(2r-1)B_2r(2r)!. Consequently, v_2(β_r)=s_2(r)-1. Every R_n is a dyadic unit.
Exact mathematical passage
\boxed{ \vTwo(\beta_r)=\sTwo(r)-1. }
\succeqUse in source · 4 passages
Search aliases: succeq
Use in source · 03_supplement_02.tex, line 203
Matrix multiplication distributes across addition and scalar multiplication, which gives complex linearity. If ρsucceq0, each term K_aρ K_a^* is positive semidefinite, hence so is their finite sum.
Exact mathematical passage
\rho\succeq0
\sup_{0\leq r\leq t}Use in source · 1 passage
Search aliases: sup_0≤ r≤ t sup leq leq
Use in source · Paper_VI_Traditional_Final.tex, line 4858
Let E(t)=sup_0≤ r≤ tnormx(r)-widetilde x(r). Subtract the two mild equations. Fubini's theorem and the triangle inequality give E(t) ≤ M_TL∫_0^tE(r) d r + M_TBtε. For t≤ T, replace the last factor t by T and apply Gronwall's inequality.
Exact mathematical passage
E(t)=\sup_{0\leq r\leq t}\norm{x(r)-\widetilde x(r)}.
\sup_{0\leq s,t\leq T}Use in source · 1 passage
Search aliases: sup_0≤ s,t≤ T sup leq leq
Use in source · Paper_VI_Traditional_Final.tex, line 2654
[Asymptotic closure criterion] Assume the weighted convergence in [source reference] holds for every bounded time interval, and define P_m,n=R_n,mC_m,n on X_n. Suppose that for every T>0, ρ_m,n(T) = sup_0≤ s,t≤ T norm C_m,nT_n(s)(I-P_m,n)T_n(t)R_n,m ⟶0. Then U_m(s+t)=U_m(s)U_m(t) for all s,t≥0. If the convergence in [source reference] is locally uniform and every T_n is norm continuous, then U_m is a norm continuous semigroup.
Exact mathematical passage
\rho_{m,n}(T) = \sup_{0\leq s,t\leq T} \norm{ C_{m,n}T_n(s)(I-P_{m,n})T_n(t)R_{n,m} } \longrightarrow0.
\sup_{0\le t\le T}Use in source · 2 passages
Search aliases: sup_0≤ t≤ T sup le le
Use in source · atlas-pages.tex, line 724
2^-(n+1), ∑_j=n^∞|A_j+1-A_j|=2^-n. vfigext_tails platecaptionThe bars show the certified tail 2^-n at horizon T=1. They are bounds on a whole time interval, not errors sampled at the right endpoint. headThe estimate follows before any graph is drawn For 0≤ t≤ T, the derivative of a↦ e^-at has absolute value at most T when a≥0. Thus sup_0≤ t≤ T|T_n(t)-e^-t|≤ T2^-n. The limit is e^-t, which does have the semigroup law. Here every reconstruction is the identity, so the hidden-range excursion is ide…
Exact mathematical passage
\sup_{0\le t\le T}|T_n(t)-e^{-t}|\le T2^{-n}.
\sup_{0\leq t\leq T}Use in source · 12 passages
Search aliases: sup_0≤ t≤ T sup leq leq
Use in source · Paper_VI_Traditional_Final.tex, line 2555
[Weighted summable defect completion] Define the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_j=n^∞w_m,j sup_0≤ t≤ TnormE_j(t).
Exact mathematical passage
\sum_{n=m}^{\infty} w_{m,n} \sup_{0\leq t\leq T}\norm{E_n(t)}<\infty,
\sup_{\abs t\leq T}Use in source · 2 passages
Search aliases: sup_abs t≤ T sup abs leq
Use in source · 02_strengthened.tex, line 2428
[Norm summable Hamiltonian tower] Let (H_n) be bounded self adjoint operators on one Hilbert space and assume ∑_n=0^∞normH_n+1-H_n<∞. Then H_n converges in norm to a bounded self adjoint operator H. For every T>0, sup_abs t≤ T norm e^- i tH_n- e^- i tH ≤ TnormH_n-H. If every H_n≥0, then sup_0≤ t≤ T norm e^-tH_n- e^-tH ≤ TnormH_n-H.
Exact mathematical passage
\sup_{\abs t\leq T} \norm{\e^{-\ii tH_n}-\e^{-\ii tH}} \leq T\norm{H_n-H}.
\sup_{F,a}Use in source · 1 passage
Search aliases: sup_F,a sup
Use in source · 03_packets_and_graphs.tex, line 443
Each P_nX=Ker(I-P_n) is closed. For x∈ X, A_ Px _u = sup_F,a ∑_n∈ Fa_nP_nx ≤ K x . For y=(y_n)∈⊕^uP_nX, define S_ Py=∑_ny_n. Since the finite partial sums with every coefficient equal to one are among those used in y _u, S_ Py ≤ y _u. The assumed expansion gives S_ P A_ Px=x. Conversely, boundedness of P_m permits termwise application to the unconditionally convergent series: P_m S_ Py = ∑_nP_my_n =y_m. Thus A_ P S_ P=I. Completeness of the coordinate space is [source reference].
Exact mathematical passage
\|\mathcal A_{\Pcal}x\|_u = \sup_{F,a} \left\|\sum_{n\in F}a_nP_nx\right\| \leq K\|x\|.
\sup_mUse in source · 2 passages
Search aliases: sup_m sup
Use in source · Paper_V_Traditional_Final.tex, line 3427
[Estimated packet Gibbs bound] Let Q_m be linear and uniformly bounded on L^p, with sup_m Q_m ≤ C_Q. For f=s+j and an estimated packet widehat j_m, define widehat f_m =Q_m(f-widehat j_m)+widehat j_m. Then f-widehat f_m _p ≤ s-Q_ms _p +(1+C_Q) j-widehat j_m _p. Consequently, smooth approximation and packet recovery may be improved independently. No claim of Gibbs removal is justified unless both terms are controlled in the norm under consideration.
Exact mathematical passage
\sup_m\|Q_m\|\leq C_Q.
\sup_{s\in L}Use in source · 1 passage
Search aliases: sup_s∈ L sup in
Use in source · Paper_IV_Traditional_Final.tex, line 2950
For N≥3, [source reference] gives | D_N'(ρ)|^-1 ≤ C e^-c_N|ρ| for every sufficiently large simple zero. Let L be a compact subset of the s plane. Because the chosen arguments of the zeros lie in a fixed bounded interval, there are constants C_L and M_L such that sup_s∈ L|e^-sLogρ| ≤ C_L|ρ|^M_L for every sufficiently large zero. Hence the absolute value of the corresponding summand is at most C_L'|ρ|^M_Le^-q c_N|ρ|. The linear zero counting estimate makes this majorant summable. The Weierstrass t…
Exact mathematical passage
\sup_{s\in L}|e^{-s\Log\rho}| \leq C_L|\rho|^{M_L}
\sup_uUse in source · 1 passage
Search aliases: sup_u sup
Use in source · Paper_V_Traditional_Final.tex, line 2767
Assume τ<widehatτ. Differentiation with respect to the location parameter gives frac∂∂ uϕ_m,u(t) =-ϕ_m-1,u(t) for almost every t. Hence ϕ_m,τ-ϕ_m,widehatτ = ∫_τ^widehatτϕ_m-1,u d u. Minkowski's inequality gives ϕ_m,τ-ϕ_m,widehatτ _p ≤ |τ-widehatτ| sup_u ϕ_m-1,u _p. The supremum is bounded by the norm of t^m-1/(m-1)! on [0,1], which is the stated constant. The supremum norm argument is identical.
Exact mathematical passage
\|\phi_{m,\tau}-\phi_{m,\widehat\tau}\|_p \leq |\tau-\widehat\tau| \sup_u\|\phi_{m-1,u}\|_p.
\sup_{z\in\Gamma}Use in source · 2 passages
Search aliases: sup_z∈Γ sup in Gamma
Use in source · 05_defects.tex, line 158
olomorphic on an open set U containing both spectra. Let Γ be an admissible finite rectifiable Cauchy cycle in U and in both resolvent sets, with winding number one on the union of the spectra and zero outside U. The length of a cycle counts multiplicities. Then Ch(T_f)-h(T_c)C = frac12π i ∫_Γ h(z)(z-T_c)^-1Δ(z-T_f)^-1 dz. Consequently, Ch(T_f)-h(T_c)C · ≤ fraclength(Γ)2π sup_z∈Γ|h(z)| Δ · × sup_z∈Γ (z-T_c)^-1 sup_z∈Γ (z-T_f)^-1 .
Exact mathematical passage
\|Ch(T_f)-h(T_c)C\| &\leq \frac{\operatorname{length}(\Gamma)}{2\pi} \sup_{z\in\Gamma}|h(z)|\,\|\Delta\|\\ &\quad\times \sup_{z\in\Gamma}\|(z-T_c)^{-1}\| \sup_{z\in\Gamma}\|(z-T_f)^{-1}\|.
\sup_{\Lambda\in\mathfrak P(\mathcal T)}Use in source · 1 passage
Search aliases: sup_Λ∈ P( T) sup Lambda in mathfrak mathcal
Use in source · Paper_V_Traditional_Final.tex, line 950
imizer is unique and at least one competing pruning exists, its gap Γ_y = min_Λ≠Λ_y^* (J_y(Λ)-J_y(Λ_y^*)) is attained and strictly positive. If the pruning set is a singleton, its only element minimizes every criterion. No real-valued minimum over an empty competitor set is introduced. item In the case with competitors, if new data y' satisfy sup_Λ∈ P( T) |J_y'(Λ)-J_y(Λ)| <fracΓ_y2, then Λ_y'^*=Λ_y^*. item Every selected minimizer preserves the exact synthesis identity of the preceding theorem.
Exact mathematical passage
\sup_{\Lambda\in\mathfrak P(\mathcal T)} |J_{y'}(\Lambda)-J_y(\Lambda)| <\frac{\Gamma_y}{2},
\sup_{\Xi}Use in source · 1 passage
Search aliases: sup_Ξ sup Xi
Use in source · Paper_V_Traditional_Final.tex, line 962
set, so the feasible set is nonempty and finite. Thus the minimum is attained. If competitors exist, their finitely many differences from a unique optimum are strictly positive, so their minimum is positive and is achieved by a competitor. In the singleton case there is no alternative selector under any objective change. In the remaining case, for any competing pruning Λ≠Λ_y^*, J_y'(Λ)-J_y'(Λ_y^*) · ≥ J_y(Λ)-J_y(Λ_y^*) -2sup_Ξ|J_y'(Ξ)-J_y(Ξ)| · >Γ_y-Γ_y=0. Hence the same pruning remains uniquely…
Exact mathematical passage
\begin{aligned} J_{y'}(\Lambda)-J_{y'}(\Lambda_y^*) &\ge J_y(\Lambda)-J_y(\Lambda_y^*) -2\sup_{\Xi}|J_{y'}(\Xi)-J_y(\Xi)|\\ &>\Gamma_y-\Gamma_y=0. \end{aligned}
\sup_\ellUse in source · 7 passages
Search aliases: sup_ℓ sup ell
Use in source · Paper_VI_Traditional_Final.tex, line 1329
[Fractional refinement towers] Let D_ℓ=C_ℓ A_ℓ+1-A_ℓ C_ℓ in a bounded nonnegative self-adjoint tower, with sup_ℓnormC_ℓ≤ M. If ∑_ℓnormD_ℓ^α<∞, then the fractional defects C_ℓ A_ℓ+1^α-A_ℓ^αC_ℓ have summable operator norms. If the tower has a uniform gap m>0, it is enough that ∑_ℓnormD_ℓ<∞. On every bounded time interval, the series of corresponding fractional-semigroup defect norms converges uniformly. The maps may have different domain and codomain spaces. No operator sum across unrelated spaces…
Exact mathematical passage
\sup_\ell\norm{C_\ell}\le M
\supp_nUse in source · 1 passage
Search aliases: supp_n supp operatorname
Use in source · 03_nonlinear_coordinates.tex, line 391
ernal profiles (δ;α_1,…,α_r) for which the restricted jet J^r_δmidα_1,…,α_rH(x) is nonzero. A profile belongs to the admissible composition set if it admits a partition π={B_1,…,B_k} and intermediate labels γ_1,…,γ_k such that the outer jet of F and every inner block jet of G in [source reference] are nonzero. Then supp_n(F∘ G)(x) is contained in this admissible set. In particular, a summand vanishes whenever one of its constituent packet jets vanishes.
Exact mathematical passage
\supp_n(F\circ G)(x)
\supp_rUse in source · 1 passage
Search aliases: supp_r supp operatorname
Use in source · 03_nonlinear_coordinates.tex, line 391
[Structural pruning] For a map H with fixed input and output packet systems, let supp_r H(x) be the set of external profiles (δ;α_1,…,α_r) for which the restricted jet J^r_δmidα_1,…,α_rH(x) is nonzero. A profile belongs to the admissible composition set if it admits a partition π={B_1,…,B_k} and intermediate labels γ_1,…,γ_k such that the outer jet of F and every inner block jet of G in [source reference] are nonzero. Then supp_n(F∘ G)(x) is contained in this admissible set. In particular, a sum…
Exact mathematical passage
\supp_r H(x)
\Supp_{R/I}Use in source · 2 passages
Search aliases: Supp_R/I Supp operatorname
Use in source · Paper_XII_Traditional_Final.tex, line 707
[Support after a quotient] Let R be Noetherian, let I⊆ R be an ideal, and let M be a finite R module. Under the canonical identification Spec(R/I)≅ V(I)⊆Spec R, one has Supp_R/I(M/IM) = Supp_R(M)∩ V(I).
Exact mathematical passage
\boxed{ \Supp_{R/I}(M/IM) = \Supp_R(M)\cap V(I). }
\operatorname{Sw}Use in source · 3 passages
Search aliases: Sw operatorname
Use in source · Paper_XI_Traditional_Final.tex, line 1829
The dimension and orbit follow from [source reference]. The cycle polynomial is [source reference]. For rank greater than one, the conductor is [source reference]. In rank one, the Swan conductor is zero, and the Artin conductor is zero for trivial inertia and one otherwise, by the formula Art=dim V-dim V^I_K+Sw. The local polynomial is [source reference]. Tensor refinement is [source reference]. The Gauss sum is its defining finite sum. For a nontrivial multiplicative character its unit phase i…
Exact mathematical passage
\operatorname{Art}=\dim V-\dim V^{I_K}+\operatorname{Sw}
\mathrm{symp}Use in source · 2 passages
Search aliases: symp mathrm
Use in source · Paper_XI_Traditional_Final.tex, line 983
[Radical free envelope] For an arbitrary scalar projective action of A, the quotient A^symp=A/rad b_c is the maximal quotient detected by noncommuting projective intertwiners. If A is an elementary abelian p group, then A^symp is a symplectic F_p space.
Exact mathematical passage
\boxed{ A^{\mathrm{symp}}=A/\rad b_c }
\operatorname{synthesize}Use in source · 1 passage
Search aliases: synthesize operatorname
Use in source · 06_models.tex, line 214
The verified contract is synthesize(analyze(x,N))=x. The Lean counterpart uses packet coordinates restricted to projector ranges and proves both inverse laws. The finite Python routine checks the concrete array realization.
Exact mathematical passage
\operatorname{synthesize}(\operatorname{analyze}(x,N))=x
\SysUse in source · 1 passage
Search aliases: Sys mathscr S mathscr
Use in source · Paper_X_Block27_Amended.tex, line 775
The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.
Exact mathematical passage
\boxed{ \Sys_{\mathrm{sys}} = \Sys_2^{\mathrm{wt}} \times_{\Compat} \prod_{\lambda\nmid2}'\Sys_\lambda. }
\mathrm{sys}Use in source · 1 passage
Search aliases: sys mathrm
Use in source · Paper_X_Block27_Amended.tex, line 775
The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.
Exact mathematical passage
\boxed{ \Sys_{\mathrm{sys}} = \Sys_2^{\mathrm{wt}} \times_{\Compat} \prod_{\lambda\nmid2}'\Sys_\lambda. }
\Sys_2^{\mathrm{wt}}Use in source · 1 passage
Search aliases: Sys_2^wt mathscr S_2^wt Sys mathrm mathscr mathrm
Use in source · Paper_X_Block27_Amended.tex, line 775
The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.
Exact mathematical passage
\boxed{ \Sys_{\mathrm{sys}} = \Sys_2^{\mathrm{wt}} \times_{\Compat} \prod_{\lambda\nmid2}'\Sys_\lambda. }
\Sys_{\mathrm{sys}}Use in source · 1 passage
Search aliases: Sys_sys mathscr S_sys Sys mathrm mathscr mathrm
Use in source · Paper_X_Block27_Amended.tex, line 775
The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.
Exact mathematical passage
\boxed{ \Sys_{\mathrm{sys}} = \Sys_2^{\mathrm{wt}} \times_{\Compat} \prod_{\lambda\nmid2}'\Sys_\lambda. }
\Sys_\lambdaUse in source · 1 passage
Search aliases: Sys_λ mathscr S_λ Sys lambda mathscr lambda
Use in source · Paper_X_Block27_Amended.tex, line 775
The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.
Exact mathematical passage
\boxed{ \Sys_{\mathrm{sys}} = \Sys_2^{\mathrm{wt}} \times_{\Compat} \prod_{\lambda\nmid2}'\Sys_\lambda. }
t'Use in source · 2 passages
Search aliases: t'
Use in source · continuation_09.tex, line 84
At depth N≥1, suppose F(a)=q^Nr and a desired next integer parameter target satisfies t'-free(a)=q^Nw, with w∈ Z^k. Surjectivity supplies a solved digit u∈ S_q^m satisfying J_a u=-bar r-K_abar w. Choose an integer representative widetilde u, and set a'=a+q^N(widetilde u,w). The parameter quotient is the supplied integer w. Therefore free(a')=t' exactly. The checked polynomial Taylor identity gives F(a')≡ F(a)+q^NDF(a)(widetilde u,w) ≡0pmodq^N+1. Its quadratic remainder is divisible by q^2N, and …
Exact mathematical passage
t'-\operatorname{free}(a)=q^Nw
T''Use in source · 1 passage
Search aliases: T''
Use in source · atlas-pages.tex, line 224
retain the reduced logarithmic information: T_N(z)=N∑_m≥1κ_N,mz^Nm-1, T_N(n)=N(Nn+N-1)!κ_N,n+1. For order four, κ_4,1=1/24 and κ_4,2=17/10080. These give T_4(0)=1 and T_4(1)=34, explaining the last line of the plate. headA nonlinear equation with a precise origin If D=e^g locally, then g'=-T. Differentiating e^g three times gives D'''/D=-T''+3TT'-T^3. Setting D'''/D=-1 yields the cubic Riccati identity shown above. No additional nonlinear model has been assumed. The reciprocal family also satisf…
Exact mathematical passage
D'''/D=-T''+3TT'-T^3
T^*Use in source · 2 passages
Search aliases: T^*
Use in source · 03_packets_and_graphs.tex, line 582
[Adjoint reversal] If X is Hilbert, every P_α is orthogonal, and T∈ B(X), then (T_βα)^*=(T^*)_αβ. [source reference]
Exact mathematical passage
\boxed{(T_{\beta\alpha})^*=(T^*)_{\alpha\beta}.}
T^{-1}Use in source · 6 passages
Search aliases: T^-1
Use in source · 03_weighted.tex, line 58
B be invertible with D,D^-1∈ F^0 B. Let E be a finite set and suppose T=D+∑_e∈ EN_e, N_e∈ F^w(e) B, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 = ∑_γ(-1)^|γ|A_γ D^-1. The series converges in the filtration, and the term indexed by γ belongs to F^W(γ) B.
Exact mathematical passage
T^{-1} = \sum_\gamma(-1)^{|\gamma|}A_\gamma D^{-1}.
t^{-1}Use in source · 3 passages
Search aliases: t^-1
Use in source · Paper_XII_Traditional_Final.tex, line 440
For detection on every irreducible submodule, one can check the stronger assertion that canonical projectors detect every nonzero matrix. In addition to the two projectors above, use P_s,-1=1 · 3 0 · 0, P_tst^-1,1=1 · 0 4 · 0. The four trace pairings with these matrices have trivial common kernel in M_2( F_7), by row reduction. Every irreducible adjoint submodule contains a nonzero matrix, so some canonical projector has nonzero trace pairing on that submodule. This verifies the last adequacy cl…
Exact mathematical passage
P_{s,-1}=\begin{pmatrix}1&3\\0&0\end{pmatrix}, \qquad P_{tst^{-1},1}=\begin{pmatrix}1&0\\4&0\end{pmatrix}.
t^{-c}Use in source · 2 passages
Search aliases: t^-c
Use in source · supplement_10.tex, line 113
Application to the actual singular ratios. The preceding v0.9.0 source independently constructs H_N,c(t)=∑_n≥0fract^Nn+c(Nn+c)!, A_c(t)=t^-cH_N,c(t), R_c=fracA_c-1A_c. The notation t^-cH_N,c means removing its proved initial zero coefficients. For each c=ap, 1≤ a<m, that source proves R_ap(log(1+T))∈ W_p. The new transfer consequently proves flatness of the same rational ratio after scalar extension to Q_p.
Exact mathematical passage
H_{N,c}(t)=\sum_{n\ge0}\frac{t^{Nn+c}}{(Nn+c)!},\quad A_c(t)=t^{-c}H_{N,c}(t),\quad R_c=\frac{A_{c-1}}{A_c}.
T^0Use in source · 2 passages
Search aliases: T^0
Use in source · 20_arithmetic.tex, line 285
^d=(p^lfloor d/prfloora_d) (T^p/p)^lfloor d/prfloorT^dbmod p. The operator preserves these monomials and hence finite weighted polynomials. Each fixed output coefficient depends on only finitely many input coefficients, which extends the result to the formal series. Substitution T=e^t-1 intertwines D with Θ and maps W_p into H_p, because d![t^d]F(e^t-1)=[T^0]Θ^dF. It follows that the singular ratios are flat. This transfer is proved in leanKomorebiPaperIX.WeightedDworkTransfer.singular_ratio_fla…
Exact mathematical passage
d![t^d]F(e^t-1)=[T^0]\Theta^dF.
T^2Use in source · 9 passages
Search aliases: T^2
Use in source · visual_atlas.tex, line 207
headComposition sums over intermediate packets The (1,2) entry of the square has two contributing routes: (T^2)_12=T_11T_12+T_12T_22+T_13T_32 =1·2+2·1+0=4. For input 3 and output 1, only the route through packet 2 survives: (T^2)_13=T_12T_23=2·3=6. For packets of higher dimension, the same law uses actual linear maps as entries. The order of their composition still matters.
Exact mathematical passage
(T^2)_{12}=T_{11}T_{12}+T_{12}T_{22}+T_{13}T_{32} =1\cdot2+2\cdot1+0=4.
t^{2N}Use in source · 1 passage
Search aliases: t^2N
Use in source · root_scope.tex, line 51
m≥1, differentiation of zcoth z gives β_m=-frac2^2m(2m-1)B_2m(2m)!, κ_m=-2^4m(2m-1)B_2m. The classical dyadic Bernoulli denominator law v_2(B_2m)=-1 and Legendre's formula yield v_2(β_m)=s_2(m)-1, v_2(κ_m)=4m-1. The source identity ∑_N≥0A(N)fract^2N(2N)! =fracsinh(2t)sinh t2t^2 implies, after multiplication by 4t^2/sinh^2(2t), E_2N=∑_m=0^N κ_mbinom2N2mA(N-m). For a retained depth M≥0, put T_M(N)= E_2N -∑_m=0^min(M,N)κ_mbinom2N2mA(N-m). An empty tail, N≤ M, is exactly zero.
Exact mathematical passage
\sum_{N\ge0}A(N)\frac{t^{2N}}{(2N)!} =\frac{\sinh(2t)\sinh t}{2t^2}
t^{2n}Use in source · 7 passages
Search aliases: t^2n
Use in source · 20_arithmetic.tex, line 749
The classical signed Euler numbers have generating function secht=∑_n≥0E_2nt^2n/(2n)!. They are the N=2, j=0 member of [source reference]. The coordinate theorem reconstructs them from finite alternating sums of odd powers.
Exact mathematical passage
\operatorname{sech}t=\sum_{n\ge0}E_{2n}t^{2n}/(2n)!
t^{2r}Use in source · 7 passages
Search aliases: t^2r
Use in source · 72_recovered.tex, line 948
[Zigzag inverse kernel] sech^2t = ∑_r≥0 (-1)^rZ_2r+1 fract^2r(2r)!.
Exact mathematical passage
\boxed{ \operatorname{sech}^2t = \sum_{r\geq0} (-1)^rZ_{2r+1} \frac{t^{2r}}{(2r)!}. }
T^3Use in source · 2 passages
Search aliases: T^3
Use in source · atlas-pages.tex, line 224
retain the reduced logarithmic information: T_N(z)=N∑_m≥1κ_N,mz^Nm-1, T_N(n)=N(Nn+N-1)!κ_N,n+1. For order four, κ_4,1=1/24 and κ_4,2=17/10080. These give T_4(0)=1 and T_4(1)=34, explaining the last line of the plate. headA nonlinear equation with a precise origin If D=e^g locally, then g'=-T. Differentiating e^g three times gives D'''/D=-T''+3TT'-T^3. Setting D'''/D=-1 yields the cubic Riccati identity shown above. No additional nonlinear model has been assumed. The reciprocal family also satisf…
Exact mathematical passage
D'''/D=-T''+3TT'-T^3
t^3Use in source · 12 passages
Search aliases: t^3
Use in source · 02_foundations.tex, line 257
[title=A cyclic nilpotent chain] Let A=𝕜[t] and M=𝕜[t]/(t^3). The residue class s=[1] is one seed and ε_s:A→ M, f(t)↦ f(t)[1] has kernel (t^3). Hence M≅ A/(t^3), and the addressed vectors s,ts,t^2s are the visible orbit chain.
Exact mathematical passage
M=\bbk[t]/(t^3)
t^{3n}Use in source · 1 passage
Search aliases: t^3n
Use in source · atlas-pages.tex, line 347
V: Theorem 11.7, Theorem 11.8, Corollary 11.9. The root finder is used only to suggest intervals. Each displayed interval is accepted by exact rational arithmetic: opposite endpoint signs prove existence, and an interval bound for D_3' excludes zero throughout the interval, proving uniqueness inside it. vfigcubic_certificates headHow the certificate works Evaluate the Taylor polynomial S_90(t)=∑_n=0^90frac(-1)^nt^3n(3n)! at the two rational endpoints. From the next term onward, magnitudes decrea…
Exact mathematical passage
S_{90}(t)=\sum_{n=0}^{90}\frac{(-1)^nt^{3n}}{(3n)!}
t^4Use in source · 6 passages
Search aliases: t^4
Use in source · visual_atlas.tex, line 363
headThe missing separating operation is now a polynomial Define p_2(t)=frac3t^5-15t^4+20t^316, p_0(t)=1-p_2(t). These polynomials satisfy p_2≡0pmodt^3, p_2≡1pmod(t-2)^3. Consequently p_0(T)=P_e and p_2(T)=P_f. Applying them to s extracts both chain heads, after which polynomial powers generate all six directions. Unlike the model on pagerefV3, the separating maps already belong to the polynomial grammar.
Exact mathematical passage
p_2(t)=\frac{3t^5-15t^4+20t^3}{16},\qquad p_0(t)=1-p_2(t).
t^{4i}Use in source · 3 passages
Search aliases: t^4i
Use in source · strengthening.tex, line 452
G_4,1(t)F_0(t) =frac12(fractanh(at)at +fractanh(bt)bt). For example, its numerator follows from sinh t+sin t =2bsinh(at)cosh(bt)+2acosh(at)sinh(bt), while cosh t+cos t=2cosh(at)cosh(bt). Only degrees divisible by four survive. Its actual Hurwitz packets are D_1(t)=∑_i≥0d_ifract^4i(4i)!, d_i=(-frac14)^i frac2^4i+2(2^4i+2-1)B_4i+2(4i+2)(4i+1). The constant is one. Since v_2(4i+2)=1, the elementary Bernoulli bound shows v_2(d_i)≥2i. Hurwitz inversion therefore gives D_1(t)^-1=∑_i≥0h_ifract^4i(4i)!,…
Exact mathematical passage
D_1(t)=\sum_{i\geq0}d_i\frac{t^{4i}}{(4i)!},\qquad d_i=\left(-\frac14\right)^i \frac{2^{4i+2}(2^{4i+2}-1)B_{4i+2}}{(4i+2)(4i+1)}. \label{eq:address-one-packets}
t^{4n}Use in source · 1 passage
Search aliases: t^4n
Use in source · supplement_11.tex, line 81
Pack g_j into degrees divisible by four and form its actual EGF. The formal coefficient identity is G_j(t)=∑_n≥0g_j(n)fract^4n(4n)! =j!∑_n≥0fract^4n(4n+j)!. Its inverse is therefore the original normalized packet EGF. The source Kummer property gives flatness of G_j. The proved reciprocal closure gives flatness of G_j^-1.
Exact mathematical passage
G_j(t)=\sum_{n\ge0}g_j(n)\frac{t^{4n}}{(4n)!} =j!\sum_{n\ge0}\frac{t^{4n}}{(4n+j)!}.
t^5Use in source · 3 passages
Search aliases: t^5
Use in source · visual_atlas.tex, line 363
headThe missing separating operation is now a polynomial Define p_2(t)=frac3t^5-15t^4+20t^316, p_0(t)=1-p_2(t). These polynomials satisfy p_2≡0pmodt^3, p_2≡1pmod(t-2)^3. Consequently p_0(T)=P_e and p_2(T)=P_f. Applying them to s extracts both chain heads, after which polynomial powers generate all six directions. Unlike the model on pagerefV3, the separating maps already belong to the polynomial grammar.
Exact mathematical passage
p_2(t)=\frac{3t^5-15t^4+20t^3}{16},\qquad p_0(t)=1-p_2(t).
t^6Use in source · 2 passages
Search aliases: t^6
Use in source · 03_weighted.tex, line 581
[A computed transfer zeta] For the graph used in the numerical companion, the walk series from packet 0 to packet 2 is Z_0,2(t) = fract^31-t^2-t^3-2t^4+t^5+t^6+t^7-t^9. The first nonzero coefficients are 1,1,1,3,1,5,4,10,7,19,17,37,… at their corresponding weights. Direct enumeration through weight 30 agrees coefficient by coefficient with the rational expansion.
Exact mathematical passage
Z_{0,2}(t) = \frac{t^3}{1-t^2-t^3-2t^4+t^5+t^6+t^7-t^9}.
t^7Use in source · 2 passages
Search aliases: t^7
Use in source · 03_weighted.tex, line 581
[A computed transfer zeta] For the graph used in the numerical companion, the walk series from packet 0 to packet 2 is Z_0,2(t) = fract^31-t^2-t^3-2t^4+t^5+t^6+t^7-t^9. The first nonzero coefficients are 1,1,1,3,1,5,4,10,7,19,17,37,… at their corresponding weights. Direct enumeration through weight 30 agrees coefficient by coefficient with the rational expansion.
Exact mathematical passage
Z_{0,2}(t) = \frac{t^3}{1-t^2-t^3-2t^4+t^5+t^6+t^7-t^9}.
t^8Use in source · 1 passage
Search aliases: t^8
Use in source · strengthening.tex, line 918
For the first contribution, independence gives L(γ_k(P+Q))=∑_a=0^kμ_aμ_k-a. The parity generating series of these convolution moments is (1+t+t^2+t^4+t^8+⋯)^2 =1+t^2+t^4+t^8+⋯ in F_2[[t]]. Its coefficient is one at every k=2^r≥4. Thus the normalized moment in [source reference] is odd and v_2(Δ_1^kC_2(0))=2k+v_2(k!)+k/2. Inserting U^4n=(1+4H_2)^n introduces only corrections with strictly higher valuation, by the filtered bound at orders k+s for s>0. The equality consequently holds at every start…
Exact mathematical passage
(1+t+t^2+t^4+t^8+\cdots)^2 =1+t^2+t^4+t^8+\cdots\quad\text{in }\mathbb F_2[[t]].
t^9Use in source · 2 passages
Search aliases: t^9
Use in source · 03_weighted.tex, line 581
[A computed transfer zeta] For the graph used in the numerical companion, the walk series from packet 0 to packet 2 is Z_0,2(t) = fract^31-t^2-t^3-2t^4+t^5+t^6+t^7-t^9. The first nonzero coefficients are 1,1,1,3,1,5,4,10,7,19,17,37,… at their corresponding weights. Direct enumeration through weight 30 agrees coefficient by coefficient with the rational expansion.
Exact mathematical passage
Z_{0,2}(t) = \frac{t^3}{1-t^2-t^3-2t^4+t^5+t^6+t^7-t^9}.
T^cUse in source · 3 passages
Search aliases: T^c
Use in source · 20_arithmetic.tex, line 252
To pass from the section to the singular ratio, factor out its initial zero: for c=ap, the section has the form H_c(log(1+T))=fracT^cc! V(T), V(0)=1, V∈ W_p. The weight of c! compensates for the coefficient shift by c. The weighted ring is closed under inversion of series with constant coefficient one and under L_T=log(1+T)(1+T)d/dT. With ℓ(T)=log(1+T)/T, interpreted by coefficient shifting, differentiation of the displayed factorization gives R_c(log(1+T))=c(1+T)ℓ(T)+(L_TV)V^-1. Since pℓ(T)∈ W_…
Exact mathematical passage
H_c(\log(1+T))=\frac{T^c}{c!}\,V(T),\qquad V(0)=1, \qquad V\in\mathcal W_p.
T^dUse in source · 7 passages
Search aliases: T^d
Use in source · 20_arithmetic.tex, line 231
At the singular addresses c=ap, 1≤ a<m, use the weighted ring W_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping the falling factorial (x)_d=∏_i=0^d-1(x-i) into consecutive blocks of length p then gives an integral residue-coordinate factor p^v_p(lfloor d/prfloor!)…
Exact mathematical passage
\mathcal W_p=\left\{\sum_{d\ge0}a_dT^d: p^{\lfloor d/p\rfloor}a_d\in\Z_p\right\}.
t^{d-1}Use in source · 2 passages
Search aliases: t^d-1
Use in source · residue_continuation_body.tex, line 38
There is a constant B≥0, depending on K,M,J, such that, for every a∈ J/M and every real t≥1, |#{x∈ J:xbmod M=a, ι(x)∈ tD} -fracvol(D)q covol(L(J))t^d| ≤ B(t^d-1+1). The same constant works for all residues a.
Exact mathematical passage
\left|\#\{x\in J:x\bmod M=a,\ \iota(x)\in tD\} -\frac{\vol(D)}{q\,\covol(L(J))}t^d\right| \leq B(t^{d-1}+1).
T^{d\bmod p}Use in source · 1 passage
Search aliases: T^dbmod p bmod
Use in source · 20_arithmetic.tex, line 276
mate, use Θ=(1+T)d/dT. Both Θ and (Θ^p^2-Θ^p)/p preserve W_p. For the first operator, its coefficient formula is (d+1)a_d+1+da_d. Whenever the floor weight rises, the factor d+1 provides the missing p. For the second operator, the proof uses its full Leibniz correction and the scalar congruence p^2mid b^p^2-b^p. Weighted monomials factor as a_dT^d=(p^lfloor d/prfloora_d) (T^p/p)^lfloor d/prfloorT^dbmod p. The operator preserves these monomials and hence finite weighted polynomials. Each fixed ou…
Exact mathematical passage
a_dT^d=(p^{\lfloor d/p\rfloor}a_d) (T^p/p)^{\lfloor d/p\rfloor}T^{d\bmod p}.
T^eUse in source · 1 passage
Search aliases: T^e
Use in source · Paper_X_Block27_Amended.tex, line 592
[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.
Exact mathematical passage
\boxed{ Z_e(T) = \frac{\sum_{r=1}^{e}q^{r-1}T^r} {1-q^{e-1}T^e}. }
T^iUse in source · 1 passage
Search aliases: T^i
Use in source · dyadic_synthesis.tex, line 361
There is also a recovery from compact data explicitly recording that root. With A_0=Λ(ϖ_K) and X=c - Ind_J^Gη, define (Tf)(g)=A_0^-1f(ϖ_Kg). Then X/(T-1)X≃π. After imposing the fixed central character, N/(F^× J) is cyclic of order N, and the finite average N^-1∑_i=0^N-1T^i selects this same extension. These operators act on complex representation spaces, so their scalar division by N is valid. This is a different recovery from the Block159 projector onto a sector of η. The exact kernel and finit…
Exact mathematical passage
N^{-1}\sum_{i=0}^{N-1}T^i
t^jUse in source · 3 passages
Search aliases: t^j
Use in source · 72_recovered.tex, line 582
For integers N≥1 and 0≤ j<N, define E_N,j(t) = ∑_n≥0 fract^Nn+j(Nn+j)!. The forced factor t^j may be removed without changing the residue address.
Exact mathematical passage
t^j
t^kUse in source · 7 passages
Search aliases: t^k
Use in source · 03_weighted.tex, line 557
[Linear recurrence] With the canonical polynomials from [source reference], write Q(t)=∑_kq_kt^k, Z_s,t(t)=∑_n≥0a_n(s,t)t^n. Then the walk counts satisfy the recurrence with constant coefficients determined by Q after the finite initial range contributed by P_s,t, with deg 0=-∞: ∑_k=0^nq_k a_n-k(s,t)=0 (n>deg P_s,t).
Exact mathematical passage
Q(t)=\sum_kq_kt^k, \qquad Z_{s,t}(t)=\sum_{n\ge0}a_n(s,t)t^n.
t^mUse in source · 3 passages
Search aliases: t^m
Use in source · Paper_IV_Traditional_Final.tex, line 980
The polynomial dependence on r can be read directly from the formal logarithm. We use the complete exponential Bell polynomials B_n defined by exp(∑_m≥1x_mfract^mm!) = ∑_n≥0B_n(x_1,…,x_n)fract^nn!.
Exact mathematical passage
\exp\left(\sum_{m\geq1}x_m\frac{t^m}{m!}\right) = \sum_{n\geq0}B_n(x_1,\ldots,x_n)\frac{t^n}{n!}.
t^{m-1}Use in source · 1 passage
Search aliases: t^m-1
Use in source · Paper_V_Traditional_Final.tex, line 2773
ion with respect to the location parameter gives frac∂∂ uϕ_m,u(t) =-ϕ_m-1,u(t) for almost every t. Hence ϕ_m,τ-ϕ_m,widehatτ = ∫_τ^widehatτϕ_m-1,u d u. Minkowski's inequality gives ϕ_m,τ-ϕ_m,widehatτ _p ≤ |τ-widehatτ| sup_u ϕ_m-1,u _p. The supremum is bounded by the norm of t^m-1/(m-1)! on [0,1], which is the stated constant. The supremum norm argument is identical.
Exact mathematical passage
t^{m-1}/(m-1)!
t^{m_0}Use in source · 2 passages
Search aliases: t^m_0
Use in source · Paper_VI_Traditional_Final.tex, line 5233
in Part II. A directed edge from α to β means that P_β A P_α is nonzero, with α≠β. A packet always has the length-zero path to itself. [Minimum transfer order] Suppose every directed packet path from α to β uses at least m_0 off-diagonal transfers. Then P_β e^tAP_α =O(t^m_0) as tdownarrow0. If there is no directed path from α to β, the block vanishes for all t.
Exact mathematical passage
P_\beta\e^{tA}P_\alpha =O(t^{m_0}) \quad\text{as }t\downarrow0.
t^NUse in source · 4 passages
Search aliases: t^N
Use in source · supplement_01.tex, line 103
The formalization does not stop at a recurrence. In the variable z=t^N, construct K_N,j(z)=∑_n≥0fracj!(Nn+j)!z^n, F_N,j(z)=∑_n≥0fracC_N,j(n)(Nn)!z^n. It proves the factorial coefficient identity, then F_N,jK_N,j=1 and therefore F_N,j=K_N,j^-1.
Exact mathematical passage
z=t^N
T^{n+1}Use in source · 1 passage
Search aliases: T^n+1
Use in source · 01_finite_coordinates.tex, line 233
The zero case is the identity matrix. If T^n= W_n, then the left-successor power convention gives T^n+1=TT^n=T W_n= W_n+1.
Exact mathematical passage
T^{n+1}=TT^n=T\cW_n=\cW_{n+1}
T^{n_j}Use in source · 2 passages
Search aliases: T^n_j
Use in source · monograph_atlas.tex, line 717
tive lower bound on the gap. Ground multiplicity does not cause a problem: P_0 selects the entire eigenspace at the minimum energy. For a Hamiltonian whose minimum is E_0≠0, the normalized transfer is e^-t(H-E_0I) before taking the limit. headPositivity is retained at several Euclidean times For arbitrary vectors u_j and nonnegative integer times n_j, the reflected kernel obeys ∑_j,k⟨ u_j,T^n_j+n_ku_k⟩ = ∑_jT^n_ju_j ^2. This is a finite Gram factorization. The retained run checks the integer sem…
Exact mathematical passage
\sum_{j,k}\langle u_j,T^{n_j+n_k}u_k\rangle =\left\|\sum_jT^{n_j}u_j\right\|^2.
T^{n_j+n_k}Use in source · 5 passages
Search aliases: T^n_j+n_k
Use in source · monograph_atlas.tex, line 717
tive lower bound on the gap. Ground multiplicity does not cause a problem: P_0 selects the entire eigenspace at the minimum energy. For a Hamiltonian whose minimum is E_0≠0, the normalized transfer is e^-t(H-E_0I) before taking the limit. headPositivity is retained at several Euclidean times For arbitrary vectors u_j and nonnegative integer times n_j, the reflected kernel obeys ∑_j,k⟨ u_j,T^n_j+n_ku_k⟩ = ∑_jT^n_ju_j ^2. This is a finite Gram factorization. The retained run checks the integer sem…
Exact mathematical passage
\sum_{j,k}\langle u_j,T^{n_j+n_k}u_k\rangle =\left\|\sum_jT^{n_j}u_j\right\|^2.
T^{n_k}Use in source · 2 passages
Search aliases: T^n_k
Use in source · 02_strengthened.tex, line 1890
Since T is positive and self adjoint, ipv_jT^n_j+n_kv_k = ipT^n_jv_jT^n_kv_k. The quadratic form is the squared norm of ∑_kc_kT^n_kv_k.
Exact mathematical passage
\ip{v_j}{T^{n_j+n_k}v_k} = \ip{T^{n_j}v_j}{T^{n_k}v_k}.
t^{Nn+c}Use in source · 3 passages
Search aliases: t^Nn+c
Use in source · 20_arithmetic.tex, line 216
Set H_c(t)=∑_n≥0fract^Nn+c(Nn+c)!, A_c(t)=∑_n≥0fract^Nn(Nn+c)!, R_c(t)=fracA_c-1(t)A_c(t) (1≤ c<N). The denominator of this last quotient has nonzero constant coefficient. The section identities give R_cH_c=tH_c-1, R_c(0)=c, j! P_N,j= P_N,0∏_c=1^jR_c. Cancellation of consecutive source sections proves the product identity and ties the ratios directly to the family under study.
Exact mathematical passage
H_c(t)=\sum_{n\ge0}\frac{t^{Nn+c}}{(Nn+c)!},\qquad A_c(t)=\sum_{n\ge0}\frac{t^{Nn}}{(Nn+c)!},\qquad R_c(t)=\frac{A_{c-1}(t)}{A_c(t)}\quad(1\le c<N).
t^{Nn+j}Use in source · 2 passages
Search aliases: t^Nn+j
Use in source · 72_recovered.tex, line 576
For integers N≥1 and 0≤ j<N, define E_N,j(t) = ∑_n≥0 fract^Nn+j(Nn+j)!. The forced factor t^j may be removed without changing the residue address.
Exact mathematical passage
\Ecal_{N,j}(t) = \sum_{n\geq0} \frac{t^{Nn+j}}{(Nn+j)!}.
t^{Nr}Use in source · 1 passage
Search aliases: t^Nr
Use in source · supplement_08.tex, line 150
pageheading4. From these sections to the original Euler packetsActual reciprocal, actual rational sequence, every order and step multiple The factorial-ratio identity is proved coefficient by coefficient: ∑_d≥0g_j(d)fract^dd! =j!∑_r≥0fract^Nr(Nr+j)!. The right side is precisely the normalized source kernel from the first block, not a replacement family. The earlier rational identity for its reciprocal is transported to Q_p[[t]]. A separate encoding theorem identifies the reciprocal's coefficient…
Exact mathematical passage
\sum_{d\ge0}g_j(d)\frac{t^d}{d!} =j!\sum_{r\ge0}\frac{t^{Nr}}{(Nr+j)!}.
T^pUse in source · 4 passages
Search aliases: T^p
Use in source · 20_arithmetic.tex, line 276
mate, use Θ=(1+T)d/dT. Both Θ and (Θ^p^2-Θ^p)/p preserve W_p. For the first operator, its coefficient formula is (d+1)a_d+1+da_d. Whenever the floor weight rises, the factor d+1 provides the missing p. For the second operator, the proof uses its full Leibniz correction and the scalar congruence p^2mid b^p^2-b^p. Weighted monomials factor as a_dT^d=(p^lfloor d/prfloora_d) (T^p/p)^lfloor d/prfloorT^dbmod p. The operator preserves these monomials and hence finite weighted polynomials. Each fixed ou…
Exact mathematical passage
a_dT^d=(p^{\lfloor d/p\rfloor}a_d) (T^p/p)^{\lfloor d/p\rfloor}T^{d\bmod p}.
T^{\Pcal}_{ba}Use in source · 2 passages
Search aliases: T^Pcal_ba T^ P_ba Pcal mathcal
Use in source · 02_coordinates.tex, line 224
[Refinement of operator blocks] Suppose that Q refines P through ρ. Then T^ P_ba = ∑_substackρ(β)=b ρ(α)=a T^ Q_βα. The matrix of T in fine coordinates is related to the matrix in coarse coordinates by [T]_ P = C_ P← Q[T]_ QJ_ Q← P.
Exact mathematical passage
T^{\Pcal}_{ba} = \sum_{\substack{\rho(\beta)=b\\\rho(\alpha)=a}} T^{\Qcal}_{\beta\alpha}.
T^{\Pcal}_{\beta\alpha}Use in source · 1 passage
Search aliases: T^Pcal_βα T^ P_βα Pcal beta alpha mathcal beta alpha
Use in source · 02_coordinates.tex, line 218
For T∈End(X) and a packet system P, write T^ P_βα=P_β T P_α.
Exact mathematical passage
T^{\Pcal}_{\beta\alpha}=P_\beta T P_\alpha.
t^qUse in source · 1 passage
Search aliases: t^q
Use in source · Paper_XI_Traditional_Final.tex, line 330
The first assertion is the Fourier idempotent calculation for a cyclic group. For the geometric Frobenius convention of [source reference], the relation is Φ^-1tΦ=t^q. Equivalently, Φ tΦ^-1=t^q^-1, with inverse taken modulo M since gcd(q,M)=1. Conjugating the Fourier sum and changing its summation index therefore gives Φ P_jΦ^-1=P_qj. The inverse Frobenius convention reverses the label direction while preserving the same orbits.
Exact mathematical passage
\Phi^{-1}t\Phi=t^q
t^{q^{-1}}Use in source · 1 passage
Search aliases: t^q^-1
Use in source · Paper_XI_Traditional_Final.tex, line 330
The first assertion is the Fourier idempotent calculation for a cyclic group. For the geometric Frobenius convention of [source reference], the relation is Φ^-1tΦ=t^q. Equivalently, Φ tΦ^-1=t^q^-1, with inverse taken modulo M since gcd(q,M)=1. Conjugating the Fourier sum and changing its summation index therefore gives Φ P_jΦ^-1=P_qj. The inverse Frobenius convention reverses the label direction while preserving the same orbits.
Exact mathematical passage
\Phi t\Phi^{-1}=t^{q^{-1}}
T^{\Qcal}_{\beta\alpha}Use in source · 2 passages
Search aliases: T^Qcal_βα T^ Q_βα Qcal beta alpha mathcal beta alpha
Use in source · 02_coordinates.tex, line 224
[Refinement of operator blocks] Suppose that Q refines P through ρ. Then T^ P_ba = ∑_substackρ(β)=b ρ(α)=a T^ Q_βα. The matrix of T in fine coordinates is related to the matrix in coarse coordinates by [T]_ P = C_ P← Q[T]_ QJ_ Q← P.
Exact mathematical passage
T^{\Pcal}_{ba} = \sum_{\substack{\rho(\beta)=b\\\rho(\alpha)=a}} T^{\Qcal}_{\beta\alpha}.
T^rUse in source · 3 passages
Search aliases: T^r
Use in source · 72_recovered.tex, line 1798
[Weighted convergence] If widehat F(T)=∑_r≥0c_rT^r and v_p(c_r)+rα⟶+∞, then the series converges whenever v_p(T)≥α.
Exact mathematical passage
\widehat F(T)=\sum_{r\geq0}c_rT^r
t^rUse in source · 5 passages
Search aliases: t^r
Use in source · 03_nonlinear_coordinates.tex, line 432
Let X be a Banach space. Fix s∈ X and continuous multilinear maps B_r:X^r⟶ X, 2≤ r≤ d. Set a= s , b_r= B_r , ϕ(t)=a+∑_r=2^db_rt^r. An ordered rooted tree has leaves labeled by s and every internal vertex of arity r labeled by B_r. Its value is obtained by evaluating from the leaves toward the root.
Exact mathematical passage
a=\|s\|, \qquad b_r=\|B_r\|, \qquad \phi(t)=a+\sum_{r=2}^{d}b_rt^r.
T^UUse in source · 2 passages
Search aliases: T^U
Use in source · visual_atlas.tex, line 275
vpage11Move the packets with the coordinatesCONJUGACY / STRUCTURE VERSUS RAW SPARSITYMain text: [source reference] Start with T=diag(1,2) and the two coordinate projections. Apply the invertible shear U. The transformed operator is T^U=UTU^-1, and the transformed projectors are Q_i=UP_iU^-1.
Exact mathematical passage
T^U=UTU^{-1}
t^\alphaUse in source · 2 passages
Search aliases: t^α alpha
Use in source · Paper_VI_Traditional_Final.tex, line 1203
[Fractional intertwining theorem] For bounded nonnegative self-adjoint A and B, D_α = fracsin(πα)π ∫_0^∞ t^α(tI+B)^-1D(tI+A)^-1 dt. The improper integral converges in operator norm. Moreover, normD_α ≤ K_αnormC^1-αnormD^α, K_α= frac2^1-αsin(πα) πα(1-α). If A,B≥ mI for some m>0, then the stronger linear estimate holds: normD_α ≤ α m^α-1normD. In particular, exact generator intertwining CA=BC implies exact fractional intertwining CA^α=B^α C.
Exact mathematical passage
\boxed{ D_\alpha = \frac{\sin(\pi\alpha)}{\pi} \int_0^\infty t^\alpha(tI+B)^{-1}D(tI+A)^{-1}\,dt. }
t^{\alpha-1}Use in source · 2 passages
Search aliases: t^α-1 alpha
Use in source · Paper_VI_Traditional_Final.tex, line 1237
For 0<α<1, the Balakrishnan formula gives A^α = fracsin(πα)π ∫_0^∞ t^α-1A(tI+A)^-1 dt, and likewise for B. Using A(tI+A)^-1=I-t(tI+A)^-1 and the rectangular resolvent identity (tI+B)^-1C-C(tI+A)^-1 = (tI+B)^-1(CA-BC)(tI+A)^-1, the difference of the two Balakrishnan integrals becomes the displayed resolvent-defect formula.
Exact mathematical passage
A^\alpha = \frac{\sin(\pi\alpha)}{\pi} \int_0^\infty t^{\alpha-1}A(tI+A)^{-1}\,dt,
t^{\alpha-2}Use in source · 1 passage
Search aliases: t^α-2 alpha
Use in source · Paper_VI_Traditional_Final.tex, line 1265
For small t, normCA(tI+A)^-1-B(tI+B)^-1C ≤ 2norm C, because the two positive resolvent factors are contractions. For large t, the resolvent identity gives the bound norm D/t. Splitting the integral at r=fracnorm D2norm C when C and D are nonzero yields normD_α · ≤ fracsin(πα)π ( 2norm C∫_0^r t^α-1 dt + norm D∫_r^∞ t^α-2 dt ) · = K_αnorm C^1-αnorm D^α. The zero cases are immediate.
Exact mathematical passage
\begin{aligned} \norm{D_\alpha} &\le \frac{\sin(\pi\alpha)}{\pi} \left( 2\norm C\int_0^r t^{\alpha-1}\,dt + \norm D\int_r^\infty t^{\alpha-2}\,dt \right)\\ &= K_\alpha\norm C^{1-\alpha}\norm D^\alpha. \end{aligned}
T_0Use in source · 7 passages
Search aliases: T_0
Use in source · 05_defects.tex, line 228
[Multilevel defect decomposition] The total defect from level L to level zero is C_0,LT_L-T_0C_0,L = ∑_ℓ=0^L-1 C_0,ℓΔ_ℓ C_ℓ+1,L. For bounded maps, C_0,LT_L-T_0C_0,L ≤ ∑_ℓ=0^L-1 C_0,ℓ Δ_ℓ C_ℓ+1,L .
Exact mathematical passage
C_{0,L}T_L-T_0C_{0,L} = \sum_{\ell=0}^{L-1} C_{0,\ell}\Delta_\ell C_{\ell+1,L}.
T_{11}Use in source · 1 passage
Search aliases: T_11
Use in source · visual_atlas.tex, line 207
headComposition sums over intermediate packets The (1,2) entry of the square has two contributing routes: (T^2)_12=T_11T_12+T_12T_22+T_13T_32 =1·2+2·1+0=4. For input 3 and output 1, only the route through packet 2 survives: (T^2)_13=T_12T_23=2·3=6. For packets of higher dimension, the same law uses actual linear maps as entries. The order of their composition still matters.
Exact mathematical passage
(T^2)_{12}=T_{11}T_{12}+T_{12}T_{22}+T_{13}T_{32} =1\cdot2+2\cdot1+0=4.
T_{12}Use in source · 2 passages
Search aliases: T_12
Use in source · visual_atlas.tex, line 207
headComposition sums over intermediate packets The (1,2) entry of the square has two contributing routes: (T^2)_12=T_11T_12+T_12T_22+T_13T_32 =1·2+2·1+0=4. For input 3 and output 1, only the route through packet 2 survives: (T^2)_13=T_12T_23=2·3=6. For packets of higher dimension, the same law uses actual linear maps as entries. The order of their composition still matters.
Exact mathematical passage
(T^2)_{12}=T_{11}T_{12}+T_{12}T_{22}+T_{13}T_{32} =1\cdot2+2\cdot1+0=4.
T_{13}Use in source · 1 passage
Search aliases: T_13
Use in source · visual_atlas.tex, line 207
headComposition sums over intermediate packets The (1,2) entry of the square has two contributing routes: (T^2)_12=T_11T_12+T_12T_22+T_13T_32 =1·2+2·1+0=4. For input 3 and output 1, only the route through packet 2 survives: (T^2)_13=T_12T_23=2·3=6. For packets of higher dimension, the same law uses actual linear maps as entries. The order of their composition still matters.
Exact mathematical passage
(T^2)_{12}=T_{11}T_{12}+T_{12}T_{22}+T_{13}T_{32} =1\cdot2+2\cdot1+0=4.
t_1^{w_1(e)}Use in source · 1 passage
Search aliases: t_1^w_1(e)
Use in source · 03_weighted.tex, line 449
[Several independent weights] Suppose every edge has a weight vector w(e)=(w_1(e),…,w_r(e))∈ N^r∖{0}. Replace t^w(e) by t_1^w_1(e)⋯ t_r^w_r(e). The resulting multigraded transfer series is rational in t_1,…,t_r.
Exact mathematical passage
t_1^{w_1(e)}\cdots t_r^{w_r(e)}.
T_2Use in source · 2 passages
Search aliases: T_2
Use in source · supplement_02.tex, line 185
The displayed manuscript row is kernel-checked. The scalar anchors are derived from the same general coefficient recurrence: (E_0,E_2,E_4,E_6,E_8)=(1,-1,5,-61,1385), (b_0,b_1,b_2,b_3,b_4)=(1,-2,16,-272,7936). The literal source sums at p=5 give (S_0,S_2,S_4,S_6,S_8)=(1,49,4705,428689,37665985). Consequently, T_2(5,8)=2 731 001 385, E_8-T_2(5,8)=-2 731 000 000. The source proves both 5^6mid E_8-T_2(5,8), 5^7nmid E_8-T_2(5,8). Thus the guaranteed power is attained in this example. No stronger unif…
Exact mathematical passage
T_2(5,8)=2\,731\,001\,385,\qquad E_8-T_2(5,8)=-2\,731\,000\,000.
T_{2,1}Use in source · 2 passages
Search aliases: T_2,1
Use in source · 06_models.tex, line 59
y any phase ω_N^k stays in the disk and has a finite inverse-chart image. The phase action is Γ↦ω_N^kΓ. Outside the open disk, the same chart formula requires 1-ω_N^kΓ≠0. Pulling the phase action back on this domain gives T_N,k(z) = frac(1+ω_N^k)z+(1-ω_N^k) (1-ω_N^k)z+(1+ω_N^k). For N=2 and k=1, T_2,1(z)=frac1z, z≠0,-1. Thus the nontrivial action at the first phase level exchanges normalized impedance and admittance. Calling these values ``packets'' is geometric terminology for the sampled orbit…
Exact mathematical passage
T_{2,1}(z)=\frac1z, \qquad z\ne0,-1. \label{eq:smith-inversion}
T_{2,M}Use in source · 3 passages
Search aliases: T_2,M
Use in source · 72_recovered.tex, line 1396
[Optimal dyadic truncation] Let T_2,M(n) = ∑_r=0^min(M,n/2) κ_2,rbinom n2rR_n-2r. Then E_n≡ T_2,M(n)pmod 2^4M+3. More precisely, v_2(E_n-T_2,M(n)) ≥ min_M+1≤ r≤ n/2 ( 4r-1+v_2binom n2r ). The uniform exponent 4M+3 is best possible.
Exact mathematical passage
T_{2,M}(n) = \sum_{r=0}^{\min(M,n/2)} \kappa_{2,r}\binom n{2r}R_{n-2r}.
T_{22}Use in source · 1 passage
Search aliases: T_22
Use in source · visual_atlas.tex, line 207
headComposition sums over intermediate packets The (1,2) entry of the square has two contributing routes: (T^2)_12=T_11T_12+T_12T_22+T_13T_32 =1·2+2·1+0=4. For input 3 and output 1, only the route through packet 2 survives: (T^2)_13=T_12T_23=2·3=6. For packets of higher dimension, the same law uses actual linear maps as entries. The order of their composition still matters.
Exact mathematical passage
(T^2)_{12}=T_{11}T_{12}+T_{12}T_{22}+T_{13}T_{32} =1\cdot2+2\cdot1+0=4.
T_{23}Use in source · 1 passage
Search aliases: T_23
Use in source · visual_atlas.tex, line 212
headComposition sums over intermediate packets The (1,2) entry of the square has two contributing routes: (T^2)_12=T_11T_12+T_12T_22+T_13T_32 =1·2+2·1+0=4. For input 3 and output 1, only the route through packet 2 survives: (T^2)_13=T_12T_23=2·3=6. For packets of higher dimension, the same law uses actual linear maps as entries. The order of their composition still matters.
Exact mathematical passage
(T^2)_{13}=T_{12}T_{23}=2\cdot3=6.
T_3Use in source · 1 passage
Search aliases: T_3
Use in source · supplement_03.tex, line 189
bottomrule The checked nonzero error valuations are E_8-T_0(8)=frac5494445, v_2(E_8-T_0(8))=5, E_8-T_3(8)=frac22937615, v_2(E_8-T_3(8))=15. They exceed the uniform guarantees. These two examples do not establish a universally stronger exponent.
Exact mathematical passage
E_8-T_3(8)=\frac{229376}{15},\qquad v_2(E_8-T_3(8))=15.
T_{32}Use in source · 1 passage
Search aliases: T_32
Use in source · visual_atlas.tex, line 207
headComposition sums over intermediate packets The (1,2) entry of the square has two contributing routes: (T^2)_12=T_11T_12+T_12T_22+T_13T_32 =1·2+2·1+0=4. For input 3 and output 1, only the route through packet 2 survives: (T^2)_13=T_12T_23=2·3=6. For packets of higher dimension, the same law uses actual linear maps as entries. The order of their composition still matters.
Exact mathematical passage
(T^2)_{12}=T_{11}T_{12}+T_{12}T_{22}+T_{13}T_{32} =1\cdot2+2\cdot1+0=4.
T_4Use in source · 2 passages
Search aliases: T_4
Use in source · atlas-pages.tex, line 222
/D_N. It compresses the linear differential equation into a nonlinear scalar identity. This division is allowed only on the zero-free set of the denominator. vfigriccati The coefficients retain the reduced logarithmic information: T_N(z)=N∑_m≥1κ_N,mz^Nm-1, T_N(n)=N(Nn+N-1)!κ_N,n+1. For order four, κ_4,1=1/24 and κ_4,2=17/10080. These give T_4(0)=1 and T_4(1)=34, explaining the last line of the plate. headA nonlinear equation with a precise origin If D=e^g locally, then g'=-T. Differentiating e^g…
Exact mathematical passage
T_4(0)=1
t_4Use in source · 1 passage
Search aliases: t_4
Use in source · root_scope.tex, line 277
For example, j=5 has (t_0,…,t_4)=(6,3,4,3,6). The exact root counts at distances 1,2,3,4,5,≥6 are 0,0,2,3,3,5. The older sufficient stages 3,5,7 are not the sharp emergence stages, which are 3,4,6. The first irregular label, j=17, has V≡1+Xpmod2. The regular-wall theorem does not apply. Complete coefficient certificates close distances 1 through 19, and the general separation theorem gives all seventeen simple roots for δ≥20. Those two kinds of evidence together resolve that family.
Exact mathematical passage
(t_0,\ldots,t_4)=(6,3,4,3,6)
t_aUse in source · 3 passages
Search aliases: t_a
Use in source · monograph_atlas.tex, line 430
ens also to have nonnegative entries and column sums one. Conjugate by the unitary Z. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become -1/4. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation. headA useful finite positive kernel For this nonnegative H, vectors u_a and times t_a≥0 satisfy ∑_a,b⟨ u_a,e^-(t_a+t_b)Hu_b⟩ = ∑_a e^-t_aHu_a ^2≥0. The factorization makes positivity transparent…
Exact mathematical passage
t_a\ge0
t_bUse in source · 11 passages
Search aliases: t_b
Use in source · monograph_atlas.tex, line 431
nnegative entries and column sums one. Conjugate by the unitary Z. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become -1/4. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation. headA useful finite positive kernel For this nonnegative H, vectors u_a and times t_a≥0 satisfy ∑_a,b⟨ u_a,e^-(t_a+t_b)Hu_b⟩ = ∑_a e^-t_aHu_a ^2≥0. The factorization makes positivity transparent. It remains a fini…
Exact mathematical passage
\sum_{a,b}\langle u_a,e^{-(t_a+t_b)H}u_b\rangle =\left\|\sum_a e^{-t_aH}u_a\right\|^2\ge0.
T_{ba}Use in source · 1 passage
Search aliases: T_ba
Use in source · visual_atlas.tex, line 29
T_ba · Input packet a, output packet b: P_bTP_a. · Blocks of a finite cyclic filter.
Exact mathematical passage
T_{ba}
t_cUse in source · 13 passages
Search aliases: t_c
Use in source · 01_manuscript.tex, line 1753
[Amortization threshold] Let s≥0 be a one time setup cost. Let t_b,t_c≥0 be the baseline and candidate times per use and let R∈ N be a reuse count. If t_b>t_c, the candidate is faster after R uses exactly when R> fracst_b-t_c. The smallest integer reuse count is R_min = lfloor fracst_b-t_c rfloor+1. If t_b≤ t_c, no positive reuse count amortizes a positive setup cost.
Exact mathematical passage
t_b,t_c\ge0
T_c'Use in source · 1 passage
Search aliases: T_c'
Use in source · visual_atlas.tex, line 547
Had we chosen the transported operator T_c'=CT_fC^-1, the defect would be zero and every polynomial action would intertwine exactly. That is a different choice of coarse operator, not a stronger conclusion from the faulty one.
Exact mathematical passage
T_c'=CT_fC^{-1}
T_c^nUse in source · 7 passages
Search aliases: T_c^n
Use in source · 05_defects.tex, line 27
[Power defect identity] For every integer n≥1, CT_f^n-T_c^nC = ∑_k=0^n-1T_c^n-1-kΔ T_f^k.
Exact mathematical passage
CT_f^n-T_c^nC = \sum_{k=0}^{n-1}T_c^{n-1-k}\Delta T_f^k.
T_c^{n+1}Use in source · 1 passage
Search aliases: T_c^n+1
Use in source · 05_defects.tex, line 36
The statement is clear for n=1. For the induction step, write CT_f^n+1-T_c^n+1C = (CT_f^n-T_c^nC)T_f+T_c^n(CT_f-T_cC) and apply the induction hypothesis.
Exact mathematical passage
CT_f^{n+1}-T_c^{n+1}C = (CT_f^n-T_c^nC)T_f+T_c^n(CT_f-T_cC)
T_c^{n-1-k}Use in source · 3 passages
Search aliases: T_c^n-1-k
Use in source · 05_defects.tex, line 27
[Power defect identity] For every integer n≥1, CT_f^n-T_c^nC = ∑_k=0^n-1T_c^n-1-kΔ T_f^k.
Exact mathematical passage
CT_f^n-T_c^nC = \sum_{k=0}^{n-1}T_c^{n-1-k}\Delta T_f^k.
T_dUse in source · 2 passages
Search aliases: T_d
Use in source · Paper_XII_Traditional_Final.tex, line 1064
[Simultaneous generalized eigenspaces] Assume that E' is algebraically closed and that T is generated by commuting operators T_1,…,T_d on V. For a maximal ideal m = (T_1-λ_1,…,T_d-λ_d), there is an integer N≥1 such that e_ mV = { v∈ V: (T_i-λ_i)^Nv=0 for every i }. Thus the canonical Artin packet is the simultaneous generalized eigenspace.
Exact mathematical passage
T_1,\ldots,T_d
t_dUse in source · 1 passage
Search aliases: t_d
Use in source · Paper_X_Block27_Amended.tex, line 294
[Smooth formal branch count] If R_bar x≅ O[[t_1,…,t_d]] as complete local O-algebras, then #X_m(bar x)=q^d(m-1). Fix a uniformizer and a section s:k→ O of reduction, with s(0)=0. Every O-valued branch is encoded by d positive-depth digit streams t_i=s(a_i,1)ϖ+s(a_i,2)ϖ^2+⋯, a_i,j∈ k,
Exact mathematical passage
R_{\bar x}\cong\Ocal[[t_1,\ldots,t_d]]\quad\text{as complete local }\Ocal\text{-algebras},
t_{\mathrm{decode}}Use in source · 1 passage
Search aliases: t_decode mathrm
Use in source · 02_supplement_02.tex, line 203
Contracts, grouping, and amortization. textitSource items 2.9, 10.2, 10.4--10.6, and 11.3 Output conversion. A decoded run returns the actual pair (decode(compressed(x)), t_run(x)+t_decode(compressed(x))). Under the declared output equality, Lean proves that this returns the same explicit result and charges the conversion time. It also proves the downstream composition identity when compressed output is consumed directly. Omitting a positive decode cost strictly understates the modeled explicit-…
Exact mathematical passage
\bigl(\operatorname{decode}(\operatorname{compressed}(x)), \;t_{\mathrm{run}}(x)+t_{\mathrm{decode}}(\operatorname{compressed}(x))\bigr).
T_eUse in source · 5 passages
Search aliases: T_e
Use in source · 03_supplement_03.tex, line 136
The actual Euclidean transfer exponential. Let H be a Hermitian matrix on a nonzero complex state space of finite dimension. The source defines T_e(t)=exp(-t(H-eI)) using Lean's exponential defined by a power series. The proof then relates this defined operator to its spectral expansion.
Exact mathematical passage
T_e(t)=\exp\bigl(-t(H-eI)\bigr)
t_eUse in source · 3 passages
Search aliases: t_e
Use in source · 01_manuscript.tex, line 1796
[Materialization threshold] Suppose a compressed representation costs t_c≥0 per downstream use, an explicit representation costs t_e≥0 per use, and one time materialization costs m≥0, with reuse count R∈ N. Materialization is profitable exactly when R> fracmt_c-t_e and t_c>t_e.
Exact mathematical passage
t_e\ge0
T_{E_0}Use in source · 1 passage
Search aliases: T_E_0
Use in source · 03_supplement_03.tex, line 161
Positivity, semigroup composition, and the interval. Every eigenvalue of T_e(t) is strictly positive, for arbitrary real e,t, since the real exponential is positive. For e=E_0 and t≥0 it is also at most one. Therefore T_E_0(t)>0, Spec T_E_0(t)⊆(0,1]. The two scalar multiples of H-eI commute, so the exponential defined by the power series obeys T_e(t+s)=T_e(t)T_e(s), T_e(0)=I. The semigroup identity is proved for all real t,s. Nonnegative time is needed for the upper spectral bound, not for the c…
Exact mathematical passage
\boxed{T_{E_0}(t)>0,\qquad \Spec T_{E_0}(t)\subseteq(0,1].}
T_{e_j}Use in source · 1 passage
Search aliases: T_e_j
Use in source · 03_supplement_02.tex, line 17
0 · x≠ a. The seed is ψ_0=ket0. Define a translation operator by (T_a v)(x)=v(x-a). At position j, the local Pauli X is X_j=T_e_j, where e_j has a single nonzero binary coordinate. The formal circuit uses the identity when a_j=0 and X_j when a_j=1.
Exact mathematical passage
X_j=T_{e_j}
T_f^kUse in source · 3 passages
Search aliases: T_f^k
Use in source · 05_defects.tex, line 27
[Power defect identity] For every integer n≥1, CT_f^n-T_c^nC = ∑_k=0^n-1T_c^n-1-kΔ T_f^k.
Exact mathematical passage
CT_f^n-T_c^nC = \sum_{k=0}^{n-1}T_c^{n-1-k}\Delta T_f^k.
T_f^nUse in source · 7 passages
Search aliases: T_f^n
Use in source · 05_defects.tex, line 27
[Power defect identity] For every integer n≥1, CT_f^n-T_c^nC = ∑_k=0^n-1T_c^n-1-kΔ T_f^k.
Exact mathematical passage
CT_f^n-T_c^nC = \sum_{k=0}^{n-1}T_c^{n-1-k}\Delta T_f^k.
T_f^{n+1}Use in source · 1 passage
Search aliases: T_f^n+1
Use in source · 05_defects.tex, line 36
The statement is clear for n=1. For the induction step, write CT_f^n+1-T_c^n+1C = (CT_f^n-T_c^nC)T_f+T_c^n(CT_f-T_cC) and apply the induction hypothesis.
Exact mathematical passage
CT_f^{n+1}-T_c^{n+1}C = (CT_f^n-T_c^nC)T_f+T_c^n(CT_f-T_cC)
T_HUse in source · 3 passages
Search aliases: T_H
Use in source · strengthening.tex, line 175
[All-row prime-power transfer] Let p be prime, a≥1, N=p^a, and 0≤ j<N. The coefficients C_N,j(n) belong to Z_p for all n≥0. Set A_N,j(n)=(-1)^n C_N,j(n) and T_H=S^H-(-1)^H. For k≥1 and positive H_1,…,H_k, v_p(Δ_H_1⋯Δ_H_kA_N,j(n)) · ≥ k+∑_ν=1^k v_p(H_ν),
Exact mathematical passage
T_H=S^H-(-1)^H
T_{H_1}Use in source · 1 passage
Search aliases: T_H_1
Use in source · strengthening.tex, line 176
[All-row prime-power transfer] Let p be prime, a≥1, N=p^a, and 0≤ j<N. The coefficients C_N,j(n) belong to Z_p for all n≥0. Set A_N,j(n)=(-1)^n C_N,j(n) and T_H=S^H-(-1)^H. For k≥1 and positive H_1,…,H_k, v_p(Δ_H_1⋯Δ_H_kA_N,j(n)) · ≥ k+∑_ν=1^k v_p(H_ν), v_p(T_H_1⋯ T_H_kE_Nn^(N,j)) · ≥ v_p(j!)+k+∑_ν=1^k v_p(H_ν). At p=2, the ordinary differences of C_N,j and of the original coefficients satisfy the same bounds. These dyadic statements also hold for N=1.
Exact mathematical passage
v_p(\Delta_{H_1}\cdots\Delta_{H_k}A_{N,j}(n)) &\geq k+\sum_{\nu=1}^k v_p(H_\nu), \label{eq:prime-power-signed-bound}\\ v_p(T_{H_1}\cdots T_{H_k}E_{Nn}^{(N,j)}) &\geq v_p(j!)+k+\sum_{\nu=1}^k v_p(H_\nu). \label{eq:prime-power-original-twist}
T_{H_k}Use in source · 1 passage
Search aliases: T_H_k
Use in source · strengthening.tex, line 176
[All-row prime-power transfer] Let p be prime, a≥1, N=p^a, and 0≤ j<N. The coefficients C_N,j(n) belong to Z_p for all n≥0. Set A_N,j(n)=(-1)^n C_N,j(n) and T_H=S^H-(-1)^H. For k≥1 and positive H_1,…,H_k, v_p(Δ_H_1⋯Δ_H_kA_N,j(n)) · ≥ k+∑_ν=1^k v_p(H_ν), v_p(T_H_1⋯ T_H_kE_Nn^(N,j)) · ≥ v_p(j!)+k+∑_ν=1^k v_p(H_ν). At p=2, the ordinary differences of C_N,j and of the original coefficients satisfy the same bounds. These dyadic statements also hold for N=1.
Exact mathematical passage
v_p(\Delta_{H_1}\cdots\Delta_{H_k}A_{N,j}(n)) &\geq k+\sum_{\nu=1}^k v_p(H_\nu), \label{eq:prime-power-signed-bound}\\ v_p(T_{H_1}\cdots T_{H_k}E_{Nn}^{(N,j)}) &\geq v_p(j!)+k+\sum_{\nu=1}^k v_p(H_\nu). \label{eq:prime-power-original-twist}
T_iUse in source · 13 passages
Search aliases: T_i
Use in source · Paper_X_Block27_Amended.tex, line 1619
Let E be a field of characteristic zero and let T_1,…,T_N be commuting operators on a finite-dimensional E-vector space A, each diagonalizable over E. Thus the selected spectra split over the coefficient field. Let a_i∈ E be a selected target scalar for T_i, possibly absent from its spectrum. Define e_i = ∏_μ∈Spec(T_i)∖{a_i} fracT_i-μ Ia_i-μ, e_T=∏_i=1^Ne_i, with e_i=0 if a_i is absent.
Exact mathematical passage
T_i
t_iUse in source · 16 passages
Search aliases: t_i
Use in source · continuation_01.tex, line 49
For every P∈ Z[X_1,…,X_d], integer a, and x,t∈ Z^d, Lean proves a^2mid P(x+at)-P(x)-a∑_i=1^d(∂_iP)(x)t_i. The proof uses polynomial induction: constants, addition, and multiplication by a variable. The product rule for the actual partial derivatives controls the multiplication step. It does not assume a Taylor identity or a liftability condition.
Exact mathematical passage
\boxed{ a^2\mid P(x+at)-P(x)-a\sum_{i=1}^{d}(\partial_iP)(x)t_i. }
T_{ij}Use in source · 3 passages
Search aliases: T_ij
Use in source · visual_atlas.tex, line 193
vpage8A transfer block has an input and an outputOPERATOR COORDINATES / MULTIPLICATION AS A SUMMain text: [source reference] Use the three coordinate packets of dimension one in Q^3. The block T_ij=P_iTP_j sends input packet j into output packet i. For this example, its nonzero scalar is the (i,j) matrix entry.
Exact mathematical passage
T_{ij}=P_iTP_j
t_jUse in source · 10 passages
Search aliases: t_j
Use in source · 04_correction_transport.tex, line 1023
Let U(t,s) be the exact propagator generated by K. Since K is continuously differentiable, the integral equation for U(t+h,t) gives a uniform estimate U(t+h,t)-I-hK(t) ≤ C_1h^2. By [source reference], the direct rotation V_j from t_j to t_j+1 therefore satisfies V_j-U(t_j+1,t_j) ≤ C_2h_j^2, h_j=t_j+1-t_j. Both V_j and the exact propagator are unitary. A telescoping product comparison gives V_π-U(T,0) ≤ ∑_j V_j-U(t_j+1,t_j) ≤ C_2∑_jh_j^2 ≤ C_2T|π|. This proves norm convergence and the stated firs…
Exact mathematical passage
t_j
T_{j+1}Use in source · 4 passages
Search aliases: T_j+1
Use in source · Paper_VI_Traditional_Final.tex, line 5249
Let C_j:X_j+1→ X_j be bounded linear maps, and put C_ℓ,m=C_ℓ⋯ C_m-1 for ℓ<m, with C_j,j=I. No common ambient space is assumed. [Multilevel telescoping identity] Let T_j be bounded operators on X_j, and set E_j=C_jT_j+1-T_jC_j. Then C_ℓ,mT_m-T_ℓ C_ℓ,m = ∑_j=ℓ^m-1 C_ℓ,jE_jC_j+1,m.
Exact mathematical passage
E_j=C_jT_{j+1}-T_jC_j.
t_{j+1}Use in source · 5 passages
Search aliases: t_j+1
Use in source · 04_correction_transport.tex, line 1023
Let U(t,s) be the exact propagator generated by K. Since K is continuously differentiable, the integral equation for U(t+h,t) gives a uniform estimate U(t+h,t)-I-hK(t) ≤ C_1h^2. By [source reference], the direct rotation V_j from t_j to t_j+1 therefore satisfies V_j-U(t_j+1,t_j) ≤ C_2h_j^2, h_j=t_j+1-t_j. Both V_j and the exact propagator are unitary. A telescoping product comparison gives V_π-U(T,0) ≤ ∑_j V_j-U(t_j+1,t_j) ≤ C_2∑_jh_j^2 ≤ C_2T|π|. This proves norm convergence and the stated firs…
Exact mathematical passage
t_{j+1}
T_{j-1}Use in source · 1 passage
Search aliases: T_j-1
Use in source · 03_nonlinear_coordinates.tex, line 317
t↦ x+th. It contains [0,1], and repeated differentiation gives g^(r)(t)=D^rF(x+th)[h,…,h]. The derivatives are continuous on the compact interval, so their Bochner integrals exist in the Banach codomain. Put T_j=frac1j!∫_0^1(1-t)^j g^(j+1)(t) dt (0≤ j≤ m). The fundamental theorem of calculus gives T_0=g(1)-g(0). For j≥1, integration by parts gives T_j=T_j-1-g^(j)(0)/j!. Iterating this relation proves the Taylor identity with remainder T_m=R_m+1(x,h) and its exact factorial normalization.
Exact mathematical passage
T_j=T_{j-1}-g^{(j)}(0)/j!
t_kUse in source · 8 passages
Search aliases: t_k
Use in source · Paper_V_Traditional_Final.tex, line 2901
Let t_k=kh, k=0,…,n, h=frac1n. Assume f(t) = s(t)+∑_q=1^Qa_qH_τ_q(t), H_τ(t)= 1_[τ,1](t), where s is Lipschitz with constant M, and Q≥1 is finite with nonzero jump amplitudes. The zero-jump case requires no minimum-amplitude condition. Suppose no jump lies at a sample point and no grid cell contains more than one jump. Observe y_k=f(t_k)+η_k. Define the adjacent contrast D_k=y_k+1-y_k.
Exact mathematical passage
t_k=kh, \qquad k=0,\ldots,n, \qquad h=\frac1n.
t_{k+1}Use in source · 2 passages
Search aliases: t_k+1
Use in source · Paper_V_Traditional_Final.tex, line 2930
[Contrast gap] Assume |η_k|≤ε and set b_h=Mh+2ε. If the cell (t_k,t_k+1) contains no jump, then |D_k|≤ b_h. If it contains one jump of amplitude a_q, then |D_k-a_q|≤ b_h and therefore |D_k|≥ |a_q|-b_h.
Exact mathematical passage
(t_k,t_{k+1})
T_{ks}Use in source · 1 passage
Search aliases: T_ks
Use in source · Paper_V_Traditional_Final.tex, line 1470
T_ks · P_N,kHP_N,s · input-to-output phase transfer
Exact mathematical passage
T_{ks}
T_LUse in source · 3 passages
Search aliases: T_L
Use in source · 05_defects.tex, line 228
[Multilevel defect decomposition] The total defect from level L to level zero is C_0,LT_L-T_0C_0,L = ∑_ℓ=0^L-1 C_0,ℓΔ_ℓ C_ℓ+1,L. For bounded maps, C_0,LT_L-T_0C_0,L ≤ ∑_ℓ=0^L-1 C_0,ℓ Δ_ℓ C_ℓ+1,L .
Exact mathematical passage
C_{0,L}T_L-T_0C_{0,L} = \sum_{\ell=0}^{L-1} C_{0,\ell}\Delta_\ell C_{\ell+1,L}.
t_LUse in source · 8 passages
Search aliases: t_L
Use in source · worked_atlas.tex, line 68
AtlasHeadA compatible root is built one digit at a timeatlas:smooth Take F(x)=x^2-7 over the 3-adic integers and start with x_1=1 modulo three. If F(x_L)=3^Lr_L, the next representative has the form x_L+1=x_L+3^Lt_L. Its correcting digit satisfies r_L+2x_Lt_L≡0pmod3. AtlasImagesmooth_tower For example, F(1)/3=-2 and 2x_1=2, so the next digit is one and x_2=4. Then F(4)/9=1, again giving digit one and x_3=13. At the next stage the residual is divisible by three, so digit zero is correct. A zero d…
Exact mathematical passage
x_{L+1}=x_L+3^Lt_L
t_L^{-q}Use in source · 2 passages
Search aliases: t_L^-q
Use in source · dyadic_synthesis.tex, line 46
At a recursive coefficient field L of characteristic two, with residue field F_2 and uniformizer t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two, and z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the ex…
Exact mathematical passage
E=L(\alpha),\qquad \alpha^2-\alpha=t_L^{-q},\qquad a=(q+1)/2,\qquad z=t_L^a\alpha.
t_L^{-q_d}Use in source · 1 passage
Search aliases: t_L^-q_d
Use in source · odd_synthesis.tex, line 51
At a recursive base (L,t_L), introduce E=L(α), α^p-α=t_L^-q_d. This is a separable totally ramified extension of degree p, with v_E(α)=-q_d and Tr_E/Lα=0. Choose 1≤ b<p with q_db≡-1pmod p, set a=(q_db+1)/p, and put z=t_L^aα^b. Then z is an explicit uniformizer. Write its Eisenstein polynomial as f(X)=X^p+⋯+a_0.
Exact mathematical passage
E=L(\alpha),\qquad \alpha^p-\alpha=t_L^{-q_d}.
t_L^aUse in source · 3 passages
Search aliases: t_L^a
Use in source · dyadic_synthesis.tex, line 46
At a recursive coefficient field L of characteristic two, with residue field F_2 and uniformizer t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two, and z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the ex…
Exact mathematical passage
E=L(\alpha),\qquad \alpha^2-\alpha=t_L^{-q},\qquad a=(q+1)/2,\qquad z=t_L^a\alpha.
T_mUse in source · 4 passages
Search aliases: T_m
Use in source · 03_nonlinear_coordinates.tex, line 317
h,…,h]. The derivatives are continuous on the compact interval, so their Bochner integrals exist in the Banach codomain. Put T_j=frac1j!∫_0^1(1-t)^j g^(j+1)(t) dt (0≤ j≤ m). The fundamental theorem of calculus gives T_0=g(1)-g(0). For j≥1, integration by parts gives T_j=T_j-1-g^(j)(0)/j!. Iterating this relation proves the Taylor identity with remainder T_m=R_m+1(x,h) and its exact factorial normalization.
Exact mathematical passage
T_m=R_{m+1}(x,h)
\mathcal T_MUse in source · 2 passages
Search aliases: T_M mathcal
Use in source · root_scope.tex, line 62
tion v_2(β_m)=s_2(m)-1, v_2(κ_m)=4m-1. The source identity ∑_N≥0A(N)fract^2N(2N)! =fracsinh(2t)sinh t2t^2 implies, after multiplication by 4t^2/sinh^2(2t), E_2N=∑_m=0^N κ_mbinom2N2mA(N-m). For a retained depth M≥0, put T_M(N)= E_2N -∑_m=0^min(M,N)κ_mbinom2N2mA(N-m). An empty tail, N≤ M, is exactly zero.
Exact mathematical passage
\mathcal T_M(N)=\mathcal E_{2N} -\sum_{m=0}^{\min(M,N)}\kappa_m\binom{2N}{2m}A(N-m).
\mathfrak T_mUse in source · 3 passages
Search aliases: T_m mathfrak
Use in source · 03_nonlinear_coordinates.tex, line 467
Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a to a leaf and applying the scalar map (t_1,…,t_r)↦ b_rt_1⋯ t_r at a vertex of arity r. A second induction gives ∑_τ∈ T_m B_τ(s) ≤ ∑_τ∈ T_mw(τ) =t_m, wh…
Exact mathematical passage
\mathfrak T_m
t_mUse in source · 42 passages
Search aliases: t_m
Use in source · 03_nonlinear_coordinates.tex, line 476
_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a to a leaf and applying the scalar map (t_1,…,t_r)↦ b_rt_1⋯ t_r at a vertex of arity r. A second induction gives ∑_τ∈ T_m B_τ(s) ≤ ∑_τ∈ T_mw(τ) =t_m, where t_0=0 and t_m+1=ϕ(t_m). The sequence (t_m) is increasing because ϕ has nonnegative coefficients. It is bounded by R because t_0≤ R and ϕ(R)≤ R. Th…
Exact mathematical passage
\sum_{\tau\in\mathfrak T_m}\|B_\tau(s)\| \le \sum_{\tau\in\mathfrak T_m}w(\tau) =t_m,
t_{m+1}Use in source · 1 passage
Search aliases: t_m+1
Use in source · 03_nonlinear_coordinates.tex, line 482
terate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a to a leaf and applying the scalar map (t_1,…,t_r)↦ b_rt_1⋯ t_r at a vertex of arity r. A second induction gives ∑_τ∈ T_m B_τ(s) ≤ ∑_τ∈ T_mw(τ) =t_m, where t_0=0 and t_m+1=ϕ(t_m). The sequence (t_m) is increasing because ϕ has nonnegative coefficients. It is bounded by R because t_0≤ R and ϕ(R)≤ R. The sum of the norms of all tree values is therefore finite. The tree series converg…
Exact mathematical passage
t_{m+1}=\phi(t_m)
T_{m,p}Use in source · 6 passages
Search aliases: T_m,p
Use in source · 72_recovered.tex, line 823
[Fourier invariance of the standard window operator] Let p be odd, let mmid p-1, and let h=(p-1)/m. Define (T_m,pf)_n = ∑_q=0^nbinommnmqf_q in F_p. Let V_h be the vector space of sequences satisfying f_n+h=f_n (n≥1). Then T_m,p(V_h)=V_h. The restriction of T_m,p to V_h is an automorphism.
Exact mathematical passage
(T_{m,p}f)_n = \sum_{q=0}^{n}\binom{mn}{mq}f_q \quad\text{in }\FF_p.
t_m^{-1}Use in source · 1 passage
Search aliases: t_m^-1
Use in source · Paper_IV_Traditional_Final.tex, line 2818
For N≥3, [source reference] writes all but finitely many reduced zeros as λ_m = (A_N(m+frac12))^N(1+ε_m) where λ_m>0, 1+ε_m>0, and ε_m = O(t_m^-1e^-γ t_m). On every compact set of s values, (1+ε_m)^-s-1 = O(|ε_m|) uniformly for large m. The difference λ_m^-s - A_N^-Ns(m+frac12)^-Ns therefore has a normally convergent sum on every compact subset of C. Indeed, the model power has at most polynomial growth in m on a compact set, while |ε_m| decays exponentially. The finitely many exceptional terms …
Exact mathematical passage
\varepsilon_m = O\left(t_m^{-1}e^{-\gamma t_m}\right).
t_m^2Use in source · 1 passage
Search aliases: t_m^2
Use in source · Paper_IV_Traditional_Final.tex, line 2232
he nonzero zero multiset of Δ_N. All but finitely many members of Λ_N are positive and simple. They can be indexed as λ_m=τ_m^N for all sufficiently large m. If N≥3, the notation may be chosen as in [source reference], and λ_m = t_m^N +O(t_m^N-1e^-γ t_m). For N=2, take τ_m=t_m=π(m+1/2) for m≥0. Then λ_m=t_m^2 exactly.
Exact mathematical passage
\lambda_m=t_m^2
t_m^NUse in source · 1 passage
Search aliases: t_m^N
Use in source · Paper_IV_Traditional_Final.tex, line 2225
[Reduced zeros] Let Λ_N be the nonzero zero multiset of Δ_N. All but finitely many members of Λ_N are positive and simple. They can be indexed as λ_m=τ_m^N for all sufficiently large m. If N≥3, the notation may be chosen as in [source reference], and λ_m = t_m^N +O(t_m^N-1e^-γ t_m). For N=2, take τ_m=t_m=π(m+1/2) for m≥0. Then λ_m=t_m^2 exactly.
Exact mathematical passage
\lambda_m = t_m^N +O\left(t_m^{N-1}e^{-\gamma t_m}\right).
t_m^{N-1}Use in source · 1 passage
Search aliases: t_m^N-1
Use in source · Paper_IV_Traditional_Final.tex, line 2225
[Reduced zeros] Let Λ_N be the nonzero zero multiset of Δ_N. All but finitely many members of Λ_N are positive and simple. They can be indexed as λ_m=τ_m^N for all sufficiently large m. If N≥3, the notation may be chosen as in [source reference], and λ_m = t_m^N +O(t_m^N-1e^-γ t_m). For N=2, take τ_m=t_m=π(m+1/2) for m≥0. Then λ_m=t_m^2 exactly.
Exact mathematical passage
\lambda_m = t_m^N +O\left(t_m^{N-1}e^{-\gamma t_m}\right).
t_NUse in source · 5 passages
Search aliases: t_N
Use in source · 04_correction_transport.tex, line 998
[Direct rotations converge to packet transport] Assume the complete packet family P(t) is twice continuously differentiable in operator norm on [0,T]. For a partition 0=t_0<t_1<⋯<t_N=T with sufficiently small mesh, define V_π = U( P(t_N), P(t_N-1)) ⋯ U( P(t_1), P(t_0)). Then V_π⟶ U(T) in operator norm, where U is the canonical transport from [source reference]. On a fixed compact interval the error is bounded by a constant times the mesh.
Exact mathematical passage
0=t_0<t_1<\cdots<t_N=T
t_nUse in source · 8 passages
Search aliases: t_n
Use in source · 30_lifting.tex, line 591
[Every compatible integer parameter path] In the preceding notation, assume J_x is surjective. For every supplied integer path t_n∈ Z^k satisfying t_0=free(x), t_n+1-t_n∈ q^L+n Z^k, there is a compatible polynomial tower z_n∈ Z^m+k with z_0=x, free(z_n)=t_n, F(z_n)∈ q^L+n Z^e, z_n+1-z_n∈ q^L+n Z^m+k. The free-coordinate equality is exact in integers. Both reduced Jacobian blocks remain fixed, and z_b-z_a∈ q^L+a Z^m+k whenever a≤ b.
Exact mathematical passage
t_n\in\Z^k
T_{n+1}Use in source · 2 passages
Search aliases: T_n+1
Use in source · Paper_VI_Traditional_Final.tex, line 2544
[Weighted summable defect completion] Define the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_j=n^∞w_m,j sup_0≤ t≤ TnormE_j(t).
Exact mathematical passage
E_n(t)=C_nT_{n+1}(t)-T_n(t)C_n.
t_{n+1}Use in source · 2 passages
Search aliases: t_n+1
Use in source · 30_lifting.tex, line 592
[Every compatible integer parameter path] In the preceding notation, assume J_x is surjective. For every supplied integer path t_n∈ Z^k satisfying t_0=free(x), t_n+1-t_n∈ q^L+n Z^k, there is a compatible polynomial tower z_n∈ Z^m+k with z_0=x, free(z_n)=t_n, F(z_n)∈ q^L+n Z^e, z_n+1-z_n∈ q^L+n Z^m+k. The free-coordinate equality is exact in integers. Both reduced Jacobian blocks remain fixed, and z_b-z_a∈ q^L+a Z^m+k whenever a≤ b.
Exact mathematical passage
t_0=\operatorname{free}(x),\qquad t_{n+1}-t_n\in q^{L+n}\Z^k,
T_{N,k}Use in source · 4 passages
Search aliases: T_N,k
Use in source · 06_models.tex, line 51
t z=Z/Z_0, and assume z≠-1. Write Γ=fracz-1z+1. On the open reflection disk |Γ|<1, multiplication by any phase ω_N^k stays in the disk and has a finite inverse-chart image. The phase action is Γ↦ω_N^kΓ. Outside the open disk, the same chart formula requires 1-ω_N^kΓ≠0. Pulling the phase action back on this domain gives T_N,k(z) = frac(1+ω_N^k)z+(1-ω_N^k) (1-ω_N^k)z+(1+ω_N^k). For N=2 and k=1, T_2,1(z)=frac1z, z≠0,-1. Thus the nontrivial action at the first phase level exchanges normalized impeda…
Exact mathematical passage
T_{N,k}(z) = \frac{(1+\omega_N^k)z+(1-\omega_N^k)} {(1-\omega_N^k)z+(1+\omega_N^k)}.
T_{N,k+\ell\, (\mathrm{mod}\,N)}Use in source · 1 passage
Search aliases: T_N,k+ℓ (mod N) ell mathrm
Use in source · Paper_V_Traditional_Final.tex, line 1153
obius action] Regard normalized impedance as the projective line P^1( C)= C∪{∞}. For N≥1, the pullback is a globally defined projective action. In the finite chart, wherever the denominator is nonzero, it is T_N,k(z) = frac(1+ω_N^k)z+(1-ω_N^k) (1-ω_N^k)z+(1+ω_N^k). As maps on P^1( C) it satisfies T_N,k∘ T_N,ℓ=T_N,k+ℓ (mod N).
Exact mathematical passage
T_{N,k}\circ T_{N,\ell}=T_{N,k+\ell\, (\mathrm{mod}\,N)}.
t_{n,\delta}Use in source · 3 passages
Search aliases: t_n,δ delta
Use in source · Paper_V_Traditional_Final.tex, line 3099
[Probability recovery bound] Fix 0<δ<1 and define t_n,δ = 2σ sqrtlog(frac2nδ) and b_h,δ=Mh+t_n,δ. With probability at least 1-δ, every contrast satisfies the conclusions of [source reference] with b_h replaced by b_h,δ.
Exact mathematical passage
t_{n,\delta} = 2\sigma \sqrt{\log\left(\frac{2n}{\delta}\right)}
T_{N,\ell}Use in source · 1 passage
Search aliases: T_N,ℓ ell
Use in source · Paper_V_Traditional_Final.tex, line 1153
obius action] Regard normalized impedance as the projective line P^1( C)= C∪{∞}. For N≥1, the pullback is a globally defined projective action. In the finite chart, wherever the denominator is nonzero, it is T_N,k(z) = frac(1+ω_N^k)z+(1-ω_N^k) (1-ω_N^k)z+(1+ω_N^k). As maps on P^1( C) it satisfies T_N,k∘ T_N,ℓ=T_N,k+ℓ (mod N).
Exact mathematical passage
T_{N,k}\circ T_{N,\ell}=T_{N,k+\ell\, (\mathrm{mod}\,N)}.
t_{N-1}Use in source · 2 passages
Search aliases: t_N-1
Use in source · 04_correction_transport.tex, line 1002
[Direct rotations converge to packet transport] Assume the complete packet family P(t) is twice continuously differentiable in operator norm on [0,T]. For a partition 0=t_0<t_1<⋯<t_N=T with sufficiently small mesh, define V_π = U( P(t_N), P(t_N-1)) ⋯ U( P(t_1), P(t_0)). Then V_π⟶ U(T) in operator norm, where U is the canonical transport from [source reference]. On a fixed compact interval the error is bounded by a constant times the mesh.
Exact mathematical passage
V_\pi = \Ucal(\Pcal(t_N),\Pcal(t_{N-1})) \cdots \Ucal(\Pcal(t_1),\Pcal(t_0)).
t_{n-1}Use in source · 1 passage
Search aliases: t_n-1
Use in source · 02_strengthened.tex, line 804
[Convergence of short packet rotations] Let (P_a(t)) be twice continuously differentiable on [0,T]. For a partition 0=t_0<t_1<⋯<t_n=T, let W_π = U( P(t_n), P(t_n-1)) ⋯ U( P(t_1), P(t_0)). For sufficiently fine partitions the factors are defined, and normW_π-W(T) ≤ C_Tabsπ for a constant depending on the packet path but not on the partition. Here the size of the partition is measured by its mesh, which is the length of its largest subinterval.
Exact mathematical passage
W_\pi = \Ucal\bigl(\Pcal(t_n),\Pcal(t_{n-1})\bigr) \cdots \Ucal\bigl(\Pcal(t_1),\Pcal(t_0)\bigr).
T_{nn}Use in source · 6 passages
Search aliases: T_nn
Use in source · 10_foundations.tex, line 180
[Forward substitution and denominator clearing] Let T_nq be coefficients over a field with T_nn≠0 for every n. For any sequence b, define a_n=fracb_n-∑_q<nT_nqa_qT_nn. This constructs the unique solution of ∑_q=0^nT_nqa_q=b_n for all n. If T_nq and b_n are integers and the solution is taken over Q, then (∏_r=0^nT_rr)a_n∈ Z.
Exact mathematical passage
T_{nn}\ne0
T_{nq}Use in source · 7 passages
Search aliases: T_nq
Use in source · 10_foundations.tex, line 180
[Forward substitution and denominator clearing] Let T_nq be coefficients over a field with T_nn≠0 for every n. For any sequence b, define a_n=fracb_n-∑_q<nT_nqa_qT_nn. This constructs the unique solution of ∑_q=0^nT_nqa_q=b_n for all n. If T_nq and b_n are integers and the solution is taken over Q, then (∏_r=0^nT_rr)a_n∈ Z.
Exact mathematical passage
T_{nq}
t_oUse in source · 3 passages
Search aliases: t_o
Use in source · 01_manuscript.tex, line 1784
[Graph replay threshold] If graph capture costs c≥0, ordinary execution costs t_o≥0, and replay costs 0≤ t_r<t_o, then capture is profitable when R> fracct_o-t_r.
Exact mathematical passage
t_o\ge0
T_PUse in source · 3 passages
Search aliases: T_P
Use in source · 03_supplement_04.tex, line 10
Deriving the estimate by differentiation. For a skew-adjoint generator G, define the actual conjugation path T_P(t)=exp(tG)Pexp(-tG). The formalization derives its derivative from the exponential power series and the product rule: T_P'(t)=exp(tG)(GP-PG)exp(-tG). Both exponential factors are proved unitary. Left and right multiplication by a unitary preserve the C-star norm. Therefore T_P'(t) = GP-PG , T_P(t)-T_P(s) ≤ |t-s| GP-PG . The second inequality follows from the mean value inequality in n…
Exact mathematical passage
T_P(t)=\exp(tG)P\exp(-tG).
T_P'Use in source · 2 passages
Search aliases: T_P'
Use in source · 03_supplement_04.tex, line 14
Deriving the estimate by differentiation. For a skew-adjoint generator G, define the actual conjugation path T_P(t)=exp(tG)Pexp(-tG). The formalization derives its derivative from the exponential power series and the product rule: T_P'(t)=exp(tG)(GP-PG)exp(-tG). Both exponential factors are proved unitary. Left and right multiplication by a unitary preserve the C-star norm. Therefore T_P'(t) = GP-PG , T_P(t)-T_P(s) ≤ |t-s| GP-PG . The second inequality follows from the mean value inequality in n…
Exact mathematical passage
T_P'(t)=\exp(tG)(GP-PG)\exp(-tG).
T_rUse in source · 1 passage
Search aliases: T_r
Use in source · visual_atlas.tex, line 114
vpage5Relations identify words, not just outputsPRESENTATIONS / WHY A LEFT IDEAL APPEARSMain text: [source reference] Let F=𝕜⟨ z_1,…,z_r⟩ be the algebra of noncommuting operator words. Evaluation at a fixed seed is Φ_s:F→ X, f↦ f(T_1,…,T_r)s. Two words represent the same generated vector exactly when their difference lies in I_s=KerΦ_s. Consequently F/I_s≅Hull_F(s) as left modules. The relation f(T)s=0 implies g(T)f(T)s=0 for every word g, which explains closure under multiplication on the left.
Exact mathematical passage
\Phi_s:F\to X,\qquad f\mapsto f(T_1,\ldots,T_r)s.
t_rUse in source · 6 passages
Search aliases: t_r
Use in source · 03_weighted.tex, line 452
[Several independent weights] Suppose every edge has a weight vector w(e)=(w_1(e),…,w_r(e))∈ N^r∖{0}. Replace t^w(e) by t_1^w_1(e)⋯ t_r^w_r(e). The resulting multigraded transfer series is rational in t_1,…,t_r.
Exact mathematical passage
t_1,\ldots,t_r
\mathcal T_{r-1}Use in source · 2 passages
Search aliases: T_r-1 mathcal
Use in source · root_scope.tex, line 117
Set r=M+1≥1 and define P_q(x)=∏_i=0^q-1(x-i)(2x-(2i+1)), P_0=1, H_r(x)=∑_q≥08^qfracβ_r+qβ_r P_q(x)mathscr A(x-q). The coefficient valuations tend to infinity. At any nonnegative integer x, terms with q>x vanish. Cancelling the factorials in [source reference] therefore gives the exact identity T_r-1(N)=κ_rbinom2N2rH_r(N-r) (N≥ r). This is a finite rational identity at every original Euler row.
Exact mathematical passage
\mathcal T_{r-1}(N)=\kappa_r\binom{2N}{2r}H_r(N-r) \qquad(N\ge r).\label{eq:root:factor}
t_r^{w_r(e)}Use in source · 1 passage
Search aliases: t_r^w_r(e)
Use in source · 03_weighted.tex, line 449
[Several independent weights] Suppose every edge has a weight vector w(e)=(w_1(e),…,w_r(e))∈ N^r∖{0}. Replace t^w(e) by t_1^w_1(e)⋯ t_r^w_r(e). The resulting multigraded transfer series is rational in t_1,…,t_r.
Exact mathematical passage
t_1^{w_1(e)}\cdots t_r^{w_r(e)}.
T_{rr}Use in source · 2 passages
Search aliases: T_rr
Use in source · 10_foundations.tex, line 188
em[Forward substitution and denominator clearing] Let T_nq be coefficients over a field with T_nn≠0 for every n. For any sequence b, define a_n=fracb_n-∑_q<nT_nqa_qT_nn. This constructs the unique solution of ∑_q=0^nT_nqa_q=b_n for all n. If T_nq and b_n are integers and the solution is taken over Q, then (∏_r=0^nT_rr)a_n∈ Z.
Exact mathematical passage
\left(\prod_{r=0}^{n}T_{rr}\right)a_n\in\Z.
t_{\mathrm{run}}Use in source · 1 passage
Search aliases: t_run mathrm
Use in source · 02_supplement_02.tex, line 203
Contracts, grouping, and amortization. textitSource items 2.9, 10.2, 10.4--10.6, and 11.3 Output conversion. A decoded run returns the actual pair (decode(compressed(x)), t_run(x)+t_decode(compressed(x))). Under the declared output equality, Lean proves that this returns the same explicit result and charges the conversion time. It also proves the downstream composition identity when compressed output is consumed directly. Omitting a positive decode cost strictly understates the modeled explicit-…
Exact mathematical passage
\bigl(\operatorname{decode}(\operatorname{compressed}(x)), \;t_{\mathrm{run}}(x)+t_{\mathrm{decode}}(\operatorname{compressed}(x))\bigr).
T_tUse in source · 1 passage
Search aliases: T_t
Use in source · Paper_VI_Traditional_Final.tex, line 717
[Gaussian chaos realization] For the standard Gaussian measure, the normalized probabilists' Hermite polynomials form orthonormal chaos packets. The Ornstein--Uhlenbeck semigroup acts diagonally: T_tH_n=e^-ntH_n. Thus conditional expectation, martingale differences, and Gaussian chaos are all projector realizations of stochastic refinement.
Exact mathematical passage
T_tH_n=e^{-nt}H_n.
T_{uv}Use in source · 1 passage
Search aliases: T_uv
Use in source · 04_correction_transport.tex, line 636
Let Γ=(V,E) be a finite connected graph and fix a vector space X. At every vertex take one labeled copy of X. Every oriented edge u→ v carries an invertible transport T_vu∈GL(X), T_uv=T_vu^-1. Equivalently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber. The matrices below are then the transports in those frames. For a path γ=(v_0,…,v_r), set T_γ = T_v_rv_r-1⋯ T_v_1v_0. For a loop, this product is its holonomy. In the based loop group, γ_2γ_1 means fi…
Exact mathematical passage
T_{vu}\in\operatorname{GL}(X), \qquad T_{uv}=T_{vu}^{-1}.
T_{v_1v_0}Use in source · 1 passage
Search aliases: T_v_1v_0
Use in source · 04_correction_transport.tex, line 642
or space X. At every vertex take one labeled copy of X. Every oriented edge u→ v carries an invertible transport T_vu∈GL(X), T_uv=T_vu^-1. Equivalently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber. The matrices below are then the transports in those frames. For a path γ=(v_0,…,v_r), set T_γ = T_v_rv_r-1⋯ T_v_1v_0. For a loop, this product is its holonomy. In the based loop group, γ_2γ_1 means first traverse γ_1 and then γ_2, so T_γ_2γ_1=T_γ_2T_γ_1.…
Exact mathematical passage
T_\gamma = T_{v_rv_{r-1}}\cdots T_{v_1v_0}.
T_{v_rv_{r-1}}Use in source · 1 passage
Search aliases: T_v_rv_r-1
Use in source · 04_correction_transport.tex, line 642
or space X. At every vertex take one labeled copy of X. Every oriented edge u→ v carries an invertible transport T_vu∈GL(X), T_uv=T_vu^-1. Equivalently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber. The matrices below are then the transports in those frames. For a path γ=(v_0,…,v_r), set T_γ = T_v_rv_r-1⋯ T_v_1v_0. For a loop, this product is its holonomy. In the based loop group, γ_2γ_1 means first traverse γ_1 and then γ_2, so T_γ_2γ_1=T_γ_2T_γ_1.…
Exact mathematical passage
T_\gamma = T_{v_rv_{r-1}}\cdots T_{v_1v_0}.
T_{vu}^{-1}Use in source · 1 passage
Search aliases: T_vu^-1
Use in source · 04_correction_transport.tex, line 636
Let Γ=(V,E) be a finite connected graph and fix a vector space X. At every vertex take one labeled copy of X. Every oriented edge u→ v carries an invertible transport T_vu∈GL(X), T_uv=T_vu^-1. Equivalently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber. The matrices below are then the transports in those frames. For a path γ=(v_0,…,v_r), set T_γ = T_v_rv_r-1⋯ T_v_1v_0. For a loop, this product is its holonomy. In the based loop group, γ_2γ_1 means fi…
Exact mathematical passage
T_{vu}\in\operatorname{GL}(X), \qquad T_{uv}=T_{vu}^{-1}.
t_wUse in source · 2 passages
Search aliases: t_w
Use in source · 01_finite_core.tex, line 802
We call a recovery complete when it is a Kraus map R(X)=∑_jK_jXK_j^*, ∑_jK_j^*K_j=I. The family is exactly correctable when one such recovery works for every coherent error combination A_w=∑_aw_aE_a: for every w there is a real t_w≥0, independent of the logical input, such that PXP=X Longrightarrow R(A_wXA_w^*)=t_wX. The coefficient is an unnormalized weight. It is not necessarily a probability for an arbitrary matrix A_w.
Exact mathematical passage
t_w\ge0
\mathcal T_xUse in source · 6 passages
Search aliases: T_x mathcal
Use in source · 30_lifting.tex, line 277
At positive polynomial depth, restrict it to the correction fiber T_x={t∈ S_q^d:overliner_x+J_xt=0} and the lift set L_x,L= {y∈( Z/q^L+1 Z)^d:y≡ xpmodq^L, F(y)=0}. The polynomial predicate in the second display is evaluation in the residue ring. Evaluation on canonical integer representatives gives the same predicate, because integer polynomial evaluation respects congruence.
Exact mathematical passage
\mathcal T_x=\{t\in S_q^d:\overline{r_x}+J_xt=0\}
T_zUse in source · 7 passages
Search aliases: T_z
Use in source · Paper_VI_Traditional_Final.tex, line 1557
Put B=A_QQ and T_z(t)=e^-zte^tB=e^t(B-zI). The generalized-eigenspace decomposition in finite dimensions writes T_z as a finite sum of constant matrix coefficients times t^k e^(λ-z)t. This follows by expanding the nilpotent part on each generalized eigenspace. Equivalently, apply the decomposition to left multiplication by B on the matrix algebra and evaluate the exponential at I. The strict spectral inequality makes every such term absolutely integrable on (0,∞) and forces it to tend to zero. H…
Exact mathematical passage
T_z(t)=e^{-zt}e^{tB}=e^{t(B-zI)}
T_z'Use in source · 1 passage
Search aliases: T_z'
Use in source · Paper_VI_Traditional_Final.tex, line 1568
Let J=∫_0^∞ T_z(t) dt. Integrating both derivative identities T_z'=(B-zI)T_z=T_z(B-zI) and using T_z(0)=I gives (zI-B)J=J(zI-B)=I. Thus J=(zI-B)^-1. Continuous left and right multiplication carry the Bochner integral through the two rectangular transfer blocks, yielding the claim.
Exact mathematical passage
T_z'=(B-zI)T_z=T_z(B-zI)
T_\alphaUse in source · 6 passages
Search aliases: T_α alpha
Use in source · Paper_VI_Traditional_Final.tex, line 1130
Let L=L^*≥0 on a finite-dimensional Hilbert space. Its spectral resolution is L=∑_jλ_jP_j. For α>0, define L^α=∑_jλ_j^α P_j. The associated fractional semigroup is T_α(t)=e^-tL^α =∑_je^-tλ_j^αP_j.
Exact mathematical passage
T_\alpha(t)=e^{-tL^\alpha} =\sum_je^{-t\lambda_j^\alpha}P_j.
T_{\beta_1\beta_0}Use in source · 1 passage
Search aliases: T_β_1β_0 beta beta
Use in source · 03_packets_and_graphs.tex, line 562
[Packet powers and paths] For m≥1, (T^m)_β_mβ_0 = ∑_β_1,…,β_m-1 T_β_mβ_m-1 ⋯ T_β_1β_0. The same formula holds for a product of nonidentical operators with the corresponding blocks in temporal order. [source reference]
Exact mathematical passage
(T^m)_{\beta_m\beta_0} = \sum_{\beta_1,\ldots,\beta_{m-1}} T_{\beta_m\beta_{m-1}} \cdots T_{\beta_1\beta_0}.
T_{\beta_m\beta_{m-1}}Use in source · 1 passage
Search aliases: T_β_mβ_m-1 beta beta
Use in source · 03_packets_and_graphs.tex, line 562
[Packet powers and paths] For m≥1, (T^m)_β_mβ_0 = ∑_β_1,…,β_m-1 T_β_mβ_m-1 ⋯ T_β_1β_0. The same formula holds for a product of nonidentical operators with the corresponding blocks in temporal order. [source reference]
Exact mathematical passage
(T^m)_{\beta_m\beta_0} = \sum_{\beta_1,\ldots,\beta_{m-1}} T_{\beta_m\beta_{m-1}} \cdots T_{\beta_1\beta_0}.
T_{\beta\gamma}Use in source · 4 passages
Search aliases: T_βγ beta gamma
Use in source · Paper_VI_Traditional_Final.tex, line 389
[Packet semigroup convolution] For every s,t and every pair of packets α,β, T_βα(s+t) =∑_γ∈ IT_βγ(s)T_γα(t). Thus the semigroup law becomes ordinary matrix multiplication in the packet indices.
Exact mathematical passage
\boxed{ T_{\beta\alpha}(s+t) =\sum_{\gamma\in I}T_{\beta\gamma}(s)T_{\gamma\alpha}(t). }
T_\gammaUse in source · 1 passage
Search aliases: T_γ gamma
Use in source · 04_correction_transport.tex, line 642
or space X. At every vertex take one labeled copy of X. Every oriented edge u→ v carries an invertible transport T_vu∈GL(X), T_uv=T_vu^-1. Equivalently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber. The matrices below are then the transports in those frames. For a path γ=(v_0,…,v_r), set T_γ = T_v_rv_r-1⋯ T_v_1v_0. For a loop, this product is its holonomy. In the based loop group, γ_2γ_1 means first traverse γ_1 and then γ_2, so T_γ_2γ_1=T_γ_2T_γ_1.…
Exact mathematical passage
T_\gamma = T_{v_rv_{r-1}}\cdots T_{v_1v_0}.
T_{\gamma_1}Use in source · 1 passage
Search aliases: T_γ_1 gamma
Use in source · 04_correction_transport.tex, line 647
valently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber. The matrices below are then the transports in those frames. For a path γ=(v_0,…,v_r), set T_γ = T_v_rv_r-1⋯ T_v_1v_0. For a loop, this product is its holonomy. In the based loop group, γ_2γ_1 means first traverse γ_1 and then γ_2, so T_γ_2γ_1=T_γ_2T_γ_1. Standard graph terminology may be found in [source reference].
Exact mathematical passage
T_{\gamma_2\gamma_1}=T_{\gamma_2}T_{\gamma_1}
T_{\gamma_2}Use in source · 1 passage
Search aliases: T_γ_2 gamma
Use in source · 04_correction_transport.tex, line 647
valently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber. The matrices below are then the transports in those frames. For a path γ=(v_0,…,v_r), set T_γ = T_v_rv_r-1⋯ T_v_1v_0. For a loop, this product is its holonomy. In the based loop group, γ_2γ_1 means first traverse γ_1 and then γ_2, so T_γ_2γ_1=T_γ_2T_γ_1. Standard graph terminology may be found in [source reference].
Exact mathematical passage
T_{\gamma_2\gamma_1}=T_{\gamma_2}T_{\gamma_1}
T_{\gamma_2\gamma_1}Use in source · 1 passage
Search aliases: T_γ_2γ_1 gamma gamma
Use in source · 04_correction_transport.tex, line 647
valently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber. The matrices below are then the transports in those frames. For a path γ=(v_0,…,v_r), set T_γ = T_v_rv_r-1⋯ T_v_1v_0. For a loop, this product is its holonomy. In the based loop group, γ_2γ_1 means first traverse γ_1 and then γ_2, so T_γ_2γ_1=T_γ_2T_γ_1. Standard graph terminology may be found in [source reference].
Exact mathematical passage
T_{\gamma_2\gamma_1}=T_{\gamma_2}T_{\gamma_1}
T_{\gamma\alpha}Use in source · 4 passages
Search aliases: T_γα gamma alpha
Use in source · Paper_VI_Traditional_Final.tex, line 389
[Packet semigroup convolution] For every s,t and every pair of packets α,β, T_βα(s+t) =∑_γ∈ IT_βγ(s)T_γα(t). Thus the semigroup law becomes ordinary matrix multiplication in the packet indices.
Exact mathematical passage
\boxed{ T_{\beta\alpha}(s+t) =\sum_{\gamma\in I}T_{\beta\gamma}(s)T_{\gamma\alpha}(t). }
T_{\delta\gamma}Use in source · 2 passages
Search aliases: T_δγ delta gamma
Use in source · 03_packets_and_graphs.tex, line 601
[Tensor product transfer blocks] Let S∈End(X) and T∈End(Y), with packet systems P and Q. Relative to the tensor product packet system of [source reference], (S⊗ T)_(β,δ),(α,γ) = S_βα⊗ T_δγ. [source reference] ExportsTensor network, quantum circuit, and multivariate packet calculations.
Exact mathematical passage
\boxed{ (S\otimes T)_{(\beta,\delta),(\alpha,\gamma)} = S_{\beta\alpha}\otimes T_{\delta\gamma}. }
t_\varepsilonUse in source · 4 passages
Search aliases: t_ε varepsilon
Use in source · monograph_atlas.tex, line 738
4 · (0,32,32,64) · 25/16 · 20 · 8/5 These are unnormalized traces. Normalizing by N changes Z_N but leaves the logarithmic derivative at that fixed grid unchanged. xfigheat_limits headThe theorem uses two separate choices For each ε>0, first choose a fixed positive time t_ε so that the continuum torus dimension differs from one by less than ε/2. Then choose N_ε so that every N≥ N_ε differs from that continuum value by less than ε/2 at the same time. This proves one controlled intermediate time f…
Exact mathematical passage
t_\varepsilon
T_{\ell+1}Use in source · 3 passages
Search aliases: T_ℓ+1 ell
Use in source · 05_defects.tex, line 217
Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.
Exact mathematical passage
\Delta_\ell=C_\ell T_{\ell+1}-T_\ell C_\ell.
T_\inftyUse in source · 5 passages
Search aliases: T_∞ infty
Use in source · Paper_VI_Traditional_Final.tex, line 2471
[Exact projective limit semigroup] For each ℓ, let T_ℓ(t) be a strongly continuous semigroup on X_ℓ. Assume C_ℓ T_ℓ+1(t)=T_ℓ(t)C_ℓ for every ℓ and t≥0, and assume the uniform bound sup_ℓnormT_ℓ(t)≤ M e^ω t. Define T_∞(t)(x_ℓ)_ℓ = (T_ℓ(t)x_ℓ)_ℓ. Then T_∞(t) is a bounded semigroup on X_∞. It is strongly continuous if lim_tdownarrow0 sup_ℓ normT_ℓ(t)x_ℓ-x_ℓ=0 for every (x_ℓ)_ℓ∈ X_∞.
Exact mathematical passage
T_\infty(t)(x_\ell)_\ell = (T_\ell(t)x_\ell)_\ell.
\takeawayUse in source · 1 passage
Search aliases: takeaway
Use in source · visual_atlas.tex, line 19
[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…
Exact mathematical passage
4pt] \textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt] \textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology? \end{tabularx} A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. \takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''} \vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}} Most of the chapter can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix. {\small \begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule \textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule $X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\ $S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\ $\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\ $\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\ $P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\ $X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\ $T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\ $F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\ $U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\ $\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\ $\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule \end{tabularx}} \head{Read products from right to left} In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column. \head{Where to begin} For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}. \vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$} Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define \[ Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0, \qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.
\tanUse in source · 8 passages
Search aliases: tan
Use in source · Paper_IV_Traditional_Final.tex, line 1370
For N=1, T_1=1. For N=2, D_2=cos z and T_2=tan z. The Riccati equation becomes T_2'=1+ T_2^2.
Exact mathematical passage
\Tfun_2=\tan z
\tanhUse in source · 3 passages
Search aliases: tanh
Use in source · 72_recovered.tex, line 960
The identity tanh t=-itan(it) gives the odd zigzag expansion of tanh t. Differentiate.
Exact mathematical passage
\tanh t=-i\tan(it)
\mathrm{target}Use in source · 1 passage
Search aliases: target mathrm
Use in source · visual_atlas.tex, line 289
headThe first-edge decomposition Let a_n(s,t) count walks of total weight n from s to t. Removing the first edge gives a_n(s,t)=1_s=t, n=0 +∑_substacke: source(e)=s w(e)≤ n a_n-w(e)(target(e),t). The empty walk is allowed precisely on the diagonal at weight zero. Positive weights ensure that every contributing walk has length at most n, so enumeration at each fixed weight is finite.
Exact mathematical passage
a_n(s,t)=\mathbf1_{s=t,\,n=0} +\sum_{\substack{e:\,\mathrm{source}(e)=s\\w(e)\le n}} a_{n-w(e)}(\mathrm{target}(e),t).
\operatorname{target}Use in source · 1 passage
Search aliases: target operatorname
Use in source · 03_weighted.tex, line 499
[Automatic finite-graph transfer-zeta rationality] Assume the vertex and edge sets are finite and every edge has positive integer weight. For every n, exact enumeration of edge tuples of lengths at most n, followed by endpoint and total-weight filtering, returns a_n(s,t). These coefficients obey the recurrence obtained by removing the first edge a_n(s,t) = 1_s=t, n=0 +∑_substacke: source(e)=s w(e)≤ n a_n-w(e)(target(e),t). Let T(t)=I-A(t), Q(t)=det T(t), P_s,t(t)=adj(T(t))_s,t. Then Q(0)=1, henc…
Exact mathematical passage
a_n(s,t) = \mathbf 1_{s=t,\,n=0} +\sum_{\substack{e:\,\operatorname{source}(e)=s\\w(e)\le n}} a_{n-w(e)}(\operatorname{target}(e),t).
\TcalUse in source · 1 passage
Search aliases: Tcal T mathcal
Use in source · 04_correction_transport.tex, line 673
[Spanning tree normalization] Choose a root and a spanning tree T of Γ. There is a gauge in which every tree edge transport is the identity. In that gauge, the transport on each edge outside the tree is the holonomy of its fundamental cycle, with the orientation used to define that cycle.
Exact mathematical passage
\Tcal
\operatorname{Teich}Use in source · 3 passages
Search aliases: Teich operatorname
Use in source · odd_synthesis.tex, line 190
Indeed, for residue cardinality q, let B=ω(Teich(k^×)). The actual inertia hypothesis gives χ(C)^N⊂ B. Since N is a power of p and χ(C) has order prime to p, this implies χ(C)⊂ B. Choose k_0 inverse to N modulo q-1, and set δ(u)=ω(Teich(u))^k_0, widetilde C={(c,u):χ(c)=δ(u)}. The matrix Teich(u^k_0)I_N acts by χ(c). The kernel acts trivially on the entire sector induction and ambient compact induction, because these matrices are central. This preserves the fixed Λ, its canonical beta-extension f…
Exact mathematical passage
B=\omega(\operatorname{Teich}(k^\times))
\Tfun_1Use in source · 1 passage
Search aliases: Tfun_1 T_1 Tfun mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 1370
For N=1, T_1=1. For N=2, D_2=cos z and T_2=tan z. The Riccati equation becomes T_2'=1+ T_2^2.
Exact mathematical passage
\Tfun_1=1
\Tfun_2Use in source · 1 passage
Search aliases: Tfun_2 T_2 Tfun mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 1370
For N=1, T_1=1. For N=2, D_2=cos z and T_2=tan z. The Riccati equation becomes T_2'=1+ T_2^2.
Exact mathematical passage
\Tfun_2=\tan z
\Tfun_2'Use in source · 1 passage
Search aliases: Tfun_2' T_2' Tfun mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 1371
For N=1, T_1=1. For N=2, D_2=cos z and T_2=tan z. The Riccati equation becomes T_2'=1+ T_2^2.
Exact mathematical passage
\Tfun_2'=1+\Tfun_2^2.
\Tfun_2^2Use in source · 1 passage
Search aliases: Tfun_2^2 T_2^2 Tfun mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 1371
For N=1, T_1=1. For N=2, D_2=cos z and T_2=tan z. The Riccati equation becomes T_2'=1+ T_2^2.
Exact mathematical passage
\Tfun_2'=1+\Tfun_2^2.
\Tfun_N'Use in source · 1 passage
Search aliases: Tfun_N' T_N' Tfun mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 1351
[Scalar higher Riccati identity] At every point of the open set {z: D_N(z)≠0}, Y_N( - T_N, - T_N', …, - T_N^(N-1) ) =-1.
Exact mathematical passage
Y_N\bigl( -\Tfun_N, -\Tfun_N', \ldots, -\Tfun_N^{(N-1)} \bigr) =-1.
\Tfun_N^{(N-1)}Use in source · 1 passage
Search aliases: Tfun_N^(N-1) T_N^(N-1) Tfun mathsf
Use in source · Paper_IV_Traditional_Final.tex, line 1351
[Scalar higher Riccati identity] At every point of the open set {z: D_N(z)≠0}, Y_N( - T_N, - T_N', …, - T_N^(N-1) ) =-1.
Exact mathematical passage
Y_N\bigl( -\Tfun_N, -\Tfun_N', \ldots, -\Tfun_N^{(N-1)} \bigr) =-1.
\Thetafun_{2,1}Use in source · 1 passage
Search aliases: Thetafun_2,1 symupΘ_2,1 Thetafun symup Theta
Use in source · Paper_IV_Traditional_Final.tex, line 1721
For N=2, symupΘ_2,1(u)=-u(u+1) and symupΘ_2,2(u) = u(u+1)(5u^2+u+2). The two factors predicted by [source reference] are visible.
Exact mathematical passage
\Thetafun_{2,1}(u)=-u(u+1)
\Thetafun_{2,2}Use in source · 1 passage
Search aliases: Thetafun_2,2 symupΘ_2,2 Thetafun symup Theta
Use in source · Paper_IV_Traditional_Final.tex, line 1725
For N=2, symupΘ_2,1(u)=-u(u+1) and symupΘ_2,2(u) = u(u+1)(5u^2+u+2). The two factors predicted by [source reference] are visible.
Exact mathematical passage
\Thetafun_{2,2}(u) = u(u+1)(5u^2+u+2).
\Thetafun_{N,1}Use in source · 1 passage
Search aliases: Thetafun_N,1 symupΘ_N,1 Thetafun symup Theta
Use in source · Paper_IV_Traditional_Final.tex, line 1697
[Theta Newton recurrence] Put θ_N,n(u)=fracsymupΘ_N,n(u)(Nn)!. Then θ_N,0(u)=1 and nθ_N,n(u) = ∑_m=1^n (-1)^mκ_N,m (u^Nm+u) θ_N,n-m(u). In particular, symupΘ_N,1(u)=-(u^N+u).
Exact mathematical passage
\Thetafun_{N,1}(u)=-(u^N+u).
\topruleUse in source · 4 passages
Search aliases: toprule
Use in source · visual_atlas.tex, line 19
[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…
Exact mathematical passage
4pt] \textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt] \textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology? \end{tabularx} A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. \takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''} \vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}} Most of the chapter can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix. {\small \begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule \textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule $X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\ $S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\ $\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\ $\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\ $P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\ $X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\ $T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\ $F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\ $U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\ $\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\ $\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule \end{tabularx}} \head{Read products from right to left} In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column. \head{Where to begin} For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}. \vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$} Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define \[ Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0, \qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.
\operatorname{Tr}Use in source · 6 passages
Search aliases: Tr operatorname
Use in source · Paper_X_Block27_Amended.tex, line 1059
cible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficient-field embedding σ_λ:E↪ E_λ therefore sends C to an invertible matrix. Its entries agree with the ordinary representation traces Tr(W_i,λ(Frob_v_j)) when the selected places satisfy the unramified compatibility convention for that coefficient prime. In particular, places above the residue characte…
Exact mathematical passage
\operatorname{Tr}(W_{i,\lambda}(\operatorname{Frob}_{v_j}))
\operatorname{tr}Use in source · 11 passages
Search aliases: tr operatorname
Use in source · visual_atlas.tex, line 279
The usual scalar invariants can be misleading: trH=2, det H=1, χ_H(t)=(t-1)^2. These are also the invariants of the 2×2 identity matrix. Nevertheless, Ob=H-I=mat0 · 1
Exact mathematical passage
\operatorname{tr}H=2,\qquad \det H=1,\qquad \chi_H(t)=(t-1)^2.
\trUse in source · 11 passages
Search aliases: tr operatorname
Use in source · Paper_IV_Traditional_Final.tex, line 851
D_N,N-1(z) . Then D_N(z)=e^zJ_Ne_0 and D_N(z)=frac1Ntr(e^zJ_N). The eigenvalues of J_N are the N roots of -1, each with multiplicity one.
Exact mathematical passage
\Dfun_N(z)=\frac1N\tr(e^{zJ_N}).
\Tr_AUse in source · 1 passage
Search aliases: Tr_A Tr operatorname
Use in source · 01_finite_core.tex, line 137
[Local orbit and Schmidt rank] Let ψ∈ H_A⊗ H_B, with dim H_A=d_A≥1, and let r be its Schmidt rank. The reduced positive operator on B is ρ_B=Tr_A(ketψbraψ). Normalization of ψ does not affect its support. Then dimspan{(A⊗ I_B)ψ:A∈ M_d_A( C)} =d_A r. In particular, local operators on A generate the complete bipartite Hilbert space exactly when the reduced state on B has full support.
Exact mathematical passage
\rho_B=\Tr_A(\ket\psi\bra\psi)
\operatorname{tr}_EUse in source · 2 passages
Search aliases: tr_E operatorname
Use in source · dyadic_synthesis.tex, line 58
_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact filtration valuation, and give the identity tr_LLift_E(C) =Tr_E/L (αtr_E C). The actual recursive matrix is B(L,t_L;h_1,…,h_d) =α I_2^d-1+ Lift_E(B(E,z;h_1,…,h_d-1)), with multiplication by α as the rank-two base case.
Exact mathematical passage
\operatorname{tr}_L\operatorname{Lift}_E(C) =\operatorname{Tr}_{E/L} \bigl(\alpha\operatorname{tr}_E C\bigr).
\Tr_{E'}Use in source · 1 passage
Search aliases: Tr_E' Tr operatorname
Use in source · Paper_XII_Traditional_Final.tex, line 1093
[Rank and trace detection in characteristic zero] Assume that E' has characteristic zero. For every maximal Hecke packet, Tr_E'(e_ m|V) = rank_E'(e_ m|V) = dim_E'e_ mV. In particular, the packet is occupied exactly when its projector has positive rank, equivalently nonzero trace.
Exact mathematical passage
\boxed{ \Tr_{E'}(e_{\frakm}|V) = \rank_{E'}(e_{\frakm}|V) = \dim_{E'}e_{\frakm}V. }
\operatorname{Tr}_{E/L}Use in source · 4 passages
Search aliases: Tr_E/L operatorname
Use in source · dyadic_synthesis.tex, line 50
At a recursive coefficient field L of characteristic two, with residue field F_2 and uniformizer t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two, and z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the ex…
Exact mathematical passage
z^2+t_L^az+t_L=0,\qquad \operatorname{Tr}_{E/L}(\alpha)=1.
\operatorname{tr}_LUse in source · 2 passages
Search aliases: tr_L operatorname
Use in source · dyadic_synthesis.tex, line 58
_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact filtration valuation, and give the identity tr_LLift_E(C) =Tr_E/L (αtr_E C). The actual recursive matrix is B(L,t_L;h_1,…,h_d) =α I_2^d-1+ Lift_E(B(E,z;h_1,…,h_d-1)), with multiplication by α as the rank-two base case.
Exact mathematical passage
\operatorname{tr}_L\operatorname{Lift}_E(C) =\operatorname{Tr}_{E/L} \bigl(\alpha\operatorname{tr}_E C\bigr).
\Tr_{\rm out}Use in source · 1 passage
Search aliases: Tr_rm out Tr rm operatorname rm
Use in source · 02_strengthened.tex, line 1083
labelstrong:standardfact:001 A linear map Φ:M_n( C)→ M_m( C) is completely positive exactly when it has a Kraus representation Φ(X)=∑_rK_rXK_r^*. It is trace preserving exactly when ∑_rK_r^*K_r=I. Equivalently, its Choi matrix J(Φ) = ∑_i,jΦ(ketibraj)⊗ketibraj is positive, and trace preservation is Tr_rm outJ(Φ)=I_rm in.
Exact mathematical passage
\Tr_{\rm out}J(\Phi)=I_{\rm in}.
\triangleUse in source · 11 passages
Search aliases: triangle
Use in source · Paper_V_Traditional_Final.tex, line 713
[Packet perturbation bound] For 1≤ p<∞, interpreted first as an inequality of extended L^p seminorms, j-widehat j _L^p ≤ |a-widehat a| μ(Ω)^1/p + |widehat a| μ(ΩtrianglewidehatΩ)^1/p. If μ(Ω) and μ(ΩtrianglewidehatΩ) are finite, all terms in this displayed jump bound are finite and it is an ordinary real-valued L^p inequality. For the pointwise supremum, and consequently also the essential L^∞ seminorm, j-widehat j _∞ ≤ |a-widehat a|+|widehat a| 1_ΩtrianglewidehatΩ≠∅.
Exact mathematical passage
\boxed{ \|j-\widehat j\|_{L^p} \le |a-\widehat a|\,\mu(\Omega)^{1/p} + |\widehat a|\,\mu(\Omega\triangle\widehat\Omega)^{1/p}. }
\TT_{\frakm}Use in source · 3 passages
Search aliases: TT_frakm T_ m TT frakm mathbb mathfrak
Use in source · Paper_XII_Traditional_Final.tex, line 1025
[Artin decomposition] Let E' be a field and let T be a finite dimensional commutative E' algebra. Then T ≅ ∏_ m∈Max T T_ m. Each factor is an Artin local algebra. Equivalently, there is a unique complete family of central idempotents (e_ m) such that e_ me_ n=0 ( m≠ n), ∑_ me_ m=1, and e_ m T≅ T_ m [source reference].
Exact mathematical passage
\boxed{ \TT \cong \prod_{\frakm\in\Max\TT}\TT_{\frakm}. }
\mathcal UUse in source · 6 passages
Search aliases: U mathcal
Use in source · 04_correction_transport.tex, line 115
[Local Lyapunov--Schmidt reduction] Let U⊆ X be an open neighborhood of zero and let F: U→ Y be a C^r map between Banach spaces with r≥1, F(0)=0, and L=DF(0). Assume there are closed decompositions X=K⊕ X_1, Y=R⊕ O, where K=Ker L, R=im L, and L_1=L|_X_1:X_1⟶ R is a Banach space isomorphism. Let π_R and π_O be the two projections on Y. Then there are neighborhoods U_K⊆ K and U_1⊆ X_1 of zero and a unique C^r map ψ:U_K⟶ U_1 with ψ(0)=0 such that π_RF(k+ψ(k))=0 for every k∈ U_K. After the neighborh…
Exact mathematical passage
\mathcal U\subseteq X
U'Use in source · 1 passage
Search aliases: U'
Use in source · 03_nonlinear_coordinates.tex, line 953
[Naturality of obstruction classes] Suppose ccc U · xrightarrowD · V mathllapadownarrow · · downarrowmathrlapb U' · xrightarrowD' · V' is a commuting square. Then there is a unique induced map overline b:coker D⟶coker D' satisfying overline b Ob_D(r)=Ob_D'(br).
Exact mathematical passage
\begin{array}{ccc} U & \xrightarrow{D} & V\\ \mathllap{a}\downarrow & & \downarrow\mathrlap{b}\\ U' & \xrightarrow{D'} & V' \end{array}
u''Use in source · 3 passages
Search aliases: u''
Use in source · Paper_VI_Traditional_Final.tex, line 1358
On the interval (0,1) with Dirichlet boundary conditions, the Green kernel for -u''=f is G(x,y)=min(x,y)-xy. Thus the point source δ_y generates the solution kernel, and a general source is reconstructed by u(x)=∫_0^1G(x,y)f(y) dy.
Exact mathematical passage
-u''=f
u^{\mathsf T}Use in source · 1 passage
Search aliases: u^ T mathsf
Use in source · 03_weighted.tex, line 395
[Rationality of the transfer series] The weighted transfer series satisfies Z_u,v(t) = u^ T(I-A(t))^-1v. It is a rational function whose denominator divides det(I-A(t)).
Exact mathematical passage
Z_{u,v}(t) = u^{\mathsf T}(I-A(t))^{-1}v.
u^{(4)}Use in source · 2 passages
Search aliases: u^(4)
Use in source · Paper_VI_Traditional_Final.tex, line 4251
[Second order Poisson solution error] Suppose u∈ C^4[0,1] solves the Dirichlet problem above. Let f^h_i=f(x_i) and let u^h solve L_hu^h=f^h. Then max_iabsu^h_i-u(x_i) ≤ frach^296 normu^(4)_∞.
Exact mathematical passage
\max_i\abs{u^h_i-u(x_i)} \leq \frac{h^2}{96} \norm{u^{(4)}}_\infty.
u^{(p-1)p^a}Use in source · 2 passages
Search aliases: u^(p-1)p^a
Use in source · 72_recovered.tex, line 1589
[Basic odd and dyadic difference gains] For odd p, Δ_(p-1)p^a^ ru^n = u^n(u^(p-1)p^a-1)^r, v_p(Δ_(p-1)p^a^ ru^n)≥(a+1)r. For odd u∈ Z_2^×, Δ_2^a+1^ ru^n = u^n(u^2^a+1-1)^r, v_2(Δ_2^a+1^ ru^n)≥(a+3)r.
Exact mathematical passage
\boxed{ \Delta_{(p-1)p^a}^{\,r}u^n = u^n\left(u^{(p-1)p^a}-1\right)^r, \qquad \vp\left(\Delta_{(p-1)p^a}^{\,r}u^n\right)\geq(a+1)r. }
U^*Use in source · 24 passages
Search aliases: U^*
Use in source · Paper_XI_Traditional_Final.tex, line 1412
[Hyperbolic phase formula] Let U be a finite dimensional F_p space, let a∈GL(U) have no eigenvalue one, and let g act on H(U)=U⊕ U^* by g=(a,(a^-1)^*). Then Φ_p(g,H(U)) = (fracdet ap).
Exact mathematical passage
H(U)=U\oplus U^*
U^{-1}Use in source · 15 passages
Search aliases: U^-1
Use in source · 04_transport.tex, line 10
[Coordinate transport] Let U:X→ Y be a vector space isomorphism and let P=(P_α) be a packet system on X. Define Q_α:=UP_α U^-1. Then Q=(Q_α) is a packet system on Y. For widetilde T=UTU^-1, Q_βwidetilde TQ_α = U(P_β TP_α)U^-1. Hence packet dimensions, zero patterns, and support preorders are invariant under coordinate transport. If a declared algebra generating family G is transported to U G U^-1, then its transfer graph and reachability relation are also preserved. [source reference]
Exact mathematical passage
Q_\alpha:=UP_\alpha U^{-1}.
U^1Use in source · 2 passages
Search aliases: U^1
Use in source · odd_capacity.tex, line 190
The right side is the Heisenberg commutator pairing for the minimal simple stratum of gamma. Its quotient J^1(γ)/H^1(γ) is precisely U^m(A)/H^m(γ), because J^1(γ)=U^1(B_order)U^m(A), with U^1(B_order)⊂ H^1(γ). BK's nondegeneracy theorem for simple-character commutators therefore says that the form on S is nondegenerate.
Exact mathematical passage
J^1(\gamma)=U^1(B_{\mathrm{order}})U^m(A)
U^2Use in source · 1 passage
Search aliases: U^2
Use in source · strengthening.tex, line 326
, and b=(1-i)/2. The exact identity F_0(t)=frac2cosh t+cos t =sech(at)sech(bt) gives, for U=a E+b F, L(U^4n)=C_0(n). Although a and b have negative valuation, their denominators disappear from the following exact polynomial identities: U^2 · = E F+i(P-Q),nonumber H_0:=fracU^4-12 · =P+Q-fracP^22-fracQ^22 +3PQ+i E F(P-Q).
Exact mathematical passage
U^2&=\mathcal E\mathcal F+i(P-Q),\nonumber\\ H_0:=\frac{U^4-1}{2} &=P+Q-\frac{P^2}{2}-\frac{Q^2}{2} +3PQ+i\mathcal E\mathcal F(P-Q). \label{eq:four-euler-H}
u^2Use in source · 15 passages
Search aliases: u^2
Use in source · 03_nonlinear_coordinates.tex, line 548
[Catalan parity packets] Over a field of characteristic zero, the formal equation u=z+u^2 has the branch u(z)=frac1-sqrt1-4z2. Write u=u_0+u_1, where u_0 contains the even powers of z and u_1 contains the odd powers. Exact packet projection gives u_0=u_0^2+u_1^2, u_1=z+2u_0u_1. The two equations remain coupled. Their sum is the original fixed point equation.
Exact mathematical passage
u=z+u^2
u^{2^{a+1}}Use in source · 3 passages
Search aliases: u^2^a+1
Use in source · 72_recovered.tex, line 1599
[Basic odd and dyadic difference gains] For odd p, Δ_(p-1)p^a^ ru^n = u^n(u^(p-1)p^a-1)^r, v_p(Δ_(p-1)p^a^ ru^n)≥(a+1)r. For odd u∈ Z_2^×, Δ_2^a+1^ ru^n = u^n(u^2^a+1-1)^r, v_2(Δ_2^a+1^ ru^n)≥(a+3)r.
Exact mathematical passage
\boxed{ \Delta_{2^{a+1}}^{\,r}u^n = u^n\left(u^{2^{a+1}}-1\right)^r, \qquad \vTwo\left(\Delta_{2^{a+1}}^{\,r}u^n\right)\geq(a+3)r. }
U^{2m}Use in source · 1 passage
Search aliases: U^2m
Use in source · odd_capacity.tex, line 166
For u,v in U m(A), their group commutator is in U^2m(A)=U^N(A), which lies in U (m+1)(A). The terminal approximation satisfies
Exact mathematical passage
U^{2m}(A)=U^N(A)
u^{2N}Use in source · 1 passage
Search aliases: u^2N
Use in source · Paper_IV_Traditional_Final.tex, line 2530
[Theta Bell polynomial formula] Put widetildeσ_N,m=(-1)^mσ_N,m. Then symupΘ_N,n(u) = frac(Nn)!n! B_n( widetildeσ_N,1(u^N+u), 1!widetildeσ_N,2(u^2N+u), …, (n-1)!widetildeσ_N,n(u^nN+u) ).
Exact mathematical passage
\Thetafun_{N,n}(u) = \frac{(Nn)!}{n!} B_n\bigl( \widetilde{\sigma}_{N,1}(u^N+u), 1!\widetilde{\sigma}_{N,2}(u^{2N}+u), \ldots, (n-1)!\widetilde{\sigma}_{N,n}(u^{nN}+u) \bigr).
U^4Use in source · 5 passages
Search aliases: U^4
Use in source · supplement_11.tex, line 88
For a packed coefficient sequence e, the exact extraction formula is (U^8-U^4)^kpack_4(e)(4n-4k) =Δ_1^k e(n), k≤ n. Here U is the coefficient shift. The nonnegative extraction index is why the first estimate covers k≤ n. The existing saturation lemma extends it to all k with exponent min(k,n), and the step-multiple lemma extends step one to every natural u.
Exact mathematical passage
\bigl(U^8-U^4\bigr)^k\operatorname{pack}_4(e)(4n-4k) =\Delta_1^k e(n),\qquad k\le n.
U^{4n}Use in source · 4 passages
Search aliases: U^4n
Use in source · strengthening.tex, line 323
Put K= Q_2(i), R= Z_2[i], a=(1+i)/2, and b=(1-i)/2. The exact identity F_0(t)=frac2cosh t+cos t =sech(at)sech(bt) gives, for U=a E+b F, L(U^4n)=C_0(n). Although a and b have negative valuation, their denominators disappear from the following exact polynomial identities: U^2 · = E F+i(P-Q),nonumber
Exact mathematical passage
\mathcal L(U^{4n})=C_0(n)
U^8Use in source · 1 passage
Search aliases: U^8
Use in source · supplement_11.tex, line 88
For a packed coefficient sequence e, the exact extraction formula is (U^8-U^4)^kpack_4(e)(4n-4k) =Δ_1^k e(n), k≤ n. Here U is the coefficient shift. The nonnegative extraction index is why the first estimate covers k≤ n. The existing saturation lemma extends it to all k with exponent min(k,n), and the step-multiple lemma extends step one to every natural u.
Exact mathematical passage
\bigl(U^8-U^4\bigr)^k\operatorname{pack}_4(e)(4n-4k) =\Delta_1^k e(n),\qquad k\le n.
U^aUse in source · 4 passages
Search aliases: U^a
Use in source · supplement_07.tex, line 33
pageheading1. Coefficient products and reciprocal differentiationA characteristic-p argument without factorial division For sequences f,g: N→ K, where K is a field, define (f⋆ g)(d)=∑_i=0^dbinom di f(i)g(d-i), (U^a f)(d)=f(d+a). These are Hurwitz coefficients. The notation does not divide by d! in F_p. In particular, the implementation is not ordinary power-series multiplication with undefined inverse factorials in positive characteristic.
Exact mathematical passage
(f\star g)(d)=\sum_{i=0}^{d}\binom di f(i)g(d-i), \qquad (U^a f)(d)=f(d+a).
U^{a-i}Use in source · 1 passage
Search aliases: U^a-i
Use in source · supplement_07.tex, line 40
The identity sequence is 1(d)=δ_d0. The new source proves the actual higher product rule U^a(f⋆ g)=∑_i=0^abinom ai (U^i f)⋆(U^a-ig). The argument starts with Pascal's identity for the first shift, then inducts on a. All sums are finite at each coefficient.
Exact mathematical passage
U^a(f\star g)=\sum_{i=0}^{a}\binom ai (U^i f)\star(U^{a-i}g).
u^{ab}Use in source · 2 passages
Search aliases: u^ab
Use in source · Paper_X_Block27_Amended.tex, line 2282
For actual finite original inertia images the common kernel sample is derived. Suppose u∈ H satisfies ν(u)=1 and t(u)≠0 and both r_D(H) and r_E(H) are finite. Their sampled actions have positive finite orders a,b. At u^ab both original actions are trivial, and t(u^ab)=(ab)t(u)≠0 by the cocycle identity and characteristic zero. Thus the same full intertwiner equivalence holds using the actual finite-image hypotheses, without a separately supplied common trivial-action sample. The initial nonzero …
Exact mathematical passage
u^{ab}
u^HUse in source · 3 passages
Search aliases: u^H
Use in source · 72_recovered.tex, line 1549
[One unit mode] Let u∈ Z_p^×, let H=Λ_p,L, and put q_u,L=u^H-1∈ p^L Z_p. For every integer address j and every x∈ Z_p, u^Hx+j = u^j∑_r≥0q_u,L^ rbinom xr. The rth Mahler coefficient has valuation at least Lr.
Exact mathematical passage
q_{u,L}=u^H-1\in p^L\Zp.
u^hUse in source · 7 passages
Search aliases: u^h
Use in source · 10_foundations.tex, line 334
[Differences of unit modes] Over every commutative ring, Δ_h^s(u^n)=u^n(u^h-1)^s. For any prime p, units u_i∈ Z_p^×, coefficients a_i∈ Z_p, and a finite index set I, the sequence f(n)=∑_i∈ Ia_i u_i^n satisfies Δ_p-1^s f(n)∈ p^s Z_p (n,s∈ N).
Exact mathematical passage
\Delta_h^s(u^n)=u^n(u^h-1)^s.
u^{H_a}Use in source · 2 passages
Search aliases: u^H_a
Use in source · 72_recovered.tex, line 2338
2^a+1, · p=2. Then Δ_H_a^ ru^n = u^n(u^H_a-1)^r. For odd p, v_p(Δ_H_a^ ru^n) ≥(a+1)r. For p=2 and odd u, v_2(Δ_H_a^ ru^n) ≥(a+3)r. Equivalently, every additional prime power digit increases the certified Mahler slope by one.
Exact mathematical passage
\boxed{ \Delta_{H_a}^{\,r}u^n = u^n\left(u^{H_a}-1\right)^r. }
u^h_iUse in source · 1 passage
Search aliases: u^h_i
Use in source · Paper_VI_Traditional_Final.tex, line 4251
[Second order Poisson solution error] Suppose u∈ C^4[0,1] solves the Dirichlet problem above. Let f^h_i=f(x_i) and let u^h solve L_hu^h=f^h. Then max_iabsu^h_i-u(x_i) ≤ frach^296 normu^(4)_∞.
Exact mathematical passage
\max_i\abs{u^h_i-u(x_i)} \leq \frac{h^2}{96} \norm{u^{(4)}}_\infty.
u^{Hx+j}Use in source · 2 passages
Search aliases: u^Hx+j
Use in source · 72_recovered.tex, line 1553
[One unit mode] Let u∈ Z_p^×, let H=Λ_p,L, and put q_u,L=u^H-1∈ p^L Z_p. For every integer address j and every x∈ Z_p, u^Hx+j = u^j∑_r≥0q_u,L^ rbinom xr. The rth Mahler coefficient has valuation at least Lr.
Exact mathematical passage
\boxed{ u^{Hx+j} = u^j\sum_{r\geq0}q_{u,L}^{\,r}\binom xr. }
U^iUse in source · 1 passage
Search aliases: U^i
Use in source · supplement_07.tex, line 40
The identity sequence is 1(d)=δ_d0. The new source proves the actual higher product rule U^a(f⋆ g)=∑_i=0^abinom ai (U^i f)⋆(U^a-ig). The argument starts with Pascal's identity for the first shift, then inducts on a. All sums are finite at each coefficient.
Exact mathematical passage
U^a(f\star g)=\sum_{i=0}^{a}\binom ai (U^i f)\star(U^{a-i}g).
u^jUse in source · 3 passages
Search aliases: u^j
Use in source · 72_recovered.tex, line 1553
[One unit mode] Let u∈ Z_p^×, let H=Λ_p,L, and put q_u,L=u^H-1∈ p^L Z_p. For every integer address j and every x∈ Z_p, u^Hx+j = u^j∑_r≥0q_u,L^ rbinom xr. The rth Mahler coefficient has valuation at least Lr.
Exact mathematical passage
\boxed{ u^{Hx+j} = u^j\sum_{r\geq0}q_{u,L}^{\,r}\binom xr. }
u^{j+(p-1)s}Use in source · 1 passage
Search aliases: u^j+(p-1)s
Use in source · 72_recovered.tex, line 2470
[Local phase lattice] For a unit mode u^n, the odd prime coordinate gives u^n = u^j+(p-1)s (u^(p-1)p^a)^q. The dyadic coordinate gives u^n = u^j+2s (u^2^a+1)^q. Thus finite character phase, local digit address, and Mahler depth are separate coordinates.
Exact mathematical passage
\boxed{ u^n = u^{j+(p-1)s} \left(u^{(p-1)p^a}\right)^q. }
u^{j+2s}Use in source · 1 passage
Search aliases: u^j+2s
Use in source · 72_recovered.tex, line 2479
[Local phase lattice] For a unit mode u^n, the odd prime coordinate gives u^n = u^j+(p-1)s (u^(p-1)p^a)^q. The dyadic coordinate gives u^n = u^j+2s (u^2^a+1)^q. Thus finite character phase, local digit address, and Mahler depth are separate coordinates.
Exact mathematical passage
\boxed{ u^n = u^{j+2s} \left(u^{2^{a+1}}\right)^q. }
U^kUse in source · 3 passages
Search aliases: U^k
Use in source · 02_coordinates.tex, line 418
0, · r≠ j, 0≤ j,r<N. Then P_N,j = η∑_k=0^N-1(ω^vee)^jkU^k. In particular, if R is a field, N>0, and ω is a primitive Nth root of unity, the certificate holds with ω^vee=ω^-1 and η=N^-1.
Exact mathematical passage
\boxed{ P_{N,j} = \eta\sum_{k=0}^{N-1}(\omega^\vee)^{jk}U^k. }
u^{k_0}Use in source · 1 passage
Search aliases: u^k_0
Use in source · odd_synthesis.tex, line 197
y q, let B=ω(Teich(k^×)). The actual inertia hypothesis gives χ(C)^N⊂ B. Since N is a power of p and χ(C) has order prime to p, this implies χ(C)⊂ B. Choose k_0 inverse to N modulo q-1, and set δ(u)=ω(Teich(u))^k_0, widetilde C={(c,u):χ(c)=δ(u)}. The matrix Teich(u^k_0)I_N acts by χ(c). The kernel acts trivially on the entire sector induction and ambient compact induction, because these matrices are central. This preserves the fixed Λ, its canonical beta-extension factor, and all completion choi…
Exact mathematical passage
\operatorname{Teich}(u^{k_0})I_N
U^mUse in source · 3 passages
Search aliases: U^m
Use in source · odd_capacity.tex, line 144
Here H m means H^1∩ U^m(A). Therefore the image S of U m(A) in W is canonically
Exact mathematical passage
H^1\cap U^m(A)
u^mUse in source · 2 passages
Search aliases: u^m
Use in source · 72_recovered.tex, line 441
The coefficients c_u in the proof of [source reference] are constant on the cosets of H_m. For u≠0, the value u^m depends only on the coset, and every element of G_m arises. Grouping the modes by cosets gives the displayed expansion.
Exact mathematical passage
u^m
U^NUse in source · 3 passages
Search aliases: U^N
Use in source · 02_coordinates.tex, line 402
Let U act on coefficient sequences by (UF)(z)=F(ω z). If ω^N=1, then U^N=I.
Exact mathematical passage
U^N=I
u^NUse in source · 3 passages
Search aliases: u^N
Use in source · Paper_IV_Traditional_Final.tex, line 1697
[Theta Newton recurrence] Put θ_N,n(u)=fracsymupΘ_N,n(u)(Nn)!. Then θ_N,0(u)=1 and nθ_N,n(u) = ∑_m=1^n (-1)^mκ_N,m (u^Nm+u) θ_N,n-m(u). In particular, symupΘ_N,1(u)=-(u^N+u).
Exact mathematical passage
\Thetafun_{N,1}(u)=-(u^N+u).
u^nUse in source · 21 passages
Search aliases: u^n
Use in source · 10_foundations.tex, line 334
[Differences of unit modes] Over every commutative ring, Δ_h^s(u^n)=u^n(u^h-1)^s. For any prime p, units u_i∈ Z_p^×, coefficients a_i∈ Z_p, and a finite index set I, the sequence f(n)=∑_i∈ Ia_i u_i^n satisfies Δ_p-1^s f(n)∈ p^s Z_p (n,s∈ N).
Exact mathematical passage
\Delta_h^s(u^n)=u^n(u^h-1)^s.
u^{n+p-1}Use in source · 1 passage
Search aliases: u^n+p-1
Use in source · 72_recovered.tex, line 343
For positive n, the zero mode contributes nothing. Every nonzero mode satisfies u^p-1=1. Hence b_n+p-1^(m,p) = ∑_u≠0c_uu^n+p-1 = ∑_u≠0c_uu^n = b_n^(m,p).
Exact mathematical passage
b_{n+p-1}^{(m,p)} = \sum_{u\neq0}c_uu^{n+p-1} = \sum_{u\neq0}c_uu^n = b_n^{(m,p)}.
u^{Nm}Use in source · 5 passages
Search aliases: u^Nm
Use in source · Paper_IV_Traditional_Final.tex, line 1688
[Theta Newton recurrence] Put θ_N,n(u)=fracsymupΘ_N,n(u)(Nn)!. Then θ_N,0(u)=1 and nθ_N,n(u) = ∑_m=1^n (-1)^mκ_N,m (u^Nm+u) θ_N,n-m(u). In particular, symupΘ_N,1(u)=-(u^N+u).
Exact mathematical passage
n\theta_{N,n}(u) = \sum_{m=1}^{n} (-1)^m\kappa_{N,m} \bigl(u^{Nm}+u\bigr) \theta_{N,n-m}(u).
u^{nN}Use in source · 1 passage
Search aliases: u^nN
Use in source · Paper_IV_Traditional_Final.tex, line 2530
[Theta Bell polynomial formula] Put widetildeσ_N,m=(-1)^mσ_N,m. Then symupΘ_N,n(u) = frac(Nn)!n! B_n( widetildeσ_N,1(u^N+u), 1!widetildeσ_N,2(u^2N+u), …, (n-1)!widetildeσ_N,n(u^nN+u) ).
Exact mathematical passage
\Thetafun_{N,n}(u) = \frac{(Nn)!}{n!} B_n\bigl( \widetilde{\sigma}_{N,1}(u^N+u), 1!\widetilde{\sigma}_{N,2}(u^{2N}+u), \ldots, (n-1)!\widetilde{\sigma}_{N,n}(u^{nN}+u) \bigr).
u^{Nq}Use in source · 1 passage
Search aliases: u^Nq
Use in source · Paper_IV_Traditional_Final.tex, line 1654
[Theta reconstruction] For every n≥0, symupΘ_N,n(u) = (-1)^n ∑_q=0^n binomNnNq P_N,q(1)u^NqP_N,n-q(u). Thus symupΘ_N,n is a polynomial of degree at most Nn.
Exact mathematical passage
\Thetafun_{N,n}(u) = (-1)^n \sum_{q=0}^{n} \binom{Nn}{Nq} P_{N,q}(1)u^{Nq}P_{N,n-q}(u).
u^pUse in source · 1 passage
Search aliases: u^p
Use in source · odd_synthesis.tex, line 96
The full-character argument uses a specific order of operations. First construct a full character at native rank p^j, trivial on U_F^1. Transfer its full beta-determinant unit character to multiplicity p, using BK 3.6.13(ii) and 3.6.14. On a scalar u∈ U_F^1, this gives θ_j(u^p)=1. Next add the actual trace-zero correction, using BK 3.3.18. It remains trivial on the scalar intersection of its domain. Finally extend at the next maximal native parameter field: the forced deep character and the triv…
Exact mathematical passage
\theta_j(u^p)=1
U^{p+L}Use in source · 1 passage
Search aliases: U^p+L
Use in source · supplement_07.tex, line 162
0, · Nnmid d, N=mp. Let L=p(p-1) and δ=U^p+L-U^p=U^p^2-U^p. The source proves the operator dictionary (δ^sX)(d)=Δ_L^s X(d+ps). At the valid nonnegative coefficient degree d=mpn-ps, with s≤ mn, this becomes (δ^sX)(mpn-ps)=Δ_h^s e(n), h=(p-1)/m. The packed sequence and difference operations are definitions. Their coefficient correspondence is proved before any divisibility assumption is introduced.
Exact mathematical passage
\delta=U^{p+L}-U^p=U^{p^2}-U^p
u^{p-1}Use in source · 3 passages
Search aliases: u^p-1
Use in source · 72_recovered.tex, line 342
For positive n, the zero mode contributes nothing. Every nonzero mode satisfies u^p-1=1. Hence b_n+p-1^(m,p) = ∑_u≠0c_uu^n+p-1 = ∑_u≠0c_uu^n = b_n^(m,p).
Exact mathematical passage
u^{p-1}=1
U^{p^2}Use in source · 3 passages
Search aliases: U^p^2
Use in source · supplement_07.tex, line 162
0, · Nnmid d, N=mp. Let L=p(p-1) and δ=U^p+L-U^p=U^p^2-U^p. The source proves the operator dictionary (δ^sX)(d)=Δ_L^s X(d+ps). At the valid nonnegative coefficient degree d=mpn-ps, with s≤ mn, this becomes (δ^sX)(mpn-ps)=Δ_h^s e(n), h=(p-1)/m. The packed sequence and difference operations are definitions. Their coefficient correspondence is proved before any divisibility assumption is introduced.
Exact mathematical passage
\delta=U^{p+L}-U^p=U^{p^2}-U^p
U^{q-a}Use in source · 1 passage
Search aliases: U^q-a
Use in source · supplement_08.tex, line 33
pageheading1. The characteristic-zero product ruleThe normalized defect is not an ordinary derivation Let A be a commutative algebra over a characteristic-zero field K, and let U be an actual K-derivation. The source defines B=fracU^p^2-U^pp, M_q(f,g)=∑_a=1^q-1frac1pbinom qa (U^a f)(U^q-ag), q=p,p^2. The coefficients of each M_q are proved integral: the middle entries of a prime-power binomial row are divisible by p. In Lean they are implemented as natural quotients, with exact division proved b…
Exact mathematical passage
B=\frac{U^{p^2}-U^p}{p},\qquad M_q(f,g)=\sum_{a=1}^{q-1}\frac{1}{p}\binom qa (U^a f)(U^{q-a}g),\quad q=p,p^2.
u^rUse in source · 4 passages
Search aliases: u^r
Use in source · 72_recovered.tex, line 164
The nth coefficient is ∑_r=0^n binom nr u^rv^n-r = (u+v)^n by the binomial theorem.
Exact mathematical passage
\sum_{r=0}^{n} \binom nr u^rv^{n-r} = (u+v)^n
u^{r_i}Use in source · 1 passage
Search aliases: u^r_i
Use in source · Paper_XI_Traditional_Final.tex, line 654
[Tame tensor local factor] The local factor of the tame tensor product is the product of (1-a_i u^r_i)^-1 over precisely those refined packets whose inertia exponent vanishes.
Exact mathematical passage
(1-a_i u^{r_i})^{-1}
u^TUse in source · 2 passages
Search aliases: u^T
Use in source · pages.tex, line 163
-1 · 3. Hence D^-1u=u/4 and S=1+u^TD^-1u=frac52. The Woodbury identity reduces the entire inverse to A^-1=D^-1-D^-1uS^-1u^TD^-1 =D^-1-frac140uu^T. vfigwoodbury headTwo hypotheses, not one Invertibility of D alone does not suffice. For D=I_2, u=e_1, and v=-e_1, the update is diag(0,1). The small matrix S is zero, exactly where the correction formula would require an invalid division.
Exact mathematical passage
S=1+u^TD^{-1}u=\frac52.
u^{\alpha-1}Use in source · 1 passage
Search aliases: u^α-1 alpha
Use in source · Paper_VI_Traditional_Final.tex, line 3969
It is enough to prove the formula on an eigenvector with eigenvalue x≥0. For x>0, the substitution s=xu gives fracsin(πα)π ∫_0^∞ s^α-1fracxs+x d s =x^α fracsin(πα)π ∫_0^∞fracu^α-11+u d u =x^α. The value at x=0 is zero. There are finitely many eigenvalues, so the operator integral converges. This is the finite spectral form of the fractional power construction in [source reference].
Exact mathematical passage
\frac{\sin(\pi\alpha)}{\pi} \int_0^\infty s^{\alpha-1}\frac{x}{s+x}\dd s =x^\alpha \frac{\sin(\pi\alpha)}{\pi} \int_0^\infty\frac{u^{\alpha-1}}{1+u}\dd u =x^\alpha.
U_{-Q}Use in source · 1 passage
Search aliases: U_-Q
Use in source · Paper_V_Traditional_Final.tex, line 2281
[Exact lifting update] The lifting matrices are invertible with L_P^-1=L_-P, U_Q^-1=U_-Q. If R_0E_0=I, then every finite product V of lifting matrices gives another exact bank by E=VE_0, R=R_0V^-1. Every lifting step has determinant one.
Exact mathematical passage
L_P^{-1}=L_{-P}, \qquad U_Q^{-1}=U_{-Q}.
U_0Use in source · 4 passages
Search aliases: U_0
Use in source · 03_nonlinear_coordinates.tex, line 1044
[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is a natural exact sequence 0 · ⟶Ker D_0 ⟶Ker D_f ⟶Ker D_c xrightarrowδcoker D_0
Exact mathematical passage
\begin{array}{ccccccccc} 0&\longrightarrow&U_0&\xrightarrow{i_U}&U_f&\xrightarrow{p_U}&U_c&\longrightarrow&0\\[4pt] &&\mathllap{D_0}\downarrow&&\mathllap{D_f}\downarrow&&\mathllap{D_c}\downarrow&&\\[4pt] 0&\longrightarrow&V_0&\xrightarrow{i_V}&V_f&\xrightarrow{p_V}&V_c&\longrightarrow&0. \end{array}
u_0Use in source · 21 passages
Search aliases: u_0
Use in source · 03_nonlinear_coordinates.tex, line 471
Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a to a leaf and applying the scalar map (t_1,…,t_r)↦ b_rt_1⋯ t_r at a vertex of arity r. A second induction gives ∑_τ∈ T_m B_τ(s) ≤ ∑_τ∈ T_mw(τ) =t_m, wh…
Exact mathematical passage
u_0=0
U_{0,n}Use in source · 1 passage
Search aliases: U_0,n
Use in source · Paper_VI_Traditional_Final.tex, line 2705
sin t · cos t . Only the first adjacent defect is nonzero, so every defect sum is finite. Yet for n≥1, U_0,n(t)=C_0T_n(t)R_0=cos t. The limiting family is not a semigroup because in general cos(s+t)≠cos scos t. The failure is exactly the excursion from the reconstructed line into its orthogonal complement between two time steps.
Exact mathematical passage
U_{0,n}(t)=C_0T_n(t)R_0=\cos t.
u_0^{-1}Use in source · 1 passage
Search aliases: u_0^-1
Use in source · odd_synthesis.tex, line 62
Let P_0 extract the constant coefficient in the basis 1,z,…,z^p-1. The normalized corestriction s=κpr, where κ=-f'(z)/z^p-1, satisfies u_0=s(P_0)=a_0/z^p∈ O_E^×, Lift_E(C)=u_0^-1CP_0, s(Lift_E(C))=C. The lift preserves the exact filtration valuation. Its crucial additional identity is tr_L(Lift_E(C)) =Tr_E/L (κ^-1tr_E(C)). Define B(L,t_L;h_1,…,h_d) =α I_p^d-1+ Lift_E(B(E,z;h_1,…,h_d-1)), with the degree-p multiplication matrix as the base case. Induction proves trace zero at every native rank. T…
Exact mathematical passage
u_0=s(P_0)=a_0/z^p\in\mathcal O_E^\times, \qquad \operatorname{Lift}_E(C)=u_0^{-1}CP_0, \qquad s(\operatorname{Lift}_E(C))=C.
u_0^2Use in source · 1 passage
Search aliases: u_0^2
Use in source · 03_nonlinear_coordinates.tex, line 556
[Catalan parity packets] Over a field of characteristic zero, the formal equation u=z+u^2 has the branch u(z)=frac1-sqrt1-4z2. Write u=u_0+u_1, where u_0 contains the even powers of z and u_1 contains the odd powers. Exact packet projection gives u_0=u_0^2+u_1^2, u_1=z+2u_0u_1. The two equations remain coupled. Their sum is the original fixed point equation.
Exact mathematical passage
u_0=u_0^2+u_1^2, \qquad u_1=z+2u_0u_1.
U_1Use in source · 11 passages
Search aliases: U_1
Use in source · 04_correction_transport.tex, line 125
ro and let F: U→ Y be a C^r map between Banach spaces with r≥1, F(0)=0, and L=DF(0). Assume there are closed decompositions X=K⊕ X_1, Y=R⊕ O, where K=Ker L, R=im L, and L_1=L|_X_1:X_1⟶ R is a Banach space isomorphism. Let π_R and π_O be the two projections on Y. Then there are neighborhoods U_K⊆ K and U_1⊆ X_1 of zero and a unique C^r map ψ:U_K⟶ U_1 with ψ(0)=0 such that π_RF(k+ψ(k))=0 for every k∈ U_K. After the neighborhoods are made smaller if needed, the solutions of F(x)=0 near zero are exa…
Exact mathematical passage
U_1\subseteq X_1
u_1Use in source · 8 passages
Search aliases: u_1
Use in source · 05_worked_models.tex, line 25
Let the quiver have vertices U,V and arrows R:U→ U, A:U→ V, B:V→ U, with the relation in the path algebra BA=e_U, whose realization is the operator identity BA=I_U. Realize U=𝕜^2, V=𝕜^3, by R= 0 · -1 1 · 0, A(u_1,u_2)=(u_1,u_2,0), B(v_1,v_2,v_3)=(v_1,v_2). Starting from the seed e_1∈ U, repeated application of R spans U, and A carries that span onto the first two coordinates of V. Any path from U to V ends with A and therefore lands in im A. The third coordinate direction, which spans ker B, is …
Exact mathematical passage
R= \begin{pmatrix}0&-1\\1&0\end{pmatrix}, \qquad A(u_1,u_2)=(u_1,u_2,0), \qquad B(v_1,v_2,v_3)=(v_1,v_2).
u_1^2Use in source · 1 passage
Search aliases: u_1^2
Use in source · 03_nonlinear_coordinates.tex, line 556
[Catalan parity packets] Over a field of characteristic zero, the formal equation u=z+u^2 has the branch u(z)=frac1-sqrt1-4z2. Write u=u_0+u_1, where u_0 contains the even powers of z and u_1 contains the odd powers. Exact packet projection gives u_0=u_0^2+u_1^2, u_1=z+2u_0u_1. The two equations remain coupled. Their sum is the original fixed point equation.
Exact mathematical passage
u_0=u_0^2+u_1^2, \qquad u_1=z+2u_0u_1.
U_2Use in source · 1 passage
Search aliases: U_2
Use in source · 20_arithmetic.tex, line 736
For example, U_2(n)≡2pmod32 and U_3(n)≡6pmod8 for every row. At precision 2^10 the least packet periods are 512,512,32,128. A packet displacement T has original degree displacement 4T, as proved by leanKomorebiPaperIX.DyadicFourLeastPeriods.original_degree_exact_displacement. Minimality here concerns packet shifts. Shifts of the full degree-indexed sequence, including its zero coefficients off the packet support, require a separate analysis. The new strengthening chapters determine all repeated-…
Exact mathematical passage
U_2(n)\equiv2\pmod{32}
u_2Use in source · 1 passage
Search aliases: u_2
Use in source · 05_worked_models.tex, line 25
Let the quiver have vertices U,V and arrows R:U→ U, A:U→ V, B:V→ U, with the relation in the path algebra BA=e_U, whose realization is the operator identity BA=I_U. Realize U=𝕜^2, V=𝕜^3, by R= 0 · -1 1 · 0, A(u_1,u_2)=(u_1,u_2,0), B(v_1,v_2,v_3)=(v_1,v_2). Starting from the seed e_1∈ U, repeated application of R spans U, and A carries that span onto the first two coordinates of V. Any path from U to V ends with A and therefore lands in im A. The third coordinate direction, which spans ker B, is …
Exact mathematical passage
R= \begin{pmatrix}0&-1\\1&0\end{pmatrix}, \qquad A(u_1,u_2)=(u_1,u_2,0), \qquad B(v_1,v_2,v_3)=(v_1,v_2).
U_3Use in source · 1 passage
Search aliases: U_3
Use in source · 20_arithmetic.tex, line 736
For example, U_2(n)≡2pmod32 and U_3(n)≡6pmod8 for every row. At precision 2^10 the least packet periods are 512,512,32,128. A packet displacement T has original degree displacement 4T, as proved by leanKomorebiPaperIX.DyadicFourLeastPeriods.original_degree_exact_displacement. Minimality here concerns packet shifts. Shifts of the full degree-indexed sequence, including its zero coefficients off the packet support, require a separate analysis. The new strengthening chapters determine all repeated-…
Exact mathematical passage
U_3(n)\equiv6\pmod8
U_{4,j}Use in source · 2 passages
Search aliases: U_4,j
Use in source · 20_arithmetic.tex, line 467
For the remainder of the four-packet discussion abbreviate C_j=C_4,j and U_j=U_4,j. Removing the original row cap requires an unshifted coefficient operator.
Exact mathematical passage
U_j=U_{4,j}
U_{6,3}Use in source · 1 passage
Search aliases: U_6,3
Use in source · 20_arithmetic.tex, line 369
C_6,3(1)=-frac184, C_6,3(2)=frac7035460, C_6,3(2)-C_6,3(1)=frac64455. The last difference has 3-adic valuation zero. Here h=1, so at n=1 the order-one bound without factorial loss would require valuation at least one. The corrected normalized bound is sharp, since v_3(C_6,3(2)-C_6,3(1))=0=1-v_3(3!). Multiplication by 3! gives U_6,3(2)-U_6,3(1)=frac384455, v_3(U_6,3(2)-U_6,3(1))=1, consistent with the unnormalized theorem. Thus the factorial correction is already necessary at a single step in the…
Exact mathematical passage
U_{6,3}(2)-U_{6,3}(1)=\frac{384}{455},\qquad v_3\bigl(U_{6,3}(2)-U_{6,3}(1)\bigr)=1,
U_AUse in source · 6 passages
Search aliases: U_A
Use in source · 01_manuscript.tex, line 669
[Exact BDLR product] Let A=D_A+U_AV_A^*, B=D_B+U_BV_B^* use the same diagonal packet layout. Then AB=D_AD_B+U_CV_C^*, where U_C = D_AU_B+U_A(V_A^*U_B) · U_A and V_C = V_B · D_B^*V_A . The unrecompressed correction rank is at most r_A+r_B.
Exact mathematical passage
A=D_A+U_AV_A^*, \qquad B=D_B+U_BV_B^*
u_aUse in source · 15 passages
Search aliases: u_a
Use in source · visual_atlas.tex, line 185
1 · 1. The same ambient vector x=(3,2)^ T has chart coordinates u_a=F_a^-1x.
Exact mathematical passage
u_a=F_a^{-1}x
u_{a+b}Use in source · 1 passage
Search aliases: u_a+b
Use in source · Paper_XI_Traditional_Final.tex, line 919
Let A be a finite abelian group, and let c:A× A→ C^× be a normalized two cocycle. The twisted group algebra C^c[A] has basis (u_a)_a∈ A and multiplication u_au_b=c(a,b)u_a+b. Define the commutator bicharacter b_c(a,b)=c(a,b)c(b,a)^-1.
Exact mathematical passage
u_au_b=c(a,b)u_{a+b}.
u_a^{-1}Use in source · 1 passage
Search aliases: u_a^-1
Use in source · Paper_XI_Traditional_Final.tex, line 975
Let z=∑_a∈ Az_au_a be central. Commuting with u_x gives z_ab_c(x,a)=z_a for every x. Nondegeneracy forces z_a=0 for every nonzero a. Thus the center is C. To prove semisimplicity directly, choose a linear projection P onto a submodule of a finite dimensional C^c[A]-module and form overline P=frac1abs A∑_a∈ Au_a P u_a^-1. Every summand maps into the submodule and is the identity there. Conjugation by any u_x permutes the summands: the cocycle scalars cancel between each operator and its inverse. …
Exact mathematical passage
\overline P=\frac{1}{\abs A}\sum_{a\in A}u_a P u_a^{-1}.
U_A^1Use in source · 2 passages
Search aliases: U_A^1
Use in source · rank2.tex, line 97
K× normalizes A and every power of its radical, so U_K 1 normalizes U_A m. Therefore P is a subgroup and is compact open. It is contained in U_A 1 because mtextgreater=1 and U_K^1⊂ U_A^1.
Exact mathematical passage
U_K^1\subset U_A^1
U_A^jUse in source · 1 passage
Search aliases: U_A^j
Use in source · rank4.tex, line 56
Use U_A^j=1+P_A^j and U_B^j=1+Q^j. BK 3.1.7--9 give the actual groups
Exact mathematical passage
U_A^j=1+P_A^j
U_A^mUse in source · 3 passages
Search aliases: U_A^m
Use in source · Paper_XI_Traditional_Final.tex, line 2119
At rank two, with q=2m+1, enlarge J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=2, and P/H_0≃ F_2^4, with carrier Ind_J^Pη of dimension four ([source reference]). The finite scalar center can be larger than μ_2. No literal extrasp…
Exact mathematical passage
P=U_K^1U_A^m
U_A^{m+1}Use in source · 3 passages
Search aliases: U_A^m+1
Use in source · Paper_XI_Traditional_Final.tex, line 2119
At rank two, with q=2m+1, enlarge J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=2, and P/H_0≃ F_2^4, with carrier Ind_J^Pη of dimension four ([source reference]). The finite scalar center can be larger than μ_2. No literal extrasp…
Exact mathematical passage
J=U_K^1U_A^{m+1}
U_A^{n-r+m}Use in source · 1 passage
Search aliases: U_A^n-r+m
Use in source · dyadic_synthesis.tex, line 338
At both low ranks, K^× normalizes the new P. In rank two this follows directly from normalization of A and its radical. In rank four, a K-uniformizer is approximated by the derived chain shift R times a unit of B, with relative error in P_A^n-r. Its conjugation error on U_B^m lies in U_A^n-r+m⊂ J, which proves normalization. The exact argument is pathlowrank/RANK4_INDUCTION_BY_STAGES167.md.
Exact mathematical passage
U_A^{n-r+m}\subset J
U_{ar}Use in source · 2 passages
Search aliases: U_ar
Use in source · 01_finite_core.tex, line 843
textitSufficiency. Diagonalize C=Udiag(λ_r)U^* with U unitary and λ_r≥0. Define compressed transformed errors F_r=∑_aU_arE_aP. Then F_rP=F_r and F_r^*F_s=λ_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal projector. Use the recovery branches K_r=V_r^*, K_unused=I-S. They are complete on the whole ambient space because ∑_rK_r^*K_r+K_unused^*K_unused =S+(…
Exact mathematical passage
F_r=\sum_aU_{ar}E_aP.
U_BUse in source · 5 passages
Search aliases: U_B
Use in source · 01_manuscript.tex, line 669
[Exact BDLR product] Let A=D_A+U_AV_A^*, B=D_B+U_BV_B^* use the same diagonal packet layout. Then AB=D_AD_B+U_CV_C^*, where U_C = D_AU_B+U_A(V_A^*U_B) · U_A and V_C = V_B · D_B^*V_A . The unrecompressed correction rank is at most r_A+r_B.
Exact mathematical passage
A=D_A+U_AV_A^*, \qquad B=D_B+U_BV_B^*
U_bUse in source · 1 passage
Search aliases: U_b
Use in source · 03_supplement_06.tex, line 49
Smooth overlap maps and rank. The coordinate transition is g_ab(x)=A_b(x)B_a(x). It is smooth on U_a∩ U_b, agrees with the actual change between local homeomorphisms, and satisfies g_aa=I, g_bcg_ab=g_ac. A continuous linear equivalence between E_a and E_x also gives local constancy of fiber dimension. No single constant rank across disconnected components is assumed.
Exact mathematical passage
U_a\cap U_b
u_bUse in source · 7 passages
Search aliases: u_b
Use in source · Paper_XI_Traditional_Final.tex, line 919
Let A be a finite abelian group, and let c:A× A→ C^× be a normalized two cocycle. The twisted group algebra C^c[A] has basis (u_a)_a∈ A and multiplication u_au_b=c(a,b)u_a+b. Define the commutator bicharacter b_c(a,b)=c(a,b)c(b,a)^-1.
Exact mathematical passage
u_au_b=c(a,b)u_{a+b}.
U_B^jUse in source · 1 passage
Search aliases: U_B^j
Use in source · rank4.tex, line 56
Use U_A^j=1+P_A^j and U_B^j=1+Q^j. BK 3.1.7--9 give the actual groups
Exact mathematical passage
U_B^j=1+Q^j
U_B^mUse in source · 4 passages
Search aliases: U_B^m
Use in source · Paper_XI_Traditional_Final.tex, line 2125
two, with q=2m+1, enlarge J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=2, and P/H_0≃ F_2^4, with carrier Ind_J^Pη of dimension four ([source reference]). The finite scalar center can be larger than μ_2. No literal extraspecial qu…
Exact mathematical passage
P=J U_B^m
U_B^{m+1}Use in source · 1 passage
Search aliases: U_B^m+1
Use in source · rank4.tex, line 145
The quotient U_B^m/U_B^m+1 has dimension two. By Section~2 the image of J cap U_B m on this grade is exactly its centralizing line, of dimension one. For the converse inclusion needed here, the B-approximation of a K-element of grade m differs from it by grade m+n-rtextgreater a. That error lies in U_A a subset J. Thus the whole centralizing line really occurs in the intersection, not only in an abstract associated graded space. Consequently
Exact mathematical passage
U_B^m/U_B^{m+1}
U_{ba}Use in source · 7 passages
Search aliases: U_ba
Use in source · monograph_atlas.tex, line 159
The packetwise unitarity identity is likewise an operator identity, including the off-diagonal cancellations: ∑_b U_ba^*U_bc=δ_acP_a, U_ba=P_bUP_a. It cannot be replaced by the assertion that every path individually has unit norm. The exact checks verify all four choices of a,c for the two rank-one packets, as well as the two path products leading to the opposite output.
Exact mathematical passage
\sum_b U_{ba}^*U_{bc}=\delta_{ac}P_a, \qquad U_{ba}=P_bUP_a.
U_{ba}^*Use in source · 3 passages
Search aliases: U_ba^*
Use in source · monograph_atlas.tex, line 159
The packetwise unitarity identity is likewise an operator identity, including the off-diagonal cancellations: ∑_b U_ba^*U_bc=δ_acP_a, U_ba=P_bUP_a. It cannot be replaced by the assertion that every path individually has unit norm. The exact checks verify all four choices of a,c for the two rank-one packets, as well as the two path products leading to the opposite output.
Exact mathematical passage
\sum_b U_{ba}^*U_{bc}=\delta_{ac}P_a, \qquad U_{ba}=P_bUP_a.
U_{bc}Use in source · 5 passages
Search aliases: U_bc
Use in source · monograph_atlas.tex, line 159
The packetwise unitarity identity is likewise an operator identity, including the off-diagonal cancellations: ∑_b U_ba^*U_bc=δ_acP_a, U_ba=P_bUP_a. It cannot be replaced by the assertion that every path individually has unit norm. The exact checks verify all four choices of a,c for the two rank-one packets, as well as the two path products leading to the opposite output.
Exact mathematical passage
\sum_b U_{ba}^*U_{bc}=\delta_{ac}P_a, \qquad U_{ba}=P_bUP_a.
U_CUse in source · 8 passages
Search aliases: U_C
Use in source · pages.tex, line 143
vpage9The cross term has somewhere to goCLOSED LOW-RANK PRODUCT ALGEBRAChapter VIII: Theorem 5.1, Corollary 5.2. Let A=D_A+u_av_a^T be the running operator. Take D_B=diag(1,2,3,4,5,6), u_b=(1,0,1,0,1,0)^T, and v_b=(0,1,0,1,0,1)^T. Define B=D_B+u_bv_b^T. U_C=[D_Au_b+u_a(v_a^Tu_b) u_a], V_C=[v_b D_B^Tv_a]. vfigbdlr_product headWhy there are only two columns Expanding AB gives three correction terms, but two share the right factor v_b^T: D_Au_bv_b^T+u_a(v_a^Tu_b)v_b^T =(D_Au_b+u_a(v_a^Tu_b))v_b^T. …
Exact mathematical passage
U_C=[D_Au_b+u_a(v_a^Tu_b)\ \ u_a],\qquad V_C=[v_b\ \ D_B^Tv_a].
U_cUse in source · 6 passages
Search aliases: U_c
Use in source · 03_nonlinear_coordinates.tex, line 1044
[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is a natural exact sequence 0 · ⟶Ker D_0 ⟶Ker D_f ⟶Ker D_c xrightarrowδcoker D_0
Exact mathematical passage
\begin{array}{ccccccccc} 0&\longrightarrow&U_0&\xrightarrow{i_U}&U_f&\xrightarrow{p_U}&U_c&\longrightarrow&0\\[4pt] &&\mathllap{D_0}\downarrow&&\mathllap{D_f}\downarrow&&\mathllap{D_c}\downarrow&&\\[4pt] 0&\longrightarrow&V_0&\xrightarrow{i_V}&V_f&\xrightarrow{p_V}&V_c&\longrightarrow&0. \end{array}
u_cUse in source · 13 passages
Search aliases: u_c
Use in source · 03_nonlinear_coordinates.tex, line 1063
· ⟶coker D_f ⟶coker D_c ⟶0. For u_c∈Ker D_c, choose u_f∈ U_f with p_Uu_f=u_c. There is a unique v_0∈ V_0 such that D_fu_f=i_Vv_0, and the connecting map is δ(u_c)=[v_0].
Exact mathematical passage
u_c\in\Ker D_c
U_dUse in source · 1 passage
Search aliases: U_d
Use in source · 30_lifting.tex, line 38
[Polynomial Taylor remainder] Let R be a commutative ring, P∈ R[X_1,…,X_d], a∈ R^d, and c∈ R. There is a polynomial Q∈ R[U_1,…,U_d], with Q(0)=0, such that for every u∈ R^d, P(a+cu)=P(a)+c∑_j=1^d(∂_jP)(a)u_j+c^2Q(u). In particular, when R= Z and a,u,c are integral, subtracting the first two terms on the right from P(a+cu) gives a multiple of c^2.
Exact mathematical passage
Q\in R[U_1,\ldots,U_d]
U_{da}Use in source · 1 passage
Search aliases: U_da
Use in source · 03_supplement_01.tex, line 35
Adjoint conservation: source Theorem 4.2. Assume a star operation with P_a^*=P_a. Then (U_ba)^*=P_aU^*P_b. The two checked conservation laws are U^*U=1 · Longrightarrow ∑_b(U_ba)^*U_bc=δ_acP_a, UU^*=1 · Longrightarrow ∑_aU_ba(U_da)^*=δ_bdP_b. For the first law, expand each corner, collapse P_b^2 and sum: ∑_bP_aU^*P_bUP_c =P_aU^*(∑_bP_b)UP_c=P_aP_c. The second law follows by the analogous calculation and has its own formal proof. For complex matrices, the adjoint is the conjugate transpose. The o…
Exact mathematical passage
U^*U=1&\quad\Longrightarrow\quad \sum_b(U_{ba})^*U_{bc}=\delta_{ac}P_a,\\ UU^*=1&\quad\Longrightarrow\quad \sum_aU_{ba}(U_{da})^*=\delta_{bd}P_b.
U_{da}^*Use in source · 1 passage
Search aliases: U_da^*
Use in source · 02_strengthened.tex, line 424
[Packet unitarity identities] If U is unitary and U_ba=P_bUP_a, then ∑_bU_ba^*U_bc=δ_acP_a and ∑_aU_baU_da^*=δ_bdP_b.
Exact mathematical passage
\sum_aU_{ba}U_{da}^*=\delta_{bd}P_b.
u_eUse in source · 7 passages
Search aliases: u_e
Use in source · 02_supplement_04.tex, line 182
Constructive parameterization. Write t=1/λ and set u_e(t)=min{m_e,tc_e/s_e}, Φ(t)=∑_e s_eu_e(t). Lean proves that Φ is continuous and nondecreasing, with Φ(0)=0. At the finite positive scale T=∑_efracm_es_ec_e, every coordinate is saturated, so u(T)=m and Φ(T)=∑_es_em_e. The intermediate value theorem produces t>0 with Φ(t)=τ. The multiplier is not assumed to exist.
Exact mathematical passage
u_e(t)=\min\{m_e,tc_e/s_e\},\qquad \Phi(t)=\sum_e s_eu_e(t).
U_E^{(q+1)/2}Use in source · 1 passage
Search aliases: U_E^(q+1)/2
Use in source · dyadic_synthesis.tex, line 98
At the minimal rank-two base, extend the forced character on U_E^(q+1)/2 to U_E^1, imposing triviality on U_F^1. The prescriptions agree on their intersection because the parameter trace is one and ψ_F(u-1)=1. This is an actual finite abelian character extension and gives the full minimal simple character.
Exact mathematical passage
U_E^{(q+1)/2}
U_E^1Use in source · 3 passages
Search aliases: U_E^1
Use in source · dyadic_character.tex, line 268
Section~4 constructs chi_E on U_E 1 extending phi and trivial on U_F 1. Define theta_1 on H^1=U_E^1 U_(A_1) a by chi_E on the first factor and psi_alpha on the second. The restrictions agree on the intersection. BK 3.2.1 gives the actual full simple character, with theta_1textbar U_F 1=1.
Exact mathematical passage
H^1=U_E^1
U_fUse in source · 3 passages
Search aliases: U_f
Use in source · 03_nonlinear_coordinates.tex, line 1044
[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is a natural exact sequence 0 · ⟶Ker D_0 ⟶Ker D_f ⟶Ker D_c xrightarrowδcoker D_0
Exact mathematical passage
\begin{array}{ccccccccc} 0&\longrightarrow&U_0&\xrightarrow{i_U}&U_f&\xrightarrow{p_U}&U_c&\longrightarrow&0\\[4pt] &&\mathllap{D_0}\downarrow&&\mathllap{D_f}\downarrow&&\mathllap{D_c}\downarrow&&\\[4pt] 0&\longrightarrow&V_0&\xrightarrow{i_V}&V_f&\xrightarrow{p_V}&V_c&\longrightarrow&0. \end{array}
u_fUse in source · 16 passages
Search aliases: u_f
Use in source · 03_nonlinear_coordinates.tex, line 1063
· ⟶coker D_f ⟶coker D_c ⟶0. For u_c∈Ker D_c, choose u_f∈ U_f with p_Uu_f=u_c. There is a unique v_0∈ V_0 such that D_fu_f=i_Vv_0, and the connecting map is δ(u_c)=[v_0].
Exact mathematical passage
u_f\in U_f
U_F^sUse in source · 1 passage
Search aliases: U_F^s
Use in source · dyadic_synthesis.tex, line 86
The replacement for trace zero is the depth inequality v_F(trδ)+s≥1. It makes the additive correction trivial on scalar U_F^s, because (u-1)trδ∈ t O_F. Our native corrections have trace one, so the inequality holds at s=1. This is the exact source-to-target transfer of the weighted-visibility principle. The Euler coefficient theorem is motivation, not an input to the arithmetic proof. In characteristic two, the weights are uniformizer depths, rather than integer powers of two in the coefficient …
Exact mathematical passage
U_F^s
U_GUse in source · 2 passages
Search aliases: U_G
Use in source · 72_recovered.tex, line 515
[Composition of coefficient transports] If U_H and U_G are the triangular transforms associated with unit kernels H and G, then U_HU_G=U_HG. In particular, the inverse of U_H is U_H^-1.
Exact mathematical passage
U_G
U_HUse in source · 9 passages
Search aliases: U_H
Use in source · 10_foundations.tex, line 90
[Constructed unit-triangular coordinates] Let h: N→ K satisfy h_0=1, write H= E(h) and H^-1= E(k), and define U_Ha= D(H E(a)). Then U_H is an equivalence, with inverse U_H^-1a= D(H^-1 E(a)). Its coefficient formulas are (U_Ha)_n=∑_r=0^nbinom nr h_r a_n-r, (U_H^-1a)_n=∑_r=0^nbinom nr k_r a_n-r. The resulting triangular matrix has diagonal entries 1. The transported selectors widehat P_N,j=U_HP_N,jU_H^-1 are idempotent and pairwise orthogonal, and their sum is the identity.
Exact mathematical passage
U_Ha=\mathcal D(H\mathcal E(a)).
u_hUse in source · 2 passages
Search aliases: u_h
Use in source · atlas-pages.tex, line 427
nzero exact errorFINITE DIFFERENCES / A QUARTIC TEST PROBLEMChapter VI, Part I: Theorems 11.1--11.2. Exact nodal inversion and the scale of low-mode errors. Use u(x)=x-x^4, which vanishes at both endpoints and solves -u''=12x^2. Its centered second difference is not exact: L_hu|_rm nodes=12x_i^2+2h^2. Since the stencil differentiates the quadratic x(1-x) exactly, the solution u_h to the sampled forcing 12x_i^2 satisfies u_h(x_i)-u(x_i)=-h^2x_i(1-x_i). This gives both the sign and size of the err…
Exact mathematical passage
u_h
U_H^{-1}Use in source · 5 passages
Search aliases: U_H^-1
Use in source · 10_foundations.tex, line 94
[Constructed unit-triangular coordinates] Let h: N→ K satisfy h_0=1, write H= E(h) and H^-1= E(k), and define U_Ha= D(H E(a)). Then U_H is an equivalence, with inverse U_H^-1a= D(H^-1 E(a)). Its coefficient formulas are (U_Ha)_n=∑_r=0^nbinom nr h_r a_n-r, (U_H^-1a)_n=∑_r=0^nbinom nr k_r a_n-r. The resulting triangular matrix has diagonal entries 1. The transported selectors widehat P_N,j=U_HP_N,jU_H^-1 are idempotent and pairwise orthogonal, and their sum is the identity.
Exact mathematical passage
U_H^{-1}a=\mathcal D(H^{-1}\mathcal E(a))
U_{H^{-1}}Use in source · 1 passage
Search aliases: U_H^-1
Use in source · 72_recovered.tex, line 521
[Composition of coefficient transports] If U_H and U_G are the triangular transforms associated with unit kernels H and G, then U_HU_G=U_HG. In particular, the inverse of U_H is U_H^-1.
Exact mathematical passage
U_{H^{-1}}
U_{HG}Use in source · 1 passage
Search aliases: U_HG
Use in source · 72_recovered.tex, line 516
[Composition of coefficient transports] If U_H and U_G are the triangular transforms associated with unit kernels H and G, then U_HU_G=U_HG. In particular, the inverse of U_H is U_H^-1.
Exact mathematical passage
\boxed{ U_HU_G=U_{HG}. }
U_iUse in source · 7 passages
Search aliases: U_i
Use in source · 30_lifting.tex, line 52
Apply polynomial induction using constants, addition, and multiplication by a variable. Constants have zero remainder, and remainders add. For the multiplication step, suppose the assertion is known for P and write D_P(U)=∑_j(∂_jP)(a)U_j. Multiplication of its expansion by a_i+cU_i gives constant term P(a)a_i, linear term c(a_iD_P(U)+P(a)U_i), and remainder c^2(Q(U)(a_i+cU_i)+D_P(U)U_i). The product rule identifies the linear term with the derivative of PX_i. The displayed remainder vanishes at …
Exact mathematical passage
a_i+cU_i
u_i^hUse in source · 2 passages
Search aliases: u_i^h
Use in source · supplement_04.tex, line 148
Finite integral linear combinations. For any finite index set S, coefficients a_i∈ Z_p and units u_i∈ Z_p^×, linearity of the actual iterated operator gives Δ_h^s (∑_i∈ Sa_i u_i^n) =∑_i∈ Sa_i u_i^n(u_i^h-1)^s. At h=p-1, the result is divisible by p^s. The empty sum is permitted. This does not prove that the general congruential Euler packets of Theorem 7.5 admit such a representation.
Exact mathematical passage
\Delta_h^s\!\left(\sum_{i\in S}a_i u_i^n\right) =\sum_{i\in S}a_i u_i^n(u_i^h-1)^s.
u_i^nUse in source · 2 passages
Search aliases: u_i^n
Use in source · 10_foundations.tex, line 338
[Differences of unit modes] Over every commutative ring, Δ_h^s(u^n)=u^n(u^h-1)^s. For any prime p, units u_i∈ Z_p^×, coefficients a_i∈ Z_p, and a finite index set I, the sequence f(n)=∑_i∈ Ia_i u_i^n satisfies Δ_p-1^s f(n)∈ p^s Z_p (n,s∈ N).
Exact mathematical passage
f(n)=\sum_{i\in I}a_i u_i^n
u_i^{p-1}Use in source · 1 passage
Search aliases: u_i^p-1
Use in source · 10_foundations.tex, line 347
Induction gives the geometric identity. Reduction of a unit to F_p^× gives u_i^p-1=1 modulo p. Taking s powers supplies p^s in each summand. The difference operator is linear over finite sums.
Exact mathematical passage
u_i^{p-1}=1
U_jUse in source · 13 passages
Search aliases: U_j
Use in source · 20_arithmetic.tex, line 467
For the remainder of the four-packet discussion abbreviate C_j=C_4,j and U_j=U_4,j. Removing the original row cap requires an unshifted coefficient operator.
Exact mathematical passage
U_j=U_{4,j}
u_jUse in source · 5 passages
Search aliases: u_j
Use in source · 30_lifting.tex, line 40
[Polynomial Taylor remainder] Let R be a commutative ring, P∈ R[X_1,…,X_d], a∈ R^d, and c∈ R. There is a polynomial Q∈ R[U_1,…,U_d], with Q(0)=0, such that for every u∈ R^d, P(a+cu)=P(a)+c∑_j=1^d(∂_jP)(a)u_j+c^2Q(u). In particular, when R= Z and a,u,c are integral, subtracting the first two terms on the right from P(a+cu) gives a multiple of c^2.
Exact mathematical passage
\label{eq:polynomial-taylor} P(a+cu)=P(a)+c\sum_{j=1}^d(\partial_jP)(a)u_j+c^2Q(u).
u_j^{\mathsf T}Use in source · 1 passage
Search aliases: u_j^ T mathsf
Use in source · atlas-pages.tex, line 79
0 · 2 · 1 · -3. The graph shows rates between actual states. The four normalized sign vectors on the right form an orthonormal basis u_j. Their projectors P_j=u_ju_j^ T sum to the identity and satisfy P_iP_j=0 for i≠ j. vfiggenerator_modes platecaptionState coordinates have off-diagonal transfers. Spectral packets reduce the generator and evolve independently. The two dashed state groups are the coarse partition used next. headThe same evolution has a closed formula Writing q=e^-2t gives e^tQ=P_…
Exact mathematical passage
P_j=u_ju_j^{\mathsf T}
U_KUse in source · 3 passages
Search aliases: U_K
Use in source · 04_correction_transport.tex, line 125
en neighborhood of zero and let F: U→ Y be a C^r map between Banach spaces with r≥1, F(0)=0, and L=DF(0). Assume there are closed decompositions X=K⊕ X_1, Y=R⊕ O, where K=Ker L, R=im L, and L_1=L|_X_1:X_1⟶ R is a Banach space isomorphism. Let π_R and π_O be the two projections on Y. Then there are neighborhoods U_K⊆ K and U_1⊆ X_1 of zero and a unique C^r map ψ:U_K⟶ U_1 with ψ(0)=0 such that π_RF(k+ψ(k))=0 for every k∈ U_K. After the neighborhoods are made smaller if needed, the solutions of F(x…
Exact mathematical passage
U_K\subseteq K
U_kUse in source · 2 passages
Search aliases: U_k
Use in source · root_scope.tex, line 176
For preparation, start from the monic polynomial g=overline G_r and its zero-one lift W_1, with U_1=1. Suppose G_r-W_kU_k is coefficientwise divisible by 2^k. Its divided reduction is a finite polynomial because G_r is restricted. Divide it by g in F_2[X]: overline(G_r-W_kU_k)/2^k=gq+s, deg s<d_r. With zero-one coefficient lifts, set W_k+1=W_k+2^ks and U_k+1=U_k+2^kq. The error is now divisible by 2^k+1, the polynomial remains monic of degree d_r, and the unit reduces to one. The sequences conve…
Exact mathematical passage
G_r-W_kU_k
u_kUse in source · 15 passages
Search aliases: u_k
Use in source · 72_recovered.tex, line 2865
[Residue subsequences of a second order recurrence] Suppose x_n+2=Px_n+1-Qx_n over a commutative ring. Define V_0=2, V_1=P, and V_n+2=PV_n+1-QV_n. For every positive integer N and every address j, the subsequence u_k=x_Nk+j satisfies u_k+2=V_Nu_k+1-Q^Nu_k.
Exact mathematical passage
u_k=x_{Nk+j}
U_{k+1}Use in source · 1 passage
Search aliases: U_k+1
Use in source · root_scope.tex, line 183
or preparation, start from the monic polynomial g=overline G_r and its zero-one lift W_1, with U_1=1. Suppose G_r-W_kU_k is coefficientwise divisible by 2^k. Its divided reduction is a finite polynomial because G_r is restricted. Divide it by g in F_2[X]: overline(G_r-W_kU_k)/2^k=gq+s, deg s<d_r. With zero-one coefficient lifts, set W_k+1=W_k+2^ks and U_k+1=U_k+2^kq. The error is now divisible by 2^k+1, the polynomial remains monic of degree d_r, and the unit reduces to one. The sequences conver…
Exact mathematical passage
U_{k+1}=U_k+2^kq
u_{k+1}Use in source · 5 passages
Search aliases: u_k+1
Use in source · 72_recovered.tex, line 2869
[Residue subsequences of a second order recurrence] Suppose x_n+2=Px_n+1-Qx_n over a commutative ring. Define V_0=2, V_1=P, and V_n+2=PV_n+1-QV_n. For every positive integer N and every address j, the subsequence u_k=x_Nk+j satisfies u_k+2=V_Nu_k+1-Q^Nu_k.
Exact mathematical passage
\boxed{ u_{k+2}=V_Nu_{k+1}-Q^Nu_k.}
u_{k+1}^{(j)}Use in source · 1 passage
Search aliases: u_k+1^(j)
Use in source · 72_recovered.tex, line 2792
[Companion jump recurrence] Every residue state packet satisfies u_k+1^(j)=C^Nu_k^(j). More explicitly, u_k^(j)=C^j(C^N)^kv_0.
Exact mathematical passage
\boxed{ u_{k+1}^{(j)}=C^Nu_k^{(j)}.}
u_{k+2}Use in source · 3 passages
Search aliases: u_k+2
Use in source · 72_recovered.tex, line 2869
[Residue subsequences of a second order recurrence] Suppose x_n+2=Px_n+1-Qx_n over a commutative ring. Define V_0=2, V_1=P, and V_n+2=PV_n+1-QV_n. For every positive integer N and every address j, the subsequence u_k=x_Nk+j satisfies u_k+2=V_Nu_k+1-Q^Nu_k.
Exact mathematical passage
\boxed{ u_{k+2}=V_Nu_{k+1}-Q^Nu_k.}
u_{k+a}Use in source · 4 passages
Search aliases: u_k+a
Use in source · supplement_06.tex, line 47
The jump formula. Associativity and the laws for powers of one matrix give u_k=C^j(C^N)^kv_0=(C^N)^k(C^jv_0). Only powers of the same matrix are interchanged. The argument does not assume that arbitrary matrices commute. In particular, u_k+1=C^Nu_k, u_k+a=(C^N)^a u_k. The step matrix is C^N, not C. A checked example with the Fibonacci matrix and N=2 rejects replacing the jump with a single original step.
Exact mathematical passage
u_{k+1}=C^Nu_k,\qquad u_{k+a}=(C^N)^a u_k.
u_{k+d}Use in source · 2 passages
Search aliases: u_k+d
Use in source · supplement_06.tex, line 82
The complete vector recurrence. Apply the matrix identity to the actual state u_k. The already proved advance identity identifies each D^a u_k with u_k+a. Consequently, ∑_a=0^dχ_D[a]u_k+a=0, u_k+d=-∑_a=0^d-1c_a u_k+a. This is the desired recurrence, with its monic leading coefficient and all earlier coefficients supplied by χ_D. It is not an assumption on the packet sequence.
Exact mathematical passage
\sum_{a=0}^{d}\chi_D[a]u_{k+a}=0, \qquad \boxed{u_{k+d}=-\sum_{a=0}^{d-1}c_a u_{k+a}.}
u_k^{(j)}Use in source · 3 passages
Search aliases: u_k^(j)
Use in source · 72_recovered.tex, line 2786
Let R be a commutative ring. Let v_n+1=Cv_n, C∈ M_d(R). For a positive integer N and an address j∈{0,…,N-1}, define the residue state packet u_k^(j)=v_Nk+j.
Exact mathematical passage
u_k^{(j)}=v_{Nk+j}.
U_K^1Use in source · 8 passages
Search aliases: U_K^1
Use in source · Paper_XI_Traditional_Final.tex, line 2119
At rank two, with q=2m+1, enlarge J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=2, and P/H_0≃ F_2^4, with carrier Ind_J^Pη of dimension four ([source reference]). The finite scalar center can be larger than μ_2. No literal extrasp…
Exact mathematical passage
J=U_K^1U_A^{m+1}
U_K^MUse in source · 2 passages
Search aliases: U_K^M
Use in source · dyadic_character.tex, line 241
It is well defined by the intersection condition. Choose Mtextgreater=a with chitextbar U_K M=1. Then chi_0 is a character of a subgroup of the finite abelian group U_K^1/U_K^M. Extend it by adjoining generators and choosing the required complex roots, and pull it back. The resulting chi_K extends chi and is trivial on U_F 1.
Exact mathematical passage
U_K^1/U_K^M
U_K^mUse in source · 1 passage
Search aliases: U_K^m
Use in source · rank2.tex, line 130
For if ku∈U_A m with k∈U_K 1 and u∈U_A (m+1), then k∈ K^×∩ U_A^m=U_K^m. The reverse inclusion is immediate. The complete period-two A-grade R^m/R^m+1 has F2-dimension 2, and its field grade p_K^m/ p_K^m+1 has dimension 1. Thus
Exact mathematical passage
k\in K^\times\cap U_A^m=U_K^m
U_mUse in source · 22 passages
Search aliases: U_m
Use in source · Paper_X_Block27_Amended.tex, line 1389
Explicit summands inside a supplied decomposition. Let K be a field, let G be a monoid, and let ρ_i:G→End_K(W_i) be actual representations on finite-dimensional spaces, indexed by a finite set I. For natural numbers m_i≤ c_i, set U_m=∏_i∈ I∏_k<m_iW_i, U_c=∏_i∈ I∏_k<c_iW_i. These finite products realize the corresponding finite direct sums. The representation on U_m acts componentwise. Define j:U_m→ U_c by retaining the first m_i copies and filling every remaining coordinate with zero. Define r:U…
Exact mathematical passage
U_m=\prod_{i\in I}\prod_{k<m_i}W_i, \qquad U_c=\prod_{i\in I}\prod_{k<c_i}W_i.
u_mUse in source · 2 passages
Search aliases: u_m
Use in source · 03_nonlinear_coordinates.tex, line 471
Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a to a leaf and applying the scalar map (t_1,…,t_r)↦ b_rt_1⋯ t_r at a vertex of arity r. A second induction gives ∑_τ∈ T_m B_τ(s) ≤ ∑_τ∈ T_mw(τ) =t_m, wh…
Exact mathematical passage
u_{m+1}=\Phi(u_m)
u_{m+1}Use in source · 1 passage
Search aliases: u_m+1
Use in source · 03_nonlinear_coordinates.tex, line 471
Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a to a leaf and applying the scalar map (t_1,…,t_r)↦ b_rt_1⋯ t_r at a vertex of arity r. A second induction gives ∑_τ∈ T_m B_τ(s) ≤ ∑_τ∈ T_mw(τ) =t_m, wh…
Exact mathematical passage
u_{m+1}=\Phi(u_m)
U_{m,n+1}Use in source · 2 passages
Search aliases: U_m,n+1
Use in source · Paper_VI_Traditional_Final.tex, line 2572
Since C_nR_n=I, U_m,n+1(t)-U_m,n(t) = · C_m,n (C_nT_n+1(t)-T_n(t)C_n) R_nR_n,m. Therefore normU_m,n+1(t)-U_m,n(t) ≤ w_m,nnormE_n(t). The weighted summability hypothesis makes the sequence uniformly Cauchy. Summing the tail estimates gives the stated bound.
Exact mathematical passage
\begin{aligned} U_{m,n+1}(t)-U_{m,n}(t) ={}& C_{m,n} \bigl(C_nT_{n+1}(t)-T_n(t)C_n\bigr) R_nR_{n,m}. \end{aligned}
U_{m-1}Use in source · 2 passages
Search aliases: U_m-1
Use in source · 01_finite_core.tex, line 276
Let U=U_mU_m-1⋯ U_1 be a circuit with m≥1, and let {P_α} be a complete packet system. For the empty circuit m=0, set U=I. Its boundary block is P_β P_α=δ_βαP_α.
Exact mathematical passage
U=U_mU_{m-1}\cdots U_1
u_{Mk+s}Use in source · 1 passage
Search aliases: u_Mk+s
Use in source · supplement_06.tex, line 54
Refinement of an arithmetic address. Taking another subsequence with step M and offset s yields u_Mk+s=v_(NM)k+(Ns+j). The formal proof establishes this identity for the actual packet function by checking the index arithmetic. It is not inferred from a diagram or a naming convention.
Exact mathematical passage
u_{Mk+s}=v_{(NM)k+(Ns+j)}.
U_{mp,j}Use in source · 1 passage
Search aliases: U_mp,j
Use in source · 20_arithmetic.tex, line 571
[Step-sensitive divisor-branch bound] Under the hypotheses of Theorem~[source reference], for all u,k,n∈ N, Δ_u(p-1)/m^kU_mp,j(n) ∈ p^min(k(1+ν_p(u)),mn) Z_p. For C_mp,j the corresponding valuation lower bound is min(k(1+ν_p(u)),mn)-v_p(j!).
Exact mathematical passage
\Delta_{u(p-1)/m}^{k}U_{mp,j}(n) \in p^{\min(k(1+\nu_p(u)),mn)}\Z_p.
U_NUse in source · 2 passages
Search aliases: U_N
Use in source · Paper_IV_Traditional_Final.tex, line 920
[Addition and inversion laws for the companion matrix] Assume N≥2. The matrix U_N(z)=e^zJ_N satisfies U_N(z+w)=U_N(z)U_N(w), U_N(z)^-1=U_N(-z), det U_N(z)=1. Its first column is D_N(z), and all of its entries are given by [source reference].
Exact mathematical passage
U_N(z)=e^{zJ_N}
U_nUse in source · 1 passage
Search aliases: U_n
Use in source · atlas-pages.tex, line 689
not close the dynamicsREFINEMENT / A PERSISTENT RANGE EXCURSIONChapter VI, Part II: Theorem E4.8 and Example E4.10. Summability and closure are different tests. Let the coarse space be R and every finer space be R^2. The first connecting maps are C(x,y)=x and Rx=(x,0). All later maps are identities. Let the coarse evolution be the identity and all fine evolutions be rotations: T(t)=matcos t · -sin t sin t · cos t, U_n(t)=CT(t)R=cos t (n≥1). The reconstructed sequence is already constant above it…
Exact mathematical passage
T(t)=\mat{\cos t&-\sin t\\\sin t&\cos t},\qquad U_n(t)=CT(t)R=\cos t\quad(n\ge1).
u_NUse in source · 3 passages
Search aliases: u_N
Use in source · monograph_atlas.tex, line 283
n·σ). This is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form P( d Pwedge d P)P. With the paper's scalar convention, its component is F_θϕ=- iTr (P[∂_θ P,∂_ϕ P]) =frac12sinθ. vfigbloch_curvature headA local frame gives the same answer Away from the south pole, use u_N=(cos(θ/2),e^ iϕsin(θ/2))^T. Then P=u_Nu_N^* and the scalar one-form is A_N=- i u_N^* d u_N=frac1-cosθ2 dϕ, d A_N=frac12sinθ dθwedge dϕ. Thus the curvature flux through the cap 0≤θ…
Exact mathematical passage
u_N=(\cos(\theta/2),e^{\ii\phi}\sin(\theta/2))^T
u_nUse in source · 2 passages
Search aliases: u_n
Use in source · Paper_VI_Traditional_Final.tex, line 1447
Let K be a column-stochastic matrix, let V be a potential on the states, and let f be terminal data. Define M_V=diag(e^-V)K^T. The backward Feynman--Kac value after n steps is u_n=M_V^nf.
Exact mathematical passage
u_n=M_V^nf.
u_N^*Use in source · 2 passages
Search aliases: u_N^*
Use in source · monograph_atlas.tex, line 283
ojector. The intrinsic curvature is the endomorphism-valued two-form P( d Pwedge d P)P. With the paper's scalar convention, its component is F_θϕ=- iTr (P[∂_θ P,∂_ϕ P]) =frac12sinθ. vfigbloch_curvature headA local frame gives the same answer Away from the south pole, use u_N=(cos(θ/2),e^ iϕsin(θ/2))^T. Then P=u_Nu_N^* and the scalar one-form is A_N=- i u_N^* d u_N=frac1-cosθ2 dϕ, d A_N=frac12sinθ dθwedge dϕ. Thus the curvature flux through the cap 0≤θ≤θ_0 is π(1-cosθ_0). At θ_0=π/3, it is exactl…
Exact mathematical passage
P=u_Nu_N^*
U_QUse in source · 1 passage
Search aliases: U_Q
Use in source · Paper_V_Traditional_Final.tex, line 2263
Let P(z) and Q(z) be Laurent polynomials. Define L_P(z) = 1 · 0 P(z) · 1 , U_Q(z) = 1 · Q(z) 0 · 1 .
Exact mathematical passage
L_P(z) = \begin{pmatrix} 1&0\\ P(z)&1 \end{pmatrix}, \qquad U_Q(z) = \begin{pmatrix} 1&Q(z)\\ 0&1 \end{pmatrix}.
U_Q^{-1}Use in source · 1 passage
Search aliases: U_Q^-1
Use in source · Paper_V_Traditional_Final.tex, line 2281
[Exact lifting update] The lifting matrices are invertible with L_P^-1=L_-P, U_Q^-1=U_-Q. If R_0E_0=I, then every finite product V of lifting matrices gives another exact bank by E=VE_0, R=R_0V^-1. Every lifting step has determinant one.
Exact mathematical passage
L_P^{-1}=L_{-P}, \qquad U_Q^{-1}=U_{-Q}.
U_rUse in source · 18 passages
Search aliases: U_r
Use in source · 04_correction_transport.tex, line 704
Let A be a transport system on copies of a fixed vector space V_0, and let B be a transport system on copies of a fixed vector space W_0. Fix a root r and an isomorphism U_r:V_0⟶ W_0.
Exact mathematical passage
U_r:V_0\longrightarrow W_0.
\mathcal U_rUse in source · 4 passages
Search aliases: U_r mathcal
Use in source · root_scope.tex, line 382
[Exact quadratic compression] There is a unique preparation K_r(Z)= U_r(Z) W_r(Z), W_r(Z)=Z^2+2β_rZ+α_r, U_r≡ u_rpmod256. With δ_r=β_r^2-α_r and τ_r= U_r'(c_r)/ U_r(c_r), δ_r=-fracΦ_r^act U_r(c_r) (1-fracτ_r^2Φ_r^act4 U_r(c_r)), v_2(δ_r)=v_2(Φ_r^act). The equality includes the zero case. The same successive monic polynomial division used in Theorem~[source reference], started at precision eight, preserves the quadratic and unit reductions of [source reference]. Write W=Z^2+2β_rZ+α_r and w=W(c_r)…
Exact mathematical passage
K_r(Z)=\mathcal U_r(Z)\mathcal W_r(Z),\qquad \mathcal W_r(Z)=Z^2+2\beta_rZ+\alpha_r,\qquad \mathcal U_r\equiv u_r\pmod{256}.
u_rUse in source · 5 passages
Search aliases: u_r
Use in source · 04_arithmetic.tex, line 819
Let M be a matrix over a DVR. Smith normal form gives invertible matrices U,V such that UMV = diag(π^s_1u_1,…,π^s_ru_r,0,…,0), where each u_i is a unit and 0≤ s_1≤⋯≤ s_r. The exponents s_i are directional lifting weights.
Exact mathematical passage
UMV = \diag(\pi^{s_1}u_1,\ldots,\pi^{s_r}u_r,0,\ldots,0),
\mathcal U_r'Use in source · 1 passage
Search aliases: U_r' mathcal
Use in source · root_scope.tex, line 388
[Exact quadratic compression] There is a unique preparation K_r(Z)= U_r(Z) W_r(Z), W_r(Z)=Z^2+2β_rZ+α_r, U_r≡ u_rpmod256. With δ_r=β_r^2-α_r and τ_r= U_r'(c_r)/ U_r(c_r), δ_r=-fracΦ_r^act U_r(c_r) (1-fracτ_r^2Φ_r^act4 U_r(c_r)), v_2(δ_r)=v_2(Φ_r^act). The equality includes the zero case. The same successive monic polynomial division used in Theorem~[source reference], started at precision eight, preserves the quadratic and unit reductions of [source reference]. Write W=Z^2+2β_rZ+α_r and w=W(c_r)…
Exact mathematical passage
\tau_r=\mathcal U_r'(c_r)/\mathcal U_r(c_r)
U_{r,q}Use in source · 3 passages
Search aliases: U_r,q
Use in source · supplement_02.tex, line 138
The row action is connected to the matrix entries. Reversing the finite sum and using binomial symmetry gives the actual source-index row action E_2r=∑_q=0^rU_r,q(p)S_2q(p), where U_r,q(p)= displaystylebinom2r2qb_r-qp^2(r-q), · q≤ r,
Exact mathematical passage
E_{2r}=\sum_{q=0}^{r}U_{r,q}(p)S_{2q}(p),
U_{r,r}Use in source · 1 passage
Search aliases: U_r,r
Use in source · supplement_02.tex, line 149
0, · q>r. The row-action equality, the diagonal identity U_r,r=1, and vanishing above the diagonal are all checked. The arithmetic weight is also explicit: p^2(r-q)mid U_r,q(p) in Z, 0≤ q≤ r. No inverse-coordinate identity, unit diagonal or weight bound is assumed for these rows.
Exact mathematical passage
U_{r,r}=1
U_r^{(q)}Use in source · 1 passage
Search aliases: U_r^(q)
Use in source · 02_strengthened.tex, line 485
[Charge pruning of circuit paths] Suppose every gate has a finite charge decomposition U_r=∑_qU_r^(q), SU_r^(q)S^*=ω^qU_r^(q). A term with gate charges q_1,…,q_m can contribute from source charge a_0 only to target charge a_m=a_0+q_1+⋯+q_m. Every packet path violating one of the intermediate charge updates vanishes.
Exact mathematical passage
U_r=\sum_qU_r^{(q)}, \qquad SU_r^{(q)}S^*=\omega^qU_r^{(q)}.
U_{\rm root}Use in source · 1 passage
Search aliases: U_rm root rm
Use in source · Paper_V_Traditional_Final.tex, line 2357
Proceed by induction on the depth of the retained subtree. If the root is retained, the statement is immediate. If the root is split, apply the unitary map U_rm root, then apply the induction hypothesis independently in each child subtree. A direct sum of unitary maps is unitary. Reversing the maps with adjoints gives synthesis.
Exact mathematical passage
U_{\rm root}
U_uUse in source · 6 passages
Search aliases: U_u
Use in source · 04_correction_transport.tex, line 712
[Fundamental holonomy intertwining criterion] The following conditions are equivalent. item There is a family of isomorphisms U_v:V_0→ W_0 with the prescribed root value U_r and satisfying U_vA_vu=B_vuU_u for every oriented edge u→ v. item For every fundamental loop γ based at r, U_rHol_A(γ)=Hol_B(γ)U_r. item The same identity holds for every loop based at r. When these conditions hold, the family (U_v) is uniquely determined by U_r.
Exact mathematical passage
U_vA_{vu}=B_{vu}U_u
u_xUse in source · 2 passages
Search aliases: u_x
Use in source · Paper_XI_Traditional_Final.tex, line 970
Let z=∑_a∈ Az_au_a be central. Commuting with u_x gives z_ab_c(x,a)=z_a for every x. Nondegeneracy forces z_a=0 for every nonzero a. Thus the center is C. To prove semisimplicity directly, choose a linear projection P onto a submodule of a finite dimensional C^c[A]-module and form overline P=frac1abs A∑_a∈ Au_a P u_a^-1. Every summand maps into the submodule and is the identity there. Conjugation by any u_x permutes the summands: the cocycle scalars cancel between each operator and its inverse. …
Exact mathematical passage
u_x
U_\alphaUse in source · 3 passages
Search aliases: U_α alpha
Use in source · 03_nonlinear_coordinates.tex, line 976
[Packet decomposition of an obstruction] Suppose U=⊕_α∈ IU_α, V=⊕_α∈ IV_α, and D(U_α)⊆ V_α for every α. Let D_α:U_α⟶ V_α be the restriction. Then coker D ≅ ⊕_α∈ Icoker D_α. Under this isomorphism, the obstruction class of r=∑_α r_α is the tuple ([r_α])_α. Hence r is correctable if and only if every packet obstruction vanishes.
Exact mathematical passage
U=\bigoplus_{\alpha\in I}U_\alpha, \qquad V=\bigoplus_{\alpha\in I}V_\alpha,
U_{\beta\alpha}Use in source · 2 passages
Search aliases: U_βα beta alpha
Use in source · 01_finite_core.tex, line 301
[Packetwise unitarity conservation] If U is unitary and U_βα=P_β UP_α, then ∑_β U_βα^*U_βγ = δ_αγP_α. Similarly, ∑_α U_βαU_δα^* = δ_βδP_β.
Exact mathematical passage
U_{\beta\alpha}=P_\beta UP_\alpha
U_{\beta\alpha}^*Use in source · 2 passages
Search aliases: U_βα^* beta alpha
Use in source · 01_finite_core.tex, line 302
[Packetwise unitarity conservation] If U is unitary and U_βα=P_β UP_α, then ∑_β U_βα^*U_βγ = δ_αγP_α. Similarly, ∑_α U_βαU_δα^* = δ_βδP_β.
Exact mathematical passage
\boxed{ \sum_\beta U_{\beta\alpha}^*U_{\beta\gamma} = \delta_{\alpha\gamma}P_\alpha. }
U_{\beta\gamma}Use in source · 2 passages
Search aliases: U_βγ beta gamma
Use in source · 01_finite_core.tex, line 302
[Packetwise unitarity conservation] If U is unitary and U_βα=P_β UP_α, then ∑_β U_βα^*U_βγ = δ_αγP_α. Similarly, ∑_α U_βαU_δα^* = δ_βδP_β.
Exact mathematical passage
\boxed{ \sum_\beta U_{\beta\alpha}^*U_{\beta\gamma} = \delta_{\alpha\gamma}P_\alpha. }
U_\gammaUse in source · 2 passages
Search aliases: U_γ gamma
Use in source · 02_strengthened.tex, line 410
For a packet path γ=(a_0,a_1,…,a_m), write U_γ = P_a_mU_mP_a_m-1 ⋯ P_a_1U_1P_a_0.
Exact mathematical passage
U_\gamma = P_{a_m}U_mP_{a_{m-1}} \cdots P_{a_1}U_1P_{a_0}.
U_{\gamma\alpha}Use in source · 1 passage
Search aliases: U_γα gamma alpha
Use in source · 03_packets_and_graphs.tex, line 626
[Blockwise intertwining equations] Let T∈End(X), S∈End(Y), let P=(P_α)_α∈ I and Q=(Q_γ)_γ∈ J be finite packet systems, and let U:X→ Y be linear. Define T_βα=P_β TP_α, S_γδ=Q_γ SQ_δ, U_γα=Q_γ UP_α. Then UT=SU if and only if, for every γ∈ J and α∈ I, ∑_β∈ IU_γβT_βα = ∑_δ∈ JS_γδU_δα. [source reference] ExportsPacketwise equivalence, gluing, symmetry, and local representation comparison.
Exact mathematical passage
T_{\beta\alpha}=P_\beta TP_\alpha, \qquad S_{\gamma\delta}=Q_\gamma SQ_\delta, \qquad U_{\gamma\alpha}=Q_\gamma UP_\alpha.
U_{\gamma\beta}Use in source · 2 passages
Search aliases: U_γβ gamma beta
Use in source · 03_packets_and_graphs.tex, line 638
et T∈End(X), S∈End(Y), let P=(P_α)_α∈ I and Q=(Q_γ)_γ∈ J be finite packet systems, and let U:X→ Y be linear. Define T_βα=P_β TP_α, S_γδ=Q_γ SQ_δ, U_γα=Q_γ UP_α. Then UT=SU if and only if, for every γ∈ J and α∈ I, ∑_β∈ IU_γβT_βα = ∑_δ∈ JS_γδU_δα. [source reference] ExportsPacketwise equivalence, gluing, symmetry, and local representation comparison.
Exact mathematical passage
\boxed{ \sum_{\beta\in I}U_{\gamma\beta}T_{\beta\alpha} = \sum_{\delta\in J}S_{\gamma\delta}U_{\delta\alpha}. }
U_{\delta\alpha}Use in source · 2 passages
Search aliases: U_δα delta alpha
Use in source · 03_packets_and_graphs.tex, line 638
et T∈End(X), S∈End(Y), let P=(P_α)_α∈ I and Q=(Q_γ)_γ∈ J be finite packet systems, and let U:X→ Y be linear. Define T_βα=P_β TP_α, S_γδ=Q_γ SQ_δ, U_γα=Q_γ UP_α. Then UT=SU if and only if, for every γ∈ J and α∈ I, ∑_β∈ IU_γβT_βα = ∑_δ∈ JS_γδU_δα. [source reference] ExportsPacketwise equivalence, gluing, symmetry, and local representation comparison.
Exact mathematical passage
\boxed{ \sum_{\beta\in I}U_{\gamma\beta}T_{\beta\alpha} = \sum_{\delta\in J}S_{\gamma\delta}U_{\delta\alpha}. }
U_{\delta\alpha}^*Use in source · 1 passage
Search aliases: U_δα^* delta alpha
Use in source · 01_finite_core.tex, line 310
[Packetwise unitarity conservation] If U is unitary and U_βα=P_β UP_α, then ∑_β U_βα^*U_βγ = δ_αγP_α. Similarly, ∑_α U_βαU_δα^* = δ_βδP_β.
Exact mathematical passage
\sum_\alpha U_{\beta\alpha}U_{\delta\alpha}^* = \delta_{\beta\delta}P_\beta.
U_\piUse in source · 3 passages
Search aliases: U_π pi
Use in source · 04_correction_transport.tex, line 1062
[Convergence of sampled transports] Assume K is Lipschitz on [0,T]. For a partition 0=t_0<t_1<⋯<t_N=T, define U_π = exp((t_N-t_N-1)K(t_N-1)) ⋯ exp((t_1-t_0)K(t_0)). As the mesh |π| tends to zero, U_π⟶ U(T) in operator norm. There is a constant C, depending only on T, the supremum norm of K, and its Lipschitz constant, such that U_π-U(T) ≤ C|π|.
Exact mathematical passage
U_\pi = \exp\big((t_N-t_{N-1})K(t_{N-1})\big) \cdots \exp\big((t_1-t_0)K(t_0)\big).
\underline qUse in source · 1 passage
Search aliases: underline q underline
Use in source · root_scope.tex, line 140
ad J_r={q:λ_r(q)=μ_r}, G_r=2^-μ_rH_r. Since λ_r(0)=0 and λ_r(q)≥3q+1-s_2(r), every minimizing address satisfies 0≤ q≤lfloorfracs_2(r)-13rfloor. In particular μ_r≤0, and the minimum is found by a finite carry calculation. Reduction of the whole series gives overline G_r(X)=∑_q∈ J_rX^underline q in F_2[X]. Its highest degree is max J_r, with leading coefficient one. Thus the normalization is the actual Gauss normalization.
Exact mathematical passage
\overline G_r(X)=\sum_{q\in J_r}X^{\underline q} \quad\text{in }\mathbb F_2[X]. \label{eq:root:reduction}
\mathrm{unused}Use in source · 2 passages
Search aliases: unused mathrm
Use in source · 01_finite_core.tex, line 851
efine compressed transformed errors F_r=∑_aU_arE_aP. Then F_rP=F_r and F_r^*F_s=λ_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal projector. Use the recovery branches K_r=V_r^*, K_unused=I-S. They are complete on the whole ambient space because ∑_rK_r^*K_r+K_unused^*K_unused =S+(I-S)^2=I. The inverse unitary change of coordinates reconstructs each E_aP…
Exact mathematical passage
K_r=V_r^*,\qquad K_{\mathrm{unused}}=I-S.
\operatorname{updateValue}Use in source · 1 passage
Search aliases: updateValue operatorname
Use in source · 02_supplement_02.tex, line 101
Correctness of the actual update. For any edited subtree T' and valid context C, Lean proves updateValue(C,F(T'))=F(fill(C,T')). The proof inducts through the context. At each frame, the updated child contributes the new value, and each other child contributes its unchanged cached optimum. The frame recurrence is therefore exactly the full-tree recurrence.
Exact mathematical passage
\operatorname{updateValue}\bigl(C,F(T')\bigr)=F\bigl(\operatorname{fill}(C,T')\bigr).
V'Use in source · 4 passages
Search aliases: V'
Use in source · 03_nonlinear_coordinates.tex, line 953
[Naturality of obstruction classes] Suppose ccc U · xrightarrowD · V mathllapadownarrow · · downarrowmathrlapb U' · xrightarrowD' · V' is a commuting square. Then there is a unique induced map overline b:coker D⟶coker D' satisfying overline b Ob_D(r)=Ob_D'(br).
Exact mathematical passage
\begin{array}{ccc} U & \xrightarrow{D} & V\\ \mathllap{a}\downarrow & & \downarrow\mathrlap{b}\\ U' & \xrightarrow{D'} & V' \end{array}
v'Use in source · 2 passages
Search aliases: v'
Use in source · 03_nonlinear_coordinates.tex, line 1031
If v-v'∈im D, then Q_α(v-v')∈im D. Hence [v]↦[Q_α v] is well defined on the quotient. The idempotent, annihilation, and reconstruction identities descend from V to V/im D. The formula for an obstruction class follows from the definition.
Exact mathematical passage
v-v'\in\im D
v^{(k)}Use in source · 2 passages
Search aliases: v^(k)
Use in source · 01_manuscript.tex, line 548
For each frequency k, let v^(k)_j = frac1sqrt n exp(frac2π i jkn). Then (C(c)v^(k))_j = ∑_r=0^n-1c_rv^(k)_j-r = ( ∑_r=0^n-1 c_rexp(-frac2π i rkn) )v^(k)_j. The eigenvalue is sqrt n(Fc)_k. The vectors v^(k) are the columns of F^*, which gives the formula.
Exact mathematical passage
(C(c)v^{(k)})_j = \sum_{r=0}^{n-1}c_rv^{(k)}_{j-r} = \left( \sum_{r=0}^{n-1} c_r\exp\left(-\frac{2\pi\ii rk}{n}\right) \right)v^{(k)}_j.
v^{(k)}_jUse in source · 2 passages
Search aliases: v^(k)_j
Use in source · 01_manuscript.tex, line 541
For each frequency k, let v^(k)_j = frac1sqrt n exp(frac2π i jkn). Then (C(c)v^(k))_j = ∑_r=0^n-1c_rv^(k)_j-r = ( ∑_r=0^n-1 c_rexp(-frac2π i rkn) )v^(k)_j. The eigenvalue is sqrt n(Fc)_k. The vectors v^(k) are the columns of F^*, which gives the formula.
Exact mathematical passage
v^{(k)}_j = \frac{1}{\sqrt n} \exp\left(\frac{2\pi\ii jk}{n}\right).
v^{(k)}_{j-r}Use in source · 1 passage
Search aliases: v^(k)_j-r
Use in source · 01_manuscript.tex, line 548
For each frequency k, let v^(k)_j = frac1sqrt n exp(frac2π i jkn). Then (C(c)v^(k))_j = ∑_r=0^n-1c_rv^(k)_j-r = ( ∑_r=0^n-1 c_rexp(-frac2π i rkn) )v^(k)_j. The eigenvalue is sqrt n(Fc)_k. The vectors v^(k) are the columns of F^*, which gives the formula.
Exact mathematical passage
(C(c)v^{(k)})_j = \sum_{r=0}^{n-1}c_rv^{(k)}_{j-r} = \left( \sum_{r=0}^{n-1} c_r\exp\left(-\frac{2\pi\ii rk}{n}\right) \right)v^{(k)}_j.
V^{-1}Use in source · 3 passages
Search aliases: V^-1
Use in source · 20_arithmetic.tex, line 261
the form H_c(log(1+T))=fracT^cc! V(T), V(0)=1, V∈ W_p. The weight of c! compensates for the coefficient shift by c. The weighted ring is closed under inversion of series with constant coefficient one and under L_T=log(1+T)(1+T)d/dT. With ℓ(T)=log(1+T)/T, interpreted by coefficient shifting, differentiation of the displayed factorization gives R_c(log(1+T))=c(1+T)ℓ(T)+(L_TV)V^-1. Since pℓ(T)∈ W_p and pmid c, both terms are weighted. This is the singular estimate leanKomorebiPaperIX.SingularRatioA…
Exact mathematical passage
R_c(\log(1+T))=c(1+T)\ell(T)+(L_TV)V^{-1}.
v^2Use in source · 3 passages
Search aliases: v^2
Use in source · visual_atlas.tex, line 476
For a canonical coordinate change with input and output chart v=2x, the target formula is not still v^2. It is widehat f_b(v)=2f(v/2)=fracv^22. Consequently widehat f_b(2x)=2x^2, and the exact coordinate defect vanishes. This is the same distinction illustrated in the example in two dimensions on Plate III.10.
Exact mathematical passage
v^2
v^3Use in source · 1 passage
Search aliases: v^3
Use in source · Paper_XII_Traditional_Final.tex, line 1129
[A nonsemisimple Hecke packet] Let T = E'[u]/(u^2) × E'[v]/(v^3) and let V= T be the regular module. The two canonical central idempotents split V=V_u⊕ V_v with dimensions two and three. Multiplication by u is nilpotent on V_u, and multiplication by v is nilpotent on V_v. Neither local factor is semisimple, but the two maximal Hecke packets are isolated exactly by the Artin idempotents.
Exact mathematical passage
\TT = E'[u]/(u^2) \times E'[v]/(v^3)
V^HUse in source · 1 passage
Search aliases: V^H
Use in source · Paper_X_Block27_Amended.tex, line 1493
Checked group, inertia and monodromy finite sums. Let K be a field, G a group, and ν:G→ K^× a character. The new datum consists of an actual representation r and nilpotent N with r(g)N=ν(g)Nr(g) for every g. For a supplied subgroup H, define E(D,H)=ker N∩ V^H. This is the intersection of independently defined kernel and actual invariant subspaces. For normal H, the restricted whole-group representation and its inverse automorphisms are constructed. A change from φ to φ h, with h∈ H, leaves the r…
Exact mathematical passage
E(D,H)=\ker N\cap V^H.
V^{I_K}Use in source · 3 passages
Search aliases: V^I_K
Use in source · Paper_XI_Traditional_Final.tex, line 634
Let V be one interaction packet. If its inertia character is nontrivial, then V^I_K=0. If the inertia character is trivial, Frobenius acts on the invariant packet by a companion cycle.
Exact mathematical passage
V^{I_K}=0
V^{m-1}Use in source · 1 passage
Search aliases: V^m-1
Use in source · 02_connections_obstructions.tex, line 610
[Packet path expansion for different matrices] For i,j∈ V and m≥1, set i_0=i and i_m=j. Then (B_m⋯ B_1)_ji = ∑_(i_1,…,i_m-1)∈ V^m-1 B_m[i_m,i_m-1]⋯ B_1[i_1,i_0]. When m=1, the sum is over the unique empty tuple and its summand is B_1[j,i]. For m=0, the product is the identity and the entry is the Kronecker delta δ_ji. If every B_r[q,p] vanishes outside a declared relation R_r(p,q), then the endpoint entry is zero whenever there is no packet path compatible with the successive relations. statusle…
Exact mathematical passage
\boxed{ (B_m\cdots B_1)_{ji} = \sum_{(i_1,\ldots,i_{m-1})\in V^{m-1}} B_m[i_m,i_{m-1}]\cdots B_1[i_1,i_0]. }
v^nUse in source · 1 passage
Search aliases: v^n
Use in source · 72_recovered.tex, line 429
[Exponential form and minimal period] Let G_m=( F_p^×)^m. This cyclic group has order h. There are uniquely determined coefficients d_v∈ F_p, indexed by v∈ G_m, such that e_n^(m,p) = ∑_v∈ G_md_vv^n (n≥1). The minimal positive period is the least common multiple of the orders of those v for which d_v≠0. In particular, it divides h.
Exact mathematical passage
\boxed{ e_n^{(m,p)} = \sum_{v\in G_m}d_vv^n \qquad(n\geq1). }
v^{n-r}Use in source · 1 passage
Search aliases: v^n-r
Use in source · 72_recovered.tex, line 164
The nth coefficient is ∑_r=0^n binom nr u^rv^n-r = (u+v)^n by the binomial theorem.
Exact mathematical passage
\sum_{r=0}^{n} \binom nr u^rv^{n-r} = (u+v)^n
V^qUse in source · 1 passage
Search aliases: V^q
Use in source · strengthening.tex, line 887
For k=2^r≥4, put q=k/2, which is even. Normalize the exact expansion [source reference] by 2^q before reducing modulo two. Its uth auxiliary term has excess u+lfloor(k-u)/2rfloor-q=lfloor u/2rfloor. Only auxiliary indices zero and one can therefore survive: fracγ_k(H_2)2^q ≡ V^qγ_k(X) +ϵ V^q-1Gγ_k-1(X) pmod2 A_2. Here V≡τpmod2 and τ^2≡1pmod2. Since q is even and ϵτ≡ϵpmod2, this reduces to γ_k(X)+ϵ Gγ_k-1(X). Moreover, G≡γ_2(X)+τ Qpmod2. After evaluation, every term with a single τ vanishes, so t…
Exact mathematical passage
\frac{\gamma_k(H_2)}{2^q} \equiv V^q\gamma_k(X) +\epsilon V^{q-1}G\gamma_{k-1}(X) \pmod{2\mathcal A_2}. \label{eq:two-power-leading-normalized}
V^{q-1}Use in source · 1 passage
Search aliases: V^q-1
Use in source · strengthening.tex, line 887
For k=2^r≥4, put q=k/2, which is even. Normalize the exact expansion [source reference] by 2^q before reducing modulo two. Its uth auxiliary term has excess u+lfloor(k-u)/2rfloor-q=lfloor u/2rfloor. Only auxiliary indices zero and one can therefore survive: fracγ_k(H_2)2^q ≡ V^qγ_k(X) +ϵ V^q-1Gγ_k-1(X) pmod2 A_2. Here V≡τpmod2 and τ^2≡1pmod2. Since q is even and ϵτ≡ϵpmod2, this reduces to γ_k(X)+ϵ Gγ_k-1(X). Moreover, G≡γ_2(X)+τ Qpmod2. After evaluation, every term with a single τ vanishes, so t…
Exact mathematical passage
\frac{\gamma_k(H_2)}{2^q} \equiv V^q\gamma_k(X) +\epsilon V^{q-1}G\gamma_{k-1}(X) \pmod{2\mathcal A_2}. \label{eq:two-power-leading-normalized}
v^rUse in source · 1 passage
Search aliases: v^r
Use in source · 72_recovered.tex, line 443
Distinct elements of G_m define linearly independent functions on one full period, by the Vandermonde determinant. Therefore a positive integer r is a period exactly when v^r=1 for every active mode. The least such r is the stated least common multiple.
Exact mathematical passage
v^r=1
v_{(NM)k+(Ns+j)}Use in source · 1 passage
Search aliases: v_(NM)k+(Ns+j)
Use in source · supplement_06.tex, line 54
Refinement of an arithmetic address. Taking another subsequence with step M and offset s yields u_Mk+s=v_(NM)k+(Ns+j). The formal proof establishes this identity for the actual packet function by checking the index arithmetic. It is not inferred from a diagram or a naming convention.
Exact mathematical passage
u_{Mk+s}=v_{(NM)k+(Ns+j)}.
v_3Use in source · 18 passages
Search aliases: v_3
Use in source · 05_worked_models.tex, line 25
Let the quiver have vertices U,V and arrows R:U→ U, A:U→ V, B:V→ U, with the relation in the path algebra BA=e_U, whose realization is the operator identity BA=I_U. Realize U=𝕜^2, V=𝕜^3, by R= 0 · -1 1 · 0, A(u_1,u_2)=(u_1,u_2,0), B(v_1,v_2,v_3)=(v_1,v_2). Starting from the seed e_1∈ U, repeated application of R spans U, and A carries that span onto the first two coordinates of V. Any path from U to V ends with A and therefore lands in im A. The third coordinate direction, which spans ker B, is …
Exact mathematical passage
R= \begin{pmatrix}0&-1\\1&0\end{pmatrix}, \qquad A(u_1,u_2)=(u_1,u_2,0), \qquad B(v_1,v_2,v_3)=(v_1,v_2).
V_AUse in source · 3 passages
Search aliases: V_A
Use in source · 01_manuscript.tex, line 687
[Exact BDLR product] Let A=D_A+U_AV_A^*, B=D_B+U_BV_B^* use the same diagonal packet layout. Then AB=D_AD_B+U_CV_C^*, where U_C = D_AU_B+U_A(V_A^*U_B) · U_A and V_C = V_B · D_B^*V_A . The unrecompressed correction rank is at most r_A+r_B.
Exact mathematical passage
V_C = \begin{bmatrix} V_B&D_B^*V_A \end{bmatrix}.
V_aUse in source · 11 passages
Search aliases: V_a
Use in source · 72_recovered.tex, line 2741
[Packet decomposition of a lift obstruction] Suppose F_p^d = ⊕_a∈ AU_a, F_p^r = ⊕_a∈ AV_a, and suppose a linear correction map D: F_p^d→ F_p^r satisfies D(U_a)⊆ V_a for every a. Write D_a=D|_U_a. Then coker D ≅ ⊕_a∈ Acoker D_a. A residual lifts if and only if every packet obstruction vanishes.
Exact mathematical passage
\FF_p^d = \bigoplus_{a\in A}U_a, \qquad \FF_p^r = \bigoplus_{a\in A}V_a,
v_aUse in source · 2 passages
Search aliases: v_a
Use in source · 03_supplement_03.tex, line 53
An independent definition of Schmidt rank. The source defines Schmidt rank as the least number of summands that are product vectors. A decomposition of length s is the concrete equality C_ij=∑_a=1^su_a(i)v_a(j). It factors C=UV through an s-dimensional intermediate space, so rank C≤ s. Conversely, choose a basis of the row space of C. Expanding each row in this basis constructs a decomposition with exactly rank C product terms. Thus r(ψ)=rank C. The definition uses a least natural number, and th…
Exact mathematical passage
C_{ij}=\sum_{a=1}^{s}u_a(i)v_a(j).
V_A^*Use in source · 5 passages
Search aliases: V_A^*
Use in source · 01_manuscript.tex, line 669
[Exact BDLR product] Let A=D_A+U_AV_A^*, B=D_B+U_BV_B^* use the same diagonal packet layout. Then AB=D_AD_B+U_CV_C^*, where U_C = D_AU_B+U_A(V_A^*U_B) · U_A and V_C = V_B · D_B^*V_A . The unrecompressed correction rank is at most r_A+r_B.
Exact mathematical passage
A=D_A+U_AV_A^*, \qquad B=D_B+U_BV_B^*
V_a^*Use in source · 7 passages
Search aliases: V_a^*
Use in source · 03_supplement_04.tex, line 95
A constructive and total syndrome recovery. Let P be an orthogonal code projector. The input certificate consists of a finite family V_a satisfying V_aP=V_a, V_a^*V_b=δ_abP. These hypotheses state that the operators are partial isometries with orthogonal syndromes. The next supplement derives such a family from arbitrary Knill--Laflamme Gram data.
Exact mathematical passage
V_aP=V_a, \qquad V_a^*V_b=\delta_{ab}P.
v_a^TUse in source · 5 passages
Search aliases: v_a^T
Use in source · pages.tex, line 142
vpage9The cross term has somewhere to goCLOSED LOW-RANK PRODUCT ALGEBRAChapter VIII: Theorem 5.1, Corollary 5.2. Let A=D_A+u_av_a^T be the running operator. Take D_B=diag(1,2,3,4,5,6), u_b=(1,0,1,0,1,0)^T, and v_b=(0,1,0,1,0,1)^T. Define B=D_B+u_bv_b^T. U_C=[D_Au_b+u_a(v_a^Tu_b) u_a], V_C=[v_b D_B^Tv_a]. vfigbdlr_product headWhy there are only two columns Expanding AB gives three correction terms, but two share the right factor v_b^T: D_Au_bv_b^T+u_a(v_a^Tu_b)v_b^T =(D_Au_b+u_a(v_a^Tu_b))v_b^T. …
Exact mathematical passage
A=D_A+u_av_a^T
V_BUse in source · 2 passages
Search aliases: V_B
Use in source · 01_manuscript.tex, line 687
[Exact BDLR product] Let A=D_A+U_AV_A^*, B=D_B+U_BV_B^* use the same diagonal packet layout. Then AB=D_AD_B+U_CV_C^*, where U_C = D_AU_B+U_A(V_A^*U_B) · U_A and V_C = V_B · D_B^*V_A . The unrecompressed correction rank is at most r_A+r_B.
Exact mathematical passage
V_C = \begin{bmatrix} V_B&D_B^*V_A \end{bmatrix}.
V_bUse in source · 1 passage
Search aliases: V_b
Use in source · 03_supplement_04.tex, line 95
A constructive and total syndrome recovery. Let P be an orthogonal code projector. The input certificate consists of a finite family V_a satisfying V_aP=V_a, V_a^*V_b=δ_abP. These hypotheses state that the operators are partial isometries with orthogonal syndromes. The next supplement derives such a family from arbitrary Knill--Laflamme Gram data.
Exact mathematical passage
V_aP=V_a, \qquad V_a^*V_b=\delta_{ab}P.
v_bUse in source · 2 passages
Search aliases: v_b
Use in source · pages.tex, line 142
vpage9The cross term has somewhere to goCLOSED LOW-RANK PRODUCT ALGEBRAChapter VIII: Theorem 5.1, Corollary 5.2. Let A=D_A+u_av_a^T be the running operator. Take D_B=diag(1,2,3,4,5,6), u_b=(1,0,1,0,1,0)^T, and v_b=(0,1,0,1,0,1)^T. Define B=D_B+u_bv_b^T. U_C=[D_Au_b+u_a(v_a^Tu_b) u_a], V_C=[v_b D_B^Tv_a]. vfigbdlr_product headWhy there are only two columns Expanding AB gives three correction terms, but two share the right factor v_b^T: D_Au_bv_b^T+u_a(v_a^Tu_b)v_b^T =(D_Au_b+u_a(v_a^Tu_b))v_b^T. …
Exact mathematical passage
v_b=(0,1,0,1,0,1)^T
V_B^*Use in source · 4 passages
Search aliases: V_B^*
Use in source · 01_manuscript.tex, line 669
[Exact BDLR product] Let A=D_A+U_AV_A^*, B=D_B+U_BV_B^* use the same diagonal packet layout. Then AB=D_AD_B+U_CV_C^*, where U_C = D_AU_B+U_A(V_A^*U_B) · U_A and V_C = V_B · D_B^*V_A . The unrecompressed correction rank is at most r_A+r_B.
Exact mathematical passage
A=D_A+U_AV_A^*, \qquad B=D_B+U_BV_B^*
V_b^*Use in source · 1 passage
Search aliases: V_b^*
Use in source · 03_supplement_04.tex, line 122
Exact correction, including coherent errors. For every operator supported on the code, Pρ P=ρ, the central identity is R(V_aρ V_b^*)=δ_abρ. Consequently normalized mixtures satisfy R (∑_ap_aV_aρ V_a^*)=ρ when ∑_ap_a=1. The identity is algebraic and is proved for complex weights. Physical mixtures use the nonnegative real case. For a coherent error E_z=∑_a z_aV_a, the calculation of the cross terms yields R(E_zρ E_z^*)=(∑_a z_aoverlinez_a)ρ. Thus the same map also corrects normalized coherent sup…
Exact mathematical passage
\boxed{\mathcal R(V_a\rho V_b^*)=\delta_{ab}\rho.}
v_b^TUse in source · 4 passages
Search aliases: v_b^T
Use in source · pages.tex, line 142
vpage9The cross term has somewhere to goCLOSED LOW-RANK PRODUCT ALGEBRAChapter VIII: Theorem 5.1, Corollary 5.2. Let A=D_A+u_av_a^T be the running operator. Take D_B=diag(1,2,3,4,5,6), u_b=(1,0,1,0,1,0)^T, and v_b=(0,1,0,1,0,1)^T. Define B=D_B+u_bv_b^T. U_C=[D_Au_b+u_a(v_a^Tu_b) u_a], V_C=[v_b D_B^Tv_a]. vfigbdlr_product headWhy there are only two columns Expanding AB gives three correction terms, but two share the right factor v_b^T: D_Au_bv_b^T+u_a(v_a^Tu_b)v_b^T =(D_Au_b+u_a(v_a^Tu_b))v_b^T. …
Exact mathematical passage
B=D_B+u_bv_b^T
V_CUse in source · 4 passages
Search aliases: V_C
Use in source · pages.tex, line 143
vpage9The cross term has somewhere to goCLOSED LOW-RANK PRODUCT ALGEBRAChapter VIII: Theorem 5.1, Corollary 5.2. Let A=D_A+u_av_a^T be the running operator. Take D_B=diag(1,2,3,4,5,6), u_b=(1,0,1,0,1,0)^T, and v_b=(0,1,0,1,0,1)^T. Define B=D_B+u_bv_b^T. U_C=[D_Au_b+u_a(v_a^Tu_b) u_a], V_C=[v_b D_B^Tv_a]. vfigbdlr_product headWhy there are only two columns Expanding AB gives three correction terms, but two share the right factor v_b^T: D_Au_bv_b^T+u_a(v_a^Tu_b)v_b^T =(D_Au_b+u_a(v_a^Tu_b))v_b^T. …
Exact mathematical passage
U_C=[D_Au_b+u_a(v_a^Tu_b)\ \ u_a],\qquad V_C=[v_b\ \ D_B^Tv_a].
V_cUse in source · 2 passages
Search aliases: V_c
Use in source · 03_nonlinear_coordinates.tex, line 1044
[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is a natural exact sequence 0 · ⟶Ker D_0 ⟶Ker D_f ⟶Ker D_c xrightarrowδcoker D_0
Exact mathematical passage
\begin{array}{ccccccccc} 0&\longrightarrow&U_0&\xrightarrow{i_U}&U_f&\xrightarrow{p_U}&U_c&\longrightarrow&0\\[4pt] &&\mathllap{D_0}\downarrow&&\mathllap{D_f}\downarrow&&\mathllap{D_c}\downarrow&&\\[4pt] 0&\longrightarrow&V_0&\xrightarrow{i_V}&V_f&\xrightarrow{p_V}&V_c&\longrightarrow&0. \end{array}
V_C^*Use in source · 2 passages
Search aliases: V_C^*
Use in source · 01_manuscript.tex, line 675
[Exact BDLR product] Let A=D_A+U_AV_A^*, B=D_B+U_BV_B^* use the same diagonal packet layout. Then AB=D_AD_B+U_CV_C^*, where U_C = D_AU_B+U_A(V_A^*U_B) · U_A and V_C = V_B · D_B^*V_A . The unrecompressed correction rank is at most r_A+r_B.
Exact mathematical passage
AB=D_AD_B+U_CV_C^*,
V_C^TUse in source · 2 passages
Search aliases: V_C^T
Use in source · pages.tex, line 154
0,1,0,1,0)^T, and v_b=(0,1,0,1,0,1)^T. Define B=D_B+u_bv_b^T. U_C=[D_Au_b+u_a(v_a^Tu_b) u_a], V_C=[v_b D_B^Tv_a]. vfigbdlr_product headWhy there are only two columns Expanding AB gives three correction terms, but two share the right factor v_b^T: D_Au_bv_b^T+u_a(v_a^Tu_b)v_b^T =(D_Au_b+u_a(v_a^Tu_b))v_b^T. The remaining term is u_av_a^TD_B. Thus AB=D_AD_B+U_CV_C^T with at most two correction directions. Counting the three expanded summands as independent ranks would overestimate this representat…
Exact mathematical passage
AB=D_AD_B+U_CV_C^T
V_{ca}Use in source · 1 passage
Search aliases: V_ca
Use in source · 03_supplement_01.tex, line 12
0, · a≠ b, ∑_a∈ IP_a=1. Adjoint conditions are added when needed. For an operator U, define the transfer corner U_ba=P_bUP_a. The target index comes first. Completeness and distributivity give ∑_b,aU_ba=U, (UV)_ba=∑_c U_bcV_ca. The second identity uses P_c^2=P_c when adjacent transfer corners are multiplied.
Exact mathematical passage
\sum_{b,a}U_{ba}=U,\qquad (UV)_{ba}=\sum_c U_{bc}V_{ca}.
v_EUse in source · 1 passage
Search aliases: v_E
Use in source · odd_synthesis.tex, line 54
At a recursive base (L,t_L), introduce E=L(α), α^p-α=t_L^-q_d. This is a separable totally ramified extension of degree p, with v_E(α)=-q_d and Tr_E/Lα=0. Choose 1≤ b<p with q_db≡-1pmod p, set a=(q_db+1)/p, and put z=t_L^aα^b. Then z is an explicit uniformizer. Write its Eisenstein polynomial as f(X)=X^p+⋯+a_0.
Exact mathematical passage
v_E(\alpha)=-q_d
V_fUse in source · 3 passages
Search aliases: V_f
Use in source · 03_nonlinear_coordinates.tex, line 1044
[Relative obstruction exact sequence] Suppose the following diagram of modules commutes and has exact rows: ccccccccc 0 · ⟶ · U_0 · xrightarrowi_U · U_f · xrightarrowp_U · U_c · ⟶ · 0 [4pt] · · mathllapD_0downarrow · · mathllapD_fdownarrow · · mathllapD_cdownarrow · · [4pt] 0 · ⟶ · V_0 · xrightarrowi_V · V_f · xrightarrowp_V · V_c · ⟶ · 0. There is a natural exact sequence 0 · ⟶Ker D_0 ⟶Ker D_f ⟶Ker D_c xrightarrowδcoker D_0
Exact mathematical passage
\begin{array}{ccccccccc} 0&\longrightarrow&U_0&\xrightarrow{i_U}&U_f&\xrightarrow{p_U}&U_c&\longrightarrow&0\\[4pt] &&\mathllap{D_0}\downarrow&&\mathllap{D_f}\downarrow&&\mathllap{D_c}\downarrow&&\\[4pt] 0&\longrightarrow&V_0&\xrightarrow{i_V}&V_f&\xrightarrow{p_V}&V_c&\longrightarrow&0. \end{array}
v_FUse in source · 2 passages
Search aliases: v_F
Use in source · Paper_XI_Traditional_Final.tex, line 2105
The Artin--Schreier filtered lift has native trace one. Consequently each native correction δ satisfies v_F(trδ)+s≥1 (s=1), which makes its additive scalar correction trivial on U_F^1. The actual field and defining-sequence proof is [source reference]. Full beta-determinant transfer and character extension at each new maximal native field prove the full-character claim, including the odd-critical endpoint 2(lfloor r/2rfloor+1)-r=1. See [source reference]. The exact original quotient has dimensio…
Exact mathematical passage
v_F(\operatorname{tr}\delta)+s\ge1 \quad (s=1),
v_fUse in source · 5 passages
Search aliases: v_f
Use in source · 03_nonlinear_coordinates.tex, line 1090
If a class in coker D_f comes from coker D_0, its image in coker D_c is zero. Conversely, suppose [v_f] maps to zero in coker D_c. Choose u_c with D_cu_c=p_Vv_f and lift it to u_f∈ U_f. Then p_V(v_f-D_fu_f)=0, so v_f-D_fu_f=i_Vv_0 for some v_0∈ V_0. Thus [v_f] comes from [v_0]. Surjectivity at the last term follows from surjectivity of p_V.
Exact mathematical passage
[v_f]
V_{\frakm}Use in source · 2 passages
Search aliases: V_frakm V_ m frakm mathfrak
Use in source · Paper_XII_Traditional_Final.tex, line 1052
[Canonical Hecke packet decomposition] Let T be a finite dimensional commutative algebra over a field E', and let V be a finite T module. Then V = ⊕_ m∈Max Te_ mV. For every maximal ideal m, the canonical map gives e_ mV≅ V_ m. The packet e_ mV is nonzero exactly when m∈Supp_ T(V).
Exact mathematical passage
\boxed{ e_{\frakm}V\cong V_{\frakm}.}
V_hUse in source · 6 passages
Search aliases: V_h
Use in source · 72_recovered.tex, line 829
[Fourier invariance of the standard window operator] Let p be odd, let mmid p-1, and let h=(p-1)/m. Define (T_m,pf)_n = ∑_q=0^nbinommnmqf_q in F_p. Let V_h be the vector space of sequences satisfying f_n+h=f_n (n≥1). Then T_m,p(V_h)=V_h. The restriction of T_m,p to V_h is an automorphism.
Exact mathematical passage
V_h
V_iUse in source · 3 passages
Search aliases: V_i
Use in source · Paper_X_Block27_Amended.tex, line 1530
The remaining arithmetic comparison is WD(⊕_i V_i^⊕ m_i)≅ ⊕_iWD(V_i)^⊕ m_i for the actual attached functor, intertwining the Weil action and monodromy, together with the intended coefficient and Hodge--Tate comparisons. These are not supplied by the abstract datum. Claim 9.2 remains partial at its full arithmetic scope.
Exact mathematical passage
\operatorname{WD}(\bigoplus_i V_i^{\oplus m_i})\cong \bigoplus_i\operatorname{WD}(V_i)^{\oplus m_i}
v_iUse in source · 8 passages
Search aliases: v_i
Use in source · Paper_X_Block27_Amended.tex, line 2189
The actual invariant images U_i=imE_i give an equivariant linear isomorphism ⊕_i∈ JU_i≃ R^n, (v_i)_i⟼∑_i v_i, v⟼(E_iv)_i. Lean represents the finite direct sum by the finite product of the image submodules. The existing coordinatewise reduction theorem maps every U_i onto the corresponding reduced projector image. No chosen bases or unproved freeness of the images are used.
Exact mathematical passage
\bigoplus_{i\in J}U_i\simeq R^n, \qquad (v_i)_i\longmapsto\sum_i v_i, \qquad v\longmapsto(E_iv)_i.
V_{i,\lambda}Use in source · 2 passages
Search aliases: V_i,λ lambda
Use in source · Paper_X_Block27_Amended.tex, line 1224
For a compatible family ρ_i,λ of representations of the fixed finite group, set V_i,λ=ρ_i,λ∘ q_L. Suppose the actual absolute-Galois members are irreducible and the systems are pairwise componentwise nonisomorphic. The finite prime separator now yields distinct permitted rational primes p_j, actual prime ideals Q_j and arithmetic Frobenius witnesses γ_j as above, outside any prescribed finite set, such that the common algebraic trace matrix has nonzero determinant. Every coefficient embedding pr…
Exact mathematical passage
V_{i,\lambda}=\rho_{i,\lambda}\circ q_L
V_i^{\oplus m_i}Use in source · 1 passage
Search aliases: V_i^⊕ m_i oplus
Use in source · Paper_X_Block27_Amended.tex, line 1530
The remaining arithmetic comparison is WD(⊕_i V_i^⊕ m_i)≅ ⊕_iWD(V_i)^⊕ m_i for the actual attached functor, intertwining the Weil action and monodromy, together with the intended coefficient and Hodge--Tate comparisons. These are not supplied by the abstract datum. Claim 9.2 remains partial at its full arithmetic scope.
Exact mathematical passage
\operatorname{WD}(\bigoplus_i V_i^{\oplus m_i})\cong \bigoplus_i\operatorname{WD}(V_i)^{\oplus m_i}
V_j^*Use in source · 8 passages
Search aliases: V_j^*
Use in source · 02_strengthened.tex, line 1331
the Knill--Laflamme identities. Then the coefficient matrix c is positive semidefinite. Choose a unitary matrix W such that WcW^*=diag(λ_1,…,λ_r), λ_j≥0, and define F_j=∑_aoverlineW_jaE_a. For every j with λ_j>0, the operator V_j=λ_j^-1/2F_jP is an isometry from the code into H. Its range projector Q_j=V_jV_j^* is a syndrome packet, and the nonzero Q_j are pairwise orthogonal.
Exact mathematical passage
Q_j=V_jV_j^*
v_j^*Use in source · 1 passage
Search aliases: v_j^*
Use in source · monograph_atlas.tex, line 93
vpage6Four phases resolve a cyclic shiftCHARACTER PROJECTORS YOU CAN MULTIPLYChapter VII: Corollary II.3.3. On four sites, let Sket k=ketk+1bmod4 and let ω= i. The character projectors are the exact matrices P_j=frac14∑_r=0^3 i^-jrS^r, v_j(k)=frac12 i^-jk, P_j=v_jv_j^*. All entries belong to the Gaussian rationals. Thus no approximation to a root of unity is needed in this model. vfigcyclic_four headReading the columns Each column v_j is a normalized wave around the four sites. Advancing the add…
Exact mathematical passage
P_j=\frac14\sum_{r=0}^3\ii^{-jr}S^r, \qquad v_j(k)=\frac12\ii^{-jk},\qquad P_j=v_jv_j^*.
v_j^{-1}Use in source · 1 passage
Search aliases: v_j^-1
Use in source · 02_foundations.tex, line 842
The formula gives lceil1/drceil=1. Directly, let v≠0 and choose a coordinate v_j≠0. For any target w=(w_i), the matrix whose jth column has entries w_iv_j^-1 and whose other columns vanish sends v to w.
Exact mathematical passage
w_iv_j^{-1}
v_kUse in source · 9 passages
Search aliases: v_k
Use in source · monograph_atlas.tex, line 688
ts memory kernel is K(t)=frac925e^- i t+frac1625e^-2 i t, K(0)=1. Each oscillatory coefficient is positive although K(t) need not be real or pointwise nonnegative. At t=π/2, it equals -16/25-9 i/25. xfigmemory_positive headPositive definite means a matrix test at arbitrary times For retained scalars v_j and real times t_j, the structural identity is ∑_j,koverline v_jK(t_j-t_k)v_k = ∑_k e^ i t_kDB^*v_k ^2≥0. The exact worked check specializes to t=(0,π/2,π) and v=(1,2,-1). Its phases lie in QQ( i…
Exact mathematical passage
\sum_{j,k}\overline v_jK(t_j-t_k)v_k =\left\|\sum_k e^{\ii t_kD}B^*v_k\right\|^2\geq0.
v_{k,h}Use in source · 1 passage
Search aliases: v_k,h
Use in source · Paper_VI_Traditional_Final.tex, line 4277
The discrete eigenpairs are λ_k,h = frac4h^2 sin^2(frackπ h2), v_k,h(i)=sin(kπ x_i).
Exact mathematical passage
\lambda_{k,h} = \frac{4}{h^2} \sin^2\left(\frac{k\pi h}{2}\right), \qquad v_{k,h}(i)=\sin(k\pi x_i).
V_{k\ell}Use in source · 1 passage
Search aliases: V_kℓ ell
Use in source · Paper_IV_Traditional_Final.tex, line 520
At the origin, the derivative matrix is diagonal with entries 1,λ,…,λ^N-1. For an explicit determinant proof, write each basis function as a finite sum of exponentials with exponents a_ℓ=λω^ℓ. Its derivative matrix factors as W(z)=V diag(e^a_ℓ z) C, V_kℓ=a_ℓ^k, where C is the constant matrix of Fourier weights. Thus det W(z)=e^z∑_ℓ a_ℓdet W(0)=det W(0), because N≥2 implies ∑_ℓω^ℓ=0. The product of the diagonal initial values is the stated power of λ.
Exact mathematical passage
W(z)=V\,\operatorname{diag}(e^{a_\ell z})\,C, \qquad V_{k\ell}=a_\ell^k,
V_{m+1}Use in source · 3 passages
Search aliases: V_m+1
Use in source · Paper_XII_Traditional_Final.tex, line 1205
[Surjectivity on stable cores] For every m, the transition map V_m+1→ V_m induces a surjection Y_m+1↠ Y_m.
Exact mathematical passage
V_{m+1}\to V_m
V_m^*Use in source · 1 passage
Search aliases: V_m^*
Use in source · 02_supplement_03.tex, line 20
Construction. Suppose P_m-1=D^(m-1)+L_m-1R_m-1^*. The new factors are L_m · =[D_mL_m-1+U_m(V_m^*L_m-1) U_m], R_m · =[R_m-1 (D^(m-1))^*V_m]. Substitution into the already checked two-factor product identity proves exactness. This is not a rank-only existence argument. The recursively defined factors are themselves audited.
Exact mathematical passage
L_m&=\bigl[D_mL_{m-1}+U_m(V_m^*L_{m-1})\quad U_m\bigr],\\ R_m&=\bigl[R_{m-1}\quad (D^{(m-1)})^*V_m\bigr].
V_NUse in source · 4 passages
Search aliases: V_N
Use in source · 72_recovered.tex, line 2869
[Residue subsequences of a second order recurrence] Suppose x_n+2=Px_n+1-Qx_n over a commutative ring. Define V_0=2, V_1=P, and V_n+2=PV_n+1-QV_n. For every positive integer N and every address j, the subsequence u_k=x_Nk+j satisfies u_k+2=V_Nu_k+1-Q^Nu_k.
Exact mathematical passage
\boxed{ u_{k+2}=V_Nu_{k+1}-Q^Nu_k.}
v_nUse in source · 10 passages
Search aliases: v_n
Use in source · 72_recovered.tex, line 1667
Let v_n+1=Cv_n, a_n=w^TC^nv_0, with C∈ M_d( Z_p) invertible modulo p.
Exact mathematical passage
v_{n+1}=Cv_n, \qquad a_n=w^TC^nv_0,
V_{n+1}Use in source · 1 passage
Search aliases: V_n+1
Use in source · 72_recovered.tex, line 2861
[Residue subsequences of a second order recurrence] Suppose x_n+2=Px_n+1-Qx_n over a commutative ring. Define V_0=2, V_1=P, and V_n+2=PV_n+1-QV_n. For every positive integer N and every address j, the subsequence u_k=x_Nk+j satisfies u_k+2=V_Nu_k+1-Q^Nu_k.
Exact mathematical passage
V_{n+2}=PV_{n+1}-QV_n.
v_{n+1}Use in source · 5 passages
Search aliases: v_n+1
Use in source · 72_recovered.tex, line 1667
Let v_n+1=Cv_n, a_n=w^TC^nv_0, with C∈ M_d( Z_p) invertible modulo p.
Exact mathematical passage
v_{n+1}=Cv_n, \qquad a_n=w^TC^nv_0,
V_{n+2}Use in source · 1 passage
Search aliases: V_n+2
Use in source · 72_recovered.tex, line 2861
[Residue subsequences of a second order recurrence] Suppose x_n+2=Px_n+1-Qx_n over a commutative ring. Define V_0=2, V_1=P, and V_n+2=PV_n+1-QV_n. For every positive integer N and every address j, the subsequence u_k=x_Nk+j satisfies u_k+2=V_Nu_k+1-Q^Nu_k.
Exact mathematical passage
V_{n+2}=PV_{n+1}-QV_n.
v_{Nk+j}Use in source · 3 passages
Search aliases: v_Nk+j
Use in source · 72_recovered.tex, line 2786
Let R be a commutative ring. Let v_n+1=Cv_n, C∈ M_d(R). For a positive integer N and an address j∈{0,…,N-1}, define the residue state packet u_k^(j)=v_Nk+j.
Exact mathematical passage
u_k^{(j)}=v_{Nk+j}.
V_{nn}Use in source · 1 passage
Search aliases: V_nn
Use in source · 20_arithmetic.tex, line 809
egintheorem[Classical odd-length reconstruction] For every positive odd integer p and every n∈ N, E_2n=∑_m=0^nbinom2n2mγ_m p^2mS_2(n-m)(p). Equivalently, E_2n=∑_q=0^n V_nqS_2q(p) for the lower triangular matrix V_nq=binom2n2qγ_n-qp^2(n-q) (q≤ n), V_nq=0 (q>n). It has V_nn=1 and p^2(n-q)mid V_nq when q≤ n.
Exact mathematical passage
V_{nn}=1
V_{nq}Use in source · 3 passages
Search aliases: V_nq
Use in source · 20_arithmetic.tex, line 803
[Classical odd-length reconstruction] For every positive odd integer p and every n∈ N, E_2n=∑_m=0^nbinom2n2mγ_m p^2mS_2(n-m)(p). Equivalently, E_2n=∑_q=0^n V_nqS_2q(p) for the lower triangular matrix V_nq=binom2n2qγ_n-qp^2(n-q) (q≤ n), V_nq=0 (q>n). It has V_nn=1 and p^2(n-q)mid V_nq when q≤ n.
Exact mathematical passage
E_{2n}=\sum_{q=0}^n V_{nq}S_{2q}(p)
v_rUse in source · 5 passages
Search aliases: v_r
Use in source · 03_nonlinear_coordinates.tex, line 330
The relation to derivatives of the complete coordinate representation is also literal. For an open domain U⊆ X, put widehat F= A_ Q∘ F∘ S_ P on the open coordinate set A_ P(U). Boundedness and linearity of analysis and synthesis imply, for 1≤ r≤ m, D^rwidehat F(u)[v_1,…,v_r] = A_ Q(D^rF( S_ Pu) [ S_ Pv_1,…, S_ Pv_r]). Indeed, the first derivative follows from the chain rule. Repeating it adds no higher derivatives of the fixed linear coordinate maps. If each v_j is supported in one coordinate sl…
Exact mathematical passage
D^r\widehat F(u)[v_1,\ldots,v_r] =\Acal_{\Qcal}\bigl(D^rF(\Scal_{\Pcal}u) [\Scal_{\Pcal}v_1,\ldots,\Scal_{\Pcal}v_r]\bigr).
v_{r-1}Use in source · 1 passage
Search aliases: v_r-1
Use in source · 04_correction_transport.tex, line 642
or space X. At every vertex take one labeled copy of X. Every oriented edge u→ v carries an invertible transport T_vu∈GL(X), T_uv=T_vu^-1. Equivalently, one may begin with separate but isomorphic vertex fibers and choose one frame in each fiber. The matrices below are then the transports in those frames. For a path γ=(v_0,…,v_r), set T_γ = T_v_rv_r-1⋯ T_v_1v_0. For a loop, this product is its holonomy. In the based loop group, γ_2γ_1 means first traverse γ_1 and then γ_2, so T_γ_2γ_1=T_γ_2T_γ_1.…
Exact mathematical passage
T_\gamma = T_{v_rv_{r-1}}\cdots T_{v_1v_0}.
V_{\mathrm{src}}Use in source · 3 passages
Search aliases: V_src mathrm
Use in source · odd_synthesis.tex, line 120
Assume the one-vertex scalar-pushout contract: there are d labeled, actually supplied full character extensions with whole symplectic quotients V_i of dimension two over F_p. Their balanced C^×-central product has quotient V_src=⊕_iV_i and Heisenberg carrier dimension p^d=N. Initially take the declared stabilizer C={1}.
Exact mathematical passage
V_{\mathrm{src}}=\bigoplus_iV_i
V_{\mathrm{src}}^\perpUse in source · 1 passage
Search aliases: V_src^⊥ mathrm perp
Use in source · odd_synthesis.tex, line 150
Write W=V_src⊥ V_src^⊥. Finite Stone--von Neumann gives η_W≃η_src⊗η_⊥. Choose an isotropic eigenspace giving a line in the complementary factor. Its actual projector in the Heisenberg image algebra e has rank N. The retained induction/idempotent theorem recovers the same inducing representation Λ from eΛ, and then the same π by compact induction. No candidate is changed at recovery.
Exact mathematical passage
W=V_{\mathrm{src}}\perp V_{\mathrm{src}}^\perp
V_uUse in source · 3 passages
Search aliases: V_u
Use in source · Paper_XI_Traditional_Final.tex, line 1608
Let Γ be a connected finite graph. At every vertex v, place finite dimensional complex vector spaces V_v and W_v of the same dimension. For every oriented edge e:u→ v, let A_e:V_u⟶ V_v, B_e:W_u⟶ W_v be invertible. The reversed edge carries the inverse map.
Exact mathematical passage
A_e:V_u\longrightarrow V_v, \qquad B_e:W_u\longrightarrow W_v
V_vUse in source · 5 passages
Search aliases: V_v
Use in source · Paper_XI_Traditional_Final.tex, line 1607
Let Γ be a connected finite graph. At every vertex v, place finite dimensional complex vector spaces V_v and W_v of the same dimension. For every oriented edge e:u→ v, let A_e:V_u⟶ V_v, B_e:W_u⟶ W_v be invertible. The reversed edge carries the inverse map.
Exact mathematical passage
V_v
V_yUse in source · 2 passages
Search aliases: V_y
Use in source · Paper_V_Traditional_Final.tex, line 982
[Dynamic-programming best packet basis] Suppose J_y(Λ)=∑_v∈Λc_v(y). For a node v, define recursively V_y(v) = c_v(y), · v terminal, [2mm] displaystyle min{c_v(y),∑_u∈ch(v)V_y(u)}, · v internal. Stop at v when c_v(y)≤∑_uV_y(u), including equality, and refine otherwise. This deterministic rule produces a globally optimal complete pruning. With the node costs supplied and child values cached, the evaluator uses one comparison per internal node, one visit per node, and at most | T|-1 additions. Its …
Exact mathematical passage
V_y(v) = \begin{cases} c_v(y),&v\text{ terminal},\\[2mm] \displaystyle \min\left\{c_v(y),\sum_{u\in\operatorname{ch}(v)}V_y(u)\right\}, &v\text{ internal}. \end{cases}
v_\alphaUse in source · 4 passages
Search aliases: v_α alpha
Use in source · 01_manuscript.tex, line 1110
[Invariant packet Krylov decomposition] Assume [A,P_α]=0 for every packet. If v=∑_α v_α, v_α=P_α v, then A^jv = ∑_α A_α^jv_α for every j≥0, where A_α=P_α AP_α. Hence K_m(A,v) ⊆ ⊕_α K_m(A_α,v_α).
Exact mathematical passage
v=\sum_\alpha v_\alpha, \qquad v_\alpha=P_\alpha v,
V_\lambdaUse in source · 2 passages
Search aliases: V_λ lambda
Use in source · Paper_X_Block27_Amended.tex, line 1350
arrier summand completion] Suppose a semisimple strictly compatible carrier decomposes as mathscr W ≅ ⊕_i=1^r mathscr W_i^⊕ c_i, with the mathscr W_i as above. At one coefficient prime λ_0, suppose a semisimple summand has the form V_λ_0 ≅ ⊕_i=1^r W_i,λ_0^⊕ m_i, 0≤ m_i≤ c_i. Then V_λ = ⊕_i=1^r W_i,λ^⊕ m_i defines a strictly compatible summand at every coefficient prime. The vector (m_i) is uniquely recoverable from finitely many Frobenius traces supplied by the separation theorem.
Exact mathematical passage
\boxed{ V_\lambda = \bigoplus_{i=1}^{r} W_{i,\lambda}^{\oplus m_i} }
V_{\lambda_0}Use in source · 1 passage
Search aliases: V_λ_0 lambda
Use in source · Paper_X_Block27_Amended.tex, line 1341
[Compatible-carrier summand completion] Suppose a semisimple strictly compatible carrier decomposes as mathscr W ≅ ⊕_i=1^r mathscr W_i^⊕ c_i, with the mathscr W_i as above. At one coefficient prime λ_0, suppose a semisimple summand has the form V_λ_0 ≅ ⊕_i=1^r W_i,λ_0^⊕ m_i, 0≤ m_i≤ c_i. Then V_λ = ⊕_i=1^r W_i,λ^⊕ m_i defines a strictly compatible summand at every coefficient prime. The vector (m_i) is uniquely recoverable from finitely many Frobenius traces supplied by the separation theorem.
Exact mathematical passage
V_{\lambda_0} \cong \bigoplus_{i=1}^{r} W_{i,\lambda_0}^{\oplus m_i}, \qquad 0\le m_i\le c_i.
V_\muUse in source · 1 passage
Search aliases: V_μ mu
Use in source · 02_strengthened.tex, line 225
Complete reducibility gives a unitary decomposition H ≅ ⊕_μ∈widehat G V_μ⊗ M_μ, U(g) ≅ ⊕_μ π_μ(g)⊗ I_M_μ. On the μ summand, the displayed average is central in the irreducible representation π_μ, so it is a scalar by Schur's lemma. Taking its trace and applying character orthogonality shows that the scalar is one when μ=λ and zero otherwise. The asserted identities follow on the direct sum.
Exact mathematical passage
\Hcal \cong \bigoplus_{\mu\in\widehat G} V_\mu\otimes M_\mu, \qquad U(g) \cong \bigoplus_\mu \pi_\mu(g)\otimes I_{M_\mu}.
V_\piUse in source · 3 passages
Search aliases: V_π pi
Use in source · 04_correction_transport.tex, line 1002
[Direct rotations converge to packet transport] Assume the complete packet family P(t) is twice continuously differentiable in operator norm on [0,T]. For a partition 0=t_0<t_1<⋯<t_N=T with sufficiently small mesh, define V_π = U( P(t_N), P(t_N-1)) ⋯ U( P(t_1), P(t_0)). Then V_π⟶ U(T) in operator norm, where U is the canonical transport from [source reference]. On a fixed compact interval the error is bounded by a constant times the mesh.
Exact mathematical passage
V_\pi = \Ucal(\Pcal(t_N),\Pcal(t_{N-1})) \cdots \Ucal(\Pcal(t_1),\Pcal(t_0)).
v_\piUse in source · 1 passage
Search aliases: v_π pi
Use in source · Paper_X_Block27_Amended.tex, line 2055
For a DVR, e=v_π(|Γ|) is derived from the actual group order. The single precision e+1, independent of rank, detects integral isomorphism. It makes reduction injective on integral isomorphism classes, and finite coefficient quotients give finitely many integral classes in fixed rank. This is a formalization of classical finite-group rigidity (Maranda, 1953, Theorem~2), with an explicit proof of the comparison's precision loss.
Exact mathematical passage
e=v_\pi(|\Gamma|)
v_\varpiUse in source · 1 passage
Search aliases: v_ϖ varpi
Use in source · Paper_X_Block27_Amended.tex, line 604
The congruence is equivalent to e v_ϖ(x)≥ m, so x must lie in ϖ^lceil m/erceil O/ϖ^m, which has the stated cardinality. Write m=ek+r, 1≤ r≤ e. The exponent becomes (e-1)k+r-1. Summing over k≥0 and 1≤ r≤ e gives the rational expression.
Exact mathematical passage
e\,v_\varpi(x)\ge m,
V_\ellUse in source · 1 passage
Search aliases: V_ℓ ell
Use in source · 02_strengthened.tex, line 1349
For z∈ C^r, let F_z=∑_az_aE_a. Then PF_z^*F_zP=(z^*cz)P≥0, so c≥0. The unitary change of error coordinates gives PF_j^*F_ℓ P = λ_jδ_jℓP. Thus V_j^*V_ℓ=δ_jℓP. Each V_j is an isometry on P H, and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.
Exact mathematical passage
V_j^*V_\ell=\delta_{j\ell}P.
\mathrm{val}Use in source · 4 passages
Search aliases: val mathrm
Use in source · 04_arithmetic.tex, line 838
[Concrete Smith-weighted congruence solve] Let p>1 and consider the diagonal system over ( Z/(p^L) Z)^d diag(p^s_1,…,p^s_d)x=b pmodp^L. The system is solvable if and only if p^min(s_i,L)mid (b_i)_val for every i, where (b_i)_val is the canonical natural representative. Conditional on this admissibility condition, the number of solutions is p^∑_i=1^dmin(s_i,L). If any coordinate is inadmissible, the solution set is empty and its cardinality is zero.
Exact mathematical passage
p^{\min(s_i,L)}\mid (b_i)_{\mathrm{val}}
\operatorname{val}Use in source · 5 passages
Search aliases: val operatorname
Use in source · 03_supplement_03.tex, line 93
Constructed cyclic character projectors. Let N>0 and let S be a unitary complex matrix with S^N=I. The implementation uses the standard additive character of Z/N Z and proves its expression through a root of unity: ω=exp(2π i/N), χ(r)=ω^val(r). Here val(r) is the canonical representative in {0,…,N-1}. The representation is constructed as ρ(r)=S^val(r). Reduction of exponents modulo N is proved from S^N=I, giving ρ(r+s)=ρ(r)ρ(s) and ρ(0)=I. The final matrix theorem derives the representation law …
Exact mathematical passage
\omega=\exp(2\pi i/N),\qquad \chi(r)=\omega^{\operatorname{val}(r)}.
\VarUse in source · 10 passages
Search aliases: Var operatorname
Use in source · Paper_VI_Traditional_Final.tex, line 580
[Symmetric random walk] Starting from δ_0 and applying the transfer T=frac12(S+S^-1) for n steps gives P(X_n=k)= 2^-nbinomn(n+k)/2 when k≡ npmod2, and zero otherwise. Hence E[X_n]=0, Var(X_n)=n. The parity packet is exact: all mass remains in the one residue class permitted by n.
Exact mathematical passage
\Exp[X_n]=0, \qquad \Var(X_n)=n.
\operatorname{Var}Use in source · 1 passage
Search aliases: Var operatorname
Use in source · atlas-pages.tex, line 184
o. The vectors below specify the corresponding directions, not only their sizes. headThe energy identity follows from actual orthogonality The tower identities give E_0E_1=E_1E_0=E_0. Together with weighted self-adjointness, these imply D_iD_j=0 (i≠ j), ∑_j=0^2 D_j=I. For the chosen f, the total weighted energy is f _π^2=13 and the mean is 16/5. Consequently Var_π(f) =13-(frac165)^2 =frac6925 =frac108175+frac157. The first term is the energy of the between-block contrast, and the second is the w…
Exact mathematical passage
\operatorname{Var}_\pi(f) =13-\left(\frac{16}{5}\right)^2 =\frac{69}{25} =\frac{108}{175}+\frac{15}{7}.
\Var_{p(t)}Use in source · 4 passages
Search aliases: Var_p(t) Var operatorname
Use in source · Paper_VI_Traditional_Final.tex, line 3809
[Evidence cumulants and score monotonicity] Set ψ(t) = log( ∑_i e^tg_ip_i(0) ). Then ψ'(t)=ipgp(t), ψ''(t)=Var_p(t)(g)≥0. Consequently, frac d d tipgp(t) = Var_p(t)(g).
Exact mathematical passage
\psi'(t)=\ip{g}{p(t)}, \qquad \psi''(t)=\Var_{p(t)}(g)\geq0.
\Var_\gammaUse in source · 2 passages
Search aliases: Var_γ Var gamma operatorname gamma
Use in source · Paper_VI_Traditional_Final.tex, line 3278
[Gaussian spectral gap] If f∈ L^2(γ) has mean zero, then norm O_tf_2 ≤ e^-tnorm f_2. For every polynomial f, Var_γ(f) ≤ ∫_ Rabsf'(x)^2γ( d x).
Exact mathematical passage
\Var_\gamma(f) \leq \int_\RR\abs{f'(x)}^2\gamma(\dd x).
\Var_\piUse in source · 1 passage
Search aliases: Var_π Var pi operatorname pi
Use in source · Paper_VI_Traditional_Final.tex, line 3093
[Law of total variance] For a real valued f, let P_0f= E_π[f] 1. Then Var_π(f) = Var_π(P_ Qf) + normf-P_ Qf_π^2.
Exact mathematical passage
\Var_\pi(f) = \Var_\pi(P_\mathcal Qf) + \norm{f-P_\mathcal Qf}_\pi^2.
\operatorname{Var}_\piUse in source · 1 passage
Search aliases: Var_π operatorname pi
Use in source · atlas-pages.tex, line 184
o. The vectors below specify the corresponding directions, not only their sizes. headThe energy identity follows from actual orthogonality The tower identities give E_0E_1=E_1E_0=E_0. Together with weighted self-adjointness, these imply D_iD_j=0 (i≠ j), ∑_j=0^2 D_j=I. For the chosen f, the total weighted energy is f _π^2=13 and the mean is 16/5. Consequently Var_π(f) =13-(frac165)^2 =frac6925 =frac108175+frac157. The first term is the energy of the between-block contrast, and the second is the w…
Exact mathematical passage
\operatorname{Var}_\pi(f) =13-\left(\frac{16}{5}\right)^2 =\frac{69}{25} =\frac{108}{175}+\frac{15}{7}.
\varprojlim_mUse in source · 6 passages
Search aliases: varprojlim_m varprojlim
Use in source · 03_nonlinear_coordinates.tex, line 72
ng conventions All packet systems are finite unless a statement says otherwise. In a normed setting every packet projector and every coarsening operator is bounded. Differentiability of operator-valued maps is taken in operator norm. A Banach space isomorphism is a bounded linear bijection with bounded inverse. For a decreasing filtration (F^mX), completeness means that the canonical map X⟶varprojlim_m X/F^mX is an isomorphism and separatedness means ⋂_mF^mX=0. Statements about congruence classe…
Exact mathematical passage
X\longrightarrow\varprojlim_m X/F^mX
\varprojlim_{m\ge0}Use in source · 1 passage
Search aliases: varprojlim_m≥0 varprojlim ge
Use in source · Paper_X_Block27_Amended.tex, line 2355
Let I⊆ R and suppose that R is I-adically complete. Using the checked ring-map reconstruction on the quadratic coordinate ring, Lean proves P_q(R)≃varprojlim_m≥0 P_q(R/I^m). The zero-precision quotient is included only for the formal indexing. The inverse constructs an actual arithmetic character, with exactly the prescribed reductions. It is unique, and its continuity is derived from its factorization through the same finite quotient.
Exact mathematical passage
\mathcal P_q(R)\simeq\varprojlim_{m\ge0}\mathcal P_q(R/I^m).
\VcalUse in source · 2 passages
Search aliases: Vcal V mathcal
Use in source · 03_nonlinear_coordinates.tex, line 169
[Change of packet coordinates] Let P and P' be packet systems on X. Let Q and Q' be packet systems on Y. For any two packet systems U and V on the same space, write J_ U← V = A_ U S_ V. Then F_ Q'← P' = J_ Q'← Q F_ Q← P J_ P← P'.
Exact mathematical passage
\Vcal
\vdotsUse in source · 3 passages
Search aliases: vdots
Use in source · Paper_IV_Traditional_Final.tex, line 836
[Companion matrix realization] Put D_N(z) = D_N,0(z) D_N,1(z) vdots D_N,N-1(z) . Then D_N(z)=e^zJ_Ne_0 and D_N(z)=frac1Ntr(e^zJ_N). The eigenvalues of J_N are the N roots of -1, each with multiplicity one.
Exact mathematical passage
\mathbf D_N(z) = \begin{pmatrix} \Dfun_{N,0}(z)\\ \Dfun_{N,1}(z)\\ \vdots\\ \Dfun_{N,N-1}(z) \end{pmatrix}.
\operatorname{vec}Use in source · 4 passages
Search aliases: vec operatorname
Use in source · pages.tex, line 107
vpage7A transpose, not an adjointKRONECKER ACTION AND ARRAY ORDERChapter VIII: Theorem 4.5, Corollary 4.6. With column-major vectorization, the tensor identity is (A⊗ B)vec(X)=vec(BXA^T). Here the transpose records index order, not a Hermitian inner product. Use the explicitly complex matrix A=mat1 · i
Exact mathematical passage
(A\otimes B)\operatorname{vec}(X)=\operatorname{vec}(BXA^T).
\volUse in source · 2 passages
Search aliases: vol operatorname
Use in source · residue_continuation_body.tex, line 38
There is a constant B≥0, depending on K,M,J, such that, for every a∈ J/M and every real t≥1, |#{x∈ J:xbmod M=a, ι(x)∈ tD} -fracvol(D)q covol(L(J))t^d| ≤ B(t^d-1+1). The same constant works for all residues a.
Exact mathematical passage
\left|\#\{x\in J:x\bmod M=a,\ \iota(x)\in tD\} -\frac{\vol(D)}{q\,\covol(L(J))}t^d\right| \leq B(t^{d-1}+1).
\vpUse in source · 20 passages
Search aliases: vp v_p
Use in source · 72_recovered.tex, line 553
w(r)<L binom nr c_rp^w(r)S_n-r pmod p^L. When the omitted set is nonempty, the tail has valuation at least min_substack0≤ r≤ n w(r)≥ L ( w(r)+v_pbinom nr+v_p(S_n-r) ).
Exact mathematical passage
\boxed{ \min_{\substack{0\leq r\leq n\\w(r)\geq L}} \left( w(r)+\vp\binom nr+\vp(S_{n-r}) \right). }
\vpageUse in source · 1 passage
Search aliases: vpage
Use in source · visual_atlas.tex, line 19
[4pt] Coordinates · Can every vector be split into packets and then reconstructed exactly? [4pt] Completion · When finite words do not suffice, does their span become dense in the specified topology? A packet projector can be available as a way to inspect a vector without belonging to the generation grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. takeawayThe useful question is not ``How powerful is this seed?'' It i…
Exact mathematical passage
4pt] \textbf{Coordinates} & Can every vector be split into packets and then reconstructed exactly?\\[4pt] \textbf{Completion} & When finite words do not suffice, does their span become dense in the specified topology? \end{tabularx} A packet projector can be available as a way to \emph{inspect} a vector without belonging to the \emph{generation} grammar. Admitting it as a new primitive can lower seed rank. That distinction explains the second and third rows of the figure. \takeaway{The useful question is not ``How powerful is this seed?'' It is ``What does this seed generate under this particular, previously fixed algebra of operators?''} \vpage{2}{An atlas of the objects}{ORIENTATION / SYMBOLS WITH JOBS}{Main text: \cref{def:typed-grammar,def:seed-system,def:seed-rank,def:packet-system,def:transfer-block}} Most of the chapter can be read by tracking a small collection of objects. The table distinguishes their mathematical roles and names a concrete instance used in this appendix. {\small \begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.15\linewidth}Y>{\raggedright\arraybackslash}p{.28\linewidth}@{}}\toprule \textbf{Object} & \textbf{What it means} & \textbf{Instance here}\\\midrule $X$ & The ambient vector space or module. It specifies the objects that might be generated. & $\QQ^6$ with two chains of length three.\\ $S\subseteq X$ & The selected seed family. A family can contain vectors with components in several packets. & $\{e_0+f_0\}$ or $\{e_0,f_0\}$.\\ $\AA$ & The fixed unital algebra of admissible operators. It is not chosen anew for each target. & $\QQ[N]$ versus $\QQ\langle N,P_e,P_f\rangle$.\\ $\Hull_{\AA}(S)$ & The smallest invariant linear subspace containing the seeds. & $\Span\{e_j+f_j:0\le j\le2\}$.\\ $P_\alpha$ & A finite family of pairwise annihilating idempotents whose sum is $I$. & Chain projections, oblique projections, or residue masks.\\ $X_\alpha$ & The packet subspace $P_\alpha X$. It can have dimension greater than one. & One chain of dimension three.\\ $T_{\beta\alpha}$ & The block $P_\beta T P_\alpha$: input $\alpha$, output $\beta$. & A numerical entry when each coordinate packet has dimension one.\\ $F/I_s$ & A module of operator words modulo relations that kill the seed. & A quotient of left modules. It need not be a quotient algebra.\\ $U$ & An invertible change of realization coordinates. & A shear that makes orthogonal packets oblique.\\ $\Phi$ & An intertwiner that need not be invertible. & Projection from two chains to the first chain.\\ $\tau$ & The topology used when taking the closure of an algebraic hull. & The $\ell^2$ norm for a bilateral shift.\\\bottomrule \end{tabularx}} \head{Read products from right to left} In $P_\beta T P_\alpha x$, first select the input packet, then apply $T$, then select the output packet. In a matrix, this is why the output label belongs to the row and the input label to the column. \head{Where to begin} For seed rank, start with \pagerefV{3}. For exact reconstruction, start with \pagerefV{7}. For graphs, use \pagerefV{15} and \pagerefV{16}. For completion in infinite dimensions, use \pagerefV{18} and \pagerefV{19}. The scope of the formal proofs is collected on \pagerefV{22}. \vpage{3}{Two chains can stay linked}{MODEL A / AN OPERATOR IN SIX DIMENSIONS}{Main text: \cref{thm:single-operator}. Matrices over $\QQ$} Use the ordered basis $(e_0,e_1,e_2,f_0,f_1,f_2)$. Define \[ Ne_0=e_1,\quad Ne_1=e_2,\quad Ne_2=0, \qquad Nf_0=f_1,\quad Nf_1=f_2,\quad Nf_2=0.
\vTwoUse in source · 17 passages
Search aliases: vTwo v_2
Use in source · 72_recovered.tex, line 90
The dyadic kernel is substantially sharper than the separate coefficient bound alone suggests. If fracz^2sinh^2z = ∑_r≥0β_rz^2r and κ_2,r=β_r2^2r(2r)!, then v_2(κ_2,r)=4r-1 (r≥1). Consequently, truncation after depth M is uniformly valid modulo 2^4M+3, and this exponent is optimal. The rowwise estimate records the additional binary carries in the binomial coefficient. Classical congruence theory for Euler families is extensive [source reference]. The formulas here are stated as exact coordinate …
Exact mathematical passage
\vTwo(\kappa_{2,r})=4r-1 \qquad(r\geq1).
\mathcal WUse in source · 9 passages
Search aliases: W mathcal
Use in source · 20_arithmetic.tex, line 231
At the singular addresses c=ap, 1≤ a<m, use the weighted ring W_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping the falling factorial (x)_d=∏_i=0^d-1(x-i) into consecutive blocks of length p then gives an integral residue-coordinate factor p^v_p(lfloor d/prfloor!)…
Exact mathematical passage
\mathcal W_p=\left\{\sum_{d\ge0}a_dT^d: p^{\lfloor d/p\rfloor}a_d\in\Z_p\right\}.
W'Use in source · 2 passages
Search aliases: W'
Use in source · root_scope.tex, line 403
athrmact). The equality includes the zero case. The same successive monic polynomial division used in Theorem~[source reference], started at precision eight, preserves the quadratic and unit reductions of [source reference]. Write W=Z^2+2β_rZ+α_r and w=W(c_r). Stationarity gives W'(c_r)=-τ_rw, while w=Φ_r^act/ U_r(c_r). The quadratic identity β_r^2-α_r=W'(c_r)^2/4-W(c_r) proves the formula. Since τ_r∈256 Z_2, its bracket is a unit.
Exact mathematical passage
W'(c_r)=-\tau_rw
W^*Use in source · 7 passages
Search aliases: W^*
Use in source · monograph_atlas.tex, line 603
0 · 0 · 0, PE_a^*E_bP=c_abP. The errors are redundant rather than three distinct syndromes. A real unitary change of error coordinates gives W=mat3/5 · 4/5 · 0 -4/5 · 3/5 · 0 0 · 0 · 1, WcW^*=diag(1,0,0), (F_1,F_2,F_3)=(P,0,0). xfiggram headNormalize only positive eigenvalues The positive branch has V_1=P as an isometry on the code, and syndrome support Q=P. No factor λ^-1/2 is formed on either zero branch: those branches vanish on the code. Treating them as extra recoverable packets would divid…
Exact mathematical passage
W=\mat{3/5&4/5&0\\-4/5&3/5&0\\0&0&1}, \quad WcW^*=\diag(1,0,0), \quad(F_1,F_2,F_3)=(P,0,0).
w^*Use in source · 2 passages
Search aliases: w^*
Use in source · monograph_atlas.tex, line 338
0 · 0 · 0 · 1. Its middle block has one negative eigenvalue. More concretely, the normalized vector w=(ket01-ket10)/sqrt2 satisfies w^*(id⊗ T)(ρ_rm Bell)w=-frac12. This exact negative value of the quadratic form certifies failure of positivity on the composite system. No numerical eigenvalue tolerance is needed. headWhy the Kraus form avoids this problem For every auxiliary dimension and positive X, each term (I⊗ K_a)X(I⊗ K_a)^* is positive. Their sum is positive as well. This argument explains …
Exact mathematical passage
w^*(\id\otimes\mathsf T)(\rho_{\rm Bell})w=-\frac12.
w^{-1}Use in source · 2 passages
Search aliases: w^-1
Use in source · 02_connections_obstructions.tex, line 677
[A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly when vu=w. If the three coordinate systems are literally the same and canonical transport is used, then u=v=w=1 and the obstruction vanishes. Choosing …
Exact mathematical passage
H_{2,1,0}(x)=w^{-1}vux.
w^2Use in source · 11 passages
Search aliases: w^2
Use in source · visual_atlas.tex, line 495
Here e=1+3w+5w^2 and o=2+4w+6w^2, so direct polynomial multiplication gives y_rm even=1+8w+16w^2+17w^3, y_rm odd=4+12w+20w^2+6w^3. This is the same result as ordinary convolution followed by residue extraction. No numerical transform or limiting argument is needed because every sequence is finitely supported.
Exact mathematical passage
e=1+3w+5w^2
W^{2,p}Use in source · 1 passage
Search aliases: W^2,p
Use in source · atlas-pages.tex, line 590
l smooth remainder. headThe normalization makes the trace coefficient literal The general packet is ϕ_m,τ(t)=(t-τ)_+^m/m!. Its derivative of order m has a unit trace jump, while its other classical trace jumps vanish. Thus its multiplier is exactly the jump being removed, with no hidden factorial. Here the remainder has matching value and first-derivative traces and belongs to W^2,p(0,1) for every 1≤ p≤∞. Its second weak derivative is 2. The singular packet accounts for the distributional bounda…
Exact mathematical passage
W^{2,p}(0,1)
w^3Use in source · 3 passages
Search aliases: w^3
Use in source · visual_atlas.tex, line 496
Here e=1+3w+5w^2 and o=2+4w+6w^2, so direct polynomial multiplication gives y_rm even=1+8w+16w^2+17w^3, y_rm odd=4+12w+20w^2+6w^3. This is the same result as ordinary convolution followed by residue extraction. No numerical transform or limiting argument is needed because every sequence is finitely supported.
Exact mathematical passage
y_{\rm even}=1+8w+16w^2+17w^3, \qquad y_{\rm odd}=4+12w+20w^2+6w^3.
w^bUse in source · 1 passage
Search aliases: w^b
Use in source · Paper_IV_Traditional_Final.tex, line 544
Expand e^λ(z+w)=e^λ ze^λ w as a bivariate power series. A term z^a w^b contributes to the kth section exactly when a+b≡ kbmod N. Grouping first by i≡ abmod N and j≡ bbmod N gives the displayed sum.
Exact mathematical passage
z^a w^b
W^kUse in source · 1 passage
Search aliases: W^k
Use in source · strengthening.tex, line 376
For every starting row, Δ_1^k C_0(n) = L(U^4nW^k) =∑_s=0^nbinom ns L(W^k+s). Each summand lies in 2^k k! Z_2, proving the factorial bound for address zero from row zero onward.
Exact mathematical passage
\Delta_1^k C_0(n) =\mathcal L(U^{4n}W^k) =\sum_{s=0}^n\binom ns\mathcal L(W^{k+s}). \label{eq:base-row-finite-expansion}
W^{k+s}Use in source · 1 passage
Search aliases: W^k+s
Use in source · strengthening.tex, line 376
For every starting row, Δ_1^k C_0(n) = L(U^4nW^k) =∑_s=0^nbinom ns L(W^k+s). Each summand lies in 2^k k! Z_2, proving the factorial bound for address zero from row zero onward.
Exact mathematical passage
\Delta_1^k C_0(n) =\mathcal L(U^{4n}W^k) =\sum_{s=0}^n\binom ns\mathcal L(W^{k+s}). \label{eq:base-row-finite-expansion}
w^{k-1}Use in source · 1 passage
Search aliases: w^k-1
Use in source · Paper_IV_Traditional_Final.tex, line 1154
Let B_r(w)=Δ_N(w)^-r = ∑_n≥0b_N,n(r)w^n. Logarithmic differentiation gives fracB_r'(w)B_r(w) = r∑_k≥1κ_N,kw^k-1. Comparison of the coefficient of w^n-1 proves the recurrence.
Exact mathematical passage
\frac{B_r'(w)}{B_r(w)} = r\sum_{k\geq1}\kappa_{N,k}w^{k-1}.
W^mUse in source · 1 passage
Search aliases: W^m
Use in source · strengthening.tex, line 366
he first scalar counts partitions into pairs and is an integer. For the last term of H_0, multiply the third identity by (i E F)^m and reduce the Euler powers using the integral quadratic relations. Finally, γ_k(X+Y)=∑_r=0^kγ_r(X)γ_k-r(Y) proves γ_k(H_0)∈ A_0. Applying the functional and putting W=U^4-1=2H_0 gives L(W^m)∈2^m m!R (m≥0). These moments are rational. The identity R∩ Q_2= Z_2, obtained from the linear independence of 1,i over Q_2, descends every such divisibility statement to Z_2. No…
Exact mathematical passage
\mathcal L(W^m)\in2^m m!R\qquad(m\geq0). \label{eq:euler-W-factorial}
w^{m-1}Use in source · 2 passages
Search aliases: w^m-1
Use in source · Paper_IV_Traditional_Final.tex, line 2484
Differentiate the product identity in logarithmic form to obtain Δ_N'(w) = -Δ_N(w) ∑_m≥1σ_N,mw^m-1. Comparison of the coefficient of w^m-1 gives ma_m = -∑_k=0^m-1a_kσ_N,m-k. Solving for σ_N,m gives the recurrence.
Exact mathematical passage
\Delta_N'(w) = -\Delta_N(w) \sum_{m\geq1}\sigma_{N,m}w^{m-1}.
w^{n-1}Use in source · 1 passage
Search aliases: w^n-1
Use in source · Paper_IV_Traditional_Final.tex, line 1159
Let B_r(w)=Δ_N(w)^-r = ∑_n≥0b_N,n(r)w^n. Logarithmic differentiation gives fracB_r'(w)B_r(w) = r∑_k≥1κ_N,kw^k-1. Comparison of the coefficient of w^n-1 proves the recurrence.
Exact mathematical passage
w^{n-1}
w^pUse in source · 1 passage
Search aliases: w^p
Use in source · supplement_05.tex, line 116
Uniqueness requires a lifting argument. The source proves the actual kernel statement bar x=0 Longleftrightarrow pmid x in Z/p^ℓ Z. For x≡ ypmod p, the checked ideal-power lifting theorem gives x^p^ℓ-1-y^p^ℓ-1∈(p)^ℓ=(0). If w^p-1=1, then w^p=w and hence w^p^k=w for every k. Therefore two roots with the same residue are equal. The decomposition is unique whenever its second factor reduces to one. Multiplicativity of both factors is also proved.
Exact mathematical passage
w^p=w
w^{p-1}Use in source · 1 passage
Search aliases: w^p-1
Use in source · supplement_05.tex, line 116
Uniqueness requires a lifting argument. The source proves the actual kernel statement bar x=0 Longleftrightarrow pmid x in Z/p^ℓ Z. For x≡ ypmod p, the checked ideal-power lifting theorem gives x^p^ℓ-1-y^p^ℓ-1∈(p)^ℓ=(0). If w^p-1=1, then w^p=w and hence w^p^k=w for every k. Therefore two roots with the same residue are equal. The decomposition is unique whenever its second factor reduces to one. Multiplicativity of both factors is also proved.
Exact mathematical passage
w^{p-1}=1
w^{p^k}Use in source · 1 passage
Search aliases: w^p^k
Use in source · supplement_05.tex, line 116
Uniqueness requires a lifting argument. The source proves the actual kernel statement bar x=0 Longleftrightarrow pmid x in Z/p^ℓ Z. For x≡ ypmod p, the checked ideal-power lifting theorem gives x^p^ℓ-1-y^p^ℓ-1∈(p)^ℓ=(0). If w^p-1=1, then w^p=w and hence w^p^k=w for every k. Therefore two roots with the same residue are equal. The decomposition is unique whenever its second factor reduces to one. Multiplicativity of both factors is also proved.
Exact mathematical passage
w^{p^k}=w
w^qUse in source · 2 passages
Search aliases: w^q
Use in source · 10_foundations.tex, line 486
The recovered Annals text develops a broader polyphase vocabulary. We explain it through elementary identities with explicit algebraic derivations. These identities have not been separately formalized in Lean for this consolidation. For a sequence a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient and remainder give A(z)=∑_r=0^M-1z^rA_r(z^M), (a_r^[M])_s^[N]=a_r+Ms^[MN]. For ordinary Cauchy multiplicati…
Exact mathematical passage
A_r(w)=\sum_q a_{Mq+r}w^q
w^TUse in source · 3 passages
Search aliases: w^T
Use in source · 72_recovered.tex, line 1667
Let v_n+1=Cv_n, a_n=w^TC^nv_0, with C∈ M_d( Z_p) invertible modulo p.
Exact mathematical passage
v_{n+1}=Cv_n, \qquad a_n=w^TC^nv_0,
w_{(1)}Use in source · 1 passage
Search aliases: w_(1)
Use in source · Paper_VI_Traditional_Final.tex, line 4740
Under the stable assumptions of [source reference], define the primary block weights w_j = ∑_k=0^ν_j-1 fracnormB_j,k (-Reμ_j)^k+1. Among all sets J containing k primary blocks, the certified upper bound ∑_j∉ Jw_j is minimized by retaining the k largest weights. If w_(1)≥ w_(2)≥⋯≥ w_(r), then the least number of retained primary blocks certified to give L^1 error at most ε is the least k for which ∑_j>kw_(j)≤ε.
Exact mathematical passage
w_{(1)}\geq w_{(2)}\geq\cdots\geq w_{(r)},
w_{(2)}Use in source · 1 passage
Search aliases: w_(2)
Use in source · Paper_VI_Traditional_Final.tex, line 4740
Under the stable assumptions of [source reference], define the primary block weights w_j = ∑_k=0^ν_j-1 fracnormB_j,k (-Reμ_j)^k+1. Among all sets J containing k primary blocks, the certified upper bound ∑_j∉ Jw_j is minimized by retaining the k largest weights. If w_(1)≥ w_(2)≥⋯≥ w_(r), then the least number of retained primary blocks certified to give L^1 error at most ε is the least k for which ∑_j>kw_(j)≤ε.
Exact mathematical passage
w_{(1)}\geq w_{(2)}\geq\cdots\geq w_{(r)},
w_{(j)}Use in source · 1 passage
Search aliases: w_(j)
Use in source · Paper_VI_Traditional_Final.tex, line 4744
normB_j,k (-Reμ_j)^k+1. Among all sets J containing k primary blocks, the certified upper bound ∑_j∉ Jw_j is minimized by retaining the k largest weights. If w_(1)≥ w_(2)≥⋯≥ w_(r), then the least number of retained primary blocks certified to give L^1 error at most ε is the least k for which ∑_j>kw_(j)≤ε.
Exact mathematical passage
\sum_{j>k}w_{(j)}\leq\varepsilon.
w_{(r)}Use in source · 1 passage
Search aliases: w_(r)
Use in source · Paper_VI_Traditional_Final.tex, line 4740
Under the stable assumptions of [source reference], define the primary block weights w_j = ∑_k=0^ν_j-1 fracnormB_j,k (-Reμ_j)^k+1. Among all sets J containing k primary blocks, the certified upper bound ∑_j∉ Jw_j is minimized by retaining the k largest weights. If w_(1)≥ w_(2)≥⋯≥ w_(r), then the least number of retained primary blocks certified to give L^1 error at most ε is the least k for which ∑_j>kw_(j)≤ε.
Exact mathematical passage
w_{(1)}\geq w_{(2)}\geq\cdots\geq w_{(r)},
W_0Use in source · 6 passages
Search aliases: W_0
Use in source · 05_worked_models.tex, line 527
[Closure specification for the orbit hull] Let V have finite dimension, let (O_i)_i∈ I be a finite operator family, and let S⊆ V. Define W_0=span_𝕜S, W_n+1=W_n+∑_i∈ IO_i(W_n). Then there is an n∈ N such that W_n+1=W_n and W_n=Hull_(O_i)(S), where the right side is the least subspace containing S and invariant under every O_i. Thus the ascending subspace iteration terminates and its stable value is exactly the finite operator hull.
Exact mathematical passage
W_0=\Span_{\bbk}S, \qquad W_{n+1}=W_n+\sum_{i\in I}O_i(W_n).
W_1Use in source · 2 passages
Search aliases: W_1
Use in source · root_scope.tex, line 175
For preparation, start from the monic polynomial g=overline G_r and its zero-one lift W_1, with U_1=1. Suppose G_r-W_kU_k is coefficientwise divisible by 2^k. Its divided reduction is a finite polynomial because G_r is restricted. Divide it by g in F_2[X]: overline(G_r-W_kU_k)/2^k=gq+s, deg s<d_r. With zero-one coefficient lifts, set W_k+1=W_k+2^ks and U_k+1=U_k+2^kq. The error is now divisible by 2^k+1, the polynomial remains monic of degree d_r, and the unit reduces to one. The sequences conve…
Exact mathematical passage
W_1
w_1Use in source · 5 passages
Search aliases: w_1
Use in source · 03_weighted.tex, line 445
[Several independent weights] Suppose every edge has a weight vector w(e)=(w_1(e),…,w_r(e))∈ N^r∖{0}. Replace t^w(e) by t_1^w_1(e)⋯ t_r^w_r(e). The resulting multigraded transfer series is rational in t_1,…,t_r.
Exact mathematical passage
w(e)=(w_1(e),\ldots,w_r(e))\in\mathbb N^r\setminus\{0\}.
W_2Use in source · 1 passage
Search aliases: W_2
Use in source · 72_recovered.tex, line 1381
egative index are omitted. An empty tail is exactly zero, and the minimum over an empty set of term valuations is +∞. These conventions include M≥ n/2. For every r≥1, v_2(κ_2,r)=4r-1. The rth reconstruction term has exact valuation v_2( κ_2,rbinom n2rR_n-2r ) = 4r-1+v_2binom n2r. Equivalently, W_2(n,r) = 4r-1+s_2(r)+s_2(n-2r)-s_2(n).
Exact mathematical passage
\boxed{ W_2(n,r) = 4r-1+\sTwo(r)+\sTwo(n-2r)-\sTwo(n). }
w_2Use in source · 1 passage
Search aliases: w_2
Use in source · atlas-pages.tex, line 854
A DEFINED COSTChapter VI, Part II: Theorem E15.1 and Corollary E15.2. Stable primary blocks. Block count is the budget. Consider the scalar memory kernel K(t)=tfrac14e^-t+(3+2t)e^-2t+tfrac13e^-3t. It has three primary blocks. The middle block carries a Jordan term and must be retained or omitted as a whole for this selection rule. Their certified integrated weights are w_1=frac14, w_2=frac32+frac22^2=2, w_3=frac1/33=frac19. vfigext_selection platecaptionWith one primary block retained, choose th…
Exact mathematical passage
w_1=\frac14,\qquad w_2=\frac32+\frac{2}{2^2}=2, \qquad w_3=\frac{1/3}{3}=\frac19.
w_3Use in source · 1 passage
Search aliases: w_3
Use in source · atlas-pages.tex, line 854
A DEFINED COSTChapter VI, Part II: Theorem E15.1 and Corollary E15.2. Stable primary blocks. Block count is the budget. Consider the scalar memory kernel K(t)=tfrac14e^-t+(3+2t)e^-2t+tfrac13e^-3t. It has three primary blocks. The middle block carries a Jordan term and must be retained or omitted as a whole for this selection rule. Their certified integrated weights are w_1=frac14, w_2=frac32+frac22^2=2, w_3=frac1/33=frac19. vfigext_selection platecaptionWith one primary block retained, choose th…
Exact mathematical passage
w_1=\frac14,\qquad w_2=\frac32+\frac{2}{2^2}=2, \qquad w_3=\frac{1/3}{3}=\frac19.
W_aUse in source · 11 passages
Search aliases: W_a
Use in source · monograph_atlas.tex, line 45
vpage3One seed, eight addressesCYCLICITY DEPENDS ON THE GRAMMARChapter VII: Proposition II.2.5. Start with ψ_0=ket000 and allow the three bit flip operators X_1,X_2,X_3. For a binary address a=(a_1,a_2,a_3), put W_a=X_1^a_1X_2^a_2X_3^a_3. W_aψ_0=keta_1a_2a_3. The drawing shows the actual eight states reached by these words. Dashed edges connect the two squares by changing the first bit. The other edges change one of the remaining bits. vfigaddresses headTwo different meanings of ``reach'' The ei…
Exact mathematical passage
W_a=X_1^{a_1}X_2^{a_2}X_3^{a_3}. \qquad W_a\psi_0=\ket{a_1a_2a_3}.
w_aUse in source · 3 passages
Search aliases: w_a
Use in source · monograph_atlas.tex, line 351
II.8.4, Example II.8.8. Encode a qubit by ket0_L=ket000 and ket1_L=ket111. Let P project onto this code and take errors E_a∈{I,X_1,X_2,X_3}. Their images are four orthogonal rank-two packets Q_a=E_aPE_a^* that sum to the full identity. vfigsyndromes Use the logical state ψ=(3ket0_L+4ket1_L)/5 and channel probabilities w=(1,2,3,4)/10. The noisy state is ∑_aw_aE_aψψ^*E_a^*. A syndrome measurement returns label a with probability w_a, independent of the logical amplitudes. headThe recovery is an ac…
Exact mathematical passage
\sum_aw_aE_a\psi\psi^*E_a^*
W_BUse in source · 3 passages
Search aliases: W_B
Use in source · 02_supplement_03.tex, line 217
Let W_B(t) be the declared unit-cost count of table-cell constructions and tested min-plus candidates. The model charges the empty child-combination table as well as the output cells and merge candidates. Lean proves W_B(t)≤2(B+1)N+((B+1)+(B+1)^2)(N-1) and hence W_B(t)≤ N(3(B+1)+(B+1)^2). The root minimum scan adds a linear pass. Thus the declared tabulation model is pseudo-polynomial in N and the numeric budget B.
Exact mathematical passage
W_B(t)
w_dUse in source · 4 passages
Search aliases: w_d
Use in source · Paper_X_Block27_Amended.tex, line 503
[Weighted smooth-chart count] For weights w=(w_1,…,w_d), N_ w(m) = q^∑_i=1^dmax(m-w_i,0).
Exact mathematical passage
\bm w=(w_1,\ldots,w_d)
W_iUse in source · 5 passages
Search aliases: W_i
Use in source · Paper_X_Block27_Amended.tex, line 1101
System isomorphism and good-place overlaps. The system equivalence used here is componentwise representation isomorphism: mathscr W_i≃mathscr W_j Longleftrightarrow ∀λ, W_i,λ≃ W_j,λ. For finite-dimensional irreducible representations of any monoid over a characteristic-zero field, the new character-rigidity theorem proves that equality of characters is equivalent to an actual equivariant linear isomorphism. Neither a finite group nor an algebraically closed coefficient field is needed.
Exact mathematical passage
\mathscr W_i\simeq\mathscr W_j \quad\Longleftrightarrow\quad \forall\lambda,\ W_{i,\lambda}\simeq W_{j,\lambda}.
w_iUse in source · 22 passages
Search aliases: w_i
Use in source · 02_foundations.tex, line 842
The formula gives lceil1/drceil=1. Directly, let v≠0 and choose a coordinate v_j≠0. For any target w=(w_i), the matrix whose jth column has entries w_iv_j^-1 and whose other columns vanish sends v to w.
Exact mathematical passage
w=(w_i)
W_{i,\lambda}Use in source · 2 passages
Search aliases: W_i,λ lambda
Use in source · Paper_X_Block27_Amended.tex, line 1059
cible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficient-field embedding σ_λ:E↪ E_λ therefore sends C to an invertible matrix. Its entries agree with the ordinary representation traces Tr(W_i,λ(Frob_v_j)) when the selected places satisfy the unramified compatibility convention for that coefficient prime. In particular, places above the residue characte…
Exact mathematical passage
\operatorname{Tr}(W_{i,\lambda}(\operatorname{Frob}_{v_j}))
W_{i,\lambda}^{\oplus m_i}Use in source · 1 passage
Search aliases: W_i,λ^⊕ m_i lambda oplus
Use in source · Paper_X_Block27_Amended.tex, line 1350
arrier summand completion] Suppose a semisimple strictly compatible carrier decomposes as mathscr W ≅ ⊕_i=1^r mathscr W_i^⊕ c_i, with the mathscr W_i as above. At one coefficient prime λ_0, suppose a semisimple summand has the form V_λ_0 ≅ ⊕_i=1^r W_i,λ_0^⊕ m_i, 0≤ m_i≤ c_i. Then V_λ = ⊕_i=1^r W_i,λ^⊕ m_i defines a strictly compatible summand at every coefficient prime. The vector (m_i) is uniquely recoverable from finitely many Frobenius traces supplied by the separation theorem.
Exact mathematical passage
\boxed{ V_\lambda = \bigoplus_{i=1}^{r} W_{i,\lambda}^{\oplus m_i} }
W_{i,\lambda_0}^{\oplus m_i}Use in source · 1 passage
Search aliases: W_i,λ_0^⊕ m_i lambda oplus
Use in source · Paper_X_Block27_Amended.tex, line 1341
[Compatible-carrier summand completion] Suppose a semisimple strictly compatible carrier decomposes as mathscr W ≅ ⊕_i=1^r mathscr W_i^⊕ c_i, with the mathscr W_i as above. At one coefficient prime λ_0, suppose a semisimple summand has the form V_λ_0 ≅ ⊕_i=1^r W_i,λ_0^⊕ m_i, 0≤ m_i≤ c_i. Then V_λ = ⊕_i=1^r W_i,λ^⊕ m_i defines a strictly compatible summand at every coefficient prime. The vector (m_i) is uniquely recoverable from finitely many Frobenius traces supplied by the separation theorem.
Exact mathematical passage
V_{\lambda_0} \cong \bigoplus_{i=1}^{r} W_{i,\lambda_0}^{\oplus m_i}, \qquad 0\le m_i\le c_i.
W_i^{\oplus c_i}Use in source · 1 passage
Search aliases: W_i^⊕ c_i oplus
Use in source · Paper_X_Block27_Amended.tex, line 1334
[Compatible-carrier summand completion] Suppose a semisimple strictly compatible carrier decomposes as mathscr W ≅ ⊕_i=1^r mathscr W_i^⊕ c_i, with the mathscr W_i as above. At one coefficient prime λ_0, suppose a semisimple summand has the form V_λ_0 ≅ ⊕_i=1^r W_i,λ_0^⊕ m_i, 0≤ m_i≤ c_i. Then V_λ = ⊕_i=1^r W_i,λ^⊕ m_i defines a strictly compatible summand at every coefficient prime. The vector (m_i) is uniquely recoverable from finitely many Frobenius traces supplied by the separation theorem.
Exact mathematical passage
\mathscr W \cong \bigoplus_{i=1}^{r} \mathscr W_i^{\oplus c_i},
W_jUse in source · 1 passage
Search aliases: W_j
Use in source · Paper_X_Block27_Amended.tex, line 1101
System isomorphism and good-place overlaps. The system equivalence used here is componentwise representation isomorphism: mathscr W_i≃mathscr W_j Longleftrightarrow ∀λ, W_i,λ≃ W_j,λ. For finite-dimensional irreducible representations of any monoid over a characteristic-zero field, the new character-rigidity theorem proves that equality of characters is equivalent to an actual equivariant linear isomorphism. Neither a finite group nor an algebraically closed coefficient field is needed.
Exact mathematical passage
\mathscr W_i\simeq\mathscr W_j \quad\Longleftrightarrow\quad \forall\lambda,\ W_{i,\lambda}\simeq W_{j,\lambda}.
w_jUse in source · 2 passages
Search aliases: w_j
Use in source · Paper_VI_Traditional_Final.tex, line 4728
[Optimal selection for the certified primary block bound] Under the stable assumptions of [source reference], define the primary block weights w_j = ∑_k=0^ν_j-1 fracnormB_j,k (-Reμ_j)^k+1. Among all sets J containing k primary blocks, the certified upper bound ∑_j∉ Jw_j is minimized by retaining the k largest weights. If w_(1)≥ w_(2)≥⋯≥ w_(r), then the least number of retained primary blocks certified to give L^1 error at most ε is the least k for which ∑_j>kw_(j)≤ε.
Exact mathematical passage
w_j = \sum_{k=0}^{\nu_j-1} \frac{\norm{B_{j,k}}} {(-\operatorname{Re}\mu_j)^{k+1}}.
w_{j,k}Use in source · 2 passages
Search aliases: w_j,k
Use in source · 01_manuscript.tex, line 1438
ct discrete precision program] Let the total integer error budget be B. Define F_j(b) as the minimum cost for assigning precisions to the first j edges with total weight exactly b. Then F_0(0)=0, F_0(b)=+∞ for b≠0. Set F_j-1(u)=+∞ whenever u<0 or the state is unreachable. Then F_j(b) = min_k { F_j-1(b-w_j,k)+c_j,k }. The minimum of F_m(b) over 0≤ b≤ B is the exact minimum cost for the quantized model.
Exact mathematical passage
\boxed{ F_j(b) = \min_k \left\{ F_{j-1}(b-w_{j,k})+c_{j,k} \right\}. }
W_{j,\lambda}Use in source · 1 passage
Search aliases: W_j,λ lambda
Use in source · Paper_X_Block27_Amended.tex, line 1101
System isomorphism and good-place overlaps. The system equivalence used here is componentwise representation isomorphism: mathscr W_i≃mathscr W_j Longleftrightarrow ∀λ, W_i,λ≃ W_j,λ. For finite-dimensional irreducible representations of any monoid over a characteristic-zero field, the new character-rigidity theorem proves that equality of characters is equivalent to an actual equivariant linear isomorphism. Neither a finite group nor an algebraically closed coefficient field is needed.
Exact mathematical passage
\mathscr W_i\simeq\mathscr W_j \quad\Longleftrightarrow\quad \forall\lambda,\ W_{i,\lambda}\simeq W_{j,\lambda}.
W_KUse in source · 14 passages
Search aliases: W_K
Use in source · Paper_XI_Traditional_Final.tex, line 230
Let W_K be the Weil group of K, let I_K be inertia, let P_K be wild inertia, and choose a geometric Frobenius lift Φ. A Weil Deligne representation is a pair (r,N) with r(w)Nr(w)^-1=absw_K N.
Exact mathematical passage
W_K
W_kUse in source · 5 passages
Search aliases: W_k
Use in source · root_scope.tex, line 176
For preparation, start from the monic polynomial g=overline G_r and its zero-one lift W_1, with U_1=1. Suppose G_r-W_kU_k is coefficientwise divisible by 2^k. Its divided reduction is a finite polynomial because G_r is restricted. Divide it by g in F_2[X]: overline(G_r-W_kU_k)/2^k=gq+s, deg s<d_r. With zero-one coefficient lifts, set W_k+1=W_k+2^ks and U_k+1=U_k+2^kq. The error is now divisible by 2^k+1, the polynomial remains monic of degree d_r, and the unit reduces to one. The sequences conve…
Exact mathematical passage
G_r-W_kU_k
W_{k+1}Use in source · 2 passages
Search aliases: W_k+1
Use in source · root_scope.tex, line 183
For preparation, start from the monic polynomial g=overline G_r and its zero-one lift W_1, with U_1=1. Suppose G_r-W_kU_k is coefficientwise divisible by 2^k. Its divided reduction is a finite polynomial because G_r is restricted. Divide it by g in F_2[X]: overline(G_r-W_kU_k)/2^k=gq+s, deg s<d_r. With zero-one coefficient lifts, set W_k+1=W_k+2^ks and U_k+1=U_k+2^kq. The error is now divisible by 2^k+1, the polynomial remains monic of degree d_r, and the unit reduces to one. The sequences conve…
Exact mathematical passage
W_{k+1}=W_k+2^ks
W_{K_f}Use in source · 6 passages
Search aliases: W_K_f
Use in source · Paper_XI_Traditional_Final.tex, line 334
[Tame orbit induction] Let τ be an irreducible complex representation of W_K with trivial wild inertia. Then the nonzero tame character packets form one transitive Frobenius orbit of length f. If K_f/K is the unramified extension of degree f, there is a character ξ of W_K_f such that τ≅Ind_W_K_f^W_Kξ.
Exact mathematical passage
W_{K_f}
W_{K_L}Use in source · 3 passages
Search aliases: W_K_L
Use in source · Paper_XI_Traditional_Final.tex, line 565
[Greatest common divisor and least common multiple packets] There is a decomposition τ_ξ⊗τ_η ≅ ⊕_i=0^d-1 Ind_W_K_L^W_K ( ξ_L,nη_L,m^Φ^i ), where ξ_L,n and η_L,m are the norm lifts to K_L.
Exact mathematical passage
\boxed{ \tau_\xi\otimes\tau_\eta \cong \bigoplus_{i=0}^{d-1} \Ind_{W_{K_L}}^{W_K} \left( \xi_{L,n}\eta_{L,m}^{\Phi^i} \right), }
W_{K_m}Use in source · 2 passages
Search aliases: W_K_m
Use in source · Paper_XI_Traditional_Final.tex, line 551
Let τ_ξ=Ind_W_K_n^W_Kξ, τ_η=Ind_W_K_m^W_Kη, where K_n/K and K_m/K are unramified. Put d=gcd(n,m), L=lcm(n,m).
Exact mathematical passage
\tau_\xi=\Ind_{W_{K_n}}^{W_K}\xi, \qquad \tau_\eta=\Ind_{W_{K_m}}^{W_K}\eta,
W_{K_n}Use in source · 2 passages
Search aliases: W_K_n
Use in source · Paper_XI_Traditional_Final.tex, line 551
Let τ_ξ=Ind_W_K_n^W_Kξ, τ_η=Ind_W_K_m^W_Kη, where K_n/K and K_m/K are unramified. Put d=gcd(n,m), L=lcm(n,m).
Exact mathematical passage
\tau_\xi=\Ind_{W_{K_n}}^{W_K}\xi, \qquad \tau_\eta=\Ind_{W_{K_m}}^{W_K}\eta,
w_{m,j}Use in source · 1 passage
Search aliases: w_m,j
Use in source · Paper_VI_Traditional_Final.tex, line 2561
fine the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_j=n^∞w_m,j sup_0≤ t≤ TnormE_j(t).
Exact mathematical passage
\sup_{0\leq t\leq T} \norm{U_m(t)-U_{m,n}(t)} \leq \sum_{j=n}^{\infty}w_{m,j} \sup_{0\leq t\leq T}\norm{E_j(t)}.
w_{m,n}Use in source · 4 passages
Search aliases: w_m,n
Use in source · Paper_VI_Traditional_Final.tex, line 2548
[Weighted summable defect completion] Define the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_j=n^∞w_m,j sup_0≤ t≤ TnormE_j(t).
Exact mathematical passage
w_{m,n} = \norm{C_{m,n}}\, \norm{R_nR_{n,m}}.
W_nUse in source · 3 passages
Search aliases: W_n
Use in source · 05_worked_models.tex, line 527
[Closure specification for the orbit hull] Let V have finite dimension, let (O_i)_i∈ I be a finite operator family, and let S⊆ V. Define W_0=span_𝕜S, W_n+1=W_n+∑_i∈ IO_i(W_n). Then there is an n∈ N such that W_n+1=W_n and W_n=Hull_(O_i)(S), where the right side is the least subspace containing S and invariant under every O_i. Thus the ascending subspace iteration terminates and its stable value is exactly the finite operator hull.
Exact mathematical passage
W_0=\Span_{\bbk}S, \qquad W_{n+1}=W_n+\sum_{i\in I}O_i(W_n).
W_{n+1}Use in source · 3 passages
Search aliases: W_n+1
Use in source · 05_worked_models.tex, line 527
[Closure specification for the orbit hull] Let V have finite dimension, let (O_i)_i∈ I be a finite operator family, and let S⊆ V. Define W_0=span_𝕜S, W_n+1=W_n+∑_i∈ IO_i(W_n). Then there is an n∈ N such that W_n+1=W_n and W_n=Hull_(O_i)(S), where the right side is the least subspace containing S and invariant under every O_i. Thus the ascending subspace iteration terminates and its stable value is exactly the finite operator hull.
Exact mathematical passage
W_0=\Span_{\bbk}S, \qquad W_{n+1}=W_n+\sum_{i\in I}O_i(W_n).
W_pUse in source · 1 passage
Search aliases: W_p
Use in source · 72_recovered.tex, line 1029
[Odd row weight] For an odd prime p, the rth nonzero term in the reconstruction has valuation W_p(n,r) = 2r+v_p(Z_2r+1) +v_pbinom n2r +v_p(S_n-2r(p)). The binomial contribution is v_pbinom n2r = fracs_p(2r)+s_p(n-2r)-s_p(n)p-1, where s_p is the base p digit sum.
Exact mathematical passage
\boxed{ W_p(n,r) = 2r+\vp(Z_{2r+1}) +\vp\binom n{2r} +\vp\bigl(S_{n-2r}(p)\bigr). }
\mathcal W_pUse in source · 7 passages
Search aliases: W_p mathcal
Use in source · 20_arithmetic.tex, line 231
At the singular addresses c=ap, 1≤ a<m, use the weighted ring W_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping the falling factorial (x)_d=∏_i=0^d-1(x-i) into consecutive blocks of length p then gives an integral residue-coordinate factor p^v_p(lfloor d/prfloor!)…
Exact mathematical passage
\mathcal W_p=\left\{\sum_{d\ge0}a_dT^d: p^{\lfloor d/p\rfloor}a_d\in\Z_p\right\}.
W_rUse in source · 9 passages
Search aliases: W_r
Use in source · root_scope.tex, line 156
[Exact classical tail valuation] For every r≥1 and N≥ r, v_2( T_r-1(N)) =4r-1+v_2binom Nr+μ_r+v_2G_r(N-r). There are unique factors G_r(X)=W_r(X)U_r(X), where W_r∈ Z_2[X] is monic of degree d_r=max J_r and U_r∈ Z_2⟨ X⟩^× satisfies U_r≡1pmod2. Consequently v_2G_r(x)=v_2W_r(x) for every x∈ Z_2, and deg W_r≤lfloorfracs_2(r)-13rfloor. Equation~[source reference], the exact weight in [source reference], and the digit-sum identity v_2binom2N2r=v_2binom Nr prove [source reference]. The binomial coeffic…
Exact mathematical passage
G_r(X)=W_r(X)U_r(X),
\mathcal W_rUse in source · 2 passages
Search aliases: W_r mathcal
Use in source · root_scope.tex, line 382
[Exact quadratic compression] There is a unique preparation K_r(Z)= U_r(Z) W_r(Z), W_r(Z)=Z^2+2β_rZ+α_r, U_r≡ u_rpmod256. With δ_r=β_r^2-α_r and τ_r= U_r'(c_r)/ U_r(c_r), δ_r=-fracΦ_r^act U_r(c_r) (1-fracτ_r^2Φ_r^act4 U_r(c_r)), v_2(δ_r)=v_2(Φ_r^act). The equality includes the zero case. The same successive monic polynomial division used in Theorem~[source reference], started at precision eight, preserves the quadratic and unit reductions of [source reference]. Write W=Z^2+2β_rZ+α_r and w=W(c_r)…
Exact mathematical passage
K_r(Z)=\mathcal U_r(Z)\mathcal W_r(Z),\qquad \mathcal W_r(Z)=Z^2+2\beta_rZ+\alpha_r,\qquad \mathcal U_r\equiv u_r\pmod{256}.
w_rUse in source · 2 passages
Search aliases: w_r
Use in source · 03_weighted.tex, line 445
[Several independent weights] Suppose every edge has a weight vector w(e)=(w_1(e),…,w_r(e))∈ N^r∖{0}. Replace t^w(e) by t_1^w_1(e)⋯ t_r^w_r(e). The resulting multigraded transfer series is rational in t_1,…,t_r.
Exact mathematical passage
w(e)=(w_1(e),\ldots,w_r(e))\in\mathbb N^r\setminus\{0\}.
w_{\rm Smith}Use in source · 1 passage
Search aliases: w_rm Smith rm
Use in source · Paper_V_Traditional_Final.tex, line 1254
Away from its pole Γ=1, the inverse Cayley map has derivative fracdzdΓ=frac2(1-Γ)^2, while normalized admittance has derivative proportional to (1+Γ)^-2. Away from Γ=±1, this motivates the symmetric sensitivity w_rm Smith(Γ) = frac1|1-Γ|^2 + frac1|1+Γ|^2.
Exact mathematical passage
\boxed{ w_{\rm Smith}(\Gam) = \frac1{|1-\Gam|^2} + \frac1{|1+\Gam|^2}. }
W_{\mathrm{sgn}}Use in source · 2 passages
Search aliases: W_sgn mathrm
Use in source · Paper_XII_Traditional_Final.tex, line 423
The sign witness is W_sgn=Z, while W_std=W(6,4,3)=3A+3B belongs to the standard summand. The corresponding projectors and trace pairings are P_t,2=2 · 4
Exact mathematical passage
W_{\mathrm{sgn}}=Z
w_{\mathrm{Smith}}Use in source · 1 passage
Search aliases: w_Smith mathrm
Use in source · 06_models.tex, line 79
The symmetric weight w_Smith(Γ) = frac1|1-Γ|^2 + frac1|1+Γ|^2 indicates where small errors in reflection coordinates produce large impedance or admittance errors.
Exact mathematical passage
w_{\mathrm{Smith}}(\Gamma) = \frac1{|1-\Gamma|^2} + \frac1{|1+\Gamma|^2}
W_{\mathrm{std}}Use in source · 2 passages
Search aliases: W_std mathrm
Use in source · Paper_XII_Traditional_Final.tex, line 424
The sign witness is W_sgn=Z, while W_std=W(6,4,3)=3A+3B belongs to the standard summand. The corresponding projectors and trace pairings are P_t,2=2 · 4
Exact mathematical passage
W_{\mathrm{std}}=W(6,4,3)=3A+3B
W_uUse in source · 1 passage
Search aliases: W_u
Use in source · Paper_XI_Traditional_Final.tex, line 1608
Let Γ be a connected finite graph. At every vertex v, place finite dimensional complex vector spaces V_v and W_v of the same dimension. For every oriented edge e:u→ v, let A_e:V_u⟶ V_v, B_e:W_u⟶ W_v be invertible. The reversed edge carries the inverse map.
Exact mathematical passage
A_e:V_u\longrightarrow V_v, \qquad B_e:W_u\longrightarrow W_v
W_vUse in source · 4 passages
Search aliases: W_v
Use in source · Paper_XI_Traditional_Final.tex, line 1607
Let Γ be a connected finite graph. At every vertex v, place finite dimensional complex vector spaces V_v and W_v of the same dimension. For every oriented edge e:u→ v, let A_e:V_u⟶ V_v, B_e:W_u⟶ W_v be invertible. The reversed edge carries the inverse map.
Exact mathematical passage
W_v
w_vUse in source · 4 passages
Search aliases: w_v
Use in source · 01_manuscript.tex, line 1658
[Quantized budget dynamic program] Assume every keep error is a nonnegative integer w_v. Let B∈ N. For each node v, define F_v(b) as the minimum cost of a pruning of the subtree at v with total error exactly b. The keep option contributes F_v(w_v)≤κ_v. The split option is obtained by min plus convolution of the child tables, followed by addition of σ_v. The smallest value among F_rm root(b), 0≤ b≤ B, is the exact minimum cost among prunings whose quantized error is at most B.
Exact mathematical passage
w_v
W_v^*Use in source · 1 passage
Search aliases: W_v^*
Use in source · atlas-pages.tex, line 199
al positions below are terminal packet addresses, not time intervals in the original signal. vfigcomplete_cuts headThree candidate selections The complete selection {0,10,11} covers every terminal branch once. The selection {0,00,1} covers the 00 branch twice: it is not an antichain. The selection {00,01,10} misses the 11 branch entirely. For actual Haar packets, set P_v=W_v^*W_v. Parent-to-child conservation gives P_v=P_v0+P_v1. Replacing unselected parents by their children until the complete …
Exact mathematical passage
P_v=W_v^*W_v
W_\lambdaUse in source · 1 passage
Search aliases: W_λ lambda
Use in source · Paper_X_Block27_Amended.tex, line 1362
Use the same multiplicities in every member of the compatible carrier. Direct sums preserve strict compatibility, and m_i≤ c_i makes V_λ a summand of W_λ. At the separating places, the target trace vector is C m. Full column rank makes m the unique solution.
Exact mathematical passage
W_\lambda
W_\piUse in source · 2 passages
Search aliases: W_π pi
Use in source · 02_strengthened.tex, line 804
[Convergence of short packet rotations] Let (P_a(t)) be twice continuously differentiable on [0,T]. For a partition 0=t_0<t_1<⋯<t_n=T, let W_π = U( P(t_n), P(t_n-1)) ⋯ U( P(t_1), P(t_0)). For sufficiently fine partitions the factors are defined, and normW_π-W(T) ≤ C_Tabsπ for a constant depending on the packet path but not on the partition. Here the size of the partition is measured by its mesh, which is the length of its largest subinterval.
Exact mathematical passage
W_\pi = \Ucal\bigl(\Pcal(t_n),\Pcal(t_{n-1})\bigr) \cdots \Ucal\bigl(\Pcal(t_1),\Pcal(t_0)\bigr).
W_{\ell,\alpha}^*Use in source · 1 passage
Search aliases: W_ℓ,α^* ell alpha
Use in source · Paper_V_Traditional_Final.tex, line 624
Set A=∑_α W_ℓ,α^*W_ℓ,α. This is a finite sum of bounded operators. The energy result says ⟨ x,Ax⟩=⟨ x,x⟩ for every x. Complex polarization, applied also to x+y and x+ i y, gives ⟨ y,(A-I)x⟩=0 for all x,y, hence A=I. Applying the operator identity to x proves synthesis. This argument uses the actual Hilbert adjoints of the decimation operators, not a separately postulated inverse symbol.
Exact mathematical passage
A=\sum_\alpha W_{\ell,\alpha}^*W_{\ell,\alpha}
W_{\ell,\alpha}^{*}Use in source · 1 passage
Search aliases: W_ℓ,α^* ell alpha
Use in source · Paper_V_Traditional_Final.tex, line 605
[Wavelet-packet reconstruction and energy] At every finite depth ℓ, x=∑_α∈{0,1}^ℓW_ℓ,α^*W_ℓ,αx, and x _2^2 = ∑_α∈{0,1}^ℓ W_ℓ,αx _2^2.
Exact mathematical passage
\boxed{ x=\sum_{\alpha\in\{0,1\}^{\ell}}W_{\ell,\alpha}^{*}W_{\ell,\alpha}x, }
\operatorname{Walsh}Use in source · 1 passage
Search aliases: Walsh operatorname
Use in source · 02_supplement_04.tex, line 140
Actual butterfly. For the two halves x_0,x_1 of the input, the independently defined procedure returns Walsh_2N(x_0,x_1) =(Walsh_N(x_0+x_1), Walsh_N(x_0-x_1)). Lean proves that this vector equals H_2Nx. The argument expands the two first-coordinate cases and applies finite-sum distributivity. It does not assume that the recursive algorithm already represents the matrix.
Exact mathematical passage
\operatorname{Walsh}_{2N}(x_0,x_1) =\bigl(\operatorname{Walsh}_N(x_0+x_1), \operatorname{Walsh}_N(x_0-x_1)\bigr).
\operatorname{Walsh}_{2N}Use in source · 1 passage
Search aliases: Walsh_2N operatorname
Use in source · 02_supplement_04.tex, line 140
Actual butterfly. For the two halves x_0,x_1 of the input, the independently defined procedure returns Walsh_2N(x_0,x_1) =(Walsh_N(x_0+x_1), Walsh_N(x_0-x_1)). Lean proves that this vector equals H_2Nx. The argument expands the two first-coordinate cases and applies finite-sum distributivity. It does not assume that the recursive algorithm already represents the matrix.
Exact mathematical passage
\operatorname{Walsh}_{2N}(x_0,x_1) =\bigl(\operatorname{Walsh}_N(x_0+x_1), \operatorname{Walsh}_N(x_0-x_1)\bigr).
\operatorname{Walsh}_NUse in source · 1 passage
Search aliases: Walsh_N operatorname
Use in source · 02_supplement_04.tex, line 140
Actual butterfly. For the two halves x_0,x_1 of the input, the independently defined procedure returns Walsh_2N(x_0,x_1) =(Walsh_N(x_0+x_1), Walsh_N(x_0-x_1)). Lean proves that this vector equals H_2Nx. The argument expands the two first-coordinate cases and applies finite-sum distributivity. It does not assume that the recursive algorithm already represents the matrix.
Exact mathematical passage
\operatorname{Walsh}_{2N}(x_0,x_1) =\bigl(\operatorname{Walsh}_N(x_0+x_1), \operatorname{Walsh}_N(x_0-x_1)\bigr).
\operatorname{WD}Use in source · 1 passage
Search aliases: WD operatorname
Use in source · Paper_X_Block27_Amended.tex, line 1530
The remaining arithmetic comparison is WD(⊕_i V_i^⊕ m_i)≅ ⊕_iWD(V_i)^⊕ m_i for the actual attached functor, intertwining the Weil action and monodromy, together with the intended coefficient and Hodge--Tate comparisons. These are not supplied by the abstract datum. Claim 9.2 remains partial at its full arithmetic scope.
Exact mathematical passage
\operatorname{WD}(\bigoplus_i V_i^{\oplus m_i})\cong \bigoplus_i\operatorname{WD}(V_i)^{\oplus m_i}
\widehat AUse in source · 22 passages
Search aliases: widehat A widehat
Use in source · pages.tex, line 333
For orthogonal coordinate blocks, the squared Frobenius norms add exactly. Operator norms instead require a suitable bound or further spectral information. The maximum omitted block norm is generally insufficient. headInput sensitivity is additional information The packet perturbation estimate is (widehat A-A)x ≤∑_e E_e P_α(e)x . It can be much tighter for an input that scarcely occupies an affected source packet. To obtain a uniform operator bound, the input dependence must be bounded uniformly…
Exact mathematical passage
\|(\widehat A-A)x\|\le\sum_e\|E_e\|\,\|P_{\alpha(e)}x\|.
\widehat aUse in source · 15 passages
Search aliases: widehat a widehat
Use in source · Paper_V_Traditional_Final.tex, line 702
Let (X,Σ,μ) be a measure space, with measurable masks Ω,widehatΩ⊆ X and scalar amplitudes a,widehat a∈ C (real amplitudes are a special case). Suppose a scalar jump packet is j=a 1_Ω, and an estimated packet is widehat j=widehat a 1_widehatΩ.
Exact mathematical passage
a,\widehat a\in\bbC
\widehat A_1Use in source · 3 passages
Search aliases: widehat A_1 widehat
Use in source · 01_manuscript.tex, line 864
[Product chain perturbation] Let A_1,…,A_m be exact factors and let widehat A_j=A_j+E_j. Then widehat A_m⋯widehat A_1 - A_m⋯ A_1 ≤ ∑_j=1^m ( ∏_i=j+1^mnormwidehat A_i ) normE_j ( ∏_i=1^j-1normA_i ). Empty products are one.
Exact mathematical passage
\boxed{ \left\| \widehat A_m\cdots\widehat A_1 - A_m\cdots A_1 \right\| \leq \sum_{j=1}^m \left( \prod_{i=j+1}^m\norm{\widehat A_i} \right) \norm{E_j} \left( \prod_{i=1}^{j-1}\norm{A_i} \right). }
\widehat A_iUse in source · 2 passages
Search aliases: widehat A_i widehat
Use in source · 01_manuscript.tex, line 864
[Product chain perturbation] Let A_1,…,A_m be exact factors and let widehat A_j=A_j+E_j. Then widehat A_m⋯widehat A_1 - A_m⋯ A_1 ≤ ∑_j=1^m ( ∏_i=j+1^mnormwidehat A_i ) normE_j ( ∏_i=1^j-1normA_i ). Empty products are one.
Exact mathematical passage
\boxed{ \left\| \widehat A_m\cdots\widehat A_1 - A_m\cdots A_1 \right\| \leq \sum_{j=1}^m \left( \prod_{i=j+1}^m\norm{\widehat A_i} \right) \norm{E_j} \left( \prod_{i=1}^{j-1}\norm{A_i} \right). }
\widehat A_jUse in source · 7 passages
Search aliases: widehat A_j widehat
Use in source · 01_manuscript.tex, line 860
[Product chain perturbation] Let A_1,…,A_m be exact factors and let widehat A_j=A_j+E_j. Then widehat A_m⋯widehat A_1 - A_m⋯ A_1 ≤ ∑_j=1^m ( ∏_i=j+1^mnormwidehat A_i ) normE_j ( ∏_i=1^j-1normA_i ). Empty products are one.
Exact mathematical passage
\widehat A_j=A_j+E_j.
\widehat A_{j+1}Use in source · 2 passages
Search aliases: widehat A_j+1 widehat
Use in source · 01_manuscript.tex, line 887
Use the telescoping identity widehat A_m⋯widehat A_1 -A_m⋯ A_1 = ∑_j=1^m widehat A_m⋯widehat A_j+1 E_j A_j-1⋯ A_1. Take norms term by term.
Exact mathematical passage
\widehat A_m\cdots\widehat A_1 -A_m\cdots A_1 = \sum_{j=1}^m \widehat A_m\cdots\widehat A_{j+1} E_j A_{j-1}\cdots A_1.
\widehat A_mUse in source · 3 passages
Search aliases: widehat A_m widehat
Use in source · 01_manuscript.tex, line 864
[Product chain perturbation] Let A_1,…,A_m be exact factors and let widehat A_j=A_j+E_j. Then widehat A_m⋯widehat A_1 - A_m⋯ A_1 ≤ ∑_j=1^m ( ∏_i=j+1^mnormwidehat A_i ) normE_j ( ∏_i=1^j-1normA_i ). Empty products are one.
Exact mathematical passage
\boxed{ \left\| \widehat A_m\cdots\widehat A_1 - A_m\cdots A_1 \right\| \leq \sum_{j=1}^m \left( \prod_{i=j+1}^m\norm{\widehat A_i} \right) \norm{E_j} \left( \prod_{i=1}^{j-1}\norm{A_i} \right). }
\widehat a_qUse in source · 7 passages
Search aliases: widehat a_q widehat
Use in source · Paper_V_Traditional_Final.tex, line 2971
[Deterministic jump cell recovery] Let a_min=min_q|a_q| and assume a_min>2b_h. Choose a threshold θ satisfying b_h<θ<a_min-b_h. Then widehat K = {k:|D_k|>θ} is exactly the set of grid cells containing jumps. If widehatτ_q is the midpoint of the detected cell and widehat a_q=D_k, then after matching by order, |widehatτ_q-τ_q| ≤ frac h2 and |widehat a_q-a_q| ≤ b_h.
Exact mathematical passage
\widehat a_q=D_k
\widehat CUse in source · 8 passages
Search aliases: widehat C widehat
Use in source · supplement_07.tex, line 119
pageheading3. Actual packet congruences and sharp boundariesConsequences for the standard window; not the full Theorem 7.5 The fifth block constructed integral lifts widehat C_mp,j(n)∈ Z_p of the original rational coefficients when 0≤ j<p, and proved that their residues equal e_n^(m,p).
Exact mathematical passage
\widehat C_{mp,j}(n)\in\Z_p
\widehat cUse in source · 2 passages
Search aliases: widehat c widehat
Use in source · Paper_V_Traditional_Final.tex, line 2361
[Coefficient perturbations in a unitary tree] Let c= A_ Lx be the leaf coefficients of an admissible unitary pruning, and let widehat c=c+e. If widehat x is synthesized from widehat c, then widehat x-x = e . Thus coefficient quantization and coefficient noise are neither amplified nor attenuated by exact unitary synthesis.
Exact mathematical passage
\widehat c=c+e
\widehat C_AUse in source · 2 passages
Search aliases: widehat C_A widehat
Use in source · 01_finite_core.tex, line 1198
At one finite level, let widehat C_A be a finite family of matrix constraints, with fixed scalar structure coefficients, and let Π^phys be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients f_AB^ C, define the anomaly operator A_AB = [widehat C_A,widehat C_B] -i∑_C f_AB^ Cwidehat C_C. Its physical packet is Π^phys A_ABΠ^phys. In this exact finite setting, widehat C_CΠ^phys=0 for every C. Associativity gives A_ABΠ^phys=0, Π^phys…
Exact mathematical passage
\widehat C_A
\widehat C_BUse in source · 1 passage
Search aliases: widehat C_B widehat
Use in source · 01_finite_core.tex, line 1199
At one finite level, let widehat C_A be a finite family of matrix constraints, with fixed scalar structure coefficients, and let Π^phys be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients f_AB^ C, define the anomaly operator A_AB = [widehat C_A,widehat C_B] -i∑_C f_AB^ Cwidehat C_C. Its physical packet is Π^phys A_ABΠ^phys. In this exact finite setting, widehat C_CΠ^phys=0 for every C. Associativity gives A_ABΠ^phys=0, Π^phys…
Exact mathematical passage
\boxed{ \mathfrak A_{AB} = [\widehat C_A,\widehat C_B] -i\sum_C f_{AB}^{\ \ C}\widehat C_C. }
\widehat C_CUse in source · 2 passages
Search aliases: widehat C_C widehat
Use in source · 01_finite_core.tex, line 1199
At one finite level, let widehat C_A be a finite family of matrix constraints, with fixed scalar structure coefficients, and let Π^phys be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients f_AB^ C, define the anomaly operator A_AB = [widehat C_A,widehat C_B] -i∑_C f_AB^ Cwidehat C_C. Its physical packet is Π^phys A_ABΠ^phys. In this exact finite setting, widehat C_CΠ^phys=0 for every C. Associativity gives A_ABΠ^phys=0, Π^phys…
Exact mathematical passage
\boxed{ \mathfrak A_{AB} = [\widehat C_A,\widehat C_B] -i\sum_C f_{AB}^{\ \ C}\widehat C_C. }
\widehat C_{mp,j}Use in source · 3 passages
Search aliases: widehat C_mp,j widehat
Use in source · supplement_07.tex, line 119
pageheading3. Actual packet congruences and sharp boundariesConsequences for the standard window; not the full Theorem 7.5 The fifth block constructed integral lifts widehat C_mp,j(n)∈ Z_p of the original rational coefficients when 0≤ j<p, and proved that their residues equal e_n^(m,p).
Exact mathematical passage
\widehat C_{mp,j}(n)\in\Z_p
\widehat EUse in source · 5 passages
Search aliases: widehat E widehat
Use in source · 02_foundations.tex, line 343
_1,…,s_r). Put E:= A^⊕ r/N_S algebraically and let j:E→ X be the injective map induced by ε_S. Give E the locally convex topology transported from j(E), equivalently the topology generated by widetilde p(e):=p(j(e)) for the continuous seminorms p on X. Every a∈ A acts continuously on E in the transported topology and therefore extends to widehat E. If Hull_ A(S) is dense in X, then j extends uniquely to a topological isomorphism of modules over A widehat E≅ X. [source reference]
Exact mathematical passage
\widehat E
\widehat FUse in source · 14 passages
Search aliases: widehat F widehat
Use in source · 01_finite_coordinates.tex, line 511
If F:X→ Y is linear and P^a, P^b are source packet systems while Q^r, Q^s are target packet systems, put widehat F_r← a = A^Y_rF S^X_a.
Exact mathematical passage
\widehat F_{r\leftarrow a} = \cA^Y_rF\cS^X_a.
\widehat fUse in source · 16 passages
Search aliases: widehat f widehat
Use in source · visual_atlas.tex, line 477
For a canonical coordinate change with input and output chart v=2x, the target formula is not still v^2. It is widehat f_b(v)=2f(v/2)=fracv^22. Consequently widehat f_b(2x)=2x^2, and the exact coordinate defect vanishes. This is the same distinction illustrated in the example in two dimensions on Plate III.10.
Exact mathematical passage
\widehat f_b(v)=2f(v/2)=\frac{v^2}{2}.
\widehat F_bUse in source · 2 passages
Search aliases: widehat F_b widehat
Use in source · visual_atlas.tex, line 217
Now represent the nonlinear operation F(x,y)=(x+y,xy) in the new chart. Set w=u+v-u^2. Its correct coordinate formula is widehat F_b(u,v)= A F S(u,v) =(w, u(v-u^2)+w^2). For (x,y)=(1,2), applying F first gives (3,2) and then (3,11) in the new chart. Changing the input first gives (u,v)=(1,3), and widehat F_b(1,3)=(3,11) as well.
Exact mathematical passage
\widehat F_b(u,v)=\mathcal A F\mathcal S(u,v) =\bigl(w,\ u(v-u^2)+w^2\bigr).
\widehat f_bUse in source · 2 passages
Search aliases: widehat f_b widehat
Use in source · visual_atlas.tex, line 477
For a canonical coordinate change with input and output chart v=2x, the target formula is not still v^2. It is widehat f_b(v)=2f(v/2)=fracv^22. Consequently widehat f_b(2x)=2x^2, and the exact coordinate defect vanishes. This is the same distinction illustrated in the example in two dimensions on Plate III.10.
Exact mathematical passage
\widehat f_b(v)=2f(v/2)=\frac{v^2}{2}.
\widehat F_iUse in source · 2 passages
Search aliases: widehat F_i widehat
Use in source · 02_supplement_02.tex, line 86
Let v be the new optimum of the edited subtree, and let j be the edited child slot at one frame. The update performed at that frame is step(v)= min{K, σ+ ∑_i v, · i=j, widehat F_i, · i≠ j. }. This result is passed to the next ancestor. The update function does not recursively evaluate sibling subtrees. It reads their stored values widehat F_i.
Exact mathematical passage
\boxed{ \operatorname{step}(v)= \min\left\{K,\;\sigma+ \sum_i\begin{cases} v,&i=j,\\ \widehat F_i,&i\ne j. \end{cases}\right\}. }
\widehat f_mUse in source · 10 passages
Search aliases: widehat f_m widehat
Use in source · Paper_V_Traditional_Final.tex, line 1680
The reconstruction formula with an estimated packet is f-widehat f_m = (I-Q_m)s+(I-Q_m)(j-widehat j). This identity identifies the two terms that must be controlled. The exact packet formulas in [source reference] then evaluate or bound j-widehat j from amplitude error, region overlap, boundary displacement, and the order of a derivative jump. A value jump has location modulus |τ-widehatτ|^1/p in finite L^p, while every positive order truncated power packet is Lipschitz in its location.
Exact mathematical passage
f-\widehat f_m = (I-Q_m)s+(I-Q_m)(j-\widehat j).
\widehat F_{r\leftarrow a}Use in source · 5 passages
Search aliases: widehat F_r← a widehat leftarrow
Use in source · 01_finite_coordinates.tex, line 511
If F:X→ Y is linear and P^a, P^b are source packet systems while Q^r, Q^s are target packet systems, put widehat F_r← a = A^Y_rF S^X_a.
Exact mathematical passage
\widehat F_{r\leftarrow a} = \cA^Y_rF\cS^X_a.
\widehat F_{s\leftarrow b}Use in source · 2 passages
Search aliases: widehat F_s← b widehat leftarrow
Use in source · 01_finite_coordinates.tex, line 520
[Canonical coordinate naturality] The exact coordinate representation preserves typed composition and satisfies J^Y_s← rwidehat F_r← a = widehat F_s← bJ^X_b← a. statusleanpriorKomorebi.PaperII.NestedRefinement.coordinateOperator_comp; Komorebi.PaperII.NestedRefinement.coordinateOperator_naturality
Exact mathematical passage
J^Y_{s\leftarrow r}\widehat F_{r\leftarrow a} = \widehat F_{s\leftarrow b}J^X_{b\leftarrow a}.
\widehat G_{t\leftarrow r}Use in source · 1 passage
Search aliases: widehat G_t← r widehat leftarrow
Use in source · 02_connections_obstructions.tex, line 740
tarrow r= A^Y_s∘ S^Y_r, and define widehat F_r← a = A^Y_r∘ F∘ S^X_a. Then J^Y_s← r∘widehat F_r← a = widehat F_s← b∘ J^X_b← a. If A^Z_t:Zxrightarrow∼Z_t is another exact coordinate equivalence and G:Y→ Z is any function, then exact coordinate representation also preserves composition: widehatG∘ F_t← a = widehat G_t← r∘widehat F_r← a. statusleanthreeKomorebi.PaperIII.Transport.exactCoordinateMap; Komorebi.PaperIII.Transport.exactCoordinateMap_comp; Komorebi.PaperIII.Transport.exactCoordinateMap_na…
Exact mathematical passage
\widehat{G\circ F}_{t\leftarrow a} = \widehat G_{t\leftarrow r}\circ\widehat F_{r\leftarrow a}.
\widehat hUse in source · 4 passages
Search aliases: widehat h widehat
Use in source · 04_correction_transport.tex, line 572
[Second Schur derivative] For retained directions h,k, set widehat h=(h,Dφ h), widehat k=(k,Dφ k). Then D^2φ[h,k] = -A^-1D^2F_Q[widehat h,widehat k] and D^2R[h,k] = D^2F_P[widehat h,widehat k] - D_vF_P A^-1D^2F_Q[widehat h,widehat k]. All derivatives on the right are evaluated at (u_0,v_0).
Exact mathematical passage
\widehat h=(h,D\varphi h), \qquad \widehat k=(k,D\varphi k).
\widehat j_mUse in source · 4 passages
Search aliases: widehat j_m widehat
Use in source · Paper_V_Traditional_Final.tex, line 3430
[Estimated packet Gibbs bound] Let Q_m be linear and uniformly bounded on L^p, with sup_m Q_m ≤ C_Q. For f=s+j and an estimated packet widehat j_m, define widehat f_m =Q_m(f-widehat j_m)+widehat j_m. Then f-widehat f_m _p ≤ s-Q_ms _p +(1+C_Q) j-widehat j_m _p. Consequently, smooth approximation and packet recovery may be improved independently. No claim of Gibbs removal is justified unless both terms are controlled in the norm under consideration.
Exact mathematical passage
\widehat j_m
\widehat kUse in source · 4 passages
Search aliases: widehat k widehat
Use in source · 04_correction_transport.tex, line 572
[Second Schur derivative] For retained directions h,k, set widehat h=(h,Dφ h), widehat k=(k,Dφ k). Then D^2φ[h,k] = -A^-1D^2F_Q[widehat h,widehat k] and D^2R[h,k] = D^2F_P[widehat h,widehat k] - D_vF_P A^-1D^2F_Q[widehat h,widehat k]. All derivatives on the right are evaluated at (u_0,v_0).
Exact mathematical passage
\widehat h=(h,D\varphi h), \qquad \widehat k=(k,D\varphi k).
\widehat PUse in source · 6 passages
Search aliases: widehat P widehat
Use in source · 10_foundations.tex, line 101
cal E(h) and H^-1= E(k), and define U_Ha= D(H E(a)). Then U_H is an equivalence, with inverse U_H^-1a= D(H^-1 E(a)). Its coefficient formulas are (U_Ha)_n=∑_r=0^nbinom nr h_r a_n-r, (U_H^-1a)_n=∑_r=0^nbinom nr k_r a_n-r. The resulting triangular matrix has diagonal entries 1. The transported selectors widehat P_N,j=U_HP_N,jU_H^-1 are idempotent and pairwise orthogonal, and their sum is the identity.
Exact mathematical passage
\widehat P_{N,j}=U_HP_{N,j}U_H^{-1}
\widehat pUse in source · 7 passages
Search aliases: widehat p widehat
Use in source · Paper_VI_Traditional_Final.tex, line 1093
Let K be column stochastic and let λ^(r) be the likelihood associated with the rth observation. The filter recursion is widehat p_r=Kp_r-1, p_r= B_λ^(r)(widehat p_r).
Exact mathematical passage
\widehat p_r=Kp_{r-1}, \qquad p_r=\cB_{\lambda^{(r)}}(\widehat p_r).
\widehat P_0Use in source · 1 passage
Search aliases: widehat P_0 widehat
Use in source · 01_manuscript.tex, line 907
[Certified repeated recompression] Let P_0=I, P_j=A_jP_j-1, be exact partial products. Suppose an implemented chain satisfies widehat P_0=I, widehat P_j = A_jwidehat P_j-1+E_j, where E_j is the additive error introduced after the jth multiplication, for example by factor recompression. Then normwidehat P_m-P_m ≤ ∑_j=1^m ( ∏_i=j+1^mnormA_i ) normE_j. If normE_j≤δ_j, the same formula with δ_j gives a certified final error.
Exact mathematical passage
\widehat P_0=I, \qquad \widehat P_j = A_j\widehat P_{j-1}+E_j,
\widehat p_3Use in source · 1 passage
Search aliases: widehat p_3 widehat
Use in source · pages.tex, line 217
vpage13Errors enter at different placesFACTOR PERTURBATION VERSUS RECOMPRESSIONChapter VIII: Theorem 5.8, Theorem 5.9. Let the exact scalar factors be (2,3,4) and start from p_0=1. If additive errors (1/10,1/5,1/20) enter after each partial multiplication, then widehat p_j=a_jwidehat p_j-1+e_j, widehat p_3-p_3=12e_1+4e_2+e_3=frac4120. vfigerror_chain headThe same local numbers, a different model If instead the factors themselves are perturbed, the computed product is (4+e_3)(3+e_2)(2+e_1). Its e…
Exact mathematical passage
\widehat p_j=a_j\widehat p_{j-1}+e_j,\qquad \widehat p_3-p_3=12e_1+4e_2+e_3=\frac{41}{20}.
\widehat P_jUse in source · 3 passages
Search aliases: widehat P_j widehat
Use in source · 01_manuscript.tex, line 907
[Certified repeated recompression] Let P_0=I, P_j=A_jP_j-1, be exact partial products. Suppose an implemented chain satisfies widehat P_0=I, widehat P_j = A_jwidehat P_j-1+E_j, where E_j is the additive error introduced after the jth multiplication, for example by factor recompression. Then normwidehat P_m-P_m ≤ ∑_j=1^m ( ∏_i=j+1^mnormA_i ) normE_j. If normE_j≤δ_j, the same formula with δ_j gives a certified final error.
Exact mathematical passage
\widehat P_0=I, \qquad \widehat P_j = A_j\widehat P_{j-1}+E_j,
\widehat p_jUse in source · 1 passage
Search aliases: widehat p_j widehat
Use in source · pages.tex, line 217
vpage13Errors enter at different placesFACTOR PERTURBATION VERSUS RECOMPRESSIONChapter VIII: Theorem 5.8, Theorem 5.9. Let the exact scalar factors be (2,3,4) and start from p_0=1. If additive errors (1/10,1/5,1/20) enter after each partial multiplication, then widehat p_j=a_jwidehat p_j-1+e_j, widehat p_3-p_3=12e_1+4e_2+e_3=frac4120. vfigerror_chain headThe same local numbers, a different model If instead the factors themselves are perturbed, the computed product is (4+e_3)(3+e_2)(2+e_1). Its e…
Exact mathematical passage
\widehat p_j=a_j\widehat p_{j-1}+e_j,\qquad \widehat p_3-p_3=12e_1+4e_2+e_3=\frac{41}{20}.
\widehat P_{j+1}Use in source · 1 passage
Search aliases: widehat P_j+1 widehat
Use in source · 02_supplement_03.tex, line 36
Actual noncommutative products. The error module defines the left-accumulated product recursively, with P_0=I and P_j+1=A_j+1P_j. For a perturbed sequence widehat A_j, it derives the discrepancy recursion widehat P_j+1-P_j+1 =widehat A_j+1(widehat P_j-P_j) +(widehat A_j+1-A_j+1)P_j. The identity follows from the actual products. It is not supplied as an assumption to a norm estimate, and it does not commute the operators. Lean uses zero-based indices internally, so index j in source code corresp…
Exact mathematical passage
\widehat P_{j+1}-P_{j+1} =\widehat A_{j+1}(\widehat P_j-P_j) +(\widehat A_{j+1}-A_{j+1})P_j.
\widehat P_{j-1}Use in source · 1 passage
Search aliases: widehat P_j-1 widehat
Use in source · 01_manuscript.tex, line 907
[Certified repeated recompression] Let P_0=I, P_j=A_jP_j-1, be exact partial products. Suppose an implemented chain satisfies widehat P_0=I, widehat P_j = A_jwidehat P_j-1+E_j, where E_j is the additive error introduced after the jth multiplication, for example by factor recompression. Then normwidehat P_m-P_m ≤ ∑_j=1^m ( ∏_i=j+1^mnormA_i ) normE_j. If normE_j≤δ_j, the same formula with δ_j gives a certified final error.
Exact mathematical passage
\widehat P_0=I, \qquad \widehat P_j = A_j\widehat P_{j-1}+E_j,
\widehat p_{j-1}Use in source · 1 passage
Search aliases: widehat p_j-1 widehat
Use in source · pages.tex, line 217
vpage13Errors enter at different placesFACTOR PERTURBATION VERSUS RECOMPRESSIONChapter VIII: Theorem 5.8, Theorem 5.9. Let the exact scalar factors be (2,3,4) and start from p_0=1. If additive errors (1/10,1/5,1/20) enter after each partial multiplication, then widehat p_j=a_jwidehat p_j-1+e_j, widehat p_3-p_3=12e_1+4e_2+e_3=frac4120. vfigerror_chain headThe same local numbers, a different model If instead the factors themselves are perturbed, the computed product is (4+e_3)(3+e_2)(2+e_1). Its e…
Exact mathematical passage
\widehat p_j=a_j\widehat p_{j-1}+e_j,\qquad \widehat p_3-p_3=12e_1+4e_2+e_3=\frac{41}{20}.
\widehat P_mUse in source · 1 passage
Search aliases: widehat P_m widehat
Use in source · 01_manuscript.tex, line 915
ntheorem[Certified repeated recompression] Let P_0=I, P_j=A_jP_j-1, be exact partial products. Suppose an implemented chain satisfies widehat P_0=I, widehat P_j = A_jwidehat P_j-1+E_j, where E_j is the additive error introduced after the jth multiplication, for example by factor recompression. Then normwidehat P_m-P_m ≤ ∑_j=1^m ( ∏_i=j+1^mnormA_i ) normE_j. If normE_j≤δ_j, the same formula with δ_j gives a certified final error.
Exact mathematical passage
\boxed{ \norm{\widehat P_m-P_m} \leq \sum_{j=1}^{m} \left( \prod_{i=j+1}^{m}\norm{A_i} \right) \norm{E_j}. }
\widehat P_{N,j}Use in source · 2 passages
Search aliases: widehat P_N,j widehat
Use in source · 10_foundations.tex, line 101
cal E(h) and H^-1= E(k), and define U_Ha= D(H E(a)). Then U_H is an equivalence, with inverse U_H^-1a= D(H^-1 E(a)). Its coefficient formulas are (U_Ha)_n=∑_r=0^nbinom nr h_r a_n-r, (U_H^-1a)_n=∑_r=0^nbinom nr k_r a_n-r. The resulting triangular matrix has diagonal entries 1. The transported selectors widehat P_N,j=U_HP_N,jU_H^-1 are idempotent and pairwise orthogonal, and their sum is the identity.
Exact mathematical passage
\widehat P_{N,j}=U_HP_{N,j}U_H^{-1}
\widehat p_rUse in source · 6 passages
Search aliases: widehat p_r widehat
Use in source · Paper_VI_Traditional_Final.tex, line 1093
Let K be column stochastic and let λ^(r) be the likelihood associated with the rth observation. The filter recursion is widehat p_r=Kp_r-1, p_r= B_λ^(r)(widehat p_r).
Exact mathematical passage
\widehat p_r=Kp_{r-1}, \qquad p_r=\cB_{\lambda^{(r)}}(\widehat p_r).
\widehat SUse in source · 2 passages
Search aliases: widehat S widehat
Use in source · Paper_XII_Traditional_Final.tex, line 285
Here completion means the actual Hausdorff completion for the additive uniformity of the indicated adic topology. For every adic ring S and ideal L, there is a canonical bicontinuous ring isomorphism widehat S/overlineLwidehat S≅widehatS/L. Indeed, the completed quotient map kills the closed extended ideal. In the other direction, the quotient map from S/L to widehat S/overlineLwidehat S extends to its completion. The latter quotient is complete: the adic uniformity has a countable base, and the…
Exact mathematical passage
\widehat S\big/\overline{L\widehat S}\cong\widehat{S/L}.
\widehat sUse in source · 3 passages
Search aliases: widehat s widehat
Use in source · Paper_V_Traditional_Final.tex, line 745
If s,widehat s are measurable smooth-background models, the same triangle inequality gives, for finite p, f-(widehat s+widehat j) _p ≤ s-widehat s _p +|a-widehat a|μ(Ω)^1/p +|widehat a|μ(ΩtrianglewidehatΩ)^1/p. This is the correct separation of errors. Detection error is not hidden inside a generic reconstruction score.
Exact mathematical passage
s,\widehat s
\widehat TUse in source · 4 passages
Search aliases: widehat T widehat
Use in source · 02_coordinates.tex, line 189
The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_Q_b,Q_a widehat T_Q_a,P_a = widehat T_Q_b,P_b J^X_P_b,P_a. The identity, cocycle, composition, inverse, and naturality laws are all proved in Lean. Th…
Exact mathematical passage
\widehat T_{Q,P}=\cA_QT\cS_P
\widehat T_2Use in source · 1 passage
Search aliases: widehat T_2 widehat
Use in source · visual_atlas.tex, line 117
The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not merely reuse the same numerical matrix under a new ordering.
Exact mathematical passage
J_{4,2}\widehat T_2=\widehat T_4J_{4,2},\qquad J_{8,4}J_{4,2}=J_{8,2}.
\widehat T_4Use in source · 1 passage
Search aliases: widehat T_4 widehat
Use in source · visual_atlas.tex, line 117
The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not merely reuse the same numerical matrix under a new ordering.
Exact mathematical passage
J_{4,2}\widehat T_2=\widehat T_4J_{4,2},\qquad J_{8,4}J_{4,2}=J_{8,2}.
\widehat T_NUse in source · 1 passage
Search aliases: widehat T_N widehat
Use in source · visual_atlas.tex, line 113
The reordered operators and change of coordinates are widehat T_N=R_NTR_N^-1, J_4,2=R_4R_2^-1. Multiplication cancels the intermediate coordinate maps: J_4,2widehat T_2=widehat T_4J_4,2, J_8,4J_4,2=J_8,2. Thus it makes no difference whether the operator is applied before or after a consistent coordinate change. One must transport the operator, not merely reuse the same numerical matrix under a new ordering.
Exact mathematical passage
\widehat T_N=R_NTR_N^{-1},\qquad J_{4,2}=R_4R_2^{-1}.
\widehat T_{Q,P}Use in source · 1 passage
Search aliases: widehat T_Q,P widehat
Use in source · 02_coordinates.tex, line 189
The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_Q_b,Q_a widehat T_Q_a,P_a = widehat T_Q_b,P_b J^X_P_b,P_a. The identity, cocycle, composition, inverse, and naturality laws are all proved in Lean. Th…
Exact mathematical passage
\widehat T_{Q,P}=\cA_QT\cS_P
\widehat T_{Q_a,P_a}Use in source · 1 passage
Search aliases: widehat T_Q_a,P_a widehat
Use in source · 02_coordinates.tex, line 193
The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_Q_b,Q_a widehat T_Q_a,P_a = widehat T_Q_b,P_b J^X_P_b,P_a. The identity, cocycle, composition, inverse, and naturality laws are all proved in Lean. Th…
Exact mathematical passage
\boxed{ J^Y_{Q_b,Q_a}\,\widehat T_{Q_a,P_a} = \widehat T_{Q_b,P_b}\,J^X_{P_b,P_a}. }
\widehat T_{Q_b,P_b}Use in source · 1 passage
Search aliases: widehat T_Q_b,P_b widehat
Use in source · 02_coordinates.tex, line 193
The same coherence applies to operators. Let P_a,P_b be packet systems on X, let Q_a,Q_b be packet systems on Y, where X and Y are modules over the same scalar ring, and let T:X→ Y be a linear map over that ring. Its exact coordinate representation widehat T_Q,P= A_QT S_P preserves identities and typed composition, and the naturality square is J^Y_Q_b,Q_a widehat T_Q_a,P_a = widehat T_Q_b,P_b J^X_P_b,P_a. The identity, cocycle, composition, inverse, and naturality laws are all proved in Lean. Th…
Exact mathematical passage
\boxed{ J^Y_{Q_b,Q_a}\,\widehat T_{Q_a,P_a} = \widehat T_{Q_b,P_b}\,J^X_{P_b,P_a}. }
\widehat uUse in source · 1 passage
Search aliases: widehat u widehat
Use in source · continuation_08.tex, line 44
inite solved-block cokernel and its parameter-dependent class by C_x=S^e/imJ, o_x(v)=[bar r+Kv]∈ C_x. Then L_v≠∅iff o_x(v)=0. More precisely, the checked digit-to-lift equivalence is L_v≅{u∈ S^m:Ju=-bar r-Kv}. It sends a solved digit u to the actual next residue represented by x+q^L(widehat u,widehat v). The obstruction depends on the solved block J, not on the range of the full map (J K).
Exact mathematical passage
x+q^L(\widehat u,\widehat v)
\widehat vUse in source · 3 passages
Search aliases: widehat v widehat
Use in source · continuation_08.tex, line 20
x=(y_0,t_0)∈ Z^m+k, F(x)≡0pmodq^L. The actual lower root supplies the unique integral residual r=q^-LF(x)∈ Z^e, bar r=rbmod q∈ S^e. Let the actual reduced Jacobian blocks be J=(D_YF)(x)bmod q:S^m→ S^e, K=(D_TF)(x)bmod q:S^k→ S^e. For a prescribed parameter digit v∈ S^k, choose its canonical integral representative widehat v. Define L_v to be the set of actual residue vectors z∈( Z/q^L+1 Z)^m+k satisfying z≡ xpmodq^L, F(z)≡0pmodq^L+1, z_T≡ t_0+q^Lwidehat vpmodq^L+1. The formal predicate uses cano…
Exact mathematical passage
\widehat v
\widehat xUse in source · 5 passages
Search aliases: widehat x widehat
Use in source · Paper_V_Traditional_Final.tex, line 2165
[Coefficient noise under canonical synthesis] Under the hypotheses of [source reference], suppose the observed coefficient vector is c= A_Ex+ν. Canonical synthesis gives widehat x=R_canc and widehat x-x _2 ≤ frac1sqrt A ν _2.
Exact mathematical passage
\widehat x=R_{\mathrm{can}}c
\widehat{\bm X}Use in source · 1 passage
Search aliases: widehat X widehat bm
Use in source · Paper_V_Traditional_Final.tex, line 492
with the monomial z^-m acting on a bilateral sequence by x[n]↦ x[n-m]. Let E∈Mat_N× N( L) and R∈Mat_N× N( L) be analysis and synthesis matrices. Their composition acts on all bilateral channel sequences by finite convolutions. At a nonzero z where the input transforms are absolutely convergent, this gives widehat X(z)= R(z) E(z) X(z).
Exact mathematical passage
\widehat{\bm X}(z)=\bm R(z)\bm E(z)\bm X(z).
\widehat{\mathbb Z}Use in source · 1 passage
Search aliases: widehat Z widehat mathbb
Use in source · 71_native.tex, line 219
The reason is exact: the CRT ring isomorphism preserves polynomial evaluation, and zero in a finite product is equivalent to zero in every component. For F empty, passing to all compatible finite congruences identifies the profinite completion widehat Z with ∏_p Z_p. The same finite reconstruction is compatible with coefficient projections and integer recurrence matrices because these operations commute with reduction and CRT.
Exact mathematical passage
\widehat{\mathbb Z}
\widehat{F'}Use in source · 1 passage
Search aliases: widehatF' widehat
Use in source · 72_recovered.tex, line 1783
[Derivative conjugacy] widehatF'(T) = (1+T)fracddTwidehat F(T).
Exact mathematical passage
\boxed{ \widehat{F'}(T) = (1+T)\frac{d}{dT}\widehat F(T). }
\widehat{G\circ F}Use in source · 1 passage
Search aliases: widehatG∘ F widehat circ
Use in source · 02_connections_obstructions.tex, line 740
tarrow r= A^Y_s∘ S^Y_r, and define widehat F_r← a = A^Y_r∘ F∘ S^X_a. Then J^Y_s← r∘widehat F_r← a = widehat F_s← b∘ J^X_b← a. If A^Z_t:Zxrightarrow∼Z_t is another exact coordinate equivalence and G:Y→ Z is any function, then exact coordinate representation also preserves composition: widehatG∘ F_t← a = widehat G_t← r∘widehat F_r← a. statusleanthreeKomorebi.PaperIII.Transport.exactCoordinateMap; Komorebi.PaperIII.Transport.exactCoordinateMap_comp; Komorebi.PaperIII.Transport.exactCoordinateMap_na…
Exact mathematical passage
\widehat{G\circ F}_{t\leftarrow a} = \widehat G_{t\leftarrow r}\circ\widehat F_{r\leftarrow a}.
\widehat{G\circ F}_{t\leftarrow a}Use in source · 1 passage
Search aliases: widehatG∘ F_t← a widehat circ leftarrow
Use in source · 02_connections_obstructions.tex, line 740
tarrow r= A^Y_s∘ S^Y_r, and define widehat F_r← a = A^Y_r∘ F∘ S^X_a. Then J^Y_s← r∘widehat F_r← a = widehat F_s← b∘ J^X_b← a. If A^Z_t:Zxrightarrow∼Z_t is another exact coordinate equivalence and G:Y→ Z is any function, then exact coordinate representation also preserves composition: widehatG∘ F_t← a = widehat G_t← r∘widehat F_r← a. statusleanthreeKomorebi.PaperIII.Transport.exactCoordinateMap; Komorebi.PaperIII.Transport.exactCoordinateMap_comp; Komorebi.PaperIII.Transport.exactCoordinateMap_na…
Exact mathematical passage
\widehat{G\circ F}_{t\leftarrow a} = \widehat G_{t\leftarrow r}\circ\widehat F_{r\leftarrow a}.
\widehat\KcalUse in source · 2 passages
Search aliases: widehatKcal widehat K widehat Kcal widehat mathcal
Use in source · Paper_V_Traditional_Final.tex, line 2966
[Deterministic jump cell recovery] Let a_min=min_q|a_q| and assume a_min>2b_h. Choose a threshold θ satisfying b_h<θ<a_min-b_h. Then widehat K = {k:|D_k|>θ} is exactly the set of grid cells containing jumps. If widehatτ_q is the midpoint of the detected cell and widehat a_q=D_k, then after matching by order, |widehatτ_q-τ_q| ≤ frac h2 and |widehat a_q-a_q| ≤ b_h.
Exact mathematical passage
\widehat\Kcal = \{k:|D_k|>\theta\}
\widehat{S/L}Use in source · 1 passage
Search aliases: widehatS/L widehat
Use in source · Paper_XII_Traditional_Final.tex, line 285
Here completion means the actual Hausdorff completion for the additive uniformity of the indicated adic topology. For every adic ring S and ideal L, there is a canonical bicontinuous ring isomorphism widehat S/overlineLwidehat S≅widehatS/L. Indeed, the completed quotient map kills the closed extended ideal. In the other direction, the quotient map from S/L to widehat S/overlineLwidehat S extends to its completion. The latter quotient is complete: the adic uniformity has a countable base, and the…
Exact mathematical passage
\widehat S\big/\overline{L\widehat S}\cong\widehat{S/L}.
\widehat\ZUse in source · 1 passage
Search aliases: widehatZ widehat Z widehat mathbb
Use in source · 50_open_questions.tex, line 149
Finite CRT reconstruction leads to the familiar completed object widehat Z=∏_p Z_p, but does not identify which local tuples come from a single global integer. A complete form of Native 13.3 must therefore specify finite-level compatibility, transition maps and completion, and any additional global descent condition. If recurrence, spectral, or Frobenius information is to supply that condition, it must be expressed through defined maps or constraints on the local data, with a precise account of …
Exact mathematical passage
\widehat\Z=\prod_p\Z_p
\widehat\tauUse in source · 20 passages
Search aliases: widehatτ widehat tau
Use in source · Paper_V_Traditional_Final.tex, line 1685
reconstruction formula with an estimated packet is f-widehat f_m = (I-Q_m)s+(I-Q_m)(j-widehat j). This identity identifies the two terms that must be controlled. The exact packet formulas in [source reference] then evaluate or bound j-widehat j from amplitude error, region overlap, boundary displacement, and the order of a derivative jump. A value jump has location modulus |τ-widehatτ|^1/p in finite L^p, while every positive order truncated power packet is Lipschitz in its location.
Exact mathematical passage
|\tau-\widehat\tau|^{1/p}
\widehat\tau_qUse in source · 4 passages
Search aliases: widehatτ_q widehat tau
Use in source · Paper_V_Traditional_Final.tex, line 2971
[Deterministic jump cell recovery] Let a_min=min_q|a_q| and assume a_min>2b_h. Choose a threshold θ satisfying b_h<θ<a_min-b_h. Then widehat K = {k:|D_k|>θ} is exactly the set of grid cells containing jumps. If widehatτ_q is the midpoint of the detected cell and widehat a_q=D_k, then after matching by order, |widehatτ_q-τ_q| ≤ frac h2 and |widehat a_q-a_q| ≤ b_h.
Exact mathematical passage
\widehat\tau_q
\widehat\PhiUse in source · 1 passage
Search aliases: widehatΦ widehat Phi
Use in source · 02_connections_obstructions.tex, line 889
For coordinate models of the source and target colors, an additive target coordinate space, and a chosen noncanonical connection, a pointwise node defect can be defined by D^Γ_b,a(Φ) = Γ^Y_b← a∘widehatΦ_a - widehatΦ_b∘ (∏_qΓ^X_q_b← a). This expression compares two chosen routes. With canonical exact transport in every slot it vanishes by [source reference].
Exact mathematical passage
\mathfrak D^\Gamma_{b,a}(\Phi) = \Gamma^Y_{b\leftarrow a}\circ\widehat\Phi_a - \widehat\Phi_b\circ \left(\prod_q\Gamma^{X_q}_{b\leftarrow a}\right).
\widehat\Phi_aUse in source · 1 passage
Search aliases: widehatΦ_a widehat Phi
Use in source · 02_connections_obstructions.tex, line 889
For coordinate models of the source and target colors, an additive target coordinate space, and a chosen noncanonical connection, a pointwise node defect can be defined by D^Γ_b,a(Φ) = Γ^Y_b← a∘widehatΦ_a - widehatΦ_b∘ (∏_qΓ^X_q_b← a). This expression compares two chosen routes. With canonical exact transport in every slot it vanishes by [source reference].
Exact mathematical passage
\mathfrak D^\Gamma_{b,a}(\Phi) = \Gamma^Y_{b\leftarrow a}\circ\widehat\Phi_a - \widehat\Phi_b\circ \left(\prod_q\Gamma^{X_q}_{b\leftarrow a}\right).
\widehat\Phi_bUse in source · 1 passage
Search aliases: widehatΦ_b widehat Phi
Use in source · 02_connections_obstructions.tex, line 889
For coordinate models of the source and target colors, an additive target coordinate space, and a chosen noncanonical connection, a pointwise node defect can be defined by D^Γ_b,a(Φ) = Γ^Y_b← a∘widehatΦ_a - widehatΦ_b∘ (∏_qΓ^X_q_b← a). This expression compares two chosen routes. With canonical exact transport in every slot it vanishes by [source reference].
Exact mathematical passage
\mathfrak D^\Gamma_{b,a}(\Phi) = \Gamma^Y_{b\leftarrow a}\circ\widehat\Phi_a - \widehat\Phi_b\circ \left(\prod_q\Gamma^{X_q}_{b\leftarrow a}\right).
\widehat\OmegaUse in source · 14 passages
Search aliases: widehatΩ widehat Omega
Use in source · Paper_V_Traditional_Final.tex, line 702
Let (X,Σ,μ) be a measure space, with measurable masks Ω,widehatΩ⊆ X and scalar amplitudes a,widehat a∈ C (real amplitudes are a special case). Suppose a scalar jump packet is j=a 1_Ω, and an estimated packet is widehat j=widehat a 1_widehatΩ.
Exact mathematical passage
\Omega,\widehat\Omega\subseteq X
\wideotimes_{\OO}Use in source · 2 passages
Search aliases: wideotimes_OO mathbinwidehat⊗_ O wideotimes OO mathbin widehat otimes mathcal
Use in source · Paper_XII_Traditional_Final.tex, line 259
[Completed tensor quotient] Let R_1,…,R_s be complete Noetherian local O algebras with common residue field k. Let I_i⊆ R_i be closed ideals. Put R^loc=R_1mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ OR_s and let J be the closed ideal generated by the images of the I_i. Then there is a canonical isomorphism (R_1/I_1)mathbinwidehat⊗_ O⋯mathbinwidehat⊗_ O(R_s/I_s) ≅ R^loc/J.
Exact mathematical passage
R^{\mathrm{loc}}=R_1\wideotimes_{\OO}\cdots\wideotimes_{\OO}R_s
\widetilde BUse in source · 8 passages
Search aliases: widetilde B widetilde
Use in source · monograph_atlas.tex, line 660
In its elimination domain, Σ(z)=1/(z-2). At z=3+ i, Σ(3+ i)=frac1- i2, ImΣ(z)=-(Im z)|(z-2)^-1|^2≤0. The sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix. xfigstability headA stability estimate with a common spectral gap Compare (B,D)=(1,2) with (widetilde B,widetilde D)=(9/8,17/8) at z=3. Both hidden spectral distances are at least δ=3/4, and both couplings are bounded by g=9/8. Then |Σ-widetildeΣ|=|1-frac81…
Exact mathematical passage
(\widetilde B,\widetilde D)=(9/8,17/8)
\widetilde bUse in source · 5 passages
Search aliases: widetilde b widetilde
Use in source · 04_arithmetic.tex, line 754
[Smith weighted congruence solve] Let M∈ M_m,n( O) have Smith exponents s_1,…,s_r. Assume the residue field is finite of cardinality q, fix L≥1, and consider Mx≡ bpmodπ^L. Put widetilde b=Ub for a Smith matrix U as in [source reference]. The congruence is solvable if and only if widetilde b_i≡0 pmodπ^min(s_i,L) (1≤ i≤ r) and widetilde b_i≡0 pmodπ^L (r<i≤ m). When it is solvable, the number of solutions in ( O/π^L O)^n is q^(n-r)L+∑_i=1^rmin(s_i,L).
Exact mathematical passage
\widetilde b=Ub
\widetilde B^*Use in source · 2 passages
Search aliases: widetilde B^* widetilde
Use in source · 02_strengthened.tex, line 1787
[Stability of the self energy] Let D=D^* and widetilde D=widetilde D^* act on one fixed hidden Hilbert space, and let B,widetilde B map that hidden space to one fixed retained Hilbert space. Define Σ(z)=B(z-D)^-1B^*, widetildeΣ(z)=widetilde B(z-widetilde D)^-1widetilde B^*. Assume norm B,normwidetilde B≤ g and dist(z,Spec D), dist(z,Specwidetilde D) ≥δ>0. Then normΣ(z)-widetildeΣ(z) ≤ frac2gδnormB-widetilde B + fracg^2δ^2normD-widetilde D.
Exact mathematical passage
\Sigma(z)=B(z-D)^{-1}B^*, \qquad \widetilde\Sigma(z)=\widetilde B(z-\widetilde D)^{-1}\widetilde B^*.
\widetilde b_iUse in source · 3 passages
Search aliases: widetilde b_i widetilde
Use in source · 04_arithmetic.tex, line 758
[Smith weighted congruence solve] Let M∈ M_m,n( O) have Smith exponents s_1,…,s_r. Assume the residue field is finite of cardinality q, fix L≥1, and consider Mx≡ bpmodπ^L. Put widetilde b=Ub for a Smith matrix U as in [source reference]. The congruence is solvable if and only if widetilde b_i≡0 pmodπ^min(s_i,L) (1≤ i≤ r) and widetilde b_i≡0 pmodπ^L (r<i≤ m). When it is solvable, the number of solutions in ( O/π^L O)^n is q^(n-r)L+∑_i=1^rmin(s_i,L).
Exact mathematical passage
\widetilde b_i\equiv0 \pmod{\pi^{\min(s_i,L)}} \qquad(1\leq i\leq r)
\widetilde cUse in source · 3 passages
Search aliases: widetilde c widetilde
Use in source · Paper_V_Traditional_Final.tex, line 2455
[Stability of the optimal cost] Let c and widetilde c be two node cost functions. For an admissible pruning L, set ε( L) = ∑_v∈ L|c(v)-widetilde c(v)| and let ε_* = max_ Lε( L). If F_* and widetilde F_* are the two optimal root costs, then |F_*-widetilde F_*| ≤ ε_*.
Exact mathematical passage
\widetilde c
\widetilde C_{6,3}Use in source · 1 passage
Search aliases: widetilde C_6,3 widetilde
Use in source · supplement_01.tex, line 180
For the unnormalized convention, Δ_1widetilde C_6,3(1)=3! frac64455=frac384455, v_3 (frac384455)=1. The factorial's valuation is also proved: v_3(3!)=1, v_3(Δ_1C_6,3(1)) =1-v_3(3!)=0. This proves sharpness at the displayed normalization boundary. It does not establish the corrected global inequality for all parameters, and does not assert that every nonunit-factorial parameter tuple must violate every no-loss specialization. All valuation calculations here concern nonzero rationals.
Exact mathematical passage
\Delta_1\widetilde C_{6,3}(1)=3!\,\frac{64}{455}=\frac{384}{455}, \qquad v_3\!\left(\frac{384}{455}\right)=1.
\widetilde C_aUse in source · 3 passages
Search aliases: widetilde C_a widetilde
Use in source · monograph_atlas.tex, line 753
ull algebra closes: [C_x,C_y]=2 i C_z, [C_y,C_z]=2 i C_x, [C_z,C_x]=2 i C_y. These are uncompressed identities. They supply an actual closure test in the quotient by the constraint span, unlike the automatic compressed zero on the physical kernel. xfigconstraint_image headExact transport holds on the coarse image Embed by the coordinate isometry R: C^3→ C^4 and set widetilde C_a=C_a⊕1. Then widetilde C_aR=RC_a, so [widetilde C_a,widetilde C_b]R=R[C_a,C_b]. The coarse physical vector maps into th…
Exact mathematical passage
\widetilde C_a=C_a\oplus1
\widetilde C_bUse in source · 1 passage
Search aliases: widetilde C_b widetilde
Use in source · monograph_atlas.tex, line 754
C_x, [C_z,C_x]=2 i C_y. These are uncompressed identities. They supply an actual closure test in the quotient by the constraint span, unlike the automatic compressed zero on the physical kernel. xfigconstraint_image headExact transport holds on the coarse image Embed by the coordinate isometry R: C^3→ C^4 and set widetilde C_a=C_a⊕1. Then widetilde C_aR=RC_a, so [widetilde C_a,widetilde C_b]R=R[C_a,C_b]. The coarse physical vector maps into the fine common kernel. However, the same prescribed Pa…
Exact mathematical passage
[\widetilde C_a,\widetilde C_b]R=R[C_a,C_b].
\widetilde C_{N,j}Use in source · 1 passage
Search aliases: widetilde C_N,j widetilde
Use in source · supplement_01.tex, line 168
Principal targets. targetFiltered.padic_filtered_reconstruction targetStandardWindow.lucas_row_reduction targetStandardWindow.standard_diagonal_unit targetStandardWindow.factorial_valuation_zero pageheading4. Sharp normalization boundarySource item 13.1 The construction fixes the two coefficient conventions: C_N,j(n), widetilde C_N,j(n)=j! C_N,j(n). The witness uses p=3, m=2, N=mp=6, j=3, n=1 and difference step h=(p-1)/m=1. The parameter conditions are checked explicitly, including odd primalit…
Exact mathematical passage
C_{N,j}(n),\qquad \widetilde C_{N,j}(n)=j!\,C_{N,j}(n).
\widetilde C_xUse in source · 1 passage
Search aliases: widetilde C_x widetilde
Use in source · monograph_atlas.tex, line 759
e coarse physical vector maps into the fine common kernel. However, the same prescribed Pauli structure does not close on the whole fine level: on the added coordinate the commutator is zero, whereas 2 iwidetilde C_z is 2 i. Exact transport constrains the image of R and leaves its orthogonal complement as separate data. headAn approximate residual is also explicit Perturb only widetilde C_x by (1/10)ket0bra0. For the unit coarse physical vector, the fine constraint residual is exactly 1/10. The …
Exact mathematical passage
\widetilde C_x
\widetilde C_zUse in source · 1 passage
Search aliases: widetilde C_z widetilde
Use in source · monograph_atlas.tex, line 757
the coarse image Embed by the coordinate isometry R: C^3→ C^4 and set widetilde C_a=C_a⊕1. Then widetilde C_aR=RC_a, so [widetilde C_a,widetilde C_b]R=R[C_a,C_b]. The coarse physical vector maps into the fine common kernel. However, the same prescribed Pauli structure does not close on the whole fine level: on the added coordinate the commutator is zero, whereas 2 iwidetilde C_z is 2 i. Exact transport constrains the image of R and leaves its orthogonal complement as separate data. headAn approx…
Exact mathematical passage
2\ii\widetilde C_z
\widetilde DUse in source · 7 passages
Search aliases: widetilde D widetilde
Use in source · monograph_atlas.tex, line 660
In its elimination domain, Σ(z)=1/(z-2). At z=3+ i, Σ(3+ i)=frac1- i2, ImΣ(z)=-(Im z)|(z-2)^-1|^2≤0. The sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix. xfigstability headA stability estimate with a common spectral gap Compare (B,D)=(1,2) with (widetilde B,widetilde D)=(9/8,17/8) at z=3. Both hidden spectral distances are at least δ=3/4, and both couplings are bounded by g=9/8. Then |Σ-widetildeΣ|=|1-frac81…
Exact mathematical passage
(\widetilde B,\widetilde D)=(9/8,17/8)
\widetilde D^*Use in source · 1 passage
Search aliases: widetilde D^* widetilde
Use in source · 02_strengthened.tex, line 1786
[Stability of the self energy] Let D=D^* and widetilde D=widetilde D^* act on one fixed hidden Hilbert space, and let B,widetilde B map that hidden space to one fixed retained Hilbert space. Define Σ(z)=B(z-D)^-1B^*, widetildeΣ(z)=widetilde B(z-widetilde D)^-1widetilde B^*. Assume norm B,normwidetilde B≤ g and dist(z,Spec D), dist(z,Specwidetilde D) ≥δ>0. Then normΣ(z)-widetildeΣ(z) ≤ frac2gδnormB-widetilde B + fracg^2δ^2normD-widetilde D.
Exact mathematical passage
\widetilde D=\widetilde D^*
\widetilde EUse in source · 8 passages
Search aliases: widetilde E widetilde
Use in source · Paper_V_Traditional_Final.tex, line 2097
[Stability under filter perturbation] Assume RE=z^-dI_N. Let widetilde E=E+Δ E, widetilde R=R+Δ R. Set e_0 · =*ess sup E _op, · r_0 · =*ess sup R _op,
Exact mathematical passage
\widetilde E=E+\Delta E, \qquad \widetilde R=R+\Delta R.
\widetilde eUse in source · 6 passages
Search aliases: widetilde e widetilde
Use in source · 72_recovered.tex, line 690
Fix a prime p, an integer m≥1, and an address 0≤ j<p. Put N=mp. For arbitrary m, let widetilde e_n^(m,p)∈ F_p be the unique sequence satisfying ∑_q=0^n binommnmq widetilde e_q^(m,p) = δ_n0. When mmid p-1, this sequence agrees with the compressed Hurwitz inverse in [source reference].
Exact mathematical passage
\widetilde e_n^{(m,p)}\in\FF_p
\widetilde e_n^{(m,p)}Use in source · 4 passages
Search aliases: widetilde e_n^(m,p) widetilde
Use in source · 72_recovered.tex, line 690
Fix a prime p, an integer m≥1, and an address 0≤ j<p. Put N=mp. For arbitrary m, let widetilde e_n^(m,p)∈ F_p be the unique sequence satisfying ∑_q=0^n binommnmq widetilde e_q^(m,p) = δ_n0. When mmid p-1, this sequence agrees with the compressed Hurwitz inverse in [source reference].
Exact mathematical passage
\widetilde e_n^{(m,p)}\in\FF_p
\widetilde e_q^{(m,p)}Use in source · 1 passage
Search aliases: widetilde e_q^(m,p) widetilde
Use in source · 72_recovered.tex, line 691
Fix a prime p, an integer m≥1, and an address 0≤ j<p. Put N=mp. For arbitrary m, let widetilde e_n^(m,p)∈ F_p be the unique sequence satisfying ∑_q=0^n binommnmq widetilde e_q^(m,p) = δ_n0. When mmid p-1, this sequence agrees with the compressed Hurwitz inverse in [source reference].
Exact mathematical passage
\boxed{ \sum_{q=0}^{n} \binom{mn}{mq} \widetilde e_q^{(m,p)} = \delta_{n0}. }
\widetilde e_r^{(m,p)}Use in source · 1 passage
Search aliases: widetilde e_r^(m,p) widetilde
Use in source · 72_recovered.tex, line 805
[Rational residual generating function] Under the hypotheses of [source reference], for every standard address j, ∑_n≥0 (C_mp,j(n)bmod p)z^n = 1+ frac ∑_r=1^hwidetilde e_r^(m,p)z^r 1-z^h.
Exact mathematical passage
\sum_{n\geq0} \left(C_{mp,j}(n)\bmod p\right)z^n = 1+ \frac{ \sum_{r=1}^{h}\widetilde e_r^{(m,p)}z^r }{1-z^h}.
\widetilde FUse in source · 4 passages
Search aliases: widetilde F widetilde
Use in source · Paper_V_Traditional_Final.tex, line 2467
[Stability of the optimal cost] Let c and widetilde c be two node cost functions. For an admissible pruning L, set ε( L) = ∑_v∈ L|c(v)-widetilde c(v)| and let ε_* = max_ Lε( L). If F_* and widetilde F_* are the two optimal root costs, then |F_*-widetilde F_*| ≤ ε_*.
Exact mathematical passage
\widetilde F_*
\widetilde fUse in source · 15 passages
Search aliases: widetilde f widetilde
Use in source · Paper_V_Traditional_Final.tex, line 656
Let f=s+j where s belongs to a smooth approximation class and j is a known discontinuity packet. Let Q_m be any approximation operator applied only to s, and define widetilde f_m=Q_ms+j.
Exact mathematical passage
\widetilde f_m=Q_ms+j.
\widetilde F_*Use in source · 4 passages
Search aliases: widetilde F_* widetilde
Use in source · Paper_V_Traditional_Final.tex, line 2467
[Stability of the optimal cost] Let c and widetilde c be two node cost functions. For an admissible pruning L, set ε( L) = ∑_v∈ L|c(v)-widetilde c(v)| and let ε_* = max_ Lε( L). If F_* and widetilde F_* are the two optimal root costs, then |F_*-widetilde F_*| ≤ ε_*.
Exact mathematical passage
\widetilde F_*
\widetilde f_mUse in source · 7 passages
Search aliases: widetilde f_m widetilde
Use in source · Paper_V_Traditional_Final.tex, line 656
Let f=s+j where s belongs to a smooth approximation class and j is a known discontinuity packet. Let Q_m be any approximation operator applied only to s, and define widetilde f_m=Q_ms+j.
Exact mathematical passage
\widetilde f_m=Q_ms+j.
\widetilde KUse in source · 9 passages
Search aliases: widetilde K widetilde
Use in source · Paper_VI_Traditional_Final.tex, line 3877
[One step filter stability] Let K and widetilde K be column stochastic. Assume 0<a≤λ_i≤ b. Then norm B_λ(Kp)- B_λ(widetilde Kq) _1 ≤ frac2ba ( normp-q_1 + normK-widetilde K_1→1 ).
Exact mathematical passage
\widetilde K
\widetilde LUse in source · 2 passages
Search aliases: widetilde L widetilde
Use in source · Paper_XI_Traditional_Final.tex, line 866
Let L⊆ V be maximal isotropic. Choose a subgroup lift widetilde L⊆ H(V) on which the central coordinate is zero.
Exact mathematical passage
\widetilde L\subseteq\frakH(V)
\widetilde PUse in source · 3 passages
Search aliases: widetilde P widetilde
Use in source · 02_supplement_03.tex, line 58
Post-product recompression is different. Define the implemented iterates by widetilde P_0=I, widetilde P_j=A_jwidetilde P_j-1+E_j. Here E_j is a residual introduced after multiplication. Lean derives widetilde P_j-P_j=A_j(widetilde P_j-1-P_j-1)+E_j and proves normwidetilde P_m-P_m ≤∑_j=1^m(∏_i=j+1^mnormA_i)δ_j, normE_j≤δ_j. This theorem works in a normed ring without the extra norm-one condition. It includes the empty iteration. There is no exact-prefix factor in the local residual term, because…
Exact mathematical passage
\widetilde P_0=I,\qquad \widetilde P_j=A_j\widetilde P_{j-1}+E_j.
\widetilde pUse in source · 2 passages
Search aliases: widetilde p widetilde
Use in source · 02_foundations.tex, line 340
free presentation] Let X be a complete Hausdorff locally convex space, let A⊆End(X) act continuously, and let S=(s_1,…,s_r). Put E:= A^⊕ r/N_S algebraically and let j:E→ X be the injective map induced by ε_S. Give E the locally convex topology transported from j(E), equivalently the topology generated by widetilde p(e):=p(j(e)) for the continuous seminorms p on X. Every a∈ A acts continuously on E in the transported topology and therefore extends to widehat E. If Hull_ A(S) is dense in X, then j…
Exact mathematical passage
\widetilde p(e):=p(j(e))
\widetilde P_0Use in source · 1 passage
Search aliases: widetilde P_0 widetilde
Use in source · 02_supplement_03.tex, line 58
Post-product recompression is different. Define the implemented iterates by widetilde P_0=I, widetilde P_j=A_jwidetilde P_j-1+E_j. Here E_j is a residual introduced after multiplication. Lean derives widetilde P_j-P_j=A_j(widetilde P_j-1-P_j-1)+E_j and proves normwidetilde P_m-P_m ≤∑_j=1^m(∏_i=j+1^mnormA_i)δ_j, normE_j≤δ_j. This theorem works in a normed ring without the extra norm-one condition. It includes the empty iteration. There is no exact-prefix factor in the local residual term, because…
Exact mathematical passage
\widetilde P_0=I,\qquad \widetilde P_j=A_j\widetilde P_{j-1}+E_j.
\widetilde P_jUse in source · 2 passages
Search aliases: widetilde P_j widetilde
Use in source · 02_supplement_03.tex, line 58
Post-product recompression is different. Define the implemented iterates by widetilde P_0=I, widetilde P_j=A_jwidetilde P_j-1+E_j. Here E_j is a residual introduced after multiplication. Lean derives widetilde P_j-P_j=A_j(widetilde P_j-1-P_j-1)+E_j and proves normwidetilde P_m-P_m ≤∑_j=1^m(∏_i=j+1^mnormA_i)δ_j, normE_j≤δ_j. This theorem works in a normed ring without the extra norm-one condition. It includes the empty iteration. There is no exact-prefix factor in the local residual term, because…
Exact mathematical passage
\widetilde P_0=I,\qquad \widetilde P_j=A_j\widetilde P_{j-1}+E_j.
\widetilde P_{j-1}Use in source · 2 passages
Search aliases: widetilde P_j-1 widetilde
Use in source · 02_supplement_03.tex, line 58
Post-product recompression is different. Define the implemented iterates by widetilde P_0=I, widetilde P_j=A_jwidetilde P_j-1+E_j. Here E_j is a residual introduced after multiplication. Lean derives widetilde P_j-P_j=A_j(widetilde P_j-1-P_j-1)+E_j and proves normwidetilde P_m-P_m ≤∑_j=1^m(∏_i=j+1^mnormA_i)δ_j, normE_j≤δ_j. This theorem works in a normed ring without the extra norm-one condition. It includes the empty iteration. There is no exact-prefix factor in the local residual term, because…
Exact mathematical passage
\widetilde P_0=I,\qquad \widetilde P_j=A_j\widetilde P_{j-1}+E_j.
\widetilde P_mUse in source · 1 passage
Search aliases: widetilde P_m widetilde
Use in source · 02_supplement_03.tex, line 67
Post-product recompression is different. Define the implemented iterates by widetilde P_0=I, widetilde P_j=A_jwidetilde P_j-1+E_j. Here E_j is a residual introduced after multiplication. Lean derives widetilde P_j-P_j=A_j(widetilde P_j-1-P_j-1)+E_j and proves normwidetilde P_m-P_m ≤∑_j=1^m(∏_i=j+1^mnormA_i)δ_j, normE_j≤δ_j. This theorem works in a normed ring without the extra norm-one condition. It includes the empty iteration. There is no exact-prefix factor in the local residual term, because…
Exact mathematical passage
\boxed{\norm{\widetilde P_m-P_m} \leq\sum_{j=1}^m\left(\prod_{i=j+1}^m\norm{A_i}\right)\delta_j, \quad \norm{E_j}\leq\delta_j.}
\widetilde RUse in source · 2 passages
Search aliases: widetilde R widetilde
Use in source · Paper_V_Traditional_Final.tex, line 2097
[Stability under filter perturbation] Assume RE=z^-dI_N. Let widetilde E=E+Δ E, widetilde R=R+Δ R. Set e_0 · =*ess sup E _op, · r_0 · =*ess sup R _op,
Exact mathematical passage
\widetilde E=E+\Delta E, \qquad \widetilde R=R+\Delta R.
\widetilde TUse in source · 7 passages
Search aliases: widetilde T widetilde
Use in source · 04_transport.tex, line 13
[Coordinate transport] Let U:X→ Y be a vector space isomorphism and let P=(P_α) be a packet system on X. Define Q_α:=UP_α U^-1. Then Q=(Q_α) is a packet system on Y. For widetilde T=UTU^-1, Q_βwidetilde TQ_α = U(P_β TP_α)U^-1. Hence packet dimensions, zero patterns, and support preorders are invariant under coordinate transport. If a declared algebra generating family G is transported to U G U^-1, then its transfer graph and reachability relation are also preserved. [source reference]
Exact mathematical passage
\widetilde T=UTU^{-1}
\widetilde tUse in source · 5 passages
Search aliases: widetilde t widetilde
Use in source · 30_lifting.tex, line 265
Integer representatives and residue classes at the next level must also be distinguished. For q≥1 and L≥0, the map S_q^d⟶ {y∈( Z/q^L+1 Z)^d:y≡ xpmodq^L}, t⟼ x+q^Lwidetilde t is a bijection, where widetilde t is any integer representative. Independence of the representative and injectivity both follow by cancelling q^L in integer divisibility. Surjectivity follows by expressing the lower congruence as a multiple of q^L. This is the equivalence leanKomorebi.ResidueLiftDigits.digitEquivResidue.
Exact mathematical passage
S_q^d\longrightarrow \{y\in(\Z/q^{L+1}\Z)^d:y\equiv x\pmod{q^L}\}, \qquad t\longmapsto x+q^L\widetilde t
\widetilde T^{-1}Use in source · 2 passages
Search aliases: widetilde T^-1 widetilde
Use in source · 03_weighted.tex, line 170
[Stability of inversion at finite precision] Let T and widetilde T be invertible elements of F^0 B. If T-widetilde T∈ F^L B, then T^-1-widetilde T^-1∈ F^L B.
Exact mathematical passage
T^{-1}-\widetilde T^{-1}\in F^L\Bcal.
\widetilde UUse in source · 2 passages
Search aliases: widetilde U widetilde
Use in source · 20_arithmetic.tex, line 1091
[Exact image of the original coefficients] The continuous extension of the original sequence is widetilde U_j(x)=j!+2^d_jΦ_j(x). It is injective, its image is exactly j!+2^d_j Z_2, and |widetilde U_j(x)-widetilde U_j(y)|_2 =2^-d_j|x-y|_2. Every point of that affine image has exactly one preimage in Z_2.
Exact mathematical passage
\widetilde U_j(x)=j!+2^{d_j}\Phi_j(x).
\widetilde uUse in source · 6 passages
Search aliases: widetilde u widetilde
Use in source · 30_lifting.tex, line 567
A candidate for the next lift has the form x+q^L(widetilde u,widetilde v), with solved digit u∈ S_q^m. Its Taylor correction equation is overliner_x+J_xu+K_xv=0. Solvability is exactly vanishing of the displayed cokernel class. Restrict the residue-digit equivalence to the prescribed parameter coordinates. This identifies the lift fiber with the solved-digit fiber. Translation by ker J_x is free and transitive on any nonempty such fiber. The prime-field count follows from rank-nullity as before.
Exact mathematical passage
x+q^L(\widetilde u,\widetilde v)
\widetilde U_jUse in source · 2 passages
Search aliases: widetilde U_j widetilde
Use in source · 20_arithmetic.tex, line 1091
[Exact image of the original coefficients] The continuous extension of the original sequence is widetilde U_j(x)=j!+2^d_jΦ_j(x). It is injective, its image is exactly j!+2^d_j Z_2, and |widetilde U_j(x)-widetilde U_j(y)|_2 =2^-d_j|x-y|_2. Every point of that affine image has exactly one preimage in Z_2.
Exact mathematical passage
\widetilde U_j(x)=j!+2^{d_j}\Phi_j(x).
\widetilde vUse in source · 2 passages
Search aliases: widetilde v widetilde
Use in source · 30_lifting.tex, line 545
ixed in advance. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0), and assume F(x)=q^Lr_x with q≥1 and L≥1. Write the reduced Jacobian as DF(x)bmod q=(J_x K_x), J_x:S_q^m→ S_q^e, K_x:S_q^k→ S_q^e. For a prescribed parameter digit v∈ S_q^k, let L_x,L,v be the set of next residue roots over x whose parameter coordinates are t_0+q^Lwidetilde v modulo q^L+1.
Exact mathematical passage
t_0+q^L\widetilde v
\widetilde xUse in source · 15 passages
Search aliases: widetilde x widetilde
Use in source · Paper_V_Traditional_Final.tex, line 2064
[Quantitative imperfect reconstruction] Define the delay compensated defect D( e^ iω) = e^ i dωR( e^ iω)E( e^ iω)-I_N. If ε_rec = *ess sup_ω D( e^ iω) _op<∞, then the delay compensated reconstruction widetilde x satisfies widetilde x-x _2 ≤ ε_rec x _2. The constant is sharp as an operator norm.
Exact mathematical passage
\widetilde x
\widetilde x'Use in source · 1 passage
Search aliases: widetilde x' widetilde
Use in source · atlas-pages.tex, line 870
0 · 0, so N^2=0 and N _2=1. On [0,T], compare the memory kernels K(t)=N and widetilde K(t)=0, with identity free semigroup, no forcing, and initial state x_0=e_1=(0,1)^ T. The equations are x'(t)=∫_0^t N x(r) dr, widetilde x'(t)=0. Their exact solutions are x(t)=(t^2/2,1)^ T and widetilde x(t)=(0,1)^ T. vfigext_trajectory platecaptionThe first-coordinate error is exactly t^2/2. The certificate bounds its maximum on the entire interval. The bound is deliberately shown separately from the trajecto…
Exact mathematical passage
x'(t)=\int_0^t N x(r)\,dr,\qquad \widetilde x'(t)=0.
\widetilde\KcalUse in source · 3 passages
Search aliases: widetildeKcal widetilde K widetilde Kcal widetilde mathcal
Use in source · Paper_VI_Traditional_Final.tex, line 5674
FloatBarrier Memory compression, rational realization, and phase-type consequences. Fix T>0. For the next estimate let A be a bounded operator on a Banach space X, let K,widetilde K∈ L^1((0,T), B(X)), and let f be continuous. Take continuously differentiable solutions of the displayed equations. This direct bounded-generator estimate retains a sharper exponential constant than the general mild-semigroup estimate in Part II. [Stability of memory compression] Let x and widetilde x solve dot x(t) =…
Exact mathematical passage
\Kcal,\widetilde\Kcal\in L^1((0,T),\mathcal B(X))
\widetilde\lambdaUse in source · 3 passages
Search aliases: widetildeλ widetilde lambda
Use in source · Paper_VI_Traditional_Final.tex, line 3609
[Approximate coarse Bayesian rule] Choose one positive number ℓ_α for every packet and define the block constant fine likelihood widetildeλ_i=ℓ_α (i∈ I_α). Then norm C B_λ(p)- B_ℓ(Cp) _1 ≤ frac 2∑_i p_iabsλ_i-widetildeλ_i max{Z_λ(p),Z_widetildeλ(p)} .
Exact mathematical passage
\widetilde\lambda_i=\ell_\alpha \quad(i\in I_\alpha).
\widetilde\lambda_iUse in source · 2 passages
Search aliases: widetildeλ_i widetilde lambda
Use in source · Paper_VI_Traditional_Final.tex, line 3609
[Approximate coarse Bayesian rule] Choose one positive number ℓ_α for every packet and define the block constant fine likelihood widetildeλ_i=ℓ_α (i∈ I_α). Then norm C B_λ(p)- B_ℓ(Cp) _1 ≤ frac 2∑_i p_iabsλ_i-widetildeλ_i max{Z_λ(p),Z_widetildeλ(p)} .
Exact mathematical passage
\widetilde\lambda_i=\ell_\alpha \quad(i\in I_\alpha).
\widetilde\SigmaUse in source · 3 passages
Search aliases: widetildeΣ widetilde Sigma
Use in source · monograph_atlas.tex, line 661
The sign in the upper half plane comes from self-adjoint hidden data. It is not an entrywise sign condition on an arbitrary effective matrix. xfigstability headA stability estimate with a common spectral gap Compare (B,D)=(1,2) with (widetilde B,widetilde D)=(9/8,17/8) at z=3. Both hidden spectral distances are at least δ=3/4, and both couplings are bounded by g=9/8. Then |Σ-widetildeΣ|=|1-frac8156|=frac2556, frac2gδ|B-widetilde B| +fracg^2δ^2|D-widetilde D|=frac2132. The theorem bounds the diff…
Exact mathematical passage
|\Sigma-\widetilde\Sigma|=\left|1-\frac{81}{56}\right|=\frac{25}{56}, \qquad \frac{2g}{\delta}|B-\widetilde B| +\frac{g^2}{\delta^2}|D-\widetilde D|=\frac{21}{32}.
\widetilde{\sigma}Use in source · 4 passages
Search aliases: widetildeσ widetilde sigma
Use in source · Paper_IV_Traditional_Final.tex, line 2526
[Theta Bell polynomial formula] Put widetildeσ_N,m=(-1)^mσ_N,m. Then symupΘ_N,n(u) = frac(Nn)!n! B_n( widetildeσ_N,1(u^N+u), 1!widetildeσ_N,2(u^2N+u), …, (n-1)!widetildeσ_N,n(u^nN+u) ).
Exact mathematical passage
\widetilde{\sigma}_{N,m}=(-1)^m\sigma_{N,m}.
\widetilde{\sigma}_{N,1}Use in source · 1 passage
Search aliases: widetildeσ_N,1 widetilde sigma
Use in source · Paper_IV_Traditional_Final.tex, line 2530
[Theta Bell polynomial formula] Put widetildeσ_N,m=(-1)^mσ_N,m. Then symupΘ_N,n(u) = frac(Nn)!n! B_n( widetildeσ_N,1(u^N+u), 1!widetildeσ_N,2(u^2N+u), …, (n-1)!widetildeσ_N,n(u^nN+u) ).
Exact mathematical passage
\Thetafun_{N,n}(u) = \frac{(Nn)!}{n!} B_n\bigl( \widetilde{\sigma}_{N,1}(u^N+u), 1!\widetilde{\sigma}_{N,2}(u^{2N}+u), \ldots, (n-1)!\widetilde{\sigma}_{N,n}(u^{nN}+u) \bigr).
\widetilde{\sigma}_{N,2}Use in source · 1 passage
Search aliases: widetildeσ_N,2 widetilde sigma
Use in source · Paper_IV_Traditional_Final.tex, line 2530
[Theta Bell polynomial formula] Put widetildeσ_N,m=(-1)^mσ_N,m. Then symupΘ_N,n(u) = frac(Nn)!n! B_n( widetildeσ_N,1(u^N+u), 1!widetildeσ_N,2(u^2N+u), …, (n-1)!widetildeσ_N,n(u^nN+u) ).
Exact mathematical passage
\Thetafun_{N,n}(u) = \frac{(Nn)!}{n!} B_n\bigl( \widetilde{\sigma}_{N,1}(u^N+u), 1!\widetilde{\sigma}_{N,2}(u^{2N}+u), \ldots, (n-1)!\widetilde{\sigma}_{N,n}(u^{nN}+u) \bigr).
\widetilde{\sigma}_{N,m}Use in source · 3 passages
Search aliases: widetildeσ_N,m widetilde sigma
Use in source · Paper_IV_Traditional_Final.tex, line 2526
[Theta Bell polynomial formula] Put widetildeσ_N,m=(-1)^mσ_N,m. Then symupΘ_N,n(u) = frac(Nn)!n! B_n( widetildeσ_N,1(u^N+u), 1!widetildeσ_N,2(u^2N+u), …, (n-1)!widetildeσ_N,n(u^nN+u) ).
Exact mathematical passage
\widetilde{\sigma}_{N,m}=(-1)^m\sigma_{N,m}.
\widetilde{\sigma}_{N,n}Use in source · 1 passage
Search aliases: widetildeσ_N,n widetilde sigma
Use in source · Paper_IV_Traditional_Final.tex, line 2530
[Theta Bell polynomial formula] Put widetildeσ_N,m=(-1)^mσ_N,m. Then symupΘ_N,n(u) = frac(Nn)!n! B_n( widetildeσ_N,1(u^N+u), 1!widetildeσ_N,2(u^2N+u), …, (n-1)!widetildeσ_N,n(u^nN+u) ).
Exact mathematical passage
\Thetafun_{N,n}(u) = \frac{(Nn)!}{n!} B_n\bigl( \widetilde{\sigma}_{N,1}(u^N+u), 1!\widetilde{\sigma}_{N,2}(u^{2N}+u), \ldots, (n-1)!\widetilde{\sigma}_{N,n}(u^{nN}+u) \bigr).
\widetilde\PhiUse in source · 4 passages
Search aliases: widetildeΦ widetilde Phi
Use in source · 04_transport.tex, line 336
[Quotient topology and completed realization] Let Φ:X↠ Y be a continuous linear quotient map between Hausdorff topological vector spaces, and put K=KerΦ. Then the induced map widetildeΦ:X/K⟶ Y is a topological isomorphism. If D⊆ X is a linear subspace whose image Φ(D) is dense in a complete space Y, then the completion of D/(D∩ K) in the topology transported from Y is canonically isomorphic to Y. [source reference]
Exact mathematical passage
\widetilde\Phi:X/K\longrightarrow Y
\WpUse in source · 22 passages
Search aliases: Wp W_p mathcal
Use in source · supplement_09.tex, line 111
Weighted membership. The class used in the formalization is W_p={F∈ Q_p[[T]]: p^lfloor d/prfloorcoefTdF∈ Z_p for every d}. The integral-coordinate estimate from page 2 now proves H_r(log(1+T))∈ W_p for every residue r, including the singular ones. Rational coefficients are mapped into Q_p by the actual coefficient ring homomorphism.
Exact mathematical passage
\Wp=\left\{F\in\Q_p[[T]]: p^{\lfloor d/p\rfloor}\coef{T}{d}F\in\Z_p\ \text{for every }d\right\}.
\mathrm{wt}Use in source · 2 passages
Search aliases: wt mathrm
Use in source · 50_open_questions.tex, line 95
The original proposed form is X≃ X_2^wt×_C∏_p>2'X_p. It leaves X, the distinguished subsets defining the restricted product, the maps to C, the ambient category, and the universal meaning of ``smallest'' unspecified. In particular, one must decide whether X is a global object or a completed object.
Exact mathematical passage
X\simeq X_2^{\mathrm{wt}}\times_C\prod_{p>2}'X_p.
\bm XUse in source · 4 passages
Search aliases: X bm
Use in source · Paper_V_Traditional_Final.tex, line 439
[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.
Exact mathematical passage
\bm Y(z)=\bm H^{(N)}(z)\bm X(z), \qquad H^{(N)}_{ks}(z)=\sum_m h[Nm+k-s]z^{-m}.
x'Use in source · 21 passages
Search aliases: x'
Use in source · 04_correction_transport.tex, line 83
· ≤ (1-q)ρ+qρ =ρ. Also, Ψ(x)-Ψ(x') ≤ q x-x' . The contraction mapping theorem gives a unique fixed point in the stated set. Applying D to the fixed point identity gives the equation. Conversely, a solution in X_R equals R(Dx) and is therefore a fixed point of Ψ.
Exact mathematical passage
\|\Psi(x)-\Psi(x')\| \le q\|x-x'\|.
x^{(0)}Use in source · 1 passage
Search aliases: x^(0)
Use in source · 02_supplement_05.tex, line 58
Parallel stages. A second, independent definition is an explicit staged evaluator. Set x^(0)=0 and x^(k+1)_v= D_v^-1 (b_v-displaystyle∑_u→ vT_vux^(k)_u), · ℓ(v)=k+1,
Exact mathematical passage
x^{(0)}=0
x^{(3)}Use in source · 1 passage
Search aliases: x^(3)
Use in source · visual_atlas.tex, line 451
vfig[.93]shift_tail atlasCaptionA finite window of the actual sequence x[n]=2^-|n|. Outside every finite window, coefficients remain nonzero. The shaded window illustrates the truncation x^(3), not a periodic wraparound.
Exact mathematical passage
x^{(3)}
x^{(d)}Use in source · 1 passage
Search aliases: x^(d)
Use in source · 02_supplement_05.tex, line 72
[5pt] x^(k)_v, · ℓ(v)≠ k+1. It reads the previous-stage vector, not partially updated coordinates. The invariant is ℓ(v)≤ k Longrightarrow x^(k)_v=x_v. Every predecessor of a level-k+1 vertex has level at most k, so the recurrence uses only coordinates already proved correct. At the canonical depth d this yields Ax^(d)=b. The local independence theorem separately proves that changing coordinates outside the strict earlier levels cannot change a vertex's local solve.
Exact mathematical passage
\boxed{Ax^{(d)}=b.}
x^{(k)}Use in source · 6 passages
Search aliases: x^(k)
Use in source · 03_packets_and_graphs.tex, line 377
Let x^(k)=(x_n^(k)) be Cauchy in · _u. For every fixed n, x_n^(k)-x_n^(ℓ) ≤ x^(k)-x^(ℓ) _u, so x_n^(k) converges to some x_n∈ X_n because X_n is closed. Given ε>0, choose k_0 such that x^(k)-x^(ℓ) _u<ε (k,ℓ≥ k_0). For finite F and |a_n|≤1, pass ℓ→∞ in the finite sum to obtain ∑_n∈ Fa_n(x_n^(k)-x_n) ≤ε (k≥ k_0). Hence x^(k)-x _u≤ε once x is known to be unconditionally summable.
Exact mathematical passage
x^{(k)}=(x_n^{(k)})
x^{(k)}_uUse in source · 1 passage
Search aliases: x^(k)_u
Use in source · 02_supplement_05.tex, line 59
Parallel stages. A second, independent definition is an explicit staged evaluator. Set x^(0)=0 and x^(k+1)_v= D_v^-1 (b_v-displaystyle∑_u→ vT_vux^(k)_u), · ℓ(v)=k+1, [5pt] x^(k)_v, · ℓ(v)≠ k+1. It reads the previous-stage vector, not partially updated coordinates. The invariant is ℓ(v)≤ k Longrightarrow x^(k)_v=x_v. Every predecessor of a level-k+1 vertex has level at most k, so the recurrence uses only coordinates already proved correct. At the canonical depth d this yields Ax^(d)=b. The local …
Exact mathematical passage
x^{(k+1)}_v= \begin{cases} D_v^{-1}\!\left(b_v-\displaystyle\sum_{u\to v}T_{vu}x^{(k)}_u\right), &\ell(v)=k+1,\\[5pt] x^{(k)}_v,&\ell(v)\ne k+1. \end{cases}
x^{(k)}_vUse in source · 2 passages
Search aliases: x^(k)_v
Use in source · 02_supplement_05.tex, line 59
Parallel stages. A second, independent definition is an explicit staged evaluator. Set x^(0)=0 and x^(k+1)_v= D_v^-1 (b_v-displaystyle∑_u→ vT_vux^(k)_u), · ℓ(v)=k+1, [5pt] x^(k)_v, · ℓ(v)≠ k+1. It reads the previous-stage vector, not partially updated coordinates. The invariant is ℓ(v)≤ k Longrightarrow x^(k)_v=x_v. Every predecessor of a level-k+1 vertex has level at most k, so the recurrence uses only coordinates already proved correct. At the canonical depth d this yields Ax^(d)=b. The local …
Exact mathematical passage
x^{(k+1)}_v= \begin{cases} D_v^{-1}\!\left(b_v-\displaystyle\sum_{u\to v}T_{vu}x^{(k)}_u\right), &\ell(v)=k+1,\\[5pt] x^{(k)}_v,&\ell(v)\ne k+1. \end{cases}
x^{(k+1)}_vUse in source · 1 passage
Search aliases: x^(k+1)_v
Use in source · 02_supplement_05.tex, line 59
Parallel stages. A second, independent definition is an explicit staged evaluator. Set x^(0)=0 and x^(k+1)_v= D_v^-1 (b_v-displaystyle∑_u→ vT_vux^(k)_u), · ℓ(v)=k+1, [5pt] x^(k)_v, · ℓ(v)≠ k+1. It reads the previous-stage vector, not partially updated coordinates. The invariant is ℓ(v)≤ k Longrightarrow x^(k)_v=x_v. Every predecessor of a level-k+1 vertex has level at most k, so the recurrence uses only coordinates already proved correct. At the canonical depth d this yields Ax^(d)=b. The local …
Exact mathematical passage
x^{(k+1)}_v= \begin{cases} D_v^{-1}\!\left(b_v-\displaystyle\sum_{u\to v}T_{vu}x^{(k)}_u\right), &\ell(v)=k+1,\\[5pt] x^{(k)}_v,&\ell(v)\ne k+1. \end{cases}
x^{(N)}Use in source · 2 passages
Search aliases: x^(N)
Use in source · visual_atlas.tex, line 453
This x cannot be a finite linear combination of shifted impulses: it has infinite support. But if x^(N) agrees with x on |n|≤ N and is zero elsewhere, then normx-x^(N)_2^2 =2∑_n>N4^-n =frac23 4^-N⟶0. The exact squared errors for N=1,3,5 are 1/6, 1/96, and 1/1536. These are identities from a geometric series, not floating residuals.
Exact mathematical passage
x^{(N)}
x^{(\ell)}Use in source · 2 passages
Search aliases: x^(ℓ) ell
Use in source · 03_packets_and_graphs.tex, line 378
Let x^(k)=(x_n^(k)) be Cauchy in · _u. For every fixed n, x_n^(k)-x_n^(ℓ) ≤ x^(k)-x^(ℓ) _u, so x_n^(k) converges to some x_n∈ X_n because X_n is closed. Given ε>0, choose k_0 such that x^(k)-x^(ℓ) _u<ε (k,ℓ≥ k_0). For finite F and |a_n|≤1, pass ℓ→∞ in the finite sum to obtain ∑_n∈ Fa_n(x_n^(k)-x_n) ≤ε (k≥ k_0). Hence x^(k)-x _u≤ε once x is known to be unconditionally summable.
Exact mathematical passage
\|x_n^{(k)}-x_n^{(\ell)}\| \leq \|x^{(k)}-x^{(\ell)}\|_u,
X^*Use in source · 11 passages
Search aliases: X^*
Use in source · 03_packets_and_graphs.tex, line 237
[Dual packet systems] Let X be finite dimensional and let P=(P_α) be a finite packet system. The algebraic dual maps P_α^*:X^*→ X^*, P_α^*(φ)=φ∘ P_α, form a finite packet system on X^*. Its packet with index α is canonically isomorphic to X_α^*, and X^*≅⊕_α X_α^*. [source reference] ExportsDual observables, covectors, trace pairings, and adjoint packet calculations.
Exact mathematical passage
P_\alpha^*:X^*\to X^*, \qquad P_\alpha^*(\varphi)=\varphi\circ P_\alpha,
x^*Use in source · 2 passages
Search aliases: x^*
Use in source · 03_supplement_05.tex, line 135
Complex polarization reaches the whole operator space. Write q(x)=xx^*. The module proves the exact identity 4xy^*=q(x+y)-q(x-y) +i(q(x+iy)-q(x-iy)). Agreement on all positive pure outer products therefore implies agreement on every outer product. Actual matrix units then span the full matrix space. Applying this argument to the sandwich by P proves equality on every code-supported operator, including non-Hermitian ones.
Exact mathematical passage
q(x)=xx^*
X^{-1}Use in source · 1 passage
Search aliases: X^-1
Use in source · 02_strengthened.tex, line 1668
Use the inverse difference identity X^-1-Y^-1=X^-1(Y-X)Y^-1 with X=F_P(z) and Y=z-A.
Exact mathematical passage
X^{-1}-Y^{-1}=X^{-1}(Y-X)Y^{-1}
x^{-1}Use in source · 1 passage
Search aliases: x^-1
Use in source · 72_recovered.tex, line 235
[Finite inverse formula] If x has augmentation one and n=x-1, then x^-1 = ∑_r=0^p-1(-n)^r.
Exact mathematical passage
\boxed{ x^{-1} = \sum_{r=0}^{p-1}(-n)^r. }
x^{-2}Use in source · 1 passage
Search aliases: x^-2
Use in source · residue_continuation_body.tex, line 169
An actual Mellin endpoint. The counting functions are constant on the integer norm steps. Their measurability, and hence that of S_w, is proved from this fact. For zero-sum w, define F_w(h)=(1+h)∫_1^∞x^-2-hS_w(x) dx (h≥0). Since 1-1/d<1, the logarithmic first moment M_w=∫_1^∞(1+log x) absnx^-2S_w(x) dx is finite. Lean proves the explicit estimate absnF_w(h)-F_w(0)≤ hM_w for every h≥0, and therefore F_w(h)=F_w(0)+O(h) as hdownarrow0. The function here is the displayed integral of the actual ideal…
Exact mathematical passage
M_w=\int_1^{\infty}(1+\log x)\,\absn{x^{-2}S_w(x)}\,dx
x^{-2-h}Use in source · 1 passage
Search aliases: x^-2-h
Use in source · residue_continuation_body.tex, line 164
An actual Mellin endpoint. The counting functions are constant on the integer norm steps. Their measurability, and hence that of S_w, is proved from this fact. For zero-sum w, define F_w(h)=(1+h)∫_1^∞x^-2-hS_w(x) dx (h≥0). Since 1-1/d<1, the logarithmic first moment M_w=∫_1^∞(1+log x) absnx^-2S_w(x) dx is finite. Lean proves the explicit estimate absnF_w(h)-F_w(0)≤ hM_w for every h≥0, and therefore F_w(h)=F_w(0)+O(h) as hdownarrow0. The function here is the displayed integral of the actual ideal…
Exact mathematical passage
F_w(h)=(1+h)\int_1^{\infty}x^{-2-h}S_w(x)\,dx \qquad(h\geq0).
X^{1-1/d}Use in source · 4 passages
Search aliases: X^1-1/d
Use in source · residue_continuation_body.tex, line 138
Ordinary class characters: an unconditional power saving. The preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl( O)→ C, define the actual finite ideal sum S_w(X)=∑_substackI≠0
Exact mathematical passage
A_C(X)=c_KX+O_{K,C}(X^{1-1/d}),\qquad c_K=\frac{2^{r_1}(2\pi)^{r_2}R_K}{\#\mu(K)\sqrt{\absn{\Delta_K}}}.
X^2Use in source · 25 passages
Search aliases: X^2
Use in source · 30_lifting.tex, line 126
The affine statement is formalized by leanKomorebi.NativeAffine.finite_affine_obstruction. Positive depth is essential in the nonlinear theorem: for F(X)=X^2-1, q=2, L=0, and x=0, the Jacobian is zero and the residual class is nonzero, yet y=1 is a root modulo two in the same residue modulo one.
Exact mathematical passage
F(X)=X^2-1
X^3Use in source · 2 passages
Search aliases: X^3
Use in source · atlas-pages.tex, line 95
ordinates and columns are input coordinates. The upper-triangular minus signs record crossing the signed edge while moving from the column to the row. They are not arbitrary orientation choices. headTwo checks connect the matrix and the scalar equation The initial derivative matrix is the identity. Each initial jet (y(0),y'(0),y''(0)) therefore determines a unique linear combination. Also, frac13Tr(e^zJ_3)=D_3(z), det(XI-J_3)=X^3+1. The roots of -1 appearing later in the exponential sum are alre…
Exact mathematical passage
\frac13\Tr(e^{zJ_3})=D_3(z),\qquad \det(XI-J_3)=X^3+1.
x^4Use in source · 13 passages
Search aliases: x^4
Use in source · atlas-pages.tex, line 267
ymbol acts finitelyAPPELL POLYNOMIALSChapter IV: Theorem 9.6, Theorem 9.8, Proposition 9.9. For C_2,0(z)=sech z, define E_m(x) by sech z e^xz=∑ E_m(x)z^m/m!. The formal differential-operator formula is E_m(x)=sech(∂_x)x^m. Although the symbol is an infinite series, derivatives above degree m vanish. vfigappell_operator For m=4, only three terms survive: (1-frac∂_x^22!+5frac∂_x^44!)x^4 =x^4-6x^2+5. The four real roots of this polynomial are ±1,±sqrt5. They are not the zeros of the original analyt…
Exact mathematical passage
\left(1-\frac{\partial_x^2}{2!}+5\frac{\partial_x^4}{4!}\right)x^4 =x^4-6x^2+5.
x^dUse in source · 4 passages
Search aliases: x^d
Use in source · strengthening.tex, line 478
Here is the polynomial estimate that supplies one additional bit. For i≥1, put P_i(x)=4^ibinom4x4i. Its constant is zero, and its degree-d coefficient is [x^d]P_i(x)=frac4^i4^d s(4i,d)(4i)!, where s(m,d) is a signed Stirling number of the first kind. In each product contributing to s(4i,d), at most d-1 of the 2i-1 even integers in 1,…,4i-1 can be omitted. Thus v_2(s(4i,d))≥max(0,2i-d). Writing s_2(i) for the number of ones in the binary expansion of i, and using v_2((4i)!)=4i-s_2(i), gives v_2([…
Exact mathematical passage
[x^d]P_i(x)=\frac{4^i4^d s(4i,d)}{(4i)!}, \label{eq:scaled-binomial-coefficients}
x^eUse in source · 5 passages
Search aliases: x^e
Use in source · Paper_X_Block27_Amended.tex, line 582
[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.
Exact mathematical passage
x^e=0
x^iUse in source · 2 passages
Search aliases: x^i
Use in source · 05_worked_models.tex, line 82
Take the seed s=1 and let M_x,M_y be multiplication by x and y. Then M_x^iM_y^js=x^iy^j, so the algebra, regarded as a module over its multiplication algebra, is generated by one seed.
Exact mathematical passage
M_x^iM_y^js=x^iy^j,
X^mUse in source · 1 passage
Search aliases: X^m
Use in source · strengthening.tex, line 580
Expand X^m=∑_r=0^mT(m,r)P_r(X). Since XP_r(X)=P_r+1(X)+r(r+1)P_r(X), these integer coefficients satisfy T(m+1,r)=T(m,r-1)+r(r+1)T(m,r), T(0,0)=1. Every r(r+1) is even. Induction, including the boundary indices, proves 2^m-rmid T(m,r) for 0≤ r≤ m. For m>0 the r=0 term vanishes. Each remaining summand of L(P^m/m!) has valuation at least m-r+2v_2(r!)-v_2(m!) =r+s_2(m)-2s_2(r)≥ s_2(m)-1≥0. Here r≥2s_2(r)-1 follows from r≥2^s_2(r)-1 and 2^u≥2u for u≥1. This proves integrality and the stronger bound.
Exact mathematical passage
X^m=\sum_{r=0}^mT(m,r)P_r(X)
x^mUse in source · 4 passages
Search aliases: x^m
Use in source · Paper_IV_Traditional_Final.tex, line 1558
[Formal differential-operator representation on polynomials] Let a formal series A(z)=∑_q≥0a_qz^q/q! act on polynomials by A(∂_x)f(x) = ∑_q≥0fraca_qq!∂_x^qf(x). The power r is applied to the formal symbol before the substitution z=∂_x. Then E_m^(N,j,r)(x) = C_N,j(∂_x)^r x^m. The inverse relation is x^m = Φ_N,j(∂_x^N)^r E_m^(N,j,r)(x). Both sums are finite on every polynomial.
Exact mathematical passage
E_m^{(N,j,r)}(x) = \Cfun_{N,j}(\partial_x)^r x^m.
x^{m-Nn}Use in source · 1 passage
Search aliases: x^m-Nn
Use in source · Paper_IV_Traditional_Final.tex, line 1514
[Explicit Appell polynomial] For every m≥0, E_m^(N,j,r)(x) = ∑_0≤ n≤ m/N binommNn C_N,j,n(r)x^m-Nn.
Exact mathematical passage
E_m^{(N,j,r)}(x) = \sum_{0\leq n\leq m/N} \binom{m}{Nn} C_{N,j,n}(r)x^{m-Nn}.
X^NUse in source · 3 passages
Search aliases: X^N
Use in source · Paper_IV_Traditional_Final.tex, line 3034
6.1--6.6 · Quotient-algebra power bases and actual exponential-series convergence in the finite-dimensional coefficient topology. Both pointwise section-coordinate identities. Signed-shift powers, all matrix-exponential entries, group laws and trace are checked. The characteristic polynomial X^N+1, its distinct roots, exact spectrum and determinant one are proved.
Exact mathematical passage
X^N+1
X^nUse in source · 2 passages
Search aliases: X^n
Use in source · Paper_XI_Traditional_Final.tex, line 525
[Frobenius cycle polynomial] The Frobenius transfer matrix has characteristic polynomial X^n-a and determinant detτ(Φ)=(-1)^n-1a.
Exact mathematical passage
X^n-a
x^NUse in source · 4 passages
Search aliases: x^N
Use in source · Paper_IV_Traditional_Final.tex, line 866
The vectors e_0,J_Ne_0,…,J_N^N-1e_0 are the standard basis. Thus e_0 is cyclic for J_N, so the minimal polynomial of J_N has degree N. Since J_N^N=-I, that minimal polynomial divides x^N+1 and therefore equals x^N+1. The characteristic polynomial has the same degree and is divisible by the minimal polynomial, so it too equals x^N+1. This polynomial has distinct roots. Consequently J_N is diagonalizable and its spectrum consists of all N roots of -1, each once. Hence tr(e^zJ_N) = ∑_η^N=-1e^η z = …
Exact mathematical passage
x^N+1
x^nUse in source · 2 passages
Search aliases: x^n
Use in source · Paper_X_Block27_Amended.tex, line 1206
When pnmid n, the comparison with the finite cyclotomic sample is proved: q_L(γ_p)=σ_p, σ_p(ζ_n)=ζ_n^p. Reduction distinguishes n-th roots of unity at such a prime. Indeed, if x^n=y^n=1 and their reductions agree, the factorization of x^n-y^n gives (x-y)∑_i=0^n-1x^i y^n-1-i=0. The second factor reduces to noverline y^ n-1≠0, so the domain property gives x=y. This proves both Frobenius recognition on the cyclotomic generator and triviality of its inertia action. Consequently, every representation…
Exact mathematical passage
x^n=y^n=1
X^{N-1}Use in source · 1 passage
Search aliases: X^N-1
Use in source · dyadic_synthesis.tex, line 80
At an inductive field step d≥2, set n=2^d-1q_d, R=2^d-2q_d-1. Then n>2R and -2R>-n-q_1. BK Remark 2.2.7 gives the maximal field generated by the actual displayed matrix. Literal truncation and the filtered lift give the defining sequence. Nonzero native trace also proves separability: the irreducible characteristic polynomial has a nonzero coefficient of X^N-1, so its derivative is nonzero. See pathtower/DYADIC_AS_TOWER167.md for all field and filtration details.
Exact mathematical passage
X^{N-1}
X^pUse in source · 3 passages
Search aliases: X^p
Use in source · supplement_09.tex, line 62
Now take the integral cyclic group ring R_N= Z[ Z/N Z], x=[1], x^N=1, and assume Nmid p(p-1). Thus (x^p-1)^p=1. Each genuine consecutive block ∏_i=0^p-1(x-(pb+i)) belongs to (p,x^p-1-1). This follows from the actual polynomial identity ∏_i∈ F_p(X-i)=X^p-X modulo p, not from a hypothesized valuation of the product.
Exact mathematical passage
\prod_{i\in\F_p}(X-i)=X^p-X
X^{p-1}Use in source · 1 passage
Search aliases: X^p-1
Use in source · odd_tower.tex, line 58
Then v_E(z)=pa-qb=1, so z is a uniformizer and E=L(z). Its monic minimal polynomial f(X)=X^p+a_p-1X^p-1+⋯+a_0 is Eisenstein. It is an explicit polynomial: compute the product over the p conjugates
Exact mathematical passage
f(X)=X^p+a_{p-1}X^{p-1}+\cdots+a_0
x^{p-1}Use in source · 4 passages
Search aliases: x^p-1
Use in source · 20_arithmetic.tex, line 237
At the singular addresses c=ap, 1≤ a<m, use the weighted ring W_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping the falling factorial (x)_d=∏_i=0^d-1(x-i) into consecutive blocks of length p then gives an integral residue-coordinate factor p^v_p(lfloor d/prfloor!)…
Exact mathematical passage
I=(p,x^{p-1}-1)
X^{p^a}Use in source · 2 passages
Search aliases: X^p^a
Use in source · 72_recovered.tex, line 2373
r Editorial hypothesis correction. Require L≥1. At L=0 the standalone claim is false: for p=3, d=1 and X=2=1+B, one has X^3-1=7∉3 Z_3. With L≥1, every higher binomial term in the retained proof has the required valuation, including p=2. If X=I+p^LB with B∈ M_d( Z_p), then for every a≥0 there is B_a∈ M_d( Z_p) such that X^p^a=I+p^L+aB_a.
Exact mathematical passage
\boxed{X^{p^a}=I+p^{L+a}B_a.}
x^{p^{\ell-1}}Use in source · 1 passage
Search aliases: x^p^ℓ-1 ell
Use in source · supplement_05.tex, line 113
Uniqueness requires a lifting argument. The source proves the actual kernel statement bar x=0 Longleftrightarrow pmid x in Z/p^ℓ Z. For x≡ ypmod p, the checked ideal-power lifting theorem gives x^p^ℓ-1-y^p^ℓ-1∈(p)^ℓ=(0). If w^p-1=1, then w^p=w and hence w^p^k=w for every k. Therefore two roots with the same residue are equal. The decomposition is unique whenever its second factor reduces to one. Multiplicativity of both factors is also proved.
Exact mathematical passage
x^{p^{\ell-1}}-y^{p^{\ell-1}}\in(p)^\ell=(0).
X^rUse in source · 4 passages
Search aliases: X^r
Use in source · 03_nonlinear_coordinates.tex, line 208
R or C and let the packet projectors on both spaces be bounded. In the complex case, derivatives are complex Fr echet derivatives and the Taylor integral below has a real parameter. Assume that F:X→ Y is m times continuously Fr echet differentiable. The same local statements hold on open domains containing the points and segments in question. For 1≤ r≤ m, the derivative D^rF(x):X^r⟶ Y is a continuous symmetric multilinear map.
Exact mathematical passage
D^rF(x):X^r\longrightarrow Y
X^{r_i}Use in source · 1 passage
Search aliases: X^r_i
Use in source · Paper_XI_Traditional_Final.tex, line 674
A nonzero fixed vector must lie in an inertia trivial packet, so c_i=0. On that packet, Frobenius is a companion cycle with characteristic polynomial X^r_i-a_i. Its fixed space is the eigenspace for eigenvalue one. This eigenspace is one dimensional exactly when a_i=1, and it is zero otherwise. Summing over the refined packets proves the formula.
Exact mathematical passage
X^{r_i}-a_i
x^TUse in source · 1 passage
Search aliases: x^T
Use in source · pages.tex, line 308
1/2 · 1. Its exact eigenvalues are 1/2 and 3/2, giving condition number three. vfigpreconditioning headThe unit circles of the energy norms The left ellipse is x^TAx=1. Writing x=By gives the right ellipse y^T(BAB)y=1. The drawing samples these explicitly defined curves. The norm and eigenvalue equalities are checked symbolically.
Exact mathematical passage
x^TAx=1
X^{\underline q}Use in source · 1 passage
Search aliases: X^underline q underline
Use in source · root_scope.tex, line 140
ad J_r={q:λ_r(q)=μ_r}, G_r=2^-μ_rH_r. Since λ_r(0)=0 and λ_r(q)≥3q+1-s_2(r), every minimizing address satisfies 0≤ q≤lfloorfracs_2(r)-13rfloor. In particular μ_r≤0, and the minimum is found by a finite carry calculation. Reduction of the whole series gives overline G_r(X)=∑_q∈ J_rX^underline q in F_2[X]. Its highest degree is max J_r, with leading coefficient one. Thus the normalization is the actual Gauss normalization.
Exact mathematical passage
\overline G_r(X)=\sum_{q\in J_r}X^{\underline q} \quad\text{in }\mathbb F_2[X]. \label{eq:root:reduction}
x^\alphaUse in source · 6 passages
Search aliases: x^α alpha
Use in source · Paper_VI_Traditional_Final.tex, line 3969
It is enough to prove the formula on an eigenvector with eigenvalue x≥0. For x>0, the substitution s=xu gives fracsin(πα)π ∫_0^∞ s^α-1fracxs+x d s =x^α fracsin(πα)π ∫_0^∞fracu^α-11+u d u =x^α. The value at x=0 is zero. There are finitely many eigenvalues, so the operator integral converges. This is the finite spectral form of the fractional power construction in [source reference].
Exact mathematical passage
\frac{\sin(\pi\alpha)}{\pi} \int_0^\infty s^{\alpha-1}\frac{x}{s+x}\dd s =x^\alpha \frac{\sin(\pi\alpha)}{\pi} \int_0^\infty\frac{u^{\alpha-1}}{1+u}\dd u =x^\alpha.
x^\ellUse in source · 1 passage
Search aliases: x^ℓ ell
Use in source · strengthening.tex, line 1245
[Taylor coefficient bounds and radii] For j=0,1,2,3, the dyadic interpolation has an ordinary power series widetilde C_j(x)=1+∑_ℓ≥1d_j,ℓx^ℓ, v_2(d_j,ℓ)≥ℓ+β_j. Its Taylor series converges for |x|_2<2. At addresses zero and one, v_2(d_j,ℓ)=ℓ (ℓ≥1), R_0=R_1=2. At address two, v_2(d_2,ℓ)≥ max(2ℓ+lfloorfracℓ2rfloor, ℓ+3), R_2=2^5/2. The radius at address three is at least two. Its exact radius is not asserted here. The stated exact radii at addresses zero, one, and two hold at every center in Z_2.
Exact mathematical passage
\widetilde C_j(x)=1+\sum_{\ell\geq1}d_{j,\ell}x^\ell, \qquad v_2(d_{j,\ell})\geq\ell+\beta_j.
X_{(i,\cdot),(j,\cdot)}Use in source · 1 passage
Search aliases: X_(i,·),(j,·) cdot cdot
Use in source · 03_supplement_02.tex, line 206
To prove complete positivity rather than positivity alone, let J be an arbitrary finite auxiliary index set. For a matrix X indexed by J× n, define its blockwise amplification by [Φ^(J)(X)]_(i,u),(j,v) =Φ(X_(i,·),(j,·))_u,v. The explicit index calculation proves the identity Φ^(J)(X)=∑_a (I_J⊗ K_a)X(I_J⊗ K_a)^*. The displayed identity identifies the defined amplification with tensoring by the identity, which is the operation needed for complete positivity. Every summand on the right is positive …
Exact mathematical passage
[\Phi^{(J)}(X)]_{(i,u),(j,v)} =\Phi\bigl(X_{(i,\cdot),(j,\cdot)}\bigr)_{u,v}.
x_*Use in source · 1 passage
Search aliases: x_*
Use in source · 01_manuscript.tex, line 1217
[Preconditioned conjugate gradient convergence] Let A and M be Hermitian positive definite on a nonempty finite coordinate space, and set κ = cond_2(M^-1/2AM^-1/2). In exact arithmetic, the energy norm error of preconditioned conjugate gradient satisfies normx_m-x_*_A ≤ 2 ( fracsqrtκ-1sqrtκ+1 )^m normx_0-x_*_A.
Exact mathematical passage
\norm{x_m-x_*}_A \leq 2 \left( \frac{\sqrt\kappa-1}{\sqrt\kappa+1} \right)^m \norm{x_0-x_*}_A.
X_0Use in source · 4 passages
Search aliases: X_0
Use in source · 01_orientation.tex, line 56
A simple two packet model already contains all four notions. Let X=X_0⊕ X_1, D=D_0 · 0 0 · D_1, N=0 · N_01{ N_10 · 0. A finer family may split X_0 and X_1 into smaller summands. The blocks N_01 and N_10 may enter at different filtration weights. Over a fraction field, the rational packet spaces may fail to split the chosen lattice. A coarse operator on one selected summand may also fail to intertwine with the full matrix. The later sections isolate these four effects one at a time.
Exact mathematical passage
X=X_0\oplus X_1, \qquad D=\begin{pmatrix}D_0&0\\0&D_1\end{pmatrix}, \qquad N=\begin{pmatrix}0&N_{01}\\N_{10}&0\end{pmatrix}.
x_0Use in source · 21 passages
Search aliases: x_0
Use in source · visual_atlas.tex, line 477
headWhat one shift does in these coordinates Let S=S_1 and z=Sx. Substituting 3k+j into the shift definition gives z_1[k]=x_0[k], z_2[k]=x_1[k], z_0[k]=x_2[k-1]. The passage from phase 2 to phase 0 carries a shift of the coarse address. If Lh[k]=h[k-1], the exact packet matrix is S_packets=mat0 · 0 · L
Exact mathematical passage
z_1[k]=x_0[k],\qquad z_2[k]=x_1[k],\qquad z_0[k]=x_2[k-1].
X_{00}Use in source · 1 passage
Search aliases: X_00
Use in source · 02_coordinates.tex, line 348
[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coarse direct summands. Fine analysis recovers the four entries exactly. Nothing has been approximated.
Exact mathematical passage
X=X_{00}\oplus X_{01}\oplus X_{10}\oplus X_{11}.
x_{00}Use in source · 2 passages
Search aliases: x_00
Use in source · 02_coordinates.tex, line 357
[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coarse direct summands. Fine analysis recovers the four entries exactly. Nothing has been approximated.
Exact mathematical passage
x=(x_{00},x_{01},x_{10},x_{11})
X_{01}Use in source · 1 passage
Search aliases: X_01
Use in source · 02_coordinates.tex, line 348
[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coarse direct summands. Fine analysis recovers the four entries exactly. Nothing has been approximated.
Exact mathematical passage
X=X_{00}\oplus X_{01}\oplus X_{10}\oplus X_{11}.
x_{01}Use in source · 2 passages
Search aliases: x_01
Use in source · 02_coordinates.tex, line 357
[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coarse direct summands. Fine analysis recovers the four entries exactly. Nothing has been approximated.
Exact mathematical passage
x=(x_{00},x_{01},x_{10},x_{11})
x_0^{m-\ell}Use in source · 1 passage
Search aliases: x_0^m-ℓ ell
Use in source · strengthening.tex, line 1290
For j=0,1, the term m=ℓ in [source reference] has exact valuation ℓ, since s(ℓ,ℓ)=1 and the Newton valuation is exact. Every later term has valuation at least ℓ+1. Thus v_2(d_j,ℓ)=ℓ, and the coefficient radius is exactly two. At a center x_0∈ Z_2, the coefficient of y^ℓ in the translated series is ∑_m≥ℓd_j,mbinom mℓ x_0^m-ℓ. Its initial term again has exact valuation ℓ, and all later terms have at least one more bit. The exact radius is therefore two at every integral center.
Exact mathematical passage
\sum_{m\geq\ell}d_{j,m}\binom m\ell x_0^{m-\ell}. \label{eq:taylor-translation-coefficient}
x_1Use in source · 22 passages
Search aliases: x_1
Use in source · 03_packets_and_graphs.tex, line 1150
[title=Why mixed seeds change the answer] Let the acting algebra be diagonal on X=X_1⊕ X_2, with no transfer between the packets. Generation by packet pure seeds requires one seed in each source component, so psrank=2. A single mixed vector x_1+x_2 may generate both packets of dimension one under the diagonal action, so unrestricted seed rank can be 1.
Exact mathematical passage
x_1+x_2
X_{10}Use in source · 1 passage
Search aliases: X_10
Use in source · 02_coordinates.tex, line 348
[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coarse direct summands. Fine analysis recovers the four entries exactly. Nothing has been approximated.
Exact mathematical passage
X=X_{00}\oplus X_{01}\oplus X_{10}\oplus X_{11}.
x_{10}Use in source · 2 passages
Search aliases: x_10
Use in source · 02_coordinates.tex, line 357
[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coarse direct summands. Fine analysis recovers the four entries exactly. Nothing has been approximated.
Exact mathematical passage
x=(x_{00},x_{01},x_{10},x_{11})
X_{11}Use in source · 1 passage
Search aliases: X_11
Use in source · 02_coordinates.tex, line 348
[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coarse direct summands. Fine analysis recovers the four entries exactly. Nothing has been approximated.
Exact mathematical passage
X=X_{00}\oplus X_{01}\oplus X_{10}\oplus X_{11}.
x_{11}Use in source · 2 passages
Search aliases: x_11
Use in source · 02_coordinates.tex, line 357
[A two level split] Let X=X_00⊕ X_01⊕ X_10⊕ X_11. Assume that all four summands are nonzero. The four coordinate projections form a fine packet system. The two sums P_0=Q_00+Q_01, P_1=Q_10+Q_11 form a coarse packet system. If x=(x_00,x_01,x_10,x_11), the coarse coordinates are (x_00+x_01,x_10+x_11) when the sums are interpreted inside the two coarse direct summands. Fine analysis recovers the four entries exactly. Nothing has been approximated.
Exact mathematical passage
x=(x_{00},x_{01},x_{10},x_{11})
x_{14}Use in source · 1 passage
Search aliases: x_14
Use in source · visual_atlas.tex, line 46
vpage3Follow one coefficient through the towerMODEL A / LOSSLESS ADDRESS REFINEMENTChapter II, mathematical text: Theorem 3.3, Theorem 4.6 Take a vector x=(x_0,…,x_15). A packet is not a single summary number: it retains every coefficient at its selected addresses. For example, P_2,0x=(x_0,0,x_2,0,…,x_14,0). The rail below lists the addresses in each packet. Its display order groups descendants under their parents. The matrix example on Plate II.5 instead uses increasing residue order.
Exact mathematical passage
P_{2,0}x=(x_0,0,x_2,0,\ldots,x_{14},0).
x_{15}Use in source · 1 passage
Search aliases: x_15
Use in source · visual_atlas.tex, line 45
vpage3Follow one coefficient through the towerMODEL A / LOSSLESS ADDRESS REFINEMENTChapter II, mathematical text: Theorem 3.3, Theorem 4.6 Take a vector x=(x_0,…,x_15). A packet is not a single summary number: it retains every coefficient at its selected addresses. For example, P_2,0x=(x_0,0,x_2,0,…,x_14,0). The rail below lists the addresses in each packet. Its display order groups descendants under their parents. The matrix example on Plate II.5 instead uses increasing residue order.
Exact mathematical passage
x=(x_0,\ldots,x_{15})
x_1^2Use in source · 1 passage
Search aliases: x_1^2
Use in source · 03_nonlinear_coordinates.tex, line 195
[A mixed quadratic map] Let X=Y=𝕜^2 with coordinate packets. For F(x_1,x_2)=(x_1x_2,x_1^2+x_2), the first output packet depends on both input packets. The second output packet contains a quadratic contribution from the first input packet and a linear contribution from the second. Packet coordinates record these interactions exactly. Replacing F by the two diagonal restrictions would lose the product x_1x_2.
Exact mathematical passage
F(x_1,x_2)=(x_1x_2,x_1^2+x_2),
x_1^3Use in source · 1 passage
Search aliases: x_1^3
Use in source · atlas-pages.tex, line 184
r=exp (r∑_m≥1κ_N,mfracw^mm), the coefficient of w^n is a polynomial of degree n in r. Bell polynomials collect the possible ways logarithmic terms contribute total degree n. vfigbell_partitions For N=2, the first three logarithmic coefficients are κ_1=frac12, κ_2=frac16, κ_3=frac115. The third Bell polynomial is B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3. Substituting x_1=rκ_1, x_2=rκ_2, x_3=2rκ_3 and multiplying by 6!/3! gives P_2,3(r)=15r^3+30r^2+16r. At r=1 this is 61. At r=2 it is 272. At r=0 it van…
Exact mathematical passage
B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3
x_1^nUse in source · 1 passage
Search aliases: x_1^n
Use in source · Paper_IV_Traditional_Final.tex, line 1022
e identity Δ_N(w)^-r = exp( r∑_m≥1κ_N,mfracw^mm ) fits the defining generating function for B_n with x_m=r(m-1)!κ_N,m. The coefficient of w^n is therefore the displayed Bell polynomial divided by n!. Multiplication by (Nn)! gives P_N,n(r). Since Δ_N(w)=1-fracwN!+O(w^2), we have κ_N,1=1/N!. The term x_1^n gives the stated leading coefficient, which is nonzero.
Exact mathematical passage
x_1^n
X_1^{p^{n_1}}Use in source · 1 passage
Search aliases: X_1^p^n_1
Use in source · Paper_XII_Traditional_Final.tex, line 495
[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.
Exact mathematical passage
\boxed{ k[\Delta] \cong k[X_1,\ldots,X_r] \big/ (X_1^{p^{n_1}},\ldots,X_r^{p^{n_r}}). }
X_2Use in source · 9 passages
Search aliases: X_2
Use in source · 01_orientation.tex, line 105
[title=R4: a directed packet system] Let X=X_1⊕ X_2⊕ X_3. Suppose the generating operators permit transfers 1→2 and 2→3, with no path entering packet 1. Packet 1 is the unique source component. If the corner action on X_1 is cyclic and the propagation hypotheses of [source reference] hold, one seed in X_1 can generate all three packets.
Exact mathematical passage
X=X_1\oplus X_2\oplus X_3
x_2Use in source · 11 passages
Search aliases: x_2
Use in source · 03_packets_and_graphs.tex, line 1150
[title=Why mixed seeds change the answer] Let the acting algebra be diagonal on X=X_1⊕ X_2, with no transfer between the packets. Generation by packet pure seeds requires one seed in each source component, so psrank=2. A single mixed vector x_1+x_2 may generate both packets of dimension one under the diagonal action, so unrestricted seed rank can be 1.
Exact mathematical passage
x_1+x_2
x_2'Use in source · 1 passage
Search aliases: x_2'
Use in source · atlas-pages.tex, line 680
path, one-edge paths, and the two-edge path exist. The coefficient 3t^2 includes both the edge product 2·3 and the ordered-time simplex area t^2/2. headPath length counts transfers, not elapsed time Starting at e_0, the complete state is (1,2t,3t^2)^ T. No fourth packet or repeated cycle can contribute. Direct differentiation gives x'(t)=Ax(t), including the relation x_2'(t)=3x_1(t).
Exact mathematical passage
x_2'(t)=3x_1(t)
X_2^{a_2}Use in source · 1 passage
Search aliases: X_2^a_2
Use in source · monograph_atlas.tex, line 45
vpage3One seed, eight addressesCYCLICITY DEPENDS ON THE GRAMMARChapter VII: Proposition II.2.5. Start with ψ_0=ket000 and allow the three bit flip operators X_1,X_2,X_3. For a binary address a=(a_1,a_2,a_3), put W_a=X_1^a_1X_2^a_2X_3^a_3. W_aψ_0=keta_1a_2a_3. The drawing shows the actual eight states reached by these words. Dashed edges connect the two squares by changing the first bit. The other edges change one of the remaining bits. vfigaddresses headTwo different meanings of ``reach'' The ei…
Exact mathematical passage
W_a=X_1^{a_1}X_2^{a_2}X_3^{a_3}. \qquad W_a\psi_0=\ket{a_1a_2a_3}.
X_2^{\mathrm{wt}}Use in source · 1 passage
Search aliases: X_2^wt mathrm
Use in source · 50_open_questions.tex, line 95
The original proposed form is X≃ X_2^wt×_C∏_p>2'X_p. It leaves X, the distinguished subsets defining the restricted product, the maps to C, the ambient category, and the universal meaning of ``smallest'' unspecified. In particular, one must decide whether X is a global object or a completed object.
Exact mathematical passage
X\simeq X_2^{\mathrm{wt}}\times_C\prod_{p>2}'X_p.
x_{2m}Use in source · 1 passage
Search aliases: x_2m
Use in source · visual_atlas.tex, line 481
vpage20Filter first or split first: same outputMODEL F / A FINITE POLYPHASE REALIZATIONChapter II, mathematical text: Section B.1 Use the finitely supported signal x=(1,2,3,4,5,6) and the filter h=(1,2,1). Outside the displayed support, the input is zero. Let e_m=x_2m and o_m=x_2m+1.
Exact mathematical passage
e_m=x_{2m}
x_{2m+1}Use in source · 1 passage
Search aliases: x_2m+1
Use in source · visual_atlas.tex, line 481
vpage20Filter first or split first: same outputMODEL F / A FINITE POLYPHASE REALIZATIONChapter II, mathematical text: Section B.1 Use the finitely supported signal x=(1,2,3,4,5,6) and the filter h=(1,2,1). Outside the displayed support, the input is zero. Let e_m=x_2m and o_m=x_2m+1.
Exact mathematical passage
o_m=x_{2m+1}
X_3Use in source · 4 passages
Search aliases: X_3
Use in source · 01_orientation.tex, line 105
[title=R4: a directed packet system] Let X=X_1⊕ X_2⊕ X_3. Suppose the generating operators permit transfers 1→2 and 2→3, with no path entering packet 1. Packet 1 is the unique source component. If the corner action on X_1 is cyclic and the propagation hypotheses of [source reference] hold, one seed in X_1 can generate all three packets.
Exact mathematical passage
X=X_1\oplus X_2\oplus X_3
x_3Use in source · 4 passages
Search aliases: x_3
Use in source · atlas-pages.tex, line 184
r=exp (r∑_m≥1κ_N,mfracw^mm), the coefficient of w^n is a polynomial of degree n in r. Bell polynomials collect the possible ways logarithmic terms contribute total degree n. vfigbell_partitions For N=2, the first three logarithmic coefficients are κ_1=frac12, κ_2=frac16, κ_3=frac115. The third Bell polynomial is B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3. Substituting x_1=rκ_1, x_2=rκ_2, x_3=2rκ_3 and multiplying by 6!/3! gives P_2,3(r)=15r^3+30r^2+16r. At r=1 this is 61. At r=2 it is 272. At r=0 it van…
Exact mathematical passage
B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3
X_3^{a_3}Use in source · 1 passage
Search aliases: X_3^a_3
Use in source · monograph_atlas.tex, line 45
vpage3One seed, eight addressesCYCLICITY DEPENDS ON THE GRAMMARChapter VII: Proposition II.2.5. Start with ψ_0=ket000 and allow the three bit flip operators X_1,X_2,X_3. For a binary address a=(a_1,a_2,a_3), put W_a=X_1^a_1X_2^a_2X_3^a_3. W_aψ_0=keta_1a_2a_3. The drawing shows the actual eight states reached by these words. Dashed edges connect the two squares by changing the first bit. The other edges change one of the remaining bits. vfigaddresses headTwo different meanings of ``reach'' The ei…
Exact mathematical passage
W_a=X_1^{a_1}X_2^{a_2}X_3^{a_3}. \qquad W_a\psi_0=\ket{a_1a_2a_3}.
x_4Use in source · 1 passage
Search aliases: x_4
Use in source · pages.tex, line 249
, put two on the diagonal and -1 on each declared incoming edge. The edges are 0→2, 1→2, 1→3, 2→4, 3→4, 3→5, 4→6, and 5→6. Solve (2I+K)x=ones. vfigtriangular headRead the stages as dependencies Packets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance, x_4=frac1+x_2+x_32=frac118, x_6=frac1+x_4+x_52=frac138. All entries in the displayed solution are checked by multiplying the original triangular …
Exact mathematical passage
x_4=\frac{1+x_2+x_3}{2}=\frac{11}{8},\qquad x_6=\frac{1+x_4+x_5}{2}=\frac{13}{8}.
x_5Use in source · 1 passage
Search aliases: x_5
Use in source · pages.tex, line 249
, put two on the diagonal and -1 on each declared incoming edge. The edges are 0→2, 1→2, 1→3, 2→4, 3→4, 3→5, 4→6, and 5→6. Solve (2I+K)x=ones. vfigtriangular headRead the stages as dependencies Packets zero and one are ready first. Packets two and three use those values. Packets four and five then become available, followed by packet six. For instance, x_4=frac1+x_2+x_32=frac118, x_6=frac1+x_4+x_52=frac138. All entries in the displayed solution are checked by multiplying the original triangular …
Exact mathematical passage
x_4=\frac{1+x_2+x_3}{2}=\frac{11}{8},\qquad x_6=\frac{1+x_4+x_5}{2}=\frac{13}{8}.
x_6Use in source · 2 passages
Search aliases: x_6
Use in source · visual_atlas.tex, line 10
headAddress resolution A coefficient such as x_6 can sit in the even packet, then the residue-2 packet modulo 4, then the residue-6 packet modulo 8. Its value does not become more accurate. Its address becomes more specific, while the full object remains exactly reconstructible.
Exact mathematical passage
x_6
x_AUse in source · 1 passage
Search aliases: x_A
Use in source · worked_atlas.tex, line 100
AtlasHeadA derivative weight can cost one digit of root precisionatlas:weighted Compare the linear equations F(x)=3x=0 and G_A(x)=3x+3^A=0, where A≥2. Their coefficient difference has 3-adic valuation A, but their roots differ by x_A-0=-3^A-1, v_3(x_A)=A-1. AtlasImageweighted_motion The derivative has valuation one. Solving the correction equation divides by three, so one digit of coefficient precision is lost in root displacement. The calculation is exact for every displayed A and also follows …
Exact mathematical passage
x_A-0=-3^{A-1},\qquad v_3(x_A)=A-1.
X_bUse in source · 1 passage
Search aliases: X_b
Use in source · 02_connections_obstructions.tex, line 714
[Exact naturality for arbitrary functions] Let A^X_a:Xxrightarrow∼X_a and A^X_b:Xxrightarrow∼X_b be exact coordinate equivalences for X, with inverses S^X_a and S^X_b. Likewise, let A^Y_r:Yxrightarrow∼Y_r and A^Y_s:Yxrightarrow∼Y_s be exact coordinate equivalences for Y, with inverses S^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity or differentiability hypothesis. Set J^X_b← a= A^X_b∘ S^X_a, J^Y_s← r= A^Y_s∘ S^Y_r, and define widehat F_r← a = A^Y_r∘ F∘ S^X_a. Then J^Y_s← r∘widehat…
Exact mathematical passage
\cA^X_b:X\xrightarrow{\sim}X_b
X_cUse in source · 19 passages
Search aliases: X_c
Use in source · 05_defects.tex, line 5
Let X_f and X_c be vector spaces or Banach spaces over a common field, and let C:X_f⟶ X_c be a linear map, and let T_f∈End(X_f), T_c∈End(X_c). Define Δ=CT_f-T_cC.
Exact mathematical passage
X_c
x_cUse in source · 3 passages
Search aliases: x_c
Use in source · 03_nonlinear_coordinates.tex, line 822
[Fixed point stability] Assume F_c is a contraction with constant q<1. Let x_f and x_c be fixed points of F_f and F_c. If Δ_F(x_f) ≤ε, then Cx_f-x_c ≤ fracε1-q. When the defect at x_f is zero, the fixed points are compatible.
Exact mathematical passage
x_c
X_DUse in source · 1 passage
Search aliases: X_D
Use in source · 03_packets_and_graphs.tex, line 1097
a:X_α↠ X_β. Because X_α⊆ A S and A S is invariant under A, surjectivity implies X_β⊆ A S. For any other vertex γ∈ D, choose a directed path inside the strongly connected component from β to γ. Successive surjective edge blocks propagate the full packet space along the path, so X_γ⊆ A S. Hence X_D⊆ A S. Induction over the condensation graph gives X= A S, proving the upper bound.
Exact mathematical passage
X_D\subseteq\A S
X_{e+k}Use in source · 1 passage
Search aliases: X_e+k
Use in source · 30_lifting.tex, line 451
[Continuous charts with a selected Jacobian minor] Let p be prime, L≥1, F∈ Z[X_1,…,X_e+k]^e, and x∈ Z^e+k satisfy F(x)∈ p^L Z^e. Suppose the full reduced Jacobian is surjective. There is a permutation σ of the coordinates such that, writing σ x=(y_0,t_0) with e solved coordinates and k parameter coordinates, the parameter ball T=t_0+p^L Z_p^k is homeomorphic to the entire root locus R={z∈ B_L(x):F(z)=0}. The inverse homeomorphism is the actual free-coordinate projection z↦free(σ z). If h:T→ R is…
Exact mathematical passage
F\in\Z[X_1,\ldots,X_{e+k}]^e
X_{e+q}Use in source · 1 passage
Search aliases: X_e+q
Use in source · continuation_05.tex, line 8
Let p be prime, L≥1, e,q≥0, and F∈ Z[X_1,…,X_e+q]^e, x∈ Z^e+q, F(x)≡0pmodp^L. Assume the actual reduced Jacobian A_x=DF(x)bmod p: F_p^e+q⟶ F_p^e is surjective. All vector congruences are coordinatewise.
Exact mathematical passage
F\in\mathbb Z[X_1,\ldots,X_{e+q}]^e,\qquad x\in\mathbb Z^{e+q},\qquad F(x)\equiv0\pmod{p^L}.
X_fUse in source · 17 passages
Search aliases: X_f
Use in source · 05_defects.tex, line 5
Let X_f and X_c be vector spaces or Banach spaces over a common field, and let C:X_f⟶ X_c be a linear map, and let T_f∈End(X_f), T_c∈End(X_c). Define Δ=CT_f-T_cC.
Exact mathematical passage
X_f
x_FUse in source · 2 passages
Search aliases: x_F
Use in source · Paper_V_Traditional_Final.tex, line 291
The shift orbit is the standard orthonormal basis {δ_n}_n∈ Z. For x∈ℓ^2, the finite truncations x_F=∑_n∈ Fx[n]δ_n, F⊂ Z finite, satisfy x-x_F _2^2=∑_n∉ F|x[n]|^2→0 as the finite sets exhaust Z. Thus the unordered Hilbert-space sum ∑_nx[n]δ_n converges to x, and the finite-support statement is exact. The shift is a surjective isometry because n↦ n-1 is a bijection of Z, with inverse shift n↦ n+1. This proves the closed-span assertion.
Exact mathematical passage
x_F=\sum_{n\in F}x[n]\delta_n,\qquad F\subset\bbZ\text{ finite},
x_fUse in source · 5 passages
Search aliases: x_f
Use in source · 03_nonlinear_coordinates.tex, line 822
[Fixed point stability] Assume F_c is a contraction with constant q<1. Let x_f and x_c be fixed points of F_f and F_c. If Δ_F(x_f) ≤ε, then Cx_f-x_c ≤ fracε1-q. When the defect at x_f is zero, the fixed points are compatible.
Exact mathematical passage
x_f
x_{g,a,b}Use in source · 2 passages
Search aliases: x_g,a,b
Use in source · Paper_XII_Traditional_Final.tex, line 1486
Finally, put B_λ= O_λ[x_g,a,b:g∈Γ, a,b∈ D]. For each declared local condition index u, supply an ideal J_λ,u⊆ B_λ. Evaluation at a representation sends x_g,a,b to its (a,b) matrix entry at g. Assume that evaluation at every admissible finite representative kills J_λ,u, with polynomial coefficients reduced modulo I_λ^n. For this class based theorem, the equations must hold in every admissible frame. A condition imposed only in one chosen frame needs a separate compatible frame argument. No arbitr…
Exact mathematical passage
B_\lambda=\OO_\lambda[x_{g,a,b}:g\in\Gamma,\ a,b\in D].
X_iUse in source · 10 passages
Search aliases: X_i
Use in source · 01_finite_coordinates.tex, line 144
[Current fiber scope] The compiled Chapter III model above has one scalar coordinate per packet. For a family of modules (X_i)_i∈ V, an intrinsic edge map would instead have type ℓ_e:X_src(e)→ X_tgt(e), and its totalization on ⊕_iX_i would be ι_tgt(e)ℓ_eπ_src(e). The same support proof applies, but this generalization with dependent fibers is not attributed to the released Chapter III modules for scalar edge systems.
Exact mathematical passage
(X_i)_{i\in V}
x_iUse in source · 26 passages
Search aliases: x_i
Use in source · visual_atlas.tex, line 512
vpage21Tensor products require every cross termOPERATIONS ON SEED RANK / A RECTANGLE, NOT A DIAGONALMain text: [source reference]. Algebraic tensor products If (x_i) generates X under A and (y_j) generates Y under B, then the candidate tensor seeds are all x_i⊗ y_j. For finite expansions, x=∑_i A_ix_i, y=∑_j B_jy_j, bilinearity gives the required double sum: x⊗ y=∑_i∑_j(A_i⊗ B_j)(x_i⊗ y_j).
Exact mathematical passage
(x_i)
x_i^2Use in source · 2 passages
Search aliases: x_i^2
Use in source · atlas-pages.tex, line 424
vpage19Convergence with a nonzero exact errorFINITE DIFFERENCES / A QUARTIC TEST PROBLEMChapter VI, Part I: Theorems 11.1--11.2. Exact nodal inversion and the scale of low-mode errors. Use u(x)=x-x^4, which vanishes at both endpoints and solves -u''=12x^2. Its centered second difference is not exact: L_hu|_rm nodes=12x_i^2+2h^2. Since the stencil differentiates the quadratic x(1-x) exactly, the solution u_h to the sampled forcing 12x_i^2 satisfies u_h(x_i)-u(x_i)=-h^2x_i(1-x_i). This gives both …
Exact mathematical passage
L_hu|_{\rm nodes}=12x_i^2+2h^2.
X_i^{p^{n_i}}Use in source · 1 passage
Search aliases: X_i^p^n_i
Use in source · Paper_XII_Traditional_Final.tex, line 509
In characteristic p, (1+X_i)^p^n_i-1=X_i^p^n_i. Thus the assignment X_i↦γ_i-1 gives a surjective homomorphism from the displayed truncated polynomial ring to k[Δ]. Both sides have dimension ∏_ip^n_i=absΔ over k, so the map is an isomorphism. The quotient by the ideal generated by the X_i is k. Since each X_i is nilpotent, that ideal is the unique maximal ideal.
Exact mathematical passage
(1+X_i)^{p^{n_i}}-1=X_i^{p^{n_i}}.
X_{ij}Use in source · 1 passage
Search aliases: X_ij
Use in source · 02_supplement_04.tex, line 77
Kronecker application. For A∈ C^m× m, B∈ C^n× n and X∈ C^n× m, Lean proves (A⊗ B)vec(X)=vec(BXA^T). The formal vectorization is explicitly column-major: its pair coordinate (j,i) stores X_ij. The right factor is the ordinary transpose, not the conjugate transpose. This distinction is essential over C.
Exact mathematical passage
X_{ij}
X_j^{a_j}Use in source · 1 passage
Search aliases: X_j^a_j
Use in source · 03_supplement_02.tex, line 21
For every binary address a, W_a=∏_j=0^n-1X_j^a_j=T_a, W_aψ_0=ket a. The family (W_aψ_0)_a∈ A_n is linearly independent, spans V_n, and has exactly 2^n elements. It is the standard computational basis.
Exact mathematical passage
W_a=\prod_{j=0}^{n-1}X_j^{a_j}=T_a, \qquad W_a\psi_0=\ket a.
X_{k\ell}Use in source · 1 passage
Search aliases: X_kℓ ell
Use in source · 01_manuscript.tex, line 606
The (i,j) entry of BXA^T is ∑_k=1^n∑_ℓ=1^m B_ikX_kℓA_jℓ. This is the coordinate formula for (A⊗ B)vec(X). Multiplying first by B and then by A^T gives the stated count.
Exact mathematical passage
\sum_{k=1}^{n}\sum_{\ell=1}^{m} B_{ik}X_{k\ell}A_{j\ell}.
x_{L+1}Use in source · 2 passages
Search aliases: x_L+1
Use in source · 72_recovered.tex, line 2552
[One level linear lifting] A class represented by x_L lifts to a solution modulo p^L+1 if and only if Ob^lin_L(x_L)=0. More precisely, a correction y∈ F_p^d gives a lift x_L+1=x_L+p^Ly pmod p^L+1 if and only if overline My=-ρ_L(x_L). When one correction exists, the set of all correction classes is a coset of keroverline M.
Exact mathematical passage
x_{L+1}=x_L+p^Ly \pmod {p^{L+1}}
x_m'Use in source · 3 passages
Search aliases: x_m'
Use in source · 04_correction_transport.tex, line 364
[Naturality of filtered obstructions] Suppose a morphism of filtered lifting problems sends x_m to x_m', intertwines L_m with L_m', and sends Φ(x_m) to Φ'(x_m') on the associated graded pieces. Then the induced cokernel map sends Ob_m(x_m) to Ob_m'(x_m').
Exact mathematical passage
x_m'
X_{m+1}Use in source · 1 passage
Search aliases: X_m+1
Use in source · Paper_X_Block27_Amended.tex, line 250
Fix bar x∈ M(k). For m≥1, set X_m(bar x)= M_bar x( O/ϖ^m). Reduction gives transition maps r_m+1,m:X_m+1(bar x)⟶ X_m(bar x). A compatible branch is a sequence (x_m)_m≥1 satisfying r_m+1,m(x_m+1)=x_m.
Exact mathematical passage
r_{m+1,m}:X_{m+1}(\bar x)\longrightarrow X_m(\bar x).
x_{m+1}Use in source · 6 passages
Search aliases: x_m+1
Use in source · 04_correction_transport.tex, line 319
Choose representatives recursively. Once x_m represents the selected depth m vertex, choose a representative x_m+1 of its selected child with x_m+1-x_m∈ F^mX. The resulting sequence is Cauchy for the filtration topology. Each representative satisfies Φ(x_m)∈ F^mY. The conclusion now follows from [source reference].
Exact mathematical passage
x_{m+1}
x_{m-1}Use in source · 1 passage
Search aliases: x_m-1
Use in source · Paper_X_Block27_Amended.tex, line 1978
For canonical representatives 0≤ x_m<p^m, set x_0=0. Compatibility also gives a_m-1=(x_m-x_m-1)/p^m-1∈{0,…,p-1}. The floor in the first formula is essential. For example, the compatible base-five residues 4,19 give the second digit 3, not the rational number 19/5. The executable companion uses the same integer floor division.
Exact mathematical passage
a_{m-1}=(x_m-x_{m-1})/p^{m-1}\in\{0,\ldots,p-1\}
x_{n+1}Use in source · 2 passages
Search aliases: x_n+1
Use in source · 10_foundations.tex, line 436
begingroupsmallraggedright Formal correspondence. leanCompanion.companion_jump, leanCompanion.packet_charpoly_recurrence, and leanCompanion.packet_observable_recurrence. The source-sequence versions derive the packet recurrence from the relation x_n+1=Cx_n. For the Fibonacci matrix C=(1 · 1
Exact mathematical passage
x_{n+1}=Cx_n
x_{n+2}Use in source · 1 passage
Search aliases: x_n+2
Use in source · 72_recovered.tex, line 2857
[Residue subsequences of a second order recurrence] Suppose x_n+2=Px_n+1-Qx_n over a commutative ring. Define V_0=2, V_1=P, and V_n+2=PV_n+1-QV_n. For every positive integer N and every address j, the subsequence u_k=x_Nk+j satisfies u_k+2=V_Nu_k+1-Q^Nu_k.
Exact mathematical passage
x_{n+2}=Px_{n+1}-Qx_n
x_{N-1}Use in source · 3 passages
Search aliases: x_N-1
Use in source · Paper_V_Traditional_Final.tex, line 351
Define the analysis and synthesis maps A_Nx=(x_0,…,x_N-1), S_N(x_0,…,x_N-1)[Nq+r]=x_r[q].
Exact mathematical passage
\cA_Nx=(x_0,\ldots,x_{N-1}), \qquad \cS_N(x_0,\ldots,x_{N-1})[Nq+r]=x_r[q].
x_{n-r}Use in source · 1 passage
Search aliases: x_n-r
Use in source · 06_models.tex, line 23
A banded operator need not remain banded after residue reordering, but its nonzero couplings become a structured packet graph. With the convention (S_rx)_n=x_n-r, the only active transfer is P_N,j+rS_rP_N,j, with indices taken modulo N. More general finite stencils produce a finite set of active phase differences.
Exact mathematical passage
(S_rx)_n=x_{n-r}
x_n^{(k)}Use in source · 5 passages
Search aliases: x_n^(k)
Use in source · 03_packets_and_graphs.tex, line 377
Let x^(k)=(x_n^(k)) be Cauchy in · _u. For every fixed n, x_n^(k)-x_n^(ℓ) ≤ x^(k)-x^(ℓ) _u, so x_n^(k) converges to some x_n∈ X_n because X_n is closed. Given ε>0, choose k_0 such that x^(k)-x^(ℓ) _u<ε (k,ℓ≥ k_0). For finite F and |a_n|≤1, pass ℓ→∞ in the finite sum to obtain ∑_n∈ Fa_n(x_n^(k)-x_n) ≤ε (k≥ k_0). Hence x^(k)-x _u≤ε once x is known to be unconditionally summable.
Exact mathematical passage
x^{(k)}=(x_n^{(k)})
x_n^{(\ell)}Use in source · 1 passage
Search aliases: x_n^(ℓ) ell
Use in source · 03_packets_and_graphs.tex, line 378
Let x^(k)=(x_n^(k)) be Cauchy in · _u. For every fixed n, x_n^(k)-x_n^(ℓ) ≤ x^(k)-x^(ℓ) _u, so x_n^(k) converges to some x_n∈ X_n because X_n is closed. Given ε>0, choose k_0 such that x^(k)-x^(ℓ) _u<ε (k,ℓ≥ k_0). For finite F and |a_n|≤1, pass ℓ→∞ in the finite sum to obtain ∑_n∈ Fa_n(x_n^(k)-x_n) ≤ε (k≥ k_0). Hence x^(k)-x _u≤ε once x is known to be unconditionally summable.
Exact mathematical passage
\|x_n^{(k)}-x_n^{(\ell)}\| \leq \|x^{(k)}-x^{(\ell)}\|_u,
X_N^{\rm pkt}Use in source · 1 passage
Search aliases: X_N^rm pkt rm
Use in source · visual_atlas.tex, line 79
Define X_N^rm pkt=⊕_j P_N,jX, A_Nx=(P_N,jx)_j, S_N(x_j)_j=∑_jx_j. Completeness and mutual annihilation give both inverse laws. Consequently, J= A_4 S_2, C= A_2 S_4, CJ=I, JC=I. The larger list of fine packets has smaller individual packet spaces.
Exact mathematical passage
X_N^{\rm pkt}=\bigoplus_j P_{N,j}X, \qquad \mathcal A_Nx=(P_{N,j}x)_j, \qquad \mathcal S_N(x_j)_j=\sum_jx_j.
x_{Nk+j}Use in source · 2 passages
Search aliases: x_Nk+j
Use in source · 10_foundations.tex, line 411
[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalization hypothesis is required.
Exact mathematical passage
b_k=x_{Nk+j}
X_PUse in source · 1 passage
Search aliases: X_P
Use in source · Paper_VI_Traditional_Final.tex, line 4460
Let X=X_P⊕ X_Q and write A= A_PP · A_PQ
Exact mathematical passage
X=X_P\oplus X_Q
x_PUse in source · 6 passages
Search aliases: x_P
Use in source · Paper_VI_Traditional_Final.tex, line 1513
Solve the hidden block equation (zI-A_QQ)x_Q=A_QPx_P+y_Q for x_Q and substitute it into the retained equation. The resulting retained coefficient is F_P(z). Invertibility of the full operator and of the hidden block implies invertibility of this Schur complement. Setting y_Q=0 gives the stated identity as an operator on Ran P. Equivalently, this is the block inverse formula.
Exact mathematical passage
(zI-A_{QQ})x_Q=A_{QP}x_P+y_Q
x_pUse in source · 2 passages
Search aliases: x_p
Use in source · Paper_X_Block27_Amended.tex, line 1803
An admissible realization here includes continuous coefficient-field members and membership of every finite reduction in its designated admissible class. Under the completion hypotheses and these supplied carrier hypotheses, Lean proves · ∃ ρ, (x_p)_p: ρ is an admissible realization of the full chosen spine, 0≠ x_p∈im(e_p) for every p · Longleftrightarrow (∀λ, X_λ,1( C)≠∅) land (∀λ, ∀ m≥2, X_λ,m( C)≠∅) land (∀ p, n_p>0). The packet index may be infinite. Each x_p belongs to its own supplied carr…
Exact mathematical passage
\begin{aligned} &\exists\,\rho,\ (x_p)_p:\quad \rho\text{ is an admissible realization of the full chosen spine},\quad 0\ne x_p\in\operatorname{im}(e_p)\text{ for every }p\\ &\quad\Longleftrightarrow\quad \bigl(\forall\lambda,\ X_{\lambda,1}(\Compat)\ne\varnothing\bigr) \ \land\ \bigl(\forall\lambda,\ \forall m\ge2, X_{\lambda,m}(\Compat)\ne\varnothing\bigr) \ \land\ \bigl(\forall p,\ n_p>0\bigr). \end{aligned}
X_qUse in source · 1 passage
Search aliases: X_q
Use in source · 02_connections_obstructions.tex, line 889
For coordinate models of the source and target colors, an additive target coordinate space, and a chosen noncanonical connection, a pointwise node defect can be defined by D^Γ_b,a(Φ) = Γ^Y_b← a∘widehatΦ_a - widehatΦ_b∘ (∏_qΓ^X_q_b← a). This expression compares two chosen routes. With canonical exact transport in every slot it vanishes by [source reference].
Exact mathematical passage
\mathfrak D^\Gamma_{b,a}(\Phi) = \Gamma^Y_{b\leftarrow a}\circ\widehat\Phi_a - \widehat\Phi_b\circ \left(\prod_q\Gamma^{X_q}_{b\leftarrow a}\right).
x_QUse in source · 7 passages
Search aliases: x_Q
Use in source · Paper_VI_Traditional_Final.tex, line 1513
Solve the hidden block equation (zI-A_QQ)x_Q=A_QPx_P+y_Q for x_Q and substitute it into the retained equation. The resulting retained coefficient is F_P(z). Invertibility of the full operator and of the hidden block implies invertibility of this Schur complement. Setting y_Q=0 gives the stated identity as an operator on Ran P. Equivalently, this is the block inverse formula.
Exact mathematical passage
(zI-A_{QQ})x_Q=A_{QP}x_P+y_Q
X_RUse in source · 12 passages
Search aliases: X_R
Use in source · 04_correction_transport.tex, line 43
[Right inverse correction theorem] Let D:X→ Y be a bounded linear map between Banach spaces. Assume Z=im D is closed and there is a bounded linear right inverse R:Z⟶ X, DR=I_Z. Put X_R=R(Z). Let N:overline B_ρ(0)∩ X_R⟶ Z be L-Lipschitz. Let y∈ Z and put q= R L. Assume q<1 and R(y-N(0)) ≤ (1-q)ρ. Then the equation Dx+N(x)=y has a unique solution in overline B_ρ(0)∩ X_R.
Exact mathematical passage
X_R=R(Z)
X_rUse in source · 5 passages
Search aliases: X_r
Use in source · Paper_XII_Traditional_Final.tex, line 495
[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.
Exact mathematical passage
\boxed{ k[\Delta] \cong k[X_1,\ldots,X_r] \big/ (X_1^{p^{n_1}},\ldots,X_r^{p^{n_r}}). }
x_rUse in source · 11 passages
Search aliases: x_r
Use in source · Paper_V_Traditional_Final.tex, line 322
Fix an integer N≥1. For 0≤ r<N, define (P_N,rx)[n] = 1_n≡ r (mod N)x[n]. The corresponding decimated channel is x_r[q]=x[Nq+r].
Exact mathematical passage
x_r[q]=x[Nq+r].
x_{r-1}Use in source · 1 passage
Search aliases: x_r-1
Use in source · Paper_V_Traditional_Final.tex, line 3579
[Quadratic nearest neighbor interaction] For y(n)=x(n)x(n-1) and modulus N, y_r(q) = x_r(q)x_r-1(q), · r≥1, x_0(q)x_N-1(q-1), · r=0. The carry at residue zero is the nonlinear analogue of a transfer crossing a coarse cell boundary.
Exact mathematical passage
y_r(q) = \begin{cases} x_r(q)x_{r-1}(q),&r\geq1,\\ x_0(q)x_{N-1}(q-1),&r=0. \end{cases}
X_r^{p^{n_r}}Use in source · 1 passage
Search aliases: X_r^p^n_r
Use in source · Paper_XII_Traditional_Final.tex, line 495
[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.
Exact mathematical passage
\boxed{ k[\Delta] \cong k[X_1,\ldots,X_r] \big/ (X_1^{p^{n_1}},\ldots,X_r^{p^{n_r}}). }
x_{r_a}Use in source · 1 passage
Search aliases: x_r_a
Use in source · Paper_V_Traditional_Final.tex, line 1839
[Translation covariance] Fix a∈ Z^d. For every r∈ R, write uniquely r-a=r_a+λ_a,r, r_a∈ R, λ_a,r∈ L. Then ( A_LT_ax)_r(ℓ) = x_r_a(ℓ+λ_a,r). Thus a fine translation permutes the lattice packets and applies an internal translation on L.
Exact mathematical passage
(\Acal_LT_ax)_r(\ell) = x_{r_a}(\ell+\lambda_{a,r}).
x_{r_j}Use in source · 1 passage
Search aliases: x_r_j
Use in source · Paper_V_Traditional_Final.tex, line 3545
0, · r≥ b_j. Then the output channel is y_r(q) = ∏_j=1^m x_r_j (q-c_j-κ_r,b_j).
Exact mathematical passage
\boxed{ y_r(q) = \prod_{j=1}^{m} x_{r_j} \bigl(q-c_j-\kappa_{r,b_j}\bigr). }
x_{r_k}Use in source · 1 passage
Search aliases: x_r_k
Use in source · Paper_V_Traditional_Final.tex, line 512
Original-sample delay. For an arbitrary original-sample delay D∈ Z, the output phase k has y_k[n]=x_r_k[n+q_k], r_k=(k-D)bmod N, q_k=(k-D)mathbindivN. Its channel matrix has the sole nonzero entry z^q_k in row k, column r_k. In particular, a one-sample delay at N=2 has matrix 0 · z^-1
Exact mathematical passage
y_k[n]=x_{r_k}[n+q_k],\qquad r_k=(k-D)\bmod N,\qquad q_k=(k-D)\mathbin{\mathrm{div}}N.
X_sUse in source · 4 passages
Search aliases: X_s
Use in source · Paper_V_Traditional_Final.tex, line 438
[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.
Exact mathematical passage
X_s(z)=\sum_nx_s[n]z^{-n}
x_sUse in source · 9 passages
Search aliases: x_s
Use in source · Paper_V_Traditional_Final.tex, line 426
[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.
Exact mathematical passage
\boxed{ y_k[n] = \sum_{s=0}^{N-1}\sum_{m\in\bbZ} h_{ks}[m]x_s[n-m]. }
X_{\src(e)}Use in source · 1 passage
Search aliases: X_src(e) src operatorname
Use in source · 01_finite_coordinates.tex, line 145
[Current fiber scope] The compiled Chapter III model above has one scalar coordinate per packet. For a family of modules (X_i)_i∈ V, an intrinsic edge map would instead have type ℓ_e:X_src(e)→ X_tgt(e), and its totalization on ⊕_iX_i would be ι_tgt(e)ℓ_eπ_src(e). The same support proof applies, but this generalization with dependent fibers is not attributed to the released Chapter III modules for scalar edge systems.
Exact mathematical passage
\ell_e:X_{\src(e)}\to X_{\tgt(e)}
X_tUse in source · 2 passages
Search aliases: X_t
Use in source · Paper_VI_Traditional_Final.tex, line 4381
[Continuous time finite state formula] Let Q be a finite Markov generator and let (X_t)_t≥0 be the corresponding chain. Then ( e^t(Q^T-D_V)f )_i = E_i[ exp( -∫_0^tV(X_s) d s ) f(X_t) ]. If V≥ v_0, then norm e^t(Q^T-D_V)_∞→∞ ≤ e^-v_0t.
Exact mathematical passage
(X_t)_{t\geq0}
X_{\tgt(e)}Use in source · 1 passage
Search aliases: X_tgt(e) tgt operatorname
Use in source · 01_finite_coordinates.tex, line 145
[Current fiber scope] The compiled Chapter III model above has one scalar coordinate per packet. For a family of modules (X_i)_i∈ V, an intrinsic edge map would instead have type ℓ_e:X_src(e)→ X_tgt(e), and its totalization on ⊕_iX_i would be ι_tgt(e)ℓ_eπ_src(e). The same support proof applies, but this generalization with dependent fibers is not attributed to the released Chapter III modules for scalar edge systems.
Exact mathematical passage
\ell_e:X_{\src(e)}\to X_{\tgt(e)}
X_\alpha^*Use in source · 4 passages
Search aliases: X_α^* alpha
Use in source · 03_packets_and_graphs.tex, line 242
[Dual packet systems] Let X be finite dimensional and let P=(P_α) be a finite packet system. The algebraic dual maps P_α^*:X^*→ X^*, P_α^*(φ)=φ∘ P_α, form a finite packet system on X^*. Its packet with index α is canonically isomorphic to X_α^*, and X^*≅⊕_α X_α^*. [source reference] ExportsDual observables, covectors, trace pairings, and adjoint packet calculations.
Exact mathematical passage
X_\alpha^*
X_{\alpha_i}Use in source · 1 passage
Search aliases: X_α_i alpha
Use in source · 03_nonlinear_coordinates.tex, line 352
[Packet Fa a di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n]
Exact mathematical passage
h_i\in X_{\alpha_i}
X_{\alpha_j}Use in source · 1 passage
Search aliases: X_α_j alpha
Use in source · 03_nonlinear_coordinates.tex, line 277
For h_j∈ X_α_j, the inherited norm gives Q_γ D^rF(x)[h_1,…,h_r] ≤ Q_γ D^rF(x) ∏_j h_j . Taking the multilinear operator norm proves the sharper restricted estimate. For ambient inputs, replace h_j by P_α_jh_j and apply the operator norm bound to each projector. Every nonzero idempotent has norm at least one, since it fixes a nonzero vector in its range. The displayed product bound therefore also follows for the restricted jet. If a zero packet is retained as a redundant index, multilinearity mak…
Exact mathematical passage
h_j\in X_{\alpha_j}
x_\betaUse in source · 4 passages
Search aliases: x_β beta
Use in source · 03_packets_and_graphs.tex, line 114
For x∈ X, S_ P A_ Px = ∑_α P_α x =x. If x_α∈ X_α, then P_β x_α=δ_αβx_α, so A_ P S_ P(x_α)_α = (P_β∑_α x_α)_β =(x_β)_β. Thus analysis and synthesis are inverse linear maps.
Exact mathematical passage
\mathcal A_{\Pcal}\mathcal S_{\Pcal}(x_\alpha)_\alpha = \left(P_\beta\sum_\alpha x_\alpha\right)_\beta =(x_\beta)_\beta.
X_\beta^*Use in source · 1 passage
Search aliases: X_β^* beta
Use in source · 03_packets_and_graphs.tex, line 829
Conversely, suppose transitivity fails. Choose α,β,γ with (β,α)∈ R, (γ,β)∈ R, (γ,α)∉ R. Choose nonzero vectors u∈ X_α, v∈ X_β, and w∈ X_γ. Choose linear functionals f∈ X_α^* and g∈ X_β^* with f(u)=g(v)=1. Let T have the sole block x↦ f(x)v from X_α to X_β, and let S have the sole block y↦ g(y)w from X_β to X_γ. Then S,T∈ M_R, but ST has a nonzero forbidden (γ,α) block. This proves necessity. Combining reflexivity and transitivity gives (iii).
Exact mathematical passage
g\in X_\beta^*
X_\gammaUse in source · 4 passages
Search aliases: X_γ gamma
Use in source · 03_packets_and_graphs.tex, line 829
Conversely, suppose transitivity fails. Choose α,β,γ with (β,α)∈ R, (γ,β)∈ R, (γ,α)∉ R. Choose nonzero vectors u∈ X_α, v∈ X_β, and w∈ X_γ. Choose linear functionals f∈ X_α^* and g∈ X_β^* with f(u)=g(v)=1. Let T have the sole block x↦ f(x)v from X_α to X_β, and let S have the sole block y↦ g(y)w from X_β to X_γ. Then S,T∈ M_R, but ST has a nonzero forbidden (γ,α) block. This proves necessity. Combining reflexivity and transitivity gives (iii).
Exact mathematical passage
w\in X_\gamma
x_\gammaUse in source · 1 passage
Search aliases: x_γ gamma
Use in source · 02_coordinates.tex, line 172
Every coarse packet is also uniquely reconstructed from its fine children. If vectors x_β∈ Q_β X sum to a coarse packet x, applying Q_γ recovers x_γ=Q_γ x. This is the uniqueness clause of the canonical nested-refinement statement.
Exact mathematical passage
x_\gamma=Q_\gamma x
X_\lambdaUse in source · 1 passage
Search aliases: X_λ lambda
Use in source · 02_foundations.tex, line 987
The irreducibles are p=t-λ. On the primary component for λ, the quotient by (T-λ I)X_λ has dimension equal to the number of Jordan blocks, which is also dimKer(T-λ I).
Exact mathematical passage
(T-\lambda I)X_\lambda
x_\lambdaUse in source · 3 passages
Search aliases: x_λ lambda
Use in source · 04_transport.tex, line 323
Given y=Φ(x), choose a net x_λ∈Hull_ A(S) converging to x. Then Φ(x_λ)→ y, and naturality places every Φ(x_λ) in Hull_ B(Φ(S)). Hence the latter hull is dense. For packets, one inclusion follows from compatibility. Conversely, if y=Q_jy, choose x with Φ(x)=y. Then y=Q_jΦ(x)=Φ P_σ^-1(j)x.
Exact mathematical passage
x_\lambda\in\Hull_{\A}(S)
X_{\lambda,n}Use in source · 3 passages
Search aliases: X_λ,n lambda
Use in source · Paper_XII_Traditional_Final.tex, line 1473
For every positive integer n, let X_λ,n be a finite nonempty set of conjugacy classes of actual continuous representations Γ⟶GL_D( O_λ/I_λ^n). Conjugacy means change of basis by an invertible matrix over that quotient. Assume reduction preserves these sets and that X_λ,1 consists of the declared residual class. Each admissible class must have a continuous representative. The auxiliary level zero is the unique class over the trivial quotient and imposes no extra occupancy hypothesis.
Exact mathematical passage
X_{\lambda,n}
x_{\rho(\beta)}Use in source · 2 passages
Search aliases: x_ρ(β) rho beta
Use in source · 02_coordinates.tex, line 121
w QJ_ Q← P=I_X^ P, J_ Q← PC_ P← Q=I_X^ Q. For three packet systems P, Q, and R on X, J_ R← QJ_ Q← P = J_ R← P, C_ P← QC_ Q← R = C_ P← R. If Q refines P through ρ, then (J_ Q← Px)_β = Q_β x_ρ(β), (C_ P← Qy)_α = ∑_ρ(β)=αy_β.
Exact mathematical passage
\bigl(J_{\Qcal\leftarrow\Pcal}x\bigr)_\beta = Q_\beta x_{\rho(\beta)}, \qquad \bigl(C_{\Pcal\leftarrow\Qcal}y\bigr)_\alpha = \sum_{\rho(\beta)=\alpha}y_\beta.
X_\ellUse in source · 7 passages
Search aliases: X_ℓ ell
Use in source · 05_defects.tex, line 213
Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.
Exact mathematical passage
C_\ell:X_{\ell+1}\longrightarrow X_\ell
x_\ellUse in source · 12 passages
Search aliases: x_ℓ ell
Use in source · 03_nonlinear_coordinates.tex, line 852
[Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.
Exact mathematical passage
x_\ell
X_{\ell+1}Use in source · 4 passages
Search aliases: X_ℓ+1 ell
Use in source · 05_defects.tex, line 213
Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.
Exact mathematical passage
C_\ell:X_{\ell+1}\longrightarrow X_\ell
x_{\ell+1}Use in source · 3 passages
Search aliases: x_ℓ+1 ell
Use in source · 03_nonlinear_coordinates.tex, line 852
[Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.
Exact mathematical passage
x_{\ell+1}
x_\inftyUse in source · 6 passages
Search aliases: x_∞ infty
Use in source · 04_correction_transport.tex, line 343
[Complete filtered lifting] Assume there is a compatible sequence x_m+1≡ x_mpmodF^mX with Φ(x_m)∈ F^mY for every m. Then x_m converges to an element x_∞∈ X and Φ(x_∞)=0. In particular, if the obstruction vanishes at every depth and compatible corrections are chosen, an exact solution exists.
Exact mathematical passage
x_\infty\in X
\xleftarrowUse in source · 1 passage
Search aliases: xleftarrow
Use in source · Paper_VI_Traditional_Final.tex, line 2437
Let X_0xleftarrowC_0X_1xleftarrowC_1X_2xleftarrow⋯ be Banach spaces with bounded connecting maps. Define the bounded inverse limit X_∞ = { (x_ℓ)_ℓ≥0: C_ℓ x_ℓ+1=x_ℓ, sup_ℓnormx_ℓ<∞ } with the supremum norm.
Exact mathematical passage
X_0\xleftarrow{C_0}X_1\xleftarrow{C_1}X_2\xleftarrow{}\cdots
\xmapstoUse in source · 1 passage
Search aliases: xmapsto
Use in source · monograph_atlas.tex, line 126
vpage8Track the Bell state through the circuitINTERMEDIATE PACKETS ARE NOT FINAL PACKETSChapter VII: Theorem II.4.1, Example II.4.6. The input is ket00. First apply a Hadamard to the first qubit, then a controlled flip: ket00 xmapstoH_rm Had⊗ I fracket00+ket10sqrt2 xmapstorm CNOT fracket00+ket11sqrt2. The displayed columns keep amplitudes, including their normalization, at every step. The middle vector occupies both parity packets. The final vector occupies only the even packet. vfigbell_circuit…
Exact mathematical passage
\ket{00}\ \xmapsto{H_{\rm Had}\otimes I}\ \frac{\ket{00}+\ket{10}}{\sqrt2} \ \xmapsto{\rm CNOT}\ \frac{\ket{00}+\ket{11}}{\sqrt2}.
\bm YUse in source · 2 passages
Search aliases: Y bm
Use in source · Paper_V_Traditional_Final.tex, line 439
[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.
Exact mathematical passage
\bm Y(z)=\bm H^{(N)}(z)\bm X(z), \qquad H^{(N)}_{ks}(z)=\sum_m h[Nm+k-s]z^{-m}.
y'Use in source · 10 passages
Search aliases: y'
Use in source · 04_correction_transport.tex, line 95
[Stability with respect to the target] Under the hypotheses of [source reference], let x and x' be the fixed points associated with targets y and y'. If both fixed point maps preserve the same ball, then x-x' ≤ frac R 1-q y-y' .
Exact mathematical passage
y'
y''Use in source · 1 passage
Search aliases: y''
Use in source · atlas-pages.tex, line 94
z^66!-⋯, D_3,1=z-fracz^44!+⋯. Rows are output coordinates and columns are input coordinates. The upper-triangular minus signs record crossing the signed edge while moving from the column to the row. They are not arbitrary orientation choices. headTwo checks connect the matrix and the scalar equation The initial derivative matrix is the identity. Each initial jet (y(0),y'(0),y''(0)) therefore determines a unique linear combination. Also, frac13Tr(e^zJ_3)=D_3(z), det(XI-J_3)=X^3+1. The roots of -1…
Exact mathematical passage
(y(0),y'(0),y''(0))
y^{(N)}Use in source · 6 passages
Search aliases: y^(N)
Use in source · Paper_IV_Traditional_Final.tex, line 204
Fix a positive integer N. Root of unity projection of the exponential produces residue-class sections that, for λ≠0, form a basis for y^(N)=λ^N y. The signed sections for y^(N)=-y yield the denominator D_N and its reduced form Δ_N. We derive reciprocal coefficient polynomials, Bell and Newton recurrences, an additive parameter law, and residue section Appell and theta families. For N≥2, Δ_N has order 1/N. We specialize classical two term zero theory for exponential sums to the regular exponent p…
Exact mathematical passage
y^{(N)}=\lambda^N y
y^*Use in source · 1 passage
Search aliases: y^*
Use in source · 03_supplement_05.tex, line 136
Complex polarization reaches the whole operator space. Write q(x)=xx^*. The module proves the exact identity 4xy^*=q(x+y)-q(x-y) +i(q(x+iy)-q(x-iy)). Agreement on all positive pure outer products therefore implies agreement on every outer product. Actual matrix units then span the full matrix space. Applying this argument to the sandwich by P proves equality on every code-supported operator, including non-Hermitian ones.
Exact mathematical passage
4xy^*=q(x+y)-q(x-y) +i\bigl(q(x+iy)-q(x-iy)\bigr).
Y^{-1}Use in source · 1 passage
Search aliases: Y^-1
Use in source · 02_strengthened.tex, line 1668
Use the inverse difference identity X^-1-Y^-1=X^-1(Y-X)Y^-1 with X=F_P(z) and Y=z-A.
Exact mathematical passage
X^{-1}-Y^{-1}=X^{-1}(Y-X)Y^{-1}
y^{-1}Use in source · 2 passages
Search aliases: y^-1
Use in source · Paper_X_Block27_Amended.tex, line 2481
The inertia theorem uses actual residue actions. For a valuation subring A of an extension field L/K, let D_A be its decomposition subgroup in Aut_K(L) and put I_A=ker(D_A⟶Aut(k_A)), where k_A is the residue field. If 2 is a unit in A, every member of I_A fixes every y∈ L with y^2=-1. Indeed y^-1=-y, so y belongs to A and is a unit there. An automorphism sends y to y or -y. Their residues are distinct because 2 and y are units, whereas inertia fixes residues. The second possibility is excluded.
Exact mathematical passage
y^{-1}=-y
Y^{1-1/d}Use in source · 2 passages
Search aliases: Y^1-1/d
Use in source · residue_continuation_body.tex, line 56
attice discrepancy theorem. The index factor comes from the covolume identity covol(L(M))=q covol(L(J)). For a complex weight w:J/M→ C, summing the estimates gives the leading term fracvol(D)t^dq covol(L(J))∑_a∈ J/Mw(a), with error at most B∑_aabsnw(a)(t^d-1+1). If the finite sum of weights is zero, only that error remains. Writing Y=t^d gives an error O(Y^1-1/d+1) for the actual norm threshold. These assertions include d=1.
Exact mathematical passage
O(Y^{1-1/d}+1)
Y^2Use in source · 3 passages
Search aliases: Y^2
Use in source · 30_lifting.tex, line 904
The error exponent cannot uniformly be increased. For F(Y,T)=Y+Y^2+T, H=0, a=0, p=2, s=0, M=1, and v=1, there is an even root y of y+y^2+2=0. Its equation forces y≡2pmod4 and y+2=-y^2, so 4mid y+2, 8nmid y+2. Thus the error depth 2M-3s=2 is attained even for parameter-only motion. The same quadratic with a constant coefficient perturbation gives the coefficient-only boundary. These examples refute a uniform extra error digit. They do not assert optimality for every individual choice of p,s,M.
Exact mathematical passage
F(Y,T)=Y+Y^2+T
y^2Use in source · 14 passages
Search aliases: y^2
Use in source · 05_worked_models.tex, line 62
Let char𝕜≠2 and let A=𝕜[x,y]/(y^2-x^3+x). The affine equation y^2=x^3-x has classical discriminant 64, which is nonzero under this hypothesis. Standard algebraic geometry then identifies its smooth projective completion as an elliptic curve [source reference]. The Lean development checks the algebraic implication 2≠0⇒64≠0, the quotient normal form, and the operator identities below. It does not formalize projective smoothness, genus, or the geometric identification of the discriminant. The packe…
Exact mathematical passage
A=\bbk[x,y]/(y^2-x^3+x).
y^{2}Use in source · 1 passage
Search aliases: y^2
Use in source · 02_supplement_06.tex, line 101
Monotonicity of the diagonal entries gives normDy^2 =∑_i=1^k+1σ_i^2|y_i|^2 ≥σ_k+1^2norm y^2. Because (D-B)y=Dy, the operator-norm inequality implies σ_k+1norm y ≤norm(D-B)y ≤normD-B_2norm y. Cancellation of the strictly positive norm of y proves σ_k+1≤normD-B_2. The norm of the actual diagonal residual D-D_k equals σ_k+1, so D_k attains the lower bound. If k≥ d, the residual is zero and optimality follows from norm nonnegativity.
Exact mathematical passage
\norm{Dy}^{2} =\sum_{i=1}^{k+1}\sigma_i^{2}|y_i|^{2} \geq\sigma_{k+1}^{2}\norm y^{2}.
y^jUse in source · 1 passage
Search aliases: y^j
Use in source · 05_worked_models.tex, line 82
Take the seed s=1 and let M_x,M_y be multiplication by x and y. Then M_x^iM_y^js=x^iy^j, so the algebra, regarded as a module over its multiplication algebra, is generated by one seed.
Exact mathematical passage
M_x^iM_y^js=x^iy^j,
y^kUse in source · 1 passage
Search aliases: y^k
Use in source · 72_recovered.tex, line 1641
[Polynomial exponential Mahler transfer] Let f(n)=P(n)u^n with P∈ Z_p[n] of degree d. Let H=Λ_p,L, fix an address j, and write Q_j(x)=P(Hx+j), y=u^H, q=y-1. Then f(Hx+j)=∑_r≥0c_j,rbinom xr, where c_j,r = u^j ∑_k=0^min(d,r) binom rk Δ^kQ_j(0)y^kq^r-k. Consequently, v_p(c_j,r)≥ Lmax(r-d,0). For R≥ d, truncation at depth R is valid modulo p^L(R+1-d) uniformly on Z_p.
Exact mathematical passage
\boxed{ c_{j,r} = u^j \sum_{k=0}^{\min(d,r)} \binom rk \Delta^kQ_j(0)y^kq^{r-k}. }
y^mUse in source · 4 passages
Search aliases: y^m
Use in source · 05_worked_models.tex, line 70
alize projective smoothness, genus, or the geometric identification of the discriminant. The packet computation itself uses only the displayed relation in the quotient ring. Every class has a unique normal form f(x,y)=a(x)+y b(x). Indeed, y^2-(x^3-x) is monic of degree two in y, so division in 𝕜[x][y] gives a unique remainder of degree less than two in y. Concretely, every power y^m with m≥2 can be reduced using y^2=x^3-x.
Exact mathematical passage
y^m
y^{m-q}Use in source · 2 passages
Search aliases: y^m-q
Use in source · Paper_IV_Traditional_Final.tex, line 1499
[Appell laws] For m≥1, the polynomials satisfy fracddxE_m^(N,j,r)(x) = mE_m-1^(N,j,r)(x) and, for m≥0, E_m^(N,j,r)(x+y) = ∑_q=0^m binom mq E_q^(N,j,r)(x)y^m-q.
Exact mathematical passage
E_m^{(N,j,r)}(x+y) = \sum_{q=0}^{m} \binom mq E_q^{(N,j,r)}(x)y^{m-q}.
y^NUse in source · 1 passage
Search aliases: y^N
Use in source · 72_recovered.tex, line 2828
[Characteristic polynomial of the jump] Let χ_C(y)=det(yI-C). Then χ_C^N(x) = Res_y(χ_C(y),x-y^N). The identity holds over the coefficient ring after interpreting the resultant universally.
Exact mathematical passage
\boxed{ \chi_{C^N}(x) = \Res_y\bigl(\chi_C(y),x-y^N\bigr). }
y^nUse in source · 2 passages
Search aliases: y^n
Use in source · Paper_X_Block27_Amended.tex, line 1206
When pnmid n, the comparison with the finite cyclotomic sample is proved: q_L(γ_p)=σ_p, σ_p(ζ_n)=ζ_n^p. Reduction distinguishes n-th roots of unity at such a prime. Indeed, if x^n=y^n=1 and their reductions agree, the factorization of x^n-y^n gives (x-y)∑_i=0^n-1x^i y^n-1-i=0. The second factor reduces to noverline y^ n-1≠0, so the domain property gives x=y. This proves both Frobenius recognition on the cyclotomic generator and triviality of its inertia action. Consequently, every representation…
Exact mathematical passage
x^n=y^n=1
y^{n-1-i}Use in source · 1 passage
Search aliases: y^n-1-i
Use in source · Paper_X_Block27_Amended.tex, line 1207
When pnmid n, the comparison with the finite cyclotomic sample is proved: q_L(γ_p)=σ_p, σ_p(ζ_n)=ζ_n^p. Reduction distinguishes n-th roots of unity at such a prime. Indeed, if x^n=y^n=1 and their reductions agree, the factorization of x^n-y^n gives (x-y)∑_i=0^n-1x^i y^n-1-i=0. The second factor reduces to noverline y^ n-1≠0, so the domain property gives x=y. This proves both Frobenius recognition on the cyclotomic generator and triviality of its inertia action. Consequently, every representation…
Exact mathematical passage
(x-y)\sum_{i=0}^{n-1}x^i y^{n-1-i}=0.
y^pUse in source · 1 passage
Search aliases: y^p
Use in source · 72_recovered.tex, line 223
Choose a generator g of the cyclic additive group. Then F_p[ F_p^+] ≅ F_p[y]/(y^p-1) = F_p[y]/(y-1)^p. The augmentation ideal is generated by y-1, so its pth power is zero. Every element has the form a+n with a∈ F_p and n∈ J. It is invertible exactly when a≠0, which is the augmentation condition.
Exact mathematical passage
\FF_p[\FF_p^+] \cong \FF_p[y]/(y^p-1) = \FF_p[y]/(y-1)^p.
y^{p^{\ell-1}}Use in source · 1 passage
Search aliases: y^p^ℓ-1 ell
Use in source · supplement_05.tex, line 113
Uniqueness requires a lifting argument. The source proves the actual kernel statement bar x=0 Longleftrightarrow pmid x in Z/p^ℓ Z. For x≡ ypmod p, the checked ideal-power lifting theorem gives x^p^ℓ-1-y^p^ℓ-1∈(p)^ℓ=(0). If w^p-1=1, then w^p=w and hence w^p^k=w for every k. Therefore two roots with the same residue are equal. The decomposition is unique whenever its second factor reduces to one. Multiplicativity of both factors is also proved.
Exact mathematical passage
x^{p^{\ell-1}}-y^{p^{\ell-1}}\in(p)^\ell=(0).
y^TUse in source · 1 passage
Search aliases: y^T
Use in source · pages.tex, line 308
1/2 · 1. Its exact eigenvalues are 1/2 and 3/2, giving condition number three. vfigpreconditioning headThe unit circles of the energy norms The left ellipse is x^TAx=1. Writing x=By gives the right ellipse y^T(BAB)y=1. The drawing samples these explicitly defined curves. The norm and eigenvalue equalities are checked symbolically.
Exact mathematical passage
y^T(BAB)y=1
y^\ellUse in source · 1 passage
Search aliases: y^ℓ ell
Use in source · strengthening.tex, line 1288
For j=0,1, the term m=ℓ in [source reference] has exact valuation ℓ, since s(ℓ,ℓ)=1 and the Newton valuation is exact. Every later term has valuation at least ℓ+1. Thus v_2(d_j,ℓ)=ℓ, and the coefficient radius is exactly two. At a center x_0∈ Z_2, the coefficient of y^ℓ in the translated series is ∑_m≥ℓd_j,mbinom mℓ x_0^m-ℓ. Its initial term again has exact valuation ℓ, and all later terms have at least one more bit. The exact radius is therefore two at every integral center.
Exact mathematical passage
y^\ell
y_1Use in source · 9 passages
Search aliases: y_1
Use in source · 72_recovered.tex, line 2621
0 · 0 . The class x_1=(0,0) solves Mx_1≡ bpmod3 for both b_1=(-3,0) and b_2=(0,-3). For b_1, the reduced residual is (1,0), which belongs to the image of overline M. There are three correction classes, given by y_1+y_2=2 pmod3. For b_2, the residual is (0,1), whose cokernel class is nonzero. No lift modulo nine exists because the second row of M remains zero.
Exact mathematical passage
y_1+y_2=2 \pmod3.
Y_2Use in source · 2 passages
Search aliases: Y_2
Use in source · 30_lifting.tex, line 673
nomialPrescribedPadicTower.exists_prescribed_padic_root_with_tower and leanKomorebi.PolynomialPrescribedPadicTower.exists_unique_prescribed_padic_root. The separate supplied-path theorem leanKomorebi.PolynomialPrescribedPadicTower.exists_root_with_prescribed_path accepts any compatible integer parameter path with limit τ. For m>e, endpoint uniqueness is false in general: the system F(Y_1,Y_2,T)=Y_1 has two distinct roots (0,0,0) and (0,2,0) in the same dyadic ball with identical parameter value.
Exact mathematical passage
F(Y_1,Y_2,T)=Y_1
y_2Use in source · 4 passages
Search aliases: y_2
Use in source · 72_recovered.tex, line 2621
0 · 0 . The class x_1=(0,0) solves Mx_1≡ bpmod3 for both b_1=(-3,0) and b_2=(0,-3). For b_1, the reduced residual is (1,0), which belongs to the image of overline M. There are three correction classes, given by y_1+y_2=2 pmod3. For b_2, the residual is (0,1), whose cokernel class is nonzero. No lift modulo nine exists because the second row of M remains zero.
Exact mathematical passage
y_1+y_2=2 \pmod3.
Y_cUse in source · 5 passages
Search aliases: Y_c
Use in source · 03_nonlinear_coordinates.tex, line 722
Let F_f:X_f⟶ Y_f, F_c:X_c⟶ Y_c be fine and coarse maps. Let C_X:X_f⟶ X_c, C_Y:Y_f⟶ Y_c be linear coarsening maps. Whenever a norm, a Lipschitz constant, or a Fr echet derivative is used, the spaces are Banach spaces and the coarsening maps are bounded.
Exact mathematical passage
F_f:X_f\longrightarrow Y_f, \qquad F_c:X_c\longrightarrow Y_c
Y_fUse in source · 3 passages
Search aliases: Y_f
Use in source · 03_nonlinear_coordinates.tex, line 722
Let F_f:X_f⟶ Y_f, F_c:X_c⟶ Y_c be fine and coarse maps. Let C_X:X_f⟶ X_c, C_Y:Y_f⟶ Y_c be linear coarsening maps. Whenever a norm, a Lipschitz constant, or a Fr echet derivative is used, the spaces are Banach spaces and the coarsening maps are bounded.
Exact mathematical passage
F_f:X_f\longrightarrow Y_f, \qquad F_c:X_c\longrightarrow Y_c
y_iUse in source · 3 passages
Search aliases: y_i
Use in source · 04_arithmetic.tex, line 779
Set x=Vy. Multiplying the congruence by U reduces it to independent equations π^s_iy_i≡widetilde b_ipmodπ^L (1≤ i≤ r) and zero equations in the remaining rows. Multiplication by π^s_i on O/π^L O has image equal to the elements divisible by π^min(s_i,L) and kernel of size q^min(s_i,L). Each of the n-r unused variables is free and contributes q^L choices. Multiplying the independent counts gives the formula.
Exact mathematical passage
\pi^{s_i}y_i\equiv\widetilde b_i\pmod{\pi^L} \qquad(1\leq i\leq r)
y_jUse in source · 4 passages
Search aliases: y_j
Use in source · visual_atlas.tex, line 512
vpage21Tensor products require every cross termOPERATIONS ON SEED RANK / A RECTANGLE, NOT A DIAGONALMain text: [source reference]. Algebraic tensor products If (x_i) generates X under A and (y_j) generates Y under B, then the candidate tensor seeds are all x_i⊗ y_j. For finite expansions, x=∑_i A_ix_i, y=∑_j B_jy_j, bilinearity gives the required double sum: x⊗ y=∑_i∑_j(A_i⊗ B_j)(x_i⊗ y_j).
Exact mathematical passage
(y_j)
y_kUse in source · 8 passages
Search aliases: y_k
Use in source · Paper_V_Traditional_Final.tex, line 426
[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.
Exact mathematical passage
\boxed{ y_k[n] = \sum_{s=0}^{N-1}\sum_{m\in\bbZ} h_{ks}[m]x_s[n-m]. }
y_{k+1}Use in source · 2 passages
Search aliases: y_k+1
Use in source · Paper_V_Traditional_Final.tex, line 2921
me f(t) = s(t)+∑_q=1^Qa_qH_τ_q(t), H_τ(t)= 1_[τ,1](t), where s is Lipschitz with constant M, and Q≥1 is finite with nonzero jump amplitudes. The zero-jump case requires no minimum-amplitude condition. Suppose no jump lies at a sample point and no grid cell contains more than one jump. Observe y_k=f(t_k)+η_k. Define the adjacent contrast D_k=y_k+1-y_k.
Exact mathematical passage
D_k=y_{k+1}-y_k.
y_mUse in source · 4 passages
Search aliases: y_m
Use in source · 03_packets_and_graphs.tex, line 460
For y=(y_n)∈⊕^uP_nX, define S_ Py=∑_ny_n. Since the finite partial sums with every coefficient equal to one are among those used in y _u, S_ Py ≤ y _u. The assumed expansion gives S_ P A_ Px=x. Conversely, boundedness of P_m permits termwise application to the unconditionally convergent series: P_m S_ Py = ∑_nP_my_n =y_m. Thus A_ P S_ P=I. Completeness of the coordinate space is [source reference].
Exact mathematical passage
P_m\mathcal S_{\Pcal}y = \sum_nP_my_n =y_m.
Y_{m+1}Use in source · 3 passages
Search aliases: Y_m+1
Use in source · Paper_XII_Traditional_Final.tex, line 1206
[Surjectivity on stable cores] For every m, the transition map V_m+1→ V_m induces a surjection Y_m+1↠ Y_m.
Exact mathematical passage
\boxed{ Y_{m+1}\twoheadrightarrow Y_m. }
y_{m+1}Use in source · 1 passage
Search aliases: y_m+1
Use in source · Paper_XII_Traditional_Final.tex, line 1244
Choose 0≠ y_1∈ Y_1. By [source reference], choose recursively y_m+1∈ Y_m+1 mapping to y_m. Every y_m is nonzero because it maps to the nonzero vector y_1. The sequence (y_m) is a nonzero element of the inverse limit.
Exact mathematical passage
y_{m+1}\in Y_{m+1}
y_nUse in source · 3 passages
Search aliases: y_n
Use in source · 03_packets_and_graphs.tex, line 450
Each P_nX=Ker(I-P_n) is closed. For x∈ X, A_ Px _u = sup_F,a ∑_n∈ Fa_nP_nx ≤ K x . For y=(y_n)∈⊕^uP_nX, define S_ Py=∑_ny_n. Since the finite partial sums with every coefficient equal to one are among those used in y _u, S_ Py ≤ y _u. The assumed expansion gives S_ P A_ Px=x. Conversely, boundedness of P_m permits termwise application to the unconditionally convergent series: P_m S_ Py = ∑_nP_my_n =y_m. Thus A_ P S_ P=I. Completeness of the coordinate space is [source reference].
Exact mathematical passage
y=(y_n)\in\bigoplus^uP_nX
Y_PUse in source · 2 passages
Search aliases: Y_P
Use in source · 04_correction_transport.tex, line 497
Let U,V,Y_P,Y_Q be Banach spaces. On an open neighborhood of (u_0,v_0)∈ U× V, split the variables and equations as x=(u,v), F(u,v)= F_P(u,v)
Exact mathematical passage
U,V,Y_P,Y_Q
Y_QUse in source · 3 passages
Search aliases: Y_Q
Use in source · 04_correction_transport.tex, line 497
Let U,V,Y_P,Y_Q be Banach spaces. On an open neighborhood of (u_0,v_0)∈ U× V, split the variables and equations as x=(u,v), F(u,v)= F_P(u,v)
Exact mathematical passage
U,V,Y_P,Y_Q
y_QUse in source · 2 passages
Search aliases: y_Q
Use in source · Paper_VI_Traditional_Final.tex, line 1513
Solve the hidden block equation (zI-A_QQ)x_Q=A_QPx_P+y_Q for x_Q and substitute it into the retained equation. The resulting retained coefficient is F_P(z). Invertibility of the full operator and of the hidden block implies invertibility of this Schur complement. Setting y_Q=0 gives the stated identity as an operator on Ran P. Equivalently, this is the block inverse formula.
Exact mathematical passage
(zI-A_{QQ})x_Q=A_{QP}x_P+y_Q
Y_rUse in source · 1 passage
Search aliases: Y_r
Use in source · 02_connections_obstructions.tex, line 716
[Exact naturality for arbitrary functions] Let A^X_a:Xxrightarrow∼X_a and A^X_b:Xxrightarrow∼X_b be exact coordinate equivalences for X, with inverses S^X_a and S^X_b. Likewise, let A^Y_r:Yxrightarrow∼Y_r and A^Y_s:Yxrightarrow∼Y_s be exact coordinate equivalences for Y, with inverses S^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity or differentiability hypothesis. Set J^X_b← a= A^X_b∘ S^X_a, J^Y_s← r= A^Y_s∘ S^Y_r, and define widehat F_r← a = A^Y_r∘ F∘ S^X_a. Then J^Y_s← r∘widehat…
Exact mathematical passage
\cA^Y_r:Y\xrightarrow{\sim}Y_r
y_rUse in source · 7 passages
Search aliases: y_r
Use in source · supplement_10.tex, line 153
Now define explicit iterations y_0=0, y_r+1=Λ_c^-1(cq+y_rq-y_r^2). If q is flat and p-supported, every iterate has those properties. The crucial link to the actual ratio is an exact finite-prefix theorem: [t^d]y_r=[t^d]y (d≤ r). Indeed, (cq+zq-z^2)-(cq+yq-y^2)=(z-y)(q-z-y). The second factor has zero constant term. Multiplication therefore raises the established agreement prefix by one. Coefficientwise division by d+c preserves that prefix.
Exact mathematical passage
y_0=0,\qquad y_{r+1}=\Lambda_c^{-1}(cq+y_rq-y_r^2).
y_{r+1}Use in source · 1 passage
Search aliases: y_r+1
Use in source · supplement_10.tex, line 153
Now define explicit iterations y_0=0, y_r+1=Λ_c^-1(cq+y_rq-y_r^2). If q is flat and p-supported, every iterate has those properties. The crucial link to the actual ratio is an exact finite-prefix theorem: [t^d]y_r=[t^d]y (d≤ r). Indeed, (cq+zq-z^2)-(cq+yq-y^2)=(z-y)(q-z-y). The second factor has zero constant term. Multiplication therefore raises the established agreement prefix by one. Coefficientwise division by d+c preserves that prefix.
Exact mathematical passage
y_0=0,\qquad y_{r+1}=\Lambda_c^{-1}(cq+y_rq-y_r^2).
y_r^2Use in source · 1 passage
Search aliases: y_r^2
Use in source · supplement_10.tex, line 153
Now define explicit iterations y_0=0, y_r+1=Λ_c^-1(cq+y_rq-y_r^2). If q is flat and p-supported, every iterate has those properties. The crucial link to the actual ratio is an exact finite-prefix theorem: [t^d]y_r=[t^d]y (d≤ r). Indeed, (cq+zq-z^2)-(cq+yq-y^2)=(z-y)(q-z-y). The second factor has zero constant term. Multiplication therefore raises the established agreement prefix by one. Coefficientwise division by d+c preserves that prefix.
Exact mathematical passage
y_0=0,\qquad y_{r+1}=\Lambda_c^{-1}(cq+y_rq-y_r^2).
Y_sUse in source · 1 passage
Search aliases: Y_s
Use in source · 02_connections_obstructions.tex, line 717
[Exact naturality for arbitrary functions] Let A^X_a:Xxrightarrow∼X_a and A^X_b:Xxrightarrow∼X_b be exact coordinate equivalences for X, with inverses S^X_a and S^X_b. Likewise, let A^Y_r:Yxrightarrow∼Y_r and A^Y_s:Yxrightarrow∼Y_s be exact coordinate equivalences for Y, with inverses S^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity or differentiability hypothesis. Set J^X_b← a= A^X_b∘ S^X_a, J^Y_s← r= A^Y_s∘ S^Y_r, and define widehat F_r← a = A^Y_r∘ F∘ S^X_a. Then J^Y_s← r∘widehat…
Exact mathematical passage
\cA^Y_s:Y\xrightarrow{\sim}Y_s
y_sUse in source · 2 passages
Search aliases: y_s
Use in source · 20_arithmetic.tex, line 302
At a nonsingular address 2≤ c<N with pnmid c, differentiation of the section identities gives tR_c'=R_c(1+R_c-1-R_c). Writing q=R_c-1-(c-1) and y=R_c-c, this becomes (tD+c)y=cq+yq-y^2. On degrees divisible by p, the inverse of tD+c divides the degree-d coefficient by the unit d+c. Starting with y_0=0, form y_s+1=(tD+c)^-1(cq+y_sq-y_s^2). Flatness and support are preserved. Indeed, put E_c=tD+c and C_p=pD^p^2-D^p. The commutator is [B_p,E_c]=C_p, and iteration gives B_p^rE_cy=E_cB_p^ry+rC_pB_p^r-…
Exact mathematical passage
y_{s+1}=(tD+c)^{-1}(cq+y_sq-y_s^2).
y_{s+1}Use in source · 1 passage
Search aliases: y_s+1
Use in source · 20_arithmetic.tex, line 302
At a nonsingular address 2≤ c<N with pnmid c, differentiation of the section identities gives tR_c'=R_c(1+R_c-1-R_c). Writing q=R_c-1-(c-1) and y=R_c-c, this becomes (tD+c)y=cq+yq-y^2. On degrees divisible by p, the inverse of tD+c divides the degree-d coefficient by the unit d+c. Starting with y_0=0, form y_s+1=(tD+c)^-1(cq+y_sq-y_s^2). Flatness and support are preserved. Indeed, put E_c=tD+c and C_p=pD^p^2-D^p. The commutator is [B_p,E_c]=C_p, and iteration gives B_p^rE_cy=E_cB_p^ry+rC_pB_p^r-…
Exact mathematical passage
y_{s+1}=(tD+c)^{-1}(cq+y_sq-y_s^2).
y_s^2Use in source · 1 passage
Search aliases: y_s^2
Use in source · 20_arithmetic.tex, line 302
At a nonsingular address 2≤ c<N with pnmid c, differentiation of the section identities gives tR_c'=R_c(1+R_c-1-R_c). Writing q=R_c-1-(c-1) and y=R_c-c, this becomes (tD+c)y=cq+yq-y^2. On degrees divisible by p, the inverse of tD+c divides the degree-d coefficient by the unit d+c. Starting with y_0=0, form y_s+1=(tD+c)^-1(cq+y_sq-y_s^2). Flatness and support are preserved. Indeed, put E_c=tD+c and C_p=pD^p^2-D^p. The commutator is [B_p,E_c]=C_p, and iteration gives B_p^rE_cy=E_cB_p^ry+rC_pB_p^r-…
Exact mathematical passage
y_{s+1}=(tD+c)^{-1}(cq+y_sq-y_s^2).
y_\alphaUse in source · 1 passage
Search aliases: y_α alpha
Use in source · 02_coordinates.tex, line 161
[Orthogonal Hilbert packets] If X is a real or complex Hilbert space and every packet system in the tower consists of orthogonal projections, equip each finite packet direct sum with the Hilbert inner product ⟨(x_α),(y_α)⟩=∑_α⟨ x_α,y_α⟩. Then analysis and synthesis are unitary. Every coordinate change in [source reference] is unitary.
Exact mathematical passage
\langle(x_\alpha),(y_\alpha)\rangle=\sum_\alpha\langle x_\alpha,y_\alpha\rangle
y_\betaUse in source · 1 passage
Search aliases: y_β beta
Use in source · 02_coordinates.tex, line 121
w QJ_ Q← P=I_X^ P, J_ Q← PC_ P← Q=I_X^ Q. For three packet systems P, Q, and R on X, J_ R← QJ_ Q← P = J_ R← P, C_ P← QC_ Q← R = C_ P← R. If Q refines P through ρ, then (J_ Q← Px)_β = Q_β x_ρ(β), (C_ P← Qy)_α = ∑_ρ(β)=αy_β.
Exact mathematical passage
\bigl(J_{\Qcal\leftarrow\Pcal}x\bigr)_\beta = Q_\beta x_{\rho(\beta)}, \qquad \bigl(C_{\Pcal\leftarrow\Qcal}y\bigr)_\alpha = \sum_{\rho(\beta)=\alpha}y_\beta.
\mathbf ZUse in source · 1 passage
Search aliases: Z mathbf
Use in source · Paper_X_Block27_Amended.tex, line 1551
[Multiplicity recovery] The original companion matrix was not retrieved or rerun. A new exact realization uses the trivial character and two independent sign characters of ( Z/2 Z)^2, evaluated at (0,0),(1,0),(0,1): C=1 · 1 · 1
Exact mathematical passage
(\mathbf Z/2\mathbf Z)^2
Z^*Use in source · 5 passages
Search aliases: Z^*
Use in source · 03_supplement_05.tex, line 111
Trace-preserving recovery supplies the original Gram data. Let Z_ra=z_ra. The recovery normalization gives PE_a^*E_bP · =PE_a^*(∑_rR_r^*R_r)E_bP · =∑_r(R_rE_aP)^*(R_rE_bP) · =(∑_roverlinez_raz_rb)P =(Z^*Z)_abP. Thus C=Z^*Z is positive semidefinite and satisfies the required compressed identities. The theorem pathgram_of_recovery implements this calculation with the original error and recovery matrices.
Exact mathematical passage
PE_a^*E_bP &=PE_a^*\Bigl(\sum_rR_r^*R_r\Bigr)E_bP\\ &=\sum_r(R_rE_aP)^*(R_rE_bP)\\ &=\Bigl(\sum_r\overline{z_{ra}}z_{rb}\Bigr)P =(Z^*Z)_{ab}P.
z^*Use in source · 1 passage
Search aliases: z^*
Use in source · 02_strengthened.tex, line 1339
For z∈ C^r, let F_z=∑_az_aE_a. Then PF_z^*F_zP=(z^*cz)P≥0, so c≥0. The unitary change of error coordinates gives PF_j^*F_ℓ P = λ_jδ_jℓP. Thus V_j^*V_ℓ=δ_jℓP. Each V_j is an isometry on P H, and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.
Exact mathematical passage
PF_z^*F_zP=(z^*cz)P\geq0,
z^{-1}Use in source · 7 passages
Search aliases: z^-1
Use in source · Paper_V_Traditional_Final.tex, line 491
Work with finite impulse-response banks. Let L= C[z,z^-1] be the Laurent-polynomial algebra, with the monomial z^-m acting on a bilateral sequence by x[n]↦ x[n-m]. Let E∈Mat_N× N( L) and R∈Mat_N× N( L) be analysis and synthesis matrices. Their composition acts on all bilateral channel sequences by finite convolutions. At a nonzero z where the input transforms are absolutely convergent, this gives widehat X(z)= R(z) E(z) X(z).
Exact mathematical passage
\mathcal L=\bbC[z,z^{-1}]
z^{-1/2}Use in source · 1 passage
Search aliases: z^-1/2
Use in source · 04_correction_transport.tex, line 987
P_α(t)+hdot P_α(t)+E_α(t,h), E_α(t,h) ≤ C_1h^2, uniformly in t and α. Hence S( P(t+h), P(t)) = I+hK(t)+E(t,h), E(t,h) ≤ C_2h^2. Since K(t)^*=-K(t), S^*S=I+E_2(t,h), E_2(t,h) ≤ C_3h^2. For small h, the spectrum of S^*S lies in a fixed compact subinterval of (0,∞). The functional calculus for z^-1/2 gives (S^*S)^-1/2=I+E_3(t,h), E_3(t,h) ≤ C_4h^2. Multiplying the last two expansions proves the estimate.
Exact mathematical passage
z^{-1/2}
z^{-j}Use in source · 1 passage
Search aliases: z^-j
Use in source · Paper_V_Traditional_Final.tex, line 459
The last factor is x_s[n-m], giving the time-domain formula. Under the additional transform hypothesis, each delayed channel has transform ∑_nx_s[n-m]z^-n =z^-m∑_jx_s[j]z^-j. Absolute summability permits this bijective reindexing, and only finitely many delayed channels are combined. Taking their finite linear combination gives the matrix formula. The time-domain theorem therefore remains valid even when no numerical bilateral transform exists.
Exact mathematical passage
\sum_nx_s[n-m]z^{-n} =z^{-m}\sum_jx_s[j]z^{-j}.
z^{-m}Use in source · 4 passages
Search aliases: z^-m
Use in source · Paper_V_Traditional_Final.tex, line 439
[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.
Exact mathematical passage
\bm Y(z)=\bm H^{(N)}(z)\bm X(z), \qquad H^{(N)}_{ks}(z)=\sum_m h[Nm+k-s]z^{-m}.
z^{-n}Use in source · 3 passages
Search aliases: z^-n
Use in source · Paper_V_Traditional_Final.tex, line 435
[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.
Exact mathematical passage
\sum_{n\in\bbZ}|x_s[n]z^{-n}|<\infty\quad(0\le s<N).
Z^1Use in source · 1 passage
Search aliases: Z^1
Use in source · Paper_X_Block27_Amended.tex, line 339
[Tangent and obstruction spaces] For a framed representation problem, first-order lifts are described by Z^1(G,adbarρ). For an unframed Schur problem, tangent directions are described by H^1(G,adbarρ). Relations in a presentation of a deformation ring map into H^2(G,adbarρ), which contains the obstruction classes.
Exact mathematical passage
Z^1(G,\ad\bar\rho)
z^{10}Use in source · 1 passage
Search aliases: z^10
Use in source · atlas-pages.tex, line 53
r rotated arguments and attach the phase i^-jℓ: ( B_4,jF)(z)=frac14∑_ℓ=0^3i^-jℓF(i^ℓ z). A monomial a_mz^m is multiplied by the finite average of i^(m-j)ℓ. Matching phases all vote 1. Nonmatching phases cancel exactly. vfigroot_filter headAn example that can be checked coefficient by coefficient Take F(z)=∑_m=0^12(m+1)z^m. Then B_4,2F=3z^2+7z^6+11z^10. Applying the same selector again changes nothing. Applying a different selector gives zero. Summing the four selected polynomials returns the ori…
Exact mathematical passage
\mathcal B_{4,2}F=3z^2+7z^6+11z^{10}.
Z^2Use in source · 4 passages
Search aliases: Z^2
Use in source · root_scope.tex, line 365
The current focused obstruction is the admitted ordinary-index family r=(17659+2^18T)2^640-161, T∈ Z_≥0, j_r=161. On its surviving chart define the actual, undepleted function K_r(Z)=2^-178G_r(10202+2^15Z). The complete coefficient theorem gives K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z), a_r,b_r∈ Z_2, u_r∈ Z_2^×, R_r∈ Z_2⟨ Z⟩. Its half derivative is a dyadic isometry. It therefore has a unique critical point c_r∈ Z_2. Set Φ_r^act=K_r(c_r). The earlier independent-undepletion notation for this same sc…
Exact mathematical passage
K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z), \quad a_r,b_r\in\mathbb Z_2,\quad u_r\in\mathbb Z_2^\times,\quad R_r\in\mathbb Z_2\langle Z\rangle. \label{eq:root:quadraticclass}
z^2Use in source · 16 passages
Search aliases: z^2
Use in source · Paper_IV_Traditional_Final.tex, line 2804
For N=2, [source reference] and D_2(z)=Δ_2(z^2) give λ_m=π^2(m+frac12)^2, m≥0. Thus the first formula holds with Q_2^red=0.
Exact mathematical passage
\Dfun_2(z)=\Delta_2(z^2)
z^{2n}Use in source · 1 passage
Search aliases: z^2n
Use in source · 20_arithmetic.tex, line 85
[The Bernoulli boundary address] At N=2, the address j=1 gives a second classical family. Define the Bernoulli numbers by B(z)=fracze^z-1=∑_r≥0B_rfracz^rr!, B_0=1, B_1=-frac12. All quotients here are regular formal series. The identity 2B(z)-B(2z)=fraczsinh z =∑_n≥0C_2,1(n)fracz^2n(2n)! follows by putting the two fractions on a common denominator and using [source reference]. Comparing even coefficients yields C_2,1(n)=(2-2^2n)B_2n (n∈ N). This classical derivation restores the boundary example …
Exact mathematical passage
2B(z)-B(2z)=\frac{z}{\sinh z} =\sum_{n\ge0}C_{2,1}(n)\frac{z^{2n}}{(2n)!}
z^{2r}Use in source · 2 passages
Search aliases: z^2r
Use in source · 72_recovered.tex, line 80
The dyadic kernel is substantially sharper than the separate coefficient bound alone suggests. If fracz^2sinh^2z = ∑_r≥0β_rz^2r and κ_2,r=β_r2^2r(2r)!, then v_2(κ_2,r)=4r-1 (r≥1). Consequently, truncation after depth M is uniformly valid modulo 2^4M+3, and this exponent is optimal. The rowwise estimate records the additional binary carries in the binomial coefficient. Classical congruence theory for Euler families is extensive [source reference]. The formulas here are stated as exact coordinate …
Exact mathematical passage
\frac{z^2}{\sinh^2z} = \sum_{r\geq0}\beta_rz^{2r}
Z^3Use in source · 1 passage
Search aliases: Z^3
Use in source · root_scope.tex, line 365
The current focused obstruction is the admitted ordinary-index family r=(17659+2^18T)2^640-161, T∈ Z_≥0, j_r=161. On its surviving chart define the actual, undepleted function K_r(Z)=2^-178G_r(10202+2^15Z). The complete coefficient theorem gives K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z), a_r,b_r∈ Z_2, u_r∈ Z_2^×, R_r∈ Z_2⟨ Z⟩. Its half derivative is a dyadic isometry. It therefore has a unique critical point c_r∈ Z_2. Set Φ_r^act=K_r(c_r). The earlier independent-undepletion notation for this same sc…
Exact mathematical passage
K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z), \quad a_r,b_r\in\mathbb Z_2,\quad u_r\in\mathbb Z_2^\times,\quad R_r\in\mathbb Z_2\langle Z\rangle. \label{eq:root:quadraticclass}
z^3Use in source · 7 passages
Search aliases: z^3
Use in source · atlas-pages.tex, line 65
vpage4Products mix the residue lanesCYCLIC CONVOLUTIONChapter IV: Theorem 3.3. Selection separates a function additively. It does not make multiplication diagonal. A residue-i term multiplied by a residue-j term lands in residue i+jpmod N. For N=3, use F=1+2z+3z^2+z^3, G=2+z+z^2. The table displays every product of their three selected sections. vfigcyclic_product The shaded cells contribute to residue zero: 2(1+z^3)+(2z)(z^2)+(3z^2)z=2+7z^3. Keeping only the product of the residue-zero inputs w…
Exact mathematical passage
F=1+2z+3z^2+z^3,\qquad G=2+z+z^2.
z^4Use in source · 6 passages
Search aliases: z^4
Use in source · Paper_IV_Traditional_Final.tex, line 1739
ion gives a polygon of exponents. Large zeros can occur where two dominant terms nearly cancel. The proof first localizes those zeros and then uses symmetry and uniqueness to place them on the rays. Finite exceptional zeros remain part of the later products and sums. modelfigurezero_orbitsThe exact order-four model. The first three zeros on each of four rays are mapped by w=z^4 to three positive reduced zeros. The reduced axis uses a logarithmic scale. This complete order-four example does not c…
Exact mathematical passage
w=z^4
z^6Use in source · 3 passages
Search aliases: z^6
Use in source · atlas-pages.tex, line 53
r rotated arguments and attach the phase i^-jℓ: ( B_4,jF)(z)=frac14∑_ℓ=0^3i^-jℓF(i^ℓ z). A monomial a_mz^m is multiplied by the finite average of i^(m-j)ℓ. Matching phases all vote 1. Nonmatching phases cancel exactly. vfigroot_filter headAn example that can be checked coefficient by coefficient Take F(z)=∑_m=0^12(m+1)z^m. Then B_4,2F=3z^2+7z^6+11z^10. Applying the same selector again changes nothing. Applying a different selector gives zero. Summing the four selected polynomials returns the ori…
Exact mathematical passage
\mathcal B_{4,2}F=3z^2+7z^6+11z^{10}.
z^aUse in source · 1 passage
Search aliases: z^a
Use in source · Paper_IV_Traditional_Final.tex, line 544
Expand e^λ(z+w)=e^λ ze^λ w as a bivariate power series. A term z^a w^b contributes to the kth section exactly when a+b≡ kbmod N. Grouping first by i≡ abmod N and j≡ bbmod N gives the displayed sum.
Exact mathematical passage
z^a w^b
\Z^dUse in source · 19 passages
Search aliases: Z^d Z mathbb
Use in source · 30_lifting.tex, line 69
[Finite polynomial obstruction] Let q≥1, L≥1, x∈ Z^d, and suppose F(x)∈ q^L Z^e. Define the exact integer residual r_x by F(x)=q^Lr_x, and put O_x=S_q^e/im J_x, o_x=[overliner_x]. Then O_x is finite and o_x=0 Longleftrightarrow ∃ y∈ Z^d: y≡ xpmodq^L, F(y)≡0pmodq^L+1. No surjectivity or nonsingularity assumption is required.
Exact mathematical passage
x\in\Z^d
\Z^eUse in source · 8 passages
Search aliases: Z^e Z mathbb
Use in source · 30_lifting.tex, line 70
[Finite polynomial obstruction] Let q≥1, L≥1, x∈ Z^d, and suppose F(x)∈ q^L Z^e. Define the exact integer residual r_x by F(x)=q^Lr_x, and put O_x=S_q^e/im J_x, o_x=[overliner_x]. Then O_x is finite and o_x=0 Longleftrightarrow ∃ y∈ Z^d: y≡ xpmodq^L, F(y)≡0pmodq^L+1. No surjectivity or nonsingularity assumption is required.
Exact mathematical passage
F(x)\in q^L\Z^e
\Z^{e+k}Use in source · 1 passage
Search aliases: Z^e+k Z mathbb
Use in source · 30_lifting.tex, line 451
[Continuous charts with a selected Jacobian minor] Let p be prime, L≥1, F∈ Z[X_1,…,X_e+k]^e, and x∈ Z^e+k satisfy F(x)∈ p^L Z^e. Suppose the full reduced Jacobian is surjective. There is a permutation σ of the coordinates such that, writing σ x=(y_0,t_0) with e solved coordinates and k parameter coordinates, the parameter ball T=t_0+p^L Z_p^k is homeomorphic to the entire root locus R={z∈ B_L(x):F(z)=0}. The inverse homeomorphism is the actual free-coordinate projection z↦free(σ z). If h:T→ R is…
Exact mathematical passage
x\in\Z^{e+k}
\mathbb Z^{e+q}Use in source · 1 passage
Search aliases: Z^e+q mathbb
Use in source · continuation_05.tex, line 8
Let p be prime, L≥1, e,q≥0, and F∈ Z[X_1,…,X_e+q]^e, x∈ Z^e+q, F(x)≡0pmodp^L. Assume the actual reduced Jacobian A_x=DF(x)bmod p: F_p^e+q⟶ F_p^e is surjective. All vector congruences are coordinatewise.
Exact mathematical passage
F\in\mathbb Z[X_1,\ldots,X_{e+q}]^e,\qquad x\in\mathbb Z^{e+q},\qquad F(x)\equiv0\pmod{p^L}.
\Z^fUse in source · 1 passage
Search aliases: Z^f Z mathbb
Use in source · 30_lifting.tex, line 242
For affine integer systems F(X)=AX-b and G(X)=BX-c, where A: Z^d→ Z^e and B: Z^d→ Z^f are linear, Corollary~[source reference] holds for q≥1 and every L≥0, including depth zero. Here x is a common root modulo q^L, and J and K are the reductions of A and B with domain Z^d. Given separately valid corrections u,v, a simultaneous next lift exists if and only if their classes in Z^d/(ker J+ker K) agree. The quotient is finite, and each label is independent of the chosen valid correction. Indeed, Prop…
Exact mathematical passage
B:\Z^d\to\Z^f
z^hUse in source · 2 passages
Search aliases: z^h
Use in source · 72_recovered.tex, line 407
[Rational ordinary generating function] In F_p[[z]], ∑_n≥0e_n^(m,p)z^n = 1+ frac ∑_r=1^he_r^(m,p)z^r 1-z^h.
Exact mathematical passage
\boxed{ \sum_{n\geq0}e_n^{(m,p)}z^n = 1+ \frac{ \sum_{r=1}^{h}e_r^{(m,p)}z^r }{1-z^h}. }
\Z^kUse in source · 3 passages
Search aliases: Z^k Z mathbb
Use in source · 30_lifting.tex, line 591
[Every compatible integer parameter path] In the preceding notation, assume J_x is surjective. For every supplied integer path t_n∈ Z^k satisfying t_0=free(x), t_n+1-t_n∈ q^L+n Z^k, there is a compatible polynomial tower z_n∈ Z^m+k with z_0=x, free(z_n)=t_n, F(z_n)∈ q^L+n Z^e, z_n+1-z_n∈ q^L+n Z^m+k. The free-coordinate equality is exact in integers. Both reduced Jacobian blocks remain fixed, and z_b-z_a∈ q^L+a Z^m+k whenever a≤ b.
Exact mathematical passage
t_n\in\Z^k
\mathbb Z^kUse in source · 3 passages
Search aliases: Z^k mathbb
Use in source · continuation_09.tex, line 27
Integer prescribed-path theorem. For every supplied path t: N→ Z^k satisfying t(0)=t_0, t(n+1)-t(n)∈ q^L+n Z^k, there is a single tower z: N→ Z^m+k with z(0)=x, free(z(n))=t(n), F(z(n))∈ q^L+n Z^e, z(n+1)-z(n)∈ q^L+n Z^m+k. The parameter equality is exact in integers. For a≤ b, z(b)-z(a)∈ q^L+a Z^m+k, DF(z(n))bmod q=DF(x)bmod q. Consequently J_z(n)=J_x at every level. No later surjectivity or per-step solvability is assumed. This theorem includes positive composite bases and q=1.
Exact mathematical passage
t:\mathbb N\to\mathbb Z^k
z^kUse in source · 3 passages
Search aliases: z^k
Use in source · Paper_IV_Traditional_Final.tex, line 488
[Differential cycle and initial data] For 1≤ j<N, fracddz E_N,j(λ z) = λ E_N,j-1(λ z), while fracddz E_N,0(λ z) = λ E_N,N-1(λ z). Thus fracd^Ndz^N E_N,j(λ z) = λ^N E_N,j(λ z). For 0≤ k<N, .fracd^kdz^k E_N,j(λ z)|_z=0 = δ_kjλ^j. If λ≠0, the N functions E_N,j(λ z) form a basis of the solution space of y^(N)=λ^Ny.
Exact mathematical passage
\left.\frac{d^k}{dz^k}\Efun_{N,j}(\lambda z)\right|_{z=0} = \delta_{kj}\lambda^j.
z^{k+1}Use in source · 1 passage
Search aliases: z^k+1
Use in source · Paper_VI_Traditional_Final.tex, line 4616
For large z, Σ_P(z) = ∑_k=0^∞ fracA_PQA_QQ^kA_QPz^k+1. Thus the second condition implies the moment identities. Conversely, the first d identities imply all later ones by the Cayley--Hamilton theorem, so the series vanishes.
Exact mathematical passage
\Sigma_P(z) = \sum_{k=0}^{\infty} \frac{A_{PQ}A_{QQ}^kA_{QP}}{z^{k+1}}.
z^{k+j}Use in source · 1 passage
Search aliases: z^k+j
Use in source · 72_recovered.tex, line 2199
The lowest degree term of Φ_M(z) is Mz, and its highest degree term is z^M. Multiplication by (1+z)^s gives the two edge coefficients. When M=2 and s=0, Φ_2(z)^k = (2z+z^2)^k = z^k(2+z)^k. Taking the coefficient of z^k+j gives the binary formula.
Exact mathematical passage
z^{k+j}
z^MUse in source · 3 passages
Search aliases: z^M
Use in source · 10_foundations.tex, line 488
polyphase vocabulary. We explain it through elementary identities with explicit algebraic derivations. These identities have not been separately formalized in Lean for this consolidation. For a sequence a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient and remainder give A(z)=∑_r=0^M-1z^rA_r(z^M), (a_r^[M])_s^[N]=a_r+Ms^[MN]. For ordinary Cauchy multiplication, the phase-r product has a term without a …
Exact mathematical passage
A(z)=\sum_{r=0}^{M-1}z^rA_r(z^M),\qquad \bigl(a_r^{[M]}\bigr)_s^{[N]}=a_{r+Ms}^{[MN]}.
\Z^{m+k}Use in source · 3 passages
Search aliases: Z^m+k Z mathbb
Use in source · 30_lifting.tex, line 596
[Every compatible integer parameter path] In the preceding notation, assume J_x is surjective. For every supplied integer path t_n∈ Z^k satisfying t_0=free(x), t_n+1-t_n∈ q^L+n Z^k, there is a compatible polynomial tower z_n∈ Z^m+k with z_0=x, free(z_n)=t_n, F(z_n)∈ q^L+n Z^e, z_n+1-z_n∈ q^L+n Z^m+k. The free-coordinate equality is exact in integers. Both reduced Jacobian blocks remain fixed, and z_b-z_a∈ q^L+a Z^m+k whenever a≤ b.
Exact mathematical passage
z_n\in\Z^{m+k}
\mathbb Z^{m+k}Use in source · 5 passages
Search aliases: Z^m+k mathbb
Use in source · continuation_08.tex, line 10
Let q>0, L≥1, and m,k,e≥0. Write S= Z/q Z. Let F∈ Z[Y_1,…,Y_m,T_1,…,T_k]^e, x=(y_0,t_0)∈ Z^m+k, F(x)≡0pmodq^L. The actual lower root supplies the unique integral residual r=q^-LF(x)∈ Z^e, bar r=rbmod q∈ S^e. Let the actual reduced Jacobian blocks be J=(D_YF)(x)bmod q:S^m→ S^e, K=(D_TF)(x)bmod q:S^k→ S^e. For a prescribed parameter digit v∈ S^k, choose its canonical integral representative widehat v. Define L_v to be the set of actual residue vectors z∈( Z/q^L+1 Z)^m+k satisfying z≡ xpmodq^L, F(z…
Exact mathematical passage
F\in\mathbb Z[Y_1,\ldots,Y_m,T_1,\ldots,T_k]^e, \qquad x=(y_0,t_0)\in\mathbb Z^{m+k}, \qquad F(x)\equiv0\pmod{q^L}.
z^{N-1}Use in source · 4 passages
Search aliases: z^N-1
Use in source · Paper_IV_Traditional_Final.tex, line 1292
Since D_N(z)=Δ_N(z^N), -frac D_N'(z) D_N(z) = -Nz^N-1fracΔ_N'(z^N)Δ_N(z^N). Differentiate the defining series for -logΔ_N and compare coefficients.
Exact mathematical passage
-\frac{\Dfun_N'(z)}{\Dfun_N(z)} = -Nz^{N-1}\frac{\Delta_N'(z^N)}{\Delta_N(z^N)}.
z^{Nm}Use in source · 1 passage
Search aliases: z^Nm
Use in source · Paper_VI_Traditional_Final.tex, line 5911
This section presents the operator-substitution result transferred from Chapter IV. Fix an integer N≥1 and the entire scalar function D_N(z)=∑_m=0^∞frac(-1)^m z^Nm(Nm)!. More generally, the result applies to any specified holomorphic scalar function that is nonzero on the declared domain. On a simply connected zero-free domain U, choose g=log D_N once and define S_N^(r)(z)=exp(-r g(z)) for complex r. The bounded Banach-space result below uses classical holomorphic functional calculus. It is sepa…
Exact mathematical passage
\Dfun_N(z)=\sum_{m=0}^{\infty}\frac{(-1)^m z^{Nm}}{(Nm)!}.
z^{Nm-1}Use in source · 2 passages
Search aliases: z^Nm-1
Use in source · Paper_IV_Traditional_Final.tex, line 1277
[Logarithmic coefficient formula] The transfer tangent satisfies T_N(z) = N∑_m≥1κ_N,mz^Nm-1. Equivalently, T_N(n) = N(Nn+N-1)!κ_N,n+1.
Exact mathematical passage
\Tfun_N(z) = N\sum_{m\geq1}\kappa_{N,m}z^{Nm-1}.
z^{Nn+d_{pq}}Use in source · 1 passage
Search aliases: z^Nn+d_pq
Use in source · Paper_IV_Traditional_Final.tex, line 908
For every nonnegative integer n, J_N^Nn+d_pqe_q = (-1)^nε_pqe_p. Only powers whose exponent is congruent to d_pq modulo N contribute to the (p,q) entry of the exponential. Therefore (e^zJ_N)_pq = ε_pq ∑_n≥0(-1)^n fracz^Nn+d_pq(Nn+d_pq)!. This is the stated formula.
Exact mathematical passage
\bigl(e^{zJ_N}\bigr)_{pq} = \varepsilon_{pq} \sum_{n\geq0}(-1)^n \frac{z^{Nn+d_{pq}}}{(Nn+d_{pq})!}.
z^{Nn-1}Use in source · 1 passage
Search aliases: z^Nn-1
Use in source · Paper_IV_Traditional_Final.tex, line 1052
Set S= D_N^-r. Logarithmic differentiation gives D_NS'+r D_N'S=0. Taking the coefficient of z^Nn-1/(Nn-1)! gives ∑_k=0^n-1(-1)^k binomNn-1NkP_N,n-k(r) +r∑_k=1^n(-1)^k binomNn-1Nk-1P_N,n-k(r)=0. The term with k=0 in the first sum is P_N,n(r), and every other term has a smaller block index. Rearrangement gives the formula.
Exact mathematical passage
z^{Nn-1}/(Nn-1)!
z^pUse in source · 2 passages
Search aliases: z^p
Use in source · supplement_09.tex, line 31
sumes primality, the modulus divisibility and the specified residue bounds. It does not assume weighted section membership, reciprocal closure, the ratio identity, a Mahler pairing or the desired higher-difference bound. Pages 2--5 describe the constructed objects and proved bridges. 1. Integral cyclic coefficients. Work first in an arbitrary commutative ring R. If z^p=1, set I=(p,z-1), P=(p). Expanding ((z-1)+1)^p=1 and using divisibility of every interior binomial coefficient in prime row p gi…
Exact mathematical passage
z^p=1
z^{p-1}Use in source · 2 passages
Search aliases: z^p-1
Use in source · odd_synthesis.tex, line 60
Let P_0 extract the constant coefficient in the basis 1,z,…,z^p-1. The normalized corestriction s=κpr, where κ=-f'(z)/z^p-1, satisfies u_0=s(P_0)=a_0/z^p∈ O_E^×, Lift_E(C)=u_0^-1CP_0, s(Lift_E(C))=C. The lift preserves the exact filtration valuation. Its crucial additional identity is tr_L(Lift_E(C)) =Tr_E/L (κ^-1tr_E(C)). Define B(L,t_L;h_1,…,h_d) =α I_p^d-1+ Lift_E(B(E,z;h_1,…,h_d-1)), with the degree-p multiplication matrix as the base case. Induction proves trace zero at every native rank. T…
Exact mathematical passage
1,z,\ldots,z^{p-1}
z^{p^a}Use in source · 1 passage
Search aliases: z^p^a
Use in source · 72_recovered.tex, line 2313
In characteristic p, (1+z)^p^a-1≡ z^p^a. Substitute this congruence in the definition of Θ_p^a,s.
Exact mathematical passage
(1+z)^{p^a}-1\equiv z^{p^a}.
z^qUse in source · 1 passage
Search aliases: z^q
Use in source · Paper_IV_Traditional_Final.tex, line 1551
[Formal differential-operator representation on polynomials] Let a formal series A(z)=∑_q≥0a_qz^q/q! act on polynomials by A(∂_x)f(x) = ∑_q≥0fraca_qq!∂_x^qf(x). The power r is applied to the formal symbol before the substitution z=∂_x. Then E_m^(N,j,r)(x) = C_N,j(∂_x)^r x^m. The inverse relation is x^m = Φ_N,j(∂_x^N)^r E_m^(N,j,r)(x). Both sums are finite on every polynomial.
Exact mathematical passage
A(z)=\sum_{q\geq0}a_qz^q/q!
z^{q_k}Use in source · 1 passage
Search aliases: z^q_k
Use in source · Paper_V_Traditional_Final.tex, line 516
Original-sample delay. For an arbitrary original-sample delay D∈ Z, the output phase k has y_k[n]=x_r_k[n+q_k], r_k=(k-D)bmod N, q_k=(k-D)mathbindivN. Its channel matrix has the sole nonzero entry z^q_k in row k, column r_k. In particular, a one-sample delay at N=2 has matrix 0 · z^-1
Exact mathematical passage
z^{q_k}
Z^rUse in source · 2 passages
Search aliases: Z^r
Use in source · strengthening.tex, line 539
[Bernoulli divided-power moments] Let Z be the ordinary Bernoulli variable, L(Z^r)=B_r, with exponential transform t/(e^t-1). For P=Z(Z+1) and μ_m= L(P^m/m!), μ_m∈ Z_2, v_2(μ_m)≥ s_2(m)-1 (m≥1). Moreover, μ_m is odd exactly when m=0 or m is a positive power of two.
Exact mathematical passage
\mathcal L(Z^r)=B_r
z^rUse in source · 9 passages
Search aliases: z^r
Use in source · 10_foundations.tex, line 488
polyphase vocabulary. We explain it through elementary identities with explicit algebraic derivations. These identities have not been separately formalized in Lean for this consolidation. For a sequence a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient and remainder give A(z)=∑_r=0^M-1z^rA_r(z^M), (a_r^[M])_s^[N]=a_r+Ms^[MN]. For ordinary Cauchy multiplication, the phase-r product has a term without a …
Exact mathematical passage
A(z)=\sum_{r=0}^{M-1}z^rA_r(z^M),\qquad \bigl(a_r^{[M]}\bigr)_s^{[N]}=a_{r+Ms}^{[MN]}.
Z^{\otimes n}Use in source · 1 passage
Search aliases: Z^⊗ n otimes
Use in source · 01_finite_core.tex, line 228
For parity, S=Z^⊗ n, P_even=fracI+S2, P_odd=fracI-S2.
Exact mathematical passage
S=Z^{\otimes n}, \qquad P_{\mathrm{even}}=\frac{I+S}{2}, \qquad P_{\mathrm{odd}}=\frac{I-S}{2}.
Z^{\otimes3}Use in source · 1 passage
Search aliases: Z^⊗3 otimes
Use in source · 01_finite_core.tex, line 270
[Exact parity refinement] For three qubits, the local computational projectors were grouped by the parity of their Hamming weight. The resulting coarse projectors agreed with (I± Z^⊗3)/2 with residual 0.
Exact mathematical passage
(I\pm Z^{\otimes3})/2
Z_{\mathrm A}Use in source · 1 passage
Search aliases: Z_ A mathrm
Use in source · atlas-pages.tex, line 204
im. Multiplying all likelihoods by one common positive factor leaves this posterior unchanged. vfigevidence platecaptionThe A packet becomes heavier and its internal shape changes. The B packet retains an equal internal split but loses total mass. These are different effects of the same likelihood. headReconstruct from packet mass times packet shape The unnormalized packet evidences are Z_ A=5/4 and Z_ B=1/2. The resulting masses and conditional shapes are m_ A=5/7, q_ A=(4/5,1/5,0,0)^ T, m_ B=2…
Exact mathematical passage
Z_{\mathrm A}=5/4
Z_{\mathrm B}Use in source · 1 passage
Search aliases: Z_ B mathrm
Use in source · atlas-pages.tex, line 204
lihoods by one common positive factor leaves this posterior unchanged. vfigevidence platecaptionThe A packet becomes heavier and its internal shape changes. The B packet retains an equal internal split but loses total mass. These are different effects of the same likelihood. headReconstruct from packet mass times packet shape The unnormalized packet evidences are Z_ A=5/4 and Z_ B=1/2. The resulting masses and conditional shapes are m_ A=5/7, q_ A=(4/5,1/5,0,0)^ T, m_ B=2/7, q_ B=(0,0,1/2,1/2)^ …
Exact mathematical passage
Z_{\mathrm B}=1/2
Z_{(1,2,4)}Use in source · 1 passage
Search aliases: Z_(1,2,4)
Use in source · Paper_X_Block27_Amended.tex, line 1859
For weights (1,2,4), the counts are 1,2,8,32,256,2048,…. The first coordinate is active immediately, the second one level later, and the third at depth five. The exact rational function generated by the code is Z_(1,2,4)(T) = fracT(1-6T-8T^2-32T^3)1-8T. The negative signs in this presentation arise from moving the denominator to 1-8T. Every power-series coefficient is positive.
Exact mathematical passage
Z_{(1,2,4)}(T) = \frac{T(1-6T-8T^2-32T^3)}{1-8T}.
Z_0Use in source · 4 passages
Search aliases: Z_0
Use in source · 06_models.tex, line 42
Let Z and Z_0≠0 be complex impedances, put z=Z/Z_0, and assume z≠-1. Write Γ=fracz-1z+1. On the open reflection disk |Γ|<1, multiplication by any phase ω_N^k stays in the disk and has a finite inverse-chart image. The phase action is Γ↦ω_N^kΓ. Outside the open disk, the same chart formula requires 1-ω_N^kΓ≠0. Pulling the phase action back on this domain gives T_N,k(z) = frac(1+ω_N^k)z+(1-ω_N^k) (1-ω_N^k)z+(1+ω_N^k). For N=2 and k=1, T_2,1(z)=frac1z, z≠0,-1. Thus the nontrivial action at the firs…
Exact mathematical passage
Z_0\ne0
z_0Use in source · 5 passages
Search aliases: z_0
Use in source · visual_atlas.tex, line 477
headWhat one shift does in these coordinates Let S=S_1 and z=Sx. Substituting 3k+j into the shift definition gives z_1[k]=x_0[k], z_2[k]=x_1[k], z_0[k]=x_2[k-1]. The passage from phase 2 to phase 0 carries a shift of the coarse address. If Lh[k]=h[k-1], the exact packet matrix is S_packets=mat0 · 0 · L
Exact mathematical passage
z_1[k]=x_0[k],\qquad z_2[k]=x_1[k],\qquad z_0[k]=x_2[k-1].
Z_{0,1}Use in source · 1 passage
Search aliases: Z_0,1
Use in source · visual_atlas.tex, line 310
From QZ_0,1=2t^2 one obtains a_0=a_1=0, a_2=2, a_n=a_n-1+2a_n-3 (n≥3). The numerator is responsible for the initial exception. Omitting it would incorrectly force the entire series to vanish.
Exact mathematical passage
QZ_{0,1}=2t^2
Z_{0,2}Use in source · 2 passages
Search aliases: Z_0,2
Use in source · 03_weighted.tex, line 463
[A transfer series with one loop] Take vertices 0, 1, and 2. Give the edge 0→1 weight one, the loop 1→1 weight two, and the edge 1→2 weight one. Every path from 0 to 2 uses the first and last edges and may traverse the loop any number of times. Hence Z_0,2(t) = t^2(1+t^2+t^4+⋯) = fract^21-t^2. There is one path at every positive even weight and no path at an odd weight. The denominator records the increase of two units produced by the loop.
Exact mathematical passage
Z_{0,2}(t) = t^2(1+t^2+t^4+\cdots) = \frac{t^2}{1-t^2}.
Z_1Use in source · 8 passages
Search aliases: Z_1
Use in source · 72_recovered.tex, line 1313
[The first odd reconstruction] For p=3 and n=2, S_2(3)=1-9+25=17. Since Z_1=1, Z_3=2, and S_0(3)=1, the odd coordinate formula gives E_2 =17-2·9 =-1. The large source value and the depth one correction are both necessary.
Exact mathematical passage
Z_1=1
z_1Use in source · 2 passages
Search aliases: z_1
Use in source · visual_atlas.tex, line 113
vpage5Relations identify words, not just outputsPRESENTATIONS / WHY A LEFT IDEAL APPEARSMain text: [source reference] Let F=𝕜⟨ z_1,…,z_r⟩ be the algebra of noncommuting operator words. Evaluation at a fixed seed is Φ_s:F→ X, f↦ f(T_1,…,T_r)s. Two words represent the same generated vector exactly when their difference lies in I_s=KerΦ_s. Consequently F/I_s≅Hull_F(s) as left modules. The relation f(T)s=0 implies g(T)f(T)s=0 for every word g, which explains closure under multiplication on the left.
Exact mathematical passage
F=\kk\langle z_1,\ldots,z_r\rangle
Z_2Use in source · 2 passages
Search aliases: Z_2
Use in source · atlas-pages.tex, line 608
,3,2,0)^ T, p_2=tfrac1104(41,15,40,8)^ T. vfigintegrated_filter platecaptionWithin-packet probabilities change during the two observations, but the packet totals agree exactly with an independent two-state filter. The second coarse posterior is (7/13,6/13). headCheck the normalizers, not only the final vector The first and second predictive evidences are Z_1=frac58, Z_2=frac1320, Z_1Z_2=frac1332. The unnormalized product D_2KD_1Kp_0 has mass 13/32 and normalizes to p_2. Because the likelihood en…
Exact mathematical passage
Z_1=\frac58,\qquad Z_2=\frac{13}{20}, \qquad Z_1Z_2=\frac{13}{32}.
z_2Use in source · 1 passage
Search aliases: z_2
Use in source · visual_atlas.tex, line 477
headWhat one shift does in these coordinates Let S=S_1 and z=Sx. Substituting 3k+j into the shift definition gives z_1[k]=x_0[k], z_2[k]=x_1[k], z_0[k]=x_2[k-1]. The passage from phase 2 to phase 0 carries a shift of the coarse address. If Lh[k]=h[k-1], the exact packet matrix is S_packets=mat0 · 0 · L
Exact mathematical passage
z_1[k]=x_0[k],\qquad z_2[k]=x_1[k],\qquad z_0[k]=x_2[k-1].
\Z_2^2Use in source · 1 passage
Search aliases: Z_2^2 Z mathbb
Use in source · 04_arithmetic.tex, line 650
[Odd-adic versus dyadic behavior] For every prime p≠2, every entry of E_+ and E_- is p-integral, and each projector has exact local denominator weight zero. At p=2, each projector has exact local denominator weight one. Multiplication by 2 makes it integral, while the unscaled matrix is not integral: s_2, Z_2^2(E_+)=s_2, Z_2^2(E_-)=1. The characteristic factors x-1 and x+1 have resultant 2. Its 2-adic valuation is one, matching both local projector weights above, and its p-adic valuation is zero…
Exact mathematical passage
s_{2,\Z_2^2}(E_+)=s_{2,\Z_2^2}(E_-)=1.
\mathbb Z_2^2Use in source · 1 passage
Search aliases: Z_2^2 mathbb
Use in source · 04_arithmetic.tex, line 436
[Strict growth under refinement] Take the standard lattice Z_2^2 and the involution from [source reference]. The one packet system (I) has weight zero. Refining it into the two rational sign packets produces weight one. The coarse lattice is already split because there is only one summand. The fine eigenspaces are glued together by one dyadic digit.
Exact mathematical passage
\mathbb Z_2^2
\mathfrak Z_2^{\mathrm{abs}}Use in source · 1 passage
Search aliases: Z_2^abs mathfrak mathrm
Use in source · Paper_IV_Traditional_Final.tex, line 2866
[Exact zeta models in orders two and four] For N=2, Z_2^red(s) = π^-2sζ(2s,frac12) and Z_2^abs(s) = 2π^-sζ(s,frac12). For N=4, put A_4=sqrt2π. Then Z_4^red(s) = A_4^-4sζ(4s,frac12) and Z_4^abs(s) = 4A_4^-sζ(s,frac12). In these two cases the entire correction terms in [source reference] vanish identically.
Exact mathematical passage
\mathfrak Z_2^{\mathrm{abs}}(s) = 2\pi^{-s}\zeta\left(s,\frac12\right).
\mathfrak Z_2^{\mathrm{red}}Use in source · 1 passage
Search aliases: Z_2^red mathfrak mathrm
Use in source · Paper_IV_Traditional_Final.tex, line 2860
[Exact zeta models in orders two and four] For N=2, Z_2^red(s) = π^-2sζ(2s,frac12) and Z_2^abs(s) = 2π^-sζ(s,frac12). For N=4, put A_4=sqrt2π. Then Z_4^red(s) = A_4^-4sζ(4s,frac12) and Z_4^abs(s) = 4A_4^-sζ(s,frac12). In these two cases the entire correction terms in [source reference] vanish identically.
Exact mathematical passage
\mathfrak Z_2^{\mathrm{red}}(s) = \pi^{-2s}\zeta\left(2s,\frac12\right)
\mathfrak Z_2^{\rm red}Use in source · 1 passage
Search aliases: Z_2^rm red mathfrak rm
Use in source · atlas-pages.tex, line 441
vpage24Convergence is not continuationZERO ZETA FUNCTIONSChapter IV: Theorem 14.2, Theorem 14.3, Corollary 14.4. Reduced zeros grow like m^N. Their absolute zero-zeta series therefore has half-plane of absolute convergence Re s>1/N. The absolute zero zeta in the original variable instead requires Re s>1. vfigzeta_plane For N=2 and N=4, the lattices are exact: Z_2^rm red(s)=π^-2sζ(2s,1/2), Z_4^rm red(s)=(sqrt2π)^-4sζ(4s,1/2). Each identity first holds where its defining series converges. The righ…
Exact mathematical passage
\mathfrak Z_2^{\rm red}(s)=\pi^{-2s}\zeta(2s,1/2),\qquad \mathfrak Z_4^{\rm red}(s)=(\sqrt2\pi)^{-4s}\zeta(4s,1/2).
\mathbb Z_2^\timesUse in source · 3 passages
Search aliases: Z_2^× mathbb times
Use in source · 72_recovered.tex, line 1598
[Basic odd and dyadic difference gains] For odd p, Δ_(p-1)p^a^ ru^n = u^n(u^(p-1)p^a-1)^r, v_p(Δ_(p-1)p^a^ ru^n)≥(a+1)r. For odd u∈ Z_2^×, Δ_2^a+1^ ru^n = u^n(u^2^a+1-1)^r, v_2(Δ_2^a+1^ ru^n)≥(a+3)r.
Exact mathematical passage
u\in\mathbb Z_2^\times
\mathbb Z_2^{\times}Use in source · 1 passage
Search aliases: Z_2^× mathbb times
Use in source · strengthening.tex, line 307
dd moments vanish. Frobenius's congruence is v_2(∑_r=0^m(-1)^m-rbinom mr Euler_2n+2r) =m+v_2(m!) (n,m≥0). For m=0, the assertion is that every even Euler number is odd. With P=( E^2-1)/2 and γ_m(P)=P^m/m!, the case n=0 says L(γ_m(P))∈ Z_2^× (m≥0). Introduce an independent Euler variable F and Q=( F^2-1)/2.
Exact mathematical passage
\mathcal L(\gamma_m(P))\in\mathbb Z_2^{\times} \qquad(m\geq0). \label{eq:euler-pd-odd-moments}
Z_{2m+1}Use in source · 4 passages
Search aliases: Z_2m+1
Use in source · 20_arithmetic.tex, line 786
The same inverse coefficients can be written in the signed zigzag notation of the original manuscript. Define Z_r by sec t+tan t=∑_r≥0Z_rfract^rr!. Then γ_m=(-1)^m Z_2m+1 (m∈ N). Indeed, the odd coefficients come from tan t, whose derivative is sec^2t. Changing the sign of the squared variable in the even series for sech^2t gives sec^2t, so its degree-2m Hurwitz coefficient is (-1)^mγ_m. Differentiation identifies this coefficient with Z_2m+1, proving the formula. The Lean bridge is leanKomorebi…
Exact mathematical passage
\label{eq:arith-signed-zigzag} \gamma_m=(-1)^m Z_{2m+1}\qquad(m\in\N).
Z_{2r+1}Use in source · 7 passages
Search aliases: Z_2r+1
Use in source · 72_recovered.tex, line 948
[Zigzag inverse kernel] sech^2t = ∑_r≥0 (-1)^rZ_2r+1 fract^2r(2r)!.
Exact mathematical passage
\boxed{ \operatorname{sech}^2t = \sum_{r\geq0} (-1)^rZ_{2r+1} \frac{t^{2r}}{(2r)!}. }
Z_3Use in source · 1 passage
Search aliases: Z_3
Use in source · 72_recovered.tex, line 1313
[The first odd reconstruction] For p=3 and n=2, S_2(3)=1-9+25=17. Since Z_1=1, Z_3=2, and S_0(3)=1, the odd coordinate formula gives E_2 =17-2·9 =-1. The large source value and the depth one correction are both necessary.
Exact mathematical passage
Z_3=2
\Z_3Use in source · 3 passages
Search aliases: Z_3 Z mathbb
Use in source · 72_recovered.tex, line 1675
:matrix_carmichael_mahler Editorial hypothesis correction. The original displayed statement leaves the vectors untyped. Its integrality conclusion requires v_0,w∈ Z_p^d, L≥1, and H,j,n∈ N with H>0. The value at Hx+j for x∈ Z_p means the matrix-binomial interpolation. With p=3, C=4, L=1, w=1 and v_0=1/3, one has Δ_1a_0=1, which is not in 3 Z_3. The printed claim without these restrictions is therefore not a certificate as written. The full original proof follows under the corrected hypotheses. As…
Exact mathematical passage
3\Z_3
\mathbb Z_3Use in source · 3 passages
Search aliases: Z_3 mathbb
Use in source · Paper_X_Block27_Amended.tex, line 924
An actual Z_3 example applies the class-level completion theorem for the cyclic group of order two and polynomial X^2-1, producing a nonidentity generator. Another checked example has compatible classes represented by diag(1,-1) modulo 3 and diag(-1,1) modulo 9: these chosen matrices disagree under reduction, and an actual swap of basis repairs them. Arbitrary independent representatives therefore do not replace the framing argument.
Exact mathematical passage
\mathbb Z_3
\mathfrak Z_4^{\mathrm{abs}}Use in source · 1 passage
Search aliases: Z_4^abs mathfrak mathrm
Use in source · Paper_IV_Traditional_Final.tex, line 2878
[Exact zeta models in orders two and four] For N=2, Z_2^red(s) = π^-2sζ(2s,frac12) and Z_2^abs(s) = 2π^-sζ(s,frac12). For N=4, put A_4=sqrt2π. Then Z_4^red(s) = A_4^-4sζ(4s,frac12) and Z_4^abs(s) = 4A_4^-sζ(s,frac12). In these two cases the entire correction terms in [source reference] vanish identically.
Exact mathematical passage
\mathfrak Z_4^{\mathrm{abs}}(s) = 4A_4^{-s}\zeta\left(s,\frac12\right).
\mathfrak Z_4^{\mathrm{red}}Use in source · 1 passage
Search aliases: Z_4^red mathfrak mathrm
Use in source · Paper_IV_Traditional_Final.tex, line 2872
[Exact zeta models in orders two and four] For N=2, Z_2^red(s) = π^-2sζ(2s,frac12) and Z_2^abs(s) = 2π^-sζ(s,frac12). For N=4, put A_4=sqrt2π. Then Z_4^red(s) = A_4^-4sζ(4s,frac12) and Z_4^abs(s) = 4A_4^-sζ(s,frac12). In these two cases the entire correction terms in [source reference] vanish identically.
Exact mathematical passage
\mathfrak Z_4^{\mathrm{red}}(s) = A_4^{-4s}\zeta\left(4s,\frac12\right)
\mathfrak Z_4^{\rm red}Use in source · 1 passage
Search aliases: Z_4^rm red mathfrak rm
Use in source · atlas-pages.tex, line 441
vpage24Convergence is not continuationZERO ZETA FUNCTIONSChapter IV: Theorem 14.2, Theorem 14.3, Corollary 14.4. Reduced zeros grow like m^N. Their absolute zero-zeta series therefore has half-plane of absolute convergence Re s>1/N. The absolute zero zeta in the original variable instead requires Re s>1. vfigzeta_plane For N=2 and N=4, the lattices are exact: Z_2^rm red(s)=π^-2sζ(2s,1/2), Z_4^rm red(s)=(sqrt2π)^-4sζ(4s,1/2). Each identity first holds where its defining series converges. The righ…
Exact mathematical passage
\mathfrak Z_2^{\rm red}(s)=\pi^{-2s}\zeta(2s,1/2),\qquad \mathfrak Z_4^{\rm red}(s)=(\sqrt2\pi)^{-4s}\zeta(4s,1/2).
\Z_5Use in source · 1 passage
Search aliases: Z_5 Z mathbf
Use in source · Paper_X_Block27_Amended.tex, line 2461
d is therefore integral. Reduction modulo three rules out a root in Q_3, and reduction modulo four rules out a root in Q_2. At an odd prime, any residue root of X^2+1 has nonzero derivative and lifts by the pinned Mathlib Hensel theorem. The finite-field square criterion then gives the stated congruence. An explicit Hensel application at the residue root 2 modulo 5 supplies a root in Z_5. Consequently LocallyTrivial_ Q_p(χ) Longleftrightarrow p≡1pmod4 lor χ=1. The arithmetic prime p is independe…
Exact mathematical passage
\Z_5
\Z_{>0}Use in source · 1 passage
Search aliases: Z_>0 Z mathbb
Use in source · 03_weighted.tex, line 249
A packet decomposition resolves H=∑_e∈ EH_e and assigns a certified filtration gain D^-1H_e∈ F^w(e) B, w(e)∈ Z_>0. Each e is a directed transfer edge. For an ordered transfer word γ=(e_1,…,e_r), define W(γ)=∑_j=1^rw(e_j), A_γ=A_e_1⋯ A_e_r, A_e=D^-1H_e.
Exact mathematical passage
D^{-1}H_e\in F^{w(e)}\cB, \qquad w(e)\in\Z_{>0}.
Z_aUse in source · 1 passage
Search aliases: Z_a
Use in source · Paper_VI_Traditional_Final.tex, line 5420
The evidence in block B_a is Z_a =q_a∑_i∈ B_aL_ir_i^(a) =q_aoverline L_a. Apply [source reference].
Exact mathematical passage
Z_a =q_a\sum_{i\in B_a}L_ir_i^{(a)} =q_a\overline L_a.
z_aUse in source · 8 passages
Search aliases: z_a
Use in source · 30_lifting.tex, line 342
z_n+1-z_n · ∈ q^L+n Z^d, · J_z_n · =J_x. For a≤ b, it also satisfies z_b-z_a∈ q^L+a Z^d.
Exact mathematical passage
z_b-z_a\in q^{L+a}\Z^d
z_bUse in source · 2 passages
Search aliases: z_b
Use in source · 30_lifting.tex, line 342
z_n+1-z_n · ∈ q^L+n Z^d, · J_z_n · =J_x. For a≤ b, it also satisfies z_b-z_a∈ q^L+a Z^d.
Exact mathematical passage
z_b-z_a\in q^{L+a}\Z^d
Z_{\bar\lambda}Use in source · 4 passages
Search aliases: Z_barλ bar lambda
Use in source · Paper_VI_Traditional_Final.tex, line 926
[Bayesian information-loss packet] Assume nonnegative fine and declared coarse likelihoods, with both Z_r>0 and Z_barλ>0. The coarsening defect is C_π B_λ(p) - B_barλ(P) = frac1Z_r (diag(P)-bar bP^ T) δ(p). The factor matrix annihilates barλ. This records the common-scale null direction at the declared coarse likelihood. The full exact defect remains homogeneous under simultaneous likelihood rescaling because the numerator and evidence denominator scale together. Moreover, C_π B_λ(p)- B_barλ(P) …
Exact mathematical passage
Z_{\bar\lambda}>0
Z_cUse in source · 2 passages
Search aliases: Z_c
Use in source · 03_nonlinear_coordinates.tex, line 749
[Exact defect under composition] Let G_f:Y_f→ Z_f and G_c:Y_c→ Z_c, with coarsening C_Z:Z_f→ Z_c. Then Δ_G∘ F(x) = Δ_G(F_f(x)) + G_c(F_c(C_Xx)+Δ_F(x)) - G_c(F_c(C_Xx)). If G_c is L-Lipschitz on the relevant set, then Δ_G∘ F(x) ≤ Δ_G(F_f(x)) + L Δ_F(x) .
Exact mathematical passage
G_c:Y_c\to Z_c
Z_dUse in source · 4 passages
Search aliases: Z_d
Use in source · Paper_X_Block27_Amended.tex, line 543
[Smooth seed zeta function] For a smooth formal component of dimension d, Z_d(T)=fracT1-q^dT.
Exact mathematical passage
\boxed{ Z_d(T)=\frac{T}{1-q^dT}. }
Z_eUse in source · 1 passage
Search aliases: Z_e
Use in source · Paper_X_Block27_Amended.tex, line 592
[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.
Exact mathematical passage
\boxed{ Z_e(T) = \frac{\sum_{r=1}^{e}q^{r-1}T^r} {1-q^{e-1}T^e}. }
Z_fUse in source · 2 passages
Search aliases: Z_f
Use in source · 03_nonlinear_coordinates.tex, line 749
[Exact defect under composition] Let G_f:Y_f→ Z_f and G_c:Y_c→ Z_c, with coarsening C_Z:Z_f→ Z_c. Then Δ_G∘ F(x) = Δ_G(F_f(x)) + G_c(F_c(C_Xx)+Δ_F(x)) - G_c(F_c(C_Xx)). If G_c is L-Lipschitz on the relevant set, then Δ_G∘ F(x) ≤ Δ_G(F_f(x)) + L Δ_F(x) .
Exact mathematical passage
G_f:Y_f\to Z_f
Z_jUse in source · 1 passage
Search aliases: Z_j
Use in source · atlas-pages.tex, line 294
_order platecaptionBoth routes produce valid probability vectors. Their difference is therefore not a normalization bug. It is a different conditioning problem. headKeep an unnormalized path alongside the filter For a sequence of observations, let D_r=diag(λ^(r)). The forward recursion is widehat p_r=Kp_r-1, p_r=D_rwidehat p_r/Z_r. Its unnormalized companion is v_r=D_rK⋯ D_1Kp_0, p_r=fracv_rones^ Tv_r, ones^ Tv_r=∏_j=1^r Z_j. The product of evidences records the normalization lost at each indivi…
Exact mathematical passage
v_r=D_rK\cdots D_1Kp_0, \qquad p_r=\frac{v_r}{\ones^{\mathsf T}v_r}, \qquad \ones^{\mathsf T}v_r=\prod_{j=1}^r Z_j.
z_jUse in source · 1 passage
Search aliases: z_j
Use in source · Paper_X_Block27_Amended.tex, line 390
For i≠ j, orthogonality gives H(p_i)H(i_j)=0. Idempotence gives H(p_j)H(i_j)=I, and completeness gives ∑_j H(i_j)H(p_j)=I. Thus the maps [z]⟼([P_j^r z])_j, ([z_j])_j⟼[∑_j i_j z_j] are inverse linear maps. The finite product equals the direct sum, proving the assertion. In the formalization, both sides are also identified with Mathlib's standard homology objects of the genuine graded complexes.
Exact mathematical passage
[z]\longmapsto([P_j^r z])_j, \qquad ([z_j])_j\longmapsto\left[\sum_j i_j z_j\right]
Z_j^rUse in source · 1 passage
Search aliases: Z_j^r
Use in source · Paper_X_Block27_Amended.tex, line 381
For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet are cochain maps, so they induce maps on these genuine quotients.
Exact mathematical passage
Z_j^r,B_j^r,H_j^r
Z_kUse in source · 1 passage
Search aliases: Z_k
Use in source · 01_manuscript.tex, line 825
R=Q_RR_R, where the columns of Q_L and Q_R are orthonormal. Form the small core G=R_LR_R^* and let G=WΣ Z^* be an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer k≥0, retain min{k,d} columns, where d is the number of supplied thin singular directions. Tail singular values beyond d are zero. Define L_k=Q_LW_kΣ_k^1/2, R_k=Q_RZ_kΣ_k^1/2. Then L_kR_k^* is a best rank at most k approximation to LR^*. Its spectral and Frobenius errors are the s…
Exact mathematical passage
L_k=Q_LW_k\Sigma_k^{1/2}, \qquad R_k=Q_RZ_k\Sigma_k^{1/2}.
Z_{L,a}Use in source · 3 passages
Search aliases: Z_L,a
Use in source · 02_strengthened.tex, line 1993
[Heat packet decomposition] If [L,P_a]=0 for every packet, then Z_L(t)=∑_aZ_L,a(t), Z_L,a(t)=Tr(P_a e^-tL). The global spectral dimension is the average of the packet dimensions weighted by their heat traces: d_L(t) = ∑_a:Z_L,a(t)>0 fracZ_L,a(t)Z_L(t) d_L,a(t), where d_L,a(t) =2t fracTr(P_aL e^-tL)Z_L,a(t) for nonempty packet heat traces.
Exact mathematical passage
Z_L(t)=\sum_aZ_{L,a}(t), \qquad Z_{L,a}(t)=\Tr(P_a\e^{-tL}).
z_mUse in source · 1 passage
Search aliases: z_m
Use in source · continuation_03.tex, line 72
Compatibility implies, for m≥ n, p^L+nmid z_m-z_n. After casting to Z_p, this bounds each coordinate distance by p^-(L+n), so the sequence is Cauchy. Completeness supplies its limit. Closed norm bounds show that the limit retains the congruence at every stage.
Exact mathematical passage
p^{L+n}\mid z_m-z_n.
Z_NUse in source · 4 passages
Search aliases: Z_N
Use in source · monograph_atlas.tex, line 725
vpage37The dimension window has an order of limitsFINITE HEAT DATA AND THE QUANTIFIERSChapter VII: Theorem II.11.5, Corollary II.11.8. For the unit circle grid with N sites, the scaled eigenvalues are λ_N(k)=4N^2sin^2(π k/N). Put Z_N(t)=∑_ke^-tλ_N(k) and d_N(t)=2t∑_kλ_N(k)e^-tλ_N(k)/Z_N(t).
Exact mathematical passage
Z_N(t)=\sum_ke^{-t\lambda_N(k)}
Z_nUse in source · 4 passages
Search aliases: Z_n
Use in source · 72_recovered.tex, line 910
Let q be a positive odd integer. Define S_n(q) = ∑_a=0^q-1 (-1)^a(2a+1)^n. Let the signed Euler numbers and Euler zigzag numbers be defined by secht = ∑_n≥0E_nfract^nn! and sec t+tan t = ∑_n≥0Z_nfract^nn!.
Exact mathematical passage
\sec t+\tan t = \sum_{n\geq0}Z_n\frac{t^n}{n!}.
z_nUse in source · 27 passages
Search aliases: z_n
Use in source · 04_correction_transport.tex, line 65
The space X_R is closed. Indeed, if Rz_n→ x, then z_n=DRz_n→ Dx, so x=R(Dx). Define Ψ(x)=R(y-N(x)). This map takes X_R to itself. For x∈overline B_ρ(0)∩ X_R, Ψ(x) · ≤ R(y-N(0)) + R N(x)-N(0)
Exact mathematical passage
Rz_n\to x
z_{n+1}Use in source · 8 passages
Search aliases: z_n+1
Use in source · 30_lifting.tex, line 338
[Compatible polynomial tower] Let q≥1, L≥1, F∈ Z[X_1,…,X_d]^e, and x∈ Z^d satisfy F(x)∈ q^L Z^e. Suppose J_x:S_q^d→ S_q^e is surjective. There is a single sequence z_n∈ Z^d such that z_0 · =x, · F(z_n) · ∈ q^L+n Z^e, z_n+1-z_n · ∈ q^L+n Z^d, · J_z_n · =J_x. For a≤ b, it also satisfies z_b-z_a∈ q^L+a Z^d.
Exact mathematical passage
z_0&=x,& F(z_n)&\in q^{L+n}\Z^e,\\ z_{n+1}-z_n&\in q^{L+n}\Z^d,& J_{z_n}&=J_x.
\mathfrak Z_N^{\rm abs}Use in source · 1 passage
Search aliases: Z_N^rm abs mathfrak rm
Use in source · atlas-pages.tex, line 448
k Z_2^rm red(s)=π^-2sζ(2s,1/2), Z_4^rm red(s)=(sqrt2π)^-4sζ(4s,1/2). Each identity first holds where its defining series converges. The right side supplies its meromorphic continuation. There is a single pole, at 1/2 or 1/4, with residue 1/(2π) or 1/(4sqrt2π) respectively. headThe factor N matters twice For every order in this chapter, Z_N^rm abs(s)=N Z_N^rm red(s/N). For general N, exponentially accurate zero locations leave an entire correction to the Hurwitz model. A finite exceptional zero s…
Exact mathematical passage
\mathfrak Z_N^{\rm abs}(s)=N\mathfrak Z_N^{\rm red}(s/N).
\mathfrak Z_N^{\rm red}Use in source · 1 passage
Search aliases: Z_N^rm red mathfrak rm
Use in source · atlas-pages.tex, line 448
k Z_2^rm red(s)=π^-2sζ(2s,1/2), Z_4^rm red(s)=(sqrt2π)^-4sζ(4s,1/2). Each identity first holds where its defining series converges. The right side supplies its meromorphic continuation. There is a single pole, at 1/2 or 1/4, with residue 1/(2π) or 1/(4sqrt2π) respectively. headThe factor N matters twice For every order in this chapter, Z_N^rm abs(s)=N Z_N^rm red(s/N). For general N, exponentially accurate zero locations leave an entire correction to the Hurwitz model. A finite exceptional zero s…
Exact mathematical passage
\mathfrak Z_N^{\rm abs}(s)=N\mathfrak Z_N^{\rm red}(s/N).
\mathbb Z_p^2Use in source · 2 passages
Search aliases: Z_p^2 mathbb
Use in source · 04_arithmetic.tex, line 562
0 · -1 act on Z_p^2. Over Q_p, the sign projectors are E_+=fracI+A2 = 1 · 1/2
Exact mathematical passage
\mathbb Z_p^2
\Z_p^dUse in source · 2 passages
Search aliases: Z_p^d Z mathbb
Use in source · 72_recovered.tex, line 1675
[Matrix Carmichael and Mahler transfer] Editorial hypothesis correction. The original displayed statement leaves the vectors untyped. Its integrality conclusion requires v_0,w∈ Z_p^d, L≥1, and H,j,n∈ N with H>0. The value at Hx+j for x∈ Z_p means the matrix-binomial interpolation. With p=3, C=4, L=1, w=1 and v_0=1/3, one has Δ_1a_0=1, which is not in 3 Z_3. The printed claim without these restrictions is therefore not a certificate as written. The full original proof follows under the corrected …
Exact mathematical passage
v_0,w\in\Z_p^d
\mathbb Z_p^dUse in source · 8 passages
Search aliases: Z_p^d mathbb
Use in source · continuation_03.tex, line 20
Let p be prime, L≥1, and F∈ Z[X_1,…,X_d]^e, x∈ Z^d, F(x)≡0pmodp^L. Assume the actual reduced Jacobian A_x=DF(x)bmod p: F_p^d⟶ F_p^e is surjective. Then Lean proves that there is a point a∈ Z_p^d, F(a)=0, a≡ xpmodp^L. Here the coefficients of the original integer polynomials are cast into Z_p, and their multivariable evaluation is exactly zero. This is an existence theorem for a fixed polynomial presentation with the stated Jacobian hypothesis.
Exact mathematical passage
\boxed{a\in\mathbb Z_p^d,\qquad F(a)=0,\qquad a\equiv x\pmod{p^L}.}
\mathbb Z_p^eUse in source · 2 passages
Search aliases: Z_p^e mathbb
Use in source · continuation_05.tex, line 26
((σ z)_i=z_σ(i). Write σ z=(y,t), using e solved coordinates and q free coordinates. Define (y_0,t_0)=σ x, B=x+p^L Z_p^e+q, T=t_0+p^L Z_p^q, R={z∈ B:F(z)=0}. There is a homeomorphism h:Txrightarrow ∼ R, h^-1(z)=free(σ z). Equivalently, a continuous function ϕ:T→ y_0+p^L Z_p^e gives the entire root locus through σ h(t)=(ϕ(t),t). Every parameter has exactly one corresponding root in B. For every N≥0, h(t)≡ h(u)pmodp^Niff t≡ upmodp^N.
Exact mathematical passage
\phi:T\to y_0+p^L\mathbb Z_p^e
\mathbb Z_p^{e+k}Use in source · 2 passages
Search aliases: Z_p^e+k mathbb
Use in source · continuation_09.tex, line 72
Square solved-block uniqueness. If m=e, every prescribed τ has exactly one root in the original ball: ∃! b∈ x+p^L Z_p^e+k: F(b)=0, free(b)=τ. Thus all valid integer parameter approximations and all lifting choices with the same target have the same endpoint. This does not assert uniqueness of the integer representatives in the tower. All prime statements include p=2.
Exact mathematical passage
\exists!\,b\in x+p^L\mathbb Z_p^{e+k}:\quad F(b)=0,\qquad\operatorname{free}(b)=\tau.
\mathbb Z_p^{e+q}Use in source · 1 passage
Search aliases: Z_p^e+q mathbb
Use in source · continuation_05.tex, line 17
There is a coordinate permutation σ, with (σ z)_i=z_σ(i). Write σ z=(y,t), using e solved coordinates and q free coordinates. Define (y_0,t_0)=σ x, B=x+p^L Z_p^e+q, T=t_0+p^L Z_p^q, R={z∈ B:F(z)=0}. There is a homeomorphism h:Txrightarrow ∼ R, h^-1(z)=free(σ z). Equivalently, a continuous function ϕ:T→ y_0+p^L Z_p^e gives the entire root locus through σ h(t)=(ϕ(t),t). Every parameter has exactly one corresponding root in B. For every N≥0, h(t)≡ h(u)pmodp^Niff t≡ upmodp^N.
Exact mathematical passage
(y_0,t_0)=\sigma x,\qquad B=x+p^L\mathbb Z_p^{e+q},\qquad T=t_0+p^L\mathbb Z_p^q,\qquad \mathcal R=\{z\in B:F(z)=0\}.
\mathbb Z_p^kUse in source · 2 passages
Search aliases: Z_p^k mathbb
Use in source · continuation_09.tex, line 42
Arbitrary p-adic target theorem. Now let q=p be prime. For every τ∈ t_0+p^L Z_p^k, there are a compatible integer tower z as above and b∈ Z_p^m+k such that z(n)⟶ b, F(b)=0, free(b)=τ, b-z(n)∈ p^L+n Z_p^m+k (n≥0). A compatible integer parameter path converging to τ is constructed in the proof, with its initial representative fixed exactly. A separate theorem accepts every supplied compatible integer path with limit τ and retains its exact coordinates in the constructed tower.
Exact mathematical passage
\tau\in t_0+p^L\mathbb Z_p^k,
\mathbb Z_p^{m+k}Use in source · 2 passages
Search aliases: Z_p^m+k mathbb
Use in source · continuation_09.tex, line 45
Arbitrary p-adic target theorem. Now let q=p be prime. For every τ∈ t_0+p^L Z_p^k, there are a compatible integer tower z as above and b∈ Z_p^m+k such that z(n)⟶ b, F(b)=0, free(b)=τ, b-z(n)∈ p^L+n Z_p^m+k (n≥0). A compatible integer parameter path converging to τ is constructed in the proof, with its initial representative fixed exactly. A separate theorem accepts every supplied compatible integer path with limit τ and retains its exact coordinates in the constructed tower.
Exact mathematical passage
b\in\mathbb Z_p^{m+k}
\mathbb Z_p^qUse in source · 1 passage
Search aliases: Z_p^q mathbb
Use in source · continuation_05.tex, line 17
There is a coordinate permutation σ, with (σ z)_i=z_σ(i). Write σ z=(y,t), using e solved coordinates and q free coordinates. Define (y_0,t_0)=σ x, B=x+p^L Z_p^e+q, T=t_0+p^L Z_p^q, R={z∈ B:F(z)=0}. There is a homeomorphism h:Txrightarrow ∼ R, h^-1(z)=free(σ z). Equivalently, a continuous function ϕ:T→ y_0+p^L Z_p^e gives the entire root locus through σ h(t)=(ϕ(t),t). Every parameter has exactly one corresponding root in B. For every N≥0, h(t)≡ h(u)pmodp^Niff t≡ upmodp^N.
Exact mathematical passage
(y_0,t_0)=\sigma x,\qquad B=x+p^L\mathbb Z_p^{e+q},\qquad T=t_0+p^L\mathbb Z_p^q,\qquad \mathcal R=\{z\in B:F(z)=0\}.
\Z_p^\timesUse in source · 2 passages
Search aliases: Z_p^× Z times mathbb times
Use in source · supplement_04.tex, line 125
Why a p-adic unit supplies one factor of p. Now let u∈ Z_p^×. The actual ring homomorphism π: Z_p⟶ F_p sends u to a nonzero unit. Fermat's finite-field theorem therefore gives π(u)^p-1=1, π(u^p-1-1)=0. Using the proved kernel identity kerπ=p Z_p, the source derives pmid u^p-1-1 inside Z_p. The conclusion is not supplied as an abstract certificate. It follows from the unit hypothesis and the standard residue-field construction.
Exact mathematical passage
u\in\Z_p^\times
\mathbb Z_p^\timesUse in source · 1 passage
Search aliases: Z_p^× mathbb times
Use in source · 72_recovered.tex, line 2329
[Scalar polyphase precision amplification] Let u∈ Z_p^×, let a,r≥0, and put H_a= (p-1)p^a, · p odd,
Exact mathematical passage
u\in\mathbb Z_p^\times
Z_rUse in source · 11 passages
Search aliases: Z_r
Use in source · 20_arithmetic.tex, line 781
The same inverse coefficients can be written in the signed zigzag notation of the original manuscript. Define Z_r by sec t+tan t=∑_r≥0Z_rfract^rr!. Then γ_m=(-1)^m Z_2m+1 (m∈ N). Indeed, the odd coefficients come from tan t, whose derivative is sec^2t. Changing the sign of the squared variable in the even series for sech^2t gives sec^2t, so its degree-2m Hurwitz coefficient is (-1)^mγ_m. Differentiation identifies this coefficient with Z_2m+1, proving the formula. The Lean bridge is leanKomorebi…
Exact mathematical passage
Z_r
z_rUse in source · 1 passage
Search aliases: z_r
Use in source · visual_atlas.tex, line 113
vpage5Relations identify words, not just outputsPRESENTATIONS / WHY A LEFT IDEAL APPEARSMain text: [source reference] Let F=𝕜⟨ z_1,…,z_r⟩ be the algebra of noncommuting operator words. Evaluation at a fixed seed is Φ_s:F→ X, f↦ f(T_1,…,T_r)s. Two words represent the same generated vector exactly when their difference lies in I_s=KerΦ_s. Consequently F/I_s≅Hull_F(s) as left modules. The relation f(T)s=0 implies g(T)f(T)s=0 for every word g, which explains closure under multiplication on the left.
Exact mathematical passage
F=\kk\langle z_1,\ldots,z_r\rangle
Z_{ra}Use in source · 1 passage
Search aliases: Z_ra
Use in source · 03_supplement_05.tex, line 110
Trace-preserving recovery supplies the original Gram data. Let Z_ra=z_ra. The recovery normalization gives PE_a^*E_bP · =PE_a^*(∑_rR_r^*R_r)E_bP
Exact mathematical passage
Z_{ra}=z_{ra}
z_{ra}Use in source · 3 passages
Search aliases: z_ra
Use in source · 03_supplement_05.tex, line 103
Applied to the corrected Kraus family, this yields scalars z_ra with R_rE_aP=z_raP. The essential declarations are pathrank_one_gram_rigidity, pathkraus_rank_one_rigidity and pathcorrected_kraus_family. No rank-one Choi characterization is inserted as an axiom or as the desired conclusion in hypothesis form.
Exact mathematical passage
z_{ra}
z_{rb}Use in source · 1 passage
Search aliases: z_rb
Use in source · 03_supplement_05.tex, line 111
Trace-preserving recovery supplies the original Gram data. Let Z_ra=z_ra. The recovery normalization gives PE_a^*E_bP · =PE_a^*(∑_rR_r^*R_r)E_bP · =∑_r(R_rE_aP)^*(R_rE_bP) · =(∑_roverlinez_raz_rb)P =(Z^*Z)_abP. Thus C=Z^*Z is positive semidefinite and satisfies the required compressed identities. The theorem pathgram_of_recovery implements this calculation with the original error and recovery matrices.
Exact mathematical passage
PE_a^*E_bP &=PE_a^*\Bigl(\sum_rR_r^*R_r\Bigr)E_bP\\ &=\sum_r(R_rE_aP)^*(R_rE_bP)\\ &=\Bigl(\sum_r\overline{z_{ra}}z_{rb}\Bigr)P =(Z^*Z)_{ab}P.
Z_tUse in source · 1 passage
Search aliases: Z_t
Use in source · 02_connections_obstructions.tex, line 737
^Y_r and S^Y_s. Let F:X→ Y be any function, with no linearity or differentiability hypothesis. Set J^X_b← a= A^X_b∘ S^X_a, J^Y_s← r= A^Y_s∘ S^Y_r, and define widehat F_r← a = A^Y_r∘ F∘ S^X_a. Then J^Y_s← r∘widehat F_r← a = widehat F_s← b∘ J^X_b← a. If A^Z_t:Zxrightarrow∼Z_t is another exact coordinate equivalence and G:Y→ Z is any function, then exact coordinate representation also preserves composition: widehatG∘ F_t← a = widehat G_t← r∘widehat F_r← a. statusleanthreeKomorebi.PaperIII.Transport…
Exact mathematical passage
\cA^Z_t:Z\xrightarrow{\sim}Z_t
z_TUse in source · 1 passage
Search aliases: z_T
Use in source · continuation_08.tex, line 22
LF(x)∈ Z^e, bar r=rbmod q∈ S^e. Let the actual reduced Jacobian blocks be J=(D_YF)(x)bmod q:S^m→ S^e, K=(D_TF)(x)bmod q:S^k→ S^e. For a prescribed parameter digit v∈ S^k, choose its canonical integral representative widehat v. Define L_v to be the set of actual residue vectors z∈( Z/q^L+1 Z)^m+k satisfying z≡ xpmodq^L, F(z)≡0pmodq^L+1, z_T≡ t_0+q^Lwidehat vpmodq^L+1. The formal predicate uses canonical integer representatives for evaluation and the lower congruence. The previously checked polyno…
Exact mathematical passage
z\equiv x\pmod{q^L},\qquad F(z)\equiv0\pmod{q^{L+1}},\qquad z_T\equiv t_0+q^L\widehat v\pmod{q^{L+1}}.
Z_{\widetilde\lambda}Use in source · 1 passage
Search aliases: Z_widetildeλ widetilde lambda
Use in source · Paper_VI_Traditional_Final.tex, line 3614
[Approximate coarse Bayesian rule] Choose one positive number ℓ_α for every packet and define the block constant fine likelihood widetildeλ_i=ℓ_α (i∈ I_α). Then norm C B_λ(p)- B_ℓ(Cp) _1 ≤ frac 2∑_i p_iabsλ_i-widetildeλ_i max{Z_λ(p),Z_widetildeλ(p)} .
Exact mathematical passage
\norm{ C\Bcal_\lambda(p)-\Bcal_\ell(Cp) }_1 \leq \frac{ 2\sum_i p_i\abs{\lambda_i-\widetilde\lambda_i} }{ \max\{Z_\lambda(p),Z_{\widetilde\lambda}(p)\} }.
Z_\alphaUse in source · 9 passages
Search aliases: Z_α alpha
Use in source · Paper_VI_Traditional_Final.tex, line 763
Let the state indices be partitioned into packets I_α. Define Z_α=∑_i∈ I_αλ_ip_i, m_α=fracZ_α∑_β Z_β, q_α,i= λ_ip_i/Z_α, · Z_α>0, i∈ I_α, 0, · otherwise.
Exact mathematical passage
Z_\alpha=\sum_{i\in I_\alpha}\lambda_ip_i, \qquad m_\alpha=\frac{Z_\alpha}{\sum_\beta Z_\beta}, \qquad q_{\alpha,i}=\begin{cases} \lambda_ip_i/Z_\alpha,&Z_\alpha>0,\ i\in I_\alpha,\\ 0,&\text{otherwise}. \end{cases}
Z_\betaUse in source · 3 passages
Search aliases: Z_β beta
Use in source · Paper_VI_Traditional_Final.tex, line 763
Let the state indices be partitioned into packets I_α. Define Z_α=∑_i∈ I_αλ_ip_i, m_α=fracZ_α∑_β Z_β, q_α,i= λ_ip_i/Z_α, · Z_α>0, i∈ I_α, 0, · otherwise.
Exact mathematical passage
Z_\alpha=\sum_{i\in I_\alpha}\lambda_ip_i, \qquad m_\alpha=\frac{Z_\alpha}{\sum_\beta Z_\beta}, \qquad q_{\alpha,i}=\begin{cases} \lambda_ip_i/Z_\alpha,&Z_\alpha>0,\ i\in I_\alpha,\\ 0,&\text{otherwise}. \end{cases}
Z_\lambdaUse in source · 20 passages
Search aliases: Z_λ lambda
Use in source · Paper_VI_Traditional_Final.tex, line 727
Let p=(p_i) be a probability vector and let λ=(λ_i) be a nonnegative likelihood with positive evidence Z_λ(p)=∑_iλ_ip_i. Define B_λ(p) = fracλodot pZ_λ(p). This map is nonlinear solely because of its scalar reconstruction factor.
Exact mathematical passage
Z_\lambda(p)=\sum_i\lambda_ip_i.
Z_{\lambda\odot\mu}Use in source · 1 passage
Search aliases: Z_λodotμ lambda odot mu
Use in source · Paper_VI_Traditional_Final.tex, line 741
[Independent-likelihood composition] For nonnegative likelihoods λ and μ, assume Z_λ(p)>0 and Z_λodotμ(p)>0. Equivalently, both successive updates have positive evidence. Then B_μ( B_λ(p)) = B_λodotμ(p).
Exact mathematical passage
Z_{\lambda\odot\mu}(p)>0
Z_\muUse in source · 6 passages
Search aliases: Z_μ mu
Use in source · Paper_VI_Traditional_Final.tex, line 805
[Likelihood perturbation bound] Let p be fixed and let λ,μ≥0 have positive evidences Z_λ,Z_μ. Then norm B_λ(p)- B_μ(p)_1 ≤ frac2∑_i|λ_i-μ_i|p_imin(Z_λ,Z_μ).
Exact mathematical passage
Z_\lambda,Z_\mu
z_{\sigma(i)}Use in source · 1 passage
Search aliases: z_σ(i) sigma
Use in source · continuation_05.tex, line 16
There is a coordinate permutation σ, with (σ z)_i=z_σ(i). Write σ z=(y,t), using e solved coordinates and q free coordinates. Define (y_0,t_0)=σ x, B=x+p^L Z_p^e+q, T=t_0+p^L Z_p^q, R={z∈ B:F(z)=0}. There is a homeomorphism h:Txrightarrow ∼ R, h^-1(z)=free(σ z). Equivalently, a continuous function ϕ:T→ y_0+p^L Z_p^e gives the entire root locus through σ h(t)=(ϕ(t),t). Every parameter has exactly one corresponding root in B. For every N≥0, h(t)≡ h(u)pmodp^Niff t≡ upmodp^N.
Exact mathematical passage
(\sigma z)_i=z_{\sigma(i)}
\Z_{\ge0}Use in source · 1 passage
Search aliases: Z_≥0 Z ge mathbf ge
Use in source · Paper_X_Block27_Amended.tex, line 1638
[Exact finite occupancy trace] The operator e_T is the projector onto the joint eigenspace A_T = {v:T_iv=a_iv for every i}. Moreover, Tr(e_T)=rank(e_T)=dim A_T∈ Z_≥0. The trace is identified with its unique nonnegative integer preimage under Z↪ E. Thus Tr(e_T)>0 refers to that integer, not to an ordering of an arbitrary field E. The finite packet is occupied exactly when this integer is positive. In positive characteristic the field-valued trace can vanish for a nonzero projector. The rank test,…
Exact mathematical passage
\boxed{ \Tr(e_T)=\rank(e_T)=\dim A_T\in\Z_{\ge0}. }
\mathbb Z_{\ge0}Use in source · 1 passage
Search aliases: Z_≥0 mathbb ge
Use in source · root_scope.tex, line 355
The current focused obstruction is the admitted ordinary-index family r=(17659+2^18T)2^640-161, T∈ Z_≥0, j_r=161. On its surviving chart define the actual, undepleted function K_r(Z)=2^-178G_r(10202+2^15Z). The complete coefficient theorem gives K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z), a_r,b_r∈ Z_2, u_r∈ Z_2^×, R_r∈ Z_2⟨ Z⟩. Its half derivative is a dyadic isometry. It therefore has a unique critical point c_r∈ Z_2. Set Φ_r^act=K_r(c_r). The earlier independent-undepletion notation for this same sc…
Exact mathematical passage
r=(17659+2^{18}T)2^{640}-161, \qquad T\in\mathbb Z_{\ge0},\qquad j_r=161. \label{eq:root:focusedfamily}
\mathbb Z_{\geq0}Use in source · 1 passage
Search aliases: Z_≥0 mathbb geq
Use in source · strengthening.tex, line 970
[A first-difference criterion] Let C: Z_≥0→ Z_2 have Newton coefficients a_k=Δ_1^kC(0). Suppose for an integer b≥0 that v_2(a_1)=b, v_2(a_k)≥ b+1+lfloorlog_2krfloor (k≥2). Then for every n≥0 and H>0, v_2(C(n+H)-C(n))=b+v_2(H). The Newton interpolation extends continuously to Z_2 and satisfies the same identity for any two distinct dyadic arguments.
Exact mathematical passage
C:\mathbb Z_{\geq0}\to\mathbb Z_2
\Zcal_RUse in source · 4 passages
Search aliases: Zcal_R Z_R Zcal mathcal
Use in source · Paper_IV_Traditional_Final.tex, line 2619
[Nearest-pole coefficient expansion] Let r∈ Z_>0. Suppose the zeros of D_N of least modulus R are simple and form the finite set Z_R. Suppose the next zero modulus is R_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_ρ,ℓ binomm+ℓ-1ℓ-1 ρ^-m + O_R_*(m!R_*^-m). The leading Laurent coefficient is a_ρ,r = (-ρ D_N'(ρ))^-r.
Exact mathematical passage
\Zcal_R
\Zcal_R^{\mathrm{orb}}Use in source · 2 passages
Search aliases: Zcal_R^orb Z_R^orb Zcal mathrm mathcal mathrm
Use in source · Paper_IV_Traditional_Final.tex, line 2696
[Orbit reduced nearest circle term] Assume the hypotheses of [source reference]. Choose one representative from each orbit of the nearest zeros under multiplication by ω, and call the resulting set Z_R^orb. With m=Nn, P_N,n(r) = · N m!binomm+r-1r-1 ∑_τ∈ Z_R^orb fracτ^-m(-τ D_N'(τ))^r
Exact mathematical passage
\Zcal_R^{\mathrm{orb}}
\Zp^eUse in source · 2 passages
Search aliases: Zp^e Z_p^e Zp mathbb
Use in source · 30_lifting.tex, line 468
R={z∈ B_L(x):F(z)=0}. The inverse homeomorphism is the actual free-coordinate projection z↦free(σ z). If h:T→ R is the forward map, then for every N≥0, h(t)≡ h(u)pmodp^N Longleftrightarrow t≡ upmodp^N. Equivalently, σ h(t)=(ϕ(t),t) for a continuous ϕ:T→ y_0+p^L Z_p^e whose graph is the entire reordered root locus.
Exact mathematical passage
\phi:T\to y_0+p^L\Zp^e
\Zp^{e+k}Use in source · 2 passages
Search aliases: Zp^e+k Z_p^e+k Zp mathbb
Use in source · 30_lifting.tex, line 507
Now suppose a chosen set of e columns of DF(x)bmod p is independent, and put those coordinates first. Apply the square result to the augmented polynomial map Ψ(Y,T)=(F(Y,T),T). Its reduced Jacobian has upper blocks (D_YF,D_TF) and lower blocks (0,I_k), so it is invertible. For t∈ T, the target (0,t) belongs to Ψ(x)+p^L Z_p^e+k because F(x)≡0pmodp^L. The inverse of Ψ at that target gives the unique root with parameters t. The inverse congruence property gives one direction of [source reference]. …
Exact mathematical passage
\Psi(x)+p^L\Zp^{e+k}
\Zp^kUse in source · 3 passages
Search aliases: Zp^k Z_p^k Zp mathbb
Use in source · 30_lifting.tex, line 456
arts with a selected Jacobian minor] Let p be prime, L≥1, F∈ Z[X_1,…,X_e+k]^e, and x∈ Z^e+k satisfy F(x)∈ p^L Z^e. Suppose the full reduced Jacobian is surjective. There is a permutation σ of the coordinates such that, writing σ x=(y_0,t_0) with e solved coordinates and k parameter coordinates, the parameter ball T=t_0+p^L Z_p^k is homeomorphic to the entire root locus R={z∈ B_L(x):F(z)=0}. The inverse homeomorphism is the actual free-coordinate projection z↦free(σ z). If h:T→ R is the forward m…
Exact mathematical passage
T=t_0+p^L\Zp^k
\Zp^{m+k}Use in source · 2 passages
Search aliases: Zp^m+k Z_p^m+k Zp mathbb
Use in source · 30_lifting.tex, line 629
[Arbitrary p-adic parameter endpoints] Let q=p be prime in Theorem~[source reference]. For every τ∈ t_0+p^L Z_p^k, there are an integer tower z_n and b∈ Z_p^m+k such that F(b)=0, free(b)=τ, z_n⟶ b, b-z_n∈ p^L+n Z_p^m+k. The initial value is z_0=x. At every stage and for every proposed parameter digit v∈ F_p^k, the corresponding next residue fiber has exactly p^m-e elements. If m=e, then for each τ there is exactly one such root in B_L(x), independently of all choices of integer approximants and …
Exact mathematical passage
b\in\Zp^{m+k}
\Zp^\timesUse in source · 2 passages
Search aliases: Zp^× Z_p^× Zp times mathbb times
Use in source · 10_foundations.tex, line 337
[Differences of unit modes] Over every commutative ring, Δ_h^s(u^n)=u^n(u^h-1)^s. For any prime p, units u_i∈ Z_p^×, coefficients a_i∈ Z_p, and a finite index set I, the sequence f(n)=∑_i∈ Ia_i u_i^n satisfies Δ_p-1^s f(n)∈ p^s Z_p (n,s∈ N).
Exact mathematical passage
u_i\in\Zp^\times
\ZZ^2Use in source · 3 passages
Search aliases: ZZ^2 ZZ ZZ^2 Z^2 ZZ mathbb
Use in source · visual_atlas.tex, line 350
vpage15Where the dyadic digit is lostMODEL D / THE EIGENLATTICE HAS INDEX TWOChapter II, mathematical text: Proposition 9.3 The eigenspaces are spanned by (1,0) and (-1,2). Their integral eigenvectors do not form an integral basis of ZZ^2. The change-of-basis determinant is two.
Exact mathematical passage
\ZZ^2
\ZZ_{(p)}^2Use in source · 1 passage
Search aliases: ZZ_(p)^2 ZZ
Use in source · visual_atlas.tex, line 361
0 · 2, AB=Bdiag(1,-1), E_+=Bdiag(1,0)B^-1. The index-two eigenlattice explains the denominator in B^-1 and in the projector. Over the localized lattice Λ_p=ZZ_(p)^2, the number 2 is a unit for every odd prime p. Hence the local denominator weight is zero there and one at p=2.
Exact mathematical passage
\Lambda_p=\ZZ_{(p)}^2
\ZZ_2Use in source · 2 passages
Search aliases: ZZ_2 Z_2 ZZ mathbb
Use in source · Paper_XII_Traditional_Final.tex, line 1588
[Failure of integral sign projectors] Let C_2=⟨ g⟩. Over Q_2, the two sign projectors are e_+=frac1+g2, e_-=frac1-g2. Neither belongs to Z_2[C_2]. After reduction modulo two, F_2[C_2] ≅ F_2[X]/(X^2), X=g-1. This ring is local and has no nontrivial idempotents.
Exact mathematical passage
\ZZ_2[C_2]
\ZZ_{>0}Use in source · 1 passage
Search aliases: ZZ_>0 Z_>0 ZZ mathbb
Use in source · Paper_IV_Traditional_Final.tex, line 2618
[Nearest-pole coefficient expansion] Let r∈ Z_>0. Suppose the zeros of D_N of least modulus R are simple and form the finite set Z_R. Suppose the next zero modulus is R_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_ρ,ℓ binomm+ℓ-1ℓ-1 ρ^-m + O_R_*(m!R_*^-m). The leading Laurent coefficient is a_ρ,r = (-ρ D_N'(ρ))^-r.
Exact mathematical passage
r\in\ZZ_{>0}
\ZZ_p^rUse in source · 1 passage
Search aliases: ZZ_p^r Z_p^r ZZ mathbb
Use in source · Paper_XII_Traditional_Final.tex, line 547
[Iwasawa limit] For Δ_m=( Z/p^m Z)^r, the inverse limit of O[Δ_m] is the Iwasawa algebra O[[ Z_p^r]]≅ O[[X_1,…,X_r]], where X_i corresponds to a chosen topological generator minus one [source reference].
Exact mathematical passage
\OO[[\ZZ_p^r]]\cong\OO[[X_1,\ldots,X_r]],
|^{-1}Use in source · 6 passages
Search aliases: |^-1
Use in source · Paper_IV_Traditional_Final.tex, line 2317
By [source reference], the order of Δ_N is 1/N<1. By [source reference], it has infinitely many zeros, and [source reference] gives ∑_λ∈Λ_N|λ|^-1<∞. Hadamard factorization therefore gives a canonical product of genus zero. The only possible exponential factor is a nonzero constant because the order is below one. Since Δ_N(0)=1, that constant is one. Substitution of w=z^N gives the second formula.
Exact mathematical passage
\sum_{\lambda\in\Lambda_N}|\lambda|^{-1}<\infty.
|^{-2}Use in source · 1 passage
Search aliases: |^-2
Use in source · atlas-pages.tex, line 479
arepsilon=1/100, the actual increment is 20/9, while derivative times increment is 2. Both are calculated exactly. Their difference comes from using a differential in place of a finite increment. The derivative gives a first order approximation. headWhat a proposed weight can and cannot certify Impedance is sensitive near Γ=1, and admittance near Γ=-1. The symmetric weight w(Γ)=|1-Γ|^-2+|1+Γ|^-2 reflects both chart sensitivities. Its growth motivates denser sampling near the endpoints, but does …
Exact mathematical passage
w(\Gamma)=|1-\Gamma|^{-2}+|1+\Gamma|^{-2}
|^{-q}Use in source · 1 passage
Search aliases: |^-q
Use in source · atlas-pages.tex, line 421
out added regularizing terms. takeawayA global analytic product theorem connects the infinite zero sum to a finite algebraic recurrence. The recurrence alone does not prove the product theorem. platecaptionDisplayed moments are checked against formal logarithm coefficients and exact order-two/order-four zeta values. Bar widths are numerical shares. General power sums use λ^-q, not |λ|^-q when exceptional nonpositive zeros are present.
Exact mathematical passage
|\lambda|^{-q}
|^{1/N}Use in source · 1 passage
Search aliases: |^1/N
Use in source · Paper_IV_Traditional_Final.tex, line 2845
For every N≥2, each nonzero zero λ of Δ_N gives N zeros of D_N, each of modulus |λ|^1/N. Hence Z_N^abs(s) = N Z_N^red(s/N). Substitution gives the second formula with Q_N^abs(s)=NQ_N^red(s/N), which is entire.
Exact mathematical passage
|\lambda|^{1/N}
|^{1/p}Use in source · 2 passages
Search aliases: |^1/p
Use in source · Paper_V_Traditional_Final.tex, line 1685
reconstruction formula with an estimated packet is f-widehat f_m = (I-Q_m)s+(I-Q_m)(j-widehat j). This identity identifies the two terms that must be controlled. The exact packet formulas in [source reference] then evaluate or bound j-widehat j from amplitude error, region overlap, boundary displacement, and the order of a derivative jump. A value jump has location modulus |τ-widehatτ|^1/p in finite L^p, while every positive order truncated power packet is Lipschitz in its location.
Exact mathematical passage
|\tau-\widehat\tau|^{1/p}
|^2Use in source · 9 passages
Search aliases: |^2
Use in source · 06_models.tex, line 79
The symmetric weight w_Smith(Γ) = frac1|1-Γ|^2 + frac1|1+Γ|^2 indicates where small errors in reflection coordinates produce large impedance or admittance errors.
Exact mathematical passage
w_{\mathrm{Smith}}(\Gamma) = \frac1{|1-\Gamma|^2} + \frac1{|1+\Gamma|^2}
|^{2}Use in source · 1 passage
Search aliases: |^2
Use in source · 02_supplement_06.tex, line 101
Monotonicity of the diagonal entries gives normDy^2 =∑_i=1^k+1σ_i^2|y_i|^2 ≥σ_k+1^2norm y^2. Because (D-B)y=Dy, the operator-norm inequality implies σ_k+1norm y ≤norm(D-B)y ≤normD-B_2norm y. Cancellation of the strictly positive norm of y proves σ_k+1≤normD-B_2. The norm of the actual diagonal residual D-D_k equals σ_k+1, so D_k attains the lower bound. If k≥ d, the residual is zero and optimality follows from norm nonnegativity.
Exact mathematical passage
\norm{Dy}^{2} =\sum_{i=1}^{k+1}\sigma_i^{2}|y_i|^{2} \geq\sigma_{k+1}^{2}\norm y^{2}.
|^3Use in source · 3 passages
Search aliases: |^3
Use in source · 01_finite_core.tex, line 1154
Put x=kh/2. The inequalities |sin x|≤|x| and |x-sin x|≤ |x|^3/6, which hold for every real argument, give 0≤ x^2-sin^2x =|x-sin x| |x+sin x| ≤frac|x|^36 2|x| =fracx^43. Multiplying by 4/h^2 proves both bounds. The estimate holds for every frequency. Its interpretation as convergence of second order keeps k fixed while h→0, so it does not assert a uniformly small relative error across all grid modes.
Exact mathematical passage
|x-\sin x|\le |x|^3/6
|^{M_L}Use in source · 2 passages
Search aliases: |^M_L
Use in source · Paper_IV_Traditional_Final.tex, line 2950
For N≥3, [source reference] gives | D_N'(ρ)|^-1 ≤ C e^-c_N|ρ| for every sufficiently large simple zero. Let L be a compact subset of the s plane. Because the chosen arguments of the zeros lie in a fixed bounded interval, there are constants C_L and M_L such that sup_s∈ L|e^-sLogρ| ≤ C_L|ρ|^M_L for every sufficiently large zero. Hence the absolute value of the corresponding summand is at most C_L'|ρ|^M_Le^-q c_N|ρ|. The linear zero counting estimate makes this majorant summable. The Weierstrass t…
Exact mathematical passage
\sup_{s\in L}|e^{-s\Log\rho}| \leq C_L|\rho|^{M_L}
|^pUse in source · 8 passages
Search aliases: |^p
Use in source · Paper_V_Traditional_Final.tex, line 1803
Every address has a unique representation r+ℓ. For finite p, x _p^p = ∑_r∈ R∑_ℓ∈ L|x(r+ℓ)|^p = ∑_r∈ R x_r _p^p. The corresponding supremum identity is immediate when p=∞. Unique address representation proves that analysis and interleaving are inverse maps. The projector identities follow from the disjoint coset partition.
Exact mathematical passage
\|x\|_p^p = \sum_{r\in R}\sum_{\ell\in L}|x(r+\ell)|^p = \sum_{r\in R}\|x_r\|_p^p.
|_2Use in source · 7 passages
Search aliases: |_2
Use in source · 20_arithmetic.tex, line 1051
[Isometric packet bijections] For 0≤ j<4, define on natural numbers f_j(n)=fracU_j(n)-j!2^d_j, (d_0,d_1,d_2,d_3)=(1,1,5,3). There is a unique continuous map Φ_j: Z_2→ Z_2 whose value at every n∈ N equals the embedded rational f_j(n). It is an isometric bijection and Φ_j(0)=0: |Φ_j(x)-Φ_j(y)|_2=|x-y|_2 (x,y∈ Z_2). In particular, every element of Z_2 has exactly one preimage.
Exact mathematical passage
|\Phi_j(x)-\Phi_j(y)|_2=|x-y|_2\qquad(x,y\in\Z_2).
|_DUse in source · 1 passage
Search aliases: |_D
Use in source · dyadic_synthesis.tex, line 111
nit character to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar units, Θ_j(uI)=θ_j(u^2)=1. For odd critical level r, use m=lfloor r/2rfloor and D=H^m+1(β)=H^m+1(γ). The endpoint is valid because 2(m+1)-r=1: the cross term for the additive character lies in the radical on which the matrix additive character is trivial. BK 3.3.18 supplies ϑ=Θ_j|_D ψ_β-γ. Integral trace makes it trivial on D∩ U_F^1.
Exact mathematical passage
\vartheta=\Theta_j|_D\,\psi_{\beta-\gamma}
|_{E_\mu}Use in source · 1 passage
Search aliases: |_E_μ mu
Use in source · Paper_VI_Traditional_Final.tex, line 1658
For clarity, the finite-dimensional holomorphic calculus used below has the following canonical definition. For A acting on C^d, its generalized eigenspaces E_μ=ker(A-μ I)^d form a direct-sum decomposition over the eigenvalues. Define the unique linear operator F(A) by .F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent of a basis and of increasing the truncation index: the extra nilpotent powers vanish. It depends only on the finite Taylor jets of F at the spectrum. I…
Exact mathematical passage
\left.F(A)\right|_{E_\mu} =\left.\sum_{k=0}^{d-1}\frac{F^{(k)}(\mu)}{k!}(A-\mu I)^k\right|_{E_\mu}.
|_{H_0}Use in source · 1 passage
Search aliases: |_H_0
Use in source · dyadic_synthesis.tex, line 297
⊂ U_A^m+1 lies in the forced domain of the additive character, so λ has values {±1}. Put H_0=kerλ. Then P/H_0≃ F_2^2, E=Ind_J^Pθ is irreducible of dimension 2. The actual represented generators are a diagonal sign and an off-diagonal matrix that anticommute. Thus their commutator form is nondegenerate, and θ|_H_0 is the full character of an actual new arithmetic Heisenberg pair. No abstract operators create that group action.
Exact mathematical passage
\theta|_{H_0}
|_KUse in source · 2 passages
Search aliases: |_K
Use in source · 04_transport.tex, line 197
[Packet exact sequence for an invariant subspace] Let K≤ X satisfy P_α K⊆ K for every α. Define K_α:=P_α K=K∩ X_α and overline P_α(x+K):=P_α x+K. Then: [label=(roman*)] item The nonzero restrictions P_α|_K form a packet system on K. item The nonzero overline P_α form a packet system on X/K. item For every α there is a short exact sequence 0⟶ K_α ⟶ X_α ⟶ overline P_α(X/K) ⟶0, and overline P_α(X/K) ≅ X_α/(X_α∩ K) ≅ (X_α+K)/K. item The full exact sequence decomposes packetwise as the direct sum of …
Exact mathematical passage
P_\alpha|_K
|_{\ker N}Use in source · 1 passage
Search aliases: |_ker N ker
Use in source · Paper_X_Block27_Amended.tex, line 1485
These statements concern supplied algebraic pairs. No Weil-group or inertia action, Hodge--Tate object, arithmetic comparison, or realization of the pair from a Galois representation is constructed here. The polynomial convention is det(XI-F|_ker N). No identification with an inertia-dependent local Euler factor is asserted. Claim 9.2 remains partial at its full arithmetic scope.
Exact mathematical passage
\det(XI-F|_{\ker N})
|_{\ker N_i}Use in source · 2 passages
Search aliases: |_ker N_i ker
Use in source · Paper_X_Block27_Amended.tex, line 1454
,N_m on U_m satisfy the same relation. F_m is invertible and N_m is nilpotent. The split maps constructed above intertwine both operators. The proof first identifies the independently defined kernel through an equivariant linear isomorphism ker N_m≃∏_i∈ I∏_k<m_iker N_i. The relation, with q≠0, proves that both F_i and its inverse preserve ker N_i. Thus F_i|_ker N_i is an actual automorphism, and Lean proves det(XI-F_m|_ker N_m) =∏_i∈ Idet(XI-F_i|_ker N_i)^m_i. The corresponding kernel dimensions…
Exact mathematical passage
F_i|_{\ker N_i}
|_{\ker N_m}Use in source · 1 passage
Search aliases: |_ker N_m ker
Use in source · Paper_X_Block27_Amended.tex, line 1455
and N_m is nilpotent. The split maps constructed above intertwine both operators. The proof first identifies the independently defined kernel through an equivariant linear isomorphism ker N_m≃∏_i∈ I∏_k<m_iker N_i. The relation, with q≠0, proves that both F_i and its inverse preserve ker N_i. Thus F_i|_ker N_i is an actual automorphism, and Lean proves det(XI-F_m|_ker N_m) =∏_i∈ Idet(XI-F_i|_ker N_i)^m_i. The corresponding kernel dimensions and traces are additive with multiplicity. Common consti…
Exact mathematical passage
\det(XI-F_m|_{\ker N_m}) =\prod_{i\in I}\det(XI-F_i|_{\ker N_i})^{m_i}.
|_MUse in source · 1 passage
Search aliases: |_M
Use in source · Paper_XI_Traditional_Final.tex, line 1781
[Breakwise phase product] In odd residue characteristic, let g∈Sp(V) preserve every break packet and suppose its restriction to M=im(g-I) is fixed point free. Put g_s=g|_V_s and M_s=im(g_s-I). The moving restrictions are symplectic, and Φ_p(g|_M,M) = ∏_s∈ SΦ_p(g_s|_M_s,M_s). In residue characteristic two, the analogous normalized Gauss sign is the product of the signs determined by the breakwise quadratic refinements.
Exact mathematical passage
\boxed{ \Phi_p(g|_M,M) = \prod_{s\in S}\Phi_p(g_s|_{M_s},M_s). }
|_{M_s}Use in source · 1 passage
Search aliases: |_M_s
Use in source · Paper_XI_Traditional_Final.tex, line 1781
[Breakwise phase product] In odd residue characteristic, let g∈Sp(V) preserve every break packet and suppose its restriction to M=im(g-I) is fixed point free. Put g_s=g|_V_s and M_s=im(g_s-I). The moving restrictions are symplectic, and Φ_p(g|_M,M) = ∏_s∈ SΦ_p(g_s|_M_s,M_s). In residue characteristic two, the analogous normalized Gauss sign is the product of the signs determined by the breakwise quadratic refinements.
Exact mathematical passage
\boxed{ \Phi_p(g|_M,M) = \prod_{s\in S}\Phi_p(g_s|_{M_s},M_s). }
|_{n=0}Use in source · 2 passages
Search aliases: |_n=0
Use in source · supplement_04.tex, line 184
A sharp unit-mode gain. For p=5 and u=2, Δ_4^2(u^n)|_n=0=(2^4-1)^2=225. The source proves 5^2mid225 and 5^3nmid225. This example attains the guaranteed gain. No additional uniform power of p is claimed.
Exact mathematical passage
\Delta_4^2(u^n)\big|_{n=0}=(2^4-1)^2=225.
|_PUse in source · 1 passage
Search aliases: |_P
Use in source · dyadic_synthesis.tex, line 344
Consequently M=K^× P is an actual open group compact modulo the center, N=K^× J is normal of index two in M, and Σ=Ind_ N^ MΛ, Σ|_P≃ E, π≃c - Ind_ M^GΣ. Restriction to P is irreducible, so Σ is irreducible. Induction in stages proves the last equality for the original Λ, including its fixed central character and noncompact root. This is an actual enlarged inducing representation. It changes neither the candidate nor its original canonical beta-extension data.
Exact mathematical passage
\Sigma=\operatorname{Ind}_{\mathcal N}^{\mathcal M}\Lambda, \qquad \Sigma|_P\simeq E, \qquad \boxed{\pi\simeq\operatorname{c\!-\!Ind}_{\mathcal M}^G\Sigma.}
|_pUse in source · 3 passages
Search aliases: |_p
Use in source · 30_lifting.tex, line 15
Unless a different coefficient ring is specified, a polynomial system is a finite list of integer polynomials. All divisibility conditions on vectors are coordinatewise. For a positive integer q, write S_q= Z/q Z and use an overline for reduction modulo q. For a prime p, write F_p=S_p, normalize |p|_p=p^-1, and put B_L(x)=x+p^L Z_p^d. An integer center is included in this notation by its natural image in Z_p^d. All dimensions below are finite and may be zero. A statement over S_q allows composit…
Exact mathematical passage
|p|_p=p^{-1}
|_{P_\alpha X}Use in source · 1 passage
Search aliases: |_P_α X alpha
Use in source · Paper_VI_Traditional_Final.tex, line 2190
thm:reducing_packets The following conditions are equivalent. item AP_α=P_α A for every α. item A_βα=0 whenever β≠α. item T_βα(t)=0 whenever β≠α and t∈ R. item Every packet space P_α X is invariant under T(t) for every t. When these conditions hold, T(t)|_P_α X = e^tA_αα.
Exact mathematical passage
T(t)|_{P_\alpha X} = \e^{tA_{\alpha\alpha}}.
|_{QX}Use in source · 2 passages
Search aliases: |_QX
Use in source · Paper_VI_Traditional_Final.tex, line 2750
[Generator control of range leakage] Let T(t)= e^tA on a Banach space X, let P be a bounded projection, and set Q=I-P. Assume normT(t)≤ M e^ω t and norm e^t(QAQ)|_QX ≤ M_Q e^ω_Qt. Then normQT(t)P ≤ M M_Qnorm P^2normQAP ∫_0^t e^ω_Q(t-s)+ω s d s. In particular, for orthogonal projections and contraction bounds, sup_0≤ t≤ T normQT(t)P ≤ TnormQAP.
Exact mathematical passage
\norm{\e^{t(QAQ)|_{QX}}} \leq M_Q\e^{\omega_Qt}.
|_{\Ran P}Use in source · 1 passage
Search aliases: |_Ran P Ran operatorname
Use in source · Paper_VI_Traditional_Final.tex, line 1504
[Schur--Feshbach projected resolvent] Assume both zI-A and zI-A_QQ are invertible. Define F_P(z) =zI-A_PP-A_PQ(zI-A_QQ)^-1A_QP. Then .P(zI-A)^-1|_Ran P=F_P(z)^-1.
Exact mathematical passage
\boxed{ \left.P(zI-A)^{-1}\right|_{\Ran P}=F_P(z)^{-1}. }
|_{\rm nodes}Use in source · 1 passage
Search aliases: |_rm nodes rm
Use in source · atlas-pages.tex, line 424
vpage19Convergence with a nonzero exact errorFINITE DIFFERENCES / A QUARTIC TEST PROBLEMChapter VI, Part I: Theorems 11.1--11.2. Exact nodal inversion and the scale of low-mode errors. Use u(x)=x-x^4, which vanishes at both endpoints and solves -u''=12x^2. Its centered second difference is not exact: L_hu|_rm nodes=12x_i^2+2h^2. Since the stencil differentiates the quadratic x(1-x) exactly, the solution u_h to the sampled forcing 12x_i^2 satisfies u_h(x_i)-u(x_i)=-h^2x_i(1-x_i). This gives both …
Exact mathematical passage
L_hu|_{\rm nodes}=12x_i^2+2h^2.
|_{t=0}Use in source · 1 passage
Search aliases: |_t=0
Use in source · Paper_VI_Traditional_Final.tex, line 3253
= · f(x)+t(f''(x)-xf'(x))+o(t). Every omitted term is a polynomial in the Gaussian variable multiplied by a quantity that is o(t), so the expansion may be integrated term by term. Therefore .frac d d t O_tf|_t=0 =f''-xf'. The Hermite eigenvalue identity also follows by differentiating O_th_n= e^-nth_n at zero. Gaussian integration by parts gives ∫_ R f'(x)overlineg(x)γ( d x) = ∫_ R f(x)overlinexg(x)-g'(x)γ( d x). Set g=f' and rearrange.
Exact mathematical passage
\left.\frac{\dd}{\dd t}\Ocal_tf\right|_{t=0} =f''-xf'.
|_{U_a}Use in source · 2 passages
Search aliases: |_U_a
Use in source · 72_recovered.tex, line 2754
[Packet decomposition of a lift obstruction] Suppose F_p^d = ⊕_a∈ AU_a, F_p^r = ⊕_a∈ AV_a, and suppose a linear correction map D: F_p^d→ F_p^r satisfies D(U_a)⊆ V_a for every a. Write D_a=D|_U_a. Then coker D ≅ ⊕_a∈ Acoker D_a. A residual lifts if and only if every packet obstruction vanishes.
Exact mathematical passage
D_a=D|_{U_a}
|_{U_F^1}Use in source · 1 passage
Search aliases: |_U_F^1
Use in source · odd_synthesis.tex, line 30
For this actual stratum there exists a full simple character θ with θ|_U_F^1=1. Retaining its canonical beta extension, the residue character and noncompact extension can be selected so that the resulting irreducible admissible supercuspidal π has any prescribed smooth central quasicharacter ω trivial on U_F^1. These completion choices must be free or already compatible. Incompatible immutably supplied choices are not replaced.
Exact mathematical passage
\theta|_{U_F^1}=1
|_{V_j}Use in source · 2 passages
Search aliases: |_V_j
Use in source · Paper_XI_Traditional_Final.tex, line 1336
[Orthogonal multiplicativity] Suppose V=V_1mathop⊥limits⋯mathop⊥limits V_s and g preserves every summand. If no g|_V_j has a fixed vector, then Φ_p(g,V) = ∏_j=1^s Φ_p(g|_V_j,V_j).
Exact mathematical passage
g|_{V_j}
|_{V_s}Use in source · 1 passage
Search aliases: |_V_s
Use in source · Paper_XI_Traditional_Final.tex, line 1780
[Breakwise phase product] In odd residue characteristic, let g∈Sp(V) preserve every break packet and suppose its restriction to M=im(g-I) is fixed point free. Put g_s=g|_V_s and M_s=im(g_s-I). The moving restrictions are symplectic, and Φ_p(g|_M,M) = ∏_s∈ SΦ_p(g_s|_M_s,M_s). In residue characteristic two, the analogous normalized Gauss sign is the product of the signs determined by the breakwise quadratic refinements.
Exact mathematical passage
g_s=g|_{V_s}
|_{W_{K_f}}Use in source · 1 passage
Search aliases: |_W_K_f
Use in source · Paper_XI_Traditional_Final.tex, line 379
[Unramified twist] Let ν be an unramified character of W_K. Then τ_ξ⊗ν ≅ Ind_W_K_f^W_K ( ξ ν|_W_K_f ). The tame inertia orbit is unchanged. If the original Frobenius cycle has closing scalar a, the twisted cycle has closing scalar aν(Φ)^f.
Exact mathematical passage
\tau_\xi\otimes\nu \cong \Ind_{W_{K_f}}^{W_K} \left( \xi\,\nu|_{W_{K_f}} \right).
|_{X_1}Use in source · 1 passage
Search aliases: |_X_1
Use in source · 04_correction_transport.tex, line 122
[Local Lyapunov--Schmidt reduction] Let U⊆ X be an open neighborhood of zero and let F: U→ Y be a C^r map between Banach spaces with r≥1, F(0)=0, and L=DF(0). Assume there are closed decompositions X=K⊕ X_1, Y=R⊕ O, where K=Ker L, R=im L, and L_1=L|_X_1:X_1⟶ R is a Banach space isomorphism. Let π_R and π_O be the two projections on Y. Then there are neighborhoods U_K⊆ K and U_1⊆ X_1 of zero and a unique C^r map ψ:U_K⟶ U_1 with ψ(0)=0 such that π_RF(k+ψ(k))=0 for every k∈ U_K. After the neighborh…
Exact mathematical passage
L_1=L|_{X_1}:X_1\longrightarrow R
|_{X_\alpha}Use in source · 1 passage
Search aliases: |_X_α alpha
Use in source · 03_packets_and_graphs.tex, line 150
[Universal mapping property of finite packet coordinates] Let P=(P_α)_α∈ I be a finite packet system on X, and let Y be a vector space over 𝕜. item Every family of linear maps f_α:X_α→ Y determines a unique linear map f:X→ Y with f|_X_α=f_α, namely f=∑_α f_α P_α. item Every family of linear maps g_α:Y→ X_α determines a unique linear map g:Y→ X with P_α g=g_α, namely g=∑_α g_α. Equivalently, Hom(X,Y)≅∏_αHom(X_α,Y), Hom(Y,X)≅∏_αHom(Y,X_α), where the products are finite. [source reference] ExportsA…
Exact mathematical passage
f|_{X_\alpha}=f_\alpha
|_{z=0}Use in source · 1 passage
Search aliases: |_z=0
Use in source · Paper_IV_Traditional_Final.tex, line 488
[Differential cycle and initial data] For 1≤ j<N, fracddz E_N,j(λ z) = λ E_N,j-1(λ z), while fracddz E_N,0(λ z) = λ E_N,N-1(λ z). Thus fracd^Ndz^N E_N,j(λ z) = λ^N E_N,j(λ z). For 0≤ k<N, .fracd^kdz^k E_N,j(λ z)|_z=0 = δ_kjλ^j. If λ≠0, the N functions E_N,j(λ z) form a basis of the solution space of y^(N)=λ^Ny.
Exact mathematical passage
\left.\frac{d^k}{dz^k}\Efun_{N,j}(\lambda z)\right|_{z=0} = \delta_{kj}\lambda^j.
\pmUse in source · 19 passages
Search aliases: ± pm
Use in source · 04_arithmetic.tex, line 685
The failure of the character formula frac12(I± A) to be integral does not invalidate coefficient selectors such as even and odd residue extraction. It shows that two distinct constructions must be distinguished: integral address selector ≠ character-average spectral projector when the group order is not a unit. The first can remain exact over Z_2. The second can acquire a denominator weight.
Exact mathematical passage
\frac12(I\pm A)
\times_CUse in source · 1 passage
Search aliases: ×_C times
Use in source · 50_open_questions.tex, line 95
The original proposed form is X≃ X_2^wt×_C∏_p>2'X_p. It leaves X, the distinguished subsets defining the restricted product, the maps to C, the ambient category, and the universal meaning of ``smallest'' unspecified. In particular, one must decide whether X is a global object or a completed object.
Exact mathematical passage
X\simeq X_2^{\mathrm{wt}}\times_C\prod_{p>2}'X_p.
\times_{\Compat}Use in source · 1 passage
Search aliases: ×_Compat ×_ C times Compat times mathfrak
Use in source · Paper_X_Block27_Amended.tex, line 775
The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.
Exact mathematical passage
\boxed{ \Sys_{\mathrm{sys}} = \Sys_2^{\mathrm{wt}} \times_{\Compat} \prod_{\lambda\nmid2}'\Sys_\lambda. }
\alpha'Use in source · 2 passages
Search aliases: α' alpha
Use in source · 02_coordinates.tex, line 303
Commutativity gives R_αβ^2=P_α^2Q_β^2=R_αβ. If (α,β)≠(α',β'), then either P_α P_α'=0 or Q_β Q_β'=0, so the corresponding products annihilate one another. Finally, ∑_α,βR_αβ = (∑_α P_α) (∑_β Q_β) =I_X. For a fixed α, P_α =P_α∑_β Q_β = ∑_β R_αβ, and the same argument works for a fixed β. Since each P_α and Q_β is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite f…
Exact mathematical passage
(\alpha,\beta)\ne(\alpha',\beta')
\alpha^{-1}Use in source · 1 passage
Search aliases: α^-1 alpha
Use in source · dyadic_synthesis.tex, line 53
(t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two, and z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the exact filtration valuation, and give the identity tr_LLift_E(C) =Tr_E/L (αtr_E C). The actual recu…
Exact mathematical passage
\kappa=\alpha^{-1},\qquad s=\kappa\operatorname{pr},\qquad s(P_0)=t_L/z^2,\qquad \operatorname{Lift}_E(C)=(z^2/t_L)CP_0.
\alpha^2Use in source · 1 passage
Search aliases: α^2 alpha
Use in source · dyadic_synthesis.tex, line 46
At a recursive coefficient field L of characteristic two, with residue field F_2 and uniformizer t_L, put E=L(α), α^2-α=t_L^-q, a=(q+1)/2, z=t_L^aα. The extension is separable and totally ramified of degree two, and z is a uniformizer and z^2+t_L^az+t_L=0, Tr_E/L(α)=1. For constant-coefficient projection P_0 in the basis 1,z, the normalized trace corestriction and exact filtered right inverse are κ=α^-1, s=κpr, s(P_0)=t_L/z^2, Lift_E(C)=(z^2/t_L)CP_0. They satisfy s(Lift_E(C))=C, preserve the ex…
Exact mathematical passage
E=L(\alpha),\qquad \alpha^2-\alpha=t_L^{-q},\qquad a=(q+1)/2,\qquad z=t_L^a\alpha.
\alpha^bUse in source · 1 passage
Search aliases: α^b alpha
Use in source · odd_synthesis.tex, line 56
At a recursive base (L,t_L), introduce E=L(α), α^p-α=t_L^-q_d. This is a separable totally ramified extension of degree p, with v_E(α)=-q_d and Tr_E/Lα=0. Choose 1≤ b<p with q_db≡-1pmod p, set a=(q_db+1)/p, and put z=t_L^aα^b. Then z is an explicit uniformizer. Write its Eisenstein polynomial as f(X)=X^p+⋯+a_0.
Exact mathematical passage
z=t_L^a\alpha^b
\alpha^pUse in source · 2 passages
Search aliases: α^p alpha
Use in source · odd_synthesis.tex, line 51
At a recursive base (L,t_L), introduce E=L(α), α^p-α=t_L^-q_d. This is a separable totally ramified extension of degree p, with v_E(α)=-q_d and Tr_E/Lα=0. Choose 1≤ b<p with q_db≡-1pmod p, set a=(q_db+1)/p, and put z=t_L^aα^b. Then z is an explicit uniformizer. Write its Eisenstein polynomial as f(X)=X^p+⋯+a_0.
Exact mathematical passage
E=L(\alpha),\qquad \alpha^p-\alpha=t_L^{-q_d}.
\alpha_0Use in source · 5 passages
Search aliases: α_0 alpha
Use in source · Paper_VI_Traditional_Final.tex, line 432
[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.
Exact mathematical passage
P_{\alpha_m}A^mP_{\alpha_0} = \sum_{\alpha_1,\ldots,\alpha_{m-1}} P_{\alpha_m}AP_{\alpha_{m-1}}\cdots P_{\alpha_1}AP_{\alpha_0}.
\alpha_2Use in source · 1 passage
Search aliases: α_2 alpha
Use in source · 03_nonlinear_coordinates.tex, line 401
[The second-order composition law] At order two, the partition formula has two shapes. For packet directions h_1 and h_2, R_δ D^2(F∘ G)(x)[h_1,h_2] = · ∑_γ_1,γ_2 J^2_δmidγ_1,γ_2F(Gx) [J^1_γ_1midα_1G(x)h_1,J^1_γ_2midα_2G(x)h_2] · + ∑_γ J^1_δmidγF(Gx) [J^2_γmidα_1,α_2G(x)[h_1,h_2]]. The first line records two first-order changes entering a quadratic interaction of F. The second records a genuine second-order interaction created inside G before F is applied.
Exact mathematical passage
\begin{aligned} R_\delta D^2(F\circ G)(x)[h_1,h_2] ={}& \sum_{\gamma_1,\gamma_2} J^2_{\delta\mid\gamma_1,\gamma_2}F(Gx) [J^1_{\gamma_1\mid\alpha_1}G(x)h_1,J^1_{\gamma_2\mid\alpha_2}G(x)h_2]\\ &+ \sum_\gamma J^1_{\delta\mid\gamma}F(Gx) [J^2_{\gamma\mid\alpha_1,\alpha_2}G(x)[h_1,h_2]]. \end{aligned}
\alpha_dUse in source · 4 passages
Search aliases: α_d alpha
Use in source · 72_recovered.tex, line 2839
After passage to an algebraic closure, let α_1,…,α_d be the eigenvalues of C, counted with algebraic multiplicity. The resultant is ∏_i=1^d(x-α_i^N), which is the characteristic polynomial of C^N. Both sides are polynomial expressions with integer coefficients in the entries of C, so the identity descends to every commutative coefficient ring.
Exact mathematical passage
\alpha_1,\ldots,\alpha_d
\alpha_iUse in source · 4 passages
Search aliases: α_i alpha
Use in source · 03_nonlinear_coordinates.tex, line 352
[Packet Fa a di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n]
Exact mathematical passage
h_i\in X_{\alpha_i}
\alpha_i^NUse in source · 1 passage
Search aliases: α_i^N alpha
Use in source · 72_recovered.tex, line 2840
After passage to an algebraic closure, let α_1,…,α_d be the eigenvalues of C, counted with algebraic multiplicity. The resultant is ∏_i=1^d(x-α_i^N), which is the characteristic polynomial of C^N. Both sides are polynomial expressions with integer coefficients in the entries of C, so the identity descends to every commutative coefficient ring.
Exact mathematical passage
\prod_{i=1}^{d}(x-\alpha_i^N),
\alpha_{i_1}Use in source · 1 passage
Search aliases: α_i_1 alpha
Use in source · 03_nonlinear_coordinates.tex, line 340
For a block B={i_1<⋯<i_s} and an intermediate packet γ, write G_γ,B(xmid h) = J^s_γmidα_i_1,…,α_i_sG(x) [h_i_1,…,h_i_s].
Exact mathematical passage
G_{\gamma,B}(x\mid h) = J^s_{\gamma\mid\alpha_{i_1},\ldots,\alpha_{i_s}}G(x) [h_{i_1},\ldots,h_{i_s}].
\alpha_{i_s}Use in source · 1 passage
Search aliases: α_i_s alpha
Use in source · 03_nonlinear_coordinates.tex, line 340
For a block B={i_1<⋯<i_s} and an intermediate packet γ, write G_γ,B(xmid h) = J^s_γmidα_i_1,…,α_i_sG(x) [h_i_1,…,h_i_s].
Exact mathematical passage
G_{\gamma,B}(x\mid h) = J^s_{\gamma\mid\alpha_{i_1},\ldots,\alpha_{i_s}}G(x) [h_{i_1},\ldots,h_{i_s}].
\alpha_jUse in source · 6 passages
Search aliases: α_j alpha
Use in source · 03_nonlinear_coordinates.tex, line 264
[Jet norm bound] Give every packet range its inherited subspace norm. The restricted jet satisfies the sharper estimate J^r_γmidα_1,…,α_rF(x) ≤ Q_γ D^rF(x) . Its ambient extension inserts P_α_j in the jth slot and has norm at most Q_γ D^rF(x) ∏_j P_α_j . Consequently the operator norm of the restricted packet jet also satisfies J^r_γmidα_1,…,α_rF(x) ≤ Q_γ D^rF(x) ∏_j=1^r P_α_j . When all packets are orthogonal projections on Hilbert spaces, every nonzero projector has norm one and the bound redu…
Exact mathematical passage
P_{\alpha_j}
\alpha_mUse in source · 5 passages
Search aliases: α_m alpha
Use in source · Paper_VI_Traditional_Final.tex, line 432
[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.
Exact mathematical passage
P_{\alpha_m}A^mP_{\alpha_0} = \sum_{\alpha_1,\ldots,\alpha_{m-1}} P_{\alpha_m}AP_{\alpha_{m-1}}\cdots P_{\alpha_1}AP_{\alpha_0}.
\alpha_{m-1}Use in source · 3 passages
Search aliases: α_m-1 alpha
Use in source · Paper_VI_Traditional_Final.tex, line 432
[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.
Exact mathematical passage
P_{\alpha_m}A^mP_{\alpha_0} = \sum_{\alpha_1,\ldots,\alpha_{m-1}} P_{\alpha_m}AP_{\alpha_{m-1}}\cdots P_{\alpha_1}AP_{\alpha_0}.
\alpha_nUse in source · 2 passages
Search aliases: α_n alpha
Use in source · 03_nonlinear_coordinates.tex, line 352
[Packet Fa a di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n]
Exact mathematical passage
\alpha_1,\ldots,\alpha_n
\beta'Use in source · 2 passages
Search aliases: β' beta
Use in source · 02_coordinates.tex, line 303
Commutativity gives R_αβ^2=P_α^2Q_β^2=R_αβ. If (α,β)≠(α',β'), then either P_α P_α'=0 or Q_β Q_β'=0, so the corresponding products annihilate one another. Finally, ∑_α,βR_αβ = (∑_α P_α) (∑_β Q_β) =I_X. For a fixed α, P_α =P_α∑_β Q_β = ∑_β R_αβ, and the same argument works for a fixed β. Since each P_α and Q_β is nonzero, these identities ensure that every parent has a nonzero product child. The two parent maps on the supported product family are therefore surjective. Iteration proves the finite f…
Exact mathematical passage
(\alpha,\beta)\ne(\alpha',\beta')
\beta_1Use in source · 4 passages
Search aliases: β_1 beta
Use in source · 03_packets_and_graphs.tex, line 562
[Packet powers and paths] For m≥1, (T^m)_β_mβ_0 = ∑_β_1,…,β_m-1 T_β_mβ_m-1 ⋯ T_β_1β_0. The same formula holds for a product of nonidentical operators with the corresponding blocks in temporal order. [source reference]
Exact mathematical passage
(T^m)_{\beta_m\beta_0} = \sum_{\beta_1,\ldots,\beta_{m-1}} T_{\beta_m\beta_{m-1}} \cdots T_{\beta_1\beta_0}.
\beta_2Use in source · 2 passages
Search aliases: β_2 beta
Use in source · supplement_03.tex, line 158
Principal targets. targetDyadicReconstruction.dyadic_reconstruction targetDyadicReconstruction.dyadic_supercongruence targetDyadicReconstruction.normalized_error_integral pageheading4. Exact examples and adversarial checksExample 5.5 and controls against incorrect normalizations The actual reciprocal recurrence gives (β_0,β_1,β_2,β_3,β_4) =(1,-frac13,frac115,-frac2189,frac1675). These values are obtained from the general formal-series definition. They are not substituted as independent input dat…
Exact mathematical passage
(\beta_0,\beta_1,\beta_2,\beta_3,\beta_4) =\left(1,-\frac13,\frac1{15},-\frac2{189},\frac1{675}\right).
\beta_3Use in source · 2 passages
Search aliases: β_3 beta
Use in source · supplement_03.tex, line 158
Principal targets. targetDyadicReconstruction.dyadic_reconstruction targetDyadicReconstruction.dyadic_supercongruence targetDyadicReconstruction.normalized_error_integral pageheading4. Exact examples and adversarial checksExample 5.5 and controls against incorrect normalizations The actual reciprocal recurrence gives (β_0,β_1,β_2,β_3,β_4) =(1,-frac13,frac115,-frac2189,frac1675). These values are obtained from the general formal-series definition. They are not substituted as independent input dat…
Exact mathematical passage
(\beta_0,\beta_1,\beta_2,\beta_3,\beta_4) =\left(1,-\frac13,\frac1{15},-\frac2{189},\frac1{675}\right).
\beta_4Use in source · 1 passage
Search aliases: β_4 beta
Use in source · supplement_03.tex, line 158
Principal targets. targetDyadicReconstruction.dyadic_reconstruction targetDyadicReconstruction.dyadic_supercongruence targetDyadicReconstruction.normalized_error_integral pageheading4. Exact examples and adversarial checksExample 5.5 and controls against incorrect normalizations The actual reciprocal recurrence gives (β_0,β_1,β_2,β_3,β_4) =(1,-frac13,frac115,-frac2189,frac1675). These values are obtained from the general formal-series definition. They are not substituted as independent input dat…
Exact mathematical passage
(\beta_0,\beta_1,\beta_2,\beta_3,\beta_4) =\left(1,-\frac13,\frac1{15},-\frac2{189},\frac1{675}\right).
\beta_iUse in source · 1 passage
Search aliases: β_i beta
Use in source · odd_capacity.tex, line 132
For H, BK 3.1.9(ii) states that H^a(β_i)=H^a(β_i+1) as soon as
Exact mathematical passage
H^a(\beta_i)=H^a(\beta_{i+1})
\beta_{i+1}Use in source · 1 passage
Search aliases: β_i+1 beta
Use in source · odd_capacity.tex, line 132
For H, BK 3.1.9(ii) states that H^a(β_i)=H^a(β_i+1) as soon as
Exact mathematical passage
H^a(\beta_i)=H^a(\beta_{i+1})
\beta_jUse in source · 4 passages
Search aliases: β_j beta
Use in source · strengthening.tex, line 260
[Four-address factorial bound] For j∈{0,1,2,3}, n≥0, k≥1, and positive mixed steps, v_2(Δ_H_1⋯Δ_H_kC_j(n)) ≥ k+v_2(k!)+∑_ν=1^k v_2(H_ν)+β_j, (β_0,β_1,β_2,β_3)=(0,0,3,1). Address two also satisfies v_2(Δ_H_1⋯Δ_H_kC_2(n)) ≥ 2k+v_2(k!)+lfloorfrack2rfloor +∑_ν=1^k v_2(H_ν). For the original coefficients, the offsets in [source reference] are (0,0,4,2). No positive-order offset is asserted for the empty difference.
Exact mathematical passage
v_2(\Delta_{H_1}\cdots\Delta_{H_k}C_j(n)) \geq k+v_2(k!)+\sum_{\nu=1}^k v_2(H_\nu)+\beta_j, \quad (\beta_0,\beta_1,\beta_2,\beta_3)=(0,0,3,1). \label{eq:factorial-four-lower}
\beta_{m-1}Use in source · 1 passage
Search aliases: β_m-1 beta
Use in source · 03_packets_and_graphs.tex, line 562
[Packet powers and paths] For m≥1, (T^m)_β_mβ_0 = ∑_β_1,…,β_m-1 T_β_mβ_m-1 ⋯ T_β_1β_0. The same formula holds for a product of nonidentical operators with the corresponding blocks in temporal order. [source reference]
Exact mathematical passage
(T^m)_{\beta_m\beta_0} = \sum_{\beta_1,\ldots,\beta_{m-1}} T_{\beta_m\beta_{m-1}} \cdots T_{\beta_1\beta_0}.
\beta_rUse in source · 13 passages
Search aliases: β_r beta
Use in source · 72_recovered.tex, line 80
The dyadic kernel is substantially sharper than the separate coefficient bound alone suggests. If fracz^2sinh^2z = ∑_r≥0β_rz^2r and κ_2,r=β_r2^2r(2r)!, then v_2(κ_2,r)=4r-1 (r≥1). Consequently, truncation after depth M is uniformly valid modulo 2^4M+3, and this exponent is optimal. The rowwise estimate records the additional binary carries in the binomial coefficient. Classical congruence theory for Euler families is extensive [source reference]. The formulas here are stated as exact coordinate …
Exact mathematical passage
\frac{z^2}{\sinh^2z} = \sum_{r\geq0}\beta_rz^{2r}
\beta_{r+q}Use in source · 1 passage
Search aliases: β_r+q beta
Use in source · root_scope.tex, line 110
Set r=M+1≥1 and define P_q(x)=∏_i=0^q-1(x-i)(2x-(2i+1)), P_0=1, H_r(x)=∑_q≥08^qfracβ_r+qβ_r P_q(x)mathscr A(x-q). The coefficient valuations tend to infinity. At any nonnegative integer x, terms with q>x vanish. Cancelling the factorials in [source reference] therefore gives the exact identity T_r-1(N)=κ_rbinom2N2rH_r(N-r) (N≥ r). This is a finite rational identity at every original Euler row.
Exact mathematical passage
H_r(x)=\sum_{q\ge0}8^q\frac{\beta_{r+q}}{\beta_r} P_q(x)\mathscr A(x-q).
\beta_r^2Use in source · 2 passages
Search aliases: β_r^2 beta
Use in source · root_scope.tex, line 387
[Exact quadratic compression] There is a unique preparation K_r(Z)= U_r(Z) W_r(Z), W_r(Z)=Z^2+2β_rZ+α_r, U_r≡ u_rpmod256. With δ_r=β_r^2-α_r and τ_r= U_r'(c_r)/ U_r(c_r), δ_r=-fracΦ_r^act U_r(c_r) (1-fracτ_r^2Φ_r^act4 U_r(c_r)), v_2(δ_r)=v_2(Φ_r^act). The equality includes the zero case. The same successive monic polynomial division used in Theorem~[source reference], started at precision eight, preserves the quadratic and unit reductions of [source reference]. Write W=Z^2+2β_rZ+α_r and w=W(c_r)…
Exact mathematical passage
\delta_r=\beta_r^2-\alpha_r
\gamma'Use in source · 1 passage
Search aliases: γ' gamma
Use in source · 02_strengthened.tex, line 457
[Path interference formula] Fix a source vector ψ∈ P_a_0 H and a target packet b. Let Γ_b be the set of packet paths from a_0 to b, and set η_γ=U_γψ. Choose any total order on this finite path set. Then normP_bUψ^2 = ∑_γ∈Γ_bnormη_γ^2 + 2Re ∑_substackγ,γ'∈Γ_b γ<γ' ipη_γη_γ'. Consequently, normP_bUψ ≤ ∑_γ∈Γ_bnormη_γ. If the path outputs are pairwise orthogonal, the target probability is the sum of the path probabilities.
Exact mathematical passage
\norm{P_bU\psi}^2 = \sum_{\gamma\in\Gamma_b}\norm{\eta_\gamma}^2 + 2\operatorname{Re} \sum_{\substack{\gamma,\gamma'\in\Gamma_b\\ \gamma<\gamma'}} \ip{\eta_\gamma}{\eta_{\gamma'}}.
\Gamma^GUse in source · 1 passage
Search aliases: Γ^G Gamma
Use in source · 02_connections_obstructions.tex, line 340
Substitute the definition of Γ^G. Adjacent factors G_b^-1G_b and G_c^-1G_c cancel. In the defect, both the composite and direct terms acquire the same outer factors G_c and G_a^-1. In the loop, all inner factors cancel and only conjugation at the base remains.
Exact mathematical passage
\Gamma^G
\Gamma^{X_q}_{b\leftarrow a}Use in source · 1 passage
Search aliases: Γ^X_q_b← a Gamma leftarrow
Use in source · 02_connections_obstructions.tex, line 889
For coordinate models of the source and target colors, an additive target coordinate space, and a chosen noncanonical connection, a pointwise node defect can be defined by D^Γ_b,a(Φ) = Γ^Y_b← a∘widehatΦ_a - widehatΦ_b∘ (∏_qΓ^X_q_b← a). This expression compares two chosen routes. With canonical exact transport in every slot it vanishes by [source reference].
Exact mathematical passage
\mathfrak D^\Gamma_{b,a}(\Phi) = \Gamma^Y_{b\leftarrow a}\circ\widehat\Phi_a - \widehat\Phi_b\circ \left(\prod_q\Gamma^{X_q}_{b\leftarrow a}\right).
\Gamma^Y_{b\leftarrow a}Use in source · 1 passage
Search aliases: Γ^Y_b← a Gamma leftarrow
Use in source · 02_connections_obstructions.tex, line 889
For coordinate models of the source and target colors, an additive target coordinate space, and a chosen noncanonical connection, a pointwise node defect can be defined by D^Γ_b,a(Φ) = Γ^Y_b← a∘widehatΦ_a - widehatΦ_b∘ (∏_qΓ^X_q_b← a). This expression compares two chosen routes. With canonical exact transport in every slot it vanishes by [source reference].
Exact mathematical passage
\mathfrak D^\Gamma_{b,a}(\Phi) = \Gamma^Y_{b\leftarrow a}\circ\widehat\Phi_a - \widehat\Phi_b\circ \left(\prod_q\Gamma^{X_q}_{b\leftarrow a}\right).
\Gamma_{0\leftarrow2}Use in source · 2 passages
Search aliases: Γ_0←2 Gamma leftarrow
Use in source · visual_atlas.tex, line 228
eal hypothesisMODEL D / THE MINIMAL AND REVERSIBLE INTERFACESChapter III, mathematical text: Definition 5.1, Definition 5.5 Choose the forward scalar transports Γ_1←0=2, Γ_2←1=3, and Γ_2←0=5. Their multiplicative comparison between transport through two steps and direct transport is D^×=6/5. A loop also needs a declared return map Γ_0←2.
Exact mathematical passage
\Gamma_{0\leftarrow2}
\gamma_1Use in source · 8 passages
Search aliases: γ_1 gamma
Use in source · 03_packets_and_graphs.tex, line 957
It is enough to treat a word in the generators and packet projectors. Insert I=∑_γ∈ IP_γ between consecutive nonprojector generators. Every resulting (β,α) block is a finite sum of products P_β G_mP_γ_m-1 G_m-1⋯ P_γ_1G_1P_α. A nonzero product requires every consecutive block to be nonzero, hence gives a directed path α→γ_1→⋯→γ_m-1→β. Packet projectors only insert stationary steps. Thus a word has no block from α to an unreachable β. Linear combinations preserve the same zero pattern.
Exact mathematical passage
P_\beta G_mP_{\gamma_{m-1}} G_{m-1}\cdots P_{\gamma_1}G_1P_\alpha.
\Gamma_{1\leftarrow0}Use in source · 5 passages
Search aliases: Γ_1←0 Gamma leftarrow
Use in source · 02_connections_obstructions.tex, line 667
[A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly when vu=w. If the three coordinate systems are literally the same and canonical transport is used, then u=v=w=1 and the obstruction vanishes. Choosing …
Exact mathematical passage
\Gamma_{1\leftarrow0}(x)=ux,\qquad \Gamma_{2\leftarrow1}(x)=vx,\qquad \Gamma_{2\leftarrow0}(x)=wx,
\gamma_2Use in source · 11 passages
Search aliases: γ_2 gamma
Use in source · 03_nonlinear_coordinates.tex, line 401
[The second-order composition law] At order two, the partition formula has two shapes. For packet directions h_1 and h_2, R_δ D^2(F∘ G)(x)[h_1,h_2] = · ∑_γ_1,γ_2 J^2_δmidγ_1,γ_2F(Gx) [J^1_γ_1midα_1G(x)h_1,J^1_γ_2midα_2G(x)h_2] · + ∑_γ J^1_δmidγF(Gx) [J^2_γmidα_1,α_2G(x)[h_1,h_2]]. The first line records two first-order changes entering a quadratic interaction of F. The second records a genuine second-order interaction created inside G before F is applied.
Exact mathematical passage
\begin{aligned} R_\delta D^2(F\circ G)(x)[h_1,h_2] ={}& \sum_{\gamma_1,\gamma_2} J^2_{\delta\mid\gamma_1,\gamma_2}F(Gx) [J^1_{\gamma_1\mid\alpha_1}G(x)h_1,J^1_{\gamma_2\mid\alpha_2}G(x)h_2]\\ &+ \sum_\gamma J^1_{\delta\mid\gamma}F(Gx) [J^2_{\gamma\mid\alpha_1,\alpha_2}G(x)[h_1,h_2]]. \end{aligned}
\gamma_{2k}Use in source · 1 passage
Search aliases: γ_2k gamma
Use in source · strengthening.tex, line 803
wers of G are integral by the same component formulas used above. Thus modulo 2 A_0, γ_k(H_0) equals the divided power of γ_2(X)+ϵ X. The identity γ_r(γ_2(X)) =frac(2r)!2^r r!γ_2r(X) has an odd integer scalar. Every term involving ϵ^j for j≥2 vanishes modulo two. The surviving zero and first powers give γ_k(H_0)≡ γ_2k(X)+ϵγ_2k-1(X) pmod2 A_0. The multiplication coefficient in the second term is 2k-1, also odd. By the odd moments [source reference] and independence, L(γ_m(P+Q))≡ m+1pmod2. The τ-w…
Exact mathematical passage
\gamma_k(H_0)\equiv \gamma_{2k}(X)+\epsilon\gamma_{2k-1}(X) \pmod{2\mathcal A_0}. \label{eq:zero-parity-leading-pd}
\gamma_{2k-1}Use in source · 1 passage
Search aliases: γ_2k-1 gamma
Use in source · strengthening.tex, line 803
wers of G are integral by the same component formulas used above. Thus modulo 2 A_0, γ_k(H_0) equals the divided power of γ_2(X)+ϵ X. The identity γ_r(γ_2(X)) =frac(2r)!2^r r!γ_2r(X) has an odd integer scalar. Every term involving ϵ^j for j≥2 vanishes modulo two. The surviving zero and first powers give γ_k(H_0)≡ γ_2k(X)+ϵγ_2k-1(X) pmod2 A_0. The multiplication coefficient in the second term is 2k-1, also odd. By the odd moments [source reference] and independence, L(γ_m(P+Q))≡ m+1pmod2. The τ-w…
Exact mathematical passage
\gamma_k(H_0)\equiv \gamma_{2k}(X)+\epsilon\gamma_{2k-1}(X) \pmod{2\mathcal A_0}. \label{eq:zero-parity-leading-pd}
\gamma_{2m}Use in source · 1 passage
Search aliases: γ_2m gamma
Use in source · strengthening.tex, line 349
Every displayed component of H_0 has all its divided powers in A_0. Besides the generators, the needed identities are γ_m(P^2/2) · =frac(2m)!2^m m!γ_2m(P),nonumber γ_m(3PQ) · =3^m m!γ_m(P)γ_m(Q),nonumber γ_m(P-Q) · =∑_r=0^m(-1)^r γ_m-r(P)γ_r(Q). The first scalar counts partitions into pairs and is an integer. For the last term of H_0, multiply the third identity by (i E F)^m and reduce the Euler powers using the integral quadratic relations. Finally, γ_k(X+Y)=∑_r=0^kγ_r(X)γ_k-r(Y) proves γ_k(H_0…
Exact mathematical passage
\gamma_m(P^2/2) &=\frac{(2m)!}{2^m m!}\gamma_{2m}(P),\nonumber\\ \gamma_m(3PQ)&=3^m m!\gamma_m(P)\gamma_m(Q),\nonumber\\ \gamma_m(P-Q)&=\sum_{r=0}^m(-1)^r \gamma_{m-r}(P)\gamma_r(Q). \label{eq:components-divided-powers}
\gamma_{2r}Use in source · 1 passage
Search aliases: γ_2r gamma
Use in source · strengthening.tex, line 797
:four-euler-H gives H_0=γ_2(X)+ϵ X+2G, G=-γ_2(P)-γ_2(Q)+PQ-τ Q, ϵ^2=2(τ-P-Q-2PQ). The divided powers of G are integral by the same component formulas used above. Thus modulo 2 A_0, γ_k(H_0) equals the divided power of γ_2(X)+ϵ X. The identity γ_r(γ_2(X)) =frac(2r)!2^r r!γ_2r(X) has an odd integer scalar. Every term involving ϵ^j for j≥2 vanishes modulo two. The surviving zero and first powers give γ_k(H_0)≡ γ_2k(X)+ϵγ_2k-1(X) pmod2 A_0. The multiplication coefficient in the second term is 2k-1, …
Exact mathematical passage
\gamma_r(\gamma_2(X)) =\frac{(2r)!}{2^r r!}\gamma_{2r}(X)
\Gamma_{2\leftarrow0}Use in source · 5 passages
Search aliases: Γ_2←0 Gamma leftarrow
Use in source · 02_connections_obstructions.tex, line 667
[A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly when vu=w. If the three coordinate systems are literally the same and canonical transport is used, then u=v=w=1 and the obstruction vanishes. Choosing …
Exact mathematical passage
\Gamma_{1\leftarrow0}(x)=ux,\qquad \Gamma_{2\leftarrow1}(x)=vx,\qquad \Gamma_{2\leftarrow0}(x)=wx,
\Gamma_{2\leftarrow1}Use in source · 5 passages
Search aliases: Γ_2←1 Gamma leftarrow
Use in source · 02_connections_obstructions.tex, line 667
[A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly when vu=w. If the three coordinate systems are literally the same and canonical transport is used, then u=v=w=1 and the obstruction vanishes. Choosing …
Exact mathematical passage
\Gamma_{1\leftarrow0}(x)=ux,\qquad \Gamma_{2\leftarrow1}(x)=vx,\qquad \Gamma_{2\leftarrow0}(x)=wx,
\Gamma_{a_0\leftarrow a_0}Use in source · 1 passage
Search aliases: Γ_a_0← a_0 Gamma leftarrow
Use in source · 02_connections_obstructions.tex, line 296
Induct on the path length. Appending a_n+1 left-multiplies the inductive expression by Γ_a_n+1← a_n, and flatness replaces the two factors by Γ_a_n+1← a_0. For a loop, normalization gives Γ_a_0← a_0=id.
Exact mathematical passage
\Gamma_{a_0\leftarrow a_0}=\id
\Gamma_{a_{n+1}\leftarrow a_0}Use in source · 1 passage
Search aliases: Γ_a_n+1← a_0 Gamma leftarrow
Use in source · 02_connections_obstructions.tex, line 295
Induct on the path length. Appending a_n+1 left-multiplies the inductive expression by Γ_a_n+1← a_n, and flatness replaces the two factors by Γ_a_n+1← a_0. For a loop, normalization gives Γ_a_0← a_0=id.
Exact mathematical passage
\Gamma_{a_{n+1}\leftarrow a_0}
\Gamma_{a_{n+1}\leftarrow a_n}Use in source · 1 passage
Search aliases: Γ_a_n+1← a_n Gamma leftarrow
Use in source · 02_connections_obstructions.tex, line 293
Induct on the path length. Appending a_n+1 left-multiplies the inductive expression by Γ_a_n+1← a_n, and flatness replaces the two factors by Γ_a_n+1← a_0. For a loop, normalization gives Γ_a_0← a_0=id.
Exact mathematical passage
\Gamma_{a_{n+1}\leftarrow a_n}
\Gamma_{a_n\leftarrow a_0}Use in source · 1 passage
Search aliases: Γ_a_n← a_0 Gamma leftarrow
Use in source · 02_connections_obstructions.tex, line 284
[Flat pair transport collapses to endpoints] If Γ is flat, then every path satisfies Γ_p=Γ_a_n← a_0. In particular, every loop has identity holonomy. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.pathTransport_eq_direct_of_flat; Komorebi.PaperIII.Transport.ReversibleConnection.indexedPathTransport_eq_direct_of_flat; Komorebi.PaperIII.Transport.ReversibleConnection.indexedLoopTransport_eq_refl_of_flat
Exact mathematical passage
\Gamma_p=\Gamma_{a_n\leftarrow a_0}.
\Gamma_{a\leftarrow o}Use in source · 2 passages
Search aliases: Γ_a← o Gamma leftarrow
Use in source · 02_connections_obstructions.tex, line 351
[Flat connections are gauges of the canonical pair transport] Suppose the C_a are exact packet coordinate modules for one ambient module, so the canonical J_b← a is available. If Γ is a flat normalized reversible pair connection, choose o∈ B and set G_a=Γ_a← oJ_o← a. Then every pair transport satisfies Γ_b← a =G_bJ_b← aG_a^-1. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.flatGauge; Komorebi.PaperIII.Transport.ReversibleConnection.flat_isGaugeOfCanonical
Exact mathematical passage
G_a=\Gamma_{a\leftarrow o}J_{o\leftarrow a}.
\Gamma_bUse in source · 4 passages
Search aliases: Γ_b Gamma
Use in source · 02_strengthened.tex, line 452
[Path interference formula] Fix a source vector ψ∈ P_a_0 H and a target packet b. Let Γ_b be the set of packet paths from a_0 to b, and set η_γ=U_γψ. Choose any total order on this finite path set. Then normP_bUψ^2 = ∑_γ∈Γ_bnormη_γ^2 + 2Re ∑_substackγ,γ'∈Γ_b
Exact mathematical passage
\Gamma_b
\Gamma_{b\leftarrow o}Use in source · 1 passage
Search aliases: Γ_b← o Gamma leftarrow
Use in source · 02_connections_obstructions.tex, line 367
Reversibility gives G_a^-1=J_a← oΓ_o← a. Therefore G_bJ_b← aG_a^-1 · = Γ_b← o J_o← bJ_b← a J_a← o Γ_o← a · = Γ_b← oΓ_o← a · = Γ_b← a. The first reduction uses the canonical cocycle twice. The second uses flatness of Γ.
Exact mathematical passage
\begin{aligned} G_bJ_{b\leftarrow a}G_a^{-1} &= \Gamma_{b\leftarrow o} J_{o\leftarrow b}J_{b\leftarrow a} J_{a\leftarrow o} \Gamma_{o\leftarrow a}\\ &= \Gamma_{b\leftarrow o}\Gamma_{o\leftarrow a}\\ &= \Gamma_{b\leftarrow a}. \end{aligned}
\Gamma_{c\leftarrow a}^{-1}Use in source · 2 passages
Search aliases: Γ_c← a^-1 Gamma leftarrow
Use in source · 02_connections_obstructions.tex, line 33
At a triple (a,b,c), the compiled cocycle predicate is Γ_c← bΓ_b← a =Γ_c← a. Flatness means this equality holds for every triple. The formal multiplicative relative defect and triangle holonomy are D^×_c,b,a = (Γ_c← bΓ_b← a) Γ_c← a^-1, H_c,b,a = Γ_a← c Γ_c← b Γ_b← a. The defect is based at the target c. The holonomy is based at a. statusleanthreeKomorebi.PaperIII.Transport.Connection.CocycleAt; Komorebi.PaperIII.Transport.Connection.Flat; Komorebi.PaperIII.Transport.Connection.cocycleDefect; Kom…
Exact mathematical passage
D^\times_{c,b,a} = \bigl(\Gamma_{c\leftarrow b}\Gamma_{b\leftarrow a}\bigr) \Gamma_{c\leftarrow a}^{-1}, \qquad H_{c,b,a} = \Gamma_{a\leftarrow c} \Gamma_{c\leftarrow b} \Gamma_{b\leftarrow a}.
\Gamma_eUse in source · 1 passage
Search aliases: Γ_e Gamma
Use in source · 02_connections_obstructions.tex, line 161
[A connection on a chart graph] One may specify Γ_e only on the oriented edges of a chart graph, together with inverse maps on reversed edges. Path transport and loop holonomy below still make sense. A direct cocycle defect for (a,b,c), however, requires a declared direct transport from a to c. The pairwise API in [source reference] keeps this comparison total.
Exact mathematical passage
\Gamma_e
\gamma_iUse in source · 3 passages
Search aliases: γ_i gamma
Use in source · Paper_XII_Traditional_Final.tex, line 494
[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.
Exact mathematical passage
\gamma_i
\gamma_jUse in source · 2 passages
Search aliases: γ_j gamma
Use in source · Paper_X_Block27_Amended.tex, line 1228
compatible family ρ_i,λ of representations of the fixed finite group, set V_i,λ=ρ_i,λ∘ q_L. Suppose the actual absolute-Galois members are irreducible and the systems are pairwise componentwise nonisomorphic. The finite prime separator now yields distinct permitted rational primes p_j, actual prime ideals Q_j and arithmetic Frobenius witnesses γ_j as above, outside any prescribed finite set, such that the common algebraic trace matrix has nonzero determinant. Every coefficient embedding preserve…
Exact mathematical passage
\gamma_j
\Gamma_kUse in source · 1 passage
Search aliases: Γ_k Gamma
Use in source · visual_atlas.tex, line 510
A NONLINEAR PHASE ORBITChapter II, mathematical text: Section 12.3, Smith-phase identity (p. 43) Smith normal form concerns linear congruences. The Smith impedance chart uses the nonlinear coordinate change Γ=fracz-1z+1, z=frac1+Γ1-Γ. They share a name, not a mathematical definition. Sample the reflection circle at the exact algebraic points Γ_k=tfrac12 e^ikπ/4.
Exact mathematical passage
\Gamma_k=\tfrac12 e^{ik\pi/4}
\gamma_kUse in source · 11 passages
Search aliases: γ_k gamma
Use in source · 03_nonlinear_coordinates.tex, line 353
di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n] · = ∑_π={B_1,…,B_k} ∑_γ_1,…,γ_k J^k_δmidγ_1,…,γ_kF(Gx) [ G_γ_1,B_1(xmid h),…, G_γ_k,B_k(xmid h) ]. The first sum runs over all set partitions of {1,…,n}.
Exact mathematical passage
\begin{aligned} &J^{n}_{\delta\mid\alpha_1,\ldots,\alpha_n}(F\circ G)(x) [h_1,\ldots,h_n]\\ &= \sum_{\pi=\{B_1,\ldots,B_k\}} \sum_{\gamma_1,\ldots,\gamma_k} J^{k}_{\delta\mid\gamma_1,\ldots,\gamma_k}F(Gx) \big[ G_{\gamma_1,B_1}(x\mid h),\ldots, G_{\gamma_k,B_k}(x\mid h) \big]. \end{aligned}
\gamma_{k-1}Use in source · 4 passages
Search aliases: γ_k-1 gamma
Use in source · strengthening.tex, line 887
For k=2^r≥4, put q=k/2, which is even. Normalize the exact expansion [source reference] by 2^q before reducing modulo two. Its uth auxiliary term has excess u+lfloor(k-u)/2rfloor-q=lfloor u/2rfloor. Only auxiliary indices zero and one can therefore survive: fracγ_k(H_2)2^q ≡ V^qγ_k(X) +ϵ V^q-1Gγ_k-1(X) pmod2 A_2. Here V≡τpmod2 and τ^2≡1pmod2. Since q is even and ϵτ≡ϵpmod2, this reduces to γ_k(X)+ϵ Gγ_k-1(X). Moreover, G≡γ_2(X)+τ Qpmod2. After evaluation, every term with a single τ vanishes, so t…
Exact mathematical passage
\frac{\gamma_k(H_2)}{2^q} \equiv V^q\gamma_k(X) +\epsilon V^{q-1}G\gamma_{k-1}(X) \pmod{2\mathcal A_2}. \label{eq:two-power-leading-normalized}
\gamma_{k-r}Use in source · 2 passages
Search aliases: γ_k-r gamma
Use in source · strengthening.tex, line 361
γ_m(P-Q) · =∑_r=0^m(-1)^r γ_m-r(P)γ_r(Q). The first scalar counts partitions into pairs and is an integer. For the last term of H_0, multiply the third identity by (i E F)^m and reduce the Euler powers using the integral quadratic relations. Finally, γ_k(X+Y)=∑_r=0^kγ_r(X)γ_k-r(Y) proves γ_k(H_0)∈ A_0. Applying the functional and putting W=U^4-1=2H_0 gives L(W^m)∈2^m m!R (m≥0). These moments are rational. The identity R∩ Q_2= Z_2, obtained from the linear independence of 1,i over Q_2, descends e…
Exact mathematical passage
\gamma_k(X+Y)=\sum_{r=0}^k\gamma_r(X)\gamma_{k-r}(Y)
\gamma_{m-1}Use in source · 2 passages
Search aliases: γ_m-1 gamma
Use in source · 03_packets_and_graphs.tex, line 957
It is enough to treat a word in the generators and packet projectors. Insert I=∑_γ∈ IP_γ between consecutive nonprojector generators. Every resulting (β,α) block is a finite sum of products P_β G_mP_γ_m-1 G_m-1⋯ P_γ_1G_1P_α. A nonzero product requires every consecutive block to be nonzero, hence gives a directed path α→γ_1→⋯→γ_m-1→β. Packet projectors only insert stationary steps. Thus a word has no block from α to an unreachable β. Linear combinations preserve the same zero pattern.
Exact mathematical passage
P_\beta G_mP_{\gamma_{m-1}} G_{m-1}\cdots P_{\gamma_1}G_1P_\alpha.
\gamma_{m-r}Use in source · 1 passage
Search aliases: γ_m-r gamma
Use in source · strengthening.tex, line 349
Every displayed component of H_0 has all its divided powers in A_0. Besides the generators, the needed identities are γ_m(P^2/2) · =frac(2m)!2^m m!γ_2m(P),nonumber γ_m(3PQ) · =3^m m!γ_m(P)γ_m(Q),nonumber γ_m(P-Q) · =∑_r=0^m(-1)^r γ_m-r(P)γ_r(Q). The first scalar counts partitions into pairs and is an integer. For the last term of H_0, multiply the third identity by (i E F)^m and reduce the Euler powers using the integral quadratic relations. Finally, γ_k(X+Y)=∑_r=0^kγ_r(X)γ_k-r(Y) proves γ_k(H_0…
Exact mathematical passage
\gamma_m(P^2/2) &=\frac{(2m)!}{2^m m!}\gamma_{2m}(P),\nonumber\\ \gamma_m(3PQ)&=3^m m!\gamma_m(P)\gamma_m(Q),\nonumber\\ \gamma_m(P-Q)&=\sum_{r=0}^m(-1)^r \gamma_{m-r}(P)\gamma_r(Q). \label{eq:components-divided-powers}
\gamma_NUse in source · 16 passages
Search aliases: γ_N gamma
Use in source · Paper_IV_Traditional_Final.tex, line 1864
The zero strings are governed by two adjacent exponents. Put a_N=fracπ N, s_N=sin a_N, c_N=cos a_N, and A_N=fracπs_N, t_m=A_N(m+frac12). For N≥3, also put γ_N=c_N-cos(3a_N)>0.
Exact mathematical passage
\gamma_N=c_N-\cos(3a_N)>0.
\gamma_{n-q}Use in source · 1 passage
Search aliases: γ_n-q gamma
Use in source · 20_arithmetic.tex, line 805
[Classical odd-length reconstruction] For every positive odd integer p and every n∈ N, E_2n=∑_m=0^nbinom2n2mγ_m p^2mS_2(n-m)(p). Equivalently, E_2n=∑_q=0^n V_nqS_2q(p) for the lower triangular matrix V_nq=binom2n2qγ_n-qp^2(n-q) (q≤ n), V_nq=0 (q>n). It has V_nn=1 and p^2(n-q)mid V_nq when q≤ n.
Exact mathematical passage
V_{nq}=\binom{2n}{2q}\gamma_{n-q}p^{2(n-q)}\quad(q\le n), \qquad V_{nq}=0\quad(q>n).
\Gamma_{o\leftarrow a}Use in source · 2 passages
Search aliases: Γ_o← a Gamma leftarrow
Use in source · 02_connections_obstructions.tex, line 366
Reversibility gives G_a^-1=J_a← oΓ_o← a. Therefore G_bJ_b← aG_a^-1 · = Γ_b← o J_o← bJ_b← a J_a← o Γ_o← a
Exact mathematical passage
G_a^{-1}=J_{a\leftarrow o}\Gamma_{o\leftarrow a}
\gamma_pUse in source · 6 passages
Search aliases: γ_p gamma
Use in source · Paper_X_Block27_Amended.tex, line 1187
The arithmetic witnesses use the integral closure A of Z in overline Q. For each rational prime p, Lean constructs a maximal ideal Q of A above (p) and an actual element γ_p∈ G_ Q such that γ_p∈Stab_G_ Q( Q), overlineγ_p(a)=overline a^ p (a∈ A). The existence argument applies Mathlib's proved profinite stabilizer-surjectivity theorem to this actual invariant ring. The residue p-power map is proved bijective by taking p-th roots in overline Q and showing that they remain integral when their p-th …
Exact mathematical passage
\gamma_p\in G_{\Q}
\Gamma_RUse in source · 2 passages
Search aliases: Γ_R Gamma
Use in source · strengthening.tex, line 334
Let Γ_R(P,Q) be the R-subalgebra generated by all γ_m(P) and γ_r(Q). Adjoin E, F with E^2=1+2P, F^2=1+2Q, and call the resulting algebra A_0. Its elements are finite expressions. No completion is being presumed. It is generated as a Γ_R(P,Q)-module by 1, E, F, E F. The functional maps this module to R. For an even basis term this is [source reference] and independence. For a basis term with an odd Euler factor the moment is zero, since P and Q are even polynomials in their respective variables.
Exact mathematical passage
\Gamma_R(P,Q)
\gamma_rUse in source · 5 passages
Search aliases: γ_r gamma
Use in source · strengthening.tex, line 335
Let Γ_R(P,Q) be the R-subalgebra generated by all γ_m(P) and γ_r(Q). Adjoin E, F with E^2=1+2P, F^2=1+2Q, and call the resulting algebra A_0. Its elements are finite expressions. No completion is being presumed. It is generated as a Γ_R(P,Q)-module by 1, E, F, E F. The functional maps this module to R. For an even basis term this is [source reference] and independence. For a basis term with an odd Euler factor the moment is zero, since P and Q are even polynomials in their respective variables.
Exact mathematical passage
\gamma_r(Q)
\gamma_uUse in source · 2 passages
Search aliases: γ_u gamma
Use in source · Paper_X_Block27_Amended.tex, line 1281
The choice of absolute prime and Frobenius lift is also controlled. Let ( Q_u,γ_u) and ( Q_v,γ_v) be any two actual arithmetic witnesses at the same rational prime. Transitivity on primes of A above (p) gives a∈ G_ Q with Q_v=a Q_u, γ_v^-1aγ_u a^-1∈ I_ Q_v. Consequently, any representation trivial on I_ Q_v has the same trace on the two witnesses. For representations factoring through L, triviality on this actual absolute inertia is derived whenever p∉ S_L. The restrictions of all absolute witne…
Exact mathematical passage
(\mathfrak Q_u,\gamma_u)
\gamma_vUse in source · 4 passages
Search aliases: γ_v gamma
Use in source · Paper_X_Block27_Amended.tex, line 1281
The choice of absolute prime and Frobenius lift is also controlled. Let ( Q_u,γ_u) and ( Q_v,γ_v) be any two actual arithmetic witnesses at the same rational prime. Transitivity on primes of A above (p) gives a∈ G_ Q with Q_v=a Q_u, γ_v^-1aγ_u a^-1∈ I_ Q_v. Consequently, any representation trivial on I_ Q_v has the same trace on the two witnesses. For representations factoring through L, triviality on this actual absolute inertia is derived whenever p∉ S_L. The restrictions of all absolute witne…
Exact mathematical passage
(\mathfrak Q_v,\gamma_v)
\gamma_v^{-1}Use in source · 1 passage
Search aliases: γ_v^-1 gamma
Use in source · Paper_X_Block27_Amended.tex, line 1284
The choice of absolute prime and Frobenius lift is also controlled. Let ( Q_u,γ_u) and ( Q_v,γ_v) be any two actual arithmetic witnesses at the same rational prime. Transitivity on primes of A above (p) gives a∈ G_ Q with Q_v=a Q_u, γ_v^-1aγ_u a^-1∈ I_ Q_v. Consequently, any representation trivial on I_ Q_v has the same trace on the two witnesses. For representations factoring through L, triviality on this actual absolute inertia is derived whenever p∉ S_L. The restrictions of all absolute witne…
Exact mathematical passage
\mathfrak Q_v=a\mathfrak Q_u,\qquad \gamma_v^{-1}a\gamma_u a^{-1}\in I_{\mathfrak Q_v}.
\Gamma_yUse in source · 4 passages
Search aliases: Γ_y Gamma
Use in source · Paper_V_Traditional_Final.tex, line 942
[label=(roman*)] item A global minimizer Λ_y^*∈*argmin_Λ∈ P( T)J_y(Λ) always exists. item If the minimizer is unique and at least one competing pruning exists, its gap Γ_y = min_Λ≠Λ_y^* (J_y(Λ)-J_y(Λ_y^*)) is attained and strictly positive. If the pruning set is a singleton, its only element minimizes every criterion. No real-valued minimum over an empty competitor set is introduced. item In the case with competitors, if new data y' satisfy sup_Λ∈ P( T) |J_y'(Λ)-J_y(Λ)| <fracΓ_y2, then Λ_y'^*=Λ_…
Exact mathematical passage
\Gamma_y = \min_{\Lambda\ne\Lambda_y^*} \bigl(J_y(\Lambda)-J_y(\Lambda_y^*)\bigr)
\Delta'Use in source · 3 passages
Search aliases: Δ' Delta
Use in source · Paper_XII_Traditional_Final.tex, line 474
[Functorial group algebra transfer] A group homomorphism q:Δ'→Δ induces an O algebra homomorphism q_*: O[Δ']⟶ O[Δ]. If q is surjective, then q_* is surjective. Augmentation commutes with the transfer: ϵ_Δ q_*=ϵ_Δ'.
Exact mathematical passage
q:\Delta'\to\Delta
\Delta^2Use in source · 1 passage
Search aliases: Δ^2 Delta
Use in source · atlas-pages.tex, line 734
vpage39Higher differences annihilate polynomial interiorsBOUNDARY STENCILS / SIGNAL RESPONSE AND NOISE GAINChapter V: Theorem E10.6, Proposition E10.7, Corollary E10.8. Let x(n)=n+2(n-3)_+ on the integer line. Its slope changes from one to three at the breakpoint 3. The second forward difference is Δ^2x(n)=x(n+2)-2x(n+1)+x(n). vfig[.87]ext_differences headAn exact small stencil calculation The values at n=0,…,6 are (0,1,2,3,6,9,12). The second differences at n=0,…,4 are (0,0,2,0,0). The only non…
Exact mathematical passage
\Delta^2x(n)=x(n+2)-2x(n+1)+x(n).
\delta^2Use in source · 7 passages
Search aliases: δ^2 delta
Use in source · Paper_V_Traditional_Final.tex, line 2873
[A perturbed disk] Let E and F be concentric disks of radii r and r+δ in R^2, with δ>0. Then μ(Emathbintriangle F) = π((r+δ)^2-r^2) = 2π rδ+πδ^2. For equal amplitude a, the exact L^2 packet error is |a|sqrt2π rδ+πδ^2. This gives the expected square root dependence on boundary displacement.
Exact mathematical passage
\mu(E\mathbin\triangle F) = \pi\bigl((r+\delta)^2-r^2\bigr) = 2\pi r\delta+\pi\delta^2.
\delta^3Use in source · 1 passage
Search aliases: δ^3 delta
Use in source · atlas-pages.tex, line 649
y on the interval between its two locations: H_τ-H_widehatτ _2=sqrtδ=tfrac14, H_τ-H_widehatτ _∞=1. vfig[.89]ext_location headFor a ramp, the error height shrinks too Write r_τ(t)=(t-τ)_+. The difference grows linearly from zero to δ on [τ,widehatτ] and stays at δ on [widehatτ,1]. Direct integration gives r_τ-r_widehatτ _2^2 =fracδ^33+(1-widehatτ)δ^2 =frac176144<frac1256=δ^2. Thus the L^2 ramp error is at most δ, and its supremum error is exactly δ. Positive-order truncated powers are Lipschitz i…
Exact mathematical passage
\|r_\tau-r_{\widehat\tau}\|_2^2 =\frac{\delta^3}{3}+(1-\widehat\tau)\delta^2 =\frac{17}{6144}<\frac1{256}=\delta^2.
\Delta^{d+1}Use in source · 3 passages
Search aliases: Δ^d+1 Delta
Use in source · Paper_V_Traditional_Final.tex, line 3337
[Polynomial annihilation and boundary support] Let x be a sequence that agrees with a polynomial of degree at most d on each interval of a finite partition of Z. Then Δ^d+1x(n)=0 whenever the complete stencil {n,n+1,…,n+d+1} lies inside one polynomial piece. Hence Δ^d+1x is supported in a band of at most d+1 addresses before each breakpoint.
Exact mathematical passage
\Delta^{d+1}x(n)=0
\Delta^G_{c,b,a}Use in source · 1 passage
Search aliases: Δ^G_c,b,a Delta
Use in source · 02_connections_obstructions.tex, line 328
[Gauge covariance] The defect and triangle holonomy transform as Δ^G_c,b,a =G_cΔ_c,b,aG_a^-1, H^G_c,b,a =G_aH_c,b,aG_a^-1. Thus flatness and triviality of triangle holonomy are gauge invariant. statusleanthreeKomorebi.PaperIII.Transport.ReversibleConnection.gauge_cocycleDefect; Komorebi.PaperIII.Transport.ReversibleConnection.gauge_triangleHolonomy; Komorebi.PaperIII.Transport.ReversibleConnection.gauge_flat_iff; Komorebi.PaperIII.Transport.ReversibleConnection.gauge_triangleHolonomy_eq_refl_iff
Exact mathematical passage
\Delta^G_{c,b,a} =G_c\Delta_{c,b,a}G_a^{-1}, \qquad H^G_{c,b,a} =G_aH_{c,b,a}G_a^{-1}.
\Delta^kUse in source · 2 passages
Search aliases: Δ^k Delta
Use in source · 72_recovered.tex, line 1641
[Polynomial exponential Mahler transfer] Let f(n)=P(n)u^n with P∈ Z_p[n] of degree d. Let H=Λ_p,L, fix an address j, and write Q_j(x)=P(Hx+j), y=u^H, q=y-1. Then f(Hx+j)=∑_r≥0c_j,rbinom xr, where c_j,r = u^j ∑_k=0^min(d,r) binom rk Δ^kQ_j(0)y^kq^r-k. Consequently, v_p(c_j,r)≥ Lmax(r-d,0). For R≥ d, truncation at depth R is valid modulo p^L(R+1-d) uniformly on Z_p.
Exact mathematical passage
\boxed{ c_{j,r} = u^j \sum_{k=0}^{\min(d,r)} \binom rk \Delta^kQ_j(0)y^kq^{r-k}. }
\Delta^mUse in source · 5 passages
Search aliases: Δ^m Delta
Use in source · strengthening.tex, line 390
The conversion from unit steps to arbitrary mixed steps is used several times, so it is useful to make it explicit. Write Δ=Δ_1. For H>0, Δ_H=∑_m=1^Hbinom HmΔ^m. A mixed term with local orders m_1,…,m_k≥1 and M=∑_ν m_ν therefore has valuation at least M+v_2(M!)+β +∑_ν(v_2(H_ν)-v_2(m_ν)) if the unit-order estimate has offset β for every positive order. The inequalities m-1≥ v_2(m), M≥ k, v_2(M!)≥ v_2(k!) make this at least k+v_2(k!)+β+∑_ν v_2(H_ν). Every expansion is finite. This proves the desir…
Exact mathematical passage
\Delta_H=\sum_{m=1}^H\binom Hm\Delta^m. \label{eq:finite-step-expansion}
\Delta^rUse in source · 1 passage
Search aliases: Δ^r Delta
Use in source · 72_recovered.tex, line 1461
[Finite Mahler reconstruction] Given f(0),…,f(N), put c_r=Δ^rf(0). Then f(n)=∑_r=0^nc_rbinom nr (0≤ n≤ N). For every s with 0≤ s≤ N, Δ^sf(n) = ∑_r=s^Nc_rbinom nr-s (0≤ n≤ N-s).
Exact mathematical passage
c_r=\Delta^rf(0)
\Delta^sUse in source · 6 passages
Search aliases: Δ^s Delta
Use in source · 72_recovered.tex, line 1469
[Finite Mahler reconstruction] Given f(0),…,f(N), put c_r=Δ^rf(0). Then f(n)=∑_r=0^nc_rbinom nr (0≤ n≤ N). For every s with 0≤ s≤ N, Δ^sf(n) = ∑_r=s^Nc_rbinom nr-s (0≤ n≤ N-s).
Exact mathematical passage
\boxed{ \Delta^sf(n) = \sum_{r=s}^{N}c_r\binom n{r-s} \qquad(0\leq n\leq N-s). }
\delta^sUse in source · 5 passages
Search aliases: δ^s delta
Use in source · supplement_07.tex, line 163
0, · Nnmid d, N=mp. Let L=p(p-1) and δ=U^p+L-U^p=U^p^2-U^p. The source proves the operator dictionary (δ^sX)(d)=Δ_L^s X(d+ps). At the valid nonnegative coefficient degree d=mpn-ps, with s≤ mn, this becomes (δ^sX)(mpn-ps)=Δ_h^s e(n), h=(p-1)/m. The packed sequence and difference operations are definitions. Their coefficient correspondence is proved before any divisibility assumption is introduced.
Exact mathematical passage
(\delta^sX)(d)=\Delta_L^s X(d+ps).
\delta^{s-1}Use in source · 1 passage
Search aliases: δ^s-1 delta
Use in source · supplement_08.tex, line 113
pageheading3. A nonsingular Euler-operator argumentThe standard interval without an analytic power-series-in-the-index premise On Hurwitz coefficients, the actual Euler operator plus c is (Λ_c a)(d)=(d+c)a(d). Direct induction proves its commutator with every defect iterate: δ^sΛ_c a =Λ_cδ^s a +sp(pU^p^2-U^p)δ^s-1a, s≥1. Suppose a is integral and supported at degrees divisible by p, and pnmid c. Every supported pivot d+c is proved to be a unit of Z_p. The defect preserves this support. Induction…
Exact mathematical passage
\delta^s\Lambda_c a =\Lambda_c\delta^s a +sp\bigl(pU^{p^2}-U^p\bigr)\delta^{s-1}a, \qquad s\ge1.
\Delta_{(p-1)p^a}^{\,r}Use in source · 1 passage
Search aliases: Δ_(p-1)p^a^ r Delta
Use in source · 72_recovered.tex, line 1589
[Basic odd and dyadic difference gains] For odd p, Δ_(p-1)p^a^ ru^n = u^n(u^(p-1)p^a-1)^r, v_p(Δ_(p-1)p^a^ ru^n)≥(a+1)r. For odd u∈ Z_2^×, Δ_2^a+1^ ru^n = u^n(u^2^a+1-1)^r, v_2(Δ_2^a+1^ ru^n)≥(a+3)r.
Exact mathematical passage
\boxed{ \Delta_{(p-1)p^a}^{\,r}u^n = u^n\left(u^{(p-1)p^a}-1\right)^r, \qquad \vp\left(\Delta_{(p-1)p^a}^{\,r}u^n\right)\geq(a+1)r. }
\delta_1Use in source · 1 passage
Search aliases: δ_1 delta
Use in source · atlas-pages.tex, line 95
N IMPULSE MAKES THE DIFFERENCE VISIBLEChapter V: Theorem 4.1. A one-sample delay swaps the two phases and wraps one of them into the preceding channel index. Its matrix is J(D), not DI_2. vfigsample_delay headTrace one impulse The input δ_0 has a single nonzero value in even channel address zero. Under J(D) it appears in odd channel address zero: this is the original sample δ_1. Under DI_2 it stays even but moves to channel address one: this is δ_2. Thus J(D)^2=DI_2, H(D)=I_2+2J(D)-J(D)^2+J(D)^3…
Exact mathematical passage
\delta_1
\Delta_{12\cdot2^a}^{\,r}Use in source · 1 passage
Search aliases: Δ_12·2^a^ r Delta cdot
Use in source · 72_recovered.tex, line 2433
18 · 11 . Therefore Δ_12·2^a^ rF_n ≡0 pmod 2^(a+3)r. Every dyadic refinement doubles the step and adds one guaranteed binary digit at each Mahler depth.
Exact mathematical passage
\boxed{ \Delta_{12\cdot2^a}^{\,r}F_n \equiv0 \pmod {2^{(a+3)r}}. }
\Delta_{13}Use in source · 1 passage
Search aliases: Δ_13 Delta
Use in source · atlas-pages.tex, line 948
0 · 1 · 0. The generator defect is Δ=C_η A-BC_η, with only one nonzero entry, Δ_13=7η (matrix indices begin at one). vfigext_substitution platecaptionThe perturbation lets the discarded eigenvalue enter the comparison. Exact naturality holds at η=0. A nonzero defect is transformed, not erased, by the rational calculus. headA resolvent identity supplies the sign and both inverse factors For F(z)=1/(1+z), all required inverses exist, and C_η F(A)-F(B)C_η =-(I+B)^-1Δ(I+A)^-1 =mat0 · 0 · -7η/30
Exact mathematical passage
\Delta_{13}=7\eta
\Delta_1^2Use in source · 3 passages
Search aliases: Δ_1^2 Delta
Use in source · supplement_08.tex, line 189
pageheading5. Examples, verification and the remaining gapThe coverage count is unchanged; the unresolved theorem now has a stronger proved part The original rational recurrence gives C_6,2(1)=-frac128, C_6,2(2)=frac425364, C_6,2(3)=-frac20397592660. Consequently, the new source proves Δ_1^2 C_6,2(1)=-frac9499591235, v_3 (Δ_1^2 C_6,2(1))=2. This attains the bound min(2,2·1)=2. A claim of a third power is rejected. The separate cubic example proves that the normalized differential defect is not a…
Exact mathematical passage
\Delta_1^2 C_{6,2}(1)=-\frac{949959}{1235},\qquad v_3\!\left(\Delta_1^2 C_{6,2}(1)\right)=2.
\Delta_1^iUse in source · 3 passages
Search aliases: Δ_1^i Delta
Use in source · 20_arithmetic.tex, line 498
g_2(n) · =frac1(4n+1)(2n+1), · g_3(n) · =frac3(4n+1)(2n+1)(4n+3). Every denominator is odd. An affine sequence with even slope satisfies 2^kmidΔ_1^k f(n) for every k,n. The exact product identity Δ_1^s(fg)(n)=∑_i=0^sbinom si Δ_1^if(n)Δ_1^s-ig(n+i) proves closure of this property under products. Applying the formula with the factors interchanged to fg=1 isolates f(n+s)Δ_1^sg(n). Multiplication by f(n+s)^-1 and induction then prove closure under pointwise unit inverses. Thus all four g_j have the …
Exact mathematical passage
\Delta_1^s(fg)(n)=\sum_{i=0}^s\binom si \Delta_1^if(n)\Delta_1^{s-i}g(n+i)
\Delta_1^kUse in source · 27 passages
Search aliases: Δ_1^k Delta
Use in source · 10_foundations.tex, line 512
affine Mahler kernel by Θ_M,s(r,k)=[z^r](1+z)^s((1+z)^M-1)^k =∑_j=0^k(-1)^k-jbinom kjbinomMj+sr. It vanishes unless k≤ r≤ Mk+s. For a finite binomial expansion f(n)=∑_r=0^Dc_rbinom nr, the kth Newton coefficient of g(q)=f(Mq+s) is therefore c_k(g)=∑_r=k^min(D,Mk+s)c_rΘ_M,s(r,k). Indeed, insert the finite stencil for Δ_1^k g(0) and interchange its two finite sums. Coefficient extraction of the binomial expansion of ((1+z)^M-1)^k gives the kernel. In the binary zero-address case, Θ_2,0(k+j,k)=2^k-…
Exact mathematical passage
\Delta_1^k g(0)
\Delta_1^{k-r}Use in source · 1 passage
Search aliases: Δ_1^k-r Delta
Use in source · strengthening.tex, line 159
ence identity is Δ_1^kfrac11+sn =frac(-s)^k k!∏_h=0^k(1+s(n+h)). It follows by induction, subtracting adjacent rational functions. Each successive subtraction adds the factor -(k+1)s in the numerator. Its denominator is a unit, so it belongs to p^k Z_p. The discrete Leibniz formula Δ_1^k(fg)(n) =∑_r=0^kbinom kr (Δ_1^r f)(n)(Δ_1^k-rg)(n+r) then proves Δ_1^k g_N,j(n)∈ p^k Z_p for every k,n. The coefficient at degree Nn of L_1^kG_N,j is precisely p^-kΔ_1^k g_N,j(n), and the other residue classes va…
Exact mathematical passage
\Delta_1^k(fg)(n) =\sum_{r=0}^k\binom kr (\Delta_1^r f)(n)(\Delta_1^{k-r}g)(n+r) \label{eq:discrete-leibniz-final}
\Delta_1^mUse in source · 4 passages
Search aliases: Δ_1^m Delta
Use in source · strengthening.tex, line 203
To retain the valuation of a step, let B be any sequence satisfying Δ_1^m B∈ p^m Z_p at every order and row. Use the finite operator identity Δ_H=∑_i=1^Hbinom HiΔ_1^i. The relation ibinom Hi=HbinomH-1i-1 implies v_p(p^ibinom Hi) ≥ i+v_p(H)-v_p(i)≥1+v_p(H). Multiplying these identities for the k steps, every term of total order M has its scalar multiplied by p^M, and is divisible by p^k+∑_ν v_p(H_ν). This proves the mixed-step bound. Finally, Δ_H A_N,j(n)=(-1)^n+HT_H C_N,j(n). Iteration contribut…
Exact mathematical passage
\Delta_1^m B\in p^m\mathbb Z_p
\Delta_1^rUse in source · 5 passages
Search aliases: Δ_1^r Delta
Use in source · 10_foundations.tex, line 249
For f: N→ R, define the step-h forward difference and its iterates by Δ_h f(n)=f(n+h)-f(n), Δ_h^0 f=f, and put c_r(f)=Δ_1^r f(0). Thus the coefficients are determined by the differences of f.
Exact mathematical passage
c_r(f)=\Delta_1^r f(0)
\Delta_1^{s-i}Use in source · 2 passages
Search aliases: Δ_1^s-i Delta
Use in source · 20_arithmetic.tex, line 498
g_2(n) · =frac1(4n+1)(2n+1), · g_3(n) · =frac3(4n+1)(2n+1)(4n+3). Every denominator is odd. An affine sequence with even slope satisfies 2^kmidΔ_1^k f(n) for every k,n. The exact product identity Δ_1^s(fg)(n)=∑_i=0^sbinom si Δ_1^if(n)Δ_1^s-ig(n+i) proves closure of this property under products. Applying the formula with the factors interchanged to fg=1 isolates f(n+s)Δ_1^sg(n). Multiplication by f(n+s)^-1 and induction then prove closure under pointwise unit inverses. Thus all four g_j have the …
Exact mathematical passage
\Delta_1^s(fg)(n)=\sum_{i=0}^s\binom si \Delta_1^if(n)\Delta_1^{s-i}g(n+i)
\Delta_2Use in source · 11 passages
Search aliases: Δ_2 Delta
Use in source · Paper_IV_Traditional_Final.tex, line 1245
[title=A coefficient calculation at order two] Write Δ_2(w)^-r=∑_n≥0b_n(r)w^n. Since Δ_2(w)=1-frac w2+fracw^224+O(w^3), -logΔ_2(w)=frac w2+fracw^212+O(w^3), the first logarithmic coefficients are κ_2,1=1/2 and κ_2,2=1/6. Newton's recurrence gives b_1(r)=frac r2, 2b_2(r)=r(frac r4+frac16). Restoring the factorial normalization yields P_2,1(r)=r and P_2,2(r)=r(3r+2). At r=1, the second value is five. No zero-location assumption is needed for this finite calculation. modelfigureparameter_polynomial…
Exact mathematical passage
\Delta_2(w)^{-r}=\sum_{n\geq0}b_n(r)w^n
\delta_2Use in source · 1 passage
Search aliases: δ_2 delta
Use in source · atlas-pages.tex, line 95
ay swaps the two phases and wraps one of them into the preceding channel index. Its matrix is J(D), not DI_2. vfigsample_delay headTrace one impulse The input δ_0 has a single nonzero value in even channel address zero. Under J(D) it appears in odd channel address zero: this is the original sample δ_1. Under DI_2 it stays even but moves to channel address one: this is δ_2. Thus J(D)^2=DI_2, H(D)=I_2+2J(D)-J(D)^2+J(D)^3 recovers the previous page's four-tap transfer matrix. The same delay algebra…
Exact mathematical passage
\delta_2
\Delta_{2,1,0}Use in source · 3 passages
Search aliases: Δ_2,1,0 Delta
Use in source · 02_connections_obstructions.tex, line 673
[A scalar triangle] Let B={0,1,2} and C_0=C_1=C_2=𝕜 for a field 𝕜. Choose u,v,w∈𝕜^× and set Γ_1←0(x)=ux, Γ_2←1(x)=vx, Γ_2←0(x)=wx, with reverse transports given by inverse scalars. Then Δ_2,1,0(x)=(vu-w)x, and the triangle holonomy based at 0 is H_2,1,0(x)=w^-1vux. Therefore H_2,1,0-id =w^-1Δ_2,1,0. The connection is flat on this triangle exactly when vu=w. If the three coordinate systems are literally the same and canonical transport is used, then u=v=w=1 and the obstruction vanishes. Choosing …
Exact mathematical passage
\Delta_{2,1,0}(x)=(vu-w)x,
\Delta_{24}^{\,r}Use in source · 1 passage
Search aliases: Δ_24^ r Delta
Use in source · 72_recovered.tex, line 1759
1932 · 1194 . For every scalar coordinate of the state sequence C^nv_0, Δ_24^ ra_n ≡0 pmod 24^r. One integral jump therefore gives simultaneous congruence ladders at the primes two and three.
Exact mathematical passage
\boxed{ \Delta_{24}^{\,r}a_n \equiv0 \pmod {24^r}. }
\Delta_{2^{a+1}}^{\,r}Use in source · 1 passage
Search aliases: Δ_2^a+1^ r Delta
Use in source · 72_recovered.tex, line 1599
[Basic odd and dyadic difference gains] For odd p, Δ_(p-1)p^a^ ru^n = u^n(u^(p-1)p^a-1)^r, v_p(Δ_(p-1)p^a^ ru^n)≥(a+1)r. For odd u∈ Z_2^×, Δ_2^a+1^ ru^n = u^n(u^2^a+1-1)^r, v_2(Δ_2^a+1^ ru^n)≥(a+3)r.
Exact mathematical passage
\boxed{ \Delta_{2^{a+1}}^{\,r}u^n = u^n\left(u^{2^{a+1}}-1\right)^r, \qquad \vTwo\left(\Delta_{2^{a+1}}^{\,r}u^n\right)\geq(a+3)r. }
\Delta_4Use in source · 2 passages
Search aliases: Δ_4 Delta
Use in source · atlas-pages.tex, line 139
vpage8Four zeros become one orbit zeroTHE REDUCED VARIABLEChapter IV: Definition 5.7, Example 11.12, Corollary 11.13. For N=4 the factorization is exact: D_4(z)=cos(z/sqrt2)cosh(z/sqrt2), D_4(z)=Δ_4(z^4). Set A_4=sqrt2π. All the zeros are A_4(m+1/2)i^k, with m≥0 and 0≤ k<4. Each zero is simple. vfigorbit_reduction The drawing uses u=z/A_4 and v=w/A_4^4, so its marked coordinates are rational. The first four-point orbit has fourth power 1/16. The second has fourth power 81/16. headMultiplicity is…
Exact mathematical passage
D_4(z)=\cos(z/\sqrt2)\cosh(z/\sqrt2),\qquad D_4(z)=\Delta_4(z^4).
\Delta_4^{-1}Use in source · 1 passage
Search aliases: Δ_4^-1 Delta
Use in source · atlas-pages.tex, line 459
tion: the number 69COEFFICIENT AND ZERO WORKED MODELChapter IV: Corollary 7.7, Example 11.12, Theorem 12.6. Use N=4 throughout this page. Its denominator, its exact zero lattice, and the reciprocal recurrences provide two routes to the same coefficient. All symbols in the plate refer to this one order. vfigworked_circuit headRoute one: invert the denominator The w^2 coefficient of Δ_4^-1 is frac124^2-frac18!=frac698!. Thus P_4,2(1)=69. Keeping r variable gives P_4,2(r)=35r^2+34r. headRoute two: …
Exact mathematical passage
\Delta_4^{-1}
\Delta_4^2Use in source · 1 passage
Search aliases: Δ_4^2 Delta
Use in source · supplement_04.tex, line 184
A sharp unit-mode gain. For p=5 and u=2, Δ_4^2(u^n)|_n=0=(2^4-1)^2=225. The source proves 5^2mid225 and 5^3nmid225. This example attains the guaranteed gain. No additional uniform power of p is claimed.
Exact mathematical passage
\Delta_4^2(u^n)\big|_{n=0}=(2^4-1)^2=225.
\Delta_4^sUse in source · 1 passage
Search aliases: Δ_4^s Delta
Use in source · 20_arithmetic.tex, line 541
Packing g_j into degrees divisible by four gives K_4,j. Its source property is precisely charge-one flatness. Inversion gives charge -1, and the charge shift restores charge one for the reciprocal P_4,j. If dec(F)(d)=d![t^d]F, the exact coefficient identity is dec(A_1^sF)(d) =2^-sΔ_4^sdec(F)(d). At d=4n, packing turns the right side into 2^-sΔ_1^sC_j(n). This extraction requires no negative degree and therefore holds at every n, including zero. It proves the unit-step bound leanKomorebiPaperIX.D…
Exact mathematical passage
\operatorname{dec}(A_1^sF)(d) =2^{-s}\Delta_4^s\operatorname{dec}(F)(d).
\delta_aUse in source · 4 passages
Search aliases: δ_a delta
Use in source · Paper_V_Traditional_Final.tex, line 1742
The translation orbit is the standard coordinate family (δ_a)_a∈ Z^d. Its finite span is the space of finitely supported sequences. Finitely supported sequences are dense in ℓ^p for finite p and in c_0. Their closure in the supremum norm is c_0, which is a proper closed subspace of ℓ^∞.
Exact mathematical passage
(\delta_a)_{a\in\ZZ^d}
\delta_{ab}Use in source · 8 passages
Search aliases: δ_ab delta
Use in source · monograph_atlas.tex, line 355
The Knill--Laflamme compressions are PE_a^*E_bP=δ_abP. Their scalar form says that the error labels do not resolve the encoded vector. The exact checks evaluate all sixteen compressions, all sixteen syndrome products, the four probabilities, the recovery normalization, and the full recovered matrix.
Exact mathematical passage
PE_a^*E_bP=\delta_{ab}P
\delta_{ac}Use in source · 4 passages
Search aliases: δ_ac delta
Use in source · monograph_atlas.tex, line 159
The packetwise unitarity identity is likewise an operator identity, including the off-diagonal cancellations: ∑_b U_ba^*U_bc=δ_acP_a, U_ba=P_bUP_a. It cannot be replaced by the assertion that every path individually has unit norm. The exact checks verify all four choices of a,c for the two rank-one packets, as well as the two path products leading to the opposite output.
Exact mathematical passage
\sum_b U_{ba}^*U_{bc}=\delta_{ac}P_a, \qquad U_{ba}=P_bUP_a.
\delta_{ba}Use in source · 1 passage
Search aliases: δ_ba delta
Use in source · 02_strengthened.tex, line 386
Let U=U_mU_m-1⋯ U_1 be a finite circuit with m≥1, and let (P_a)_a∈ I be a complete orthogonal packet family. For the empty circuit, U=I and its transfer block is P_bP_a=δ_baP_a, whose endpoint sum reconstructs I.
Exact mathematical passage
P_bP_a=\delta_{ba}P_a
\delta_{bd}Use in source · 3 passages
Search aliases: δ_bd delta
Use in source · 02_strengthened.tex, line 424
[Packet unitarity identities] If U is unitary and U_ba=P_bUP_a, then ∑_bU_ba^*U_bc=δ_acP_a and ∑_aU_baU_da^*=δ_bdP_b.
Exact mathematical passage
\sum_aU_{ba}U_{da}^*=\delta_{bd}P_b.
\delta_{d0}Use in source · 3 passages
Search aliases: δ_d0 delta
Use in source · supplement_07.tex, line 39
The identity sequence is 1(d)=δ_d0. The new source proves the actual higher product rule U^a(f⋆ g)=∑_i=0^abinom ai (U^i f)⋆(U^a-ig). The argument starts with Pascal's identity for the first shift, then inducts on a. All sums are finite at each coefficient.
Exact mathematical passage
\mathbf1(d)=\delta_{d0}
\Delta_eUse in source · 8 passages
Search aliases: Δ_e Delta
Use in source · 02_supplement_01.tex, line 127
Packet perturbation certificates. textitSource item 8.1 and two uniform consequences Let X be a normed space over a nontrivially normed field. The formal objects are actual continuous linear maps X→ X, with their operator norm. Let a(e) identify the source packet of edge e, and suppose Δ_e=Δ_e P_a(e). Set A=∑_e A_e, widehat A=∑_e(A_e+Δ_e). The exact operator and vector identities are proved first: widehat A-A=∑_eΔ_e P_a(e), (widehat A-A)x=∑_eΔ_e(P_a(e)x). Applying the finite triangle inequality …
Exact mathematical passage
\Delta_e=\Delta_e P_{a(e)}.
\delta_EUse in source · 2 passages
Search aliases: δ_E delta
Use in source · Paper_V_Traditional_Final.tex, line 2103
[Stability under filter perturbation] Assume RE=z^-dI_N. Let widetilde E=E+Δ E, widetilde R=R+Δ R. Set e_0 · =*ess sup E _op, · r_0 · =*ess sup R _op, δ_E · =*ess sup Δ E _op, · δ_R · =*ess sup Δ R _op. Then the delay compensated reconstruction defect obeys ε_rec ≤ r_0δ_E+e_0δ_R+δ_Rδ_E.
Exact mathematical passage
\begin{aligned} e_0&=\esssup\|E\|_{\mathrm{op}}, & r_0&=\esssup\|R\|_{\mathrm{op}},\\ \delta_E&=\esssup\|\Delta E\|_{\mathrm{op}}, & \delta_R&=\esssup\|\Delta R\|_{\mathrm{op}}. \end{aligned}
\Delta_GUse in source · 3 passages
Search aliases: Δ_G Delta
Use in source · 03_nonlinear_coordinates.tex, line 750
[Exact defect under composition] Let G_f:Y_f→ Z_f and G_c:Y_c→ Z_c, with coarsening C_Z:Z_f→ Z_c. Then Δ_G∘ F(x) = Δ_G(F_f(x)) + G_c(F_c(C_Xx)+Δ_F(x)) - G_c(F_c(C_Xx)). If G_c is L-Lipschitz on the relevant set, then Δ_G∘ F(x) ≤ Δ_G(F_f(x)) + L Δ_F(x) .
Exact mathematical passage
\Delta_{G\circ F}(x) = \Delta_G(F_f(x)) + G_c\big(F_c(C_Xx)+\Delta_F(x)\big) - G_c\big(F_c(C_Xx)\big).
\Delta_{GF}Use in source · 1 passage
Search aliases: Δ_GF Delta
Use in source · 03_nonlinear_coordinates.tex, line 779
[Linear product rule] If all maps in [source reference] are linear, then Δ_GF = Δ_G F_f+G_cΔ_F.
Exact mathematical passage
\Delta_{GF} = \Delta_G F_f+G_c\Delta_F.
\Delta_{G\circ F}Use in source · 2 passages
Search aliases: Δ_G∘ F Delta circ
Use in source · 03_nonlinear_coordinates.tex, line 750
[Exact defect under composition] Let G_f:Y_f→ Z_f and G_c:Y_c→ Z_c, with coarsening C_Z:Z_f→ Z_c. Then Δ_G∘ F(x) = Δ_G(F_f(x)) + G_c(F_c(C_Xx)+Δ_F(x)) - G_c(F_c(C_Xx)). If G_c is L-Lipschitz on the relevant set, then Δ_G∘ F(x) ≤ Δ_G(F_f(x)) + L Δ_F(x) .
Exact mathematical passage
\Delta_{G\circ F}(x) = \Delta_G(F_f(x)) + G_c\big(F_c(C_Xx)+\Delta_F(x)\big) - G_c\big(F_c(C_Xx)\big).
\Delta_HUse in source · 4 passages
Search aliases: Δ_H Delta
Use in source · strengthening.tex, line 31
Write S f(n)=f(n+1) and Δ_H=S^H-1. A mixed difference Δ_H_1⋯Δ_H_k acts on the row variable n, rather than on the full Euler subscript. We use v_p(0)=+∞. Orders and starting rows are nonnegative integers. Steps are positive integers unless stated otherwise. An empty mixed difference is the original value, and a zero ordinary step makes every nonempty product containing it zero.
Exact mathematical passage
\Delta_H=S^H-1
\Delta_H^kUse in source · 3 passages
Search aliases: Δ_H^k Delta
Use in source · strengthening.tex, line 232
[Threshold-free dyadic congruence] For N=2^a, a≥0, all permitted j, and n≥0, v_2(Δ_H_1⋯Δ_H_kE_Nn^(N,j)) ≥ v_2(j!)+k+∑_ν=1^k v_2(H_ν). In particular, Δ_H^k gains k(1+v_2(H)) bits before normalization. For r>v_2(j!), the row displacement H=2^max(0,r-v_2(j!)-1) is a sufficient period modulo 2^r from row zero. If r≤ v_2(j!), all original coefficients vanish modulo 2^r.
Exact mathematical passage
\Delta_H^k
\Delta_h^{k-mn}Use in source · 1 passage
Search aliases: Δ_h^k-mn Delta
Use in source · 20_arithmetic.tex, line 339
Here is the coefficient extraction explicitly. If a_d=d![t^d]F, then d![t^d]B_p^kF=p^-kΔ_p(p-1)^ka_d+pk. For F=j! P_N,j and k≤ mn, take d=mpn-pk. Support in multiples of mp turns this into p^-kΔ_h^kU_N,j(n), so flatness gives divisibility by p^k. For k>mn, expand Δ_h^k-mn applied to Δ_h^mnU_N,j(n). Every shifted row is at least n, so each summand is divisible by p^mn. Finally Δ_uh=((I+Δ_h)^u-I), whose kth power for u>0 is an integral combination of powers of Δ_h of order at least k. The case u=0…
Exact mathematical passage
\Delta_h^{k-mn}
\Delta_h^{mn}Use in source · 1 passage
Search aliases: Δ_h^mn Delta
Use in source · 20_arithmetic.tex, line 340
Here is the coefficient extraction explicitly. If a_d=d![t^d]F, then d![t^d]B_p^kF=p^-kΔ_p(p-1)^ka_d+pk. For F=j! P_N,j and k≤ mn, take d=mpn-pk. Support in multiples of mp turns this into p^-kΔ_h^kU_N,j(n), so flatness gives divisibility by p^k. For k>mn, expand Δ_h^k-mn applied to Δ_h^mnU_N,j(n). Every shifted row is at least n, so each summand is divisible by p^mn. Finally Δ_uh=((I+Δ_h)^u-I), whose kth power for u>0 is an integral combination of powers of Δ_h of order at least k. The case u=0…
Exact mathematical passage
\Delta_h^{mn}U_{N,j}(n)
\Delta_H^rUse in source · 2 passages
Search aliases: Δ_H^r Delta
Use in source · 72_recovered.tex, line 1691
0=1, which is not in 3 Z_3. The printed claim without these restrictions is therefore not a certificate as written. The full original proof follows under the corrected hypotheses. Assume C^H=I+p^LB with B∈ M_d( Z_p). For every address j and every x∈ Z_p, a_Hx+j = ∑_r≥0 p^Lrbinom xr w^TC^jB^rv_0. The series converges on Z_p, and Δ_H^ra_n ≡0 pmod p^Lr.
Exact mathematical passage
\boxed{ \Delta_H^ra_n \equiv0 \pmod {p^{Lr}}. }
\Delta_h^sUse in source · 8 passages
Search aliases: Δ_h^s Delta
Use in source · 10_foundations.tex, line 255
[Finite Newton--Mahler identities] For every n,s,h∈ N, f(n) · =∑_r=0^nc_r(f)binom nr, c_r(Δ_1^s f) · =c_s+r(f), Δ_1^s f(n) · =∑_j=0^nc_s+j(f)binom nj, Δ_h^s f(n) · =∑_j=0^s(-1)^s-jbinom sj f(n+jh). The finite interpolant I_Nf(n)=∑_r=0^Nc_r(f)binom nr agrees with f for n≤ N. It has the same order-s unit-step difference at n whenever n+s≤ N. For arbitrary coefficients c_0,…,c_N, the corresponding binomial interpolant has exactly those coefficients and zero coefficients above N.
Exact mathematical passage
\begin{aligned} f(n)&=\sum_{r=0}^{n}c_r(f)\binom nr,\\ c_r(\Delta_1^s f)&=c_{s+r}(f),\\ \Delta_1^s f(n)&=\sum_{j=0}^{n}c_{s+j}(f)\binom nj,\\ \Delta_h^s f(n)&=\sum_{j=0}^{s}(-1)^{s-j}\binom sj f(n+jh). \end{aligned}
\Delta_{H_1}Use in source · 7 passages
Search aliases: Δ_H_1 Delta
Use in source · strengthening.tex, line 32
Write S f(n)=f(n+1) and Δ_H=S^H-1. A mixed difference Δ_H_1⋯Δ_H_k acts on the row variable n, rather than on the full Euler subscript. We use v_p(0)=+∞. Orders and starting rows are nonnegative integers. Steps are positive integers unless stated otherwise. An empty mixed difference is the original value, and a zero ordinary step makes every nonempty product containing it zero.
Exact mathematical passage
\Delta_{H_1}\cdots\Delta_{H_k}
\Delta_{H_a}^{\,r}Use in source · 4 passages
Search aliases: Δ_H_a^ r Delta
Use in source · 72_recovered.tex, line 2338
2^a+1, · p=2. Then Δ_H_a^ ru^n = u^n(u^H_a-1)^r. For odd p, v_p(Δ_H_a^ ru^n) ≥(a+1)r. For p=2 and odd u, v_2(Δ_H_a^ ru^n) ≥(a+3)r. Equivalently, every additional prime power digit increases the certified Mahler slope by one.
Exact mathematical passage
\boxed{ \Delta_{H_a}^{\,r}u^n = u^n\left(u^{H_a}-1\right)^r. }
\Delta_{H_k}Use in source · 7 passages
Search aliases: Δ_H_k Delta
Use in source · strengthening.tex, line 32
Write S f(n)=f(n+1) and Δ_H=S^H-1. A mixed difference Δ_H_1⋯Δ_H_k acts on the row variable n, rather than on the full Euler subscript. We use v_p(0)=+∞. Orders and starting rows are nonnegative integers. Steps are positive integers unless stated otherwise. An empty mixed difference is the original value, and a zero ordinary step makes every nonempty product containing it zero.
Exact mathematical passage
\Delta_{H_1}\cdots\Delta_{H_k}
\delta_iUse in source · 1 passage
Search aliases: δ_i delta
Use in source · Paper_VI_Traditional_Final.tex, line 1640
Markov · point mass δ_i · K^n or e^tQ · state lumping and CK^n=bar K^nC
Exact mathematical passage
\delta_i
\delta_{ij}Use in source · 7 passages
Search aliases: δ_ij delta
Use in source · 02_coordinates.tex, line 383
[Residue packet algebra] For every commutative ring R, the operators (P_N,j)_0≤ j<N satisfy the identities below. If R is nonzero, they form a packet system on R[[z]]: P_N,iP_N,j=δ_ijP_N,j, ∑_j=0^N-1P_N,j=I.
Exact mathematical passage
P_{N,i}P_{N,j}=\delta_{ij}P_{N,j}, \qquad \sum_{j=0}^{N-1}P_{N,j}=I.
\delta_jUse in source · 13 passages
Search aliases: δ_j delta
Use in source · Paper_VI_Traditional_Final.tex, line 5275
[Summable generator defects] Suppose C_j are contractions and (T_j(t)) are contraction semigroups with bounded generators A_j. Set δ_j=normC_jA_j+1-A_jC_j. Then normC_ℓ,mT_m(t)-T_ℓ(t)C_ℓ,m ≤ t∑_j=ℓ^m-1δ_j. If ∑_j≥0δ_j<∞, the right side tends to zero as ℓ→∞, uniformly in m>ℓ and uniformly for t in a fixed bounded interval.
Exact mathematical passage
\delta_j=\norm{C_jA_{j+1}-A_jC_j}.
\Delta_{j,e}Use in source · 5 passages
Search aliases: Δ_j,e Delta
Use in source · 01_manuscript.tex, line 1322
[Mixed precision product chain] If each fixed linear factor A_j is replaced by a fixed packetwise linear approximation widehat A_j, write E_j=widehat A_j-A_j=∑_eΔ_j,eP_α(j,e). A uniform certificate is E_j ≤δ_j:=∑_e Δ_j,e P_α(j,e) . For orthogonal packet projectors, δ_j≤∑_e Δ_j,e . The final operator error is bounded by [source reference] with these uniform δ_j. The corresponding vector error is this operator bound multiplied by the input norm.
Exact mathematical passage
E_j=\widehat A_j-A_j=\sum_e\Delta_{j,e}P_{\alpha(j,e)}
\delta_{ji}Use in source · 2 passages
Search aliases: δ_ji delta
Use in source · 02_connections_obstructions.tex, line 621
pansion for different matrices] For i,j∈ V and m≥1, set i_0=i and i_m=j. Then (B_m⋯ B_1)_ji = ∑_(i_1,…,i_m-1)∈ V^m-1 B_m[i_m,i_m-1]⋯ B_1[i_1,i_0]. When m=1, the sum is over the unique empty tuple and its summand is B_1[j,i]. For m=0, the product is the identity and the entry is the Kronecker delta δ_ji. If every B_r[q,p] vanishes outside a declared relation R_r(p,q), then the endpoint entry is zero whenever there is no packet path compatible with the successive relations. statusleanthreeKomorebi…
Exact mathematical passage
\delta_{ji}
\delta_{jk}Use in source · 6 passages
Search aliases: δ_jk delta
Use in source · Paper_IV_Traditional_Final.tex, line 651
[Signed derivative cycle] For 1≤ j<N, D_N,j'= D_N,j-1, and D_N,0'=- D_N,N-1. Every signed section satisfies D_N,j^(N)=- D_N,j. The N signed sections form the basis of y^(N)=-y determined by D_N,j^(k)(0)=δ_jk for 0≤ j,k<N.
Exact mathematical passage
\Dfun_{N,j}^{(k)}(0)=\delta_{jk}
\delta_{j\ell}Use in source · 2 passages
Search aliases: δ_jℓ delta ell
Use in source · 02_strengthened.tex, line 1343
For z∈ C^r, let F_z=∑_az_aE_a. Then PF_z^*F_zP=(z^*cz)P≥0, so c≥0. The unitary change of error coordinates gives PF_j^*F_ℓ P = λ_jδ_jℓP. Thus V_j^*V_ℓ=δ_jℓP. Each V_j is an isometry on P H, and distinct ranges are orthogonal. Their range projectors are therefore pairwise orthogonal.
Exact mathematical passage
PF_j^*F_\ell P = \lambda_j\delta_{j\ell}P.
\Delta_KUse in source · 7 passages
Search aliases: Δ_K Delta
Use in source · residue_continuation_body.tex, line 138
Ordinary class characters: an unconditional power saving. The preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl( O)→ C, define the actual finite ideal sum S_w(X)=∑_substackI≠0
Exact mathematical passage
A_C(X)=c_KX+O_{K,C}(X^{1-1/d}),\qquad c_K=\frac{2^{r_1}(2\pi)^{r_2}R_K}{\#\mu(K)\sqrt{\absn{\Delta_K}}}.
\delta_{kj}Use in source · 1 passage
Search aliases: δ_kj delta
Use in source · Paper_IV_Traditional_Final.tex, line 488
[Differential cycle and initial data] For 1≤ j<N, fracddz E_N,j(λ z) = λ E_N,j-1(λ z), while fracddz E_N,0(λ z) = λ E_N,N-1(λ z). Thus fracd^Ndz^N E_N,j(λ z) = λ^N E_N,j(λ z). For 0≤ k<N, .fracd^kdz^k E_N,j(λ z)|_z=0 = δ_kjλ^j. If λ≠0, the N functions E_N,j(λ z) form a basis of the solution space of y^(N)=λ^Ny.
Exact mathematical passage
\left.\frac{d^k}{dz^k}\Efun_{N,j}(\lambda z)\right|_{z=0} = \delta_{kj}\lambda^j.
\Delta_LUse in source · 6 passages
Search aliases: Δ_L Delta
Use in source · Paper_VI_Traditional_Final.tex, line 4010
[Fractional defect bound] Let 0<α<1. Suppose L_f≥ mI, L_c≥ mI for some m>0. Define Δ_L=CL_f-L_cC. Then normCL_f^α-L_c^α C ≤ α m^α-1normΔ_L. Consequently, norm C e^-tL_f^α - e^-tL_c^αC ≤ α t m^α-1normΔ_L.
Exact mathematical passage
\Delta_L=CL_f-L_cC.
\Delta_L^sUse in source · 1 passage
Search aliases: Δ_L^s Delta
Use in source · supplement_07.tex, line 163
0, · Nnmid d, N=mp. Let L=p(p-1) and δ=U^p+L-U^p=U^p^2-U^p. The source proves the operator dictionary (δ^sX)(d)=Δ_L^s X(d+ps). At the valid nonnegative coefficient degree d=mpn-ps, with s≤ mn, this becomes (δ^sX)(mpn-ps)=Δ_h^s e(n), h=(p-1)/m. The packed sequence and difference operations are definitions. Their coefficient correspondence is proved before any divisibility assumption is introduced.
Exact mathematical passage
(\delta^sX)(d)=\Delta_L^s X(d+ps).
\Delta_mUse in source · 2 passages
Search aliases: Δ_m Delta
Use in source · Paper_XII_Traditional_Final.tex, line 546
[Iwasawa limit] For Δ_m=( Z/p^m Z)^r, the inverse limit of O[Δ_m] is the Iwasawa algebra O[[ Z_p^r]]≅ O[[X_1,…,X_r]], where X_i corresponds to a chosen topological generator minus one [source reference].
Exact mathematical passage
\Delta_m=(\ZZ/p^m\ZZ)^r
\delta_mUse in source · 6 passages
Search aliases: δ_m delta
Use in source · 05_worked_models.tex, line 206
Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. Therefore sequences with finite support lie in its algebraic orbit hull and are dense in ℓ^2( Z).
Exact mathematical passage
S_m\delta_0=\delta_m
\delta_m^{(1)}Use in source · 1 passage
Search aliases: δ_m^(1) delta
Use in source · atlas-pages.tex, line 333
cos(sqrt3t/2)3. Divide by the nonzero factor 2e^t/2/3. The resulting equation separates the dominant cosine from the small third exponential. vfigtwo_term Ignoring the last term gives model zeros t_m=frac2πsqrt3(m+1/2). At t_m, the full normalized expression equals tfrac12e^-3t_m/2>0, so t_m is not an exact zero. Linearizing the cosine at that point predicts δ_m^(1)=frac(-1)^msqrt3e^-3t_m/2. The alternating slope explains why the true zero is to the right of the first model zero, then to the lef…
Exact mathematical passage
\delta_m^{(1)}=\frac{(-1)^m}{\sqrt3}e^{-3t_m/2}.
\delta_m^2Use in source · 1 passage
Search aliases: δ_m^2 delta
Use in source · Paper_IV_Traditional_Final.tex, line 2114
Put δ_m=τ_m-t_m. The large zero theorem gives δ_m=O(e^-γ_Nt_m). Since e^2is_Nt_m=-1, 1+e^2is_N(t_m+δ_m) = -2is_Nδ_m+O(δ_m^2). The remainder estimate through the first derivative gives R_N(t_m+δ_m) = R_N(t_m)+O(e^-γ_Nt_m|δ_m|). Substitution in the zero equation gives the displacement formula.
Exact mathematical passage
1+e^{2is_N(t_m+\delta_m)} = -2is_N\delta_m+O(\delta_m^2).
\delta_{mn}Use in source · 1 passage
Search aliases: δ_mn delta
Use in source · Paper_VI_Traditional_Final.tex, line 3187
· = ∑_n=0^∞ e^-ntH_n(x)fracu^nn!. Comparison of coefficients proves the eigenvalue formula. The same generating function shows ∫ H_mH_n dγ=n!δ_mn. The Hermite polynomials span all polynomials, and polynomials are dense in L^2(γ). Hence the normalized family is an orthonormal basis. See [source reference] for the general Gaussian Hilbert space setting.
Exact mathematical passage
\int H_mH_n\dd\gamma=n!\delta_{mn}.
\Delta_N'Use in source · 4 passages
Search aliases: Δ_N' Delta
Use in source · Paper_IV_Traditional_Final.tex, line 1292
Since D_N(z)=Δ_N(z^N), -frac D_N'(z) D_N(z) = -Nz^N-1fracΔ_N'(z^N)Δ_N(z^N). Differentiate the defining series for -logΔ_N and compare coefficients.
Exact mathematical passage
-\frac{\Dfun_N'(z)}{\Dfun_N(z)} = -Nz^{N-1}\frac{\Delta_N'(z^N)}{\Delta_N(z^N)}.
\delta_{n,r}Use in source · 1 passage
Search aliases: δ_n,r delta
Use in source · 06_models.tex, line 6
Let x=(x_n)_n∈ Z∈ c_00( Z;R) and let the seed be δ_0 under the bilateral shifts (S^rδ_0)_n=δ_n,r. This makes the algebraic span and every displayed sum finite. The N packets are x_j[m]=x[Nm+j]. Synthesis interleaves these sequences. A linear filter with a finite impulse response that is invariant under shifts becomes an N× N matrix of transfer blocks. Refining from N to M with Nmid M splits each coarse phase into its descendants without changing the reconstructed signal. The same formulas extend…
Exact mathematical passage
(S^r\delta_0)_n=\delta_{n,r}
\Delta_{n-1}Use in source · 2 passages
Search aliases: Δ_n-1 Delta
Use in source · Paper_VI_Traditional_Final.tex, line 3331
Let Δ_n-1 = {p∈ R^n: p_i≥0, ∑_ip_i=1 } be the probability simplex. Let λ∈[0,∞)^n have positive evidence Z_λ(p)=∑_iλ_ip_i. The Bayesian transfer is B_λ(p) = fracλodot pZ_λ(p).
Exact mathematical passage
\Delta_{n-1} = \left\{p\in\RR^n: p_i\geq0, \quad \sum_ip_i=1 \right\}
\delta_{nm}Use in source · 1 passage
Search aliases: δ_nm delta
Use in source · 03_packets_and_graphs.tex, line 410
[Countable unconditional packet isomorphism] Let X be a real or complex Banach space and let (P_n)_n≥1⊆ B(X) satisfy P_nP_m=δ_nmP_n and x=∑_n≥1P_nx unconditionally for every x∈ X. Assume K:= sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nP_n <∞. Then analysis and synthesis define inverse Banach space isomorphisms X≅⊕_n≥1^uP_nX. More precisely, A_ P ≤ K, S_ P ≤1. [source reference] ExportsCountable analytic packet decompositions under explicit convergence control.
Exact mathematical passage
P_nP_m=\delta_{nm}P_n
\Delta_{p(p-1)}^kUse in source · 1 passage
Search aliases: Δ_p(p-1)^k Delta
Use in source · 20_arithmetic.tex, line 333
Here is the coefficient extraction explicitly. If a_d=d![t^d]F, then d![t^d]B_p^kF=p^-kΔ_p(p-1)^ka_d+pk. For F=j! P_N,j and k≤ mn, take d=mpn-pk. Support in multiples of mp turns this into p^-kΔ_h^kU_N,j(n), so flatness gives divisibility by p^k. For k>mn, expand Δ_h^k-mn applied to Δ_h^mnU_N,j(n). Every shifted row is at least n, so each summand is divisible by p^mn. Finally Δ_uh=((I+Δ_h)^u-I), whose kth power for u>0 is an integral combination of powers of Δ_h of order at least k. The case u=0…
Exact mathematical passage
d![t^d]B_p^kF=p^{-k}\Delta_{p(p-1)}^ka_{d+pk}.
\Delta_{p-1}^sUse in source · 2 passages
Search aliases: Δ_p-1^s Delta
Use in source · 10_foundations.tex, line 339
[Differences of unit modes] Over every commutative ring, Δ_h^s(u^n)=u^n(u^h-1)^s. For any prime p, units u_i∈ Z_p^×, coefficients a_i∈ Z_p, and a finite index set I, the sequence f(n)=∑_i∈ Ia_i u_i^n satisfies Δ_p-1^s f(n)∈ p^s Z_p (n,s∈ N).
Exact mathematical passage
\Delta_{p-1}^s f(n)\in p^s\Zp \qquad(n,s\in\N).
\Delta_{\Pcal}Use in source · 4 passages
Search aliases: Δ_Pcal Δ_ P Delta Pcal Delta mathcal
Use in source · 03_packets_and_graphs.tex, line 757
[Block diagonal projection and packet commutant] Define Δ_ P(T):=∑_αP_α TP_α. Then Δ_ P is a linear idempotent on End(X) whose range is the block diagonal subalgebra. For T∈End(X), the following are equivalent: [label=(roman*)] item TP_α=P_α T for every α. item T_βα=0 whenever β≠α. item Δ_ P(T)=T. [source reference] ExportsSymmetry sector invariance and packet leakage decompositions.
Exact mathematical passage
\Delta_{\Pcal}(T):=\sum_{\alpha}P_\alpha TP_\alpha.
\Delta_{\Pcal}^2Use in source · 1 passage
Search aliases: Δ_Pcal^2 Δ_ P^2 Delta Pcal Delta mathcal
Use in source · 03_packets_and_graphs.tex, line 772
Pairwise annihilation gives Δ_ P^2=Δ_ P and identifies its range with diagonal block families. If T commutes with all P_α, then for β≠α, P_β TP_α=P_β P_α T=0. Thus (i) implies (ii). Exact reconstruction shows (ii) implies (iii). If T=∑_α P_α TP_α, then TP_γ=P_γ TP_γ=P_γ T, so (iii) implies (i).
Exact mathematical passage
\Delta_{\Pcal}^2=\Delta_{\Pcal}
\delta_RUse in source · 2 passages
Search aliases: δ_R delta
Use in source · Paper_V_Traditional_Final.tex, line 2103
[Stability under filter perturbation] Assume RE=z^-dI_N. Let widetilde E=E+Δ E, widetilde R=R+Δ R. Set e_0 · =*ess sup E _op, · r_0 · =*ess sup R _op, δ_E · =*ess sup Δ E _op, · δ_R · =*ess sup Δ R _op. Then the delay compensated reconstruction defect obeys ε_rec ≤ r_0δ_E+e_0δ_R+δ_Rδ_E.
Exact mathematical passage
\begin{aligned} e_0&=\esssup\|E\|_{\mathrm{op}}, & r_0&=\esssup\|R\|_{\mathrm{op}},\\ \delta_E&=\esssup\|\Delta E\|_{\mathrm{op}}, & \delta_R&=\esssup\|\Delta R\|_{\mathrm{op}}. \end{aligned}
\delta_rUse in source · 4 passages
Search aliases: δ_r delta
Use in source · root_scope.tex, line 387
[Exact quadratic compression] There is a unique preparation K_r(Z)= U_r(Z) W_r(Z), W_r(Z)=Z^2+2β_rZ+α_r, U_r≡ u_rpmod256. With δ_r=β_r^2-α_r and τ_r= U_r'(c_r)/ U_r(c_r), δ_r=-fracΦ_r^act U_r(c_r) (1-fracτ_r^2Φ_r^act4 U_r(c_r)), v_2(δ_r)=v_2(Φ_r^act). The equality includes the zero case. The same successive monic polynomial division used in Theorem~[source reference], started at precision eight, preserves the quadratic and unit reductions of [source reference]. Write W=Z^2+2β_rZ+α_r and w=W(c_r)…
Exact mathematical passage
\delta_r=\beta_r^2-\alpha_r
\delta_{rs}Use in source · 8 passages
Search aliases: δ_rs delta
Use in source · 72_recovered.tex, line 2985
[Complete residue packet system] For fixed N, the projectors P_N,r satisfy P_N,rP_N,s = δ_rsP_N,r and ∑_rbmod NP_N,r=I.
Exact mathematical passage
P_{N,r}P_{N,s} = \delta_{rs}P_{N,r}
\delta_{su}Use in source · 1 passage
Search aliases: δ_su delta
Use in source · 03_nonlinear_coordinates.tex, line 705
[Matrix difference grading] Let A=M_N(𝕜) and assign the matrix unit E_rs the label r-s modulo N. Since E_rsE_uv=δ_suE_rv, the output label is (r-s)+(u-v)=r-v whenever the product is nonzero. Matrix multiplication therefore obeys the cyclic convolution law exactly. The same grading resolves commutators and polynomial matrix equations.
Exact mathematical passage
E_{rs}E_{uv}=\delta_{su}E_{rv},
\Delta_{u(p-1)/m}^{k}Use in source · 1 passage
Search aliases: Δ_u(p-1)/m^k Delta
Use in source · 20_arithmetic.tex, line 571
[Step-sensitive divisor-branch bound] Under the hypotheses of Theorem~[source reference], for all u,k,n∈ N, Δ_u(p-1)/m^kU_mp,j(n) ∈ p^min(k(1+ν_p(u)),mn) Z_p. For C_mp,j the corresponding valuation lower bound is min(k(1+ν_p(u)),mn)-v_p(j!).
Exact mathematical passage
\Delta_{u(p-1)/m}^{k}U_{mp,j}(n) \in p^{\min(k(1+\nu_p(u)),mn)}\Z_p.
\Delta_{uh}Use in source · 1 passage
Search aliases: Δ_uh Delta
Use in source · 20_arithmetic.tex, line 342
]F, then d![t^d]B_p^kF=p^-kΔ_p(p-1)^ka_d+pk. For F=j! P_N,j and k≤ mn, take d=mpn-pk. Support in multiples of mp turns this into p^-kΔ_h^kU_N,j(n), so flatness gives divisibility by p^k. For k>mn, expand Δ_h^k-mn applied to Δ_h^mnU_N,j(n). Every shifted row is at least n, so each summand is divisible by p^mn. Finally Δ_uh=((I+Δ_h)^u-I), whose kth power for u>0 is an integral combination of powers of Δ_h of order at least k. The case u=0 gives zero for k>0, and order zero is the original integral…
Exact mathematical passage
\Delta_{uh}=((I+\Delta_h)^u-I)
\Delta_{uh}^kUse in source · 1 passage
Search aliases: Δ_uh^k Delta
Use in source · 20_arithmetic.tex, line 346
pplied to Δ_h^mnU_N,j(n). Every shifted row is at least n, so each summand is divisible by p^mn. Finally Δ_uh=((I+Δ_h)^u-I), whose kth power for u>0 is an integral combination of powers of Δ_h of order at least k. The case u=0 gives zero for k>0, and order zero is the original integral coefficient. This proves the bound for all u and k. Finally Δ_uh^kU_N,j=j!Δ_uh^kC_N,j gives [source reference], with the zero case retained. The resulting statements are leanKomorebiPaperIX.AllResidueEuler.all_res…
Exact mathematical passage
\Delta_{uh}^kU_{N,j}=j!\Delta_{uh}^kC_{N,j}
\delta_yUse in source · 2 passages
Search aliases: δ_y delta
Use in source · Paper_VI_Traditional_Final.tex, line 1362
On the interval (0,1) with Dirichlet boundary conditions, the Green kernel for -u''=f is G(x,y)=min(x,y)-xy. Thus the point source δ_y generates the solution kernel, and a general source is reconstructed by u(x)=∫_0^1G(x,y)f(y) dy.
Exact mathematical passage
\delta_y
\delta_{\alpha\gamma}Use in source · 1 passage
Search aliases: δ_αγ delta alpha gamma
Use in source · 01_finite_core.tex, line 302
[Packetwise unitarity conservation] If U is unitary and U_βα=P_β UP_α, then ∑_β U_βα^*U_βγ = δ_αγP_α. Similarly, ∑_α U_βαU_δα^* = δ_βδP_β.
Exact mathematical passage
\boxed{ \sum_\beta U_{\beta\alpha}^*U_{\beta\gamma} = \delta_{\alpha\gamma}P_\alpha. }
\delta_{\beta\alpha}Use in source · 2 passages
Search aliases: δ_βα delta beta alpha
Use in source · Paper_VI_Traditional_Final.tex, line 2103
[Block differential system] The packet blocks are differentiable in operator norm and satisfy frac d d tT_βα(t) = ∑_γ∈ I A_βγT_γα(t) = ∑_γ∈ I T_βγ(t)A_γα. Their initial values are T_βα(0)=δ_βαP_α.
Exact mathematical passage
T_{\beta\alpha}(0)=\delta_{\beta\alpha}P_\alpha.
\delta_{\beta\delta}Use in source · 1 passage
Search aliases: δ_βδ delta beta delta
Use in source · 01_finite_core.tex, line 310
[Packetwise unitarity conservation] If U is unitary and U_βα=P_β UP_α, then ∑_β U_βα^*U_βγ = δ_αγP_α. Similarly, ∑_α U_βαU_δα^* = δ_βδP_β.
Exact mathematical passage
\sum_\alpha U_{\beta\alpha}U_{\delta\alpha}^* = \delta_{\beta\delta}P_\beta.
\delta_{\lambda\mu}Use in source · 2 passages
Search aliases: δ_λμ delta lambda mu
Use in source · 02_strengthened.tex, line 215
[Finite group isotypic projectors] The operators P_λ = fracd_λabsG ∑_g∈ G overlineχ_λ(g)U(g) are orthogonal projectors. They satisfy P_λ P_μ=δ_λμP_λ, ∑_λ∈widehat GP_λ=I_ H. The range of P_λ is the λ isotypic component of H.
Exact mathematical passage
P_\lambda P_\mu=\delta_{\lambda\mu}P_\lambda, \qquad \sum_{\lambda\in\widehat G}P_\lambda=I_{\Hcal}.
\Delta_\ellUse in source · 9 passages
Search aliases: Δ_ℓ Delta ell
Use in source · 05_defects.tex, line 217
Let C_ℓ:X_ℓ+1⟶ X_ℓ be linear maps for 0≤ℓ<L, and let T_ℓ∈End(X_ℓ). Put Δ_ℓ=C_ℓ T_ℓ+1-T_ℓ C_ℓ. For a<b, write C_a,b=C_aC_a+1⋯ C_b-1, with C_a,a=I.
Exact mathematical passage
\Delta_\ell=C_\ell T_{\ell+1}-T_\ell C_\ell.
\varepsilon^*Use in source · 3 passages
Search aliases: ε^* varepsilon
Use in source · 02_supplement_02.tex, line 141
Global comparison. For any other feasible ε, set r_e=ε_e/ε_e^*>0. The budget identities give ∑_ec_er_e=C. Using the standard analytic inequality log r≤ r-1 yields ∑_ec_elog r_e≤∑_ec_e(r_e-1)=0. Lean separately proves the exact objective-difference identity J(ε)-J(ε^*)=-∑_ec_elog r_e. Therefore J(ε)≥ J(ε^*).
Exact mathematical passage
J(\varepsilon)-J(\varepsilon^*)=-\sum_ec_e\log r_e.
\epsilon^2Use in source · 2 passages
Search aliases: ϵ^2 epsilon
Use in source · strengthening.tex, line 683
There is a still stronger filtration. Put X=P+Q, τ=i B D, ϵ=1+τ. Then the exact identities are H_2=ϵ X+2G, G=-γ_2(P)-γ_2(Q)+3PQ-τ Q, ϵ^2=2V, V=τ-2X-8PQ. Every divided power of G is integral component by component. The finite addition identity gives γ_k(H_2)=∑_r=0^k2^rγ_r(G) ϵ^k-rγ_k-r(X) ∈2^lfloor k/2rfloor A_2, because the rth term belongs to 2^r+lfloor(k-r)/2rfloor A_2. Applying the functional, rational descent, and the base-index expansion proves the unit-step version of [source reference]. F…
Exact mathematical passage
H_2=\epsilon X+2G,\quad G=-\gamma_2(P)-\gamma_2(Q)+3PQ-\tau Q,\quad \epsilon^2=2V,\quad V=\tau-2X-8PQ. \label{eq:address-two-filtration-algebra}
\varepsilon^2Use in source · 3 passages
Search aliases: ε^2 varepsilon
Use in source · 02_foundations.tex, line 705
[title=A local nilpotent algebra] Let A=𝕜[ε]/(ε^3) and M=A^⊕2. The radical is J=(ε), and M/JM≅𝕜^2. Therefore two seeds are necessary and sufficient. The extra nilpotent layers ε M and ε^2M are generated automatically once the two top directions are seeded.
Exact mathematical passage
\varepsilon^2M
\varepsilon^3Use in source · 1 passage
Search aliases: ε^3 varepsilon
Use in source · 02_foundations.tex, line 701
[title=A local nilpotent algebra] Let A=𝕜[ε]/(ε^3) and M=A^⊕2. The radical is J=(ε), and M/JM≅𝕜^2. Therefore two seeds are necessary and sufficient. The extra nilpotent layers ε M and ε^2M are generated automatically once the two top directions are seeded.
Exact mathematical passage
A=\bbk[\varepsilon]/(\varepsilon^3)
\epsilon^jUse in source · 1 passage
Search aliases: ϵ^j epsilon
Use in source · strengthening.tex, line 801
ϵ^2=2(τ-P-Q-2PQ). The divided powers of G are integral by the same component formulas used above. Thus modulo 2 A_0, γ_k(H_0) equals the divided power of γ_2(X)+ϵ X. The identity γ_r(γ_2(X)) =frac(2r)!2^r r!γ_2r(X) has an odd integer scalar. Every term involving ϵ^j for j≥2 vanishes modulo two. The surviving zero and first powers give γ_k(H_0)≡ γ_2k(X)+ϵγ_2k-1(X) pmod2 A_0. The multiplication coefficient in the second term is 2k-1, also odd. By the odd moments [source reference] and independence…
Exact mathematical passage
\epsilon^j
\epsilon^{k-r}Use in source · 1 passage
Search aliases: ϵ^k-r epsilon
Use in source · strengthening.tex, line 691
Put X=P+Q, τ=i B D, ϵ=1+τ. Then the exact identities are H_2=ϵ X+2G, G=-γ_2(P)-γ_2(Q)+3PQ-τ Q, ϵ^2=2V, V=τ-2X-8PQ. Every divided power of G is integral component by component. The finite addition identity gives γ_k(H_2)=∑_r=0^k2^rγ_r(G) ϵ^k-rγ_k-r(X) ∈2^lfloor k/2rfloor A_2, because the rth term belongs to 2^r+lfloor(k-r)/2rfloor A_2. Applying the functional, rational descent, and the base-index expansion proves the unit-step version of [source reference]. For a mixed term of total order M, its …
Exact mathematical passage
\gamma_k(H_2)=\sum_{r=0}^k2^r\gamma_r(G) \epsilon^{k-r}\gamma_{k-r}(X) \in2^{\lfloor k/2\rfloor}\mathcal A_2, \label{eq:address-two-filtration-pd}
\varepsilon_*Use in source · 8 passages
Search aliases: ε_* varepsilon
Use in source · Paper_V_Traditional_Final.tex, line 2462
[Stability of the optimal cost] Let c and widetilde c be two node cost functions. For an admissible pruning L, set ε( L) = ∑_v∈ L|c(v)-widetilde c(v)| and let ε_* = max_ Lε( L). If F_* and widetilde F_* are the two optimal root costs, then |F_*-widetilde F_*| ≤ ε_*.
Exact mathematical passage
\varepsilon_* = \max_{\Lcal}\varepsilon(\Lcal).
\varepsilon_aUse in source · 5 passages
Search aliases: ε_a varepsilon
Use in source · Paper_V_Traditional_Final.tex, line 1016
[1mm] displaystyle max{ε_v, ∑_u∈ch(v)E(u)}, · v internal. For the complete prunings P_v below v, the envelope is exactly E(v)=max_Λ∈ P_v∑_a∈Λε_a. The maximum is attained. Every complete-pruning cost changes by at most E(∅), and the two globally optimal values differ by at most this quantity. Consequently, for a unique optimum with competitors, Γ_y>2E(∅) is a deterministic certificate that the globally optimal pruning is unchanged. A singleton pruning set is stable without a gap condition. Assume…
Exact mathematical passage
E(v)=\max_{\Lambda\in\mathfrak P_v}\sum_{a\in\Lambda}\varepsilon_a.
\varepsilon_bUse in source · 1 passage
Search aliases: ε_b varepsilon
Use in source · 02_strengthened.tex, line 2309
[Approximate constraint transport] Assume normC_ℓ+1,aR_ℓ-R_ℓ C_ℓ,a ≤ε_a and normC_ℓ,a,normC_ℓ+1,a≤ M. Then a coarse physical unit vector ψ satisfies normC_ℓ+1,aR_ℓψ≤ε_a. Also, · norm [C_ℓ+1,a,C_ℓ+1,b]R_ℓ -R_ℓ[C_ℓ,a,C_ℓ,b] · ≤ 2M(ε_a+ε_b).
Exact mathematical passage
\begin{aligned} &\norm{ [C_{\ell+1,a},C_{\ell+1,b}]R_\ell -R_\ell[C_{\ell,a},C_{\ell,b}] }\\ &\hspace{4em}\leq 2M(\varepsilon_a+\varepsilon_b). \end{aligned}
\epsilon_eUse in source · 5 passages
Search aliases: ϵ_e epsilon
Use in source · pages.tex, line 25
s_e,ϵ_e · Sensitivity and tolerance · Is the budget input-dependent or uniform over unit inputs?
Exact mathematical passage
s_e,\epsilon_e
\varepsilon_eUse in source · 13 passages
Search aliases: ε_e varepsilon
Use in source · 02_supplement_01.tex, line 146
_a(e). Set A=∑_e A_e, widehat A=∑_e(A_e+Δ_e). The exact operator and vector identities are proved first: widehat A-A=∑_eΔ_e P_a(e), (widehat A-A)x=∑_eΔ_e(P_a(e)x). Applying the finite triangle inequality and the defining operator-norm estimate gives norm(widehat A-A)x ≤∑_enormΔ_e normP_a(e)x. If normΔ_e≤ε_enormA_e, then the formal relative certificate is norm(widehat A-A)x ≤∑_eε_enormA_e normP_a(e)x. The source additionally assumes that each exact block is source supported. The proof of this bou…
Exact mathematical passage
\norm{\Delta_e}\le\varepsilon_e\norm{A_e}
\varepsilon_e^*Use in source · 2 passages
Search aliases: ε_e^* varepsilon
Use in source · 02_supplement_02.tex, line 126
orem textitSource item 8.4, with an explicit positive budget Let E be a finite nonempty edge set. Assume s_e>0, c_e>0, τ>0, and define C=∑_ec_e. The optimization problem is min_ε_e>0∑_ec_elogfrac1ε_e subject to ∑_es_eε_e=τ. Lean constructs the displayed candidate ε_e^*=fracτ c_es_eC and proves strict positivity and the exact budget equality. It then proves global optimality and uniqueness for the actual logarithmic objective.
Exact mathematical passage
\boxed{\varepsilon_e^*=\frac{\tau c_e}{s_eC}}
\varepsilon_hUse in source · 1 passage
Search aliases: ε_h varepsilon
Use in source · Paper_V_Traditional_Final.tex, line 3060
[Deterministic consistency] Assume h→0 along grids satisfying the preceding sampling assumptions, ε_h→0, the number of jumps is fixed, and the nonzero jump amplitudes and positive minimum separation remain fixed. Then the detected packet converges to the true packet in every finite L^p norm. If Q_m has a vanishing smooth approximation error and uniformly bounded norm, the complete reconstruction also converges in L^p.
Exact mathematical passage
\varepsilon_h\to0
\varepsilon_jUse in source · 3 passages
Search aliases: ε_j varepsilon
Use in source · 03_nonlinear_coordinates.tex, line 792
[Iterated orbit defect] Let F_f:X_f→ X_f and F_c:X_c→ X_c, with coarsening C:X_f→ X_c. Suppose F_c is q-Lipschitz. For a fine orbit starting at x, assume Δ_F(F_f^j x) ≤ε_j. Then CF_f^n(x)-F_c^n(Cx) ≤ ∑_j=0^n-1q^n-1-jε_j. If 0≤ q<1 and every ε_j≤ε, then CF_f^n(x)-F_c^n(Cx) ≤ εfrac1-q^n1-q.
Exact mathematical passage
\|\Delta_F(F_f^j x)\|\le\varepsilon_j.
\varepsilon_MUse in source · 3 passages
Search aliases: ε_M varepsilon
Use in source · Paper_IV_Traditional_Final.tex, line 1936
[Near cancellation of two exponentials] For every M>0, there are constants ε_M>0 and C_M>0 such that the following holds. If |Re w|≤ M and |1+e^w|≤ε_M, then there is a unique integer m with |w-(2m+1)π i| ≤ C_M|1+e^w|.
Exact mathematical passage
\varepsilon_M>0
\varepsilon_mUse in source · 5 passages
Search aliases: ε_m varepsilon
Use in source · Paper_IV_Traditional_Final.tex, line 2812
For N≥3, [source reference] writes all but finitely many reduced zeros as λ_m = (A_N(m+frac12))^N(1+ε_m) where λ_m>0, 1+ε_m>0, and ε_m = O(t_m^-1e^-γ t_m). On every compact set of s values, (1+ε_m)^-s-1 = O(|ε_m|) uniformly for large m. The difference λ_m^-s - A_N^-Ns(m+frac12)^-Ns therefore has a normally convergent sum on every compact subset of C. Indeed, the model power has at most polynomial growth in m on a compact set, while |ε_m| decays exponentially. The finitely many exceptional terms …
Exact mathematical passage
\lambda_m = \left(A_N\left(m+\frac12\right)\right)^N(1+\varepsilon_m)
\varepsilon_nUse in source · 1 passage
Search aliases: ε_n varepsilon
Use in source · 03_nonlinear_coordinates.tex, line 815
Let e_n= CF_f^n(x)-F_c^n(Cx) . The definition of the defect and the Lipschitz property give e_n+1≤ε_n+qe_n. Iterating this recurrence proves the first estimate. The second is the geometric sum.
Exact mathematical passage
e_{n+1}\le\varepsilon_n+qe_n.
\varepsilon_{pq}Use in source · 4 passages
Search aliases: ε_pq varepsilon
Use in source · Paper_IV_Traditional_Final.tex, line 883
[Entries of the fundamental matrix] For 0≤ p,q<N, let d_pq be the member of {0,…,N-1} congruent to p-q modulo N. Put ε_pq = 1, · p≥ q, -1, · p<q. Then (e^zJ_N)_pq = ε_pq D_N,d_pq(z). Thus every entry of the fundamental matrix is one of the signed sections, with the sign recording whether the cyclic path from q to p crosses the signed edge.
Exact mathematical passage
\varepsilon_{pq} = \begin{cases} 1,&p\geq q,\\ -1,&p<q. \end{cases}
\varepsilon_{\mathrm{rec}}Use in source · 3 passages
Search aliases: ε_rec varepsilon mathrm
Use in source · Paper_V_Traditional_Final.tex, line 2059
[Quantitative imperfect reconstruction] Define the delay compensated defect D( e^ iω) = e^ i dωR( e^ iω)E( e^ iω)-I_N. If ε_rec = *ess sup_ω D( e^ iω) _op<∞, then the delay compensated reconstruction widetilde x satisfies widetilde x-x _2 ≤ ε_rec x _2. The constant is sharp as an operator norm.
Exact mathematical passage
\varepsilon_{\mathrm{rec}} = \esssup_{\omega}\|D(\e^{\ii\omega})\|_{\mathrm{op}}<\infty,
\varepsilon_sUse in source · 1 passage
Search aliases: ε_s varepsilon
Use in source · 02_foundations.tex, line 258
[title=A cyclic nilpotent chain] Let A=𝕜[t] and M=𝕜[t]/(t^3). The residue class s=[1] is one seed and ε_s:A→ M, f(t)↦ f(t)[1] has kernel (t^3). Hence M≅ A/(t^3), and the addressed vectors s,ts,t^2s are the visible orbit chain.
Exact mathematical passage
\varepsilon_s:A\to M, \qquad f(t)\mapsto f(t)[1]
\epsilon_uUse in source · 8 passages
Search aliases: ϵ_u epsilon
Use in source · 72_recovered.tex, line 40
This article begins with the Hurwitz product (amathbin⋆ b)_n = ∑_r=0^n binom nr a_rb_n-r. Over a field of characteristic zero it is multiplication of exponential generating functions. Over a finite field it realizes the additive group algebra through the exponential modes ϵ_u=(u^n)_n≥0. This finite algebra gives an exact proof of the standard window period. Hurwitz series rings are treated in [source reference]. The finite difference and umbral background may be found in [source reference].
Exact mathematical passage
\epsilon_u=(u^n)_{n\geq0}
\epsilon_{u+v}Use in source · 1 passage
Search aliases: ϵ_u+v epsilon
Use in source · 72_recovered.tex, line 153
[Addition law for modes] For every u,v∈ R, ϵ_umathbin⋆ϵ_v = ϵ_u+v.
Exact mathematical passage
\boxed{ \epsilon_u\starh\epsilon_v = \epsilon_{u+v}. }
\varepsilon_vUse in source · 5 passages
Search aliases: ε_v varepsilon
Use in source · Paper_V_Traditional_Final.tex, line 1001
[Local perturbation certificate] Assume |c_v(y')-c_v(y)|≤ε_v. Define E(v) = ε_v, · v terminal,
Exact mathematical passage
|c_v(y')-c_v(y)|\le\varepsilon_v.
\epsilon_v^{(j)}Use in source · 2 passages
Search aliases: ϵ_v^(j) epsilon
Use in source · 01_manuscript.tex, line 1588
[Shared compilation for an operator family] Let operators A^(1),…,A^(m) use the same candidate tree. Give operator j a nonnegative reuse weight r_j. Replace the local keep cost and error by κ_v = κ_v^rm setup + ∑_j=1^m r_jκ_v^(j) and ϵ_v = ∑_j=1^m r_jϵ_v^(j). Then [source reference] gives the exact optimal shared pruning for the additive weighted objective.
Exact mathematical passage
\epsilon_v = \sum_{j=1}^m r_j\epsilon_v^{(j)}.
\epsilon_\DeltaUse in source · 1 passage
Search aliases: ϵ_Δ epsilon Delta
Use in source · Paper_XII_Traditional_Final.tex, line 479
[Functorial group algebra transfer] A group homomorphism q:Δ'→Δ induces an O algebra homomorphism q_*: O[Δ']⟶ O[Δ]. If q is surjective, then q_* is surjective. Augmentation commutes with the transfer: ϵ_Δ q_*=ϵ_Δ'.
Exact mathematical passage
\epsilon_\Delta q_*=\epsilon_{\Delta'}.
\epsilon_{\Delta'}Use in source · 1 passage
Search aliases: ϵ_Δ' epsilon Delta
Use in source · Paper_XII_Traditional_Final.tex, line 479
[Functorial group algebra transfer] A group homomorphism q:Δ'→Δ induces an O algebra homomorphism q_*: O[Δ']⟶ O[Δ]. If q is surjective, then q_* is surjective. Augmentation commutes with the transfer: ϵ_Δ q_*=ϵ_Δ'.
Exact mathematical passage
\epsilon_\Delta q_*=\epsilon_{\Delta'}.
\epsilon_\zetaUse in source · 1 passage
Search aliases: ϵ_ζ epsilon zeta
Use in source · 72_recovered.tex, line 272
[Subgroup average] The multiple selector has the mode expansion a^(m) = frac1m ∑_ζ∈ H_m ϵ_ζ.
Exact mathematical passage
\boxed{ a^{(m)} = \frac1m \sum_{\zeta\in H_m} \epsilon_\zeta. }
\epsilon_{\eta u}Use in source · 1 passage
Search aliases: ϵ_η u epsilon eta
Use in source · 72_recovered.tex, line 323
Multiplication by an element η∈ H_m gives an automorphism of the additive group and therefore an automorphism ϵ_u⟼ϵ_η u of the mode algebra. The selector a^(m) is fixed by every such automorphism, so its unique inverse is fixed as well.
Exact mathematical passage
\epsilon_u\longmapsto\epsilon_{\eta u}
\varepsilon_\ellUse in source · 4 passages
Search aliases: ε_ℓ varepsilon ell
Use in source · 03_nonlinear_coordinates.tex, line 852
[Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.
Exact mathematical passage
\varepsilon_\ell
\zetaUse in source · 28 passages
Search aliases: ζ zeta
Use in source · 05_worked_models.tex, line 383
ery n≥3, two is the least finite number of adjacency seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any field. For the lower bound over C, let ζ be a primitive nth root of unity and define q(x)=(∑_j=0^n-1ζ^j x_j, ∑_j=0^n-1ζ^-jx_j). Then q(Ax)=(ζ+ζ^-1)q(x). The images of e_0 and e_1 are (1,1) and (ζ,ζ^-1), which are ind…
Exact mathematical passage
\zeta
\zeta^{-1}Use in source · 3 passages
Search aliases: ζ^-1 zeta
Use in source · 05_worked_models.tex, line 388
[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any field. For the lower bound over C, let ζ be a primitive nth root of unity and define q(x)=(∑_j=0^n-1ζ^j x_j, ∑_j=0^n-1ζ^-jx_j). Then q(Ax)=(ζ+ζ^-1)q(x). The images of e_0 and e_1 are (1,1) and (ζ,ζ^-1), which are independent because ζ≠ζ^-1 for n≥3. Thus q is a scalar quotient of dimension two, and [source reference] gives the m…
Exact mathematical passage
q(Ax)=(\zeta+\zeta^{-1})q(x)
\zeta^{-j}Use in source · 1 passage
Search aliases: ζ^-j zeta
Use in source · 05_worked_models.tex, line 384
y seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any field. For the lower bound over C, let ζ be a primitive nth root of unity and define q(x)=(∑_j=0^n-1ζ^j x_j, ∑_j=0^n-1ζ^-jx_j). Then q(Ax)=(ζ+ζ^-1)q(x). The images of e_0 and e_1 are (1,1) and (ζ,ζ^-1), which are independent because ζ≠ζ^-1 for n≥3. Thus q is a scalar…
Exact mathematical passage
q(x)=\left(\sum_{j=0}^{n-1}\zeta^j x_j, \sum_{j=0}^{n-1}\zeta^{-j}x_j\right).
\zeta^jUse in source · 1 passage
Search aliases: ζ^j zeta
Use in source · 05_worked_models.tex, line 384
y seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any field. For the lower bound over C, let ζ be a primitive nth root of unity and define q(x)=(∑_j=0^n-1ζ^j x_j, ∑_j=0^n-1ζ^-jx_j). Then q(Ax)=(ζ+ζ^-1)q(x). The images of e_0 and e_1 are (1,1) and (ζ,ζ^-1), which are independent because ζ≠ζ^-1 for n≥3. Thus q is a scalar…
Exact mathematical passage
q(x)=\left(\sum_{j=0}^{n-1}\zeta^j x_j, \sum_{j=0}^{n-1}\zeta^{-j}x_j\right).
\zeta^mUse in source · 1 passage
Search aliases: ζ^m zeta
Use in source · 72_recovered.tex, line 251
Fix a divisor m of p-1. Let H_m = {ζ∈ F_p^×:ζ^m=1}. This group has order m.
Exact mathematical passage
H_m = \{\zeta\in\FF_p^\times:\zeta^m=1\}.
\zeta^nUse in source · 1 passage
Search aliases: ζ^n zeta
Use in source · 72_recovered.tex, line 285
The nth coefficient on the right is frac1m ∑_ζ∈ H_mζ^n. The character sum is one when mmid n and zero otherwise.
Exact mathematical passage
\frac1m \sum_{\zeta\in H_m}\zeta^n.
\zeta_3Use in source · 1 passage
Search aliases: ζ_3 zeta
Use in source · odd_synthesis.tex, line 216
The old fixed-field central argument genuinely failed. At p=3, (h_1,h_2)=(1,5), its naturally transferred full character takes the value ζ_3≠1 on 1+t^2, although the desired wild center is trivial. A shallow character choice cannot repair a forced value. The two-stage changing-field repair is retained in pathcentral/CENTRAL_REPAIR166.md. The trace-zero tower above resolves this central issue uniformly in every length.
Exact mathematical passage
\zeta_3\ne1
\zeta_4Use in source · 1 passage
Search aliases: ζ_4 zeta
Use in source · Paper_X_Block27_Amended.tex, line 1162
ite set, with nonzero determinant. Every coefficient embedding preserves its nonvanishing. Actual member trace comparisons retain that member's good-prime condition. No coverage or character-continuity hypothesis is added in this cyclotomic specialization: the prime coverage is proved, and the finite extension's Krull topology is discrete. A concrete conductor-four instance uses the actual field Q(ζ_4).
Exact mathematical passage
\Q(\zeta_4)
\zeta_MUse in source · 1 passage
Search aliases: ζ_M zeta
Use in source · Paper_XI_Traditional_Final.tex, line 313
Assume that P_K acts trivially. After passing to a finite quotient, tame inertia is cyclic. Let t generate a quotient of order M, and put ζ_M= e^2π i/M. For j∈ Z/M Z, define P_j = frac1M ∑_a=0^M-1 ζ_M^-jar(t)^a.
Exact mathematical passage
\zeta_M=\e^{2\pi i/M}
\zeta_M^{-ja}Use in source · 1 passage
Search aliases: ζ_M^-ja zeta
Use in source · Paper_XI_Traditional_Final.tex, line 314
Assume that P_K acts trivially. After passing to a finite quotient, tame inertia is cyclic. Let t generate a quotient of order M, and put ζ_M= e^2π i/M. For j∈ Z/M Z, define P_j = frac1M ∑_a=0^M-1 ζ_M^-jar(t)^a.
Exact mathematical passage
P_j = \frac1M \sum_{a=0}^{M-1} \zeta_M^{-ja}r(t)^a.
\zeta_nUse in source · 5 passages
Search aliases: ζ_n zeta
Use in source · Paper_X_Block27_Amended.tex, line 1129
Arithmetic coverage for a fixed cyclotomic extension. Let n>0 and let L= Q(ζ_n) be a cyclotomic number field. The canonical isomorphism Gal(L/ Q)≃( Z/n Z)^× defines σ_p by σ_p(ζ_n)=ζ_n^p when pnmid n. For every automorphism σ, finite set B⊂ N and bound b, Lean proves that there is a rational prime p>b, with p∉ B and pnmid n, such that σ_p=σ. The prime-existence step uses the proved Dirichlet theorem in the pinned Mathlib dependency.
Exact mathematical passage
L=\Q(\zeta_n)
\zeta_n^pUse in source · 2 passages
Search aliases: ζ_n^p zeta
Use in source · Paper_X_Block27_Amended.tex, line 1134
Arithmetic coverage for a fixed cyclotomic extension. Let n>0 and let L= Q(ζ_n) be a cyclotomic number field. The canonical isomorphism Gal(L/ Q)≃( Z/n Z)^× defines σ_p by σ_p(ζ_n)=ζ_n^p when pnmid n. For every automorphism σ, finite set B⊂ N and bound b, Lean proves that there is a rational prime p>b, with p∉ B and pnmid n, such that σ_p=σ. The prime-existence step uses the proved Dirichlet theorem in the pinned Mathlib dependency.
Exact mathematical passage
\sigma_p(\zeta_n)=\zeta_n^p
\eta^NUse in source · 5 passages
Search aliases: η^N eta
Use in source · Paper_IV_Traditional_Final.tex, line 269
The second half of this chapter concerns the zeros of D_N. The finite exponential sum D_N(z)=frac1N∑_η^N=-1e^η z has a regular polygon of exponents. The general relation between an exponent polygon and the zero strips of an exponential sum goes back to P olya and Langer. Later developments include the work of Moreno and the treatment in Levin's monograph [source reference]. Bleher and Mallison give a particularly close general result: for convex-polygon exponential sums they locate all sufficien…
Exact mathematical passage
\Dfun_N(z)=\frac1N\sum_{\eta^N=-1}e^{\eta z}
\eta^nUse in source · 2 passages
Search aliases: η^n eta
Use in source · 72_recovered.tex, line 335
Write b^(m,p) = ∑_u∈ F_pc_uϵ_u. Invariance gives c_η u=c_u. After the change of variable u↦η u, b_n^(m,p) = η^n b_n^(m,p) for every η∈ H_m. If mnmid n, some η satisfies η^n≠1, so the coefficient vanishes.
Exact mathematical passage
b_n^{(m,p)} = \eta^n b_n^{(m,p)}
\eta_0Use in source · 2 passages
Search aliases: η_0 eta
Use in source · Paper_IV_Traditional_Final.tex, line 1782
The finite exponential representation gives | D_N(z)|≤ e^|z|, so the type is at most one. Choose a root η_0 and put z=toverlineη_0 with t>0. The term e^η_0z=e^t has strictly larger real exponent than every other term. Hence D_N(toverlineη_0) = frac1Ne^t(1+o(1)). The type is therefore one.
Exact mathematical passage
\eta_0
\eta_1Use in source · 3 passages
Search aliases: η_1 eta
Use in source · 02_connections_obstructions.tex, line 795
[Input locality for operation trees] Let η_1,η_2:I→ Y be two input environments. If η_1(i)=η_2(i) for every i∈inputs(t), then the two evaluations of t are equal under the same constant, unary, and binary semantics. In particular, if inputs(t)=∅, evaluation is independent of the input environment. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.OperationTree.eval_congr_on_inputs; Komorebi.PaperIII.NonlinearObstructions.OperationTree.eval_eq_of_inputs_eq_empty
Exact mathematical passage
\eta_1,\eta_2:I\to Y
\eta_2Use in source · 2 passages
Search aliases: η_2 eta
Use in source · 02_connections_obstructions.tex, line 795
[Input locality for operation trees] Let η_1,η_2:I→ Y be two input environments. If η_1(i)=η_2(i) for every i∈inputs(t), then the two evaluations of t are equal under the same constant, unary, and binary semantics. In particular, if inputs(t)=∅, evaluation is independent of the input environment. statusleanthreeKomorebi.PaperIII.NonlinearObstructions.OperationTree.eval_congr_on_inputs; Komorebi.PaperIII.NonlinearObstructions.OperationTree.eval_eq_of_inputs_eq_empty
Exact mathematical passage
\eta_1,\eta_2:I\to Y
\eta_gUse in source · 4 passages
Search aliases: η_g eta
Use in source · 03_nonlinear_coordinates.tex, line 624
[Packet Maurer Cartan equations] Let L=⊕_g∈ G L_g be a differential graded Lie algebra over a ring in which two is invertible. Assume d L_g⊆ L_g, [ L_g, L_h]⊆ L_g+h. For a degree one element η=∑_gη_g, the Maurer Cartan equation dη+frac12[η,η]=0 is equivalent to dη_k + frac12 ∑_g+h=k[η_g,η_h] =0 for every k∈ G.
Exact mathematical passage
\eta=\sum_g\eta_g
\eta_{g_1}Use in source · 1 passage
Search aliases: η_g_1 eta
Use in source · 03_nonlinear_coordinates.tex, line 651
belprop:packet_l_infinity Let L be complete and carry operations ℓ_r that respect a finite abelian group grading. Assume the packet projectors are continuous for the topology in which the series below converges. Assume the Maurer Cartan series ∑_r≥1frac1r!ℓ_r(η,…,η) converges and the required factorials are invertible. Then its packet of label k is ∑_r≥1frac1r! ∑_g_1+⋯+g_r=k ℓ_r(η_g_1,…,η_g_r). The full Maurer Cartan equation holds if and only if every packet equation vanishes.
Exact mathematical passage
\sum_{r\ge1}\frac1{r!} \sum_{g_1+\cdots+g_r=k} \ell_r(\eta_{g_1},\ldots,\eta_{g_r}).
\eta_{g_r}Use in source · 1 passage
Search aliases: η_g_r eta
Use in source · 03_nonlinear_coordinates.tex, line 651
belprop:packet_l_infinity Let L be complete and carry operations ℓ_r that respect a finite abelian group grading. Assume the packet projectors are continuous for the topology in which the series below converges. Assume the Maurer Cartan series ∑_r≥1frac1r!ℓ_r(η,…,η) converges and the required factorials are invertible. Then its packet of label k is ∑_r≥1frac1r! ∑_g_1+⋯+g_r=k ℓ_r(η_g_1,…,η_g_r). The full Maurer Cartan equation holds if and only if every packet equation vanishes.
Exact mathematical passage
\sum_{r\ge1}\frac1{r!} \sum_{g_1+\cdots+g_r=k} \ell_r(\eta_{g_1},\ldots,\eta_{g_r}).
\eta_hUse in source · 3 passages
Search aliases: η_h eta
Use in source · 03_nonlinear_coordinates.tex, line 629
[Packet Maurer Cartan equations] Let L=⊕_g∈ G L_g be a differential graded Lie algebra over a ring in which two is invertible. Assume d L_g⊆ L_g, [ L_g, L_h]⊆ L_g+h. For a degree one element η=∑_gη_g, the Maurer Cartan equation dη+frac12[η,η]=0 is equivalent to dη_k + frac12 ∑_g+h=k[η_g,η_h] =0 for every k∈ G.
Exact mathematical passage
d\eta_k + \frac12 \sum_{g+h=k}[\eta_g,\eta_h] =0
\eta_kUse in source · 13 passages
Search aliases: η_k eta
Use in source · 03_nonlinear_coordinates.tex, line 629
[Packet Maurer Cartan equations] Let L=⊕_g∈ G L_g be a differential graded Lie algebra over a ring in which two is invertible. Assume d L_g⊆ L_g, [ L_g, L_h]⊆ L_g+h. For a degree one element η=∑_gη_g, the Maurer Cartan equation dη+frac12[η,η]=0 is equivalent to dη_k + frac12 ∑_g+h=k[η_g,η_h] =0 for every k∈ G.
Exact mathematical passage
d\eta_k + \frac12 \sum_{g+h=k}[\eta_g,\eta_h] =0
\eta_{k+1}Use in source · 4 passages
Search aliases: η_k+1 eta
Use in source · Paper_V_Traditional_Final.tex, line 2946
In a cell without a jump, D_k=s(t_k+1)-s(t_k)+η_k+1-η_k. The Lipschitz and noise bounds give |D_k|≤ Mh+2ε. In a cell with one jump, the same formula has the additional term a_q, which proves the second estimate.
Exact mathematical passage
D_k=s(t_{k+1})-s(t_k)+\eta_{k+1}-\eta_k.
\eta_{L,m}Use in source · 1 passage
Search aliases: η_L,m eta
Use in source · Paper_XI_Traditional_Final.tex, line 576
[Greatest common divisor and least common multiple packets] There is a decomposition τ_ξ⊗τ_η ≅ ⊕_i=0^d-1 Ind_W_K_L^W_K ( ξ_L,nη_L,m^Φ^i ), where ξ_L,n and η_L,m are the norm lifts to K_L.
Exact mathematical passage
\eta_{L,m}
\eta_{L,m}^{\Phi^i}Use in source · 1 passage
Search aliases: η_L,m^Φ^i eta Phi
Use in source · Paper_XI_Traditional_Final.tex, line 565
[Greatest common divisor and least common multiple packets] There is a decomposition τ_ξ⊗τ_η ≅ ⊕_i=0^d-1 Ind_W_K_L^W_K ( ξ_L,nη_L,m^Φ^i ), where ξ_L,n and η_L,m are the norm lifts to K_L.
Exact mathematical passage
\boxed{ \tau_\xi\otimes\tau_\eta \cong \bigoplus_{i=0}^{d-1} \Ind_{W_{K_L}}^{W_K} \left( \xi_{L,n}\eta_{L,m}^{\Phi^i} \right), }
\eta_{m,n}Use in source · 4 passages
Search aliases: η_m,n eta
Use in source · Paper_VI_Traditional_Final.tex, line 2717
[Range leakage criterion] Under the contractive assumptions of [source reference], define η_m,n(T) = sup_0≤ t≤ T norm(I-P_m,n)T_n(t)R_n,m. Then ρ_m,n(T)≤η_m,n(T). Consequently, η_m,n(T)→0 for every T>0 is sufficient for the limiting family U_m to satisfy the semigroup law.
Exact mathematical passage
\eta_{m,n}(T) = \sup_{0\leq t\leq T} \norm{(I-P_{m,n})T_n(t)R_{n,m}}.
\eta_{\mathrm{src}}Use in source · 1 passage
Search aliases: η_src eta mathrm
Use in source · odd_synthesis.tex, line 152
Write W=V_src⊥ V_src^⊥. Finite Stone--von Neumann gives η_W≃η_src⊗η_⊥. Choose an isotropic eigenspace giving a line in the complementary factor. Its actual projector in the Heisenberg image algebra e has rank N. The retained induction/idempotent theorem recovers the same inducing representation Λ from eΛ, and then the same π by compact induction. No candidate is changed at recovery.
Exact mathematical passage
\eta_W\simeq\eta_{\mathrm{src}}\otimes\eta_\perp
\eta_tUse in source · 6 passages
Search aliases: η_t eta
Use in source · atlas-pages.tex, line 375
vpage17A random diffusion clockSUBORDINATION / A DISTRIBUTION OF ORDINARY TIMESChapter VI, Part I: Theorem 10.2. The Poisson subordination formula for the finite positive model. Square-root evolution can be written as an average of ordinary diffusion over a positive clock variable s: e^-tsqrt L=∫_0^∞ η_t(s)e^-sL ds, η_t(s)=fract2sqrtπs^-3/2e^-t^2/(4s), t>0. Here t is the external time, and s is the ordinary diffusion time being mixed. They are not interchangeable variables. vfigrandom_clock plat…
Exact mathematical passage
e^{-t\sqrt L}=\int_0^\infty \eta_t(s)e^{-sL}\,ds, \qquad \eta_t(s)=\frac{t}{2\sqrt\pi}s^{-3/2}e^{-t^2/(4s)},\quad t>0.
\eta_WUse in source · 1 passage
Search aliases: η_W eta
Use in source · odd_synthesis.tex, line 152
Write W=V_src⊥ V_src^⊥. Finite Stone--von Neumann gives η_W≃η_src⊗η_⊥. Choose an isotropic eigenspace giving a line in the complementary factor. Its actual projector in the Heisenberg image algebra e has rank N. The retained induction/idempotent theorem recovers the same inducing representation Λ from eΛ, and then the same π by compact induction. No candidate is changed at recovery.
Exact mathematical passage
\eta_W\simeq\eta_{\mathrm{src}}\otimes\eta_\perp
\eta_{\alpha,s}Use in source · 1 passage
Search aliases: η_α,s eta alpha
Use in source · Paper_VI_Traditional_Final.tex, line 4091
d t>0. There is a probability measure η_α,t on [0,∞) such that ∫_0^∞ e^-sλη_α,t( d s) = e^-tλ^α for every λ≥0. Consequently, for a finite dimensional positive self-adjoint operator, e^-tL^α = ∫_0^∞ e^-sLη_α,t( d s) in operator norm. The measures satisfy the convolution law η_α,s*η_α,t = η_α,s+t.
Exact mathematical passage
\eta_{\alpha,s}*\eta_{\alpha,t} = \eta_{\alpha,s+t}.
\eta_{\alpha,s+t}Use in source · 1 passage
Search aliases: η_α,s+t eta alpha
Use in source · Paper_VI_Traditional_Final.tex, line 4091
d t>0. There is a probability measure η_α,t on [0,∞) such that ∫_0^∞ e^-sλη_α,t( d s) = e^-tλ^α for every λ≥0. Consequently, for a finite dimensional positive self-adjoint operator, e^-tL^α = ∫_0^∞ e^-sLη_α,t( d s) in operator norm. The measures satisfy the convolution law η_α,s*η_α,t = η_α,s+t.
Exact mathematical passage
\eta_{\alpha,s}*\eta_{\alpha,t} = \eta_{\alpha,s+t}.
\eta_{\alpha,t}Use in source · 5 passages
Search aliases: η_α,t eta alpha
Use in source · Paper_VI_Traditional_Final.tex, line 4077
[Stable subordination] Let 0<α<1 and t>0. There is a probability measure η_α,t on [0,∞) such that ∫_0^∞ e^-sλη_α,t( d s) = e^-tλ^α for every λ≥0. Consequently, for a finite dimensional positive self-adjoint operator, e^-tL^α = ∫_0^∞ e^-sLη_α,t( d s) in operator norm. The measures satisfy the convolution law η_α,s*η_α,t = η_α,s+t.
Exact mathematical passage
\eta_{\alpha,t}
\eta_\gammaUse in source · 4 passages
Search aliases: η_γ eta gamma
Use in source · 02_strengthened.tex, line 453
[Path interference formula] Fix a source vector ψ∈ P_a_0 H and a target packet b. Let Γ_b be the set of packet paths from a_0 to b, and set η_γ=U_γψ. Choose any total order on this finite path set. Then normP_bUψ^2 = ∑_γ∈Γ_bnormη_γ^2 + 2Re ∑_substackγ,γ'∈Γ_b
Exact mathematical passage
\eta_\gamma=U_\gamma\psi.
\eta_{\gamma'}Use in source · 1 passage
Search aliases: η_γ' eta gamma
Use in source · 02_strengthened.tex, line 457
[Path interference formula] Fix a source vector ψ∈ P_a_0 H and a target packet b. Let Γ_b be the set of packet paths from a_0 to b, and set η_γ=U_γψ. Choose any total order on this finite path set. Then normP_bUψ^2 = ∑_γ∈Γ_bnormη_γ^2 + 2Re ∑_substackγ,γ'∈Γ_b γ<γ' ipη_γη_γ'. Consequently, normP_bUψ ≤ ∑_γ∈Γ_bnormη_γ. If the path outputs are pairwise orthogonal, the target probability is the sum of the path probabilities.
Exact mathematical passage
\norm{P_bU\psi}^2 = \sum_{\gamma\in\Gamma_b}\norm{\eta_\gamma}^2 + 2\operatorname{Re} \sum_{\substack{\gamma,\gamma'\in\Gamma_b\\ \gamma<\gamma'}} \ip{\eta_\gamma}{\eta_{\gamma'}}.
\eta_\thetaUse in source · 2 passages
Search aliases: η_θ eta theta
Use in source · Paper_XI_Traditional_Final.tex, line 1166
[Type side Heisenberg quotient] The pairing h_θ is nondegenerate and alternating. There is one irreducible representation η_θ of J_θ^1 containing θ, and its dimension is sqrtabsJ_θ^1/H_θ^1.
Exact mathematical passage
\eta_\theta
\eta_\ellUse in source · 1 passage
Search aliases: η_ℓ eta ell
Use in source · Paper_IV_Traditional_Final.tex, line 1906
ojection estimate gives the stronger bound Y=fraclog(N-1)2s_N for every zero in the sector, without a large-radius assumption. Write z=re^iθ with 0≤θ≤ a_N and r≥0. For 0≤ℓ≤ N-2, the angle θ+(2ℓ+1)a_N lies between a_N+θ and 2π-a_N-θ. Monotonicity of cosine on [0,π], followed by reflection about π, therefore gives Re(η_ℓ z)≤ rcos(a_N+θ). The remaining vertex e^-ia_N has projection rcos(a_N-θ). Their difference is at least 2s_Nrsinθ=2s_Ny. If D_N(z)=0, divide its finite exponential sum by the term …
Exact mathematical passage
\Re(\eta_\ell z)\leq r\cos(a_N+\theta).
\eta_\perpUse in source · 1 passage
Search aliases: η_⊥ eta perp
Use in source · odd_synthesis.tex, line 152
Write W=V_src⊥ V_src^⊥. Finite Stone--von Neumann gives η_W≃η_src⊗η_⊥. Choose an isotropic eigenspace giving a line in the complementary factor. Its actual projector in the Heisenberg image algebra e has rank N. The retained induction/idempotent theorem recovers the same inducing representation Λ from eΛ, and then the same π by compact induction. No candidate is changed at recovery.
Exact mathematical passage
\eta_W\simeq\eta_{\mathrm{src}}\otimes\eta_\perp
\ThetaUse in source · 45 passages
Search aliases: Θ Theta
Use in source · 03_packets_and_graphs.tex, line 678
[Full block algebra representation] The map Θ_ P:End_𝕜(X) ⟶ ∏_β,α∈ IHom_𝕜(X_α,X_β), T⟼(T_βα)_β,α, is an algebra isomorphism when the codomain is equipped with matrix multiplication whose entries are operators. DependsnolinkurlPSC-I-T-OPERATOR-RECONSTRUCTION and nolinkurlPSC-I-T-BLOCK-MULTIPLICATION.
Exact mathematical passage
\Theta_{\Pcal}:\End_{\bbk}(X) \longrightarrow \prod_{\beta,\alpha\in I}\Hom_{\bbk}(X_\alpha,X_\beta), \qquad T\longmapsto(T_{\beta\alpha})_{\beta,\alpha},
\varthetaUse in source · 1 passage
Search aliases: ϑ vartheta
Use in source · dyadic_synthesis.tex, line 111
nit character to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar units, Θ_j(uI)=θ_j(u^2)=1. For odd critical level r, use m=lfloor r/2rfloor and D=H^m+1(β)=H^m+1(γ). The endpoint is valid because 2(m+1)-r=1: the cross term for the additive character lies in the radical on which the matrix additive character is trivial. BK 3.3.18 supplies ϑ=Θ_j|_D ψ_β-γ. Integral trace makes it trivial on D∩ U_F^1.
Exact mathematical passage
\vartheta=\Theta_j|_D\,\psi_{\beta-\gamma}
\theta^2Use in source · 2 passages
Search aliases: θ^2 theta
Use in source · atlas-pages.tex, line 309
vpage14Conjugacy tracks exponentsBETA MODEL / POSTERIOR AND EVIDENCEChapter VI, Part I: Example 9.2. A finite-parameter conjugate family, not a universal posterior model. For a small exact example, take the Beta(2,3) density ρ(θ)=12θ(1-θ)^2,qquad0≤θ≤1. Observe a specified ordered sequence with two successes and one failure. Its likelihood is θ^2(1-θ). Multiplication adds exponents, so the posterior belongs to Beta(4,4) with density 140θ^3(1-θ)^3. vfigbeta platecaptionThe dashed prior is asymmetr…
Exact mathematical passage
\theta^2(1-\theta)
\theta^3Use in source · 2 passages
Search aliases: θ^3 theta
Use in source · atlas-pages.tex, line 309
ite-parameter conjugate family, not a universal posterior model. For a small exact example, take the Beta(2,3) density ρ(θ)=12θ(1-θ)^2,qquad0≤θ≤1. Observe a specified ordered sequence with two successes and one failure. Its likelihood is θ^2(1-θ). Multiplication adds exponents, so the posterior belongs to Beta(4,4) with density 140θ^3(1-θ)^3. vfigbeta platecaptionThe dashed prior is asymmetric. The solid posterior is symmetric with mean 1/2. Polynomial integration verifies the normalizers exactl…
Exact mathematical passage
140\theta^3(1-\theta)^3
\Theta^dUse in source · 1 passage
Search aliases: Θ^d Theta
Use in source · 20_arithmetic.tex, line 285
^d=(p^lfloor d/prfloora_d) (T^p/p)^lfloor d/prfloorT^dbmod p. The operator preserves these monomials and hence finite weighted polynomials. Each fixed output coefficient depends on only finitely many input coefficients, which extends the result to the formal series. Substitution T=e^t-1 intertwines D with Θ and maps W_p into H_p, because d![t^d]F(e^t-1)=[T^0]Θ^dF. It follows that the singular ratios are flat. This transfer is proved in leanKomorebiPaperIX.WeightedDworkTransfer.singular_ratio_fla…
Exact mathematical passage
d![t^d]F(e^t-1)=[T^0]\Theta^dF.
\theta^gUse in source · 2 passages
Search aliases: θ^g theta
Use in source · Paper_XI_Traditional_Final.tex, line 2120
At rank two, with q=2m+1, enlarge J=U_K^1U_A^m+1 to P=U_K^1U_A^m. The actual character ratio λ=θ^g/θ for g∉ J is nontrivial and sign-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=2, and P/H_0≃ F_2^4, with carrier Ind_J^Pη of dimension four ([source reference]). The finite scalar center can be larger than μ_2. No literal extrasp…
Exact mathematical passage
\lambda=\theta^g/\theta
\Theta^nUse in source · 1 passage
Search aliases: Θ^n Theta
Use in source · supplement_10.tex, line 101
The relevant integral subring is the Hurwitz subring H( Z_p)={f∈ Q_p[[t]]: n![t^n]f∈ Z_p for every n}. It is not coefficientwise integrality of ordinary power-series coefficients. The precise connection is n![t^n] E(F)=[T^0]Θ^nF. Since every iterate of Θ preserves W_p, the right side is integral. Therefore E( W_p)⊆ H( Z_p). Applying the same statement to every B_Θ^sF proves f(log(1+T))∈ W_p Longrightarrow B_D^sf∈ H( Z_p) for every s. This is the proved transfer that was missing in the earlier at…
Exact mathematical passage
n![t^n]\mathcal E(F)=[T^0]\Theta^nF.
\Theta^pUse in source · 2 passages
Search aliases: Θ^p Theta
Use in source · 20_arithmetic.tex, line 270
To deduce flatness from the weighted estimate, use Θ=(1+T)d/dT. Both Θ and (Θ^p^2-Θ^p)/p preserve W_p. For the first operator, its coefficient formula is (d+1)a_d+1+da_d. Whenever the floor weight rises, the factor d+1 provides the missing p. For the second operator, the proof uses its full Leibniz correction and the scalar congruence p^2mid b^p^2-b^p. Weighted monomials factor as a_dT^d=(p^lfloor d/prfloora_d) (T^p/p)^lfloor d/prfloorT^dbmod p. The operator preserves these monomials and hence f…
Exact mathematical passage
(\Theta^{p^2}-\Theta^p)/p
\Theta^{p^2}Use in source · 2 passages
Search aliases: Θ^p^2 Theta
Use in source · 20_arithmetic.tex, line 270
To deduce flatness from the weighted estimate, use Θ=(1+T)d/dT. Both Θ and (Θ^p^2-Θ^p)/p preserve W_p. For the first operator, its coefficient formula is (d+1)a_d+1+da_d. Whenever the floor weight rises, the factor d+1 provides the missing p. For the second operator, the proof uses its full Leibniz correction and the scalar congruence p^2mid b^p^2-b^p. Weighted monomials factor as a_dT^d=(p^lfloor d/prfloora_d) (T^p/p)^lfloor d/prfloorT^dbmod p. The operator preserves these monomials and hence f…
Exact mathematical passage
(\Theta^{p^2}-\Theta^p)/p
\theta^uUse in source · 1 passage
Search aliases: θ^u theta
Use in source · dyadic_synthesis.tex, line 318
eEnd_E_0(F^4). The odd lower critical level is r=3h_1=2m+1. The half leading level is a=3h_2>r. Define P=J U_B^m, where J=J^1 is the actual compact type group. The filtered centralizer estimate for the actual final field, without assuming E_0⊂ K, proves that P is a compact group with J normal of index two. For a concrete new coset representative u, λ=θ^u/θ on the actual H=H^1 is nontrivial with values {±1}. Put H_0=kerλ⊂ H. The calculation proves P/H_0≃ F_2^4, E=Ind_J^Pη is irreducible of dimens…
Exact mathematical passage
\lambda=\theta^u/\theta
\theta_0Use in source · 3 passages
Search aliases: θ_0 theta
Use in source · monograph_atlas.tex, line 288
∂_ϕ P]) =frac12sinθ. vfigbloch_curvature headA local frame gives the same answer Away from the south pole, use u_N=(cos(θ/2),e^ iϕsin(θ/2))^T. Then P=u_Nu_N^* and the scalar one-form is A_N=- i u_N^* d u_N=frac1-cosθ2 dϕ, d A_N=frac12sinθ dθwedge dϕ. Thus the curvature flux through the cap 0≤θ≤θ_0 is π(1-cosθ_0). At θ_0=π/3, it is exactly π/2. The latitude drawn in the plate illustrates this cap. The exact flux comes from the formula above.
Exact mathematical passage
0\le\theta\le\theta_0
\Theta_{2,0}Use in source · 3 passages
Search aliases: Θ_2,0 Theta
Use in source · 10_foundations.tex, line 515
or a finite binomial expansion f(n)=∑_r=0^Dc_rbinom nr, the kth Newton coefficient of g(q)=f(Mq+s) is therefore c_k(g)=∑_r=k^min(D,Mk+s)c_rΘ_M,s(r,k). Indeed, insert the finite stencil for Δ_1^k g(0) and interchange its two finite sums. Coefficient extraction of the binomial expansion of ((1+z)^M-1)^k gives the kernel. In the binary zero-address case, Θ_2,0(k+j,k)=2^k-jbinom kj (0≤ j≤ k), since (2z+z^2)^k=z^k(2+z)^k. Exact valuations for the general kernel remain to be determined. The recovered …
Exact mathematical passage
\Theta_{2,0}(k+j,k)=2^{k-j}\binom kj\qquad(0\leq j\leq k),
\Theta_{2,2}Use in source · 1 passage
Search aliases: Θ_2,2 Theta
Use in source · atlas-pages.tex, line 296
efficient (-1)^mκ_N,mfracu^Nm+um. At the first block this gives Θ_N,1(u)=-(u^N+u). Every positive-index theta polynomial vanishes at u=0. For even N, H_N(-z)=H_N(z), so the defining product is also 1 at u=-1. headThe odd-order boundary is visible For N=3, Θ_3,1(-1)=2, not zero. The factor u+1 is not a uniform law across orders. In contrast, Θ_2,2(u)=u(u+1)(5u^2+u+2). The two guaranteed factors follow from the whole generating function, not from guessing the first few polynomial roots. takeawayA …
Exact mathematical passage
\Theta_{2,2}(u)=u(u+1)(5u^2+u+2).
\Theta_{3,1}Use in source · 1 passage
Search aliases: Θ_3,1 Theta
Use in source · atlas-pages.tex, line 295
garithmic block m. Reciprocal order contributes u. vfigtheta Their combined logarithm has coefficient (-1)^mκ_N,mfracu^Nm+um. At the first block this gives Θ_N,1(u)=-(u^N+u). Every positive-index theta polynomial vanishes at u=0. For even N, H_N(-z)=H_N(z), so the defining product is also 1 at u=-1. headThe odd-order boundary is visible For N=3, Θ_3,1(-1)=2, not zero. The factor u+1 is not a uniform law across orders. In contrast, Θ_2,2(u)=u(u+1)(5u^2+u+2). The two guaranteed factors follow from…
Exact mathematical passage
\Theta_{3,1}(-1)=2
\theta_iUse in source · 2 passages
Search aliases: θ_i theta
Use in source · odd_synthesis.tex, line 141
Choose a labeled symplectic embedding V_src→ S. For each original full character form P_i=( C^×× J_i^1)/ {(θ_i(h)^-1,h):h∈ H_i^1}. The balanced product and the corresponding host restriction have the same alternating commutator. The center-fixing extension isomorphism, obtained with matched Darboux lifts and complex roots, therefore preserves every supplied value θ_i(h), including the shallow values.
Exact mathematical passage
\mathcal P_i=(\mathbb C^\times\times J_i^1)/ \{(\theta_i(h)^{-1},h):h\in H_i^1\}.
\Theta_jUse in source · 2 passages
Search aliases: Θ_j Theta
Use in source · dyadic_synthesis.tex, line 105
For a later native step, first transfer the full beta-determinant unit character to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar units, Θ_j(uI)=θ_j(u^2)=1. For odd critical level r, use m=lfloor r/2rfloor and D=H^m+1(β)=H^m+1(γ). The endpoint is valid because 2(m+1)-r=1: the cross term for the additive character lies in the radical on which the matrix additive character is trivial. BK 3.3.18 supplies ϑ=Θ_j|_D ψ_β-γ. Integral trace makes it trivial on D∩ U_F^1.
Exact mathematical passage
\Theta_j(uI)=\theta_j(u^2)=1.
\theta_jUse in source · 2 passages
Search aliases: θ_j theta
Use in source · dyadic_synthesis.tex, line 105
For a later native step, first transfer the full beta-determinant unit character to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar units, Θ_j(uI)=θ_j(u^2)=1. For odd critical level r, use m=lfloor r/2rfloor and D=H^m+1(β)=H^m+1(γ). The endpoint is valid because 2(m+1)-r=1: the cross term for the additive character lies in the radical on which the matrix additive character is trivial. BK 3.3.18 supplies ϑ=Θ_j|_D ψ_β-γ. Integral trace makes it trivial on D∩ U_F^1.
Exact mathematical passage
\Theta_j(uI)=\theta_j(u^2)=1.
\Theta_{M,s}Use in source · 9 passages
Search aliases: Θ_M,s Theta
Use in source · 10_foundations.tex, line 502
For M≥1 and s,r,k∈ N, define the affine Mahler kernel by Θ_M,s(r,k)=[z^r](1+z)^s((1+z)^M-1)^k =∑_j=0^k(-1)^k-jbinom kjbinomMj+sr. It vanishes unless k≤ r≤ Mk+s. For a finite binomial expansion f(n)=∑_r=0^Dc_rbinom nr, the kth Newton coefficient of g(q)=f(Mq+s) is therefore c_k(g)=∑_r=k^min(D,Mk+s)c_rΘ_M,s(r,k). Indeed, insert the finite stencil for Δ_1^k g(0) and interchange its two finite sums. Coefficient extraction of the binomial expansion of ((1+z)^M-1)^k gives the kernel. In the binary zer…
Exact mathematical passage
\Theta_{M,s}(r,k)=[z^r](1+z)^s\bigl((1+z)^M-1\bigr)^k =\sum_{j=0}^{k}(-1)^{k-j}\binom kj\binom{Mj+s}{r}.
\Theta_{MN,Mt+s}Use in source · 1 passage
Search aliases: Θ_MN,Mt+s Theta
Use in source · 72_recovered.tex, line 2242
[Kernel composition] For positive integers M,N, ∑_k≥0 Θ_M,s(r,k) Θ_N,t(k,ℓ) = Θ_MN,Mt+s(r,ℓ). The sum is finite.
Exact mathematical passage
\boxed{ \sum_{k\geq0} \Theta_{M,s}(r,k) \Theta_{N,t}(k,\ell) = \Theta_{MN,Mt+s}(r,\ell). }
\theta_{N,0}Use in source · 1 passage
Search aliases: θ_N,0 theta
Use in source · Paper_IV_Traditional_Final.tex, line 1687
[Theta Newton recurrence] Put θ_N,n(u)=fracsymupΘ_N,n(u)(Nn)!. Then θ_N,0(u)=1 and nθ_N,n(u) = ∑_m=1^n (-1)^mκ_N,m (u^Nm+u) θ_N,n-m(u). In particular, symupΘ_N,1(u)=-(u^N+u).
Exact mathematical passage
\theta_{N,0}(u)=1
\Theta_{N,1}Use in source · 1 passage
Search aliases: Θ_N,1 Theta
Use in source · atlas-pages.tex, line 293
E_N,0(z). The theta family uses the same parameter u twice, but in two different roles: H_N(uz)^-1H_N(z)^-u =∑_n≥0Θ_N,n(u)fracz^Nn(Nn)!. Argument scaling contributes u^Nm at logarithmic block m. Reciprocal order contributes u. vfigtheta Their combined logarithm has coefficient (-1)^mκ_N,mfracu^Nm+um. At the first block this gives Θ_N,1(u)=-(u^N+u). Every positive-index theta polynomial vanishes at u=0. For even N, H_N(-z)=H_N(z), so the defining product is also 1 at u=-1. headThe odd-order bound…
Exact mathematical passage
\Theta_{N,1}(u)=-(u^N+u)
\Theta_{N,n}Use in source · 1 passage
Search aliases: Θ_N,n Theta
Use in source · atlas-pages.tex, line 283
vpage16Scale and order enter differentlyTHETA COUPLINGChapter IV: Theorem 10.2, Proposition 10.3, Proposition 10.4. Let H_N(z)=E_N,0(z). The theta family uses the same parameter u twice, but in two different roles: H_N(uz)^-1H_N(z)^-u =∑_n≥0Θ_N,n(u)fracz^Nn(Nn)!. Argument scaling contributes u^Nm at logarithmic block m. Reciprocal order contributes u. vfigtheta Their combined logarithm has coefficient (-1)^mκ_N,mfracu^Nm+um. At the first block this gives Θ_N,1(u)=-(u^N+u). Every positive-index t…
Exact mathematical passage
H_N(uz)^{-1}H_N(z)^{-u} =\sum_{n\ge0}\Theta_{N,n}(u)\frac{z^{Nn}}{(Nn)!}.
\theta_{N,n}Use in source · 2 passages
Search aliases: θ_N,n theta
Use in source · Paper_IV_Traditional_Final.tex, line 1684
[Theta Newton recurrence] Put θ_N,n(u)=fracsymupΘ_N,n(u)(Nn)!. Then θ_N,0(u)=1 and nθ_N,n(u) = ∑_m=1^n (-1)^mκ_N,m (u^Nm+u) θ_N,n-m(u). In particular, symupΘ_N,1(u)=-(u^N+u).
Exact mathematical passage
\theta_{N,n}(u)=\frac{\Thetafun_{N,n}(u)}{(Nn)!}.
\theta_{N,n-m}Use in source · 1 passage
Search aliases: θ_N,n-m theta
Use in source · Paper_IV_Traditional_Final.tex, line 1688
[Theta Newton recurrence] Put θ_N,n(u)=fracsymupΘ_N,n(u)(Nn)!. Then θ_N,0(u)=1 and nθ_N,n(u) = ∑_m=1^n (-1)^mκ_N,m (u^Nm+u) θ_N,n-m(u). In particular, symupΘ_N,1(u)=-(u^N+u).
Exact mathematical passage
n\theta_{N,n}(u) = \sum_{m=1}^{n} (-1)^m\kappa_{N,m} \bigl(u^{Nm}+u\bigr) \theta_{N,n-m}(u).
\Theta_{N,t}Use in source · 1 passage
Search aliases: Θ_N,t Theta
Use in source · 72_recovered.tex, line 2242
[Kernel composition] For positive integers M,N, ∑_k≥0 Θ_M,s(r,k) Θ_N,t(k,ℓ) = Θ_MN,Mt+s(r,ℓ). The sum is finite.
Exact mathematical passage
\boxed{ \sum_{k\geq0} \Theta_{M,s}(r,k) \Theta_{N,t}(k,\ell) = \Theta_{MN,Mt+s}(r,\ell). }
\Theta_{p^a,s}Use in source · 5 passages
Search aliases: Θ_p^a,s Theta
Use in source · 72_recovered.tex, line 2286
but its consequence for transformed Mahler coefficients requires c_r∈ Z_p for every r. Convergence of a Q_p-valued Mahler expansion alone is insufficient. For p=2, M=2, s=0 and f(x)=x/2, one has c_1=1/2, c_2=0, while g(q)=q has d_1=1. Thus d_1not≡ c_2pmod2. The coefficient consequence is read with integral original coefficients. Let M=p^a. Then Θ_p^a,s(r,k) ≡ binomsr-p^ak pmod p, where the binomial coefficient is zero outside its natural range. Consequently, d_s,k ≡ ∑_u=0^s binom su c_p^ak+u pmo…
Exact mathematical passage
\boxed{ \Theta_{p^a,s}(r,k) \equiv \binom{s}{r-p^ak} \pmod p, }
\Theta_{\Pcal}Use in source · 2 passages
Search aliases: Θ_Pcal Θ_ P Theta Pcal Theta mathcal
Use in source · 03_packets_and_graphs.tex, line 678
[Full block algebra representation] The map Θ_ P:End_𝕜(X) ⟶ ∏_β,α∈ IHom_𝕜(X_α,X_β), T⟼(T_βα)_β,α, is an algebra isomorphism when the codomain is equipped with matrix multiplication whose entries are operators. DependsnolinkurlPSC-I-T-OPERATOR-RECONSTRUCTION and nolinkurlPSC-I-T-BLOCK-MULTIPLICATION.
Exact mathematical passage
\Theta_{\Pcal}:\End_{\bbk}(X) \longrightarrow \prod_{\beta,\alpha\in I}\Hom_{\bbk}(X_\alpha,X_\beta), \qquad T\longmapsto(T_{\beta\alpha})_{\beta,\alpha},
\iota^{-1}Use in source · 1 passage
Search aliases: ι^-1 iota
Use in source · Paper_XI_Traditional_Final.tex, line 1321
[Symplectic invariance] If ι:(V,b)⟶(V',b') is a symplectic isometry and g'=ι gι^-1, then Φ_p(g,V)=Φ_p(g',V').
Exact mathematical passage
g'=\iota g\iota^{-1}
\iota_HUse in source · 3 passages
Search aliases: ι_H iota
Use in source · Paper_XI_Traditional_Final.tex, line 1214
[Restricted Heisenberg bridge] A restricted bridge datum determines an isomorphism between the two Heisenberg representations, unique up to a nonzero scalar. More precisely, there is an invertible map U: H_ρ⟶ H_θ such that Uω_ρ(h) = ω_θ(ι_H(h))U for every Heisenberg element h, and every other such map is a scalar multiple of U.
Exact mathematical passage
U\omega_\rho(h) = \omega_\theta(\iota_H(h))U
\iota_{\tgt(e)}Use in source · 1 passage
Search aliases: ι_tgt(e) iota tgt iota operatorname
Use in source · 01_finite_coordinates.tex, line 147
[Current fiber scope] The compiled Chapter III model above has one scalar coordinate per packet. For a family of modules (X_i)_i∈ V, an intrinsic edge map would instead have type ℓ_e:X_src(e)→ X_tgt(e), and its totalization on ⊕_iX_i would be ι_tgt(e)ℓ_eπ_src(e). The same support proof applies, but this generalization with dependent fibers is not attributed to the released Chapter III modules for scalar edge systems.
Exact mathematical passage
\iota_{\tgt(e)}\ell_e\pi_{\src(e)}
\iota_\lambdaUse in source · 1 passage
Search aliases: ι_λ iota lambda
Use in source · Paper_X_Block27_Amended.tex, line 1231
le and the systems are pairwise componentwise nonisomorphic. The finite prime separator now yields distinct permitted rational primes p_j, actual prime ideals Q_j and arithmetic Frobenius witnesses γ_j as above, outside any prescribed finite set, such that the common algebraic trace matrix has nonzero determinant. Every coefficient embedding preserves this nonvanishing, and ι_λ(t_i(p_j))= tr(V_i,λ(γ_j)) whenever p_j is good for that coefficient member. The operator and trace values agree for eve…
Exact mathematical passage
\iota_\lambda(t_i(p_j))= \operatorname{tr}(V_{i,\lambda}(\gamma_j))
\iota_\ellUse in source · 6 passages
Search aliases: ι_ℓ iota ell
Use in source · 02_strengthened.tex, line 2361
Let H_∞ be the Hilbert direct limit of the isometric system, and let ι_ℓ: H_ℓ⟶ H_∞ be the canonical isometries.
Exact mathematical passage
\iota_\ell:\Hcal_\ell\longrightarrow\Hcal_\infty
\kappa^{-1}Use in source · 1 passage
Search aliases: κ^-1 kappa
Use in source · odd_synthesis.tex, line 68
(P_0 extract the constant coefficient in the basis 1,z,…,z^p-1. The normalized corestriction s=κpr, where κ=-f'(z)/z^p-1, satisfies u_0=s(P_0)=a_0/z^p∈ O_E^×, Lift_E(C)=u_0^-1CP_0, s(Lift_E(C))=C. The lift preserves the exact filtration valuation. Its crucial additional identity is tr_L(Lift_E(C)) =Tr_E/L (κ^-1tr_E(C)). Define B(L,t_L;h_1,…,h_d) =α I_p^d-1+ Lift_E(B(E,z;h_1,…,h_d-1)), with the degree-p multiplication matrix as the base case. Induction proves trace zero at every native rank. This…
Exact mathematical passage
\operatorname{tr}_L(\operatorname{Lift}_E(C)) =\operatorname{Tr}_{E/L} \bigl(\kappa^{-1}\operatorname{tr}_E(C)\bigr).
\kappa_0Use in source · 1 passage
Search aliases: κ_0 kappa
Use in source · root_scope.tex, line 37
racte^t-1=∑_n≥0B_nfract^nn!, B_1=-frac12. Throughout this appendix, v_2(2)=1 and v_2(0)=∞. Let s_2(n) be the number of ones in the binary expansion of the nonnegative integer n. Define A(N)=frac3^2N+2-14(2N+1)(2N+2), (frac zsinh z)^2=∑_m≥0β_m z^2m, κ_m=2^2m(2m)!β_m. The initial values are β_0=κ_0=1. For m≥1, differentiation of zcoth z gives β_m=-frac2^2m(2m-1)B_2m(2m)!, κ_m=-2^4m(2m-1)B_2m. The classical dyadic Bernoulli denominator law v_2(B_2m)=-1 and Legendre's formula yield v_2(β_m)=s_2(m)-1…
Exact mathematical passage
\beta_0=\kappa_0=1
\kappa_1Use in source · 2 passages
Search aliases: κ_1 kappa
Use in source · atlas-pages.tex, line 181
rem 7.2, Corollary 7.7. Let r vary instead of fixing the reciprocal power. Since Δ_N(w)^-r=exp (r∑_m≥1κ_N,mfracw^mm), the coefficient of w^n is a polynomial of degree n in r. Bell polynomials collect the possible ways logarithmic terms contribute total degree n. vfigbell_partitions For N=2, the first three logarithmic coefficients are κ_1=frac12, κ_2=frac16, κ_3=frac115. The third Bell polynomial is B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3. Substituting x_1=rκ_1, x_2=rκ_2, x_3=2rκ_3 and multiplying by…
Exact mathematical passage
\kappa_1=\frac12,\qquad\kappa_2=\frac16,\qquad\kappa_3=\frac1{15}.
\kappa_2Use in source · 6 passages
Search aliases: κ_2 kappa
Use in source · atlas-pages.tex, line 181
rem 7.2, Corollary 7.7. Let r vary instead of fixing the reciprocal power. Since Δ_N(w)^-r=exp (r∑_m≥1κ_N,mfracw^mm), the coefficient of w^n is a polynomial of degree n in r. Bell polynomials collect the possible ways logarithmic terms contribute total degree n. vfigbell_partitions For N=2, the first three logarithmic coefficients are κ_1=frac12, κ_2=frac16, κ_3=frac115. The third Bell polynomial is B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3. Substituting x_1=rκ_1, x_2=rκ_2, x_3=2rκ_3 and multiplying by…
Exact mathematical passage
\kappa_1=\frac12,\qquad\kappa_2=\frac16,\qquad\kappa_3=\frac1{15}.
\kappa_{2,1}Use in source · 2 passages
Search aliases: κ_2,1 kappa
Use in source · Paper_IV_Traditional_Final.tex, line 1250
[title=A coefficient calculation at order two] Write Δ_2(w)^-r=∑_n≥0b_n(r)w^n. Since Δ_2(w)=1-frac w2+fracw^224+O(w^3), -logΔ_2(w)=frac w2+fracw^212+O(w^3), the first logarithmic coefficients are κ_2,1=1/2 and κ_2,2=1/6. Newton's recurrence gives b_1(r)=frac r2, 2b_2(r)=r(frac r4+frac16). Restoring the factorial normalization yields P_2,1(r)=r and P_2,2(r)=r(3r+2). At r=1, the second value is five. No zero-location assumption is needed for this finite calculation. modelfigureparameter_polynomial…
Exact mathematical passage
\kappa_{2,1}=1/2
\kappa_{2,2}Use in source · 2 passages
Search aliases: κ_2,2 kappa
Use in source · Paper_IV_Traditional_Final.tex, line 1250
[title=A coefficient calculation at order two] Write Δ_2(w)^-r=∑_n≥0b_n(r)w^n. Since Δ_2(w)=1-frac w2+fracw^224+O(w^3), -logΔ_2(w)=frac w2+fracw^212+O(w^3), the first logarithmic coefficients are κ_2,1=1/2 and κ_2,2=1/6. Newton's recurrence gives b_1(r)=frac r2, 2b_2(r)=r(frac r4+frac16). Restoring the factorial normalization yields P_2,1(r)=r and P_2,2(r)=r(3r+2). At r=1, the second value is five. No zero-location assumption is needed for this finite calculation. modelfigureparameter_polynomial…
Exact mathematical passage
\kappa_{2,2}=1/6
\kappa_{2,r}Use in source · 8 passages
Search aliases: κ_2,r kappa
Use in source · 72_recovered.tex, line 86
The dyadic kernel is substantially sharper than the separate coefficient bound alone suggests. If fracz^2sinh^2z = ∑_r≥0β_rz^2r and κ_2,r=β_r2^2r(2r)!, then v_2(κ_2,r)=4r-1 (r≥1). Consequently, truncation after depth M is uniformly valid modulo 2^4M+3, and this exponent is optimal. The rowwise estimate records the additional binary carries in the binomial coefficient. Classical congruence theory for Euler families is extensive [source reference]. The formulas here are stated as exact coordinate …
Exact mathematical passage
\kappa_{2,r}=\beta_r2^{2r}(2r)!,
\kappa_3Use in source · 2 passages
Search aliases: κ_3 kappa
Use in source · atlas-pages.tex, line 181
rem 7.2, Corollary 7.7. Let r vary instead of fixing the reciprocal power. Since Δ_N(w)^-r=exp (r∑_m≥1κ_N,mfracw^mm), the coefficient of w^n is a polynomial of degree n in r. Bell polynomials collect the possible ways logarithmic terms contribute total degree n. vfigbell_partitions For N=2, the first three logarithmic coefficients are κ_1=frac12, κ_2=frac16, κ_3=frac115. The third Bell polynomial is B_3(x_1,x_2,x_3)=x_1^3+3x_1x_2+x_3. Substituting x_1=rκ_1, x_2=rκ_2, x_3=2rκ_3 and multiplying by…
Exact mathematical passage
\kappa_1=\frac12,\qquad\kappa_2=\frac16,\qquad\kappa_3=\frac1{15}.
\kappa_{4,1}Use in source · 1 passage
Search aliases: κ_4,1 kappa
Use in source · atlas-pages.tex, line 222
ollary 8.4, Theorem 8.5. The transfer tangent is T_N=-D_N'/D_N. It compresses the linear differential equation into a nonlinear scalar identity. This division is allowed only on the zero-free set of the denominator. vfigriccati The coefficients retain the reduced logarithmic information: T_N(z)=N∑_m≥1κ_N,mz^Nm-1, T_N(n)=N(Nn+N-1)!κ_N,n+1. For order four, κ_4,1=1/24 and κ_4,2=17/10080. These give T_4(0)=1 and T_4(1)=34, explaining the last line of the plate. headA nonlinear equation with a precis…
Exact mathematical passage
\kappa_{4,1}=1/24
\kappa_{4,2}Use in source · 1 passage
Search aliases: κ_4,2 kappa
Use in source · atlas-pages.tex, line 222
The transfer tangent is T_N=-D_N'/D_N. It compresses the linear differential equation into a nonlinear scalar identity. This division is allowed only on the zero-free set of the denominator. vfigriccati The coefficients retain the reduced logarithmic information: T_N(z)=N∑_m≥1κ_N,mz^Nm-1, T_N(n)=N(Nn+N-1)!κ_N,n+1. For order four, κ_4,1=1/24 and κ_4,2=17/10080. These give T_4(0)=1 and T_4(1)=34, explaining the last line of the plate. headA nonlinear equation with a precise origin If D=e^g locally…
Exact mathematical passage
\kappa_{4,2}=17/10080
\kappa_mUse in source · 11 passages
Search aliases: κ_m kappa
Use in source · 20_arithmetic.tex, line 951
[Exact weighted inverse coefficient] Let κ_m=β_m2^2m(2m)!. For every m≥1, κ_m=-2^4m(2m-1)B_2m≠0, v_2(κ_m)=4m-1, where B_2m is the usual even Bernoulli number.
Exact mathematical passage
\kappa_m=\beta_m2^{2m}(2m)!
\kappa_{M+1}Use in source · 1 passage
Search aliases: κ_M+1 kappa
Use in source · 20_arithmetic.tex, line 1006
At n=M+1 there is exactly one omitted summand. Here binom2n2n=1 and A(0)=1, so the error is κ_M+1. Lemma~[source reference] gives its exact valuation 4M+3 and nonvanishing. The statements are leanKomorebiPaperIX.DyadicSharpTower.dyadic_supercongruence_sharp, leanKomorebiPaperIX.DyadicSharpTower.first_omitted_error, and leanKomorebiPaperIX.DyadicSharpTower.first_omitted_error_valuation.
Exact mathematical passage
\kappa_{M+1}
\kappa_{N,1}Use in source · 4 passages
Search aliases: κ_N,1 kappa
Use in source · Paper_IV_Traditional_Final.tex, line 988
[Bell polynomial formula] For every n≥0, P_N,n(r) = frac(Nn)!n! B_n( rκ_N,1, r 1! κ_N,2, …, r (n-1)! κ_N,n ). For n≥1, this is a polynomial in r of exact degree n. Its leading coefficient is frac(Nn)!n!(N!)^n.
Exact mathematical passage
P_{N,n}(r) = \frac{(Nn)!}{n!} B_n\bigl( r\kappa_{N,1}, r\,1!\,\kappa_{N,2}, \ldots, r\,(n-1)!\,\kappa_{N,n} \bigr).
\kappa_{N,2}Use in source · 2 passages
Search aliases: κ_N,2 kappa
Use in source · Paper_IV_Traditional_Final.tex, line 988
[Bell polynomial formula] For every n≥0, P_N,n(r) = frac(Nn)!n! B_n( rκ_N,1, r 1! κ_N,2, …, r (n-1)! κ_N,n ). For n≥1, this is a polynomial in r of exact degree n. Its leading coefficient is frac(Nn)!n!(N!)^n.
Exact mathematical passage
P_{N,n}(r) = \frac{(Nn)!}{n!} B_n\bigl( r\kappa_{N,1}, r\,1!\,\kappa_{N,2}, \ldots, r\,(n-1)!\,\kappa_{N,n} \bigr).
\kappa_{N,3}Use in source · 1 passage
Search aliases: κ_N,3 kappa
Use in source · Paper_IV_Traditional_Final.tex, line 1207
The first logarithmic coefficients are κ_N,1=frac1N!, κ_N,2=frac1(N!)^2-frac2(2N)!, and κ_N,3 = frac1(N!)^3 - frac3N!(2N)! + frac3(3N)!. Insert these values into [source reference] and simplify.
Exact mathematical passage
\kappa_{N,3} = \frac1{(N!)^3} - \frac3{N!(2N)!} + \frac3{(3N)!}.
\kappa_{N,k}Use in source · 5 passages
Search aliases: κ_N,k kappa
Use in source · Paper_IV_Traditional_Final.tex, line 1116
[Logarithmic Newton recurrence] Set b_N,n(r)=fracP_N,n(r)(Nn)!. Then b_N,0(r)=1 and for every n≥1, nb_N,n(r) = r∑_k=1^nκ_N,kb_N,n-k(r). Equivalently, P_N,n(r) = frac r n ∑_k=1^n frac(Nn)!(N(n-k))! κ_N,kP_N,n-k(r). If p_k=rκ_N,k, then, for n≥1, b_N,n(r) = frac1n! det p_1 · -1 · 0 · ⋯ · 0
Exact mathematical passage
nb_{N,n}(r) = r\sum_{k=1}^{n}\kappa_{N,k}b_{N,n-k}(r).
\kappa_{N,n}Use in source · 1 passage
Search aliases: κ_N,n kappa
Use in source · Paper_IV_Traditional_Final.tex, line 988
[Bell polynomial formula] For every n≥0, P_N,n(r) = frac(Nn)!n! B_n( rκ_N,1, r 1! κ_N,2, …, r (n-1)! κ_N,n ). For n≥1, this is a polynomial in r of exact degree n. Its leading coefficient is frac(Nn)!n!(N!)^n.
Exact mathematical passage
P_{N,n}(r) = \frac{(Nn)!}{n!} B_n\bigl( r\kappa_{N,1}, r\,1!\,\kappa_{N,2}, \ldots, r\,(n-1)!\,\kappa_{N,n} \bigr).
\kappa_{N,n+1}Use in source · 2 passages
Search aliases: κ_N,n+1 kappa
Use in source · Paper_IV_Traditional_Final.tex, line 1283
[Logarithmic coefficient formula] The transfer tangent satisfies T_N(z) = N∑_m≥1κ_N,mz^Nm-1. Equivalently, T_N(n) = N(Nn+N-1)!κ_N,n+1.
Exact mathematical passage
T_N(n) = N(Nn+N-1)!\kappa_{N,n+1}.
\kappa_{N,q}Use in source · 1 passage
Search aliases: κ_N,q kappa
Use in source · atlas-pages.tex, line 402
vpage22Finite coefficients give infinite momentsZERO POWER SUMSChapter IV: Theorem 12.6, Corollary 12.7, Corollary 12.9, Corollary 12.10. Taking the logarithm of the canonical product identifies κ_N,q=σ_N,q:=∑_λ∈Λ_Nλ^-q =Tr(K_N^q), q≥1. The same exact numbers can be computed from finitely many denominator coefficients by Newton's recurrence. No numerical zero list is needed for that calculation. vfigmoments For order two, σ_2,1=frac12, σ_2,2=frac16, σ_2,3=frac115, σ_2,4=frac17630. The plate show…
Exact mathematical passage
\kappa_{N,q}=\sigma_{N,q}:=\sum_{\lambda\in\Lambda_N}\lambda^{-q} =\Tr(K_N^q),\qquad q\ge1.
\kappa_{p,r}Use in source · 3 passages
Search aliases: κ_p,r kappa
Use in source · 72_recovered.tex, line 1166
0, · p=2 and n odd, and κ_p,r = (-1)^rZ_2r+1p^2r, · p odd, β_r2^2r(2r)!, · p=2.
Exact mathematical passage
\kappa_{p,r} = \begin{cases} (-1)^rZ_{2r+1}p^{2r},&p\text{ odd},\\ \beta_r2^{2r}(2r)!,&p=2. \end{cases}
\kappa_rUse in source · 1 passage
Search aliases: κ_r kappa
Use in source · root_scope.tex, line 117
Set r=M+1≥1 and define P_q(x)=∏_i=0^q-1(x-i)(2x-(2i+1)), P_0=1, H_r(x)=∑_q≥08^qfracβ_r+qβ_r P_q(x)mathscr A(x-q). The coefficient valuations tend to infinity. At any nonnegative integer x, terms with q>x vanish. Cancelling the factorials in [source reference] therefore gives the exact identity T_r-1(N)=κ_rbinom2N2rH_r(N-r) (N≥ r). This is a finite rational identity at every original Euler row.
Exact mathematical passage
\mathcal T_{r-1}(N)=\kappa_r\binom{2N}{2r}H_r(N-r) \qquad(N\ge r).\label{eq:root:factor}
\kappa_{r,b_j}Use in source · 2 passages
Search aliases: κ_r,b_j kappa
Use in source · Paper_V_Traditional_Final.tex, line 3536
[Shifted pointwise monomial law] Let y(n)=∏_j=1^mx(n-a_j) and write a_j=Nc_j+b_j, 0≤ b_j<N. For output residue r, define r_j≡ r-b_jpmod N, 0≤ r_j<N, and let κ_r,b_j = 1, · r<b_j, 0, · r≥ b_j. Then the output channel is y_r(q) = ∏_j=1^m x_r_j (q-c_j-κ_r,b_j).
Exact mathematical passage
\kappa_{r,b_j} = \begin{cases} 1,&r<b_j,\\ 0,&r\geq b_j. \end{cases}
\kappa_v^{(j)}Use in source · 2 passages
Search aliases: κ_v^(j) kappa
Use in source · 01_manuscript.tex, line 1580
[Shared compilation for an operator family] Let operators A^(1),…,A^(m) use the same candidate tree. Give operator j a nonnegative reuse weight r_j. Replace the local keep cost and error by κ_v = κ_v^rm setup + ∑_j=1^m r_jκ_v^(j) and ϵ_v = ∑_j=1^m r_jϵ_v^(j). Then [source reference] gives the exact optimal shared pruning for the additive weighted objective.
Exact mathematical passage
\kappa_v = \kappa_v^{\rm setup} + \sum_{j=1}^m r_j\kappa_v^{(j)}
\kappa_v^{\rm setup}Use in source · 1 passage
Search aliases: κ_v^rm setup kappa rm
Use in source · 01_manuscript.tex, line 1580
[Shared compilation for an operator family] Let operators A^(1),…,A^(m) use the same candidate tree. Give operator j a nonnegative reuse weight r_j. Replace the local keep cost and error by κ_v = κ_v^rm setup + ∑_j=1^m r_jκ_v^(j) and ϵ_v = ∑_j=1^m r_jϵ_v^(j). Then [source reference] gives the exact optimal shared pruning for the additive weighted objective.
Exact mathematical passage
\kappa_v = \kappa_v^{\rm setup} + \sum_{j=1}^m r_j\kappa_v^{(j)}
\kappa_v^{\mathrm{setup}}Use in source · 1 passage
Search aliases: κ_v^setup kappa mathrm
Use in source · 02_supplement_02.tex, line 29
Shared operator families. For reuse weights r_j, the local keep model can aggregate setup and per-operator costs: κ_v=κ_v^setup+∑_jr_jκ_v^(j), ϵ_v=∑_jr_jϵ_v^(j). Lean proves the aggregation identity at the cut level, including split overhead, before applying the compiler theorem. The proof is valid over an ordered field, including Q and R. The intended nonnegative model is a specialization.
Exact mathematical passage
\kappa_v=\kappa_v^{\mathrm{setup}}+\sum_jr_j\kappa_v^{(j)}, \qquad \epsilon_v=\sum_jr_j\epsilon_v^{(j)}.
\kappa_\thetaUse in source · 6 passages
Search aliases: κ_θ kappa theta
Use in source · Paper_XI_Traditional_Final.tex, line 459
[Regular elliptic torus characters] A regular character θ:T_n→ C^× determines an irreducible cuspidal representation κ_θ of G_n. Frobenius conjugate characters determine isomorphic representations, and every irreducible cuspidal representation of G_n occurs in this way.
Exact mathematical passage
\kappa_\theta
\lambda^{\mathsf T}Use in source · 1 passage
Search aliases: λ^ T lambda mathsf
Use in source · atlas-pages.tex, line 35
B_λ · diag(λ)p/(λ^ Tp). · Nonnegative likelihood and positive total evidence.
Exact mathematical passage
\operatorname{diag}(\lambda)p/(\lambda^{\mathsf T}p)
\lambda^{(1)}Use in source · 2 passages
Search aliases: λ^(1) lambda
Use in source · atlas-pages.tex, line 595
vpage26One complete coarse filterEND-TO-END MODEL / TWO OBSERVATIONSChapter VI, Part I: Proposition 9.3 with Theorems 5.1 and 8.2. Markov closure and likelihood sufficiency are separate. Use the four-state kernel K and aggregation C from pagerefV5, starting at p_0=e_0. Observe two likelihood vectors constant inside both packets: λ^(1)=(3/4,3/4,1/4,1/4), λ^(2)=(1/2,1/2,1,1). Predict, then update at each observation. The fine calculation gives p_1=tfrac120(15,3,2,0)^ T, p_2=tfrac1104(41,15,40,8)^ …
Exact mathematical passage
\lambda^{(1)}=(3/4,3/4,1/4,1/4),\qquad \lambda^{(2)}=(1/2,1/2,1,1).
\lambda^{(2)}Use in source · 2 passages
Search aliases: λ^(2) lambda
Use in source · atlas-pages.tex, line 595
vpage26One complete coarse filterEND-TO-END MODEL / TWO OBSERVATIONSChapter VI, Part I: Proposition 9.3 with Theorems 5.1 and 8.2. Markov closure and likelihood sufficiency are separate. Use the four-state kernel K and aggregation C from pagerefV5, starting at p_0=e_0. Observe two likelihood vectors constant inside both packets: λ^(1)=(3/4,3/4,1/4,1/4), λ^(2)=(1/2,1/2,1,1). Predict, then update at each observation. The fine calculation gives p_1=tfrac120(15,3,2,0)^ T, p_2=tfrac1104(41,15,40,8)^ …
Exact mathematical passage
\lambda^{(1)}=(3/4,3/4,1/4,1/4),\qquad \lambda^{(2)}=(1/2,1/2,1,1).
\lambda^{(r)}Use in source · 5 passages
Search aliases: λ^(r) lambda
Use in source · Paper_VI_Traditional_Final.tex, line 1092
Let K be column stochastic and let λ^(r) be the likelihood associated with the rth observation. The filter recursion is widehat p_r=Kp_r-1, p_r= B_λ^(r)(widehat p_r).
Exact mathematical passage
\lambda^{(r)}
\lambda^{(y)}Use in source · 1 passage
Search aliases: λ^(y) lambda
Use in source · Paper_VI_Traditional_Final.tex, line 989
[Canonical coarsest likelihood-resolving partition] For a finite family of likelihoods {λ^(y)}_y∈ Y, define i∼ j Longleftrightarrow λ_i^(y)=λ_j^(y) for every y∈ Y. The equivalence classes form the unique coarsest likelihood-resolving partition: every likelihood is constant on each packet. Every update therefore closes on this partition whenever its evidence is positive. If all likelihoods are strictly positive and there are at least two profile classes, this is also the unique coarsest nontrivia…
Exact mathematical passage
\{\lambda^{(y)}\}_{y\in Y}
\lambda^{-1/2}Use in source · 1 passage
Search aliases: λ^-1/2 lambda
Use in source · monograph_atlas.tex, line 610
0 · 0 · 1, WcW^*=diag(1,0,0), (F_1,F_2,F_3)=(P,0,0). xfiggram headNormalize only positive eigenvalues The positive branch has V_1=P as an isometry on the code, and syndrome support Q=P. No factor λ^-1/2 is formed on either zero branch: those branches vanish on the code. Treating them as extra recoverable packets would divide by zero.
Exact mathematical passage
\lambda^{-1/2}
\lambda^{-q}Use in source · 2 passages
Search aliases: λ^-q lambda
Use in source · atlas-pages.tex, line 402
vpage22Finite coefficients give infinite momentsZERO POWER SUMSChapter IV: Theorem 12.6, Corollary 12.7, Corollary 12.9, Corollary 12.10. Taking the logarithm of the canonical product identifies κ_N,q=σ_N,q:=∑_λ∈Λ_Nλ^-q =Tr(K_N^q), q≥1. The same exact numbers can be computed from finitely many denominator coefficients by Newton's recurrence. No numerical zero list is needed for that calculation. vfigmoments For order two, σ_2,1=frac12, σ_2,2=frac16, σ_2,3=frac115, σ_2,4=frac17630. The plate show…
Exact mathematical passage
\kappa_{N,q}=\sigma_{N,q}:=\sum_{\lambda\in\Lambda_N}\lambda^{-q} =\Tr(K_N^q),\qquad q\ge1.
\lambda^2Use in source · 2 passages
Search aliases: λ^2 lambda
Use in source · Paper_V_Traditional_Final.tex, line 3091
Assume the noise variables are independent, centered, and σ sub-Gaussian: E e^λη_k ≤ e^σ^2λ^2/2.
Exact mathematical passage
\mathbb E\e^{\lambda\eta_k} \leq \e^{\sigma^2\lambda^2/2}.
\Lambda^dUse in source · 1 passage
Search aliases: Λ^d Lambda
Use in source · Paper_XII_Traditional_Final.tex, line 542
Choose a Λ basis of M. Tensoring the resulting isomorphism M≅Λ^d along augmentation gives O^d. Tensoring by Λ/ a is the same as quotienting by a M.
Exact mathematical passage
M\cong\Lambda^d
\lambda^gUse in source · 1 passage
Search aliases: λ^g lambda
Use in source · rank2.tex, line 204
As g²∈J, conjugation by g² fixes the character theta. Thus λ^g=λ^-1=λ. It follows that
Exact mathematical passage
\lambda^g=\lambda^{-1}=\lambda
\lambda^jUse in source · 1 passage
Search aliases: λ^j lambda
Use in source · Paper_IV_Traditional_Final.tex, line 488
[Differential cycle and initial data] For 1≤ j<N, fracddz E_N,j(λ z) = λ E_N,j-1(λ z), while fracddz E_N,0(λ z) = λ E_N,N-1(λ z). Thus fracd^Ndz^N E_N,j(λ z) = λ^N E_N,j(λ z). For 0≤ k<N, .fracd^kdz^k E_N,j(λ z)|_z=0 = δ_kjλ^j. If λ≠0, the N functions E_N,j(λ z) form a basis of the solution space of y^(N)=λ^Ny.
Exact mathematical passage
\left.\frac{d^k}{dz^k}\Efun_{N,j}(\lambda z)\right|_{z=0} = \delta_{kj}\lambda^j.
\lambda^NUse in source · 4 passages
Search aliases: λ^N lambda
Use in source · Paper_IV_Traditional_Final.tex, line 204
Fix a positive integer N. Root of unity projection of the exponential produces residue-class sections that, for λ≠0, form a basis for y^(N)=λ^N y. The signed sections for y^(N)=-y yield the denominator D_N and its reduced form Δ_N. We derive reciprocal coefficient polynomials, Bell and Newton recurrences, an additive parameter law, and residue section Appell and theta families. For N≥2, Δ_N has order 1/N. We specialize classical two term zero theory for exponential sums to the regular exponent p…
Exact mathematical passage
y^{(N)}=\lambda^N y
\lambda^{N(N-1)/2}Use in source · 1 passage
Search aliases: λ^N(N-1)/2 lambda
Use in source · Paper_IV_Traditional_Final.tex, line 507
[Wronskian of the transfer basis] Assume N≥2. For λ≠0, the Wronskian of the ordered basis E_N,0(λ z),…, E_N,N-1(λ z) is the constant λ^N(N-1)/2.
Exact mathematical passage
\lambda^{N(N-1)/2}.
\lambda^{N-1}Use in source · 1 passage
Search aliases: λ^N-1 lambda
Use in source · Paper_IV_Traditional_Final.tex, line 514
At the origin, the derivative matrix is diagonal with entries 1,λ,…,λ^N-1. For an explicit determinant proof, write each basis function as a finite sum of exponentials with exponents a_ℓ=λω^ℓ. Its derivative matrix factors as W(z)=V diag(e^a_ℓ z) C, V_kℓ=a_ℓ^k, where C is the constant matrix of Fourier weights. Thus det W(z)=e^z∑_ℓ a_ℓdet W(0)=det W(0), because N≥2 implies ∑_ℓω^ℓ=0. The product of the diagonal initial values is the stated power of λ.
Exact mathematical passage
1,\lambda,\ldots,\lambda^{N-1}.
\lambda^uUse in source · 1 passage
Search aliases: λ^u lambda
Use in source · rank4.tex, line 232
First, lambda is invariant under u. Indeed u 2 belongs to J and theta is J-invariant, so λ^u=λ^-1=λ by (3). Consequently H0 is normal in P, and theta0 is P-invariant.
Exact mathematical passage
\lambda^u=\lambda^{-1}=\lambda
\lambda^\alphaUse in source · 2 passages
Search aliases: λ^α lambda alpha
Use in source · Paper_VI_Traditional_Final.tex, line 4078
[Stable subordination] Let 0<α<1 and t>0. There is a probability measure η_α,t on [0,∞) such that ∫_0^∞ e^-sλη_α,t( d s) = e^-tλ^α for every λ≥0. Consequently, for a finite dimensional positive self-adjoint operator, e^-tL^α = ∫_0^∞ e^-sLη_α,t( d s) in operator norm. The measures satisfy the convolution law η_α,s*η_α,t = η_α,s+t.
Exact mathematical passage
\int_0^\infty \e^{-s\lambda}\eta_{\alpha,t}(\dd s) = \e^{-t\lambda^\alpha}
\Lambda_*Use in source · 2 passages
Search aliases: Λ_* Lambda
Use in source · atlas-pages.tex, line 218
)+V(v1)}, V(v)=c_v at a terminal node. headRead the calculation from the bottom upward At 10, the children cost 1+1=2, better than stopping for 7. At 11, stopping costs 4, better than 2+3=5. The right subtree therefore costs 6, better than its stop cost 8. The left subtree costs 1+2=3, better than 5. At the root, 3+6=9 beats 20. The unique optimum is Λ_* = {00,01,100,101,11}, J(Λ_*)=9. All 26 complete prunings are separately enumerated and give the same minimum. At equality, the rule stops, maki…
Exact mathematical passage
\Lambda_* = \{00,01,100,101,11\},\qquad J(\Lambda_*)=9.
\lambda_*Use in source · 3 passages
Search aliases: λ_* lambda
Use in source · 05_worked_models.tex, line 292
r] Let 𝕜 be a field and let Λ be a nonempty finite set. Suppose there are pairwise distinct scalars a_λ∈𝕜, multiplicities m_λ∈ N, and an eigenbasis {b_λ,j:λ∈Λ, 1≤ j≤ m_λ} of 𝕜^V(G) satisfying A_Gb_λ,j=a_λb_λ,j. If m_λ≤ R for all λ and m_λ_*=R for some λ_*, then R is the least natural number of adjacency seeds: R=min{r∈ N: l there exist s_1,…,s_r with
Exact mathematical passage
m_{\lambda_*}=R
\Lambda_0Use in source · 1 passage
Search aliases: Λ_0 Lambda
Use in source · 04_arithmetic.tex, line 505
[The same rational split with two lattices] Let V=Ke_1⊕ Ke_2 and let E project onto Ke_1. On the split lattice Λ_0= O e_1⊕ O e_2, the weight is zero. Fix s≥1 and consider Λ_s = O e_1+ O(π^-se_1+e_2). Then Λ_s∩ Ke_1= O e_1, Λ_s∩ Ke_2=π^s O e_2. The gluing module is isomorphic to O/π^s O, so s_Λ_s(E)=s. The rational decomposition has not changed. Only the lattice has changed.
Exact mathematical passage
\Lambda_0=\OO e_1\oplus\OO e_2,
\lambda_0Use in source · 8 passages
Search aliases: λ_0 lambda
Use in source · Paper_X_Block27_Amended.tex, line 1111
Suppose two members are isomorphic at λ_0. Injectivity of the common coefficient embedding gives t_i(v)=t_j(v) at every place good for λ_0. At another coefficient prime λ, this equality compares actual member traces on the intersection of the two good-place sets. Conjugacy density of that intersection and continuity extend the equality to the whole group. Character rigidity then gives an actual isomorphism at λ. Thus the nonisomorphism of systems supplies the required nonisomorphism of members. …
Exact mathematical passage
\lambda_0
\Lambda_1Use in source · 5 passages
Search aliases: Λ_1 Lambda
Use in source · 04_arithmetic.tex, line 349
[Two complementary packets] Suppose V=V_1⊕ V_2 with projector E onto V_1 and complementary projector I-E. Then Λ/(Λ_1⊕Λ_2) ≅ EΛ/Λ_1 ≅ (I-E)Λ/Λ_2. Consequently s_Λ(E)=s_Λ(I-E) = exp_π(Λ/(Λ_1⊕Λ_2)).
Exact mathematical passage
\Lambda/(\Lambda_1\oplus\Lambda_2) \cong E\Lambda/\Lambda_1 \cong (I-E)\Lambda/\Lambda_2.
\Lambda_2Use in source · 4 passages
Search aliases: Λ_2 Lambda
Use in source · 04_arithmetic.tex, line 349
[Two complementary packets] Suppose V=V_1⊕ V_2 with projector E onto V_1 and complementary projector I-E. Then Λ/(Λ_1⊕Λ_2) ≅ EΛ/Λ_1 ≅ (I-E)Λ/Λ_2. Consequently s_Λ(E)=s_Λ(I-E) = exp_π(Λ/(Λ_1⊕Λ_2)).
Exact mathematical passage
\Lambda/(\Lambda_1\oplus\Lambda_2) \cong E\Lambda/\Lambda_1 \cong (I-E)\Lambda/\Lambda_2.
\lambda_2Use in source · 5 passages
Search aliases: λ_2 lambda
Use in source · 01_manuscript.tex, line 1601
[Monotone Lagrangian frontier] Let 0≤λ_1<λ_2. Choose optimal prunings L_1 and L_2 for the two Lagrangian parameters. Then err( L_2) ≤ err( L_1) and cost( L_2) ≥ cost( L_1).
Exact mathematical passage
0\leq\lambda_1<\lambda_2
\Lambda_{2,L}Use in source · 1 passage
Search aliases: Λ_2,L Lambda
Use in source · 72_recovered.tex, line 1542
For odd p and L≥1, put Λ_p,L=(p-1)p^L-1. For p=2 and L≥3, put Λ_2,L=2^L-2. These are exponents of the unit groups modulo the declared prime powers.
Exact mathematical passage
\Lambda_{2,L}=2^{L-2}.
\lambda_aUse in source · 6 passages
Search aliases: λ_a lambda
Use in source · Paper_XI_Traditional_Final.tex, line 928
[Scalar choice invariance] Replacing u_a by λ_au_a, with λ_a∈ C^×, changes c by a coboundary and leaves b_c unchanged.
Exact mathematical passage
\lambda_au_a
\lambda_{a+b}^{-1}Use in source · 1 passage
Search aliases: λ_a+b^-1 lambda
Use in source · Paper_XI_Traditional_Final.tex, line 933
The new cocycle is c'(a,b) = λ_aλ_bλ_a+b^-1c(a,b). The scalar coboundary factor is symmetric in a and b, so it cancels from c'(a,b)c'(b,a)^-1.
Exact mathematical passage
c'(a,b) = \lambda_a\lambda_b\lambda_{a+b}^{-1}c(a,b).
\lambda_{a,r}Use in source · 3 passages
Search aliases: λ_a,r lambda
Use in source · Paper_V_Traditional_Final.tex, line 1831
[Translation covariance] Fix a∈ Z^d. For every r∈ R, write uniquely r-a=r_a+λ_a,r, r_a∈ R, λ_a,r∈ L. Then ( A_LT_ax)_r(ℓ) = x_r_a(ℓ+λ_a,r). Thus a fine translation permutes the lattice packets and applies an internal translation on L.
Exact mathematical passage
r-a=r_a+\lambda_{a,r}, \qquad r_a\in R, \quad \lambda_{a,r}\in L.
\lambda_bUse in source · 2 passages
Search aliases: λ_b lambda
Use in source · Paper_XI_Traditional_Final.tex, line 933
The new cocycle is c'(a,b) = λ_aλ_bλ_a+b^-1c(a,b). The scalar coboundary factor is symmetric in a and b, so it cancels from c'(a,b)c'(b,a)^-1.
Exact mathematical passage
c'(a,b) = \lambda_a\lambda_b\lambda_{a+b}^{-1}c(a,b).
\Lambda_cUse in source · 5 passages
Search aliases: Λ_c Lambda
Use in source · supplement_08.tex, line 109
pageheading3. A nonsingular Euler-operator argumentThe standard interval without an analytic power-series-in-the-index premise On Hurwitz coefficients, the actual Euler operator plus c is (Λ_c a)(d)=(d+c)a(d). Direct induction proves its commutator with every defect iterate: δ^sΛ_c a =Λ_cδ^s a +sp(pU^p^2-U^p)δ^s-1a, s≥1. Suppose a is integral and supported at degrees divisible by p, and pnmid c. Every supported pivot d+c is proved to be a unit of Z_p. The defect preserves this support. Induction…
Exact mathematical passage
(\Lambda_c a)(d)=(d+c)a(d).
\Lambda_c^{-1}Use in source · 2 passages
Search aliases: Λ_c^-1 Lambda
Use in source · supplement_10.tex, line 147
When pnmid c, define the actual inverse coefficientwise: [t^d]Λ_c^-1f=frac[t^d]fd+c. On p-supported degrees, d+c is a genuine p-adic unit. The proof establishes Hurwitz integrality of this solution, its support, both inverse identities, and preservation of full flatness using the earlier all-order Euler commutator theorem. The singular pivots c=ap are never declared to be units.
Exact mathematical passage
[t^d]\Lambda_c^{-1}f=\frac{[t^d]f}{d+c}.
\lambda_dUse in source · 1 passage
Search aliases: λ_d lambda
Use in source · Paper_XII_Traditional_Final.tex, line 1065
[Simultaneous generalized eigenspaces] Assume that E' is algebraically closed and that T is generated by commuting operators T_1,…,T_d on V. For a maximal ideal m = (T_1-λ_1,…,T_d-λ_d), there is an integer N≥1 such that e_ mV = { v∈ V: (T_i-λ_i)^Nv=0 for every i }. Thus the canonical Artin packet is the simultaneous generalized eigenspace.
Exact mathematical passage
\frakm = (T_1-\lambda_1,\ldots,T_d-\lambda_d),
\lambda_{i'}Use in source · 2 passages
Search aliases: λ_i' lambda
Use in source · Paper_VI_Traditional_Final.tex, line 899
sely, fix two fine states i,i'∈ I_a and one state j∈ I_b in another coarse packet. Vary the distribution of a fixed coarse mass inside I_a between i and i' while leaving every coarse packet mass unchanged. The right side depends only on those coarse masses and is therefore unchanged. The left-side posterior mass of packet a depends on the conditional average of λ_i and λ_i'. It can be independent of that variation only if λ_i=λ_i'. Hence λ is constant on every packet. Write the value on I_a as λ…
Exact mathematical passage
\lambda_{i'}
\lambda_i^{(y)}Use in source · 1 passage
Search aliases: λ_i^(y) lambda
Use in source · Paper_VI_Traditional_Final.tex, line 990
[Canonical coarsest likelihood-resolving partition] For a finite family of likelihoods {λ^(y)}_y∈ Y, define i∼ j Longleftrightarrow λ_i^(y)=λ_j^(y) for every y∈ Y. The equivalence classes form the unique coarsest likelihood-resolving partition: every likelihood is constant on each packet. Every update therefore closes on this partition whenever its evidence is positive. If all likelihoods are strictly positive and there are at least two profile classes, this is also the unique coarsest nontrivia…
Exact mathematical passage
i\sim j \quad\Longleftrightarrow\quad \lambda_i^{(y)}=\lambda_j^{(y)} \text{ for every }y\in Y.
\lambda_jUse in source · 21 passages
Search aliases: λ_j lambda
Use in source · Paper_VI_Traditional_Final.tex, line 753
The first normalization appears in both numerator and denominator of the second update and cancels: fracμ_i(λ_ip_i/Z_λ)∑_jμ_jλ_jp_j/Z_λ = fracμ_iλ_ip_i∑_jμ_jλ_jp_j.
Exact mathematical passage
\frac{\mu_i(\lambda_ip_i/Z_\lambda)}{\sum_j\mu_j\lambda_jp_j/Z_\lambda} = \frac{\mu_i\lambda_ip_i}{\sum_j\mu_j\lambda_jp_j}.
\Lambda_{j+1}Use in source · 1 passage
Search aliases: Λ_j+1 Lambda
Use in source · supplement_08.tex, line 137
For j=0, g_0 is the literal N-divisibility indicator. Because Nmid p(p-1), the two shifts in δ g_0 agree and δ g_0=0. It is flat at every order. The actual binomial identity gives Λ_j+1g_j+1=(j+1)g_j. As 1≤ j+1<p, the nonsingular reflection theorem inductively proves flatness of every g_j in the standard window. There is no uniform-convergence assumption in this argument.
Exact mathematical passage
\Lambda_{j+1}g_{j+1}=(j+1)g_j.
\lambda_j^{(y)}Use in source · 1 passage
Search aliases: λ_j^(y) lambda
Use in source · Paper_VI_Traditional_Final.tex, line 990
[Canonical coarsest likelihood-resolving partition] For a finite family of likelihoods {λ^(y)}_y∈ Y, define i∼ j Longleftrightarrow λ_i^(y)=λ_j^(y) for every y∈ Y. The equivalence classes form the unique coarsest likelihood-resolving partition: every likelihood is constant on each packet. Every update therefore closes on this partition whenever its evidence is positive. If all likelihoods are strictly positive and there are at least two profile classes, this is also the unique coarsest nontrivia…
Exact mathematical passage
i\sim j \quad\Longleftrightarrow\quad \lambda_i^{(y)}=\lambda_j^{(y)} \text{ for every }y\in Y.
\lambda_j^{-1/2}Use in source · 1 passage
Search aliases: λ_j^-1/2 lambda
Use in source · 02_strengthened.tex, line 1327
syndrome construction] Assume the Knill--Laflamme identities. Then the coefficient matrix c is positive semidefinite. Choose a unitary matrix W such that WcW^*=diag(λ_1,…,λ_r), λ_j≥0, and define F_j=∑_aoverlineW_jaE_a. For every j with λ_j>0, the operator V_j=λ_j^-1/2F_jP is an isometry from the code into H. Its range projector Q_j=V_jV_j^* is a syndrome packet, and the nonzero Q_j are pairwise orthogonal.
Exact mathematical passage
V_j=\lambda_j^{-1/2}F_jP
\lambda_j^\alphaUse in source · 5 passages
Search aliases: λ_j^α lambda alpha
Use in source · Paper_VI_Traditional_Final.tex, line 1126
Let L=L^*≥0 on a finite-dimensional Hilbert space. Its spectral resolution is L=∑_jλ_jP_j. For α>0, define L^α=∑_jλ_j^α P_j. The associated fractional semigroup is T_α(t)=e^-tL^α =∑_je^-tλ_j^αP_j.
Exact mathematical passage
\boxed{L^\alpha=\sum_j\lambda_j^\alpha P_j.}
\lambda_j^{\alpha+\beta}Use in source · 1 passage
Search aliases: λ_j^α+β lambda alpha beta
Use in source · Paper_VI_Traditional_Final.tex, line 1149
All statements hold coefficientwise on the mutually orthogonal spectral projectors. For example, L^α L^β =∑_j,kλ_j^αλ_k^β P_jP_k =∑_jλ_j^α+βP_j.
Exact mathematical passage
L^\alpha L^\beta =\sum_{j,k}\lambda_j^\alpha\lambda_k^\beta P_jP_k =\sum_j\lambda_j^{\alpha+\beta}P_j.
\lambda_kUse in source · 5 passages
Search aliases: λ_k lambda
Use in source · 05_worked_models.tex, line 340
For every field 𝕜 and every n≥1, an endpoint of P_n is a cyclic vector for the adjacency operator. Consequently one is the least finite number of adjacency seeds: 1=min{r∈ N: ∃ s_1,…,s_r, 𝕜^n=span_𝕜{p(A_P_n)s_j:p∈𝕜[t], 1≤ j≤ r} }. Over C, the familiar spectral description has the distinct eigenvalues λ_k=2cosfrackπn+1, 1≤ k≤ n, and the normalized eigenvectors have endpoint coordinate proportional to sinfrackπn+1≠0. There is also a combinatorial proof valid over every field. Let K_n=[e_1,Ae_1,…,A…
Exact mathematical passage
\lambda_k=2\cos\frac{k\pi}{n+1}, \qquad 1\le k\le n,
\lambda_{k,h}Use in source · 21 passages
Search aliases: λ_k,h lambda
Use in source · Paper_VI_Traditional_Final.tex, line 1411
The kth discrete eigenvalue is λ_k,h=frac4h^2sin^2(frackπ h2), whereas the continuum eigenvalue is (kπ)^2.
Exact mathematical passage
\lambda_{k,h}=\frac4{h^2}\sin^2\left(\frac{k\pi h}{2}\right),
\lambda_{k,h}^\alphaUse in source · 5 passages
Search aliases: λ_k,h^α lambda alpha
Use in source · Paper_VI_Traditional_Final.tex, line 4320
[Heat and fractional mode transfer] Let λ_k=(kπ)^2. For t≥0, 0≤ e^-tλ_k,h- e^-tλ_k ≤ fract(kπ)^412h^2 e^-tλ_k,h. For 0<α≤1 and fixed k, λ_k,h^α = λ_k^α+O(h^2) and e^-tλ_k,h^α = e^-tλ_k^α+O(th^2) uniformly for t in bounded intervals.
Exact mathematical passage
\lambda_{k,h}^\alpha = \lambda_k^\alpha+O(h^2)
\lambda_{k,h}^{\alpha-1}Use in source · 1 passage
Search aliases: λ_k,h^α-1 lambda alpha
Use in source · Paper_VI_Traditional_Final.tex, line 5607
The chord bound sin x≥2x/π on [0,π/2] gives λ_k,h≥4k^2. Since x↦ x^α is increasing and has decreasing derivative, (kπ)^2α-λ_k,h^α ≤ αλ_k,h^α-1 ((kπ)^2-λ_k,h). Use [source reference] and the lower bound for λ_k,h. The exponential estimate follows from abs e^-ta- e^-tb ≤ tabsa-b for nonnegative a and b.
Exact mathematical passage
(k\pi)^{2\alpha}-\lambda_{k,h}^\alpha \leq \alpha\lambda_{k,h}^{\alpha-1} \bigl((k\pi)^2-\lambda_{k,h}\bigr).
\lambda_k^\alphaUse in source · 2 passages
Search aliases: λ_k^α lambda alpha
Use in source · Paper_VI_Traditional_Final.tex, line 4320
[Heat and fractional mode transfer] Let λ_k=(kπ)^2. For t≥0, 0≤ e^-tλ_k,h- e^-tλ_k ≤ fract(kπ)^412h^2 e^-tλ_k,h. For 0<α≤1 and fixed k, λ_k,h^α = λ_k^α+O(h^2) and e^-tλ_k,h^α = e^-tλ_k^α+O(th^2) uniformly for t in bounded intervals.
Exact mathematical passage
\lambda_{k,h}^\alpha = \lambda_k^\alpha+O(h^2)
\lambda_k^\betaUse in source · 1 passage
Search aliases: λ_k^β lambda beta
Use in source · Paper_VI_Traditional_Final.tex, line 1149
All statements hold coefficientwise on the mutually orthogonal spectral projectors. For example, L^α L^β =∑_j,kλ_j^αλ_k^β P_jP_k =∑_jλ_j^α+βP_j.
Exact mathematical passage
L^\alpha L^\beta =\sum_{j,k}\lambda_j^\alpha\lambda_k^\beta P_jP_k =\sum_j\lambda_j^{\alpha+\beta}P_j.
\lambda_mUse in source · 10 passages
Search aliases: λ_m lambda
Use in source · Paper_IV_Traditional_Final.tex, line 2220
[Reduced zeros] Let Λ_N be the nonzero zero multiset of Δ_N. All but finitely many members of Λ_N are positive and simple. They can be indexed as λ_m=τ_m^N for all sufficiently large m. If N≥3, the notation may be chosen as in [source reference], and λ_m = t_m^N +O(t_m^N-1e^-γ t_m). For N=2, take τ_m=t_m=π(m+1/2) for m≥0. Then λ_m=t_m^2 exactly.
Exact mathematical passage
\lambda_m=\tau_m^N
\lambda_m^{-1}Use in source · 1 passage
Search aliases: λ_m^-1 lambda
Use in source · atlas-pages.tex, line 391
on ℓ^2(Λ_N) gives the corresponding diagonal Fredholm determinant. Repeated zeros require repeated diagonal entries. vfigproduct headFinite compression is an approximation For N=2, λ_m=π^2(m+1/2)^2. The finite determinant P_L is exactly the displayed finite product, but it is not exactly Δ_2. For 0≤ w≤1 and L≥1, an elementary tail estimate gives 0≤1-fracΔ_2(w)P_L(w) ≤ w∑_m≥ Lλ_m^-1 ≤frac19(L-1/2). The last bound uses π>3 and integral comparison. Thus truncation has an explicit error mechanism. t…
Exact mathematical passage
0\le1-\frac{\Delta_2(w)}{P_L(w)} \le w\sum_{m\ge L}\lambda_m^{-1} \le\frac1{9(L-1/2)}.
\lambda_m^{-s}Use in source · 1 passage
Search aliases: λ_m^-s lambda
Use in source · Paper_IV_Traditional_Final.tex, line 2830
For N≥3, [source reference] writes all but finitely many reduced zeros as λ_m = (A_N(m+frac12))^N(1+ε_m) where λ_m>0, 1+ε_m>0, and ε_m = O(t_m^-1e^-γ t_m). On every compact set of s values, (1+ε_m)^-s-1 = O(|ε_m|) uniformly for large m. The difference λ_m^-s - A_N^-Ns(m+frac12)^-Ns therefore has a normally convergent sum on every compact subset of C. Indeed, the model power has at most polynomial growth in m on a compact set, while |ε_m| decays exponentially. The finitely many exceptional terms …
Exact mathematical passage
\lambda_m^{-s} - A_N^{-Ns}\left(m+\frac12\right)^{-Ns}
\lambda_{\min}Use in source · 1 passage
Search aliases: λ_min lambda min
Use in source · 01_finite_core.tex, line 1131
d_s(τ) is a diagnostic, not a topological invariant. Indeed, finite differentiation gives d_s(τ)=2τ frac∑_jλ_j e^-τλ_j∑_j e^-τλ_j. It tends to zero as τdownarrow0 and, under the stated assumption of a zero mode, as τ→∞. If a general positive operator has no zero mode, its behavior at large times is instead asymptotic to 2τλ_min. The conclusion for operators with a zero mode therefore does not apply to that case. Inferring a geometric dimension requires a stable intermediate window and a controll…
Exact mathematical passage
2\tau\lambda_{\min}
\Lambda_{N,j}Use in source · 3 passages
Search aliases: Λ_N,j Lambda
Use in source · Paper_IV_Traditional_Final.tex, line 2397
[Products for the normalized sections] For every 0≤ j<N, the entire function Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)! has order 1/N. If N≥2 and Λ_N,j is its nonzero zero multiset, then Φ_N,j(w) = ∏_μ∈Λ_N,j (1-frac wμ). For m≥1, put σ_N,j,m = ∑_μ∈Λ_N,jμ^-m. Then C_N,j,n(r) = frac(Nn)!n! B_n( rσ_N,j,1, r1!σ_N,j,2, …, r(n-1)!σ_N,j,n ).
Exact mathematical passage
\Lambda_{N,j}
\Lambda_N^*Use in source · 2 passages
Search aliases: Λ_N^* Lambda
Use in source · 02_strengthened.tex, line 2061
[Fixed time heat trace convergence] For every t>0, ∑_k∈Λ_N^* e^-tλ_N(k) ⟶ ∑_k∈ Z^d e^-4π^2tabsk^2 as N→∞. The first spectral moments also converge: ∑_k∈Λ_N^* λ_N(k) e^-tλ_N(k) ⟶ ∑_k∈ Z^d 4π^2absk^2 e^-4π^2tabsk^2. Consequently, the discrete spectral dimensions converge to the continuum torus spectral dimension at every fixed positive time.
Exact mathematical passage
\sum_{k\in\Lambda_N^*} \e^{-t\lambda_N(k)} \longrightarrow \sum_{k\in\ZZ^d} \e^{-4\pi^2t\abs{k}^2}
\Lambda_pUse in source · 1 passage
Search aliases: Λ_p Lambda
Use in source · visual_atlas.tex, line 361
0 · 2, AB=Bdiag(1,-1), E_+=Bdiag(1,0)B^-1. The index-two eigenlattice explains the denominator in B^-1 and in the projector. Over the localized lattice Λ_p=ZZ_(p)^2, the number 2 is a unit for every odd prime p. Hence the local denominator weight is zero there and one at p=2.
Exact mathematical passage
\Lambda_p=\ZZ_{(p)}^2
\Lambda_{p,L}Use in source · 3 passages
Search aliases: Λ_p,L Lambda
Use in source · 72_recovered.tex, line 1538
For odd p and L≥1, put Λ_p,L=(p-1)p^L-1. For p=2 and L≥3, put Λ_2,L=2^L-2. These are exponents of the unit groups modulo the declared prime powers.
Exact mathematical passage
\Lambda_{p,L}=(p-1)p^{L-1}.
\lambda_{p,r}Use in source · 2 passages
Search aliases: λ_p,r lambda
Use in source · 72_recovered.tex, line 1223
[Unified inverse local transfer] Define λ_p,r = 1, · p odd and r=0, 2^2r-1p^2r, · p odd and r≥1, dfrac2^4r+1(2r+1)(2r+2), · p=2. Then for every prime p and every even n≥0, A_p,n = ∑_r=0^n/2 binom n2r λ_p,r E_n-2r. Equivalently, if L_p(t)=K_p(t)^-1, then H_p(t)=L_p(t)secht, where L_p(t)=cosh^2(pt) for odd p, and L_2(t)=(fracsinh(2t)2t)^2 for p=2.
Exact mathematical passage
\lambda_{p,r} = \begin{cases} 1,&p\text{ odd and }r=0,\\ 2^{2r-1}p^{2r},&p\text{ odd and }r\geq1,\\ \dfrac{2^{4r+1}}{(2r+1)(2r+2)},&p=2. \end{cases}
\Lambda_{\Pcal}Use in source · 4 passages
Search aliases: Λ_Pcal Λ_ P Lambda Pcal Lambda mathcal
Use in source · 04_arithmetic.tex, line 399
[Gluing exact sequence under refinement] Let Q=(Q_β) refine P=(P_α) on V, and let Λ be a lattice. Define Λ_ P = ⊕_α(Λ∩ P_α V), Λ_ Q = ⊕_β(Λ∩ Q_β V). Then Λ_ Q⊆Λ_ P⊆Λ and there is a short exact sequence 0 ⟶ Λ_ P/Λ_ Q ⟶ Λ/Λ_ Q ⟶ Λ/Λ_ P ⟶0. In particular, s_Λ( P)≤ s_Λ( Q).
Exact mathematical passage
\Lambda_{\Pcal} = \bigoplus_\alpha(\Lambda\cap P_\alpha V), \qquad \Lambda_{\Qcal} = \bigoplus_\beta(\Lambda\cap Q_\beta V).
\Lambda_{\Qcal}Use in source · 4 passages
Search aliases: Λ_Qcal Λ_ Q Lambda Qcal Lambda mathcal
Use in source · 04_arithmetic.tex, line 399
[Gluing exact sequence under refinement] Let Q=(Q_β) refine P=(P_α) on V, and let Λ be a lattice. Define Λ_ P = ⊕_α(Λ∩ P_α V), Λ_ Q = ⊕_β(Λ∩ Q_β V). Then Λ_ Q⊆Λ_ P⊆Λ and there is a short exact sequence 0 ⟶ Λ_ P/Λ_ Q ⟶ Λ/Λ_ Q ⟶ Λ/Λ_ P ⟶0. In particular, s_Λ( P)≤ s_Λ( Q).
Exact mathematical passage
\Lambda_{\Pcal} = \bigoplus_\alpha(\Lambda\cap P_\alpha V), \qquad \Lambda_{\Qcal} = \bigoplus_\beta(\Lambda\cap Q_\beta V).
\lambda_rUse in source · 16 passages
Search aliases: λ_r lambda
Use in source · root_scope.tex, line 124
The scalar packet weights have exact valuations λ_r(q)=3q+s_2(r+q)-s_2(r). Put μ_r=min_q≥0λ_r(q), J_r={q:λ_r(q)=μ_r}, G_r=2^-μ_rH_r. Since λ_r(0)=0 and λ_r(q)≥3q+1-s_2(r), every minimizing address satisfies 0≤ q≤lfloorfracs_2(r)-13rfloor. In particular μ_r≤0, and the minimum is found by a finite carry calculation. Reduction of the whole series gives overline G_r(X)=∑_q∈ J_rX^underline q in F_2[X]. Its highest degree is max J_r, with leading coefficient one. Thus the normalization is the actual G…
Exact mathematical passage
\lambda_r(q)=3q+s_2(r+q)-s_2(r).
\lambda_r^{-1/2}Use in source · 1 passage
Search aliases: λ_r^-1/2 lambda
Use in source · 03_supplement_05.tex, line 61
Null directions are removed by proof. If λ_r=0, then B_r^*B_r=0. Positivity of the ordinary complex matrix inner product gives B_r=0. This is the checked declaration pathDiagonalErrors.zero_weight_error. The normalized syndrome operators are defined only on A_+={r:λ_r>0}, V_r=λ_r^-1/2B_r. Consequently no inverse square root at a null eigenvalue is needed. Lean proves V_rP=V_r, V_r^*V_s=δ_rsP. The recovery from the previous block is instantiated from these derived syndromes, rather than from an a…
Exact mathematical passage
A_+=\{r:\lambda_r>0\},\qquad V_r=\lambda_r^{-1/2}B_r.
\Lambda_sUse in source · 4 passages
Search aliases: Λ_s Lambda
Use in source · 04_arithmetic.tex, line 509
[The same rational split with two lattices] Let V=Ke_1⊕ Ke_2 and let E project onto Ke_1. On the split lattice Λ_0= O e_1⊕ O e_2, the weight is zero. Fix s≥1 and consider Λ_s = O e_1+ O(π^-se_1+e_2). Then Λ_s∩ Ke_1= O e_1, Λ_s∩ Ke_2=π^s O e_2. The gluing module is isomorphic to O/π^s O, so s_Λ_s(E)=s. The rational decomposition has not changed. Only the lattice has changed.
Exact mathematical passage
\Lambda_s = \OO e_1+ \OO(\pi^{-s}e_1+e_2).
\Lambda_{y'}^*Use in source · 1 passage
Search aliases: Λ_y'^* Lambda
Use in source · Paper_V_Traditional_Final.tex, line 955
da_y^* (J_y(Λ)-J_y(Λ_y^*)) is attained and strictly positive. If the pruning set is a singleton, its only element minimizes every criterion. No real-valued minimum over an empty competitor set is introduced. item In the case with competitors, if new data y' satisfy sup_Λ∈ P( T) |J_y'(Λ)-J_y(Λ)| <fracΓ_y2, then Λ_y'^*=Λ_y^*. item Every selected minimizer preserves the exact synthesis identity of the preceding theorem.
Exact mathematical passage
\Lambda_{y'}^*=\Lambda_y^*
\Lambda_y^*Use in source · 5 passages
Search aliases: Λ_y^* Lambda
Use in source · Paper_V_Traditional_Final.tex, line 937
[label=(roman*)] item A global minimizer Λ_y^*∈*argmin_Λ∈ P( T)J_y(Λ) always exists. item If the minimizer is unique and at least one competing pruning exists, its gap Γ_y = min_Λ≠Λ_y^* (J_y(Λ)-J_y(Λ_y^*)) is attained and strictly positive. If the pruning set is a singleton, its only element minimizes every criterion. No real-valued minimum over an empty competitor set is introduced. item In the case with competitors, if new data y' satisfy sup_Λ∈ P( T) |J_y'(Λ)-J_y(Λ)| <fracΓ_y2, then Λ_y'^*=Λ_…
Exact mathematical passage
\Lambda_y^*\in\operatorname*{argmin}_{\Lambda\in\mathfrak P(\mathcal T)}J_y(\Lambda)
\mu^{-m}Use in source · 1 passage
Search aliases: μ^-m mu
Use in source · Paper_IV_Traditional_Final.tex, line 2405
[Products for the normalized sections] For every 0≤ j<N, the entire function Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)! has order 1/N. If N≥2 and Λ_N,j is its nonzero zero multiset, then Φ_N,j(w) = ∏_μ∈Λ_N,j (1-frac wμ). For m≥1, put σ_N,j,m = ∑_μ∈Λ_N,jμ^-m. Then C_N,j,n(r) = frac(Nn)!n! B_n( rσ_N,j,1, r1!σ_N,j,2, …, r(n-1)!σ_N,j,n ).
Exact mathematical passage
\sigma_{N,j,m} = \sum_{\mu\in\Lambda_{N,j}}\mu^{-m}.
\mu_0Use in source · 1 passage
Search aliases: μ_0 mu
Use in source · strengthening.tex, line 614
odd: T(m,1)=2^m-1 and v_2(m!)=m-1. For m≥4 a power of two, the r=3 term is even. More explicitly, fracT(m,3)2^m-3=h_m-3(1,3,6), ∑_d≥0h_d(1,3,6)t^d =frac1(1-t)(1-3t)(1-6t). Reduction modulo two gives (1-t)^-2, whose degree m-3 coefficient has parity m-2=0. At m=1,2 the r=3 term is absent, and the r=1 term is again odd. Finally μ_0=1.
Exact mathematical passage
\mu_0=1
\mu_1Use in source · 1 passage
Search aliases: μ_1 mu
Use in source · Paper_VI_Traditional_Final.tex, line 4645
[Polynomial exponential memory of a finite hidden system] Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt ∑_k=0^ν_j-1 fract^kk!B_j,k and Σ_P(z) = ∑_j=1^r ∑_k=0^ν_j-1 fracB_j,k(z-μ_j)^k+1. Suppose Reμ_j<0 for every j. If J⊆{1,…,r}…
Exact mathematical passage
\mu_1,\ldots,\mu_r
\mu_2Use in source · 4 passages
Search aliases: μ_2 mu
Use in source · Paper_XI_Traditional_Final.tex, line 2128
-valued. With H_0=kerλ, the quotient is P/H_0≃ F_2^2, and Ind_J^Pθ is an irreducible two-dimensional Heisenberg carrier ([source reference]). At rank four, the actual outer quadratic centralizer gives P=J U_B^m, [P:J]=2, and P/H_0≃ F_2^4, with carrier Ind_J^Pη of dimension four ([source reference]). The finite scalar center can be larger than μ_2. No literal extraspecial quotient is claimed.
Exact mathematical passage
\mu_2
\mu_aUse in source · 3 passages
Search aliases: μ_a mu
Use in source · strengthening.tex, line 905
this reduces to γ_k(X)+ϵ Gγ_k-1(X). Moreover, G≡γ_2(X)+τ Qpmod2. After evaluation, every term with a single τ vanishes, so the second contribution reduces to L((γ_2(X)+Q)γ_k-1(X)). The γ_2(X) product has scalar binomk+12, which is even because 4mid k. The other contribution expands as L(Qγ_k-1(P+Q)) =∑_a=0^k-1(k-a)μ_aμ_k-a. An odd summand would require k-a to be odd and μ_k-a to be odd. By Lemma~[source reference], this requires k-a=1. Then a=k-1 is not a power of two and μ_a is even. Hence the …
Exact mathematical passage
\mathcal L(Q\gamma_{k-1}(P+Q)) =\sum_{a=0}^{k-1}(k-a)\mu_a\mu_{k-a}. \label{eq:two-power-Q-evaluation}
\mu_{\frakp}Use in source · 2 passages
Search aliases: μ_frakp μ_ p mu frakp mu mathfrak
Use in source · Paper_XII_Traditional_Final.tex, line 629
[Higher Fitting strata] Let R be Noetherian and let M be finite. For r≥1, put μ_ p(M) = dim_κ( p) (M⊗_Rκ( p)). Then { p∈Spec R: μ_ p(M)≥ r } = V(Fitt_r-1^R(M)).
Exact mathematical passage
\mu_{\frakp}(M) = \dim_{\kappa(\frakp)} \left(M\otimes_R\kappa(\frakp)\right).
\mu_hUse in source · 2 passages
Search aliases: μ_h mu
Use in source · 72_recovered.tex, line 843
Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding term vanishes for positive n. Otherwise ((1+ζ y)^m)^h = (1+ζ y)^p-1 =1. Every summand is therefore h periodic for positive n. Thus T_m,p preserves V_h.…
Exact mathematical passage
d_q=\sum_{\chi\in\mu_h}c_\chi\chi^q.
\mu_iUse in source · 3 passages
Search aliases: μ_i mu
Use in source · Paper_VI_Traditional_Final.tex, line 753
The first normalization appears in both numerator and denominator of the second update and cancels: fracμ_i(λ_ip_i/Z_λ)∑_jμ_jλ_jp_j/Z_λ = fracμ_iλ_ip_i∑_jμ_jλ_jp_j.
Exact mathematical passage
\frac{\mu_i(\lambda_ip_i/Z_\lambda)}{\sum_j\mu_j\lambda_jp_j/Z_\lambda} = \frac{\mu_i\lambda_ip_i}{\sum_j\mu_j\lambda_jp_j}.
\mu_jUse in source · 19 passages
Search aliases: μ_j mu
Use in source · Paper_VI_Traditional_Final.tex, line 753
The first normalization appears in both numerator and denominator of the second update and cancels: fracμ_i(λ_ip_i/Z_λ)∑_jμ_jλ_jp_j/Z_λ = fracμ_iλ_ip_i∑_jμ_jλ_jp_j.
Exact mathematical passage
\frac{\mu_i(\lambda_ip_i/Z_\lambda)}{\sum_j\mu_j\lambda_jp_j/Z_\lambda} = \frac{\mu_i\lambda_ip_i}{\sum_j\mu_j\lambda_jp_j}.
\mu_{k-a}Use in source · 3 passages
Search aliases: μ_k-a mu
Use in source · strengthening.tex, line 905
this reduces to γ_k(X)+ϵ Gγ_k-1(X). Moreover, G≡γ_2(X)+τ Qpmod2. After evaluation, every term with a single τ vanishes, so the second contribution reduces to L((γ_2(X)+Q)γ_k-1(X)). The γ_2(X) product has scalar binomk+12, which is even because 4mid k. The other contribution expands as L(Qγ_k-1(P+Q)) =∑_a=0^k-1(k-a)μ_aμ_k-a. An odd summand would require k-a to be odd and μ_k-a to be odd. By Lemma~[source reference], this requires k-a=1. Then a=k-1 is not a power of two and μ_a is even. Hence the …
Exact mathematical passage
\mathcal L(Q\gamma_{k-1}(P+Q)) =\sum_{a=0}^{k-1}(k-a)\mu_a\mu_{k-a}. \label{eq:two-power-Q-evaluation}
\mu_mUse in source · 5 passages
Search aliases: μ_m mu
Use in source · 72_recovered.tex, line 847
Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding term vanishes for positive n. Otherwise ((1+ζ y)^m)^h = (1+ζ y)^p-1 =1. Every summand is therefore h periodic for positive n. Thus T_m,p preserves V_h.…
Exact mathematical passage
T_{m,p}(\chi^\bullet)_n = \frac1m \sum_{\zeta\in\mu_m} (1+\zeta y)^{mn}.
\mu_pUse in source · 11 passages
Search aliases: μ_p mu
Use in source · 02_foundations.tex, line 896
[Seed rank for a single operator] Let X have finite dimension over 𝕜 and let T∈End_𝕜(X). Regard X as a module over 𝕜[t] by t· x=Tx. For every monic irreducible p, let X_p be the primary component for p and set μ_p(T):= dim_𝕜[t]/(p)(X_p/p(T)X_p). Then srank_𝕜[T](X)=max_pμ_p(T). Moreover, μ_p(T) is the number of cyclic primary summands for p in the decomposition into elementary divisors of X_p. DependsnolinkurlPSC-I-CI-PID-PRIMARY-DECOMPOSITION, nolinkurlPSC-I-C-LOCAL-ALGEBRA-FORMULA, and nolinkur…
Exact mathematical passage
\mu_p(T):= \dim_{\bbk[t]/(p)}\bigl(X_p/p(T)X_p\bigr).
\mu_rUse in source · 5 passages
Search aliases: μ_r mu
Use in source · root_scope.tex, line 128
The scalar packet weights have exact valuations λ_r(q)=3q+s_2(r+q)-s_2(r). Put μ_r=min_q≥0λ_r(q), J_r={q:λ_r(q)=μ_r}, G_r=2^-μ_rH_r. Since λ_r(0)=0 and λ_r(q)≥3q+1-s_2(r), every minimizing address satisfies 0≤ q≤lfloorfracs_2(r)-13rfloor. In particular μ_r≤0, and the minimum is found by a finite carry calculation. Reduction of the whole series gives overline G_r(X)=∑_q∈ J_rX^underline q in F_2[X]. Its highest degree is max J_r, with leading coefficient one. Thus the normalization is the actual G…
Exact mathematical passage
\mu_r=\min_{q\ge0}\lambda_r(q),\qquad J_r=\{q:\lambda_r(q)=\mu_r\},\qquad G_r=2^{-\mu_r}H_r.
\nu_2Use in source · 1 passage
Search aliases: ν_2 nu
Use in source · 20_arithmetic.tex, line 472
[All-row dyadic higher differences] For 0≤ j<4 and all u,k,n∈ N, C_j(n) · ∈ Z_2, Δ_u^kC_j(n) · ∈ 2^k(1+ν_2(u)) Z_2, Δ_u^kU_j(n) · ∈ 2^k(1+ν_2(u))+v_2(j!) Z_2. There is no restriction k≤ n and no minimum with the row n.
Exact mathematical passage
C_j(n)&\in\Z_2,\\ \Delta_u^kC_j(n)&\in 2^{k(1+\nu_2(u))}\Z_2,\label{eq:arith-allrow-c}\\ \Delta_u^kU_j(n)&\in 2^{k(1+\nu_2(u))+v_2(j!)}\Z_2.\label{eq:arith-allrow-u}
\nu_jUse in source · 12 passages
Search aliases: ν_j nu
Use in source · Paper_VI_Traditional_Final.tex, line 4649
[Polynomial exponential memory of a finite hidden system] Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt ∑_k=0^ν_j-1 fract^kk!B_j,k and Σ_P(z) = ∑_j=1^r ∑_k=0^ν_j-1 fracB_j,k(z-μ_j)^k+1. Suppose Reμ_j<0 for every j. If J⊆{1,…,r}…
Exact mathematical passage
\nu_j
\XiUse in source · 1 passage
Search aliases: Ξ Xi
Use in source · Paper_V_Traditional_Final.tex, line 962
set, so the feasible set is nonempty and finite. Thus the minimum is attained. If competitors exist, their finitely many differences from a unique optimum are strictly positive, so their minimum is positive and is achieved by a competitor. In the singleton case there is no alternative selector under any objective change. In the remaining case, for any competing pruning Λ≠Λ_y^*, J_y'(Λ)-J_y'(Λ_y^*) · ≥ J_y(Λ)-J_y(Λ_y^*) -2sup_Ξ|J_y'(Ξ)-J_y(Ξ)| · >Γ_y-Γ_y=0. Hence the same pruning remains uniquely…
Exact mathematical passage
\begin{aligned} J_{y'}(\Lambda)-J_{y'}(\Lambda_y^*) &\ge J_y(\Lambda)-J_y(\Lambda_y^*) -2\sup_{\Xi}|J_{y'}(\Xi)-J_y(\Xi)|\\ &>\Gamma_y-\Gamma_y=0. \end{aligned}
\xi^{-1}Use in source · 3 passages
Search aliases: ξ^-1 xi
Use in source · Paper_XI_Traditional_Final.tex, line 361
[Dual and self dual tame packets] For a tame atom τ_ξ=Ind_W_K_f^W_Kξ, we have τ_ξ^vee ≅ Ind_W_K_f^W_Kξ^-1. The atom is self dual exactly when ξ^-1=ξ^Φ^j for some j∈ Z/f Z.
Exact mathematical passage
\tau_\xi^\vee \cong \Ind_{W_{K_f}}^{W_K}\xi^{-1}.
\xi^2Use in source · 2 passages
Search aliases: ξ^2 xi
Use in source · atlas-pages.tex, line 103
vpage6Addition is quotient multiplicationFINITE ALGEBRAChapter IV: Proposition 5.5, Theorem 6.2. Work in C[ξ]/(ξ^3+1). Every element has coordinates in 1,ξ,ξ^2. The entire exponential has the finite-coordinate form e^zξ=D_3,0(z)+D_3,1(z)ξ+D_3,2(z)ξ^2. The multiplication table makes the carry sign unavoidable. vfigsigned_addition Multiply the coordinate forms of e^zξ and e^wξ. Their product is e^(z+w)ξ. The coefficient of 1 receives one contribution without a wrap and two contributions with a wra…
Exact mathematical passage
1,\xi,\xi^2
\xi^3Use in source · 2 passages
Search aliases: ξ^3 xi
Use in source · atlas-pages.tex, line 103
vpage6Addition is quotient multiplicationFINITE ALGEBRAChapter IV: Proposition 5.5, Theorem 6.2. Work in C[ξ]/(ξ^3+1). Every element has coordinates in 1,ξ,ξ^2. The entire exponential has the finite-coordinate form e^zξ=D_3,0(z)+D_3,1(z)ξ+D_3,2(z)ξ^2. The multiplication table makes the carry sign unavoidable. vfigsigned_addition Multiply the coordinate forms of e^zξ and e^wξ. Their product is e^(z+w)ξ. The coefficient of 1 receives one contribution without a wrap and two contributions with a wra…
Exact mathematical passage
\CC[\xi]/(\xi^3+1)
\xi^jUse in source · 4 passages
Search aliases: ξ^j xi
Use in source · Paper_IV_Traditional_Final.tex, line 799
[Finite algebra exponential identities] In A_N^+ one has e^λ zξ = ∑_j=0^N-1 E_N,j(λ z)ξ^j. In A_N^- one has e^zξ = ∑_j=0^N-1 D_N,j(z)ξ^j. Differentiation and multiplication of these identities recover the cyclic derivative systems and the two addition laws.
Exact mathematical passage
e^{\lambda z\xi} = \sum_{j=0}^{N-1}\Efun_{N,j}(\lambda z)\xi^j.
\xi^{N-1}Use in source · 1 passage
Search aliases: ξ^N-1 xi
Use in source · Paper_IV_Traditional_Final.tex, line 820
Differentiation multiplies by λξ in the first algebra and by ξ in the second. Reduction of the product by the defining relation gives the derivative cycles. The identity e^(z+w)ξ=e^zξe^wξ gives the addition laws after comparison of the coefficients of 1,ξ,…,ξ^N-1.
Exact mathematical passage
1,\xi,\ldots,\xi^{N-1}
\xi^{Nn+j}Use in source · 2 passages
Search aliases: ξ^Nn+j xi
Use in source · Paper_IV_Traditional_Final.tex, line 814
Expand the exponential and write every exponent uniquely as Nn+j. In A_N^+, the relation ξ^Nn+j=ξ^j gives the transfer sections. In A_N^-, the relation ξ^Nn+j=(-1)^nξ^j gives the signed sections.
Exact mathematical passage
\xi^{Nn+j}=\xi^j
\xi^{\Phi^j}Use in source · 1 passage
Search aliases: ξ^Φ^j xi Phi
Use in source · Paper_XI_Traditional_Final.tex, line 367
[Dual and self dual tame packets] For a tame atom τ_ξ=Ind_W_K_f^W_Kξ, we have τ_ξ^vee ≅ Ind_W_K_f^W_Kξ^-1. The atom is self dual exactly when ξ^-1=ξ^Φ^j for some j∈ Z/f Z.
Exact mathematical passage
\xi^{-1}=\xi^{\Phi^j}
\xi_{L,n}Use in source · 2 passages
Search aliases: ξ_L,n xi
Use in source · Paper_XI_Traditional_Final.tex, line 565
[Greatest common divisor and least common multiple packets] There is a decomposition τ_ξ⊗τ_η ≅ ⊕_i=0^d-1 Ind_W_K_L^W_K ( ξ_L,nη_L,m^Φ^i ), where ξ_L,n and η_L,m are the norm lifts to K_L.
Exact mathematical passage
\boxed{ \tau_\xi\otimes\tau_\eta \cong \bigoplus_{i=0}^{d-1} \Ind_{W_{K_L}}^{W_K} \left( \xi_{L,n}\eta_{L,m}^{\Phi^i} \right), }
\pi^{-2s}Use in source · 2 passages
Search aliases: π^-2s pi
Use in source · Paper_IV_Traditional_Final.tex, line 2860
[Exact zeta models in orders two and four] For N=2, Z_2^red(s) = π^-2sζ(2s,frac12) and Z_2^abs(s) = 2π^-sζ(s,frac12). For N=4, put A_4=sqrt2π. Then Z_4^red(s) = A_4^-4sζ(4s,frac12) and Z_4^abs(s) = 4A_4^-sζ(s,frac12). In these two cases the entire correction terms in [source reference] vanish identically.
Exact mathematical passage
\mathfrak Z_2^{\mathrm{red}}(s) = \pi^{-2s}\zeta\left(2s,\frac12\right)
\pi^{-s}Use in source · 3 passages
Search aliases: π^-s pi
Use in source · 04_arithmetic.tex, line 509
[The same rational split with two lattices] Let V=Ke_1⊕ Ke_2 and let E project onto Ke_1. On the split lattice Λ_0= O e_1⊕ O e_2, the weight is zero. Fix s≥1 and consider Λ_s = O e_1+ O(π^-se_1+e_2). Then Λ_s∩ Ke_1= O e_1, Λ_s∩ Ke_2=π^s O e_2. The gluing module is isomorphic to O/π^s O, so s_Λ_s(E)=s. The rational decomposition has not changed. Only the lattice has changed.
Exact mathematical passage
\Lambda_s = \OO e_1+ \OO(\pi^{-s}e_1+e_2).
\pi^2Use in source · 18 passages
Search aliases: π^2 pi
Use in source · Paper_IV_Traditional_Final.tex, line 2291
ct uses every reduced zero with its multiplicity. Its logarithm turns multiplication into addition, so reciprocal-zero power sums become coefficients. This step depends on the global product theorem. A truncated zero list provides a numerical partial sum, while the identity uses the entire multiset. modelfigurezero_momentsOrder-two moment sums over the first forty reduced zeros λ_k=π^2(k+1/2)^2. The curves are finite numerical partial sums. The dashed limits 1/2 and 1/6 follow from the exact log…
Exact mathematical passage
\lambda_k=\pi^2(k+1/2)^2
\varpi^2Use in source · 1 passage
Search aliases: ϖ^2 varpi
Use in source · Paper_X_Block27_Amended.tex, line 304
[Smooth formal branch count] If R_bar x≅ O[[t_1,…,t_d]] as complete local O-algebras, then #X_m(bar x)=q^d(m-1). Fix a uniformizer and a section s:k→ O of reduction, with s(0)=0. Every O-valued branch is encoded by d positive-depth digit streams t_i=s(a_i,1)ϖ+s(a_i,2)ϖ^2+⋯, a_i,j∈ k,
Exact mathematical passage
t_i=s(a_{i,1})\varpi+s(a_{i,2})\varpi^2+\cdots, \qquad a_{i,j}\in k,
\pi^{2q}Use in source · 1 passage
Search aliases: π^2q pi
Use in source · Paper_IV_Traditional_Final.tex, line 2605
[The cosine orbit sums] For N=2, Δ_2(w)=cossqrt w. Its zeros are λ_m=π^2(m+frac12)^2, m≥0. Hence σ_2,q = frac2^2q-1π^2qζ(2q). The identity in [source reference] recovers the classical tangent coefficients from these zero power sums.
Exact mathematical passage
\sigma_{2,q} = \frac{2^{2q}-1}{\pi^{2q}}\zeta(2q).
\pi^4Use in source · 3 passages
Search aliases: π^4 pi
Use in source · atlas-pages.tex, line 148
st four-point orbit has fourth power 1/16. The second has fourth power 81/16. headMultiplicity is not multiplied in the reduced function At a nonzero point, z↦ z^4 has nonzero derivative. It preserves the local multiplicity of a zero. Four distinct simple zeros of D_4 therefore correspond to one simple zero of Δ_4, not one zero of multiplicity four. The reduced zeros are λ_m=A_4^4(m+1/2)^4=4π^4(m+1/2)^4. This is why an orbit factor in the z plane is 1-z^4/λ_m. It is the product of the four corre…
Exact mathematical passage
\lambda_m=A_4^4(m+1/2)^4=4\pi^4(m+1/2)^4.
\pi^aUse in source · 2 passages
Search aliases: π^a pi
Use in source · 03_weighted.tex, line 211
[A triangular transfer] Let O be complete with uniformizer π, and consider T= 1 · π^a 0 · 1 with a≥1. The only nontrivial transfer has weight a, and its square is zero. Hence T^-1 = 1 · -π^a
Exact mathematical passage
T= \begin{pmatrix} 1&\pi^a\\ 0&1 \end{pmatrix}
\pi^{a_1}Use in source · 1 passage
Search aliases: π^a_1 pi
Use in source · 04_arithmetic.tex, line 729
[Computing a packet weight from an inclusion matrix] Choose an O basis of Λ and an O basis of Λ_split. Let B be the matrix of the inclusion Λ_split↪Λ. If the Smith form of B is diag(π^a_1,…,π^a_n), 0≤ a_1≤⋯≤ a_n, then G_Λ, P ≅ ⊕_j=1^n O/π^a_j O, s_Λ( P)=max({0}∪{a_j:1≤ j≤ n}).
Exact mathematical passage
\diag(\pi^{a_1},\ldots,\pi^{a_n}), \qquad 0\leq a_1\leq\cdots\leq a_n,
\pi^{a_j}Use in source · 3 passages
Search aliases: π^a_j pi
Use in source · 04_arithmetic.tex, line 284
ght] With the notation of [source reference], the map Φ:G_Λ, P ⟶ ⊕_α∈ IE_αΛ/Λ_α, [x]⟼(E_α x+Λ_α)_α is injective, and every coordinate projection of its image is surjective. We also have exp_π(G_Λ, P) = s_Λ( P). If G_Λ, P ≅ ⊕_j=1^r O/π^a_j O with a_1≤⋯≤ a_r, then s_Λ( P)=max({0}∪{a_j:1≤ j≤ r}).
Exact mathematical passage
G_{\Lambda,\Pcal} \cong \bigoplus_{j=1}^r\OO/\pi^{a_j}\OO
\pi^{a_n}Use in source · 1 passage
Search aliases: π^a_n pi
Use in source · 04_arithmetic.tex, line 729
[Computing a packet weight from an inclusion matrix] Choose an O basis of Λ and an O basis of Λ_split. Let B be the matrix of the inclusion Λ_split↪Λ. If the Smith form of B is diag(π^a_1,…,π^a_n), 0≤ a_1≤⋯≤ a_n, then G_Λ, P ≅ ⊕_j=1^n O/π^a_j O, s_Λ( P)=max({0}∪{a_j:1≤ j≤ n}).
Exact mathematical passage
\diag(\pi^{a_1},\ldots,\pi^{a_n}), \qquad 0\leq a_1\leq\cdots\leq a_n,
\pi^cUse in source · 2 passages
Search aliases: π^c pi
Use in source · Paper_X_Block27_Amended.tex, line 1013
For a second sector let R be a DVR with uniformizer π, fraction field K and matrix rank n. If a fixed template τ has full K-linear action span, finite generation and denominator clearing give a fixed c such that π^c M_n(R)⊆span_R{τ(g):g∈ G}. Normalize a field comparison to an integral P with a unit entry. The displayed span constructs an integral B with BP=π^cI, so det Pmidπ^cn, det Pnot≡0pmodπ^cn+1. One c works for all integral representations in the fixed field-isomorphism class. Jacobson dens…
Exact mathematical passage
\pi^c M_n(R)\subseteq\operatorname{span}_R\{\tau(g):g\in G\}.
\pi^{cn}Use in source · 1 passage
Search aliases: π^cn pi
Use in source · Paper_X_Block27_Amended.tex, line 1018
R with uniformizer π, fraction field K and matrix rank n. If a fixed template τ has full K-linear action span, finite generation and denominator clearing give a fixed c such that π^c M_n(R)⊆span_R{τ(g):g∈ G}. Normalize a field comparison to an integral P with a unit entry. The displayed span constructs an integral B with BP=π^cI, so det Pmidπ^cn, det Pnot≡0pmodπ^cn+1. One c works for all integral representations in the fixed field-isomorphism class. Jacobson density derives full span from actual…
Exact mathematical passage
\det P\mid\pi^{cn},\qquad \det P\not\equiv0\pmod{\pi^{cn+1}}.
\pi^{cn+1}Use in source · 1 passage
Search aliases: π^cn+1 pi
Use in source · Paper_X_Block27_Amended.tex, line 1018
R with uniformizer π, fraction field K and matrix rank n. If a fixed template τ has full K-linear action span, finite generation and denominator clearing give a fixed c such that π^c M_n(R)⊆span_R{τ(g):g∈ G}. Normalize a field comparison to an integral P with a unit entry. The displayed span constructs an integral B with BP=π^cI, so det Pmidπ^cn, det Pnot≡0pmodπ^cn+1. One c works for all integral representations in the fixed field-isomorphism class. Jacobson density derives full span from actual…
Exact mathematical passage
\det P\mid\pi^{cn},\qquad \det P\not\equiv0\pmod{\pi^{cn+1}}.
\pi^dUse in source · 5 passages
Search aliases: π^d pi
Use in source · 04_arithmetic.tex, line 315
Let s=s_Λ( P). For every x∈Λ and every α, π^sE_α x∈Λ_α. Hence π^sx=∑_απ^sE_α x∈Λ_split, so π^sG_Λ, P=0. This proves exp_π(G_Λ, P)≤ s. If d<s, then for some α there is an x∈Λ with π^dE_α x∉Λ_α. The class π^d[x] is therefore nonzero in the gluing module. Thus π^d does not annihilate the gluing module, and equality follows. The final formula is the usual description of the exponent through Smith normal form.
Exact mathematical passage
\pi^dE_\alpha x\notin\Lambda_\alpha.
\pi^eUse in source · 3 passages
Search aliases: π^e pi
Use in source · Paper_X_Block27_Amended.tex, line 2043
Finite-group comparison with precision loss. Let R be a commutative domain, Γ a finite group, and ρ,τ actual integral matrix representations. Suppose a=|Γ|≠0 in R and amidπ^e. An equivariant matrix modulo (π)^e+k lifts to an integral equivariant matrix with the prescribed reduction modulo (π)^k. Indeed, lift its entries to A and form S=∑_g∈Γρ(g)Aτ(g^-1). The finite equations imply S-aA=π^e+kE with E integral. Choose b with π^e=ab and put P=A+bπ^kE. Then aP=S, so cancellation of a≠0 proves equiva…
Exact mathematical passage
a\mid\pi^e
\pi^{e+k}Use in source · 1 passage
Search aliases: π^e+k pi
Use in source · Paper_X_Block27_Amended.tex, line 2049
a commutative domain, Γ a finite group, and ρ,τ actual integral matrix representations. Suppose a=|Γ|≠0 in R and amidπ^e. An equivariant matrix modulo (π)^e+k lifts to an integral equivariant matrix with the prescribed reduction modulo (π)^k. Indeed, lift its entries to A and form S=∑_g∈Γρ(g)Aτ(g^-1). The finite equations imply S-aA=π^e+kE with E integral. Choose b with π^e=ab and put P=A+bπ^kE. Then aP=S, so cancellation of a≠0 proves equivariance. Over a local ring, if π is a nonunit, k>0 and …
Exact mathematical passage
S-aA=\pi^{e+k}E
\varpi^{em}Use in source · 1 passage
Search aliases: ϖ^em varpi
Use in source · Paper_X_Block27_Amended.tex, line 631
Counting exponent versus compatible branches. The preceding exponent counts all finite-level solutions, including nodes which cannot lift indefinitely. It is not the cardinality or dimension of the space of infinite branches. In fact every compatible tower of solutions to x^e=0, for e≥1, is the zero tower. For each m, a solution at depth em lies in ϖ^m O/ϖ^em. Its reduction at depth m is therefore zero. Compatibility proves the assertion at every depth. The formal companion identifies proofs of …
Exact mathematical passage
\varpi^m\Ocal/\varpi^{em}
\varpi^jUse in source · 1 passage
Search aliases: ϖ^j varpi
Use in source · Paper_X_Block27_Amended.tex, line 316
For the digit claim, subtract the chosen representative of the residue and divide the remainder by the uniformizer. Iterating gives unique digits. Conversely, the partial sums of any digit stream form a compatible Cauchy family and reconstruct uniquely by completeness and separatedness. More precisely, for each parameter there is a unique stream (a_i,j)_j≥1 such that ϖ^n+1mid t_i-∑_j=1^ns(a_i,j)ϖ^j (n≥0). The section is a set-theoretic section of reduction, not an asserted embedding of residue f…
Exact mathematical passage
\varpi^{n+1}\mid t_i-\sum_{j=1}^{n}s(a_{i,j})\varpi^j\qquad(n\ge0).
\pi^kUse in source · 1 passage
Search aliases: π^k pi
Use in source · Paper_X_Block27_Amended.tex, line 2050
tegral matrix representations. Suppose a=|Γ|≠0 in R and amidπ^e. An equivariant matrix modulo (π)^e+k lifts to an integral equivariant matrix with the prescribed reduction modulo (π)^k. Indeed, lift its entries to A and form S=∑_g∈Γρ(g)Aτ(g^-1). The finite equations imply S-aA=π^e+kE with E integral. Choose b with π^e=ab and put P=A+bπ^kE. Then aP=S, so cancellation of a≠0 proves equivariance. Over a local ring, if π is a nonunit, k>0 and the supplied comparison is invertible, P is an integral u…
Exact mathematical passage
P=A+b\pi^kE
\pi^LUse in source · 8 passages
Search aliases: π^L pi
Use in source · 03_weighted.tex, line 29
If O is a complete discrete valuation ring with uniformizer π and B is a separated and complete O algebra for the filtration F^L B=π^L B, then this is a standard example. Finite matrix algebras over O satisfy these conditions. Completeness of O alone does not imply completeness of every O algebra. Basic facts about completeness in this setting are reviewed in [source reference].
Exact mathematical passage
F^L\Bcal=\pi^L\Bcal,
\varpi^{\lceil m/e\rceil}Use in source · 1 passage
Search aliases: ϖ^lceil m/erceil varpi lceil rceil
Use in source · Paper_X_Block27_Amended.tex, line 607
The congruence is equivalent to e v_ϖ(x)≥ m, so x must lie in ϖ^lceil m/erceil O/ϖ^m, which has the stated cardinality. Write m=ek+r, 1≤ r≤ e. The exponent becomes (e-1)k+r-1. Summing over k≥0 and 1≤ r≤ e gives the rational expression.
Exact mathematical passage
\varpi^{\lceil m/e\rceil}\Ocal/\varpi^m
\pi^mUse in source · 8 passages
Search aliases: π^m pi
Use in source · 03_weighted.tex, line 237
Let O be a complete discrete valuation ring with uniformizer π, fraction field K, and filtration F^m B=π^m B on an O-algebra B. The finite quotient statements below do not require B itself to be complete. Suppose T=D+H, where D is invertible and the perturbation H raises filtration. The symbol K is reserved for the fraction field.
Exact mathematical passage
F^m\cB=\pi^m\cB
\pi^{m+d}Use in source · 3 passages
Search aliases: π^m+d pi
Use in source · 04_arithmetic.tex, line 460
n delay] Let E be a rational projector on V, let Λ be a lattice, and let s=s_Λ(E). For every m≥0, the rule [x]_m+s⟼[Ex]_m from Λ/π^m+sΛ to V/π^mΛ is well defined. Here m is an integer. If d is an integer with 0≤ d<s, then for every integer m≥0 there are x,y∈Λ such that x≡ ypmodπ^m+dΛ but Exnot≡ Eypmodπ^mΛ. Thus s extra digits are both sufficient and necessary for a uniform precision guarantee.
Exact mathematical passage
x\equiv y\pmod{\pi^{m+d}\Lambda}
\pi^{m+s}Use in source · 5 passages
Search aliases: π^m+s pi
Use in source · 04_arithmetic.tex, line 209
[Projector precision delay] Let E be a rational projector with π^sE(Λ)⊆Λ. If x≡ ypmodπ^m+sΛ, then Ex≡ Eypmodπ^mΛ. Thus m+s input digits suffice to determine m output digits of the projected packet.
Exact mathematical passage
x\equiv y\pmod{\pi^{m+s}\Lambda},
\pi^{\min(s_i,L)}Use in source · 2 passages
Search aliases: π^min(s_i,L) pi min
Use in source · 04_arithmetic.tex, line 758
[Smith weighted congruence solve] Let M∈ M_m,n( O) have Smith exponents s_1,…,s_r. Assume the residue field is finite of cardinality q, fix L≥1, and consider Mx≡ bpmodπ^L. Put widetilde b=Ub for a Smith matrix U as in [source reference]. The congruence is solvable if and only if widetilde b_i≡0 pmodπ^min(s_i,L) (1≤ i≤ r) and widetilde b_i≡0 pmodπ^L (r<i≤ m). When it is solvable, the number of solutions in ( O/π^L O)^n is q^(n-r)L+∑_i=1^rmin(s_i,L).
Exact mathematical passage
\widetilde b_i\equiv0 \pmod{\pi^{\min(s_i,L)}} \qquad(1\leq i\leq r)
\pi^nUse in source · 2 passages
Search aliases: π^n pi
Use in source · Paper_X_Block27_Amended.tex, line 981
0 · 1, every integral intertwiner with ρ_1 has determinant aP_22^2. Taking a=π^n for a nonzero π∈ I rules out every fixed detecting depth, even though all the field-valued approximants are isomorphic and their integral limit is trivial. This does not refute Claim 8.2, which already assumes inverse-limit closure of its chosen local conditions.
Exact mathematical passage
a=\pi^n
\varpi^{n+1}Use in source · 1 passage
Search aliases: ϖ^n+1 varpi
Use in source · Paper_X_Block27_Amended.tex, line 316
For the digit claim, subtract the chosen representative of the residue and divide the remainder by the uniformizer. Iterating gives unique digits. Conversely, the partial sums of any digit stream form a compatible Cauchy family and reconstruct uniquely by completeness and separatedness. More precisely, for each parameter there is a unique stream (a_i,j)_j≥1 such that ϖ^n+1mid t_i-∑_j=1^ns(a_i,j)ϖ^j (n≥0). The section is a set-theoretic section of reduction, not an asserted embedding of residue f…
Exact mathematical passage
\varpi^{n+1}\mid t_i-\sum_{j=1}^{n}s(a_{i,j})\varpi^j\qquad(n\ge0).
\Pi^{\mathrm{phys}}Use in source · 4 passages
Search aliases: Π^phys Pi mathrm
Use in source · 01_finite_core.tex, line 1198
At one finite level, let widehat C_A be a finite family of matrix constraints, with fixed scalar structure coefficients, and let Π^phys be the orthogonal projector onto their exact common kernel. If the classical constraint algebra has structure coefficients f_AB^ C, define the anomaly operator A_AB = [widehat C_A,widehat C_B] -i∑_C f_AB^ Cwidehat C_C. Its physical packet is Π^phys A_ABΠ^phys. In this exact finite setting, widehat C_CΠ^phys=0 for every C. Associativity gives A_ABΠ^phys=0, Π^phys…
Exact mathematical passage
\Pi^{\mathrm{phys}}
\pi^{s_1}Use in source · 2 passages
Search aliases: π^s_1 pi
Use in source · 04_arithmetic.tex, line 711
[Smith normal form over a discrete valuation ring] Let M∈ M_m,n( O). There are invertible matrices U∈GL_m( O), V∈GL_n( O) such that UMV = diag(π^s_1,…,π^s_r,0,…,0), where 0≤ s_1≤⋯≤ s_r and r is the rank over K. The exponents are uniquely determined. See [source reference].
Exact mathematical passage
UMV = \diag(\pi^{s_1},\ldots,\pi^{s_r},0,\ldots,0),
\pi^{s_i}Use in source · 2 passages
Search aliases: π^s_i pi
Use in source · 04_arithmetic.tex, line 779
Set x=Vy. Multiplying the congruence by U reduces it to independent equations π^s_iy_i≡widetilde b_ipmodπ^L (1≤ i≤ r) and zero equations in the remaining rows. Multiplication by π^s_i on O/π^L O has image equal to the elements divisible by π^min(s_i,L) and kernel of size q^min(s_i,L). Each of the n-r unused variables is free and contributes q^L choices. Multiplying the independent counts gives the formula.
Exact mathematical passage
\pi^{s_i}y_i\equiv\widetilde b_i\pmod{\pi^L} \qquad(1\leq i\leq r)
\pi^{s_r}Use in source · 2 passages
Search aliases: π^s_r pi
Use in source · 04_arithmetic.tex, line 711
[Smith normal form over a discrete valuation ring] Let M∈ M_m,n( O). There are invertible matrices U∈GL_m( O), V∈GL_n( O) such that UMV = diag(π^s_1,…,π^s_r,0,…,0), where 0≤ s_1≤⋯≤ s_r and r is the rank over K. The exponents are uniquely determined. See [source reference].
Exact mathematical passage
UMV = \diag(\pi^{s_1},\ldots,\pi^{s_r},0,\ldots,0),
\pi^{\val_\pi(\Res(f,g))}Use in source · 1 passage
Search aliases: π^val_π(Res(f,g)) pi val pi Res pi operatorname pi operatorname
Use in source · 04_arithmetic.tex, line 89
[Resultant denominator bound] Under the hypotheses of [source reference], one may take r=Res(f,g) up to multiplication by a unit. Therefore π^val_π(Res(f,g))E_fΛ⊆Λ.
Exact mathematical passage
\pi^{\val_\pi(\Res(f,g))}E_f\Lambda\subseteq\Lambda.
\varpi^{w_i}Use in source · 2 passages
Search aliases: ϖ^w_i varpi
Use in source · Paper_X_Block27_Amended.tex, line 497
A simple weighted formal model already captures delayed visibility. Let w_i≥1 and constrain t_i∈(ϖ^w_i O+ϖ^m O)/ϖ^m O. Then coordinate i is invisible until m>w_i.
Exact mathematical passage
t_i\in(\varpi^{w_i}\Ocal+\varpi^m\Ocal)/\varpi^m\Ocal.
\pi_1Use in source · 4 passages
Search aliases: π_1 pi
Use in source · 04_correction_transport.tex, line 745
[Gauge classification of transports] Let G⊆GL(X) be the chosen structure group and fix a root r. Restrict both edge transports and gauge values to G. Gauge equivalence classes of G valued transport systems are naturally identified with Hom(π_1(Γ,r),G)/G, where G acts by conjugation. After a spanning tree and orientations of the remaining edges are chosen, this set is identified with simultaneous conjugacy classes of b_1=|E|-|V|+1 tuples in G.
Exact mathematical passage
\Hom(\pi_1(\Gamma,r),G)/G,
\varpi_KUse in source · 2 passages
Search aliases: ϖ_K varpi
Use in source · dyadic_synthesis.tex, line 356
There is also a recovery from compact data explicitly recording that root. With A_0=Λ(ϖ_K) and X=c - Ind_J^Gη, define (Tf)(g)=A_0^-1f(ϖ_Kg). Then X/(T-1)X≃π. After imposing the fixed central character, N/(F^× J) is cyclic of order N, and the finite average N^-1∑_i=0^N-1T^i selects this same extension. These operators act on complex representation spaces, so their scalar division by N is valid. This is a different recovery from the Block159 projector onto a sector of η. The exact kernel and finit…
Exact mathematical passage
A_0=\Lambda(\varpi_K)
\pi_OUse in source · 6 passages
Search aliases: π_O pi
Use in source · 04_correction_transport.tex, line 125
ion] Let U⊆ X be an open neighborhood of zero and let F: U→ Y be a C^r map between Banach spaces with r≥1, F(0)=0, and L=DF(0). Assume there are closed decompositions X=K⊕ X_1, Y=R⊕ O, where K=Ker L, R=im L, and L_1=L|_X_1:X_1⟶ R is a Banach space isomorphism. Let π_R and π_O be the two projections on Y. Then there are neighborhoods U_K⊆ K and U_1⊆ X_1 of zero and a unique C^r map ψ:U_K⟶ U_1 with ψ(0)=0 such that π_RF(k+ψ(k))=0 for every k∈ U_K. After the neighborhoods are made smaller if needed…
Exact mathematical passage
\pi_O
\Pi_{\Pcal}Use in source · 3 passages
Search aliases: Π_Pcal Π_ P Pi Pcal Pi mathcal
Use in source · 01_manuscript.tex, line 1469
[Exact block discovery certificate] For an orthogonal packet system (P_α), define the block diagonal compression Π_ P(A) = ∑_α P_α AP_α. Then normA-Π_ P(A)_rm F^2 = ∑_α≠β normP_β AP_α_rm F^2. In operator norm, normA-Π_ P(A)_2 ≤ ∑_α≠β normP_β AP_α_2. Thus any proposed packet discovery has an exact finite residual certificate after its projectors are known.
Exact mathematical passage
\Pi_{\Pcal}(A) = \sum_\alpha P_\alpha AP_\alpha.
\pi_RUse in source · 3 passages
Search aliases: π_R pi
Use in source · 04_correction_transport.tex, line 125
hmidt reduction] Let U⊆ X be an open neighborhood of zero and let F: U→ Y be a C^r map between Banach spaces with r≥1, F(0)=0, and L=DF(0). Assume there are closed decompositions X=K⊕ X_1, Y=R⊕ O, where K=Ker L, R=im L, and L_1=L|_X_1:X_1⟶ R is a Banach space isomorphism. Let π_R and π_O be the two projections on Y. Then there are neighborhoods U_K⊆ K and U_1⊆ X_1 of zero and a unique C^r map ψ:U_K⟶ U_1 with ψ(0)=0 such that π_RF(k+ψ(k))=0 for every k∈ U_K. After the neighborhoods are made small…
Exact mathematical passage
\pi_R
\pi_{\src(e)}Use in source · 1 passage
Search aliases: π_src(e) pi src pi operatorname
Use in source · 01_finite_coordinates.tex, line 147
[Current fiber scope] The compiled Chapter III model above has one scalar coordinate per packet. For a family of modules (X_i)_i∈ V, an intrinsic edge map would instead have type ℓ_e:X_src(e)→ X_tgt(e), and its totalization on ⊕_iX_i would be ι_tgt(e)ℓ_eπ_src(e). The same support proof applies, but this generalization with dependent fibers is not attributed to the released Chapter III modules for scalar edge systems.
Exact mathematical passage
\iota_{\tgt(e)}\ell_e\pi_{\src(e)}
\pi_xUse in source · 6 passages
Search aliases: π_x pi
Use in source · Paper_XII_Traditional_Final.tex, line 1350
tion below to be continuous. Assume the following data. item A represented deformation ring R with universal representation ρ_R:G_F,S⟶GL_n(R). item An isomorphism R≅ T. item An automorphic Hecke datum ( T,H,(P_v)). item A characteristic zero point x:R→ E' lying in Supp_R(H). item A classical automorphic representation π_x occurring in the fiber H_x, together with a semisimple attached representation ρ_π_x:G_F,S⟶GL_n(E') whose unramified characteristic polynomial is x(P_v(X)). item Equality det(X…
Exact mathematical passage
\pi_x
\pi_YUse in source · 4 passages
Search aliases: π_Y pi
Use in source · 40_regularity.tex, line 112
ather At the base parameter, the strict Fr echet derivative is Dψ(t_0)=-J^-1K. In particular, ψ is differentiable on U. Let g be the analytic local inverse of Φ supplied by Lemma~[source reference]. Since F(a)=0, Φ(a)=(0,t_0) and g(0,t_0)=a. Define ψ(t)=π_Yg(0,t), where π_Y projects to the solved coordinates. This gives the base value and analyticity at t_0. The chain rule and [source reference] give Dψ(t_0)v =π_YJ^-1 · -J^-1K
Exact mathematical passage
\psi(t)=\pi_Yg(0,t),
\pi_\betaUse in source · 1 passage
Search aliases: π_β pi beta
Use in source · 01_manuscript.tex, line 103
[Exact evaluation] Let A∈End(X). Set B_βα=π_β Aι_α and let E contain exactly the nonzero blocks. Then A= ∑_(β,α)∈ E P_β AP_α. For every matrix of right hand sides Z, AZ = ∑_(β,α)∈ E ι_β B_βαπ_α Z.
Exact mathematical passage
B_{\beta\alpha}=\pi_\beta A\iota_\alpha
\pi_\lambdaUse in source · 2 passages
Search aliases: π_λ pi lambda
Use in source · 02_strengthened.tex, line 242
[Representation matrix units] Fix an irreducible unitary representation π_λ:G→ U(d_λ) and an orthonormal basis of its representation space. Define E_ij^λ = fracd_λabsG ∑_g∈ G [π_λ(g^-1)]_jiU(g). Then E_ij^λE_kℓ^μ = δ_λμδ_jkE_iℓ^λ, (E_ij^λ)^*=E_ji^λ, and P_λ=∑_i=1^d_λE_ii^λ.
Exact mathematical passage
\pi_\lambda:G\to U(d_\lambda)
\Pi_\muUse in source · 1 passage
Search aliases: Π_μ Pi mu
Use in source · visual_atlas.tex, line 34
Π_μObΠ_ν · Block from input to output of Ob=H-I. · pagerefV16 to pagerefV18
Exact mathematical passage
\Pi_\mu\Ob\Pi_\nu
\pi_\muUse in source · 2 passages
Search aliases: π_μ pi mu
Use in source · 02_strengthened.tex, line 225
Complete reducibility gives a unitary decomposition H ≅ ⊕_μ∈widehat G V_μ⊗ M_μ, U(g) ≅ ⊕_μ π_μ(g)⊗ I_M_μ. On the μ summand, the displayed average is central in the irreducible representation π_μ, so it is a scalar by Schur's lemma. Taking its trace and applying character orthogonality shows that the scalar is one when μ=λ and zero otherwise. The asserted identities follow on the direct sum.
Exact mathematical passage
\Hcal \cong \bigoplus_{\mu\in\widehat G} V_\mu\otimes M_\mu, \qquad U(g) \cong \bigoplus_\mu \pi_\mu(g)\otimes I_{M_\mu}.
\Pi_\nuUse in source · 1 passage
Search aliases: Π_ν Pi nu
Use in source · visual_atlas.tex, line 34
Π_μObΠ_ν · Block from input to output of Ob=H-I. · pagerefV16 to pagerefV18
Exact mathematical passage
\Pi_\mu\Ob\Pi_\nu
\Pi_\ell^{\mathrm{phys}}Use in source · 1 passage
Search aliases: Π_ℓ^phys Pi ell mathrm
Use in source · 01_finite_core.tex, line 1003
At resolution ℓ, define Q_ℓ = ( A_ℓ, H_ℓ, D_ℓ, {P_ℓ,α}, E_ℓ, C_ℓ, Π_ℓ^phys, T_ℓ ). In order, the entries specify the observable algebra, Hilbert space, geometric or Dirac-type operator, packet projectors, embedding into the fine space, reconstruction onto the coarse space, physical constraint projector, and transfer operator. The maps have types E_ℓ: H_ℓ→ H_ℓ+1 and C_ℓ: H_ℓ+1→ H_ℓ. The left inverse relation C_ℓ E_ℓ=I alone does not imply C_ℓ=E_ℓ^* or make the embedding isometric.
Exact mathematical passage
\boxed{ \mathfrak Q_\ell = \left( \cA_\ell, \cH_\ell, D_\ell, \{P_{\ell,\alpha}\}, E_\ell, C_\ell, \Pi_\ell^{\mathrm{phys}}, T_\ell \right). }
\rho^{-j}Use in source · 4 passages
Search aliases: ρ^-j rho
Use in source · Paper_IV_Traditional_Final.tex, line 621
[Relation with the transfer exponential] For every j, D_N,j(z) = ρ^-j E_N,j(ρ z).
Exact mathematical passage
\Dfun_{N,j}(z) = \rho^{-j}\Efun_{N,j}(\rho z).
\rho^{-m}Use in source · 4 passages
Search aliases: ρ^-m rho
Use in source · Paper_IV_Traditional_Final.tex, line 2632
rcle_expansion Let r∈ Z_>0. Suppose the zeros of D_N of least modulus R are simple and form the finite set Z_R. Suppose the next zero modulus is R_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_ρ,ℓ binomm+ℓ-1ℓ-1 ρ^-m + O_R_*(m!R_*^-m). The leading Laurent coefficient is a_ρ,r = (-ρ D_N'(ρ))^-r.
Exact mathematical passage
P_{N,n}(r) = m! \sum_{\rho\in\Zcal_R} \sum_{\ell=1}^{r} a_{\rho,\ell} \binom{m+\ell-1}{\ell-1} \rho^{-m} + O_{R_*}(m!R_*^{-m}).
\rho^2Use in source · 1 passage
Search aliases: ρ^2 rho
Use in source · Paper_V_Traditional_Final.tex, line 1200
For 0≤ρ<1 and angle θ, Γ=ρ e^ iθ gives z(ρ,θ) =frac1+ρ e^ iθ1-ρ e^ iθ, with r(ρ,θ)=frac1-ρ^21-2ρcosθ+ρ^2, x(ρ,θ)=frac2ρsinθ1-2ρcosθ+ρ^2. A fixed ρ is a constant-mismatch circle. Refining N increases angular resolution without changing ρ.
Exact mathematical passage
r(\rho,\theta)=\frac{1-\rho^2}{1-2\rho\cos\theta+\rho^2}, \qquad x(\rho,\theta)=\frac{2\rho\sin\theta}{1-2\rho\cos\theta+\rho^2}.
\rho^{i+j-k}Use in source · 1 passage
Search aliases: ρ^i+j-k rho
Use in source · Paper_IV_Traditional_Final.tex, line 690
Use [source reference]. The factor introduced by a pair (i,j) is ρ^i+j-k. Since 0≤ i+j≤2N-2 and i+j≡ kbmod N, this factor is one when i+j=k and -1 when i+j=k+N.
Exact mathematical passage
\rho^{i+j-k}.
\rho^NUse in source · 2 passages
Search aliases: ρ^N rho
Use in source · Paper_IV_Traditional_Final.tex, line 617
Fix any ρ∈ C with ρ^N=-1.
Exact mathematical passage
\rho^N=-1
\rho^{n-1}Use in source · 1 passage
Search aliases: ρ^n-1 rho
Use in source · 05_defects.tex, line 70
[Norm bound] For a submultiplicative operator norm, CT_f^n-T_c^nC ≤ Δ ∑_k=0^n-1 T_c ^n-1-k T_f ^k. If T_f , T_c ≤ρ, then CT_f^n-T_c^nC ≤ nρ^n-1 Δ .
Exact mathematical passage
\boxed{ \|CT_f^n-T_c^nC\| \le n\rho^{n-1}\|\Delta\|. }
\rho^{Nn+j}Use in source · 1 passage
Search aliases: ρ^Nn+j rho
Use in source · Paper_IV_Traditional_Final.tex, line 630
The coefficient of z^Nn+j on the right is ρ^-jρ^Nn+jfrac1(Nn+j)! = (-1)^nfrac1(Nn+j)!.
Exact mathematical passage
\rho^{-j}\rho^{Nn+j}\frac1{(Nn+j)!} = (-1)^n\frac1{(Nn+j)!}.
\rho_+Use in source · 2 passages
Search aliases: ρ_+ rho
Use in source · monograph_atlas.tex, line 320
For ρ_+=frac12mat1 · 1 1 · 1, the exact output is Φ(ρ_+)=mat17/25 · 2/5
Exact mathematical passage
\rho_+=\frac12\mat{1&1\\1&1}
\rho_0Use in source · 2 passages
Search aliases: ρ_0 rho
Use in source · 02_strengthened.tex, line 1385
The map R_0 is completely positive. It is trace preserving on operators supported in Q H because Tr(V_j^*Q_jXQ_jV_j) = Tr(Q_jXQ_j). Fix a code state ρ_0 and write Q^⊥=I-Q. Then R(X) = R_0(QXQ) + Tr(Q^⊥ X)ρ_0 is completely positive and trace preserving. It extends R_0 on operators supported in Q H.
Exact mathematical passage
\rho_0
\rho_{00}Use in source · 1 passage
Search aliases: ρ_00 rho
Use in source · monograph_atlas.tex, line 318
0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter keeps every displayed coefficient rational. vfigdamping The superoperator uses the entry order (ρ_00,ρ_01,ρ_10,ρ_11). Population leaves the excited state with factor 9/25, while coherences contract by 4/5. These are not the same rate or the same coordinate.
Exact mathematical passage
(\rho_{00},\rho_{01},\rho_{10},\rho_{11})
\rho_{01}Use in source · 1 passage
Search aliases: ρ_01 rho
Use in source · monograph_atlas.tex, line 318
0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter keeps every displayed coefficient rational. vfigdamping The superoperator uses the entry order (ρ_00,ρ_01,ρ_10,ρ_11). Population leaves the excited state with factor 9/25, while coherences contract by 4/5. These are not the same rate or the same coordinate.
Exact mathematical passage
(\rho_{00},\rho_{01},\rho_{10},\rho_{11})
\rho_1Use in source · 3 passages
Search aliases: ρ_1 rho
Use in source · Paper_X_Block27_Amended.tex, line 981
0 · 1, every integral intertwiner with ρ_1 has determinant aP_22^2. Taking a=π^n for a nonzero π∈ I rules out every fixed detecting depth, even though all the field-valued approximants are isomorphic and their integral limit is trivial. This does not refute Claim 8.2, which already assumes inverse-limit closure of its chosen local conditions.
Exact mathematical passage
\rho_1
\rho_{10}Use in source · 1 passage
Search aliases: ρ_10 rho
Use in source · monograph_atlas.tex, line 318
0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter keeps every displayed coefficient rational. vfigdamping The superoperator uses the entry order (ρ_00,ρ_01,ρ_10,ρ_11). Population leaves the excited state with factor 9/25, while coherences contract by 4/5. These are not the same rate or the same coordinate.
Exact mathematical passage
(\rho_{00},\rho_{01},\rho_{10},\rho_{11})
\rho_{11}Use in source · 1 passage
Search aliases: ρ_11 rho
Use in source · monograph_atlas.tex, line 318
0 · 0, K_0^*K_0+K_1^*K_1=I. The channel Φ(ρ)=∑_aK_aρ K_a^* is trace preserving and completely positive. The special parameter keeps every displayed coefficient rational. vfigdamping The superoperator uses the entry order (ρ_00,ρ_01,ρ_10,ρ_11). Population leaves the excited state with factor 9/25, while coherences contract by 4/5. These are not the same rate or the same coordinate.
Exact mathematical passage
(\rho_{00},\rho_{01},\rho_{10},\rho_{11})
\rho_2Use in source · 2 passages
Search aliases: ρ_2 rho
Use in source · Paper_XII_Traditional_Final.tex, line 1306
[Chebotarev and Brauer Nesbitt identification] Let E' be a Hausdorff topological field of characteristic zero, with continuous ring operations. Let ρ_1,ρ_2:G_F,S⟶GL_n(E') be continuous semisimple representations. If their characteristic polynomials agree at every Frobenius element outside a density zero exceptional set, then ρ_1≅ρ_2. This follows from Chebotarev density and the Brauer Nesbitt theorem [source reference].
Exact mathematical passage
\rho_1,\rho_2:G_{F,S}\longrightarrow\GL_n(E')
\rho_aUse in source · 1 passage
Search aliases: ρ_a rho
Use in source · Paper_X_Block27_Amended.tex, line 980
The bound is substantive: for the unipotent actions ρ_a(g)=1 · ag 0 · 1, every integral intertwiner with ρ_1 has determinant aP_22^2. Taking a=π^n for a nonzero π∈ I rules out every fixed detecting depth, even though all the field-valued approximants are isomorphic and their integral limit is trivial. This does not refute Claim 8.2, which already assumes inverse-limit closure of its chosen local conditions.
Exact mathematical passage
\rho_a(g)=\begin{pmatrix}1&ag\\0&1\end{pmatrix}
\rho_cUse in source · 1 passage
Search aliases: ρ_c rho
Use in source · Paper_X_Block27_Amended.tex, line 1397
≤ c_i, set U_m=∏_i∈ I∏_k<m_iW_i, U_c=∏_i∈ I∏_k<c_iW_i. These finite products realize the corresponding finite direct sums. The representation on U_m acts componentwise. Define j:U_m→ U_c by retaining the first m_i copies and filling every remaining coordinate with zero. Define r:U_c→ U_m by restriction to the selected copies. Lean proves rj=1_U_m, jρ_m(g)=ρ_c(g)j, rρ_c(g)=ρ_m(g)r. The image of j is exactly the subspace whose unused coordinates vanish. Consequently jr is an equivariant idempotent…
Exact mathematical passage
rj=1_{U_m},\qquad j\rho_m(g)=\rho_c(g)j,\qquad r\rho_c(g)=\rho_m(g)r.
\rho_DUse in source · 2 passages
Search aliases: ρ_D rho
Use in source · Paper_X_Block27_Amended.tex, line 2271
For supplied covariance data and the twisted cocycle of the hyperref[book:x-construction-13]finite-exponential reconstruction, let ρ_D(g)=exp(t(g)N_D)r_D(g). If a common element u satisfies r_D(u)=r_E(u)=1 and t(u)≠0, every intertwiner of ρ_D and ρ_E recovers both the original r-intertwiner and the N-intertwiner. There is no cubic nilpotence restriction. This recovers equivariant projectors for both original components. Reconstructed-action isomorphisms induce actual isomorphisms of the inertia-…
Exact mathematical passage
\rho_D(g)=\exp(t(g)N_D)r_D(g)
\rho_EUse in source · 1 passage
Search aliases: ρ_E rho
Use in source · Paper_X_Block27_Amended.tex, line 2273
For supplied covariance data and the twisted cocycle of the hyperref[book:x-construction-13]finite-exponential reconstruction, let ρ_D(g)=exp(t(g)N_D)r_D(g). If a common element u satisfies r_D(u)=r_E(u)=1 and t(u)≠0, every intertwiner of ρ_D and ρ_E recovers both the original r-intertwiner and the N-intertwiner. There is no cubic nilpotence restriction. This recovers equivariant projectors for both original components. Reconstructed-action isomorphisms induce actual isomorphisms of the inertia-…
Exact mathematical passage
\rho_E
\rho_{i,\lambda}Use in source · 2 passages
Search aliases: ρ_i,λ rho lambda
Use in source · Paper_X_Block27_Amended.tex, line 1223
For a compatible family ρ_i,λ of representations of the fixed finite group, set V_i,λ=ρ_i,λ∘ q_L. Suppose the actual absolute-Galois members are irreducible and the systems are pairwise componentwise nonisomorphic. The finite prime separator now yields distinct permitted rational primes p_j, actual prime ideals Q_j and arithmetic Frobenius witnesses γ_j as above, outside any prescribed finite set, such that the common algebraic trace matrix has nonzero determinant. Every coefficient embedding pr…
Exact mathematical passage
\rho_{i,\lambda}
\rho_mUse in source · 4 passages
Search aliases: ρ_m rho
Use in source · Paper_X_Block27_Amended.tex, line 865
Checked representation-level completion and its arithmetic boundary. The representation-level completion construction works with actual matrix representations of a fixed group. For a complete separated ring R and ideal I, compatible maps ρ_m:G→GL_d(R/I^m) reconstruct a unique integral representation for that fixed compatible family. Multiplication and inverses are verified after every reduction. If all finite representations are continuous for the discrete quotient topology, the integral represe…
Exact mathematical passage
\rho_m:G\to\operatorname{GL}_d(R/I^m)
\rho_{m,n}Use in source · 4 passages
Search aliases: ρ_m,n rho
Use in source · Paper_VI_Traditional_Final.tex, line 2654
[Asymptotic closure criterion] Assume the weighted convergence in [source reference] holds for every bounded time interval, and define P_m,n=R_n,mC_m,n on X_n. Suppose that for every T>0, ρ_m,n(T) = sup_0≤ s,t≤ T norm C_m,nT_n(s)(I-P_m,n)T_n(t)R_n,m ⟶0. Then U_m(s+t)=U_m(s)U_m(t) for all s,t≥0. If the convergence in [source reference] is locally uniform and every T_n is norm continuous, then U_m is a norm continuous semigroup.
Exact mathematical passage
\rho_{m,n}(T) = \sup_{0\leq s,t\leq T} \norm{ C_{m,n}T_n(s)(I-P_{m,n})T_n(t)R_{n,m} } \longrightarrow0.
\rho_{m,\ell}Use in source · 3 passages
Search aliases: ρ_m,ℓ rho ell
Use in source · 02_coordinates.tex, line 69
A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.
Exact mathematical passage
\rho_{m,\ell}:I_m\longrightarrow I_\ell \qquad(m\geq\ell)
\rho_nUse in source · 1 passage
Search aliases: ρ_n rho
Use in source · Paper_X_Block27_Amended.tex, line 1776
Local properties need their own limit argument. Over a domain, for a nonzero π∈ I, the representations of the additive integer group given by ρ_n(g)=1 · π^n g 0 · 1 are all nontrivial and converge pointwise in the ideal-adic topology to the trivial representation. The convergence and the failure of limit preservation for nontriviality are formally checked. Thus not every property of approximating integral representations can be asserted for a limit. This example is not claimed to disprove stabil…
Exact mathematical passage
\rho_n(g)=\begin{pmatrix}1&\pi^n g\\0&1\end{pmatrix}
\rho_{n,m}Use in source · 1 passage
Search aliases: ρ_n,m rho
Use in source · 02_coordinates.tex, line 74
A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.
Exact mathematical passage
\rho_{\ell,\ell}=\id_{I_\ell}, \qquad \rho_{n,\ell}=\rho_{m,\ell}\circ\rho_{n,m} \qquad(n\geq m\geq\ell).
\rho_{n,\ell}Use in source · 1 passage
Search aliases: ρ_n,ℓ rho ell
Use in source · 02_coordinates.tex, line 74
A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.
Exact mathematical passage
\rho_{\ell,\ell}=\id_{I_\ell}, \qquad \rho_{n,\ell}=\rho_{m,\ell}\circ\rho_{n,m} \qquad(n\geq m\geq\ell).
\rho_pUse in source · 1 passage
Search aliases: ρ_p rho
Use in source · continuation_02.tex, line 102
The earlier obstruction used the integer-domain map J_x=ρ_p∘ DF(x): Z^d→ F_p^e. The new count uses A_x: F_p^d→ F_p^e. Reduction of integer vectors onto finite vectors is surjective, so the two maps have the same image after restricting scalars. Lean constructs a class-preserving integer-linear equivalence F_p^e/imJ_x ≃ F_p^e/imA_x. Consequently the cardinality theorem counts the original obstruction group, preserving its distinguished residual class. It is not merely a formula for a separately i…
Exact mathematical passage
J_x=\rho_p\circ DF(x):\mathbb Z^d\to\mathbb F_p^e.
\rho_{\rm Bell}Use in source · 2 passages
Search aliases: ρ_rm Bell rho rm
Use in source · monograph_atlas.tex, line 333
lassical input II.7.7. Ordinary transpose T sends a positive matrix on one system to another positive matrix and preserves its trace. Those properties are insufficient for a quantum channel on a subsystem of an entangled state. The test must include id⊗ T on the larger system. vfigcomplete_positivity Apply this extended map to the Bell density matrix. The result is (id⊗ T)(ρ_rm Bell) =frac12mat1 · 0 · 0 · 0 0 · 0 · 1 · 0 0 · 1 · 0 · 0 0 · 0 · 0 · 1. Its middle block has one negative eigenvalue. …
Exact mathematical passage
(\id\otimes\mathsf T)(\rho_{\rm Bell}) =\frac12\mat{1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1}.
\rho_tUse in source · 2 passages
Search aliases: ρ_t rho
Use in source · Paper_VI_Traditional_Final.tex, line 3236
Write ρ_t= e^-t=1-t+O(t^2), σ_t^2=1- e^-2t=2t+O(t^2). For a polynomial f, Taylor expansion at x and Gaussian integration give O_tf(x) = · f(x) +f'(x)(ρ_t-1)x +frac12f''(x)σ_t^2 +o(t)
Exact mathematical passage
\rho_t=\e^{-t}=1-t+O(t^2), \qquad \sigma_t^2=1-\e^{-2t}=2t+O(t^2).
\rho_x^{\mathrm{ss}}Use in source · 3 passages
Search aliases: ρ_x^ss rho mathrm
Use in source · Paper_XII_Traditional_Final.tex, line 1362
tomorphic representation π_x occurring in the fiber H_x, together with a semisimple attached representation ρ_π_x:G_F,S⟶GL_n(E') whose unramified characteristic polynomial is x(P_v(X)). item Equality det(X-ρ_x(Frob_v))=x(P_v(X)) for ρ_x=x∘ρ_R at every unramified place in a Chebotarev dense set. Then ρ_x^ss ≅ ρ_π_x. If ρ_x is irreducible, then ρ_x≅ρ_π_x.
Exact mathematical passage
\boxed{ \rho_x^{\mathrm{ss}} \cong \rho_{\pi_x}. }
\rho_\etaUse in source · 1 passage
Search aliases: ρ_η rho eta
Use in source · monograph_atlas.tex, line 503
actual amplitude vector is (- i,1, i,1)^T/2, giving uniform site probabilities. The subsequent phases recombine those four amplitudes into a single output at the opposite site. Uniform intermediate probabilities do not imply an incoherent intermediate state. headA channel changes the conclusion predictably Before evolving, mix the seed density matrix with its complete momentum dephasing: ρ_η=(1-η)ket0bra0+η∑_jP_jket0bra0P_j =(1-η)ket0bra0+η I/4, quad0≤η≤1. At transfer time, the probability at th…
Exact mathematical passage
\rho_\eta=(1-\eta)\ket0\bra0+\eta\sum_jP_j\ket0\bra0P_j =(1-\eta)\ket0\bra0+\eta I/4, \quad0\le\eta\le1.
\rho_\lambdaUse in source · 11 passages
Search aliases: ρ_λ rho lambda
Use in source · Paper_X_Block27_Amended.tex, line 795
pine C over a number field M is fixed; item for every finite place λ of M and every m≥1, the companion level X_λ,m( C) is finite and nonempty; item the reduction forests are finitely branching; item the selected local conditions are closed under inverse limits. Then for every λ there is an infinite compatible branch ρ_λ∈varprojlim_mX_λ,m( C), and the collection {ρ_λ}_λ realizes the prescribed strictly compatible spine.
Exact mathematical passage
\rho_\lambda\in\varprojlim_mX_{\lambda,m}(\Compat),
\rho_{\lambda_0}Use in source · 1 passage
Search aliases: ρ_λ_0 rho lambda
Use in source · Paper_XII_Traditional_Final.tex, line 1527
[One member propagation in a fixed system] Let {ρ_λ}_λ∈Λ be a semisimple strictly compatible system with Frobenius spine (P_v). Suppose that for one coefficient place λ_0, the representation ρ_λ_0 is identified with the Galois representation attached to an automorphic representation π. Assume that the compatible system attached to π has the same polynomials P_v(X). Then for every λ∈Λ, ρ_λ ≅ ρ_π,λ^ss.
Exact mathematical passage
\rho_{\lambda_0}
\rho_{\pi,\lambda}Use in source · 1 passage
Search aliases: ρ_π,λ rho pi lambda
Use in source · Paper_XII_Traditional_Final.tex, line 1748
For propagation, suppose both a semisimple compatible family {ρ_λ} containing ρ_x and a family {ρ_π,λ} attached to this same π already exist. Require the distinguished members to be ρ_x and its attached representation, respectively. Assume the coefficient, continuity, density, semisimplification and common algebraic Frobenius spine hypotheses of [source reference], with that spine agreeing with the specialized Hecke polynomials at the distinguished place. Then ρ_λ≅ρ_π,λ^ss for every λ.
Exact mathematical passage
\{\rho_{\pi,\lambda}\}
\rho_{\pi,\lambda}^{\mathrm{ss}}Use in source · 3 passages
Search aliases: ρ_π,λ^ss rho pi lambda mathrm
Use in source · Paper_XII_Traditional_Final.tex, line 1528
be a semisimple strictly compatible system with Frobenius spine (P_v). Suppose that for one coefficient place λ_0, the representation ρ_λ_0 is identified with the Galois representation attached to an automorphic representation π. Assume that the compatible system attached to π has the same polynomials P_v(X). Then for every λ∈Λ, ρ_λ ≅ ρ_π,λ^ss.
Exact mathematical passage
\boxed{ \rho_\lambda \cong \rho_{\pi,\lambda}^{\mathrm{ss}}. }
\rho_{\pi_x}Use in source · 5 passages
Search aliases: ρ_π_x rho pi
Use in source · Paper_XII_Traditional_Final.tex, line 1351
deformation ring R with universal representation ρ_R:G_F,S⟶GL_n(R). item An isomorphism R≅ T. item An automorphic Hecke datum ( T,H,(P_v)). item A characteristic zero point x:R→ E' lying in Supp_R(H). item A classical automorphic representation π_x occurring in the fiber H_x, together with a semisimple attached representation ρ_π_x:G_F,S⟶GL_n(E') whose unramified characteristic polynomial is x(P_v(X)). item Equality det(X-ρ_x(Frob_v))=x(P_v(X)) for ρ_x=x∘ρ_R at every unramified place in a Chebot…
Exact mathematical passage
\rho_{\pi_x}:G_{F,S}\longrightarrow\GL_n(E')
\rho_{\ell,k}Use in source · 3 passages
Search aliases: ρ_ℓ,k rho ell
Use in source · supplement_05.tex, line 132
Principal targets. targetFiniteUnits.finite_unit_decomposition, .power_lift_congruence targetFiniteUnits.omega_unique, .decomposition_unique targetFiniteUnits.omega_mul, .principal_mul pageheading3. Precision compatibility and the dyadic signFinite refinements and Proposition 6.2's dyadic branch For 1≤ k≤ℓ, let ρ_ℓ,k: Z/p^ℓ Z⟶ Z/p^k Z be the actual reduction homomorphism. It induces a map on units. The explicit construction satisfies ρ_ℓ,k(ω_ℓ(a)) =ω_k(ρ_ℓ,k(a)), ρ_ℓ,k(⟨ a⟩_ℓ) =⟨ρ_ℓ,k(a)⟩_k. The…
Exact mathematical passage
\rho_{\ell,k}:\Z/p^\ell\Z\longrightarrow\Z/p^k\Z
\rho_{\ell,\ell}Use in source · 1 passage
Search aliases: ρ_ℓ,ℓ rho ell ell
Use in source · 02_coordinates.tex, line 74
A refinement tower is a sequence ( P_ℓ)_ℓ≥0 with parent maps ρ_m,ℓ:I_m⟶ I_ℓ (m≥ℓ) such that P_m refines P_ℓ through ρ_m,ℓ for every m≥ℓ, and ρ_ℓ,ℓ=id_I_ℓ, ρ_n,ℓ=ρ_m,ℓ∘ρ_n,m (n≥ m≥ℓ). Thus every parent map witnesses the stated refinement and the maps compose coherently. The next result shows that the coordinate changes form a coherent system.
Exact mathematical passage
\rho_{\ell,\ell}=\id_{I_\ell}, \qquad \rho_{n,\ell}=\rho_{m,\ell}\circ\rho_{n,m} \qquad(n\geq m\geq\ell).
\bm\sigmaUse in source · 2 passages
Search aliases: σ bm sigma
Use in source · 02_strengthened.tex, line 891
[Bloch sphere curvature] Let P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). With the convention F_θϕ =- iTr(P[∂_θ P,∂_ϕ P]), one has F_θϕ=frac12sinθ and frac12π ∫_0^2π∫_0^π F_θϕ dθ dϕ =1.
Exact mathematical passage
P(\theta,\phi) = \frac12\left(I+\bm n(\theta,\phi)\cdot\bm\sigma\right),
\boldsymbol\sigmaUse in source · 3 passages
Search aliases: σ boldsymbol sigma
Use in source · 01_finite_core.tex, line 637
For an orthogonal rank-one complex projector, use the scalar connection convention A=-i⟨ u,du⟩ in a local unit frame. The associated scalar curvature density is F_μν =-iTr(P[∂_μ P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)·σ. Using Tr(P a·σ)= n· a therefore yields F_θϕ =frac12 n·(∂_θ nmathbin×∂_ϕ n) =frac12sinθ. The scalar sign changes if the opposite convention for A i…
Exact mathematical passage
P(\theta,\phi) = \frac12\left(I+\mathbf n(\theta,\phi)\cdot\boldsymbol\sigma\right),
\Sigma'Use in source · 2 passages
Search aliases: Σ' Sigma
Use in source · 02_strengthened.tex, line 1608
[Uniform self energy bounds] Let δ=dist(z,Spec D)>0, g=norm B. Then normΣ(z)≤fracg^2δ and normΣ'(z)≤fracg^2δ^2. For real E,E' in an interval at distance at least δ from Spec D, normΣ(E)-Σ(E') ≤ fracg^2δ^2absE-E'.
Exact mathematical passage
\norm{\Sigma'(z)}\leq\frac{g^2}{\delta^2}.
\sigma^2Use in source · 4 passages
Search aliases: σ^2 sigma
Use in source · Paper_V_Traditional_Final.tex, line 3091
Assume the noise variables are independent, centered, and σ sub-Gaussian: E e^λη_k ≤ e^σ^2λ^2/2.
Exact mathematical passage
\mathbb E\e^{\lambda\eta_k} \leq \e^{\sigma^2\lambda^2/2}.
\sigma_1Use in source · 5 passages
Search aliases: σ_1 sigma
Use in source · atlas-pages.tex, line 465
cient. All symbols in the plate refer to this one order. vfigworked_circuit headRoute one: invert the denominator The w^2 coefficient of Δ_4^-1 is frac124^2-frac18!=frac698!. Thus P_4,2(1)=69. Keeping r variable gives P_4,2(r)=35r^2+34r. headRoute two: use the zero moments The exact reduced zeros are 4π^4(m+1/2)^4. Their first two power sums are σ_1=1/24 and σ_2=17/10080. The Bell/Newton identity gives the same reciprocal coefficient (σ_1^2+σ_2)/2. The tangent independently reads off the second …
Exact mathematical passage
\sigma_1=1/24
\sigma_1^2Use in source · 2 passages
Search aliases: σ_1^2 sigma
Use in source · atlas-pages.tex, line 465
he denominator The w^2 coefficient of Δ_4^-1 is frac124^2-frac18!=frac698!. Thus P_4,2(1)=69. Keeping r variable gives P_4,2(r)=35r^2+34r. headRoute two: use the zero moments The exact reduced zeros are 4π^4(m+1/2)^4. Their first two power sums are σ_1=1/24 and σ_2=17/10080. The Bell/Newton identity gives the same reciprocal coefficient (σ_1^2+σ_2)/2. The tangent independently reads off the second moment as 4·7!σ_2=34. This links coefficients, geometry, and derivatives without a fitted parameter…
Exact mathematical passage
(\sigma_1^2+\sigma_2)/2
\sigma_2Use in source · 4 passages
Search aliases: σ_2 sigma
Use in source · atlas-pages.tex, line 465
n the plate refer to this one order. vfigworked_circuit headRoute one: invert the denominator The w^2 coefficient of Δ_4^-1 is frac124^2-frac18!=frac698!. Thus P_4,2(1)=69. Keeping r variable gives P_4,2(r)=35r^2+34r. headRoute two: use the zero moments The exact reduced zeros are 4π^4(m+1/2)^4. Their first two power sums are σ_1=1/24 and σ_2=17/10080. The Bell/Newton identity gives the same reciprocal coefficient (σ_1^2+σ_2)/2. The tangent independently reads off the second moment as 4·7!σ_2=34…
Exact mathematical passage
\sigma_2=17/10080
\sigma_{2,1}Use in source · 1 passage
Search aliases: σ_2,1 sigma
Use in source · atlas-pages.tex, line 409
Corollary 12.7, Corollary 12.9, Corollary 12.10. Taking the logarithm of the canonical product identifies κ_N,q=σ_N,q:=∑_λ∈Λ_Nλ^-q =Tr(K_N^q), q≥1. The same exact numbers can be computed from finitely many denominator coefficients by Newton's recurrence. No numerical zero list is needed for that calculation. vfigmoments For order two, σ_2,1=frac12, σ_2,2=frac16, σ_2,3=frac115, σ_2,4=frac17630. The plate shows how successive inverse powers emphasize the nearest zero. Its shaded pieces are contrib…
Exact mathematical passage
\sigma_{2,1}=\frac12,\quad\sigma_{2,2}=\frac16,\quad \sigma_{2,3}=\frac1{15},\quad\sigma_{2,4}=\frac{17}{630}.
\sigma_{2,2}Use in source · 1 passage
Search aliases: σ_2,2 sigma
Use in source · atlas-pages.tex, line 409
Corollary 12.7, Corollary 12.9, Corollary 12.10. Taking the logarithm of the canonical product identifies κ_N,q=σ_N,q:=∑_λ∈Λ_Nλ^-q =Tr(K_N^q), q≥1. The same exact numbers can be computed from finitely many denominator coefficients by Newton's recurrence. No numerical zero list is needed for that calculation. vfigmoments For order two, σ_2,1=frac12, σ_2,2=frac16, σ_2,3=frac115, σ_2,4=frac17630. The plate shows how successive inverse powers emphasize the nearest zero. Its shaded pieces are contrib…
Exact mathematical passage
\sigma_{2,1}=\frac12,\quad\sigma_{2,2}=\frac16,\quad \sigma_{2,3}=\frac1{15},\quad\sigma_{2,4}=\frac{17}{630}.
\sigma_{2,3}Use in source · 1 passage
Search aliases: σ_2,3 sigma
Use in source · atlas-pages.tex, line 409
Corollary 12.7, Corollary 12.9, Corollary 12.10. Taking the logarithm of the canonical product identifies κ_N,q=σ_N,q:=∑_λ∈Λ_Nλ^-q =Tr(K_N^q), q≥1. The same exact numbers can be computed from finitely many denominator coefficients by Newton's recurrence. No numerical zero list is needed for that calculation. vfigmoments For order two, σ_2,1=frac12, σ_2,2=frac16, σ_2,3=frac115, σ_2,4=frac17630. The plate shows how successive inverse powers emphasize the nearest zero. Its shaded pieces are contrib…
Exact mathematical passage
\sigma_{2,1}=\frac12,\quad\sigma_{2,2}=\frac16,\quad \sigma_{2,3}=\frac1{15},\quad\sigma_{2,4}=\frac{17}{630}.
\sigma_{2,4}Use in source · 1 passage
Search aliases: σ_2,4 sigma
Use in source · atlas-pages.tex, line 409
Corollary 12.7, Corollary 12.9, Corollary 12.10. Taking the logarithm of the canonical product identifies κ_N,q=σ_N,q:=∑_λ∈Λ_Nλ^-q =Tr(K_N^q), q≥1. The same exact numbers can be computed from finitely many denominator coefficients by Newton's recurrence. No numerical zero list is needed for that calculation. vfigmoments For order two, σ_2,1=frac12, σ_2,2=frac16, σ_2,3=frac115, σ_2,4=frac17630. The plate shows how successive inverse powers emphasize the nearest zero. Its shaded pieces are contrib…
Exact mathematical passage
\sigma_{2,1}=\frac12,\quad\sigma_{2,2}=\frac16,\quad \sigma_{2,3}=\frac1{15},\quad\sigma_{2,4}=\frac{17}{630}.
\sigma_{2,q}Use in source · 1 passage
Search aliases: σ_2,q sigma
Use in source · Paper_IV_Traditional_Final.tex, line 2605
[The cosine orbit sums] For N=2, Δ_2(w)=cossqrt w. Its zeros are λ_m=π^2(m+frac12)^2, m≥0. Hence σ_2,q = frac2^2q-1π^2qζ(2q). The identity in [source reference] recovers the classical tangent coefficients from these zero power sums.
Exact mathematical passage
\sigma_{2,q} = \frac{2^{2q}-1}{\pi^{2q}}\zeta(2q).
\sigma_aUse in source · 1 passage
Search aliases: σ_a sigma
Use in source · monograph_atlas.tex, line 746
vpage38Transport constraints without hiding closureTHE IMAGE OF AN ISOMETRY IS NOT THE WHOLE LEVELChapter VII: Theorem II.12.2, Theorem II.12.5, Proposition II.12.6. On C^3= Cket0⊕ C^2, use C_a=0⊕σ_a for a=x,y,z. The common kernel is Cket0, and the full algebra closes: [C_x,C_y]=2 i C_z, [C_y,C_z]=2 i C_x, [C_z,C_x]=2 i C_y. These are uncompressed identities. They supply an actual closure test in the quotient by the constraint span, unlike the automatic compressed zero on the physical kernel. xf…
Exact mathematical passage
C_a=0\oplus\sigma_a
\sigma_dUse in source · 3 passages
Search aliases: σ_d sigma
Use in source · 02_supplement_06.tex, line 87
Spectral optimality from a kernel witness. textitModules DiagonalTruncation and SpectralOptimality Let σ_1≥⋯≥σ_d≥0 and D=diag(σ_1,…,σ_d), D_k=diag(σ_1,…,σ_min(k,d),0,…,0). Lean uses indices beginning at zero, while the formulas on this page use the conventional one-based notation. The rank of D_k is proved to be at most k, even when retained singular values are zero.
Exact mathematical passage
\sigma_1\geq\cdots\geq\sigma_d\geq0
\sigma_d^2Use in source · 1 passage
Search aliases: σ_d^2 sigma
Use in source · 02_supplement_06.tex, line 168
05textitModules CompactTruncation and SmallCoreRecompression For the actual core G=R_LR_R^*, supply ordered thin SVD data G=Wdiag(σ)Z^*, W^*W=I, Z^*Z=I, σ_1≥⋯≥σ_d≥0. These are precisely the fields of the formal SVD structure. It contains no rank-optimality or tail-error field. A derived theorem checks the Gram eigenvector equation G^*GZ=Zdiag(σ_1^2,…,σ_d^2). Let h=min(d,k) and retain exactly the first h column indices. The rectangular square-root mask has entries M_ij=sqrtσ_j, · i=j,
Exact mathematical passage
G^*GZ=Z\diag(\sigma_1^2,\ldots,\sigma_d^2).
\sigma_i^2Use in source · 3 passages
Search aliases: σ_i^2 sigma
Use in source · 02_supplement_06.tex, line 147
Set a_i=σ_i^2. Right compression cannot increase the Frobenius norm, and BQ=0, hence normD-B_ F^2 · ≥norm(D-B)Q_ F^2
Exact mathematical passage
a_i=\sigma_i^2
\sigma_i^{2}Use in source · 1 passage
Search aliases: σ_i^2 sigma
Use in source · 02_supplement_06.tex, line 101
Monotonicity of the diagonal entries gives normDy^2 =∑_i=1^k+1σ_i^2|y_i|^2 ≥σ_k+1^2norm y^2. Because (D-B)y=Dy, the operator-norm inequality implies σ_k+1norm y ≤norm(D-B)y ≤normD-B_2norm y. Cancellation of the strictly positive norm of y proves σ_k+1≤normD-B_2. The norm of the actual diagonal residual D-D_k equals σ_k+1, so D_k attains the lower bound. If k≥ d, the residual is zero and optimality follows from norm nonnegativity.
Exact mathematical passage
\norm{Dy}^{2} =\sum_{i=1}^{k+1}\sigma_i^{2}|y_i|^{2} \geq\sigma_{k+1}^{2}\norm y^{2}.
\sigma_jUse in source · 1 passage
Search aliases: σ_j sigma
Use in source · 02_supplement_06.tex, line 172
W^*W=I, Z^*Z=I, σ_1≥⋯≥σ_d≥0. These are precisely the fields of the formal SVD structure. It contains no rank-optimality or tail-error field. A derived theorem checks the Gram eigenvector equation G^*GZ=Zdiag(σ_1^2,…,σ_d^2). Let h=min(d,k) and retain exactly the first h column indices. The rectangular square-root mask has entries M_ij=sqrtσ_j, · i=j, 0, · i≠ j, 1≤ i≤ d, 1≤ j≤ h. Lean proves its index cardinality and MM^*=D_k. Define L_k=Q_LWM, R_k=Q_RZM. The product equals Q_LWD_kZ^*Q_R^* and has…
Exact mathematical passage
M_{ij}=\begin{cases}\sqrt{\sigma_j},&i=j,\\0,&i\ne j,\end{cases} \qquad 1\leq i\leq d,\quad 1\leq j\leq h.
\sigma_j^2Use in source · 1 passage
Search aliases: σ_j^2 sigma
Use in source · 01_manuscript.tex, line 792
[Eckart--Young--Mirsky theorem] If M has singular values σ_1≥σ_2≥⋯, pad this finite sequence with zeros. For each integer k≥0, its truncated singular value decomposition M_k is a best rank-at-most-k approximation. It satisfies normM-M_k_2=σ_k+1 and normM-M_k_rm F = ( ∑_j>kσ_j^2 )^1/2.
Exact mathematical passage
\norm{M-M_k}_{\rm F} = \left( \sum_{j>k}\sigma_j^2 \right)^{1/2}.
\sigma_{k+1}Use in source · 6 passages
Search aliases: σ_k+1 sigma
Use in source · pages.tex, line 210
For a d-dimensional diagonal core and 0≤ k<d, the spectral lower bound has a concrete witness. A rank-at-most-k competitor has a nonzero kernel vector in the first k+1 coordinates. On that vector the diagonal matrix acts with norm at least σ_k+1 times the vector norm. The competitor cannot cancel that direction.
Exact mathematical passage
\sigma_{k+1}
\sigma_{k+1}^{2}Use in source · 1 passage
Search aliases: σ_k+1^2 sigma
Use in source · 02_supplement_06.tex, line 101
Monotonicity of the diagonal entries gives normDy^2 =∑_i=1^k+1σ_i^2|y_i|^2 ≥σ_k+1^2norm y^2. Because (D-B)y=Dy, the operator-norm inequality implies σ_k+1norm y ≤norm(D-B)y ≤normD-B_2norm y. Cancellation of the strictly positive norm of y proves σ_k+1≤normD-B_2. The norm of the actual diagonal residual D-D_k equals σ_k+1, so D_k attains the lower bound. If k≥ d, the residual is zero and optimality follows from norm nonnegativity.
Exact mathematical passage
\norm{Dy}^{2} =\sum_{i=1}^{k+1}\sigma_i^{2}|y_i|^{2} \geq\sigma_{k+1}^{2}\norm y^{2}.
\Sigma_k^{1/2}Use in source · 1 passage
Search aliases: Σ_k^1/2 Sigma
Use in source · 01_manuscript.tex, line 825
R=Q_RR_R, where the columns of Q_L and Q_R are orthonormal. Form the small core G=R_LR_R^* and let G=WΣ Z^* be an ordered thin singular value decomposition, with nonnegative nonincreasing singular values. For an integer k≥0, retain min{k,d} columns, where d is the number of supplied thin singular directions. Tail singular values beyond d are zero. Define L_k=Q_LW_kΣ_k^1/2, R_k=Q_RZ_kΣ_k^1/2. Then L_kR_k^* is a best rank at most k approximation to LR^*. Its spectral and Frobenius errors are the s…
Exact mathematical passage
L_k=Q_LW_k\Sigma_k^{1/2}, \qquad R_k=Q_RZ_k\Sigma_k^{1/2}.
\sigma_{\min(k,d)}Use in source · 1 passage
Search aliases: σ_min(k,d) sigma min
Use in source · 02_supplement_06.tex, line 88
Spectral optimality from a kernel witness. textitModules DiagonalTruncation and SpectralOptimality Let σ_1≥⋯≥σ_d≥0 and D=diag(σ_1,…,σ_d), D_k=diag(σ_1,…,σ_min(k,d),0,…,0). Lean uses indices beginning at zero, while the formulas on this page use the conventional one-based notation. The rank of D_k is proved to be at most k, even when retained singular values are zero.
Exact mathematical passage
D=\diag(\sigma_1,\ldots,\sigma_d),\qquad D_k=\diag(\sigma_1,\ldots,\sigma_{\min(k,d)},0,\ldots,0).
\sigma_nUse in source · 2 passages
Search aliases: σ_n sigma
Use in source · Paper_V_Traditional_Final.tex, line 3156
[Probability consistency] Let σ_nsqrtlog n→0 and h=1/n. For every fixed 0<δ<1, along grids satisfying the preceding sampling assumptions, all jump cells are recovered with probability at least 1-δ for sufficiently large n. For the fixed finite jump model above, along grids satisfying the preceding sampling assumptions, the packet error converges to zero in probability in every finite L^p norm.
Exact mathematical passage
\sigma_n\sqrt{\log n}\to0
\sigma_{N,1}Use in source · 1 passage
Search aliases: σ_N,1 sigma
Use in source · Paper_IV_Traditional_Final.tex, line 2577
[Reciprocal coefficients from zero data] For every n≥0, P_N,n(r) = frac(Nn)!n! B_n( rσ_N,1, r1!σ_N,2, …, r(n-1)!σ_N,n ).
Exact mathematical passage
P_{N,n}(r) = \frac{(Nn)!}{n!} B_n\bigl( r\sigma_{N,1}, r1!\sigma_{N,2}, \ldots, r(n-1)!\sigma_{N,n} \bigr).
\sigma_{N,2}Use in source · 1 passage
Search aliases: σ_N,2 sigma
Use in source · Paper_IV_Traditional_Final.tex, line 2577
[Reciprocal coefficients from zero data] For every n≥0, P_N,n(r) = frac(Nn)!n! B_n( rσ_N,1, r1!σ_N,2, …, r(n-1)!σ_N,n ).
Exact mathematical passage
P_{N,n}(r) = \frac{(Nn)!}{n!} B_n\bigl( r\sigma_{N,1}, r1!\sigma_{N,2}, \ldots, r(n-1)!\sigma_{N,n} \bigr).
\sigma_{N,j,1}Use in source · 1 passage
Search aliases: σ_N,j,1 sigma
Use in source · Paper_IV_Traditional_Final.tex, line 2411
ized sections] For every 0≤ j<N, the entire function Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)! has order 1/N. If N≥2 and Λ_N,j is its nonzero zero multiset, then Φ_N,j(w) = ∏_μ∈Λ_N,j (1-frac wμ). For m≥1, put σ_N,j,m = ∑_μ∈Λ_N,jμ^-m. Then C_N,j,n(r) = frac(Nn)!n! B_n( rσ_N,j,1, r1!σ_N,j,2, …, r(n-1)!σ_N,j,n ).
Exact mathematical passage
C_{N,j,n}(r) = \frac{(Nn)!}{n!} B_n\bigl( r\sigma_{N,j,1}, r1!\sigma_{N,j,2}, \ldots, r(n-1)!\sigma_{N,j,n} \bigr).
\sigma_{N,j,2}Use in source · 1 passage
Search aliases: σ_N,j,2 sigma
Use in source · Paper_IV_Traditional_Final.tex, line 2411
ized sections] For every 0≤ j<N, the entire function Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)! has order 1/N. If N≥2 and Λ_N,j is its nonzero zero multiset, then Φ_N,j(w) = ∏_μ∈Λ_N,j (1-frac wμ). For m≥1, put σ_N,j,m = ∑_μ∈Λ_N,jμ^-m. Then C_N,j,n(r) = frac(Nn)!n! B_n( rσ_N,j,1, r1!σ_N,j,2, …, r(n-1)!σ_N,j,n ).
Exact mathematical passage
C_{N,j,n}(r) = \frac{(Nn)!}{n!} B_n\bigl( r\sigma_{N,j,1}, r1!\sigma_{N,j,2}, \ldots, r(n-1)!\sigma_{N,j,n} \bigr).
\sigma_{N,j,m}Use in source · 2 passages
Search aliases: σ_N,j,m sigma
Use in source · Paper_IV_Traditional_Final.tex, line 2405
[Products for the normalized sections] For every 0≤ j<N, the entire function Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)! has order 1/N. If N≥2 and Λ_N,j is its nonzero zero multiset, then Φ_N,j(w) = ∏_μ∈Λ_N,j (1-frac wμ). For m≥1, put σ_N,j,m = ∑_μ∈Λ_N,jμ^-m. Then C_N,j,n(r) = frac(Nn)!n! B_n( rσ_N,j,1, r1!σ_N,j,2, …, r(n-1)!σ_N,j,n ).
Exact mathematical passage
\sigma_{N,j,m} = \sum_{\mu\in\Lambda_{N,j}}\mu^{-m}.
\sigma_{N,j,n}Use in source · 1 passage
Search aliases: σ_N,j,n sigma
Use in source · Paper_IV_Traditional_Final.tex, line 2411
ized sections] For every 0≤ j<N, the entire function Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)! has order 1/N. If N≥2 and Λ_N,j is its nonzero zero multiset, then Φ_N,j(w) = ∏_μ∈Λ_N,j (1-frac wμ). For m≥1, put σ_N,j,m = ∑_μ∈Λ_N,jμ^-m. Then C_N,j,n(r) = frac(Nn)!n! B_n( rσ_N,j,1, r1!σ_N,j,2, …, r(n-1)!σ_N,j,n ).
Exact mathematical passage
C_{N,j,n}(r) = \frac{(Nn)!}{n!} B_n\bigl( r\sigma_{N,j,1}, r1!\sigma_{N,j,2}, \ldots, r(n-1)!\sigma_{N,j,n} \bigr).
\sigma_{N,m-k}Use in source · 2 passages
Search aliases: σ_N,m-k sigma
Use in source · Paper_IV_Traditional_Final.tex, line 2466
[Newton sums for the reduced zeros] For |w| below the least zero modulus, with the analytic logarithm normalized to zero at the origin, -logΔ_N(w) = ∑_m≥1σ_N,mfracw^mm. Thus κ_N,m=σ_N,m. If a_m=frac(-1)^m(Nm)!, a_0=1, then σ_N,m = -ma_m- ∑_k=1^m-1a_kσ_N,m-k.
Exact mathematical passage
\sigma_{N,m} = -ma_m- \sum_{k=1}^{m-1}a_k\sigma_{N,m-k}.
\sigma_{N,n}Use in source · 1 passage
Search aliases: σ_N,n sigma
Use in source · Paper_IV_Traditional_Final.tex, line 2577
[Reciprocal coefficients from zero data] For every n≥0, P_N,n(r) = frac(Nn)!n! B_n( rσ_N,1, r1!σ_N,2, …, r(n-1)!σ_N,n ).
Exact mathematical passage
P_{N,n}(r) = \frac{(Nn)!}{n!} B_n\bigl( r\sigma_{N,1}, r1!\sigma_{N,2}, \ldots, r(n-1)!\sigma_{N,n} \bigr).
\sigma_{N,n+1}Use in source · 1 passage
Search aliases: σ_N,n+1 sigma
Use in source · Paper_IV_Traditional_Final.tex, line 2564
[Tangent coefficients as zero sums] For every n≥0, T_N(n) = N(Nn+N-1)!σ_N,n+1.
Exact mathematical passage
T_N(n) = N(Nn+N-1)!\sigma_{N,n+1}.
\sigma_{N,q}Use in source · 1 passage
Search aliases: σ_N,q sigma
Use in source · atlas-pages.tex, line 402
vpage22Finite coefficients give infinite momentsZERO POWER SUMSChapter IV: Theorem 12.6, Corollary 12.7, Corollary 12.9, Corollary 12.10. Taking the logarithm of the canonical product identifies κ_N,q=σ_N,q:=∑_λ∈Λ_Nλ^-q =Tr(K_N^q), q≥1. The same exact numbers can be computed from finitely many denominator coefficients by Newton's recurrence. No numerical zero list is needed for that calculation. vfigmoments For order two, σ_2,1=frac12, σ_2,2=frac16, σ_2,3=frac115, σ_2,4=frac17630. The plate show…
Exact mathematical passage
\kappa_{N,q}=\sigma_{N,q}:=\sum_{\lambda\in\Lambda_N}\lambda^{-q} =\Tr(K_N^q),\qquad q\ge1.
\Sigma_PUse in source · 12 passages
Search aliases: Σ_P Sigma
Use in source · Paper_VI_Traditional_Final.tex, line 1520
The self-energy Σ_P(z)=A_PQ(zI-A_QQ)^-1A_QP contains the exact effect of the eliminated packets. A bare truncation A_PP discards this term and is generally not exact.
Exact mathematical passage
\Sigma_P(z)=A_{PQ}(zI-A_{QQ})^{-1}A_{QP}
\sigma_pUse in source · 13 passages
Search aliases: σ_p sigma
Use in source · 72_recovered.tex, line 1815
For an odd prime p, the source S_n(p) contains one nonunit base. Put σ_p=(-1)^(p-1)/2 and S_n^×(p)=S_n(p)-σ_pp^n.
Exact mathematical passage
\sigma_p=(-1)^{(p-1)/2}
\sigma_t^2Use in source · 2 passages
Search aliases: σ_t^2 sigma
Use in source · Paper_VI_Traditional_Final.tex, line 3236
Write ρ_t= e^-t=1-t+O(t^2), σ_t^2=1- e^-2t=2t+O(t^2). For a polynomial f, Taylor expansion at x and Gaussian integration give O_tf(x) = · f(x) +f'(x)(ρ_t-1)x +frac12f''(x)σ_t^2 +o(t)
Exact mathematical passage
\rho_t=\e^{-t}=1-t+O(t^2), \qquad \sigma_t^2=1-\e^{-2t}=2t+O(t^2).
\sigma_uUse in source · 1 passage
Search aliases: σ_u sigma
Use in source · 01_manuscript.tex, line 1514
The cost and error of a pruning L are cost( L) = ∑_v∈ Lκ_v + ∑_u splitσ_u and err( L)=∑_v∈ Lϵ_v. The model can include arithmetic, storage, launch, transfer, and precision costs inside κ_v. It remains exact only when the declared totals are additive.
Exact mathematical passage
\cost(\Lcal) = \sum_{v\in\Lcal}\kappa_v + \sum_{u\text{ split}}\sigma_u
\sigma_\lambdaUse in source · 1 passage
Search aliases: σ_λ sigma lambda
Use in source · Paper_X_Block27_Amended.tex, line 1056
ion] Let mathscr W_1,…,mathscr W_r be pairwise nonisomorphic strictly compatible systems whose members are absolutely irreducible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficient-field embedding σ_λ:E↪ E_λ therefore sends C to an invertible matrix. Its entries agree with the ordinary representation traces Tr(W_i,λ(Frob_v_j)) when the selected places …
Exact mathematical passage
\sigma_\lambda:E\hookrightarrow E_\lambda
\sigma_\tauUse in source · 3 passages
Search aliases: σ_τ sigma tau
Use in source · Paper_XI_Traditional_Final.tex, line 294
[Formal atomic reconstruction] Assume that every irreducible Weil representation τ has been assigned a supercuspidal representation σ_τ, compatibly with isomorphism and unramified twist. Then the rules Sp_m(τ) ⟼ St_m(σ_τ) and normalized parabolic induction determine a candidate correspondence on all Frobenius semisimple Weil Deligne parameters.
Exact mathematical passage
\sigma_\tau
\tau^{-1}Use in source · 1 passage
Search aliases: τ^-1 tau
Use in source · 04_transport.tex, line 157
Let (Φ,φ,σ):(X, A, P)→(Y, B, Q) and (Ψ,ψ,τ):(Y, B, Q)→(Z, C, R) be morphisms. Intertwining is preserved because ΨΦ(Ax) = Ψ(φ(A)Φ x) = (ψ∘φ)(A)ΨΦ x. For k in the target packet set, R_kΨΦ · =Ψ Q_τ^-1(k)Φ · =ΨΦ ∑_τ(σ(α))=kP_α. Thus the composite packet map is τ∘σ. If the morphisms are seeded, then Φ(S)⊆Hull_ B(S'), Ψ(S')⊆Hull_ C(S''). By [source reference], ΨΦ(S) ⊆ Ψ(Hull_ B(S')) ⊆ Hull_ C(Ψ(S')) ⊆ Hull_ C(S''). Identity maps satisfy the definitions with identity algebra and packet maps.
Exact mathematical passage
R_k\Psi\Phi &=\Psi Q_{\tau^{-1}(k)}\Phi\\ &=\Psi\Phi \sum_{\tau(\sigma(\alpha))=k}P_\alpha.
\tau^{-m}Use in source · 2 passages
Search aliases: τ^-m tau
Use in source · Paper_IV_Traditional_Final.tex, line 2697
[Orbit reduced nearest circle term] Assume the hypotheses of [source reference]. Choose one representative from each orbit of the nearest zeros under multiplication by ω, and call the resulting set Z_R^orb. With m=Nn, P_N,n(r) = · N m!binomm+r-1r-1 ∑_τ∈ Z_R^orb fracτ^-m(-τ D_N'(τ))^r · +O(m!m^r-2R^-m) +O_R_*(m!R_*^-m). For r=1, the first error term is absent.
Exact mathematical passage
\begin{aligned} P_{N,n}(r) ={}&N m!\binom{m+r-1}{r-1} \sum_{\tau\in\Zcal_R^{\mathrm{orb}}} \frac{\tau^{-m}}{(-\tau\Dfun_N'(\tau))^r} \\ &+O\left(m!m^{r-2}R^{-m}\right) +O_{R_*}\left(m!R_*^{-m}\right). \end{aligned}
\tau^2Use in source · 2 passages
Search aliases: τ^2 tau
Use in source · strengthening.tex, line 894
ulo two. Its uth auxiliary term has excess u+lfloor(k-u)/2rfloor-q=lfloor u/2rfloor. Only auxiliary indices zero and one can therefore survive: fracγ_k(H_2)2^q ≡ V^qγ_k(X) +ϵ V^q-1Gγ_k-1(X) pmod2 A_2. Here V≡τpmod2 and τ^2≡1pmod2. Since q is even and ϵτ≡ϵpmod2, this reduces to γ_k(X)+ϵ Gγ_k-1(X). Moreover, G≡γ_2(X)+τ Qpmod2. After evaluation, every term with a single τ vanishes, so the second contribution reduces to L((γ_2(X)+Q)γ_k-1(X)). The γ_2(X) product has scalar binomk+12, which is even be…
Exact mathematical passage
\tau^2\equiv1\pmod2
\tau^{I_K}Use in source · 2 passages
Search aliases: τ^I_K tau
Use in source · Paper_XI_Traditional_Final.tex, line 693
[Conductor of a nontrivial tame atom] Let τ be an irreducible tame Weil representation of dimension n>1. Then Swan(τ)=0, dimτ^I_K=0, Art(τ)=n.
Exact mathematical passage
\Swan(\tau)=0, \qquad \dim\tau^{I_K}=0, \qquad \Art(\tau)=n.
\tau^mUse in source · 1 passage
Search aliases: τ^m tau
Use in source · Paper_VI_Traditional_Final.tex, line 4988
[Laplace transform and moments] Assume the spectrum of A lies in the open left half plane. Then absorption occurs with probability one. For s≥0, E[ e^-sτ] = a^T(sI-A)^-1p_0. For every integer m≥1, E[τ^m] = m! 1^T(-A)^-mp_0.
Exact mathematical passage
\Exp[\tau^m] = m!\one^T(-A)^{-m}p_0.
\tau_0Use in source · 2 passages
Search aliases: τ_0 tau
Use in source · atlas-pages.tex, line 356
1)^nt^3n(3n)! at the two rational endpoints. From the next term onward, magnitudes decrease, because their successive ratio is fract^3(3n+1)(3n+2)(3n+3)<1 there. The alternating-series remainder is bounded by the first omitted term. A separate outward interval evaluation of the differentiated polynomial, plus its tail bound, certifies the derivative sign. The first bracket is 1.84981279919014347629<τ_0<1.84981279919014347630. Its displacement differs from the leading asymptotic prediction by abo…
Exact mathematical passage
1.84981279919014347629<\tau_0<1.84981279919014347630.
\tau_MUse in source · 1 passage
Search aliases: τ_M tau
Use in source · 02_foundations.tex, line 437
onotonicity] Let φ:A→ B be a unital ring homomorphism and let Φ:M↠ N be a surjective additive map satisfying Φ(am)=φ(a)Φ(m) (a∈ A, m∈ M). Then srank_B(N)≤srank_A(M). If M and N are topological modules, Φ is continuous and surjective, and the actions are continuous, then srank_B,τ_N(N)≤srank_A,τ_M(M). [source reference] ExportsAll quotient lower bounds and realization inequalities.
Exact mathematical passage
\srank_{B,\tau_N}(N)\leq\srank_{A,\tau_M}(M).
\tau_m^NUse in source · 2 passages
Search aliases: τ_m^N tau
Use in source · Paper_IV_Traditional_Final.tex, line 2220
[Reduced zeros] Let Λ_N be the nonzero zero multiset of Δ_N. All but finitely many members of Λ_N are positive and simple. They can be indexed as λ_m=τ_m^N for all sufficiently large m. If N≥3, the notation may be chosen as in [source reference], and λ_m = t_m^N +O(t_m^N-1e^-γ t_m). For N=2, take τ_m=t_m=π(m+1/2) for m≥0. Then λ_m=t_m^2 exactly.
Exact mathematical passage
\lambda_m=\tau_m^N
\tau_NUse in source · 1 passage
Search aliases: τ_N tau
Use in source · 02_foundations.tex, line 437
onotonicity] Let φ:A→ B be a unital ring homomorphism and let Φ:M↠ N be a surjective additive map satisfying Φ(am)=φ(a)Φ(m) (a∈ A, m∈ M). Then srank_B(N)≤srank_A(M). If M and N are topological modules, Φ is continuous and surjective, and the actions are continuous, then srank_B,τ_N(N)≤srank_A,τ_M(M). [source reference] ExportsAll quotient lower bounds and realization inequalities.
Exact mathematical passage
\srank_{B,\tau_N}(N)\leq\srank_{A,\tau_M}(M).
\tau_pUse in source · 1 passage
Search aliases: τ_p tau
Use in source · 10_foundations.tex, line 12
The original arithmetic seed notation S=(A,a_0, O, R,{P_N,j},w,τ_p) records a coefficient ring or module, a seed, allowed operators, relations, residue selectors, a depth weight, and a topology. This notation describes the ingredients of the construction. A category of seed presentations and a notion of equivalence still require separate definitions. We therefore state each theorem for a concrete object. Section~[source reference] discusses the additional structure needed for the original native…
Exact mathematical passage
\mathcal S=(A,a_0,\mathcal O,\mathcal R,\{P_{N,j}\},w,\tau_p)
\tau_{Q+1}Use in source · 1 passage
Search aliases: τ_Q+1 tau
Use in source · Paper_V_Traditional_Final.tex, line 2534
[Higher order singular packet decomposition] Let 0=τ_0<τ_1<⋯<τ_Q<τ_Q+1=1. Suppose the restriction of f to every interval (τ_q,τ_q+1) belongs to W^r,p, and choose the one-dimensional Sobolev representatives whose derivatives through order r-1 have one-sided traces at the endpoints. Define a_q,m = jumpf^(m)_τ_q = f^(m)(τ_q+)-f^(m)(τ_q-) and j_f(t) = ∑_q=1^Q∑_m=0^r-1a_q,mϕ_m,τ_q(t). Then s_f=f-j_f has matching one-sided derivatives through order r-1 at every breakpoint and belongs to W^r,p(0,1). It…
Exact mathematical passage
0=\tau_0<\tau_1<\cdots<\tau_Q<\tau_{Q+1}=1.
\tau_{q+1}Use in source · 1 passage
Search aliases: τ_q+1 tau
Use in source · Paper_V_Traditional_Final.tex, line 2537
[Higher order singular packet decomposition] Let 0=τ_0<τ_1<⋯<τ_Q<τ_Q+1=1. Suppose the restriction of f to every interval (τ_q,τ_q+1) belongs to W^r,p, and choose the one-dimensional Sobolev representatives whose derivatives through order r-1 have one-sided traces at the endpoints. Define a_q,m = jumpf^(m)_τ_q = f^(m)(τ_q+)-f^(m)(τ_q-) and j_f(t) = ∑_q=1^Q∑_m=0^r-1a_q,mϕ_m,τ_q(t). Then s_f=f-j_f has matching one-sided derivatives through order r-1 at every breakpoint and belongs to W^r,p(0,1). It…
Exact mathematical passage
(\tau_q,\tau_{q+1})
\tau_rUse in source · 5 passages
Search aliases: τ_r tau
Use in source · 03_nonlinear_coordinates.tex, line 467
Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a to a leaf and applying the scalar map (t_1,…,t_r)↦ b_rt_1⋯ t_r at a vertex of arity r. A second induction gives ∑_τ∈ T_m B_τ(s) ≤ ∑_τ∈ T_mw(τ) =t_m, wh…
Exact mathematical passage
\tau_1,\ldots,\tau_r
\tau_r^2Use in source · 1 passage
Search aliases: τ_r^2 tau
Use in source · root_scope.tex, line 389
[Exact quadratic compression] There is a unique preparation K_r(Z)= U_r(Z) W_r(Z), W_r(Z)=Z^2+2β_rZ+α_r, U_r≡ u_rpmod256. With δ_r=β_r^2-α_r and τ_r= U_r'(c_r)/ U_r(c_r), δ_r=-fracΦ_r^act U_r(c_r) (1-fracτ_r^2Φ_r^act4 U_r(c_r)), v_2(δ_r)=v_2(Φ_r^act). The equality includes the zero case. The same successive monic polynomial division used in Theorem~[source reference], started at precision eight, preserves the quadratic and unit reductions of [source reference]. Write W=Z^2+2β_rZ+α_r and w=W(c_r)…
Exact mathematical passage
\delta_r=-\frac{\Phi_r^{\mathrm{act}}}{\mathcal U_r(c_r)} \left(1-\frac{\tau_r^2\Phi_r^{\mathrm{act}}}{4\mathcal U_r(c_r)}\right), \qquad v_2(\delta_r)=v_2(\Phi_r^{\mathrm{act}}). \label{eq:root:discriminant}
\tau_sUse in source · 1 passage
Search aliases: τ_s tau
Use in source · Paper_X_Block27_Amended.tex, line 946
Actual local comparisons from fixed-depth certificates. Let R be an I-adically complete separated commutative ring with finite coefficient quotients and an injective map into a field K. Fix a finite matrix rank. For finitely many local group maps ϕ_s:H_s→ G and integral template representations τ_s, fix detecting depths d_s. At precision m retain matrices P_s,m as part of the admissible object and require ρ_m(ϕ_s(h))P_s,m=P_s,mτ_s,m(h). For m≥ d_s require their determinants to remain nonzero aft…
Exact mathematical passage
\tau_s
\tau_{s,m}Use in source · 1 passage
Search aliases: τ_s,m tau
Use in source · Paper_X_Block27_Amended.tex, line 948
x-construction-15 Let R be an I-adically complete separated commutative ring with finite coefficient quotients and an injective map into a field K. Fix a finite matrix rank. For finitely many local group maps ϕ_s:H_s→ G and integral template representations τ_s, fix detecting depths d_s. At precision m retain matrices P_s,m as part of the admissible object and require ρ_m(ϕ_s(h))P_s,m=P_s,mτ_s,m(h). For m≥ d_s require their determinants to remain nonzero after reduction to R/I^d_s. Earlier level…
Exact mathematical passage
\rho_m(\phi_s(h))P_{s,m}=P_{s,m}\tau_{s,m}(h).
\tau_\etaUse in source · 5 passages
Search aliases: τ_η tau eta
Use in source · Paper_XI_Traditional_Final.tex, line 551
Let τ_ξ=Ind_W_K_n^W_Kξ, τ_η=Ind_W_K_m^W_Kη, where K_n/K and K_m/K are unramified. Put d=gcd(n,m), L=lcm(n,m).
Exact mathematical passage
\tau_\xi=\Ind_{W_{K_n}}^{W_K}\xi, \qquad \tau_\eta=\Ind_{W_{K_m}}^{W_K}\eta,
\tau_\xiUse in source · 6 passages
Search aliases: τ_ξ tau xi
Use in source · Paper_XI_Traditional_Final.tex, line 357
[Dual and self dual tame packets] For a tame atom τ_ξ=Ind_W_K_f^W_Kξ, we have τ_ξ^vee ≅ Ind_W_K_f^W_Kξ^-1. The atom is self dual exactly when ξ^-1=ξ^Φ^j for some j∈ Z/f Z.
Exact mathematical passage
\tau_\xi=\Ind_{W_{K_f}}^{W_K}\xi,
\tau_\xi^\veeUse in source · 2 passages
Search aliases: τ_ξ^vee tau xi vee
Use in source · Paper_XI_Traditional_Final.tex, line 361
[Dual and self dual tame packets] For a tame atom τ_ξ=Ind_W_K_f^W_Kξ, we have τ_ξ^vee ≅ Ind_W_K_f^W_Kξ^-1. The atom is self dual exactly when ξ^-1=ξ^Φ^j for some j∈ Z/f Z.
Exact mathematical passage
\tau_\xi^\vee \cong \Ind_{W_{K_f}}^{W_K}\xi^{-1}.
\Phi'Use in source · 1 passage
Search aliases: Φ' Phi
Use in source · 04_correction_transport.tex, line 364
[Naturality of filtered obstructions] Suppose a morphism of filtered lifting problems sends x_m to x_m', intertwines L_m with L_m', and sends Φ(x_m) to Φ'(x_m') on the associated graded pieces. Then the induced cokernel map sends Ob_m(x_m) to Ob_m'(x_m').
Exact mathematical passage
\Phi'(x_m')
\Phi^{(J)}Use in source · 2 passages
Search aliases: Φ^(J) Phi
Use in source · 03_supplement_02.tex, line 206
To prove complete positivity rather than positivity alone, let J be an arbitrary finite auxiliary index set. For a matrix X indexed by J× n, define its blockwise amplification by [Φ^(J)(X)]_(i,u),(j,v) =Φ(X_(i,·),(j,·))_u,v. The explicit index calculation proves the identity Φ^(J)(X)=∑_a (I_J⊗ K_a)X(I_J⊗ K_a)^*. The displayed identity identifies the defined amplification with tensoring by the identity, which is the operation needed for complete positivity. Every summand on the right is positive …
Exact mathematical passage
[\Phi^{(J)}(X)]_{(i,u),(j,v)} =\Phi\bigl(X_{(i,\cdot),(j,\cdot)}\bigr)_{u,v}.
\varphi^*Use in source · 2 passages
Search aliases: φ^* varphi
Use in source · Paper_XII_Traditional_Final.tex, line 923
[Support and Fitting transport] Let φ:R→ T be an isomorphism of Noetherian rings. Let M be a finite T module, viewed as an R module through φ. Then the homeomorphism φ^*:Spec T⟶Spec R satisfies φ^*(Supp_ T(M)) = Supp_R(M). For every j≥0, φ ( Fitt_j^R(M) ) = Fitt_j^ T(M).
Exact mathematical passage
\varphi^*:\Spec\TT\longrightarrow\Spec R
\Phi^+Use in source · 1 passage
Search aliases: Φ^+ Phi
Use in source · monograph_atlas.tex, line 294
vpage16Which coherence can a measurement see?THE ANSWER DEPENDS ON ITS EFFECTSChapter VII: Theorem II.7.2, Theorem II.7.4, Corollary II.7.5. For the Bell density matrix ρ=ketΦ^+braΦ^+, parity dephasing does nothing: both nonzero amplitudes already lie in the even subspace. Dephasing in the computational basis removes the entries between addresses 00 and 11. vfigdephasing Every effect in the computational basis E_a=ket abra a has the same expectation on both matrices, namely 1/2 at 00 and 11 and …
Exact mathematical passage
\rho=\ket{\Phi^+}\bra{\Phi^+}
\Phi^{-1}Use in source · 4 passages
Search aliases: Φ^-1 Phi
Use in source · Paper_XI_Traditional_Final.tex, line 324
[Tame character projectors] The operators P_j are pairwise orthogonal idempotents whose sum is the identity. Frobenius permutes them by Φ P_jΦ^-1=P_qj.
Exact mathematical passage
\Phi P_j\Phi^{-1}=P_{qj}.
\Phi^fUse in source · 2 passages
Search aliases: Φ^f Phi
Use in source · Paper_XI_Traditional_Final.tex, line 343
The nonzero tame packets are permuted by Frobenius. A union of Frobenius orbits is stable under W_K, so irreducibility forces one orbit. Choose one packet V_θ. Its stabilizer is W_K_f=⟨ I_K,Φ^f⟩. A nonzero proper stable subspace of V_θ would generate a nonzero proper W_K stable subspace after summing its Frobenius translates. Thus V_θ is irreducible for the stabilizer. Inertia acts on it by the scalar character θ, and the remaining action is generated by Φ^f. The stabilizer action on this packet…
Exact mathematical passage
W_{K_f}=\langle I_K,\Phi^f\rangle
\Phi^iUse in source · 1 passage
Search aliases: Φ^i Phi
Use in source · Paper_XI_Traditional_Final.tex, line 565
[Greatest common divisor and least common multiple packets] There is a decomposition τ_ξ⊗τ_η ≅ ⊕_i=0^d-1 Ind_W_K_L^W_K ( ξ_L,nη_L,m^Φ^i ), where ξ_L,n and η_L,m are the norm lifts to K_L.
Exact mathematical passage
\boxed{ \tau_\xi\otimes\tau_\eta \cong \bigoplus_{i=0}^{d-1} \Ind_{W_{K_L}}^{W_K} \left( \xi_{L,n}\eta_{L,m}^{\Phi^i} \right), }
\Phi^jUse in source · 1 passage
Search aliases: Φ^j Phi
Use in source · Paper_XI_Traditional_Final.tex, line 367
[Dual and self dual tame packets] For a tame atom τ_ξ=Ind_W_K_f^W_Kξ, we have τ_ξ^vee ≅ Ind_W_K_f^W_Kξ^-1. The atom is self dual exactly when ξ^-1=ξ^Φ^j for some j∈ Z/f Z.
Exact mathematical passage
\xi^{-1}=\xi^{\Phi^j}
\phi_{0,1/3}Use in source · 1 passage
Search aliases: ϕ_0,1/3 phi
Use in source · atlas-pages.tex, line 583
vpage31Derivative jumps have their own packetsTRACE DATA / A STEP AND A RAMPChapter V: Theorem E7.2, Corollary E7.3. On [0,1], let H_τ=1_[τ,1] and take f(t)=t^2+2H_1/3(t)-3(t-2/3)_+. The function has a value jump 2 at 1/3 and a first-derivative jump -3 at 2/3. Its packet and remainder are j_f(t)=2ϕ_0,1/3(t)-3ϕ_1,2/3(t), s_f(t)=t^2. vfig[.92]ext_trace platecaptionThe displayed curves are exact polynomial pieces. The solid jump is not connected across its discontinuity. The dashed curve is the glo…
Exact mathematical passage
j_f(t)=2\phi_{0,1/3}(t)-3\phi_{1,2/3}(t),\qquad s_f(t)=t^2.
\phi_{0,\tau}Use in source · 1 passage
Search aliases: ϕ_0,τ phi tau
Use in source · Paper_V_Traditional_Final.tex, line 2518
For m≥0 and 0<τ<1, define ϕ_m,τ(t) = frac(t-τ)_+^mm!. Thus ϕ_0,τ= 1_[τ,1]. On each side of τ, the packet is a polynomial. Its derivative of order m has a unit jump at τ.
Exact mathematical passage
\phi_{0,\tau}=\one_{[\tau,1]}
\phi_{1,2/3}Use in source · 1 passage
Search aliases: ϕ_1,2/3 phi
Use in source · atlas-pages.tex, line 583
vpage31Derivative jumps have their own packetsTRACE DATA / A STEP AND A RAMPChapter V: Theorem E7.2, Corollary E7.3. On [0,1], let H_τ=1_[τ,1] and take f(t)=t^2+2H_1/3(t)-3(t-2/3)_+. The function has a value jump 2 at 1/3 and a first-derivative jump -3 at 2/3. Its packet and remainder are j_f(t)=2ϕ_0,1/3(t)-3ϕ_1,2/3(t), s_f(t)=t^2. vfig[.92]ext_trace platecaptionThe displayed curves are exact polynomial pieces. The solid jump is not connected across its discontinuity. The dashed curve is the glo…
Exact mathematical passage
j_f(t)=2\phi_{0,1/3}(t)-3\phi_{1,2/3}(t),\qquad s_f(t)=t^2.
\Phi_2Use in source · 1 passage
Search aliases: Φ_2 Phi
Use in source · 72_recovered.tex, line 2192
The lowest degree term of Φ_M(z) is Mz, and its highest degree term is z^M. Multiplication by (1+z)^s gives the two edge coefficients. When M=2 and s=0, Φ_2(z)^k = (2z+z^2)^k = z^k(2+z)^k. Taking the coefficient of z^k+j gives the binary formula.
Exact mathematical passage
\Phi_2(z)^k = (2z+z^2)^k = z^k(2+z)^k.
\phi_3Use in source · 3 passages
Search aliases: ϕ_3 phi
Use in source · atlas-pages.tex, line 271
id and dashed curves sum to one. The curvature is caused by the shared evidence denominator, not by a nonlinear likelihood multiplication law. headTwo properties that must not be conflated If a>0 multiplies the odds, the first-coordinate map is ϕ_a(x)=fracax1+(a-1)x. Direct algebra gives ϕ_a(ϕ_b(x))=ϕ_ab(x). Thus a=e^t gives a nonlinear semigroup. Nevertheless, ϕ_3(1/2)=3/4, while the mean of ϕ_3(0) and ϕ_3(1) is 1/2. Superposition fails.
Exact mathematical passage
\phi_3(1/2)=3/4
\phi_aUse in source · 2 passages
Search aliases: ϕ_a phi
Use in source · atlas-pages.tex, line 268
nentially, while the probabilities remain inside the simplex. At t=log 2 the posterior is exactly (2/3,1/3). vfigodds platecaptionThe solid and dashed curves sum to one. The curvature is caused by the shared evidence denominator, not by a nonlinear likelihood multiplication law. headTwo properties that must not be conflated If a>0 multiplies the odds, the first-coordinate map is ϕ_a(x)=fracax1+(a-1)x. Direct algebra gives ϕ_a(ϕ_b(x))=ϕ_ab(x). Thus a=e^t gives a nonlinear semigroup. Nevertheless,…
Exact mathematical passage
\phi_a(x)=\frac{ax}{1+(a-1)x}.
\phi_{ab}Use in source · 1 passage
Search aliases: ϕ_ab phi
Use in source · atlas-pages.tex, line 271
lex. At t=log 2 the posterior is exactly (2/3,1/3). vfigodds platecaptionThe solid and dashed curves sum to one. The curvature is caused by the shared evidence denominator, not by a nonlinear likelihood multiplication law. headTwo properties that must not be conflated If a>0 multiplies the odds, the first-coordinate map is ϕ_a(x)=fracax1+(a-1)x. Direct algebra gives ϕ_a(ϕ_b(x))=ϕ_ab(x). Thus a=e^t gives a nonlinear semigroup. Nevertheless, ϕ_3(1/2)=3/4, while the mean of ϕ_3(0) and ϕ_3(1) is 1/2…
Exact mathematical passage
\phi_a(\phi_b(x))=\phi_{ab}(x)
\phi_bUse in source · 1 passage
Search aliases: ϕ_b phi
Use in source · atlas-pages.tex, line 271
lex. At t=log 2 the posterior is exactly (2/3,1/3). vfigodds platecaptionThe solid and dashed curves sum to one. The curvature is caused by the shared evidence denominator, not by a nonlinear likelihood multiplication law. headTwo properties that must not be conflated If a>0 multiplies the odds, the first-coordinate map is ϕ_a(x)=fracax1+(a-1)x. Direct algebra gives ϕ_a(ϕ_b(x))=ϕ_ab(x). Thus a=e^t gives a nonlinear semigroup. Nevertheless, ϕ_3(1/2)=3/4, while the mean of ϕ_3(0) and ϕ_3(1) is 1/2…
Exact mathematical passage
\phi_a(\phi_b(x))=\phi_{ab}(x)
\Phi_{cd,ab}Use in source · 3 passages
Search aliases: Φ_cd,ab Phi
Use in source · 02_strengthened.tex, line 1043
Let Φ:End( H_X)→End( H_Y) be linear. Choose packet systems (P_a) on H_X and (Q_c) on H_Y. Define Φ_cd,ab = Q_cdΦ P_ab.
Exact mathematical passage
\Phi_{cd,ab} = \Qcal_{cd}\Phi\Pcal_{ab}.
\Phi_jUse in source · 6 passages
Search aliases: Φ_j Phi
Use in source · 20_arithmetic.tex, line 1048
[Isometric packet bijections] For 0≤ j<4, define on natural numbers f_j(n)=fracU_j(n)-j!2^d_j, (d_0,d_1,d_2,d_3)=(1,1,5,3). There is a unique continuous map Φ_j: Z_2→ Z_2 whose value at every n∈ N equals the embedded rational f_j(n). It is an isometric bijection and Φ_j(0)=0: |Φ_j(x)-Φ_j(y)|_2=|x-y|_2 (x,y∈ Z_2). In particular, every element of Z_2 has exactly one preimage.
Exact mathematical passage
\Phi_j:\Z_2\to\Z_2
\Phi_MUse in source · 4 passages
Search aliases: Φ_M Phi
Use in source · 72_recovered.tex, line 2137
Let Φ_M(z)=(1+z)^M-1. For M≥1, s≥0, and r,k≥0, define Θ_M,s(r,k) = [z^r](1+z)^sΦ_M(z)^k. Equivalently, Θ_M,s(r,k) = ∑_j=0^k (-1)^k-jbinom kjbinomMj+sr. It vanishes unless k≤ r≤ Mk+s.
Exact mathematical passage
\Phi_M(z)=(1+z)^M-1.
\Phi_mUse in source · 2 passages
Search aliases: Φ_m Phi
Use in source · Paper_IV_Traditional_Final.tex, line 2037
Fix a large m, put a=2is_N, and consider the closed disk B_m={z:|z-t_m|≤ 1/(2s_N)}. The remainder bounds hold uniformly on these fixed-radius disks. For sufficiently large m, |R_N|≤ 1/2 and |R_N'|≤ s_N/2 there. Define Φ_m(z)=t_m+fraclog(1+R_N(z))2is_N, using the principal logarithm. The estimate |log(1+u)|≤(3/2)|u| shows that Φ_m maps B_m into itself. The derivative bound |Φ_m'|≤1/2 makes it a contraction. Banach's fixed-point theorem gives a unique fixed point. On B_m, |a(z-t_m)|≤1<π, so the pr…
Exact mathematical passage
\Phi_m(z)=t_m+\frac{\log(1+R_N(z))}{2is_N},
\Phi_m'Use in source · 1 passage
Search aliases: Φ_m' Phi
Use in source · Paper_IV_Traditional_Final.tex, line 2042
_N, and consider the closed disk B_m={z:|z-t_m|≤ 1/(2s_N)}. The remainder bounds hold uniformly on these fixed-radius disks. For sufficiently large m, |R_N|≤ 1/2 and |R_N'|≤ s_N/2 there. Define Φ_m(z)=t_m+fraclog(1+R_N(z))2is_N, using the principal logarithm. The estimate |log(1+u)|≤(3/2)|u| shows that Φ_m maps B_m into itself. The derivative bound |Φ_m'|≤1/2 makes it a contraction. Banach's fixed-point theorem gives a unique fixed point. On B_m, |a(z-t_m)|≤1<π, so the principal logarithm invert…
Exact mathematical passage
|\Phi_m'|\leq1/2
\phi_{m,u}Use in source · 1 passage
Search aliases: ϕ_m,u phi
Use in source · Paper_V_Traditional_Final.tex, line 2756
Assume τ<widehatτ. Differentiation with respect to the location parameter gives frac∂∂ uϕ_m,u(t) =-ϕ_m-1,u(t) for almost every t. Hence ϕ_m,τ-ϕ_m,widehatτ = ∫_τ^widehatτϕ_m-1,u d u. Minkowski's inequality gives ϕ_m,τ-ϕ_m,widehatτ _p ≤ |τ-widehatτ| sup_u ϕ_m-1,u _p. The supremum is bounded by the norm of t^m-1/(m-1)! on [0,1], which is the stated constant. The supremum norm argument is identical.
Exact mathematical passage
\frac{\partial}{\partial u}\phi_{m,u}(t) =-\phi_{m-1,u}(t)
\phi_{m,\widehat\tau}Use in source · 4 passages
Search aliases: ϕ_m,widehatτ phi widehat tau
Use in source · Paper_V_Traditional_Final.tex, line 2739
[Location stability for higher order packets] Let m≥1, 1≤ p<∞, and 0≤τ,widehatτ≤1. Then ϕ_m,τ-ϕ_m,widehatτ _L^p(0,1) ≤ frac|τ-widehatτ| (m-1)!((m-1)p+1)^1/p. For p=∞, ϕ_m,τ-ϕ_m,widehatτ _∞ ≤ frac|τ-widehatτ|(m-1)!. Thus location error is Lipschitz for every derivative jump packet of positive order, while a value jump has only the finite L^p modulus in [source reference].
Exact mathematical passage
\|\phi_{m,\tau}-\phi_{m,\widehat\tau}\|_{L^p(0,1)} \leq \frac{|\tau-\widehat\tau|} {(m-1)!((m-1)p+1)^{1/p}}.
\phi_{m,\tau}^{(\ell)}Use in source · 1 passage
Search aliases: ϕ_m,τ^(ℓ) phi tau ell
Use in source · Paper_V_Traditional_Final.tex, line 2560
For 0≤ ℓ≤ m, the classical derivative of ϕ_m,τ on the two open sides is ϕ_m,τ^(ℓ)(t) = frac(t-τ)_+^m-ℓ(m-ℓ)!. This derivative has zero jump when ℓ<m and a unit jump when ℓ=m. Derivatives of order larger than m vanish on both open sides. It follows that subtracting the term a_q,mϕ_m,τ_q removes exactly the jump in the derivative of order m without changing the other classical trace jumps. Hence s_f has matching traces through order r-1.
Exact mathematical passage
\phi_{m,\tau}^{(\ell)}(t) = \frac{(t-\tau)_+^{m-\ell}}{(m-\ell)!}.
\phi_{m,\tau_q}Use in source · 4 passages
Search aliases: ϕ_m,τ_q phi tau
Use in source · Paper_V_Traditional_Final.tex, line 2546
_packet_decomposition Let 0=τ_0<τ_1<⋯<τ_Q<τ_Q+1=1. Suppose the restriction of f to every interval (τ_q,τ_q+1) belongs to W^r,p, and choose the one-dimensional Sobolev representatives whose derivatives through order r-1 have one-sided traces at the endpoints. Define a_q,m = jumpf^(m)_τ_q = f^(m)(τ_q+)-f^(m)(τ_q-) and j_f(t) = ∑_q=1^Q∑_m=0^r-1a_q,mϕ_m,τ_q(t). Then s_f=f-j_f has matching one-sided derivatives through order r-1 at every breakpoint and belongs to W^r,p(0,1). Its weak derivative of or…
Exact mathematical passage
j_f(t) = \sum_{q=1}^{Q}\sum_{m=0}^{r-1}a_{q,m}\phi_{m,\tau_q}(t).
\phi_{m-1,u}Use in source · 3 passages
Search aliases: ϕ_m-1,u phi
Use in source · Paper_V_Traditional_Final.tex, line 2756
Assume τ<widehatτ. Differentiation with respect to the location parameter gives frac∂∂ uϕ_m,u(t) =-ϕ_m-1,u(t) for almost every t. Hence ϕ_m,τ-ϕ_m,widehatτ = ∫_τ^widehatτϕ_m-1,u d u. Minkowski's inequality gives ϕ_m,τ-ϕ_m,widehatτ _p ≤ |τ-widehatτ| sup_u ϕ_m-1,u _p. The supremum is bounded by the norm of t^m-1/(m-1)! on [0,1], which is the stated constant. The supremum norm argument is identical.
Exact mathematical passage
\frac{\partial}{\partial u}\phi_{m,u}(t) =-\phi_{m-1,u}(t)
\Phi_rUse in source · 1 passage
Search aliases: Φ_r Phi
Use in source · root_scope.tex, line 376
K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z), a_r,b_r∈ Z_2, u_r∈ Z_2^×, R_r∈ Z_2⟨ Z⟩. Its half derivative is a dyadic isometry. It therefore has a unique critical point c_r∈ Z_2. Set Φ_r^act=K_r(c_r). The earlier independent-undepletion notation for this same scalar is Φ_r(2^2r-25). Excluding a multiple root on the surviving chart is exactly the assertion Φ_r^act≠0.
Exact mathematical passage
\Phi_r(2^{2r-25})
\Phi_{r,m}^{\mathrm{act}}Use in source · 2 passages
Search aliases: Φ_r,m^act Phi mathrm
Use in source · root_scope.tex, line 431
The latest continuation retains the exact Bernoulli undepletion factor e_r,q=frac1-2^2r-11-2^2r-14^q, q≥0. It is a dyadic unit, including e_r,0=1. Multiplication by this factor preserves the all-q error of the shifted Jacobi moment model. Consequently the actual model critical values satisfy, for every m≥25, v_2(Φ_r,m^act-Φ_r^act) ≥ L_m=8m-4s_2(m)-178, L_m⟶∞. The estimate includes every packet and controls motion of the critical point. There is no ceiling at the undepletion exponent 2r-25. For e…
Exact mathematical passage
v_2\bigl(\Phi_{r,m}^{\mathrm{act}}-\Phi_r^{\mathrm{act}}\bigr) \ge L_m=8m-4s_2(m)-178, \qquad L_m\longrightarrow\infty. \label{eq:root:actualapprox}
\Phi_r^{\mathrm{act}}Use in source · 6 passages
Search aliases: Φ_r^act Phi mathrm
Use in source · root_scope.tex, line 374
ion K_r(Z)=2^-178G_r(10202+2^15Z). The complete coefficient theorem gives K_r(Z)=a_r+2b_rZ+u_rZ^2+256Z^3R_r(Z), a_r,b_r∈ Z_2, u_r∈ Z_2^×, R_r∈ Z_2⟨ Z⟩. Its half derivative is a dyadic isometry. It therefore has a unique critical point c_r∈ Z_2. Set Φ_r^act=K_r(c_r). The earlier independent-undepletion notation for this same scalar is Φ_r(2^2r-25). Excluding a multiple root on the surviving chart is exactly the assertion Φ_r^act≠0.
Exact mathematical passage
\Phi_r^{\mathrm{act}}=K_r(c_r)
\Phi_sUse in source · 4 passages
Search aliases: Φ_s Phi
Use in source · visual_atlas.tex, line 114
vpage5Relations identify words, not just outputsPRESENTATIONS / WHY A LEFT IDEAL APPEARSMain text: [source reference] Let F=𝕜⟨ z_1,…,z_r⟩ be the algebra of noncommuting operator words. Evaluation at a fixed seed is Φ_s:F→ X, f↦ f(T_1,…,T_r)s. Two words represent the same generated vector exactly when their difference lies in I_s=KerΦ_s. Consequently F/I_s≅Hull_F(s) as left modules. The relation f(T)s=0 implies g(T)f(T)s=0 for every word g, which explains closure under multiplication on the left.
Exact mathematical passage
\Phi_s:F\to X,\qquad f\mapsto f(T_1,\ldots,T_r)s.
\phi_sUse in source · 2 passages
Search aliases: ϕ_s phi
Use in source · Paper_X_Block27_Amended.tex, line 945
Actual local comparisons from fixed-depth certificates. Let R be an I-adically complete separated commutative ring with finite coefficient quotients and an injective map into a field K. Fix a finite matrix rank. For finitely many local group maps ϕ_s:H_s→ G and integral template representations τ_s, fix detecting depths d_s. At precision m retain matrices P_s,m as part of the admissible object and require ρ_m(ϕ_s(h))P_s,m=P_s,mτ_s,m(h). For m≥ d_s require their determinants to remain nonzero aft…
Exact mathematical passage
\phi_s:H_s\to G
\Phi_{s+t}Use in source · 2 passages
Search aliases: Φ_s+t Phi
Use in source · Paper_VI_Traditional_Final.tex, line 1041
[Bayesian nonlinear semigroup] Φ_s+t=Φ_s∘Φ_t. Its generator is the centered multiplicative field fracddtp_i(t) = (g_i-⟨ g,p(t)⟩)p_i(t).
Exact mathematical passage
\boxed{ \Phi_{s+t}=\Phi_s\circ\Phi_t. }
\Phi_tUse in source · 5 passages
Search aliases: Φ_t Phi
Use in source · Paper_VI_Traditional_Final.tex, line 1034
Fix a score vector g. The likelihood flow is λ_i(t)=e^tg_i, and the Bayesian flow is Φ_t(p)_i = frace^tg_ip_i∑_je^tg_jp_j.
Exact mathematical passage
\Phi_t(p)_i = \frac{e^{tg_i}p_i}{\sum_je^{tg_j}p_j}.
\Phi_{x,L}Use in source · 2 passages
Search aliases: Φ_x,L Phi
Use in source · continuation_02.tex, line 48
For every positive integer q, including composite moduli and modulus one, and every L≥0, the checked map Φ_x,L:( Z/q Z)^d⟶ {ybmod q^L+1:y≡ xpmodq^L}, t⟼ x+q^Lwidetilde t is a bijection. Here widetilde t is the canonical integer representative of the digit vector. The entire residue fiber therefore has q^d elements. Injectivity is proved by cancelling q^L in the corresponding integer divisibility statement. Surjectivity comes from the lower-residue congruence.
Exact mathematical passage
\Phi_{x,L}:(\mathbb Z/q\mathbb Z)^d\longrightarrow \{y\bmod q^{L+1}:y\equiv x\pmod{q^L}\}, \qquad t\longmapsto x+q^L\widetilde t
\Phi_yUse in source · 3 passages
Search aliases: Φ_y Phi
Use in source · Paper_VI_Traditional_Final.tex, line 5535
For stochastic matrices K,overline K and positive likelihoods L_y,overline L_y, put Φ_y= B_L_y∘ K and overlineΦ_y= B_overline L_y∘overline K. [Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯∘Φ_y_1 = overlineΦ_y_r∘⋯∘overlineΦ_y_1C.
Exact mathematical passage
\Phi_y=\Bcal_{L_y}\circ K
\Phi_{y_1}Use in source · 1 passage
Search aliases: Φ_y_1 Phi
Use in source · Paper_VI_Traditional_Final.tex, line 5546
[Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯∘Φ_y_1 = overlineΦ_y_r∘⋯∘overlineΦ_y_1C.
Exact mathematical passage
C\Phi_{y_r}\circ\cdots\circ\Phi_{y_1} = \overline\Phi_{y_r}\circ\cdots\circ\overline\Phi_{y_1}C.
\Phi_{y_r}Use in source · 1 passage
Search aliases: Φ_y_r Phi
Use in source · Paper_VI_Traditional_Final.tex, line 5546
[Exact coarse hidden Markov filter] Assume CK=overline KC and every likelihood L_y is constant on each coarse block with block values overline L_y. Then CΦ_y(p)=overlineΦ_y(Cp) for every prior. For any observation sequence y_1,…,y_r, the complete filtering recursion commutes with aggregation: CΦ_y_r∘⋯∘Φ_y_1 = overlineΦ_y_r∘⋯∘overlineΦ_y_1C.
Exact mathematical passage
C\Phi_{y_r}\circ\cdots\circ\Phi_{y_1} = \overline\Phi_{y_r}\circ\cdots\circ\overline\Phi_{y_1}C.
\Phi_{\alpha\beta;\gamma\delta}Use in source · 1 passage
Search aliases: Φ_αβ;γδ Phi alpha beta gamma delta
Use in source · 01_finite_core.tex, line 751
Packet superoperator blocks are defined by Φ_αβ;γδ(X) = P_α Φ(P_γ XP_δ) P_β. Summing all four packet indices reconstructs Φ(X) exactly.
Exact mathematical passage
\Phi_{\alpha\beta;\gamma\delta}(X) = P_\alpha \Phi(P_\gamma XP_\delta) P_\beta.
\Phi_\gammaUse in source · 1 passage
Search aliases: Φ_γ Phi gamma
Use in source · 02_strengthened.tex, line 1196
0 · 0 . The channel Φ_γ(ρ)=K_0ρ K_0^*+K_1ρ K_1^* is trace preserving. It sends the off diagonal packet ket0bra1 to sqrt1-γket0bra1, while the excited population transfers partly into the ground population.
Exact mathematical passage
\Phi_\gamma(\rho)=K_0\rho K_0^*+K_1\rho K_1^*
\Phi_\thetaUse in source · 5 passages
Search aliases: Φ_θ Phi theta
Use in source · Paper_V_Traditional_Final.tex, line 3669
[Cyclic M obius action] The pullback Φ_θ =C^-1R_θ C is the fractional linear map Φ_θ(z) = frac(1+ e^ iθ)z+(1- e^ iθ) (1- e^ iθ)z+(1+ e^ iθ). It preserves H and satisfies Φ_θΦ_φ = Φ_θ+φ. For an integer N≥1 and θ=2π/N, the map has order N.
Exact mathematical passage
\Phi_\theta =C^{-1}R_\theta C
\Phi_{\theta+\varphi}Use in source · 1 passage
Search aliases: Φ_θ+φ Phi theta varphi
Use in source · Paper_V_Traditional_Final.tex, line 3683
[Cyclic M obius action] The pullback Φ_θ =C^-1R_θ C is the fractional linear map Φ_θ(z) = frac(1+ e^ iθ)z+(1- e^ iθ) (1- e^ iθ)z+(1+ e^ iθ). It preserves H and satisfies Φ_θΦ_φ = Φ_θ+φ. For an integer N≥1 and θ=2π/N, the map has order N.
Exact mathematical passage
\Phi_\theta\Phi_\varphi = \Phi_{\theta+\varphi}.
\Phi_\piUse in source · 1 passage
Search aliases: Φ_π Phi pi
Use in source · Paper_V_Traditional_Final.tex, line 3697
[Impedance and admittance] For N=2, θ=π and Φ_π(z)=frac1z. Thus the two phase action is normalized impedance and admittance inversion.
Exact mathematical passage
\Phi_\pi(z)=\frac1z.
\Phi_\varphiUse in source · 1 passage
Search aliases: Φ_φ Phi varphi
Use in source · Paper_V_Traditional_Final.tex, line 3683
[Cyclic M obius action] The pullback Φ_θ =C^-1R_θ C is the fractional linear map Φ_θ(z) = frac(1+ e^ iθ)z+(1- e^ iθ) (1- e^ iθ)z+(1+ e^ iθ). It preserves H and satisfies Φ_θΦ_φ = Φ_θ+φ. For an integer N≥1 and θ=2π/N, the map has order N.
Exact mathematical passage
\Phi_\theta\Phi_\varphi = \Phi_{\theta+\varphi}.
\chi^{-1}Use in source · 2 passages
Search aliases: χ^-1 chi
Use in source · Paper_XI_Traditional_Final.tex, line 751
[Duality of Gauss phases] For nontrivial χ, γ(χ,ψ)γ(χ^-1,ψ) = χ(-1).
Exact mathematical passage
\boxed{ \gamma(\chi,\psi)\gamma(\chi^{-1},\psi) = \chi(-1). }
\chi^\bulletUse in source · 1 passage
Search aliases: χ^bullet chi bullet
Use in source · 72_recovered.tex, line 847
Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding term vanishes for positive n. Otherwise ((1+ζ y)^m)^h = (1+ζ y)^p-1 =1. Every summand is therefore h periodic for positive n. Thus T_m,p preserves V_h.…
Exact mathematical passage
T_{m,p}(\chi^\bullet)_n = \frac1m \sum_{\zeta\in\mu_m} (1+\zeta y)^{mn}.
\chi^qUse in source · 1 passage
Search aliases: χ^q chi
Use in source · 72_recovered.tex, line 843
Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding term vanishes for positive n. Otherwise ((1+ζ y)^m)^h = (1+ζ y)^p-1 =1. Every summand is therefore h periodic for positive n. Thus T_m,p preserves V_h.…
Exact mathematical passage
d_q=\sum_{\chi\in\mu_h}c_\chi\chi^q.
\chi_+Use in source · 1 passage
Search aliases: χ_+ chi
Use in source · Paper_X_Block27_Amended.tex, line 483
[The two sign branches] The characters χ_+(g)=1 and χ_-(g)=-1 have the same reduction modulo 2, but 1not≡-1pmod4. Their first separation depth is therefore 2. One mod-two root contains two branches that become distinct at the next level.
Exact mathematical passage
\chi_+(g)=1
\chi_-Use in source · 1 passage
Search aliases: χ_- chi
Use in source · Paper_X_Block27_Amended.tex, line 483
[The two sign branches] The characters χ_+(g)=1 and χ_-(g)=-1 have the same reduction modulo 2, but 1not≡-1pmod4. Their first separation depth is therefore 2. One mod-two root contains two branches that become distinct at the next level.
Exact mathematical passage
\chi_-(g)=-1
\chi_AUse in source · 6 passages
Search aliases: χ_A chi
Use in source · 04_arithmetic.tex, line 11
[Bézout spectral projector] Assume that χ_A(x)=f(x)g(x) with monic polynomials f,g∈ O[x] that are coprime in K[x]. Choose u,v∈ O[x] and a nonzero r∈ O such that u(x)f(x)+v(x)g(x)=r. Then E_f=r^-1v(A)g(A) is the projector of V onto Ker f(A) along Ker g(A). In particular, E_f^2=E_f, rE_f∈End_ O(Λ).
Exact mathematical passage
\chi_A(x)=f(x)g(x)
\chi_CUse in source · 3 passages
Search aliases: χ_C chi
Use in source · 72_recovered.tex, line 2824
[Characteristic polynomial of the jump] Let χ_C(y)=det(yI-C). Then χ_C^N(x) = Res_y(χ_C(y),x-y^N). The identity holds over the coefficient ring after interpreting the resultant universally.
Exact mathematical passage
\chi_C(y)=\det(yI-C).
\chi_{C^N}Use in source · 4 passages
Search aliases: χ_C^N chi
Use in source · 10_foundations.tex, line 415
[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalization hypothesis is required.
Exact mathematical passage
\chi_{C^N}(X)=X^d+\sum_{r<d}q_rX^r
\chi_HUse in source · 1 passage
Search aliases: χ_H chi
Use in source · visual_atlas.tex, line 279
The usual scalar invariants can be misleading: trH=2, det H=1, χ_H(t)=(t-1)^2. These are also the invariants of the 2×2 identity matrix. Nevertheless, Ob=H-I=mat0 · 1
Exact mathematical passage
\operatorname{tr}H=2,\qquad \det H=1,\qquad \chi_H(t)=(t-1)^2.
\chi_iUse in source · 2 passages
Search aliases: χ_i chi
Use in source · residue_continuation_body.tex, line 181
For a finite family of nontrivial ordinary class characters, suppose functions p,Z satisfy h p(h)→ P≠0, h Z(h)→ Q≠0, and, for all sufficiently small h>0, Z(h)=(p(h)∏_i F_χ_i(h))^e, e∈ N. The checked real-axis criterion then gives e=1 and F_χ_i(0)≠0 for every i. The pole limits and product identity remain hypotheses of this application.
Exact mathematical passage
Z(h)=\left(p(h)\prod_i F_{\chi_i}(h)\right)^e, \qquad e\in\NN.
\chi_KUse in source · 2 passages
Search aliases: χ_K chi
Use in source · Paper_X_Block27_Amended.tex, line 2433
For every commutative coefficient ring B and χ∈ P_q(B), define χ_K=χ∘ j_K. The local character is continuous and commutes with coefficient change. The precise local admissibility condition is LocallyTrivial_K(χ) Longleftrightarrow χ_K=1. Descent through the fixed quadratic quotient proves LocallyTrivial_K(χ) Longleftrightarrow (∃ x∈ K, x^2=-1) lor χ=1. No reducedness, characteristic-zero or domain hypothesis is imposed on B. In particular, the formula applies to the finite coefficient quotients …
Exact mathematical passage
\chi_K=\chi\circ j_K
\chi_{\Q_{p_s}}Use in source · 1 passage
Search aliases: χ_Q_p_s χ_ Q_p_s chi Q chi mathbf
Use in source · Paper_X_Block27_Amended.tex, line 2522
Suppose now that R is I-adically complete and that every positive quotient R/I^m is finite. Let C_m⊆ P_q(R/I^m) be allowed subsets preserved by reduction, and let (p_s)_s∈ S be any indexed family of arithmetic primes. The index set S need not be finite. Lean proves the simultaneous arithmetic completion criterion · ∃χ∈ P_q(R), χ continuous, χbmod I^m∈ C_m (m≥1), χ_ Q_p_s=1 (s∈ S) · Longleftrightarrow ∀ m≥1, ∃ψ∈ C_m, ∀ s∈ S, p_s≡1pmod4 lor ψ=1. The constructed character also satisfies every odd-p…
Exact mathematical passage
\begin{split} &\exists\chi\in\mathcal P_q(R),\quad \chi\text{ continuous},\quad \chi\bmod I^m\in C_m\ (m\ge1),\quad \chi_{\Q_{p_s}}=1\ (s\in S)\\ &\quad\Longleftrightarrow\quad \forall m\ge1,\ \exists\psi\in C_m,\quad \forall s\in S,\ p_s\equiv1\pmod4\ \lor\ \psi=1. \end{split}
\chi_\thetaUse in source · 5 passages
Search aliases: χ_θ chi theta
Use in source · Paper_XI_Traditional_Final.tex, line 465
[Primitive cuspidal idempotent] Let χ_θ be the irreducible character of κ_θ, and let d_θ=χ_θ(1). Then e_θ = fracd_θabsG_n ∑_g∈ G_n χ_θ(g^-1)g is the primitive central idempotent selecting the κ_θ block in C[G_n].
Exact mathematical passage
\chi_\theta
\PsiUse in source · 19 passages
Search aliases: Ψ Psi
Use in source · 04_transport.tex, line 148
Let (Φ,φ,σ):(X, A, P)→(Y, B, Q) and (Ψ,ψ,τ):(Y, B, Q)→(Z, C, R) be morphisms. Intertwining is preserved because ΨΦ(Ax) = Ψ(φ(A)Φ x) = (ψ∘φ)(A)ΨΦ x. For k in the target packet set, R_kΨΦ · =Ψ Q_τ^-1(k)Φ
Exact mathematical passage
(\Psi,\psi,\tau):(Y,\B,\Qcal)\to(Z,\Ccal,\Rcal)
\psi'Use in source · 1 passage
Search aliases: ψ' psi
Use in source · Paper_VI_Traditional_Final.tex, line 3809
[Evidence cumulants and score monotonicity] Set ψ(t) = log( ∑_i e^tg_ip_i(0) ). Then ψ'(t)=ipgp(t), ψ''(t)=Var_p(t)(g)≥0. Consequently, frac d d tipgp(t) = Var_p(t)(g).
Exact mathematical passage
\psi'(t)=\ip{g}{p(t)}, \qquad \psi''(t)=\Var_{p(t)}(g)\geq0.
\psi''Use in source · 1 passage
Search aliases: ψ'' psi
Use in source · Paper_VI_Traditional_Final.tex, line 3809
[Evidence cumulants and score monotonicity] Set ψ(t) = log( ∑_i e^tg_ip_i(0) ). Then ψ'(t)=ipgp(t), ψ''(t)=Var_p(t)(g)≥0. Consequently, frac d d tipgp(t) = Var_p(t)(g).
Exact mathematical passage
\psi'(t)=\ip{g}{p(t)}, \qquad \psi''(t)=\Var_{p(t)}(g)\geq0.
\psi^*Use in source · 4 passages
Search aliases: ψ^* psi
Use in source · monograph_atlas.tex, line 10
to follow when its objects remain visible. This appendix follows a state vector, a density matrix, and a retained part of a dynamical system. They may describe the same experiment, but they do not answer the same question. vfigthree_objects headAmplitudes retain relative phase For the Bell vector ψ=(ket00+ket11)/sqrt2, the two nonzero entries are amplitudes. The density matrix ρ=ψψ^* also records coherence in its off-diagonal entries. Measurements that compare the two addresses can detect this c…
Exact mathematical passage
\rho=\psi\psi^*
\psi^{-1}Use in source · 1 passage
Search aliases: ψ^-1 psi
Use in source · 40_regularity.tex, line 131
rhoods as follows. The source of the open partial homeomorphism defined by Φ is open around a. Its inverse is g. Choose open sets V and U_0 containing y_0 and t_0 such that V× U_0 lies in that source. Restrict U_0 further by the preimage of an open analytic inverse neighborhood under t↦(0,t), the open locus where this same ψ is analytic, and the interior of ψ^-1(V). All of these sets contain t_0. For the last set, use the continuity of ψ at t_0 and ψ(t_0)=y_0∈ V. Their intersection is an open ne…
Exact mathematical passage
\psi^{-1}(V)
\Psi_{ef,cd}Use in source · 1 passage
Search aliases: Ψ_ef,cd Psi
Use in source · 02_strengthened.tex, line 1055
[Exact channel block calculus] Every linear map has the exact reconstruction Φ=∑_a,b,c,dΦ_cd,ab. If Ψ:End( H_Y)→End( H_Z), then (ΨΦ)_ef,ab = ∑_c,d Ψ_ef,cdΦ_cd,ab.
Exact mathematical passage
(\Psi\Phi)_{ef,ab} = \sum_{c,d} \Psi_{ef,cd}\Phi_{cd,ab}.
\psi_FUse in source · 2 passages
Search aliases: ψ_F psi
Use in source · dyadic_synthesis.tex, line 13
Let F= F_2((t)), fix an additive character ψ_F of conductor t O_F, and take increasing odd positive integers h_1<⋯<h_d. Put q_i=3h_i, N=2^d, n_j=2^d-jq_d-j+1. There is an explicit β∈ M_N(F) generating a separable totally ramified degree-N field that normalizes the complete lattice chain. It gives a maximal simple stratum with leading level n_1, final critical exponent -q_1, and successive approximation-field degrees 2,4,…,2^d, with critical levels n_2,…,n_d.
Exact mathematical passage
\psi_F
\Psi_{g,\ell}Use in source · 1 passage
Search aliases: Ψ_g,ℓ Psi ell
Use in source · 01_finite_core.tex, line 1239
[Einstein infrared limit] There exist semiclassical packet states Ψ_g,ℓ such that the effective action converges to frac116π G ∫sqrt-g (R-2Λ) d^4x with controlled higher-order corrections.
Exact mathematical passage
\Psi_{g,\ell}
\psi_iUse in source · 1 passage
Search aliases: ψ_i psi
Use in source · 01_finite_core.tex, line 92
A ≅ ⊕_i=1^r (M_d_i( C)⊗ I_m_i) on H ≅ ⊕_i=1^r ( C^d_i⊗ C^m_i). Here every d_i is a positive integer and every multiplicity m_i is a nonnegative integer. An empty block family represents the zero space and has seed rank zero. The maximum over an empty block family is also assigned the value zero. A vector ψ_i∈ C^d_i⊗ C^m_i is identified with a d_i× m_i coefficient matrix C_i.
Exact mathematical passage
\psi_i\in\bbC^{d_i}\otimes\bbC^{m_i}
\psi_{j,k}Use in source · 7 passages
Search aliases: ψ_j,k psi
Use in source · Paper_V_Traditional_Final.tex, line 3196
Let I_j,k = [k2^-j,(k+1)2^-j) and let ψ_j,k be the normalized Haar wavelet supported on I_j,k, positive on the left half and negative on the right half.
Exact mathematical passage
\psi_{j,k}
\Psi_NUse in source · 2 passages
Search aliases: Ψ_N Psi
Use in source · Paper_IV_Traditional_Final.tex, line 2545
The reduced unsigned denominator is Ψ_N(w)=∑_n≥0fracw^n(Nn)!=Δ_N(-w). By [source reference], -logΨ_N(w) = ∑_m≥1widetildeσ_N,mfracw^mm. With w=z^N, the logarithm of the theta generating function is ∑_m≥1 widetildeσ_N,m(u^Nm+u)fracw^mm. The complete Bell polynomial formula follows.
Exact mathematical passage
\Psi_N(w)=\sum_{n\geq0}\frac{w^n}{(Nn)!}=\Delta_N(-w).
\psi_{\beta-\gamma}Use in source · 1 passage
Search aliases: ψ_β-γ psi beta gamma
Use in source · dyadic_synthesis.tex, line 111
nit character to multiplicity two. BK 3.6.13(ii) and 3.6.14 give, on scalar units, Θ_j(uI)=θ_j(u^2)=1. For odd critical level r, use m=lfloor r/2rfloor and D=H^m+1(β)=H^m+1(γ). The endpoint is valid because 2(m+1)-r=1: the cross term for the additive character lies in the radical on which the matrix additive character is trivial. BK 3.3.18 supplies ϑ=Θ_j|_D ψ_β-γ. Integral trace makes it trivial on D∩ U_F^1.
Exact mathematical passage
\vartheta=\Theta_j|_D\,\psi_{\beta-\gamma}
\omega^{(k-j)r}Use in source · 1 passage
Search aliases: ω^(k-j)r omega
Use in source · 01_finite_core.tex, line 217
Using S^r+s=S^r+sbmod N and the finite character orthogonality relation, frac1N∑_r=0^N-1ω^(k-j)r=δ_jk, we obtain P_jP_k=δ_jkP_j. Summing over j cancels every term indexed by r≠0 and retains the r=0 term, which proves completeness. Unitarity and the change of residue r↦-r give P_j^*=frac1N∑_r=0^N-1ω^jrS^-r=P_j. Finally, multiplication by S shifts the summation index and contributes the factor ω^j.
Exact mathematical passage
\frac1N\sum_{r=0}^{N-1}\omega^{(k-j)r}=\delta_{jk},
\omega^{(m-j)\ell}Use in source · 1 passage
Search aliases: ω^(m-j)ℓ omega ell
Use in source · Paper_IV_Traditional_Final.tex, line 394
Substitution of the power series gives B_N,jF(z) = ∑_m≥0a_mz^m ( frac1N∑_ℓ=0^N-1ω^(m-j)ℓ ). The inner average is one when m≡ jbmod N and zero otherwise. The operator identities follow from the disjoint residue classes.
Exact mathematical passage
\Bcal_{N,j}F(z) = \sum_{m\geq0}a_mz^m \left( \frac1N\sum_{\ell=0}^{N-1}\omega^{(m-j)\ell} \right).
\omega^{-1}Use in source · 2 passages
Search aliases: ω^-1 omega
Use in source · 02_coordinates.tex, line 427
0, · r≠ j, 0≤ j,r<N. Then P_N,j = η∑_k=0^N-1(ω^vee)^jkU^k. In particular, if R is a field, N>0, and ω is a primitive Nth root of unity, the certificate holds with ω^vee=ω^-1 and η=N^-1.
Exact mathematical passage
\omega^\vee=\omega^{-1}
\omega^{-jk}Use in source · 2 passages
Search aliases: ω^-jk omega
Use in source · 02_coordinates.tex, line 457
The field specialization. [Character realization] Let bbk be a field, assume that N is nonzero in bbk, and suppose that bbk contains a primitive Nth root of unity ω. Define (UF)(z)=F(ω z). Then P_N,j = frac1N∑_k=0^N-1ω^-jkU^k.
Exact mathematical passage
P_{N,j} = \frac1N\sum_{k=0}^{N-1}\omega^{-jk}U^k.
\omega^{-jr}Use in source · 2 passages
Search aliases: ω^-jr omega
Use in source · 01_finite_core.tex, line 196
[Cyclic quantum projectors] The operators P_j = frac1N∑_r=0^N-1ω^-jrS^r, 0≤ j<N, are self-adjoint, pairwise orthogonal projectors satisfying P_jP_k=δ_jkP_j, ∑_j=0^N-1P_j=I, SP_j=ω^jP_j.
Exact mathematical passage
\boxed{ P_j = \frac1N\sum_{r=0}^{N-1}\omega^{-jr}S^r, \qquad 0\le j<N, }
\omega^{-j\ell}Use in source · 2 passages
Search aliases: ω^-jℓ omega ell
Use in source · Paper_IV_Traditional_Final.tex, line 363
Fix N≥1 and write ω=e^2π i/N. Let Ω be a disk centered at the origin, with the whole plane allowed, and assume that Ω is invariant under multiplication by ω. For a holomorphic function F on Ω, define ( B_N,jF)(z) = frac1N∑_ℓ=0^N-1ω^-jℓF(ω^ℓ z), 0≤ j<N. The same formula is valid for formal power series.
Exact mathematical passage
(\Bcal_{N,j}F)(z) = \frac1N\sum_{\ell=0}^{N-1}\omega^{-j\ell}F(\omega^\ell z), \qquad 0\leq j<N.
\omega^{-k}Use in source · 2 passages
Search aliases: ω^-k omega
Use in source · Paper_IV_Traditional_Final.tex, line 1983
The derivatives satisfy D_N'(ω^kτ_m) = ω^-k D_N'(τ_m) and | D_N'(τ_m)| = frac2s_NNe^c_Nτ_m (1+O(e^-γτ_m)).
Exact mathematical passage
\Dfun_N'(\omega^k\tau_m) = \omega^{-k}\Dfun_N'(\tau_m)
\omega^{-qr}Use in source · 1 passage
Search aliases: ω^-qr omega
Use in source · 02_strengthened.tex, line 1099
[Coherence order projectors] For q∈ Z/N Z, define L_q(X) = frac1N ∑_r=0^N-1 ω^-qrS^rXS^-r. Then the L_q form a complete orthogonal packet system on Hilbert--Schmidt operator space. Their ranges are the operators satisfying SXS^*=ω^qX. If P_j are the state charge projectors, then Ran L_q = ⊕_jP_j+qEnd( H)P_j.
Exact mathematical passage
\Lcal_q(X) = \frac1N \sum_{r=0}^{N-1} \omega^{-qr}S^rXS^{-r}.
\omega^{i+j}Use in source · 1 passage
Search aliases: ω^i+j omega
Use in source · Paper_IV_Traditional_Final.tex, line 444
Write F=∑_iF_i and G=∑_jG_j, where F_i= B_N,iF and G_j= B_N,jG. Their covariance gives (F_iG_j)(ω z)=ω^i+jF_i(z)G_j(z), so F_iG_j belongs to the packet labeled by i+j modulo N. Expanding the finite product and applying B_N,k gives the formula.
Exact mathematical passage
(F_iG_j)(\omega z)=\omega^{i+j}F_i(z)G_j(z),
\omega^jUse in source · 8 passages
Search aliases: ω^j omega
Use in source · Paper_IV_Traditional_Final.tex, line 407
[Covariance characterization] A holomorphic function F belongs to the range of B_N,j if and only if F(ω z)=ω^jF(z).
Exact mathematical passage
F(\omega z)=\omega^jF(z).
\omega^{jr}Use in source · 2 passages
Search aliases: ω^jr omega
Use in source · 01_finite_core.tex, line 221
Using S^r+s=S^r+sbmod N and the finite character orthogonality relation, frac1N∑_r=0^N-1ω^(k-j)r=δ_jk, we obtain P_jP_k=δ_jkP_j. Summing over j cancels every term indexed by r≠0 and retains the r=0 term, which proves completeness. Unitarity and the change of residue r↦-r give P_j^*=frac1N∑_r=0^N-1ω^jrS^-r=P_j. Finally, multiplication by S shifts the summation index and contributes the factor ω^j.
Exact mathematical passage
P_j^*=\frac1N\sum_{r=0}^{N-1}\omega^{jr}S^{-r}=P_j.
\omega^kUse in source · 7 passages
Search aliases: ω^k omega
Use in source · Paper_IV_Traditional_Final.tex, line 414
Every monomial selected by B_N,j has the stated covariance. Conversely, write F=∑_k B_N,kF. Replacing z by ω z gives ∑_kω^k B_N,kF = ω^j∑_k B_N,kF. Application of B_N,k gives (ω^k-ω^j) B_N,kF=0. The powers 1,ω,…,ω^N-1 are distinct, so B_N,kF=0 for k≠ j.
Exact mathematical passage
\sum_k\omega^k\Bcal_{N,k}F = \omega^j\sum_k\Bcal_{N,k}F.
\omega^{k(n-j)}Use in source · 1 passage
Search aliases: ω^k(n-j) omega
Use in source · 02_coordinates.tex, line 466
On z^n, the right side acts by the scalar frac1N∑_k=0^N-1ω^k(n-j). This is one when n≡ jpmod N and zero otherwise by the geometric sum identity in the field bbk.
Exact mathematical passage
\frac1N\sum_{k=0}^{N-1}\omega^{k(n-j)}.
\omega^mUse in source · 2 passages
Search aliases: ω^m omega
Use in source · Paper_IV_Traditional_Final.tex, line 1595
[Appell symmetry] Each polynomial E_m^(N,j,r) is monic of degree m, and E_m^(N,j,r)(ω x) = ω^mE_m^(N,j,r)(x).
Exact mathematical passage
E_m^{(N,j,r)}(\omega x) = \omega^mE_m^{(N,j,r)}(x).
\omega^NUse in source · 3 passages
Search aliases: ω^N omega
Use in source · 02_coordinates.tex, line 402
Let U act on coefficient sequences by (UF)(z)=F(ω z). If ω^N=1, then U^N=I.
Exact mathematical passage
\omega^N=1
\omega^{N-1}Use in source · 1 passage
Search aliases: ω^N-1 omega
Use in source · Paper_IV_Traditional_Final.tex, line 423
Every monomial selected by B_N,j has the stated covariance. Conversely, write F=∑_k B_N,kF. Replacing z by ω z gives ∑_kω^k B_N,kF = ω^j∑_k B_N,kF. Application of B_N,k gives (ω^k-ω^j) B_N,kF=0. The powers 1,ω,…,ω^N-1 are distinct, so B_N,kF=0 for k≠ j.
Exact mathematical passage
1,\omega,\ldots,\omega^{N-1}
\omega^{nk}Use in source · 1 passage
Search aliases: ω^nk omega
Use in source · 02_coordinates.tex, line 432
On the monomial z^n, the right-hand side acts by η∑_k=0^N-1(ω^vee)^jkω^nk. The identity ω^N=1 reduces this expression to the table entry indexed by r=nbmod N. The finite certificate therefore gives exactly the coefficient mask defining P_N,j. Over a field, the primitive-root geometric sum supplies the certificate.
Exact mathematical passage
\eta\sum_{k=0}^{N-1}(\omega^\vee)^{jk}\omega^{nk}.
\omega^{q+j}Use in source · 1 passage
Search aliases: ω^q+j omega
Use in source · 02_strengthened.tex, line 313
For x∈ P_j H, S(Ax) =(SAS^*)Sx =ω^q+jAx. Thus Ax∈ P_j+q H. The block statement follows.
Exact mathematical passage
S(Ax) =(SAS^*)Sx =\omega^{q+j}Ax.
\omega^{q_r}Use in source · 1 passage
Search aliases: ω^q_r omega
Use in source · 02_strengthened.tex, line 1237
[Charge preservation by a Lindblad generator] Let S be unitary and N≥1 an integer with S^N=I. Assume SHS^*=H and, for every jump operator, there is a charge q_r such that SL_rS^*=ω^q_rL_r. Then the Lindblad generator G preserves every coherence order: G L_p= L_p G. The quantum dynamical semigroup e^t G has the same property.
Exact mathematical passage
SL_rS^*=\omega^{q_r}L_r.
\omega^{rk}Use in source · 1 passage
Search aliases: ω^rk omega
Use in source · 02_coordinates.tex, line 408
[Root-of-unity realization] Let R be a commutative semiring, let N>0, and suppose ω,ω^vee,η∈ R satisfy ω^N=1 and the finite N× N Fourier orthogonality table η∑_k=0^N-1(ω^vee)^jkω^rk = 1, · r=j, 0, · r≠ j, 0≤ j,r<N. Then P_N,j = η∑_k=0^N-1(ω^vee)^jkU^k. In particular, if R is a field, N>0, and ω is a primitive Nth root of unity, the certificate holds with ω^vee=ω^-1 and η=N^-1.
Exact mathematical passage
\eta\sum_{k=0}^{N-1}(\omega^\vee)^{jk}\omega^{rk} = \begin{cases} 1,&r=j,\\ 0,&r\ne j, \end{cases} \qquad 0\le j,r<N.
\omega^{\operatorname{val}(r)}Use in source · 1 passage
Search aliases: ω^val(r) omega operatorname
Use in source · 03_supplement_03.tex, line 93
Constructed cyclic character projectors. Let N>0 and let S be a unitary complex matrix with S^N=I. The implementation uses the standard additive character of Z/N Z and proves its expression through a root of unity: ω=exp(2π i/N), χ(r)=ω^val(r). Here val(r) is the canonical representative in {0,…,N-1}. The representation is constructed as ρ(r)=S^val(r). Reduction of exponents modulo N is proved from S^N=I, giving ρ(r+s)=ρ(r)ρ(s) and ρ(0)=I. The final matrix theorem derives the representation law …
Exact mathematical passage
\omega=\exp(2\pi i/N),\qquad \chi(r)=\omega^{\operatorname{val}(r)}.
\omega^{\operatorname{val}(r)\operatorname{val}(j)}Use in source · 1 passage
Search aliases: ω^val(r)val(j) omega operatorname operatorname
Use in source · 03_supplement_03.tex, line 109
Exact coefficient formula. The packet definition is a finite sum over residue classes: P_j=frac1N∑_r∈ Z/N Zχ(-rj)ρ(r). The source proves χ(-rj)=(ω^val(r)val(j))^-1, so this is precisely the manuscript's N-term Fourier projector formula, with canonical representatives.
Exact mathematical passage
\chi(-rj)=\bigl(\omega^{\operatorname{val}(r)\operatorname{val}(j)}\bigr)^{-1},
\omega^\veeUse in source · 5 passages
Search aliases: ω^vee omega vee
Use in source · 02_coordinates.tex, line 406
[Root-of-unity realization] Let R be a commutative semiring, let N>0, and suppose ω,ω^vee,η∈ R satisfy ω^N=1 and the finite N× N Fourier orthogonality table η∑_k=0^N-1(ω^vee)^jkω^rk = 1, · r=j,
Exact mathematical passage
\omega,\omega^\vee,\eta\in R
\omega^\ellUse in source · 5 passages
Search aliases: ω^ℓ omega ell
Use in source · Paper_IV_Traditional_Final.tex, line 363
Fix N≥1 and write ω=e^2π i/N. Let Ω be a disk centered at the origin, with the whole plane allowed, and assume that Ω is invariant under multiplication by ω. For a holomorphic function F on Ω, define ( B_N,jF)(z) = frac1N∑_ℓ=0^N-1ω^-jℓF(ω^ℓ z), 0≤ j<N. The same formula is valid for formal power series.
Exact mathematical passage
(\Bcal_{N,j}F)(z) = \frac1N\sum_{\ell=0}^{N-1}\omega^{-j\ell}F(\omega^\ell z), \qquad 0\leq j<N.
\omega_1Use in source · 1 passage
Search aliases: ω_1 omega
Use in source · Paper_XII_Traditional_Final.tex, line 1547
[Finite Frobenius separator] Let t_1,…,t_r be linearly independent functions on a set Ω with values in a field E'. Then there are points ω_1,…,ω_r∈Ω such that the matrix (t_i(ω_j))_j,i is invertible. In particular, when the trace functions of finitely many compatible constituents are linearly independent on the unramified Frobenius set, finitely many Frobenius traces recover their multiplicities in any known semisimple carrier.
Exact mathematical passage
\omega_1,\ldots,\omega_r\in\Omega
\omega_2Use in source · 3 passages
Search aliases: ω_2 omega
Use in source · supplement_05.tex, line 119
For p=3,ℓ=2,a=2, the checked values are ω_2(2)=8, ⟨2⟩_2=7, 8^2=1,quad8·7=2 in Z/9 Z. The unlifted representative 2 is not a square root of one modulo nine.
Exact mathematical passage
\omega_2(2)=8,\qquad \langle2\rangle_2=7,\qquad 8^2=1,\quad8\cdot7=2\quad\text{in }\Z/9\Z.
\omega_aUse in source · 3 passages
Search aliases: ω_a omega
Use in source · Paper_VI_Traditional_Final.tex, line 5491
[Oscillation bound] Suppose overline L_a is any representative likelihood for block B_a, and set ω_a = max_i∈ B_aabsL_i-overline L_a. Then d(pmid L,overline L) ≤ ∑_aq_aω_a. Thus small within block likelihood oscillation gives a small coarse posterior defect whenever the evidence is bounded away from zero.
Exact mathematical passage
\omega_a = \max_{i\in B_a}\abs{L_i-\overline L_a}.
\omega_cUse in source · 4 passages
Search aliases: ω_c omega
Use in source · Paper_VI_Traditional_Final.tex, line 487
[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.
Exact mathematical passage
\norm{e^{tA_f}}\le M_f e^{\omega_f t}, \qquad \norm{e^{tA_c}}\le M_c e^{\omega_c t}.
\omega_fUse in source · 4 passages
Search aliases: ω_f omega
Use in source · Paper_VI_Traditional_Final.tex, line 487
[Norm bound] Suppose norme^tA_f≤ M_f e^ω_f t, norme^tA_c≤ M_c e^ω_c t. Then norm D(t) ≤ M_fM_cnormΔ ∫_0^t e^ω_c(t-r)e^ω_fr dr. For contraction semigroups, norm D(t)≤ tnormΔ.
Exact mathematical passage
\norm{e^{tA_f}}\le M_f e^{\omega_f t}, \qquad \norm{e^{tA_c}}\le M_c e^{\omega_c t}.
\omega_jUse in source · 1 passage
Search aliases: ω_j omega
Use in source · Paper_XII_Traditional_Final.tex, line 1548
[Finite Frobenius separator] Let t_1,…,t_r be linearly independent functions on a set Ω with values in a field E'. Then there are points ω_1,…,ω_r∈Ω such that the matrix (t_i(ω_j))_j,i is invertible. In particular, when the trace functions of finitely many compatible constituents are linearly independent on the unramified Frobenius set, finitely many Frobenius traces recover their multiplicities in any known semisimple carrier.
Exact mathematical passage
\left(t_i(\omega_j)\right)_{j,i}
\omega_kUse in source · 1 passage
Search aliases: ω_k omega
Use in source · supplement_05.tex, line 136
a_unique, .decomposition_unique targetFiniteUnits.omega_mul, .principal_mul pageheading3. Precision compatibility and the dyadic signFinite refinements and Proposition 6.2's dyadic branch For 1≤ k≤ℓ, let ρ_ℓ,k: Z/p^ℓ Z⟶ Z/p^k Z be the actual reduction homomorphism. It induces a map on units. The explicit construction satisfies ρ_ℓ,k(ω_ℓ(a)) =ω_k(ρ_ℓ,k(a)), ρ_ℓ,k(⟨ a⟩_ℓ) =⟨ρ_ℓ,k(a)⟩_k. The proof uses the exponent factorization p^ℓ-1=p^k-1p^ℓ-k and the already proved Frobenius invariance of a lift…
Exact mathematical passage
\boxed{\rho_{\ell,k}(\omega_\ell(a)) =\omega_k(\rho_{\ell,k}(a)),}
\omega_LUse in source · 3 passages
Search aliases: ω_L omega
Use in source · 10_foundations.tex, line 364
[Finite unit coordinates] Let p be prime, L≥0, and a∈( Z/p^L+1 Z)^×. Define ω_L(a)=a^p^L, ⟨ a⟩_L=aω_L(a)^-1. Then a=ω_L(a)⟨ a⟩_L, ω_L(a)^p-1=1, ⟨ a⟩_L≡1pmod p. The two factors are uniquely determined by these properties and are compatible with reduction to lower precision. For a∈( Z/2^L+2 Z)^×, there is instead a unique decomposition a=ϵ(a)⟨ a⟩_4, ϵ(a)∈{1,-1}, ⟨ a⟩_4≡1pmod4.
Exact mathematical passage
\omega_L(a)=a^{p^L},\qquad \langle a\rangle_L=a\omega_L(a)^{-1}.
\omega_NUse in source · 1 passage
Search aliases: ω_N omega
Use in source · Paper_V_Traditional_Final.tex, line 1134
Fix ω_N=e^2π i/N and rotate the reflection coefficient by R_N,k(Γ)=ω_N^kΓ.
Exact mathematical passage
\omega_N=e^{2\pi\ii/N}
\omega_N^kUse in source · 7 passages
Search aliases: ω_N^k omega
Use in source · 06_models.tex, line 46
Let Z and Z_0≠0 be complex impedances, put z=Z/Z_0, and assume z≠-1. Write Γ=fracz-1z+1. On the open reflection disk |Γ|<1, multiplication by any phase ω_N^k stays in the disk and has a finite inverse-chart image. The phase action is Γ↦ω_N^kΓ. Outside the open disk, the same chart formula requires 1-ω_N^kΓ≠0. Pulling the phase action back on this domain gives T_N,k(z) = frac(1+ω_N^k)z+(1-ω_N^k) (1-ω_N^k)z+(1+ω_N^k). For N=2 and k=1, T_2,1(z)=frac1z, z≠0,-1. Thus the nontrivial action at the firs…
Exact mathematical passage
\omega_N^k
\omega_N^NUse in source · 1 passage
Search aliases: ω_N^N omega
Use in source · Paper_V_Traditional_Final.tex, line 1167
1-w · 1+w. It has determinant w≠0, preserves nonzero pairs, and commutes with rescaling, so it induces a projective map. Direct multiplication gives A(w)A(w')=A(ww') and A(1)=I. Since ω_N^N=1, this proves the cyclic composition law modulo N. The Cayley coordinate of [u:v] is [u-v:u+v], and applying A(w) changes this pair to [w(u-v):u+v]. This proves the conjugacy also at poles. For [z:1], taking the coordinate ratio gives the displayed finite-chart formula.
Exact mathematical passage
\omega_N^N=1
\Omega_{P/\mathbb Z}Use in source · 1 passage
Search aliases: Ω_P/ Z Omega mathbb
Use in source · 71_native.tex, line 134
coker(Hom_B(Ω_P/ Z⊗_P B, F_p)⟶Hom_B(I/I^2, F_p)).
Exact mathematical passage
\operatorname{coker}\left(\operatorname{Hom}_B(\Omega_{P/\mathbb Z}\otimes_P B,\mathbb F_p)\longrightarrow\operatorname{Hom}_B(I/I^2,\mathbb F_p)\right).
\Omega_qUse in source · 1 passage
Search aliases: Ω_q Omega
Use in source · Paper_V_Traditional_Final.tex, line 1295
[Edge-packet approximation class] There should exist natural function spaces in which a signal is represented as f=s+∑_q a_q1_Ω_q, with separate norms for smooth approximation, boundary geometry, packet amplitude, and packet identification. A projector-resolved approximation theory on such spaces should yield rates that do not mix smooth truncation error with discontinuity-location error.
Exact mathematical passage
f=s+\sum_q a_q\mathbf1_{\Omega_q},
\omega_QUse in source · 2 passages
Search aliases: ω_Q omega
Use in source · Paper_VI_Traditional_Final.tex, line 2750
[Generator control of range leakage] Let T(t)= e^tA on a Banach space X, let P be a bounded projection, and set Q=I-P. Assume normT(t)≤ M e^ω t and norm e^t(QAQ)|_QX ≤ M_Q e^ω_Qt. Then normQT(t)P ≤ M M_Qnorm P^2normQAP ∫_0^t e^ω_Q(t-s)+ω s d s. In particular, for orthogonal projections and contraction bounds, sup_0≤ t≤ T normQT(t)P ≤ TnormQAP.
Exact mathematical passage
\norm{\e^{t(QAQ)|_{QX}}} \leq M_Q\e^{\omega_Qt}.
\omega_rUse in source · 1 passage
Search aliases: ω_r omega
Use in source · Paper_XII_Traditional_Final.tex, line 1547
[Finite Frobenius separator] Let t_1,…,t_r be linearly independent functions on a set Ω with values in a field E'. Then there are points ω_1,…,ω_r∈Ω such that the matrix (t_i(ω_j))_j,i is invertible. In particular, when the trace functions of finitely many compatible constituents are linearly independent on the unramified Frobenius set, finitely many Frobenius traces recover their multiplicities in any known semisimple carrier.
Exact mathematical passage
\omega_1,\ldots,\omega_r\in\Omega
\omega_VUse in source · 1 passage
Search aliases: ω_V omega
Use in source · Paper_XI_Traditional_Final.tex, line 1760
[Heisenberg tensor factorization] Fix compatible central characters. The irreducible Heisenberg representation of an orthogonal sum is the tensor product of the irreducible Heisenberg representations of its summands: ω_V ≅ ⊗_s∈ Sω_V_s.
Exact mathematical passage
\boxed{ \omega_V \cong \bigotimes_{s\in S}\omega_{V_s}. }
\omega_{V_s}Use in source · 1 passage
Search aliases: ω_V_s omega
Use in source · Paper_XI_Traditional_Final.tex, line 1760
[Heisenberg tensor factorization] Fix compatible central characters. The irreducible Heisenberg representation of an orthogonal sum is the tensor product of the irreducible Heisenberg representations of its summands: ω_V ≅ ⊗_s∈ Sω_V_s.
Exact mathematical passage
\boxed{ \omega_V \cong \bigotimes_{s\in S}\omega_{V_s}. }
\Omega_\thetaUse in source · 2 passages
Search aliases: Ω_θ Omega theta
Use in source · Paper_XI_Traditional_Final.tex, line 1234
[Cyclic discrepancy character] Assume that the two projective stabilizer actions have been linearized to honest representations of the cyclic group C. Then there is a character δ:C→ C^× such that UΩ_ρ(g) = δ(g)Ω_θ(g)U for every g∈ C.
Exact mathematical passage
U\Omega_\rho(g) = \delta(g)\Omega_\theta(g)U
\omega_\thetaUse in source · 1 passage
Search aliases: ω_θ omega theta
Use in source · Paper_XI_Traditional_Final.tex, line 1214
[Restricted Heisenberg bridge] A restricted bridge datum determines an isomorphism between the two Heisenberg representations, unique up to a nonzero scalar. More precisely, there is an invertible map U: H_ρ⟶ H_θ such that Uω_ρ(h) = ω_θ(ι_H(h))U for every Heisenberg element h, and every other such map is a scalar multiple of U.
Exact mathematical passage
U\omega_\rho(h) = \omega_\theta(\iota_H(h))U
\Omega_\lambdaUse in source · 3 passages
Search aliases: Ω_λ Omega lambda
Use in source · Paper_XII_Traditional_Final.tex, line 1479
Choose elements γ_v∈Γ on a sample set Ω, with declared good subsets Ω_λ⊆Ω. In the arithmetic application these are the chosen Frobenius elements at the permitted places. For each v∈Ω_λ, supply an integral polynomial P_λ,v∈ O_λ[X] such that i_λ(P_λ,v)=e_λ(P_v) in K_λ[X]. Here and below a ring homomorphism applied to a polynomial acts on its coefficients. Assume every representative of every class in X_λ,n has characteristic polynomial P_λ,vbmod I_λ^n at γ_v. Also supply a subgroup H_λ≤Γ on which …
Exact mathematical passage
\Omega_\lambda\subseteq\Omega
\Omega_\rhoUse in source · 2 passages
Search aliases: Ω_ρ Omega rho
Use in source · Paper_XI_Traditional_Final.tex, line 1234
[Cyclic discrepancy character] Assume that the two projective stabilizer actions have been linearized to honest representations of the cyclic group C. Then there is a character δ:C→ C^× such that UΩ_ρ(g) = δ(g)Ω_θ(g)U for every g∈ C.
Exact mathematical passage
U\Omega_\rho(g) = \delta(g)\Omega_\theta(g)U
\omega_\rhoUse in source · 1 passage
Search aliases: ω_ρ omega rho
Use in source · Paper_XI_Traditional_Final.tex, line 1214
[Restricted Heisenberg bridge] A restricted bridge datum determines an isomorphism between the two Heisenberg representations, unique up to a nonzero scalar. More precisely, there is an invertible map U: H_ρ⟶ H_θ such that Uω_ρ(h) = ω_θ(ι_H(h))U for every Heisenberg element h, and every other such map is a scalar multiple of U.
Exact mathematical passage
U\omega_\rho(h) = \omega_\theta(\iota_H(h))U
\omega_\ellUse in source · 4 passages
Search aliases: ω_ℓ omega ell
Use in source · supplement_05.tex, line 81
ffPadic_coe, .standard_window_integral targetWindowIndependence.residue_independence, .reduced_unique targetWindowIndependence.two_window_addresses_congruent pageheading2. Explicit finite prime-power decompositionProposition 6.2; odd-prime branch Let p be prime, ℓ≥1, and a∈( Z/p^ℓ Z)^×. Define actual units in the same ring by ω_ℓ(a)=a^p^ℓ-1, ⟨ a⟩_ℓ=a ω_ℓ(a)^-1. The Lean source writes ℓ=L+1, so zero precision is excluded without truncated natural-number subtraction.
Exact mathematical passage
\boxed{\omega_\ell(a)=a^{p^{\ell-1}},\qquad \langle a\rangle_\ell=a\,\omega_\ell(a)^{-1}.}
\daggerUse in source · 5 passages
Search aliases: † dagger
Use in source · 04_correction_transport.tex, line 6
[Orthogonal obstruction projector] Let D:H_1→ H_2 be a bounded operator between Hilbert spaces with closed range. Let D^† be its Moore--Penrose inverse. Then P_ob=I_H_2-DD^† is the orthogonal projector onto (im D)^⊥=Ker D^*. For r∈ H_2, the following are equivalent: item Du=-r is solvable item [r]=0 in coker D item P_obr=0. When these conditions hold, -D^† r is the least norm correction. For every r, dist(r,im D)= P_obr .
Exact mathematical passage
D^\dagger
\ell'Use in source · 3 passages
Search aliases: ℓ' ell
Use in source · Paper_V_Traditional_Final.tex, line 1943
Evaluate the output at r+ℓ. Write every input address as s+ℓ'. The convolution offset is (r+ℓ)-(s+ℓ') =(r-s)+(ℓ-ℓ'). Setting m=ℓ-ℓ' gives the formula.
Exact mathematical passage
s+\ell'
\ell^1Use in source · 12 passages
Search aliases: ℓ^1 ell
Use in source · Paper_V_Traditional_Final.tex, line 306
[Several completions] The same algebraic seed description applies to finite sequences, ℓ^1, ℓ^2, periodic sequences, tempered distributions, and multidimensional lattices. The topology decides which infinite reconstructions are legitimate. Finite-dimensional experiments in this chapter use ordinary Euclidean norms. Theorem statements specify the ambient norm when it matters.
Exact mathematical passage
\ell^1
\ell^mUse in source · 1 passage
Search aliases: ℓ^m ell
Use in source · 72_recovered.tex, line 1905
Put y=x+1 and let ℓ=log 9. Since v_2(ℓ)=3, frac9^y-18y = fracℓ8 ∑_m≥0 fracℓ^m(m+1)!y^m. The coefficient of y^m has valuation 3m-v_2((m+1)!), which is nonnegative and tends to positive infinity. The quotient therefore belongs to Z_2⟨ y⟩. The factor 2x+1=1+2x is a unit in Z_2⟨ x⟩, with inverse ∑_m≥0(-2x)^m. Hence mathscr R is integral and analytic. The integer values follow from the definition of R_2n.
Exact mathematical passage
\frac{9^y-1}{8y} = \frac{\ell}{8} \sum_{m\geq0} \frac{\ell^m}{(m+1)!}y^m.
\ell^\inftyUse in source · 3 passages
Search aliases: ℓ^∞ ell infty
Use in source · Paper_V_Traditional_Final.tex, line 1738
[Impulse generation] For 1≤ p<∞, overlinespan{T_aδ_0:a∈ Z^d}^ℓ^p = ℓ^p( Z^d). The same statement holds in c_0( Z^d) with the supremum norm. In ℓ^∞( Z^d), the closed span of the translation orbit is exactly c_0( Z^d).
Exact mathematical passage
\ell^\infty(\ZZ^d)
\ell_1Use in source · 1 passage
Search aliases: ℓ_1 ell
Use in source · Paper_VI_Traditional_Final.tex, line 5850
[A visible likelihood loss] Let B_1={1,2}, let B_2={3}, and take p=(3/10,2/10,5/10), L=(1,3,2). The coarse prior is (1/2,1/2). The exact block likelihoods are ℓ_1(p)=frac(3/10)+3(2/10)1/2=frac95, ℓ_2(p)=2. Thus the exact coarse posterior is (9/19,10/19). If the first block is replaced by the fixed representative value 2, the approximate coarse posterior is (1/2,1/2). The ℓ^1 defect is 1/19, while [source reference] gives the valid upper bound 1/10.
Exact mathematical passage
\ell_1(p)=\frac{(3/10)+3(2/10)}{1/2}=\frac95, \qquad \ell_2(p)=2.
\ell_2Use in source · 1 passage
Search aliases: ℓ_2 ell
Use in source · Paper_VI_Traditional_Final.tex, line 5850
[A visible likelihood loss] Let B_1={1,2}, let B_2={3}, and take p=(3/10,2/10,5/10), L=(1,3,2). The coarse prior is (1/2,1/2). The exact block likelihoods are ℓ_1(p)=frac(3/10)+3(2/10)1/2=frac95, ℓ_2(p)=2. Thus the exact coarse posterior is (9/19,10/19). If the first block is replaced by the fixed representative value 2, the approximate coarse posterior is (1/2,1/2). The ℓ^1 defect is 1/19, while [source reference] gives the valid upper bound 1/10.
Exact mathematical passage
\ell_1(p)=\frac{(3/10)+3(2/10)}{1/2}=\frac95, \qquad \ell_2(p)=2.
\ell_aUse in source · 6 passages
Search aliases: ℓ_a ell
Use in source · Paper_VI_Traditional_Final.tex, line 5452
For an arbitrary prior p, put q=Cp and Z(p,L)=ipLp. Define ℓ_a(p)=∑_i∈ B_aL_ip_i/q_a when q_a>0. Choose any positive value when q_a=0. Only the product q_aℓ_a(p) enters the formulas. [Exact information loss formula] For every prior, C B_L(p) = B_ℓ(p)(q). Let overline L be any fixed positive coarse likelihood and set d(pmid L,overline L) = ∑_a=1^mq_aabsℓ_a(p)-overline L_a. Then norm C B_L(p)- B_overline L(q) _1 ≤ frac2d(pmid L,overline L) max{Z(p,L),ipoverline Lq}.
Exact mathematical passage
\ell_a(p)=\sum_{i\in B_a}L_ip_i/q_a
\ell_eUse in source · 2 passages
Search aliases: ℓ_e ell
Use in source · 01_finite_coordinates.tex, line 145
[Current fiber scope] The compiled Chapter III model above has one scalar coordinate per packet. For a family of modules (X_i)_i∈ V, an intrinsic edge map would instead have type ℓ_e:X_src(e)→ X_tgt(e), and its totalization on ⊕_iX_i would be ι_tgt(e)ℓ_eπ_src(e). The same support proof applies, but this generalization with dependent fibers is not attributed to the released Chapter III modules for scalar edge systems.
Exact mathematical passage
\ell_e:X_{\src(e)}\to X_{\tgt(e)}
\ell_iUse in source · 2 passages
Search aliases: ℓ_i ell
Use in source · 02_foundations.tex, line 493
The lower inequality is quotient monotonicity. For the upper inequality, choose generating families (ℓ_i)_i∈ I of L and (n_j)_j∈ J of N with the least possible cardinalities. Choose a lift m_j∈ M of each n_j. For any m∈ M, a finite subset F⊆ J and coefficients a_j∈ A give π(m)=∑_j∈ Fa_jn_j. The vector m-∑_j∈ Fa_jm_j belongs to ι(L), so it is a finite linear combination of the ι(ℓ_i). The union of the lifted family and the image of the generating family of L therefore generates M. Its cardinality…
Exact mathematical passage
(\ell_i)_{i\in I}
\ell_rUse in source · 4 passages
Search aliases: ℓ_r ell
Use in source · 03_nonlinear_coordinates.tex, line 646
[Packet equations for L_∞ operations] Let L be complete and carry operations ℓ_r that respect a finite abelian group grading. Assume the packet projectors are continuous for the topology in which the series below converges. Assume the Maurer Cartan series ∑_r≥1frac1r!ℓ_r(η,…,η) converges and the required factorials are invertible. Then its packet of label k is ∑_r≥1frac1r! ∑_g_1+⋯+g_r=k ℓ_r(η_g_1,…,η_g_r). The full Maurer Cartan equation holds if and only if every packet equation vanishes.
Exact mathematical passage
\ell_r
\ell_yUse in source · 1 passage
Search aliases: ℓ_y ell
Use in source · Paper_VI_Traditional_Final.tex, line 5558
[One step filter defect] Let r=Kp, q=Cr, and let ℓ_y(r) be the exact coarse likelihood obtained by averaging L_y inside each block with respect to r. Assume 0<overline m≤(overline L_y)_a≤overline M. Then normCΦ_y(p)-overlineΦ_y(Cp)_1 ≤ · frac2d(rmid L_y,overline L_y) max{ipL_yr,ipoverline L_yq}
Exact mathematical passage
\ell_y(r)
\ell_\alphaUse in source · 3 passages
Search aliases: ℓ_α ell alpha
Use in source · Paper_VI_Traditional_Final.tex, line 3523
ons are equivalent. item There is a positive coarse likelihood ℓ∈(0,∞)^m such that C B_λ(p) = B_ℓ(Cp) for every strictly positive prior p, and the posterior conditional distribution inside every packet equals its prior conditional distribution. item The likelihood is constant on every packet. When these conditions hold, ℓ_α can be chosen as the common value of λ on I_α, and it is unique up to a common positive scale.
Exact mathematical passage
\ell_\alpha
\leftrightarrowUse in source · 4 passages
Search aliases: ↔ leftrightarrow
Use in source · 01_orientation.tex, line 48
How are vectors and operators resolved? · Complete idempotents (P_α) · X≅⊕_α P_α X and T↔(P_β TP_α).
Exact mathematical passage
T\leftrightarrow(P_\beta TP_\alpha)
\RightarrowUse in source · 3 passages
Search aliases: ⇒ Rightarrow
Use in source · 05_worked_models.tex, line 65
Let char𝕜≠2 and let A=𝕜[x,y]/(y^2-x^3+x). The affine equation y^2=x^3-x has classical discriminant 64, which is nonzero under this hypothesis. Standard algebraic geometry then identifies its smooth projective completion as an elliptic curve [source reference]. The Lean development checks the algebraic implication 2≠0⇒64≠0, the quotient normal form, and the operator identities below. It does not formalize projective smoothness, genus, or the geometric identification of the discriminant. The packe…
Exact mathematical passage
2\ne0\Rightarrow64\ne0
\LeftrightarrowUse in source · 1 passage
Search aliases: ⇔ Leftrightarrow
Use in source · 02_foundations.tex, line 1006
ii) holds exactly when every μ_p(T) belonging to a nonzero primary component equals one, which is (iii). Relatively prime primary cyclic modules combine into one cyclic module by the Chinese remainder theorem, so (iii) is equivalent to (iv). In rational canonical form, one invariant factor means that the unique invariant factor is both the minimal and characteristic polynomial, proving (iv)⇔(v). Scalar extension preserves the equality of minimal and characteristic polynomials. Over an algebraic …
Exact mathematical passage
\Leftrightarrow
\partialUse in source · 28 passages
Search aliases: ∂ partial
Use in source · Paper_IV_Traditional_Final.tex, line 1552
[Formal differential-operator representation on polynomials] Let a formal series A(z)=∑_q≥0a_qz^q/q! act on polynomials by A(∂_x)f(x) = ∑_q≥0fraca_qq!∂_x^qf(x). The power r is applied to the formal symbol before the substitution z=∂_x. Then E_m^(N,j,r)(x) = C_N,j(∂_x)^r x^m. The inverse relation is x^m = Φ_N,j(∂_x^N)^r E_m^(N,j,r)(x). Both sums are finite on every polynomial.
Exact mathematical passage
A(\partial_x)f(x) = \sum_{q\geq0}\frac{a_q}{q!}\partial_x^qf(x).
\partial_iUse in source · 1 passage
Search aliases: ∂_i partial
Use in source · continuation_01.tex, line 49
For every P∈ Z[X_1,…,X_d], integer a, and x,t∈ Z^d, Lean proves a^2mid P(x+at)-P(x)-a∑_i=1^d(∂_iP)(x)t_i. The proof uses polynomial induction: constants, addition, and multiplication by a variable. The product rule for the actual partial derivatives controls the multiplication step. It does not assume a Taylor identity or a liftability condition.
Exact mathematical passage
\boxed{ a^2\mid P(x+at)-P(x)-a\sum_{i=1}^{d}(\partial_iP)(x)t_i. }
\partial_jUse in source · 5 passages
Search aliases: ∂_j partial
Use in source · 30_lifting.tex, line 28
For F=(F_1,…,F_e)∈ Z[X_1,…,X_d]^e, define its evaluated Jacobian by the formal partial derivatives: DF(x)_ij=(∂_jF_i)(x), J_x=DF(x)bmod q:S_q^d⟶ S_q^e. The obstruction therefore uses the Jacobian of the given polynomials. The following algebraic Taylor identity relates this linear map to polynomial evaluation.
Exact mathematical passage
DF(x)_{ij}=(\partial_jF_i)(x),\qquad J_x=DF(x)\bmod q:S_q^d\longrightarrow S_q^e.
\partial_xUse in source · 9 passages
Search aliases: ∂_x partial
Use in source · Paper_IV_Traditional_Final.tex, line 1552
[Formal differential-operator representation on polynomials] Let a formal series A(z)=∑_q≥0a_qz^q/q! act on polynomials by A(∂_x)f(x) = ∑_q≥0fraca_qq!∂_x^qf(x). The power r is applied to the formal symbol before the substitution z=∂_x. Then E_m^(N,j,r)(x) = C_N,j(∂_x)^r x^m. The inverse relation is x^m = Φ_N,j(∂_x^N)^r E_m^(N,j,r)(x). Both sums are finite on every polynomial.
Exact mathematical passage
A(\partial_x)f(x) = \sum_{q\geq0}\frac{a_q}{q!}\partial_x^qf(x).
\partial_x^2Use in source · 1 passage
Search aliases: ∂_x^2 partial
Use in source · atlas-pages.tex, line 267
ymbol acts finitelyAPPELL POLYNOMIALSChapter IV: Theorem 9.6, Theorem 9.8, Proposition 9.9. For C_2,0(z)=sech z, define E_m(x) by sech z e^xz=∑ E_m(x)z^m/m!. The formal differential-operator formula is E_m(x)=sech(∂_x)x^m. Although the symbol is an infinite series, derivatives above degree m vanish. vfigappell_operator For m=4, only three terms survive: (1-frac∂_x^22!+5frac∂_x^44!)x^4 =x^4-6x^2+5. The four real roots of this polynomial are ±1,±sqrt5. They are not the zeros of the original analyt…
Exact mathematical passage
\left(1-\frac{\partial_x^2}{2!}+5\frac{\partial_x^4}{4!}\right)x^4 =x^4-6x^2+5.
\partial_x^4Use in source · 1 passage
Search aliases: ∂_x^4 partial
Use in source · atlas-pages.tex, line 267
ymbol acts finitelyAPPELL POLYNOMIALSChapter IV: Theorem 9.6, Theorem 9.8, Proposition 9.9. For C_2,0(z)=sech z, define E_m(x) by sech z e^xz=∑ E_m(x)z^m/m!. The formal differential-operator formula is E_m(x)=sech(∂_x)x^m. Although the symbol is an infinite series, derivatives above degree m vanish. vfigappell_operator For m=4, only three terms survive: (1-frac∂_x^22!+5frac∂_x^44!)x^4 =x^4-6x^2+5. The four real roots of this polynomial are ±1,±sqrt5. They are not the zeros of the original analyt…
Exact mathematical passage
\left(1-\frac{\partial_x^2}{2!}+5\frac{\partial_x^4}{4!}\right)x^4 =x^4-6x^2+5.
\partial_x^NUse in source · 2 passages
Search aliases: ∂_x^N partial
Use in source · Paper_IV_Traditional_Final.tex, line 1564
ferential-operator representation on polynomials] Let a formal series A(z)=∑_q≥0a_qz^q/q! act on polynomials by A(∂_x)f(x) = ∑_q≥0fraca_qq!∂_x^qf(x). The power r is applied to the formal symbol before the substitution z=∂_x. Then E_m^(N,j,r)(x) = C_N,j(∂_x)^r x^m. The inverse relation is x^m = Φ_N,j(∂_x^N)^r E_m^(N,j,r)(x). Both sums are finite on every polynomial.
Exact mathematical passage
x^m = \Phi_{N,j}(\partial_x^N)^r E_m^{(N,j,r)}(x).
\partial_x^qUse in source · 1 passage
Search aliases: ∂_x^q partial
Use in source · Paper_IV_Traditional_Final.tex, line 1552
[Formal differential-operator representation on polynomials] Let a formal series A(z)=∑_q≥0a_qz^q/q! act on polynomials by A(∂_x)f(x) = ∑_q≥0fraca_qq!∂_x^qf(x). The power r is applied to the formal symbol before the substitution z=∂_x. Then E_m^(N,j,r)(x) = C_N,j(∂_x)^r x^m. The inverse relation is x^m = Φ_N,j(∂_x^N)^r E_m^(N,j,r)(x). Both sums are finite on every polynomial.
Exact mathematical passage
A(\partial_x)f(x) = \sum_{q\geq0}\frac{a_q}{q!}\partial_x^qf(x).
\partial_\thetaUse in source · 5 passages
Search aliases: ∂_θ partial theta
Use in source · monograph_atlas.tex, line 277
EXACT BLOCH PROJECTOR CALCULATIONChapter VII: Theorem II.6.7, Corollary II.6.8. On the unit sphere, set n=(sinθcosϕ,sinθsinϕ,cosθ) and P(θ,ϕ)=frac12(I+n·σ). This is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form P( d Pwedge d P)P. With the paper's scalar convention, its component is F_θϕ=- iTr (P[∂_θ P,∂_ϕ P]) =frac12sinθ. vfigbloch_curvature headA local frame gives the same answer Away from the south pole, use u_N=(cos(θ/2),e^ iϕsin(θ/2))^T. …
Exact mathematical passage
F_{\theta\phi}=-\ii\Tr\!\left(P[\partial_\theta P,\partial_\phi P]\right) =\frac12\sin\theta.
\partial_\muUse in source · 1 passage
Search aliases: ∂_μ partial mu
Use in source · 01_finite_core.tex, line 632
For an orthogonal rank-one complex projector, use the scalar connection convention A=-i⟨ u,du⟩ in a local unit frame. The associated scalar curvature density is F_μν =-iTr(P[∂_μ P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)·σ. Using Tr(P a·σ)= n· a therefore yields F_θϕ =frac12 n·(∂_θ nmathbin×∂_ϕ n) =frac12sinθ. The scalar sign changes if the opposite convention for A i…
Exact mathematical passage
\cF_{\mu\nu} =-i\Tr\left(P[\partial_\mu P,\partial_\nu P]\right).
\partial_\nuUse in source · 1 passage
Search aliases: ∂_ν partial nu
Use in source · 01_finite_core.tex, line 632
For an orthogonal rank-one complex projector, use the scalar connection convention A=-i⟨ u,du⟩ in a local unit frame. The associated scalar curvature density is F_μν =-iTr(P[∂_μ P,∂_ν P]). On the Bloch sphere, P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). The Pauli multiplication law gives [∂_θ P,∂_ϕ P] =fraci2(∂_θ nmathbin×∂_ϕ n)·σ. Using Tr(P a·σ)= n· a therefore yields F_θϕ =frac12 n·(∂_θ nmathbin×∂_ϕ n) =frac12sinθ. The scalar sign changes if the opposite convention for A i…
Exact mathematical passage
\cF_{\mu\nu} =-i\Tr\left(P[\partial_\mu P,\partial_\nu P]\right).
\partial_\phiUse in source · 5 passages
Search aliases: ∂_ϕ partial phi
Use in source · monograph_atlas.tex, line 277
EXACT BLOCH PROJECTOR CALCULATIONChapter VII: Theorem II.6.7, Corollary II.6.8. On the unit sphere, set n=(sinθcosϕ,sinθsinϕ,cosθ) and P(θ,ϕ)=frac12(I+n·σ). This is a smooth rank-one orthogonal projector. The intrinsic curvature is the endomorphism-valued two-form P( d Pwedge d P)P. With the paper's scalar convention, its component is F_θϕ=- iTr (P[∂_θ P,∂_ϕ P]) =frac12sinθ. vfigbloch_curvature headA local frame gives the same answer Away from the south pole, use u_N=(cos(θ/2),e^ iϕsin(θ/2))^T. …
Exact mathematical passage
F_{\theta\phi}=-\ii\Tr\!\left(P[\partial_\theta P,\partial_\phi P]\right) =\frac12\sin\theta.
\existsUse in source · 24 passages
Search aliases: ∃ exists
Use in source · 03_packets_and_graphs.tex, line 706
[Propagation of packet support] Let T∈End(X) and x∈ X. Then Supp_ P(Tx) ⊆ {β:∃α∈Supp_ P(x) with T_βα≠0}. Equality holds whenever, for every target β in the set on the right, the active contributions (T_βαP_α x: α∈Supp_ P(x), T_βα≠0) have nonzero sum. In particular, equality holds if at most one active source packet feeds each such β and that source coordinate does not lie in the kernel of the corresponding block. [source reference] ExportsGraph reachability, leakage bounds, and support pruning.
Exact mathematical passage
\boxed{ \Supp_{\Pcal}(Tx) \subseteq \left\{\beta:\exists\alpha\in\Supp_{\Pcal}(x) \text{ with }T_{\beta\alpha}\neq0\right\}. }
\nabla^{(a)}_XUse in source · 2 passages
Search aliases: ∇^(a)_X nabla
Use in source · 03_supplement_06.tex, line 88
The local connection and its change of coordinates. For a coordinate function u valued in E_a, define ∇^(a)_Xu=A_a(P D_M(B_a u)[X]). For every overlap point, Lean proves the operator identity ∇^(b)_X(g_abu)=g_ab∇^(a)_Xu. This is proved for arbitrary local coordinate functions. It does not assume that u was obtained from a globally smooth section or that a smooth extension has been supplied. Together with the local smoothness criterion, this closes the gluing step left open in the fifth block.
Exact mathematical passage
\nabla^{(a)}_Xu=A_a\bigl(P\,D_M(B_a u)[X]\bigr).
\nabla^{(b)}_XUse in source · 1 passage
Search aliases: ∇^(b)_X nabla
Use in source · 03_supplement_06.tex, line 92
The local connection and its change of coordinates. For a coordinate function u valued in E_a, define ∇^(a)_Xu=A_a(P D_M(B_a u)[X]). For every overlap point, Lean proves the operator identity ∇^(b)_X(g_abu)=g_ab∇^(a)_Xu. This is proved for arbitrary local coordinate functions. It does not assume that u was obtained from a globally smooth section or that a smooth extension has been supplied. Together with the local smoothness criterion, this closes the gluing step left open in the fifth block.
Exact mathematical passage
\boxed{\nabla^{(b)}_X(g_{ab}u)=g_{ab}\nabla^{(a)}_Xu.}
\nabla_{[X,Y]}Use in source · 3 passages
Search aliases: ∇_[X,Y] nabla
Use in source · 01_finite_core.tex, line 580
The rank of P is locally constant. On each open rank region, its ranges form a smooth vector bundle. For a smooth range section s, so that Ps=s, define ∇_Xs=P ds(X). This is a connection on smooth range sections. For smooth vector fields X,Y, its curvature satisfies (∇_X∇_Y-∇_Y∇_X-∇_[X,Y])s =P[dP(X),dP(Y)]P s. Equivalently, its endomorphism-valued curvature is F=P(dPwedge dP)P. No global frame or globally constant rank is required, and the vector fields need not commute. The geometric formalizat…
Exact mathematical passage
\boxed{ (\nabla_X\nabla_Y-\nabla_Y\nabla_X-\nabla_{[X,Y]})s =P[dP(X),dP(Y)]P\,s. }
\nabla_{fX}Use in source · 1 passage
Search aliases: ∇_fX nabla
Use in source · 01_finite_core.tex, line 606
textitConnection rules. The smoothness of P ds(X) and the identity P^2=P show that ∇_Xs is a smooth range section. Differentiation and projection give ∇_X+Ys · =∇_Xs+∇_Ys, · ∇_fXs · =f∇_Xs, ∇_X(s+t) · =∇_Xs+∇_Xt, · ∇_X(fs) · =f∇_Xs+(Xf)s for smooth real f. The range coordinates above identify ambient smooth range sections with smooth sections of the constructed bundle.
Exact mathematical passage
\begin{aligned} \nabla_{X+Y}s&=\nabla_Xs+\nabla_Ys,& \nabla_{fX}s&=f\nabla_Xs,\\ \nabla_X(s+t)&=\nabla_Xs+\nabla_Xt,& \nabla_X(fs)&=f\nabla_Xs+(Xf)s \end{aligned}
\nabla_{X+Y}Use in source · 1 passage
Search aliases: ∇_X+Y nabla
Use in source · 01_finite_core.tex, line 606
textitConnection rules. The smoothness of P ds(X) and the identity P^2=P show that ∇_Xs is a smooth range section. Differentiation and projection give ∇_X+Ys · =∇_Xs+∇_Ys, · ∇_fXs · =f∇_Xs, ∇_X(s+t) · =∇_Xs+∇_Xt, · ∇_X(fs) · =f∇_Xs+(Xf)s for smooth real f. The range coordinates above identify ambient smooth range sections with smooth sections of the constructed bundle.
Exact mathematical passage
\begin{aligned} \nabla_{X+Y}s&=\nabla_Xs+\nabla_Ys,& \nabla_{fX}s&=f\nabla_Xs,\\ \nabla_X(s+t)&=\nabla_Xs+\nabla_Xt,& \nabla_X(fs)&=f\nabla_Xs+(Xf)s \end{aligned}
\nabla_YUse in source · 4 passages
Search aliases: ∇_Y nabla
Use in source · 01_finite_core.tex, line 580
The rank of P is locally constant. On each open rank region, its ranges form a smooth vector bundle. For a smooth range section s, so that Ps=s, define ∇_Xs=P ds(X). This is a connection on smooth range sections. For smooth vector fields X,Y, its curvature satisfies (∇_X∇_Y-∇_Y∇_X-∇_[X,Y])s =P[dP(X),dP(Y)]P s. Equivalently, its endomorphism-valued curvature is F=P(dPwedge dP)P. No global frame or globally constant rank is required, and the vector fields need not commute. The geometric formalizat…
Exact mathematical passage
\boxed{ (\nabla_X\nabla_Y-\nabla_Y\nabla_X-\nabla_{[X,Y]})s =P[dP(X),dP(Y)]P\,s. }
\prod_{a=0}^{r-1}Use in source · 2 passages
Search aliases: ∏_a=0^r-1 prod
Use in source · strengthening.tex, line 552
Define the central-factorial basis polynomials P_r(X)=∏_a=0^r-1(X-a(a+1)). At X=Z(Z+1), they have the exact factorization P_r(Z(Z+1)) =∏_a=0^r-1(Z-a)(Z+a+1) =(2r)!binomZ+r2r. The Bernoulli transform gives L((1+T)^Z)=log(1+T)/T, whence L(binom Zj)=(-1)^j/(j+1). Finite Vandermonde expansion therefore gives L(binomZ+r2r) · =(-1)^r∑_q=0^r(-1)^qbinom rqfrac1r+q+1nonumber
Exact mathematical passage
P_r(X)=\prod_{a=0}^{r-1}(X-a(a+1)).
\prod_{a\in A_{N,j}}Use in source · 2 passages
Search aliases: ∏_a∈ A_N,j prod in
Use in source · Paper_IV_Traditional_Final.tex, line 580
The multiplication formula for the gamma function gives (Nn+j)! = j!N^Nnn! ∏_a∈ A_N,j(a)_n. Indeed, the N Pochhammer factors supplied by the multiplication formula are ((j+m)/N)_n for 1≤ m≤ N. The excluded factor is ((j+N-j)/N)_n=(1)_n=n!. Insertion into the defining series gives the stated hypergeometric expansion.
Exact mathematical passage
(Nn+j)! = j!N^{Nn}n! \prod_{a\in A_{N,j}}(a)_n.
\prod_CUse in source · 1 passage
Search aliases: ∏_C prod
Use in source · 03_packets_and_graphs.tex, line 884
Let P_C=∑_α∈ CP_α. Define ρ: M_R⟶∏_CEnd(X_C), ρ(T)=(P_CTP_C)_C. If an intermediate class D contributes to P_CSTP_C, then the preorder contains transfers C→ D and D→ C, forcing C=D in the quotient poset. Hence P_CSTP_C=(P_CSP_C)(P_CTP_C), so ρ is an algebra homomorphism. Every collection of endomorphisms of the X_C defines an allowed operator that is block diagonal with respect to the classes, so ρ is surjective. Its kernel is exactly N_R. The first isomorphism theorem gives the quotient formula.
Exact mathematical passage
\rho:\mathfrak M_R\longrightarrow\prod_C\End(X_C), \qquad \rho(T)=(P_CTP_C)_C.
\prod_{c=1}^{j}Use in source · 2 passages
Search aliases: ∏_c=1^j prod
Use in source · 20_arithmetic.tex, line 223
Set H_c(t)=∑_n≥0fract^Nn+c(Nn+c)!, A_c(t)=∑_n≥0fract^Nn(Nn+c)!, R_c(t)=fracA_c-1(t)A_c(t) (1≤ c<N). The denominator of this last quotient has nonzero constant coefficient. The section identities give R_cH_c=tH_c-1, R_c(0)=c, j! P_N,j= P_N,0∏_c=1^jR_c. Cancellation of consecutive source sections proves the product identity and ties the ratios directly to the family under study.
Exact mathematical passage
R_cH_c=tH_{c-1},\qquad R_c(0)=c,\qquad j!\mathcal P_{N,j}=\mathcal P_{N,0}\prod_{c=1}^{j}R_c.
\prod_{C\in\overline I}Use in source · 1 passage
Search aliases: ∏_C∈overline I prod in overline
Use in source · 03_packets_and_graphs.tex, line 867
k M_R consist of operators whose blocks vanish whenever source and target belong to the same equivalence class. Then: [label=(roman*)] item N_R is a nilpotent ideal on both sides. item If h is the maximal number of strict steps in a chain of overline I, then N_R^ h+1=0. item The map to the diagonal classes induces an algebra isomorphism M_R/ N_R ≅ ∏_C∈overline IEnd_𝕜(X_C). [source reference] ExportsFiltered transfer algebras and triangular packet inverses in Paper II.
Exact mathematical passage
\boxed{ \mathfrak M_R/\mathfrak N_R \cong \prod_{C\in\overline I}\End_{\bbk}(X_C). }
\prod_{f(\alpha)=0}Use in source · 1 passage
Search aliases: ∏_f(α)=0 prod alpha
Use in source · 72_recovered.tex, line 2815
We use the resultant convention Res_y(f,g) = ∏_f(α)=0g(α) when f is monic, with roots counted with multiplicity.
Exact mathematical passage
\Res_y(f,g) = \prod_{f(\alpha)=0}g(\alpha)
\prod_{\frakm\in\Max\TT}Use in source · 1 passage
Search aliases: ∏_frakm∈MaxTT ∏_ m∈Max T prod frakm in Max TT prod mathfrak in operatorname mathbb
Use in source · Paper_XII_Traditional_Final.tex, line 1025
[Artin decomposition] Let E' be a field and let T be a finite dimensional commutative E' algebra. Then T ≅ ∏_ m∈Max T T_ m. Each factor is an Artin local algebra. Equivalently, there is a unique complete family of central idempotents (e_ m) such that e_ me_ n=0 ( m≠ n), ∑_ me_ m=1, and e_ m T≅ T_ m [source reference].
Exact mathematical passage
\boxed{ \TT \cong \prod_{\frakm\in\Max\TT}\TT_{\frakm}. }
\prod_{h=0}^kUse in source · 1 passage
Search aliases: ∏_h=0^k prod
Use in source · strengthening.tex, line 150
For s∈ p Z_p, an exact finite-difference identity is Δ_1^kfrac11+sn =frac(-s)^k k!∏_h=0^k(1+s(n+h)). It follows by induction, subtracting adjacent rational functions. Each successive subtraction adds the factor -(k+1)s in the numerator. Its denominator is a unit, so it belongs to p^k Z_p. The discrete Leibniz formula Δ_1^k(fg)(n) =∑_r=0^kbinom kr (Δ_1^r f)(n)(Δ_1^k-rg)(n+r) then proves Δ_1^k g_N,j(n)∈ p^k Z_p for every k,n. The coefficient at degree Nn of L_1^kG_N,j is precisely p^-kΔ_1^k g_N,j(…
Exact mathematical passage
\Delta_1^k\frac1{1+sn} =\frac{(-s)^k k!}{\prod_{h=0}^k(1+s(n+h))}. \label{eq:affine-reciprocal-exact}
\prod_{h=1}^{2^a-1}Use in source · 1 passage
Search aliases: ∏_h=1^2^a-1 prod
Use in source · strengthening.tex, line 1139
at least two, since i≤ j<N/2. Hence g(1)≡1pmod4 and g(2)≡1pmod8. The first two Newton coefficients from the reciprocal recurrence are a_1=-g(1)-1, a_2=binom2NNg(1)^2-g(2)+2g(1)+1. The first has valuation exactly one. Also binom2NN≡6pmod8 for every N=2^a≥2. To see this, write B_a=binom2^a+12^a =2∏_h=1^2^a-1(1+frac2^ah). The even-h factors give B_a-1. The odd-h factors are one modulo four for a≥2. Since B_a-1 has valuation one, B_a≡ B_a-1pmod8, and B_1=6. Therefore a_2≡6-1+2+1=0pmod8. The global M…
Exact mathematical passage
B_a=\binom{2^{a+1}}{2^a} =2\prod_{h=1}^{2^a-1}\left(1+\frac{2^a}{h}\right).
\prod_iUse in source · 5 passages
Search aliases: ∏_i prod
Use in source · strengthening.tex, line 101
e with unspecified remainders. On the tensor of the two factors use the mutually commuting operators L_λ⊗1, 1⊗ L_μ, c_iD^i⊗ D^N-i (1≤ i<N). Applying multiplication after the kth power of their sum gives exactly L_λ+μ^k(AB). Expanding it yields terms indexed by a,b,e_1,…,e_N-1≥0, with a+b+∑_i e_i=k: frack!a!b!∏_i e_i!∏_i c_i^e_i (D^α L_λ^a A)(D^β L_μ^b B), α=∑_i i e_i, β=∑_i(N-i)e_i. Every scalar is integral. If A and B have the stated flatness types, both factors in every term belong to H_p, as …
Exact mathematical passage
\frac{k!}{a!b!\prod_i e_i!}\prod_i c_i^{e_i} (D^\alpha L_\lambda^a A)(D^\beta L_\mu^b B), \quad \alpha=\sum_i i e_i,\quad \beta=\sum_i(N-i)e_i. \label{eq:translated-multinomial}
\prod_{i=0}^{d-1}Use in source · 2 passages
Search aliases: ∏_i=0^d-1 prod
Use in source · 20_arithmetic.tex, line 240
_p={∑_d≥0a_dT^d: p^lfloor d/prfloora_d∈ Z_p}. The key estimate is R_ap(log(1+T))∈ W_p. Its arithmetic input comes from the cyclic group ring Z[ Z/N Z]. For its canonical element x and I=(p,x^p-1-1), the relation (x^p-1)^p=1 implies I^p⊆(p)I. Induction yields I^q⊆(p^v_p(q!)). Grouping the falling factorial (x)_d=∏_i=0^d-1(x-i) into consecutive blocks of length p then gives an integral residue-coordinate factor p^v_p(lfloor d/prfloor!). The identity v_p(d!)=lfloor d/prfloor+v_p(lfloor d/prfloor!) …
Exact mathematical passage
(x)_d=\prod_{i=0}^{d-1}(x-i)
\prod_{i=0}^{p-1}Use in source · 1 passage
Search aliases: ∏_i=0^p-1 prod
Use in source · supplement_09.tex, line 58
Now take the integral cyclic group ring R_N= Z[ Z/N Z], x=[1], x^N=1, and assume Nmid p(p-1). Thus (x^p-1)^p=1. Each genuine consecutive block ∏_i=0^p-1(x-(pb+i)) belongs to (p,x^p-1-1). This follows from the actual polynomial identity ∏_i∈ F_p(X-i)=X^p-X modulo p, not from a hypothesized valuation of the product.
Exact mathematical passage
\prod_{i=0}^{p-1}\bigl(x-(pb+i)\bigr)
\prod_{i=0}^{q-1}Use in source · 1 passage
Search aliases: ∏_i=0^q-1 prod
Use in source · root_scope.tex, line 107
Set r=M+1≥1 and define P_q(x)=∏_i=0^q-1(x-i)(2x-(2i+1)), P_0=1, H_r(x)=∑_q≥08^qfracβ_r+qβ_r P_q(x)mathscr A(x-q). The coefficient valuations tend to infinity. At any nonnegative integer x, terms with q>x vanish. Cancelling the factorials in [source reference] therefore gives the exact identity T_r-1(N)=κ_rbinom2N2rH_r(N-r) (N≥ r). This is a finite rational identity at every original Euler row.
Exact mathematical passage
P_q(x)=\prod_{i=0}^{q-1}(x-i)(2x-(2i+1)),\qquad P_0=1,
\prod_{i=1}^{d}Use in source · 1 passage
Search aliases: ∏_i=1^d prod
Use in source · 72_recovered.tex, line 2840
After passage to an algebraic closure, let α_1,…,α_d be the eigenvalues of C, counted with algebraic multiplicity. The resultant is ∏_i=1^d(x-α_i^N), which is the characteristic polynomial of C^N. Both sides are polynomial expressions with integer coefficients in the entries of C, so the identity descends to every commutative coefficient ring.
Exact mathematical passage
\prod_{i=1}^{d}(x-\alpha_i^N),
\prod_{i=1}^jUse in source · 1 passage
Search aliases: ∏_i=1^j prod
Use in source · strengthening.tex, line 140
The packed factorial coefficient of G_N,j is g_N,j(n)=fracj!(Nn)!(Nn+j)! =∏_i=1^j(1+frac Ni n)^-1. The empty product is one. If N=p^a and 1≤ i<N, then v_p(i)<a, so N/i∈ p Z_p. Every affine denominator in [source reference] is therefore a unit on Z_p.
Exact mathematical passage
g_{N,j}(n)=\frac{j!(Nn)!}{(Nn+j)!} =\prod_{i=1}^j\left(1+\frac Ni n\right)^{-1}. \label{eq:actual-prime-power-kernel}
\prod_{i=1}^{j-1}Use in source · 2 passages
Search aliases: ∏_i=1^j-1 prod
Use in source · 01_manuscript.tex, line 864
[Product chain perturbation] Let A_1,…,A_m be exact factors and let widehat A_j=A_j+E_j. Then widehat A_m⋯widehat A_1 - A_m⋯ A_1 ≤ ∑_j=1^m ( ∏_i=j+1^mnormwidehat A_i ) normE_j ( ∏_i=1^j-1normA_i ). Empty products are one.
Exact mathematical passage
\boxed{ \left\| \widehat A_m\cdots\widehat A_1 - A_m\cdots A_1 \right\| \leq \sum_{j=1}^m \left( \prod_{i=j+1}^m\norm{\widehat A_i} \right) \norm{E_j} \left( \prod_{i=1}^{j-1}\norm{A_i} \right). }
\prod_{i=1}^{N}Use in source · 1 passage
Search aliases: ∏_i=1^N prod
Use in source · Paper_X_Block27_Amended.tex, line 1620
Let E be a field of characteristic zero and let T_1,…,T_N be commuting operators on a finite-dimensional E-vector space A, each diagonalizable over E. Thus the selected spectra split over the coefficient field. Let a_i∈ E be a selected target scalar for T_i, possibly absent from its spectrum. Define e_i = ∏_μ∈Spec(T_i)∖{a_i} fracT_i-μ Ia_i-μ, e_T=∏_i=1^Ne_i, with e_i=0 if a_i is absent.
Exact mathematical passage
e_i = \prod_{\mu\in\Spec(T_i)\setminus\{a_i\}} \frac{T_i-\mu I}{a_i-\mu}, \qquad e_T=\prod_{i=1}^{N}e_i,
\prod_{i=1}^rUse in source · 1 passage
Search aliases: ∏_i=1^r prod
Use in source · 04_arithmetic.tex, line 56
[Several coprime spectral packets] Assume χ_A(x)=∏_i=1^rf_i(x), where the monic factors f_i∈ O[x] are pairwise coprime in K[x]. Put g_i=∏_j≠ if_j. There are complementary projectors E_i with im E_i=Ker f_i(A), ∑_iE_i=I, E_iE_j=0 (i≠ j). One may choose E_i so that Res(f_i,g_i)E_i∈End_ O(Λ) up to multiplication of the resultant by a unit.
Exact mathematical passage
\chi_A(x)=\prod_{i=1}^rf_i(x),
\prod_{i=1}^{r}Use in source · 1 passage
Search aliases: ∏_i=1^r prod
Use in source · Paper_XII_Traditional_Final.tex, line 490
[Residual augmentation algebra] Suppose Δ≅ ∏_i=1^rC_p^n_i. Choose generators γ_i, and let X_i=γ_i-1. Then k[Δ] ≅ k[X_1,…,X_r] / (X_1^p^n_1,…,X_r^p^n_r). The augmentation ideal is generated by the X_i, and k[Δ] is local.
Exact mathematical passage
\Delta\cong \prod_{i=1}^{r}C_{p^{n_i}}.
\prod_{i=1}^{\ell-1}Use in source · 1 passage
Search aliases: ∏_i=1^ℓ-1 prod ell
Use in source · 72_recovered.tex, line 2018
[Nested discrete refinement] For positive integers M,N, (a_r^[M])_s^[N] = a_r+Ms^[MN]. For a mixed radix path (M_1,…,M_ℓ) with digits 0≤ s_i<M_i, the final address is r( s) = s_1+M_1s_2+⋯+ (∏_i=1^ℓ-1M_i)s_ℓ. Only the product radix and accumulated address determine the final branch.
Exact mathematical passage
\boxed{ r(\boldsymbol s) = s_1+M_1s_2+\cdots+ \left(\prod_{i=1}^{\ell-1}M_i\right)s_\ell. }
\prod_{i=j+1}^mUse in source · 3 passages
Search aliases: ∏_i=j+1^m prod
Use in source · 01_manuscript.tex, line 864
[Product chain perturbation] Let A_1,…,A_m be exact factors and let widehat A_j=A_j+E_j. Then widehat A_m⋯widehat A_1 - A_m⋯ A_1 ≤ ∑_j=1^m ( ∏_i=j+1^mnormwidehat A_i ) normE_j ( ∏_i=1^j-1normA_i ). Empty products are one.
Exact mathematical passage
\boxed{ \left\| \widehat A_m\cdots\widehat A_1 - A_m\cdots A_1 \right\| \leq \sum_{j=1}^m \left( \prod_{i=j+1}^m\norm{\widehat A_i} \right) \norm{E_j} \left( \prod_{i=1}^{j-1}\norm{A_i} \right). }
\prod_{i=j+1}^{m}Use in source · 1 passage
Search aliases: ∏_i=j+1^m prod
Use in source · 01_manuscript.tex, line 915
ntheorem[Certified repeated recompression] Let P_0=I, P_j=A_jP_j-1, be exact partial products. Suppose an implemented chain satisfies widehat P_0=I, widehat P_j = A_jwidehat P_j-1+E_j, where E_j is the additive error introduced after the jth multiplication, for example by factor recompression. Then normwidehat P_m-P_m ≤ ∑_j=1^m ( ∏_i=j+1^mnormA_i ) normE_j. If normE_j≤δ_j, the same formula with δ_j gives a certified final error.
Exact mathematical passage
\boxed{ \norm{\widehat P_m-P_m} \leq \sum_{j=1}^{m} \left( \prod_{i=j+1}^{m}\norm{A_i} \right) \norm{E_j}. }
\prod_{i\in F}Use in source · 1 passage
Search aliases: ∏_i∈ F prod in
Use in source · 03_supplement_02.tex, line 48
Let F be an arbitrary finite set of factors and let D_i be a finite coordinate index for factor i. For matrix families A_i, define the coordinate tensor product on the product index space by T(A)(x,y)=∏_i∈ F A_i(x_i,y_i). The proof first establishes the laws for multiplication, identity, adjoints, and finite sums. These laws apply to any finite number of factors with the stated finite coordinate sets.
Exact mathematical passage
\mathsf T(A)(x,y)=\prod_{i\in F} A_i(x_i,y_i).
\prod_{i\in I}Use in source · 4 passages
Search aliases: ∏_i∈ I prod in
Use in source · Paper_X_Block27_Amended.tex, line 1389
Explicit summands inside a supplied decomposition. Let K be a field, let G be a monoid, and let ρ_i:G→End_K(W_i) be actual representations on finite-dimensional spaces, indexed by a finite set I. For natural numbers m_i≤ c_i, set U_m=∏_i∈ I∏_k<m_iW_i, U_c=∏_i∈ I∏_k<c_iW_i. These finite products realize the corresponding finite direct sums. The representation on U_m acts componentwise. Define j:U_m→ U_c by retaining the first m_i copies and filling every remaining coordinate with zero. Define r:U…
Exact mathematical passage
U_m=\prod_{i\in I}\prod_{k<m_i}W_i, \qquad U_c=\prod_{i\in I}\prod_{k<c_i}W_i.
\prod_{i\in\F_p}Use in source · 1 passage
Search aliases: ∏_i∈F_p ∏_i∈ F_p prod in F prod in mathbb
Use in source · supplement_09.tex, line 62
Now take the integral cyclic group ring R_N= Z[ Z/N Z], x=[1], x^N=1, and assume Nmid p(p-1). Thus (x^p-1)^p=1. Each genuine consecutive block ∏_i=0^p-1(x-(pb+i)) belongs to (p,x^p-1-1). This follows from the actual polynomial identity ∏_i∈ F_p(X-i)=X^p-X modulo p, not from a hypothesized valuation of the product.
Exact mathematical passage
\prod_{i\in\F_p}(X-i)=X^p-X
\prod_jUse in source · 3 passages
Search aliases: ∏_j prod
Use in source · 03_nonlinear_coordinates.tex, line 265
[Jet norm bound] Give every packet range its inherited subspace norm. The restricted jet satisfies the sharper estimate J^r_γmidα_1,…,α_rF(x) ≤ Q_γ D^rF(x) . Its ambient extension inserts P_α_j in the jth slot and has norm at most Q_γ D^rF(x) ∏_j P_α_j . Consequently the operator norm of the restricted packet jet also satisfies J^r_γmidα_1,…,α_rF(x) ≤ Q_γ D^rF(x) ∏_j=1^r P_α_j . When all packets are orthogonal projections on Hilbert spaces, every nonzero projector has norm one and the bound redu…
Exact mathematical passage
\|Q_\gamma\|\,\|D^rF(x)\|\prod_j\|P_{\alpha_j}\|
\prod_{j=0}^{n-1}Use in source · 2 passages
Search aliases: ∏_j=0^n-1 prod
Use in source · Paper_X_Block27_Amended.tex, line 1471
The exact special chain has basis e_0,…,e_n-1 and operators Ne_0=0, Ne_j=e_j-1 (j>0), Fe_j=aq^-je_j, a,q∈ K^×. Lean proves N^n=0, and for n>0 also N^n-1≠0 and ker N=Ke_0. Consequently its kernel polynomial is X-a, whereas its whole-space polynomial is ∏_j=0^n-1(X-aq^-j). The zero-length chain is included as a zero-dimensional space. The assertion ker N=Ke_0 is made only for positive length.
Exact mathematical passage
\prod_{j=0}^{n-1}(X-aq^{-j})
\prod_{j=1}^{m}Use in source · 2 passages
Search aliases: ∏_j=1^m prod
Use in source · Paper_V_Traditional_Final.tex, line 3520
[Shifted pointwise monomial law] Let y(n)=∏_j=1^mx(n-a_j) and write a_j=Nc_j+b_j, 0≤ b_j<N. For output residue r, define r_j≡ r-b_jpmod N, 0≤ r_j<N, and let κ_r,b_j = 1, · r<b_j,
Exact mathematical passage
y(n)=\prod_{j=1}^{m}x(n-a_j)
\prod_{j=1}^rUse in source · 2 passages
Search aliases: ∏_j=1^r prod
Use in source · 03_nonlinear_coordinates.tex, line 267
cket range its inherited subspace norm. The restricted jet satisfies the sharper estimate J^r_γmidα_1,…,α_rF(x) ≤ Q_γ D^rF(x) . Its ambient extension inserts P_α_j in the jth slot and has norm at most Q_γ D^rF(x) ∏_j P_α_j . Consequently the operator norm of the restricted packet jet also satisfies J^r_γmidα_1,…,α_rF(x) ≤ Q_γ D^rF(x) ∏_j=1^r P_α_j . When all packets are orthogonal projections on Hilbert spaces, every nonzero projector has norm one and the bound reduces to the norm of the full de…
Exact mathematical passage
\left\|J^{r}_{\gamma\mid\alpha_1,\ldots,\alpha_r}F(x)\right\| \le \|Q_\gamma\|\,\|D^rF(x)\| \prod_{j=1}^r\|P_{\alpha_j}\|.
\prod_{j=1}^{s}Use in source · 5 passages
Search aliases: ∏_j=1^s prod
Use in source · 72_recovered.tex, line 3027
[Several coprime moduli] Let N_1,…,N_s be pairwise coprime and put N=∏_j=1^sN_j. For compatible addresses r_j, P_N,r = ∏_j=1^sP_N_j,r_j, where r is the unique combined address modulo N.
Exact mathematical passage
N=\prod_{j=1}^{s}N_j.
\prod_{j\ne i}Use in source · 1 passage
Search aliases: ∏_j≠ i prod ne
Use in source · 04_arithmetic.tex, line 60
[Several coprime spectral packets] Assume χ_A(x)=∏_i=1^rf_i(x), where the monic factors f_i∈ O[x] are pairwise coprime in K[x]. Put g_i=∏_j≠ if_j. There are complementary projectors E_i with im E_i=Ker f_i(A), ∑_iE_i=I, E_iE_j=0 (i≠ j). One may choose E_i so that Res(f_i,g_i)E_i∈End_ O(Λ) up to multiplication of the resultant by a unit.
Exact mathematical passage
g_i=\prod_{j\ne i}f_j.
\prod_{k<c_i}Use in source · 1 passage
Search aliases: ∏_k<c_i prod
Use in source · Paper_X_Block27_Amended.tex, line 1389
Explicit summands inside a supplied decomposition. Let K be a field, let G be a monoid, and let ρ_i:G→End_K(W_i) be actual representations on finite-dimensional spaces, indexed by a finite set I. For natural numbers m_i≤ c_i, set U_m=∏_i∈ I∏_k<m_iW_i, U_c=∏_i∈ I∏_k<c_iW_i. These finite products realize the corresponding finite direct sums. The representation on U_m acts componentwise. Define j:U_m→ U_c by retaining the first m_i copies and filling every remaining coordinate with zero. Define r:U…
Exact mathematical passage
U_m=\prod_{i\in I}\prod_{k<m_i}W_i, \qquad U_c=\prod_{i\in I}\prod_{k<c_i}W_i.
\prod_{k<m_i}Use in source · 2 passages
Search aliases: ∏_k<m_i prod
Use in source · Paper_X_Block27_Amended.tex, line 1389
Explicit summands inside a supplied decomposition. Let K be a field, let G be a monoid, and let ρ_i:G→End_K(W_i) be actual representations on finite-dimensional spaces, indexed by a finite set I. For natural numbers m_i≤ c_i, set U_m=∏_i∈ I∏_k<m_iW_i, U_c=∏_i∈ I∏_k<c_iW_i. These finite products realize the corresponding finite direct sums. The representation on U_m acts componentwise. Define j:U_m→ U_c by retaining the first m_i copies and filling every remaining coordinate with zero. Define r:U…
Exact mathematical passage
U_m=\prod_{i\in I}\prod_{k<m_i}W_i, \qquad U_c=\prod_{i\in I}\prod_{k<c_i}W_i.
\prod_pUse in source · 4 passages
Search aliases: ∏_p prod
Use in source · 02_foundations.tex, line 920
Let the minimal polynomial be m_T(t)=∏_pp(t)^e_p. The acting algebra is 𝕜[T]≅𝕜[t]/(m_T) ≅∏_pR_p, R_p:=𝕜[t]/(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with maximal ideal generated by the class of p and residue division field K_p=𝕜[t]/(p). By [source reference], srank_R_p(X_p) = dim_K_pX_p/p(T)X_p = μ_p(T). …
Exact mathematical passage
\bbk[T]\cong\bbk[t]/(m_T) \cong\prod_pR_p, \qquad R_p:=\bbk[t]/(p^{e_p}),
\prod_{p}Use in source · 1 passage
Search aliases: ∏_p prod
Use in source · 02_foundations.tex, line 916
Let the minimal polynomial be m_T(t)=∏_pp(t)^e_p. The acting algebra is 𝕜[T]≅𝕜[t]/(m_T) ≅∏_pR_p, R_p:=𝕜[t]/(p^e_p), by the Chinese remainder theorem. The factor R_p acts on X_p and annihilates every other primary component. Hence the formula for a direct sum with independent actions gives srank_𝕜[T](X)=max_psrank_R_p(X_p). Each R_p is local with maximal ideal generated by the class of p and residue division field K_p=𝕜[t]/(p). By [source reference], srank_R_p(X_p) = dim_K_pX_p/p(T)X_p = μ_p(T). …
Exact mathematical passage
m_T(t)=\prod_{p}p(t)^{e_p}.
\prod_{p>2}Use in source · 2 passages
Search aliases: ∏_p>2 prod
Use in source · 50_open_questions.tex, line 138
Consequently this canonical global diagonal cannot be an equality locus {(a,b):f(a)=g(b)} ⊆ Z_2×∏_p>2 Z_p for continuous maps to a Hausdorff space C. Such an equality locus is closed, as the inverse image of the closed diagonal in C× C, whereas D is proper and dense. This topological argument, retained from the native progress note, concerns the stated topological interpretation. It has not been formalized in Lean for this manuscript.
Exact mathematical passage
\{(a,b):f(a)=g(b)\} \subseteq \Z_2\times\prod_{p>2}\Z_p
\prod_{p>2}'Use in source · 1 passage
Search aliases: ∏_p>2' prod
Use in source · 50_open_questions.tex, line 95
The original proposed form is X≃ X_2^wt×_C∏_p>2'X_p. It leaves X, the distinguished subsets defining the restricted product, the maps to C, the ambient category, and the universal meaning of ``smallest'' unspecified. In particular, one must decide whether X is a global object or a completed object.
Exact mathematical passage
X\simeq X_2^{\mathrm{wt}}\times_C\prod_{p>2}'X_p.
\prod_{p\in S}Use in source · 3 passages
Search aliases: ∏_p∈ S prod in
Use in source · 71_native.tex, line 210
N=∏_p∈ Sp^L_p, Sol_F(R)={x∈ R^d:F(x)=0}.
Exact mathematical passage
N=\prod_{p\in S}p^{L_p},\qquad\operatorname{Sol}_F(R)=\{x\in R^d:F(x)=0\}.
\prod_qUse in source · 1 passage
Search aliases: ∏_q prod
Use in source · 02_connections_obstructions.tex, line 889
For coordinate models of the source and target colors, an additive target coordinate space, and a chosen noncanonical connection, a pointwise node defect can be defined by D^Γ_b,a(Φ) = Γ^Y_b← a∘widehatΦ_a - widehatΦ_b∘ (∏_qΓ^X_q_b← a). This expression compares two chosen routes. With canonical exact transport in every slot it vanishes by [source reference].
Exact mathematical passage
\mathfrak D^\Gamma_{b,a}(\Phi) = \Gamma^Y_{b\leftarrow a}\circ\widehat\Phi_a - \widehat\Phi_b\circ \left(\prod_q\Gamma^{X_q}_{b\leftarrow a}\right).
\prod_rUse in source · 3 passages
Search aliases: ∏_r prod
Use in source · 72_recovered.tex, line 704
[Lucas congruence] If a=a_0+a_1p+⋯ and b=b_0+b_1p+⋯ are base p expansions, then binom ab ≡ ∏_rbinoma_rb_r pmod p. This is the classical theorem of Lucas [source reference].
Exact mathematical passage
\binom ab \equiv \prod_r\binom{a_r}{b_r} \pmod p.
\prod_{r<n}Use in source · 1 passage
Search aliases: ∏_r<n prod
Use in source · 10_foundations.tex, line 197
The recursion uses only smaller indices, and substitution verifies each row. If a second solution agrees through row n-1, subtracting row n and cancelling T_nn proves equality at n. For integrality, use induction and promote every earlier clearing product to ∏_r<nT_rr. Multiplication by the additional factor T_nn cancels the only new division.
Exact mathematical passage
\prod_{r<n}T_{rr}
\prod_{r=0}^{2}Use in source · 1 passage
Search aliases: ∏_r=0^2 prod
Use in source · atlas-pages.tex, line 448
1 · 3, start at state 0, use weights w=(1,1/2), and terminal observable f=(2,1). For three transitions, there are eight possible state sequences after the fixed initial state. We evaluate E_0[∏_r=0^2w(X_r) f(X_3)]. Each path contributes its transition probability times its three departure weights times its terminal value. vfigpaths platecaptionEvery possible path is included. The probability column sums to one. The contribution column sums to 311/256, agreeing exactly with the transfer calculati…
Exact mathematical passage
\EE_0\left[\prod_{r=0}^{2}w(X_r)\,f(X_3)\right].
\prod_{r=0}^{n}Use in source · 1 passage
Search aliases: ∏_r=0^n prod
Use in source · 10_foundations.tex, line 188
em[Forward substitution and denominator clearing] Let T_nq be coefficients over a field with T_nn≠0 for every n. For any sequence b, define a_n=fracb_n-∑_q<nT_nqa_qT_nn. This constructs the unique solution of ∑_q=0^nT_nqa_q=b_n for all n. If T_nq and b_n are integers and the solution is taken over Q, then (∏_r=0^nT_rr)a_n∈ Z.
Exact mathematical passage
\left(\prod_{r=0}^{n}T_{rr}\right)a_n\in\Z.
\prod_{r=1}^{n}Use in source · 3 passages
Search aliases: ∏_r=1^n prod
Use in source · 20_arithmetic.tex, line 59
[A denominator bound for every address] For N≥1, 0≤ j<N, and n∈ N, set D_N,j(n)=∏_r=1^nbinomNr+jNr, D_N,j(0)=1. Then D_N,j(n)C_N,j(n)∈ Z.
Exact mathematical passage
D_{N,j}(n)=\prod_{r=1}^{n}\binom{Nr+j}{Nr},\qquad D_{N,j}(0)=1.
\prod_sUse in source · 1 passage
Search aliases: ∏_s prod
Use in source · Paper_XI_Traditional_Final.tex, line 1771
The Heisenberg group of an orthogonal sum is the central product of the Heisenberg groups of the summands. The tensor product representation has the declared central character and has dimension ∏_ssqrtabsV_s = sqrtabs V. It is irreducible by the finite Stone von Neumann theorem, so it is the required representation.
Exact mathematical passage
\prod_s\sqrt{\abs{V_s}} = \sqrt{\abs V}.
\prod_{s\in S}Use in source · 1 passage
Search aliases: ∏_s∈ S prod in
Use in source · Paper_XI_Traditional_Final.tex, line 1781
[Breakwise phase product] In odd residue characteristic, let g∈Sp(V) preserve every break packet and suppose its restriction to M=im(g-I) is fixed point free. Put g_s=g|_V_s and M_s=im(g_s-I). The moving restrictions are symplectic, and Φ_p(g|_M,M) = ∏_s∈ SΦ_p(g_s|_M_s,M_s). In residue characteristic two, the analogous normalized Gauss sign is the product of the signs determined by the breakwise quadratic refinements.
Exact mathematical passage
\boxed{ \Phi_p(g|_M,M) = \prod_{s\in S}\Phi_p(g_s|_{M_s},M_s). }
\prod_\alphaUse in source · 1 passage
Search aliases: ∏_α prod alpha
Use in source · 03_packets_and_graphs.tex, line 160
enumerate item Every family of linear maps f_α:X_α→ Y determines a unique linear map f:X→ Y with f|_X_α=f_α, namely f=∑_α f_α P_α. item Every family of linear maps g_α:Y→ X_α determines a unique linear map g:Y→ X with P_α g=g_α, namely g=∑_α g_α. Equivalently, Hom(X,Y)≅∏_αHom(X_α,Y), Hom(Y,X)≅∏_αHom(Y,X_α), where the products are finite. [source reference] ExportsAssembly of packetwise operators, channels, corrections, and local transport data.
Exact mathematical passage
\Hom(X,Y)\cong\prod_\alpha\Hom(X_\alpha,Y), \qquad \Hom(Y,X)\cong\prod_\alpha\Hom(Y,X_\alpha),
\prod_{\beta,\alpha\in I}Use in source · 1 passage
Search aliases: ∏_β,α∈ I prod beta alpha in
Use in source · 03_packets_and_graphs.tex, line 678
[Full block algebra representation] The map Θ_ P:End_𝕜(X) ⟶ ∏_β,α∈ IHom_𝕜(X_α,X_β), T⟼(T_βα)_β,α, is an algebra isomorphism when the codomain is equipped with matrix multiplication whose entries are operators. DependsnolinkurlPSC-I-T-OPERATOR-RECONSTRUCTION and nolinkurlPSC-I-T-BLOCK-MULTIPLICATION.
Exact mathematical passage
\Theta_{\Pcal}:\End_{\bbk}(X) \longrightarrow \prod_{\beta,\alpha\in I}\Hom_{\bbk}(X_\alpha,X_\beta), \qquad T\longmapsto(T_{\beta\alpha})_{\beta,\alpha},
\prod_{\lambda\nmid2}'Use in source · 1 passage
Search aliases: ∏_λnmid2' prod lambda nmid
Use in source · Paper_X_Block27_Amended.tex, line 775
The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.
Exact mathematical passage
\boxed{ \Sys_{\mathrm{sys}} = \Sys_2^{\mathrm{wt}} \times_{\Compat} \prod_{\lambda\nmid2}'\Sys_\lambda. }
\prod_{\lambda\in\Lambda_N}Use in source · 5 passages
Search aliases: ∏_λ∈Λ_N prod lambda in Lambda
Use in source · Paper_IV_Traditional_Final.tex, line 2298
[Genus zero product] For N≥2, let Λ_N be the nonzero zero multiset of Δ_N. Then Δ_N(w) = ∏_λ∈Λ_N (1-frac wλ), where the product converges locally uniformly. Consequently, D_N(z) = ∏_λ∈Λ_N (1-fracz^Nλ).
Exact mathematical passage
\Delta_N(w) = \prod_{\lambda\in\Lambda_N} \left(1-\frac w\lambda\right),
\prod_{\mu\in\Spec(T_i)\setminus\{a_i\}}Use in source · 1 passage
Search aliases: ∏_μ∈Spec(T_i)∖{a_i} prod mu in Spec setminus prod mu in operatorname setminus
Use in source · Paper_X_Block27_Amended.tex, line 1620
Let E be a field of characteristic zero and let T_1,…,T_N be commuting operators on a finite-dimensional E-vector space A, each diagonalizable over E. Thus the selected spectra split over the coefficient field. Let a_i∈ E be a selected target scalar for T_i, possibly absent from its spectrum. Define e_i = ∏_μ∈Spec(T_i)∖{a_i} fracT_i-μ Ia_i-μ, e_T=∏_i=1^Ne_i, with e_i=0 if a_i is absent.
Exact mathematical passage
e_i = \prod_{\mu\in\Spec(T_i)\setminus\{a_i\}} \frac{T_i-\mu I}{a_i-\mu}, \qquad e_T=\prod_{i=1}^{N}e_i,
\prod_{\mu\in\Spec(T_i)\setminus\{a\}}Use in source · 1 passage
Search aliases: ∏_μ∈Spec(T_i)∖{a} prod mu in Spec setminus prod mu in operatorname setminus
Use in source · Paper_X_Block27_Amended.tex, line 1649
For each operator, its finite set of actual eigenvalues is used without multiplicity. The polynomial p_i,a(X)=∏_μ∈Spec(T_i)∖{a} (a-μ)^-1(X-μ) is used when a occurs, and the zero polynomial otherwise. For an eigenvector T_i v=bv, evaluation gives p_i,a(T_i)v=v · b=a,
Exact mathematical passage
p_{i,a}(X)=\prod_{\mu\in\Spec(T_i)\setminus\{a\}} (a-\mu)^{-1}(X-\mu)
\prod_{\mu\in\Lambda_{N,j}}Use in source · 1 passage
Search aliases: ∏_μ∈Λ_N,j prod mu in Lambda
Use in source · Paper_IV_Traditional_Final.tex, line 2398
[Products for the normalized sections] For every 0≤ j<N, the entire function Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)! has order 1/N. If N≥2 and Λ_N,j is its nonzero zero multiset, then Φ_N,j(w) = ∏_μ∈Λ_N,j (1-frac wμ). For m≥1, put σ_N,j,m = ∑_μ∈Λ_N,jμ^-m. Then C_N,j,n(r) = frac(Nn)!n! B_n( rσ_N,j,1, r1!σ_N,j,2, …, r(n-1)!σ_N,j,n ).
Exact mathematical passage
\Phi_{N,j}(w) = \prod_{\mu\in\Lambda_{N,j}} \left(1-\frac w\mu\right).
\prod_{\sigma}Use in source · 1 passage
Search aliases: ∏_σ prod sigma
Use in source · residue_continuation_body.tex, line 79
The relation to the geometric sign is also proved. If σ ranges over real embeddings and τ over one embedding from each complex pair, then n(x)=∏_σσ(x)∏_τabsnτ(x)^2. For x≠0, the second factor is positive. Thus the sign of n(x) is exactly the sign of the real-coordinate product. Cutting D into its positive and negative portions D_+,D_- adds no boundary outside the existing frontier, and both portions retain the proved boundary atlas. They are disjoint and their union is D.
Exact mathematical passage
n(x)=\prod_{\sigma}\sigma(x)\prod_{\tau}\absn{\tau(x)}^2.
\prod_{\tau}Use in source · 1 passage
Search aliases: ∏_τ prod tau
Use in source · residue_continuation_body.tex, line 79
The relation to the geometric sign is also proved. If σ ranges over real embeddings and τ over one embedding from each complex pair, then n(x)=∏_σσ(x)∏_τabsnτ(x)^2. For x≠0, the second factor is positive. Thus the sign of n(x) is exactly the sign of the real-coordinate product. Cutting D into its positive and negative portions D_+,D_- adds no boundary outside the existing frontier, and both portions retain the proved boundary atlas. They are disjoint and their union is D.
Exact mathematical passage
n(x)=\prod_{\sigma}\sigma(x)\prod_{\tau}\absn{\tau(x)}^2.
\sum_{(b,a)\in E_A}Use in source · 2 passages
Search aliases: ∑_(b,a)∈ E_A sum in
Use in source · 02_supplement_01.tex, line 13
Reconstruction and action. The checked finite sum is A=∑_(b,a)∈ E_A A_ba. The proof inserts the complete projector sum on both sides and removes only zero summands. Applying this identity to an arbitrary module gives exact action. A separate finite-matrix theorem establishes AZ=∑_(b,a)∈ E_AA_baZ for every rectangular batch of right-hand sides Z.
Exact mathematical passage
\boxed{A=\sum_{(b,a)\in E_A} A_{ba}.}
\sum_{(e_1,\ldots,e_n)\in E^n}Use in source · 1 passage
Search aliases: ∑_(e_1,…,e_n)∈ E^n sum ldots in
Use in source · 01_finite_coordinates.tex, line 307
Induction and finite distributivity give (∑_e∈ EL_e)^n = ∑_(e_1,…,e_n)∈ E^nL_e_n⋯ L_e_1. Every noncomposable summand is zero by [source reference], so inserting the composability filter does not change the sum. In Lean, tuple concatenation and the successor sum are handled by leannametupleWordOp_snoc and leannameexplicitWordExpansion_succ. The two bridge equalities then yield the displayed theorem.
Exact mathematical passage
\left(\sum_{e\in E}L_e\right)^n = \sum_{(e_1,\ldots,e_n)\in E^n}L_{e_n}\cdots L_{e_1}.
\sum_{(i_1,\ldots,i_{m-1})\in V^{m-1}}Use in source · 1 passage
Search aliases: ∑_(i_1,…,i_m-1)∈ V^m-1 sum ldots in
Use in source · 02_connections_obstructions.tex, line 610
[Packet path expansion for different matrices] For i,j∈ V and m≥1, set i_0=i and i_m=j. Then (B_m⋯ B_1)_ji = ∑_(i_1,…,i_m-1)∈ V^m-1 B_m[i_m,i_m-1]⋯ B_1[i_1,i_0]. When m=1, the sum is over the unique empty tuple and its summand is B_1[j,i]. For m=0, the product is the identity and the entry is the Kronecker delta δ_ji. If every B_r[q,p] vanishes outside a declared relation R_r(p,q), then the endpoint entry is zero whenever there is no packet path compatible with the successive relations. statusle…
Exact mathematical passage
\boxed{ (B_m\cdots B_1)_{ji} = \sum_{(i_1,\ldots,i_{m-1})\in V^{m-1}} B_m[i_m,i_{m-1}]\cdots B_1[i_1,i_0]. }
\sum_{0\leq n\leq m/N}Use in source · 1 passage
Search aliases: ∑_0≤ n≤ m/N sum leq leq
Use in source · Paper_IV_Traditional_Final.tex, line 1514
[Explicit Appell polynomial] For every m≥0, E_m^(N,j,r)(x) = ∑_0≤ n≤ m/N binommNn C_N,j,n(r)x^m-Nn.
Exact mathematical passage
E_m^{(N,j,r)}(x) = \sum_{0\leq n\leq m/N} \binom{m}{Nn} C_{N,j,n}(r)x^{m-Nn}.
\sum_{2g=k}Use in source · 1 passage
Search aliases: ∑_2g=k sum
Use in source · 03_nonlinear_coordinates.tex, line 679
[Divided square packet equations] Assume q( L_g)⊆ L_2g. Choose an ordering of the finite label set. The equation dη+q(η)=0 is equivalent to dη_k + ∑_2g=kq(η_g) + ∑_substackg<h g+h=k[η_g,η_h] =0 for every k.
Exact mathematical passage
d\eta_k + \sum_{2g=k}q(\eta_g) + \sum_{\substack{g<h\\g+h=k}}[\eta_g,\eta_h] =0
\sum_{a,b}Use in source · 5 passages
Search aliases: ∑_a,b sum
Use in source · monograph_atlas.tex, line 431
nnegative entries and column sums one. Conjugate by the unitary Z. The spectrum and positivity of the quadratic form are preserved, but both off-diagonal entries become -1/4. Thus positivity as an operator is not entrywise positivity and does not by itself give a Markov interpretation. headA useful finite positive kernel For this nonnegative H, vectors u_a and times t_a≥0 satisfy ∑_a,b⟨ u_a,e^-(t_a+t_b)Hu_b⟩ = ∑_a e^-t_aHu_a ^2≥0. The factorization makes positivity transparent. It remains a fini…
Exact mathematical passage
\sum_{a,b}\langle u_a,e^{-(t_a+t_b)H}u_b\rangle =\left\|\sum_a e^{-t_aH}u_a\right\|^2\ge0.
\sum_{a,b,c}Use in source · 1 passage
Search aliases: ∑_a,b,c sum
Use in source · monograph_atlas.tex, line 136
le vector occupies both parity packets. The final vector occupies only the even packet. vfigbell_circuit headHow the exact path expansion works Insert P_rm even+P_rm odd=I between the two gates. Each summand records a particular intermediate packet, while the boundary projectors select its input and output packets. Summing all terms recovers the original 4×4 circuit matrix: ∑_a,b,cP_b CNOT P_a (H_rm Had⊗ I)P_c =CNOT(H_rm Had⊗ I). This expansion is an identity even though the intermediate parity …
Exact mathematical passage
\sum_{a,b,c}P_b\,\mathrm{CNOT}\,P_a (H_{\rm Had}\otimes I)P_c =\mathrm{CNOT}(H_{\rm Had}\otimes I).
\sum_{a,b,c,d}Use in source · 1 passage
Search aliases: ∑_a,b,c,d sum
Use in source · 02_strengthened.tex, line 1051
[Exact channel block calculus] Every linear map has the exact reconstruction Φ=∑_a,b,c,dΦ_cd,ab. If Ψ:End( H_Y)→End( H_Z), then (ΨΦ)_ef,ab = ∑_c,d Ψ_ef,cdΦ_cd,ab.
Exact mathematical passage
\Phi=\sum_{a,b,c,d}\Phi_{cd,ab}.
\sum_{a:Z_{L,a}(t)>0}Use in source · 1 passage
Search aliases: ∑_a:Z_L,a(t)>0 sum
Use in source · 02_strengthened.tex, line 1999
[Heat packet decomposition] If [L,P_a]=0 for every packet, then Z_L(t)=∑_aZ_L,a(t), Z_L,a(t)=Tr(P_a e^-tL). The global spectral dimension is the average of the packet dimensions weighted by their heat traces: d_L(t) = ∑_a:Z_L,a(t)>0 fracZ_L,a(t)Z_L(t) d_L,a(t), where d_L,a(t) =2t fracTr(P_aL e^-tL)Z_L,a(t) for nonempty packet heat traces.
Exact mathematical passage
d_L(t) = \sum_{a:Z_{L,a}(t)>0} \frac{Z_{L,a}(t)}{Z_L(t)} d_{L,a}(t),
\sum_{a=0}^1Use in source · 1 passage
Search aliases: ∑_a=0^1 sum
Use in source · monograph_atlas.tex, line 556
vpage30Rotate whole packets by a polar factorAN EXACT LOCAL CHART AND ITS FAILUREChapter VII: Theorem II.6.5, Theorem II.6.6. On HH= C^2, put P_0=diag(1,0), P_1=I-P_0, and R=mat3/5 · -4/5 4/5 · 3/5, Q_a=RP_aR^*, S=∑_a=0^1Q_aP_a=frac35R. The exact Gram matrices give (S-I)^*(S-I)=frac1625I, S^*S=frac925I, normS-I=frac45<1. Consequently the unitary polar factor is S(S^*S)^-1/2=R, and Q_aR=RP_a for both complete packets. xfigpolar headThe norm condition selects a chart If the target rank-one packet …
Exact mathematical passage
R=\mat{3/5&-4/5\\4/5&3/5},\qquad Q_a=RP_aR^*, \qquad S=\sum_{a=0}^1Q_aP_a=\frac35R.
\sum_{a=0}^kUse in source · 1 passage
Search aliases: ∑_a=0^k sum
Use in source · strengthening.tex, line 916
For the first contribution, independence gives L(γ_k(P+Q))=∑_a=0^kμ_aμ_k-a. The parity generating series of these convolution moments is (1+t+t^2+t^4+t^8+⋯)^2 =1+t^2+t^4+t^8+⋯ in F_2[[t]]. Its coefficient is one at every k=2^r≥4. Thus the normalized moment in [source reference] is odd and v_2(Δ_1^kC_2(0))=2k+v_2(k!)+k/2. Inserting U^4n=(1+4H_2)^n introduces only corrections with strictly higher valuation, by the filtered bound at orders k+s for s>0. The equality consequently holds at every start…
Exact mathematical passage
\mathcal L(\gamma_k(P+Q))=\sum_{a=0}^k\mu_a\mu_{k-a}
\sum_{a=0}^{k-1}Use in source · 1 passage
Search aliases: ∑_a=0^k-1 sum
Use in source · strengthening.tex, line 905
this reduces to γ_k(X)+ϵ Gγ_k-1(X). Moreover, G≡γ_2(X)+τ Qpmod2. After evaluation, every term with a single τ vanishes, so the second contribution reduces to L((γ_2(X)+Q)γ_k-1(X)). The γ_2(X) product has scalar binomk+12, which is even because 4mid k. The other contribution expands as L(Qγ_k-1(P+Q)) =∑_a=0^k-1(k-a)μ_aμ_k-a. An odd summand would require k-a to be odd and μ_k-a to be odd. By Lemma~[source reference], this requires k-a=1. Then a=k-1 is not a power of two and μ_a is even. Hence the …
Exact mathematical passage
\mathcal L(Q\gamma_{k-1}(P+Q)) =\sum_{a=0}^{k-1}(k-a)\mu_a\mu_{k-a}. \label{eq:two-power-Q-evaluation}
\sum_{a=0}^{M-1}Use in source · 1 passage
Search aliases: ∑_a=0^M-1 sum
Use in source · Paper_XI_Traditional_Final.tex, line 314
Assume that P_K acts trivially. After passing to a finite quotient, tame inertia is cyclic. Let t generate a quotient of order M, and put ζ_M= e^2π i/M. For j∈ Z/M Z, define P_j = frac1M ∑_a=0^M-1 ζ_M^-jar(t)^a.
Exact mathematical passage
P_j = \frac1M \sum_{a=0}^{M-1} \zeta_M^{-ja}r(t)^a.
\sum_{a=0}^{q-1}Use in source · 2 passages
Search aliases: ∑_a=0^q-1 sum
Use in source · 72_recovered.tex, line 897
Let q be a positive odd integer. Define S_n(q) = ∑_a=0^q-1 (-1)^a(2a+1)^n. Let the signed Euler numbers and Euler zigzag numbers be defined by secht = ∑_n≥0E_nfract^nn! and sec t+tan t = ∑_n≥0Z_nfract^nn!.
Exact mathematical passage
S_n(q) = \sum_{a=0}^{q-1} (-1)^a(2a+1)^n.
\sum_{a=0}^{u-1}Use in source · 1 passage
Search aliases: ∑_a=0^u-1 sum
Use in source · supplement_08.tex, line 162
l closure therefore gives flatness of the actual packed Euler coefficient sequence. The earlier coefficient-extraction theorem now has a proved antecedent for every j<p, and yields p^min(k,mn)midΔ_h^k C_mp,j(n) in Z_p, h=(p-1)/m. A new finite telescoping argument proves the extension to arbitrary natural multiples of h. Its first-step identity is f(n+uh)-f(n)=∑_a=0^u-1Δ_h f(n+ah). Commutation of differences and induction on their order transfer any fixed ideal bound valid on a forward tail. The …
Exact mathematical passage
f(n+uh)-f(n)=\sum_{a=0}^{u-1}\Delta_h f(n+ah).
\sum_{a=1}^kUse in source · 1 passage
Search aliases: ∑_a=1^k sum
Use in source · 03_supplement_03.tex, line 18
Several seeds: both necessity and construction. Continue under the assumption above that the number of rows is positive. For seeds C_1,…,C_k, generation means the concrete statement ∀ X∈ M_d× m( C), ∃ A_1,…,A_k∈ M_d( C): X=∑_a=1^k A_aC_a. Stack the C_a vertically into a kd× m matrix D. The source identifies the action on the stacked matrix and proves the equivalence the seeds generate Longleftrightarrow rank D=m. Thus m≤ kd is necessary. It is sufficient as well: embed the m coordinate columns i…
Exact mathematical passage
\forall X\in M_{d\times m}(\C),\quad \exists A_1,\ldots,A_k\in M_d(\C):\quad X=\sum_{a=1}^k A_aC_a.
\sum_{a=1}^mUse in source · 1 passage
Search aliases: ∑_a=1^m sum
Use in source · Paper_VI_Traditional_Final.tex, line 5463
Define ℓ_a(p)=∑_i∈ B_aL_ip_i/q_a when q_a>0. Choose any positive value when q_a=0. Only the product q_aℓ_a(p) enters the formulas. [Exact information loss formula] For every prior, C B_L(p) = B_ℓ(p)(q). Let overline L be any fixed positive coarse likelihood and set d(pmid L,overline L) = ∑_a=1^mq_aabsℓ_a(p)-overline L_a. Then norm C B_L(p)- B_overline L(q) _1 ≤ frac2d(pmid L,overline L) max{Z(p,L),ipoverline Lq}.
Exact mathematical passage
d(p\mid L,\overline L) = \sum_{a=1}^mq_a\abs{\ell_a(p)-\overline L_a}.
\sum_{a=1}^{q-1}Use in source · 1 passage
Search aliases: ∑_a=1^q-1 sum
Use in source · supplement_08.tex, line 33
pageheading1. The characteristic-zero product ruleThe normalized defect is not an ordinary derivation Let A be a commutative algebra over a characteristic-zero field K, and let U be an actual K-derivation. The source defines B=fracU^p^2-U^pp, M_q(f,g)=∑_a=1^q-1frac1pbinom qa (U^a f)(U^q-ag), q=p,p^2. The coefficients of each M_q are proved integral: the middle entries of a prime-power binomial row are divisible by p. In Lean they are implemented as natural quotients, with exact division proved b…
Exact mathematical passage
B=\frac{U^{p^2}-U^p}{p},\qquad M_q(f,g)=\sum_{a=1}^{q-1}\frac{1}{p}\binom qa (U^a f)(U^{q-a}g),\quad q=p,p^2.
\sum_{a=1}^{s}Use in source · 1 passage
Search aliases: ∑_a=1^s sum
Use in source · 03_supplement_03.tex, line 53
An independent definition of Schmidt rank. The source defines Schmidt rank as the least number of summands that are product vectors. A decomposition of length s is the concrete equality C_ij=∑_a=1^su_a(i)v_a(j). It factors C=UV through an s-dimensional intermediate space, so rank C≤ s. Conversely, choose a basis of the row space of C. Expanding each row in this basis constructs a decomposition with exactly rank C product terms. Thus r(ψ)=rank C. The definition uses a least natural number, and th…
Exact mathematical passage
C_{ij}=\sum_{a=1}^{s}u_a(i)v_a(j).
\sum_{a_1,\ldots,a_{m-1}}Use in source · 2 passages
Search aliases: ∑_a_1,…,a_m-1 sum ldots
Use in source · 02_strengthened.tex, line 390
[Exact circuit path expansion] For source packet a_0 and target packet a_m, P_a_mUP_a_0 = ∑_a_1,…,a_m-1 P_a_mU_mP_a_m-1 ⋯ P_a_1U_1P_a_0. Summing over a_0 and a_m reconstructs U.
Exact mathematical passage
P_{a_m}UP_{a_0} = \sum_{a_1,\ldots,a_{m-1}} P_{a_m}U_mP_{a_{m-1}} \cdots P_{a_1}U_1P_{a_0}.
\sum_{a\in A}Use in source · 2 passages
Search aliases: ∑_a∈ A sum in
Use in source · Paper_XI_Traditional_Final.tex, line 967
Let z=∑_a∈ Az_au_a be central. Commuting with u_x gives z_ab_c(x,a)=z_a for every x. Nondegeneracy forces z_a=0 for every nonzero a. Thus the center is C. To prove semisimplicity directly, choose a linear projection P onto a submodule of a finite dimensional C^c[A]-module and form overline P=frac1abs A∑_a∈ Au_a P u_a^-1. Every summand maps into the submodule and is the identity there. Conjugation by any u_x permutes the summands: the cocycle scalars cancel between each operator and its inverse. …
Exact mathematical passage
z=\sum_{a\in A}z_au_a
\sum_{a\in I}Use in source · 2 passages
Search aliases: ∑_a∈ I sum in
Use in source · 02_strengthened.tex, line 6
The elementary coordinate object is a complete family of orthogonal projectors P_aP_b=δ_abP_a, ∑_a∈ IP_a=I. Every vector has packet coordinates P_aψ. Every operator has transfer blocks A_ba=P_bAP_a. These identities are simple. Their value appears when the same coordinates are used across representation theory, circuits, channels, error correction, effective dynamics, and geometric limits.
Exact mathematical passage
P_aP_b=\delta_{ab}P_a, \qquad \sum_{a\in I}P_a=I.
\sum_{a\in J/M}Use in source · 1 passage
Search aliases: ∑_a∈ J/M sum in
Use in source · residue_continuation_body.tex, line 51
The estimate is proved from the actual boundary charts and the fixed-lattice discrepancy theorem. The index factor comes from the covolume identity covol(L(M))=q covol(L(J)). For a complex weight w:J/M→ C, summing the estimates gives the leading term fracvol(D)t^dq covol(L(J))∑_a∈ J/Mw(a), with error at most B∑_aabsnw(a)(t^d-1+1). If the finite sum of weights is zero, only that error remains. Writing Y=t^d gives an error O(Y^1-1/d+1) for the actual norm threshold. These assertions include d=1.
Exact mathematical passage
\frac{\vol(D)t^d}{q\,\covol(L(J))}\sum_{a\in J/M}w(a),
\sum_{a\in J/mJ}Use in source · 1 passage
Search aliases: ∑_a∈ J/mJ sum in
Use in source · residue_continuation_body.tex, line 211
ty of J/mJ→ O/m O or coprimality condition on J is used. The well-defined residue norm on this quotient satisfies overline n_J(u· a)=overline n(u) overline n_J(a). For a multiplicative χ: Z/m Z→ C preserving 1, an exhibited unit with u∈( O/m O)^×, χ(overline n(u))≠1 therefore forces ∑_a∈ J/mJχ(s overline n_J(a))=0 (s∈ Z/m Z). Taking s=1,-1 gives the two finite means needed by the sign-corrected norm weights. Their vanishing follows from the actual permutation.
Exact mathematical passage
\sum_{a\in J/mJ}\chi\bigl(s\,\overline n_J(a)\bigr)=0 \qquad(s\in\ZZ/m\ZZ).
\sum_{a\in\Lambda}Use in source · 1 passage
Search aliases: ∑_a∈Λ sum in Lambda
Use in source · Paper_V_Traditional_Final.tex, line 1016
[1mm] displaystyle max{ε_v, ∑_u∈ch(v)E(u)}, · v internal. For the complete prunings P_v below v, the envelope is exactly E(v)=max_Λ∈ P_v∑_a∈Λε_a. The maximum is attained. Every complete-pruning cost changes by at most E(∅), and the two globally optimal values differ by at most this quantity. Consequently, for a unique optimum with competitors, Γ_y>2E(∅) is a deterministic certificate that the globally optimal pruning is unchanged. A singleton pruning set is stable without a gap condition. Assume…
Exact mathematical passage
E(v)=\max_{\Lambda\in\mathfrak P_v}\sum_{a\in\Lambda}\varepsilon_a.
\sum_bUse in source · 7 passages
Search aliases: ∑_b sum
Use in source · Paper_VI_Traditional_Final.tex, line 866
For P=C_π p and Z=∑_iλ_ip_i=∑_aP_ar_a(p), (C_π B_λ(p))_a = frac∑_i∈ I_aλ_ip_iZ = fracP_ar_a(p)∑_bP_br_b(p) = ( B_r(p)(P))_a.
Exact mathematical passage
\bigl(C_\pi\cB_\lambda(p)\bigr)_a = \frac{\sum_{i\in I_a}\lambda_ip_i}{Z} = \frac{P_ar_a(p)}{\sum_bP_br_b(p)} = \bigl(\cB_{r(p)}(P)\bigr)_a.
\sum_{b,a}Use in source · 1 passage
Search aliases: ∑_b,a sum
Use in source · 03_supplement_01.tex, line 12
0, · a≠ b, ∑_a∈ IP_a=1. Adjoint conditions are added when needed. For an operator U, define the transfer corner U_ba=P_bUP_a. The target index comes first. Completeness and distributivity give ∑_b,aU_ba=U, (UV)_ba=∑_c U_bcV_ca. The second identity uses P_c^2=P_c when adjacent transfer corners are multiplied.
Exact mathematical passage
\sum_{b,a}U_{ba}=U,\qquad (UV)_{ba}=\sum_c U_{bc}V_{ca}.
\sum_{b=0}^BUse in source · 1 passage
Search aliases: ∑_b=0^B sum
Use in source · 02_supplement_03.tex, line 212
Counted candidates. One full convolution table examines indices (b,k), 0≤ b≤ B,quad0≤ k≤ b. These form a concrete finite dependent-sum type. Its cardinality is proved to be ∑_b=0^B(b+1)≤(B+1)^2. For a nonempty finite tree, the number of child merges plus one equals the node count N. This identity is proved for arbitrary finite branching.
Exact mathematical passage
\sum_{b=0}^B(b+1)\leq(B+1)^2.
\sum_{B\in\mathcal Q}Use in source · 2 passages
Search aliases: ∑_B∈ Q sum in mathcal
Use in source · Paper_VI_Traditional_Final.tex, line 675
After one application, the result is constant on every block, so applying the same block average again changes nothing. Weighted self-adjointness follows by expanding both inner products block by block. Both equal ∑_B∈ Q frac1π(B) (∑_i∈ Bπ_i f_i) overline(∑_j∈ Bπ_j g_j).
Exact mathematical passage
\sum_{B\in\mathcal Q} \frac{1}{\pi(B)} \left(\sum_{i\in B}\pi_i f_i\right) \overline{\left(\sum_{j\in B}\pi_j g_j\right)}.
\sum_{b\ne a}Use in source · 3 passages
Search aliases: ∑_b≠ a sum ne
Use in source · 02_supplement_05.tex, line 160
esidual certificates for supplied packets textitSource item 9.2; module DiscoveryCertificate Let (P_a) be a complete orthogonal projector system on a finite complex coordinate space. The projectors need not be coordinate masks. For an actual matrix A, define Π(A)=∑_aP_aAP_a, E=A-Π(A). Reconstruction gives the exact off-diagonal residual E=∑_b≠ aP_bAP_a.
Exact mathematical passage
E=\sum_{b\ne a}P_bAP_a.
\sum_CUse in source · 3 passages
Search aliases: ∑_C sum
Use in source · residue_continuation_body.tex, line 148
Nm I≤ Xw([I]). Exact grouping by ideal class gives S_w(X)=∑_C w(C)A_C(X). If ∑_Cw(C)=0, the common leading coefficient cancels, so there is B_w≥0 such that absnS_w(X)≤ B_wX^1-1/d (X≥1). For every nontrivial character χ:Cl( O)→ C^×, the finite character sum is zero. Thus the displayed estimate applies without an additional arithmetic cancellation hypothesis. This conclusion concerns ordinary class characters. It does not identify them with the further norm-residue characters needed for Frobenius …
Exact mathematical passage
S_w(X)=\sum_C w(C)A_C(X)
\sum_{c,d}Use in source · 1 passage
Search aliases: ∑_c,d sum
Use in source · 02_strengthened.tex, line 1055
[Exact channel block calculus] Every linear map has the exact reconstruction Φ=∑_a,b,c,dΦ_cd,ab. If Ψ:End( H_Y)→End( H_Z), then (ΨΦ)_ef,ab = ∑_c,d Ψ_ef,cdΦ_cd,ab.
Exact mathematical passage
(\Psi\Phi)_{ef,ab} = \sum_{c,d} \Psi_{ef,cd}\Phi_{cd,ab}.
\sum_{c:\,A_{bc}\ne0}Use in source · 1 passage
Search aliases: ∑_c: A_bc≠0 sum ne
Use in source · 02_supplement_01.tex, line 49
Define the active contribution sum recursively by F_0(b,a) · =P_a · b=a, 0 · b≠ a, F_n+1(b,a) · =∑_c: A_bc≠0A_bcF_n(c,a). The checked identity is P_bA^nP_a=F_n(b,a) (n≥0). Its proof is induction using the exact composition theorem, not an assumed support recurrence. When no active walk exists, the corresponding contribution sum is zero, and hence the actual block of A^n vanishes.
Exact mathematical passage
F_0(b,a)&=\begin{cases}P_a&b=a,\\0&b\ne a,\end{cases}\\ F_{n+1}(b,a)&=\sum_{c:\,A_{bc}\ne0}A_{bc}F_n(c,a).
\sum_{C\in\mathfrak C_0}Use in source · 2 passages
Search aliases: ∑_C∈ C_0 sum in mathfrak
Use in source · 03_packets_and_graphs.tex, line 1069
[Source component formula for packet pure seed rank] Assume I is finite, every packet space is nonzero, and the module resolved into packets is surjective along edges. Let C_0 be the set of source strongly connected components of the condensation graph of Γ( G, P). Then psrank_ A, P(X) = ∑_C∈ C_0r_C. [source reference] ExportsResidual transport forests and directed source rank calculations.
Exact mathematical passage
\boxed{ \psrank_{\A,\Pcal}(X) = \sum_{C\in\mathfrak C_0}r_C. }
\sum_{c\in\{\mathrm A,\mathrm B\}}Use in source · 1 passage
Search aliases: ∑_c∈{ A, B} sum in mathrm mathrm
Use in source · atlas-pages.tex, line 90
Any other complete packet family still has exact block convolution. For the coordinate groups A and B, P_b e^(s+t)QP_a=∑_c∈{ A, B}(P_b e^sQP_c)(P_c e^tQP_a). The intermediate packet cannot generally be omitted. Only a reducing family eliminates all cross terms.
Exact mathematical passage
P_b e^{(s+t)Q}P_a=\sum_{c\in\{\mathrm A,\mathrm B\}}(P_b e^{sQ}P_c)(P_c e^{tQ}P_a).
\sum_{d\mid n}Use in source · 1 passage
Search aliases: ∑_dmid n sum mid
Use in source · Paper_XI_Traditional_Final.tex, line 410
[Möbius orbit count] The number of regular Frobenius orbits of length n is c_n(q) = frac1n ∑_dmid n μ(d) (q^n/d-1).
Exact mathematical passage
\boxed{ c_n(q) = \frac1n \sum_{d\mid n} \mu(d) \bigl(q^{n/d}-1\bigr). }
\sum_{d\equiv j\ ({\rm mod}\ N)}Use in source · 1 passage
Search aliases: ∑_d≡ j (rm mod N) sum equiv rm
Use in source · supplement_01.tex, line 111
For N>0 and 0≤ j<N, the independently defined original exponential section is E_N,j(t)=∑_d≡ j (rm mod N)fract^dd!. Using the actual power-series expansion homomorphism z↦ t^N, the code proves E_N,j(t)=fract^jj!K_N,j(t^N), F_N,j(t^N)E_N,j(t)=fract^jj!. Thus the recurrence is connected back to the original normalized EGF definition. The section vanishes at zero when j>0. No inverse of that nonunit section is assumed.
Exact mathematical passage
E_{N,j}(t)=\sum_{d\equiv j\ ({\rm mod}\ N)}\frac{t^d}{d!}.
\sum_eUse in source · 43 passages
Search aliases: ∑_e sum
Use in source · 03_weighted.tex, line 329
W(γ)<LA_γ. Lean proves directly that q_L(S^pr_L,N) equals the mapped geometric inverse whenever q_L(A^N)=0, and hence is a two-sided inverse of q_L(I+A), where A=∑_eA_e=D^-1H. With the supplied two-sided inverse D^-1 of D, this gives the explicit two-sided inverse q_L (S^pr_L,ND^-1) of q_L(D+H). If every edge weight is at least m and L≤ mN, then the nilpotence condition follows from the filtration data because m|γ|≤ W(γ). For m>0, the executable choice N=lceil L/mrceil satisfies L≤ mN. This elem…
Exact mathematical passage
A=\sum_eA_e=D^{-1}H
\sum_{e:\alpha\to\beta}Use in source · 1 passage
Search aliases: ∑_e:α→β sum alpha to beta
Use in source · 03_weighted.tex, line 99
ra of operators. Suppose D=∑_α P_α D P_α belongs to B and is invertible on every packet. Assume that the assembled inverse D^-1=∑_α D_α^-1P_α∈ B=F^0 B, D_α=P_α DP_α, where each packet inverse is extended by zero to the other packets. Let the edge set be finite, assume w(e)≥1 for every edge, and write N=∑_e:α→βN_e, N_e=P_β N_eP_α, N_e∈ F^w(e) B. For an edge e:α→β, put A_e=(P_β DP_β)^-1N_e. Then the block from P_α X to P_β X of (D+N)^-1 is P_β(D+N)^-1P_α = ∑_γ:αleadstoβ (-1)^|γ|A_γ D_α^-1, where D…
Exact mathematical passage
N=\sum_{e:\alpha\to\beta}N_e, \qquad N_e=P_\beta N_eP_\alpha, \qquad N_e\in F^{w(e)}\Bcal.
\sum_fUse in source · 2 passages
Search aliases: ∑_f sum
Use in source · 01_manuscript.tex, line 1346
[Continuous precision allocation] Let E be a finite nonempty edge set, let s_e>0 be sensitivities and c_e>0 be costs, and let τ>0. Consider min_ε_e>0 ∑_e c_elogfrac1ε_e subject to ∑_es_eε_e=τ. The unique minimizer is ε_e = fracτ c_es_e∑_fc_f.
Exact mathematical passage
\boxed{ \eps_e = \frac{\tau c_e}{s_e\sum_fc_f}. }
\sum_{\frakm}Use in source · 1 passage
Search aliases: ∑_frakm ∑_ m sum frakm sum mathfrak
Use in source · Paper_XII_Traditional_Final.tex, line 1033
[Artin decomposition] Let E' be a field and let T be a finite dimensional commutative E' algebra. Then T ≅ ∏_ m∈Max T T_ m. Each factor is an Artin local algebra. Equivalently, there is a unique complete family of central idempotents (e_ m) such that e_ me_ n=0 ( m≠ n), ∑_ me_ m=1, and e_ m T≅ T_ m [source reference].
Exact mathematical passage
e_{\frakm}e_{\mathfrak n}=0 \quad(\frakm\neq\mathfrak n), \qquad \sum_{\frakm}e_{\frakm}=1,
\sum_gUse in source · 2 passages
Search aliases: ∑_g sum
Use in source · 03_nonlinear_coordinates.tex, line 624
[Packet Maurer Cartan equations] Let L=⊕_g∈ G L_g be a differential graded Lie algebra over a ring in which two is invertible. Assume d L_g⊆ L_g, [ L_g, L_h]⊆ L_g+h. For a degree one element η=∑_gη_g, the Maurer Cartan equation dη+frac12[η,η]=0 is equivalent to dη_k + frac12 ∑_g+h=k[η_g,η_h] =0 for every k∈ G.
Exact mathematical passage
\eta=\sum_g\eta_g
\sum_{g+h=k}Use in source · 1 passage
Search aliases: ∑_g+h=k sum
Use in source · 03_nonlinear_coordinates.tex, line 629
[Packet Maurer Cartan equations] Let L=⊕_g∈ G L_g be a differential graded Lie algebra over a ring in which two is invertible. Assume d L_g⊆ L_g, [ L_g, L_h]⊆ L_g+h. For a degree one element η=∑_gη_g, the Maurer Cartan equation dη+frac12[η,η]=0 is equivalent to dη_k + frac12 ∑_g+h=k[η_g,η_h] =0 for every k∈ G.
Exact mathematical passage
d\eta_k + \frac12 \sum_{g+h=k}[\eta_g,\eta_h] =0
\sum_{g<h}Use in source · 1 passage
Search aliases: ∑_g<h sum
Use in source · 03_nonlinear_coordinates.tex, line 692
Repeated use of the polarization identity for q gives q(∑_gη_g) = ∑_gq(η_g) + ∑_g<h[η_g,η_h]. Project onto the component of label k.
Exact mathematical passage
q\left(\sum_g\eta_g\right) = \sum_gq(\eta_g) + \sum_{g<h}[\eta_g,\eta_h].
\sum_{g_1+\cdots+g_r=k}Use in source · 2 passages
Search aliases: ∑_g_1+⋯+g_r=k sum cdots
Use in source · 03_nonlinear_coordinates.tex, line 583
[Graded packet convolution] If a_j=∑_g∈ Ga_j,g and μ respects the grading, then P_kμ(a_1,…,a_r) = ∑_g_1+⋯+g_r=k μ(a_1,g_1,…,a_r,g_r).
Exact mathematical passage
P_k\mu(a_1,\ldots,a_r) = \sum_{g_1+\cdots+g_r=k} \mu(a_{1,g_1},\ldots,a_{r,g_r}).
\sum_{g\in G}Use in source · 3 passages
Search aliases: ∑_g∈ G sum in
Use in source · 03_nonlinear_coordinates.tex, line 582
[Graded packet convolution] If a_j=∑_g∈ Ga_j,g and μ respects the grading, then P_kμ(a_1,…,a_r) = ∑_g_1+⋯+g_r=k μ(a_1,g_1,…,a_r,g_r).
Exact mathematical passage
a_j=\sum_{g\in G}a_{j,g}
\sum_{g\in G_n}Use in source · 2 passages
Search aliases: ∑_g∈ G_n sum in
Use in source · Paper_XI_Traditional_Final.tex, line 466
[Primitive cuspidal idempotent] Let χ_θ be the irreducible character of κ_θ, and let d_θ=χ_θ(1). Then e_θ = fracd_θabsG_n ∑_g∈ G_n χ_θ(g^-1)g is the primitive central idempotent selecting the κ_θ block in C[G_n].
Exact mathematical passage
e_\theta = \frac{d_\theta}{\abs{G_n}} \sum_{g\in G_n} \chi_\theta(g^{-1})g
\sum_{g\in S_3}Use in source · 1 passage
Search aliases: ∑_g∈ S_3 sum in
Use in source · monograph_atlas.tex, line 534
cibility, not unproved continuum claims or a new theorem count. vpage29Nonabelian packets carry multiplicityA SIX-DIMENSIONAL REGULAR REPRESENTATIONChapter VII: Theorem II.3.1, Theorem II.3.2. Let HH= C[S_3] have the six orthonormal basis vectors ket g, with left action U(h)ket g=kethg. The irreducible characters on the identity, transposition, and three-cycle classes are c|rrr|r · e · (12) · (123) · d_λ trivial · 1 · 1 · 1 · 1 sign · 1 · -1 · 1 · 1 standard · 2 · 0 · -1 · 2 P_λ=fracd_λ6∑_g∈ S_3…
Exact mathematical passage
\begin{array}{c|rrr|r} & e & (12)&(123)&d_\lambda\\\hline \text{trivial}&1&1&1&1\\ \text{sign}&1&-1&1&1\\ \text{standard}&2&0&-1&2 \end{array} \qquad P_\lambda=\frac{d_\lambda}{6}\sum_{g\in S_3}\chi_\lambda(g^{-1})U(g).
\sum_{g\in\Gamma}Use in source · 2 passages
Search aliases: ∑_g∈Γ sum in Gamma
Use in source · Paper_X_Block27_Amended.tex, line 998
Derived comparison bounds. For a finite group Γ whose order is a unit in a commutative local coefficient ring R, any prescribed equivariant residual isomorphism lifts to an integral isomorphism. Lift its matrix to A and define P=|Γ|^-1∑_g∈Γρ(g)Aτ(g^-1). Reindexing proves equivariance, and reduction recovers the prescribed matrix. Its determinant is a unit. Complete local coefficients with finite quotients, occupied compatible finite conditions and this residual type therefore give actual integra…
Exact mathematical passage
P=|\Gamma|^{-1}\sum_{g\in\Gamma}\rho(g)A\tau(g^{-1}).
\sum_{g\in\Delta}Use in source · 1 passage
Search aliases: ∑_g∈Δ sum in Delta
Use in source · Paper_XII_Traditional_Final.tex, line 466
Let Δ be a finite abelian p group and let Λ= O[Δ]. The augmentation map is ϵ:Λ⟶ O, ∑_g∈Δa_gg⟼∑_g∈Δa_g. Its kernel is the augmentation ideal a.
Exact mathematical passage
\epsilon:\Lambda\longrightarrow\OO, \qquad \sum_{g\in\Delta}a_gg\longmapsto\sum_{g\in\Delta}a_g.
\sum_{h=0}^kUse in source · 1 passage
Search aliases: ∑_h=0^k sum
Use in source · strengthening.tex, line 1066
3 · 4 · 161644544/4849845 · 15 They are finite rational evaluations of [source reference], followed by a_k=∑_h=0^k(-1)^k-hbinom kh C_j(h). For addresses zero and one, k≥3 is covered by the generic bound. For address two, k≥7 is covered. For address three, k≥5 is covered. In each range the elementary inequality is k-lfloorlog_2krfloor≥ b_j+1. Lemma~[source reference] proves the normalized identity. Multiplication by j! proves the original identity.
Exact mathematical passage
a_k=\sum_{h=0}^k(-1)^{k-h}\binom kh C_j(h)
\sum_{h\in S_3}Use in source · 1 passage
Search aliases: ∑_h∈ S_3 sum in
Use in source · Paper_XII_Traditional_Final.tex, line 405
A trace zero matrix fixed by conjugation by both s and t is zero, as follows by comparing its four entries. This proves H^0(S_3,ad^0V)=0. For any F_7[S_3] module M and one cocycle f, put u=-6^-1∑_h∈ S_3f(h). Summing f(gh)=g f(h)+f(g) over h gives gu-u=f(g). Hence every one cocycle is a coboundary and H^1(S_3,M)=0, giving both asserted first cohomology vanishings.
Exact mathematical passage
u=-6^{-1}\sum_{h\in S_3}f(h).
\sum_{i+j=k}Use in source · 1 passage
Search aliases: ∑_i+j=k sum
Use in source · 03_nonlinear_coordinates.tex, line 603
[Cyclic product formula] For a cyclically graded algebra A=⊕_j∈ Z/N ZA_j, A_iA_j⊆ A_i+j, one has P_k(ab)= ∑_i+j=ka_ib_j.
Exact mathematical passage
P_k(ab)= \sum_{i+j=k}a_ib_j.
\sum_{i+j\equiv k\, (\mathrm{mod}\,N)}Use in source · 1 passage
Search aliases: ∑_i+j≡ k (mod N) sum equiv mathrm
Use in source · Paper_V_Traditional_Final.tex, line 1068
[Cyclic product law] P_N,k(AB) = ∑_i+j≡ k (mod N)A_iB_j. For q∈ N, the qth-power identity is P_N,k(A^q) = ∑_i_1+⋯+i_q≡ k A_i_1⋯ A_i_q.
Exact mathematical passage
\boxed{ P_{N,k}(AB) = \sum_{i+j\equiv k\, (\mathrm{mod}\,N)}A_iB_j. }
\sum_{i+j\equiv k\ (N)}Use in source · 2 passages
Search aliases: ∑_i+j≡ k (N) sum equiv
Use in source · atlas-pages.tex, line 76
1+2z+3z^2+z^3, G=2+z+z^2. The table displays every product of their three selected sections. vfigcyclic_product The shaded cells contribute to residue zero: 2(1+z^3)+(2z)(z^2)+(3z^2)z=2+7z^3. Keeping only the product of the residue-zero inputs would give 2+2z^3 and lose five units of the z^3 coefficient. The mixed cells are part of the exact result. headThe general rule B_N,k(FG)= ∑_i+j≡ k (N)( B_N,iF)( B_N,jG). Every pair belongs to exactly one output residue. No term is discarded and no coeffi…
Exact mathematical passage
\mathcal B_{N,k}(FG)= \sum_{i+j\equiv k\ (N)}(\mathcal B_{N,i}F)(\mathcal B_{N,j}G).
\sum_{i,j=1}^3Use in source · 1 passage
Search aliases: ∑_i,j=1^3 sum
Use in source · visual_atlas.tex, line 200
headReconstruction retains every block The identity P_1+P_2+P_3=I gives T=∑_i,j=1^3 P_iTP_j. Selecting only the diagonal would change the operator. The entries 2,3,4 carry transfers between different packets.
Exact mathematical passage
T=\sum_{i,j=1}^3 P_iTP_j.
\sum_{i,k}Use in source · 1 passage
Search aliases: ∑_i,k sum
Use in source · Paper_V_Traditional_Final.tex, line 2804
[Background and packet error] Suppose the true and estimated packets use the same amplitude list (a_i)_i=0^I, but possibly different regions. Then j-widehat j _p^p = ∑_i,k|a_i-a_k|^pμ(E_i∩ F_k). Only off-diagonal overlaps contribute.
Exact mathematical passage
\|j-\widehat j\|_p^p = \sum_{i,k}|a_i-a_k|^p\mu(E_i\cap F_k).
\sum_{i<d}Use in source · 1 passage
Search aliases: ∑_i<d sum
Use in source · supplement_07.tex, line 54
Suppose g(0)=1. The source constructs f by the full triangular recurrence f(d)=δ_d0-∑_i<dbinom di f(i)g(d-i) and proves f⋆ g=1. It also proves cancellation: if a⋆ g=0, then a=0, by induction on the first nonzero coefficient.
Exact mathematical passage
f(d)=\delta_{d0}-\sum_{i<d}\binom di f(i)g(d-i)
\sum_{i=0}^{a}Use in source · 1 passage
Search aliases: ∑_i=0^a sum
Use in source · supplement_07.tex, line 40
The identity sequence is 1(d)=δ_d0. The new source proves the actual higher product rule U^a(f⋆ g)=∑_i=0^abinom ai (U^i f)⋆(U^a-ig). The argument starts with Pascal's identity for the first shift, then inducts on a. All sums are finite at each coefficient.
Exact mathematical passage
U^a(f\star g)=\sum_{i=0}^{a}\binom ai (U^i f)\star(U^{a-i}g).
\sum_{i=0}^dUse in source · 1 passage
Search aliases: ∑_i=0^d sum
Use in source · strengthening.tex, line 57
Products preserve H_p because the coefficient at factorial degree d is (AB)_d=∑_i=0^dbinom di a_i b_d-i. The derivative D acts by the coefficient shift (DA)_d=a_d+1. Consequently every nonnegative power of D preserves the lattice. If A∈ H_p has unit constant term, its reciprocal belongs to the same lattice: with B=A^-1, b_0=a_0^-1, b_d=-a_0^-1∑_i=1^dbinom di a_i b_d-i. If A is supported at degrees divisible by N, this recurrence also proves that B has the same support.
Exact mathematical passage
(AB)_d=\sum_{i=0}^d\binom di a_i b_{d-i}.
\sum_{i=0}^{d}Use in source · 1 passage
Search aliases: ∑_i=0^d sum
Use in source · supplement_07.tex, line 33
pageheading1. Coefficient products and reciprocal differentiationA characteristic-p argument without factorial division For sequences f,g: N→ K, where K is a field, define (f⋆ g)(d)=∑_i=0^dbinom di f(i)g(d-i), (U^a f)(d)=f(d+a). These are Hurwitz coefficients. The notation does not divide by d! in F_p. In particular, the implementation is not ordinary power-series multiplication with undefined inverse factorials in positive characteristic.
Exact mathematical passage
(f\star g)(d)=\sum_{i=0}^{d}\binom di f(i)g(d-i), \qquad (U^a f)(d)=f(d+a).
\sum_{i=0}^{I}Use in source · 1 passage
Search aliases: ∑_i=0^I sum
Use in source · Paper_V_Traditional_Final.tex, line 2786
[Overlap matrix formula] Define M_ij=μ(E_i∩ F_j). For 1≤ p<∞, ∑_i=0^Ia_i 1_E_i - ∑_j=0^Jb_j 1_F_j _p^p = ∑_i=0^I∑_j=0^J|a_i-b_j|^pM_ij. The row sums of M are μ(E_i), and the column sums are μ(F_j).
Exact mathematical passage
\left\| \sum_{i=0}^{I}a_i\one_{E_i} - \sum_{j=0}^{J}b_j\one_{F_j} \right\|_p^p = \sum_{i=0}^{I}\sum_{j=0}^{J}|a_i-b_j|^pM_{ij}.
\sum_{i=0}^kUse in source · 2 passages
Search aliases: ∑_i=0^k sum
Use in source · edition_summary.tex, line 11
The strongest common-step conclusion in width four is compact. For n≥0 and k,H≥1, ∑_i=0^k(-1)^k-ibinom ki E_4(n+iH)^(4,j) ≡0pmod2^ v_2(k!)+kv_2(H)+A_j(k), where A_0(k)=A_1(k)=k, A_2(k)=max{k+4, 2k+lfloor k/2rfloor+1}, A_3(k)=k+2. There is no starting-row threshold. The exponent is exact in addresses zero and one for every order and positive step. It is exact in address two at every power-of-two order. The proof chapters also give mixed-step forms, the cancellation criterion in addresses zero and…
Exact mathematical passage
\sum_{i=0}^k(-1)^{k-i}\binom ki E_{4(n+iH)}^{(4,j)} \equiv0\pmod{2^{\,v_2(k!)+kv_2(H)+A_j(k)}},
\sum_{i=0}^{n}Use in source · 1 passage
Search aliases: ∑_i=0^n sum
Use in source · 20_arithmetic.tex, line 628
Finite Newton reconstruction at an arbitrary row gives Δ_1^kf(n)=∑_i=0^nbinom ni a_k+i. The monotonicity of lfloorlog_2krfloor therefore preserves the higher-coefficient bound after translation. In addition, Δ_1f(n)-a_1∈2^c+1 Z_2, so the first difference is nonzero of exact valuation c.
Exact mathematical passage
\Delta_1^kf(n)=\sum_{i=0}^{n}\binom ni a_{k+i}.
\sum_{i=0}^{N-1}Use in source · 1 passage
Search aliases: ∑_i=0^N-1 sum
Use in source · dyadic_synthesis.tex, line 361
There is also a recovery from compact data explicitly recording that root. With A_0=Λ(ϖ_K) and X=c - Ind_J^Gη, define (Tf)(g)=A_0^-1f(ϖ_Kg). Then X/(T-1)X≃π. After imposing the fixed central character, N/(F^× J) is cyclic of order N, and the finite average N^-1∑_i=0^N-1T^i selects this same extension. These operators act on complex representation spaces, so their scalar division by N is valid. This is a different recovery from the Block159 projector onto a sector of η. The exact kernel and finit…
Exact mathematical passage
N^{-1}\sum_{i=0}^{N-1}T^i
\sum_{i=0}^{n-1}Use in source · 1 passage
Search aliases: ∑_i=0^n-1 sum
Use in source · Paper_X_Block27_Amended.tex, line 1207
When pnmid n, the comparison with the finite cyclotomic sample is proved: q_L(γ_p)=σ_p, σ_p(ζ_n)=ζ_n^p. Reduction distinguishes n-th roots of unity at such a prime. Indeed, if x^n=y^n=1 and their reductions agree, the factorization of x^n-y^n gives (x-y)∑_i=0^n-1x^i y^n-1-i=0. The second factor reduces to noverline y^ n-1≠0, so the domain property gives x=y. This proves both Frobenius recognition on the cyclotomic generator and triviality of its inertia action. Consequently, every representation…
Exact mathematical passage
(x-y)\sum_{i=0}^{n-1}x^i y^{n-1-i}=0.
\sum_{i=0}^sUse in source · 2 passages
Search aliases: ∑_i=0^s sum
Use in source · 20_arithmetic.tex, line 498
g_2(n) · =frac1(4n+1)(2n+1), · g_3(n) · =frac3(4n+1)(2n+1)(4n+3). Every denominator is odd. An affine sequence with even slope satisfies 2^kmidΔ_1^k f(n) for every k,n. The exact product identity Δ_1^s(fg)(n)=∑_i=0^sbinom si Δ_1^if(n)Δ_1^s-ig(n+i) proves closure of this property under products. Applying the formula with the factors interchanged to fg=1 isolates f(n+s)Δ_1^sg(n). Multiplication by f(n+s)^-1 and induction then prove closure under pointwise unit inverses. Thus all four g_j have the …
Exact mathematical passage
\Delta_1^s(fg)(n)=\sum_{i=0}^s\binom si \Delta_1^if(n)\Delta_1^{s-i}g(n+i)
\sum_{i=1}^{d}Use in source · 1 passage
Search aliases: ∑_i=1^d sum
Use in source · continuation_01.tex, line 49
For every P∈ Z[X_1,…,X_d], integer a, and x,t∈ Z^d, Lean proves a^2mid P(x+at)-P(x)-a∑_i=1^d(∂_iP)(x)t_i. The proof uses polynomial induction: constants, addition, and multiplication by a variable. The product rule for the actual partial derivatives controls the multiplication step. It does not assume a Taylor identity or a liftability condition.
Exact mathematical passage
\boxed{ a^2\mid P(x+at)-P(x)-a\sum_{i=1}^{d}(\partial_iP)(x)t_i. }
\sum_{i=1}^{d_\lambda}Use in source · 1 passage
Search aliases: ∑_i=1^d_λ sum lambda
Use in source · 02_strengthened.tex, line 259
x an irreducible unitary representation π_λ:G→ U(d_λ) and an orthonormal basis of its representation space. Define E_ij^λ = fracd_λabsG ∑_g∈ G [π_λ(g^-1)]_jiU(g). Then E_ij^λE_kℓ^μ = δ_λμδ_jkE_iℓ^λ, (E_ij^λ)^*=E_ji^λ, and P_λ=∑_i=1^d_λE_ii^λ.
Exact mathematical passage
P_\lambda=\sum_{i=1}^{d_\lambda}E_{ii}^{\lambda}.
\sum_{i=1}^HUse in source · 1 passage
Search aliases: ∑_i=1^H sum
Use in source · strengthening.tex, line 205
To retain the valuation of a step, let B be any sequence satisfying Δ_1^m B∈ p^m Z_p at every order and row. Use the finite operator identity Δ_H=∑_i=1^Hbinom HiΔ_1^i. The relation ibinom Hi=HbinomH-1i-1 implies v_p(p^ibinom Hi) ≥ i+v_p(H)-v_p(i)≥1+v_p(H). Multiplying these identities for the k steps, every term of total order M has its scalar multiplied by p^M, and is divisible by p^k+∑_ν v_p(H_ν). This proves the mixed-step bound. Finally, Δ_H A_N,j(n)=(-1)^n+HT_H C_N,j(n). Iteration contribut…
Exact mathematical passage
\Delta_H=\sum_{i=1}^H\binom Hi\Delta_1^i.
\sum_{i=1}^kUse in source · 1 passage
Search aliases: ∑_i=1^k sum
Use in source · strengthening.tex, line 989
At a natural argument the exact Newton expansion is finite. Vandermonde gives binomn+Hk-binom nk =∑_i=1^kbinom Hibinom nk-i. For 1≤ i≤ H, v_2(binom Hi)≥ v_2(H)-v_2(i) ≥ v_2(H)-lfloorlog_2krfloor. Terms with i>H vanish. Thus each Newton summand with k≥2 has valuation at least b+v_2(H)+1. The k=1 summand is Ha_1, of exact valuation b+v_2(H), proving [source reference].
Exact mathematical passage
\binom{n+H}{k}-\binom nk =\sum_{i=1}^k\binom Hi\binom n{k-i}.
\sum_{i=1}^{k+1}Use in source · 1 passage
Search aliases: ∑_i=1^k+1 sum
Use in source · 02_supplement_06.tex, line 101
Monotonicity of the diagonal entries gives normDy^2 =∑_i=1^k+1σ_i^2|y_i|^2 ≥σ_k+1^2norm y^2. Because (D-B)y=Dy, the operator-norm inequality implies σ_k+1norm y ≤norm(D-B)y ≤normD-B_2norm y. Cancellation of the strictly positive norm of y proves σ_k+1≤normD-B_2. The norm of the actual diagonal residual D-D_k equals σ_k+1, so D_k attains the lower bound. If k≥ d, the residual is zero and optimality follows from norm nonnegativity.
Exact mathematical passage
\norm{Dy}^{2} =\sum_{i=1}^{k+1}\sigma_i^{2}|y_i|^{2} \geq\sigma_{k+1}^{2}\norm y^{2}.
\sum_{i=1}^{N-1}Use in source · 1 passage
Search aliases: ∑_i=1^N-1 sum
Use in source · strengthening.tex, line 75
[Translated product formula] For N=p^a, a≥1, and λ,μ∈ Z_p, L_λ+μ(AB) =(L_λ A)B+A(L_μ B) +∑_i=1^N-1c_i(D^iA)(D^N-iB), c_i=frac1pbinom Ni∈ Z. Flatness types add under multiplication.
Exact mathematical passage
L_{\lambda+\mu}(AB) =(L_\lambda A)B+A(L_\mu B) +\sum_{i=1}^{N-1}c_i(D^iA)(D^{N-i}B), \qquad c_i=\frac1p\binom Ni\in\mathbb Z. \label{eq:translated-product}
\sum_{i=1}^rUse in source · 3 passages
Search aliases: ∑_i=1^r sum
Use in source · 02_foundations.tex, line 614
Let S={s_1,…,s_r} generate M. Every pure tensor λ⊗ m∈ M_K has an expression m=∑_i=1^r a_is_i, so λ⊗ m = ∑_i=1^r(λ⊗ a_i)(1⊗ s_i). Pure tensors span M_K, hence 1⊗ S generates M_K over A_K. Taking the minimum over S proves the inequality.
Exact mathematical passage
m=\sum_{i=1}^r a_is_i,
\sum_{i>k}Use in source · 3 passages
Search aliases: ∑_i>k sum
Use in source · 02_supplement_06.tex, line 138
For any nonnegative decreasing sequence a_i, weights with these properties satisfy ∑_i a_iw_i≥∑_i>ka_i. This finite weighted inequality is itself proved. For k<d, use μ=a_k+1 and compare the summands to tail weights v_i=0 for i≤ k and v_i=1 otherwise: a_iv_i+μ(w_i-v_i)≤ a_iw_i. For a retained coordinate the inequality uses a_i≥μ and w_i≥0. For a tail coordinate it uses a_i≤μ and w_i≤1. Summation and ∑ w_i≥∑ v_i give the result.
Exact mathematical passage
\sum_i a_iw_i\geq\sum_{i>k}a_i.
\sum_{i_1+\cdots+i_q\equiv k}Use in source · 1 passage
Search aliases: ∑_i_1+⋯+i_q≡ k sum cdots equiv
Use in source · Paper_V_Traditional_Final.tex, line 1076
[Cyclic product law] P_N,k(AB) = ∑_i+j≡ k (mod N)A_iB_j. For q∈ N, the qth-power identity is P_N,k(A^q) = ∑_i_1+⋯+i_q≡ k A_i_1⋯ A_i_q.
Exact mathematical passage
P_{N,k}(A^q) = \sum_{i_1+\cdots+i_q\equiv k} A_{i_1}\cdots A_{i_q}.
\sum_{i\in B}Use in source · 2 passages
Search aliases: ∑_i∈ B sum in
Use in source · Paper_VI_Traditional_Final.tex, line 675
After one application, the result is constant on every block, so applying the same block average again changes nothing. Weighted self-adjointness follows by expanding both inner products block by block. Both equal ∑_B∈ Q frac1π(B) (∑_i∈ Bπ_i f_i) overline(∑_j∈ Bπ_j g_j).
Exact mathematical passage
\sum_{B\in\mathcal Q} \frac{1}{\pi(B)} \left(\sum_{i\in B}\pi_i f_i\right) \overline{\left(\sum_{j\in B}\pi_j g_j\right)}.
\sum_{i\in B_a}Use in source · 5 passages
Search aliases: ∑_i∈ B_a sum in
Use in source · Paper_VI_Traditional_Final.tex, line 5356
se Markov generator. For observables, the backward generator is Q^T. Weighted L^2 inner products use the stated stationary probability. [Reversibility descends] Assume Q is strongly lumpable and reversible with respect to a positive stationary distribution π, so Q_ijπ_j=Q_jiπ_i. Define overlineπ_a=∑_i∈ B_aπ_i. Then overlineπ is stationary for overline Q, and overline Q_aboverlineπ_b = overline Q_baoverlineπ_a. Thus the coarse chain is reversible.
Exact mathematical passage
\overline\pi_a=\sum_{i\in B_a}\pi_i.
\sum_{i\in I}Use in source · 4 passages
Search aliases: ∑_i∈ I sum in
Use in source · 05_worked_models.tex, line 527
[Closure specification for the orbit hull] Let V have finite dimension, let (O_i)_i∈ I be a finite operator family, and let S⊆ V. Define W_0=span_𝕜S, W_n+1=W_n+∑_i∈ IO_i(W_n). Then there is an n∈ N such that W_n+1=W_n and W_n=Hull_(O_i)(S), where the right side is the least subspace containing S and invariant under every O_i. Thus the ascending subspace iteration terminates and its stable value is exactly the finite operator hull.
Exact mathematical passage
W_0=\Span_{\bbk}S, \qquad W_{n+1}=W_n+\sum_{i\in I}O_i(W_n).
\sum_{i\in I_a}Use in source · 5 passages
Search aliases: ∑_i∈ I_a sum in
Use in source · 01_finite_coordinates.tex, line 438
The graph model above resolves a fixed scalar matrix. Coordinate transport is a separate construction. Let X be an R-module, and let P^a=(P^a_i)_i∈ I_a be a finite complete family of pairwise orthogonal linear idempotents: P^a_iP^a_j=δ_ijP^a_i, ∑_i∈ I_aP^a_i=I_X. Define the exact packet coordinate module Coord( P^a) = {(x_i)_i∈ I_a:P^a_ix_i=x_i}. Its analysis and synthesis maps are A_a(x)=(P^a_ix)_i, S_a((x_i)_i)=∑_ix_i. Completeness and orthogonality give S_a A_a=id_X, A_a S_a=id_Coord( P^a).
Exact mathematical passage
P^a_iP^a_j=\delta_{ij}P^a_i, \qquad \sum_{i\in I_a}P^a_i=I_X.
\sum_{i\in I_v}Use in source · 1 passage
Search aliases: ∑_i∈ I_v sum in
Use in source · 02_connections_obstructions.tex, line 861
Suppose every X_v has a finite complete packet family (P^v_i)_i∈ I_v. A packet labeling assigns an admissible packet index to each edge or node of a typed tree. For multilinear Φ_σ, inserting id_X_v=∑_i∈ I_vP^v_i at every input and output yields a finite sum over trees with packet labels. This is the branching analogue of inserting intermediate projectors in [source reference].
Exact mathematical passage
\id_{X_v}=\sum_{i\in I_v}P^v_i
\sum_{i\in J}Use in source · 2 passages
Search aliases: ∑_i∈ J sum in
Use in source · Paper_X_Block27_Amended.tex, line 2168
Keep the complete domain, supplied finite-group carrier and group-order hypotheses of the hyperref[book:x-construction-18]supplied-carrier summand-lifting construction. Let J be any finite label set and k>0. Suppose (bar E_i)_i∈ J is a complete orthogonal family of equivariant idempotents modulo (π)^e+k. There exists an integral equivariant family (E_i)_i∈ J satisfying E_i^2=E_i, E_iE_j=0 (i≠ j), ∑_i∈ JE_i=1, E_ibmod(π)^k=bar E_ibmod(π)^k. The loss is at most e, independent of |J| and the ambien…
Exact mathematical passage
E_i^2=E_i,\qquad E_iE_j=0\ (i\ne j),\qquad \sum_{i\in J}E_i=1,\qquad E_i\bmod(\pi)^k=\bar E_i\bmod(\pi)^k.
\sum_{i\in M}Use in source · 2 passages
Search aliases: ∑_i∈ M sum in
Use in source · Paper_VI_Traditional_Final.tex, line 3839
[Concentration on the maximal score packet] Let G=max{g_i:p_i(0)>0} and let M be the set of supported indices with score G. If M is the full support, then the conclusion below is immediate. Otherwise, suppose δ = min{G-g_i:p_i(0)>0, i∉ M}>0. Then 1-∑_i∈ Mp_i(t) ≤ frac1-p_M(0)p_M(0) e^-δ t, where p_M(0)=∑_i∈ Mp_i(0).
Exact mathematical passage
1-\sum_{i\in M}p_i(t) \leq \frac{1-p_M(0)}{p_M(0)}\e^{-\delta t},
\sum_{i\in S}Use in source · 1 passage
Search aliases: ∑_i∈ S sum in
Use in source · supplement_04.tex, line 148
Finite integral linear combinations. For any finite index set S, coefficients a_i∈ Z_p and units u_i∈ Z_p^×, linearity of the actual iterated operator gives Δ_h^s (∑_i∈ Sa_i u_i^n) =∑_i∈ Sa_i u_i^n(u_i^h-1)^s. At h=p-1, the result is divisible by p^s. The empty sum is permitted. This does not prove that the general congruential Euler packets of Theorem 7.5 admit such a representation.
Exact mathematical passage
\Delta_h^s\!\left(\sum_{i\in S}a_i u_i^n\right) =\sum_{i\in S}a_i u_i^n(u_i^h-1)^s.
\sum_{i\geq0}Use in source · 3 passages
Search aliases: ∑_i≥0 sum geq
Use in source · strengthening.tex, line 452
G_4,1(t)F_0(t) =frac12(fractanh(at)at +fractanh(bt)bt). For example, its numerator follows from sinh t+sin t =2bsinh(at)cosh(bt)+2acosh(at)sinh(bt), while cosh t+cos t=2cosh(at)cosh(bt). Only degrees divisible by four survive. Its actual Hurwitz packets are D_1(t)=∑_i≥0d_ifract^4i(4i)!, d_i=(-frac14)^i frac2^4i+2(2^4i+2-1)B_4i+2(4i+2)(4i+1). The constant is one. Since v_2(4i+2)=1, the elementary Bernoulli bound shows v_2(d_i)≥2i. Hurwitz inversion therefore gives D_1(t)^-1=∑_i≥0h_ifract^4i(4i)!,…
Exact mathematical passage
D_1(t)=\sum_{i\geq0}d_i\frac{t^{4i}}{(4i)!},\qquad d_i=\left(-\frac14\right)^i \frac{2^{4i+2}(2^{4i+2}-1)B_{4i+2}}{(4i+2)(4i+1)}. \label{eq:address-one-packets}
\sum_{i\geq1}Use in source · 2 passages
Search aliases: ∑_i≥1 sum geq
Use in source · strengthening.tex, line 468
the elementary Bernoulli bound shows v_2(d_i)≥2i. Hurwitz inversion therefore gives D_1(t)^-1=∑_i≥0h_ifract^4i(4i)!, h_0=1, v_2(h_i)≥2i. This follows by induction from the inverse recurrence: for packet index i, every product d_rh_i-r has valuation at least 2i, and its binomial multiplier is integral. Because F_1=F_0D_1^-1, C_1(n)-C_0(n) =∑_i≥1h_ibinom4n4iC_0(n-i). At a natural n this is a finite sum, with i≤ n.
Exact mathematical passage
C_1(n)-C_0(n) =\sum_{i\geq1}h_i\binom{4n}{4i}C_0(n-i). \label{eq:address-one-exact-transfer}
\sum_jUse in source · 52 passages
Search aliases: ∑_j sum
Use in source · 02_foundations.tex, line 245
The image consists exactly of finite sums ∑_ja_js_j, which is Hull_A(S). Because ε_S is linear over A, its kernel is a left submodule. The first isomorphism theorem gives A^⊕ r/N_S≅imε_S=Hull_A(S). The final equivalence is the equality imε_S=M.
Exact mathematical passage
\sum_ja_js_j
\sum_{j,k}Use in source · 6 passages
Search aliases: ∑_j,k sum
Use in source · Paper_VI_Traditional_Final.tex, line 1149
All statements hold coefficientwise on the mutually orthogonal spectral projectors. For example, L^α L^β =∑_j,kλ_j^αλ_k^β P_jP_k =∑_jλ_j^α+βP_j.
Exact mathematical passage
L^\alpha L^\beta =\sum_{j,k}\lambda_j^\alpha\lambda_k^\beta P_jP_k =\sum_j\lambda_j^{\alpha+\beta}P_j.
\sum_{j,k=1}^{m}Use in source · 1 passage
Search aliases: ∑_j,k=1^m sum
Use in source · 02_strengthened.tex, line 1714
[Positive definite quantum memory] The kernel K(t-s) is positive definite. For times t_1,…,t_m and retained vectors v_1,…,v_m, ∑_j,k=1^m ipv_jK(t_j-t_k)v_k ≥0.
Exact mathematical passage
\sum_{j,k=1}^{m} \ip{v_j}{K(t_j-t_k)v_k} \geq0.
\sum_{j=0}^2Use in source · 1 passage
Search aliases: ∑_j=0^2 sum
Use in source · atlas-pages.tex, line 180
ck. vfigvariance platecaptionThe strip lengths are proportional to the three exact weighted energies. The between-block component is small but not zero. The vectors below specify the corresponding directions, not only their sizes. headThe energy identity follows from actual orthogonality The tower identities give E_0E_1=E_1E_0=E_0. Together with weighted self-adjointness, these imply D_iD_j=0 (i≠ j), ∑_j=0^2 D_j=I. For the chosen f, the total weighted energy is f _π^2=13 and the mean is 16/5. Co…
Exact mathematical passage
D_iD_j=0\ (i\ne j),\qquad \sum_{j=0}^2 D_j=I.
\sum_{j=0}^dUse in source · 1 passage
Search aliases: ∑_j=0^d sum
Use in source · 01_manuscript.tex, line 978
[Polynomial support] If p(z)=∑_j=0^dc_jz^j, then supp(p(A)) is contained in the union of the support relations of A^j for 0≤ j≤ d.
Exact mathematical passage
p(z)=\sum_{j=0}^dc_jz^j,
\sum_{j=0}^{d}Use in source · 1 passage
Search aliases: ∑_j=0^d sum
Use in source · 02_supplement_01.tex, line 71
Polynomial support. For the actual finite sum p(A)=∑_j=0^dc_jA^j, the checked support lies in the union of the support relations of the powers. The formalization treats the j=0 contribution explicitly, so the constant term is not accidentally omitted.
Exact mathematical passage
p(A)=\sum_{j=0}^{d}c_jA^j,
\sum_{j=0}^{J}Use in source · 1 passage
Search aliases: ∑_j=0^J sum
Use in source · Paper_V_Traditional_Final.tex, line 2786
[Overlap matrix formula] Define M_ij=μ(E_i∩ F_j). For 1≤ p<∞, ∑_i=0^Ia_i 1_E_i - ∑_j=0^Jb_j 1_F_j _p^p = ∑_i=0^I∑_j=0^J|a_i-b_j|^pM_ij. The row sums of M are μ(E_i), and the column sums are μ(F_j).
Exact mathematical passage
\left\| \sum_{i=0}^{I}a_i\one_{E_i} - \sum_{j=0}^{J}b_j\one_{F_j} \right\|_p^p = \sum_{i=0}^{I}\sum_{j=0}^{J}|a_i-b_j|^pM_{ij}.
\sum_{j=0}^{k}Use in source · 2 passages
Search aliases: ∑_j=0^k sum
Use in source · 10_foundations.tex, line 502
For M≥1 and s,r,k∈ N, define the affine Mahler kernel by Θ_M,s(r,k)=[z^r](1+z)^s((1+z)^M-1)^k =∑_j=0^k(-1)^k-jbinom kjbinomMj+sr. It vanishes unless k≤ r≤ Mk+s. For a finite binomial expansion f(n)=∑_r=0^Dc_rbinom nr, the kth Newton coefficient of g(q)=f(Mq+s) is therefore c_k(g)=∑_r=k^min(D,Mk+s)c_rΘ_M,s(r,k). Indeed, insert the finite stencil for Δ_1^k g(0) and interchange its two finite sums. Coefficient extraction of the binomial expansion of ((1+z)^M-1)^k gives the kernel. In the binary zer…
Exact mathematical passage
\Theta_{M,s}(r,k)=[z^r](1+z)^s\bigl((1+z)^M-1\bigr)^k =\sum_{j=0}^{k}(-1)^{k-j}\binom kj\binom{Mj+s}{r}.
\sum_{j=0}^{n}Use in source · 2 passages
Search aliases: ∑_j=0^n sum
Use in source · 10_foundations.tex, line 255
[Finite Newton--Mahler identities] For every n,s,h∈ N, f(n) · =∑_r=0^nc_r(f)binom nr, c_r(Δ_1^s f) · =c_s+r(f), Δ_1^s f(n) · =∑_j=0^nc_s+j(f)binom nj, Δ_h^s f(n) · =∑_j=0^s(-1)^s-jbinom sj f(n+jh). The finite interpolant I_Nf(n)=∑_r=0^Nc_r(f)binom nr agrees with f for n≤ N. It has the same order-s unit-step difference at n whenever n+s≤ N. For arbitrary coefficients c_0,…,c_N, the corresponding binomial interpolant has exactly those coefficients and zero coefficients above N.
Exact mathematical passage
\begin{aligned} f(n)&=\sum_{r=0}^{n}c_r(f)\binom nr,\\ c_r(\Delta_1^s f)&=c_{s+r}(f),\\ \Delta_1^s f(n)&=\sum_{j=0}^{n}c_{s+j}(f)\binom nj,\\ \Delta_h^s f(n)&=\sum_{j=0}^{s}(-1)^{s-j}\binom sj f(n+jh). \end{aligned}
\sum_{j=0}^{N-1}Use in source · 7 passages
Search aliases: ∑_j=0^N-1 sum
Use in source · 02_coordinates.tex, line 383
[Residue packet algebra] For every commutative ring R, the operators (P_N,j)_0≤ j<N satisfy the identities below. If R is nonzero, they form a packet system on R[[z]]: P_N,iP_N,j=δ_ijP_N,j, ∑_j=0^N-1P_N,j=I.
Exact mathematical passage
P_{N,i}P_{N,j}=\delta_{ij}P_{N,j}, \qquad \sum_{j=0}^{N-1}P_{N,j}=I.
\sum_{j=0}^{n-1}Use in source · 5 passages
Search aliases: ∑_j=0^n-1 sum
Use in source · 05_worked_models.tex, line 384
y seeds: 2=min{r∈ N: ∃ s_1,…,s_r, C^n=span_ C{p(A_C_n)s_j:p∈ C[t], 1≤ j≤ r} }. To see the upper bound directly, index the vertices modulo n. The identity Ae_j=e_j-1+e_j+1 recovers successive vertex vectors from e_0 and e_1, over any field. For the lower bound over C, let ζ be a primitive nth root of unity and define q(x)=(∑_j=0^n-1ζ^j x_j, ∑_j=0^n-1ζ^-jx_j). Then q(Ax)=(ζ+ζ^-1)q(x). The images of e_0 and e_1 are (1,1) and (ζ,ζ^-1), which are independent because ζ≠ζ^-1 for n≥3. Thus q is a scalar…
Exact mathematical passage
q(x)=\left(\sum_{j=0}^{n-1}\zeta^j x_j, \sum_{j=0}^{n-1}\zeta^{-j}x_j\right).
\sum_{j=0}^rUse in source · 4 passages
Search aliases: ∑_j=0^r sum
Use in source · Paper_VI_Traditional_Final.tex, line 697
[Tower law and martingale packets] If Q_c is coarser than Q_f, then P_cP_f=P_fP_c=P_c. For a nested tower P_0≤ P_1≤⋯≤ P_r, define D_0=P_0, D_j=P_j-P_j-1. Then D_iD_j=δ_ijD_j, P_r=∑_j=0^rD_j.
Exact mathematical passage
D_iD_j=\delta_{ij}D_j, \qquad P_r=\sum_{j=0}^rD_j.
\sum_{j=0}^{s}Use in source · 2 passages
Search aliases: ∑_j=0^s sum
Use in source · 10_foundations.tex, line 255
[Finite Newton--Mahler identities] For every n,s,h∈ N, f(n) · =∑_r=0^nc_r(f)binom nr, c_r(Δ_1^s f) · =c_s+r(f), Δ_1^s f(n) · =∑_j=0^nc_s+j(f)binom nj, Δ_h^s f(n) · =∑_j=0^s(-1)^s-jbinom sj f(n+jh). The finite interpolant I_Nf(n)=∑_r=0^Nc_r(f)binom nr agrees with f for n≤ N. It has the same order-s unit-step difference at n whenever n+s≤ N. For arbitrary coefficients c_0,…,c_N, the corresponding binomial interpolant has exactly those coefficients and zero coefficients above N.
Exact mathematical passage
\begin{aligned} f(n)&=\sum_{r=0}^{n}c_r(f)\binom nr,\\ c_r(\Delta_1^s f)&=c_{s+r}(f),\\ \Delta_1^s f(n)&=\sum_{j=0}^{n}c_{s+j}(f)\binom nj,\\ \Delta_h^s f(n)&=\sum_{j=0}^{s}(-1)^{s-j}\binom sj f(n+jh). \end{aligned}
\sum_{j=1}^dUse in source · 1 passage
Search aliases: ∑_j=1^d sum
Use in source · 30_lifting.tex, line 40
[Polynomial Taylor remainder] Let R be a commutative ring, P∈ R[X_1,…,X_d], a∈ R^d, and c∈ R. There is a polynomial Q∈ R[U_1,…,U_d], with Q(0)=0, such that for every u∈ R^d, P(a+cu)=P(a)+c∑_j=1^d(∂_jP)(a)u_j+c^2Q(u). In particular, when R= Z and a,u,c are integral, subtracting the first two terms on the right from P(a+cu) gives a multiple of c^2.
Exact mathematical passage
\label{eq:polynomial-taylor} P(a+cu)=P(a)+c\sum_{j=1}^d(\partial_jP)(a)u_j+c^2Q(u).
\sum_{j=1}^{d}Use in source · 2 passages
Search aliases: ∑_j=1^d sum
Use in source · 02_strengthened.tex, line 2022
Let N≥2 and let Λ_N=( Z/N Z)^d be the unit flat torus grid with spacing 1/N. The standard periodic difference Laplacian has eigenvalues λ_N(k) = 4N^2 ∑_j=1^d sin^2(fracπ k_jN), where k runs through centered residue representatives. This is the standard periodic finite difference Laplacian [source reference].
Exact mathematical passage
\lambda_N(k) = 4N^2 \sum_{j=1}^{d} \sin^2\left(\frac{\pi k_j}{N}\right),
\sum_{j=1}^{d-1}Use in source · 2 passages
Search aliases: ∑_j=1^d-1 sum
Use in source · Paper_X_Block27_Amended.tex, line 2234
General finite logarithmic recovery. Let A be a rational algebra, without any commutativity assumption on A. For a unipotent element U and a positive integer d satisfying (U-1)^d=0, put L_d(U)=∑_j=1^d-1frac(-1)^j+1j(U-1)^j. The value is independent of the chosen annihilating exponent. The finite logarithm and the nilpotent exponential satisfy L(exp N)=N, exp(L(U))=U. Both maps preserve every annihilating exponent, and hence preserve the exact nilpotency class of N and exp(N)-1, or of U-1 and L(U…
Exact mathematical passage
L_d(U)=\sum_{j=1}^{d-1}\frac{(-1)^{j+1}}{j}(U-1)^j.
\sum_{j=1}^mUse in source · 9 passages
Search aliases: ∑_j=1^m sum
Use in source · 01_manuscript.tex, line 726
[Rank growth in a product chain] Let A_j=D_j+U_jV_j^*, j=1,…,m, share one diagonal packet layout. Before recompression, the product A_m⋯ A_1 has a BDLR representation whose correction rank is at most ∑_j=1^m r_j.
Exact mathematical passage
\boxed{ \sum_{j=1}^m r_j. }
\sum_{j=1}^{m}Use in source · 3 passages
Search aliases: ∑_j=1^m sum
Use in source · Paper_VI_Traditional_Final.tex, line 5760
[Exponential sum memory] If K(t)=∑_j=1^mB_j e^λ_jtC_j, then K has a finite auxiliary realization with one auxiliary block for each exponential. Repeated poles produce polynomial factors in t and are realized by Jordan blocks.
Exact mathematical passage
\Kcal(t)=\sum_{j=1}^{m}B_j\e^{\lambda_jt}C_j,
\sum_{j=1}^{m-1}Use in source · 1 passage
Search aliases: ∑_j=1^m-1 sum
Use in source · Paper_XI_Traditional_Final.tex, line 259
[Exact block reconstruction] Let P_j be the projector onto V_j. Then N=∑_j=1^m-1P_j-1NP_j, N^m=0. Every power N^r is the sum of the packet paths of length r along the chain.
Exact mathematical passage
N=\sum_{j=1}^{m-1}P_{j-1}NP_j, \qquad N^m=0.
\sum_{j=1}^nUse in source · 2 passages
Search aliases: ∑_j=1^n sum
Use in source · 40_regularity.tex, line 29
[Derivative of polynomial evaluation] For P∈ K[X_1,…,X_n], polynomial evaluation is analytic at every point. At a∈ K^n its strict Fr echet derivative is v⟼∑_j=1^n (∂_jP)(a)v_j. For a finite system, its strict derivative is the continuous linear map represented by the matrix of evaluated formal partial derivatives. For the derivative statement, use polynomial induction on constants, addition, and multiplication by a variable. Constants have zero derivative. The sum rule agrees with the additivity…
Exact mathematical passage
v\longmapsto\sum_{j=1}^n (\partial_jP)(a)v_j.
\sum_{j=1}^{n}Use in source · 1 passage
Search aliases: ∑_j=1^n sum
Use in source · Paper_X_Block27_Amended.tex, line 316
For the digit claim, subtract the chosen representative of the residue and divide the remainder by the uniformizer. Iterating gives unique digits. Conversely, the partial sums of any digit stream form a compatible Cauchy family and reconstruct uniquely by completeness and separatedness. More precisely, for each parameter there is a unique stream (a_i,j)_j≥1 such that ϖ^n+1mid t_i-∑_j=1^ns(a_i,j)ϖ^j (n≥0). The section is a set-theoretic section of reduction, not an asserted embedding of residue f…
Exact mathematical passage
\varpi^{n+1}\mid t_i-\sum_{j=1}^{n}s(a_{i,j})\varpi^j\qquad(n\ge0).
\sum_{j=1}^{q}Use in source · 1 passage
Search aliases: ∑_j=1^q sum
Use in source · 02_supplement_06.tex, line 128
Frobenius optimality from kernel weights. textitModules KernelFrame and FrobeniusOptimality For an arbitrary competitor B with rank B≤ k, construct an orthonormal matrix Q spanning its actual kernel. Then BQ=0, Q^*Q=I, q=dimker B=d-rank B. Define the row weights w_i=∑_j=1^q|Q_ij|^2. The proof derives 0≤ w_i≤1, ∑_iw_i=q≥max(d-k,0). The upper bound uses contraction of Q^* on a coordinate vector. The weight sum follows from the trace of Q^*Q and rank-nullity.
Exact mathematical passage
w_i=\sum_{j=1}^{q}|Q_{ij}|^2.
\sum_{j=1}^rUse in source · 7 passages
Search aliases: ∑_j=1^r sum
Use in source · 03_weighted.tex, line 52
[Weighted inverse] Let B be a complete filtered algebra. Let D∈ B be invertible with D,D^-1∈ F^0 B. Let E be a finite set and suppose T=D+∑_e∈ EN_e, N_e∈ F^w(e) B, w(e)≥1. Set A_e=D^-1N_e. For a word γ=(e_1,…,e_r), write A_γ=A_e_r⋯ A_e_1, W(γ)=∑_j=1^rw(e_j). The empty word has weight zero and contributes the identity. Then T is invertible and T^-1 = ∑_γ(-1)^|γ|A_γ D^-1. The series converges in the filtration, and the term indexed by γ belongs to F^W(γ) B.
Exact mathematical passage
A_\gamma=A_{e_r}\cdots A_{e_1}, \qquad W(\gamma)=\sum_{j=1}^rw(e_j).
\sum_{j=n}^{\infty}Use in source · 2 passages
Search aliases: ∑_j=n^∞ sum infty
Use in source · Paper_VI_Traditional_Final.tex, line 2561
fine the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_j=n^∞w_m,j sup_0≤ t≤ TnormE_j(t).
Exact mathematical passage
\sup_{0\leq t\leq T} \norm{U_m(t)-U_{m,n}(t)} \leq \sum_{j=n}^{\infty}w_{m,j} \sup_{0\leq t\leq T}\norm{E_j(t)}.
\sum_{j=\ell}^{m-1}Use in source · 2 passages
Search aliases: ∑_j=ℓ^m-1 sum ell
Use in source · Paper_VI_Traditional_Final.tex, line 5253
Let C_j:X_j+1→ X_j be bounded linear maps, and put C_ℓ,m=C_ℓ⋯ C_m-1 for ℓ<m, with C_j,j=I. No common ambient space is assumed. [Multilevel telescoping identity] Let T_j be bounded operators on X_j, and set E_j=C_jT_j+1-T_jC_j. Then C_ℓ,mT_m-T_ℓ C_ℓ,m = ∑_j=ℓ^m-1 C_ℓ,jE_jC_j+1,m.
Exact mathematical passage
\boxed{ C_{\ell,m}T_m-T_\ell C_{\ell,m} = \sum_{j=\ell}^{m-1} C_{\ell,j}E_jC_{j+1,m}. }
\sum_{j>k}Use in source · 2 passages
Search aliases: ∑_j>k sum
Use in source · Paper_VI_Traditional_Final.tex, line 4744
normB_j,k (-Reμ_j)^k+1. Among all sets J containing k primary blocks, the certified upper bound ∑_j∉ Jw_j is minimized by retaining the k largest weights. If w_(1)≥ w_(2)≥⋯≥ w_(r), then the least number of retained primary blocks certified to give L^1 error at most ε is the least k for which ∑_j>kw_(j)≤ε.
Exact mathematical passage
\sum_{j>k}w_{(j)}\leq\varepsilon.
\sum_{j_1+\cdots+j_m=r}Use in source · 1 passage
Search aliases: ∑_j_1+⋯+j_m=r sum cdots
Use in source · 02_strengthened.tex, line 351
[Tensor charge refinement] For r∈ Z/N Z, define P_r^rm total = ∑_j_1+⋯+j_m=r P_j_1^(1)⊗⋯⊗ P_j_m^(m). Then the P_r^rm total form a complete orthogonal packet system. Every total charge packet is the exact sum of its fine tensor descendants.
Exact mathematical passage
P_r^{\rm total} = \sum_{j_1+\cdots+j_m=r} P_{j_1}^{(1)}\otimes\cdots\otimes P_{j_m}^{(m)}.
\sum_{j_1+\cdots+j_m\equiv r\pmod N}Use in source · 1 passage
Search aliases: ∑_j_1+⋯+j_m≡ rpmod N sum cdots equiv pmod
Use in source · 01_finite_core.tex, line 247
[Tensor/polyphase refinement] For each residue r∈ Z/N Z, define P_r^coarse = ∑_j_1+⋯+j_m≡ rpmod N P^(1)_j_1⊗⋯⊗ P^(m)_j_m. Then the P_r^coarse are complete orthogonal projectors, and every coarse packet is the exact sum of its fine descendants.
Exact mathematical passage
\boxed{ P_r^{\mathrm{coarse}} = \sum_{j_1+\cdots+j_m\equiv r\pmod N} P^{(1)}_{j_1}\otimes\cdots\otimes P^{(m)}_{j_m}. }
\sum_{j\in B_b}Use in source · 1 passage
Search aliases: ∑_j∈ B_b sum in
Use in source · Paper_VI_Traditional_Final.tex, line 5376
Stationarity follows from overline Qoverlineπ =overline QCπ =CQπ=0. For reversibility, use the aggregate rate formula: overline Q_aboverlineπ_b · =∑_j∈ B_bπ_j∑_i∈ B_aQ_ij · =∑_i∈ B_a∑_j∈ B_bπ_iQ_ji · =overline Q_baoverlineπ_a.
Exact mathematical passage
\begin{aligned} \overline Q_{ab}\overline\pi_b &=\sum_{j\in B_b}\pi_j\sum_{i\in B_a}Q_{ij}\\ &=\sum_{i\in B_a}\sum_{j\in B_b}\pi_iQ_{ji}\\ &=\overline Q_{ba}\overline\pi_a. \end{aligned}
\sum_{j\in F}Use in source · 2 passages
Search aliases: ∑_j∈ F sum in
Use in source · 02_foundations.tex, line 494
The lower inequality is quotient monotonicity. For the upper inequality, choose generating families (ℓ_i)_i∈ I of L and (n_j)_j∈ J of N with the least possible cardinalities. Choose a lift m_j∈ M of each n_j. For any m∈ M, a finite subset F⊆ J and coefficients a_j∈ A give π(m)=∑_j∈ Fa_jn_j. The vector m-∑_j∈ Fa_jm_j belongs to ι(L), so it is a finite linear combination of the ι(ℓ_i). The union of the lifted family and the image of the generating family of L therefore generates M. Its cardinality…
Exact mathematical passage
\pi(m)=\sum_{j\in F}a_jn_j.
\sum_{j\in J}Use in source · 1 passage
Search aliases: ∑_j∈ J sum in
Use in source · 05_worked_models.tex, line 553
[Table specification for transfer blocks] Given finite projector systems (P_i)_i∈ I and (Q_j)_j∈ J and a linear map T, form the table B_ji=Q_jTP_i. Its assembly reconstructs the input map exactly, ∑_j∈ J∑_i∈ IB_ji=T, and its reported support relation is exactly {(i,j):B_ji≠0}. The latter equality is the specification of the transfer graph output. It does not claim that Lean extracts a routine that decides the edge list for every coefficient field.
Exact mathematical passage
\sum_{j\in J}\sum_{i\in I}B_{ji}=T,
\sum_{j\notin J}Use in source · 4 passages
Search aliases: ∑_j∉ J sum notin
Use in source · Paper_VI_Traditional_Final.tex, line 4675
j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt ∑_k=0^ν_j-1 fract^kk!B_j,k and Σ_P(z) = ∑_j=1^r ∑_k=0^ν_j-1 fracB_j,k(z-μ_j)^k+1. Suppose Reμ_j<0 for every j. If J⊆{1,…,r} and K_J retains the complete primary blocks indexed by J, then norm K- K_J_L^1(0,∞) ≤ ∑_j∉ J ∑_k=0^ν_j-1 fracnormB_j,k (-Reμ_j)^k+1.
Exact mathematical passage
\norm{\Kcal-\Kcal_J}_{L^1(0,\infty)} \leq \sum_{j\notin J} \sum_{k=0}^{\nu_j-1} \frac{\norm{B_{j,k}}} {(-\operatorname{Re}\mu_j)^{k+1}}.
\sum_{j\geq0}Use in source · 1 passage
Search aliases: ∑_j≥0 sum geq
Use in source · Paper_VI_Traditional_Final.tex, line 5284
[Summable generator defects] Suppose C_j are contractions and (T_j(t)) are contraction semigroups with bounded generators A_j. Set δ_j=normC_jA_j+1-A_jC_j. Then normC_ℓ,mT_m(t)-T_ℓ(t)C_ℓ,m ≤ t∑_j=ℓ^m-1δ_j. If ∑_j≥0δ_j<∞, the right side tends to zero as ℓ→∞, uniformly in m>ℓ and uniformly for t in a fixed bounded interval.
Exact mathematical passage
\sum_{j\geq0}\delta_j<\infty
\sum_kUse in source · 8 passages
Search aliases: ∑_k sum
Use in source · 03_weighted.tex, line 557
[Linear recurrence] With the canonical polynomials from [source reference], write Q(t)=∑_kq_kt^k, Z_s,t(t)=∑_n≥0a_n(s,t)t^n. Then the walk counts satisfy the recurrence with constant coefficients determined by Q after the finite initial range contributed by P_s,t, with deg 0=-∞: ∑_k=0^nq_k a_n-k(s,t)=0 (n>deg P_s,t).
Exact mathematical passage
Q(t)=\sum_kq_kt^k, \qquad Z_{s,t}(t)=\sum_{n\ge0}a_n(s,t)t^n.
\sum_{k=0}^2Use in source · 1 passage
Search aliases: ∑_k=0^2 sum
Use in source · visual_atlas.tex, line 166
On the monomial z^n, define UF(z)=F(2z). The proposed projector acts by 5∑_k=0^2 4^jk2^nk. The identity 2^3=1 makes this depend only on nbmod3. The nine-entry table therefore establishes the formula for every exponent once periodicity is used, rather than requiring a separate test at every coefficient.
Exact mathematical passage
5\sum_{k=0}^2 4^{jk}2^{nk}.
\sum_{k=0}^{d-1}Use in source · 1 passage
Search aliases: ∑_k=0^d-1 sum
Use in source · Paper_VI_Traditional_Final.tex, line 1658
For clarity, the finite-dimensional holomorphic calculus used below has the following canonical definition. For A acting on C^d, its generalized eigenspaces E_μ=ker(A-μ I)^d form a direct-sum decomposition over the eigenvalues. Define the unique linear operator F(A) by .F(A)|_E_μ =.∑_k=0^d-1fracF^(k)(μ)k!(A-μ I)^k|_E_μ. The formula is independent of a basis and of increasing the truncation index: the extra nilpotent powers vanish. It depends only on the finite Taylor jets of F at the spectrum. I…
Exact mathematical passage
\left.F(A)\right|_{E_\mu} =\left.\sum_{k=0}^{d-1}\frac{F^{(k)}(\mu)}{k!}(A-\mu I)^k\right|_{E_\mu}.
\sum_{k=0}^{j-1}Use in source · 1 passage
Search aliases: ∑_k=0^j-1 sum
Use in source · 01_manuscript.tex, line 1169
Use the telescoping identity A^j-D^j = ∑_k=0^j-1 A^j-1-kKD^k. Each term has norm at most M^j-1normK. The vector D^jv lies in the direct sum of local Krylov spaces.
Exact mathematical passage
A^j-D^j = \sum_{k=0}^{j-1} A^{j-1-k}KD^k.
\sum_{k=0}^{m-1}Use in source · 1 passage
Search aliases: ∑_k=0^m-1 sum
Use in source · Paper_IV_Traditional_Final.tex, line 2491
Differentiate the product identity in logarithmic form to obtain Δ_N'(w) = -Δ_N(w) ∑_m≥1σ_N,mw^m-1. Comparison of the coefficient of w^m-1 gives ma_m = -∑_k=0^m-1a_kσ_N,m-k. Solving for σ_N,m gives the recurrence.
Exact mathematical passage
ma_m = -\sum_{k=0}^{m-1}a_k\sigma_{N,m-k}.
\sum_{k=0}^{\min(d,r)}Use in source · 1 passage
Search aliases: ∑_k=0^min(d,r) sum min
Use in source · 72_recovered.tex, line 1641
[Polynomial exponential Mahler transfer] Let f(n)=P(n)u^n with P∈ Z_p[n] of degree d. Let H=Λ_p,L, fix an address j, and write Q_j(x)=P(Hx+j), y=u^H, q=y-1. Then f(Hx+j)=∑_r≥0c_j,rbinom xr, where c_j,r = u^j ∑_k=0^min(d,r) binom rk Δ^kQ_j(0)y^kq^r-k. Consequently, v_p(c_j,r)≥ Lmax(r-d,0). For R≥ d, truncation at depth R is valid modulo p^L(R+1-d) uniformly on Z_p.
Exact mathematical passage
\boxed{ c_{j,r} = u^j \sum_{k=0}^{\min(d,r)} \binom rk \Delta^kQ_j(0)y^kq^{r-k}. }
\sum_{k=0}^{n}Use in source · 3 passages
Search aliases: ∑_k=0^n sum
Use in source · 03_weighted.tex, line 564
[Linear recurrence] With the canonical polynomials from [source reference], write Q(t)=∑_kq_kt^k, Z_s,t(t)=∑_n≥0a_n(s,t)t^n. Then the walk counts satisfy the recurrence with constant coefficients determined by Q after the finite initial range contributed by P_s,t, with deg 0=-∞: ∑_k=0^nq_k a_n-k(s,t)=0 (n>deg P_s,t).
Exact mathematical passage
\boxed{ \sum_{k=0}^{n}q_k a_{n-k}(s,t)=0 \qquad\bigl(n>\deg P_{s,t}\bigr). }
\sum_{k=0}^{N-1}Use in source · 8 passages
Search aliases: ∑_k=0^N-1 sum
Use in source · 02_coordinates.tex, line 408
[Root-of-unity realization] Let R be a commutative semiring, let N>0, and suppose ω,ω^vee,η∈ R satisfy ω^N=1 and the finite N× N Fourier orthogonality table η∑_k=0^N-1(ω^vee)^jkω^rk = 1, · r=j, 0, · r≠ j, 0≤ j,r<N. Then P_N,j = η∑_k=0^N-1(ω^vee)^jkU^k. In particular, if R is a field, N>0, and ω is a primitive Nth root of unity, the certificate holds with ω^vee=ω^-1 and η=N^-1.
Exact mathematical passage
\eta\sum_{k=0}^{N-1}(\omega^\vee)^{jk}\omega^{rk} = \begin{cases} 1,&r=j,\\ 0,&r\ne j, \end{cases} \qquad 0\le j,r<N.
\sum_{k=0}^{n-1}Use in source · 8 passages
Search aliases: ∑_k=0^n-1 sum
Use in source · 05_defects.tex, line 27
[Power defect identity] For every integer n≥1, CT_f^n-T_c^nC = ∑_k=0^n-1T_c^n-1-kΔ T_f^k.
Exact mathematical passage
CT_f^n-T_c^nC = \sum_{k=0}^{n-1}T_c^{n-1-k}\Delta T_f^k.
\sum_{k=0}^{r-1}Use in source · 2 passages
Search aliases: ∑_k=0^r-1 sum
Use in source · 03_weighted.tex, line 278
[Certified weighted inverse] Let R be a possibly noncommutative ring, let D^-1 be a supplied two-sided inverse of D, and put A=D^-1H. [label=(alph*)] item If A^r=0, then D+H has the two-sided inverse (D+H)^-1=(∑_k=0^r-1(-A)^k)D^-1. item If R is a complete normed ring and lVert ArVert<1, then ∑_k≥0(-A)^k converges and (D+H)^-1=(∑_k≥0(-A)^k)D^-1.
Exact mathematical passage
\boxed{(D+H)^{-1}=\left(\sum_{k=0}^{r-1}(-A)^k\right)D^{-1}.}
\sum_{k=0}^{\nu_j-1}Use in source · 7 passages
Search aliases: ∑_k=0^ν_j-1 sum nu
Use in source · Paper_VI_Traditional_Final.tex, line 4658
xtended:thm:modal_memory Let X_Q be a finite dimensional complex space. Let μ_1,…,μ_r be the distinct eigenvalues of A_QQ, let Q_j be the corresponding primary spectral projections, and set N_j=(A_QQ-μ_jI)Q_j. Let ν_j be the least positive integer for which N_j^ν_j=0, and define B_j,k = A_PQN_j^kQ_jA_QP (0≤ k<ν_j). Then K(t) = ∑_j=1^r e^μ_jt ∑_k=0^ν_j-1 fract^kk!B_j,k and Σ_P(z) = ∑_j=1^r ∑_k=0^ν_j-1 fracB_j,k(z-μ_j)^k+1. Suppose Reμ_j<0 for every j. If J⊆{1,…,r} and K_J retains the complete pri…
Exact mathematical passage
\Kcal(t) = \sum_{j=1}^r \e^{\mu_jt} \sum_{k=0}^{\nu_j-1} \frac{t^k}{k!}B_{j,k}
\sum_{k=0}^{\infty}Use in source · 1 passage
Search aliases: ∑_k=0^∞ sum infty
Use in source · Paper_VI_Traditional_Final.tex, line 4616
For large z, Σ_P(z) = ∑_k=0^∞ fracA_PQA_QQ^kA_QPz^k+1. Thus the second condition implies the moment identities. Conversely, the first d identities imply all later ones by the Cayley--Hamilton theorem, so the series vanishes.
Exact mathematical passage
\Sigma_P(z) = \sum_{k=0}^{\infty} \frac{A_{PQ}A_{QQ}^kA_{QP}}{z^{k+1}}.
\sum_{k=1}^{m}Use in source · 1 passage
Search aliases: ∑_k=1^m sum
Use in source · 02_strengthened.tex, line 1723
Set w=∑_k=1^m e^ i t_kDB^*v_k. Then norm w^2 = ∑_j,k ipB^*v_j e^- i(t_j-t_k)DB^*v_k = ∑_j,k ipv_jK(t_j-t_k)v_k.
Exact mathematical passage
w=\sum_{k=1}^{m}\e^{\ii t_kD}B^*v_k.
\sum_{k=1}^{m-1}Use in source · 1 passage
Search aliases: ∑_k=1^m-1 sum
Use in source · Paper_IV_Traditional_Final.tex, line 2466
[Newton sums for the reduced zeros] For |w| below the least zero modulus, with the analytic logarithm normalized to zero at the origin, -logΔ_N(w) = ∑_m≥1σ_N,mfracw^mm. Thus κ_N,m=σ_N,m. If a_m=frac(-1)^m(Nm)!, a_0=1, then σ_N,m = -ma_m- ∑_k=1^m-1a_kσ_N,m-k.
Exact mathematical passage
\sigma_{N,m} = -ma_m- \sum_{k=1}^{m-1}a_k\sigma_{N,m-k}.
\sum_{k=1}^{n-1}Use in source · 1 passage
Search aliases: ∑_k=1^n-1 sum
Use in source · Paper_IV_Traditional_Final.tex, line 1029
[Parameter recurrence] We have P_N,0(r)=1. For n≥1, P_N,n(r) = · - ∑_k=1^n-1 (-1)^k binomNn-1Nk P_N,n-k(r) · -r ∑_k=1^n (-1)^k binomNn-1Nk-1 P_N,n-k(r).
Exact mathematical passage
\begin{aligned} P_{N,n}(r) ={}&- \sum_{k=1}^{n-1} (-1)^k \binom{Nn-1}{Nk} P_{N,n-k}(r) \\ &-r \sum_{k=1}^{n} (-1)^k \binom{Nn-1}{Nk-1} P_{N,n-k}(r). \end{aligned}
\sum_{k=1}^{p^{r-1}}Use in source · 2 passages
Search aliases: ∑_k=1^p^r-1 sum
Use in source · 20_arithmetic.tex, line 424
The finite Newton expansion gives f(n+p^r-1h)-f(n) =∑_k=1^p^r-1binomp^r-1kΔ_h^k f(n). Terms with k≥ r have the required factor from the difference. For 1≤ k<r, the identity kbinomp^r-1k=p^r-1binomp^r-1-1k-1 and v_p(k)<k imply p^rmidbinomp^r-1k p^k in Z. Mapping that integer divisibility to R proves the claim term by term. The argument includes r=1. The declaration is leanKomorebiPaperIX.PeriodFromDifferences.prime_power_period.
Exact mathematical passage
f(n+p^{r-1}h)-f(n) =\sum_{k=1}^{p^{r-1}}\binom{p^{r-1}}k\Delta_h^k f(n).
\sum_{k=1}^{w}Use in source · 1 passage
Search aliases: ∑_k=1^w sum
Use in source · 01_manuscript.tex, line 484
The diagonal contributes p blocks. At offset k, where 1≤ k≤ w, the upper and lower block diagonals contribute 2(p-k) blocks. Therefore p+2∑_k=1^w(p-k) = p(2w+1)-w(w+1). The cost and storage follow from [source reference].
Exact mathematical passage
p+2\sum_{k=1}^{w}(p-k) = p(2w+1)-w(w+1).
\sum_{k=2}^{T}Use in source · 1 passage
Search aliases: ∑_k=2^T sum
Use in source · 20_arithmetic.tex, line 637
Now expand the displacement: f(n+T)-f(n)=TΔ_1f(n) +∑_k=2^Tbinom TkΔ_1^kf(n). For 1≤ k≤ T, the binomial denominator identity implies v_2binom Tk≥ v_2(T)-v_2(k) ≥ v_2(T)-lfloorlog_2krfloor. Every term of the remaining sum consequently has valuation at least v_2(T)+c+1. Its sum cannot cancel the leading term, which has exact valuation v_2(T)+c. This proves nonvanishing and the claimed valuation using only finite sums. The general theorem is leanKomorebiPaperIX.NewtonDominance.exact_displacement.
Exact mathematical passage
f(n+T)-f(n)=T\Delta_1f(n) +\sum_{k=2}^{T}\binom Tk\Delta_1^kf(n).
\sum_{k\in\ZZ^d}Use in source · 3 passages
Search aliases: ∑_k∈ZZ^d ∑_k∈ Z^d sum in ZZ sum in mathbb
Use in source · 02_strengthened.tex, line 2061
[Fixed time heat trace convergence] For every t>0, ∑_k∈Λ_N^* e^-tλ_N(k) ⟶ ∑_k∈ Z^d e^-4π^2tabsk^2 as N→∞. The first spectral moments also converge: ∑_k∈Λ_N^* λ_N(k) e^-tλ_N(k) ⟶ ∑_k∈ Z^d 4π^2absk^2 e^-4π^2tabsk^2. Consequently, the discrete spectral dimensions converge to the continuum torus spectral dimension at every fixed positive time.
Exact mathematical passage
\sum_{k\in\Lambda_N^*} \e^{-t\lambda_N(k)} \longrightarrow \sum_{k\in\ZZ^d} \e^{-4\pi^2t\abs{k}^2}
\sum_{k\in\Lambda_N^*}Use in source · 2 passages
Search aliases: ∑_k∈Λ_N^* sum in Lambda
Use in source · 02_strengthened.tex, line 2061
[Fixed time heat trace convergence] For every t>0, ∑_k∈Λ_N^* e^-tλ_N(k) ⟶ ∑_k∈ Z^d e^-4π^2tabsk^2 as N→∞. The first spectral moments also converge: ∑_k∈Λ_N^* λ_N(k) e^-tλ_N(k) ⟶ ∑_k∈ Z^d 4π^2absk^2 e^-4π^2tabsk^2. Consequently, the discrete spectral dimensions converge to the continuum torus spectral dimension at every fixed positive time.
Exact mathematical passage
\sum_{k\in\Lambda_N^*} \e^{-t\lambda_N(k)} \longrightarrow \sum_{k\in\ZZ^d} \e^{-4\pi^2t\abs{k}^2}
\sum_{k\ge0}Use in source · 2 passages
Search aliases: ∑_k≥0 sum ge
Use in source · 03_weighted.tex, line 282
[Certified weighted inverse] Let R be a possibly noncommutative ring, let D^-1 be a supplied two-sided inverse of D, and put A=D^-1H. [label=(alph*)] item If A^r=0, then D+H has the two-sided inverse (D+H)^-1=(∑_k=0^r-1(-A)^k)D^-1. item If R is a complete normed ring and lVert ArVert<1, then ∑_k≥0(-A)^k converges and (D+H)^-1=(∑_k≥0(-A)^k)D^-1.
Exact mathematical passage
\sum_{k\ge0}(-A)^k
\sum_{k\geq0}Use in source · 3 passages
Search aliases: ∑_k≥0 sum geq
Use in source · 72_recovered.tex, line 2212
[Mahler polyphase transfer] Let f(x)=∑_r≥0c_rbinom xr be a convergent Mahler expansion. For g_s(q)=f(Mq+s), one has g_s(q)=∑_k≥0d_s,kbinom qk, where d_s,k = ∑_r=k^Mk+s c_rΘ_M,s(r,k). Every refined coefficient uses a finite window of the original Mahler coefficients.
Exact mathematical passage
\boxed{ g_s(q)=\sum_{k\geq0}d_{s,k}\binom qk,}
\sum_{k\geq1}Use in source · 1 passage
Search aliases: ∑_k≥1 sum geq
Use in source · Paper_IV_Traditional_Final.tex, line 1154
Let B_r(w)=Δ_N(w)^-r = ∑_n≥0b_N,n(r)w^n. Logarithmic differentiation gives fracB_r'(w)B_r(w) = r∑_k≥1κ_N,kw^k-1. Comparison of the coefficient of w^n-1 proves the recurrence.
Exact mathematical passage
\frac{B_r'(w)}{B_r(w)} = r\sum_{k\geq1}\kappa_{N,k}w^{k-1}.
\sum_mUse in source · 3 passages
Search aliases: ∑_m sum
Use in source · Paper_V_Traditional_Final.tex, line 439
[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.
Exact mathematical passage
\bm Y(z)=\bm H^{(N)}(z)\bm X(z), \qquad H^{(N)}_{ks}(z)=\sum_m h[Nm+k-s]z^{-m}.
\sum_{m,n\ge0}Use in source · 1 passage
Search aliases: ∑_m,n≥0 sum ge
Use in source · Paper_VI_Traditional_Final.tex, line 343
Both exponentials are absolutely convergent matrix power series. Since every power of A commutes with every other power, the Cauchy product gives e^sAe^tA =∑_m,n≥0fracs^mt^nm!n!A^m+n =∑_r≥0frac(s+t)^rr!A^r. Termwise differentiation is valid in finite dimension and gives the derivative identities.
Exact mathematical passage
e^{sA}e^{tA} =\sum_{m,n\ge0}\frac{s^mt^n}{m!n!}A^{m+n} =\sum_{r\ge0}\frac{(s+t)^r}{r!}A^r.
\sum_{m=0}^{12}Use in source · 1 passage
Search aliases: ∑_m=0^12 sum
Use in source · atlas-pages.tex, line 52
idue j, evaluate the function at four rotated arguments and attach the phase i^-jℓ: ( B_4,jF)(z)=frac14∑_ℓ=0^3i^-jℓF(i^ℓ z). A monomial a_mz^m is multiplied by the finite average of i^(m-j)ℓ. Matching phases all vote 1. Nonmatching phases cancel exactly. vfigroot_filter headAn example that can be checked coefficient by coefficient Take F(z)=∑_m=0^12(m+1)z^m. Then B_4,2F=3z^2+7z^6+11z^10. Applying the same selector again changes nothing. Applying a different selector gives zero. Summing the four …
Exact mathematical passage
F(z)=\sum_{m=0}^{12}(m+1)z^m
\sum_{m=0}^{\min(M,N)}Use in source · 1 passage
Search aliases: ∑_m=0^min(M,N) sum min
Use in source · root_scope.tex, line 62
tion v_2(β_m)=s_2(m)-1, v_2(κ_m)=4m-1. The source identity ∑_N≥0A(N)fract^2N(2N)! =fracsinh(2t)sinh t2t^2 implies, after multiplication by 4t^2/sinh^2(2t), E_2N=∑_m=0^N κ_mbinom2N2mA(N-m). For a retained depth M≥0, put T_M(N)= E_2N -∑_m=0^min(M,N)κ_mbinom2N2mA(N-m). An empty tail, N≤ M, is exactly zero.
Exact mathematical passage
\mathcal T_M(N)=\mathcal E_{2N} -\sum_{m=0}^{\min(M,N)}\kappa_m\binom{2N}{2m}A(N-m).
\sum_{m=0}^{\min(M,r)}Use in source · 2 passages
Search aliases: ∑_m=0^min(M,r) sum min
Use in source · supplement_02.tex, line 163
al targets. targetOddEuler.odd_reconstruction_series targetOddEuler.odd_euler_reconstruction targetOddEuler.odd_euler_reconstruction_zigzag targetOddEuler.row_action, OddEuler.row_diagonal, OddEuler.row_weight pageheading4. Uniform truncation and a sharp exact exampleCorollary 4.3 and Example 4.4 For row 2r and depth M, the actual truncation is T_M(p,2r)=∑_m=0^min(M,r)binom2r2mb_m p^2mS_2(r-m)(p). Every omitted term has m≥ M+1. Its other factors are integers, so the first-block omitted-tail theo…
Exact mathematical passage
T_M(p,2r)=\sum_{m=0}^{\min(M,r)}\binom{2r}{2m}b_m p^{2m}S_{2(r-m)}(p).
\sum_{m=0}^NUse in source · 1 passage
Search aliases: ∑_m=0^N sum
Use in source · root_scope.tex, line 56
The classical dyadic Bernoulli denominator law v_2(B_2m)=-1 and Legendre's formula yield v_2(β_m)=s_2(m)-1, v_2(κ_m)=4m-1. The source identity ∑_N≥0A(N)fract^2N(2N)! =fracsinh(2t)sinh t2t^2 implies, after multiplication by 4t^2/sinh^2(2t), E_2N=∑_m=0^N κ_mbinom2N2mA(N-m). For a retained depth M≥0, put T_M(N)= E_2N -∑_m=0^min(M,N)κ_mbinom2N2mA(N-m). An empty tail, N≤ M, is exactly zero.
Exact mathematical passage
\mathcal E_{2N}=\sum_{m=0}^N \kappa_m\binom{2N}{2m}A(N-m). \label{eq:root:packet}
\sum_{m=0}^{n}Use in source · 2 passages
Search aliases: ∑_m=0^n sum
Use in source · 20_arithmetic.tex, line 800
[Classical odd-length reconstruction] For every positive odd integer p and every n∈ N, E_2n=∑_m=0^nbinom2n2mγ_m p^2mS_2(n-m)(p). Equivalently, E_2n=∑_q=0^n V_nqS_2q(p) for the lower triangular matrix V_nq=binom2n2qγ_n-qp^2(n-q) (q≤ n), V_nq=0 (q>n). It has V_nn=1 and p^2(n-q)mid V_nq when q≤ n.
Exact mathematical passage
E_{2n}=\sum_{m=0}^{n}\binom{2n}{2m}\gamma_m p^{2m}S_{2(n-m)}(p).
\sum_{m=0}^{n/2}Use in source · 1 passage
Search aliases: ∑_m=0^n/2 sum
Use in source · supplement_02.tex, line 131
Substituting the independently proved signed-zigzag identity recovers the manuscript's notation. For every even degree n, E_n=∑_m=0^n/2(-1)^m Z_2m+1binom n2mp^2mS_n-2m(p). The Lean version uses n=2r to avoid ambiguous natural-number division. Every even natural degree has that form. Primality is not used in this equality.
Exact mathematical passage
\boxed{E_n=\sum_{m=0}^{n/2}(-1)^m Z_{2m+1}\binom n{2m}p^{2m}S_{n-2m}(p).}
\sum_{m=0}^{r}Use in source · 2 passages
Search aliases: ∑_m=0^r sum
Use in source · supplement_02.tex, line 125
ing3. Exact reconstruction and the triangular matrixTheorem 4.2; manuscript page 11 For odd p, rescaling the already proved inverse law gives sech(pt)cosh(pt)=1. Multiplying the even-source identity by sech^2(pt) therefore proves sech^2(pt)(∑_r≥0S_2r(p)fract^2r(2r)!)=sech t. Actual EGF multiplication now gives the integer coefficient formula E_2r=∑_m=0^rbinom2r2mb_m p^2mS_2(r-m)(p). The formal proof first extracts the sum over all indices and then proves that the odd summands vanish. A checked e…
Exact mathematical passage
\boxed{E_{2r}=\sum_{m=0}^{r}\binom{2r}{2m}b_m p^{2m}S_{2(r-m)}(p).}
\sum_{m=0}^{r-1}Use in source · 2 passages
Search aliases: ∑_m=0^r-1 sum
Use in source · Paper_V_Traditional_Final.tex, line 2546
_packet_decomposition Let 0=τ_0<τ_1<⋯<τ_Q<τ_Q+1=1. Suppose the restriction of f to every interval (τ_q,τ_q+1) belongs to W^r,p, and choose the one-dimensional Sobolev representatives whose derivatives through order r-1 have one-sided traces at the endpoints. Define a_q,m = jumpf^(m)_τ_q = f^(m)(τ_q+)-f^(m)(τ_q-) and j_f(t) = ∑_q=1^Q∑_m=0^r-1a_q,mϕ_m,τ_q(t). Then s_f=f-j_f has matching one-sided derivatives through order r-1 at every breakpoint and belongs to W^r,p(0,1). Its weak derivative of or…
Exact mathematical passage
j_f(t) = \sum_{q=1}^{Q}\sum_{m=0}^{r-1}a_{q,m}\phi_{m,\tau_q}(t).
\sum_{m=0}^\inftyUse in source · 1 passage
Search aliases: ∑_m=0^∞ sum infty
Use in source · Paper_VI_Traditional_Final.tex, line 2164
[Resolvent expansion over packet paths] With A=D+N as in the directed path decomposition, let R_D(z)=(zI-D)^-1. If normR_D(z)N<1, then (zI-A)^-1 = ∑_m=0^∞ (R_D(z)N)^mR_D(z). After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.
Exact mathematical passage
(zI-A)^{-1} = \sum_{m=0}^\infty \bigl(R_D(z)N\bigr)^mR_D(z).
\sum_{m=0}^{\infty}Use in source · 1 passage
Search aliases: ∑_m=0^∞ sum infty
Use in source · Paper_VI_Traditional_Final.tex, line 5911
This section presents the operator-substitution result transferred from Chapter IV. Fix an integer N≥1 and the entire scalar function D_N(z)=∑_m=0^∞frac(-1)^m z^Nm(Nm)!. More generally, the result applies to any specified holomorphic scalar function that is nonzero on the declared domain. On a simply connected zero-free domain U, choose g=log D_N once and define S_N^(r)(z)=exp(-r g(z)) for complex r. The bounded Banach-space result below uses classical holomorphic functional calculus. It is sepa…
Exact mathematical passage
\Dfun_N(z)=\sum_{m=0}^{\infty}\frac{(-1)^m z^{Nm}}{(Nm)!}.
\sum_{m=1}^HUse in source · 1 passage
Search aliases: ∑_m=1^H sum
Use in source · strengthening.tex, line 390
The conversion from unit steps to arbitrary mixed steps is used several times, so it is useful to make it explicit. Write Δ=Δ_1. For H>0, Δ_H=∑_m=1^Hbinom HmΔ^m. A mixed term with local orders m_1,…,m_k≥1 and M=∑_ν m_ν therefore has valuation at least M+v_2(M!)+β +∑_ν(v_2(H_ν)-v_2(m_ν)) if the unit-order estimate has offset β for every positive order. The inequalities m-1≥ v_2(m), M≥ k, v_2(M!)≥ v_2(k!) make this at least k+v_2(k!)+β+∑_ν v_2(H_ν). Every expansion is finite. This proves the desir…
Exact mathematical passage
\Delta_H=\sum_{m=1}^H\binom Hm\Delta^m. \label{eq:finite-step-expansion}
\sum_{m=1}^{n}Use in source · 1 passage
Search aliases: ∑_m=1^n sum
Use in source · Paper_IV_Traditional_Final.tex, line 1688
[Theta Newton recurrence] Put θ_N,n(u)=fracsymupΘ_N,n(u)(Nn)!. Then θ_N,0(u)=1 and nθ_N,n(u) = ∑_m=1^n (-1)^mκ_N,m (u^Nm+u) θ_N,n-m(u). In particular, symupΘ_N,1(u)=-(u^N+u).
Exact mathematical passage
n\theta_{N,n}(u) = \sum_{m=1}^{n} (-1)^m\kappa_{N,m} \bigl(u^{Nm}+u\bigr) \theta_{N,n-m}(u).
\sum_{m=1}^{w_d}Use in source · 1 passage
Search aliases: ∑_m=1^w_d sum
Use in source · Paper_X_Block27_Amended.tex, line 560
[Weighted seed zeta function] Let 1≤ w_1≤⋯≤ w_d, put W=∑_iw_i, and let a_m=q^∑_imax(m-w_i,0). Then Z_ w(T) = ∑_m=1^w_da_mT^m + q^-Wfrac(q^dT)^w_d+11-q^dT. In particular, every weighted smooth chart has a rational seed zeta function.
Exact mathematical passage
\boxed{ Z_{\bm w}(T) = \sum_{m=1}^{w_d}a_mT^m + q^{-W}\frac{(q^dT)^{w_d+1}}{1-q^dT}. }
\sum_{m=1}^{\infty}Use in source · 1 passage
Search aliases: ∑_m=1^∞ sum infty
Use in source · Paper_VI_Traditional_Final.tex, line 2236
[Dyson expansion over packet paths] For t≥0, e^tA = · e^tD + ∑_m=1^∞ ∫_0<s_1<⋯<s_m<t e^(t-s_m)DN e^(s_m-s_m-1)D ⋯ N e^s_1D d s_1⋯ d s_m. The series converges absolutely in operator norm. After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.
Exact mathematical passage
\begin{aligned} \e^{tA} ={}&\e^{tD} + \sum_{m=1}^{\infty} \int_{0<s_1<\cdots<s_m<t} \e^{(t-s_m)D}N\e^{(s_m-s_{m-1})D} \cdots N\e^{s_1D} \dd s_1\cdots\dd s_m. \end{aligned}
\sum_{m=k}^{n+k}Use in source · 1 passage
Search aliases: ∑_m=k^n+k sum
Use in source · strengthening.tex, line 667
eciprocal are C_2(0)=1, C_2(1)=-frac115, C_2(2)=frac1345, a_1=-frac1615, a_2=frac6445, where a_m=Δ_1^m C_2(0). Thus v_2(a_1)=4 and v_2(a_2)=6. For m≥3, [source reference] bounds a_m by 2m+v_2(m!). The finite shifted Newton formula Δ_1^kC_2(n)=∑_m=k^n+ka_mbinom nm-k therefore gives v_2(Δ_1^kC_2(n))≥ k+v_2(k!)+3 for every k≥1. For k≥3 this is 2k≥ k+3. For k=1,2 it uses the exact initial coefficients and the higher tail bound. The finite step expansion proves the offset three in [source reference].
Exact mathematical passage
\Delta_1^kC_2(n)=\sum_{m=k}^{n+k}a_m\binom n{m-k} \label{eq:shifted-newton-finite}
\sum_{m\in L}Use in source · 1 passage
Search aliases: ∑_m∈ L sum in
Use in source · Paper_V_Traditional_Final.tex, line 1928
[Exact convolution transfer law] Let y=h*x. Then y_r(ℓ) = ∑_s∈ R∑_m∈ Lh_rs(m)x_s(ℓ-m). Equivalently, A_LH A_L^-1 = (H_rs)_r,s∈ R, where H_rs is convolution on L by h_rs.
Exact mathematical passage
y_r(\ell) = \sum_{s\in R}\sum_{m\in L}h_{rs}(m)x_s(\ell-m).
\sum_{m\in\bbZ}Use in source · 1 passage
Search aliases: ∑_m∈bbZ ∑_m∈ Z sum in bbZ sum in mathbb
Use in source · Paper_V_Traditional_Final.tex, line 426
[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.
Exact mathematical passage
\boxed{ y_k[n] = \sum_{s=0}^{N-1}\sum_{m\in\bbZ} h_{ks}[m]x_s[n-m]. }
\sum_{m\in\ZZ}Use in source · 1 passage
Search aliases: ∑_m∈ZZ ∑_m∈ Z sum in ZZ sum in mathbb
Use in source · Paper_V_Traditional_Final.tex, line 1988
For L=N Z with N≥1, first take |z|=1. The assumption h∈ℓ^1( Z) makes the following symbol series absolutely convergent. Define H_ks(z)=∑_m∈ Zh(Nm+k-s)z^-m. For x∈ℓ^1( Z), the channel transforms converge absolutely on the unit circle and satisfy Y(z)= H_N(z) X(z). For x∈ℓ^2( Z), the same identity holds almost everywhere in the L^2 Fourier sense. For finite impulse-response kernels it also holds at nonzero z wherever the input channel transforms are weighted absolutely summable. This is the classi…
Exact mathematical passage
H_{ks}(z)=\sum_{m\in\ZZ}h(Nm+k-s)z^{-m}.
\sum_{m\equiv j\bmod N}Use in source · 1 passage
Search aliases: ∑_m≡ jbmod N sum equiv bmod
Use in source · Paper_IV_Traditional_Final.tex, line 377
[Root filter] If F(z)=∑_m≥0a_mz^m, then B_N,jF(z) = ∑_m≡ jbmod Na_mz^m. The operators satisfy B_N,i B_N,j=δ_ij B_N,j and ∑_j=0^N-1 B_N,j=I.
Exact mathematical passage
\Bcal_{N,j}F(z) = \sum_{m\equiv j\bmod N}a_mz^m.
\sum_{m\geq 0}Use in source · 1 passage
Search aliases: ∑_m≥ 0 sum geq
Use in source · Paper_IV_Traditional_Final.tex, line 239
For a power series F(z)=∑_m≥ 0a_mz^m and a positive integer N, the coefficients may be separated according to the residue of m modulo N. Root of unity averaging gives exact projection operators for this separation. This elementary construction is usually called a multisection or a root filter. It appears throughout the classical theory of generating functions and special functions. Background on multisections, hypergeometric functions, and related coefficient systems may be found in [source refe…
Exact mathematical passage
F(z)=\sum_{m\geq 0}a_mz^m
\sum_{m\ge L}Use in source · 1 passage
Search aliases: ∑_m≥ L sum ge
Use in source · atlas-pages.tex, line 391
on ℓ^2(Λ_N) gives the corresponding diagonal Fredholm determinant. Repeated zeros require repeated diagonal entries. vfigproduct headFinite compression is an approximation For N=2, λ_m=π^2(m+1/2)^2. The finite determinant P_L is exactly the displayed finite product, but it is not exactly Δ_2. For 0≤ w≤1 and L≥1, an elementary tail estimate gives 0≤1-fracΔ_2(w)P_L(w) ≤ w∑_m≥ Lλ_m^-1 ≤frac19(L-1/2). The last bound uses π>3 and integral comparison. Thus truncation has an explicit error mechanism. t…
Exact mathematical passage
0\le1-\frac{\Delta_2(w)}{P_L(w)} \le w\sum_{m\ge L}\lambda_m^{-1} \le\frac1{9(L-1/2)}.
\sum_{m\geq\ell}Use in source · 2 passages
Search aliases: ∑_m≥ℓ sum geq ell
Use in source · strengthening.tex, line 1272
Put a_m=Δ_1^mC_j(0) and expand the falling factorial in the Newton series. The ordinary coefficients are d_j,ℓ=∑_m≥ℓfraca_mm!s(m,ℓ). Since s(m,ℓ) is integral and v_2(a_m/m!)≥ m+β_j, the coefficient sum converges and has valuation at least ℓ+β_j. The sum exchange is justified on Z_2 by these uniform geometric coefficient bounds. The ordinary power series agrees there with the Newton interpolation and converges whenever |x|_2<2.
Exact mathematical passage
d_{j,\ell}=\sum_{m\geq\ell}\frac{a_m}{m!}s(m,\ell). \label{eq:newton-taylor-stirling}
\sum_{n\text{ even}}Use in source · 1 passage
Search aliases: ∑_n even sum text
Use in source · 72_recovered.tex, line 1064
For even n≥0, define R_n = frac3^n+2-14(n+1)(n+2). Its even exponential generating function is ∑_n evenR_nfract^nn! = fracsinh(2t)sinh t2t^2. Define β_r by fracz^2sinh^2z = ∑_r≥0β_rz^2r.
Exact mathematical passage
\boxed{ \sum_{n\text{ even}}R_n\frac{t^n}{n!} = \frac{\sinh(2t)\sinh t}{2t^2}. }
\sum_{n=0}^{90}Use in source · 1 passage
Search aliases: ∑_n=0^90 sum
Use in source · atlas-pages.tex, line 347
V: Theorem 11.7, Theorem 11.8, Corollary 11.9. The root finder is used only to suggest intervals. Each displayed interval is accepted by exact rational arithmetic: opposite endpoint signs prove existence, and an interval bound for D_3' excludes zero throughout the interval, proving uniqueness inside it. vfigcubic_certificates headHow the certificate works Evaluate the Taylor polynomial S_90(t)=∑_n=0^90frac(-1)^nt^3n(3n)! at the two rational endpoints. From the next term onward, magnitudes decrea…
Exact mathematical passage
S_{90}(t)=\sum_{n=0}^{90}\frac{(-1)^nt^{3n}}{(3n)!}
\sum_{n=0}^NUse in source · 1 passage
Search aliases: ∑_n=0^N sum
Use in source · 05_defects.tex, line 80
[Polynomial defect] For p(z)=∑_n=0^Na_nz^n, we have Cp(T_f)-p(T_c)C = ∑_n=1^Na_n ∑_k=0^n-1T_c^n-1-kΔ T_f^k.
Exact mathematical passage
p(z)=\sum_{n=0}^Na_nz^n,
\sum_{n=0}^{N-1}Use in source · 1 passage
Search aliases: ∑_n=0^N-1 sum
Use in source · 03_weighted.tex, line 321
et q_L:R→ R_L be a ring homomorphism that kills F^L R. Suppose H=∑_e∈ EH_e, D^-1H_e∈ F^w(e)R, w(e)≥1, with E finite. Then q_L(D^-1H)^L=0, the finite sum in [source reference](a) is a two-sided inverse after applying q_L, and its word expansion may discard every word of weight at least L. More explicitly, for a length cutoff N, define S^pr_L,N = ∑_n=0^N-1(-1)^n ∑_substackγ∈ E^n W(γ)<LA_γ. Lean proves directly that q_L(S^pr_L,N) equals the mapped geometric inverse whenever q_L(A^N)=0, and hence is…
Exact mathematical passage
S^{\mathrm{pr}}_{L,N} = \sum_{n=0}^{N-1}(-1)^n \sum_{\substack{\gamma\in E^n\\W(\gamma)<L}}A_\gamma.
\sum_{n=0}^\inftyUse in source · 6 passages
Search aliases: ∑_n=0^∞ sum infty
Use in source · Paper_VI_Traditional_Final.tex, line 3131
Let γ be the standard Gaussian probability measure on R. For t≥0, define the Ornstein--Uhlenbeck transfer ( O_t f)(x) = ∫_ R f( e^-tx+sqrt1- e^-2t y ) γ( d y). Let H_n be the probabilists' Hermite polynomial defined by exp(ux-fracu^22) = ∑_n=0^∞ H_n(x)fracu^nn!, and set h_n=H_n/sqrtn!.
Exact mathematical passage
\exp\left(ux-\frac{u^2}{2}\right) = \sum_{n=0}^\infty H_n(x)\frac{u^n}{n!},
\sum_{n=0}^{\infty}Use in source · 4 passages
Search aliases: ∑_n=0^∞ sum infty
Use in source · monograph_atlas.tex, line 670
|=frac2556, frac2gδ|B-widetilde B| +fracg^2δ^2|D-widetilde D|=frac2132. The theorem bounds the difference using a common spectral gap. Applying the bound uniformly to a refinement family requires uniform spectral separation and control of the couplings. headA series near zero energy and its exact remainder At z=1/2, |z| |D^-1|=1/4<1, so Σ(z)=-∑_n=0^∞fracz^n2^n+1. After the first four terms, the exact remainder has magnitude 1/384. The geometric series controls the infinite tail. The retained che…
Exact mathematical passage
\Sigma(z)=-\sum_{n=0}^{\infty}\frac{z^n}{2^{n+1}}.
\sum_{n=1}^{m}Use in source · 1 passage
Search aliases: ∑_n=1^m sum
Use in source · atlas-pages.tex, line 266
vpage16Why a Fourier cutoff cannot reproduce this jump pointwiseA CONTINUOUS PULSE / EXACT ENDPOINT VALUESChapter V: Section 6, Section 8. Use a separate period-one example on [-1/2,1/2): j(t)= 1_{|t|<1/4}, S_mj(t)=tfrac12+frac2π∑_n=1^mfracsin(π n/2)ncos(2π nt). This is the exact finite Fourier formula. The drawn curves sample it numerically. The endpoint calculation below is symbolic. vfigfourier_jump headOne value is fixed at every cutoff At t=±1/4, every summand vanishes because sin(π n/2)cos…
Exact mathematical passage
j(t)=\mathbf 1_{\{|t|<1/4\}},\qquad S_mj(t)=\tfrac12+\frac2\pi\sum_{n=1}^{m}\frac{\sin(\pi n/2)}{n}\cos(2\pi nt).
\sum_{n=1}^NUse in source · 1 passage
Search aliases: ∑_n=1^N sum
Use in source · 05_defects.tex, line 84
[Polynomial defect] For p(z)=∑_n=0^Na_nz^n, we have Cp(T_f)-p(T_c)C = ∑_n=1^Na_n ∑_k=0^n-1T_c^n-1-kΔ T_f^k.
Exact mathematical passage
Cp(T_f)-p(T_c)C = \sum_{n=1}^Na_n \sum_{k=0}^{n-1}T_c^{n-1-k}\Delta T_f^k.
\sum_{n=m}^{\infty}Use in source · 1 passage
Search aliases: ∑_n=m^∞ sum infty
Use in source · Paper_VI_Traditional_Final.tex, line 2555
[Weighted summable defect completion] Define the adjacent semigroup defect E_n(t)=C_nT_n+1(t)-T_n(t)C_n. Fix a coarse level m and a time horizon T>0. For n≥ m, set w_m,n = normC_m,n normR_nR_n,m. If ∑_n=m^∞ w_m,n sup_0≤ t≤ TnormE_n(t)<∞, then U_m,n(t) converges in operator norm uniformly for 0≤ t≤ T. Its limit is denoted by U_m(t), and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_j=n^∞w_m,j sup_0≤ t≤ TnormE_j(t).
Exact mathematical passage
\sum_{n=m}^{\infty} w_{m,n} \sup_{0\leq t\leq T}\norm{E_n(t)}<\infty,
\sum_{n>N}Use in source · 1 passage
Search aliases: ∑_n>N sum
Use in source · visual_atlas.tex, line 454
This x cannot be a finite linear combination of shifted impulses: it has infinite support. But if x^(N) agrees with x on |n|≤ N and is zero elsewhere, then normx-x^(N)_2^2 =2∑_n>N4^-n =frac23 4^-N⟶0. The exact squared errors for N=1,3,5 are 1/6, 1/96, and 1/1536. These are identities from a geometric series, not floating residuals.
Exact mathematical passage
\norm{x-x^{(N)}}_2^2 =2\sum_{n>N}4^{-n} =\frac{2}{3}\,4^{-N}\longrightarrow0.
\sum_{n\in\bbZ}Use in source · 2 passages
Search aliases: ∑_n∈bbZ ∑_n∈ Z sum in bbZ sum in mathbb
Use in source · Paper_V_Traditional_Final.tex, line 284
[Impulse seed theorem] The closed shift-generated hull of δ_0 is all of ℓ^2( Z): overlinespan{S^nδ_0:n∈ Z}^ ℓ^2 =ℓ^2( Z). Every finitely supported signal has the exact finite expression x=∑_n∈ Zx[n]S^nδ_0.
Exact mathematical passage
x=\sum_{n\in\bbZ}x[n]S^n\delta_0.
\sum_{n\in\ZZ^d}Use in source · 1 passage
Search aliases: ∑_n∈ZZ^d ∑_n∈ Z^d sum in ZZ sum in mathbb
Use in source · 02_strengthened.tex, line 2107
[Theta transformation] The continuum flat torus heat trace satisfies Z_d(t) = ∑_k∈ Z^d e^-4π^2tabsk^2 = (4π t)^-d/2 ∑_n∈ Z^d e^-absn^2/(4t). This is the Poisson summation formula for a Gaussian.
Exact mathematical passage
Z_d(t) = \sum_{k\in\ZZ^d}\e^{-4\pi^2t\abs{k}^2} = (4\pi t)^{-d/2} \sum_{n\in\ZZ^d}\e^{-\abs{n}^2/(4t)}.
\sum_{n\notin F}Use in source · 1 passage
Search aliases: ∑_n∉ F sum notin
Use in source · Paper_V_Traditional_Final.tex, line 294
The shift orbit is the standard orthonormal basis {δ_n}_n∈ Z. For x∈ℓ^2, the finite truncations x_F=∑_n∈ Fx[n]δ_n, F⊂ Z finite, satisfy x-x_F _2^2=∑_n∉ F|x[n]|^2→0 as the finite sets exhaust Z. Thus the unordered Hilbert-space sum ∑_nx[n]δ_n converges to x, and the finite-support statement is exact. The shift is a surjective isometry because n↦ n-1 is a bijection of Z, with inverse shift n↦ n+1. This proves the closed-span assertion.
Exact mathematical passage
\|x-x_F\|_2^2=\sum_{n\notin F}|x[n]|^2\to0
\sum_{n\neq0}Use in source · 1 passage
Search aliases: ∑_n≠0 sum neq
Use in source · 02_strengthened.tex, line 2140
By [source reference], Z_d(t) =(4π t)^-d/2(1+R(t)), where R(t)=∑_n≠0 e^-absn^2/(4t) =O( e^-1/(4t)). Differentiation gives tR'(t)=O(t^-1 e^-1/(4t)). Substituting this estimate into the logarithmic derivative gives the stated small time expansion.
Exact mathematical passage
R(t)=\sum_{n\neq0}\e^{-\abs{n}^2/(4t)} =O\bigl(\e^{-1/(4t)}\bigr).
\sum_{n\equiv r\ (\mathrm{mod}\,N)}Use in source · 1 passage
Search aliases: ∑_n≡ r (mod N) sum equiv mathrm
Use in source · Paper_V_Traditional_Final.tex, line 342
Each integer belongs to exactly one residue class modulo N. Multiplying the two indicator functions gives zero for distinct classes and the same indicator for equal classes. Summing all residue indicators gives one at every address. Moreover, P_N,rx _2^2=∑_n≡ r (mod N)|x[n]|^2≤ x _2^2, ⟨ P_N,rx,y⟩=⟨ x,P_N,ry⟩. The mask is real-valued, so the inner-product identity follows term by term. Hence each projector is self-adjoint, and the ranges of different residue projectors are orthogonal.
Exact mathematical passage
\|P_{N,r}x\|_2^2=\sum_{n\equiv r\ (\mathrm{mod}\,N)}|x[n]|^2\le\|x\|_2^2, \qquad \langle P_{N,r}x,y\rangle=\langle x,P_{N,r}y\rangle.
\sum_{n\geq 0}Use in source · 2 passages
Search aliases: ∑_n≥ 0 sum geq
Use in source · Paper_IV_Traditional_Final.tex, line 256
This chapter follows the consequences of applying the cyclic projections to the exponential function. The projected exponentials form a finite differential system. After one sign is inserted around the derivative cycle, they form a basis for y^(N)=-y. The section with initial value one is D_N(z)=∑_n≥ 0(-1)^nfracz^Nn(Nn)!. Its reciprocal powers produce polynomial coefficient families. Its logarithmic derivative produces a scalar higher Riccati identity. The same reciprocal construction applied to…
Exact mathematical passage
\Dfun_N(z)=\sum_{n\geq 0}(-1)^n\frac{z^{Nn}}{(Nn)!}.
\sum_{N\ge0}Use in source · 1 passage
Search aliases: ∑_N≥0 sum ge
Use in source · root_scope.tex, line 51
m≥1, differentiation of zcoth z gives β_m=-frac2^2m(2m-1)B_2m(2m)!, κ_m=-2^4m(2m-1)B_2m. The classical dyadic Bernoulli denominator law v_2(B_2m)=-1 and Legendre's formula yield v_2(β_m)=s_2(m)-1, v_2(κ_m)=4m-1. The source identity ∑_N≥0A(N)fract^2N(2N)! =fracsinh(2t)sinh t2t^2 implies, after multiplication by 4t^2/sinh^2(2t), E_2N=∑_m=0^N κ_mbinom2N2mA(N-m). For a retained depth M≥0, put T_M(N)= E_2N -∑_m=0^min(M,N)κ_mbinom2N2mA(N-m). An empty tail, N≤ M, is exactly zero.
Exact mathematical passage
\sum_{N\ge0}A(N)\frac{t^{2N}}{(2N)!} =\frac{\sinh(2t)\sinh t}{2t^2}
\sum_{n\ge1}Use in source · 1 passage
Search aliases: ∑_n≥1 sum ge
Use in source · 03_nonlinear_coordinates.tex, line 491
[Quadratic Catalan expansion] Let B:X× X→ X be bounded and consider u=s+B(u,u). Put a= s and b= B . If 4ab<1, then u= ∑_n≥1 ∑_substackτ full ordered binary tree with n leaves B_τ(s) converges absolutely. It also satisfies u ≤ frac1-sqrt1-4ab2b when b>0.
Exact mathematical passage
u= \sum_{n\ge1} \sum_{\substack{\tau\text{ full ordered binary tree}\\\text{with }n\text{ leaves}}} B_\tau(s)
\sum_{n\geq1}Use in source · 3 passages
Search aliases: ∑_n≥1 sum geq
Use in source · 03_packets_and_graphs.tex, line 414
[Countable unconditional packet isomorphism] Let X be a real or complex Banach space and let (P_n)_n≥1⊆ B(X) satisfy P_nP_m=δ_nmP_n and x=∑_n≥1P_nx unconditionally for every x∈ X. Assume K:= sup_FSubset N sup_|a_n|≤1 ∑_n∈ Fa_nP_n <∞. Then analysis and synthesis define inverse Banach space isomorphisms X≅⊕_n≥1^uP_nX. More precisely, A_ P ≤ K, S_ P ≤1. [source reference] ExportsCountable analytic packet decompositions under explicit convergence control.
Exact mathematical passage
x=\sum_{n\geq1}P_nx
\sum_{q<n}Use in source · 3 passages
Search aliases: ∑_q<n sum
Use in source · 10_foundations.tex, line 182
[Forward substitution and denominator clearing] Let T_nq be coefficients over a field with T_nn≠0 for every n. For any sequence b, define a_n=fracb_n-∑_q<nT_nqa_qT_nn. This constructs the unique solution of ∑_q=0^nT_nqa_q=b_n for all n. If T_nq and b_n are integers and the solution is taken over Q, then (∏_r=0^nT_rr)a_n∈ Z.
Exact mathematical passage
a_n=\frac{b_n-\sum_{q<n}T_{nq}a_q}{T_{nn}}.
\sum_{q=0}^mUse in source · 1 passage
Search aliases: ∑_q=0^m sum
Use in source · atlas-pages.tex, line 274
The four real roots of this polynomial are ±1,±sqrt5. They are not the zeros of the original analytic denominator. The plot distinguishes the input polynomial and its transformed output. headAn inverse operation, not a truncation error Applying cosh(∂_x) returns x^4 exactly. The general inverse uses the reciprocal formal symbol. The lowering and translation identities are E_m'=mE_m-1, E_m(x+y)=∑_q=0^mbinom mqE_q(x)y^m-q. Both follow from the generating function. No operator-series convergence as…
Exact mathematical passage
E_m'=mE_{m-1},\qquad E_m(x+y)=\sum_{q=0}^m\binom mqE_q(x)y^{m-q}.
\sum_{q=0}^{m}Use in source · 2 passages
Search aliases: ∑_q=0^m sum
Use in source · Paper_IV_Traditional_Final.tex, line 1499
[Appell laws] For m≥1, the polynomials satisfy fracddxE_m^(N,j,r)(x) = mE_m-1^(N,j,r)(x) and, for m≥0, E_m^(N,j,r)(x+y) = ∑_q=0^m binom mq E_q^(N,j,r)(x)y^m-q.
Exact mathematical passage
E_m^{(N,j,r)}(x+y) = \sum_{q=0}^{m} \binom mq E_q^{(N,j,r)}(x)y^{m-q}.
\sum_{q=0}^nUse in source · 1 passage
Search aliases: ∑_q=0^n sum
Use in source · 20_arithmetic.tex, line 803
[Classical odd-length reconstruction] For every positive odd integer p and every n∈ N, E_2n=∑_m=0^nbinom2n2mγ_m p^2mS_2(n-m)(p). Equivalently, E_2n=∑_q=0^n V_nqS_2q(p) for the lower triangular matrix V_nq=binom2n2qγ_n-qp^2(n-q) (q≤ n), V_nq=0 (q>n). It has V_nn=1 and p^2(n-q)mid V_nq when q≤ n.
Exact mathematical passage
E_{2n}=\sum_{q=0}^n V_{nq}S_{2q}(p)
\sum_{q=0}^{n-1}Use in source · 3 passages
Search aliases: ∑_q=0^n-1 sum
Use in source · Paper_IV_Traditional_Final.tex, line 1421
[Triangular inverse system] The coefficients are determined by C_N,j(0)=1 and ∑_q=0^n binomNn+jNq C_N,j(q) = δ_n0. For n≥1, C_N,j(n) = - frac displaystyle∑_q=0^n-1 binomNn+jNqC_N,j(q) displaystylebinomNn+jNn .
Exact mathematical passage
C_{N,j}(n) = - \frac{ \displaystyle\sum_{q=0}^{n-1} \binom{Nn+j}{Nq}C_{N,j}(q) }{ \displaystyle\binom{Nn+j}{Nn} }.
\sum_{q=0}^rUse in source · 1 passage
Search aliases: ∑_q=0^r sum
Use in source · strengthening.tex, line 566
X)=∏_a=0^r-1(X-a(a+1)). At X=Z(Z+1), they have the exact factorization P_r(Z(Z+1)) =∏_a=0^r-1(Z-a)(Z+a+1) =(2r)!binomZ+r2r. The Bernoulli transform gives L((1+T)^Z)=log(1+T)/T, whence L(binom Zj)=(-1)^j/(j+1). Finite Vandermonde expansion therefore gives L(binomZ+r2r) · =(-1)^r∑_q=0^r(-1)^qbinom rqfrac1r+q+1nonumber · =(-1)^rfrac(r!)^2(2r+1)!. The last equality is the rational identity obtained by expanding ∫_0^1t^r(1-t)^r dt. This use of an ordinary finite integral makes no assertion about a po…
Exact mathematical passage
\mathcal L\left(\binom{Z+r}{2r}\right) &=(-1)^r\sum_{q=0}^r(-1)^q\binom rq\frac1{r+q+1}\nonumber\\ &=(-1)^r\frac{(r!)^2}{(2r+1)!}. \label{eq:central-factorial-evaluation}
\sum_{q=0}^{r}Use in source · 1 passage
Search aliases: ∑_q=0^r sum
Use in source · supplement_02.tex, line 138
The row action is connected to the matrix entries. Reversing the finite sum and using binomial symmetry gives the actual source-index row action E_2r=∑_q=0^rU_r,q(p)S_2q(p), where U_r,q(p)= displaystylebinom2r2qb_r-qp^2(r-q), · q≤ r,
Exact mathematical passage
E_{2r}=\sum_{q=0}^{r}U_{r,q}(p)S_{2q}(p),
\sum_{q=1}^MUse in source · 1 passage
Search aliases: ∑_q=1^M sum
Use in source · Paper_V_Traditional_Final.tex, line 693
[Several exact edge packets] If f=s+∑_q=1^Mj_q and every j_q is retained exactly, then f-(Q_ms+∑_qj_q) = s-Q_ms . Curved edges, corners, and different jump amplitudes do not alter the identity. The finite sum j=∑_q=1^M j_q is a single retained packet. Applying the preceding cancellation identity gives the result. The empty family M=0 is included with sum zero.
Exact mathematical passage
j=\sum_{q=1}^M j_q
\sum_{q=1}^{M}Use in source · 1 passage
Search aliases: ∑_q=1^M sum
Use in source · Paper_V_Traditional_Final.tex, line 682
[Several exact edge packets] If f=s+∑_q=1^Mj_q and every j_q is retained exactly, then f-(Q_ms+∑_qj_q) = s-Q_ms . Curved edges, corners, and different jump amplitudes do not alter the identity. The finite sum j=∑_q=1^M j_q is a single retained packet. Applying the preceding cancellation identity gives the result. The empty family M=0 is included with sum zero.
Exact mathematical passage
f=s+\sum_{q=1}^{M}j_q
\sum_{q=1}^nUse in source · 1 passage
Search aliases: ∑_q=1^n sum
Use in source · strengthening.tex, line 1031
The actual reciprocal recurrence is C_N,j(n)=-∑_q=1^nbinomNnNq g_N,j(q)C_N,j(n-q), C_N,j(0)=1. For N=4, the four kernels are 1, frac14q+1, frac1(4q+1)(2q+1), frac3(4q+1)(2q+1)(4q+3). The threshold-free theorem gives v_2(a_k)≥ k in every address. The following exact values cover all orders where that bound does not already imply [source reference]: c r l r j · k · a_k · v_2(a_k)
Exact mathematical passage
C_{N,j}(n)=-\sum_{q=1}^n\binom{Nn}{Nq} g_{N,j}(q)C_{N,j}(n-q),\qquad C_{N,j}(0)=1. \label{eq:exact-reciprocal-row-recurrence}
\sum_{q\in J_r}Use in source · 1 passage
Search aliases: ∑_q∈ J_r sum in
Use in source · root_scope.tex, line 140
ad J_r={q:λ_r(q)=μ_r}, G_r=2^-μ_rH_r. Since λ_r(0)=0 and λ_r(q)≥3q+1-s_2(r), every minimizing address satisfies 0≤ q≤lfloorfracs_2(r)-13rfloor. In particular μ_r≤0, and the minimum is found by a finite carry calculation. Reduction of the whole series gives overline G_r(X)=∑_q∈ J_rX^underline q in F_2[X]. Its highest degree is max J_r, with leading coefficient one. Thus the normalization is the actual Gauss normalization.
Exact mathematical passage
\overline G_r(X)=\sum_{q\in J_r}X^{\underline q} \quad\text{in }\mathbb F_2[X]. \label{eq:root:reduction}
\sum_{q\in\bbZ}Use in source · 1 passage
Search aliases: ∑_q∈bbZ ∑_q∈ Z sum in bbZ sum in mathbb
Use in source · Paper_V_Traditional_Final.tex, line 411
Let h be a finite impulse response and y=h*x, y[n]=∑_q∈ Zh[q]x[n-q]. Resolve both input and output into N phases. For output phase k and input phase s, define the transfer kernel h_ks[m]=h[Nm+k-s]. Out-of-range taps are understood to be zero.
Exact mathematical passage
y=h*x, \qquad y[n]=\sum_{q\in\bbZ}h[q]x[n-q].
\sum_{q\in\ZZ^d}Use in source · 1 passage
Search aliases: ∑_q∈ZZ^d ∑_q∈ Z^d sum in ZZ sum in mathbb
Use in source · Paper_V_Traditional_Final.tex, line 1906
Let h∈ℓ^1( Z^d) and x∈ℓ^p( Z^d), with 1≤ p≤∞, and define (Hx)(n) =(h*x)(n) =∑_q∈ Z^dh(q)x(n-q). Young's inequality gives Hx _p≤ h _1 x _p for every 1≤ p≤∞.
Exact mathematical passage
(Hx)(n) =(h*x)(n) =\sum_{q\in\ZZ^d}h(q)x(n-q).
\sum_{q\ge0}Use in source · 1 passage
Search aliases: ∑_q≥0 sum ge
Use in source · root_scope.tex, line 110
Set r=M+1≥1 and define P_q(x)=∏_i=0^q-1(x-i)(2x-(2i+1)), P_0=1, H_r(x)=∑_q≥08^qfracβ_r+qβ_r P_q(x)mathscr A(x-q). The coefficient valuations tend to infinity. At any nonnegative integer x, terms with q>x vanish. Cancelling the factorials in [source reference] therefore gives the exact identity T_r-1(N)=κ_rbinom2N2rH_r(N-r) (N≥ r). This is a finite rational identity at every original Euler row.
Exact mathematical passage
H_r(x)=\sum_{q\ge0}8^q\frac{\beta_{r+q}}{\beta_r} P_q(x)\mathscr A(x-q).
\sum_{q\geq0}Use in source · 3 passages
Search aliases: ∑_q≥0 sum geq
Use in source · Paper_IV_Traditional_Final.tex, line 1551
[Formal differential-operator representation on polynomials] Let a formal series A(z)=∑_q≥0a_qz^q/q! act on polynomials by A(∂_x)f(x) = ∑_q≥0fraca_qq!∂_x^qf(x). The power r is applied to the formal symbol before the substitution z=∂_x. Then E_m^(N,j,r)(x) = C_N,j(∂_x)^r x^m. The inverse relation is x^m = Φ_N,j(∂_x^N)^r E_m^(N,j,r)(x). Both sums are finite on every polynomial.
Exact mathematical passage
A(z)=\sum_{q\geq0}a_qz^q/q!
\sum_rUse in source · 14 passages
Search aliases: ∑_r sum
Use in source · strengthening.tex, line 498
g s_2(i) for the number of ones in the binary expansion of i, and using v_2((4i)!)=4i-s_2(i), gives v_2([x^d]P_i) ≥2d-2i+s_2(i)+max(0,2i-d) ≥ d+s_2(i)≥ d+1 (d≥1). For k≥1, Δ_1^k x^d vanishes when d<k and is an integer polynomial with values divisible by k!. To see the latter, expand x^d=∑_r{d r} r!binom xr and apply Δ_1^kbinom xr=binomxr-k. It follows that Δ_1^kP_i(x)∈2^k+1k! Z_2 (k≥0, x∈ Z_2). For k=0 the bound follows directly from the coefficient estimate.
Exact mathematical passage
x^d=\sum_r\left\{\begin{smallmatrix}d\\r\end{smallmatrix}\right\} r!\binom xr
\sum_{r}Use in source · 1 passage
Search aliases: ∑_r sum
Use in source · 02_strengthened.tex, line 1740
[Finite oscillatory memory expansion] If D=∑_rE_rQ_r is the spectral decomposition of the hidden Hamiltonian, then K(t)=∑_r e^- i tE_rBQ_rB^*. Every coefficient BQ_rB^* is positive semidefinite.
Exact mathematical passage
D=\sum_{r}E_rQ_r
\sum_{r+s\equiv t\!\!\pmod N}Use in source · 1 passage
Search aliases: ∑_r+s≡ t pmod N sum equiv pmod
Use in source · Paper_V_Traditional_Final.tex, line 3505
rem[Residue law for Cauchy products] Let F(z)=∑_n≥0f_nz^n, G(z)=∑_n≥0g_nz^n be formal power series. The same statement holds for power series that converge on a common disk. Decompose both series into residue packets modulo N: F=∑_r=0^N-1F_r, G=∑_s=0^N-1G_s. If H=FG, then its residue packet H_t is H_t = ∑_r+s≡ t pmod NF_rG_s.
Exact mathematical passage
\boxed{ H_t = \sum_{r+s\equiv t\!\!\pmod N}F_rG_s. }
\sum_{r,a}Use in source · 1 passage
Search aliases: ∑_r,a sum
Use in source · 03_supplement_05.tex, line 91
Deriving the Gram matrix from recovery Suppose a finite recovery that preserves trace corrects every operator supported on the code. To derive the Knill--Laflamme condition without assuming a syndrome decomposition, the proof expands the composite channel into the finite Kraus family A_ra=R_rE_aP. For every ambient matrix ρ, correction on the code implies ∑_r,a A_raρ A_ra^*=Pρ P. This equality on all matrices follows by applying correction to the sandwich supported on the code, Pρ P, using the p…
Exact mathematical passage
\sum_{r,a} A_{ra}\rho A_{ra}^*=P\rho P.
\sum_{r<d}Use in source · 1 passage
Search aliases: ∑_r<d sum
Use in source · 10_foundations.tex, line 415
[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalization hypothesis is required.
Exact mathematical passage
\chi_{C^N}(X)=X^d+\sum_{r<d}q_rX^r
\sum_{r<n}Use in source · 1 passage
Search aliases: ∑_r<n sum
Use in source · 10_foundations.tex, line 157
Pascal's identity proves the first product rule, and induction proves the higher rule. The intermediate binomial coefficients binom pr vanish in characteristic p. For the inverse, solve row n using the coefficient of b_n, which is g_0=1: b_n=1_n-∑_r<nbinom nr b_r g_n-r. Subtract the first and pth product rules for b⋆ g=1. The assumed identity for g leaves (D^p b-Db)⋆ g=0. A second triangular induction shows that convolution with a sequence having constant coefficient 1 has trivial kernel.
Exact mathematical passage
b_n=\mathbf1_n-\sum_{r<n}\binom nr b_r g_{n-r}.
\sum_{r=0}^3Use in source · 2 passages
Search aliases: ∑_r=0^3 sum
Use in source · atlas-pages.tex, line 43
ump packet, not an estimated feature or a frequency band. headWhat has changed? Only where the values are stored The complete tuple of channels contains sixteen values, just as the original vector does. A single decimated channel is shorter, but keeping all four loses no samples. The residue projector P_4,rf is different: it keeps the original lattice and writes zeros outside phase r. S_4 A_4f=f, f _2^2=∑_r=0^3 f_r _2^2=57. The norm on the channel tuple is the sum-of-squares Hilbert norm, not th…
Exact mathematical passage
\mathcal S_4\mathcal A_4f=f, \qquad \|f\|_2^2=\sum_{r=0}^3\|f_r\|_2^2=57.
\sum_{r=0}^{D}Use in source · 1 passage
Search aliases: ∑_r=0^D sum
Use in source · 10_foundations.tex, line 507
For M≥1 and s,r,k∈ N, define the affine Mahler kernel by Θ_M,s(r,k)=[z^r](1+z)^s((1+z)^M-1)^k =∑_j=0^k(-1)^k-jbinom kjbinomMj+sr. It vanishes unless k≤ r≤ Mk+s. For a finite binomial expansion f(n)=∑_r=0^Dc_rbinom nr, the kth Newton coefficient of g(q)=f(Mq+s) is therefore c_k(g)=∑_r=k^min(D,Mk+s)c_rΘ_M,s(r,k). Indeed, insert the finite stencil for Δ_1^k g(0) and interchange its two finite sums. Coefficient extraction of the binomial expansion of ((1+z)^M-1)^k gives the kernel. In the binary zer…
Exact mathematical passage
f(n)=\sum_{r=0}^{D}c_r\binom nr
\sum_{r=0}^{d}Use in source · 2 passages
Search aliases: ∑_r=0^d sum
Use in source · 10_foundations.tex, line 140
[Hurwitz product rules] For d≥0, D^d(f⋆ g)=∑_r=0^dbinom dr(D^r f)⋆(D^d-rg). If R has prime characteristic p, then D^p(f⋆ g)=f⋆ D^p g+D^p f⋆ g. Over a field, if g_0=1, a sequence b is constructed recursively with b⋆ g=1. If also D^p g=Dg in characteristic p, then D^p b=Db.
Exact mathematical passage
D^d(f\star g)=\sum_{r=0}^{d}\binom dr(D^r f)\star(D^{d-r}g).
\sum_{r=0}^{d-1}Use in source · 2 passages
Search aliases: ∑_r=0^d-1 sum
Use in source · 10_foundations.tex, line 416
[Companion jumps and packet recurrences] Let C be a d× d matrix over a commutative ring and x_n=C^n x_0. For N,j,k∈ N, put b_k=x_Nk+j. Then b_k=C^j(C^N)^kx_0, b_k+1=C^Nb_k. If χ_C^N(X)=X^d+∑_r<dq_rX^r, then b_k+d=-∑_r=0^d-1q_r b_k+r. Every coordinate and every linear scalar observable satisfies the same recurrence. No invertibility or diagonalization hypothesis is required.
Exact mathematical passage
b_{k+d}=-\sum_{r=0}^{d-1}q_r b_{k+r}.
\sum_{r=0}^kUse in source · 3 passages
Search aliases: ∑_r=0^k sum
Use in source · strengthening.tex, line 159
ence identity is Δ_1^kfrac11+sn =frac(-s)^k k!∏_h=0^k(1+s(n+h)). It follows by induction, subtracting adjacent rational functions. Each successive subtraction adds the factor -(k+1)s in the numerator. Its denominator is a unit, so it belongs to p^k Z_p. The discrete Leibniz formula Δ_1^k(fg)(n) =∑_r=0^kbinom kr (Δ_1^r f)(n)(Δ_1^k-rg)(n+r) then proves Δ_1^k g_N,j(n)∈ p^k Z_p for every k,n. The coefficient at degree Nn of L_1^kG_N,j is precisely p^-kΔ_1^k g_N,j(n), and the other residue classes va…
Exact mathematical passage
\Delta_1^k(fg)(n) =\sum_{r=0}^k\binom kr (\Delta_1^r f)(n)(\Delta_1^{k-r}g)(n+r) \label{eq:discrete-leibniz-final}
\sum_{r=0}^{m}Use in source · 1 passage
Search aliases: ∑_r=0^m sum
Use in source · Paper_V_Traditional_Final.tex, line 3328
Let (Δ x)(n)=x(n+1)-x(n). Then Δ^m x(n) = ∑_r=0^m (-1)^m-rbinom mr x(n+r).
Exact mathematical passage
\Delta^m x(n) = \sum_{r=0}^{m} (-1)^{m-r}\binom mr x(n+r).
\sum_{r=0}^{M-1}Use in source · 2 passages
Search aliases: ∑_r=0^M-1 sum
Use in source · 10_foundations.tex, line 488
polyphase vocabulary. We explain it through elementary identities with explicit algebraic derivations. These identities have not been separately formalized in Lean for this consolidation. For a sequence a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient and remainder give A(z)=∑_r=0^M-1z^rA_r(z^M), (a_r^[M])_s^[N]=a_r+Ms^[MN]. For ordinary Cauchy multiplication, the phase-r product has a term without a …
Exact mathematical passage
A(z)=\sum_{r=0}^{M-1}z^rA_r(z^M),\qquad \bigl(a_r^{[M]}\bigr)_s^{[N]}=a_{r+Ms}^{[MN]}.
\sum_{r=0}^{\min(M,n/2)}Use in source · 2 passages
Search aliases: ∑_r=0^min(M,n/2) sum min
Use in source · 72_recovered.tex, line 1010
[Odd modulus truncation] For every M≥0, every positive odd q, and every even n, E_n ≡ ∑_r=0^min(M,n/2) (-1)^rZ_2r+1 binom n2rq^2rS_n-2r(q) pmod q^2M+2. If pmid q and a=v_p(q), the same truncation is valid modulo p^2a(M+1).
Exact mathematical passage
\boxed{ E_n \equiv \sum_{r=0}^{\min(M,n/2)} (-1)^rZ_{2r+1} \binom n{2r}q^{2r}S_{n-2r}(q) \pmod {q^{2M+2}}. }
\sum_{r=0}^{N}Use in source · 3 passages
Search aliases: ∑_r=0^N sum
Use in source · 10_foundations.tex, line 264
Δ_h^s f(n) · =∑_j=0^s(-1)^s-jbinom sj f(n+jh). The finite interpolant I_Nf(n)=∑_r=0^Nc_r(f)binom nr agrees with f for n≤ N. It has the same order-s unit-step difference at n whenever n+s≤ N. For arbitrary coefficients c_0,…,c_N, the corresponding binomial interpolant has exactly those coefficients and zero coefficients above N.
Exact mathematical passage
I_Nf(n)=\sum_{r=0}^{N}c_r(f)\binom nr
\sum_{r=0}^{N-1}Use in source · 10 passages
Search aliases: ∑_r=0^N-1 sum
Use in source · Paper_V_Traditional_Final.tex, line 328
[Polyphase projector theorem] The maps P_N,r are bounded orthogonal projectors on ℓ^2, with operator norm at most one, and satisfy P_N,rP_N,s=δ_rsP_N,r, ∑_r=0^N-1P_N,r=I. Consequently, x=∑_r=0^N-1P_N,rx with no approximation.
Exact mathematical passage
P_{N,r}P_{N,s}=\delta_{rs}P_{N,r}, \qquad \sum_{r=0}^{N-1}P_{N,r}=I.
\sum_{r=0}^{n-1}Use in source · 4 passages
Search aliases: ∑_r=0^n-1 sum
Use in source · worked_atlas.tex, line 16
tlasFigDir/#1.pdf AtlasHeadCompute the reciprocal before estimating itatlas:reciprocal For width N and address j, the normalized source and its reciprocal are G_N,j(t)=j!∑_n≥0fract^Nn(Nn+j)!, G_N,j(t)^-1=∑_n≥0C_N,j(n)fract^Nn(Nn)!. Multiplying the two series and clearing the factorials gives an exact triangular recurrence: C_N,j(0)=1, C_N,j(n)=-fracdisplaystyle∑_r=0^n-1 binomNn+jNrC_N,j(r)binomNn+jNn. AtlasImagereciprocal_rows The table is for N=4. The first two address-zero rows illustrate the …
Exact mathematical passage
C_{N,j}(0)=1,\qquad C_{N,j}(n)=-\frac{\displaystyle\sum_{r=0}^{n-1} \binom{Nn+j}{Nr}C_{N,j}(r)}{\binom{Nn+j}{Nn}}.
\sum_{r=0}^{n/2}Use in source · 6 passages
Search aliases: ∑_r=0^n/2 sum
Use in source · 72_recovered.tex, line 965
[Odd modulus Euler coordinate pair] For every positive odd q and every even n≥0, E_n = ∑_r=0^n/2 (-1)^rZ_2r+1 binom n2r q^2rS_n-2r(q). The inverse transform is S_n(q) = E_n + ∑_r=1^n/2 2^2r-1 binom n2r q^2rE_n-2r. Both transforms are lower triangular with diagonal one.
Exact mathematical passage
\boxed{ E_n = \sum_{r=0}^{n/2} (-1)^rZ_{2r+1} \binom n{2r} q^{2r}S_{n-2r}(q). }
\sum_{r=0}^{p-1}Use in source · 1 passage
Search aliases: ∑_r=0^p-1 sum
Use in source · 72_recovered.tex, line 235
[Finite inverse formula] If x has augmentation one and n=x-1, then x^-1 = ∑_r=0^p-1(-n)^r.
Exact mathematical passage
\boxed{ x^{-1} = \sum_{r=0}^{p-1}(-n)^r. }
\sum_{r=0}^{x}Use in source · 1 passage
Search aliases: ∑_r=0^x sum
Use in source · 72_recovered.tex, line 1719
/3 in the one-prime example above loses the advertised digit. Invertibility of C alone does not restore it. The original proof is retained under these hypotheses. Let Q=∏_p∈ Sp^L_p for a finite set of primes. Let C∈ M_d( Z) be invertible modulo Q, and choose H such that C^H=I+QB with B∈ M_d( Z). Then for every nonnegative integer x, a_Hx+j = ∑_r=0^xQ^rbinom xr w^TC^jB^rv_0. For every r, Δ_H^ra_n≡0pmod Q^r.
Exact mathematical passage
\boxed{ a_{Hx+j} = \sum_{r=0}^{x}Q^r\binom xr w^TC^jB^rv_0. }
\sum_{r=1}^{e}Use in source · 1 passage
Search aliases: ∑_r=1^e sum
Use in source · Paper_X_Block27_Amended.tex, line 592
[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.
Exact mathematical passage
\boxed{ Z_e(T) = \frac{\sum_{r=1}^{e}q^{r-1}T^r} {1-q^{e-1}T^e}. }
\sum_{r=1}^{h}Use in source · 2 passages
Search aliases: ∑_r=1^h sum
Use in source · 72_recovered.tex, line 407
[Rational ordinary generating function] In F_p[[z]], ∑_n≥0e_n^(m,p)z^n = 1+ frac ∑_r=1^he_r^(m,p)z^r 1-z^h.
Exact mathematical passage
\boxed{ \sum_{n\geq0}e_n^{(m,p)}z^n = 1+ \frac{ \sum_{r=1}^{h}e_r^{(m,p)}z^r }{1-z^h}. }
\sum_{r=1}^{m}Use in source · 1 passage
Search aliases: ∑_r=1^m sum
Use in source · 03_nonlinear_coordinates.tex, line 287
[Packet Taylor formula] Assume F is m+1 times continuously Fr echet differentiable on an open set containing the segment x+th for 0≤ t≤1. Then Q_γ F(x+h) = · Q_γ F(x) + ∑_r=1^mfrac1r! ∑_α_1,…,α_r J^r_γmidα_1,…,α_rF(x) [P_α_1h,…,P_α_rh] · +Q_γ R_m+1(x,h), where the Bochner integral remainder is R_m+1(x,h) = frac1m! ∫_0^1(1-t)^m D^m+1F(x+th)[h,…,h] dt.
Exact mathematical passage
\begin{aligned} Q_\gamma F(x+h) ={}&Q_\gamma F(x) + \sum_{r=1}^{m}\frac1{r!} \sum_{\alpha_1,\ldots,\alpha_r} J^{r}_{\gamma\mid\alpha_1,\ldots,\alpha_r}F(x) [P_{\alpha_1}h,\ldots,P_{\alpha_r}h]\\ &+Q_\gamma R_{m+1}(x,h), \end{aligned}
\sum_{r=1}^{n}Use in source · 1 passage
Search aliases: ∑_r=1^n sum
Use in source · 72_recovered.tex, line 649
[A denominator bound] Let D_N,j(n) = ∏_r=1^n binomNr+jNr, D_N,j(0)=1. Then D_N,j(n)C_N,j(n) ∈ Z. For every prime p, v_p(C_N,j(n)) ≥ - ∑_r=1^n v_pbinomNr+jNr.
Exact mathematical passage
\vp(C_{N,j}(n)) \geq - \sum_{r=1}^{n} \vp\binom{Nr+j}{Nr}.
\sum_{r=1}^{n/2}Use in source · 1 passage
Search aliases: ∑_r=1^n/2 sum
Use in source · 72_recovered.tex, line 976
[Odd modulus Euler coordinate pair] For every positive odd q and every even n≥0, E_n = ∑_r=0^n/2 (-1)^rZ_2r+1 binom n2r q^2rS_n-2r(q). The inverse transform is S_n(q) = E_n + ∑_r=1^n/2 2^2r-1 binom n2r q^2rE_n-2r. Both transforms are lower triangular with diagonal one.
Exact mathematical passage
\boxed{ S_n(q) = E_n + \sum_{r=1}^{n/2} 2^{2r-1} \binom n{2r} q^{2r}E_{n-2r}. }
\sum_{r=1}^qUse in source · 3 passages
Search aliases: ∑_r=1^q sum
Use in source · Paper_VI_Traditional_Final.tex, line 5039
[Competing risk packets] Suppose the exit row splits into nonnegative cause rows a^T = ∑_r=1^q a_r^T. Define f_r(t)=a_r^T e^tAp_0. Then f(t)=∑_r=1^qf_r(t). The cause specific hazard is h_r(t)=fracf_r(t)S(t), and h(t)=∑_r=1^qh_r(t). If the spectrum of A lies in the open left half-plane, the eventual probability of cause r is ∫_0^∞ f_r(t) d t = a_r^T(-A)^-1p_0.
Exact mathematical passage
a^T = \sum_{r=1}^q a_r^T.
\sum_{r=2}^{d}Use in source · 3 passages
Search aliases: ∑_r=2^d sum
Use in source · 03_nonlinear_coordinates.tex, line 432
Let X be a Banach space. Fix s∈ X and continuous multilinear maps B_r:X^r⟶ X, 2≤ r≤ d. Set a= s , b_r= B_r , ϕ(t)=a+∑_r=2^db_rt^r. An ordered rooted tree has leaves labeled by s and every internal vertex of arity r labeled by B_r. Its value is obtained by evaluating from the leaves toward the root.
Exact mathematical passage
a=\|s\|, \qquad b_r=\|B_r\|, \qquad \phi(t)=a+\sum_{r=2}^{d}b_rt^r.
\sum_{r=k}^{\min(D,Mk+s)}Use in source · 1 passage
Search aliases: ∑_r=k^min(D,Mk+s) sum min
Use in source · 10_foundations.tex, line 509
For M≥1 and s,r,k∈ N, define the affine Mahler kernel by Θ_M,s(r,k)=[z^r](1+z)^s((1+z)^M-1)^k =∑_j=0^k(-1)^k-jbinom kjbinomMj+sr. It vanishes unless k≤ r≤ Mk+s. For a finite binomial expansion f(n)=∑_r=0^Dc_rbinom nr, the kth Newton coefficient of g(q)=f(Mq+s) is therefore c_k(g)=∑_r=k^min(D,Mk+s)c_rΘ_M,s(r,k). Indeed, insert the finite stencil for Δ_1^k g(0) and interchange its two finite sums. Coefficient extraction of the binomial expansion of ((1+z)^M-1)^k gives the kernel. In the binary zer…
Exact mathematical passage
c_k(g)=\sum_{r=k}^{\min(D,Mk+s)}c_r\Theta_{M,s}(r,k).
\sum_{r=k}^{Mk+s}Use in source · 1 passage
Search aliases: ∑_r=k^Mk+s sum
Use in source · 72_recovered.tex, line 2217
[Mahler polyphase transfer] Let f(x)=∑_r≥0c_rbinom xr be a convergent Mahler expansion. For g_s(q)=f(Mq+s), one has g_s(q)=∑_k≥0d_s,kbinom qk, where d_s,k = ∑_r=k^Mk+s c_rΘ_M,s(r,k). Every refined coefficient uses a finite window of the original Mahler coefficients.
Exact mathematical passage
\boxed{ d_{s,k} = \sum_{r=k}^{Mk+s} c_r\Theta_{M,s}(r,k). }
\sum_{r=s}^{N}Use in source · 2 passages
Search aliases: ∑_r=s^N sum
Use in source · 72_recovered.tex, line 1469
[Finite Mahler reconstruction] Given f(0),…,f(N), put c_r=Δ^rf(0). Then f(n)=∑_r=0^nc_rbinom nr (0≤ n≤ N). For every s with 0≤ s≤ N, Δ^sf(n) = ∑_r=s^Nc_rbinom nr-s (0≤ n≤ N-s).
Exact mathematical passage
\boxed{ \Delta^sf(n) = \sum_{r=s}^{N}c_r\binom n{r-s} \qquad(0\leq n\leq N-s). }
\sum_{r\bmod N}Use in source · 1 passage
Search aliases: ∑_rbmod N sum bmod
Use in source · 72_recovered.tex, line 2991
[Complete residue packet system] For fixed N, the projectors P_N,r satisfy P_N,rP_N,s = δ_rsP_N,r and ∑_rbmod NP_N,r=I.
Exact mathematical passage
\boxed{ \sum_{r\bmod N}P_{N,r}=I. }
\sum_{r\in B_\alpha}Use in source · 4 passages
Search aliases: ∑_r∈ B_α sum in alpha
Use in source · Paper_VI_Traditional_Final.tex, line 2898
[Coordinate form of the lumping obstruction] For the partition aggregation above, Ker C is spanned by differences e_i-e_j with i and j in one source block. Also, Ob_C(K)(e_i-e_j) = ( ∑_r∈ B_α(K_ri-K_rj) )_α=1^m.
Exact mathematical passage
\Ob_C(K)(e_i-e_j) = \left( \sum_{r\in B_\alpha}(K_{ri}-K_{rj}) \right)_{\alpha=1}^m.
\sum_{r\in R}Use in source · 3 passages
Search aliases: ∑_r∈ R sum in
Use in source · Paper_V_Traditional_Final.tex, line 1777
For finite p, equip ⊕_r∈ Rℓ^p(L) with (u_r) ^p=∑_r∈ R u_r _p^p. For p=∞, use the maximum norm.
Exact mathematical passage
\|(u_r)\|^p=\sum_{r\in R}\|u_r\|_p^p.
\sum_{r\in\mathbb Z/N\mathbb Z}Use in source · 1 passage
Search aliases: ∑_r∈ Z/N Z sum in mathbb mathbb
Use in source · 03_supplement_03.tex, line 105
Exact coefficient formula. The packet definition is a finite sum over residue classes: P_j=frac1N∑_r∈ Z/N Zχ(-rj)ρ(r). The source proves χ(-rj)=(ω^val(r)val(j))^-1, so this is precisely the manuscript's N-term Fourier projector formula, with canonical representatives.
Exact mathematical passage
P_j=\frac1N\sum_{r\in\mathbb Z/N\mathbb Z}\chi(-rj)\rho(r).
\sum_{r\ge0}Use in source · 6 passages
Search aliases: ∑_r≥0 sum ge
Use in source · 20_arithmetic.tex, line 80
[The Bernoulli boundary address] At N=2, the address j=1 gives a second classical family. Define the Bernoulli numbers by B(z)=fracze^z-1=∑_r≥0B_rfracz^rr!, B_0=1, B_1=-frac12. All quotients here are regular formal series. The identity 2B(z)-B(2z)=fraczsinh z =∑_n≥0C_2,1(n)fracz^2n(2n)! follows by putting the two fractions on a common denominator and using [source reference]. Comparing even coefficients yields C_2,1(n)=(2-2^2n)B_2n (n∈ N). This classical derivation restores the boundary example …
Exact mathematical passage
B(z)=\frac{z}{e^z-1}=\sum_{r\ge0}B_r\frac{z^r}{r!}, \qquad B_0=1,\quad B_1=-\frac12.
\sum_{r\geq0}Use in source · 19 passages
Search aliases: ∑_r≥0 sum geq
Use in source · 03_weighted.tex, line 72
Put A=∑_e∈ EA_e. Since left multiplication by D^-1 preserves the filtration, A∈ F^1 B. Completeness gives the convergent Neumann series (I+A)^-1=∑_r≥0(-A)^r. Because T=D(I+A), T^-1=(I+A)^-1D^-1. Expanding each power of A gives the sum over words, with the rightmost edge acting first. Multiplicativity of the filtration gives A_γ∈ F^W(γ) B. For each L, only finitely many words have weight below L because E is finite and every edge has positive weight. Thus the word expansion converges to the same …
Exact mathematical passage
(I+A)^{-1}=\sum_{r\geq0}(-A)^r.
\sum_{r\ge1}Use in source · 2 passages
Search aliases: ∑_r≥1 sum ge
Use in source · 03_nonlinear_coordinates.tex, line 647
[Packet equations for L_∞ operations] Let L be complete and carry operations ℓ_r that respect a finite abelian group grading. Assume the packet projectors are continuous for the topology in which the series below converges. Assume the Maurer Cartan series ∑_r≥1frac1r!ℓ_r(η,…,η) converges and the required factorials are invertible. Then its packet of label k is ∑_r≥1frac1r! ∑_g_1+⋯+g_r=k ℓ_r(η_g_1,…,η_g_r). The full Maurer Cartan equation holds if and only if every packet equation vanishes.
Exact mathematical passage
\sum_{r\ge1}\frac1{r!}\ell_r(\eta,\ldots,\eta)
\sum_{r\geq1}Use in source · 1 passage
Search aliases: ∑_r≥1 sum geq
Use in source · 72_recovered.tex, line 1000
The even source identity gives secht = sech^2(qt) ∑_n≥0S_2n(q)fract^2n(2n)!. Insert [source reference] and compare coefficients. For the inverse, multiply secht by cosh^2(qt) = 1+ ∑_r≥12^2r-1q^2rfract^2r(2r)!.
Exact mathematical passage
\cosh^2(qt) = 1+ \sum_{r\geq1}2^{2r-1}q^{2r}\frac{t^{2r}}{(2r)!}.
\sum_sUse in source · 3 passages
Search aliases: ∑_s sum
Use in source · Paper_V_Traditional_Final.tex, line 1100
For a finite nonlinear shift operator F(x)[n]=x[n]+α x[n]x[n-1], the exact packet coordinates are simply F_r((x_s)_s)=P_N,rF (∑_sP_N,sx). Analysis followed by this packet system and synthesis reproduces the direct nonlinear output exactly. The numerical residual in the companion implementation was zero to machine representation.
Exact mathematical passage
F_r((x_s)_s)=P_{N,r}F\!\left(\sum_sP_{N,s}x\right).
\sum_{s=0}^nUse in source · 1 passage
Search aliases: ∑_s=0^n sum
Use in source · strengthening.tex, line 376
For every starting row, Δ_1^k C_0(n) = L(U^4nW^k) =∑_s=0^nbinom ns L(W^k+s). Each summand lies in 2^k k! Z_2, proving the factorial bound for address zero from row zero onward.
Exact mathematical passage
\Delta_1^k C_0(n) =\mathcal L(U^{4n}W^k) =\sum_{s=0}^n\binom ns\mathcal L(W^{k+s}). \label{eq:base-row-finite-expansion}
\sum_{s=0}^{N-1}Use in source · 3 passages
Search aliases: ∑_s=0^N-1 sum
Use in source · Paper_V_Traditional_Final.tex, line 426
[Exact FIR polyphase transfer law] The output channels satisfy y_k[n] = ∑_s=0^N-1∑_m∈ Z h_ks[m]x_s[n-m]. For the numerical channel transform, take z∈ C∖{0} and assume ∑_n∈ Z|x_s[n]z^-n|<∞ (0≤ s<N). Define X_s(z)=∑_nx_s[n]z^-n. The output series then also converge absolutely, and Y(z)= H^(N)(z) X(z), H^(N)_ks(z)=∑_m h[Nm+k-s]z^-m.
Exact mathematical passage
\boxed{ y_k[n] = \sum_{s=0}^{N-1}\sum_{m\in\bbZ} h_{ks}[m]x_s[n-m]. }
\sum_{s\succ r}Use in source · 1 passage
Search aliases: ∑_ssucc r sum succ
Use in source · Paper_V_Traditional_Final.tex, line 1359
The signal realization makes the seed, address, packet, transfer, refinement, and synthesis operations concrete. The objects and laws can be read directly: seed · =δ_0, address · =S^n, packet · =P_N,rx, transfer · =P_N,kHP_N,s, refinement · =P_N,r=∑_ssucc rP_M,s, synthesis · =∑_rP_N,rx, nonlinearity · =P_N,rF(∑_sP_N,sx). Filter banks, wavelet packets, adaptive complete prunings, edge packets, and Smith coordinates inhabit the same operator grammar, but their claims differ. Polyphase and filter-b…
Exact mathematical passage
\boxed{ \begin{aligned} \text{seed}&=\delta_0,\\ \text{address}&=S^n,\\ \text{packet}&=P_{N,r}x,\\ \text{transfer}&=P_{N,k}HP_{N,s},\\ \text{refinement}&=P_{N,r}=\sum_{s\succ r}P_{M,s},\\ \text{synthesis}&=\sum_rP_{N,r}x,\\ \text{nonlinearity}&=P_{N,r}F\left(\sum_sP_{N,s}x\right). \end{aligned} }
\sum_{\substack{0\le a<M\\a\equiv r\pmod N}}Use in source · 1 passage
Search aliases: ∑_substack0≤ a<M a≡ rpmod N sum substack le a equiv pmod
Use in source · atlas-pages.tex, line 66
leaved to recover the coarse channel. Adding them entrywise instead produces sums of different original samples. For example, at the first descendant address the values f[0]=0 and f[2]=1/2 are two separate values. Their sum 1/2 cannot tell us both of them. Exact refinement is a disjoint partition, not an averaging rule or a new measurement. headThe general formula When N divides M, P_N,r=∑_substack0≤ a<M a≡ rpmod NP_M,a. The divisibility hypothesis ensures that every fine residue class lies enti…
Exact mathematical passage
P_{N,r}=\sum_{\substack{0\le a<M\\a\equiv r\pmod N}}P_{M,a}.
\sum_{\substack{0\leq a<p\\2a+1\neq p}}Use in source · 1 passage
Search aliases: ∑_substack0≤ a<p 2a+1≠ p sum substack leq neq
Use in source · 72_recovered.tex, line 1841
Fix an even address j∈{0,2,…,p-3}. For x∈ Z_p, define mathscr E_p,j^×(x) = ∑_r≥0 · (-1)^rZ_2r+1p^2r binom(p-1)x+j2r · × ∑_substack0≤ a<p 2a+1≠ p (-1)^a (2a+1)^j-2r ((2a+1)^p-1)^x.
Exact mathematical passage
\begin{aligned} \mathscr E_{p,j}^\times(x) = \sum_{r\geq0} &(-1)^rZ_{2r+1}p^{2r} \binom{(p-1)x+j}{2r}\\ &\times \sum_{\substack{0\leq a<p\\2a+1\neq p}} (-1)^a (2a+1)^{j-2r} \left((2a+1)^{p-1}\right)^x. \end{aligned}
\sum_{\substack{0\leq i,j<N\\ i+j\equiv k\bmod N}}Use in source · 3 passages
Search aliases: ∑_substack0≤ i,j<N i+j≡ kbmod N sum substack leq equiv bmod
Use in source · Paper_IV_Traditional_Final.tex, line 433
[Cyclic product law] If F and G are holomorphic on the same rotation-invariant domain, then B_N,k(FG) = ∑_substack0≤ i,j<N i+j≡ kbmod N ( B_N,iF)( B_N,jG).
Exact mathematical passage
\Bcal_{N,k}(FG) = \sum_{\substack{0\leq i,j<N\\ i+j\equiv k\bmod N}} (\Bcal_{N,i}F)(\Bcal_{N,j}G).
\sum_{\substack{0\le r<M\\r\bmod N=j}}Use in source · 1 passage
Search aliases: ∑_substack0≤ r<M rbmod N=j sum substack le r bmod
Use in source · supplement_01.tex, line 47
0, · otherwise. The source proves idempotence for every selector, orthogonality for distinct standard residue labels, and, when N>0, ∑_j=0^N-1P_N,ja=a. If Nmid M and M>0, it also proves P_N,ja=∑_substack0≤ r<M rbmod N=jP_M,ra. The formal power-series versions are proved coefficientwise for actual mcodePowerSeries objects. There is no finite truncation of the sequence domain in these theorems.
Exact mathematical passage
P_{N,j}a=\sum_{\substack{0\le r<M\\r\bmod N=j}}P_{M,r}a.
\sum_{\substack{0\leq r<M\\r\equiv j\pmod N}}Use in source · 1 passage
Search aliases: ∑_substack0≤ r<M r≡ jpmod N sum substack leq r equiv pmod
Use in source · 10_foundations.tex, line 50
[Residue reconstruction] The selectors satisfy P_N,j^2=P_N,j, P_N,iP_N,j=0 (i≠ j), ∑_j=0^N-1P_N,j=I. If Nmid M and M>0, then P_N,ja=∑_substack0≤ r<M r≡ jpmod NP_M,ra. The same identities hold coefficientwise for formal power series.
Exact mathematical passage
P_{N,j}a=\sum_{\substack{0\leq r<M\\r\equiv j\pmod N}}P_{M,r}a.
\sum_{\substack{0\le r\le n\\w(r)<L}}Use in source · 1 passage
Search aliases: ∑_substack0≤ r≤ n w(r)<L sum substack le le w
Use in source · supplement_01.tex, line 135
0, · w=∞. The infinity convention expresses a vanishing coefficient, not an infinite exponent operation. For any finite index set, every term with weight at least L is divisible by p^L. A finite-sum proof yields p^L bigm| ∑_r=0^nbinom nr c_rp^[w(r)]S_n-r -∑_substack0≤ r≤ n w(r)<L binom nr c_rp^[w(r)]S_n-r. The result is specialized to actual p-adic integers R= Z_p for prime p. Divisibility there is precisely the asserted membership in p^L Z_p.
Exact mathematical passage
\boxed{p^L\ \bigm|\ \sum_{r=0}^{n}\binom nr c_rp^{[w(r)]}S_{n-r} -\sum_{\substack{0\le r\le n\\w(r)<L}} \binom nr c_rp^{[w(r)]}S_{n-r}.}
\sum_{\substack{0\leq r\leq n\\w(r)<L}}Use in source · 2 passages
Search aliases: ∑_substack0≤ r≤ n w(r)<L sum substack leq leq w
Use in source · 10_foundations.tex, line 215
[Filtered finite reconstruction] For every n,L∈ N and sequences c,S in that ring, · ∑_r=0^nbinom nr c_rp^[w(r)]S_n-r · ≡ ∑_substack0≤ r≤ n w(r)<L binom nr c_rp^[w(r)]S_n-rpmodp^L. Here congruence means divisibility of the difference by p^L in the coefficient ring. In Z_p, it means membership in p^L Z_p.
Exact mathematical passage
\begin{split} &\sum_{r=0}^{n}\binom nr c_rp^{[w(r)]}S_{n-r}\\ &\hspace{8mm}\equiv \sum_{\substack{0\leq r\leq n\\w(r)<L}} \binom nr c_rp^{[w(r)]}S_{n-r}\pmod{p^L}. \end{split}
\sum_{\substack{0\leq s<M\\s\equiv j\pmod N}}Use in source · 1 passage
Search aliases: ∑_substack0≤ s<M s≡ jpmod N sum substack leq s equiv pmod
Use in source · 02_coordinates.tex, line 478
[Nested residue refinement] Let N,M be positive integers with N dividing M, and let 0≤ j<N. Then P_N,j = ∑_substack0≤ s<M s≡ jpmod NP_M,s. When R is nonzero, the packet system modulo M refines the packet system modulo N.
Exact mathematical passage
P_{N,j} = \sum_{\substack{0\leq s<M\\s\equiv j\pmod N}}P_{M,s}.
\sum_{\substack{0\le s<M\\s\equiv r\, (\mathrm{mod}\,N)}}Use in source · 1 passage
Search aliases: ∑_substack0≤ s<M s≡ r (mod N) sum substack le s equiv mathrm
Use in source · Paper_V_Traditional_Final.tex, line 393
[Exact nested refinement] For N,M≥1 and Nmid M, P_N,r = ∑_substack0≤ s<M s≡ r (mod N)P_M,s.
Exact mathematical passage
\boxed{ P_{N,r} = \sum_{\substack{0\le s<M\\s\equiv r\, (\mathrm{mod}\,N)}}P_{M,s}. }
\sum_{\substack{e:\,\mathrm{source}(e)=s\\w(e)\le n}}Use in source · 1 passage
Search aliases: ∑_substacke: source(e)=s w(e)≤ n sum substack mathrm w le
Use in source · visual_atlas.tex, line 289
headThe first-edge decomposition Let a_n(s,t) count walks of total weight n from s to t. Removing the first edge gives a_n(s,t)=1_s=t, n=0 +∑_substacke: source(e)=s w(e)≤ n a_n-w(e)(target(e),t). The empty walk is allowed precisely on the diagonal at weight zero. Positive weights ensure that every contributing walk has length at most n, so enumeration at each fixed weight is finite.
Exact mathematical passage
a_n(s,t)=\mathbf1_{s=t,\,n=0} +\sum_{\substack{e:\,\mathrm{source}(e)=s\\w(e)\le n}} a_{n-w(e)}(\mathrm{target}(e),t).
\sum_{\substack{e:\,\operatorname{source}(e)=s\\w(e)\le n}}Use in source · 1 passage
Search aliases: ∑_substacke: source(e)=s w(e)≤ n sum substack operatorname w le
Use in source · 03_weighted.tex, line 499
[Automatic finite-graph transfer-zeta rationality] Assume the vertex and edge sets are finite and every edge has positive integer weight. For every n, exact enumeration of edge tuples of lengths at most n, followed by endpoint and total-weight filtering, returns a_n(s,t). These coefficients obey the recurrence obtained by removing the first edge a_n(s,t) = 1_s=t, n=0 +∑_substacke: source(e)=s w(e)≤ n a_n-w(e)(target(e),t). Let T(t)=I-A(t), Q(t)=det T(t), P_s,t(t)=adj(T(t))_s,t. Then Q(0)=1, henc…
Exact mathematical passage
a_n(s,t) = \mathbf 1_{s=t,\,n=0} +\sum_{\substack{e:\,\operatorname{source}(e)=s\\w(e)\le n}} a_{n-w(e)}(\operatorname{target}(e),t).
\sum_{\substack{g<h\\g+h=k}}Use in source · 1 passage
Search aliases: ∑_substackg<h g+h=k sum substack g
Use in source · 03_nonlinear_coordinates.tex, line 679
[Divided square packet equations] Assume q( L_g)⊆ L_2g. Choose an ordering of the finite label set. The equation dη+q(η)=0 is equivalent to dη_k + ∑_2g=kq(η_g) + ∑_substackg<h g+h=k[η_g,η_h] =0 for every k.
Exact mathematical passage
d\eta_k + \sum_{2g=k}q(\eta_g) + \sum_{\substack{g<h\\g+h=k}}[\eta_g,\eta_h] =0
\sum_{\substack{I\ne0\\\Nm I\leq X}}Use in source · 1 passage
Search aliases: ∑_substackI≠0 Nm I≤ X ∑_substackI≠0 N I≤ X sum substack ne Nm leq sum substack ne operatorname leq
Use in source · residue_continuation_body.tex, line 145
e preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl( O)→ C, define the actual finite ideal sum S_w(X)=∑_substackI≠0 N I≤ Xw([I]). Exact grouping by ideal class gives S_w(X)=∑_C w(C)A_C(X). If ∑_Cw(C)=0, the common leading coefficient cancels, so there is B_w≥0 s…
Exact mathematical passage
S_w(X)=\sum_{\substack{I\ne0\\\Nm I\leq X}}w([I]).
\sum_{\substack{I\ne0,\ [I]=C\\\Nm I\leq X}}Use in source · 1 passage
Search aliases: ∑_substackI≠0, [I]=C Nm I≤ X ∑_substackI≠0, [I]=C N I≤ X sum substack ne Nm leq sum substack ne operatorname leq
Use in source · residue_continuation_body.tex, line 225
The assembled ideal norm-character estimate. Let C∈Cl( O), let J≠0 be an integral ideal with [J]=C^-1, and let m>0. Let χ: Z/m Z→ C preserve multiplication, 0, and 1. Assume a residue-ring unit satisfying [source reference], and assume χ( N Jbmod m)≠0. Then T_C,χ(X) =∑_substackI≠0, [I]=C N I≤ X χ( N Ibmod m) =O_K,J,m,χ(X^1-1/d). More precisely, some B≥0 bounds its absolute value by BX^1-1/d for every real X≥1.
Exact mathematical passage
T_{C,\chi}(X) =\sum_{\substack{I\ne0,\ [I]=C\\\Nm I\leq X}} \chi(\Nm I\bmod m) =O_{K,J,m,\chi}(X^{1-1/d}).
\sum_{\substack{\gamma,\gamma'\in\Gamma_b\\ \gamma<\gamma'}}Use in source · 1 passage
Search aliases: ∑_substackγ,γ'∈Γ_b γ<γ' sum substack gamma gamma in Gamma gamma gamma
Use in source · 02_strengthened.tex, line 457
[Path interference formula] Fix a source vector ψ∈ P_a_0 H and a target packet b. Let Γ_b be the set of packet paths from a_0 to b, and set η_γ=U_γψ. Choose any total order on this finite path set. Then normP_bUψ^2 = ∑_γ∈Γ_bnormη_γ^2 + 2Re ∑_substackγ,γ'∈Γ_b γ<γ' ipη_γη_γ'. Consequently, normP_bUψ ≤ ∑_γ∈Γ_bnormη_γ. If the path outputs are pairwise orthogonal, the target probability is the sum of the path probabilities.
Exact mathematical passage
\norm{P_bU\psi}^2 = \sum_{\gamma\in\Gamma_b}\norm{\eta_\gamma}^2 + 2\operatorname{Re} \sum_{\substack{\gamma,\gamma'\in\Gamma_b\\ \gamma<\gamma'}} \ip{\eta_\gamma}{\eta_{\gamma'}}.
\sum_{\substack{\gamma:\ P_\alpha R_\gamma=R_\gamma\\Q_\beta R_\gamma=R_\gamma}}Use in source · 1 passage
Search aliases: ∑_substackγ: P_α R_γ=R_γ Q_β R_γ=R_γ ∑_substackγ: P_α R_γ=R_γ{ Q_β R_γ=R_γ sum substack gamma alpha gamma gamma Q beta gamma gamma sum substack gamma alpha gamma gamma mathbb beta gamma gamma
Use in source · 02_coordinates.tex, line 339
a packet in R. Because R refines P, there is a unique α with P_α R_γ=R_γ. Likewise there is a unique β with Q_β R_γ=R_γ. Therefore P_α Q_β R_γ=R_γ. Every packet of R lies in one product packet. Multiplying ∑_γ R_γ=I_X by P_α Q_β and using the parent identities gives P_α Q_β =∑_substackγ: P_α R_γ=R_γ{ Q_β R_γ=R_γR_γ. Thus the packets with the same pair of parents reconstruct the product packet. If that product packet is nonzero, the sum has a child, so the parent map onto the supported product fa…
Exact mathematical passage
P_\alpha Q_\beta =\sum_{\substack{\gamma:\ P_\alpha R_\gamma=R_\gamma\\Q_\beta R_\gamma=R_\gamma}}R_\gamma.
\sum_{\substack{\gamma\in E^n\\W(\gamma)<L}}Use in source · 1 passage
Search aliases: ∑_substackγ∈ E^n W(γ)<L sum substack gamma in W gamma
Use in source · 03_weighted.tex, line 321
et q_L:R→ R_L be a ring homomorphism that kills F^L R. Suppose H=∑_e∈ EH_e, D^-1H_e∈ F^w(e)R, w(e)≥1, with E finite. Then q_L(D^-1H)^L=0, the finite sum in [source reference](a) is a two-sided inverse after applying q_L, and its word expansion may discard every word of weight at least L. More explicitly, for a length cutoff N, define S^pr_L,N = ∑_n=0^N-1(-1)^n ∑_substackγ∈ E^n W(γ)<LA_γ. Lean proves directly that q_L(S^pr_L,N) equals the mapped geometric inverse whenever q_L(A^N)=0, and hence is…
Exact mathematical passage
S^{\mathrm{pr}}_{L,N} = \sum_{n=0}^{N-1}(-1)^n \sum_{\substack{\gamma\in E^n\\W(\gamma)<L}}A_\gamma.
\sum_{\substack{\eta^N=-1\\
\eta\ne e^{ia_N},e^{-ia_N}}}Use in source · 1 passage
Search aliases: ∑_substackη^N=-1 η≠ e^ia_N,e^-ia_N sum substack eta eta ne
Use in source · Paper_IV_Traditional_Final.tex, line 1874
On the strip x≥ R and |y|≤ Y, define R_N(z) = ∑_substackη^N=-1 η≠ e^ia_N,e^-ia_N exp((η-e^-ia_N)z). Then N e^-e^-ia_Nz D_N(z) = 1+e^2is_Nz+R_N(z), and, for q=0,1,2, |R_N^(q)(z)| ≤ C e^-γ_Nx.
Exact mathematical passage
R_N(z) = \sum_{\substack{\eta^N=-1\\ \eta\ne e^{ia_N},e^{-ia_N}}} \exp\bigl((\eta-e^{-ia_N})z\bigr).
\sum_{\substack{\rho(\beta)=b\\\rho(\alpha)=a}}Use in source · 1 passage
Search aliases: ∑_substackρ(β)=b ρ(α)=a sum substack rho beta rho alpha
Use in source · 02_coordinates.tex, line 224
[Refinement of operator blocks] Suppose that Q refines P through ρ. Then T^ P_ba = ∑_substackρ(β)=b ρ(α)=a T^ Q_βα. The matrix of T in fine coordinates is related to the matrix in coarse coordinates by [T]_ P = C_ P← Q[T]_ QJ_ Q← P.
Exact mathematical passage
T^{\Pcal}_{ba} = \sum_{\substack{\rho(\beta)=b\\\rho(\alpha)=a}} T^{\Qcal}_{\beta\alpha}.
\sum_{\substack{\sigma(\beta)=j\\\sigma(\alpha)=i}}Use in source · 2 passages
Search aliases: ∑_substackσ(β)=j σ(α)=i sum substack sigma beta sigma alpha
Use in source · 04_transport.tex, line 100
[Naturality of generated hulls and transfer blocks] Let (Φ,φ,σ) be a packet compatible operator morphism. For every subset S⊆ X, Φ(Hull_ A(S)) = Hull_φ( A)(Φ(S)) ⊆ Hull_ B(Φ(S)). For T∈ A and i,j∈ J, Q_jφ(T)Q_iΦ = ∑_substackσ(β)=j σ(α)=i Φ P_β TP_α. If Φ is surjective and S generates X, then Φ(S) generates Y under φ( A) and hence under B. If, in addition, the morphism is seeded, then S' generates Y. [source reference] ExportsThe canonical naturality theorem for every later realization.
Exact mathematical passage
\boxed{ Q_j\varphi(T)Q_i\Phi = \sum_{\substack{\sigma(\beta)=j\\\sigma(\alpha)=i}} \Phi P_\beta TP_\alpha. }
\sum_{\substack{\tau\text{ full ordered binary tree}\\\text{with }n\text{ leaves}}}Use in source · 1 passage
Search aliases: ∑_substackτ full ordered binary tree with n leaves sum substack tau text text text
Use in source · 03_nonlinear_coordinates.tex, line 491
[Quadratic Catalan expansion] Let B:X× X→ X be bounded and consider u=s+B(u,u). Put a= s and b= B . If 4ab<1, then u= ∑_n≥1 ∑_substackτ full ordered binary tree with n leaves B_τ(s) converges absolutely. It also satisfies u ≤ frac1-sqrt1-4ab2b when b>0.
Exact mathematical passage
u= \sum_{n\ge1} \sum_{\substack{\tau\text{ full ordered binary tree}\\\text{with }n\text{ leaves}}} B_\tau(s)
\sum_{s\in R}Use in source · 3 passages
Search aliases: ∑_s∈ R sum in
Use in source · Paper_V_Traditional_Final.tex, line 1928
[Exact convolution transfer law] Let y=h*x. Then y_r(ℓ) = ∑_s∈ R∑_m∈ Lh_rs(m)x_s(ℓ-m). Equivalently, A_LH A_L^-1 = (H_rs)_r,s∈ R, where H_rs is convolution on L by h_rs.
Exact mathematical passage
y_r(\ell) = \sum_{s\in R}\sum_{m\in L}h_{rs}(m)x_s(\ell-m).
\sum_{s\in\operatorname{Child}(r)}Use in source · 1 passage
Search aliases: ∑_s∈Child(r) sum in operatorname
Use in source · Paper_V_Traditional_Final.tex, line 1875
[Exact lattice refinement] For every coarse packet, P_r^L = ∑_s∈Child(r)P_s^L'. The fine packets in the sum are pairwise disjoint. Refinement through several nested sublattices is transitive.
Exact mathematical passage
P_r^{L} = \sum_{s\in\operatorname{Child}(r)}P_s^{L'}.
\sum_{t\in k^{\times}}Use in source · 1 passage
Search aliases: ∑_t∈ k^× sum in times
Use in source · Paper_XI_Traditional_Final.tex, line 736
Compute absG(χ,ψ)^2 = ∑_x,y∈ k^× χ(xy^-1)ψ(x-y). Write x=ty. The inner sum over y equals q-1 when t=1 and equals -1 otherwise. Since ∑_t∈ k^×χ(t)=0, the result is q.
Exact mathematical passage
\sum_{t\in k^{\times}}\chi(t)=0
\sum_uUse in source · 2 passages
Search aliases: ∑_u sum
Use in source · 72_recovered.tex, line 210
[Augmentation criterion] Let x=∑_u∈ F_pc_u[u] ∈ F_p[ F_p^+]. Then x is a unit if and only if aug(x) = ∑_uc_u is nonzero. If J is the augmentation ideal, then J^p=0.
Exact mathematical passage
\operatorname{aug}(x) = \sum_uc_u
\sum_{u\text{ split}}Use in source · 1 passage
Search aliases: ∑_u split sum text
Use in source · 01_manuscript.tex, line 1514
The cost and error of a pruning L are cost( L) = ∑_v∈ Lκ_v + ∑_u splitσ_u and err( L)=∑_v∈ Lϵ_v. The model can include arithmetic, storage, launch, transfer, and precision costs inside κ_v. It remains exact only when the declared totals are additive.
Exact mathematical passage
\cost(\Lcal) = \sum_{v\in\Lcal}\kappa_v + \sum_{u\text{ split}}\sigma_u
\sum_{u+v=r}Use in source · 2 passages
Search aliases: ∑_u+v=r sum
Use in source · 10_foundations.tex, line 494
a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient and remainder give A(z)=∑_r=0^M-1z^rA_r(z^M), (a_r^[M])_s^[N]=a_r+Ms^[MN]. For ordinary Cauchy multiplication, the phase-r product has a term without a carry and a term with one carry: C_r(w)=∑_u+v=rA_u(w)B_v(w) +w∑_u+v=r+MA_u(w)B_v(w), 0≤ u,v<M. Adding the two phase addresses gives the formula. It describes ordinary Cauchy multiplication. Hurwitz convo…
Exact mathematical passage
C_r(w)=\sum_{u+v=r}A_u(w)B_v(w) +w\sum_{u+v=r+M}A_u(w)B_v(w),\qquad 0\leq u,v<M.
\sum_{u+v=r+M}Use in source · 2 passages
Search aliases: ∑_u+v=r+M sum
Use in source · 10_foundations.tex, line 494
a and M≥1, let a_r^[M](q)=a_Mq+r for 0≤ r<M. If A(z)=∑_n a_nz^n and A_r(w)=∑_q a_Mq+rw^q are ordinary generating functions, then unique quotient and remainder give A(z)=∑_r=0^M-1z^rA_r(z^M), (a_r^[M])_s^[N]=a_r+Ms^[MN]. For ordinary Cauchy multiplication, the phase-r product has a term without a carry and a term with one carry: C_r(w)=∑_u+v=rA_u(w)B_v(w) +w∑_u+v=r+MA_u(w)B_v(w), 0≤ u,v<M. Adding the two phase addresses gives the formula. It describes ordinary Cauchy multiplication. Hurwitz convo…
Exact mathematical passage
C_r(w)=\sum_{u+v=r}A_u(w)B_v(w) +w\sum_{u+v=r+M}A_u(w)B_v(w),\qquad 0\leq u,v<M.
\sum_{u=0}^{s}Use in source · 1 passage
Search aliases: ∑_u=0^s sum
Use in source · 72_recovered.tex, line 2295
insufficient. For p=2, M=2, s=0 and f(x)=x/2, one has c_1=1/2, c_2=0, while g(q)=q has d_1=1. Thus d_1not≡ c_2pmod2. The coefficient consequence is read with integral original coefficients. Let M=p^a. Then Θ_p^a,s(r,k) ≡ binomsr-p^ak pmod p, where the binomial coefficient is zero outside its natural range. Consequently, d_s,k ≡ ∑_u=0^s binom su c_p^ak+u pmod p. In particular, d_0,k≡ c_p^akpmod p.
Exact mathematical passage
\boxed{ d_{s,k} \equiv \sum_{u=0}^{s} \binom su c_{p^ak+u} \pmod p. }
\sum_{u\in U}Use in source · 1 passage
Search aliases: ∑_u∈ U sum in
Use in source · Paper_XI_Traditional_Final.tex, line 488
[Parabolic annihilation] Let P=LU be a proper parabolic subgroup of G_n, and put e_U=frac1abs U∑_u∈ Uu. If κ_θ is cuspidal, then e_Ue_θ=e_θ e_U=0.
Exact mathematical passage
e_U=\frac1{\abs U}\sum_{u\in U}u.
\sum_{u\in X}Use in source · 1 passage
Search aliases: ∑_u∈ X sum in
Use in source · Paper_XI_Traditional_Final.tex, line 852
If the translation component x is nonzero, the operator permutes the standard basis without a fixed point, so its trace is zero. If x=0 but ξ≠0, the trace is ψ(z)∑_u∈ Xψ(ξ(u))=0. When x=ξ=0, the operator is the scalar ψ(z).
Exact mathematical passage
\psi(z)\sum_{u\in X}\psi(\xi(u))=0.
\sum_{u\in\FF_p}Use in source · 3 passages
Search aliases: ∑_u∈FF_p ∑_u∈ F_p sum in FF sum in mathbb
Use in source · 72_recovered.tex, line 192
The product law follows from [source reference]. To prove injectivity, suppose ∑_u∈ F_pc_uϵ_u=0. Looking at the coordinates n=0,1,…,p-1 gives a Vandermonde system with matrix (u^n)_0≤ n≤ p-1, u∈ F_p. Its determinant is the product of all differences between distinct field elements, so it is nonzero. Hence every c_u vanishes.
Exact mathematical passage
\sum_{u\in\FF_p}c_u\epsilon_u=0.
\sum_{u\neq0}Use in source · 1 passage
Search aliases: ∑_u≠0 sum neq
Use in source · 72_recovered.tex, line 343
For positive n, the zero mode contributes nothing. Every nonzero mode satisfies u^p-1=1. Hence b_n+p-1^(m,p) = ∑_u≠0c_uu^n+p-1 = ∑_u≠0c_uu^n = b_n^(m,p).
Exact mathematical passage
b_{n+p-1}^{(m,p)} = \sum_{u\neq0}c_uu^{n+p-1} = \sum_{u\neq0}c_uu^n = b_n^{(m,p)}.
\sum_vUse in source · 1 passage
Search aliases: ∑_v sum
Use in source · 02_foundations.tex, line 90
, the e_v are pairwise orthogonal, and the same identities hold in Λ_ G. Given a realization, define the action of a path by composition of its arrow maps and extend linearly. Every element of I acts as zero, so the action factors through Λ_ G. The vertex idempotents give |X|=⊕_v e_v|X|=⊕_vX_v. Conversely, if M is unital, then M=1M=∑_ve_vM. Orthogonality makes the sum direct. For an arrow a:v→ w, the identities a=e_wae_v imply a(e_vM)⊆ e_wM. The relations in I act as zero because M is a module o…
Exact mathematical passage
M=1M=\sum_ve_vM.
\sum_{v\in G_m}Use in source · 1 passage
Search aliases: ∑_v∈ G_m sum in
Use in source · 72_recovered.tex, line 429
[Exponential form and minimal period] Let G_m=( F_p^×)^m. This cyclic group has order h. There are uniquely determined coefficients d_v∈ F_p, indexed by v∈ G_m, such that e_n^(m,p) = ∑_v∈ G_md_vv^n (n≥1). The minimal positive period is the least common multiple of the orders of those v for which d_v≠0. In particular, it divides h.
Exact mathematical passage
\boxed{ e_n^{(m,p)} = \sum_{v\in G_m}d_vv^n \qquad(n\geq1). }
\sum_{v\in Q_0}Use in source · 1 passage
Search aliases: ∑_v∈ Q_0 sum in
Use in source · 02_foundations.tex, line 85
Because Q_0 is finite, 1=∑_v∈ Q_0e_v in 𝕜 Q, the e_v are pairwise orthogonal, and the same identities hold in Λ_ G. Given a realization, define the action of a path by composition of its arrow maps and extend linearly. Every element of I acts as zero, so the action factors through Λ_ G. The vertex idempotents give |X|=⊕_v e_v|X|=⊕_vX_v. Conversely, if M is unital, then M=1M=∑_ve_vM. Orthogonality makes the sum direct. For an arrow a:v→ w, the identities a=e_wae_v imply a(e_vM)⊆ e_wM. The relatio…
Exact mathematical passage
1=\sum_{v\in Q_0}e_v
\sum_{v\in\Lcal}Use in source · 5 passages
Search aliases: ∑_v∈Lcal ∑_v∈ L sum in Lcal sum in mathcal
Use in source · Paper_V_Traditional_Final.tex, line 2349
[Reconstruction on a pruned packet tree] Every admissible pruning L defines a unitary analysis map A_ L: H_rm root ⟶ ⊕_v∈ L H_v. Its inverse is obtained by applying the adjoints of the local splits from the leaves toward the root. In particular, x ^2 = ∑_v∈ L x_v ^2.
Exact mathematical passage
\|x\|^2 = \sum_{v\in\Lcal}\|x_v\|^2.
\sum_{v\in\Lambda}Use in source · 3 passages
Search aliases: ∑_v∈Λ sum in Lambda
Use in source · Paper_V_Traditional_Final.tex, line 910
[Exact synthesis from every complete pruning] For every complete pruning Λ, ∑_v∈ΛP_v=I, x=∑_v∈ΛP_vx. Thus changing the selected tree changes the coordinates but does not introduce reconstruction error.
Exact mathematical passage
\boxed{\sum_{v\in\Lambda}P_v=I,} \qquad \boxed{x=\sum_{v\in\Lambda}P_vx.}
\sum_wUse in source · 2 passages
Search aliases: ∑_w sum
Use in source · Paper_V_Traditional_Final.tex, line 2407
Assume the child subtree bounds have been proved. Then ∑_w∑_ℓ∈ L_w x_ℓ ^2 ≥ min_w a(w) ∑_w (U_vx)_w ^2 ≥ A_vmin_w a(w) x ^2. The upper bound follows in the same way with the maximum and B_v. Induction completes the proof.
Exact mathematical passage
\sum_{w}\sum_{\ell\in\Lcal_w}\|x_\ell\|^2 \geq \min_w a(w) \sum_w\|(U_vx)_w\|^2 \geq A_v\min_w a(w)\|x\|^2.
\sum_{w}Use in source · 1 passage
Search aliases: ∑_w sum
Use in source · Paper_V_Traditional_Final.tex, line 2407
Assume the child subtree bounds have been proved. Then ∑_w∑_ℓ∈ L_w x_ℓ ^2 ≥ min_w a(w) ∑_w (U_vx)_w ^2 ≥ A_vmin_w a(w) x ^2. The upper bound follows in the same way with the maximum and B_v. Induction completes the proof.
Exact mathematical passage
\sum_{w}\sum_{\ell\in\Lcal_w}\|x_\ell\|^2 \geq \min_w a(w) \sum_w\|(U_vx)_w\|^2 \geq A_v\min_w a(w)\|x\|^2.
\sum_{W(\gamma)<L}Use in source · 2 passages
Search aliases: ∑_W(γ)<L sum gamma
Use in source · 03_weighted.tex, line 187
[Exact finite precision truncation] Under the hypotheses of [source reference], define R_L = ∑_W(γ)<L(-1)^|γ|A_γ D^-1. Then T^-1-R_L∈ F^L B, TR_L-I∈ F^L B, R_LT-I∈ F^L B. Thus R_L is the exact inverse in B/F^L B.
Exact mathematical passage
R_L = \sum_{W(\gamma)<L}(-1)^{|\gamma|}A_\gamma D^{-1}.
\sum_{w\in E^n}Use in source · 1 passage
Search aliases: ∑_w∈ E^n sum in
Use in source · 01_finite_coordinates.tex, line 288
[Finite operator expansion from words to walks] For every n∈ N, T^n = ∑_w∈ E^nL_w = ∑_w∈ E^n L_w, · w composable, 0, · w not composable. Parallel edges occur as distinct summands. statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp_pow_eq_explicitWordExpansion; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp_pow_eq_walkExpansion
Exact mathematical passage
\boxed{ T^n = \sum_{w\in E^n}L_w = \sum_{w\in E^n} \begin{cases} L_w,&w\text{ composable},\\ 0,&w\text{ not composable}. \end{cases} }
\sum_{w\in\mathrm{child}(v)}Use in source · 1 passage
Search aliases: ∑_w∈child(v) sum in mathrm
Use in source · 02_supplement_02.tex, line 20
Recurrence. For terminal nodes, F_λ(v)=κ_v+λϵ_v. For internal nodes, F_λ(v)=min{κ_v+λϵ_v, σ_v+∑_w∈child(v)F_λ(w)}. The checked result has two parts. First, F_λ(v)≤ J_λ(L) for every legal cut of the subtree. Second, the constructed function codecompile returns a legal cut with objective exactly F_λ(v). Thus the minimum is attained, not merely bounded below.
Exact mathematical passage
\boxed{F_\lambda(v)=\min\left\{\kappa_v+\lambda\epsilon_v, \;\sigma_v+\sum_{w\in\mathrm{child}(v)}F_\lambda(w)\right\}.}
\sum_{w\in\operatorname{child}(v)}Use in source · 4 passages
Search aliases: ∑_w∈child(v) sum in operatorname
Use in source · Paper_V_Traditional_Final.tex, line 1662
The packet tree results treat every admissible pruning, not only complete trees of a fixed depth. Unitary local splits give a unitary global transform. Local frame bounds propagate recursively in the nonunitary case. If the cost of a pruning is additive over retained nodes, then F(v) = min{ c(v), ∑_w∈child(v)F(w) } gives the exact optimal tree. The cost and the selected basis are stable when the node costs change by less than the gap separating the optimal pruning from its competitors.
Exact mathematical passage
F(v) = \min\left\{ c(v), \sum_{w\in\operatorname{child}(v)}F(w) \right\}
\sum_{w\in\Walk_n(i,j)}Use in source · 1 passage
Search aliases: ∑_w∈Walk_n(i,j) sum in Walk sum in operatorname
Use in source · 01_finite_coordinates.tex, line 323
[Endpoint coefficient walk sum] For all n∈ N and i,j∈ V, (T^n)_ji = ∑_w∈Walk_n(i,j)c^←(w). statusleanthreeKomorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.Connects; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.endpointWalkCoefficientSum; Komorebi.PaperIII.EdgeOperators.PacketGraph.EdgeSystem.transferOp_pow_apply_eq_sum_walks
Exact mathematical passage
\boxed{ (T^n)_{ji} = \sum_{w\in\Walk_n(i,j)}c^\leftarrow(w). }
\sum_xUse in source · 2 passages
Search aliases: ∑_x sum
Use in source · Paper_XI_Traditional_Final.tex, line 1562
Using the quadratic identity, q_v(x)=q(x+v)+q(v). Therefore ∑_x(-1)^q_v(x) = (-1)^q(v) ∑_x(-1)^q(x). Compare the signs in [source reference].
Exact mathematical passage
\sum_x(-1)^{q_v(x)} = (-1)^{q(v)} \sum_x(-1)^{q(x)}.
\sum_{x,y\in k^{\times}}Use in source · 1 passage
Search aliases: ∑_x,y∈ k^× sum in times
Use in source · Paper_XI_Traditional_Final.tex, line 730
Compute absG(χ,ψ)^2 = ∑_x,y∈ k^× χ(xy^-1)ψ(x-y). Write x=ty. The inner sum over y equals q-1 when t=1 and equals -1 otherwise. Since ∑_t∈ k^×χ(t)=0, the result is q.
Exact mathematical passage
\abs{G(\chi,\psi)}^2 = \sum_{x,y\in k^{\times}} \chi(xy^{-1})\psi(x-y).
\sum_{x\in k^{\times}}Use in source · 1 passage
Search aliases: ∑_x∈ k^× sum in times
Use in source · Paper_XI_Traditional_Final.tex, line 712
Let k= F_q, let ψ:k→ C^× be a nontrivial additive character, and let χ:k^×→ C^× be a multiplicative character. Define G(χ,ψ) = ∑_x∈ k^×χ(x)ψ(x). The standard theory of Gauss and Jacobi sums will be used in the form recorded in [source reference].
Exact mathematical passage
G(\chi,\psi) = \sum_{x\in k^{\times}}\chi(x)\psi(x).
\sum_{x\in V}Use in source · 1 passage
Search aliases: ∑_x∈ V sum in
Use in source · Paper_XI_Traditional_Final.tex, line 1505
[Dyadic Gauss sign] Let dim V=2r, and let q be nondegenerate. Then ∑_x∈ V(-1)^q(x) = (-1)^Arf(q)2^r.
Exact mathematical passage
\boxed{ \sum_{x\in V}(-1)^{q(x)} = (-1)^{\Arf(q)}2^r. }
\sum_{x\in X}Use in source · 1 passage
Search aliases: ∑_x∈ X sum in
Use in source · Paper_XI_Traditional_Final.tex, line 882
The lifted isotropic subgroup is abelian, so group averaging is an idempotent. Every operator in the average is unitary, and the subgroup is closed under inversion, so the average is self adjoint. For L=X, the average is the translation average (P_Lf)(u) = frac1abs X ∑_x∈ Xf(u+x), which is the constant function with value equal to the mean of f. Its range is one dimensional.
Exact mathematical passage
(P_Lf)(u) = \frac1{\abs X} \sum_{x\in X}f(u+x),
\sum_zUse in source · 1 passage
Search aliases: ∑_z sum
Use in source · visual_atlas.tex, line 59
On a vector (x_0,x_1,x_2)^ T, the first step produces (0,2x_0,0)^ T, and the second produces (0,0,6x_0)^ T. Thus L_fL_e=6E_20, L_eL_f=0. For general edges, a summand in the entrywise product can survive only when the same intermediate packet is both tgt(e) and src(f): (L_fL_e)_yx=∑_z(L_f)_yz(L_e)_zx. This is why a broken interface annihilates the entire product, even if both edge coefficients are nonzero.
Exact mathematical passage
(L_fL_e)_{yx}=\sum_z(L_f)_{yz}(L_e)_{zx}.
\sum_{z\in V}Use in source · 1 passage
Search aliases: ∑_z∈ V sum in
Use in source · 01_finite_coordinates.tex, line 114
For a matrix entry (y,x), (L_fL_e)_yx=∑_z∈ V(L_f)_yz(L_e)_zx. A potentially nonzero summand would require simultaneously z=src(f) and z=tgt(e), contrary to the hypothesis.
Exact mathematical passage
(L_fL_e)_{yx}=\sum_{z\in V}(L_f)_{yz}(L_e)_{zx}.
\sum_{|\alpha|=2}Use in source · 1 passage
Search aliases: ∑_|α|=2 sum alpha
Use in source · atlas-pages.tex, line 171
/4, 0, 1/4, which again sum to 57. Two factors of 1/sqrt2 explain why the depth-two coefficients are rational halves and quarters. headCoefficients and projectors are different objects W_α f is the short coefficient vector. The original-space component is W_α^*W_α f. Summing these synthesized components, not the short arrays entrywise, reconstructs f: f=∑_|α|=2W_α^*W_α f. The finite verification constructs the actual analysis matrices and checks their product with their adjoints. The chapter's ℓ…
Exact mathematical passage
f=\sum_{|\alpha|=2}W_\alpha^*W_\alpha f.
\sum_{\alpha}Use in source · 1 passage
Search aliases: ∑_α sum alpha
Use in source · 03_packets_and_graphs.tex, line 757
[Block diagonal projection and packet commutant] Define Δ_ P(T):=∑_αP_α TP_α. Then Δ_ P is a linear idempotent on End(X) whose range is the block diagonal subalgebra. For T∈End(X), the following are equivalent: [label=(roman*)] item TP_α=P_α T for every α. item T_βα=0 whenever β≠α. item Δ_ P(T)=T. [source reference] ExportsSymmetry sector invariance and packet leakage decompositions.
Exact mathematical passage
\Delta_{\Pcal}(T):=\sum_{\alpha}P_\alpha TP_\alpha.
\sum_{\alpha,\beta}Use in source · 6 passages
Search aliases: ∑_α,β sum alpha beta
Use in source · 03_packets_and_graphs.tex, line 514
Insert the finite packet identity on both sides: T= (∑_β P_β) T (∑_α P_α) = ∑_α,βP_β TP_α.
Exact mathematical passage
T= \left(\sum_\beta P_\beta\right) T \left(\sum_\alpha P_\alpha\right) = \sum_{\alpha,\beta}P_\beta TP_\alpha.
\sum_{\alpha,\beta\in I}Use in source · 1 passage
Search aliases: ∑_α,β∈ I sum alpha beta in
Use in source · 03_packets_and_graphs.tex, line 505
[Exact operator reconstruction] Every T∈End_𝕜(X) satisfies T=∑_α,β∈ IT_βα. [source reference]
Exact mathematical passage
\boxed{T=\sum_{\alpha,\beta\in I}T_{\beta\alpha}.}
\sum_{\alpha,\beta\in I_0}Use in source · 1 passage
Search aliases: ∑_α,β∈ I_0 sum alpha beta in
Use in source · Paper_VI_Traditional_Final.tex, line 5192
X). The general Laplace formula is [source reference]. [Packet convolution] For all s,t≥0 and all packets α,β, T_βα(s+t) = ∑_γ∈ I_0 T_βγ(s)T_γα(t). The original semigroup is reconstructed from its blocks by T(t)=∑_α,β∈ I_0T_βα(t).
Exact mathematical passage
T(t)=\sum_{\alpha,\beta\in I_0}T_{\beta\alpha}(t).
\sum_{\alpha,\gamma}Use in source · 1 passage
Search aliases: ∑_α,γ sum alpha gamma
Use in source · 03_packets_and_graphs.tex, line 223
For every pair of packet indices, (P_α⊗ Q_γ)^2 =P_α^2⊗ Q_γ^2 =P_α⊗ Q_γ. If (α,γ)≠(β,δ), then either P_α P_β=0 or Q_γ Q_δ=0, so the corresponding tensor idempotents annihilate each other. Finally, ∑_α,γP_α⊗ Q_γ = (∑_α P_α) ⊗ (∑_γ Q_γ) =I_X⊗ I_Y. The range is generated by pure tensors P_α x⊗ Q_γ y, hence equals X_α⊗ Y_γ. Exact reconstruction follows from [source reference].
Exact mathematical passage
\sum_{\alpha,\gamma}P_\alpha\otimes Q_\gamma = \left(\sum_\alpha P_\alpha\right) \otimes \left(\sum_\gamma Q_\gamma\right) =I_X\otimes I_Y.
\sum_{\alpha:n_\alpha>0}Use in source · 1 passage
Search aliases: ∑_α:n_α>0 sum alpha alpha
Use in source · Paper_VI_Traditional_Final.tex, line 3655
on decomposition] Let p and q be probability vectors on a finite partition (I_α)_α, and assume q_i=0 whenever p_i=0. Let m=Cp, n=Cq, and let p_α and q_α be the conditional distributions inside packet I_α whenever the corresponding mass is positive. Then D(qVert p) = D(nVert m) + ∑_α:n_α>0 n_α D(q_αVert p_α). The first term is the information retained by the coarse masses. The second term is the exact information carried by changes inside the packets.
Exact mathematical passage
D(q\Vert p) = D(n\Vert m) + \sum_{\alpha:n_\alpha>0} n_\alpha D(q_\alpha\Vert p_\alpha).
\sum_{\alpha:Z_\alpha>0}Use in source · 1 passage
Search aliases: ∑_α:Z_α>0 sum alpha alpha
Use in source · Paper_VI_Traditional_Final.tex, line 3385
[Bayesian packet reconstruction] The posterior is the disjoint packet synthesis B_λ(p) = ∑_α:Z_α>0 m_α q_α. The local shape q_α depends only on the prior and likelihood inside packet I_α. Coupling between different packets occurs through the normalization of the packet evidences.
Exact mathematical passage
\Bcal_\lambda(p) = \sum_{\alpha:Z_\alpha>0} m_\alpha q_\alpha.
\sum_{\alpha_1,\ldots,\alpha_{m-1}}Use in source · 2 passages
Search aliases: ∑_α_1,…,α_m-1 sum alpha ldots alpha
Use in source · Paper_VI_Traditional_Final.tex, line 432
[Path expansion] Expanding the exponential and inserting I=∑_α P_α between every pair of generators gives P_α_mA^mP_α_0 = ∑_α_1,…,α_m-1 P_α_mAP_α_m-1⋯ P_α_1AP_α_0. Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.
Exact mathematical passage
P_{\alpha_m}A^mP_{\alpha_0} = \sum_{\alpha_1,\ldots,\alpha_{m-1}} P_{\alpha_m}AP_{\alpha_{m-1}}\cdots P_{\alpha_1}AP_{\alpha_0}.
\sum_{\alpha_1,\ldots,\alpha_r}Use in source · 2 passages
Search aliases: ∑_α_1,…,α_r sum alpha ldots alpha
Use in source · 03_nonlinear_coordinates.tex, line 236
[Exact reconstruction of derivatives] For h_1,…,h_r∈ X, D^rF(x)[h_1,…,h_r] = ∑_γ ∑_α_1,…,α_r J^r_γmidα_1,…,α_rF(x) [P_α_1h_1,…,P_α_rh_r].
Exact mathematical passage
D^rF(x)[h_1,\ldots,h_r] = \sum_{\gamma} \sum_{\alpha_1,\ldots,\alpha_r} J^{r}_{\gamma\mid\alpha_1,\ldots,\alpha_r}F(x) [P_{\alpha_1}h_1,\ldots,P_{\alpha_r}h_r].
\sum_{\alpha\to\beta}Use in source · 1 passage
Search aliases: ∑_α→β sum alpha to beta
Use in source · 01_manuscript.tex, line 1026
The equation in packet β is D_β x_β = b_β - ∑_α→βK_βαx_α. Every predecessor of β lies in an earlier level. Once those values are available, x_β is obtained from the local solve with D_β. Vertices in the same level have no directed edge between them, so their solves are independent.
Exact mathematical passage
D_\beta x_\beta = b_\beta - \sum_{\alpha\to\beta}K_{\beta\alpha}x_\alpha.
\sum_{\alpha\in\Supp_{\Pcal}(x)}Use in source · 1 passage
Search aliases: ∑_α∈Supp_Pcal(x) ∑_α∈Supp_ P(x) sum alpha in Supp Pcal sum alpha in operatorname mathcal
Use in source · 03_packets_and_graphs.tex, line 727
For every target packet β, P_β Tx = ∑_α∈Supp_ P(x)T_βαP_α x. If every block in this sum is zero, then the coordinate with index β vanishes. This proves the inclusion. Equality is exactly the assertion that the displayed active sum is nonzero for every target in the set on the right. The final sufficient condition leaves at most one nonzero summand at each target, and kernel avoidance makes that summand nonzero.
Exact mathematical passage
P_\beta Tx = \sum_{\alpha\in\Supp_{\Pcal}(x)}T_{\beta\alpha}P_\alpha x.
\sum_{\alpha\in\{0,1\}^{\ell}}Use in source · 2 passages
Search aliases: ∑_α∈{0,1}^ℓ sum alpha in ell
Use in source · Paper_V_Traditional_Final.tex, line 605
[Wavelet-packet reconstruction and energy] At every finite depth ℓ, x=∑_α∈{0,1}^ℓW_ℓ,α^*W_ℓ,αx, and x _2^2 = ∑_α∈{0,1}^ℓ W_ℓ,αx _2^2.
Exact mathematical passage
\boxed{ x=\sum_{\alpha\in\{0,1\}^{\ell}}W_{\ell,\alpha}^{*}W_{\ell,\alpha}x, }
\sum_{\alpha\neq\beta}Use in source · 2 passages
Search aliases: ∑_α≠β sum alpha neq beta
Use in source · 01_manuscript.tex, line 1475
[Exact block discovery certificate] For an orthogonal packet system (P_α), define the block diagonal compression Π_ P(A) = ∑_α P_α AP_α. Then normA-Π_ P(A)_rm F^2 = ∑_α≠β normP_β AP_α_rm F^2. In operator norm, normA-Π_ P(A)_2 ≤ ∑_α≠β normP_β AP_α_2. Thus any proposed packet discovery has an exact finite residual certificate after its projectors are known.
Exact mathematical passage
\boxed{ \norm{A-\Pi_{\Pcal}(A)}_{\rm F}^2 = \sum_{\alpha\neq\beta} \norm{P_\beta AP_\alpha}_{\rm F}^2. }
\sum_\betaUse in source · 17 passages
Search aliases: ∑_β sum beta
Use in source · 03_packets_and_graphs.tex, line 514
Insert the finite packet identity on both sides: T= (∑_β P_β) T (∑_α P_α) = ∑_α,βP_β TP_α.
Exact mathematical passage
T= \left(\sum_\beta P_\beta\right) T \left(\sum_\alpha P_\alpha\right) = \sum_{\alpha,\beta}P_\beta TP_\alpha.
\sum_{\beta_1,\ldots,\beta_{m-1}}Use in source · 1 passage
Search aliases: ∑_β_1,…,β_m-1 sum beta ldots beta
Use in source · 03_packets_and_graphs.tex, line 562
[Packet powers and paths] For m≥1, (T^m)_β_mβ_0 = ∑_β_1,…,β_m-1 T_β_mβ_m-1 ⋯ T_β_1β_0. The same formula holds for a product of nonidentical operators with the corresponding blocks in temporal order. [source reference]
Exact mathematical passage
(T^m)_{\beta_m\beta_0} = \sum_{\beta_1,\ldots,\beta_{m-1}} T_{\beta_m\beta_{m-1}} \cdots T_{\beta_1\beta_0}.
\sum_{\beta\succ b,\,\alpha\succ a}Use in source · 1 passage
Search aliases: ∑_βsucc b, αsucc a sum beta succ alpha succ
Use in source · visual_atlas.tex, line 125
headRefined blocks still reconstruct the same map A coarse block expands as P_bTP_a=∑_βsucc b, αsucc aP_β TP_α. The terms are maps embedded in the original source and target spaces. Adding their norms, or adding their matrix entries after forgetting the embeddings, is not this operator identity.
Exact mathematical passage
P_bTP_a=\sum_{\beta\succ b,\,\alpha\succ a}P_\beta TP_\alpha.
\sum_{\beta\succ\alpha}Use in source · 1 passage
Search aliases: ∑_βsuccα sum beta succ alpha
Use in source · 01_finite_core.tex, line 1026
A coherent refinement should satisfy C_ℓ E_ℓ=I and P_ℓ,α = C_ℓ ( ∑_βsuccαP_ℓ+1,β ) E_ℓ. For an operator D, define the refinement defect Δ_ℓ(D) =C_ℓ D_ℓ+1E_ℓ-D_ℓ. Exact resolution independence is Δ_ℓ(D)=0. When exact equality is too strong, the correct continuum problem is to prove convergence of the resolvents and uniform control of the effective self-energies.
Exact mathematical passage
\boxed{ P_{\ell,\alpha} = C_\ell \left( \sum_{\beta\succ\alpha}P_{\ell+1,\beta} \right) E_\ell. }
\sum_\gammaUse in source · 10 passages
Search aliases: ∑_γ sum gamma
Use in source · 03_packets_and_graphs.tex, line 223
For every pair of packet indices, (P_α⊗ Q_γ)^2 =P_α^2⊗ Q_γ^2 =P_α⊗ Q_γ. If (α,γ)≠(β,δ), then either P_α P_β=0 or Q_γ Q_δ=0, so the corresponding tensor idempotents annihilate each other. Finally, ∑_α,γP_α⊗ Q_γ = (∑_α P_α) ⊗ (∑_γ Q_γ) =I_X⊗ I_Y. The range is generated by pure tensors P_α x⊗ Q_γ y, hence equals X_α⊗ Y_γ. Exact reconstruction follows from [source reference].
Exact mathematical passage
\sum_{\alpha,\gamma}P_\alpha\otimes Q_\gamma = \left(\sum_\alpha P_\alpha\right) \otimes \left(\sum_\gamma Q_\gamma\right) =I_X\otimes I_Y.
\sum_{\gamma}Use in source · 1 passage
Search aliases: ∑_γ sum gamma
Use in source · 03_nonlinear_coordinates.tex, line 236
[Exact reconstruction of derivatives] For h_1,…,h_r∈ X, D^rF(x)[h_1,…,h_r] = ∑_γ ∑_α_1,…,α_r J^r_γmidα_1,…,α_rF(x) [P_α_1h_1,…,P_α_rh_r].
Exact mathematical passage
D^rF(x)[h_1,\ldots,h_r] = \sum_{\gamma} \sum_{\alpha_1,\ldots,\alpha_r} J^{r}_{\gamma\mid\alpha_1,\ldots,\alpha_r}F(x) [P_{\alpha_1}h_1,\ldots,P_{\alpha_r}h_r].
\sum_{\gamma:\alpha\leadsto\beta}Use in source · 1 passage
Search aliases: ∑_γ:αleadstoβ sum gamma alpha leadsto beta
Use in source · 03_weighted.tex, line 111
pha DP_α, where each packet inverse is extended by zero to the other packets. Let the edge set be finite, assume w(e)≥1 for every edge, and write N=∑_e:α→βN_e, N_e=P_β N_eP_α, N_e∈ F^w(e) B. For an edge e:α→β, put A_e=(P_β DP_β)^-1N_e. Then the block from P_α X to P_β X of (D+N)^-1 is P_β(D+N)^-1P_α = ∑_γ:αleadstoβ (-1)^|γ|A_γ D_α^-1, where D_α=P_α DP_α and the sum runs over directed paths from α to β, including the empty path when α=β.
Exact mathematical passage
P_\beta(D+N)^{-1}P_\alpha = \sum_{\gamma:\alpha\leadsto\beta} (-1)^{|\gamma|}A_\gamma D_\alpha^{-1},
\sum_{\gamma_1,\gamma_2}Use in source · 1 passage
Search aliases: ∑_γ_1,γ_2 sum gamma gamma
Use in source · 03_nonlinear_coordinates.tex, line 401
[The second-order composition law] At order two, the partition formula has two shapes. For packet directions h_1 and h_2, R_δ D^2(F∘ G)(x)[h_1,h_2] = · ∑_γ_1,γ_2 J^2_δmidγ_1,γ_2F(Gx) [J^1_γ_1midα_1G(x)h_1,J^1_γ_2midα_2G(x)h_2] · + ∑_γ J^1_δmidγF(Gx) [J^2_γmidα_1,α_2G(x)[h_1,h_2]]. The first line records two first-order changes entering a quadratic interaction of F. The second records a genuine second-order interaction created inside G before F is applied.
Exact mathematical passage
\begin{aligned} R_\delta D^2(F\circ G)(x)[h_1,h_2] ={}& \sum_{\gamma_1,\gamma_2} J^2_{\delta\mid\gamma_1,\gamma_2}F(Gx) [J^1_{\gamma_1\mid\alpha_1}G(x)h_1,J^1_{\gamma_2\mid\alpha_2}G(x)h_2]\\ &+ \sum_\gamma J^1_{\delta\mid\gamma}F(Gx) [J^2_{\gamma\mid\alpha_1,\alpha_2}G(x)[h_1,h_2]]. \end{aligned}
\sum_{\gamma_1,\ldots,\gamma_k}Use in source · 1 passage
Search aliases: ∑_γ_1,…,γ_k sum gamma ldots gamma
Use in source · 03_nonlinear_coordinates.tex, line 353
di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n] · = ∑_π={B_1,…,B_k} ∑_γ_1,…,γ_k J^k_δmidγ_1,…,γ_kF(Gx) [ G_γ_1,B_1(xmid h),…, G_γ_k,B_k(xmid h) ]. The first sum runs over all set partitions of {1,…,n}.
Exact mathematical passage
\begin{aligned} &J^{n}_{\delta\mid\alpha_1,\ldots,\alpha_n}(F\circ G)(x) [h_1,\ldots,h_n]\\ &= \sum_{\pi=\{B_1,\ldots,B_k\}} \sum_{\gamma_1,\ldots,\gamma_k} J^{k}_{\delta\mid\gamma_1,\ldots,\gamma_k}F(Gx) \big[ G_{\gamma_1,B_1}(x\mid h),\ldots, G_{\gamma_k,B_k}(x\mid h) \big]. \end{aligned}
\sum_{\gamma\in I}Use in source · 4 passages
Search aliases: ∑_γ∈ I sum gamma in
Use in source · 03_packets_and_graphs.tex, line 953
It is enough to treat a word in the generators and packet projectors. Insert I=∑_γ∈ IP_γ between consecutive nonprojector generators. Every resulting (β,α) block is a finite sum of products P_β G_mP_γ_m-1 G_m-1⋯ P_γ_1G_1P_α. A nonzero product requires every consecutive block to be nonzero, hence gives a directed path α→γ_1→⋯→γ_m-1→β. Packet projectors only insert stationary steps. Thus a word has no block from α to an unreachable β. Linear combinations preserve the same zero pattern.
Exact mathematical passage
I=\sum_{\gamma\in I}P_\gamma
\sum_{\gamma\in I_0}Use in source · 1 passage
Search aliases: ∑_γ∈ I_0 sum gamma in
Use in source · Paper_VI_Traditional_Final.tex, line 5183
apply the integrand to each x∈ X and take its Bochner integral in X. It is not asserted to be a Bochner integral in the operator-norm space B(X). The general Laplace formula is [source reference]. [Packet convolution] For all s,t≥0 and all packets α,β, T_βα(s+t) = ∑_γ∈ I_0 T_βγ(s)T_γα(t). The original semigroup is reconstructed from its blocks by T(t)=∑_α,β∈ I_0T_βα(t).
Exact mathematical passage
\boxed{ T_{\beta\alpha}(s+t) = \sum_{\gamma\in I_0} T_{\beta\gamma}(s)T_{\gamma\alpha}(t). }
\sum_{\gamma\in\Gamma_b}Use in source · 3 passages
Search aliases: ∑_γ∈Γ_b sum gamma in Gamma
Use in source · 02_strengthened.tex, line 457
[Path interference formula] Fix a source vector ψ∈ P_a_0 H and a target packet b. Let Γ_b be the set of packet paths from a_0 to b, and set η_γ=U_γψ. Choose any total order on this finite path set. Then normP_bUψ^2 = ∑_γ∈Γ_bnormη_γ^2 + 2Re ∑_substackγ,γ'∈Γ_b γ<γ' ipη_γη_γ'. Consequently, normP_bUψ ≤ ∑_γ∈Γ_bnormη_γ. If the path outputs are pairwise orthogonal, the target probability is the sum of the path probabilities.
Exact mathematical passage
\norm{P_bU\psi}^2 = \sum_{\gamma\in\Gamma_b}\norm{\eta_\gamma}^2 + 2\operatorname{Re} \sum_{\substack{\gamma,\gamma'\in\Gamma_b\\ \gamma<\gamma'}} \ip{\eta_\gamma}{\eta_{\gamma'}}.
\sum_\deltaUse in source · 2 passages
Search aliases: ∑_δ sum delta
Use in source · 03_packets_and_graphs.tex, line 659
Apply Q_γ on the left and P_α on the right. Inserting I_X=∑_β P_β gives Q_γ UTP_α = ∑_β Q_γ UP_β TP_α = ∑_β U_γβT_βα. Inserting I_Y=∑_δ Q_δ gives Q_γ SUP_α = ∑_δ Q_γ SQ_δ UP_α = ∑_δ S_γδU_δα. Thus UT=SU implies every block equation. Conversely, if every block equation holds, then all blocks of UT-SU vanish. Exact operator reconstruction yields UT-SU=0.
Exact mathematical passage
I_Y=\sum_\delta Q_\delta
\sum_{\delta\in J}Use in source · 1 passage
Search aliases: ∑_δ∈ J sum delta in
Use in source · 03_packets_and_graphs.tex, line 638
et T∈End(X), S∈End(Y), let P=(P_α)_α∈ I and Q=(Q_γ)_γ∈ J be finite packet systems, and let U:X→ Y be linear. Define T_βα=P_β TP_α, S_γδ=Q_γ SQ_δ, U_γα=Q_γ UP_α. Then UT=SU if and only if, for every γ∈ J and α∈ I, ∑_β∈ IU_γβT_βα = ∑_δ∈ JS_γδU_δα. [source reference] ExportsPacketwise equivalence, gluing, symmetry, and local representation comparison.
Exact mathematical passage
\boxed{ \sum_{\beta\in I}U_{\gamma\beta}T_{\beta\alpha} = \sum_{\delta\in J}S_{\gamma\delta}U_{\delta\alpha}. }
\sum_{\zeta\in H_m}Use in source · 2 passages
Search aliases: ∑_ζ∈ H_m sum zeta in
Use in source · 72_recovered.tex, line 272
[Subgroup average] The multiple selector has the mode expansion a^(m) = frac1m ∑_ζ∈ H_m ϵ_ζ.
Exact mathematical passage
\boxed{ a^{(m)} = \frac1m \sum_{\zeta\in H_m} \epsilon_\zeta. }
\sum_{\zeta\in\mu_m}Use in source · 1 passage
Search aliases: ∑_ζ∈μ_m sum zeta in mu
Use in source · 72_recovered.tex, line 847
Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding term vanishes for positive n. Otherwise ((1+ζ y)^m)^h = (1+ζ y)^p-1 =1. Every summand is therefore h periodic for positive n. Thus T_m,p preserves V_h.…
Exact mathematical passage
T_{m,p}(\chi^\bullet)_n = \frac1m \sum_{\zeta\in\mu_m} (1+\zeta y)^{mn}.
\sum_{\eta^N=-1}Use in source · 3 passages
Search aliases: ∑_η^N=-1 sum eta
Use in source · Paper_IV_Traditional_Final.tex, line 269
The second half of this chapter concerns the zeros of D_N. The finite exponential sum D_N(z)=frac1N∑_η^N=-1e^η z has a regular polygon of exponents. The general relation between an exponent polygon and the zero strips of an exponential sum goes back to P olya and Langer. Later developments include the work of Moreno and the treatment in Levin's monograph [source reference]. Bleher and Mallison give a particularly close general result: for convex-polygon exponential sums they locate all sufficien…
Exact mathematical passage
\Dfun_N(z)=\frac1N\sum_{\eta^N=-1}e^{\eta z}
\sum_{\eta\in\Rcal_N}Use in source · 1 passage
Search aliases: ∑_η∈Rcal_N ∑_η∈ R_N sum eta in Rcal sum eta in mathcal
Use in source · Paper_IV_Traditional_Final.tex, line 702
[Finite exponential representation] Let R_N={η∈ Cmathrel:η^N=-1}. Then D_N(z) = frac1N∑_η∈ R_Ne^η z. More generally, D_N,j(z) = fracρ^-jN ∑_ℓ=0^N-1 ω^-jℓe^ρω^ℓ z.
Exact mathematical passage
\Dfun_N(z) = \frac1N\sum_{\eta\in\Rcal_N}e^{\eta z}.
\sum_{\lambda}Use in source · 1 passage
Search aliases: ∑_λ sum lambda
Use in source · Paper_IV_Traditional_Final.tex, line 2521
Logarithmic differentiation of the genus zero product gives -fracΔ_N'(w)Δ_N(w) = ∑_λ∈Λ_Nfrac1λ-w. The series is locally uniform because ∑_λ|λ|^-1<∞. Substitute w=z^N and use the chain rule.
Exact mathematical passage
\sum_{\lambda}|\lambda|^{-1}<\infty
\sum_{\lambda_j>0}Use in source · 4 passages
Search aliases: ∑_λ_j>0 sum lambda
Use in source · 02_strengthened.tex, line 1357
[Exact recovery on the error span] Under the hypotheses of [source reference], define on the syndrome support Q=∑_λ_j>0Q_j the completely positive map R_0(X) = ∑_λ_j>0 V_j^*Q_jXQ_jV_j. It extends to a quantum channel R on End( H). If Φ(X)=∑_aE_aXE_a^* is trace preserving on code states, then RΦ(ρ)=ρ for every density matrix ρ=Pρ P.
Exact mathematical passage
Q=\sum_{\lambda_j>0}Q_j
\sum_{\lambda_r>0}Use in source · 1 passage
Search aliases: ∑_λ_r>0 sum lambda
Use in source · 01_finite_core.tex, line 850
titSufficiency. Diagonalize C=Udiag(λ_r)U^* with U unitary and λ_r≥0. Define compressed transformed errors F_r=∑_aU_arE_aP. Then F_rP=F_r and F_r^*F_s=λ_rδ_rsP. If λ_r=0, F_r^*F_r=0 forces F_r=0. For positive eigenvalues only, set V_r=F_r/sqrtλ_r. These satisfy V_rP=V_r, V_r^*V_s=δ_rsP. Let S=∑_λ_r>0V_rV_r^*. Orthogonality makes S an orthogonal projector. Use the recovery branches K_r=V_r^*, K_unused=I-S. They are complete on the whole ambient space because ∑_rK_r^*K_r+K_unused^*K_unused =S+(I-S…
Exact mathematical passage
S=\sum_{\lambda_r>0}V_rV_r^*
\sum_{\lambda\in\widehat G}Use in source · 1 passage
Search aliases: ∑_λ∈widehat G sum lambda in widehat
Use in source · 02_strengthened.tex, line 215
[Finite group isotypic projectors] The operators P_λ = fracd_λabsG ∑_g∈ G overlineχ_λ(g)U(g) are orthogonal projectors. They satisfy P_λ P_μ=δ_λμP_λ, ∑_λ∈widehat GP_λ=I_ H. The range of P_λ is the λ isotypic component of H.
Exact mathematical passage
P_\lambda P_\mu=\delta_{\lambda\mu}P_\lambda, \qquad \sum_{\lambda\in\widehat G}P_\lambda=I_{\Hcal}.
\sum_\muUse in source · 1 passage
Search aliases: ∑_μ sum mu
Use in source · 02_connections_obstructions.tex, line 513
Insert completeness on both sides: Ob = (∑_μΠ^a_μ) Ob (∑_νΠ^a_ν) = ∑_μ,νΠ^a_μObΠ^a_ν. Thus (iv) implies (iii), and (iii) implies (iv) by composition with the projectors. Conditions (ii) and (iii) are equivalent by the definition of Ob and additive cancellation. Conditions (i) and (ii) are equivalent by [source reference].
Exact mathematical passage
\Ob = \left(\sum_\mu\Pi^a_\mu\right) \Ob \left(\sum_\nu\Pi^a_\nu\right) = \sum_{\mu,\nu}\Pi^a_\mu\Ob\Pi^a_\nu.
\sum_{\mu,\nu}Use in source · 2 passages
Search aliases: ∑_μ,ν sum mu nu
Use in source · 02_connections_obstructions.tex, line 513
Insert completeness on both sides: Ob = (∑_μΠ^a_μ) Ob (∑_νΠ^a_ν) = ∑_μ,νΠ^a_μObΠ^a_ν. Thus (iv) implies (iii), and (iii) implies (iv) by composition with the projectors. Conditions (ii) and (iii) are equivalent by the definition of Ob and additive cancellation. Conditions (i) and (ii) are equivalent by [source reference].
Exact mathematical passage
\Ob = \left(\sum_\mu\Pi^a_\mu\right) \Ob \left(\sum_\nu\Pi^a_\nu\right) = \sum_{\mu,\nu}\Pi^a_\mu\Ob\Pi^a_\nu.
\sum_{\mu,\nu\in I_a}Use in source · 1 passage
Search aliases: ∑_μ,ν∈ I_a sum mu nu in
Use in source · 02_connections_obstructions.tex, line 493
[Exact reconstruction and detection by obstruction blocks] The total obstruction has the exact finite resolution Ob_c,b,a = ∑_μ,ν∈ I_a Ob^μν_c,b,a. Moreover, the following are equivalent: [label=(roman*)] item Δ_c,b,a=0; item H_c,b,a=id; item Ob_c,b,a=0; item Ob^μν_c,b,a=0 for every μ,ν. statusleanthreeKomorebi.PaperIII.HolonomyBlocks.obstruction_sum_blocks; Komorebi.PaperIII.HolonomyBlocks.obstructionBlocks_zero_iff; Komorebi.PaperIII.HolonomyBlocks.obstructionDetection_equivalences
Exact mathematical passage
\boxed{ \Ob_{c,b,a} = \sum_{\mu,\nu\in I_a} \Ob^{\mu\nu}_{c,b,a}. }
\sum_{\mu\in\Lambda_{N,j}}Use in source · 1 passage
Search aliases: ∑_μ∈Λ_N,j sum mu in Lambda
Use in source · Paper_IV_Traditional_Final.tex, line 2405
[Products for the normalized sections] For every 0≤ j<N, the entire function Φ_N,j(w) = j!∑_n≥0fracw^n(Nn+j)! has order 1/N. If N≥2 and Λ_N,j is its nonzero zero multiset, then Φ_N,j(w) = ∏_μ∈Λ_N,j (1-frac wμ). For m≥1, put σ_N,j,m = ∑_μ∈Λ_N,jμ^-m. Then C_N,j,n(r) = frac(Nn)!n! B_n( rσ_N,j,1, r1!σ_N,j,2, …, r(n-1)!σ_N,j,n ).
Exact mathematical passage
\sigma_{N,j,m} = \sum_{\mu\in\Lambda_{N,j}}\mu^{-m}.
\sum_\nuUse in source · 10 passages
Search aliases: ∑_ν sum nu
Use in source · 02_connections_obstructions.tex, line 513
Insert completeness on both sides: Ob = (∑_μΠ^a_μ) Ob (∑_νΠ^a_ν) = ∑_μ,νΠ^a_μObΠ^a_ν. Thus (iv) implies (iii), and (iii) implies (iv) by composition with the projectors. Conditions (ii) and (iii) are equivalent by the definition of Ob and additive cancellation. Conditions (i) and (ii) are equivalent by [source reference].
Exact mathematical passage
\Ob = \left(\sum_\mu\Pi^a_\mu\right) \Ob \left(\sum_\nu\Pi^a_\nu\right) = \sum_{\mu,\nu}\Pi^a_\mu\Ob\Pi^a_\nu.
\sum_{\nu=1}^kUse in source · 6 passages
Search aliases: ∑_ν=1^k sum nu
Use in source · strengthening.tex, line 176
[All-row prime-power transfer] Let p be prime, a≥1, N=p^a, and 0≤ j<N. The coefficients C_N,j(n) belong to Z_p for all n≥0. Set A_N,j(n)=(-1)^n C_N,j(n) and T_H=S^H-(-1)^H. For k≥1 and positive H_1,…,H_k, v_p(Δ_H_1⋯Δ_H_kA_N,j(n)) · ≥ k+∑_ν=1^k v_p(H_ν), v_p(T_H_1⋯ T_H_kE_Nn^(N,j)) · ≥ v_p(j!)+k+∑_ν=1^k v_p(H_ν). At p=2, the ordinary differences of C_N,j and of the original coefficients satisfy the same bounds. These dyadic statements also hold for N=1.
Exact mathematical passage
v_p(\Delta_{H_1}\cdots\Delta_{H_k}A_{N,j}(n)) &\geq k+\sum_{\nu=1}^k v_p(H_\nu), \label{eq:prime-power-signed-bound}\\ v_p(T_{H_1}\cdots T_{H_k}E_{Nn}^{(N,j)}) &\geq v_p(j!)+k+\sum_{\nu=1}^k v_p(H_\nu). \label{eq:prime-power-original-twist}
\sum_{\pi=\{B_1,\ldots,B_k\}}Use in source · 1 passage
Search aliases: ∑_π={B_1,…,B_k} sum pi ldots
Use in source · 03_nonlinear_coordinates.tex, line 353
di Bruno formula] Let n≥1, let X,Y,Z be Banach spaces over the same real or complex field, and let G:X→ Y and F:Y→ Z be n times continuously Fr echet differentiable. Let P, Q, and R be packet systems on X, Y, and Z. Fix input packets α_1,…,α_n, an output packet δ, and directions h_i∈ X_α_i. Then · J^n_δmidα_1,…,α_n(F∘ G)(x) [h_1,…,h_n] · = ∑_π={B_1,…,B_k} ∑_γ_1,…,γ_k J^k_δmidγ_1,…,γ_kF(Gx) [ G_γ_1,B_1(xmid h),…, G_γ_k,B_k(xmid h) ]. The first sum runs over all set partitions of {1,…,n}.
Exact mathematical passage
\begin{aligned} &J^{n}_{\delta\mid\alpha_1,\ldots,\alpha_n}(F\circ G)(x) [h_1,\ldots,h_n]\\ &= \sum_{\pi=\{B_1,\ldots,B_k\}} \sum_{\gamma_1,\ldots,\gamma_k} J^{k}_{\delta\mid\gamma_1,\ldots,\gamma_k}F(Gx) \big[ G_{\gamma_1,B_1}(x\mid h),\ldots, G_{\gamma_k,B_k}(x\mid h) \big]. \end{aligned}
\sum_{\rho(\alpha)=a}Use in source · 1 passage
Search aliases: ∑_ρ(α)=a sum rho alpha
Use in source · 02_coordinates.tex, line 240
Expand the two coarse projectors into their children: P_bTP_a = (∑_ρ(β)=bQ_β) T (∑_ρ(α)=aQ_α). Distributing the finite sums gives the first identity. For the coordinate identity, use [T]_ P= A_ PT S_ P and insert I_X= S_ Q A_ Q.
Exact mathematical passage
P_bTP_a = \left(\sum_{\rho(\beta)=b}Q_\beta\right) T \left(\sum_{\rho(\alpha)=a}Q_\alpha\right).
\sum_{\rho(\beta)=b}Use in source · 1 passage
Search aliases: ∑_ρ(β)=b sum rho beta
Use in source · 02_coordinates.tex, line 240
Expand the two coarse projectors into their children: P_bTP_a = (∑_ρ(β)=bQ_β) T (∑_ρ(α)=aQ_α). Distributing the finite sums gives the first identity. For the coordinate identity, use [T]_ P= A_ PT S_ P and insert I_X= S_ Q A_ Q.
Exact mathematical passage
P_bTP_a = \left(\sum_{\rho(\beta)=b}Q_\beta\right) T \left(\sum_{\rho(\alpha)=a}Q_\alpha\right).
\sum_{\rho(\gamma)=\alpha}Use in source · 1 passage
Search aliases: ∑_ρ(γ)=α sum rho gamma alpha
Use in source · 02_coordinates.tex, line 62
Multiply the identity P_α=∑_ρ(γ)=αQ_γ by Q_β on either side. Pairwise annihilation in the fine family leaves Q_β when ρ(β)=α and leaves zero otherwise. For transitivity, substitute the child expansion of each Q_β into the expansion of P_α and group the resulting terms by the composite parent.
Exact mathematical passage
P_\alpha=\sum_{\rho(\gamma)=\alpha}Q_\gamma
\sum_{\rho\in\Zcal_R}Use in source · 2 passages
Search aliases: ∑_ρ∈Zcal_R ∑_ρ∈ Z_R sum rho in Zcal sum rho in mathcal
Use in source · Paper_IV_Traditional_Final.tex, line 2632
rcle_expansion Let r∈ Z_>0. Suppose the zeros of D_N of least modulus R are simple and form the finite set Z_R. Suppose the next zero modulus is R_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_ρ,ℓ binomm+ℓ-1ℓ-1 ρ^-m + O_R_*(m!R_*^-m). The leading Laurent coefficient is a_ρ,r = (-ρ D_N'(ρ))^-r.
Exact mathematical passage
P_{N,n}(r) = m! \sum_{\rho\in\Zcal_R} \sum_{\ell=1}^{r} a_{\rho,\ell} \binom{m+\ell-1}{\ell-1} \rho^{-m} + O_{R_*}(m!R_*^{-m}).
\sum_{\tau(\sigma(\alpha))=k}Use in source · 1 passage
Search aliases: ∑_τ(σ(α))=k sum tau sigma alpha
Use in source · 04_transport.tex, line 157
Let (Φ,φ,σ):(X, A, P)→(Y, B, Q) and (Ψ,ψ,τ):(Y, B, Q)→(Z, C, R) be morphisms. Intertwining is preserved because ΨΦ(Ax) = Ψ(φ(A)Φ x) = (ψ∘φ)(A)ΨΦ x. For k in the target packet set, R_kΨΦ · =Ψ Q_τ^-1(k)Φ · =ΨΦ ∑_τ(σ(α))=kP_α. Thus the composite packet map is τ∘σ. If the morphisms are seeded, then Φ(S)⊆Hull_ B(S'), Ψ(S')⊆Hull_ C(S''). By [source reference], ΨΦ(S) ⊆ Ψ(Hull_ B(S')) ⊆ Hull_ C(Ψ(S')) ⊆ Hull_ C(S''). Identity maps satisfy the definitions with identity algebra and packet maps.
Exact mathematical passage
R_k\Psi\Phi &=\Psi Q_{\tau^{-1}(k)}\Phi\\ &=\Psi\Phi \sum_{\tau(\sigma(\alpha))=k}P_\alpha.
\sum_{\tau\in\mathfrak T_m}Use in source · 2 passages
Search aliases: ∑_τ∈ T_m sum tau in mathfrak
Use in source · 03_nonlinear_coordinates.tex, line 472
Let T_m be the finite set of ordered rooted trees of height less than m. A leaf has value s. If the root of τ has arity r and ordered subtrees τ_1,…,τ_r, define B_τ(s)=B_r(B_τ_1(s),…,B_τ_r(s)). Induction on m shows that the Picard iterate u_0=0 and u_m+1=Φ(u_m) satisfies u_m=∑_τ∈ T_mB_τ(s). Define a nonnegative scalar weight by assigning weight a to a leaf and applying the scalar map (t_1,…,t_r)↦ b_rt_1⋯ t_r at a vertex of arity r. A second induction gives ∑_τ∈ T_m B_τ(s) ≤ ∑_τ∈ T_mw(τ) =t_m, wh…
Exact mathematical passage
u_m=\sum_{\tau\in\mathfrak T_m}B_\tau(s).
\sum_{\tau\in\Zcal_R^{\mathrm{orb}}}Use in source · 1 passage
Search aliases: ∑_τ∈Zcal_R^orb ∑_τ∈ Z_R^orb sum tau in Zcal mathrm sum tau in mathcal mathrm
Use in source · Paper_IV_Traditional_Final.tex, line 2697
[Orbit reduced nearest circle term] Assume the hypotheses of [source reference]. Choose one representative from each orbit of the nearest zeros under multiplication by ω, and call the resulting set Z_R^orb. With m=Nn, P_N,n(r) = · N m!binomm+r-1r-1 ∑_τ∈ Z_R^orb fracτ^-m(-τ D_N'(τ))^r · +O(m!m^r-2R^-m) +O_R_*(m!R_*^-m). For r=1, the first error term is absent.
Exact mathematical passage
\begin{aligned} P_{N,n}(r) ={}&N m!\binom{m+r-1}{r-1} \sum_{\tau\in\Zcal_R^{\mathrm{orb}}} \frac{\tau^{-m}}{(-\tau\Dfun_N'(\tau))^r} \\ &+O\left(m!m^{r-2}R^{-m}\right) +O_{R_*}\left(m!R_*^{-m}\right). \end{aligned}
\sum_{\chi\in\mu_h}Use in source · 1 passage
Search aliases: ∑_χ∈μ_h sum chi in mu
Use in source · 72_recovered.tex, line 843
Choose d_0=f_h and d_n=f_n for n≥1. Then d is fully h periodic and f=d+cδ_0 for c=f_0-f_h. Since T_m,pδ_0 is the constant sequence one, it is enough to treat d. Expand d_q=∑_χ∈μ_hc_χχ^q. The mth power map on F_p^× has image μ_h. Choose y with y^m=χ. The root filter gives T_m,p(χ^bullet)_n = frac1m ∑_ζ∈μ_m (1+ζ y)^mn. If 1+ζ y=0, the corresponding term vanishes for positive n. Otherwise ((1+ζ y)^m)^h = (1+ζ y)^p-1 =1. Every summand is therefore h periodic for positive n. Thus T_m,p preserves V_h.…
Exact mathematical passage
d_q=\sum_{\chi\in\mu_h}c_\chi\chi^q.
\sum_\ellUse in source · 5 passages
Search aliases: ∑_ℓ sum ell
Use in source · Paper_IV_Traditional_Final.tex, line 525
igin, the derivative matrix is diagonal with entries 1,λ,…,λ^N-1. For an explicit determinant proof, write each basis function as a finite sum of exponentials with exponents a_ℓ=λω^ℓ. Its derivative matrix factors as W(z)=V diag(e^a_ℓ z) C, V_kℓ=a_ℓ^k, where C is the constant matrix of Fourier weights. Thus det W(z)=e^z∑_ℓ a_ℓdet W(0)=det W(0), because N≥2 implies ∑_ℓω^ℓ=0. The product of the diagonal initial values is the stated power of λ.
Exact mathematical passage
\det W(z)=e^{z\sum_\ell a_\ell}\det W(0)=\det W(0),
\sum_{\ell=0}^{3}Use in source · 1 passage
Search aliases: ∑_ℓ=0^3 sum ell
Use in source · atlas-pages.tex, line 46
vpage3A root filter is an exact voteCYCLIC PROJECTIONChapter IV: Theorem 3.1, Proposition 3.2. For N=4, write ω=i. To extract residue j, evaluate the function at four rotated arguments and attach the phase i^-jℓ: ( B_4,jF)(z)=frac14∑_ℓ=0^3i^-jℓF(i^ℓ z). A monomial a_mz^m is multiplied by the finite average of i^(m-j)ℓ. Matching phases all vote 1. Nonmatching phases cancel exactly. vfigroot_filter headAn example that can be checked coefficient by coefficient Take F(z)=∑_m=0^12(m+1)z^m. Then B_4,2…
Exact mathematical passage
(\mathcal B_{4,j}F)(z)=\frac14\sum_{\ell=0}^{3}i^{-j\ell}F(i^\ell z).
\sum_{\ell=0}^{L-1}Use in source · 2 passages
Search aliases: ∑_ℓ=0^L-1 sum ell
Use in source · 05_defects.tex, line 228
[Multilevel defect decomposition] The total defect from level L to level zero is C_0,LT_L-T_0C_0,L = ∑_ℓ=0^L-1 C_0,ℓΔ_ℓ C_ℓ+1,L. For bounded maps, C_0,LT_L-T_0C_0,L ≤ ∑_ℓ=0^L-1 C_0,ℓ Δ_ℓ C_ℓ+1,L .
Exact mathematical passage
C_{0,L}T_L-T_0C_{0,L} = \sum_{\ell=0}^{L-1} C_{0,\ell}\Delta_\ell C_{\ell+1,L}.
\sum_{\ell=0}^{m-1}Use in source · 1 passage
Search aliases: ∑_ℓ=0^m-1 sum ell
Use in source · 03_nonlinear_coordinates.tex, line 860
y[Several refinement levels] Let C_ℓ:X_ℓ+1→ X_ℓ for 0≤ℓ<m. Suppose x_ℓ is a fixed point at level ℓ, the coarse map at level ℓ has contraction constant q_ℓ<1, and the adjacent defect at x_ℓ+1 is at most ε_ℓ. Write C_a,b=C_aC_a+1⋯ C_b-1 for a<b, C_a,a=I. Then C_0,mx_m-x_0 ≤ ∑_ℓ=0^m-1 C_0,ℓ fracε_ℓ1-q_ℓ.
Exact mathematical passage
\|C_{0,m}x_m-x_0\| \le \sum_{\ell=0}^{m-1} \|C_{0,\ell}\| \frac{\varepsilon_\ell}{1-q_\ell}.
\sum_{\ell=0}^{N-1}Use in source · 3 passages
Search aliases: ∑_ℓ=0^N-1 sum ell
Use in source · Paper_IV_Traditional_Final.tex, line 363
Fix N≥1 and write ω=e^2π i/N. Let Ω be a disk centered at the origin, with the whole plane allowed, and assume that Ω is invariant under multiplication by ω. For a holomorphic function F on Ω, define ( B_N,jF)(z) = frac1N∑_ℓ=0^N-1ω^-jℓF(ω^ℓ z), 0≤ j<N. The same formula is valid for formal power series.
Exact mathematical passage
(\Bcal_{N,j}F)(z) = \frac1N\sum_{\ell=0}^{N-1}\omega^{-j\ell}F(\omega^\ell z), \qquad 0\leq j<N.
\sum_{\ell=0}^{\infty}Use in source · 2 passages
Search aliases: ∑_ℓ=0^∞ sum ell infty
Use in source · Paper_VI_Traditional_Final.tex, line 2600
[Contractive summable defect completion] Assume normC_ℓ≤1, normR_ℓ≤1, normT_ℓ(t)≤1 for every ℓ and t≥0. If ∑_ℓ=0^∞ sup_0≤ t≤ TnormE_ℓ(t)<∞, then the conclusion of [source reference] holds, and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_ℓ=n^∞ sup_0≤ t≤ TnormE_ℓ(t).
Exact mathematical passage
\sum_{\ell=0}^{\infty} \sup_{0\leq t\leq T}\norm{E_\ell(t)}<\infty,
\sum_{\ell=1}^{m}Use in source · 1 passage
Search aliases: ∑_ℓ=1^m sum ell
Use in source · 01_manuscript.tex, line 606
The (i,j) entry of BXA^T is ∑_k=1^n∑_ℓ=1^m B_ikX_kℓA_jℓ. This is the coordinate formula for (A⊗ B)vec(X). Multiplying first by B and then by A^T gives the stated count.
Exact mathematical passage
\sum_{k=1}^{n}\sum_{\ell=1}^{m} B_{ik}X_{k\ell}A_{j\ell}.
\sum_{\ell=1}^{r}Use in source · 2 passages
Search aliases: ∑_ℓ=1^r sum ell
Use in source · Paper_IV_Traditional_Final.tex, line 2622
[Nearest-pole coefficient expansion] Let r∈ Z_>0. Suppose the zeros of D_N of least modulus R are simple and form the finite set Z_R. Suppose the next zero modulus is R_2>R. At each ρ∈ Z_R, write the principal part D_N(z)^-r = ∑_ℓ=1^r a_ρ,ℓ (1-frac zρ)^-ℓ + O(1). Then, for every R_* with R<R_*<R_2 and m=Nn, as n→∞, P_N,n(r) = m! ∑_ρ∈ Z_R ∑_ℓ=1^r a_ρ,ℓ binomm+ℓ-1ℓ-1 ρ^-m + O_R_*(m!R_*^-m). The leading Laurent coefficient is a_ρ,r = (-ρ D_N'(ρ))^-r.
Exact mathematical passage
\Dfun_N(z)^{-r} = \sum_{\ell=1}^{r} a_{\rho,\ell} \left(1-\frac z\rho\right)^{-\ell} + O(1).
\sum_{\ell=n}^{\infty}Use in source · 2 passages
Search aliases: ∑_ℓ=n^∞ sum ell infty
Use in source · Paper_VI_Traditional_Final.tex, line 2605
ontractive summable defect completion] Assume normC_ℓ≤1, normR_ℓ≤1, normT_ℓ(t)≤1 for every ℓ and t≥0. If ∑_ℓ=0^∞ sup_0≤ t≤ TnormE_ℓ(t)<∞, then the conclusion of [source reference] holds, and sup_0≤ t≤ T normU_m(t)-U_m,n(t) ≤ ∑_ℓ=n^∞ sup_0≤ t≤ TnormE_ℓ(t).
Exact mathematical passage
\sup_{0\leq t\leq T} \norm{U_m(t)-U_{m,n}(t)} \leq \sum_{\ell=n}^{\infty} \sup_{0\leq t\leq T}\norm{E_\ell(t)}.
\sum_{\ell\in L}Use in source · 1 passage
Search aliases: ∑_ℓ∈ L sum ell in
Use in source · Paper_V_Traditional_Final.tex, line 1803
Every address has a unique representation r+ℓ. For finite p, x _p^p = ∑_r∈ R∑_ℓ∈ L|x(r+ℓ)|^p = ∑_r∈ R x_r _p^p. The corresponding supremum identity is immediate when p=∞. Unique address representation proves that analysis and interleaving are inverse maps. The projector identities follow from the disjoint coset partition.
Exact mathematical passage
\|x\|_p^p = \sum_{r\in R}\sum_{\ell\in L}|x(r+\ell)|^p = \sum_{r\in R}\|x_r\|_p^p.
\sum_{\ell\in\Lcal_w}Use in source · 1 passage
Search aliases: ∑_ℓ∈Lcal_w ∑_ℓ∈ L_w sum ell in Lcal sum ell in mathcal
Use in source · Paper_V_Traditional_Final.tex, line 2407
Assume the child subtree bounds have been proved. Then ∑_w∑_ℓ∈ L_w x_ℓ ^2 ≥ min_w a(w) ∑_w (U_vx)_w ^2 ≥ A_vmin_w a(w) x ^2. The upper bound follows in the same way with the maximum and B_v. Induction completes the proof.
Exact mathematical passage
\sum_{w}\sum_{\ell\in\Lcal_w}\|x_\ell\|^2 \geq \min_w a(w) \sum_w\|(U_vx)_w\|^2 \geq A_v\min_w a(w)\|x\|^2.
\sum_{\ell\in\widetilde L}Use in source · 1 passage
Search aliases: ∑_ℓ∈widetilde L sum ell in widetilde
Use in source · Paper_XI_Traditional_Final.tex, line 870
[Isotropic averaging projector] The operator P_L = frac1abs L ∑_ℓ∈widetilde L ρ(ℓ) is an orthogonal projector. In the Schrödinger model with L=X, it has rank one and selects the constant functions.
Exact mathematical passage
P_L = \frac1{\abs L} \sum_{\ell\in\widetilde L} \rho(\ell)
\sum_{\ell\geq1}Use in source · 1 passage
Search aliases: ∑_ℓ≥1 sum ell geq
Use in source · strengthening.tex, line 1245
[Taylor coefficient bounds and radii] For j=0,1,2,3, the dyadic interpolation has an ordinary power series widetilde C_j(x)=1+∑_ℓ≥1d_j,ℓx^ℓ, v_2(d_j,ℓ)≥ℓ+β_j. Its Taylor series converges for |x|_2<2. At addresses zero and one, v_2(d_j,ℓ)=ℓ (ℓ≥1), R_0=R_1=2. At address two, v_2(d_2,ℓ)≥ max(2ℓ+lfloorfracℓ2rfloor, ℓ+3), R_2=2^5/2. The radius at address three is at least two. Its exact radius is not asserted here. The stated exact radii at addresses zero, one, and two hold at every center in Z_2.
Exact mathematical passage
\widetilde C_j(x)=1+\sum_{\ell\geq1}d_{j,\ell}x^\ell, \qquad v_2(d_{j,\ell})\geq\ell+\beta_j.
\sum_{\sum_i a_i=r}Use in source · 1 passage
Search aliases: ∑_∑_i a_i=r sum sum
Use in source · 03_supplement_02.tex, line 54
Suppose N>0 and each factor has a complete orthogonal self-adjoint projector family (P^(i)_j)_j∈ Z/N Z. For a fine address a:F→ Z/N Z, set Q_a= T(i↦ P^(i)_a_i), C_r=∑_∑_i a_i=rQ_a.
Exact mathematical passage
Q_a=\mathsf T\bigl(i\mapsto P^{(i)}_{a_i}\bigr), \qquad C_r=\sum_{\sum_i a_i=r}Q_a.
\setminusUse in source · 12 passages
Search aliases: ∖ setminus
Use in source · 03_packets_and_graphs.tex, line 396
Choose such a k≥ k_0. Since x^(k) is unconditionally summable, there is a finite F_0 such that every finite F⊆ N∖ F_0 and every |a_n|≤1 satisfy ∑_n∈ Fa_nx_n^(k) <ε. The preceding uniform estimate then gives ∑_n∈ Fa_nx_n <2ε. Thus the tails of ∑_nx_n are uniformly small under bounded scalar changes, which is the unconditional Cauchy criterion. Therefore x∈⊕^uX_n, and the earlier estimate proves x^(k)→ x in · _u.
Exact mathematical passage
F\subseteq\mathbb N\setminus F_0
\int_{0<s_1<\cdots<s_m<t}Use in source · 1 passage
Search aliases: ∫_0<s_1<⋯<s_m<t int cdots
Use in source · Paper_VI_Traditional_Final.tex, line 2236
[Dyson expansion over packet paths] For t≥0, e^tA = · e^tD + ∑_m=1^∞ ∫_0<s_1<⋯<s_m<t e^(t-s_m)DN e^(s_m-s_m-1)D ⋯ N e^s_1D d s_1⋯ d s_m. The series converges absolutely in operator norm. After multiplication by P_β and P_α, the term of order m is a sum over directed packet paths of length m from α to β.
Exact mathematical passage
\begin{aligned} \e^{tA} ={}&\e^{tD} + \sum_{m=1}^{\infty} \int_{0<s_1<\cdots<s_m<t} \e^{(t-s_m)D}N\e^{(s_m-s_{m-1})D} \cdots N\e^{s_1D} \dd s_1\cdots\dd s_m. \end{aligned}
\int_0^1Use in source · 7 passages
Search aliases: ∫_0^1 int
Use in source · 03_nonlinear_coordinates.tex, line 300
· +Q_γ R_m+1(x,h), where the Bochner integral remainder is R_m+1(x,h) = frac1m! ∫_0^1(1-t)^m D^m+1F(x+th)[h,…,h] dt.
Exact mathematical passage
R_{m+1}(x,h) = \frac1{m!} \int_0^1(1-t)^m D^{m+1}F(x+th)[h,\ldots,h]\,dt.
\int_0^{2\pi}Use in source · 1 passage
Search aliases: ∫_0^2π int pi
Use in source · 02_strengthened.tex, line 910
[Bloch sphere curvature] Let P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). With the convention F_θϕ =- iTr(P[∂_θ P,∂_ϕ P]), one has F_θϕ=frac12sinθ and frac12π ∫_0^2π∫_0^π F_θϕ dθ dϕ =1.
Exact mathematical passage
\frac1{2\pi} \int_0^{2\pi}\int_0^\pi \Fcal_{\theta\phi}\dd\theta\dd\phi =1.
\int_0^rUse in source · 1 passage
Search aliases: ∫_0^r int
Use in source · Paper_VI_Traditional_Final.tex, line 1265
For small t, normCA(tI+A)^-1-B(tI+B)^-1C ≤ 2norm C, because the two positive resolvent factors are contractions. For large t, the resolvent identity gives the bound norm D/t. Splitting the integral at r=fracnorm D2norm C when C and D are nonzero yields normD_α · ≤ fracsin(πα)π ( 2norm C∫_0^r t^α-1 dt + norm D∫_r^∞ t^α-2 dt ) · = K_αnorm C^1-αnorm D^α. The zero cases are immediate.
Exact mathematical passage
\begin{aligned} \norm{D_\alpha} &\le \frac{\sin(\pi\alpha)}{\pi} \left( 2\norm C\int_0^r t^{\alpha-1}\,dt + \norm D\int_r^\infty t^{\alpha-2}\,dt \right)\\ &= K_\alpha\norm C^{1-\alpha}\norm D^\alpha. \end{aligned}
\int_0^sUse in source · 2 passages
Search aliases: ∫_0^s int
Use in source · Paper_VI_Traditional_Final.tex, line 1297
[Fractional semigroup refinement] For s≥0, Ce^-sA^α-e^-sB^αC = -∫_0^s e^-(s-r)B^αD_α e^-rA^α dr. Hence normCe^-sA^α-e^-sB^αC ≤ snormD_α. Under the common gap A,B≥ mI, normCe^-sA^α-e^-sB^αC ≤ s e^-sm^αα m^α-1norm D.
Exact mathematical passage
\boxed{ Ce^{-sA^\alpha}-e^{-sB^\alpha}C = -\int_0^s e^{-(s-r)B^\alpha}D_\alpha e^{-rA^\alpha}\,dr. }
\int_0^zUse in source · 1 passage
Search aliases: ∫_0^z int
Use in source · Paper_V_Traditional_Final.tex, line 3391
Let q(x)=tfrac12sign(sin x), whose jump at zero has size one. For n≥1, its partial sum through the first n positive odd frequencies is Q_n(x)=frac2π∑_k=0^n-1fracsin((2k+1)x)2k+1. At x_n=π/(2n), x_n⟶0, Q_n(x_n)⟶fracSi(π)π =frac12+G, G≈0.089489872236, where Si(z)=∫_0^zsin(t)/t dt and G=Si(π)/π-1/2>0. Thus a nonzero overshoot persists along points approaching the jump. Scaling by a real amplitude a gives an overshoot or undershoot of magnitude |a|G.
Exact mathematical passage
\operatorname{Si}(z)=\int_0^z\sin(t)/t\,dt
\int_0^\piUse in source · 1 passage
Search aliases: ∫_0^π int pi
Use in source · 02_strengthened.tex, line 910
[Bloch sphere curvature] Let P(θ,ϕ) = frac12(I+ n(θ,ϕ)·σ), where n=(sinθcosϕ,sinθsinϕ,cosθ). With the convention F_θϕ =- iTr(P[∂_θ P,∂_ϕ P]), one has F_θϕ=frac12sinθ and frac12π ∫_0^2π∫_0^π F_θϕ dθ dϕ =1.
Exact mathematical passage
\frac1{2\pi} \int_0^{2\pi}\int_0^\pi \Fcal_{\theta\phi}\dd\theta\dd\phi =1.
\int_0^\inftyUse in source · 31 passages
Search aliases: ∫_0^∞ int infty
Use in source · Paper_VI_Traditional_Final.tex, line 1167
[Poisson subordination] For t>0 and L≥0, e^-tsqrt L = fract2sqrtπ ∫_0^∞ s^-3/2e^-t^2/(4s)e^-sL ds. The integral converges in operator norm in finite dimension.
Exact mathematical passage
\boxed{ e^{-t\sqrt L} = \frac{t}{2\sqrt\pi} \int_0^\infty s^{-3/2}e^{-t^2/(4s)}e^{-sL}\,ds. }
\int_0^{\infty}Use in source · 2 passages
Search aliases: ∫_0^∞ int infty
Use in source · Paper_VI_Traditional_Final.tex, line 5212
[Packet resolvent convolution] Let z belong to the resolvent set of A. If Rez is larger than a growth bound for the semigroup, then P_β(zI-A)^-1P_α = ∫_0^∞ e^-ztT_βα(t) d t.
Exact mathematical passage
P_\beta(zI-A)^{-1}P_\alpha = \int_0^{\infty}\e^{-zt}T_{\beta\alpha}(t)\dd t.
\int_1^{\infty}Use in source · 2 passages
Search aliases: ∫_1^∞ int infty
Use in source · residue_continuation_body.tex, line 164
An actual Mellin endpoint. The counting functions are constant on the integer norm steps. Their measurability, and hence that of S_w, is proved from this fact. For zero-sum w, define F_w(h)=(1+h)∫_1^∞x^-2-hS_w(x) dx (h≥0). Since 1-1/d<1, the logarithmic first moment M_w=∫_1^∞(1+log x) absnx^-2S_w(x) dx is finite. Lean proves the explicit estimate absnF_w(h)-F_w(0)≤ hM_w for every h≥0, and therefore F_w(h)=F_w(0)+O(h) as hdownarrow0. The function here is the displayed integral of the actual ideal…
Exact mathematical passage
F_w(h)=(1+h)\int_1^{\infty}x^{-2-h}S_w(x)\,dx \qquad(h\geq0).
\int_a^{a+L/2}Use in source · 1 passage
Search aliases: ∫_a^a+L/2 int
Use in source · Paper_V_Traditional_Final.tex, line 3209
Let L=2^-j and let a be the left endpoint of the support. Pair every point t∈[a,a+L/2] with t+L/2. Then ⟨ s,ψ_j,k⟩ = L^-1/2 ∫_a^a+L/2 overlines(t)-s(t+L/2) d t. The integrand has magnitude at most ML/2, and the integration interval has length L/2. The result is ML^3/2/4.
Exact mathematical passage
\langle s,\psi_{j,k}\rangle = L^{-1/2} \int_a^{a+L/2} \overline{s(t)-s(t+L/2)}\,\dd t.
\int_r^\inftyUse in source · 1 passage
Search aliases: ∫_r^∞ int infty
Use in source · Paper_VI_Traditional_Final.tex, line 1265
For small t, normCA(tI+A)^-1-B(tI+B)^-1C ≤ 2norm C, because the two positive resolvent factors are contractions. For large t, the resolvent identity gives the bound norm D/t. Splitting the integral at r=fracnorm D2norm C when C and D are nonzero yields normD_α · ≤ fracsin(πα)π ( 2norm C∫_0^r t^α-1 dt + norm D∫_r^∞ t^α-2 dt ) · = K_αnorm C^1-αnorm D^α. The zero cases are immediate.
Exact mathematical passage
\begin{aligned} \norm{D_\alpha} &\le \frac{\sin(\pi\alpha)}{\pi} \left( 2\norm C\int_0^r t^{\alpha-1}\,dt + \norm D\int_r^\infty t^{\alpha-2}\,dt \right)\\ &= K_\alpha\norm C^{1-\alpha}\norm D^\alpha. \end{aligned}
\int_\RRUse in source · 4 passages
Search aliases: ∫_RR ∫_ R int RR int mathbb
Use in source · Paper_VI_Traditional_Final.tex, line 3121
Let γ be the standard Gaussian probability measure on R. For t≥0, define the Ornstein--Uhlenbeck transfer ( O_t f)(x) = ∫_ R f( e^-tx+sqrt1- e^-2t y ) γ( d y). Let H_n be the probabilists' Hermite polynomial defined by exp(ux-fracu^22) = ∑_n=0^∞ H_n(x)fracu^nn!, and set h_n=H_n/sqrtn!.
Exact mathematical passage
(\Ocal_t f)(x) = \int_\RR f\left( \e^{-t}x+\sqrt{1-\e^{-2t}}\,y \right) \gamma(\dd y).
\int_\GammaUse in source · 5 passages
Search aliases: ∫_Γ int Gamma
Use in source · 05_defects.tex, line 150
X_c be complex Banach spaces, and let C:X_f→ X_c, T_f, and T_c be bounded complex-linear maps. Let h be holomorphic on an open set U containing both spectra. Let Γ be an admissible finite rectifiable Cauchy cycle in U and in both resolvent sets, with winding number one on the union of the spectra and zero outside U. The length of a cycle counts multiplicities. Then Ch(T_f)-h(T_c)C = frac12π i ∫_Γ h(z)(z-T_c)^-1Δ(z-T_f)^-1 dz. Consequently, Ch(T_f)-h(T_c)C · ≤ fraclength(Γ)2π sup_z∈Γ|h(z)| Δ
Exact mathematical passage
Ch(T_f)-h(T_c)C = \frac{1}{2\pi i} \int_\Gamma h(z)(z-T_c)^{-1}\Delta(z-T_f)^{-1}\,dz.
\int_\tau^{\widehat\tau}Use in source · 1 passage
Search aliases: ∫_τ^widehatτ int tau widehat tau
Use in source · Paper_V_Traditional_Final.tex, line 2761
Assume τ<widehatτ. Differentiation with respect to the location parameter gives frac∂∂ uϕ_m,u(t) =-ϕ_m-1,u(t) for almost every t. Hence ϕ_m,τ-ϕ_m,widehatτ = ∫_τ^widehatτϕ_m-1,u d u. Minkowski's inequality gives ϕ_m,τ-ϕ_m,widehatτ _p ≤ |τ-widehatτ| sup_u ϕ_m-1,u _p. The supremum is bounded by the norm of t^m-1/(m-1)! on [0,1], which is the stated constant. The supremum norm argument is identical.
Exact mathematical passage
\phi_{m,\tau}-\phi_{m,\widehat\tau} = \int_\tau^{\widehat\tau}\phi_{m-1,u}\dd u.
\sim_{\mathcal L}Use in source · 1 passage
Search aliases: ∼_ L sim mathcal
Use in source · Paper_VI_Traditional_Final.tex, line 3545
[Coarsest fixed sufficient partition] Let L be a nonempty family of strictly positive likelihood vectors. Define i∼_ Lj Longleftrightarrow λ_i=λ_j hboxfor every λ∈ L. Let Q_ L be the partition into equivalence classes. A partition Q gives a fixed coarse sufficient description in the sense of [source reference] for every λ∈ L if and only if every block of Q is contained in a block of Q_ L. Hence Q_ L is the unique coarsest partition with this property.
Exact mathematical passage
i\sim_{\mathcal L}j \quad\Longleftrightarrow\quad \lambda_i=\lambda_j \quad\hbox{for every }\lambda\in\mathcal L.
\approxUse in source · 1 passage
Search aliases: ≈ approx
Use in source · Paper_V_Traditional_Final.tex, line 3387
[Classical square-wave Gibbs behavior] Let q(x)=tfrac12sign(sin x), whose jump at zero has size one. For n≥1, its partial sum through the first n positive odd frequencies is Q_n(x)=frac2π∑_k=0^n-1fracsin((2k+1)x)2k+1. At x_n=π/(2n), x_n⟶0, Q_n(x_n)⟶fracSi(π)π =frac12+G, G≈0.089489872236, where Si(z)=∫_0^zsin(t)/t dt and G=Si(π)/π-1/2>0. Thus a nonzero overshoot persists along points approaching the jump. Scaling by a real amplitude a gives an overshoot or undershoot of magnitude |a|G.
Exact mathematical passage
x_n\longrightarrow0,\qquad Q_n(x_n)\longrightarrow\frac{\operatorname{Si}(\pi)}{\pi} =\frac12+G,\qquad G\approx0.089489872236,
\bigoplus^uUse in source · 2 passages
Search aliases: ⊕^u bigoplus
Use in source · 03_packets_and_graphs.tex, line 404
s a finite F_0 such that every finite F⊆ N∖ F_0 and every |a_n|≤1 satisfy ∑_n∈ Fa_nx_n^(k) <ε. The preceding uniform estimate then gives ∑_n∈ Fa_nx_n <2ε. Thus the tails of ∑_nx_n are uniformly small under bounded scalar changes, which is the unconditional Cauchy criterion. Therefore x∈⊕^uX_n, and the earlier estimate proves x^(k)→ x in · _u.
Exact mathematical passage
x\in\bigoplus^uX_n
\bigoplus_{(\alpha,\gamma)\in I\times J}Use in source · 1 passage
Search aliases: ⊕_(α,γ)∈ I× J bigoplus alpha gamma in times
Use in source · 03_packets_and_graphs.tex, line 202
lpha∈ I be a finite packet system on X, and let Q=(Q_γ)_γ∈ J be a finite packet system on Y. Then Pboxtimes Q := (P_α⊗ Q_γ)_(α,γ)∈ I× J is a finite packet system on the algebraic tensor product X⊗_𝕜Y, with packet spaces (P_α⊗ Q_γ)(X⊗ Y) =X_α⊗ Y_γ. Hence X⊗ Y ≅ ⊕_(α,γ)∈ I× J X_α⊗ Y_γ. [source reference] ExportsTensor refinement for quantum, signal, arithmetic, and multivariate realizations.
Exact mathematical passage
\boxed{ X\otimes Y \cong \bigoplus_{(\alpha,\gamma)\in I\times J} X_\alpha\otimes Y_\gamma. }
\bigoplus_{a\in A}Use in source · 2 passages
Search aliases: ⊕_a∈ A bigoplus in
Use in source · 72_recovered.tex, line 2741
[Packet decomposition of a lift obstruction] Suppose F_p^d = ⊕_a∈ AU_a, F_p^r = ⊕_a∈ AV_a, and suppose a linear correction map D: F_p^d→ F_p^r satisfies D(U_a)⊆ V_a for every a. Write D_a=D|_U_a. Then coker D ≅ ⊕_a∈ Acoker D_a. A residual lifts if and only if every packet obstruction vanishes.
Exact mathematical passage
\FF_p^d = \bigoplus_{a\in A}U_a, \qquad \FF_p^r = \bigoplus_{a\in A}V_a,
\bigoplus_{\frakm\in\Max\TT}Use in source · 1 passage
Search aliases: ⊕_frakm∈MaxTT ⊕_ m∈Max T bigoplus frakm in Max TT bigoplus mathfrak in operatorname mathbb
Use in source · Paper_XII_Traditional_Final.tex, line 1044
[Canonical Hecke packet decomposition] Let T be a finite dimensional commutative algebra over a field E', and let V be a finite T module. Then V = ⊕_ m∈Max Te_ mV. For every maximal ideal m, the canonical map gives e_ mV≅ V_ m. The packet e_ mV is nonzero exactly when m∈Supp_ T(V).
Exact mathematical passage
\boxed{ V = \bigoplus_{\frakm\in\Max\TT}e_{\frakm}V. }
\bigoplus_{g\in G}Use in source · 2 passages
Search aliases: ⊕_g∈ G bigoplus in
Use in source · 03_nonlinear_coordinates.tex, line 567
Let G be a finite abelian group and let A=⊕_g∈ GA_g be a G graded module.
Exact mathematical passage
A=\bigoplus_{g\in G}A_g
\bigoplus_{i=0}^{d-1}Use in source · 1 passage
Search aliases: ⊕_i=0^d-1 bigoplus
Use in source · Paper_XI_Traditional_Final.tex, line 565
[Greatest common divisor and least common multiple packets] There is a decomposition τ_ξ⊗τ_η ≅ ⊕_i=0^d-1 Ind_W_K_L^W_K ( ξ_L,nη_L,m^Φ^i ), where ξ_L,n and η_L,m are the norm lifts to K_L.
Exact mathematical passage
\boxed{ \tau_\xi\otimes\tau_\eta \cong \bigoplus_{i=0}^{d-1} \Ind_{W_{K_L}}^{W_K} \left( \xi_{L,n}\eta_{L,m}^{\Phi^i} \right), }
\bigoplus_{i=1}^rUse in source · 2 passages
Search aliases: ⊕_i=1^r bigoplus
Use in source · 01_finite_core.tex, line 79
Let a finite-dimensional unital *-subalgebra of operators act in its standard multiplicity form A ≅ ⊕_i=1^r (M_d_i( C)⊗ I_m_i) on H ≅ ⊕_i=1^r ( C^d_i⊗ C^m_i). Here every d_i is a positive integer and every multiplicity m_i is a nonnegative integer. An empty block family represents the zero space and has seed rank zero. The maximum over an empty block family is also assigned the value zero. A vector ψ_i∈ C^d_i⊗ C^m_i is identified with a d_i× m_i coefficient matrix C_i.
Exact mathematical passage
\cA \cong \bigoplus_{i=1}^r \left(M_{d_i}(\bbC)\otimes I_{m_i}\right)
\bigoplus_{i=1}^{r}Use in source · 3 passages
Search aliases: ⊕_i=1^r bigoplus
Use in source · Paper_X_Block27_Amended.tex, line 1334
[Compatible-carrier summand completion] Suppose a semisimple strictly compatible carrier decomposes as mathscr W ≅ ⊕_i=1^r mathscr W_i^⊕ c_i, with the mathscr W_i as above. At one coefficient prime λ_0, suppose a semisimple summand has the form V_λ_0 ≅ ⊕_i=1^r W_i,λ_0^⊕ m_i, 0≤ m_i≤ c_i. Then V_λ = ⊕_i=1^r W_i,λ^⊕ m_i defines a strictly compatible summand at every coefficient prime. The vector (m_i) is uniquely recoverable from finitely many Frobenius traces supplied by the separation theorem.
Exact mathematical passage
\mathscr W \cong \bigoplus_{i=1}^{r} \mathscr W_i^{\oplus c_i},
\bigoplus_{i\in J}Use in source · 1 passage
Search aliases: ⊕_i∈ J bigoplus in
Use in source · Paper_X_Block27_Amended.tex, line 2189
The actual invariant images U_i=imE_i give an equivariant linear isomorphism ⊕_i∈ JU_i≃ R^n, (v_i)_i⟼∑_i v_i, v⟼(E_iv)_i. Lean represents the finite direct sum by the finite product of the image submodules. The existing coordinatewise reduction theorem maps every U_i onto the corresponding reduced projector image. No chosen bases or unproved freeness of the images are used.
Exact mathematical passage
\bigoplus_{i\in J}U_i\simeq R^n, \qquad (v_i)_i\longmapsto\sum_i v_i, \qquad v\longmapsto(E_iv)_i.
\bigoplus_jUse in source · 4 passages
Search aliases: ⊕_j bigoplus
Use in source · visual_atlas.tex, line 79
Define X_N^rm pkt=⊕_j P_N,jX, A_Nx=(P_N,jx)_j, S_N(x_j)_j=∑_jx_j. Completeness and mutual annihilation give both inverse laws. Consequently, J= A_4 S_2, C= A_2 S_4, CJ=I, JC=I. The larger list of fine packets has smaller individual packet spaces.
Exact mathematical passage
X_N^{\rm pkt}=\bigoplus_j P_{N,j}X, \qquad \mathcal A_Nx=(P_{N,j}x)_j, \qquad \mathcal S_N(x_j)_j=\sum_jx_j.
\bigoplus_{j=1}^{d}Use in source · 1 passage
Search aliases: ⊕_j=1^d bigoplus
Use in source · 02_foundations.tex, line 753
For one factor M_d(D), let e_jj be the diagonal matrix units. The left regular module decomposes as M_d(D)=⊕_j=1^dM_d(D)e_jj. Each minimal left ideal M_d(D)e_jj is isomorphic to the natural column module D^d. Taking the direct product of factors gives the stated decomposition.
Exact mathematical passage
M_d(D)=\bigoplus_{j=1}^{d}M_d(D)e_{jj}.
\bigoplus_{j=1}^nUse in source · 1 passage
Search aliases: ⊕_j=1^n bigoplus
Use in source · 04_arithmetic.tex, line 735
[Computing a packet weight from an inclusion matrix] Choose an O basis of Λ and an O basis of Λ_split. Let B be the matrix of the inclusion Λ_split↪Λ. If the Smith form of B is diag(π^a_1,…,π^a_n), 0≤ a_1≤⋯≤ a_n, then G_Λ, P ≅ ⊕_j=1^n O/π^a_j O, s_Λ( P)=max({0}∪{a_j:1≤ j≤ n}).
Exact mathematical passage
G_{\Lambda,\Pcal} \cong \bigoplus_{j=1}^n\OO/\pi^{a_j}\OO, \qquad s_\Lambda(\Pcal)=\max\bigl(\{0\}\cup\{a_j:1\leq j\leq n\}\bigr).
\bigoplus_{j=1}^rUse in source · 2 passages
Search aliases: ⊕_j=1^r bigoplus
Use in source · 04_arithmetic.tex, line 284
ght] With the notation of [source reference], the map Φ:G_Λ, P ⟶ ⊕_α∈ IE_αΛ/Λ_α, [x]⟼(E_α x+Λ_α)_α is injective, and every coordinate projection of its image is surjective. We also have exp_π(G_Λ, P) = s_Λ( P). If G_Λ, P ≅ ⊕_j=1^r O/π^a_j O with a_1≤⋯≤ a_r, then s_Λ( P)=max({0}∪{a_j:1≤ j≤ r}).
Exact mathematical passage
G_{\Lambda,\Pcal} \cong \bigoplus_{j=1}^r\OO/\pi^{a_j}\OO
\bigoplus_{j=1}^{r}Use in source · 1 passage
Search aliases: ⊕_j=1^r bigoplus
Use in source · 02_foundations.tex, line 954
Invariant factors, Jordan blocks, and cyclicity. [Invariant factor count] If X≠0 and the rational canonical decomposition is written with nonconstant invariant factors X≅⊕_j=1^r𝕜[t]/(f_j), f_1mid f_2mid⋯mid f_r, then srank_𝕜[T](X)=r. [source reference]
Exact mathematical passage
X\cong\bigoplus_{j=1}^{r}\bbk[t]/(f_j), \qquad f_1\mid f_2\mid\cdots\mid f_r,
\bigoplus_{j\in\mathbb Z/N\mathbb Z}Use in source · 1 passage
Search aliases: ⊕_j∈ Z/N Z bigoplus in mathbb mathbb
Use in source · 03_nonlinear_coordinates.tex, line 597
[Cyclic product formula] For a cyclically graded algebra A=⊕_j∈ Z/N ZA_j, A_iA_j⊆ A_i+j, one has P_k(ab)= ∑_i+j=ka_ib_j.
Exact mathematical passage
A=\bigoplus_{j\in\mathbb Z/N\mathbb Z}A_j, \qquad A_iA_j\subseteq A_{i+j},
\bigoplus_{r=0}^{N-1}Use in source · 1 passage
Search aliases: ⊕_r=0^N-1 bigoplus
Use in source · Paper_V_Traditional_Final.tex, line 1819
[One dimensional residue coordinates] For L=N Z and R={0,…,N-1}, x_r(k)=x(Nk+r) and ℓ^p( Z)≅⊕_r=0^N-1ℓ^p( Z) isometrically.
Exact mathematical passage
\ell^p(\ZZ)\cong\bigoplus_{r=0}^{N-1}\ell^p(\ZZ)
\bigoplus_{r=1}^{s}Use in source · 2 passages
Search aliases: ⊕_r=1^s bigoplus
Use in source · 02_strengthened.tex, line 104
Let A ≅ ⊕_r=1^s (M_d_r( C)⊗ I_m_r) act on H ≅ ⊕_r=1^s ( C^d_r⊗ C^m_r). Assume d_r,m_r≥1 for each displayed block. For an empty block family, the seed rank and the displayed finite supremum are defined as zero. A vector in one block is identified with a d_r by m_r coefficient matrix.
Exact mathematical passage
\Acal \cong \bigoplus_{r=1}^{s} \bigl(M_{d_r}(\CC)\otimes I_{m_r}\bigr)
\bigoplus_{r\in R}Use in source · 2 passages
Search aliases: ⊕_r∈ R bigoplus in
Use in source · Paper_V_Traditional_Final.tex, line 1776
For finite p, equip ⊕_r∈ Rℓ^p(L) with (u_r) ^p=∑_r∈ R u_r _p^p. For p=∞, use the maximum norm.
Exact mathematical passage
\bigoplus_{r\in R}\ell^p(L)
\bigoplus_vUse in source · 1 passage
Search aliases: ⊕_v bigoplus
Use in source · 02_foundations.tex, line 86
Because Q_0 is finite, 1=∑_v∈ Q_0e_v in 𝕜 Q, the e_v are pairwise orthogonal, and the same identities hold in Λ_ G. Given a realization, define the action of a path by composition of its arrow maps and extend linearly. Every element of I acts as zero, so the action factors through Λ_ G. The vertex idempotents give |X|=⊕_v e_v|X|=⊕_vX_v. Conversely, if M is unital, then M=1M=∑_ve_vM. Orthogonality makes the sum direct. For an arrow a:v→ w, the identities a=e_wae_v imply a(e_vM)⊆ e_wM. The relatio…
Exact mathematical passage
|X|=\bigoplus_v e_v|X|=\bigoplus_vX_v.
\bigoplus_{v\in\Lcal}Use in source · 1 passage
Search aliases: ⊕_v∈Lcal ⊕_v∈ L bigoplus in Lcal bigoplus in mathcal
Use in source · Paper_V_Traditional_Final.tex, line 2343
[Reconstruction on a pruned packet tree] Every admissible pruning L defines a unitary analysis map A_ L: H_rm root ⟶ ⊕_v∈ L H_v. Its inverse is obtained by applying the adjoints of the local splits from the leaves toward the root. In particular, x ^2 = ∑_v∈ L x_v ^2.
Exact mathematical passage
\Acal_{\Lcal}:\Hcal_{\rm root} \longrightarrow \bigoplus_{v\in\Lcal}\Hcal_v.
\bigoplus_{\alpha}Use in source · 1 passage
Search aliases: ⊕_α bigoplus alpha
Use in source · 01_orientation.tex, line 18
Λ be an O lattice in V. A rational projector E need not preserve Λ. Its separation weight is the least s such that π^sEΛ⊆Λ. A Bézout construction gives an upper bound through a polynomial resultant. That bound need not be exact because it does not see the position of the lattice. The exact answer is controlled by the finite length module G_Λ, P = Λ/⊕_α(Λ∩ V_α), where V=⊕_α V_α is the rational packet decomposition. The largest Smith exponent of this gluing module is exactly the common denominator…
Exact mathematical passage
G_{\Lambda,\Pcal} = \Lambda\Big/\bigoplus_{\alpha}(\Lambda\cap V_\alpha),
\bigoplus_{\alpha\in C}Use in source · 1 passage
Search aliases: ⊕_α∈ C bigoplus alpha in
Use in source · 03_packets_and_graphs.tex, line 856
[Preorder support algebra decomposition] Let R be a preorder on the finite set I. Define α∼β Longleftrightarrow (β,α)∈ R and (α,β)∈ R. Let overline I=I/∼, which carries the induced partial order, and put X_C:=⊕_α∈ CX_α. Let N_R⊆ M_R consist of operators whose blocks vanish whenever source and target belong to the same equivalence class. Then: [label=(roman*)] item N_R is a nilpotent ideal on both sides. item If h is the maximal number of strict steps in a chain of overline I, then N_R^ h+1=0. it…
Exact mathematical passage
X_C:=\bigoplus_{\alpha\in C}X_\alpha.
\bigoplus_{\alpha\in I_0}Use in source · 1 passage
Search aliases: ⊕_α∈ I_0 bigoplus alpha in
Use in source · 03_packets_and_graphs.tex, line 980
Reachability obstructions for seeds. [Reachability obstruction for generated hulls] Let I_0⊆ I and let S⊆⊕_α∈ I_0X_α. Then Hull_ A(S) ⊆ ⊕_β∈Reach(I_0)X_β. In particular, no seed supported outside a source strongly connected component can generate a nonzero vector in that source component. [source reference]
Exact mathematical passage
S\subseteq\bigoplus_{\alpha\in I_0}X_\alpha.
\bigoplus_\betaUse in source · 1 passage
Search aliases: ⊕_β bigoplus beta
Use in source · 04_arithmetic.tex, line 399
[Gluing exact sequence under refinement] Let Q=(Q_β) refine P=(P_α) on V, and let Λ be a lattice. Define Λ_ P = ⊕_α(Λ∩ P_α V), Λ_ Q = ⊕_β(Λ∩ Q_β V). Then Λ_ Q⊆Λ_ P⊆Λ and there is a short exact sequence 0 ⟶ Λ_ P/Λ_ Q ⟶ Λ/Λ_ Q ⟶ Λ/Λ_ P ⟶0. In particular, s_Λ( P)≤ s_Λ( Q).
Exact mathematical passage
\Lambda_{\Pcal} = \bigoplus_\alpha(\Lambda\cap P_\alpha V), \qquad \Lambda_{\Qcal} = \bigoplus_\beta(\Lambda\cap Q_\beta V).
\bigoplus_{\beta\in\Reach(I_0)}Use in source · 1 passage
Search aliases: ⊕_β∈Reach(I_0) bigoplus beta in Reach bigoplus beta in operatorname
Use in source · 03_packets_and_graphs.tex, line 984
Reachability obstructions for seeds. [Reachability obstruction for generated hulls] Let I_0⊆ I and let S⊆⊕_α∈ I_0X_α. Then Hull_ A(S) ⊆ ⊕_β∈Reach(I_0)X_β. In particular, no seed supported outside a source strongly connected component can generate a nonzero vector in that source component. [source reference]
Exact mathematical passage
\boxed{ \Hull_{\A}(S) \subseteq \bigoplus_{\beta\in\Reach(I_0)}X_\beta. }
\bigoplus_{\gamma\in R}Use in source · 1 passage
Search aliases: ⊕_γ∈ R bigoplus gamma in
Use in source · 01_manuscript.tex, line 1064
Let R be the set of packets reachable from α, including α, and set X_R=⊕_γ∈ RX_γ. No active edge leaves R, so AX_R⊆ X_R. The restriction of A to the finite dimensional space X_R is injective. It is therefore onto X_R. Hence A^-1X_R⊆ X_R. Since β∉ R, the stated block of A^-1 vanishes.
Exact mathematical passage
X_R=\bigoplus_{\gamma\in R}X_\gamma.
\bigoplus_\muUse in source · 1 passage
Search aliases: ⊕_μ bigoplus mu
Use in source · 02_strengthened.tex, line 225
Complete reducibility gives a unitary decomposition H ≅ ⊕_μ∈widehat G V_μ⊗ M_μ, U(g) ≅ ⊕_μ π_μ(g)⊗ I_M_μ. On the μ summand, the displayed average is central in the irreducible representation π_μ, so it is a scalar by Schur's lemma. Taking its trace and applying character orthogonality shows that the scalar is one when μ=λ and zero otherwise. The asserted identities follow on the direct sum.
Exact mathematical passage
\Hcal \cong \bigoplus_{\mu\in\widehat G} V_\mu\otimes M_\mu, \qquad U(g) \cong \bigoplus_\mu \pi_\mu(g)\otimes I_{M_\mu}.
\bigoplus_{\mu\in\widehat G}Use in source · 1 passage
Search aliases: ⊕_μ∈widehat G bigoplus mu in widehat
Use in source · 02_strengthened.tex, line 225
Complete reducibility gives a unitary decomposition H ≅ ⊕_μ∈widehat G V_μ⊗ M_μ, U(g) ≅ ⊕_μ π_μ(g)⊗ I_M_μ. On the μ summand, the displayed average is central in the irreducible representation π_μ, so it is a scalar by Schur's lemma. Taking its trace and applying character orthogonality shows that the scalar is one when μ=λ and zero otherwise. The asserted identities follow on the direct sum.
Exact mathematical passage
\Hcal \cong \bigoplus_{\mu\in\widehat G} V_\mu\otimes M_\mu, \qquad U(g) \cong \bigoplus_\mu \pi_\mu(g)\otimes I_{M_\mu}.
\bigoplus_{\sigma(\alpha)=j}Use in source · 1 passage
Search aliases: ⊕_σ(α)=j bigoplus sigma alpha
Use in source · 03_packets_and_graphs.tex, line 52
[Finite packet aggregation] Let σ:I↠ J be a surjection and define Q_j:=∑_σ(α)=jP_α. Then (Q_j)_j∈ J is a finite packet system and Q_jX=⊕_σ(α)=jX_α. [source reference]
Exact mathematical passage
Q_jX=\bigoplus_{\sigma(\alpha)=j}X_\alpha.
\otimes_AUse in source · 1 passage
Search aliases: ⊗_A otimes
Use in source · Paper_XII_Traditional_Final.tex, line 313
Put A=R^loc, B=R^square, and let J⊆ A be the closed local condition ideal. By [source reference], the independently constructed ring R^loc,τ is canonically A/J. The ordinary tensor identity B⊗_A(A/J)≅ B/JB carries the defining ideal of the tensor to the image of m_B. Here the structural map is local by the stated global deformation datum, so it carries m_A into m_B. Completing, and applying the completion and quotient comparison from the preceding proof, gives a bicontinuous isomorphism R^τ≅ B/o…
Exact mathematical passage
B\otimes_A(A/J)\cong B/JB
\otimes_{\bbk}Use in source · 4 passages
Search aliases: ⊗_bbk ⊗_𝕜 otimes bbk otimes Bbbk
Use in source · 02_foundations.tex, line 559
Tensor generation and witnesses for lower bounds. [Tensor product bound] Let A and B be unital algebras over 𝕜, let M be a left module over A, and let N be a left module over B. Under the natural A⊗_𝕜B action, srank_A⊗ B(M⊗_𝕜N) ≤ srank_A(M)srank_B(N). [source reference]
Exact mathematical passage
A\otimes_{\bbk}B
\otimes_\OOUse in source · 1 passage
Search aliases: ⊗_OO ⊗_ O otimes OO otimes mathcal
Use in source · 04_arithmetic.tex, line 121
[Evaluated B ezout spectral projector] Let F,G,U,V∈End_K(Λ⊗_ O K) and let r∈ K^×. Suppose UF+VG=rI, FG=0, FG=GF, GU=UG. Put N=VG. Then N^2=rN. Consequently the operator E=r^-1VG is idempotent, acts as the identity on ker F, and acts as zero on ker G. If V and G preserve Λ, then rE=VG∈End_ O(Λ).
Exact mathematical passage
F,G,U,V\in\End_K(\Lambda\otimes_\OO K)
\otimes_{\OO}Use in source · 1 passage
Search aliases: ⊗_OO ⊗_ O otimes OO otimes mathcal
Use in source · 04_arithmetic.tex, line 4
Let O be a discrete valuation ring with fraction field K and uniformizer π. Let Λ be a finite free O module and put V=K⊗_ OΛ. Suppose A∈End_ O(Λ).
Exact mathematical passage
V=K\otimes_{\OO}\Lambda.
\otimes_PUse in source · 1 passage
Search aliases: ⊗_P otimes
Use in source · 71_native.tex, line 134
coker(Hom_B(Ω_P/ Z⊗_P B, F_p)⟶Hom_B(I/I^2, F_p)).
Exact mathematical passage
\operatorname{coker}\left(\operatorname{Hom}_B(\Omega_{P/\mathbb Z}\otimes_P B,\mathbb F_p)\longrightarrow\operatorname{Hom}_B(I/I^2,\mathbb F_p)\right).
\otimes_RUse in source · 6 passages
Search aliases: ⊗_R otimes
Use in source · Paper_XII_Traditional_Final.tex, line 629
[Higher Fitting strata] Let R be Noetherian and let M be finite. For r≥1, put μ_ p(M) = dim_κ( p) (M⊗_Rκ( p)). Then { p∈Spec R: μ_ p(M)≥ r } = V(Fitt_r-1^R(M)).
Exact mathematical passage
\mu_{\frakp}(M) = \dim_{\kappa(\frakp)} \left(M\otimes_R\kappa(\frakp)\right).
\otimes_{R,x}Use in source · 1 passage
Search aliases: ⊗_R,x otimes
Use in source · Paper_XII_Traditional_Final.tex, line 978
[Field fiber criterion] Let R be a commutative ring, let M be a finite R module, and let x:R→ E' be a homomorphism to a field. Put p=ker x. Then M⊗_R,xE'≠0 Longleftrightarrow p∈Supp_R(M).
Exact mathematical passage
\boxed{ M\otimes_{R,x}E'\neq0 \quad\Longleftrightarrow\quad \frakp\in\Supp_R(M). }
\otimes_{R/I}Use in source · 1 passage
Search aliases: ⊗_R/I otimes
Use in source · Paper_XII_Traditional_Final.tex, line 853
Let q∈Spec(R/I), and let p be its inverse image in R. The composite point R→ R/I→κ(q) has kernel p, so it factors through a field embedding κ( p)→κ(q). The ordinary tensor associator in every degree gives an isomorphism of complexes κ(q)⊗_R/I((R/I)⊗_RP) ≅κ(q)⊗_RP ≅κ(q)⊗_κ( p) (κ( p)⊗_RP). Extension of fields is faithfully flat. It preserves and reflects exactness in each degree, hence also acyclicity. The left-hand complex is thus nonzero in the derived category exactly when the residue field fi…
Exact mathematical passage
\kappa(q)\otimes_{R/I}\bigl((R/I)\otimes_RP\bigr) \cong\kappa(q)\otimes_RP \cong\kappa(q)\otimes_{\kappa(\frakp)} \bigl(\kappa(\frakp)\otimes_RP\bigr).
\otimes_R^{\mathbf L}Use in source · 1 passage
Search aliases: ⊗_R^ L otimes mathbf
Use in source · Paper_XII_Traditional_Final.tex, line 850
Choose a bounded complex P of finite projective R modules representing C. Scalar extension to a commutative R algebra S is computed degree by degree as S⊗_RP. The canonical symmetry of tensor products gives the displayed order C⊗_R^ LS. Scalar extension preserves the bounds, finite generation, and projectivity of the terms. A derived isomorphism between two such projective models lifts to a homotopy equivalence, which scalar extension preserves. The resulting derived object and its residue field…
Exact mathematical passage
C\otimes_R^{\mathbf L}S
\bigotimes_{s\in S}Use in source · 1 passage
Search aliases: ⊗_s∈ S bigotimes in
Use in source · Paper_XI_Traditional_Final.tex, line 1760
[Heisenberg tensor factorization] Fix compatible central characters. The irreducible Heisenberg representation of an orthogonal sum is the tensor product of the irreducible Heisenberg representations of its summands: ω_V ≅ ⊗_s∈ Sω_V_s.
Exact mathematical passage
\boxed{ \omega_V \cong \bigotimes_{s\in S}\omega_{V_s}. }
\otimes_{\TT,x}Use in source · 3 passages
Search aliases: ⊗_TT,x ⊗_ T,x otimes TT otimes mathbb
Use in source · Paper_XII_Traditional_Final.tex, line 1000
[Supported deformation points give nonzero Hecke fibers] Let R≅ T, and let M be a finite Hecke module. If a point x:R→ E' lies in Supp_R(M), then the corresponding Hecke fiber M⊗_ T,xE' is nonzero.
Exact mathematical passage
M\otimes_{\TT,x}E'
\otimes_{\kappa(\frakp)}Use in source · 1 passage
Search aliases: ⊗_κ(frakp) ⊗_κ( p) otimes kappa frakp otimes kappa mathfrak
Use in source · Paper_XII_Traditional_Final.tex, line 853
Let q∈Spec(R/I), and let p be its inverse image in R. The composite point R→ R/I→κ(q) has kernel p, so it factors through a field embedding κ( p)→κ(q). The ordinary tensor associator in every degree gives an isomorphism of complexes κ(q)⊗_R/I((R/I)⊗_RP) ≅κ(q)⊗_RP ≅κ(q)⊗_κ( p) (κ( p)⊗_RP). Extension of fields is faithfully flat. It preserves and reflects exactness in each degree, hence also acyclicity. The left-hand complex is thus nonzero in the derived category exactly when the residue field fi…
Exact mathematical passage
\kappa(q)\otimes_{R/I}\bigl((R/I)\otimes_RP\bigr) \cong\kappa(q)\otimes_RP \cong\kappa(q)\otimes_{\kappa(\frakp)} \bigl(\kappa(\frakp)\otimes_RP\bigr).
\perpUse in source · 7 passages
Search aliases: ⊥ perp
Use in source · 04_correction_transport.tex, line 11
[Orthogonal obstruction projector] Let D:H_1→ H_2 be a bounded operator between Hilbert spaces with closed range. Let D^† be its Moore--Penrose inverse. Then P_ob=I_H_2-DD^† is the orthogonal projector onto (im D)^⊥=Ker D^*. For r∈ H_2, the following are equivalent: item Du=-r is solvable item [r]=0 in coker D item P_obr=0. When these conditions hold, -D^† r is the least norm correction. For every r, dist(r,im D)= P_obr .
Exact mathematical passage
(\im D)^\perp=\Ker D^*.
\bigcap_aUse in source · 1 passage
Search aliases: ⋂_a bigcap
Use in source · 02_strengthened.tex, line 2216
[Physical compression is not a closure test] Let H_phys=⋂_aKer C_a and let P_phys be its orthogonal projector. Then P_phys[C_a,C_b]P_phys=0 for every pair a,b, whether or not the constraint span closes.
Exact mathematical passage
\Hcal_{\phys}=\bigcap_a\Ker C_a
\bigcap_mUse in source · 1 passage
Search aliases: ⋂_m bigcap
Use in source · 03_nonlinear_coordinates.tex, line 75
a normed setting every packet projector and every coarsening operator is bounded. Differentiability of operator-valued maps is taken in operator norm. A Banach space isomorphism is a bounded linear bijection with bounded inverse. For a decreasing filtration (F^mX), completeness means that the canonical map X⟶varprojlim_m X/F^mX is an isomorphism and separatedness means ⋂_mF^mX=0. Statements about congruence classes in a filtered nonlinear problem also assume that the nonlinear map preserves filt…
Exact mathematical passage
\bigcap_mF^mX=0
\bigcap_{m\geq1}Use in source · 3 passages
Search aliases: ⋂_m≥1 bigcap geq
Use in source · Paper_XII_Traditional_Final.tex, line 1155
[Nested packets in one finite carrier] Let V be a finite dimensional vector space. Let e_1,e_2,… be idempotents such that e_m+1e_m=e_me_m+1=e_m+1 for every m. If e_mV≠0 for every m, then the descending chain stabilizes and ⋂_m≥1e_mV ≠0.
Exact mathematical passage
\boxed{ \bigcap_{m\geq1}e_mV \neq0. }
\bigcap_rUse in source · 1 passage
Search aliases: ⋂_r bigcap
Use in source · 04_correction_transport.tex, line 358
Fix r. For every m≥ r, one has Φ(x_m)∈ F^mY⊆ F^rY. The submodule F^rY is open and therefore closed in the filtration topology. Continuity of Φ gives Φ(x_m)→Φ(x_∞), so Φ(x_∞)∈ F^rY. Since this holds for every r and Y is separated, Φ(x_∞)∈⋂_rF^rY=0.
Exact mathematical passage
\Phi(x_\infty)\in\bigcap_rF^rY=0.
\bigcupUse in source · 3 passages
Search aliases: ⋃ bigcup
Use in source · 03_packets_and_graphs.tex, line 1053
Every packet pure generating family is an unrestricted generating family, which gives the first inequality. For each α, choose a minimal generating family S_α⊆ X_α for the corner action P_α A P_α. The union S=⋃_α S_α is packet pure. Since every x∈ X has the unique decomposition x=∑_α x_α and every x_α lies in (P_α A P_α)S_α⊆ AS, the union generates X. Its cardinality is at most the displayed sum.
Exact mathematical passage
S=\bigcup_\alpha S_\alpha
\bigcup_CUse in source · 1 passage
Search aliases: ⋃_C bigcup
Use in source · 03_packets_and_graphs.tex, line 1093
For the reverse inequality, choose for each source component C a packet pure family S_C of size r_C generating X_C under A_C, and put S=⋃_CS_C. Then every source component lies in A S. Traverse the finite condensation graph in a topological order. Suppose all components preceding a component D that is not a source have already been shown to lie in A S. Choose an incoming condensation edge from such a component to D, represented by a graph edge α→β with α already generated and β∈ D. Surjectivity …
Exact mathematical passage
S=\bigcup_CS_C
\bigcup_iUse in source · 1 passage
Search aliases: ⋃_i bigcup
Use in source · Paper_XII_Traditional_Final.tex, line 826
[Cohomological description of derived support] If R is Noetherian and C is perfect, then Supp_R(C) = ⋃_iSupp_R H^i(C). In particular, derived support is closed and is empty exactly when C≃0 [source reference].
Exact mathematical passage
\Supp_R(C) = \bigcup_i\Supp_R H^i(C).
\bigcup_\alphaUse in source · 1 passage
Search aliases: ⋃_α bigcup alpha
Use in source · 03_packets_and_graphs.tex, line 1053
Every packet pure generating family is an unrestricted generating family, which gives the first inequality. For each α, choose a minimal generating family S_α⊆ X_α for the corner action P_α A P_α. The union S=⋃_α S_α is packet pure. Since every x∈ X has the unique decomposition x=∑_α x_α and every x_α lies in (P_α A P_α)S_α⊆ AS, the union generates X. Its cardinality is at most the displayed sum.
Exact mathematical passage
S=\bigcup_\alpha S_\alpha
\rangle^\timesUse in source · 3 passages
Search aliases: ⟩^× rangle times
Use in source · root_scope.tex, line 103
to zero dyadically. Extend A by mathscr A(x)=frac9^x+1-18(x+1)(2x+1), mathscr A(-1)=-fraclog98. The apparent singularity at x=-1 is removable. Since v_2(log9)=3, its convergent expansion is mathscr A(x)=fraclog9/81+2x ∑_h≥0frac(log9)^h(h+1)!(x+1)^h. It follows coefficientwise that mathscr A∈ Z_2⟨ x⟩^×, mathscr A≡1pmod2, and mathscr A(N)=A(N).
Exact mathematical passage
\mathscr A\in\mathbb Z_2\langle x\rangle^\times
\rangle_2Use in source · 1 passage
Search aliases: ⟩_2 rangle
Use in source · supplement_05.tex, line 119
For p=3,ℓ=2,a=2, the checked values are ω_2(2)=8, ⟨2⟩_2=7, 8^2=1,quad8·7=2 in Z/9 Z. The unlifted representative 2 is not a square root of one modulo nine.
Exact mathematical passage
\omega_2(2)=8,\qquad \langle2\rangle_2=7,\qquad 8^2=1,\quad8\cdot7=2\quad\text{in }\Z/9\Z.
\rangle_{2,\ell}Use in source · 2 passages
Search aliases: ⟩_2,ℓ rangle ell
Use in source · supplement_05.tex, line 153
Why the mod-four branch is separate. Now let ℓ≥2 and a∈( Z/2^ℓ Z)^×. The actual units of Z/4 Z are exhausted by 1 and -1. Define ε_ℓ(a)= 1, · a≡1pmod4, -1, · a≡3pmod4, ⟨ a⟩_2,ℓ=ε_ℓ(a)a. The source uses ℓ=L+2. It proves a=ε_ℓ(a)⟨ a⟩_2,ℓ, ε_ℓ(a)∈{1,-1}, ⟨ a⟩_2,ℓ≡1pmod4. The sign is uniquely determined by the residue modulo four, and cancellation gives uniqueness of the principal factor.
Exact mathematical passage
\varepsilon_\ell(a)= \begin{cases}1,&a\equiv1\pmod4,\\-1,&a\equiv3\pmod4,\end{cases} \qquad \langle a\rangle_{2,\ell}=\varepsilon_\ell(a)a.
\rangle_4Use in source · 1 passage
Search aliases: ⟩_4 rangle
Use in source · 10_foundations.tex, line 375
Z/p^L+1 Z)^×. Define ω_L(a)=a^p^L, ⟨ a⟩_L=aω_L(a)^-1. Then a=ω_L(a)⟨ a⟩_L, ω_L(a)^p-1=1, ⟨ a⟩_L≡1pmod p. The two factors are uniquely determined by these properties and are compatible with reduction to lower precision. For a∈( Z/2^L+2 Z)^×, there is instead a unique decomposition a=ϵ(a)⟨ a⟩_4, ϵ(a)∈{1,-1}, ⟨ a⟩_4≡1pmod4.
Exact mathematical passage
a=\epsilon(a)\langle a\rangle_4, \qquad \epsilon(a)\in\{1,-1\},\qquad \langle a\rangle_4\equiv1\pmod4.
\rangle_kUse in source · 1 passage
Search aliases: ⟩_k rangle
Use in source · supplement_05.tex, line 140
pal_mul pageheading3. Precision compatibility and the dyadic signFinite refinements and Proposition 6.2's dyadic branch For 1≤ k≤ℓ, let ρ_ℓ,k: Z/p^ℓ Z⟶ Z/p^k Z be the actual reduction homomorphism. It induces a map on units. The explicit construction satisfies ρ_ℓ,k(ω_ℓ(a)) =ω_k(ρ_ℓ,k(a)), ρ_ℓ,k(⟨ a⟩_ℓ) =⟨ρ_ℓ,k(a)⟩_k. The proof uses the exponent factorization p^ℓ-1=p^k-1p^ℓ-k and the already proved Frobenius invariance of a lifted root. Principal factors also satisfy ⟨ a⟩_ℓ^ p^ℓ-1=1. At precisio…
Exact mathematical passage
\boxed{\rho_{\ell,k}(\langle a\rangle_\ell) =\langle\rho_{\ell,k}(a)\rangle_k.}
\rangle_LUse in source · 2 passages
Search aliases: ⟩_L rangle
Use in source · 10_foundations.tex, line 364
[Finite unit coordinates] Let p be prime, L≥0, and a∈( Z/p^L+1 Z)^×. Define ω_L(a)=a^p^L, ⟨ a⟩_L=aω_L(a)^-1. Then a=ω_L(a)⟨ a⟩_L, ω_L(a)^p-1=1, ⟨ a⟩_L≡1pmod p. The two factors are uniquely determined by these properties and are compatible with reduction to lower precision. For a∈( Z/2^L+2 Z)^×, there is instead a unique decomposition a=ϵ(a)⟨ a⟩_4, ϵ(a)∈{1,-1}, ⟨ a⟩_4≡1pmod4.
Exact mathematical passage
\omega_L(a)=a^{p^L},\qquad \langle a\rangle_L=a\omega_L(a)^{-1}.
\rangle_\piUse in source · 2 passages
Search aliases: ⟩_π rangle pi
Use in source · Paper_VI_Traditional_Final.tex, line 659
[Finite conditional-expectation projector] With the weighted inner product ⟨ f,g⟩_π=∑_iπ_if_ioverlineg_i, the operator P_ Q satisfies P_ Q^2=P_ Q, ⟨ P_ Qf,g⟩_π = ⟨ f,P_ Qg⟩_π. Its range is the space of functions constant on each partition block.
Exact mathematical passage
\langle f,g\rangle_\pi=\sum_i\pi_if_i\overline{g_i},
\rangle_\ellUse in source · 3 passages
Search aliases: ⟩_ℓ rangle ell
Use in source · supplement_05.tex, line 81
ffPadic_coe, .standard_window_integral targetWindowIndependence.residue_independence, .reduced_unique targetWindowIndependence.two_window_addresses_congruent pageheading2. Explicit finite prime-power decompositionProposition 6.2; odd-prime branch Let p be prime, ℓ≥1, and a∈( Z/p^ℓ Z)^×. Define actual units in the same ring by ω_ℓ(a)=a^p^ℓ-1, ⟨ a⟩_ℓ=a ω_ℓ(a)^-1. The Lean source writes ℓ=L+1, so zero precision is excluded without truncated natural-number subtraction.
Exact mathematical passage
\boxed{\omega_\ell(a)=a^{p^{\ell-1}},\qquad \langle a\rangle_\ell=a\,\omega_\ell(a)^{-1}.}
\rangle_\ell^{\,p^{\ell-1}}Use in source · 1 passage
Search aliases: ⟩_ℓ^ p^ℓ-1 rangle ell ell
Use in source · supplement_05.tex, line 146
hism. It induces a map on units. The explicit construction satisfies ρ_ℓ,k(ω_ℓ(a)) =ω_k(ρ_ℓ,k(a)), ρ_ℓ,k(⟨ a⟩_ℓ) =⟨ρ_ℓ,k(a)⟩_k. The proof uses the exponent factorization p^ℓ-1=p^k-1p^ℓ-k and the already proved Frobenius invariance of a lifted root. Principal factors also satisfy ⟨ a⟩_ℓ^ p^ℓ-1=1. At precision one the principal factor is exactly one. These are coherent finite certificates. No infinite inverse limit or topological unit-group equivalence is constructed in this release.
Exact mathematical passage
\langle a\rangle_\ell^{\,p^{\ell-1}}=1.
\mathbb{F}Use in source · 1 passage
Search aliases: 𝔽 mathbb
Use in source · 05_worked_models.tex, line 195
Let 𝔽∈{ R, C} and let X=ℓ^2( Z; F) with a family of bounded reindexing operators indexed by the integers (S_mx)[n]=x[n-m] (m,n∈ Z). Thus S_0=I, S_m+r=S_mS_r, and S_1 is the usual bilateral shift. We use the standard conventions for Hilbert spaces and bounded operators of [source reference]. The impulse δ_0 satisfies S_mδ_0=δ_m for every m∈ Z. Therefore sequences with finite support lie in its algebraic orbit hull and are dense in ℓ^2( Z).
Exact mathematical passage
\mathbb{F}\in\{\R,\C\}
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