Notation by work
Realization Calculus: twin edition
All notation · Glossary · Source coverage
1290 symbol spellings in 4806 mathematical passages across 57 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 1290 entries are available below.
*Notation table · 131 passages
Search aliases: *
Notation table · 07_concordance.tex, line 6
midrule V,F_v,f^* · A declared view category, its fibers, and contravariant restrictions.
Exact mathematical passage
\V,F_v,f^*
+Notation table · 436 passages
Search aliases: +
Notation table · 00_notation.tex, line 17
S_A · The prime set {p_2n+ 1_A(n):n∈ N}.
Exact mathematical passage
\{p_{2n+\one_A(n)}:n\in\N\}
=Notation table · 1425 passages
Search aliases: =
Notation table · 07_concordance.tex, line 20
χ_A,M=SA-BC · Split exact/residual coordinates and the mismatch between chosen repair lifts.
Exact mathematical passage
\chi_A,M=SA-BC
ANotation table · 498 passages
Search aliases: A
Notation table · 07_concordance.tex, line 20
χ_A,M=SA-BC · Split exact/residual coordinates and the mismatch between chosen repair lifts.
Exact mathematical passage
\chi_A,M=SA-BC
aNotation table · 449 passages
Search aliases: a
Notation table · 07_concordance.tex, line 13
c_f,a_v · Coherent nonabelian transition labels and gauge values.
Exact mathematical passage
c_f,a_v
a_vNotation table · 15 passages
Search aliases: a_v
Notation table · 07_concordance.tex, line 13
c_f,a_v · Coherent nonabelian transition labels and gauge values.
Exact mathematical passage
c_f,a_v
a_\alphaNotation table · 60 passages
Search aliases: a_α alpha
Notation table · 07_concordance.tex, line 25
β_T,r(M),a_α · Minimum residual at correction budget M, and the quadratic regularized correction.
Exact mathematical passage
\beta_{T,r}(M),a_\alpha
BNotation table · 341 passages
Search aliases: B
Notation table · 07_concordance.tex, line 20
χ_A,M=SA-BC · Split exact/residual coordinates and the mismatch between chosen repair lifts.
Exact mathematical passage
\chi_A,M=SA-BC
bNotation table · 191 passages
Search aliases: b
Notation table · 07_concordance.tex, line 15
E_b(R) · Realizations equipped with an actual identification of their boundary with b.
Exact mathematical passage
\mathcal E_b(R)
b_nNotation table · 9 passages
Search aliases: b_n
Notation table · 07_concordance.tex, line 21
ω,Q_n,C_n,b_n · A residual, its prefix-product rate, movement coefficient, and summable step budget. Here Q_n is not the dyadic remainder operator Q_N.
Exact mathematical passage
\omega,Q_n,C_n,b_n
CNotation table · 252 passages
Search aliases: C
Notation table · 07_concordance.tex, line 12
C_j,J_j,Q_N,S_N · Coarsener, chosen lift, restored coarse remainder, and retained finite reconstruction.
Exact mathematical passage
C_j,J_j,Q_N,S_N
cNotation table · 180 passages
Search aliases: c
Notation table · 07_concordance.tex, line 13
c_f,a_v · Coherent nonabelian transition labels and gauge values.
Exact mathematical passage
c_f,a_v
c_ANotation table · 9 passages
Search aliases: c_A
Notation table · 00_notation.tex, line 21
c_A(r,t) · The finite cocycle s_A(r)+s_A(t)-s_A(r+t).
Exact mathematical passage
c_A(r,t)
c_fNotation table · 23 passages
Search aliases: c_f
Notation table · 07_concordance.tex, line 13
c_f,a_v · Coherent nonabelian transition labels and gauge values.
Exact mathematical passage
c_f,a_v
C_jNotation table · 6 passages
Search aliases: C_j
Notation table · 07_concordance.tex, line 12
C_j,J_j,Q_N,S_N · Coarsener, chosen lift, restored coarse remainder, and retained finite reconstruction.
Exact mathematical passage
C_j,J_j,Q_N,S_N
C_nNotation table · 13 passages
Search aliases: C_n
Notation table · 07_concordance.tex, line 21
ω,Q_n,C_n,b_n · A residual, its prefix-product rate, movement coefficient, and summable step budget. Here Q_n is not the dyadic remainder operator Q_N.
Exact mathematical passage
\omega,Q_n,C_n,b_n
C_pNotation table · 22 passages
Search aliases: C_p
Notation table · 00_notation.tex, line 11
midrule C_p= F_p · The additive cyclic group of prime order p.
Exact mathematical passage
C_p=\F_p
DNotation table · 323 passages
Search aliases: D
Notation table · 07_concordance.tex, line 9
δ,∂,d,D · Respectively the view differential, tower differential, a model residual, and a Hilbert defect operator. Each target is declared where used.
Exact mathematical passage
\delta,\partial,d,D
\mathcal DNotation table · 14 passages
Search aliases: D mathcal
Notation table · 07_concordance.tex, line 27
v,g,e, D_α · Fit residual, Hilbert normal residual, optimizer error, and regularized dual objective in the solver chapters.
Exact mathematical passage
v,g,e,\mathcal D_\alpha
dNotation table · 226 passages
Search aliases: d
Notation table · 07_concordance.tex, line 9
δ,∂,d,D · Respectively the view differential, tower differential, a model residual, and a Hilbert defect operator. Each target is declared where used.
Exact mathematical passage
\delta,\partial,d,D
D_BNotation table · 20 passages
Search aliases: D_B
Notation table · 00_notation.tex, line 15
D_B · The subgroup supported on prime indices in B⊆ N.
Exact mathematical passage
D_B
d_WNotation table · 9 passages
Search aliases: d_W
Notation table · 07_concordance.tex, line 22
ν_B,d_W,d_W^sym · Residual floor, extended directed transport distance, and its symmetric maximum.
Exact mathematical passage
\nu_B,d_W,d_W^{\mathrm{sym}}
d_W^{\mathrm{sym}}Notation table · 4 passages
Search aliases: d_W^sym mathrm
Notation table · 07_concordance.tex, line 22
ν_B,d_W,d_W^sym · Residual floor, extended directed transport distance, and its symmetric maximum.
Exact mathematical passage
\nu_B,d_W,d_W^{\mathrm{sym}}
\mathcal D_\alphaNotation table · 9 passages
Search aliases: D_α mathcal alpha
Notation table · 07_concordance.tex, line 27
v,g,e, D_α · Fit residual, Hilbert normal residual, optimizer error, and regularized dual objective in the solver chapters.
Exact mathematical passage
v,g,e,\mathcal D_\alpha
\mathcal ENotation table · 4 passages
Search aliases: E mathcal
Notation table · 07_concordance.tex, line 15
E_b(R) · Realizations equipped with an actual identification of their boundary with b.
Exact mathematical passage
\mathcal E_b(R)
eNotation table · 95 passages
Search aliases: e
Notation table · 07_concordance.tex, line 27
v,g,e, D_α · Fit residual, Hilbert normal residual, optimizer error, and regularized dual objective in the solver chapters.
Exact mathematical passage
v,g,e,\mathcal D_\alpha
\mathcal E_bNotation table · 4 passages
Search aliases: E_b mathcal
Notation table · 07_concordance.tex, line 15
E_b(R) · Realizations equipped with an actual identification of their boundary with b.
Exact mathematical passage
\mathcal E_b(R)
e_pNotation table · 10 passages
Search aliases: e_p
Notation table · 00_notation.tex, line 22
e_p · The vector supported at p with value one.
Exact mathematical passage
e_p
\eqfinNotation table · 11 passages
Search aliases: eqfin =^*
Notation table · 00_notation.tex, line 24
A=^* B · The symmetric difference Atriangle B is finite.
Exact mathematical passage
A\eqfin B
\operatorname{EventualHom}Notation table · 3 passages
Search aliases: EventualHom operatorname
Notation table · 07_concordance.tex, line 19
ϕ,EventualHom · A monotone source schedule and transport classes modulo eventual equality.
Exact mathematical passage
\phi,\operatorname{EventualHom}
\ExtNotation table · 12 passages
Search aliases: Ext
Notation table · 07_concordance.tex, line 8
Γ_U(F),Ext_U(s) · Partial sections on a full patch and their global extension fiber.
Exact mathematical passage
\Gamma_U(F),\Ext_U(s)
\Ext_UNotation table · 11 passages
Search aliases: Ext_U Ext
Notation table · 07_concordance.tex, line 8
Γ_U(F),Ext_U(s) · Partial sections on a full patch and their global extension fiber.
Exact mathematical passage
\Gamma_U(F),\Ext_U(s)
FNotation table · 91 passages
Search aliases: F
Notation table · 07_concordance.tex, line 6
midrule V,F_v,f^* · A declared view category, its fibers, and contravariant restrictions.
Exact mathematical passage
\V,F_v,f^*
\FNotation table · 7 passages
Search aliases: F mathbb
Notation table · 00_notation.tex, line 11
midrule C_p= F_p · The additive cyclic group of prime order p.
Exact mathematical passage
C_p=\F_p
fNotation table · 254 passages
Search aliases: f
Notation table · 07_concordance.tex, line 6
midrule V,F_v,f^* · A declared view category, its fibers, and contravariant restrictions.
Exact mathematical passage
\V,F_v,f^*
f^*Notation table · 39 passages
Search aliases: f^*
Notation table · 07_concordance.tex, line 6
midrule V,F_v,f^* · A declared view category, its fibers, and contravariant restrictions.
Exact mathematical passage
\V,F_v,f^*
\F_pNotation table · 4 passages
Search aliases: F_p F mathbb
Notation table · 00_notation.tex, line 11
midrule C_p= F_p · The additive cyclic group of prime order p.
Exact mathematical passage
C_p=\F_p
F_vNotation table · 7 passages
Search aliases: F_v
Notation table · 07_concordance.tex, line 6
midrule V,F_v,f^* · A declared view category, its fibers, and contravariant restrictions.
Exact mathematical passage
\V,F_v,f^*
GNotation table · 183 passages
Search aliases: G
Notation table · 07_concordance.tex, line 14
K_G,J_rm comp · The cocycle patch groupoid and its composition-support topology.
Exact mathematical passage
\mathfrak K_G,J_{\rm comp}
\mathfrak GNotation table · 13 passages
Search aliases: G mathfrak
Notation table · 00_notation.tex, line 28
K_ G(G) · Least admissible finite description length, or +∞.
Exact mathematical passage
K_{\mathfrak G}(G)
gNotation table · 97 passages
Search aliases: g
Notation table · 07_concordance.tex, line 27
v,g,e, D_α · Fit residual, Hilbert normal residual, optimizer error, and regularized dual objective in the solver chapters.
Exact mathematical passage
v,g,e,\mathcal D_\alpha
G_ANotation table · 90 passages
Search aliases: G_A
Notation table · 00_notation.tex, line 19
G_A · The group q^-1( Q q(u_A)).
Exact mathematical passage
G_A
HNotation table · 179 passages
Search aliases: H
Notation table · 07_concordance.tex, line 11
o_U,H_U · The existence obstruction and the group of invisible coherent changes. Additive and nonabelian conventions are stated separately.
Exact mathematical passage
\mathfrak o_U,H_U
\mathcal HNotation table · 12 passages
Search aliases: H mathcal
Notation table · 07_concordance.tex, line 17
H,q,P · Root residual, an orthogonal kernel projector, and the explicitly defined model repair. A graph repair need not be orthogonal in the global maximum norm.
Exact mathematical passage
\mathcal H,q,P
H_UNotation table · 19 passages
Search aliases: H_U
Notation table · 07_concordance.tex, line 11
o_U,H_U · The existence obstruction and the group of invisible coherent changes. Additive and nonabelian conventions are stated separately.
Exact mathematical passage
\mathfrak o_U,H_U
H_WNotation table · 29 passages
Search aliases: H_W
Notation table · 00_notation.tex, line 16
H_W=q^-1(W) · The intermediate group associated to a rational subspace W≤ V.
Exact mathematical passage
H_W=q^{-1}(W)
iNotation table · 134 passages
Search aliases: i
Notation table · 07_concordance.tex, line 16
T_i,K_ij,R_p · Transport to the root, the corresponding root loop operator, and reconstruction from the root.
Exact mathematical passage
T_i,K_{ij},R_p
JNotation table · 61 passages
Search aliases: J
Notation table · 07_concordance.tex, line 12
C_j,J_j,Q_N,S_N · Coarsener, chosen lift, restored coarse remainder, and retained finite reconstruction.
Exact mathematical passage
C_j,J_j,Q_N,S_N
jNotation table · 163 passages
Search aliases: j
Notation table · 07_concordance.tex, line 12
C_j,J_j,Q_N,S_N · Coarsener, chosen lift, restored coarse remainder, and retained finite reconstruction.
Exact mathematical passage
C_j,J_j,Q_N,S_N
J_jNotation table · 11 passages
Search aliases: J_j
Notation table · 07_concordance.tex, line 12
C_j,J_j,Q_N,S_N · Coarsener, chosen lift, restored coarse remainder, and retained finite reconstruction.
Exact mathematical passage
C_j,J_j,Q_N,S_N
J_{\rm comp}Notation table · 1 passage
Search aliases: J_rm comp rm
Notation table · 07_concordance.tex, line 14
K_G,J_rm comp · The cocycle patch groupoid and its composition-support topology.
Exact mathematical passage
\mathfrak K_G,J_{\rm comp}
KNotation table · 102 passages
Search aliases: K
Notation table · 07_concordance.tex, line 16
T_i,K_ij,R_p · Transport to the root, the corresponding root loop operator, and reconstruction from the root.
Exact mathematical passage
T_i,K_{ij},R_p
\mathfrak KNotation table · 5 passages
Search aliases: K mathfrak
Notation table · 07_concordance.tex, line 14
K_G,J_rm comp · The cocycle patch groupoid and its composition-support topology.
Exact mathematical passage
\mathfrak K_G,J_{\rm comp}
K_{\mathfrak G}Notation table · 6 passages
Search aliases: K_ G mathfrak
Notation table · 00_notation.tex, line 28
K_ G(G) · Least admissible finite description length, or +∞.
Exact mathematical passage
K_{\mathfrak G}(G)
\mathfrak K_GNotation table · 3 passages
Search aliases: K_G mathfrak
Notation table · 07_concordance.tex, line 14
K_G,J_rm comp · The cocycle patch groupoid and its composition-support topology.
Exact mathematical passage
\mathfrak K_G,J_{\rm comp}
K_{ij}Notation table · 5 passages
Search aliases: K_ij
Notation table · 07_concordance.tex, line 16
T_i,K_ij,R_p · Transport to the root, the corresponding root loop operator, and reconstruction from the root.
Exact mathematical passage
T_i,K_{ij},R_p
\kerNotation table · 133 passages
Search aliases: ker
Notation table · 07_concordance.tex, line 24
H,T,r · In the budget chapters, H=ker p with its inherited Hilbert metric, T=d|_H, and r=df for the true raw residual. Measured data are declared separately.
Exact mathematical passage
H=\ker p
LNotation table · 134 passages
Search aliases: L
Notation table · 07_concordance.tex, line 10
p,p_U,p_L · A protected observable, restriction to prescribed views, and an innovation observable depending on the splitting.
Exact mathematical passage
p,p_U,p_L
MNotation table · 200 passages
Search aliases: M
Notation table · 07_concordance.tex, line 20
χ_A,M=SA-BC · Split exact/residual coordinates and the mismatch between chosen repair lifts.
Exact mathematical passage
\chi_A,M=SA-BC
mNotation table · 354 passages
Search aliases: m
Notation table · 07_concordance.tex, line 14
K_G,J_rm comp · The cocycle patch groupoid and its composition-support topology.
Exact mathematical passage
\mathfrak K_G,J_{\rm comp}
\midNotation table · 24 passages
Search aliases: mid
Notation table · 00_notation.tex, line 23
pmid_G g · There exists y∈ G with py=g.
Exact mathematical passage
p\mid_G g
\mid_GNotation table · 1 passage
Search aliases: mid_G mid
Notation table · 00_notation.tex, line 23
pmid_G g · There exists y∈ G with py=g.
Exact mathematical passage
p\mid_G g
NNotation table · 192 passages
Search aliases: N
Notation table · 07_concordance.tex, line 12
C_j,J_j,Q_N,S_N · Coarsener, chosen lift, restored coarse remainder, and retained finite reconstruction.
Exact mathematical passage
C_j,J_j,Q_N,S_N
\NNotation table · 36 passages
Search aliases: N N mathbb
Notation table · 00_notation.tex, line 15
D_B · The subgroup supported on prime indices in B⊆ N.
Exact mathematical passage
B\subseteq\N
nNotation table · 572 passages
Search aliases: n
Notation table · 07_concordance.tex, line 21
ω,Q_n,C_n,b_n · A residual, its prefix-product rate, movement coefficient, and summable step budget. Here Q_n is not the dyadic remainder operator Q_N.
Exact mathematical passage
\omega,Q_n,C_n,b_n
\mathfrak oNotation table · 18 passages
Search aliases: o mathfrak
Notation table · 07_concordance.tex, line 11
o_U,H_U · The existence obstruction and the group of invisible coherent changes. Additive and nonabelian conventions are stated separately.
Exact mathematical passage
\mathfrak o_U,H_U
oNotation table · 27 passages
Search aliases: o
Notation table · 07_concordance.tex, line 14
K_G,J_rm comp · The cocycle patch groupoid and its composition-support topology.
Exact mathematical passage
\mathfrak K_G,J_{\rm comp}
\mathfrak o_UNotation table · 18 passages
Search aliases: o_U mathfrak
Notation table · 07_concordance.tex, line 11
o_U,H_U · The existence obstruction and the group of invisible coherent changes. Additive and nonabelian conventions are stated separately.
Exact mathematical passage
\mathfrak o_U,H_U
\oneNotation table · 6 passages
Search aliases: one 1 mathbf
Notation table · 00_notation.tex, line 17
S_A · The prime set {p_2n+ 1_A(n):n∈ N}.
Exact mathematical passage
\{p_{2n+\one_A(n)}:n\in\N\}
\one_ANotation table · 3 passages
Search aliases: one_A 1_A one mathbf
Notation table · 00_notation.tex, line 17
S_A · The prime set {p_2n+ 1_A(n):n∈ N}.
Exact mathematical passage
\{p_{2n+\one_A(n)}:n\in\N\}
PNotation table · 230 passages
Search aliases: P
Notation table · 07_concordance.tex, line 17
H,q,P · Root residual, an orthogonal kernel projector, and the explicitly defined model repair. A graph repair need not be orthogonal in the global maximum norm.
Exact mathematical passage
\mathcal H,q,P
pNotation table · 292 passages
Search aliases: p
Notation table · 07_concordance.tex, line 10
p,p_U,p_L · A protected observable, restriction to prescribed views, and an innovation observable depending on the splitting.
Exact mathematical passage
p,p_U,p_L
p_{2n+\one_A(n)}Notation table · 1 passage
Search aliases: p_2n+one_A(n) p_2n+ 1_A(n) one mathbf
Notation table · 00_notation.tex, line 17
S_A · The prime set {p_2n+ 1_A(n):n∈ N}.
Exact mathematical passage
\{p_{2n+\one_A(n)}:n\in\N\}
p_LNotation table · 17 passages
Search aliases: p_L
Notation table · 07_concordance.tex, line 10
p,p_U,p_L · A protected observable, restriction to prescribed views, and an innovation observable depending on the splitting.
Exact mathematical passage
p,p_U,p_L
p_UNotation table · 16 passages
Search aliases: p_U
Notation table · 07_concordance.tex, line 10
p,p_U,p_L · A protected observable, restriction to prescribed views, and an innovation observable depending on the splitting.
Exact mathematical passage
p,p_U,p_L
QNotation table · 61 passages
Search aliases: Q
Notation table · 07_concordance.tex, line 12
C_j,J_j,Q_N,S_N · Coarsener, chosen lift, restored coarse remainder, and retained finite reconstruction.
Exact mathematical passage
C_j,J_j,Q_N,S_N
\QNotation table · 42 passages
Search aliases: Q Q mathbb
Notation table · 00_notation.tex, line 19
G_A · The group q^-1( Q q(u_A)).
Exact mathematical passage
q^{-1}(\Q q(u_A))
qNotation table · 216 passages
Search aliases: q
Notation table · 07_concordance.tex, line 17
H,q,P · Root residual, an orthogonal kernel projector, and the explicitly defined model repair. A graph repair need not be orthogonal in the global maximum norm.
Exact mathematical passage
\mathcal H,q,P
q^{-1}Notation table · 9 passages
Search aliases: q^-1
Notation table · 00_notation.tex, line 16
H_W=q^-1(W) · The intermediate group associated to a rational subspace W≤ V.
Exact mathematical passage
H_W=q^{-1}(W)
Q_NNotation table · 33 passages
Search aliases: Q_N
Notation table · 07_concordance.tex, line 12
C_j,J_j,Q_N,S_N · Coarsener, chosen lift, restored coarse remainder, and retained finite reconstruction.
Exact mathematical passage
C_j,J_j,Q_N,S_N
Q_nNotation table · 14 passages
Search aliases: Q_n
Notation table · 07_concordance.tex, line 21
ω,Q_n,C_n,b_n · A residual, its prefix-product rate, movement coefficient, and summable step budget. Here Q_n is not the dyadic remainder operator Q_N.
Exact mathematical passage
\omega,Q_n,C_n,b_n
RNotation table · 147 passages
Search aliases: R
Notation table · 07_concordance.tex, line 15
E_b(R) · Realizations equipped with an actual identification of their boundary with b.
Exact mathematical passage
\mathcal E_b(R)
rNotation table · 355 passages
Search aliases: r
Notation table · 07_concordance.tex, line 24
H,T,r · In the budget chapters, H=ker p with its inherited Hilbert metric, T=d|_H, and r=df for the true raw residual. Measured data are declared separately.
Exact mathematical passage
H,T,r
R_pNotation table · 19 passages
Search aliases: R_p
Notation table · 07_concordance.tex, line 16
T_i,K_ij,R_p · Transport to the root, the corresponding root loop operator, and reconstruction from the root.
Exact mathematical passage
T_i,K_{ij},R_p
\radNotation table · 13 passages
Search aliases: rad operatorname
Notation table · 00_notation.tex, line 25
rad(m) · The product of the distinct prime divisors of m, with rad(1)=1.
Exact mathematical passage
\rad(m)
\rmNotation table · 4 passages
Search aliases: rm
Notation table · 07_concordance.tex, line 14
K_G,J_rm comp · The cocycle patch groupoid and its composition-support topology.
Exact mathematical passage
\mathfrak K_G,J_{\rm comp}
SNotation table · 116 passages
Search aliases: S
Notation table · 07_concordance.tex, line 12
C_j,J_j,Q_N,S_N · Coarsener, chosen lift, restored coarse remainder, and retained finite reconstruction.
Exact mathematical passage
C_j,J_j,Q_N,S_N
sNotation table · 149 passages
Search aliases: s
Notation table · 07_concordance.tex, line 8
Γ_U(F),Ext_U(s) · Partial sections on a full patch and their global extension fiber.
Exact mathematical passage
\Gamma_U(F),\Ext_U(s)
S_ANotation table · 20 passages
Search aliases: S_A
Notation table · 00_notation.tex, line 17
S_A · The prime set {p_2n+ 1_A(n):n∈ N}.
Exact mathematical passage
S_A
s_ANotation table · 32 passages
Search aliases: s_A
Notation table · 00_notation.tex, line 20
s_A(r) · The specified rational lift of r q(u_A) to P.
Exact mathematical passage
s_A(r)
S_NNotation table · 10 passages
Search aliases: S_N
Notation table · 07_concordance.tex, line 12
C_j,J_j,Q_N,S_N · Coarsener, chosen lift, restored coarse remainder, and retained finite reconstruction.
Exact mathematical passage
C_j,J_j,Q_N,S_N
\mathrm{sym}Notation table · 4 passages
Search aliases: sym mathrm
Notation table · 07_concordance.tex, line 22
ν_B,d_W,d_W^sym · Residual floor, extended directed transport distance, and its symmetric maximum.
Exact mathematical passage
\nu_B,d_W,d_W^{\mathrm{sym}}
TNotation table · 255 passages
Search aliases: T
Notation table · 07_concordance.tex, line 16
T_i,K_ij,R_p · Transport to the root, the corresponding root loop operator, and reconstruction from the root.
Exact mathematical passage
T_i,K_{ij},R_p
tNotation table · 152 passages
Search aliases: t
Notation table · 00_notation.tex, line 21
c_A(r,t) · The finite cocycle s_A(r)+s_A(t)-s_A(r+t).
Exact mathematical passage
c_A(r,t)
T_iNotation table · 10 passages
Search aliases: T_i
Notation table · 07_concordance.tex, line 16
T_i,K_ij,R_p · Transport to the root, the corresponding root loop operator, and reconstruction from the root.
Exact mathematical passage
T_i,K_{ij},R_p
\triangleNotation table · 1 passage
Search aliases: triangle
Notation table · 00_notation.tex, line 24
A=^* B · The symmetric difference Atriangle B is finite.
Exact mathematical passage
A\triangle B
UNotation table · 163 passages
Search aliases: U
Notation table · 07_concordance.tex, line 8
Γ_U(F),Ext_U(s) · Partial sections on a full patch and their global extension fiber.
Exact mathematical passage
\Gamma_U(F),\Ext_U(s)
uNotation table · 165 passages
Search aliases: u
Notation table · 00_notation.tex, line 18
u_A · The coordinate indicator of S_A in P.
Exact mathematical passage
u_A
u_ANotation table · 42 passages
Search aliases: u_A
Notation table · 00_notation.tex, line 18
u_A · The coordinate indicator of S_A in P.
Exact mathematical passage
u_A
VNotation table · 21 passages
Search aliases: V
Notation table · 00_notation.tex, line 14
q:P→ V · The quotient map to V=P/D.
Exact mathematical passage
q:P\to V
\VNotation table · 18 passages
Search aliases: V
Notation table · 07_concordance.tex, line 6
midrule V,F_v,f^* · A declared view category, its fibers, and contravariant restrictions.
Exact mathematical passage
\V,F_v,f^*
vNotation table · 198 passages
Search aliases: v
Notation table · 07_concordance.tex, line 6
midrule V,F_v,f^* · A declared view category, its fibers, and contravariant restrictions.
Exact mathematical passage
\V,F_v,f^*
WNotation table · 109 passages
Search aliases: W
Notation table · 07_concordance.tex, line 18
π_W,U,H_U · The map forgetting part of the prescription and the corresponding invisible coherent ambiguity.
Exact mathematical passage
\pi_{W,U},H_U
\widehat GNotation table · 2 passages
Search aliases: widehat G widehat
Notation table · 00_notation.tex, line 26
widehat G · The profinite completion, not the Pontryagin dual.
Exact mathematical passage
\widehat G
XNotation table · 133 passages
Search aliases: X
Notation table · 00_notation.tex, line 27
A≤_T X · Membership in A is computable using oracle X.
Exact mathematical passage
A\le_T X
xNotation table · 568 passages
Search aliases: x
Notation table · 07_concordance.tex, line 23
c(x,y),κ(x,y) · The two-bit pair phase and its antisymmetric scalar-gauge invariant.
Exact mathematical passage
c(x,y),\kappa(x,y)
yNotation table · 229 passages
Search aliases: y
Notation table · 07_concordance.tex, line 23
c(x,y),κ(x,y) · The two-bit pair phase and its antisymmetric scalar-gauge invariant.
Exact mathematical passage
c(x,y),\kappa(x,y)
|Notation table · 53 passages
Search aliases: |
Notation table · 07_concordance.tex, line 24
H,T,r · In the budget chapters, H=ker p with its inherited Hilbert metric, T=d|_H, and r=df for the true raw residual. Measured data are declared separately.
Exact mathematical passage
T=d|_H
|_HNotation table · 3 passages
Search aliases: |_H
Notation table · 07_concordance.tex, line 24
H,T,r · In the budget chapters, H=ker p with its inherited Hilbert metric, T=d|_H, and r=df for the true raw residual. Measured data are declared separately.
Exact mathematical passage
T=d|_H
\alphaNotation table · 175 passages
Search aliases: α alpha
Notation table · 07_concordance.tex, line 25
β_T,r(M),a_α · Minimum residual at correction budget M, and the quadratic regularized correction.
Exact mathematical passage
\beta_{T,r}(M),a_\alpha
\betaNotation table · 40 passages
Search aliases: β beta
Notation table · 07_concordance.tex, line 25
β_T,r(M),a_α · Minimum residual at correction budget M, and the quadratic regularized correction.
Exact mathematical passage
\beta_{T,r}(M),a_\alpha
\beta_{T,r}Notation table · 6 passages
Search aliases: β_T,r beta
Notation table · 07_concordance.tex, line 25
β_T,r(M),a_α · Minimum residual at correction budget M, and the quadratic regularized correction.
Exact mathematical passage
\beta_{T,r}(M),a_\alpha
\GammaNotation table · 45 passages
Search aliases: Γ Gamma
Notation table · 07_concordance.tex, line 7
Γ(F) · The strict set of coherent global families.
Exact mathematical passage
\Gamma(F)
\Gamma_UNotation table · 8 passages
Search aliases: Γ_U Gamma
Notation table · 07_concordance.tex, line 8
Γ_U(F),Ext_U(s) · Partial sections on a full patch and their global extension fiber.
Exact mathematical passage
\Gamma_U(F),\Ext_U(s)
\deltaNotation table · 61 passages
Search aliases: δ delta
Notation table · 07_concordance.tex, line 9
δ,∂,d,D · Respectively the view differential, tower differential, a model residual, and a Hilbert defect operator. Each target is declared where used.
Exact mathematical passage
\delta,\partial,d,D
\kappaNotation table · 11 passages
Search aliases: κ kappa
Notation table · 07_concordance.tex, line 23
c(x,y),κ(x,y) · The two-bit pair phase and its antisymmetric scalar-gauge invariant.
Exact mathematical passage
c(x,y),\kappa(x,y)
\nuNotation table · 11 passages
Search aliases: ν nu
Notation table · 07_concordance.tex, line 22
ν_B,d_W,d_W^sym · Residual floor, extended directed transport distance, and its symmetric maximum.
Exact mathematical passage
\nu_B,d_W,d_W^{\mathrm{sym}}
\nu_BNotation table · 7 passages
Search aliases: ν_B nu
Notation table · 07_concordance.tex, line 22
ν_B,d_W,d_W^sym · Residual floor, extended directed transport distance, and its symmetric maximum.
Exact mathematical passage
\nu_B,d_W,d_W^{\mathrm{sym}}
\piNotation table · 19 passages
Search aliases: π pi
Notation table · 07_concordance.tex, line 18
π_W,U,H_U · The map forgetting part of the prescription and the corresponding invisible coherent ambiguity.
Exact mathematical passage
\pi_{W,U},H_U
\pi_{W,U}Notation table · 6 passages
Search aliases: π_W,U pi
Notation table · 07_concordance.tex, line 18
π_W,U,H_U · The map forgetting part of the prescription and the corresponding invisible coherent ambiguity.
Exact mathematical passage
\pi_{W,U},H_U
\phiNotation table · 84 passages
Search aliases: ϕ phi
Notation table · 07_concordance.tex, line 19
ϕ,EventualHom · A monotone source schedule and transport classes modulo eventual equality.
Exact mathematical passage
\phi,\operatorname{EventualHom}
\chiNotation table · 3 passages
Search aliases: χ chi
Notation table · 07_concordance.tex, line 20
χ_A,M=SA-BC · Split exact/residual coordinates and the mismatch between chosen repair lifts.
Exact mathematical passage
\chi_A,M=SA-BC
\chi_ANotation table · 2 passages
Search aliases: χ_A chi
Notation table · 07_concordance.tex, line 20
χ_A,M=SA-BC · Split exact/residual coordinates and the mismatch between chosen repair lifts.
Exact mathematical passage
\chi_A,M=SA-BC
\omegaNotation table · 33 passages
Search aliases: ω omega
Notation table · 07_concordance.tex, line 21
ω,Q_n,C_n,b_n · A residual, its prefix-product rate, movement coefficient, and summable step budget. Here Q_n is not the dyadic remainder operator Q_N.
Exact mathematical passage
\omega,Q_n,C_n,b_n
\toNotation table · 212 passages
Search aliases: → to
Notation table · 00_notation.tex, line 14
q:P→ V · The quotient map to V=P/D.
Exact mathematical passage
q:P\to V
\partialNotation table · 24 passages
Search aliases: ∂ partial
Notation table · 07_concordance.tex, line 9
δ,∂,d,D · Respectively the view differential, tower differential, a model residual, and a Hilbert defect operator. Each target is declared where used.
Exact mathematical passage
\delta,\partial,d,D
\inNotation table · 300 passages
Search aliases: ∈ in
Notation table · 00_notation.tex, line 17
S_A · The prime set {p_2n+ 1_A(n):n∈ N}.
Exact mathematical passage
\{p_{2n+\one_A(n)}:n\in\N\}
\prodNotation table · 20 passages
Search aliases: ∏ prod
Notation table · 00_notation.tex, line 13
P=∏_p C_p · All prime-coordinate vectors, with product topology.
Exact mathematical passage
P=\prod_p C_p
\prod_pNotation table · 3 passages
Search aliases: ∏_p prod
Notation table · 00_notation.tex, line 13
P=∏_p C_p · All prime-coordinate vectors, with product topology.
Exact mathematical passage
P=\prod_p C_p
\inftyNotation table · 66 passages
Search aliases: ∞ infty
Notation table · 00_notation.tex, line 28
K_ G(G) · Least admissible finite description length, or +∞.
Exact mathematical passage
+\infty
\simeqNotation table · 17 passages
Search aliases: ≃ simeq
Notation table · 07_concordance.tex, line 29
≃,≃_L · A specified equivalence, and a continuous linear equivalence. Neither notation alone asserts an isometry.
Exact mathematical passage
\simeq,\simeq_L
\simeq_LNotation table · 5 passages
Search aliases: ≃_L simeq
Notation table · 07_concordance.tex, line 29
≃,≃_L · A specified equivalence, and a continuous linear equivalence. Neither notation alone asserts an isometry.
Exact mathematical passage
\simeq,\simeq_L
\leNotation table · 303 passages
Search aliases: ≤ le
Notation table · 00_notation.tex, line 16
H_W=q^-1(W) · The intermediate group associated to a rational subspace W≤ V.
Exact mathematical passage
W\le V
\le_TNotation table · 5 passages
Search aliases: ≤_T le
Notation table · 00_notation.tex, line 27
A≤_T X · Membership in A is computable using oracle X.
Exact mathematical passage
A\le_T X
\subseteqNotation table · 49 passages
Search aliases: ⊆ subseteq
Notation table · 00_notation.tex, line 15
D_B · The subgroup supported on prime indices in B⊆ N.
Exact mathematical passage
B\subseteq\N
\bigoplusNotation table · 6 passages
Search aliases: ⊕ bigoplus
Notation table · 00_notation.tex, line 12
D=⊕_p C_p · Finite-support prime-coordinate vectors.
Exact mathematical passage
D=\bigoplus_p C_p
\bigoplus_pNotation table · 2 passages
Search aliases: ⊕_p bigoplus
Notation table · 00_notation.tex, line 12
D=⊕_p C_p · Finite-support prime-coordinate vectors.
Exact mathematical passage
D=\bigoplus_p C_p
\mathcal CDefinition passage · 7 passages
Search aliases: C mathcal
Definition passage: Marked realization groupoid · 04_symmetry.tex, line 393
[Marked realization groupoid] The objects of E_b(R) are pairs (x,ϕ), with x∈ C and ϕ:Rxxrightarrow∼b. An arrow (x,ϕ)→(y,ψ) is an arrow a:x→ y with ψ R(a)=ϕ. Composition and inverses preserve this equation, so this is a groupoid. It is the explicit groupoid homotopy fiber construction, without invoking a higher categorical interface.
Exact mathematical passage
x\in\mathcal C
c_{1_v}Definition passage · 3 passages
Search aliases: c_1_v
Definition passage: Cocycle · 04_symmetry.tex, line 8
[Cocycle] A cocycle is a family c_f∈ G_v for every f:v→ w, satisfying c_1_v=1, c_g∘ f=c_f f^*(c_g). It determines a twisted view system with maps T_c(f)(x)=c_f f^*(x). A section of the twisted system satisfies x_v=c_f f^*(x_w). The cocycle law says that the transition rules compose coherently. It does not assert that the twisted system has a section. The coherent transition system and a global trivialization of that system are different objects.
Exact mathematical passage
c_{1_v}=1,\qquad c_{g\circ f}=c_f f^*(c_g).
c_gDefinition passage · 4 passages
Search aliases: c_g
Definition passage: Cocycle · 04_symmetry.tex, line 8
[Cocycle] A cocycle is a family c_f∈ G_v for every f:v→ w, satisfying c_1_v=1, c_g∘ f=c_f f^*(c_g). It determines a twisted view system with maps T_c(f)(x)=c_f f^*(x). A section of the twisted system satisfies x_v=c_f f^*(x_w). The cocycle law says that the transition rules compose coherently. It does not assert that the twisted system has a section. The coherent transition system and a global trivialization of that system are different objects.
Exact mathematical passage
c_{1_v}=1,\qquad c_{g\circ f}=c_f f^*(c_g).
c_{g\circ f}Definition passage · 4 passages
Search aliases: c_g∘ f circ
Definition passage: Cocycle · 04_symmetry.tex, line 8
[Cocycle] A cocycle is a family c_f∈ G_v for every f:v→ w, satisfying c_1_v=1, c_g∘ f=c_f f^*(c_g). It determines a twisted view system with maps T_c(f)(x)=c_f f^*(x). A section of the twisted system satisfies x_v=c_f f^*(x_w). The cocycle law says that the transition rules compose coherently. It does not assert that the twisted system has a section. The coherent transition system and a global trivialization of that system are different objects.
Exact mathematical passage
c_{1_v}=1,\qquad c_{g\circ f}=c_f f^*(c_g).
\EndDefinition passage · 6 passages
Search aliases: End operatorname
Definition passage: Operator-seed realization · 01_contracts.tex, line 11
[Operator-seed realization] Let M be an abelian group and let O⊆End(M) be a declared collection of endomorphisms. For S⊆ M, its operator hull is the subgroup generated by all T(s), where s∈ S and T is a finite composition of members of O, including the identity. A finite operator-seed realization requires a finite S whose hull is M. The allowed addition and inverse operations are part of the group structure. An arbitrary target-dependent map is not automatically an admissible endomorphism. In pa…
Exact mathematical passage
\mathcal O\subseteq\End(M)
\operatorname{ev}Definition passage · 8 passages
Search aliases: ev operatorname
Definition passage: Strict Komorebi realization · 01_views.tex, line 22
[Strict Komorebi realization] The ambient realization set is K(F)=Γ(F)= {x∈∏_v∈ObVF_v: f^*(x_w)=x_v for every f:v→ w}. A point of K(F) is a whole coherent family. Its value in view v is the evaluation ev_v(x)=x_v. The raw product contains all families, including inconsistent ones. Equation~[source reference] cuts out the exact families. No nonemptiness hypothesis is built into the definition.
Exact mathematical passage
\operatorname{ev}_v(x)=x_v
\operatorname{ev}_vDefinition passage · 5 passages
Search aliases: ev_v operatorname
Definition passage: Strict Komorebi realization · 01_views.tex, line 22
[Strict Komorebi realization] The ambient realization set is K(F)=Γ(F)= {x∈∏_v∈ObVF_v: f^*(x_w)=x_v for every f:v→ w}. A point of K(F) is a whole coherent family. Its value in view v is the evaluation ev_v(x)=x_v. The raw product contains all families, including inconsistent ones. Equation~[source reference] cuts out the exact families. No nonemptiness hypothesis is built into the definition.
Exact mathematical passage
\operatorname{ev}_v(x)=x_v
F_wDefinition passage · 1 passage
Search aliases: F_w
Definition passage: View system · 01_views.tex, line 8
[View system] A view system consists of sets F_v and maps f^*:F_w⟶ F_v (f:v⟶ w) satisfying (1_v)^*=id and (g∘ f)^*=f^*g^*. Equivalently, it is a functor F:V^op→Set. The direction matters. An arrow v→ w induces a restriction from the fiber at w to the fiber at v. A restriction need not be invertible. A category with several arrows between two views can retain different routes between their descriptions. Replacing it by a preorder can lose those routes and the constraints they impose.
Exact mathematical passage
f^*:F_w\longrightarrow F_v\qquad(f:v\longrightarrow w)
g^*Definition passage · 4 passages
Search aliases: g^*
Definition passage: View system · 01_views.tex, line 11
[View system] A view system consists of sets F_v and maps f^*:F_w⟶ F_v (f:v⟶ w) satisfying (1_v)^*=id and (g∘ f)^*=f^*g^*. Equivalently, it is a functor F:V^op→Set. The direction matters. An arrow v→ w induces a restriction from the fiber at w to the fiber at v. A restriction need not be invertible. A category with several arrows between two views can retain different routes between their descriptions. Replacing it by a preorder can lose those routes and the constraints they impose.
Exact mathematical passage
(g\circ f)^*=f^*g^*
G_vDefinition passage · 9 passages
Search aliases: G_v
Definition passage: Cocycle · 04_symmetry.tex, line 7
[Cocycle] A cocycle is a family c_f∈ G_v for every f:v→ w, satisfying c_1_v=1, c_g∘ f=c_f f^*(c_g). It determines a twisted view system with maps T_c(f)(x)=c_f f^*(x). A section of the twisted system satisfies x_v=c_f f^*(x_w). The cocycle law says that the transition rules compose coherently. It does not assert that the twisted system has a section. The coherent transition system and a global trivialization of that system are different objects.
Exact mathematical passage
c_f\in G_v
hDefinition passage · 167 passages
Search aliases: h
Definition passage: Purity · 04_purity.tex, line 19
[Purity] A subgroup H of an abelian group M is pure when H∩ mM=mH (m≥1). Equivalently, if an equation my=h with h∈ H has a solution in M, then it has a solution in H. Purity is a standard algebraic notion [source reference]. The proof below uses only the displayed criterion. In particular, no structure theorem for mixed abelian groups is required.
Exact mathematical passage
my=h
\idDefinition passage · 7 passages
Search aliases: id
Definition passage: View system · 01_views.tex, line 11
[View system] A view system consists of sets F_v and maps f^*:F_w⟶ F_v (f:v⟶ w) satisfying (1_v)^*=id and (g∘ f)^*=f^*g^*. Equivalently, it is a functor F:V^op→Set. The direction matters. An arrow v→ w induces a restriction from the fiber at w to the fiber at v. A restriction need not be invertible. A category with several arrows between two views can retain different routes between their descriptions. Replacing it by a preorder can lose those routes and the constraints they impose.
Exact mathematical passage
(1_v)^*=\id
\infDefinition passage · 5 passages
Search aliases: inf
Definition passage: Grammar-relative description complexity · 01_contracts.tex, line 48
[Grammar-relative description complexity] For a fixed binary coding of G, let lvert drvert be its bit length and set K_ G(G)=inf{lvert drvert: d is valid and Realize_ G(d)≅ G}, with value +∞ when there is no such description. The value depends on the grammar and the coding. A fixed finite description can be long without being impossible. Conversely, a group with K_ G(G)=+∞ may have a short description in an expanded language.
Exact mathematical passage
K_{\mathfrak G}(G)=\inf\{\lvert d\rvert:\ d\text{ is valid and }\operatorname{Realize}_{\mathfrak G}(d)\cong G\},
\lvertDefinition passage · 9 passages
Search aliases: lvert
Definition passage: Grammar-relative description complexity · 01_contracts.tex, line 47
[Grammar-relative description complexity] For a fixed binary coding of G, let lvert drvert be its bit length and set K_ G(G)=inf{lvert drvert: d is valid and Realize_ G(d)≅ G}, with value +∞ when there is no such description. The value depends on the grammar and the coding. A fixed finite description can be long without being impossible. Conversely, a group with K_ G(G)=+∞ may have a short description in an expanded language.
Exact mathematical passage
\lvert d\rvert
\mathcal ODefinition passage · 3 passages
Search aliases: O mathcal
Definition passage: Operator-seed realization · 01_contracts.tex, line 11
[Operator-seed realization] Let M be an abelian group and let O⊆End(M) be a declared collection of endomorphisms. For S⊆ M, its operator hull is the subgroup generated by all T(s), where s∈ S and T is a finite composition of members of O, including the identity. A finite operator-seed realization requires a finite S whose hull is M. The allowed addition and inverse operations are part of the group structure. An arbitrary target-dependent map is not automatically an admissible endomorphism. In pa…
Exact mathematical passage
\mathcal O\subseteq\End(M)
\ObDefinition passage · 6 passages
Search aliases: Ob
Definition passage: Strict Komorebi realization · 01_views.tex, line 17
[Strict Komorebi realization] The ambient realization set is K(F)=Γ(F)= {x∈∏_v∈ObVF_v: f^*(x_w)=x_v for every f:v→ w}. A point of K(F) is a whole coherent family. Its value in view v is the evaluation ev_v(x)=x_v. The raw product contains all families, including inconsistent ones. Equation~[source reference] cuts out the exact families. No nonemptiness hypothesis is built into the definition.
Exact mathematical passage
\label{eq:ambient} \mathfrak K(F)=\Gamma(F)= \left\{x\in\prod_{v\in\Ob\V}F_v: f^*(x_w)=x_v\text{ for every }f:v\to w\right\}.
\mathrm{op}Definition passage · 1 passage
Search aliases: op mathrm
Definition passage: View system · 01_views.tex, line 11
[View system] A view system consists of sets F_v and maps f^*:F_w⟶ F_v (f:v⟶ w) satisfying (1_v)^*=id and (g∘ f)^*=f^*g^*. Equivalently, it is a functor F:V^op→Set. The direction matters. An arrow v→ w induces a restriction from the fiber at w to the fiber at v. A restriction need not be invertible. A category with several arrows between two views can retain different routes between their descriptions. Replacing it by a preorder can lose those routes and the constraints they impose.
Exact mathematical passage
F:\V^{\mathrm{op}}\to\Set
\operatorname{Realize}Definition passage · 1 passage
Search aliases: Realize operatorname
Definition passage: Grammar-relative description complexity · 01_contracts.tex, line 48
[Grammar-relative description complexity] For a fixed binary coding of G, let lvert drvert be its bit length and set K_ G(G)=inf{lvert drvert: d is valid and Realize_ G(d)≅ G}, with value +∞ when there is no such description. The value depends on the grammar and the coding. A fixed finite description can be long without being impossible. Conversely, a group with K_ G(G)=+∞ may have a short description in an expanded language.
Exact mathematical passage
K_{\mathfrak G}(G)=\inf\{\lvert d\rvert:\ d\text{ is valid and }\operatorname{Realize}_{\mathfrak G}(d)\cong G\},
\operatorname{Realize}_{\mathfrak G}Definition passage · 1 passage
Search aliases: Realize_ G operatorname mathfrak
Definition passage: Grammar-relative description complexity · 01_contracts.tex, line 48
[Grammar-relative description complexity] For a fixed binary coding of G, let lvert drvert be its bit length and set K_ G(G)=inf{lvert drvert: d is valid and Realize_ G(d)≅ G}, with value +∞ when there is no such description. The value depends on the grammar and the coding. A fixed finite description can be long without being impossible. Conversely, a group with K_ G(G)=+∞ may have a short description in an expanded language.
Exact mathematical passage
K_{\mathfrak G}(G)=\inf\{\lvert d\rvert:\ d\text{ is valid and }\operatorname{Realize}_{\mathfrak G}(d)\cong G\},
\rvertDefinition passage · 9 passages
Search aliases: rvert
Definition passage: Grammar-relative description complexity · 01_contracts.tex, line 47
[Grammar-relative description complexity] For a fixed binary coding of G, let lvert drvert be its bit length and set K_ G(G)=inf{lvert drvert: d is valid and Realize_ G(d)≅ G}, with value +∞ when there is no such description. The value depends on the grammar and the coding. A fixed finite description can be long without being impossible. Conversely, a group with K_ G(G)=+∞ may have a short description in an expanded language.
Exact mathematical passage
\lvert d\rvert
\SetDefinition passage · 1 passage
Search aliases: Set
Definition passage: View system · 01_views.tex, line 11
[View system] A view system consists of sets F_v and maps f^*:F_w⟶ F_v (f:v⟶ w) satisfying (1_v)^*=id and (g∘ f)^*=f^*g^*. Equivalently, it is a functor F:V^op→Set. The direction matters. An arrow v→ w induces a restriction from the fiber at w to the fiber at v. A restriction need not be invertible. A category with several arrows between two views can retain different routes between their descriptions. Replacing it by a preorder can lose those routes and the constraints they impose.
Exact mathematical passage
F:\V^{\mathrm{op}}\to\Set
T_cDefinition passage · 6 passages
Search aliases: T_c
Definition passage: Cocycle · 04_symmetry.tex, line 12
[Cocycle] A cocycle is a family c_f∈ G_v for every f:v→ w, satisfying c_1_v=1, c_g∘ f=c_f f^*(c_g). It determines a twisted view system with maps T_c(f)(x)=c_f f^*(x). A section of the twisted system satisfies x_v=c_f f^*(x_w). The cocycle law says that the transition rules compose coherently. It does not assert that the twisted system has a section. The coherent transition system and a global trivialization of that system are different objects.
Exact mathematical passage
T_c(f)(x)=c_f f^*(x).
\trankDefinition passage · 2 passages
Search aliases: trank operatorname
Definition passage: Topological generation · 01_contracts.tex, line 28
[Topological generation] For a topological group M, a subset S topologically generates M when the closure of the subgroup generated by S is all of M. Its least possible size is denoted trank(M). An operator version takes the closure of the operator hull instead. The closure is taken in the stated topology. A change of topology is a change of contract, not an innocuous notational adjustment.
Exact mathematical passage
\trank(M)
\V^{\mathrm{op}}Definition passage · 1 passage
Search aliases: V^op V mathrm
Definition passage: View system · 01_views.tex, line 11
[View system] A view system consists of sets F_v and maps f^*:F_w⟶ F_v (f:v⟶ w) satisfying (1_v)^*=id and (g∘ f)^*=f^*g^*. Equivalently, it is a functor F:V^op→Set. The direction matters. An arrow v→ w induces a restriction from the fiber at w to the fiber at v. A restriction need not be invertible. A category with several arrows between two views can retain different routes between their descriptions. Replacing it by a preorder can lose those routes and the constraints they impose.
Exact mathematical passage
F:\V^{\mathrm{op}}\to\Set
wDefinition passage · 95 passages
Search aliases: w
Definition passage: View system · 01_views.tex, line 8
[View system] A view system consists of sets F_v and maps f^*:F_w⟶ F_v (f:v⟶ w) satisfying (1_v)^*=id and (g∘ f)^*=f^*g^*. Equivalently, it is a functor F:V^op→Set. The direction matters. An arrow v→ w induces a restriction from the fiber at w to the fiber at v. A restriction need not be invertible. A category with several arrows between two views can retain different routes between their descriptions. Replacing it by a preorder can lose those routes and the constraints they impose.
Exact mathematical passage
f^*:F_w\longrightarrow F_v\qquad(f:v\longrightarrow w)
x_vDefinition passage · 17 passages
Search aliases: x_v
Definition passage: Strict Komorebi realization · 01_views.tex, line 17
[Strict Komorebi realization] The ambient realization set is K(F)=Γ(F)= {x∈∏_v∈ObVF_v: f^*(x_w)=x_v for every f:v→ w}. A point of K(F) is a whole coherent family. Its value in view v is the evaluation ev_v(x)=x_v. The raw product contains all families, including inconsistent ones. Equation~[source reference] cuts out the exact families. No nonemptiness hypothesis is built into the definition.
Exact mathematical passage
\label{eq:ambient} \mathfrak K(F)=\Gamma(F)= \left\{x\in\prod_{v\in\Ob\V}F_v: f^*(x_w)=x_v\text{ for every }f:v\to w\right\}.
x_wDefinition passage · 11 passages
Search aliases: x_w
Definition passage: Strict Komorebi realization · 01_views.tex, line 17
[Strict Komorebi realization] The ambient realization set is K(F)=Γ(F)= {x∈∏_v∈ObVF_v: f^*(x_w)=x_v for every f:v→ w}. A point of K(F) is a whole coherent family. Its value in view v is the evaluation ev_v(x)=x_v. The raw product contains all families, including inconsistent ones. Equation~[source reference] cuts out the exact families. No nonemptiness hypothesis is built into the definition.
Exact mathematical passage
\label{eq:ambient} \mathfrak K(F)=\Gamma(F)= \left\{x\in\prod_{v\in\Ob\V}F_v: f^*(x_w)=x_v\text{ for every }f:v\to w\right\}.
\xrightarrowDefinition passage · 9 passages
Search aliases: xrightarrow
Definition passage: Marked realization groupoid · 04_symmetry.tex, line 393
[Marked realization groupoid] The objects of E_b(R) are pairs (x,ϕ), with x∈ C and ϕ:Rxxrightarrow∼b. An arrow (x,ϕ)→(y,ψ) is an arrow a:x→ y with ψ R(a)=ϕ. Composition and inverses preserve this equation, so this is a groupoid. It is the explicit groupoid homotopy fiber construction, without invoking a higher categorical interface.
Exact mathematical passage
\phi:Rx\xrightarrow{\sim}b
\psiDefinition passage · 31 passages
Search aliases: ψ psi
Definition passage: Marked realization groupoid · 04_symmetry.tex, line 393
[Marked realization groupoid] The objects of E_b(R) are pairs (x,ϕ), with x∈ C and ϕ:Rxxrightarrow∼b. An arrow (x,ϕ)→(y,ψ) is an arrow a:x→ y with ψ R(a)=ϕ. Composition and inverses preserve this equation, so this is a groupoid. It is the explicit groupoid homotopy fiber construction, without invoking a higher categorical interface.
Exact mathematical passage
(x,\phi)\to(y,\psi)
\prod_{v\in\Ob\V}Definition passage · 1 passage
Search aliases: ∏_v∈ObV prod in Ob V
Definition passage: Strict Komorebi realization · 01_views.tex, line 17
[Strict Komorebi realization] The ambient realization set is K(F)=Γ(F)= {x∈∏_v∈ObVF_v: f^*(x_w)=x_v for every f:v→ w}. A point of K(F) is a whole coherent family. Its value in view v is the evaluation ev_v(x)=x_v. The raw product contains all families, including inconsistent ones. Equation~[source reference] cuts out the exact families. No nonemptiness hypothesis is built into the definition.
Exact mathematical passage
\label{eq:ambient} \mathfrak K(F)=\Gamma(F)= \left\{x\in\prod_{v\in\Ob\V}F_v: f^*(x_w)=x_v\text{ for every }f:v\to w\right\}.
\circDefinition passage · 9 passages
Search aliases: ∘ circ
Definition passage: View system · 01_views.tex, line 11
[View system] A view system consists of sets F_v and maps f^*:F_w⟶ F_v (f:v⟶ w) satisfying (1_v)^*=id and (g∘ f)^*=f^*g^*. Equivalently, it is a functor F:V^op→Set. The direction matters. An arrow v→ w induces a restriction from the fiber at w to the fiber at v. A restriction need not be invertible. A category with several arrows between two views can retain different routes between their descriptions. Replacing it by a preorder can lose those routes and the constraints they impose.
Exact mathematical passage
(g\circ f)^*=f^*g^*
\capDefinition passage · 18 passages
Search aliases: ∩ cap
Definition passage: Purity · 04_purity.tex, line 16
[Purity] A subgroup H of an abelian group M is pure when H∩ mM=mH (m≥1). Equivalently, if an equation my=h with h∈ H has a solution in M, then it has a solution in H. Purity is a standard algebraic notion [source reference]. The proof below uses only the displayed criterion. In particular, no structure theorem for mixed abelian groups is required.
Exact mathematical passage
H\cap mM=mH\qquad(m\ge1).
\simDefinition passage · 2 passages
Search aliases: ∼ sim
Definition passage: Marked realization groupoid · 04_symmetry.tex, line 393
[Marked realization groupoid] The objects of E_b(R) are pairs (x,ϕ), with x∈ C and ϕ:Rxxrightarrow∼b. An arrow (x,ϕ)→(y,ψ) is an arrow a:x→ y with ψ R(a)=ϕ. Composition and inverses preserve this equation, so this is a groupoid. It is the explicit groupoid homotopy fiber construction, without invoking a higher categorical interface.
Exact mathematical passage
\phi:Rx\xrightarrow{\sim}b
\congDefinition passage · 20 passages
Search aliases: ≅ cong
Definition passage: Grammar-relative description complexity · 01_contracts.tex, line 48
[Grammar-relative description complexity] For a fixed binary coding of G, let lvert drvert be its bit length and set K_ G(G)=inf{lvert drvert: d is valid and Realize_ G(d)≅ G}, with value +∞ when there is no such description. The value depends on the grammar and the coding. A fixed finite description can be long without being impossible. Conversely, a group with K_ G(G)=+∞ may have a short description in an expanded language.
Exact mathematical passage
K_{\mathfrak G}(G)=\inf\{\lvert d\rvert:\ d\text{ is valid and }\operatorname{Realize}_{\mathfrak G}(d)\cong G\},
\geDefinition passage · 103 passages
Search aliases: ≥ ge
Definition passage: Purity · 04_purity.tex, line 16
[Purity] A subgroup H of an abelian group M is pure when H∩ mM=mH (m≥1). Equivalently, if an equation my=h with h∈ H has a solution in M, then it has a solution in H. Purity is a standard algebraic notion [source reference]. The proof below uses only the displayed criterion. In particular, no structure theorem for mixed abelian groups is required.
Exact mathematical passage
H\cap mM=mH\qquad(m\ge1).
\longrightarrowDefinition passage · 31 passages
Search aliases: ⟶ longrightarrow
Definition passage: View system · 01_views.tex, line 8
[View system] A view system consists of sets F_v and maps f^*:F_w⟶ F_v (f:v⟶ w) satisfying (1_v)^*=id and (g∘ f)^*=f^*g^*. Equivalently, it is a functor F:V^op→Set. The direction matters. An arrow v→ w induces a restriction from the fiber at w to the fiber at v. A restriction need not be invertible. A category with several arrows between two views can retain different routes between their descriptions. Replacing it by a preorder can lose those routes and the constraints they impose.
Exact mathematical passage
f^*:F_w\longrightarrow F_v\qquad(f:v\longrightarrow w)
a_*Introduction passage · 7 passages
Search aliases: a_*
Introduction passage · 05f_bounded_realization.tex, line 107
[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection is empty, then m_n→∞. The nearest-point theorem for nonempty closed convex subsets of a Hilbert space, proved in Appendix~[source reference], supplies …
Exact mathematical passage
a_*\in\bigcap_n A_n
A_2Introduction passage · 4 passages
Search aliases: A_2
Introduction passage · 03b_repair_transport.tex, line 103
For composition, denote the three sections by A,B,A_2. The composite residual map is C_2C_1, so its mismatch is S_2S_1A-A_2C_2C_1. Insert and subtract S_2BC_1: S_2S_1A-A_2C_2C_1 · =S_2(S_1A-BC_1)+(S_2B-A_2C_2)C_1
Exact mathematical passage
A,B,A_2
a_mIntroduction passage · 5 passages
Search aliases: a_m
Introduction passage · 05f_bounded_realization.tex, line 104
[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection is empty, then m_n→∞. The nearest-point theorem for nonempty closed convex subsets of a Hilbert space, proved in Appendix~[source reference], supplies …
Exact mathematical passage
\label{eq:nested-energy} \norm{a_m-a_n}^2=m_m^2-m_n^2.
A_nIntroduction passage · 13 passages
Search aliases: A_n
Introduction passage · 05f_bounded_realization.tex, line 107
[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection is empty, then m_n→∞. The nearest-point theorem for nonempty closed convex subsets of a Hilbert space, proved in Appendix~[source reference], supplies …
Exact mathematical passage
a_*\in\bigcap_n A_n
a_nIntroduction passage · 49 passages
Search aliases: a_n
Introduction passage · 05f_bounded_realization.tex, line 104
[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection is empty, then m_n→∞. The nearest-point theorem for nonempty closed convex subsets of a Hilbert space, proved in Appendix~[source reference], supplies …
Exact mathematical passage
\label{eq:nested-energy} \norm{a_m-a_n}^2=m_m^2-m_n^2.
A_vIntroduction passage · 3 passages
Search aliases: A_v
Introduction passage · 02_obstructions.tex, line 31
The admissible corrections. Fix U⊆ObV. Write p_U:E→∏_v∈ UA_v for raw restriction. A correction preserving the prescribed values belongs to ker p_U. Its possible defects form the subgroup B_U=δ(ker p_U)≤ R. For a partial section s, choose any raw extension widetilde s, for example zero off U, and define Q_U=R/B_U, o_U(s)=[δwidetilde s]∈ Q_U. If another raw extension is chosen, their difference vanishes on U, so its differential belongs to B_U. Thus the class is independent of the lift.
Exact mathematical passage
p_U:E\to\prod_{v\in U}A_v
B^{\mathsf T}Introduction passage · 9 passages
Search aliases: B^ T mathsf
Introduction passage · 05k_stopping_rules.tex, line 5
The residual of the coordinate equation is a covector. In finite dimensions, use orthonormal coordinates in the ambient candidate and residual spaces. Let the columns of B form an arbitrary basis of H=ker p, not necessarily an orthonormal one. Write a=Bc, S=dB, and G=B^ TB. As before, G is positive definite. For an inexact coefficient vector c, the residual of the coordinate normal equation is q=(S^ TS+α G)c-S^ Tr. The vector q stores pairings against basis vectors. It is not the coordinate vect…
Exact mathematical passage
G=B^{\mathsf T}B
b^{-1}Introduction passage · 1 passage
Search aliases: b^-1
Introduction passage · 05_construction.tex, line 28
A section with finite arithmetic defects. For a reduced rational r=a/b, with b>0, define (s_A(a/b))_p= a b^-1pmod p, · p∈ S_A and pnmid b, 0, · p∉ S_A or pmid b. The inverse is used only when the denominator is a unit in F_p. The convention at denominator primes is essential. It is a choice of representatives, not a claim that division by p is defined in C_p.
Exact mathematical passage
\label{eq:section} (s_A(a/b))_p= \begin{cases} a b^{-1}\pmod p,&p\in S_A\text{ and }p\nmid b,\\ 0,&p\notin S_A\text{ or }p\mid b. \end{cases}
B_jIntroduction passage · 8 passages
Search aliases: B_j
Introduction passage · 10_grammar.tex, line 12
For each j whose output lies in the family, choose B_j⊆ N such that R_j≅ G_B_j. For other j, set B_j=∅. This is a set-theoretic choice of representatives. It is not a procedure for deciding whether an arbitrary description denotes a group in the family.
Exact mathematical passage
B_j\subseteq\N
B_UIntroduction passage · 15 passages
Search aliases: B_U
Introduction passage · 02_obstructions.tex, line 32
The admissible corrections. Fix U⊆ObV. Write p_U:E→∏_v∈ UA_v for raw restriction. A correction preserving the prescribed values belongs to ker p_U. Its possible defects form the subgroup B_U=δ(ker p_U)≤ R. For a partial section s, choose any raw extension widetilde s, for example zero off U, and define Q_U=R/B_U, o_U(s)=[δwidetilde s]∈ Q_U. If another raw extension is chosen, their difference vanishes on U, so its differential belongs to B_U. Thus the class is independent of the lift.
Exact mathematical passage
B_U=\delta(\ker p_U)\le R.
b_vIntroduction passage · 6 passages
Search aliases: b_v
Introduction passage · 04_symmetry.tex, line 121
Conversely, suppose x is a global twisted section extending s. Put b_v=x_v^-1q_v. This is based, and direct substitution gives (b· c^q)_f =x_v^-1c_ff^*(x_w)=1, where the last equality is precisely the section equation for x. Hence the obstruction is neutral. These arguments establish the result using group cancellation only. In particular they do not assume an additive quotient, a norm on labels, or a decision procedure for finding b. sourceKS31 The orbit is defined from actual cocycles and an a…
Exact mathematical passage
b_v=x_v^{-1}q_v
B_XIntroduction passage · 10 passages
Search aliases: B_X
Introduction passage · 12_proof_tools.tex, line 114
emma[Continuous inverse for Banach spaces] Let S:X→ Y be a bounded linear surjection between real Banach spaces. There is a constant C<∞ such that every y∈ Y has a preimage x with normx≤ Cnormy. If S is also injective, its inverse is bounded linear. If Y={0}, take the zero preimage. Otherwise write B_X for the closed unit ball. Surjectivity gives Y=⋃_n≥1overlineS(nB_X). By Lemma~[source reference], at least one of the closed sets on the right has nonempty interior. Thus there are n≥1, y_0∈ Y, an…
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B_X
c^qIntroduction passage · 5 passages
Search aliases: c^q
Introduction passage · 04_symmetry.tex, line 122
Conversely, suppose x is a global twisted section extending s. Put b_v=x_v^-1q_v. This is based, and direct substitution gives (b· c^q)_f =x_v^-1c_ff^*(x_w)=1, where the last equality is precisely the section equation for x. Hence the obstruction is neutral. These arguments establish the result using group cancellation only. In particular they do not assume an additive quotient, a norm on labels, or a decision procedure for finding b. sourceKS31 The orbit is defined from actual cocycles and an a…
Exact mathematical passage
(b\cdot c^q)_f =x_v^{-1}c_ff^*(x_w)=1,
C_1Introduction passage · 7 passages
Search aliases: C_1
Introduction passage · 03b_repair_transport.tex, line 103
For composition, denote the three sections by A,B,A_2. The composite residual map is C_2C_1, so its mismatch is S_2S_1A-A_2C_2C_1. Insert and subtract S_2BC_1: S_2S_1A-A_2C_2C_1 · =S_2(S_1A-BC_1)+(S_2B-A_2C_2)C_1
Exact mathematical passage
C_2C_1
C_2Introduction passage · 11 passages
Search aliases: C_2
Introduction passage · 03b_repair_transport.tex, line 103
For composition, denote the three sections by A,B,A_2. The composite residual map is C_2C_1, so its mismatch is S_2S_1A-A_2C_2C_1. Insert and subtract S_2BC_1: S_2S_1A-A_2C_2C_1 · =S_2(S_1A-BC_1)+(S_2B-A_2C_2)C_1
Exact mathematical passage
C_2C_1
d_0Introduction passage · 1 passage
Search aliases: d_0
Introduction passage · 10_grammar.tex, line 7
Fix the semantics before diagonalizing. Let G be a grammar with countably many finite, parameter-free descriptions. Each valid description has one group isomorphism type as its semantics. Enumerate the strings as d_0,d_1,d_2,… and write R_j for the semantic output. Invalid strings may be assigned a fixed dummy group. No decidability assumption is made about validity or evaluation.
Exact mathematical passage
d_0,d_1,d_2,\ldots
d_1Introduction passage · 1 passage
Search aliases: d_1
Introduction passage · 10_grammar.tex, line 7
Fix the semantics before diagonalizing. Let G be a grammar with countably many finite, parameter-free descriptions. Each valid description has one group isomorphism type as its semantics. Enumerate the strings as d_0,d_1,d_2,… and write R_j for the semantic output. Invalid strings may be assigned a fixed dummy group. No decidability assumption is made about validity or evaluation.
Exact mathematical passage
d_0,d_1,d_2,\ldots
d_2Introduction passage · 1 passage
Search aliases: d_2
Introduction passage · 10_grammar.tex, line 7
Fix the semantics before diagonalizing. Let G be a grammar with countably many finite, parameter-free descriptions. Each valid description has one group isomorphism type as its semantics. Enumerate the strings as d_0,d_1,d_2,… and write R_j for the semantic output. Invalid strings may be assigned a fixed dummy group. No decidability assumption is made about validity or evaluation.
Exact mathematical passage
d_0,d_1,d_2,\ldots
EIntroduction passage · 84 passages
Search aliases: E
Introduction passage · 02_obstructions.tex, line 31
The admissible corrections. Fix U⊆ObV. Write p_U:E→∏_v∈ UA_v for raw restriction. A correction preserving the prescribed values belongs to ker p_U. Its possible defects form the subgroup B_U=δ(ker p_U)≤ R. For a partial section s, choose any raw extension widetilde s, for example zero off U, and define Q_U=R/B_U, o_U(s)=[δwidetilde s]∈ Q_U. If another raw extension is chosen, their difference vanishes on U, so its differential belongs to B_U. Thus the class is independent of the lift.
Exact mathematical passage
p_U:E\to\prod_{v\in U}A_v
\operatorname{ev}_nIntroduction passage · 1 passage
Search aliases: ev_n operatorname
Introduction passage · 06_boundaries.tex, line 39
[Finite fibers and persistent images] Let (X_n,ρ_n) be a sequential inverse system of nonempty finite sets. Its inverse limit is nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) for m≥ n. Each I_n,m is nonempty. The composition law gives I_n,m+1=ρ_n,m(ρ_m(X_m+1))⊆ I_n,m. These are subsets of the finite set X_n. Every strict dec…
Exact mathematical passage
\im(\operatorname{ev}_n)=Y_n, \qquad Y_n=\bigcap_{m\ge n}\im\rho_{n,m}.
f^{(m)}Introduction passage · 8 passages
Search aliases: f^(m)
Introduction passage · 03_towers.tex, line 149
For linearity, write L=I-Ad, so f^(m)=L^mf. For fixed f,g and scalars λ,μ, L^m(λ f+μ g)=λ L^mf+μ L^mg. Continuity of addition and scalar multiplication allows passage to the norm limit and proves linearity of P_A. The estimate with m=0 gives normP_Af≤norm f+fracnorm A1-qnormdf ≤(1+fracnorm Anorm d1-q)norm f, so the linear map is bounded. If df=0, every iterate equals f, hence P_Af=f. Its image is contained in ker d and it fixes that kernel, proving surjectivity onto its stated image and P_A^2=P_…
Exact mathematical passage
f^{(m)}=L^mf
f_iIntroduction passage · 2 passages
Search aliases: f_i
Introduction passage · 03a_observation.tex, line 10
An output level may require a finer input level. Let X and Y be sequential set-valued resolutions. Write ρ^X_i,j:X_j→ X_i for composite restriction. A reindexed transport consists of a monotone schedule ϕ:N→N and maps f_n:X_ϕ(n)⟶ Y_n satisfying, for i≤ j, ρ^Y_i,jf_j=f_iρ^X_ϕ(i),ϕ(j). The schedule records how much input is used for each output level. Monotonicity means that asking for more output never forces the representation to discard input already available. It does not require strict growth…
Exact mathematical passage
\label{eq:scheduled-naturality} \rho^Y_{i,j}f_j=f_i\rho^X_{\phi(i),\phi(j)}.
f_jIntroduction passage · 4 passages
Search aliases: f_j
Introduction passage · 03a_observation.tex, line 10
An output level may require a finer input level. Let X and Y be sequential set-valued resolutions. Write ρ^X_i,j:X_j→ X_i for composite restriction. A reindexed transport consists of a monotone schedule ϕ:N→N and maps f_n:X_ϕ(n)⟶ Y_n satisfying, for i≤ j, ρ^Y_i,jf_j=f_iρ^X_ϕ(i),ϕ(j). The schedule records how much input is used for each output level. Monotonicity means that asking for more output never forces the representation to discard input already available. It does not require strict growth…
Exact mathematical passage
\label{eq:scheduled-naturality} \rho^Y_{i,j}f_j=f_i\rho^X_{\phi(i),\phi(j)}.
f_nIntroduction passage · 51 passages
Search aliases: f_n
Introduction passage · 03a_observation.tex, line 6
An output level may require a finer input level. Let X and Y be sequential set-valued resolutions. Write ρ^X_i,j:X_j→ X_i for composite restriction. A reindexed transport consists of a monotone schedule ϕ:N→N and maps f_n:X_ϕ(n)⟶ Y_n satisfying, for i≤ j, ρ^Y_i,jf_j=f_iρ^X_ϕ(i),ϕ(j). The schedule records how much input is used for each output level. Monotonicity means that asking for more output never forces the representation to discard input already available. It does not require strict growth…
Exact mathematical passage
f_n:X_{\phi(n)}\longrightarrow Y_n
G_{B_j}Introduction passage · 1 passage
Search aliases: G_B_j
Introduction passage · 10_grammar.tex, line 12
For each j whose output lies in the family, choose B_j⊆ N such that R_j≅ G_B_j. For other j, set B_j=∅. This is a set-theoretic choice of representatives. It is not a procedure for deciding whether an arbitrary description denotes a group in the family.
Exact mathematical passage
R_j\cong G_{B_j}
IIntroduction passage · 100 passages
Search aliases: I
Introduction passage · 03_towers.tex, line 57
[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.
Exact mathematical passage
\rho_{i,i}=I
i_0Introduction passage · 9 passages
Search aliases: i_0
Introduction passage · 05_geometry.tex, line 22
Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_k+1x_…
Exact mathematical passage
q=(i_0,\ldots,i_m)
i_1Introduction passage · 3 passages
Search aliases: i_1
Introduction passage · 05_geometry.tex, line 23
Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_k+1x_…
Exact mathematical passage
T_qx_{i_m}-x_{i_0} =\sum_{k=0}^{m-1} (\tau_{i_0i_1}\cdots\tau_{i_{k-1}i_k})(dx)_{i_ki_{k+1}}.
i_kIntroduction passage · 6 passages
Search aliases: i_k
Introduction passage · 05_geometry.tex, line 23
Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_k+1x_…
Exact mathematical passage
T_qx_{i_m}-x_{i_0} =\sum_{k=0}^{m-1} (\tau_{i_0i_1}\cdots\tau_{i_{k-1}i_k})(dx)_{i_ki_{k+1}}.
i_{k+1}Introduction passage · 4 passages
Search aliases: i_k+1
Introduction passage · 05_geometry.tex, line 23
Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_k+1x_…
Exact mathematical passage
T_qx_{i_m}-x_{i_0} =\sum_{k=0}^{m-1} (\tau_{i_0i_1}\cdots\tau_{i_{k-1}i_k})(dx)_{i_ki_{k+1}}.
i_{k-1}Introduction passage · 2 passages
Search aliases: i_k-1
Introduction passage · 05_geometry.tex, line 23
Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_k+1x_…
Exact mathematical passage
T_qx_{i_m}-x_{i_0} =\sum_{k=0}^{m-1} (\tau_{i_0i_1}\cdots\tau_{i_{k-1}i_k})(dx)_{i_ki_{k+1}}.
i_mIntroduction passage · 7 passages
Search aliases: i_m
Introduction passage · 05_geometry.tex, line 22
Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_k+1x_…
Exact mathematical passage
q=(i_0,\ldots,i_m)
I_{n,m}Introduction passage · 5 passages
Search aliases: I_n,m
Introduction passage · 06_boundaries.tex, line 46
8 Let (X_n,ρ_n) be a sequential inverse system of nonempty finite sets. Its inverse limit is nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) for m≥ n. Each I_n,m is nonempty. The composition law gives I_n,m+1=ρ_n,m(ρ_m(X_m+1))⊆ I_n,m. These are subsets of the finite set X_n. Every strict decrease removes an element permanently…
Exact mathematical passage
I_{n,m}=\rho_{n,m}(X_m)
I_{n,m+1}Introduction passage · 1 passage
Search aliases: I_n,m+1
Introduction passage · 06_boundaries.tex, line 47
nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) for m≥ n. Each I_n,m is nonempty. The composition law gives I_n,m+1=ρ_n,m(ρ_m(X_m+1))⊆ I_n,m. These are subsets of the finite set X_n. Every strict decrease removes an element permanently, so only finitely many strict decreases are possible. There is an index s_n≥ n such that I_n…
Exact mathematical passage
I_{n,m+1}=\rho_{n,m}\bigl(\rho_m(X_{m+1})\bigr)\subseteq I_{n,m}.
I_{n,s_n}Introduction passage · 2 passages
Search aliases: I_n,s_n
Introduction passage · 06_boundaries.tex, line 50
roposition Write I_n,m=ρ_n,m(X_m) for m≥ n. Each I_n,m is nonempty. The composition law gives I_n,m+1=ρ_n,m(ρ_m(X_m+1))⊆ I_n,m. These are subsets of the finite set X_n. Every strict decrease removes an element permanently, so only finitely many strict decreases are possible. There is an index s_n≥ n such that I_n,m=I_n,s_n for all m≥ s_n. Consequently Y_n=I_n,s_n is nonempty.
Exact mathematical passage
I_{n,m}=I_{n,s_n}
\imIntroduction passage · 3 passages
Search aliases: im
Introduction passage · 06_boundaries.tex, line 39
[Finite fibers and persistent images] Let (X_n,ρ_n) be a sequential inverse system of nonempty finite sets. Its inverse limit is nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) for m≥ n. Each I_n,m is nonempty. The composition law gives I_n,m+1=ρ_n,m(ρ_m(X_m+1))⊆ I_n,m. These are subsets of the finite set X_n. Every strict dec…
Exact mathematical passage
\im(\operatorname{ev}_n)=Y_n, \qquad Y_n=\bigcap_{m\ge n}\im\rho_{n,m}.
kIntroduction passage · 163 passages
Search aliases: k
Introduction passage · 03_towers.tex, line 57
[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.
Exact mathematical passage
\rho_{i,j}\rho_{j,k}=\rho_{i,k}
L^mIntroduction passage · 2 passages
Search aliases: L^m
Introduction passage · 03_towers.tex, line 149
For linearity, write L=I-Ad, so f^(m)=L^mf. For fixed f,g and scalars λ,μ, L^m(λ f+μ g)=λ L^mf+μ L^mg. Continuity of addition and scalar multiplication allows passage to the norm limit and proves linearity of P_A. The estimate with m=0 gives normP_Af≤norm f+fracnorm A1-qnormdf ≤(1+fracnorm Anorm d1-q)norm f, so the linear map is bounded. If df=0, every iterate equals f, hence P_Af=f. Its image is contained in ker d and it fixes that kernel, proving surjectivity onto its stated image and P_A^2=P_…
Exact mathematical passage
f^{(m)}=L^mf
L^\perpIntroduction passage · 4 passages
Search aliases: L^⊥ perp
Introduction passage · 12_proof_tools.tex, line 48
For a closed linear subspace L, denote its projection by P_L. The lemma gives a unique decomposition x=P_Lx+(x-P_Lx), P_Lx∈ L, x-P_Lx∈ L^⊥. Adding two such decompositions, or multiplying one by a scalar, proves linearity by uniqueness. The Pythagorean identity gives normP_Lx≤normx, so P_L is bounded. It fixes L, annihilates L^⊥, and satisfies P_L^2=P_L. For x,y∈ H, orthogonality gives ⟨ P_Lx,y⟩=⟨ P_Lx,P_Ly⟩ =⟨ x,P_Ly⟩. Thus P_L is self-adjoint. These properties are conclusions of the constructio…
Exact mathematical passage
x=P_Lx+(x-P_Lx),\qquad P_Lx\in L,\quad x-P_Lx\in L^\perp.
L_nIntroduction passage · 4 passages
Search aliases: L_n
Introduction passage · 05f_bounded_realization.tex, line 114
nly if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection is empty, then m_n→∞. The nearest-point theorem for nonempty closed convex subsets of a Hilbert space, proved in Appendix~[source reference], supplies a_n. Write A_n=a_n+L_n, where L_n is its closed direction subspace. For ℓ∈ L_n and every real t, the vector a_n+tℓ belongs to A_n. Minimality implies 0≤norma_n+tℓ^2-norma_n^2 =2t⟨ a_n,ℓ⟩+t^2…
Exact mathematical passage
A_n=a_n+L_n
\LongleftrightarrowIntroduction passage · 20 passages
Search aliases: Longleftrightarrow
Introduction passage · 07_classification.tex, line 11
[Tail signature] If g=s_A(r)+d has infinite order, then for all but finitely many primes, pmid_G_Ag Longleftrightarrow p∉ S_A. An explicit exception set consists of the primes dividing the nonzero numerator or denominator of r, together with supp(d). Outside that set, the reduction of r is defined and nonzero. The coordinate of g is therefore nonzero exactly at the selected primes. Apply Theorem~[source reference] with multiplier p. The full signature of the distinguished element is exact. The s…
Exact mathematical passage
p\mid_{G_A}g\quad\Longleftrightarrow\quad p\notin S_A.
\LongrightarrowIntroduction passage · 4 passages
Search aliases: Longrightarrow
Introduction passage · 03_towers.tex, line 418
[Dyadic determination and finite coordinate recovery] In the actual dyadic system, all detail coordinates jointly determine the residual. Moreover, k+1<2^N Longrightarrow (Q_Nr)_k=0, (S_Nr)_k=r_k. Thus every fixed residual coordinate is recovered exactly at a sufficiently large finite depth. The statement does not assert convergence in the supremum norm over all coordinates. Write R_j for the residual space at scale j. Its lift has the actual coordinate formula (J_jz)_2m=0, (J_jz)_2m+1=s^(j)_2mz…
Exact mathematical passage
k+1<2^N\quad\Longrightarrow\quad (Q_Nr)_k=0,\qquad(S_Nr)_k=r_k.
M_1Introduction passage · 13 passages
Search aliases: M_1
Introduction passage · 03b_repair_transport.tex, line 104
For composition, denote the three sections by A,B,A_2. The composite residual map is C_2C_1, so its mismatch is S_2S_1A-A_2C_2C_1. Insert and subtract S_2BC_1: S_2S_1A-A_2C_2C_1 · =S_2(S_1A-BC_1)+(S_2B-A_2C_2)C_1 · =S_2M_1+M_2C_1. All terms map the first residual space into the last candidate space, in the written order. Finally, for fixed bounded M, the estimate normM(dx_j)≤norm Mnormdx_j proves convergence of the discrepancy when dx_j→0. Allowing the section or representation norms to grow wit…
Exact mathematical passage
\begin{aligned} S_2S_1A-A_2C_2C_1 &=S_2(S_1A-BC_1)+(S_2B-A_2C_2)C_1\\ &=S_2M_1+M_2C_1. \end{aligned}
M_2Introduction passage · 14 passages
Search aliases: M_2
Introduction passage · 03b_repair_transport.tex, line 104
For composition, denote the three sections by A,B,A_2. The composite residual map is C_2C_1, so its mismatch is S_2S_1A-A_2C_2C_1. Insert and subtract S_2BC_1: S_2S_1A-A_2C_2C_1 · =S_2(S_1A-BC_1)+(S_2B-A_2C_2)C_1 · =S_2M_1+M_2C_1. All terms map the first residual space into the last candidate space, in the written order. Finally, for fixed bounded M, the estimate normM(dx_j)≤norm Mnormdx_j proves convergence of the discrepancy when dx_j→0. Allowing the section or representation norms to grow wit…
Exact mathematical passage
\begin{aligned} S_2S_1A-A_2C_2C_1 &=S_2(S_1A-BC_1)+(S_2B-A_2C_2)C_1\\ &=S_2M_1+M_2C_1. \end{aligned}
m_m^2Introduction passage · 2 passages
Search aliases: m_m^2
Introduction passage · 05f_bounded_realization.tex, line 104
[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection is empty, then m_n→∞. The nearest-point theorem for nonempty closed convex subsets of a Hilbert space, proved in Appendix~[source reference], supplies …
Exact mathematical passage
\label{eq:nested-energy} \norm{a_m-a_n}^2=m_m^2-m_n^2.
m_nIntroduction passage · 5 passages
Search aliases: m_n
Introduction passage · 05f_bounded_realization.tex, line 103
[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection is empty, then m_n→∞. The nearest-point theorem for nonempty closed convex subsets of a Hilbert space, proved in Appendix~[source reference], supplies …
Exact mathematical passage
m_n
m_n^2Introduction passage · 4 passages
Search aliases: m_n^2
Introduction passage · 05f_bounded_realization.tex, line 104
[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection is empty, then m_n→∞. The nearest-point theorem for nonempty closed convex subsets of a Hilbert space, proved in Appendix~[source reference], supplies …
Exact mathematical passage
\label{eq:nested-energy} \norm{a_m-a_n}^2=m_m^2-m_n^2.
\mid_{G_A}Introduction passage · 3 passages
Search aliases: mid_G_A mid
Introduction passage · 07_classification.tex, line 11
[Tail signature] If g=s_A(r)+d has infinite order, then for all but finitely many primes, pmid_G_Ag Longleftrightarrow p∉ S_A. An explicit exception set consists of the primes dividing the nonzero numerator or denominator of r, together with supp(d). Outside that set, the reduction of r is defined and nonzero. The coordinate of g is therefore nonzero exactly at the selected primes. Apply Theorem~[source reference] with multiplier p. The full signature of the distinguished element is exact. The s…
Exact mathematical passage
p\mid_{G_A}g\quad\Longleftrightarrow\quad p\notin S_A.
\nmidIntroduction passage · 4 passages
Search aliases: nmid
Introduction passage · 05_construction.tex, line 28
A section with finite arithmetic defects. For a reduced rational r=a/b, with b>0, define (s_A(a/b))_p= a b^-1pmod p, · p∈ S_A and pnmid b, 0, · p∉ S_A or pmid b. The inverse is used only when the denominator is a unit in F_p. The convention at denominator primes is essential. It is a choice of representatives, not a claim that division by p is defined in C_p.
Exact mathematical passage
\label{eq:section} (s_A(a/b))_p= \begin{cases} a b^{-1}\pmod p,&p\in S_A\text{ and }p\nmid b,\\ 0,&p\notin S_A\text{ or }p\mid b. \end{cases}
\normIntroduction passage · 334 passages
Search aliases: norm
Introduction passage · 03_towers.tex, line 154
For linearity, write L=I-Ad, so f^(m)=L^mf. For fixed f,g and scalars λ,μ, L^m(λ f+μ g)=λ L^mf+μ L^mg. Continuity of addition and scalar multiplication allows passage to the norm limit and proves linearity of P_A. The estimate with m=0 gives normP_Af≤norm f+fracnorm A1-qnormdf ≤(1+fracnorm Anorm d1-q)norm f, so the linear map is bounded. If df=0, every iterate equals f, hence P_Af=f. Its image is contained in ker d and it fixes that kernel, proving surjectivity onto its stated image and P_A^2=P_…
Exact mathematical passage
\norm{P_Af}\le\norm f+\frac{\norm A}{1-q}\norm{df} \le\left(1+\frac{\norm A\norm d}{1-q}\right)\norm f,
\ObsIntroduction passage · 3 passages
Search aliases: Obs F mathcal
Introduction passage · 09_reconstruction.tex, line 6
The observation contract. Let F(G) denote the unmarked inverse system of finite quotient groups and their transition maps, considered up to isomorphism as a pro-object. The source group G is not supplied again as part of the input. Nor do we retain evaluations of named source elements. Supplying the source itself would make a reconstruction problem vacuous, and supplying coherent marked elements is a stronger contract.
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\Obs(G)
\overline rIntroduction passage · 2 passages
Search aliases: overline r overline
Introduction passage · 07_classification.tex, line 92
The endomorphism ring and its units. Write overline r_p for reduction of a rational r at a prime not dividing its denominator. The phrase ``for almost every selected prime'' below means outside a finite set, which may include those denominator primes.
Exact mathematical passage
\overline r_p
\overline r_pIntroduction passage · 2 passages
Search aliases: overline r_p overline
Introduction passage · 07_classification.tex, line 92
The endomorphism ring and its units. Write overline r_p for reduction of a rational r at a prime not dividing its denominator. The phrase ``for almost every selected prime'' below means outside a finite set, which may include those denominator primes.
Exact mathematical passage
\overline r_p
\overline{S(nB_X)}Introduction passage · 2 passages
Search aliases: overlineS(nB_X) overline
Introduction passage · 12_proof_tools.tex, line 115
:inverse-tool Let S:X→ Y be a bounded linear surjection between real Banach spaces. There is a constant C<∞ such that every y∈ Y has a preimage x with normx≤ Cnormy. If S is also injective, its inverse is bounded linear. If Y={0}, take the zero preimage. Otherwise write B_X for the closed unit ball. Surjectivity gives Y=⋃_n≥1overlineS(nB_X). By Lemma~[source reference], at least one of the closed sets on the right has nonempty interior. Thus there are n≥1, y_0∈ Y, and ρ>0 such that B(y_0,ρ)⊆over…
Exact mathematical passage
Y=\bigcup_{n\ge1}\overline{S(nB_X)}.
P_AIntroduction passage · 33 passages
Search aliases: P_A
Introduction passage · 03_towers.tex, line 153
For linearity, write L=I-Ad, so f^(m)=L^mf. For fixed f,g and scalars λ,μ, L^m(λ f+μ g)=λ L^mf+μ L^mg. Continuity of addition and scalar multiplication allows passage to the norm limit and proves linearity of P_A. The estimate with m=0 gives normP_Af≤norm f+fracnorm A1-qnormdf ≤(1+fracnorm Anorm d1-q)norm f, so the linear map is bounded. If df=0, every iterate equals f, hence P_Af=f. Its image is contained in ker d and it fixes that kernel, proving surjectivity onto its stated image and P_A^2=P_…
Exact mathematical passage
P_A
P_LIntroduction passage · 12 passages
Search aliases: P_L
Introduction passage · 12_proof_tools.tex, line 47
For a closed linear subspace L, denote its projection by P_L. The lemma gives a unique decomposition x=P_Lx+(x-P_Lx), P_Lx∈ L, x-P_Lx∈ L^⊥. Adding two such decompositions, or multiplying one by a scalar, proves linearity by uniqueness. The Pythagorean identity gives normP_Lx≤normx, so P_L is bounded. It fixes L, annihilates L^⊥, and satisfies P_L^2=P_L. For x,y∈ H, orthogonality gives ⟨ P_Lx,y⟩=⟨ P_Lx,P_Ly⟩ =⟨ x,P_Ly⟩. Thus P_L is self-adjoint. These properties are conclusions of the constructio…
Exact mathematical passage
P_L
P_L^2Introduction passage · 1 passage
Search aliases: P_L^2
Introduction passage · 12_proof_tools.tex, line 51
sed linear subspace L, denote its projection by P_L. The lemma gives a unique decomposition x=P_Lx+(x-P_Lx), P_Lx∈ L, x-P_Lx∈ L^⊥. Adding two such decompositions, or multiplying one by a scalar, proves linearity by uniqueness. The Pythagorean identity gives normP_Lx≤normx, so P_L is bounded. It fixes L, annihilates L^⊥, and satisfies P_L^2=P_L. For x,y∈ H, orthogonality gives ⟨ P_Lx,y⟩=⟨ P_Lx,P_Ly⟩ =⟨ x,P_Ly⟩. Thus P_L is self-adjoint. These properties are conclusions of the construction, not ex…
Exact mathematical passage
P_L^2=P_L
\pmodIntroduction passage · 1 passage
Search aliases: pmod
Introduction passage · 05_construction.tex, line 28
A section with finite arithmetic defects. For a reduced rational r=a/b, with b>0, define (s_A(a/b))_p= a b^-1pmod p, · p∈ S_A and pnmid b, 0, · p∉ S_A or pmid b. The inverse is used only when the denominator is a unit in F_p. The convention at denominator primes is essential. It is a choice of representatives, not a claim that division by p is defined in C_p.
Exact mathematical passage
\label{eq:section} (s_A(a/b))_p= \begin{cases} a b^{-1}\pmod p,&p\in S_A\text{ and }p\nmid b,\\ 0,&p\notin S_A\text{ or }p\mid b. \end{cases}
Q_UIntroduction passage · 6 passages
Search aliases: Q_U
Introduction passage · 02_obstructions.tex, line 36
The admissible corrections. Fix U⊆ObV. Write p_U:E→∏_v∈ UA_v for raw restriction. A correction preserving the prescribed values belongs to ker p_U. Its possible defects form the subgroup B_U=δ(ker p_U)≤ R. For a partial section s, choose any raw extension widetilde s, for example zero off U, and define Q_U=R/B_U, o_U(s)=[δwidetilde s]∈ Q_U. If another raw extension is chosen, their difference vanishes on U, so its differential belongs to B_U. Thus the class is independent of the lift.
Exact mathematical passage
\label{eq:obstruction} Q_U=R/B_U,\qquad \mathfrak o_U(s)=[\delta\widetilde s]\in Q_U.
q_vIntroduction passage · 11 passages
Search aliases: q_v
Introduction passage · 04_symmetry.tex, line 121
Conversely, suppose x is a global twisted section extending s. Put b_v=x_v^-1q_v. This is based, and direct substitution gives (b· c^q)_f =x_v^-1c_ff^*(x_w)=1, where the last equality is precisely the section equation for x. Hence the obstruction is neutral. These arguments establish the result using group cancellation only. In particular they do not assume an additive quotient, a norm on labels, or a decision procedure for finding b. sourceKS31 The orbit is defined from actual cocycles and an a…
Exact mathematical passage
b_v=x_v^{-1}q_v
\mathcal RIntroduction passage · 12 passages
Search aliases: R mathcal
Introduction passage · 04_symmetry.tex, line 281
The topology is defined without assuming effectivity. For a composable pair q=(vxrightarrow f wxrightarrow g z), let its support be |q|={v,w,z}. A sieve R on W is covering when |q|⊆ W Longrightarrow (|q|↪ W)∈ R for every composable q. This condition depends only on the view category, not on the coefficient groups or on a promised global realization.
Exact mathematical passage
\mathcal R
R_jIntroduction passage · 7 passages
Search aliases: R_j
Introduction passage · 03_towers.tex, line 425
ery] In the actual dyadic system, all detail coordinates jointly determine the residual. Moreover, k+1<2^N Longrightarrow (Q_Nr)_k=0, (S_Nr)_k=r_k. Thus every fixed residual coordinate is recovered exactly at a sufficiently large finite depth. The statement does not assert convergence in the supremum norm over all coordinates. Write R_j for the residual space at scale j. Its lift has the actual coordinate formula (J_jz)_2m=0, (J_jz)_2m+1=s^(j)_2mz_m, where s^(j)_2m is the corresponding target se…
Exact mathematical passage
R_j
r_kIntroduction passage · 4 passages
Search aliases: r_k
Introduction passage · 03_towers.tex, line 418
[Dyadic determination and finite coordinate recovery] In the actual dyadic system, all detail coordinates jointly determine the residual. Moreover, k+1<2^N Longrightarrow (Q_Nr)_k=0, (S_Nr)_k=r_k. Thus every fixed residual coordinate is recovered exactly at a sufficiently large finite depth. The statement does not assert convergence in the supremum norm over all coordinates. Write R_j for the residual space at scale j. Its lift has the actual coordinate formula (J_jz)_2m=0, (J_jz)_2m+1=s^(j)_2mz…
Exact mathematical passage
k+1<2^N\quad\Longrightarrow\quad (Q_Nr)_k=0,\qquad(S_Nr)_k=r_k.
r_XIntroduction passage · 2 passages
Search aliases: r_X
Introduction passage · 03a_observation.tex, line 184
Write s_X,r_X and s_Y,r_Y for the sampling and recovery isomorphisms of source and target. The asserted Hom map is [f]⟼ s_Y[f]r_X, with inverse [g]↦ r_Y[g]s_X. Associativity and the two inverse equations cancel the inserted pairs, proving both round trips. Applying the realization functor transports these equations to point spaces, where sampling and recovery are the cofinal point equivalences already constructed. No norm invariance or surjectivity of source point projections is needed for this …
Exact mathematical passage
s_X,r_X
r_YIntroduction passage · 2 passages
Search aliases: r_Y
Introduction passage · 03a_observation.tex, line 184
Write s_X,r_X and s_Y,r_Y for the sampling and recovery isomorphisms of source and target. The asserted Hom map is [f]⟼ s_Y[f]r_X, with inverse [g]↦ r_Y[g]s_X. Associativity and the two inverse equations cancel the inserted pairs, proving both round trips. Applying the realization functor transports these equations to point spaces, where sampling and recovery are the cofinal point equivalences already constructed. No norm invariance or surjectivity of source point projections is needed for this …
Exact mathematical passage
s_Y,r_Y
S^{\mathsf T}Introduction passage · 4 passages
Search aliases: S^ T mathsf
Introduction passage · 05k_stopping_rules.tex, line 6
rdinate equation is a covector In finite dimensions, use orthonormal coordinates in the ambient candidate and residual spaces. Let the columns of B form an arbitrary basis of H=ker p, not necessarily an orthonormal one. Write a=Bc, S=dB, and G=B^ TB. As before, G is positive definite. For an inexact coefficient vector c, the residual of the coordinate normal equation is q=(S^ TS+α G)c-S^ Tr. The vector q stores pairings against basis vectors. It is not the coordinate vector of the Hilbert normal…
Exact mathematical passage
q=(S^{\mathsf T}S+\alpha G)c-S^{\mathsf T}r.
s^{(j)}_{2m}Introduction passage · 3 passages
Search aliases: s^(j)_2m
Introduction passage · 03_towers.tex, line 426
residual. Moreover, k+1<2^N Longrightarrow (Q_Nr)_k=0, (S_Nr)_k=r_k. Thus every fixed residual coordinate is recovered exactly at a sufficiently large finite depth. The statement does not assert convergence in the supremum norm over all coordinates. Write R_j for the residual space at scale j. Its lift has the actual coordinate formula (J_jz)_2m=0, (J_jz)_2m+1=s^(j)_2mz_m, where s^(j)_2m is the corresponding target section at that scale. The expression on the right is composition of the section …
Exact mathematical passage
(J_jz)_{2m}=0,\qquad (J_jz)_{2m+1}=s^{(j)}_{2m}z_m,
S_1Introduction passage · 3 passages
Search aliases: S_1
Introduction passage · 03b_repair_transport.tex, line 103
For composition, denote the three sections by A,B,A_2. The composite residual map is C_2C_1, so its mismatch is S_2S_1A-A_2C_2C_1. Insert and subtract S_2BC_1: S_2S_1A-A_2C_2C_1 · =S_2(S_1A-BC_1)+(S_2B-A_2C_2)C_1
Exact mathematical passage
S_2S_1A-A_2C_2C_1
S_2Introduction passage · 5 passages
Search aliases: S_2
Introduction passage · 03b_repair_transport.tex, line 103
For composition, denote the three sections by A,B,A_2. The composite residual map is C_2C_1, so its mismatch is S_2S_1A-A_2C_2C_1. Insert and subtract S_2BC_1: S_2S_1A-A_2C_2C_1 · =S_2(S_1A-BC_1)+(S_2B-A_2C_2)C_1
Exact mathematical passage
S_2S_1A-A_2C_2C_1
s_nIntroduction passage · 17 passages
Search aliases: s_n
Introduction passage · 06_boundaries.tex, line 50
ery later image. Write I_n,m=ρ_n,m(X_m) for m≥ n. Each I_n,m is nonempty. The composition law gives I_n,m+1=ρ_n,m(ρ_m(X_m+1))⊆ I_n,m. These are subsets of the finite set X_n. Every strict decrease removes an element permanently, so only finitely many strict decreases are possible. There is an index s_n≥ n such that I_n,m=I_n,s_n for all m≥ s_n. Consequently Y_n=I_n,s_n is nonempty.
Exact mathematical passage
s_n\ge n
s_XIntroduction passage · 2 passages
Search aliases: s_X
Introduction passage · 03a_observation.tex, line 184
Write s_X,r_X and s_Y,r_Y for the sampling and recovery isomorphisms of source and target. The asserted Hom map is [f]⟼ s_Y[f]r_X, with inverse [g]↦ r_Y[g]s_X. Associativity and the two inverse equations cancel the inserted pairs, proving both round trips. Applying the realization functor transports these equations to point spaces, where sampling and recovery are the cofinal point equivalences already constructed. No norm invariance or surjectivity of source point projections is needed for this …
Exact mathematical passage
s_X,r_X
s_YIntroduction passage · 2 passages
Search aliases: s_Y
Introduction passage · 03a_observation.tex, line 184
Write s_X,r_X and s_Y,r_Y for the sampling and recovery isomorphisms of source and target. The asserted Hom map is [f]⟼ s_Y[f]r_X, with inverse [g]↦ r_Y[g]s_X. Associativity and the two inverse equations cancel the inserted pairs, proving both round trips. Applying the realization functor transports these equations to point spaces, where sampling and recovery are the cofinal point equivalences already constructed. No norm invariance or surjectivity of source point projections is needed for this …
Exact mathematical passage
s_Y,r_Y
\srankIntroduction passage · 2 passages
Search aliases: srank operatorname
Introduction passage · 01_contracts.tex, line 15
If all endomorphisms are allowed, this is module generation over the unital ring End(M). We write srank_End(M)(M) for its least generating cardinal. An infinite value can therefore express a strong obstruction even when the operator collection itself is unrestricted.
Exact mathematical passage
\srank_{\End(M)}(M)
\srank_{\End(M)}Introduction passage · 1 passage
Search aliases: srank_End(M) srank End operatorname operatorname
Introduction passage · 01_contracts.tex, line 15
If all endomorphisms are allowed, this is module generation over the unital ring End(M). We write srank_End(M)(M) for its least generating cardinal. An infinite value can therefore express a strong obstruction even when the operator collection itself is unrestricted.
Exact mathematical passage
\srank_{\End(M)}(M)
\supIntroduction passage · 5 passages
Search aliases: sup
Introduction passage · 05f_bounded_realization.tex, line 107
[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection is empty, then m_n→∞. The nearest-point theorem for nonempty closed convex subsets of a Hilbert space, proved in Appendix~[source reference], supplies …
Exact mathematical passage
\sup_n m_n<\infty
\sup_nIntroduction passage · 2 passages
Search aliases: sup_n sup
Introduction passage · 05f_bounded_realization.tex, line 107
[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection is empty, then m_n→∞. The nearest-point theorem for nonempty closed convex subsets of a Hilbert space, proved in Appendix~[source reference], supplies …
Exact mathematical passage
\sup_n m_n<\infty
\suppIntroduction passage · 2 passages
Search aliases: supp operatorname
Introduction passage · 07_classification.tex, line 14
[Tail signature] If g=s_A(r)+d has infinite order, then for all but finitely many primes, pmid_G_Ag Longleftrightarrow p∉ S_A. An explicit exception set consists of the primes dividing the nonzero numerator or denominator of r, together with supp(d). Outside that set, the reduction of r is defined and nonzero. The coordinate of g is therefore nonzero exactly at the selected primes. Apply Theorem~[source reference] with multiplier p. The full signature of the distinguished element is exact. The s…
Exact mathematical passage
\supp(d)
\mathsf TIntroduction passage · 28 passages
Search aliases: T mathsf
Introduction passage · 05k_stopping_rules.tex, line 5
The residual of the coordinate equation is a covector. In finite dimensions, use orthonormal coordinates in the ambient candidate and residual spaces. Let the columns of B form an arbitrary basis of H=ker p, not necessarily an orthonormal one. Write a=Bc, S=dB, and G=B^ TB. As before, G is positive definite. For an inexact coefficient vector c, the residual of the coordinate normal equation is q=(S^ TS+α G)c-S^ Tr. The vector q stores pairings against basis vectors. It is not the coordinate vect…
Exact mathematical passage
G=B^{\mathsf T}B
T^*Introduction passage · 50 passages
Search aliases: T^*
Introduction passage · 05j_inexact_certificates.tex, line 12
There are three different residuals. Keep H=ker p, T=d|_H, and a residual datum r. Fix α>0. Write a_α for the unique minimizer of Theorem~[source reference] and let a∈ H be any proposed correction. Three quantities now have different meanings: v=r-Ta, g=T^*(Ta-r)+α a, e=a-a_α. The vector v is the fit residual. The vector g is the normal-equation residual, equal to one half of the derivative of the quadratic objective after identifying the dual with H. The vector e is the unknown correction error…
Exact mathematical passage
v=r-Ta,\qquad g=T^*(Ta-r)+\alpha a,\qquad e=a-a_\alpha.
t^2Introduction passage · 21 passages
Search aliases: t^2
Introduction passage · 05f_bounded_realization.tex, line 115
orma_n^2. If the intersection is empty, then m_n→∞. The nearest-point theorem for nonempty closed convex subsets of a Hilbert space, proved in Appendix~[source reference], supplies a_n. Write A_n=a_n+L_n, where L_n is its closed direction subspace. For ℓ∈ L_n and every real t, the vector a_n+tℓ belongs to A_n. Minimality implies 0≤norma_n+tℓ^2-norma_n^2 =2t⟨ a_n,ℓ⟩+t^2normℓ^2. Allowing both signs of arbitrarily small t forces ⟨ a_n,ℓ⟩=0.
Exact mathematical passage
0\le\norm{a_n+t\ell}^2-\norm{a_n}^2 =2t\langle a_n,\ell\rangle+t^2\norm\ell^2.
T_qIntroduction passage · 5 passages
Search aliases: T_q
Introduction passage · 05_geometry.tex, line 23
Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_k+1x_…
Exact mathematical passage
T_qx_{i_m}-x_{i_0} =\sum_{k=0}^{m-1} (\tau_{i_0i_1}\cdots\tau_{i_{k-1}i_k})(dx)_{i_ki_{k+1}}.
U_0Introduction passage · 2 passages
Search aliases: U_0
Introduction passage · 05_geometry.tex, line 34
[The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_k+1x_i_k+1-U_kx_i_k. Summing over k=0,…,m-1…
Exact mathematical passage
U_0=I
U_kIntroduction passage · 5 passages
Search aliases: U_k
Introduction passage · 05_geometry.tex, line 34
tion[The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_k+1x_i_k+1-U_kx_i_k. Summing over k=0,……
Exact mathematical passage
U_k=\tau_{i_0i_1}\cdots\tau_{i_{k-1}i_k}
U_{k+1}Introduction passage · 2 passages
Search aliases: U_k+1
Introduction passage · 05_geometry.tex, line 34
. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_k+1x_i_k+1-U_kx_i_k. Summing over k=0,…,m-1 cancels each interior value once with each sign. The remaining terms are U_mx_i_m-U_0x_i_0=T_qx_i_m-x_i_0,…
Exact mathematical passage
U_{k+1}=U_k\tau_{i_ki_{k+1}}
\widetilde sIntroduction passage · 16 passages
Search aliases: widetilde s widetilde
Introduction passage · 02_obstructions.tex, line 35
The admissible corrections. Fix U⊆ObV. Write p_U:E→∏_v∈ UA_v for raw restriction. A correction preserving the prescribed values belongs to ker p_U. Its possible defects form the subgroup B_U=δ(ker p_U)≤ R. For a partial section s, choose any raw extension widetilde s, for example zero off U, and define Q_U=R/B_U, o_U(s)=[δwidetilde s]∈ Q_U. If another raw extension is chosen, their difference vanishes on U, so its differential belongs to B_U. Thus the class is independent of the lift.
Exact mathematical passage
\widetilde s
X_iIntroduction passage · 2 passages
Search aliases: X_i
Introduction passage · 03_towers.tex, line 57
[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.
Exact mathematical passage
\rho_{i,j}:X_j\to X_i
x_{i_0}Introduction passage · 4 passages
Search aliases: x_i_0
Introduction passage · 05_geometry.tex, line 23
Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_k+1x_…
Exact mathematical passage
T_qx_{i_m}-x_{i_0} =\sum_{k=0}^{m-1} (\tau_{i_0i_1}\cdots\tau_{i_{k-1}i_k})(dx)_{i_ki_{k+1}}.
x_{i_k}Introduction passage · 1 passage
Search aliases: x_i_k
Introduction passage · 05_geometry.tex, line 35
of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_k+1x_i_k+1-U_kx_i_k. Summing over k=0,…,m-1 cancels each interior value once with each sign. The remaining terms are U_mx_i_m-U_0x_i_0=T_qx_i_m-x_i_0, proving the identity with its actual ordered products.…
Exact mathematical passage
U_k(dx)_{i_ki_{k+1}} =U_{k+1}x_{i_{k+1}}-U_kx_{i_k}.
x_{i_{k+1}}Introduction passage · 1 passage
Search aliases: x_i_k+1
Introduction passage · 05_geometry.tex, line 35
of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_k+1x_i_k+1-U_kx_i_k. Summing over k=0,…,m-1 cancels each interior value once with each sign. The remaining terms are U_mx_i_m-U_0x_i_0=T_qx_i_m-x_i_0, proving the identity with its actual ordered products.…
Exact mathematical passage
U_k(dx)_{i_ki_{k+1}} =U_{k+1}x_{i_{k+1}}-U_kx_{i_k}.
x_{i_m}Introduction passage · 4 passages
Search aliases: x_i_m
Introduction passage · 05_geometry.tex, line 23
Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_k+1x_…
Exact mathematical passage
T_qx_{i_m}-x_{i_0} =\sum_{k=0}^{m-1} (\tau_{i_0i_1}\cdots\tau_{i_{k-1}i_k})(dx)_{i_ki_{k+1}}.
X_jIntroduction passage · 2 passages
Search aliases: X_j
Introduction passage · 03_towers.tex, line 57
[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.
Exact mathematical passage
\rho_{i,j}:X_j\to X_i
X_mIntroduction passage · 4 passages
Search aliases: X_m
Introduction passage · 06_boundaries.tex, line 38
[Finite fibers and persistent images] Let (X_n,ρ_n) be a sequential inverse system of nonempty finite sets. Its inverse limit is nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) for m≥ n. Each I_n,m is nonempty. The composition law gives I_n,m+1=ρ_n,m(ρ_m(X_m+1))⊆ I_n,m. These are subsets of the finite set X_n. Every strict dec…
Exact mathematical passage
\rho_{n,m}:X_m\to X_n
X_{m+1}Introduction passage · 2 passages
Search aliases: X_m+1
Introduction passage · 06_boundaries.tex, line 47
nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) for m≥ n. Each I_n,m is nonempty. The composition law gives I_n,m+1=ρ_n,m(ρ_m(X_m+1))⊆ I_n,m. These are subsets of the finite set X_n. Every strict decrease removes an element permanently, so only finitely many strict decreases are possible. There is an index s_n≥ n such that I_n…
Exact mathematical passage
I_{n,m+1}=\rho_{n,m}\bigl(\rho_m(X_{m+1})\bigr)\subseteq I_{n,m}.
X_nIntroduction passage · 16 passages
Search aliases: X_n
Introduction passage · 06_boundaries.tex, line 38
[Finite fibers and persistent images] Let (X_n,ρ_n) be a sequential inverse system of nonempty finite sets. Its inverse limit is nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) for m≥ n. Each I_n,m is nonempty. The composition law gives I_n,m+1=ρ_n,m(ρ_m(X_m+1))⊆ I_n,m. These are subsets of the finite set X_n. Every strict dec…
Exact mathematical passage
(X_n,\rho_n)
X_{n+1}Introduction passage · 3 passages
Search aliases: X_n+1
Introduction passage · 06_boundaries.tex, line 52
We next prove that restriction induces a surjection Y_n+1→ Y_n, even though X_n+1→ X_n need not be surjective. For membership, let z∈ Y_n+1. Given m≥ n+1, write z=ρ_n+1,m(w) for some w∈ X_m. Then ρ_n z=ρ_n,m(w). Together with the automatic condition at m=n, this shows ρ_nz∈ Y_n.
Exact mathematical passage
X_{n+1}\to X_n
x_v^{-1}Introduction passage · 2 passages
Search aliases: x_v^-1
Introduction passage · 04_symmetry.tex, line 121
Conversely, suppose x is a global twisted section extending s. Put b_v=x_v^-1q_v. This is based, and direct substitution gives (b· c^q)_f =x_v^-1c_ff^*(x_w)=1, where the last equality is precisely the section equation for x. Hence the obstruction is neutral. These arguments establish the result using group cancellation only. In particular they do not assume an additive quotient, a norm on labels, or a decision procedure for finding b. sourceKS31 The orbit is defined from actual cocycles and an a…
Exact mathematical passage
b_v=x_v^{-1}q_v
X_{\phi(n)}Introduction passage · 2 passages
Search aliases: X_ϕ(n) phi
Introduction passage · 03a_observation.tex, line 6
An output level may require a finer input level. Let X and Y be sequential set-valued resolutions. Write ρ^X_i,j:X_j→ X_i for composite restriction. A reindexed transport consists of a monotone schedule ϕ:N→N and maps f_n:X_ϕ(n)⟶ Y_n satisfying, for i≤ j, ρ^Y_i,jf_j=f_iρ^X_ϕ(i),ϕ(j). The schedule records how much input is used for each output level. Monotonicity means that asking for more output never forces the representation to discard input already available. It does not require strict growth…
Exact mathematical passage
f_n:X_{\phi(n)}\longrightarrow Y_n
YIntroduction passage · 86 passages
Search aliases: Y
Introduction passage · 03a_observation.tex, line 5
An output level may require a finer input level. Let X and Y be sequential set-valued resolutions. Write ρ^X_i,j:X_j→ X_i for composite restriction. A reindexed transport consists of a monotone schedule ϕ:N→N and maps f_n:X_ϕ(n)⟶ Y_n satisfying, for i≤ j, ρ^Y_i,jf_j=f_iρ^X_ϕ(i),ϕ(j). The schedule records how much input is used for each output level. Monotonicity means that asking for more output never forces the representation to discard input already available. It does not require strict growth…
Exact mathematical passage
Y
y_0Introduction passage · 12 passages
Search aliases: y_0
Introduction passage · 12_proof_tools.tex, line 118
with normx≤ Cnormy. If S is also injective, its inverse is bounded linear. If Y={0}, take the zero preimage. Otherwise write B_X for the closed unit ball. Surjectivity gives Y=⋃_n≥1overlineS(nB_X). By Lemma~[source reference], at least one of the closed sets on the right has nonempty interior. Thus there are n≥1, y_0∈ Y, and ρ>0 such that B(y_0,ρ)⊆overlineS(nB_X). Here B(y_0,ρ) is an open ball. If normh<ρ, both y_0+h and y_0 belong to this closure. Choose approximating image sequences for both. …
Exact mathematical passage
y_0\in Y
y_mIntroduction passage · 2 passages
Search aliases: y_m
Introduction passage · 03_towers.tex, line 58
[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.
Exact mathematical passage
\rho_{\phi(n),\phi(m)}y_m=y_n\qquad(n\le m),
Y_nIntroduction passage · 16 passages
Search aliases: Y_n
Introduction passage · 03a_observation.tex, line 6
An output level may require a finer input level. Let X and Y be sequential set-valued resolutions. Write ρ^X_i,j:X_j→ X_i for composite restriction. A reindexed transport consists of a monotone schedule ϕ:N→N and maps f_n:X_ϕ(n)⟶ Y_n satisfying, for i≤ j, ρ^Y_i,jf_j=f_iρ^X_ϕ(i),ϕ(j). The schedule records how much input is used for each output level. Monotonicity means that asking for more output never forces the representation to discard input already available. It does not require strict growth…
Exact mathematical passage
f_n:X_{\phi(n)}\longrightarrow Y_n
y_nIntroduction passage · 5 passages
Search aliases: y_n
Introduction passage · 03_towers.tex, line 58
[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.
Exact mathematical passage
\rho_{\phi(n),\phi(m)}y_m=y_n\qquad(n\le m),
Y_{n+1}Introduction passage · 5 passages
Search aliases: Y_n+1
Introduction passage · 06_boundaries.tex, line 52
We next prove that restriction induces a surjection Y_n+1→ Y_n, even though X_n+1→ X_n need not be surjective. For membership, let z∈ Y_n+1. Given m≥ n+1, write z=ρ_n+1,m(w) for some w∈ X_m. Then ρ_n z=ρ_n,m(w). Together with the automatic condition at m=n, this shows ρ_nz∈ Y_n.
Exact mathematical passage
Y_{n+1}\to Y_n
zIntroduction passage · 125 passages
Search aliases: z
Introduction passage · 03_towers.tex, line 426
residual. Moreover, k+1<2^N Longrightarrow (Q_Nr)_k=0, (S_Nr)_k=r_k. Thus every fixed residual coordinate is recovered exactly at a sufficiently large finite depth. The statement does not assert convergence in the supremum norm over all coordinates. Write R_j for the residual space at scale j. Its lift has the actual coordinate formula (J_jz)_2m=0, (J_jz)_2m+1=s^(j)_2mz_m, where s^(j)_2m is the corresponding target section at that scale. The expression on the right is composition of the section …
Exact mathematical passage
(J_jz)_{2m}=0,\qquad (J_jz)_{2m+1}=s^{(j)}_{2m}z_m,
z_mIntroduction passage · 1 passage
Search aliases: z_m
Introduction passage · 03_towers.tex, line 426
residual. Moreover, k+1<2^N Longrightarrow (Q_Nr)_k=0, (S_Nr)_k=r_k. Thus every fixed residual coordinate is recovered exactly at a sufficiently large finite depth. The statement does not assert convergence in the supremum norm over all coordinates. Write R_j for the residual space at scale j. Its lift has the actual coordinate formula (J_jz)_2m=0, (J_jz)_2m+1=s^(j)_2mz_m, where s^(j)_2m is the corresponding target section at that scale. The expression on the right is composition of the section …
Exact mathematical passage
(J_jz)_{2m}=0,\qquad (J_jz)_{2m+1}=s^{(j)}_{2m}z_m,
|_LIntroduction passage · 2 passages
Search aliases: |_L
Introduction passage · 05_geometry.tex, line 327
chapQuantitative descent in a common Hilbert model Local exactness and matching are separate operators. Let E be a real Hilbert space of raw local candidates. Let D:E→ R and C:E→ Y be continuous linear maps into Banach spaces, measuring local equation defects and overlap mismatch, respectively. Put L=ker D, B=C|_L, M=ker D∩ker C. The exact matched families form M. First project to L, then project inside L to ker B. Denote the resulting map by P.
Exact mathematical passage
L=\ker D,\qquad B=C|_L, \qquad M=\ker D\cap\ker C.
\cdotIntroduction passage · 14 passages
Search aliases: · cdot
Introduction passage · 04_symmetry.tex, line 122
Conversely, suppose x is a global twisted section extending s. Put b_v=x_v^-1q_v. This is based, and direct substitution gives (b· c^q)_f =x_v^-1c_ff^*(x_w)=1, where the last equality is precisely the section equation for x. Hence the obstruction is neutral. These arguments establish the result using group cancellation only. In particular they do not assume an additive quotient, a norm on labels, or a decision procedure for finding b. sourceKS31 The orbit is defined from actual cocycles and an a…
Exact mathematical passage
(b\cdot c^q)_f =x_v^{-1}c_ff^*(x_w)=1,
\gammaIntroduction passage · 12 passages
Search aliases: γ gamma
Introduction passage · 05_geometry.tex, line 28
Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_k+1x_…
Exact mathematical passage
\gamma(q)
\lambdaIntroduction passage · 34 passages
Search aliases: λ lambda
Introduction passage · 03_towers.tex, line 149
For linearity, write L=I-Ad, so f^(m)=L^mf. For fixed f,g and scalars λ,μ, L^m(λ f+μ g)=λ L^mf+μ L^mg. Continuity of addition and scalar multiplication allows passage to the norm limit and proves linearity of P_A. The estimate with m=0 gives normP_Af≤norm f+fracnorm A1-qnormdf ≤(1+fracnorm Anorm d1-q)norm f, so the linear map is bounded. If df=0, every iterate equals f, hence P_Af=f. Its image is contained in ker d and it fixes that kernel, proving surjectivity onto its stated image and P_A^2=P_…
Exact mathematical passage
\lambda,\mu
\muIntroduction passage · 11 passages
Search aliases: μ mu
Introduction passage · 03_towers.tex, line 149
For linearity, write L=I-Ad, so f^(m)=L^mf. For fixed f,g and scalars λ,μ, L^m(λ f+μ g)=λ L^mf+μ L^mg. Continuity of addition and scalar multiplication allows passage to the norm limit and proves linearity of P_A. The estimate with m=0 gives normP_Af≤norm f+fracnorm A1-qnormdf ≤(1+fracnorm Anorm d1-q)norm f, so the linear map is bounded. If df=0, every iterate equals f, hence P_Af=f. Its image is contained in ker d and it fixes that kernel, proving surjectivity onto its stated image and P_A^2=P_…
Exact mathematical passage
\lambda,\mu
\rhoIntroduction passage · 79 passages
Search aliases: ρ rho
Introduction passage · 03_towers.tex, line 57
[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.
Exact mathematical passage
\rho_{i,j}:X_j\to X_i
\rho^X_{i,j}Introduction passage · 1 passage
Search aliases: ρ^X_i,j rho
Introduction passage · 03a_observation.tex, line 5
An output level may require a finer input level. Let X and Y be sequential set-valued resolutions. Write ρ^X_i,j:X_j→ X_i for composite restriction. A reindexed transport consists of a monotone schedule ϕ:N→N and maps f_n:X_ϕ(n)⟶ Y_n satisfying, for i≤ j, ρ^Y_i,jf_j=f_iρ^X_ϕ(i),ϕ(j). The schedule records how much input is used for each output level. Monotonicity means that asking for more output never forces the representation to discard input already available. It does not require strict growth…
Exact mathematical passage
\rho^X_{i,j}:X_j\to X_i
\rho^X_{\phi(i),\phi(j)}Introduction passage · 2 passages
Search aliases: ρ^X_ϕ(i),ϕ(j) rho phi phi
Introduction passage · 03a_observation.tex, line 10
An output level may require a finer input level. Let X and Y be sequential set-valued resolutions. Write ρ^X_i,j:X_j→ X_i for composite restriction. A reindexed transport consists of a monotone schedule ϕ:N→N and maps f_n:X_ϕ(n)⟶ Y_n satisfying, for i≤ j, ρ^Y_i,jf_j=f_iρ^X_ϕ(i),ϕ(j). The schedule records how much input is used for each output level. Monotonicity means that asking for more output never forces the representation to discard input already available. It does not require strict growth…
Exact mathematical passage
\label{eq:scheduled-naturality} \rho^Y_{i,j}f_j=f_i\rho^X_{\phi(i),\phi(j)}.
\rho^Y_{i,j}Introduction passage · 2 passages
Search aliases: ρ^Y_i,j rho
Introduction passage · 03a_observation.tex, line 10
An output level may require a finer input level. Let X and Y be sequential set-valued resolutions. Write ρ^X_i,j:X_j→ X_i for composite restriction. A reindexed transport consists of a monotone schedule ϕ:N→N and maps f_n:X_ϕ(n)⟶ Y_n satisfying, for i≤ j, ρ^Y_i,jf_j=f_iρ^X_ϕ(i),ϕ(j). The schedule records how much input is used for each output level. Monotonicity means that asking for more output never forces the representation to discard input already available. It does not require strict growth…
Exact mathematical passage
\label{eq:scheduled-naturality} \rho^Y_{i,j}f_j=f_i\rho^X_{\phi(i),\phi(j)}.
\rho_{i,i}Introduction passage · 1 passage
Search aliases: ρ_i,i rho
Introduction passage · 03_towers.tex, line 57
[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.
Exact mathematical passage
\rho_{i,i}=I
\rho_{i,j}Introduction passage · 2 passages
Search aliases: ρ_i,j rho
Introduction passage · 03_towers.tex, line 57
[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.
Exact mathematical passage
\rho_{i,j}:X_j\to X_i
\rho_{i,k}Introduction passage · 1 passage
Search aliases: ρ_i,k rho
Introduction passage · 03_towers.tex, line 57
[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.
Exact mathematical passage
\rho_{i,j}\rho_{j,k}=\rho_{i,k}
\rho_{j,k}Introduction passage · 1 passage
Search aliases: ρ_j,k rho
Introduction passage · 03_towers.tex, line 57
[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.
Exact mathematical passage
\rho_{i,j}\rho_{j,k}=\rho_{i,k}
\rho_mIntroduction passage · 1 passage
Search aliases: ρ_m rho
Introduction passage · 06_boundaries.tex, line 47
nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) for m≥ n. Each I_n,m is nonempty. The composition law gives I_n,m+1=ρ_n,m(ρ_m(X_m+1))⊆ I_n,m. These are subsets of the finite set X_n. Every strict decrease removes an element permanently, so only finitely many strict decreases are possible. There is an index s_n≥ n such that I_n…
Exact mathematical passage
I_{n,m+1}=\rho_{n,m}\bigl(\rho_m(X_{m+1})\bigr)\subseteq I_{n,m}.
\rho_nIntroduction passage · 6 passages
Search aliases: ρ_n rho
Introduction passage · 06_boundaries.tex, line 38
[Finite fibers and persistent images] Let (X_n,ρ_n) be a sequential inverse system of nonempty finite sets. Its inverse limit is nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) for m≥ n. Each I_n,m is nonempty. The composition law gives I_n,m+1=ρ_n,m(ρ_m(X_m+1))⊆ I_n,m. These are subsets of the finite set X_n. Every strict dec…
Exact mathematical passage
(X_n,\rho_n)
\rho_{n+1,m}Introduction passage · 2 passages
Search aliases: ρ_n+1,m rho
Introduction passage · 06_boundaries.tex, line 52
We next prove that restriction induces a surjection Y_n+1→ Y_n, even though X_n+1→ X_n need not be surjective. For membership, let z∈ Y_n+1. Given m≥ n+1, write z=ρ_n+1,m(w) for some w∈ X_m. Then ρ_n z=ρ_n,m(w). Together with the automatic condition at m=n, this shows ρ_nz∈ Y_n.
Exact mathematical passage
z=\rho_{n+1,m}(w)
\rho_{n,m}Introduction passage · 7 passages
Search aliases: ρ_n,m rho
Introduction passage · 06_boundaries.tex, line 38
[Finite fibers and persistent images] Let (X_n,ρ_n) be a sequential inverse system of nonempty finite sets. Its inverse limit is nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) for m≥ n. Each I_n,m is nonempty. The composition law gives I_n,m+1=ρ_n,m(ρ_m(X_m+1))⊆ I_n,m. These are subsets of the finite set X_n. Every strict dec…
Exact mathematical passage
\rho_{n,m}:X_m\to X_n
\rho_{\phi(n),\phi(m)}Introduction passage · 1 passage
Search aliases: ρ_ϕ(n),ϕ(m) rho phi phi
Introduction passage · 03_towers.tex, line 58
[Cofinal reconstruction of points] Sampling induces an equivalence between the coherent points of X and of its sampled tower. Write ρ_i,j:X_j→ X_i for the composite restriction when i≤ j, with ρ_i,i=I. These maps satisfy ρ_i,jρ_j,k=ρ_i,k. A coherent sampled point obeys ρ_ϕ(n),ϕ(m)y_m=y_n (n≤ m), by iterating its adjacent coherence equations.
Exact mathematical passage
\rho_{\phi(n),\phi(m)}y_m=y_n\qquad(n\le m),
\tauIntroduction passage · 12 passages
Search aliases: τ tau
Introduction passage · 05_geometry.tex, line 23
Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_k+1x_…
Exact mathematical passage
T_qx_{i_m}-x_{i_0} =\sum_{k=0}^{m-1} (\tau_{i_0i_1}\cdots\tau_{i_{k-1}i_k})(dx)_{i_ki_{k+1}}.
\tau_{i_0i_1}Introduction passage · 3 passages
Search aliases: τ_i_0i_1 tau
Introduction passage · 05_geometry.tex, line 23
Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_k+1x_…
Exact mathematical passage
T_qx_{i_m}-x_{i_0} =\sum_{k=0}^{m-1} (\tau_{i_0i_1}\cdots\tau_{i_{k-1}i_k})(dx)_{i_ki_{k+1}}.
\tau_{i_{k-1}i_k}Introduction passage · 2 passages
Search aliases: τ_i_k-1i_k tau
Introduction passage · 05_geometry.tex, line 23
Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_k+1x_…
Exact mathematical passage
T_qx_{i_m}-x_{i_0} =\sum_{k=0}^{m-1} (\tau_{i_0i_1}\cdots\tau_{i_{k-1}i_k})(dx)_{i_ki_{k+1}}.
\tau_{i_ki_{k+1}}Introduction passage · 1 passage
Search aliases: τ_i_ki_k+1 tau
Introduction passage · 05_geometry.tex, line 34
. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_k+1x_i_k+1-U_kx_i_k. Summing over k=0,…,m-1 cancels each interior value once with each sign. The remaining terms are U_mx_i_m-U_0x_i_0=T_qx_i_m-x_i_0,…
Exact mathematical passage
U_{k+1}=U_k\tau_{i_ki_{k+1}}
\ldotsIntroduction passage · 18 passages
Search aliases: … ldots
Introduction passage · 05_geometry.tex, line 22
Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_k+1x_…
Exact mathematical passage
q=(i_0,\ldots,i_m)
\ellIntroduction passage · 51 passages
Search aliases: ℓ ell
Introduction passage · 05f_bounded_realization.tex, line 114
to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection is empty, then m_n→∞. The nearest-point theorem for nonempty closed convex subsets of a Hilbert space, proved in Appendix~[source reference], supplies a_n. Write A_n=a_n+L_n, where L_n is its closed direction subspace. For ℓ∈ L_n and every real t, the vector a_n+tℓ belongs to A_n. Minimality implies 0≤norma_n+tℓ^2-norma_n^2 =2t⟨ a_n,ℓ⟩+t^2normℓ^2. Allowing both signs of arbitrarily small t for…
Exact mathematical passage
\ell\in L_n
\ell^2Introduction passage · 18 passages
Search aliases: ℓ^2 ell
Introduction passage · 05f_bounded_realization.tex, line 115
orma_n^2. If the intersection is empty, then m_n→∞. The nearest-point theorem for nonempty closed convex subsets of a Hilbert space, proved in Appendix~[source reference], supplies a_n. Write A_n=a_n+L_n, where L_n is its closed direction subspace. For ℓ∈ L_n and every real t, the vector a_n+tℓ belongs to A_n. Minimality implies 0≤norma_n+tℓ^2-norma_n^2 =2t⟨ a_n,ℓ⟩+t^2normℓ^2. Allowing both signs of arbitrarily small t forces ⟨ a_n,ℓ⟩=0.
Exact mathematical passage
0\le\norm{a_n+t\ell}^2-\norm{a_n}^2 =2t\langle a_n,\ell\rangle+t^2\norm\ell^2.
\mapstoIntroduction passage · 22 passages
Search aliases: ↦ mapsto
Introduction passage · 03a_observation.tex, line 188
Write s_X,r_X and s_Y,r_Y for the sampling and recovery isomorphisms of source and target. The asserted Hom map is [f]⟼ s_Y[f]r_X, with inverse [g]↦ r_Y[g]s_X. Associativity and the two inverse equations cancel the inserted pairs, proving both round trips. Applying the realization functor transports these equations to point spaces, where sampling and recovery are the cofinal point equivalences already constructed. No norm invariance or surjectivity of source point projections is needed for this …
Exact mathematical passage
[g]\mapsto r_Y[g]s_X
\hookrightarrowIntroduction passage · 5 passages
Search aliases: ↪ hookrightarrow
Introduction passage · 04_symmetry.tex, line 282
The topology is defined without assuming effectivity. For a composable pair q=(vxrightarrow f wxrightarrow g z), let its support be |q|={v,w,z}. A sieve R on W is covering when |q|⊆ W Longrightarrow (|q|↪ W)∈ R for every composable q. This condition depends only on the view category, not on the coefficient groups or on a promised global realization.
Exact mathematical passage
\label{eq:site} |q|\subseteq W\quad\Longrightarrow\quad (|q|\hookrightarrow W)\in\mathcal R \quad\text{for every composable }q.
\varnothingIntroduction passage · 6 passages
Search aliases: ∅ varnothing
Introduction passage · 10_grammar.tex, line 12
For each j whose output lies in the family, choose B_j⊆ N such that R_j≅ G_B_j. For other j, set B_j=∅. This is a set-theoretic choice of representatives. It is not a procedure for deciding whether an arbitrary description denotes a group in the family.
Exact mathematical passage
B_j=\varnothing
\notinIntroduction passage · 13 passages
Search aliases: ∉ notin
Introduction passage · 05_construction.tex, line 28
A section with finite arithmetic defects. For a reduced rational r=a/b, with b>0, define (s_A(a/b))_p= a b^-1pmod p, · p∈ S_A and pnmid b, 0, · p∉ S_A or pmid b. The inverse is used only when the denominator is a unit in F_p. The convention at denominator primes is essential. It is a choice of representatives, not a claim that division by p is defined in C_p.
Exact mathematical passage
\label{eq:section} (s_A(a/b))_p= \begin{cases} a b^{-1}\pmod p,&p\in S_A\text{ and }p\nmid b,\\ 0,&p\notin S_A\text{ or }p\mid b. \end{cases}
\prod_{v\in U}Introduction passage · 1 passage
Search aliases: ∏_v∈ U prod in
Introduction passage · 02_obstructions.tex, line 31
The admissible corrections. Fix U⊆ObV. Write p_U:E→∏_v∈ UA_v for raw restriction. A correction preserving the prescribed values belongs to ker p_U. Its possible defects form the subgroup B_U=δ(ker p_U)≤ R. For a partial section s, choose any raw extension widetilde s, for example zero off U, and define Q_U=R/B_U, o_U(s)=[δwidetilde s]∈ Q_U. If another raw extension is chosen, their difference vanishes on U, so its differential belongs to B_U. Thus the class is independent of the lift.
Exact mathematical passage
p_U:E\to\prod_{v\in U}A_v
\sumIntroduction passage · 47 passages
Search aliases: ∑ sum
Introduction passage · 05_geometry.tex, line 23
Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_k+1x_…
Exact mathematical passage
T_qx_{i_m}-x_{i_0} =\sum_{k=0}^{m-1} (\tau_{i_0i_1}\cdots\tau_{i_{k-1}i_k})(dx)_{i_ki_{k+1}}.
\sum_{k=0}^{m-1}Introduction passage · 2 passages
Search aliases: ∑_k=0^m-1 sum
Introduction passage · 05_geometry.tex, line 23
Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_k+1x_…
Exact mathematical passage
T_qx_{i_m}-x_{i_0} =\sum_{k=0}^{m-1} (\tau_{i_0i_1}\cdots\tau_{i_{k-1}i_k})(dx)_{i_ki_{k+1}}.
\perpIntroduction passage · 33 passages
Search aliases: ⊥ perp
Introduction passage · 12_proof_tools.tex, line 48
For a closed linear subspace L, denote its projection by P_L. The lemma gives a unique decomposition x=P_Lx+(x-P_Lx), P_Lx∈ L, x-P_Lx∈ L^⊥. Adding two such decompositions, or multiplying one by a scalar, proves linearity by uniqueness. The Pythagorean identity gives normP_Lx≤normx, so P_L is bounded. It fixes L, annihilates L^⊥, and satisfies P_L^2=P_L. For x,y∈ H, orthogonality gives ⟨ P_Lx,y⟩=⟨ P_Lx,P_Ly⟩ =⟨ x,P_Ly⟩. Thus P_L is self-adjoint. These properties are conclusions of the constructio…
Exact mathematical passage
x=P_Lx+(x-P_Lx),\qquad P_Lx\in L,\quad x-P_Lx\in L^\perp.
\bigcapIntroduction passage · 3 passages
Search aliases: ⋂ bigcap
Introduction passage · 05f_bounded_realization.tex, line 107
[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection is empty, then m_n→∞. The nearest-point theorem for nonempty closed convex subsets of a Hilbert space, proved in Appendix~[source reference], supplies …
Exact mathematical passage
a_*\in\bigcap_n A_n
\bigcap_{m\ge n}Introduction passage · 1 passage
Search aliases: ⋂_m≥ n bigcap ge
Introduction passage · 06_boundaries.tex, line 39
[Finite fibers and persistent images] Let (X_n,ρ_n) be a sequential inverse system of nonempty finite sets. Its inverse limit is nonempty. More precisely, if ρ_n,m:X_m→ X_n is composite restriction, then im(ev_n)=Y_n, Y_n=⋂_m≥ nimρ_n,m. Thus a prescribed value at level n extends precisely when it survives every later image. Write I_n,m=ρ_n,m(X_m) for m≥ n. Each I_n,m is nonempty. The composition law gives I_n,m+1=ρ_n,m(ρ_m(X_m+1))⊆ I_n,m. These are subsets of the finite set X_n. Every strict dec…
Exact mathematical passage
\im(\operatorname{ev}_n)=Y_n, \qquad Y_n=\bigcap_{m\ge n}\im\rho_{n,m}.
\bigcap_nIntroduction passage · 1 passage
Search aliases: ⋂_n bigcap
Introduction passage · 05f_bounded_realization.tex, line 107
[A uniform finite budget yields a strong realization] For the nested affine sets above, m_n is nondecreasing, and for m≥ n, norma_m-a_n^2=m_m^2-m_n^2. Their intersection is nonempty if and only if sup_n m_n<∞. In that case a_n converges in norm to the unique least norm point a_*∈⋂_n A_n, with norma_*-a_n^2=norma_*^2-norma_n^2. If the intersection is empty, then m_n→∞. The nearest-point theorem for nonempty closed convex subsets of a Hilbert space, proved in Appendix~[source reference], supplies …
Exact mathematical passage
a_*\in\bigcap_n A_n
\bigcupIntroduction passage · 2 passages
Search aliases: ⋃ bigcup
Introduction passage · 12_proof_tools.tex, line 115
:inverse-tool Let S:X→ Y be a bounded linear surjection between real Banach spaces. There is a constant C<∞ such that every y∈ Y has a preimage x with normx≤ Cnormy. If S is also injective, its inverse is bounded linear. If Y={0}, take the zero preimage. Otherwise write B_X for the closed unit ball. Surjectivity gives Y=⋃_n≥1overlineS(nB_X). By Lemma~[source reference], at least one of the closed sets on the right has nonempty interior. Thus there are n≥1, y_0∈ Y, and ρ>0 such that B(y_0,ρ)⊆over…
Exact mathematical passage
Y=\bigcup_{n\ge1}\overline{S(nB_X)}.
\bigcup_{n\ge1}Introduction passage · 1 passage
Search aliases: ⋃_n≥1 bigcup ge
Introduction passage · 12_proof_tools.tex, line 115
:inverse-tool Let S:X→ Y be a bounded linear surjection between real Banach spaces. There is a constant C<∞ such that every y∈ Y has a preimage x with normx≤ Cnormy. If S is also injective, its inverse is bounded linear. If Y={0}, take the zero preimage. Otherwise write B_X for the closed unit ball. Surjectivity gives Y=⋃_n≥1overlineS(nB_X). By Lemma~[source reference], at least one of the closed sets on the right has nonempty interior. Thus there are n≥1, y_0∈ Y, and ρ>0 such that B(y_0,ρ)⊆over…
Exact mathematical passage
Y=\bigcup_{n\ge1}\overline{S(nB_X)}.
\cdotsIntroduction passage · 9 passages
Search aliases: ⋯ cdots
Introduction passage · 05_geometry.tex, line 23
Transported residuals and path gains. [The path telescope] For a walk q=(i_0,…,i_m), T_qx_i_m-x_i_0 =∑_k=0^m-1 (τ_i_0i_1⋯τ_i_k-1i_k)(dx)_i_ki_k+1. The empty prefix is the identity. If γ(q) is the sum of the products of preceding edge norms, then normT_qx_i_m-x_i_0≤γ(q)normdx_∞. Write U_0=I and U_k=τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_k+1x_…
Exact mathematical passage
T_qx_{i_m}-x_{i_0} =\sum_{k=0}^{m-1} (\tau_{i_0i_1}\cdots\tau_{i_{k-1}i_k})(dx)_{i_ki_{k+1}}.
\langleIntroduction passage · 51 passages
Search aliases: ⟨ langle
Introduction passage · 05f_bounded_realization.tex, line 115
orma_n^2. If the intersection is empty, then m_n→∞. The nearest-point theorem for nonempty closed convex subsets of a Hilbert space, proved in Appendix~[source reference], supplies a_n. Write A_n=a_n+L_n, where L_n is its closed direction subspace. For ℓ∈ L_n and every real t, the vector a_n+tℓ belongs to A_n. Minimality implies 0≤norma_n+tℓ^2-norma_n^2 =2t⟨ a_n,ℓ⟩+t^2normℓ^2. Allowing both signs of arbitrarily small t forces ⟨ a_n,ℓ⟩=0.
Exact mathematical passage
0\le\norm{a_n+t\ell}^2-\norm{a_n}^2 =2t\langle a_n,\ell\rangle+t^2\norm\ell^2.
\rangleIntroduction passage · 51 passages
Search aliases: ⟩ rangle
Introduction passage · 05f_bounded_realization.tex, line 115
orma_n^2. If the intersection is empty, then m_n→∞. The nearest-point theorem for nonempty closed convex subsets of a Hilbert space, proved in Appendix~[source reference], supplies a_n. Write A_n=a_n+L_n, where L_n is its closed direction subspace. For ℓ∈ L_n and every real t, the vector a_n+tℓ belongs to A_n. Minimality implies 0≤norma_n+tℓ^2-norma_n^2 =2t⟨ a_n,ℓ⟩+t^2normℓ^2. Allowing both signs of arbitrarily small t forces ⟨ a_n,ℓ⟩=0.
Exact mathematical passage
0\le\norm{a_n+t\ell}^2-\norm{a_n}^2 =2t\langle a_n,\ell\rangle+t^2\norm\ell^2.
\longmapstoIntroduction passage · 8 passages
Search aliases: ⟼ longmapsto
Introduction passage · 03a_observation.tex, line 185
Write s_X,r_X and s_Y,r_Y for the sampling and recovery isomorphisms of source and target. The asserted Hom map is [f]⟼ s_Y[f]r_X, with inverse [g]↦ r_Y[g]s_X. Associativity and the two inverse equations cancel the inserted pairs, proving both round trips. Applying the realization functor transports these equations to point spaces, where sampling and recovery are the cofinal point equivalences already constructed. No norm invariance or surjectivity of source point projections is needed for this …
Exact mathematical passage
[f]\longmapsto s_Y[f]r_X,
\mathbf0Use in source · 1 passage
Search aliases: 0 mathbf
Use in source · 08_computability.tex, line 81
ystem presentation of Chapter~[source reference]. Choose any set A of the desired degree and apply Theorem~[source reference]. All the structural and completion properties were proved in Chapters~[source reference]--[source reference]. For A=∅, the copy is computable. For A=K, the group has no computable copy and its least degree is the halting degree 0'. Iterated jumps yield further examples. An arbitrary selector supplies any desired degree without changing the finite quotient tables or their …
Exact mathematical passage
\mathbf 0'
\mathbf0'Use in source · 1 passage
Search aliases: 0' mathbf
Use in source · 08_computability.tex, line 81
ystem presentation of Chapter~[source reference]. Choose any set A of the desired degree and apply Theorem~[source reference]. All the structural and completion properties were proved in Chapters~[source reference]--[source reference]. For A=∅, the copy is computable. For A=K, the group has no computable copy and its least degree is the halting degree 0'. Iterated jumps yield further examples. An arbitrary selector supplies any desired degree without changing the finite quotient tables or their …
Exact mathematical passage
\mathbf 0'
\mathbf1Use in source · 7 passages
Search aliases: 1 mathbf
Use in source · 03_towers.tex, line 470
Two boundary examples. On ℓ^∞(N,R), let C remove the first coordinate and let J prepend zero. Then Q_Nx is the tail of x, placed at its original coordinates with zeros before N. Every fixed coordinate is eventually zero in the remainder, and normQ_N≤1. Nevertheless the constant sequence 1 satisfies normQ_N1=1 for every N.
Exact mathematical passage
\mathbf1
\mathbf1^{\mathsf T}Use in source · 1 passage
Search aliases: 1^ T mathbf mathsf
Use in source · 06_cases.tex, line 154
-1 · 0 · -1. The ordinary cycle block satisfies A^ TA=3I-11^ T. It is zero on the constant line and equals 3I on its orthogonal complement. The reflected block satisfies B^ TB=I+vv^ T, v=(1,-1,1)^ T. Since norm v^2=3, this matrix has eigenvalue four in the direction of v and eigenvalue one on its orthogonal complement. In particular it is positive definite.
Exact mathematical passage
A^{\mathsf T}A=3I-\mathbf1\mathbf1^{\mathsf T}.
\mathbf1_NUse in source · 3 passages
Search aliases: 1_N mathbf
Use in source · 05e_escape.tex, line 49
A complete calculation of the finite repairs. Let 1_N be one in coordinates 1,…,N and zero elsewhere. Put a_N=( 1_N,0), f_N=f_0-a_N=(- 1_N,1). Then p(f_N)=1, and the first N residual coordinates vanish exactly. The remaining residual and movement satisfy d(f_N)_n=0 · n≤ N,
Exact mathematical passage
\mathbf 1_N
A'Use in source · 14 passages
Search aliases: A'
Use in source · 05_geometry.tex, line 485
Constructing the new certificate. Assume the target R is Banach and A:R→ E is a continuous linear right inverse of D, with DA=I, norm A≤ a, and aε<1. Set T=(D-D')A, Z=(I-T)^-1, A'=AZ. The inverse is supplied by the convergent Neumann series because norm T≤ aε<1.
Exact mathematical passage
T=(D-D')A,\qquad Z=(I-T)^{-1},\qquad A'=AZ.
a'_{ij}Use in source · 1 passage
Search aliases: a'_ij
Use in source · 04_symmetry.tex, line 195
a_ik(v) · =a_jk(v)a_ij(v). The triple equation implies a_ii=1 and a_ji=a_ij^-1. A morphism of descent data is a family of local gauges r_i satisfying r_j(v)a_ij(v)=a'_ij(v)r_i(v). These objects and arrows form a groupoid.
Exact mathematical passage
r_j(v)a_{ij}(v)=a'_{ij}(v)r_i(v).
A^{\mathsf T}Use in source · 1 passage
Search aliases: A^ T mathsf
Use in source · 06_cases.tex, line 154
-1 · 0 · -1. The ordinary cycle block satisfies A^ TA=3I-11^ T. It is zero on the constant line and equals 3I on its orthogonal complement. The reflected block satisfies B^ TB=I+vv^ T, v=(1,-1,1)^ T. Since norm v^2=3, this matrix has eigenvalue four in the direction of v and eigenvalue one on its orthogonal complement. In particular it is positive definite.
Exact mathematical passage
A^{\mathsf T}A=3I-\mathbf1\mathbf1^{\mathsf T}.
a^{(m)}Use in source · 5 passages
Search aliases: a^(m)
Use in source · 05i_diagonal_regularization.tex, line 6
The entire regularized family is explicit. Use Tu=(u_n/n) and r=w=(1/n) as in Section~[source reference]. Choose α=m^-2 for an integer m≥1. The normal equation in each coordinate gives a^(m)_n=fracm^2m^2+n^2, v^(m)_n=r_n-(Ta^(m))_n=fracnm^2+n^2. The repaired ambient candidate is (-a^(m),1). Its protected coordinate is one for every m.
Exact mathematical passage
a^{(m)}_n=\frac{m^2}{m^2+n^2},\qquad v^{(m)}_n=r_n-(Ta^{(m)})_n=\frac{n}{m^2+n^2}.
a^{(m)}_nUse in source · 1 passage
Search aliases: a^(m)_n
Use in source · 05i_diagonal_regularization.tex, line 6
The entire regularized family is explicit. Use Tu=(u_n/n) and r=w=(1/n) as in Section~[source reference]. Choose α=m^-2 for an integer m≥1. The normal equation in each coordinate gives a^(m)_n=fracm^2m^2+n^2, v^(m)_n=r_n-(Ta^(m))_n=fracnm^2+n^2. The repaired ambient candidate is (-a^(m),1). Its protected coordinate is one for every m.
Exact mathematical passage
a^{(m)}_n=\frac{m^2}{m^2+n^2},\qquad v^{(m)}_n=r_n-(Ta^{(m)})_n=\frac{n}{m^2+n^2}.
a^{-1}Use in source · 3 passages
Search aliases: a^-1
Use in source · 04_symmetry.tex, line 63
For the groupoid, an arrow a:c→ d means the equation a· c=d. If b:d→ e, then (ba)· c=b·(a· c)=e, giving composition. The identity label is 1 and the inverse label is a^-1. Associativity and the two inverse identities now follow from the corresponding multiplication laws in the product group. The label ba is therefore forced by the order of application, not an arbitrary notational convention. sourceKS30 The action groupoid Z^1(V;G)//∏_vG_v retains comparison witnesses. Its orbit set Z^1(V;G)/∏_vG…
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a^{-1}
a^2Use in source · 8 passages
Search aliases: a^2
Use in source · 05_geometry.tex, line 366
stable repair requires a stable matching problem Suppose ran D and ran B are closed, with residual gains a,b. Then norm u≤ anormDx, norm v≤ bnormCP_Lx ≤ b(normCx+norm C anormDx). Therefore normx-Px≤ a(1+bnorm C)normDx+bnormCx. The Pythagorean identity gives the sharper estimate normx-Px^2≤ a^2normDx^2+b^2(normCx+anorm CnormDx)^2. For the maximum norm on R× Y, two sufficient gains are therefore G_2=sqrta^2+b^2(1+anorm C)^2 ≤ G_1=a(1+bnorm C)+b. The inequality follows from sqrtu^2+v^2≤ u+v for non…
Exact mathematical passage
\norm{x-Px}^2\le a^2\norm{Dx}^2+b^2\bigl(\norm{Cx}+a\norm C\norm{Dx}\bigr)^2.
a^\daggerUse in source · 20 passages
Search aliases: a^† dagger
Use in source · 05h_regularized_repair.tex, line 147
[The regularization limit alternative] As αdownarrow0, Ta_α⟶ b, normr-Ta_α⟶ q, αnorma_α^2⟶0. If b∈ran T, let a^† be its unique minimum norm preimage. Then a_α⟶ a^† in norm. If b∉ran T, then norma_α→∞. Step 1: identify the unavoidable component. Since r-b⊥overlineran T, the adjoint identity gives T^*(r-b)=0. Thus a_α(r)=a_α(b). For every correction the orthogonal decomposition is normr-Ta^2=q^2+normb-Ta^2. In particular both the residual square and the minimum objective are at least q^2.
Exact mathematical passage
a^\dagger
a^\dagger_nUse in source · 2 passages
Search aliases: a^†_n dagger
Use in source · 05i_diagonal_regularization.tex, line 146
The same operator has an attainable datum. Change only the residual to b_n=1/n^2. Its minimum norm preimage is a^†_n=1/n∈ℓ^2. At α=m^-2, the regularized correction is c^(m)_n=fracm^2n(m^2+n^2), a^†_n-c^(m)_n=fracnm^2+n^2. The bias square is exactly the previously computed B_m. This links the two cases without conflating their existence hypotheses: the escaping residual of one problem is the regularization bias of an attainable problem.
Exact mathematical passage
a^\dagger_n=1/n\in\ell^2
a_0Use in source · 6 passages
Search aliases: a_0
Use in source · 04a_boundary_lab.tex, line 7
Use a constant group H as coefficients, with identity pullback homomorphisms. A cocycle on the duplicated category is specified by its label r∈ H on the arrow 0→1. Its reverse label is r^-1, and identity labels are one. A gauge (a_0,a_1) carries r to r'=a_0ra_1^-1. Every r is isomorphic to the trivial cocycle, for example with gauge (1,r). The automorphisms of r are the pairs (a_0,r^-1a_0r), a_0∈ H. Thus the unframed groupoid has one isomorphism class but retains a copy of H as the automorphism …
Exact mathematical passage
(a_0,a_1)
A_1Use in source · 1 passage
Search aliases: A_1
Use in source · 05f_bounded_realization.tex, line 98
Suppose now that A_1⊇ A_2⊇⋯ are nonempty closed affine subsets of a real Hilbert correction space H. A typical A_n consists of corrections satisfying the first n bounded linear observation equations. The sets are affine because the equations are linear, and closed because the observations are continuous.
Exact mathematical passage
A_1\supseteq A_2\supseteq\cdots
a_1Use in source · 12 passages
Search aliases: a_1
Use in source · 04a_boundary_lab.tex, line 7
Use a constant group H as coefficients, with identity pullback homomorphisms. A cocycle on the duplicated category is specified by its label r∈ H on the arrow 0→1. Its reverse label is r^-1, and identity labels are one. A gauge (a_0,a_1) carries r to r'=a_0ra_1^-1. Every r is isomorphic to the trivial cocycle, for example with gauge (1,r). The automorphisms of r are the pairs (a_0,r^-1a_0r), a_0∈ H. Thus the unframed groupoid has one isomorphism class but retains a copy of H as the automorphism …
Exact mathematical passage
(a_0,a_1)
a_1'Use in source · 4 passages
Search aliases: a_1'
Use in source · 05g_budget_certificates.tex, line 39
s a_1,a_2 at budgets M_1,M_2 and t∈[0,1]. The candidate a_t=ta_1+(1-t)a_2 satisfies norma_t≤ tM_1+(1-t)M_2, and its residual is t(r-Ta_1)+(1-t)(r-Ta_2). The triangle inequality therefore gives β(tM_1+(1-t)M_2)≤ tβ(M_1)+(1-t)β(M_2). Monotonicity follows from inclusion of balls. If 0≤ M_1≤ M_2 and M_2>0, choose a minimizer a_2 at M_2 and put a_1'=(M_1/M_2)a_2. Then norma_1'≤ M_1 and norma_2-a_1'≤ M_2-M_1, so β(M_1)≤normr-Ta_1' ≤β(M_2)+norm T(M_2-M_1). Together with monotonicity this proves the Lip…
Exact mathematical passage
a_1'=(M_1/M_2)a_2
a_1^{-1}Use in source · 2 passages
Search aliases: a_1^-1
Use in source · 04a_boundary_lab.tex, line 8
Use a constant group H as coefficients, with identity pullback homomorphisms. A cocycle on the duplicated category is specified by its label r∈ H on the arrow 0→1. Its reverse label is r^-1, and identity labels are one. A gauge (a_0,a_1) carries r to r'=a_0ra_1^-1. Every r is isomorphic to the trivial cocycle, for example with gauge (1,r). The automorphisms of r are the pairs (a_0,r^-1a_0r), a_0∈ H. Thus the unframed groupoid has one isomorphism class but retains a copy of H as the automorphism …
Exact mathematical passage
\label{eq:two-copy-gauge} r'=a_0ra_1^{-1}.
a_2Use in source · 10 passages
Search aliases: a_2
Use in source · 05g_budget_certificates.tex, line 31
For convexity, take minimizers a_1,a_2 at budgets M_1,M_2 and t∈[0,1]. The candidate a_t=ta_1+(1-t)a_2 satisfies norma_t≤ tM_1+(1-t)M_2, and its residual is t(r-Ta_1)+(1-t)(r-Ta_2). The triangle inequality therefore gives β(tM_1+(1-t)M_2)≤ tβ(M_1)+(1-t)β(M_2). Monotonicity follows from inclusion of balls. If 0≤ M_1≤ M_2 and M_2>0, choose a minimizer a_2 at M_2 and put a_1'=(M_1/M_2)a_2. Then norma_1'≤ M_1 and norma_2-a_1'≤ M_2-M_1, so β(M_1)≤normr-Ta_1' ≤β(M_2)+norm T(M_2-M_1). Together with mon…
Exact mathematical passage
a_1,a_2
A_aUse in source · 2 passages
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Use in source · 09_workshop.tex, line 11
2. A residual chart can change without changing the defect. Let d:R^2→R be d(u,r)=r. For a∈R, let A_a(r)=(ar,r). Compute the exact repair P_a, compare it with P_b, and decide when they commute.
Exact mathematical passage
A_a(r)=(ar,r)
a_aUse in source · 1 passage
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Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
A_CUse in source · 9 passages
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Use in source · 10_grammar.tex, line 17
Repeated diagonal coding. Fix a computable pairing bijection ⟨ j,k⟩: N^2→ N. For a freely chosen set C⊆ N, define a new selector A_C by 1_A_C(2⟨ j,k⟩) · =1- 1_B_j(2⟨ j,k⟩),
Exact mathematical passage
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a_cUse in source · 1 passage
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Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
A_{C'}Use in source · 1 passage
Search aliases: A_C'
Use in source · 10_grammar.tex, line 33
If C≠ C', choose j at which they differ. Rule [source reference] makes A_C and A_C' differ at infinitely many odd indices. Theorems~[source reference] and [source reference] then give nonisomorphism and nonembeddability. The remaining properties hold for every paired-prime group.
Exact mathematical passage
A_{C'}
a_eUse in source · 1 passage
Search aliases: a_e
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
a_fUse in source · 1 passage
Search aliases: a_f
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
a_{ii}Use in source · 3 passages
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Use in source · 04_symmetry.tex, line 194
a_ik(v) · =a_jk(v)a_ij(v). The triple equation implies a_ii=1 and a_ji=a_ij^-1. A morphism of descent data is a family of local gauges r_i satisfying r_j(v)a_ij(v)=a'_ij(v)r_i(v). These objects and arrows form a groupoid.
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a_{ii}=1
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Use in source · 04_symmetry.tex, line 189
A descent datum consists of local cocycles c_i and overlap gauges a_ij(v) with c_j(f) · =a_ij(v)c_i(f)[f^*a_ij(w)]^-1,
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a_{ij}(v)
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Use in source · 04_symmetry.tex, line 194
a_ik(v) · =a_jk(v)a_ij(v). The triple equation implies a_ii=1 and a_ji=a_ij^-1. A morphism of descent data is a family of local gauges r_i satisfying r_j(v)a_ij(v)=a'_ij(v)r_i(v). These objects and arrows form a groupoid.
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a_{ji}=a_{ij}^{-1}
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Use in source · 04_symmetry.tex, line 190
A descent datum consists of local cocycles c_i and overlap gauges a_ij(v) with c_j(f) · =a_ij(v)c_i(f)[f^*a_ij(w)]^-1, a_ik(v) · =a_jk(v)a_ij(v). The triple equation implies a_ii=1 and a_ji=a_ij^-1. A morphism of descent data is a family of local gauges r_i satisfying r_j(v)a_ij(v)=a'_ij(v)r_i(v). These objects and arrows form a groupoid.
Exact mathematical passage
c_j(f)&=a_{ij}(v)c_i(f)[f^*a_{ij}(w)]^{-1},\label{eq:localgauge}\\ a_{ik}(v)&=a_{jk}(v)a_{ij}(v).\label{eq:triple}
a_{ji}Use in source · 3 passages
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Use in source · 04_symmetry.tex, line 194
a_ik(v) · =a_jk(v)a_ij(v). The triple equation implies a_ii=1 and a_ji=a_ij^-1. A morphism of descent data is a family of local gauges r_i satisfying r_j(v)a_ij(v)=a'_ij(v)r_i(v). These objects and arrows form a groupoid.
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a_{ji}=a_{ij}^{-1}
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Use in source · 04_symmetry.tex, line 190
A descent datum consists of local cocycles c_i and overlap gauges a_ij(v) with c_j(f) · =a_ij(v)c_i(f)[f^*a_ij(w)]^-1, a_ik(v) · =a_jk(v)a_ij(v). The triple equation implies a_ii=1 and a_ji=a_ij^-1. A morphism of descent data is a family of local gauges r_i satisfying r_j(v)a_ij(v)=a'_ij(v)r_i(v). These objects and arrows form a groupoid.
Exact mathematical passage
c_j(f)&=a_{ij}(v)c_i(f)[f^*a_{ij}(w)]^{-1},\label{eq:localgauge}\\ a_{ik}(v)&=a_{jk}(v)a_{ij}(v).\label{eq:triple}
A_mUse in source · 16 passages
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Use in source · 05i_diagonal_regularization.tex, line 14
[Energy, residual, and certified tails] The displayed a^(m) belongs to ℓ^2 and is the unique regularized correction at α=m^-2. Set A_m=norma^(m)^2=∑_n≥1fracm^4(m^2+n^2)^2, B_m=normv^(m)^2=∑_n≥1fracn^2(m^2+n^2)^2. Then frac m4≤ A_m≤frac4m3, frac14(m+1)≤ B_m≤frac2m. For the sums A_m,K,B_m,K through any integer K≥ m, 0≤ A_m-A_m,K≤fracm^43K^3, 0≤ B_m-B_m,K≤frac1K. For n>m, one has (a_n^(m))^2≤ m^4/n^4, so the correction sequence is square-summable. The diagonal operator is bounded with norm one, and…
Exact mathematical passage
A_m=\norm{a^{(m)}}^2=\sum_{n\ge1}\frac{m^4}{(m^2+n^2)^2},\qquad B_m=\norm{v^{(m)}}^2=\sum_{n\ge1}\frac{n^2}{(m^2+n^2)^2}.
a_MUse in source · 8 passages
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Use in source · 05g_budget_certificates.tex, line 89
To prove sharpness, let a_M be an attaining correction from Theorem~[source reference] and put v=r-Ta_M. If v=0, then β(M)=0 and z=0 attains the dual bound. Assume henceforth v≠0 and set z=v/norm v.
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a_M
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Use in source · 05i_diagonal_regularization.tex, line 119
Step 3: move from regularization parameters to arbitrary budgets. Set M_m=sqrtA_m. Every positive coordinate m^2/(m^2+n^2) increases strictly with m, so A_m, and hence M_m, increases strictly. Since A_m/m→ c=π/4>0, these budgets tend to infinity and (fracM_m+1M_m)^2 =fracA_m+1/(m+1)A_m/mfracm+1m⟶1. Proposition~[source reference] identifies β(M_m)=sqrtB_m. Thus M_mβ(M_m)=sqrt(A_m/m)(mB_m)⟶ c. For every sufficiently large M, choose m with M_m≤ M≤ M_m+1. Monotonicity of β gives M_mβ(M_m+1)≤ Mβ(M)≤ …
Exact mathematical passage
\left(\frac{M_{m+1}}{M_m}\right)^2 =\frac{A_{m+1}/(m+1)}{A_m/m}\frac{m+1}{m}\longrightarrow1.
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Use in source · 05i_diagonal_regularization.tex, line 23
s] The displayed a^(m) belongs to ℓ^2 and is the unique regularized correction at α=m^-2. Set A_m=norma^(m)^2=∑_n≥1fracm^4(m^2+n^2)^2, B_m=normv^(m)^2=∑_n≥1fracn^2(m^2+n^2)^2. Then frac m4≤ A_m≤frac4m3, frac14(m+1)≤ B_m≤frac2m. For the sums A_m,K,B_m,K through any integer K≥ m, 0≤ A_m-A_m,K≤fracm^43K^3, 0≤ B_m-B_m,K≤frac1K. For n>m, one has (a_n^(m))^2≤ m^4/n^4, so the correction sequence is square-summable. The diagonal operator is bounded with norm one, and w=(1/n)∈ℓ^2, hence the residual sequ…
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A_{m,K},B_{m,K}
a_{m^{-2}}Use in source · 2 passages
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Use in source · 05i_diagonal_regularization.tex, line 167
lta with 0<δ≤1. Choose m=lceilδ^-2/3rceil, α=m^-2. Then the noisy regularized correction satisfies norma_α(widetilde b)-a^† ≤sqrtfrac2m+fracδ m2 ≤(sqrt2+1)δ^1/3. The exact-data bias has coordinates n/(m^2+n^2), so its squared norm is precisely B_m. By Proposition~[source reference], norma_m^-2(b)-a^†≤sqrt2/m. Theorem~[source reference] at the same parameter gives norma_m^-2(widetilde b)-a_m^-2(b) ≤fracδ2sqrtm^-2=fracδ m2. The triangle inequality proves the first bound.
Exact mathematical passage
\norm{a_{m^{-2}}(b)-a^\dagger}\le\sqrt{2/m}.
A_NUse in source · 7 passages
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Use in source · 02_ambient.tex, line 7
Finite stages and two connecting systems. For N≥1, define A_N=∏_i<NC_p_i, m_N=∏_i<Np_i. The Chinese remainder map sends kbmod m_N to (kbmod p_i)_i<N. It is an isomorphism, so A_N is cyclic and its all-one vector is a generator.
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A_N=\prod_{i<N}C_{p_i},\qquad m_N=\prod_{i<N}p_i.
a_NUse in source · 3 passages
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Use in source · 05e_escape.tex, line 50
A complete calculation of the finite repairs. Let 1_N be one in coordinates 1,…,N and zero elsewhere. Put a_N=( 1_N,0), f_N=f_0-a_N=(- 1_N,1). Then p(f_N)=1, and the first N residual coordinates vanish exactly. The remaining residual and movement satisfy d(f_N)_n=0 · n≤ N,
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a_N=(\mathbf 1_N,0),\qquad f_N=f_0-a_N=(-\mathbf 1_N,1).
A_{N+1}Use in source · 2 passages
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Use in source · 02_ambient.tex, line 24
Zero extension gives injections A_N→ A_N+1. Forgetting the last coordinate gives surjections A_N+1→ A_N. These are different diagrams. Their limits are D=varinjlim_N A_N=⊕_p C_p, P=varprojlim_N A_N=∏_p C_p. The first group contains only finite-support vectors. The second contains arbitrary compatible coordinate assignments.
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A_N\to A_{N+1}
a_n^{(m)}Use in source · 1 passage
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Use in source · 05i_diagonal_regularization.tex, line 30
fracn^2(m^2+n^2)^2. Then frac m4≤ A_m≤frac4m3, frac14(m+1)≤ B_m≤frac2m. For the sums A_m,K,B_m,K through any integer K≥ m, 0≤ A_m-A_m,K≤fracm^43K^3, 0≤ B_m-B_m,K≤frac1K. For n>m, one has (a_n^(m))^2≤ m^4/n^4, so the correction sequence is square-summable. The diagonal operator is bounded with norm one, and w=(1/n)∈ℓ^2, hence the residual sequence is also in ℓ^2. The operator is self-adjoint, and the coordinate equality frac1nfracnm^2+n^2 =m^-2fracm^2m^2+n^2 proves T^*v^(m)=α a^(m) as an equality…
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(a_n^{(m)})^2\le m^4/n^4
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Use in source · 05k_stopping_rules.tex, line 106
displayed data and normal-residual bounds, then a_n→ a^† in norm. At every index, norma_n-a^† ≤fracε_nα_n +fracδ_n2sqrtα_n +norma_α_n(r)-a^†. At index n, distinguish two exact minimizers: let widetilde a_n=a_α_n(widetilde r_n) solve the measured problem and let a_n^0=a_α_n(r) solve the intended-data problem. All three corrections, including the proposal a_n, belong to the same space H.
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a_n^0=a_{\alpha_n}(r)
a_{n_k}Use in source · 12 passages
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Use in source · 05f_bounded_realization.tex, line 27
The compactness needed for the existence argument. [Bounded sequences have weakly convergent subsequences] Every bounded sequence (a_n) in a real Hilbert space has a subsequence converging weakly to some a. Moreover, norm a≤liminf_knorma_n_k. Choose M≥0 with norma_n≤ M. If M=0, every term is zero and the conclusion is immediate. Otherwise let H_0 be the closed linear span of the countably many vectors a_n. Finite rational linear combinations of those vectors form a countable dense set, so H_0 is…
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\norm a\le\liminf_k\norm{a_{n_k}}.
A_pUse in source · 6 passages
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Use in source · 05_geometry.tex, line 239
Every factor in the bound has a meaning. Define the route gain A_p=max_iγ(overlinep_i), using reverse routes so that the error is measured in each destination view. The path telescope gives normx-R_p(x_o)_∞≤ A_pnormdx_∞. Let N_p be the residual normalization (N_pr)_ij=T_ir_ij. The exact identity H=N_pdR_p implies norm H(x_o)_∞ ≤normN_p(norm d A_p+1)normdx_∞.
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A_p=\max_i\gamma(\overline{p_i}),
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Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
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4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
a_tUse in source · 3 passages
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Use in source · 05g_budget_certificates.tex, line 31
For convexity, take minimizers a_1,a_2 at budgets M_1,M_2 and t∈[0,1]. The candidate a_t=ta_1+(1-t)a_2 satisfies norma_t≤ tM_1+(1-t)M_2, and its residual is t(r-Ta_1)+(1-t)(r-Ta_2). The triangle inequality therefore gives β(tM_1+(1-t)M_2)≤ tβ(M_1)+(1-t)β(M_2). Monotonicity follows from inclusion of balls. If 0≤ M_1≤ M_2 and M_2>0, choose a minimizer a_2 at M_2 and put a_1'=(M_1/M_2)a_2. Then norma_1'≤ M_1 and norma_2-a_1'≤ M_2-M_1, so β(M_1)≤normr-Ta_1' ≤β(M_2)+norm T(M_2-M_1). Together with mon…
Exact mathematical passage
a_t=ta_1+(1-t)a_2
A_uUse in source · 1 passage
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Use in source · 02a_prescription.tex, line 6
Three maps that must not be confused. Keep the additive view system of Section~[source reference]. For a prescribed domain U, there are three distinct maps: p_U:E⟶∏_u∈ UA_u, r_U:Γ(A)⟶Γ_U(A), o_U:Γ_U(A)⟶ Q_U. The first restricts a raw family, whether or not it is coherent. The second restricts a global coherent family. The third detects whether an already coherent partial family has a global completion. Their domains are different. In particular, surjectivity of the raw coordinate projection does…
Exact mathematical passage
p_U:E\longrightarrow\prod_{u\in U}A_u,\qquad r_U:\Gamma(A)\longrightarrow\Gamma_U(A),\qquad \mathfrak o_U:\Gamma_U(A)\longrightarrow Q_U.
a_uUse in source · 1 passage
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Use in source · 04_symmetry.tex, line 79
The relative obstruction without commutativity. Prescribe a coherent partial section s of T_c on U. The based gauge group is G_U={a∈∏_vG_v:a_u=1 for u∈ U}. Choose a raw family q equal to s on U. Identity elements can be used elsewhere. Normalize the labels by c^q_f=q_v^-1c_f f^*(q_w). Its labels on arrows internal to U are one. Define the pointed orbit obstruction o_U(c,s)=[c^q]∈ Z^1(V;G)/ G_U, whose neutral point is the class of the trivial cocycle.
Exact mathematical passage
\mathcal G_U=\{a\in\prod_vG_v:a_u=1\text{ for }u\in U\}.
a_v^{-1}Use in source · 2 passages
Search aliases: a_v^-1
Use in source · 04_symmetry.tex, line 35
Put d=a· c. Its identity label is a_v1a_v^-1=1. For a composite, expand all factors in their original order: d_f f^*(d_g) · =a_vc_f[f^*(a_w)]^-1 f^*(a_wc_g[g^*(a_z)]^-1)
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a_v1a_v^{-1}=1
a_wUse in source · 6 passages
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Use in source · 01_views.tex, line 156
a coordinate preserving equivalence Γ(B_a,b)≃ B(a,b). Under a transition cover, matching local bridge sections glue to a unique global bridge with both prescribed projections unchanged. We must first check that the displayed fibers really form a view system. Let m∈ B_a,b(w) and f:v→ w. Naturality of p and coherence of a give p_v(f_M^*m)=f_F^*p_w(m)=f_F^*a_w=a_v. The same calculation with q,b gives q_v(f_M^*m)=b_v. Therefore f_M^*m belongs to B_a,b(v). Identity and composition of these restricted…
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p_v(f_M^*m)=f_F^*p_w(m)=f_F^*a_w=a_v.
a_zUse in source · 2 passages
Search aliases: a_z
Use in source · 04_symmetry.tex, line 36
Put d=a· c. Its identity label is a_v1a_v^-1=1. For a composite, expand all factors in their original order: d_f f^*(d_g) · =a_vc_f[f^*(a_w)]^-1 f^*(a_wc_g[g^*(a_z)]^-1) · =a_vc_f[f^*(a_w)]^-1f^*(a_w)f^*(c_g) [f^*g^*(a_z)]^-1 · =a_vc_g∘ f[(g∘ f)^*(a_z)]^-1 =d_g∘ f. Thus the gauge formula preserves the cocycle condition. For two gauges a,b, substitution gives (a·(b· c))_f =a_vb_vc_f[f^*(b_w)]^-1[f^*(a_w)]^-1 =((ab)· c)_f. The reversed order in the two inverse factors is exactly the formula for th…
Exact mathematical passage
\begin{aligned} d_f f^*(d_g) &=a_vc_f[f^*(a_w)]^{-1} f^*(a_wc_g[g^*(a_z)]^{-1})\\ &=a_vc_f[f^*(a_w)]^{-1}f^*(a_w)f^*(c_g) [f^*g^*(a_z)]^{-1}\\ &=a_vc_{g\circ f}[(g\circ f)^*(a_z)]^{-1} =d_{g\circ f}. \end{aligned}
A_\alphaUse in source · 5 passages
Search aliases: A_α alpha
Use in source · 05h_regularized_repair.tex, line 97
gularized answers lie on the budget frontier] If 0<α<γ, then norma_α≥norma_γ, normr-Ta_α≤normr-Ta_γ. At M_α=norma_α, β(M_α)=normr-Ta_α. When v_α=r-Ta_α≠0, its normalization is an optimizing witness in Theorem~[source reference] at that budget. Let A_α=norma_α^2 and R_α=normr-Ta_α^2. Comparing the two admissible minimizers in each other's objectives gives R_α+α A_α≤ R_γ+α A_γ, R_γ+γ A_γ≤ R_α+γ A_α. Add these inequalities and cancel the residual terms. The result is (γ-α)(A_α-A_γ)≥0. Since γ>α, it…
Exact mathematical passage
A_\alpha=\norm{a_\alpha}^2
a_{\alpha(\delta)}Use in source · 1 passage
Search aliases: a_α(δ) alpha delta
Use in source · 05h_regularized_repair.tex, line 243
For the limiting statement, insert the exact-data correction at the selected parameter: norma_α(δ)(widetilde r)-a^† ≤fracδ2sqrtα(δ) +norma_α(δ)(r)-a^†. The first term tends to zero by the noise/parameter hypothesis. The second tends to zero by the attained-projection case of Theorem~[source reference], because the selected parameter tends to zero. Both statements require the same fixed operator and Hilbert metrics. The conclusion is a least-squares correction when the unavoidable residual compon…
Exact mathematical passage
\norm{a_{\alpha(\delta)}(\widetilde r)-a^\dagger} \le\frac{\delta}{2\sqrt{\alpha(\delta)}} +\norm{a_{\alpha(\delta)}(r)-a^\dagger}.
a_{\alpha_n}Use in source · 5 passages
Search aliases: a_α_n alpha
Use in source · 05h_regularized_repair.tex, line 188
Step 5: in the unattained case the norms tend to infinity. If b∉ran T and the norms failed to tend to infinity, there would exist M<∞ and positive α_n→0 with norma_α_n≤ M. Their images still tend to b by Step 2. Theorem~[source reference] would then put b in the actual range, a contradiction. This proves divergence of the norms, not just unboundedness somewhere along the path. writtenWR08
Exact mathematical passage
\norm{a_{\alpha_n}}\le M
a_{\alpha_{n_k}}Use in source · 3 passages
Search aliases: a_α_n_k alpha
Use in source · 05h_regularized_repair.tex, line 175
Step 4: strengthen weak convergence to norm convergence. Take an arbitrary sequence α_ndownarrow0, or more generally positive α_n→0. The bound supplies a weakly convergent subsequence a_α_n_krightharpoonup a. Since its images converge in norm to b, the adjoint pairing argument of Theorem~[source reference] yields Ta=b. Weak lower semicontinuity gives norm a≤liminf_knorma_α_n_k ≤norma^†. Minimality and uniqueness of a^† force a=a^†. Moreover the same lower bound, together with the uniform upper b…
Exact mathematical passage
a_{\alpha_{n_k}}\rightharpoonup a
A_\gammaUse in source · 4 passages
Search aliases: A_γ gamma
Use in source · 05h_regularized_repair.tex, line 98
a_α≤normr-Ta_γ. At M_α=norma_α, β(M_α)=normr-Ta_α. When v_α=r-Ta_α≠0, its normalization is an optimizing witness in Theorem~[source reference] at that budget. Let A_α=norma_α^2 and R_α=normr-Ta_α^2. Comparing the two admissible minimizers in each other's objectives gives R_α+α A_α≤ R_γ+α A_γ, R_γ+γ A_γ≤ R_α+γ A_α. Add these inequalities and cancel the residual terms. The result is (γ-α)(A_α-A_γ)≥0. Since γ>α, it follows that A_α≥ A_γ. Return to the first comparison to obtain R_α-R_γ≤α(A_γ-A_α)≤0…
Exact mathematical passage
R_\alpha+\alpha A_\alpha\le R_\gamma+\alpha A_\gamma, \qquad R_\gamma+\gamma A_\gamma\le R_\alpha+\gamma A_\alpha.
a_\gammaUse in source · 1 passage
Search aliases: a_γ gamma
Use in source · 05h_regularized_repair.tex, line 86
Parameter monotonicity and the fixed-budget problem. [Regularized answers lie on the budget frontier] If 0<α<γ, then norma_α≥norma_γ, normr-Ta_α≤normr-Ta_γ. At M_α=norma_α, β(M_α)=normr-Ta_α. When v_α=r-Ta_α≠0, its normalization is an optimizing witness in Theorem~[source reference] at that budget. Let A_α=norma_α^2 and R_α=normr-Ta_α^2. Comparing the two admissible minimizers in each other's objectives gives R_α+α A_α≤ R_γ+α A_γ, R_γ+γ A_γ≤ R_α+γ A_α. Add these inequalities and cancel the resid…
Exact mathematical passage
\norm{a_\alpha}\ge\norm{a_\gamma},\qquad \norm{r-Ta_\alpha}\le\norm{r-Ta_\gamma}.
a_{\kappa(v),i}Use in source · 1 passage
Search aliases: a_κ(v),i kappa
Use in source · 04_symmetry.tex, line 205
rded. Step 1: normalize the supplied identifications. The triple-overlap equation with all indices equal gives a_ii=a_iia_ii, hence a_ii=1. With indices i,j,i it gives a_jia_ij=1, hence a_ji=a_ij^-1. Composition coverage implies object coverage by inserting identity arrows. Choose a containing patch κ(v) for each view and put b_i(v)=a_κ(v),i(v) (v∈ U_i). At a point of U_i∩ U_j, the triple law, now on patches κ(v),i,j, gives b_j(v)=a_ij(v)b_i(v). tag* The two endpoints of an arrow may have differ…
Exact mathematical passage
b_i(v)=a_{\kappa(v),i}(v)\quad(v\in U_i).
\absUse in source · 3 passages
Search aliases: abs
Use in source · 05g_budget_certificates.tex, line 139
How measurement and model uncertainty affect the budget. [A robust budget profile] Suppose the bounded linear map widetilde T:H→ Y and the datum widetilde r∈ Y satisfy normT-widetilde T≤η, normr-widetilde r≤δ, η,δ≥0. For every M≥0, absβ_T,r(M)-β_widetilde T,widetilde r(M) ≤δ+η M. In particular, for norm z≤1, β_T,r(M)≥ ⟨widetilde r,z⟩-Mnormwidetilde T^*z -δ-η M. Fix a feasible correction a. The two residual vectors have difference (r-Ta)-(widetilde r-widetilde Ta) =(r-widetilde r)-(T-widetilde T)…
Exact mathematical passage
\abs{\beta_{T,r}(M)-\beta_{\widetilde T,\widetilde r}(M)} \le\delta+\eta M.
\alephUse in source · 4 passages
Search aliases: aleph
Use in source · 01_contracts.tex, line 37
[Countable-code bound] Let G be a finite-description grammar as above. Then its class of realized group isomorphism types has cardinal at most aleph_0. For each length n, the set of strings of length n is countable. The union over n∈ N is countable. The valid strings form a subset, and a single-valued image of a countable set is countable. This is a boundary of description, not of existence. It also says nothing about whether a particular named group lies outside a chosen grammar. The later cons…
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\aleph_0
\aleph_0Use in source · 4 passages
Search aliases: aleph_0 aleph
Use in source · 01_contracts.tex, line 37
[Countable-code bound] Let G be a finite-description grammar as above. Then its class of realized group isomorphism types has cardinal at most aleph_0. For each length n, the set of strings of length n is countable. The union over n∈ N is countable. The valid strings form a subset, and a single-valued image of a countable set is countable. This is a boundary of description, not of existence. It also says nothing about whether a particular named group lies outside a chosen grammar. The later cons…
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\aleph_0
\mathfrak BUse in source · 3 passages
Search aliases: B mathfrak
Use in source · 01_views.tex, line 143
Correspondences must also be coherent. Suppose natural maps form a span FxleftarrowpMxrightarrowqG, and fix a∈Γ(F), b∈Γ(G). At each view define the ordinary fiber B_a,b(v)={m∈ M_v:p_v(m)=a_v, q_v(m)=b_v}. Naturality and coherence provide its restrictions. A global bridge is an element of B(a,b)={z∈Γ(M):Γ(p)z=a, Γ(q)z=b}.
Exact mathematical passage
\mathfrak B(a,b)=\{z\in\Gamma(M):\Gamma(p)z=a,\ \Gamma(q)z=b\}.
B'Use in source · 3 passages
Search aliases: B'
Use in source · 05k_stopping_rules.tex, line 16
[The invariant norm of a coordinate certificate] The actual normal residual is g=BG^-1q, norm g^2=q^ TG^-1q. For any invertible change of basis B'=BC, the same proposed correction has c'=C^-1c, G'=C^ TGC, q'=C^ Tq, and (q')^ T(G')^-1q'=q^ TG^-1q. Because B has independent columns, for every nonzero coefficient vector c, c^ TGc=c^ TB^ TBc=normBc^2>0. Thus G is positive definite and invertible. The matrix P=BG^-1B^ T is symmetric and obeys P^2=BG^-1(B^ TB)G^-1B^ T=P, PB=B. Its range is precisely t…
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B'=BC
b'Use in source · 2 passages
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Use in source · 04_symmetry.tex, line 455
Replacing b by an isomorphic boundary b' also gives an equivalence, by composing the identification with b→ b'. For R=id, the marked realization groupoid has center (b,1_b), and between (x,ϕ) and (y,ψ) there is exactly one compatible arrow ψ^-1ϕ. This is a useful example of contractible realization data, not a claim that the original groupoid has no symmetries.
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b'
b'_iUse in source · 3 passages
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Use in source · 04_symmetry.tex, line 256
The correct uniqueness statement. Suppose (c,b_i) and (c',b'_i) realize the same descent datum. The only possible compatible comparison is g_v=b'_i(v)^-1b_i(v), using any patch through v. The overlap identities make it independent of the patch, and b'_ig=b_i. It is therefore the unique global comparison respecting the fixed local identifications.
Exact mathematical passage
(c',b'_i)
B^{-1}Use in source · 2 passages
Search aliases: B^-1
Use in source · 05h_regularized_repair.tex, line 43
The inverse therefore exists. Its linearity follows from uniqueness of preimages, and its bound is normB^-1y≤α^-1norm y. This construction does not assume closedness of ran T. It uses the positive term α I in the normal operator.
Exact mathematical passage
\norm{B^{-1}y}\le\alpha^{-1}\norm y
b^2Use in source · 2 passages
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Use in source · 05_geometry.tex, line 366
stable repair requires a stable matching problem Suppose ran D and ran B are closed, with residual gains a,b. Then norm u≤ anormDx, norm v≤ bnormCP_Lx ≤ b(normCx+norm C anormDx). Therefore normx-Px≤ a(1+bnorm C)normDx+bnormCx. The Pythagorean identity gives the sharper estimate normx-Px^2≤ a^2normDx^2+b^2(normCx+anorm CnormDx)^2. For the maximum norm on R× Y, two sufficient gains are therefore G_2=sqrta^2+b^2(1+anorm C)^2 ≤ G_1=a(1+bnorm C)+b. The inequality follows from sqrtu^2+v^2≤ u+v for non…
Exact mathematical passage
\norm{x-Px}^2\le a^2\norm{Dx}^2+b^2\bigl(\norm{Cx}+a\norm C\norm{Dx}\bigr)^2.
b_0Use in source · 10 passages
Search aliases: b_0
Use in source · 04_symmetry.tex, line 265
A failed triple. Let t≠1 be an involution and ask for frames b_0,b_1,b_2 satisfying b_1=tb_0, b_2=tb_1, b_2=tb_0. The first two give b_2=b_0, while the third forces t=1. Each displayed transition is individually invertible, but they cannot be the overlap identifications of a global frame. The defect lies in the triple relation.
Exact mathematical passage
b_0,b_1,b_2
B_1Use in source · 1 passage
Search aliases: B_1
Use in source · 12_proof_tools.tex, line 105
[Baire's complete-space principle] A nonempty complete metric space cannot be the union of countably many closed sets with empty interior. Suppose F_1,F_2,… are closed with empty interior. Choose a nonempty open ball whose closed ball B_1, of positive radius at most 1/2, avoids F_1. Such a choice is possible because the complement of F_1 is open and meets every nonempty open set. Recursively choose a closed ball B_n+1 of positive radius at most 2^-(n+1), contained in the open ball with the same …
Exact mathematical passage
B_1
b_1Use in source · 5 passages
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Use in source · 04_symmetry.tex, line 265
A failed triple. Let t≠1 be an involution and ask for frames b_0,b_1,b_2 satisfying b_1=tb_0, b_2=tb_1, b_2=tb_0. The first two give b_2=b_0, while the third forces t=1. Each displayed transition is individually invertible, but they cannot be the overlap identifications of a global frame. The defect lies in the triple relation.
Exact mathematical passage
b_0,b_1,b_2
b_2Use in source · 6 passages
Search aliases: b_2
Use in source · 04_symmetry.tex, line 265
A failed triple. Let t≠1 be an involution and ask for frames b_0,b_1,b_2 satisfying b_1=tb_0, b_2=tb_1, b_2=tb_0. The first two give b_2=b_0, while the third forces t=1. Each displayed transition is individually invertible, but they cannot be the overlap identifications of a global frame. The defect lies in the triple relation.
Exact mathematical passage
b_0,b_1,b_2
B_{a,b}Use in source · 7 passages
Search aliases: B_a,b
Use in source · 01_views.tex, line 139
Correspondences must also be coherent. Suppose natural maps form a span FxleftarrowpMxrightarrowqG, and fix a∈Γ(F), b∈Γ(G). At each view define the ordinary fiber B_a,b(v)={m∈ M_v:p_v(m)=a_v, q_v(m)=b_v}. Naturality and coherence provide its restrictions. A global bridge is an element of B(a,b)={z∈Γ(M):Γ(p)z=a, Γ(q)z=b}.
Exact mathematical passage
B_{a,b}(v)=\{m\in M_v:p_v(m)=a_v,\ q_v(m)=b_v\}.
b_iUse in source · 10 passages
Search aliases: b_i
Use in source · 04_symmetry.tex, line 205
rded. Step 1: normalize the supplied identifications. The triple-overlap equation with all indices equal gives a_ii=a_iia_ii, hence a_ii=1. With indices i,j,i it gives a_jia_ij=1, hence a_ji=a_ij^-1. Composition coverage implies object coverage by inserting identity arrows. Choose a containing patch κ(v) for each view and put b_i(v)=a_κ(v),i(v) (v∈ U_i). At a point of U_i∩ U_j, the triple law, now on patches κ(v),i,j, gives b_j(v)=a_ij(v)b_i(v). tag* The two endpoints of an arrow may have differ…
Exact mathematical passage
b_i(v)=a_{\kappa(v),i}(v)\quad(v\in U_i).
b_jUse in source · 4 passages
Search aliases: b_j
Use in source · 04_symmetry.tex, line 209
s equal gives a_ii=a_iia_ii, hence a_ii=1. With indices i,j,i it gives a_jia_ij=1, hence a_ji=a_ij^-1. Composition coverage implies object coverage by inserting identity arrows. Choose a containing patch κ(v) for each view and put b_i(v)=a_κ(v),i(v) (v∈ U_i). At a point of U_i∩ U_j, the triple law, now on patches κ(v),i,j, gives b_j(v)=a_ij(v)b_i(v). tag* The two endpoints of an arrow may have different reference patches. No common reference patch is being assumed.
Exact mathematical passage
b_j(v)=a_{ij}(v)b_i(v). \tag{*}
B_KUse in source · 3 passages
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Use in source · 03_spectrum.tex, line 66
Define B_K={2n:n∈ N}∪{2n+1:n∉ K}, H=D_B_K. This is an explicit mathematical definition of a countably infinite, locally cyclic abelian group.
Exact mathematical passage
B_K=\{2n:n\in\N\}\cup\{2n+1:n\notin K\}, \qquad H=D_{B_K}.
b_kUse in source · 2 passages
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Use in source · 05c_iteration.tex, line 22
[Summable movement gives exactification] Suppose b_n≥0, ∑_n=0^∞b_n<∞, and dist(f_n,f_n+1)≤ b_n, ω(f_n)⟶0. Then there is a limit x∈ M satisfying f_n⟶ x, ω(x)=0, dist(f_n,x)≤∑_k=n^∞b_k. For integers m>n, apply the triangle inequality successively along the finite chain f_n,f_n+1,…,f_m. It gives dist(f_n,f_m)≤∑_k=n^m-1dist(f_k,f_k+1) ≤∑_k=n^m-1b_k≤∑_k=n^∞b_k. Given ε>0, convergence of the nonnegative series supplies N for which the final tail is below ε whenever n≥ N. Interchanging m,n when necessa…
Exact mathematical passage
f_n\longrightarrow x,\qquad\omega(x)=0,\qquad \dist(f_n,x)\le\sum_{k=n}^{\infty}b_k.
b_lUse in source · 1 passage
Search aliases: b_l
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
B_mUse in source · 13 passages
Search aliases: B_m
Use in source · 05i_diagonal_regularization.tex, line 14
[Energy, residual, and certified tails] The displayed a^(m) belongs to ℓ^2 and is the unique regularized correction at α=m^-2. Set A_m=norma^(m)^2=∑_n≥1fracm^4(m^2+n^2)^2, B_m=normv^(m)^2=∑_n≥1fracn^2(m^2+n^2)^2. Then frac m4≤ A_m≤frac4m3, frac14(m+1)≤ B_m≤frac2m. For the sums A_m,K,B_m,K through any integer K≥ m, 0≤ A_m-A_m,K≤fracm^43K^3, 0≤ B_m-B_m,K≤frac1K. For n>m, one has (a_n^(m))^2≤ m^4/n^4, so the correction sequence is square-summable. The diagonal operator is bounded with norm one, and…
Exact mathematical passage
A_m=\norm{a^{(m)}}^2=\sum_{n\ge1}\frac{m^4}{(m^2+n^2)^2},\qquad B_m=\norm{v^{(m)}}^2=\sum_{n\ge1}\frac{n^2}{(m^2+n^2)^2}.
B_{m,K}Use in source · 2 passages
Search aliases: B_m,K
Use in source · 05i_diagonal_regularization.tex, line 23
s] The displayed a^(m) belongs to ℓ^2 and is the unique regularized correction at α=m^-2. Set A_m=norma^(m)^2=∑_n≥1fracm^4(m^2+n^2)^2, B_m=normv^(m)^2=∑_n≥1fracn^2(m^2+n^2)^2. Then frac m4≤ A_m≤frac4m3, frac14(m+1)≤ B_m≤frac2m. For the sums A_m,K,B_m,K through any integer K≥ m, 0≤ A_m-A_m,K≤fracm^43K^3, 0≤ B_m-B_m,K≤frac1K. For n>m, one has (a_n^(m))^2≤ m^4/n^4, so the correction sequence is square-summable. The diagonal operator is bounded with norm one, and w=(1/n)∈ℓ^2, hence the residual sequ…
Exact mathematical passage
A_{m,K},B_{m,K}
B_nUse in source · 13 passages
Search aliases: B_n
Use in source · 03_towers.tex, line 245
orem] The observable is a bounded projection with p_LA=0 and p_LP=p_L. It separates exact candidates. If S_L={s∈ E:p_Ls=s}, then Φ:E≃_L S_L× R, Φ(f)=(p_Lf,∂ f), Φ^-1(s,r)=Ps+Ar. Repair becomes (s,r)↦(s,0). It is the unique exact repair preserving p_Lf. Put B_n=I-s_nρ^Y_n. Since ρ^Y_ns_n=I, direct multiplication gives B_n^2=B_n, ρ^Y_nB_n=0, B_ns_n=0. Moreover, B_ny=y whenever ρ^Y_ny=0. Thus B_n really projects onto the target kernel. Its norm is at most 1+SB_Y. Coordinate zero of p_L is the ident…
Exact mathematical passage
B_n=I-s_n\rho^Y_n
B_{n+1}Use in source · 1 passage
Search aliases: B_n+1
Use in source · 12_proof_tools.tex, line 105
annot be the union of countably many closed sets with empty interior. Suppose F_1,F_2,… are closed with empty interior. Choose a nonempty open ball whose closed ball B_1, of positive radius at most 1/2, avoids F_1. Such a choice is possible because the complement of F_1 is open and meets every nonempty open set. Recursively choose a closed ball B_n+1 of positive radius at most 2^-(n+1), contained in the open ball with the same center and radius as B_n, and disjoint from F_n+1. The intersection o…
Exact mathematical passage
B_{n+1}
B_n^2Use in source · 1 passage
Search aliases: B_n^2
Use in source · 03_towers.tex, line 246
p_LP=p_L. It separates exact candidates. If S_L={s∈ E:p_Ls=s}, then Φ:E≃_L S_L× R, Φ(f)=(p_Lf,∂ f), Φ^-1(s,r)=Ps+Ar. Repair becomes (s,r)↦(s,0). It is the unique exact repair preserving p_Lf. Put B_n=I-s_nρ^Y_n. Since ρ^Y_ns_n=I, direct multiplication gives B_n^2=B_n, ρ^Y_nB_n=0, B_ns_n=0. Moreover, B_ny=y whenever ρ^Y_ny=0. Thus B_n really projects onto the target kernel. Its norm is at most 1+SB_Y. Coordinate zero of p_L is the identity, and every later coordinate applies B_n, so p_L^2=p_L and…
Exact mathematical passage
B_n^2=B_n,\qquad \rho^Y_nB_n=0,\qquad B_ns_n=0.
B_tUse in source · 2 passages
Search aliases: B_t
Use in source · 05_geometry.tex, line 598
Subtract q_0=diag(0,1), obtaining the symmetric matrix B_t=q_t-q_0=frac11+t^2 t^2 · -t -t · -t^2. The two off-diagonal entries of its square are zero, since -t^3+t^3=0. Both diagonal entries are (t^4+t^2)/(1+t^2)^2=t^2/(1+t^2). Thus B_t^ TB_t=B_t^2=fract^21+t^2I. Pairing this identity with an arbitrary vector proves the all-vector norm formula. It bounds the operator norm above by |t|/sqrt1+t^2, and evaluating any unit vector gives equality. The argument includes t=0, where both sides vanish.
Exact mathematical passage
B_t=q_t-q_0=\frac1{1+t^2} \begin{pmatrix}t^2&-t\\-t&-t^2\end{pmatrix}.
B_t^{\mathsf T}Use in source · 1 passage
Search aliases: B_t^ T mathsf
Use in source · 05_geometry.tex, line 603
-t · -t^2. The two off-diagonal entries of its square are zero, since -t^3+t^3=0. Both diagonal entries are (t^4+t^2)/(1+t^2)^2=t^2/(1+t^2). Thus B_t^ TB_t=B_t^2=fract^21+t^2I. Pairing this identity with an arbitrary vector proves the all-vector norm formula. It bounds the operator norm above by |t|/sqrt1+t^2, and evaluating any unit vector gives equality. The argument includes t=0, where both sides vanish.
Exact mathematical passage
B_t^{\mathsf T}B_t=B_t^2=\frac{t^2}{1+t^2}I.
B_t^2Use in source · 1 passage
Search aliases: B_t^2
Use in source · 05_geometry.tex, line 603
-t · -t^2. The two off-diagonal entries of its square are zero, since -t^3+t^3=0. Both diagonal entries are (t^4+t^2)/(1+t^2)^2=t^2/(1+t^2). Thus B_t^ TB_t=B_t^2=fract^21+t^2I. Pairing this identity with an arbitrary vector proves the all-vector norm formula. It bounds the operator norm above by |t|/sqrt1+t^2, and evaluating any unit vector gives equality. The argument includes t=0, where both sides vanish.
Exact mathematical passage
B_t^{\mathsf T}B_t=B_t^2=\frac{t^2}{1+t^2}I.
b_v^{-1}Use in source · 3 passages
Search aliases: b_v^-1
Use in source · 04_symmetry.tex, line 115
Suppose the orbit is neutral. Equality of orbit classes means there exists an actual based gauge b with b· c^q=1. At an arrow f:v→ w, this equation is b_vq_v^-1c_ff^*(q_w)[f^*(b_w)]^-1=1. Multiply on the left by q_vb_v^-1. Homomorphic preservation of products then gives c_ff^*(q_wb_w^-1)=q_vb_v^-1. Therefore x_v=q_vb_v^-1 is a twisted coherent family. On U, b_v=1 and q_v=s_v, so it extends the prescribed section.
Exact mathematical passage
q_vb_v^{-1}
B_WUse in source · 4 passages
Search aliases: B_W
Use in source · 02a_prescription.tex, line 17
Forgetting a prescription gives a quotient map. Let U⊆ W. Then ker p_W⊆ker p_U, B_W⊆ B_U, H_W⊆ H_U. A correction invisible on all of W is certainly invisible on U. The direction of the quotient map is consequently π_W,U:Q_W⟶ Q_U, [r]_B_W⟼[r]_B_U. The larger prescribed domain appears in the source. More protection means a smaller subgroup is divided out, so more obstruction information can remain.
Exact mathematical passage
\ker p_W\subseteq\ker p_U,\qquad B_W\subseteq B_U, \qquad H_W\subseteq H_U.
b_wUse in source · 2 passages
Search aliases: b_w
Use in source · 04_symmetry.tex, line 48
· =a_vc_g∘ f[(g∘ f)^*(a_z)]^-1 =d_g∘ f. Thus the gauge formula preserves the cocycle condition. For two gauges a,b, substitution gives (a·(b· c))_f =a_vb_vc_f[f^*(b_w)]^-1[f^*(a_w)]^-1 =((ab)· c)_f. The reversed order in the two inverse factors is exactly the formula for the inverse of a product. The identity gauge acts trivially.
Exact mathematical passage
(a\cdot(b\cdot c))_f =a_vb_vc_f[f^*(b_w)]^{-1}[f^*(a_w)]^{-1} =((ab)\cdot c)_f.
b_w^{-1}Use in source · 1 passage
Search aliases: b_w^-1
Use in source · 04_symmetry.tex, line 116
Suppose the orbit is neutral. Equality of orbit classes means there exists an actual based gauge b with b· c^q=1. At an arrow f:v→ w, this equation is b_vq_v^-1c_ff^*(q_w)[f^*(b_w)]^-1=1. Multiply on the left by q_vb_v^-1. Homomorphic preservation of products then gives c_ff^*(q_wb_w^-1)=q_vb_v^-1. Therefore x_v=q_vb_v^-1 is a twisted coherent family. On U, b_v=1 and q_v=s_v, so it extends the prescribed section.
Exact mathematical passage
c_ff^*(q_wb_w^{-1})=q_vb_v^{-1}.
B_YUse in source · 5 passages
Search aliases: B_Y
Use in source · 03_towers.tex, line 91
chapBanach repair and constructed split coordinates An actual complete candidate space. Let X_n,Y_n be Banach spaces over a complete nontrivially normed field, with continuous linear bonding maps and uniform bounds normρ^X_n≤ B_X, normρ^Y_n≤ B_Y. Use the spaces of bounded operator families E=ℓ^∞_n L(X_n,Y_n), R=ℓ^∞_n L(X_n+1,Y_n). The source formalization derives the required completeness from the operator spaces and their bounded product. The same differential satisfies norm∂ f_∞≤(B_X+B_Y)norm …
Exact mathematical passage
\norm{\rho^Y_n}\le B_Y
\bitsUse in source · 1 passage
Search aliases: bits {0,1}^ N mathbb
Use in source · 03_spectrum.tex, line 31
Give Sub(D) the topology inherited from {0,1}^D by its membership functions. Membership of a vector with finite support depends on finitely many selector bits. Conversely, membership of e_p_i recovers the ith bit. Consequently the order isomorphism above is also a homeomorphism from Cantor space {0,1}^ N to Sub(D).
Exact mathematical passage
\bits
\bmodUse in source · 2 passages
Search aliases: bmod
Use in source · 02_ambient.tex, line 10
Finite stages and two connecting systems. For N≥1, define A_N=∏_i<NC_p_i, m_N=∏_i<Np_i. The Chinese remainder map sends kbmod m_N to (kbmod p_i)_i<N. It is an isomorphism, so A_N is cyclic and its all-one vector is a generator.
Exact mathematical passage
k\bmod m_N
\bottomruleUse in source · 2 passages
Search aliases: bottomrule
Use in source · 05h_regularized_repair.tex, line 197
[3pt] q=0 · No · Residual vanishes, correction norms diverge, no exact protected answer. [3pt] q>0 · Yes · Strong convergence to the least-squares correction of minimum norm, not an exact answer. [3pt] q>0 · No · A positive residual floor and diverging correction norms occur together. The last case is not hypothetical. Let Tu=(u_n/n) on ℓ^2, define Su=(Tu,0) into ℓ^2⊕R, and take r=((1/n),1). Then b=((1/n),0) is in the closure but not the range, while q=1. The same escaping corrections as before …
Exact mathematical passage
3pt] $q=0$ & No & Residual vanishes, correction norms diverge, no exact protected answer.\\[3pt] $q>0$ & Yes & Strong convergence to the least-squares correction of minimum norm, not an exact answer.\\[3pt] $q>0$ & No & A positive residual floor and diverging correction norms occur together.\\\bottomrule \end{tabularx} \end{center} The last case is not hypothetical. Let $Tu=(u_n/n)$ on $\ell^2$, define $Su=(Tu,0)$ into $\ell^2\oplus\R$, and take $r=((1/n),1)$. Then $b=((1/n),0)$ is in the closure but not the range, while $q=1$. The same escaping corrections as before coexist with an irreducible orthogonal residual. Positive floor and unbounded correction cost are independent features. \subsection{Noise is controlled at a fixed parameter} \fixedplate{34_noise_filter}{For one singular value $\sigma$, the response to residual-data noise is $\sigma/(\sigma^2+\alpha)$. Its maximum occurs at $\sigma=\sqrt\alpha$ and equals $1/(2\sqrt\alpha)$. The plot evaluates this formula and the universal horizontal bounds at three explicit positive parameters.}{fig:noise-filter} \begin{theorem}[Data sensitivity and a convergent parameter choice]\label{WR09} For two residual data vectors $r,\widetilde r$ with $\norm{r-\widetilde r}\le\delta$ and the same $T,\alpha$, \[ \norm{a_\alpha(r)-a_\alpha(\widetilde r)} \le\frac{\delta}{2\sqrt\alpha},\qquad \norm{T(a_\alpha(r)-a_\alpha(\widetilde r))}\le\delta.
C'Use in source · 17 passages
Search aliases: C'
Use in source · 05_geometry.tex, line 421
chapControlled motion of exact spaces A two-sided condition number theorem. Let D,D':E→ R be continuous linear maps from a real Hilbert space to the same normed target. Write q=P_ker D, q'=P_ker D'. Suppose norm(I-q)x≤ CnormDx, norm(I-q')x≤ C'normD'x, C,C'≥0, and normD-D'≤ε.
Exact mathematical passage
\norm{(I-q)x}\le C\norm{Dx},\qquad \norm{(I-q')x}\le C'\norm{D'x},\qquad C,C'\ge0,
\mathcal C'Use in source · 1 passage
Search aliases: C' mathcal
Use in source · 04_symmetry.tex, line 409
Weakly commuting squares are enough. Suppose E: C→ C' and K: D→ D' are equivalences, with functors R,R' and a natural isomorphism σ:KRxRightarrow∼R'E. The square is not required to commute literally.
Exact mathematical passage
E:\mathcal C\to\mathcal C'
c'Use in source · 10 passages
Search aliases: c'
Use in source · 04_symmetry.tex, line 256
The correct uniqueness statement. Suppose (c,b_i) and (c',b'_i) realize the same descent datum. The only possible compatible comparison is g_v=b'_i(v)^-1b_i(v), using any patch through v. The overlap identities make it independent of the patch, and b'_ig=b_i. It is therefore the unique global comparison respecting the fixed local identifications.
Exact mathematical passage
(c',b'_i)
C^{\mathsf T}Use in source · 5 passages
Search aliases: C^ T mathsf
Use in source · 05k_stopping_rules.tex, line 17
[The invariant norm of a coordinate certificate] The actual normal residual is g=BG^-1q, norm g^2=q^ TG^-1q. For any invertible change of basis B'=BC, the same proposed correction has c'=C^-1c, G'=C^ TGC, q'=C^ Tq, and (q')^ T(G')^-1q'=q^ TG^-1q. Because B has independent columns, for every nonzero coefficient vector c, c^ TGc=c^ TB^ TBc=normBc^2>0. Thus G is positive definite and invertible. The matrix P=BG^-1B^ T is symmetric and obeys P^2=BG^-1(B^ TB)G^-1B^ T=P, PB=B. Its range is precisely t…
Exact mathematical passage
c'=C^{-1}c,\quad G'=C^{\mathsf T}GC,\quad q'=C^{\mathsf T}q,
c^{\mathsf T}Use in source · 1 passage
Search aliases: c^ T mathsf
Use in source · 05k_stopping_rules.tex, line 27
covector-norm g=BG^-1q, norm g^2=q^ TG^-1q. For any invertible change of basis B'=BC, the same proposed correction has c'=C^-1c, G'=C^ TGC, q'=C^ Tq, and (q')^ T(G')^-1q'=q^ TG^-1q. Because B has independent columns, for every nonzero coefficient vector c, c^ TGc=c^ TB^ TBc=normBc^2>0. Thus G is positive definite and invertible. The matrix P=BG^-1B^ T is symmetric and obeys P^2=BG^-1(B^ TB)G^-1B^ T=P, PB=B. Its range is precisely the column space H, so it is the orthogonal projection onto the en…
Exact mathematical passage
c^{\mathsf T}Gc=c^{\mathsf T}B^{\mathsf T}Bc=\norm{Bc}^2>0.
c^{(m)}_nUse in source · 1 passage
Search aliases: c^(m)_n
Use in source · 05i_diagonal_regularization.tex, line 147
The same operator has an attainable datum. Change only the residual to b_n=1/n^2. Its minimum norm preimage is a^†_n=1/n∈ℓ^2. At α=m^-2, the regularized correction is c^(m)_n=fracm^2n(m^2+n^2), a^†_n-c^(m)_n=fracnm^2+n^2. The bias square is exactly the previously computed B_m. This links the two cases without conflating their existence hypotheses: the escaping residual of one problem is the regularization bias of an attainable problem.
Exact mathematical passage
c^{(m)}_n=\frac{m^2}{n(m^2+n^2)},\qquad a^\dagger_n-c^{(m)}_n=\frac{n}{m^2+n^2}.
C^{-1}Use in source · 5 passages
Search aliases: C^-1
Use in source · 05k_stopping_rules.tex, line 17
[The invariant norm of a coordinate certificate] The actual normal residual is g=BG^-1q, norm g^2=q^ TG^-1q. For any invertible change of basis B'=BC, the same proposed correction has c'=C^-1c, G'=C^ TGC, q'=C^ Tq, and (q')^ T(G')^-1q'=q^ TG^-1q. Because B has independent columns, for every nonzero coefficient vector c, c^ TGc=c^ TB^ TBc=normBc^2>0. Thus G is positive definite and invertible. The matrix P=BG^-1B^ T is symmetric and obeys P^2=BG^-1(B^ TB)G^-1B^ T=P, PB=B. Its range is precisely t…
Exact mathematical passage
c'=C^{-1}c,\quad G'=C^{\mathsf T}GC,\quad q'=C^{\mathsf T}q,
c^q_fUse in source · 1 passage
Search aliases: c^q_f
Use in source · 04_symmetry.tex, line 83
The relative obstruction without commutativity. Prescribe a coherent partial section s of T_c on U. The based gauge group is G_U={a∈∏_vG_v:a_u=1 for u∈ U}. Choose a raw family q equal to s on U. Identity elements can be used elsewhere. Normalize the labels by c^q_f=q_v^-1c_f f^*(q_w). Its labels on arrows internal to U are one. Define the pointed orbit obstruction o_U(c,s)=[c^q]∈ Z^1(V;G)/ G_U, whose neutral point is the class of the trivial cocycle.
Exact mathematical passage
c^q_f=q_v^{-1}c_f f^*(q_w).
C_0Use in source · 1 passage
Search aliases: C_0
Use in source · 03_towers.tex, line 351
egintheorem[Finite reconstruction and nested remainders] For every N, x=Q_Nx+∑_j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of C_jJ_j=I gives D_NL_N=I_R_N. This cancellation takes place between adjacent maps with matching domains. No identification of different residual space…
Exact mathematical passage
D_N=C_{N-1}\cdots C_0:R_0\to R_N
c_0Use in source · 2 passages
Search aliases: c_0
Use in source · 03_towers.tex, line 472
[The recovery domain is c_0] For this shift cascade, S_Nx→ x in ℓ^∞ Longleftrightarrow x_n→0. For x_n=q^n, 0≤ q<1, the error is exactly q^N. The error norm is sup_n≥ N|x_n|. It tends to zero precisely when the sequence tends to zero. For a geometric sequence its largest tail value is the first one, q^N. sourceKS29 As a second example, take scalar spaces with C=tfrac12I, J=2I. The coarse value x_N=2^-Nx tends to zero, but restoration gives Q_Nx=x. Every detail is zero and the entire input is pers…
Exact mathematical passage
c_0
c_1Use in source · 1 passage
Search aliases: c_1
Use in source · 03b_repair_transport.tex, line 133
Two shears compose by adding their parameters. In the mismatch law this appears as M_c_2+c_1=S_c_2M_c_1+M_c_2, and S_c_2 fixes the exact horizontal axis. The identity is therefore a simple instance of the same composition rule, not an unrelated coordinate trick.
Exact mathematical passage
M_{c_2+c_1}=S_{c_2}M_{c_1}+M_{c_2}
C_{12}Use in source · 1 passage
Search aliases: C_12
Use in source · 11_atlas.tex, line 71
For m=12, only coordinates 2 and 3 matter. The six representatives are 0, e_2, e_3, 2e_3, e_2+e_3, e_2+2e_3. The vector e_2+e_3 generates this quotient, which is C_6, not C_12. Repeated prime powers in the multiplier do not create additional coordinates.
Exact mathematical passage
C_{12}
c_2Use in source · 2 passages
Search aliases: c_2
Use in source · 03b_repair_transport.tex, line 133
Two shears compose by adding their parameters. In the mismatch law this appears as M_c_2+c_1=S_c_2M_c_1+M_c_2, and S_c_2 fixes the exact horizontal axis. The identity is therefore a simple instance of the same composition rule, not an unrelated coordinate trick.
Exact mathematical passage
M_{c_2+c_1}=S_{c_2}M_{c_1}+M_{c_2}
C_{210}Use in source · 1 passage
Search aliases: C_210
Use in source · 02_ambient.tex, line 19
4 · 2,3,5,7 · 210 · C_210
Exact mathematical passage
C_{210}
C_{2310}Use in source · 1 passage
Search aliases: C_2310
Use in source · 02_ambient.tex, line 20
5 · 2,3,5,7,11 · 2310 · C_2310
Exact mathematical passage
C_{2310}
C_{2k}Use in source · 1 passage
Search aliases: C_2k
Use in source · 05c_iteration.tex, line 115
Residual bursts can be compatible with convergence. Set q_2k=2, q_2k+1=1/8. Then Q_2k=4^-k, Q_2k+1=2·4^-k. For f_n=(s,Q_n), the residual doubles at the start of every pair of steps. Choose C_2k=1 and C_2k+1=7/8. These give equality in the movement bound, and ∑_n=0^∞C_nQ_n =∑_k=0^∞(1+frac74)4^-k =frac113. Thus f_n→(s,0) despite infinitely many increases. Starting at an even index 2k, the remaining movement budget is frac1134^-k. At an odd index 2k+1, it is frac83 4^-k. These bounds are larger tha…
Exact mathematical passage
C_{2k}=1
C_{2k+1}Use in source · 1 passage
Search aliases: C_2k+1
Use in source · 05c_iteration.tex, line 115
Residual bursts can be compatible with convergence. Set q_2k=2, q_2k+1=1/8. Then Q_2k=4^-k, Q_2k+1=2·4^-k. For f_n=(s,Q_n), the residual doubles at the start of every pair of steps. Choose C_2k=1 and C_2k+1=7/8. These give equality in the movement bound, and ∑_n=0^∞C_nQ_n =∑_k=0^∞(1+frac74)4^-k =frac113. Thus f_n→(s,0) despite infinitely many increases. Starting at an even index 2k, the remaining movement budget is frac1134^-k. At an odd index 2k+1, it is frac83 4^-k. These bounds are larger tha…
Exact mathematical passage
C_{2k+1}=7/8
C_3Use in source · 2 passages
Search aliases: C_3
Use in source · 11_atlas.tex, line 36
At the unselected prime 3, the coordinate of u_A is zero. In this case 3(1/3,0)=(1,0). The three roots are (1/3,0), (1/3,e_3), (1/3,2e_3). Their differences form G_A[3]=C_3, as predicted by Proposition~[source reference].
Exact mathematical passage
G_A[3]=C_3
C_{30}Use in source · 3 passages
Search aliases: C_30
Use in source · 02_ambient.tex, line 18
3 · 2,3,5 · 30 · C_30
Exact mathematical passage
C_{30}
C_6Use in source · 4 passages
Search aliases: C_6
Use in source · 02_ambient.tex, line 17
2 · 2,3 · 6 · C_6
Exact mathematical passage
C_6
C_{a,b}Use in source · 2 passages
Search aliases: C_a,b
Use in source · 03b_repair_transport.tex, line 143
For a block from a to b, its value on a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a,b. The dyadic coarsener in Section~[source reference] is the case b=a+2.
Exact mathematical passage
\label{eq:block-defect-sum} (C_{a,b}r)=\sum_{k=a}^{b-1} \rho^Y_{a,k}\,r_k\,\rho^X_{k+1,b}.
c_f^{-1}Use in source · 2 passages
Search aliases: c_f^-1
Use in source · 04_symmetry.tex, line 70
Automorphisms and ambiguity are not the same question. The stabilizer of a cocycle is the group of families with a_v=c_f f^*(a_w)c_f^-1. If one gauge a:c→ d is known, all gauges from c to d are uniquely of the form ah, with h∈Stab(c). Indeed a^-1b stabilizes c for every other gauge b. This is the ordinary Hom torsor principle in a groupoid.
Exact mathematical passage
a_v=c_f f^*(a_w)c_f^{-1}.
c_f^qUse in source · 1 passage
Search aliases: c_f^q
Use in source · 04_symmetry.tex, line 96
relative extension] The obstruction is independent of the raw lift. It is neutral exactly when s extends to a global section of T_c. The normalized label c^q is the gauge transform of c by q^-1, so Proposition~[source reference] makes it a cocycle. If f:v→ w is internal to U, the partial-section equation says q_v=c_ff^*(q_w), and therefore c_f^q=1 on that arrow.
Exact mathematical passage
c_f^q=1
c_f^tUse in source · 1 passage
Search aliases: c_f^t
Use in source · 04_symmetry.tex, line 99
If t is another raw lift, define a_v=t_v^-1q_v. On U both lifts equal s, so a_v=1 and a∈ G_U. Its action is (a· c^q)_f · =t_v^-1q_v q_v^-1c_ff^*(q_w) [f^*(t_w^-1q_w)]^-1 · =t_v^-1c_ff^*(q_w) [f^*(q_w)]^-1f^*(t_w) =t_v^-1c_ff^*(t_w)=c_f^t. Thus changing the raw lift leaves the based orbit unchanged.
Exact mathematical passage
\begin{aligned} (a\cdot c^q)_f &=t_v^{-1}q_v\,q_v^{-1}c_ff^*(q_w) [f^*(t_w^{-1}q_w)]^{-1}\\ &=t_v^{-1}c_ff^*(q_w) [f^*(q_w)]^{-1}f^*(t_w) =t_v^{-1}c_ff^*(t_w)=c_f^t. \end{aligned}
c_iUse in source · 5 passages
Search aliases: c_i
Use in source · 04_symmetry.tex, line 189
A descent datum consists of local cocycles c_i and overlap gauges a_ij(v) with c_j(f) · =a_ij(v)c_i(f)[f^*a_ij(w)]^-1,
Exact mathematical passage
c_i
c_jUse in source · 3 passages
Search aliases: c_j
Use in source · 04_symmetry.tex, line 190
A descent datum consists of local cocycles c_i and overlap gauges a_ij(v) with c_j(f) · =a_ij(v)c_i(f)[f^*a_ij(w)]^-1, a_ik(v) · =a_jk(v)a_ij(v). The triple equation implies a_ii=1 and a_ji=a_ij^-1. A morphism of descent data is a family of local gauges r_i satisfying r_j(v)a_ij(v)=a'_ij(v)r_i(v). These objects and arrows form a groupoid.
Exact mathematical passage
c_j(f)&=a_{ij}(v)c_i(f)[f^*a_{ij}(w)]^{-1},\label{eq:localgauge}\\ a_{ik}(v)&=a_{jk}(v)a_{ij}(v).\label{eq:triple}
c_j^2Use in source · 1 passage
Search aliases: c_j^2
Use in source · 05f_bounded_realization.tex, line 37
For each finite J, Bessel's inequality and passage to the limit give ∑_j=1^Jc_j^2 =lim_k∑_j=1^J⟨ a_n_k,e_j⟩^2≤ M^2. The nondecreasing partial sums are therefore bounded. Orthogonality makes the partial vector sums ∑_j≤ Jc_je_j Cauchy, so completeness defines a∈ H_0 with norm a≤ M.
Exact mathematical passage
\sum_{j=1}^Jc_j^2 =\lim_k\sum_{j=1}^J\langle a_{n_k},e_j\rangle^2\le M^2.
C_kUse in source · 2 passages
Search aliases: C_k
Use in source · 05c_iteration.tex, line 59
[Variable-rate exactification] If Q_n→0 and ∑_n C_nQ_n<∞, then (f_n) has an exact limit and dist(f_n,x)≤ω(f_0)∑_k=n^∞C_kQ_k. For constant q_n=q∈[0,1) and constant C_n=C, the sharper restarted estimate is dist(f_n,x)≤fracC1-q ω(f_n). The estimate ω(f_n)≤ Q_nω(f_0) follows by induction. At n=0 it is equality. If it holds at n, nonnegativity of q_n permits multiplication and gives ω(f_n+1)≤ q_nω(f_n) ≤ q_nQ_nω(f_0)=Q_n+1ω(f_0). Because Q_n→0 and ω(f_0) is a fixed finite number, the residual tends t…
Exact mathematical passage
\label{eq:variable-tail} \dist(f_n,x)\le\omega(f_0)\sum_{k=n}^{\infty}C_kQ_k.
C_mUse in source · 2 passages
Search aliases: C_m
Use in source · 05_construction.tex, line 125
[Finitely generated subgroups] Every finitely generated subgroup of G_A is either finite cyclic or isomorphic to Z⊕ C_m for a squarefree positive integer m, with C_1 trivial. Let L≤ G_A be generated by finitely many elements. Its image in Q is a finitely generated subgroup of Q, hence either zero or infinite cyclic. If it is zero, then L≤ D and its finite support places it inside a finite cyclic group.
Exact mathematical passage
\Z\oplus C_m
c_mUse in source · 2 passages
Search aliases: c_m
Use in source · 12_proof_tools.tex, line 16
⟨ x-c,z-c⟩≤0 (z∈ C). If C=c_0+L is a closed affine subspace with direction space L, the condition is x-c⊥ L. Produce a Cauchy minimizing sequence. Put d=inf_z∈ Cnormx-z. Nonemptiness makes this a finite nonnegative real number. Choose c_n∈ C so that normx-c_n→ d. Convexity places (c_n+c_m)/2 in C. The parallelogram identity gives normc_n-c_m^2 =2normx-c_n^2+2normx-c_m^2 -4 x-fracc_n+c_m2 ^2 ≤2normx-c_n^2+2normx-c_m^2-4d^2. The right side tends to zero as both indices tend to infinity. Thus (c_n)…
Exact mathematical passage
(c_n+c_m)/2
C_{M-1}Use in source · 1 passage
Search aliases: C_M-1
Use in source · 03_towers.tex, line 367
Idempotence follows from D_NL_N=I: Q_N^2=L_N(D_NL_N)D_N=Q_N. For M≥ N, put C_N,M=C_M-1⋯ C_N and J_N,M=J_N⋯ J_M-1, using identities for M=N. Then D_M=C_N,MD_N, L_M=L_NJ_N,M, C_N,MJ_N,M=I. The identities D_NL_M=J_N,M and D_ML_N=C_N,M now give Q_NQ_M=L_NJ_N,MD_M=Q_M, Q_MQ_N=L_MC_N,MD_N=Q_M. Both orders have been checked, rather than inferred from one range inclusion.
Exact mathematical passage
C_{N,M}=C_{M-1}\cdots C_N
C_NUse in source · 1 passage
Search aliases: C_N
Use in source · 03_towers.tex, line 367
Idempotence follows from D_NL_N=I: Q_N^2=L_N(D_NL_N)D_N=Q_N. For M≥ N, put C_N,M=C_M-1⋯ C_N and J_N,M=J_N⋯ J_M-1, using identities for M=N. Then D_M=C_N,MD_N, L_M=L_NJ_N,M, C_N,MJ_N,M=I. The identities D_NL_M=J_N,M and D_ML_N=C_N,M now give Q_NQ_M=L_NJ_N,MD_M=Q_M, Q_MQ_N=L_MC_N,MD_N=Q_M. Both orders have been checked, rather than inferred from one range inclusion.
Exact mathematical passage
C_{N,M}=C_{M-1}\cdots C_N
c_nUse in source · 6 passages
Search aliases: c_n
Use in source · 03_towers.tex, line 317
The actual dyadic tower coarsener has the typed formula (Cr)_n=ρ^Y_2nr_2n+1+r_2nρ^X_2n+1. It is obtained by telescoping two adjacent compatibility squares. With target sections, a lift is (Jc)_2n=0, (Jc)_2n+1=s_2nc_n. The section identity gives CJ=I, so the missing detail is explicitly Hr, not an unspecified reconstruction error. The normed construction imposes the bounds needed to make these maps bounded on its operator family spaces. sourceKS25
Exact mathematical passage
(Jc)_{2n}=0,\qquad (Jc)_{2n+1}=s_{2n}c_n.
C_{N,M}Use in source · 4 passages
Search aliases: C_N,M
Use in source · 03_towers.tex, line 367
Idempotence follows from D_NL_N=I: Q_N^2=L_N(D_NL_N)D_N=Q_N. For M≥ N, put C_N,M=C_M-1⋯ C_N and J_N,M=J_N⋯ J_M-1, using identities for M=N. Then D_M=C_N,MD_N, L_M=L_NJ_N,M, C_N,MJ_N,M=I. The identities D_NL_M=J_N,M and D_ML_N=C_N,M now give Q_NQ_M=L_NJ_N,MD_M=Q_M, Q_MQ_N=L_MC_N,MD_N=Q_M. Both orders have been checked, rather than inferred from one range inclusion.
Exact mathematical passage
C_{N,M}=C_{M-1}\cdots C_N
C_{N-1}Use in source · 1 passage
Search aliases: C_N-1
Use in source · 03_towers.tex, line 351
egintheorem[Finite reconstruction and nested remainders] For every N, x=Q_Nx+∑_j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of C_jJ_j=I gives D_NL_N=I_R_N. This cancellation takes place between adjacent maps with matching domains. No identification of different residual space…
Exact mathematical passage
D_N=C_{N-1}\cdots C_0:R_0\to R_N
C_{p_i}Use in source · 2 passages
Search aliases: C_p_i
Use in source · 02_ambient.tex, line 7
Finite stages and two connecting systems. For N≥1, define A_N=∏_i<NC_p_i, m_N=∏_i<Np_i. The Chinese remainder map sends kbmod m_N to (kbmod p_i)_i<N. It is an isomorphism, so A_N is cyclic and its all-one vector is a generator.
Exact mathematical passage
A_N=\prod_{i<N}C_{p_i},\qquad m_N=\prod_{i<N}p_i.
C_{p_n}Use in source · 1 passage
Search aliases: C_p_n
Use in source · 03_spectrum.tex, line 84
A purely finite-state boundary. Consider a restricted grammar in which a finite automaton reads the unary word 1^n and selects the factor C_p_n exactly when it accepts.
Exact mathematical passage
C_{p_n}
C_{\rad(m)}Use in source · 4 passages
Search aliases: C_rad(m) rad operatorname
Use in source · 00_notation.tex, line 33
The main statements have the following form: Tor(G_A) · =D, · G_A/D · ≅ Q, G_A/mG_A · ≅ C_rad(m), · widehatG_A · ≅ P, G_A≅ G_B · iff A=^* B, · · G_A has an X-computable copy · iff A≤_T X. · · The extension by D does not split. Distinct eventual selector classes do not embed in one another. All of these conclusions are proved within the book, with the exact hypotheses retained.
Exact mathematical passage
\Tor(G_A)&=D,&G_A/D&\cong\Q,\\ G_A/mG_A&\cong C_{\rad(m)},&\widehat{G_A}&\cong P,\\ G_A\cong G_B&\iff A\eqfin B,&&\\ G_A\text{ has an }X\text{-computable copy}&\iff A\le_T X.&&
c_vUse in source · 9 passages
Search aliases: c_v
Use in source · 01_views.tex, line 27
[Joint determination and the universal property] The evaluations jointly determine a realization. If a set T carries maps c_v:T→ F_v satisfying f^*c_w=c_v, there is a unique map c:T→Γ(F) satisfying ev_v c=c_v for every view. We first verify that the evaluation maps themselves form a compatible family. If x∈Γ(F) and f:v→ w, the defining equation for x gives f^*(ev_w x)=f^*(x_w)=x_v=ev_v x. If x,y∈Γ(F) have equal evaluations, then x_v=y_v for every v. Equality of functions into a dependent product…
Exact mathematical passage
c_v:T\to F_v
c_wUse in source · 2 passages
Search aliases: c_w
Use in source · 01_views.tex, line 27
[Joint determination and the universal property] The evaluations jointly determine a realization. If a set T carries maps c_v:T→ F_v satisfying f^*c_w=c_v, there is a unique map c:T→Γ(F) satisfying ev_v c=c_v for every view. We first verify that the evaluation maps themselves form a compatible family. If x∈Γ(F) and f:v→ w, the defining equation for x gives f^*(ev_w x)=f^*(x_w)=x_v=ev_v x. If x,y∈Γ(F) have equal evaluations, then x_v=y_v for every v. Equality of functions into a dependent product…
Exact mathematical passage
f^*c_w=c_v
C_\varepsilonUse in source · 1 passage
Search aliases: C_ε varepsilon
Use in source · 05c_iteration.tex, line 41
It remains to prove exactness of the limit. Fix ε>0. Since ω(f_m)→0, an entire tail of the sequence belongs to the sublevel set C_ε={z:ω(z)≤ε}. Lower semicontinuity makes this set closed, so the limit of that tail belongs to it. Hence ω(x)≤ε for every positive ε. If ω(x)>0, choosing ε=ω(x)/2 contradicts this inequality. Since ω is nonnegative, ω(x)=0. We have not used, or asserted, continuity of ω or the stronger equality ω(f_m)→ω(x) for arbitrary convergent inputs. sourceKS71 The formal theorem…
Exact mathematical passage
C_\varepsilon=\{z:\omega(z)\le\varepsilon\}
C_\phiUse in source · 6 passages
Search aliases: C_ϕ phi
Use in source · 03b_repair_transport.tex, line 140
Actual tower resampling gives a defect map. Return to Banach towers with contractive source and target bonds. Let ϕ be a monotone schedule. Candidate sampling is (S_ϕ f)_n=f_ϕ(n). The sampled bonding map is the actual composite restriction between the selected levels. The residual map C_ϕ is not pointwise sampling of a scalar error score. It accumulates transported fine defects across each omitted block.
Exact mathematical passage
C_\phi
C_{\phi(n),\phi(n+1)}Use in source · 1 passage
Search aliases: C_ϕ(n),ϕ(n+1) phi phi
Use in source · 03b_repair_transport.tex, line 147
For a block from a to b, its value on a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a,b. The dyadic coarsener in Section~[source reference] is the case b=a+2.
Exact mathematical passage
(C_\phi r)_n=C_{\phi(n),\phi(n+1)}r
\codeUse in source · 1 passage
Search aliases: code
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
\mathrm{comp}Use in source · 2 passages
Search aliases: comp mathrm
Use in source · 04_symmetry.tex, line 290
[The composition topology] Condition~[source reference] defines a Grothendieck topology J_comp. The sieve generated by a family of inclusions U_i⊆ W is covering exactly when the family covers every composition support in W. In the inclusion poset, identify a sieve on W with a downward-closed collection of subsets of W. Pullback along Z⊆ W consists of those subsets of Z that were already members of the original sieve. We verify the three topology axioms using this description.
Exact mathematical passage
J_{\mathrm{comp}}
\coprodUse in source · 2 passages
Search aliases: coprod
Use in source · 02_obstructions.tex, line 145
Extension in stages. For U⊆ W⊆ObV, an extension first to W and then globally must retain the intermediate choice. There is an explicit equivalence Ext_U(s)≃ coprod_substackt∈Γ_W(A) t|_U=sExt_W(t). It sends a global extension to its restriction to W, together with itself as an extension of that restriction. The inverse forgets the intermediate coordinate. These operations are inverse because the required intermediate coordinate of a global section is its actual restriction. A coherent intermediat…
Exact mathematical passage
\label{eq:stages} \Ext_U(s)\simeq \coprod_{\substack{t\in\Gamma_W(A)\\t|_U=s}}\Ext_W(t).
\coprod_{\substack{t\in\Gamma_W(A)\\t|_U=s}}Use in source · 2 passages
Search aliases: coprod_substackt∈Γ_W(A) t|_U=s coprod substack in Gamma t
Use in source · 02_obstructions.tex, line 145
Extension in stages. For U⊆ W⊆ObV, an extension first to W and then globally must retain the intermediate choice. There is an explicit equivalence Ext_U(s)≃ coprod_substackt∈Γ_W(A) t|_U=sExt_W(t). It sends a global extension to its restriction to W, together with itself as an extension of that restriction. The inverse forgets the intermediate coordinate. These operations are inverse because the required intermediate coordinate of a global section is its actual restriction. A coherent intermediat…
Exact mathematical passage
\label{eq:stages} \Ext_U(s)\simeq \coprod_{\substack{t\in\Gamma_W(A)\\t|_U=s}}\Ext_W(t).
\cosUse in source · 1 passage
Search aliases: cos
Use in source · 05i_diagonal_regularization.tex, line 111
Step 2: evaluate the integrals. The substitution x=tanθ gives dx=sec^2θ dθ and 1+x^2=sec^2θ, so ∫_0^∞ f(x) dx=∫_0^π/2cos^2θ dθ=fracπ4, ∫_0^∞ g(x) dx=∫_0^π/2sin^2θ dθ=fracπ4. Step 1 now proves the two stated asymptotics.
Exact mathematical passage
\int_0^\infty f(x)\,dx=\int_0^{\pi/2}\cos^2\theta\,d\theta=\frac\pi4, \qquad \int_0^\infty g(x)\,dx=\int_0^{\pi/2}\sin^2\theta\,d\theta=\frac\pi4.
\cos^2Use in source · 1 passage
Search aliases: cos^2 cos
Use in source · 05i_diagonal_regularization.tex, line 111
Step 2: evaluate the integrals. The substitution x=tanθ gives dx=sec^2θ dθ and 1+x^2=sec^2θ, so ∫_0^∞ f(x) dx=∫_0^π/2cos^2θ dθ=fracπ4, ∫_0^∞ g(x) dx=∫_0^π/2sin^2θ dθ=fracπ4. Step 1 now proves the two stated asymptotics.
Exact mathematical passage
\int_0^\infty f(x)\,dx=\int_0^{\pi/2}\cos^2\theta\,d\theta=\frac\pi4, \qquad \int_0^\infty g(x)\,dx=\int_0^{\pi/2}\sin^2\theta\,d\theta=\frac\pi4.
D'Use in source · 38 passages
Search aliases: D'
Use in source · 05_geometry.tex, line 420
chapControlled motion of exact spaces A two-sided condition number theorem. Let D,D':E→ R be continuous linear maps from a real Hilbert space to the same normed target. Write q=P_ker D, q'=P_ker D'. Suppose norm(I-q)x≤ CnormDx, norm(I-q')x≤ C'normD'x, C,C'≥0, and normD-D'≤ε.
Exact mathematical passage
D,D':E\to R
\mathcal D'Use in source · 1 passage
Search aliases: D' mathcal
Use in source · 04_symmetry.tex, line 409
Weakly commuting squares are enough. Suppose E: C→ C' and K: D→ D' are equivalences, with functors R,R' and a natural isomorphism σ:KRxRightarrow∼R'E. The square is not required to commute literally.
Exact mathematical passage
K:\mathcal D\to\mathcal D'
d^{\mathsf T}Use in source · 1 passage
Search aliases: d^ T mathsf
Use in source · 05k_stopping_rules.tex, line 38
The restricted Hilbert normal residual is g=P d^ T(dBc-r)+α Bc. Substitute S=dB, G=B^ TB and the expression for P: g=BG^-1(S^ T(Sc-r)+α Gc)=BG^-1q. Equivalently q=B^ Tg records the pairings of g with the basis columns. Its squared Hilbert norm is g^ Tg =q^ TG^-1B^ TBG^-1q =q^ TG^-1q, using symmetry of G^-1. This proves both coordinate formulas.
Exact mathematical passage
g=P d^{\mathsf T}(dBc-r)+\alpha Bc.
d^*Use in source · 3 passages
Search aliases: d^*
Use in source · 05f_bounded_realization.tex, line 22
The adjoint below always means T^*:Y→ H, taken with the inherited Hilbert metric on H. In ambient coordinates this is the projection of d^* onto H. A coordinate basis of ker p need not be orthonormal. Its Gram matrix must then be included in any calculation of correction energy.
Exact mathematical passage
d^*
d^2Use in source · 2 passages
Search aliases: d^2
Use in source · 12_proof_tools.tex, line 17
If C=c_0+L is a closed affine subspace with direction space L, the condition is x-c⊥ L. Produce a Cauchy minimizing sequence. Put d=inf_z∈ Cnormx-z. Nonemptiness makes this a finite nonnegative real number. Choose c_n∈ C so that normx-c_n→ d. Convexity places (c_n+c_m)/2 in C. The parallelogram identity gives normc_n-c_m^2 =2normx-c_n^2+2normx-c_m^2 -4 x-fracc_n+c_m2 ^2 ≤2normx-c_n^2+2normx-c_m^2-4d^2. The right side tends to zero as both indices tend to infinity. Thus (c_n) is Cauchy. Completen…
Exact mathematical passage
\norm{c_n-c_m}^2 =2\norm{x-c_n}^2+2\norm{x-c_m}^2 -4\left\|x-\frac{c_n+c_m}{2}\right\|^2 \le2\norm{x-c_n}^2+2\norm{x-c_m}^2-4d^2.
D_0Use in source · 2 passages
Search aliases: D_0
Use in source · 03_towers.tex, line 351
[Finite reconstruction and nested remainders] For every N, x=Q_Nx+∑_j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of C_jJ_j=I gives D_NL_N=I_R_N. This cancellation takes place between adjacent maps with matching domains. No identification of different residual spaces is needed…
Exact mathematical passage
D_0=I
\mathcal D_1Use in source · 1 passage
Search aliases: D_1 mathcal
Use in source · 05j_inexact_certificates.tex, line 168
The proposed objective is 11/16. Its actual excess over the optimum is 1/48. The residual witness gives D_1(3/4)=frac2132, J(a)- D_1(3/4)=frac132. The duality gap is an upper bound for suboptimality, not necessarily the exact suboptimality. Both repaired candidates preserve p(f)=1.
Exact mathematical passage
\mathcal D_1(3/4)=\frac{21}{32},\qquad J(a)-\mathcal D_1(3/4)=\frac1{32}.
d_aUse in source · 1 passage
Search aliases: d_a
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
d_bUse in source · 1 passage
Search aliases: d_b
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
D_{B_K}Use in source · 1 passage
Search aliases: D_B_K
Use in source · 03_spectrum.tex, line 66
Define B_K={2n:n∈ N}∪{2n+1:n∉ K}, H=D_B_K. This is an explicit mathematical definition of a countably infinite, locally cyclic abelian group.
Exact mathematical passage
B_K=\{2n:n\in\N\}\cup\{2n+1:n\notin K\}, \qquad H=D_{B_K}.
D_{B\cap C}Use in source · 1 passage
Search aliases: D_B∩ C cap
Use in source · 03_spectrum.tex, line 25
e factors. Every element of H lies in D_B, and every finite sum of the factors in B lies in H. Hence H=D_B. Uniqueness follows by testing membership of e_p_i. Proposition~[source reference] proves full invariance. Finally σ_p_i(D_B)=1 exactly when i∈ B, and these values are isomorphism invariants. The lattice operations have literal set-theoretic descriptions: D_B+D_C=D_B∪ C, D_B∩ D_C=D_B∩ C, D=D_B⊕ D_ N∖ B. The infinite support choice is therefore already visible in an elementary algebraic latt…
Exact mathematical passage
D_B+D_C=D_{B\cup C},\qquad D_B\cap D_C=D_{B\cap C}, \qquad D=D_B\oplus D_{\N\setminus B}.
D_{B\cup C}Use in source · 1 passage
Search aliases: D_B∪ C cup
Use in source · 03_spectrum.tex, line 25
e factors. Every element of H lies in D_B, and every finite sum of the factors in B lies in H. Hence H=D_B. Uniqueness follows by testing membership of e_p_i. Proposition~[source reference] proves full invariance. Finally σ_p_i(D_B)=1 exactly when i∈ B, and these values are isomorphism invariants. The lattice operations have literal set-theoretic descriptions: D_B+D_C=D_B∪ C, D_B∩ D_C=D_B∩ C, D=D_B⊕ D_ N∖ B. The infinite support choice is therefore already visible in an elementary algebraic latt…
Exact mathematical passage
D_B+D_C=D_{B\cup C},\qquad D_B\cap D_C=D_{B\cap C}, \qquad D=D_B\oplus D_{\N\setminus B}.
D_CUse in source · 2 passages
Search aliases: D_C
Use in source · 03_spectrum.tex, line 15
[Subgroup and isomorphism classification] The map B↦ D_B is an order isomorphism from P( N) onto Sub(D). Every subgroup is fully invariant under all endomorphisms of D, and D_B≅ D_C Longleftrightarrow B=C. Let H≤ D and let x∈ H. Its support is a finite set F of primes. For p∈ F, choose by the Chinese remainder theorem an integer k that is one modulo p and zero modulo every other prime in F. Then kx is supported only at p, where it is nonzero. It generates the entire C_p coordinate. Thus H contai…
Exact mathematical passage
D_B\cong D_C\quad\Longleftrightarrow\quad B=C.
d_cUse in source · 1 passage
Search aliases: d_c
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
d_eUse in source · 1 passage
Search aliases: d_e
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
d_fUse in source · 3 passages
Search aliases: d_f
Use in source · 04_symmetry.tex, line 36
Put d=a· c. Its identity label is a_v1a_v^-1=1. For a composite, expand all factors in their original order: d_f f^*(d_g) · =a_vc_f[f^*(a_w)]^-1 f^*(a_wc_g[g^*(a_z)]^-1) · =a_vc_f[f^*(a_w)]^-1f^*(a_w)f^*(c_g) [f^*g^*(a_z)]^-1 · =a_vc_g∘ f[(g∘ f)^*(a_z)]^-1 =d_g∘ f. Thus the gauge formula preserves the cocycle condition. For two gauges a,b, substitution gives (a·(b· c))_f =a_vb_vc_f[f^*(b_w)]^-1[f^*(a_w)]^-1 =((ab)· c)_f. The reversed order in the two inverse factors is exactly the formula for th…
Exact mathematical passage
\begin{aligned} d_f f^*(d_g) &=a_vc_f[f^*(a_w)]^{-1} f^*(a_wc_g[g^*(a_z)]^{-1})\\ &=a_vc_f[f^*(a_w)]^{-1}f^*(a_w)f^*(c_g) [f^*g^*(a_z)]^{-1}\\ &=a_vc_{g\circ f}[(g\circ f)^*(a_z)]^{-1} =d_{g\circ f}. \end{aligned}
d_gUse in source · 1 passage
Search aliases: d_g
Use in source · 04_symmetry.tex, line 36
Put d=a· c. Its identity label is a_v1a_v^-1=1. For a composite, expand all factors in their original order: d_f f^*(d_g) · =a_vc_f[f^*(a_w)]^-1 f^*(a_wc_g[g^*(a_z)]^-1) · =a_vc_f[f^*(a_w)]^-1f^*(a_w)f^*(c_g) [f^*g^*(a_z)]^-1 · =a_vc_g∘ f[(g∘ f)^*(a_z)]^-1 =d_g∘ f. Thus the gauge formula preserves the cocycle condition. For two gauges a,b, substitution gives (a·(b· c))_f =a_vb_vc_f[f^*(b_w)]^-1[f^*(a_w)]^-1 =((ab)· c)_f. The reversed order in the two inverse factors is exactly the formula for th…
Exact mathematical passage
\begin{aligned} d_f f^*(d_g) &=a_vc_f[f^*(a_w)]^{-1} f^*(a_wc_g[g^*(a_z)]^{-1})\\ &=a_vc_f[f^*(a_w)]^{-1}f^*(a_w)f^*(c_g) [f^*g^*(a_z)]^{-1}\\ &=a_vc_{g\circ f}[(g\circ f)^*(a_z)]^{-1} =d_{g\circ f}. \end{aligned}
d_{g\circ f}Use in source · 1 passage
Search aliases: d_g∘ f circ
Use in source · 04_symmetry.tex, line 36
Put d=a· c. Its identity label is a_v1a_v^-1=1. For a composite, expand all factors in their original order: d_f f^*(d_g) · =a_vc_f[f^*(a_w)]^-1 f^*(a_wc_g[g^*(a_z)]^-1) · =a_vc_f[f^*(a_w)]^-1f^*(a_w)f^*(c_g) [f^*g^*(a_z)]^-1 · =a_vc_g∘ f[(g∘ f)^*(a_z)]^-1 =d_g∘ f. Thus the gauge formula preserves the cocycle condition. For two gauges a,b, substitution gives (a·(b· c))_f =a_vb_vc_f[f^*(b_w)]^-1[f^*(a_w)]^-1 =((ab)· c)_f. The reversed order in the two inverse factors is exactly the formula for th…
Exact mathematical passage
\begin{aligned} d_f f^*(d_g) &=a_vc_f[f^*(a_w)]^{-1} f^*(a_wc_g[g^*(a_z)]^{-1})\\ &=a_vc_f[f^*(a_w)]^{-1}f^*(a_w)f^*(c_g) [f^*g^*(a_z)]^{-1}\\ &=a_vc_{g\circ f}[(g\circ f)^*(a_z)]^{-1} =d_{g\circ f}. \end{aligned}
D_jUse in source · 1 passage
Search aliases: D_j
Use in source · 03_towers.tex, line 378
If all h_j vanish, the finite telescope gives Q_Nx=x for every N. Conversely, if every Q_Nx=x, then L_jh_j=Q_jx-Q_j+1x=0. Apply the left inverse D_j of L_j to obtain h_j=0. The injectivity of the restoration composite is the step that recovers the original detail from its transported value. sourceKS26
Exact mathematical passage
D_j
d_lUse in source · 1 passage
Search aliases: d_l
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
D_MUse in source · 3 passages
Search aliases: D_M
Use in source · 03_towers.tex, line 368
Idempotence follows from D_NL_N=I: Q_N^2=L_N(D_NL_N)D_N=Q_N. For M≥ N, put C_N,M=C_M-1⋯ C_N and J_N,M=J_N⋯ J_M-1, using identities for M=N. Then D_M=C_N,MD_N, L_M=L_NJ_N,M, C_N,MJ_N,M=I. The identities D_NL_M=J_N,M and D_ML_N=C_N,M now give Q_NQ_M=L_NJ_N,MD_M=Q_M, Q_MQ_N=L_MC_N,MD_N=Q_M. Both orders have been checked, rather than inferred from one range inclusion.
Exact mathematical passage
D_M=C_{N,M}D_N,\quad L_M=L_NJ_{N,M},\quad C_{N,M}J_{N,M}=I.
d_mUse in source · 3 passages
Search aliases: d_m
Use in source · 11_atlas.tex, line 62
Finite quotient representatives. For any target g and positive multiplier m, define the finite vector d_m(g)=∑_pmid mg_p e_p. Then g-d_m(g) has zero coordinates at every prime dividing m. The root theorem provides a witness that it belongs to mG_A. Thus d_m(g) is an exact finite representative of the coset of g.
Exact mathematical passage
d_m(g)=\sum_{p\mid m}g_p e_p.
D_NUse in source · 9 passages
Search aliases: D_N
Use in source · 03_towers.tex, line 351
egintheorem[Finite reconstruction and nested remainders] For every N, x=Q_Nx+∑_j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of C_jJ_j=I gives D_NL_N=I_R_N. This cancellation takes place between adjacent maps with matching domains. No identification of different residual space…
Exact mathematical passage
D_N=C_{N-1}\cdots C_0:R_0\to R_N
d_nUse in source · 1 passage
Search aliases: d_n
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
D_{\N\setminus B}Use in source · 1 passage
Search aliases: D_N∖ B D_ N∖ B N setminus mathbb setminus
Use in source · 03_spectrum.tex, line 25
e factors. Every element of H lies in D_B, and every finite sum of the factors in B lies in H. Hence H=D_B. Uniqueness follows by testing membership of e_p_i. Proposition~[source reference] proves full invariance. Finally σ_p_i(D_B)=1 exactly when i∈ B, and these values are isomorphism invariants. The lattice operations have literal set-theoretic descriptions: D_B+D_C=D_B∪ C, D_B∩ D_C=D_B∩ C, D=D_B⊕ D_ N∖ B. The infinite support choice is therefore already visible in an elementary algebraic latt…
Exact mathematical passage
D_B+D_C=D_{B\cup C},\qquad D_B\cap D_C=D_{B\cap C}, \qquad D=D_B\oplus D_{\N\setminus B}.
d_pUse in source · 3 passages
Search aliases: d_p
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
d_rUse in source · 1 passage
Search aliases: d_r
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
D_sUse in source · 1 passage
Search aliases: D_s
Use in source · 05_geometry.tex, line 580
-t · 1. The common right inverse r↦(r,0) supplies gain at most one for every t, and normD_s-D_t=|s-t|. The general theorem gives normq_s-q_t≤|s-t|.
Exact mathematical passage
\norm{D_s-D_t}=|s-t|
D_tUse in source · 6 passages
Search aliases: D_t
Use in source · 05_geometry.tex, line 575
An all-real worked model. Let D_t(x,y)=x+ty. Its kernel is the line {(-tb,b):b∈R}, and its orthogonal projector is q_t=frac11+t^2 t^2 · -t
Exact mathematical passage
D_t(x,y)=x+ty
d_tUse in source · 1 passage
Search aliases: d_t
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
D_\perpUse in source · 1 passage
Search aliases: D_⊥ perp
Use in source · 05_geometry.tex, line 135
such that every x has some exact y with normx-y≤ CnormDx. When the range is closed, the least admissible constant is the operator norm of the inverse of D:K^⊥→ran D. In the zero operator case that inverse has zero dimensional domain and its norm is zero. Let K=ker D and K^⊥ carry the induced Hilbert norm. Define D_⊥ u=Du, but regard its target as ran D with the norm inherited from R. It is injective: Du=0 puts u in both K and K^⊥, so u=0. It is surjective onto that target: if r=Dx, then u=x-qx∈ …
Exact mathematical passage
D_\perp u=Du
D_\perp^{-1}Use in source · 1 passage
Search aliases: D_⊥^-1 perp
Use in source · 05_geometry.tex, line 137
Suppose first that the range is closed. Because R is Banach, ran D is then Banach. The domain K^⊥ is also complete. The bounded inverse theorem, proved in Appendix~[source reference], gives a bounded inverse A=D_⊥^-1. Applying it to Dx=D(x-qx) yields x-qx=A(Dx), normx-qx≤norm AnormDx. Hence condition 2 holds. If any C≥0 satisfies condition 2, then for every r∈ran D apply it to u=Ar∈ K^⊥. Since qu=0, it gives normAr≤ Cnorm r. Taking the operator norm proves norm A≤ C. This establishes the least c…
Exact mathematical passage
A=D_\perp^{-1}
\degUse in source · 3 passages
Search aliases: deg
Use in source · 08_computability.tex, line 17
[Uniform construction relative to the selector] If A≤_T X, then the normal-form construction gives an X-computable copy of G_A. In its canonical normal-form presentation, the group operation has Turing degree exactly deg_T(A). To add (r,d) and (t,e), compute the prime divisors of den(r)den(t). At each of these finitely many primes, query the corresponding selector bit and evaluate the three section residues. Lemma~[source reference] guarantees that every other correction coordinate is zero. Form…
Exact mathematical passage
\deg_T(A)
\deg_TUse in source · 3 passages
Search aliases: deg_T deg
Use in source · 08_computability.tex, line 17
[Uniform construction relative to the selector] If A≤_T X, then the normal-form construction gives an X-computable copy of G_A. In its canonical normal-form presentation, the group operation has Turing degree exactly deg_T(A). To add (r,d) and (t,e), compute the prime divisors of den(r)den(t). At each of these finitely many primes, query the corresponding selector bit and evaluate the three section residues. Lemma~[source reference] guarantees that every other correction coordinate is zero. Form…
Exact mathematical passage
\deg_T(A)
\denUse in source · 3 passages
Search aliases: den operatorname
Use in source · 05_construction.tex, line 67
[A finite support bound] For all r,t∈ Q, supp c_A(r,t)⊆ {p:pmidden(r)den(t)}. At a prime not dividing either denominator, all three rationals have well-defined reductions. On a selected coordinate, reduction respects addition, so the defect is zero. On an unselected coordinate, all section values are zero. The reduced denominator of r+t has no prime divisor absent from the two original denominators.
Exact mathematical passage
\supp c_A(r,t)\subseteq \{p:p\mid\den(r)\den(t)\}.
\operatorname{diag}Use in source · 5 passages
Search aliases: diag operatorname
Use in source · 03_towers.tex, line 280
[Repair of the identity family] The bounded candidate f_n=I is not exact. Its repair is (Pf)_n=diag(2^-n,2^-(n-1),…,2^-1,1). The new last output row is retained at every stage while inherited rows are propagated by the factor 1/2. At level zero the value is one. Exactness forces every inherited row at the next level to be one half of its previous value. Preservation of the innovation forces the new row to remain the corresponding identity row. Induction gives the displayed diagonal. sourceKS22 F…
Exact mathematical passage
(Pf)_n=\operatorname{diag}(2^{-n},2^{-(n-1)},\ldots,2^{-1},1).
\dimUse in source · 2 passages
Search aliases: dim
Use in source · 06_boundaries.tex, line 88
is y with Ay=0 and Py=Px. The full answer set is y+H, where H=ker A∩ker P. item There are row vectors λ∈K^1× m and μ∈K^1× p with λ A=μ P, λ Ax=1. The negative witness annihilates every admissible correction defect, since λ AN=μ PN=0. The ambiguity dimension in the positive case is dim H=n-rankA P. Set T=A
Exact mathematical passage
\dim H=n-\operatorname{rank}\begin{pmatrix}A\\P\end{pmatrix}.
\dim_{\Q}Use in source · 1 passage
Search aliases: dim_Q dim_ Q dim Q dim mathbb
Use in source · 04_purity.tex, line 106
A splitting would embed the nonzero divisible group W into P. Theorem~[source reference] excludes every such subgroup. The rank of the torsion-free quotient is exactly dim_ QW. This gives countable examples of every finite rank and of countably infinite rank with the same finite-quotient system. The rank-one family studied next is chosen because it admits an especially sharp classification and computability theorem.
Exact mathematical passage
\dim_{\Q}W
\distUse in source · 22 passages
Search aliases: dist
Use in source · 05_geometry.tex, line 163
A small residual far from exactness. On ℓ^2(N), index the standard basis from one and define De_n=n^-1e_n. The map is bounded and injective, so its exact space is zero. Yet norme_n=1, normDe_n=1/n⟶0. No finite C can bound dist(e_n,ker D) by CnormDe_n. The range is not closed. For example finite truncations of (1/n)_n≥1 belong to the range and converge in ℓ^2, but a preimage of the limit would be the constant sequence, which does not belong to ℓ^2.
Exact mathematical passage
\dist(e_n,\ker D)
\downarrowUse in source · 5 passages
Search aliases: downarrow
Use in source · 05g_budget_certificates.tex, line 54
C and b-Ta∈ C, every correction satisfies normr-Ta^2=q^2+normb-Ta^2. Thus β(M)≥ q. Given ϵ>0, range density in C supplies one correction a with normb-Ta<ϵ. For every M≥norm a it follows that q≤β(M)≤sqrtq^2+ϵ^2. First letting the budget increase and then letting ϵdownarrow0 proves the limit.
Exact mathematical passage
\epsilon\downarrow0
E^2Use in source · 1 passage
Search aliases: E^2
Use in source · 05e_escape.tex, line 7
The complete spaces and the preservation rule. Index sequences by the positive integers. Set E=ℓ^2(N_>0;R)⊕R, R=ℓ^2(N_>0;R), norm(u,t)_E^2=∑_n≥1|u_n|^2+|t|^2. Let w=(1/n)_n≥1. It belongs to ℓ^2, because ∑_n≥1frac1n^2 ≤1+∑_n≥2frac1n(n-1)=2. Define the defect and protected observable by d(u,t)_n=fracu_n+tn, p(u,t)=t. The diagonal map T:u↦(u_n/n)_n has operator norm one. Since d(u,t)=Tu+tw, normd(u,t)≤norm u+|t|norm w ≤sqrt1+norm w^2 norm(u,t)_E. Thus both d and p are continuous linear maps between…
Exact mathematical passage
E=\ell^2(\N_{>0};\R)\oplus\R,\qquad R=\ell^2(\N_{>0};\R),\qquad \norm{(u,t)}_E^2=\sum_{n\ge1}|u_n|^2+|t|^2.
e^2Use in source · 3 passages
Search aliases: e^2
Use in source · 05j_inexact_certificates.tex, line 41
0≤ J_α(a)-J_α(a_α) · ≤fracnorm g^2α. Let e=a-a_α and B=T^*T+α I. The normal equation of the constructed optimizer gives Be=(T^*T+α I)a-T^*r=g. Pairing with e yields the exact identity normTe^2+αnorm e^2=⟨ g,e⟩ ≤norm gnorm e. If e=0, all three conclusions hold with zero error. Otherwise drop the first nonnegative term and divide by αnorm e>0 to obtain norm e≤norm g/α.
Exact mathematical passage
\label{eq:solver-energy} \norm{Te}^2+\alpha\norm e^2=\langle g,e\rangle \le\norm g\norm e.
e_1Use in source · 11 passages
Search aliases: e_1
Use in source · 04b_phase.tex, line 32
Every expression a(x)+a(y)-a(x+y) is symmetric in x,y, because G is abelian. With e_1=(1,0) and e_2=(0,1), however, c(e_1,e_2)=0, c(e_2,e_1)=1. This contradicts the symmetry required by [source reference]. sourceKS80 The proof rules out all sixteen possible scalar functions a, not just linear functions or selected trial gauges. The source codeProjectiveExample.lean proves the normalization, cocycle, asymmetry, and no-scalar-flattening statements. The matrix interpretation and group construction …
Exact mathematical passage
e_1=(1,0)
e_2Use in source · 18 passages
Search aliases: e_2
Use in source · 04b_phase.tex, line 32
Every expression a(x)+a(y)-a(x+y) is symmetric in x,y, because G is abelian. With e_1=(1,0) and e_2=(0,1), however, c(e_1,e_2)=0, c(e_2,e_1)=1. This contradicts the symmetry required by [source reference]. sourceKS80 The proof rules out all sixteen possible scalar functions a, not just linear functions or selected trial gauges. The source codeProjectiveExample.lean proves the normalization, cocycle, asymmetry, and no-scalar-flattening statements. The matrix interpretation and group construction …
Exact mathematical passage
e_2=(0,1)
e_3Use in source · 9 passages
Search aliases: e_3
Use in source · 07_classification.tex, line 52
[The same group with a different marked point] Let A=∅ and B={0}. Only the first prime pair changes. Then G_A=G_B inside P, but u_A-u_B=e_2-e_3. The distinguished element is not preserved. In particular, the statement G_A≅ G_Biff A=B would be false for unmarked groups.
Exact mathematical passage
u_A-u_B=e_2-e_3
e_7Use in source · 1 passage
Search aliases: e_7
Use in source · 11_atlas.tex, line 96
A finite tail exception. Consider an infinite-order element x=s_A(5/6)+2e_7. An exception set for its signature is {2,3,5,7}. Outside these primes, the denominator is invertible, the numerator is nonzero, and the torsion correction vanishes. Thus the root pattern of x agrees with that of u_A in every later prime pair.
Exact mathematical passage
x=s_A(5/6)+2e_7.
E_AUse in source · 4 passages
Search aliases: E_A
Use in source · 07_classification.tex, line 97
[Endomorphism ring] Under coordinate multiplication, End(G_A)≅ E_A={λ∈ P:∃ r∈ Q, λ_p=overline r_p for almost every p∈ S_A}. The rational r is unique. It is the scalar induced on G_A/D, and the map E_A→ Q sending λ to r is a surjective ring homomorphism. Apply Theorem~[source reference] with B=A. At unselected coordinates, both λ u_A and s_A(r) vanish. At selected coordinates the finite-support condition is exactly the displayed eventual equality.
Exact mathematical passage
\End(G_A)\cong E_A=\{\lambda\in P:\exists r\in\Q,\ \lambda_p=\overline r_p\text{ for almost every }p\in S_A\}.
e_bUse in source · 1 passage
Search aliases: e_b
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
e_cUse in source · 1 passage
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Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
e_eUse in source · 1 passage
Search aliases: e_e
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
e_fUse in source · 1 passage
Search aliases: e_f
Use in source · 05h_regularized_repair.tex, line 197
[3pt] q=0 · No · Residual vanishes, correction norms diverge, no exact protected answer. [3pt] q>0 · Yes · Strong convergence to the least-squares correction of minimum norm, not an exact answer. [3pt] q>0 · No · A positive residual floor and diverging correction norms occur together. The last case is not hypothetical. Let Tu=(u_n/n) on ℓ^2, define Su=(Tu,0) into ℓ^2⊕R, and take r=((1/n),1). Then b=((1/n),0) is in the closure but not the range, while q=1. The same escaping corrections as before …
Exact mathematical passage
3pt] $q=0$ & No & Residual vanishes, correction norms diverge, no exact protected answer.\\[3pt] $q>0$ & Yes & Strong convergence to the least-squares correction of minimum norm, not an exact answer.\\[3pt] $q>0$ & No & A positive residual floor and diverging correction norms occur together.\\\bottomrule \end{tabularx} \end{center} The last case is not hypothetical. Let $Tu=(u_n/n)$ on $\ell^2$, define $Su=(Tu,0)$ into $\ell^2\oplus\R$, and take $r=((1/n),1)$. Then $b=((1/n),0)$ is in the closure but not the range, while $q=1$. The same escaping corrections as before coexist with an irreducible orthogonal residual. Positive floor and unbounded correction cost are independent features. \subsection{Noise is controlled at a fixed parameter} \fixedplate{34_noise_filter}{For one singular value $\sigma$, the response to residual-data noise is $\sigma/(\sigma^2+\alpha)$. Its maximum occurs at $\sigma=\sqrt\alpha$ and equals $1/(2\sqrt\alpha)$. The plot evaluates this formula and the universal horizontal bounds at three explicit positive parameters.}{fig:noise-filter} \begin{theorem}[Data sensitivity and a convergent parameter choice]\label{WR09} For two residual data vectors $r,\widetilde r$ with $\norm{r-\widetilde r}\le\delta$ and the same $T,\alpha$, \[ \norm{a_\alpha(r)-a_\alpha(\widetilde r)} \le\frac{\delta}{2\sqrt\alpha},\qquad \norm{T(a_\alpha(r)-a_\alpha(\widetilde r))}\le\delta.
e_gUse in source · 1 passage
Search aliases: e_g
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
e_iUse in source · 3 passages
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Use in source · 05_geometry.tex, line 299
Flatness does not remove amplification. Consider a scalar chain with n+1 vertices, where n≥1, and coordinates e_i(z)=2^iz. Set τ_ij=e_ie_j^-1. This connection is flat, but reverse transport along one edge multiplies norms by two. Let x_i=8·2^i-1, y_i=7·2^i. The family y is exact and preserves x_o=7. On a forward edge the residual of x is 1/2, and on its reverse it is -1. Thus normdx_∞=1, normx-Px_∞=2^n-1. For n≥2, a route length bound of n fails. This is amplification with trivial holonomy, dist…
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e_i(z)=2^iz
e_jUse in source · 4 passages
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Use in source · 05f_bounded_realization.tex, line 32
, every term is zero and the conclusion is immediate. Otherwise let H_0 be the closed linear span of the countably many vectors a_n. Finite rational linear combinations of those vectors form a countable dense set, so H_0 is separable. Apply Gram--Schmidt to a countable dense sequence, omitting zero remainders. This gives either a finite orthonormal basis or a countable orthonormal sequence (e_j) whose closed span is H_0. The ambient Hilbert space itself need not be separable.
Exact mathematical passage
(e_j)
e_j^{-1}Use in source · 1 passage
Search aliases: e_j^-1
Use in source · 05_geometry.tex, line 299
Flatness does not remove amplification. Consider a scalar chain with n+1 vertices, where n≥1, and coordinates e_i(z)=2^iz. Set τ_ij=e_ie_j^-1. This connection is flat, but reverse transport along one edge multiplies norms by two. Let x_i=8·2^i-1, y_i=7·2^i. The family y is exact and preserves x_o=7. On a forward edge the residual of x is 1/2, and on its reverse it is -1. Thus normdx_∞=1, normx-Px_∞=2^n-1. For n≥2, a route length bound of n fails. This is amplification with trivial holonomy, dist…
Exact mathematical passage
\tau_{ij}=e_ie_j^{-1}
\mathcal E_{Kb}Use in source · 1 passage
Search aliases: E_Kb mathcal
Use in source · 04_symmetry.tex, line 420
[Invariance of the marked realization problem] The map (x,ϕ)⟼(Ex,K(ϕ)σ_x^-1), a⟼ E(a) defines an equivalence E_b(R)≃ E_Kb(R'). For an object (x,ϕ) put β_x=K(ϕ)σ_x^-1:R'Ex→ Kb. This is an isomorphism of the required type. If a:(x,ϕ)→(y,ψ), naturality of σ says R'E(a)σ_x=σ_yKR(a), σ_y^-1R'E(a)=KR(a)σ_x^-1. Consequently β_yR'E(a) =K(ψ)KR(a)σ_x^-1 =K(ψ R(a))σ_x^-1 =K(ϕ)σ_x^-1=β_x. So E(a) is a marked arrow. Identity and composition are preserved because the underlying arrow map is that of the functo…
Exact mathematical passage
\mathcal E_b(R)\simeq\mathcal E_{Kb}(R')
e_lUse in source · 1 passage
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Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
e_mUse in source · 2 passages
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Use in source · 05f_bounded_realization.tex, line 91
Existence does not make an arbitrary sequence converge strongly. Take H=Y=ℓ^2 and Te_n=e_n/n. The vectors a_n=e_n have norm one and satisfy Ta_n→0. An exact answer exists, namely zero. Yet norme_n-e_m=sqrt2 for n≠ m, so no subsequence converges in norm.
Exact mathematical passage
\norm{e_n-e_m}=\sqrt2
e_nUse in source · 9 passages
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Use in source · 05_geometry.tex, line 159
A small residual far from exactness. On ℓ^2(N), index the standard basis from one and define De_n=n^-1e_n. The map is bounded and injective, so its exact space is zero. Yet norme_n=1, normDe_n=1/n⟶0. No finite C can bound dist(e_n,ker D) by CnormDe_n. The range is not closed. For example finite truncations of (1/n)_n≥1 belong to the range and converge in ℓ^2, but a preimage of the limit would be the constant sequence, which does not belong to ℓ^2.
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De_n=n^{-1}e_n
e_oUse in source · 2 passages
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Use in source · 05_geometry.tex, line 280
For necessity, suppose a constant C satisfies dist(w,ker d)≤ Cnormdw_∞ for every vertex family w. Fix z∈ K^⊥ and take w=R_pz. Every exact family e has e_o∈ K, and root evaluation has norm at most one. Therefore norm z=dist(z,K)≤normz-e_o≤normR_pz-e_∞. Taking the infimum over e∈ker d is legitimate without assuming that this global nearest point is attained. It gives norm z≤ CnormdR_pz_∞ ≤ CnormN_p^-1norm Hz_∞. Here N_p is a continuous linear equivalence on the finite edge product because every co…
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e_o\in K
e_{p_i}Use in source · 2 passages
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Use in source · 03_spectrum.tex, line 22
Let B be the set of indices of these factors. Every element of H lies in D_B, and every finite sum of the factors in B lies in H. Hence H=D_B. Uniqueness follows by testing membership of e_p_i. Proposition~[source reference] proves full invariance. Finally σ_p_i(D_B)=1 exactly when i∈ B, and these values are isomorphism invariants. The lattice operations have literal set-theoretic descriptions: D_B+D_C=D_B∪ C, D_B∩ D_C=D_B∩ C, D=D_B⊕ D_ N∖ B. The infinite support choice is therefore already visi…
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e_{p_i}
e_rUse in source · 1 passage
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Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
e_sUse in source · 1 passage
Search aliases: e_s
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
\endheadUse in source · 1 passage
Search aliases: endhead
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
\operatorname{ev}_kUse in source · 2 passages
Search aliases: ev_k operatorname
Use in source · 03a_observation.tex, line 68
When global behavior determines the finite presentation. Say that X has surjective point projections when ∀ k ev_k:Point(X)→ X_k is onto. This means that every finite input state really occurs in some coherent global input. It is stronger than nonemptiness of the global space.
Exact mathematical passage
\forall k\quad \operatorname{ev}_k:\operatorname{Point}(X)\to X_k \quad\text{is onto}.
\operatorname{ev}_wUse in source · 1 passage
Search aliases: ev_w operatorname
Use in source · 01_views.tex, line 31
The evaluations jointly determine a realization. If a set T carries maps c_v:T→ F_v satisfying f^*c_w=c_v, there is a unique map c:T→Γ(F) satisfying ev_v c=c_v for every view. We first verify that the evaluation maps themselves form a compatible family. If x∈Γ(F) and f:v→ w, the defining equation for x gives f^*(ev_w x)=f^*(x_w)=x_v=ev_v x. If x,y∈Γ(F) have equal evaluations, then x_v=y_v for every v. Equality of functions into a dependent product gives equality of their underlying families. Mem…
Exact mathematical passage
f^*(\operatorname{ev}_w x)=f^*(x_w)=x_v=\operatorname{ev}_v x.
\Ext_{U'}^{H^*F}Use in source · 1 passage
Search aliases: Ext_U'^H^*F Ext
Use in source · 04_symmetry.tex, line 343
[Equivalence of section spaces] Pullback induces Γ(F)≃Γ(H^*F). If U⊆ObW is closed under isomorphism and U'=H^-1U, it also induces Ext_U^F(s)≃Ext_U'^H^*F(H^*s). Existence and uniqueness of the prescribed extension are preserved and reflected. Write the unit of the equivalence as η:I_V⇒ LH, and take the unit and counit with their triangle identities. Set t_w=ε_w^-1:w→ HLw. Given y∈Γ(H^*F), define (Ry)_w=F(t_w)y_Lw. For an arrow f:w→ w', naturality of the inverse counit states HL(f) t_w=t_w'f. Cont…
Exact mathematical passage
\Ext_U^F(s)\simeq\Ext_{U'}^{H^*F}(H^*s).
\Ext_U^FUse in source · 1 passage
Search aliases: Ext_U^F Ext
Use in source · 04_symmetry.tex, line 343
[Equivalence of section spaces] Pullback induces Γ(F)≃Γ(H^*F). If U⊆ObW is closed under isomorphism and U'=H^-1U, it also induces Ext_U^F(s)≃Ext_U'^H^*F(H^*s). Existence and uniqueness of the prescribed extension are preserved and reflected. Write the unit of the equivalence as η:I_V⇒ LH, and take the unit and counit with their triangle identities. Set t_w=ε_w^-1:w→ HLw. Given y∈Γ(H^*F), define (Ry)_w=F(t_w)y_Lw. For an arrow f:w→ w', naturality of the inverse counit states HL(f) t_w=t_w'f. Cont…
Exact mathematical passage
\Ext_U^F(s)\simeq\Ext_{U'}^{H^*F}(H^*s).
\Ext_WUse in source · 2 passages
Search aliases: Ext_W Ext
Use in source · 02_obstructions.tex, line 145
Extension in stages. For U⊆ W⊆ObV, an extension first to W and then globally must retain the intermediate choice. There is an explicit equivalence Ext_U(s)≃ coprod_substackt∈Γ_W(A) t|_U=sExt_W(t). It sends a global extension to its restriction to W, together with itself as an extension of that restriction. The inverse forgets the intermediate coordinate. These operations are inverse because the required intermediate coordinate of a global section is its actual restriction. A coherent intermediat…
Exact mathematical passage
\label{eq:stages} \Ext_U(s)\simeq \coprod_{\substack{t\in\Gamma_W(A)\\t|_U=s}}\Ext_W(t).
\mathbb FUse in source · 3 passages
Search aliases: F mathbb
Use in source · 02a_prescription.tex, line 104
F_2 · Only zero · Two
Exact mathematical passage
\mathbb F_2
\mathbf FUse in source · 7 passages
Search aliases: F mathbf
Use in source · 04b_phase.tex, line 7
A two-bit phase law. Write F_2=Z/2Z additively, and let G= F_2^2, x=(x_1,x_2), c(x,y)=x_2y_1. All entries and operations in this subsection lie in F_2. In particular minus and plus agree, although retaining a minus sign in the coboundary formula helps explain its origin.
Exact mathematical passage
\mathbf F_2=\Z/2\Z
\mathcal FUse in source · 3 passages
Search aliases: F mathcal
Use in source · 03_towers.tex, line 386
Convergence has its own theorem. Call x norm recoverable when Q_Nx→0. This is equivalent to S_Nx→ x, but it does not follow merely from vanishing persistent subspace. Let F=⋃_Nker Q_N be the finite core.
Exact mathematical passage
\mathcal F=\bigcup_N\ker Q_N
f'Use in source · 1 passage
Search aliases: f'
Use in source · 03a_observation.tex, line 36
We check compatibility with composition with all indices displayed. Write f,f' with schedules ϕ,ϕ' and g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applying the naturality equations and then the two witness equations gives g_nf_ψ(n)ρ^X_ϕ(ψ(n)),k · =g_nρ^Y_ψ(n),jf_jρ^X_ϕ(j),k
Exact mathematical passage
f,f'
f'_jUse in source · 2 passages
Search aliases: f'_j
Use in source · 03a_observation.tex, line 41
We check compatibility with composition with all indices displayed. Write f,f' with schedules ϕ,ϕ' and g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applying the naturality equations and then the two witness equations gives g_nf_ψ(n)ρ^X_ϕ(ψ(n)),k · =g_nρ^Y_ψ(n),jf_jρ^X_ϕ(j),k
Exact mathematical passage
f_j\rho^X_{\phi(j),k}=f'_j\rho^X_{\phi'(j),k}.
f'_{\psi'(n)}Use in source · 1 passage
Search aliases: f'_ψ'(n) psi
Use in source · 03a_observation.tex, line 45
nd g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applying the naturality equations and then the two witness equations gives g_nf_ψ(n)ρ^X_ϕ(ψ(n)),k · =g_nρ^Y_ψ(n),jf_jρ^X_ϕ(j),k · =g'_nρ^Y_ψ'(n),jf_jρ^X_ϕ(j),k · =g'_nρ^Y_ψ'(n),jf'_jρ^X_ϕ'(j),k · =g'_nf'_ψ'(n)ρ^X_ϕ'(ψ'(n)),k. This is the required equality o…
Exact mathematical passage
\begin{aligned} g_nf_{\psi(n)}\rho^X_{\phi(\psi(n)),k} &=g_n\rho^Y_{\psi(n),j}f_j\rho^X_{\phi(j),k}\\ &=g'_n\rho^Y_{\psi'(n),j}f_j\rho^X_{\phi(j),k}\\ &=g'_n\rho^Y_{\psi'(n),j}f'_j\rho^X_{\phi'(j),k}\\ &=g'_nf'_{\psi'(n)}\rho^X_{\phi'(\psi'(n)),k}. \end{aligned}
f^{(0)}Use in source · 1 passage
Search aliases: f^(0)
Use in source · 03_towers.tex, line 110
[Constructed geometric repair] Starting from f^(0)=f, define f^(m+1)=f^(m)-A(df^(m)). There is a bounded linear projection P_A:E→ker d with f^(m)⟶ P_Af, p(P_Af)=p(f), and normf^(m)-P_Af ≤fracnorm A1-qnormdf^(m). It fixes every exact candidate and satisfies normP_A≤1+fracnorm Anorm d1-q. Put r_m=df^(m). Additivity of d gives r_m+1=r_m-dAr_m. Each r_m is the residual of an actual candidate, so the stated hypothesis applies at every iteration. Induction yields, for j≥0, normr_m+j≤ q^jnormr_m, normf…
Exact mathematical passage
f^{(0)}=f
f^{(m+1)}Use in source · 2 passages
Search aliases: f^(m+1)
Use in source · 03_towers.tex, line 111
[Constructed geometric repair] Starting from f^(0)=f, define f^(m+1)=f^(m)-A(df^(m)). There is a bounded linear projection P_A:E→ker d with f^(m)⟶ P_Af, p(P_Af)=p(f), and normf^(m)-P_Af ≤fracnorm A1-qnormdf^(m). It fixes every exact candidate and satisfies normP_A≤1+fracnorm Anorm d1-q. Put r_m=df^(m). Additivity of d gives r_m+1=r_m-dAr_m. Each r_m is the residual of an actual candidate, so the stated hypothesis applies at every iteration. Induction yields, for j≥0, normr_m+j≤ q^jnormr_m, normf…
Exact mathematical passage
f^{(m+1)}=f^{(m)}-A(df^{(m)}).
f^{(m+j)}Use in source · 1 passage
Search aliases: f^(m+j)
Use in source · 03_towers.tex, line 134
nd normf^(m)-P_Af ≤fracnorm A1-qnormdf^(m). It fixes every exact candidate and satisfies normP_A≤1+fracnorm Anorm d1-q. Put r_m=df^(m). Additivity of d gives r_m+1=r_m-dAr_m. Each r_m is the residual of an actual candidate, so the stated hypothesis applies at every iteration. Induction yields, for j≥0, normr_m+j≤ q^jnormr_m, normf^(m+j+1)-f^(m+j) ≤norm A q^jnormr_m. For n>m, the finite triangle inequality therefore gives normf^(n)-f^(m) ≤norm A normr_m∑_j=0^n-m-1q^j ≤fracnorm A1-qnormr_m. Since …
Exact mathematical passage
\norm{r_{m+j}}\le q^j\norm{r_m},\qquad \norm{f^{(m+j+1)}-f^{(m+j)}} \le\norm A\,q^j\norm{r_m}.
f^{(m+j+1)}Use in source · 1 passage
Search aliases: f^(m+j+1)
Use in source · 03_towers.tex, line 134
nd normf^(m)-P_Af ≤fracnorm A1-qnormdf^(m). It fixes every exact candidate and satisfies normP_A≤1+fracnorm Anorm d1-q. Put r_m=df^(m). Additivity of d gives r_m+1=r_m-dAr_m. Each r_m is the residual of an actual candidate, so the stated hypothesis applies at every iteration. Induction yields, for j≥0, normr_m+j≤ q^jnormr_m, normf^(m+j+1)-f^(m+j) ≤norm A q^jnormr_m. For n>m, the finite triangle inequality therefore gives normf^(n)-f^(m) ≤norm A normr_m∑_j=0^n-m-1q^j ≤fracnorm A1-qnormr_m. Since …
Exact mathematical passage
\norm{r_{m+j}}\le q^j\norm{r_m},\qquad \norm{f^{(m+j+1)}-f^{(m+j)}} \le\norm A\,q^j\norm{r_m}.
f^{(n)}Use in source · 1 passage
Search aliases: f^(n)
Use in source · 03_towers.tex, line 140
Put r_m=df^(m). Additivity of d gives r_m+1=r_m-dAr_m. Each r_m is the residual of an actual candidate, so the stated hypothesis applies at every iteration. Induction yields, for j≥0, normr_m+j≤ q^jnormr_m, normf^(m+j+1)-f^(m+j) ≤norm A q^jnormr_m. For n>m, the finite triangle inequality therefore gives normf^(n)-f^(m) ≤norm A normr_m∑_j=0^n-m-1q^j ≤fracnorm A1-qnormr_m. Since normr_m≤ q^mnormdf tends to zero, this is a Cauchy estimate. Completeness of E supplies a limit, denoted P_Af. Passing n…
Exact mathematical passage
\norm{f^{(n)}-f^{(m)}} \le\norm A\,\norm{r_m}\sum_{j=0}^{n-m-1}q^j \le\frac{\norm A}{1-q}\norm{r_m}.
f_0Use in source · 23 passages
Search aliases: f_0
Use in source · 03_towers.tex, line 184
[Split tower correction] The recurrence defines a continuous linear map A:R→ E with ∂ A=I_R, norm A≤ K. The operator P=I-A∂ is an exact projection and (Pf)_0=f_0, normf-Pf_∞≤ Knorm∂ f_∞. Let θ=SB_X<1. The number K=S/(1-θ) is nonnegative and satisfies S+SB_XK=K. Fix r∈ R and define its coordinates recursively as in [source reference]. The initial coordinate has norm zero. If norm(Ar)_n≤ Knorm r_∞, submultiplicativity of operator norms gives norm(Ar)_n+1 · ≤norms_n(normr_n+norm(Ar)_nnormρ^X_n)
Exact mathematical passage
(Pf)_0=f_0,\qquad \norm{f-Pf}_\infty\le K\norm{\partial f}_\infty.
F_1Use in source · 3 passages
Search aliases: F_1
Use in source · 12_proof_tools.tex, line 105
[Baire's complete-space principle] A nonempty complete metric space cannot be the union of countably many closed sets with empty interior. Suppose F_1,F_2,… are closed with empty interior. Choose a nonempty open ball whose closed ball B_1, of positive radius at most 1/2, avoids F_1. Such a choice is possible because the complement of F_1 is open and meets every nonempty open set. Recursively choose a closed ball B_n+1 of positive radius at most 2^-(n+1), contained in the open ball with the same …
Exact mathematical passage
F_1,F_2,\ldots
F_2Use in source · 1 passage
Search aliases: F_2
Use in source · 12_proof_tools.tex, line 105
[Baire's complete-space principle] A nonempty complete metric space cannot be the union of countably many closed sets with empty interior. Suppose F_1,F_2,… are closed with empty interior. Choose a nonempty open ball whose closed ball B_1, of positive radius at most 1/2, avoids F_1. Such a choice is possible because the complement of F_1 is open and meets every nonempty open set. Recursively choose a closed ball B_n+1 of positive radius at most 2^-(n+1), contained in the open ball with the same …
Exact mathematical passage
F_1,F_2,\ldots
\mathbb F_2Use in source · 2 passages
Search aliases: F_2 mathbb
Use in source · 02a_prescription.tex, line 104
F_2 · Only zero · Two
Exact mathematical passage
\mathbb F_2
\mathbf F_2Use in source · 5 passages
Search aliases: F_2 mathbf
Use in source · 04b_phase.tex, line 7
A two-bit phase law. Write F_2=Z/2Z additively, and let G= F_2^2, x=(x_1,x_2), c(x,y)=x_2y_1. All entries and operations in this subsection lie in F_2. In particular minus and plus agree, although retaining a minus sign in the coboundary formula helps explain its origin.
Exact mathematical passage
\mathbf F_2=\Z/2\Z
\mathbf F_2^2Use in source · 2 passages
Search aliases: F_2^2 mathbf
Use in source · 04b_phase.tex, line 8
A two-bit phase law. Write F_2=Z/2Z additively, and let G= F_2^2, x=(x_1,x_2), c(x,y)=x_2y_1. All entries and operations in this subsection lie in F_2. In particular minus and plus agree, although retaining a minus sign in the coboundary formula helps explain its origin.
Exact mathematical passage
G=\mathbf F_2^2,\qquad x=(x_1,x_2),\qquad c(x,y)=x_2y_1.
\F_3Use in source · 1 passage
Search aliases: F_3 F mathbb
Use in source · 11_atlas.tex, line 58
A torsion correction that cannot be discarded. Take the purely torsion target e_3 and multiplier 2. The root criterion succeeds because the target has zero 2-coordinate. Its rational part is zero, but the answer is not the zero element: 2(0,2e_3)=(0,e_3). In F_3, multiplication by 2 sends 2 to 1. This is the simplest example showing why the finite correction in the root algorithm is necessary even when the rational coordinate causes no difficulty.
Exact mathematical passage
\F_3
\mathbb F_3Use in source · 1 passage
Search aliases: F_3 mathbb
Use in source · 02a_prescription.tex, line 105
F_3 · Every element · One
Exact mathematical passage
\mathbb F_3
\F_5Use in source · 1 passage
Search aliases: F_5 F mathbb
Use in source · 11_atlas.tex, line 20
bottomrule Each residue is exact. At p=2, the lift of 1/2 is set to zero because the denominator is not invertible. At p=5, its value is 3 because 2·3=1 in F_5.
Exact mathematical passage
\F_5
f_aUse in source · 2 passages
Search aliases: f_a
Use in source · 03b_repair_transport.tex, line 148
a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a,b. The dyadic coarsener in Section~[source reference] is the case b=a+2.
Exact mathematical passage
C_{a,b}(\partial f)=\rho^Y_{a,b}f_b-f_a\rho^X_{a,b}.
f_bUse in source · 3 passages
Search aliases: f_b
Use in source · 03b_repair_transport.tex, line 148
a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a,b. The dyadic coarsener in Section~[source reference] is the case b=a+2.
Exact mathematical passage
C_{a,b}(\partial f)=\rho^Y_{a,b}f_b-f_a\rho^X_{a,b}.
f_F^*Use in source · 2 passages
Search aliases: f_F^*
Use in source · 01_views.tex, line 55
Maps of realizations. Let α:F→ G consist of maps α_v:F_v→ G_v with α_v f_F^*=f_G^*α_w. The induced map on realizations is Γ(α)(x)_v=α_v(x_v). The naturality square is exactly what is needed to show that the image family is coherent.
Exact mathematical passage
\alpha_v f_F^*=f_G^*\alpha_w.
f_G^*Use in source · 1 passage
Search aliases: f_G^*
Use in source · 01_views.tex, line 55
Maps of realizations. Let α:F→ G consist of maps α_v:F_v→ G_v with α_v f_F^*=f_G^*α_w. The induced map on realizations is Γ(α)(x)_v=α_v(x_v). The naturality square is exactly what is needed to show that the image family is coherent.
Exact mathematical passage
\alpha_v f_F^*=f_G^*\alpha_w.
f_kUse in source · 4 passages
Search aliases: f_k
Use in source · 03b_repair_transport.tex, line 166
uad normC_ϕ≤ m, ∂_ϕ S_ϕ=C_ϕ∂. Exact candidates remain exact under every monotone common resampling, even without a gap bound. If A and B are bounded right inverses for the fine and sampled differentials, then normP_BS_ϕ f-S_ϕ P_Af ≤(norm A+mnorm B)norm∂ f. For a bounded candidate family, sup_nnormf_ϕ(n)≤sup_knormf_k, proving the sampling bound. For a bounded residual family, each summand in [source reference] is a map from X_b to Y_a and has norm at most norm r_∞, because every composite bond is…
Exact mathematical passage
\sup_n\norm{f_{\phi(n)}}\le\sup_k\norm{f_k}
f_{k+1}Use in source · 3 passages
Search aliases: f_k+1
Use in source · 03b_repair_transport.tex, line 169
To verify the commuting square, insert r_k=ρ^Y_kf_k+1-f_kρ^X_k. The k-th term becomes ρ^Y_a,k+1f_k+1ρ^X_k+1,b -ρ^Y_a,kf_kρ^X_k,b. Successive internal terms cancel in the finite sum, leaving ρ^Y_a,bf_b-f_aρ^X_a,b. This is exactly the sampled differential. No infinite series interchange is involved.
Exact mathematical passage
r_k=\rho^Y_kf_{k+1}-f_k\rho^X_k
f_mUse in source · 5 passages
Search aliases: f_m
Use in source · 05c_iteration.tex, line 28
on] Suppose b_n≥0, ∑_n=0^∞b_n<∞, and dist(f_n,f_n+1)≤ b_n, ω(f_n)⟶0. Then there is a limit x∈ M satisfying f_n⟶ x, ω(x)=0, dist(f_n,x)≤∑_k=n^∞b_k. For integers m>n, apply the triangle inequality successively along the finite chain f_n,f_n+1,…,f_m. It gives dist(f_n,f_m)≤∑_k=n^m-1dist(f_k,f_k+1) ≤∑_k=n^m-1b_k≤∑_k=n^∞b_k. Given ε>0, convergence of the nonnegative series supplies N for which the final tail is below ε whenever n≥ N. Interchanging m,n when necessary proves the Cauchy condition for al…
Exact mathematical passage
f_n,f_{n+1},\ldots,f_m
f_M^*Use in source · 3 passages
Search aliases: f_M^*
Use in source · 01_views.tex, line 156
a coordinate preserving equivalence Γ(B_a,b)≃ B(a,b). Under a transition cover, matching local bridge sections glue to a unique global bridge with both prescribed projections unchanged. We must first check that the displayed fibers really form a view system. Let m∈ B_a,b(w) and f:v→ w. Naturality of p and coherence of a give p_v(f_M^*m)=f_F^*p_w(m)=f_F^*a_w=a_v. The same calculation with q,b gives q_v(f_M^*m)=b_v. Therefore f_M^*m belongs to B_a,b(v). Identity and composition of these restricted…
Exact mathematical passage
p_v(f_M^*m)=f_F^*p_w(m)=f_F^*a_w=a_v.
F_nUse in source · 1 passage
Search aliases: F_n
Use in source · 12_proof_tools.tex, line 107
The centers form a Cauchy sequence: all centers after stage n lie in B_n, whose diameter is at most 2^1-n. Completeness gives a limit. For each fixed n, a tail of centers lies in the closed set B_n, so the limit lies there. It therefore belongs to no F_n, contradicting that these sets cover the space.
Exact mathematical passage
F_n
f_NUse in source · 5 passages
Search aliases: f_N
Use in source · 05e_escape.tex, line 50
A complete calculation of the finite repairs. Let 1_N be one in coordinates 1,…,N and zero elsewhere. Put a_N=( 1_N,0), f_N=f_0-a_N=(- 1_N,1). Then p(f_N)=1, and the first N residual coordinates vanish exactly. The remaining residual and movement satisfy d(f_N)_n=0 · n≤ N,
Exact mathematical passage
a_N=(\mathbf 1_N,0),\qquad f_N=f_0-a_N=(-\mathbf 1_N,1).
F_{n+1}Use in source · 2 passages
Search aliases: F_n+1
Use in source · 12_proof_tools.tex, line 105
interior. Choose a nonempty open ball whose closed ball B_1, of positive radius at most 1/2, avoids F_1. Such a choice is possible because the complement of F_1 is open and meets every nonempty open set. Recursively choose a closed ball B_n+1 of positive radius at most 2^-(n+1), contained in the open ball with the same center and radius as B_n, and disjoint from F_n+1. The intersection of that open ball with the complement of F_n+1 is nonempty and open, so it contains a sufficiently small closed…
Exact mathematical passage
F_{n+1}
f_{N+1}Use in source · 1 passage
Search aliases: f_N+1
Use in source · 05e_escape.tex, line 66
The residual tends to zero, but the corrections grow like sqrt N. Successive candidates differ by a unit coordinate, so normf_N+1-f_N=1. The movement series diverges and the sequence is not Cauchy. This is exactly the hypothesis missing from the summable exactification theorem.
Exact mathematical passage
\norm{f_{N+1}-f_N}=1.
f_{n+1}Use in source · 18 passages
Search aliases: f_n+1
Use in source · 03_towers.tex, line 12
A candidate transport from X to Y consists of maps f_n:X_n→ Y_n. It is exact when ρ^Y_n f_n+1=f_nρ^X_n for all n.
Exact mathematical passage
\label{eq:exacttower} \rho^Y_n f_{n+1}=f_n\rho^X_n\quad\text{for all }n.
f_{n+j}Use in source · 1 passage
Search aliases: f_n+j
Use in source · 05c_iteration.tex, line 80
For constant rates, the global estimate can be sharpened by restarting at the current index. Fix n. A new induction on j gives ω(f_n+j)≤ q^jω(f_n), dist(f_n+j,f_n+j+1)≤ Cq^jω(f_n). Sum the finite chain from n to n+m, and then let m→∞ using the already established limit. The geometric series equals 1/(1-q), including q=0, where its first term is one and all later terms vanish. This proves the restarted bound. The variable-rate proof itself requires only nonnegative q_n, decay of their cumulative …
Exact mathematical passage
\omega(f_{n+j})\le q^j\omega(f_n),\qquad \dist(f_{n+j},f_{n+j+1})\le Cq^j\omega(f_n).
f_{n+j+1}Use in source · 1 passage
Search aliases: f_n+j+1
Use in source · 05c_iteration.tex, line 80
For constant rates, the global estimate can be sharpened by restarting at the current index. Fix n. A new induction on j gives ω(f_n+j)≤ q^jω(f_n), dist(f_n+j,f_n+j+1)≤ Cq^jω(f_n). Sum the finite chain from n to n+m, and then let m→∞ using the already established limit. The geometric series equals 1/(1-q), including q=0, where its first term is one and all later terms vanish. This proves the restarted bound. The variable-rate proof itself requires only nonnegative q_n, decay of their cumulative …
Exact mathematical passage
\omega(f_{n+j})\le q^j\omega(f_n),\qquad \dist(f_{n+j},f_{n+j+1})\le Cq^j\omega(f_n).
f_n^{(a)}Use in source · 2 passages
Search aliases: f_n^(a)
Use in source · 05c_iteration.tex, line 167
ct and protected value one, but their middle coordinates differ. They are distinct points of the same feasible fiber. Constant sequences at those points meet exactness and preservation but not a vanishing comparison bound, so these conditions alone cannot imply path independence. sourceKS74 A stronger worked example uses actual convergent paths from the same initial input: f_n^(a)=(s,a(1-2^-n),2^-n) (a∈R). All paths begin at (s,0,1), keep p=s, and have residual 2^-n. Their maximum-norm movements…
Exact mathematical passage
f_n^{(a)}=(s,a(1-2^{-n}),2^{-n})\qquad(a\in\R).
f_n^{(b)}Use in source · 1 passage
Search aliases: f_n^(b)
Use in source · 05c_iteration.tex, line 171
eKS74 A stronger worked example uses actual convergent paths from the same initial input: f_n^(a)=(s,a(1-2^-n),2^-n) (a∈R). All paths begin at (s,0,1), keep p=s, and have residual 2^-n. Their maximum-norm movements are frac12max{|a|,1}2^-n, so every path satisfies geometric exactification. Their limits are (s,a,0). For different a,b, dist(f_n^(a),f_n^(b))=|a-b|(1-2^-n)⟶|a-b|, not zero. The missing uniqueness hypothesis is visible in the exact direction along which the paths separate.
Exact mathematical passage
\dist(f_n^{(a)},f_n^{(b)})=|a-b|(1-2^{-n})\longrightarrow|a-b|,
\F_{p_{b_j}}Use in source · 1 passage
Search aliases: F_p_b_j F mathbb
Use in source · 03_spectrum.tex, line 56
Conversely, suppose B is infinite and enumerable in X. List its distinct elements as b_0,b_1,…. To compute b_j, wait for the (j+1)st distinct enumerated value. This procedure terminates for every j. Use canonical finite tuples with the jth coordinate in F_p_b_j. Coordinatewise addition, normalization, and equality are computable in X. If B is finite, D_B is a finite group and has a computable copy. This last existence statement does not require a uniform way to decide when a finite enumeration h…
Exact mathematical passage
\F_{p_{b_j}}
F_tUse in source · 2 passages
Search aliases: F_t
Use in source · 01_views.tex, line 75
A global view and a useful warning. [Terminal evaluation] If t is terminal in V, evaluation induces an equivalence Γ(F)≃ F_t. For a∈ F_t, take its restriction along the unique arrow v→ t at each view. Uniqueness of the arrow to t, together with the functor laws, proves coherence. Evaluation at t recovers a. Conversely, any coherent family is the restriction of its coordinate at t. sourceKS03 This proposition is both useful and diagnostic. If the supposedly reconstructed global object has already…
Exact mathematical passage
\Gamma(F)\simeq F_t
f_{\phi(n)}Use in source · 2 passages
Search aliases: f_ϕ(n) phi
Use in source · 03b_repair_transport.tex, line 137
Actual tower resampling gives a defect map. Return to Banach towers with contractive source and target bonds. Let ϕ be a monotone schedule. Candidate sampling is (S_ϕ f)_n=f_ϕ(n). The sampled bonding map is the actual composite restriction between the selected levels. The residual map C_ϕ is not pointwise sampling of a scalar error score. It accumulates transported fine defects across each omitted block.
Exact mathematical passage
(S_\phi f)_n=f_{\phi(n)}.
f_{\psi(n)}Use in source · 3 passages
Search aliases: f_ψ(n) psi
Use in source · 03a_observation.tex, line 20
The realized map is |f|:Point(X)→Point(Y), |f|(x)_n=f_n(x_ϕ(n)). Equation~[source reference] proves that this is a coherent output. A second transport g_n:Y_ψ(n)→ Z_n composes with f using schedule ϕ∘ψ and coordinate map g_n f_ψ(n). The order of the schedules is fixed by the types, not by the order in which their letters are written in prose.
Exact mathematical passage
g_n f_{\psi(n)}
f_\inftyUse in source · 1 passage
Search aliases: f_∞ infty
Use in source · 03_towers.tex, line 97
ed field, with continuous linear bonding maps and uniform bounds normρ^X_n≤ B_X, normρ^Y_n≤ B_Y. Use the spaces of bounded operator families E=ℓ^∞_n L(X_n,Y_n), R=ℓ^∞_n L(X_n+1,Y_n). The source formalization derives the required completeness from the operator spaces and their bounded product. The same differential satisfies norm∂ f_∞≤(B_X+B_Y)norm f_∞. Thus it is a continuous linear map, and evaluation at a fixed level is continuous. These facts connect algebraic compatibility to an analytic lim…
Exact mathematical passage
\norm{\partial f}_\infty\le(B_X+B_Y)\norm f_\infty.
\fixedplateUse in source · 1 passage
Search aliases: fixedplate
Use in source · 05h_regularized_repair.tex, line 197
[3pt] q=0 · No · Residual vanishes, correction norms diverge, no exact protected answer. [3pt] q>0 · Yes · Strong convergence to the least-squares correction of minimum norm, not an exact answer. [3pt] q>0 · No · A positive residual floor and diverging correction norms occur together. The last case is not hypothetical. Let Tu=(u_n/n) on ℓ^2, define Su=(Tu,0) into ℓ^2⊕R, and take r=((1/n),1). Then b=((1/n),0) is in the closure but not the range, while q=1. The same escaping corrections as before …
Exact mathematical passage
3pt] $q=0$ & No & Residual vanishes, correction norms diverge, no exact protected answer.\\[3pt] $q>0$ & Yes & Strong convergence to the least-squares correction of minimum norm, not an exact answer.\\[3pt] $q>0$ & No & A positive residual floor and diverging correction norms occur together.\\\bottomrule \end{tabularx} \end{center} The last case is not hypothetical. Let $Tu=(u_n/n)$ on $\ell^2$, define $Su=(Tu,0)$ into $\ell^2\oplus\R$, and take $r=((1/n),1)$. Then $b=((1/n),0)$ is in the closure but not the range, while $q=1$. The same escaping corrections as before coexist with an irreducible orthogonal residual. Positive floor and unbounded correction cost are independent features. \subsection{Noise is controlled at a fixed parameter} \fixedplate{34_noise_filter}{For one singular value $\sigma$, the response to residual-data noise is $\sigma/(\sigma^2+\alpha)$. Its maximum occurs at $\sigma=\sqrt\alpha$ and equals $1/(2\sqrt\alpha)$. The plot evaluates this formula and the universal horizontal bounds at three explicit positive parameters.}{fig:noise-filter} \begin{theorem}[Data sensitivity and a convergent parameter choice]\label{WR09} For two residual data vectors $r,\widetilde r$ with $\norm{r-\widetilde r}\le\delta$ and the same $T,\alpha$, \[ \norm{a_\alpha(r)-a_\alpha(\widetilde r)} \le\frac{\delta}{2\sqrt\alpha},\qquad \norm{T(a_\alpha(r)-a_\alpha(\widetilde r))}\le\delta.
\mathcal GUse in source · 3 passages
Search aliases: G mathcal
Use in source · 04_symmetry.tex, line 79
The relative obstruction without commutativity. Prescribe a coherent partial section s of T_c on U. The based gauge group is G_U={a∈∏_vG_v:a_u=1 for u∈ U}. Choose a raw family q equal to s on U. Identity elements can be used elsewhere. Normalize the labels by c^q_f=q_v^-1c_f f^*(q_w). Its labels on arrows internal to U are one. Define the pointed orbit obstruction o_U(c,s)=[c^q]∈ Z^1(V;G)/ G_U, whose neutral point is the class of the trivial cocycle.
Exact mathematical passage
\mathcal G_U=\{a\in\prod_vG_v:a_u=1\text{ for }u\in U\}.
G'Use in source · 10 passages
Search aliases: G'
Use in source · 05_geometry.tex, line 626
Motion of the actual descent repair. Compare pairs (D,C) and (D',C') on the same raw Hilbert space. Suppose each pair has closed local range and closed overlap range on its own locally exact subspace. Let G,G' be the resulting joint residual gains. If normD-D'≤ε_D, normC-C'≤ε_C, then the maximum norm on the target product gives norm(D,C)-(D',C')≤max(ε_D,ε_C). Applying Theorem~[source reference] to the joint operators, and using the actual two-stage identity of Theorem~[source reference], gives n…
Exact mathematical passage
G,G'
g'Use in source · 1 passage
Search aliases: g'
Use in source · 03a_observation.tex, line 36
We check compatibility with composition with all indices displayed. Write f,f' with schedules ϕ,ϕ' and g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applying the naturality equations and then the two witness equations gives g_nf_ψ(n)ρ^X_ϕ(ψ(n)),k · =g_nρ^Y_ψ(n),jf_jρ^X_ϕ(j),k
Exact mathematical passage
g,g'
g'_nUse in source · 2 passages
Search aliases: g'_n
Use in source · 03a_observation.tex, line 37
We check compatibility with composition with all indices displayed. Write f,f' with schedules ϕ,ϕ' and g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applying the naturality equations and then the two witness equations gives g_nf_ψ(n)ρ^X_ϕ(ψ(n)),k · =g_nρ^Y_ψ(n),jf_jρ^X_ϕ(j),k
Exact mathematical passage
g_n\rho^Y_{\psi(n),j}=g'_n\rho^Y_{\psi'(n),j}.
g^{\mathsf T}Use in source · 1 passage
Search aliases: g^ T mathsf
Use in source · 05k_stopping_rules.tex, line 46
The restricted Hilbert normal residual is g=P d^ T(dBc-r)+α Bc. Substitute S=dB, G=B^ TB and the expression for P: g=BG^-1(S^ T(Sc-r)+α Gc)=BG^-1q. Equivalently q=B^ Tg records the pairings of g with the basis columns. Its squared Hilbert norm is g^ Tg =q^ TG^-1B^ TBG^-1q =q^ TG^-1q, using symmetry of G^-1. This proves both coordinate formulas.
Exact mathematical passage
g^{\mathsf T}g =q^{\mathsf T}G^{-1}B^{\mathsf T}BG^{-1}q =q^{\mathsf T}G^{-1}q,
G^{-1}Use in source · 12 passages
Search aliases: G^-1
Use in source · 05k_stopping_rules.tex, line 13
[The invariant norm of a coordinate certificate] The actual normal residual is g=BG^-1q, norm g^2=q^ TG^-1q. For any invertible change of basis B'=BC, the same proposed correction has c'=C^-1c, G'=C^ TGC, q'=C^ Tq, and (q')^ T(G')^-1q'=q^ TG^-1q. Because B has independent columns, for every nonzero coefficient vector c, c^ TGc=c^ TB^ TBc=normBc^2>0. Thus G is positive definite and invertible. The matrix P=BG^-1B^ T is symmetric and obeys P^2=BG^-1(B^ TB)G^-1B^ T=P, PB=B. Its range is precisely t…
Exact mathematical passage
\label{eq:covector-norm} g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
g^2Use in source · 8 passages
Search aliases: g^2
Use in source · 05j_inexact_certificates.tex, line 28
A posteriori correction and fit bounds. [Normal residual as a solver certificate] For every proposed a∈ H, norma-a_α · ≤fracnorm gα, normT(a-a_α) · ≤fracnorm g2sqrtα, 0≤ J_α(a)-J_α(a_α) · ≤fracnorm g^2α. Let e=a-a_α and B=T^*T+α I. The normal equation of the constructed optimizer gives Be=(T^*T+α I)a-T^*r=g. Pairing with e yields the exact identity normTe^2+αnorm e^2=⟨ g,e⟩ ≤norm gnorm e. If e=0, all three conclusions hold with zero error. Otherwise drop the first nonnegative term and divide by …
Exact mathematical passage
\norm{a-a_\alpha}&\le\frac{\norm g}{\alpha},\label{eq:solver-error}\\ \norm{T(a-a_\alpha)}&\le\frac{\norm g}{2\sqrt\alpha},\label{eq:solver-fit}\\ 0\le J_\alpha(a)-J_\alpha(a_\alpha)&\le\frac{\norm g^2}{\alpha}. \label{eq:solver-objective}
G_1Use in source · 2 passages
Search aliases: G_1
Use in source · 05_geometry.tex, line 371
uad norm v≤ bnormCP_Lx ≤ b(normCx+norm C anormDx). Therefore normx-Px≤ a(1+bnorm C)normDx+bnormCx. The Pythagorean identity gives the sharper estimate normx-Px^2≤ a^2normDx^2+b^2(normCx+anorm CnormDx)^2. For the maximum norm on R× Y, two sufficient gains are therefore G_2=sqrta^2+b^2(1+anorm C)^2 ≤ G_1=a(1+bnorm C)+b. The inequality follows from sqrtu^2+v^2≤ u+v for nonnegative u,v. The formal interface supplies the sum-type gain G_1, while its squared-error theorem also yields G_2. Neither is c…
Exact mathematical passage
G_2=\sqrt{a^2+b^2(1+a\norm C)^2} \le G_1=a(1+b\norm C)+b.
G_2Use in source · 2 passages
Search aliases: G_2
Use in source · 05_geometry.tex, line 371
uad norm v≤ bnormCP_Lx ≤ b(normCx+norm C anormDx). Therefore normx-Px≤ a(1+bnorm C)normDx+bnormCx. The Pythagorean identity gives the sharper estimate normx-Px^2≤ a^2normDx^2+b^2(normCx+anorm CnormDx)^2. For the maximum norm on R× Y, two sufficient gains are therefore G_2=sqrta^2+b^2(1+anorm C)^2 ≤ G_1=a(1+bnorm C)+b. The inequality follows from sqrtu^2+v^2≤ u+v for nonnegative u,v. The formal interface supplies the sum-type gain G_1, while its squared-error theorem also yields G_2. Neither is c…
Exact mathematical passage
G_2=\sqrt{a^2+b^2(1+a\norm C)^2} \le G_1=a(1+b\norm C)+b.
G_{A_C}Use in source · 3 passages
Search aliases: G_A_C
Use in source · 10_grammar.tex, line 28
[Escape within one profinite-completion class] For every C⊆ N, the group G_A_C is not isomorphic to any output of G. Distinct C give pairwise nonisomorphic groups, with no embeddings in either direction. Every group is countable, reduced, residually finite, and has profinite completion P. Fix a description index j. If R_j is outside the family, it cannot be isomorphic to G_A_C. Otherwise rule [source reference] makes A_C disagree with B_j at every index 2⟨ j,k⟩. These are infinitely many distinc…
Exact mathematical passage
G_{A_C}
G_BUse in source · 18 passages
Search aliases: G_B
Use in source · 00_notation.tex, line 33
The main statements have the following form: Tor(G_A) · =D, · G_A/D · ≅ Q, G_A/mG_A · ≅ C_rad(m), · widehatG_A · ≅ P, G_A≅ G_B · iff A=^* B, · · G_A has an X-computable copy · iff A≤_T X. · · The extension by D does not split. Distinct eventual selector classes do not embed in one another. All of these conclusions are proved within the book, with the exact hypotheses retained.
Exact mathematical passage
\Tor(G_A)&=D,&G_A/D&\cong\Q,\\ G_A/mG_A&\cong C_{\rad(m)},&\widehat{G_A}&\cong P,\\ G_A\cong G_B&\iff A\eqfin B,&&\\ G_A\text{ has an }X\text{-computable copy}&\iff A\le_T X.&&
g_dUse in source · 1 passage
Search aliases: g_d
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
G_KUse in source · 1 passage
Search aliases: G_K
Use in source · 08_computability.tex, line 83
The group G_K still has a finite mathematical definition. Its noncomputability is not a proof that it cannot be named in a finite semantic language. The theorem concerns effective copies with computable operations and equality.
Exact mathematical passage
G_K
g_nUse in source · 18 passages
Search aliases: g_n
Use in source · 03_towers.tex, line 20
[The exact tower category] Identity maps and composites of exact transports are exact. An exact transport induces a map of coherent point spaces by x↦(f_nx_n)_n, preserving identities and composition. For two exact transports, substitute ρ^Z_ng_n+1=g_nρ^Y_n and ρ^Y_nf_n+1=f_nρ^X_n into the composite square. For a coherent point, ρ^Y_n f_n+1x_n+1 =f_nρ^X_nx_n+1=f_nx_n. Identity, associativity, and composition on points are checked coordinatewise. sourceKS16 This is a category on a fixed schedule.…
Exact mathematical passage
\rho^Z_ng_{n+1}=g_n\rho^Y_n
g_{n+1}Use in source · 2 passages
Search aliases: g_n+1
Use in source · 03_towers.tex, line 20
[The exact tower category] Identity maps and composites of exact transports are exact. An exact transport induces a map of coherent point spaces by x↦(f_nx_n)_n, preserving identities and composition. For two exact transports, substitute ρ^Z_ng_n+1=g_nρ^Y_n and ρ^Y_nf_n+1=f_nρ^X_n into the composite square. For a coherent point, ρ^Y_n f_n+1x_n+1 =f_nρ^X_nx_n+1=f_nx_n. Identity, associativity, and composition on points are checked coordinatewise. sourceKS16 This is a category on a fixed schedule.…
Exact mathematical passage
\rho^Z_ng_{n+1}=g_n\rho^Y_n
g_pUse in source · 4 passages
Search aliases: g_p
Use in source · 06_roots.tex, line 17
[Root or obstruction] Let g=s_A(r)+d∈ G_A and let m≥1. Then g∈ mG_A Longleftrightarrow g_p=0 for every pmid m. If the condition fails, a prime pmid m with g_p≠0 is a nonexistence certificate. If it holds, a root is obtained by the finite procedure in the proof. Necessity follows because multiplication by m is zero in every coordinate C_p with pmid m.
Exact mathematical passage
g\in mG_A\quad\Longleftrightarrow\quad g_p=0\quad\text{for every }p\mid m.
g_rUse in source · 1 passage
Search aliases: g_r
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
\mathcal G_UUse in source · 3 passages
Search aliases: G_U mathcal
Use in source · 04_symmetry.tex, line 79
The relative obstruction without commutativity. Prescribe a coherent partial section s of T_c on U. The based gauge group is G_U={a∈∏_vG_v:a_u=1 for u∈ U}. Choose a raw family q equal to s on U. Identity elements can be used elsewhere. Normalize the labels by c^q_f=q_v^-1c_f f^*(q_w). Its labels on arrows internal to U are one. Define the pointed orbit obstruction o_U(c,s)=[c^q]∈ Z^1(V;G)/ G_U, whose neutral point is the class of the trivial cocycle.
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\mathcal G_U=\{a\in\prod_vG_v:a_u=1\text{ for }u\in U\}.
g_vUse in source · 1 passage
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Use in source · 04_symmetry.tex, line 257
The correct uniqueness statement. Suppose (c,b_i) and (c',b'_i) realize the same descent datum. The only possible compatible comparison is g_v=b'_i(v)^-1b_i(v), using any patch through v. The overlap identities make it independent of the patch, and b'_ig=b_i. It is therefore the unique global comparison respecting the fixed local identifications.
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g_v=b'_i(v)^{-1}b_i(v),
G_wUse in source · 1 passage
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Use in source · 04_symmetry.tex, line 4
descentA set of solutions forgets the witnesses relating equivalent solutions. The next layer retains those witnesses, their composition, and their automorphisms. Gluing then has to respect the overlap identifications themselves. Nonabelian coherence and relative trivialization. Coherent transition labels. Fix groups G_v and restriction homomorphisms f^*:G_w→ G_v, with the same identity and composition laws as before. The groups need not be abelian and the homomorphisms need not be bijective.
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f^*:G_w\to G_v
G_zUse in source · 1 passage
Search aliases: G_z
Use in source · 04_symmetry.tex, line 28
groupoid] Equation~[source reference] defines an action on cocycles. A gauge carrying c to d induces an equivalence of twisted section sets by x_v↦ a_vx_v. The cocycles and actual gauges form a groupoid: an arrow a:c→ d followed by b:d→ e has label ba. First, T_c really is a view system. For f:v→ w, g:w→ z, and x∈ G_z, the homomorphism and cocycle laws give T_c(f)T_c(g)(x)=c_ff^*(c_gg^*x) =c_ff^*(c_g)f^*g^*x=c_g∘ f(g∘ f)^*x. The identity equation follows from c_1_v=1. No rearrangement of factors…
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x\in G_z
\mathfrak HUse in source · 3 passages
Search aliases: H mathfrak
Use in source · 10_grammar.tex, line 56
Monotonicity under grammar extension. [Admissible expansion] Suppose every valid description in G translates to a valid description in H with the same semantic group. Then every G-realizable group is H-realizable. If every translated description has length at most f(lvert drvert), then K_ H(G)≤ f(K_ G(G)) whenever K_ G(G) is finite. Translate a shortest valid G description of G. Its semantic output remains G, and its length has the stated bound. An obstruction to a small grammar can therefore di…
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\mathfrak H
H^*Use in source · 8 passages
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Use in source · 04_symmetry.tex, line 339
chapEquivalent presentations and identified boundaries Transporting the same question. Let H:V→W be part of a category equivalence, with inverse L and counit ε:HL⇒id_W. Pull a view system F on W back along H. Sampling a global section gives (H^*x)_v=x_Hv.
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(H^*x)_v=x_{Hv}
H^{-1}Use in source · 2 passages
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Use in source · 04_symmetry.tex, line 342
[Equivalence of section spaces] Pullback induces Γ(F)≃Γ(H^*F). If U⊆ObW is closed under isomorphism and U'=H^-1U, it also induces Ext_U^F(s)≃Ext_U'^H^*F(H^*s). Existence and uniqueness of the prescribed extension are preserved and reflected. Write the unit of the equivalence as η:I_V⇒ LH, and take the unit and counit with their triangle identities. Set t_w=ε_w^-1:w→ HLw. Given y∈Γ(H^*F), define (Ry)_w=F(t_w)y_Lw. For an arrow f:w→ w', naturality of the inverse counit states HL(f) t_w=t_w'f. Cont…
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U'=H^{-1}U
H^2Use in source · 1 passage
Search aliases: H^2
Use in source · 03_towers.tex, line 307
chapMultiscale residuals and the limits of reconstruction A split coarsener retains its missing detail. Let C:R→overline R and J:overline R→ R be linear maps with CJ=I. Put H=I-JC. Then r=JCr+Hr, CH=0, H^2=H. The two coordinates (Cr,Hr) retain all information. The coarse coordinate alone generally does not.
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r=JCr+Hr,\qquad CH=0,\qquad H^2=H.
h^2Use in source · 4 passages
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Use in source · 05h_regularized_repair.tex, line 27
. It satisfies T^*(r-Ta_α)=α a_α. For every a∈ H, J_α(a)-J_α(a_α) =normT(a-a_α)^2+αnorma-a_α^2. Step 1: the normal operator has an actual bounded inverse. Let B=T^*T+α I:H→ H. It is bounded and self-adjoint. For every h∈ H, ⟨ Bh,h⟩=normTh^2+αnorm h^2 ≥αnorm h^2. If h≠0, Cauchy--Schwarz and division by norm h give normBh≥αnorm h. The same inequality holds at zero. It implies injectivity of B.
Exact mathematical passage
\langle Bh,h\rangle=\norm{Th}^2+\alpha\norm h^2 \ge\alpha\norm h^2.
h_*Use in source · 12 passages
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Use in source · 05_geometry.tex, line 180
[Nearest feasible protected repair] Every feasible x has a unique closest exact answer y_* preserving p(x). Its correction h_*=x-y_* is the unique vector in the orthogonal complement of ker B inside M satisfying Bh_*=Dx. For every other feasible answer y, normx-y^2=normx-y_*^2+normy_*-y^2. Moreover, the following conditions are equivalent: ran B is closed Longleftrightarrow ∃ C≥0 ∀ feasible x: normx-y_*≤ CnormDx. When these conditions hold, the least nonnegative constant is the transverse invers…
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h_*=x-y_*
H_0Use in source · 6 passages
Search aliases: H_0
Use in source · 05f_bounded_realization.tex, line 32
ent [Bounded sequences have weakly convergent subsequences] Every bounded sequence (a_n) in a real Hilbert space has a subsequence converging weakly to some a. Moreover, norm a≤liminf_knorma_n_k. Choose M≥0 with norma_n≤ M. If M=0, every term is zero and the conclusion is immediate. Otherwise let H_0 be the closed linear span of the countably many vectors a_n. Finite rational linear combinations of those vectors form a countable dense set, so H_0 is separable. Apply Gram--Schmidt to a countable …
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H_0
h_0Use in source · 7 passages
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Use in source · 05_geometry.tex, line 194
ormx-y_*^2+normy_*-y^2. Moreover, the following conditions are equivalent: ran B is closed Longleftrightarrow ∃ C≥0 ∀ feasible x: normx-y_*≤ CnormDx. When these conditions hold, the least nonnegative constant is the transverse inverse norm of B, including the zero-range case. Feasibility gives some h_0=x-y_0∈ M with Bh_0=Dx. Conversely, every h∈ M satisfying Bh=Dx produces an exact protected answer y=x-h. Thus the answer problem and the correction problem are equivalent, with correction set {h∈ …
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h_0=x-y_0\in M
h_1Use in source · 2 passages
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Use in source · 02_obstructions.tex, line 104
The choice of x_0 only supplies an origin for the parameters. If x_1=x_0+h_1 is used instead, the parameter of x changes from x-x_0 to x-x_1=(x-x_0)-h_1. The affine set of answers itself has not changed. sourceKS10 An extension fiber is therefore an affine copy, or torsor, of the invisible coherent group. Selecting one solution gives coordinates on the fiber. Without a selected solution, the affine structure does not designate an origin.
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x_1=x_0+h_1
h_JUse in source · 3 passages
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Use in source · 05f_bounded_realization.tex, line 43
To verify weak convergence against h∈ H_0, choose a finite basis combination h_J close to h. Then |⟨ a_n_k-a,h⟩| ≤ |⟨ a_n_k-a,h_J⟩| +2Mnormh-h_J. For a prescribed tolerance, choose h_J first to control the second term and then k to control the first by finitely many coordinate limits. For a general h in the ambient space, write h=P_H_0h+h_⊥. The orthogonal component pairs to zero with every a_n_k and with a, reducing to the preceding case. This proves weak convergence against every test vector.
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h_J
h_jUse in source · 6 passages
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Use in source · 03_towers.tex, line 329
A whole cascade. At scale j, let C_j:R_j→ R_j+1 and J_j:R_j+1→ R_j be bounded linear maps with C_jJ_j=I. For x∈ R_0, define x_0=x, x_j+1=C_jx_j, L_0=I, L_N=J_0⋯ J_N-1, h_j=x_j-J_jx_j+1, Q_Nx=L_Nx_N, S_N=I-Q_N. The map Q_N restores the surviving coarse remainder to the original space. It is this restored remainder, not just x_N, that controls reconstruction.
Exact mathematical passage
h_j=x_j-J_jx_{j+1},\qquad Q_Nx=L_Nx_N, \qquad S_N=I-Q_N.
h_mUse in source · 2 passages
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Use in source · 05h_regularized_repair.tex, line 33
To prove closedness of its range, suppose Bh_n→ y. Applying the lower bound to h_n-h_m gives normh_n-h_m≤α^-1normBh_n-Bh_m. Thus h_n is Cauchy. Completeness of H supplies a limit h, and continuity gives Bh=y. If y is orthogonal to the range of B, then for every h, 0=⟨ Bh,y⟩=⟨ h,By⟩. Taking h=By gives By=0, and injectivity gives y=0. A closed subspace with zero orthogonal complement is the whole Hilbert space, so ran B=H.
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h_n-h_m
H_nUse in source · 3 passages
Search aliases: H_n
Use in source · 05c_iteration.tex, line 198
Three hypotheses that cannot simply be dropped. Summable movement. In maximum-norm R^3, let f_n=(s,H_n,(n+1)^-1), H_n=∑_j=1^nfrac1j. The first coordinate is protected and the residual is the last coordinate. It tends to zero with the same ratios as the polynomial example. The movement equals 1/(n+1), however, and the unprotected coordinate diverges. Here C_n=1, so ∑ C_nQ_n is the divergent harmonic series. Residual decay and semantic preservation hold, but exactification of this path fails.
Exact mathematical passage
f_n=(s,H_n,(n+1)^{-1}),\qquad H_n=\sum_{j=1}^{n}\frac1j.
h_nUse in source · 4 passages
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Use in source · 05h_regularized_repair.tex, line 33
To prove closedness of its range, suppose Bh_n→ y. Applying the lower bound to h_n-h_m gives normh_n-h_m≤α^-1normBh_n-Bh_m. Thus h_n is Cauchy. Completeness of H supplies a limit h, and continuity gives Bh=y. If y is orthogonal to the range of B, then for every h, 0=⟨ Bh,y⟩=⟨ h,By⟩. Taking h=By gives By=0, and injectivity gives y=0. A closed subspace with zero orthogonal complement is the whole Hilbert space, so ran B=H.
Exact mathematical passage
Bh_n\to y
h_pUse in source · 3 passages
Search aliases: h_p
Use in source · 04_purity.tex, line 44
Root conditions do not depend on the subspace. [Ambient root criterion] For every rational subspace W, every h∈ H_W, and every m≥1, h∈ mH_W Longleftrightarrow h_p=0 for all primes pmid m. Multiplication by m on P is zero at primes dividing m and invertible on every other coordinate. Thus the right-hand condition is equivalent to h∈ mP. Apply purity from Theorem~[source reference]. The condition is independent of W, but the set of admissible h is not. This distinction is the reason identical quot…
Exact mathematical passage
h\in mH_W\quad\Longleftrightarrow\quad h_p=0\text{ for all primes }p\mid m.
H_VUse in source · 1 passage
Search aliases: H_V
Use in source · 04_purity.tex, line 38
D agree with h on the primes dividing m and vanish elsewhere. Then h-d∈ mP∩ H. Purity gives h-d=mx for some x∈ H, so q(x)=q(h)/m. Thus q(H) is closed under rational scalars. Since D⊆ H, the equality H=q^-1(q(H)) holds. This proves that the constructions are inverse. Both preserve inclusion. The bottom object is H_{0}=D, and the top object is H_V=P. Intersections of rational subspaces correspond to intersections of their preimages. Sums correspond to subgroup sums, with D serving as the bottom of…
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H_V=P
h_vUse in source · 4 passages
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Use in source · 04_symmetry.tex, line 132
Let H_U={h:f^*(h_w)=h_v for every f, h|_U=1}.
Exact mathematical passage
H_U=\{h:f^*(h_w)=h_v\text{ for every }f,\ h|_U=1\}.
h_wUse in source · 3 passages
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Use in source · 04_symmetry.tex, line 132
Let H_U={h:f^*(h_w)=h_v for every f, h|_U=1}.
Exact mathematical passage
H_U=\{h:f^*(h_w)=h_v\text{ for every }f,\ h|_U=1\}.
H_{\{0\}}Use in source · 1 passage
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Use in source · 04_purity.tex, line 38
Given h∈ H and m≥1, let d∈ D agree with h on the primes dividing m and vanish elsewhere. Then h-d∈ mP∩ H. Purity gives h-d=mx for some x∈ H, so q(x)=q(h)/m. Thus q(H) is closed under rational scalars. Since D⊆ H, the equality H=q^-1(q(H)) holds. This proves that the constructions are inverse. Both preserve inclusion. The bottom object is H_{0}=D, and the top object is H_V=P. Intersections of rational subspaces correspond to intersections of their preimages. Sums correspond to subgroup sums, with…
Exact mathematical passage
H_{\{0\}}=D
h_\perpUse in source · 1 passage
Search aliases: h_⊥ perp
Use in source · 05f_bounded_realization.tex, line 49
To verify weak convergence against h∈ H_0, choose a finite basis combination h_J close to h. Then |⟨ a_n_k-a,h⟩| ≤ |⟨ a_n_k-a,h_J⟩| +2Mnormh-h_J. For a prescribed tolerance, choose h_J first to control the second term and then k to control the first by finitely many coordinate limits. For a general h in the ambient space, write h=P_H_0h+h_⊥. The orthogonal component pairs to zero with every a_n_k and with a, reducing to the preceding case. This proves weak convergence against every test vector.
Exact mathematical passage
h=P_{H_0}h+h_\perp
\hlineUse in source · 1 passage
Search aliases: hline
Use in source · 05b_transport_cost.tex, line 49
category Consider four vertices A,B,C,D, and the free path category generated by Axrightarrow0B, Bxrightarrow0A, Axrightarrow2C, Cxrightarrow5A. There are no arrows incident to D, apart from its identity. A path costs the sum of its edge labels. Concatenation is additive and empty paths cost zero, so all cost axioms hold. The directed distances are c|rrrr d_W · A · B · C · D A · 0 · 0 · 2 · ∞ B · 0 · 0 · 2 · ∞ C · 5 · 5 · 0 · ∞ D · ∞ · ∞ · ∞ · 0 For example, the path from B through A to C costs …
Exact mathematical passage
\begin{array}{c|rrrr} d_W&A&B&C&D\\\hline A&0&0&2&\infty\\ B&0&0&2&\infty\\ C&5&5&0&\infty\\ D&\infty&\infty&\infty&0 \end{array}
\HomUse in source · 3 passages
Search aliases: Hom Hom operatorname
Use in source · 03_towers.tex, line 35
The actual residual of a transport. Assume C is preadditive, so differences of maps make sense. The tower differential is (∂ f)_n=ρ^Y_nf_n+1-f_nρ^X_n. Its target has components Hom(X_n+1,Y_n), not Hom(X_n,Y_n).
Exact mathematical passage
\Hom(X_{n+1},Y_n)
\HullUse in source · 1 passage
Search aliases: Hull operatorname
Use in source · 01_contracts.tex, line 66
[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If M_j≠0 and j is not reachable, P_j annihilates the entire hull but not M. An element of the direct sum has finite support. Applying one primitive oper…
Exact mathematical passage
\Hull_{\mathcal O}(S)\subseteq \bigoplus_{j\in\operatorname{Reach}_E(F)}M_j.
\Hull_{\mathcal O}Use in source · 1 passage
Search aliases: Hull_ O Hull mathcal operatorname mathcal
Use in source · 01_contracts.tex, line 66
[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If M_j≠0 and j is not reachable, P_j annihilates the entire hull but not M. An element of the direct sum has finite support. Applying one primitive oper…
Exact mathematical passage
\Hull_{\mathcal O}(S)\subseteq \bigoplus_{j\in\operatorname{Reach}_E(F)}M_j.
I_EUse in source · 1 passage
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Use in source · 03b_repair_transport.tex, line 6
A section is a choice of residual coordinates. Let d:E→ R be a continuous linear map between normed spaces over a nontrivially normed field. Suppose it has a continuous linear right inverse A:R→ E, so dA=I_R. Set K=ker d, P_A=I_E-Ad. Then dP_A=0, P_A fixes K, and P_A^2=P_A. The right inverse chooses where each residual is represented inside the candidate space. Different right inverses can choose different representatives of the same residual.
Exact mathematical passage
K=\ker d,\qquad P_A=I_E-Ad.
i_jUse in source · 1 passage
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Use in source · 05_geometry.tex, line 42
The triangle inequality and the operator norm bound now give normT_qx_i_m-x_i_0 ≤∑_k=0^m-1normU_knorm(dx)_i_ki_k+1 ≤(∑_k=0^m-1∏_j=0^k-1normτ_i_ji_j+1)normdx_∞. The bracket is γ(q), with the empty product equal to one. Repeated vertices or edges do not affect the cancellation. Reordering the edge maps would, in general, invalidate it. sourceKS40 For isometric maps, γ(q)≤|q|. For arbitrary bounded equivalences, route length cannot replace the amplification gain. Reversing a route can have a differ…
Exact mathematical passage
\norm{T_qx_{i_m}-x_{i_0}} \le\sum_{k=0}^{m-1}\norm{U_k}\norm{(dx)_{i_ki_{k+1}}} \le\left(\sum_{k=0}^{m-1}\prod_{j=0}^{k-1}\norm{\tau_{i_ji_{j+1}}}\right)\norm{dx}_\infty.
i_{j+1}Use in source · 1 passage
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Use in source · 05_geometry.tex, line 42
The triangle inequality and the operator norm bound now give normT_qx_i_m-x_i_0 ≤∑_k=0^m-1normU_knorm(dx)_i_ki_k+1 ≤(∑_k=0^m-1∏_j=0^k-1normτ_i_ji_j+1)normdx_∞. The bracket is γ(q), with the empty product equal to one. Repeated vertices or edges do not affect the cancellation. Reordering the edge maps would, in general, invalidate it. sourceKS40 For isometric maps, γ(q)≤|q|. For arbitrary bounded equivalences, route length cannot replace the amplification gain. Reversing a route can have a differ…
Exact mathematical passage
\norm{T_qx_{i_m}-x_{i_0}} \le\sum_{k=0}^{m-1}\norm{U_k}\norm{(dx)_{i_ki_{k+1}}} \le\left(\sum_{k=0}^{m-1}\prod_{j=0}^{k-1}\norm{\tau_{i_ji_{j+1}}}\right)\norm{dx}_\infty.
i_{m-1}Use in source · 1 passage
Search aliases: i_m-1
Use in source · 05_geometry.tex, line 15
Distinct walks are retained as arrows of the view category. The product along a walk q=(i_0,…,i_m) is T_q=τ_i_0i_1⋯τ_i_m-1i_m. Concatenation gives the restriction composition law. An exact vertex family defines a coherent walk-view section, and a coherent walk-view section satisfies every edge equation. Thus this graph construction returns to the original ambient realization type rather than creating a disconnected numerical model.
Exact mathematical passage
T_q=\tau_{i_0i_1}\cdots\tau_{i_{m-1}i_m}.
I_RUse in source · 3 passages
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Use in source · 03_towers.tex, line 180
[Split tower correction] The recurrence defines a continuous linear map A:R→ E with ∂ A=I_R, norm A≤ K. The operator P=I-A∂ is an exact projection and (Pf)_0=f_0, normf-Pf_∞≤ Knorm∂ f_∞. Let θ=SB_X<1. The number K=S/(1-θ) is nonnegative and satisfies S+SB_XK=K. Fix r∈ R and define its coordinates recursively as in [source reference]. The initial coordinate has norm zero. If norm(Ar)_n≤ Knorm r_∞, submultiplicativity of operator norms gives norm(Ar)_n+1 · ≤norms_n(normr_n+norm(Ar)_nnormρ^X_n)
Exact mathematical passage
\partial A=I_R,\qquad \norm A\le K.
I_{R_j}Use in source · 1 passage
Search aliases: I_R_j
Use in source · 03_towers.tex, line 431
Set L_j,0=I_R_j and L_j,N+1=J_jL_j+1,N. We prove, simultaneously for all starting scales j, that k+1<2^N Longrightarrow (L_j,Nz)_k=0 (z∈ R_j+N). For N=0, the inequality k+1<1 has no solution. Suppose the assertion holds at depth N. At depth N+1, an even coordinate vanishes by the displayed lift formula. For an odd coordinate k=2m+1, the hypothesis gives 2(m+1)<2^N+1, m+1<2^N. The induction hypothesis at starting scale j+1 makes (L_j+1,Nz)_m zero. Composing that zero operator with s^(j)_2m gives …
Exact mathematical passage
L_{j,0}=I_{R_j}
I_{R_N}Use in source · 1 passage
Search aliases: I_R_N
Use in source · 03_towers.tex, line 352
j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of C_jJ_j=I gives D_NL_N=I_R_N. This cancellation takes place between adjacent maps with matching domains. No identification of different residual spaces is needed.
Exact mathematical passage
D_NL_N=I_{R_N}.
I_\VUse in source · 1 passage
Search aliases: I_V V
Use in source · 04_symmetry.tex, line 349
[Equivalence of section spaces] Pullback induces Γ(F)≃Γ(H^*F). If U⊆ObW is closed under isomorphism and U'=H^-1U, it also induces Ext_U^F(s)≃Ext_U'^H^*F(H^*s). Existence and uniqueness of the prescribed extension are preserved and reflected. Write the unit of the equivalence as η:I_V⇒ LH, and take the unit and counit with their triangle identities. Set t_w=ε_w^-1:w→ HLw. Given y∈Γ(H^*F), define (Ry)_w=F(t_w)y_Lw. For an arrow f:w→ w', naturality of the inverse counit states HL(f) t_w=t_w'f. Cont…
Exact mathematical passage
\eta:I_\V\Rightarrow LH
\id_AUse in source · 1 passage
Search aliases: id_A id
Use in source · 05b_transport_cost.tex, line 9
Extended directed distance. Let C be a category and let W assign each arrow a value in the extended nonnegative reals [0,∞]. Suppose W(id_A)=0, W(gf)≤ W(f)+W(g). The order of ordinary composition is first f, then g. Define d_W(A,B)=inf_f:A→ BW(f). The infimum of an empty family is ∞. No assumption of connectedness, finite cost, symmetry, or attainment is made.
Exact mathematical passage
W(\id_A)=0,\qquad W(gf)\le W(f)+W(g).
\id_FUse in source · 1 passage
Search aliases: id_F id
Use in source · 01_views.tex, line 62
[Functorial transport] These induced maps preserve identities and composition: Γ(id_F)=id_Γ(F), Γ(βα)=Γ(β)Γ(α). Evaluate each side at an arbitrary view and apply the corresponding identity for the coordinate maps. Theorem~[source reference] then identifies the maps on the whole realization set. sourceKS02 An arbitrary collection of fiber maps need not define a map on realizations. Its compatibility equations are part of the theorem, not a convention that can be omitted after a notation has been …
Exact mathematical passage
\Gamma(\id_F)=\id_{\Gamma(F)},\qquad \Gamma(\beta\alpha)=\Gamma(\beta)\Gamma(\alpha).
\id_\WUse in source · 1 passage
Search aliases: id_W id W
Use in source · 04_symmetry.tex, line 339
chapEquivalent presentations and identified boundaries Transporting the same question. Let H:V→W be part of a category equivalence, with inverse L and counit ε:HL⇒id_W. Pull a view system F on W back along H. Sampling a global section gives (H^*x)_v=x_Hv.
Exact mathematical passage
\varepsilon:HL\Rightarrow\id_\W
\id_{\Gamma(F)}Use in source · 1 passage
Search aliases: id_Γ(F) id Gamma
Use in source · 01_views.tex, line 62
[Functorial transport] These induced maps preserve identities and composition: Γ(id_F)=id_Γ(F), Γ(βα)=Γ(β)Γ(α). Evaluate each side at an arbitrary view and apply the corresponding identity for the coordinate maps. Theorem~[source reference] then identifies the maps on the whole realization set. sourceKS02 An arbitrary collection of fiber maps need not define a map on realizations. Its compatibility equations are part of the theorem, not a convention that can be omitted after a notation has been …
Exact mathematical passage
\Gamma(\id_F)=\id_{\Gamma(F)},\qquad \Gamma(\beta\alpha)=\Gamma(\beta)\Gamma(\alpha).
\iffUse in source · 2 passages
Search aliases: iff
Use in source · 00_notation.tex, line 33
The main statements have the following form: Tor(G_A) · =D, · G_A/D · ≅ Q, G_A/mG_A · ≅ C_rad(m), · widehatG_A · ≅ P, G_A≅ G_B · iff A=^* B, · · G_A has an X-computable copy · iff A≤_T X. · · The extension by D does not split. Distinct eventual selector classes do not embed in one another. All of these conclusions are proved within the book, with the exact hypotheses retained.
Exact mathematical passage
\Tor(G_A)&=D,&G_A/D&\cong\Q,\\ G_A/mG_A&\cong C_{\rad(m)},&\widehat{G_A}&\cong P,\\ G_A\cong G_B&\iff A\eqfin B,&&\\ G_A\text{ has an }X\text{-computable copy}&\iff A\le_T X.&&
\inf_{b\in B}Use in source · 1 passage
Search aliases: inf_b∈ B inf in
Use in source · 05d_floors.tex, line 7
Distance to the actual correction image. Let R be a normed additive group and B≤ R an additive subgroup. In the protected linear model, B=d(ker p). For a residual Ω∈ R, define ν_B(Ω)=inf_b∈ BnormΩ-b. The subgroup contains zero, so this is a finite nonnegative real number. It is an infimum, not an assertion that a nearest admissible residual exists.
Exact mathematical passage
\nu_B(\Omega)=\inf_{b\in B}\norm{\Omega-b}.
\inf_{f:A\to B}Use in source · 1 passage
Search aliases: inf_f:A→ B inf to
Use in source · 05b_transport_cost.tex, line 13
Extended directed distance. Let C be a category and let W assign each arrow a value in the extended nonnegative reals [0,∞]. Suppose W(id_A)=0, W(gf)≤ W(f)+W(g). The order of ordinary composition is first f, then g. Define d_W(A,B)=inf_f:A→ BW(f). The infimum of an empty family is ∞. No assumption of connectedness, finite cost, symmetry, or attainment is made.
Exact mathematical passage
d_W(A,B)=\inf_{f:A\to B}W(f).
\inf_{\norm a\le M}Use in source · 1 passage
Search aliases: inf_norm a≤ M inf norm le
Use in source · 05g_budget_certificates.tex, line 3
chapThe attainable residual at a fixed correction budget The unrestricted floor dist(r,ran T) forgets how large a correction must be. To retain that information, fix a nonnegative budget M and define β_T,r(M)=inf_norm a≤ Mnormr-Ta. The norm is the actual correction norm in H=ker p. It is not the norm of arbitrary coordinates used to represent that kernel. The function β describes the smallest residual compatible with a declared amount of allowed change.
Exact mathematical passage
\beta_{T,r}(M)=\inf_{\norm a\le M}\norm{r-Ta}.
\inf_{z\in C}Use in source · 1 passage
Search aliases: inf_z∈ C inf in
Use in source · 12_proof_tools.tex, line 16
d convex, and fix x∈ H. There is a unique c∈ C minimizing normx-c. It is characterized by ⟨ x-c,z-c⟩≤0 (z∈ C). If C=c_0+L is a closed affine subspace with direction space L, the condition is x-c⊥ L. Produce a Cauchy minimizing sequence. Put d=inf_z∈ Cnormx-z. Nonemptiness makes this a finite nonnegative real number. Choose c_n∈ C so that normx-c_n→ d. Convexity places (c_n+c_m)/2 in C. The parallelogram identity gives normc_n-c_m^2 =2normx-c_n^2+2normx-c_m^2 -4 x-fracc_n+c_m2 ^2 ≤2normx-c_n^2+2n…
Exact mathematical passage
d=\inf_{z\in C}\norm{x-z}
J_0Use in source · 1 passage
Search aliases: J_0
Use in source · 03_towers.tex, line 325
A whole cascade. At scale j, let C_j:R_j→ R_j+1 and J_j:R_j+1→ R_j be bounded linear maps with C_jJ_j=I. For x∈ R_0, define x_0=x, x_j+1=C_jx_j, L_0=I, L_N=J_0⋯ J_N-1, h_j=x_j-J_jx_j+1, Q_Nx=L_Nx_N, S_N=I-Q_N. The map Q_N restores the surviving coarse remainder to the original space. It is this restored remainder, not just x_N, that controls reconstruction.
Exact mathematical passage
x_0=x,\quad x_{j+1}=C_jx_j,\quad L_0=I,\quad L_N=J_0\cdots J_{N-1},
J_{\mathrm{comp}}Use in source · 2 passages
Search aliases: J_comp mathrm
Use in source · 04_symmetry.tex, line 290
[The composition topology] Condition~[source reference] defines a Grothendieck topology J_comp. The sieve generated by a family of inclusions U_i⊆ W is covering exactly when the family covers every composition support in W. In the inclusion poset, identify a sieve on W with a downward-closed collection of subsets of W. Pullback along Z⊆ W consists of those subsets of Z that were already members of the original sieve. We verify the three topology axioms using this description.
Exact mathematical passage
J_{\mathrm{comp}}
J_{M-1}Use in source · 1 passage
Search aliases: J_M-1
Use in source · 03_towers.tex, line 367
Idempotence follows from D_NL_N=I: Q_N^2=L_N(D_NL_N)D_N=Q_N. For M≥ N, put C_N,M=C_M-1⋯ C_N and J_N,M=J_N⋯ J_M-1, using identities for M=N. Then D_M=C_N,MD_N, L_M=L_NJ_N,M, C_N,MJ_N,M=I. The identities D_NL_M=J_N,M and D_ML_N=C_N,M now give Q_NQ_M=L_NJ_N,MD_M=Q_M, Q_MQ_N=L_MC_N,MD_N=Q_M. Both orders have been checked, rather than inferred from one range inclusion.
Exact mathematical passage
J_{N,M}=J_N\cdots J_{M-1}
J_NUse in source · 1 passage
Search aliases: J_N
Use in source · 03_towers.tex, line 367
Idempotence follows from D_NL_N=I: Q_N^2=L_N(D_NL_N)D_N=Q_N. For M≥ N, put C_N,M=C_M-1⋯ C_N and J_N,M=J_N⋯ J_M-1, using identities for M=N. Then D_M=C_N,MD_N, L_M=L_NJ_N,M, C_N,MJ_N,M=I. The identities D_NL_M=J_N,M and D_ML_N=C_N,M now give Q_NQ_M=L_NJ_N,MD_M=Q_M, Q_MQ_N=L_MC_N,MD_N=Q_M. Both orders have been checked, rather than inferred from one range inclusion.
Exact mathematical passage
J_{N,M}=J_N\cdots J_{M-1}
J_{N,M}Use in source · 4 passages
Search aliases: J_N,M
Use in source · 03_towers.tex, line 367
Idempotence follows from D_NL_N=I: Q_N^2=L_N(D_NL_N)D_N=Q_N. For M≥ N, put C_N,M=C_M-1⋯ C_N and J_N,M=J_N⋯ J_M-1, using identities for M=N. Then D_M=C_N,MD_N, L_M=L_NJ_N,M, C_N,MJ_N,M=I. The identities D_NL_M=J_N,M and D_ML_N=C_N,M now give Q_NQ_M=L_NJ_N,MD_M=Q_M, Q_MQ_N=L_MC_N,MD_N=Q_M. Both orders have been checked, rather than inferred from one range inclusion.
Exact mathematical passage
J_{N,M}=J_N\cdots J_{M-1}
J_{N-1}Use in source · 1 passage
Search aliases: J_N-1
Use in source · 03_towers.tex, line 325
A whole cascade. At scale j, let C_j:R_j→ R_j+1 and J_j:R_j+1→ R_j be bounded linear maps with C_jJ_j=I. For x∈ R_0, define x_0=x, x_j+1=C_jx_j, L_0=I, L_N=J_0⋯ J_N-1, h_j=x_j-J_jx_j+1, Q_Nx=L_Nx_N, S_N=I-Q_N. The map Q_N restores the surviving coarse remainder to the original space. It is this restored remainder, not just x_N, that controls reconstruction.
Exact mathematical passage
x_0=x,\quad x_{j+1}=C_jx_j,\quad L_0=I,\quad L_N=J_0\cdots J_{N-1},
J_\alphaUse in source · 14 passages
Search aliases: J_α alpha
Use in source · 05h_regularized_repair.tex, line 9
The normal equation is derived in the protected space. For α>0, define on H=ker p J_α(a)=normTa-r^2+αnorm a^2. Both norms are the declared Hilbert norms. All minimizers below are corrections in H, so preservation is exact for every parameter, not just in the limit. This is ordinary quadratic Tikhonov regularization applied to the actual protected correction operator. The standard regularization background is discussed in [source reference]. The proof here does not require T to be compact.
Exact mathematical passage
J_\alpha(a)=\norm{Ta-r}^2+\alpha\norm a^2.
J_{\alpha_n}Use in source · 1 passage
Search aliases: J_α_n alpha
Use in source · 05j_inexact_certificates.tex, line 76
Why a vanishing normal residual is not enough. Let T=0 on R, let r=0, and take α_n=n^-2 for n≥1. The exact optimizer is zero for every such n. The fixed proposal a_n=1 instead has v_n=0, g_n=α_n⟶0, J_α_n(a_n)=α_n⟶0, absa_n-a_α_n=1. Thus the fit residual, the normal residual, and even the absolute objective value can all tend to zero without convergence to the selected optimizer. The certificate correctly reports absg_n/α_n=1.
Exact mathematical passage
v_n=0,\qquad g_n=\alpha_n\longrightarrow0,\qquad J_{\alpha_n}(a_n)=\alpha_n\longrightarrow0, \qquad\abs{a_n-a_{\alpha_n}}=1.
\KUse in source · 6 passages
Search aliases: K
Use in source · 06_boundaries.tex, line 72
For a finite linear system over a field, the same question admits independently checkable positive and negative certificates. Let A∈K^m× n encode compatibility, P∈K^p× n the protected values, and x∈K^n the raw candidate. Let the columns of N be a basis of ker P. The correction equation is ANz=Ax, y=x-Nz. It is equivalent to the stacked system Ay=0, Py=Px.
Exact mathematical passage
A\in\K^{m\times n}
\mathcal KUse in source · 4 passages
Search aliases: K mathcal
Use in source · 06_boundaries.tex, line 11
Internal realization and intended objects. Suppose a separate category S contains the intended semantic objects of an application, and a realization construction gives a comparison functor R: S⟶ K, where K is the relevant category or groupoid of coherent data. Defining K does not establish that R is faithful, full, or essentially surjective. These are new mathematical questions about the actual semantic comparison.
Exact mathematical passage
\mathcal R:\mathcal S\longrightarrow\mathcal K,
\K^{1\times m}Use in source · 1 passage
Search aliases: K^1× m K times
Use in source · 06_boundaries.tex, line 82
[A finite certificate alternative] Exactly one of the following occurs: item There is y with Ay=0 and Py=Px. The full answer set is y+H, where H=ker A∩ker P. item There are row vectors λ∈K^1× m and μ∈K^1× p with λ A=μ P, λ Ax=1. The negative witness annihilates every admissible correction defect, since λ AN=μ PN=0. The ambiguity dimension in the positive case is dim H=n-rankA
Exact mathematical passage
\lambda\in\K^{1\times m}
\K^{1\times p}Use in source · 1 passage
Search aliases: K^1× p K times
Use in source · 06_boundaries.tex, line 82
[A finite certificate alternative] Exactly one of the following occurs: item There is y with Ay=0 and Py=Px. The full answer set is y+H, where H=ker A∩ker P. item There are row vectors λ∈K^1× m and μ∈K^1× p with λ A=μ P, λ Ax=1. The negative witness annihilates every admissible correction defect, since λ AN=μ PN=0. The ambiguity dimension in the positive case is dim H=n-rankA
Exact mathematical passage
\mu\in\K^{1\times p}
k^2Use in source · 1 passage
Search aliases: k^2
Use in source · 05_geometry.tex, line 203
Every other correction can be written uniquely as h_*+k with k∈ker B. Orthogonality gives normh_*+k^2=normh_*^2+norm k^2. Writing y_*=x-h_* and y=x-h_*-k, we have x-y=h_*+k and y_*-y=k. Substitution proves the stated Pythagorean identity. Its last term vanishes only when y=y_*, establishing the unique closest answer.
Exact mathematical passage
\norm{h_*+k}^2=\norm{h_*}^2+\norm k^2.
K^3Use in source · 1 passage
Search aliases: K^3
Use in source · 05i_diagonal_regularization.tex, line 24
is the unique regularized correction at α=m^-2. Set A_m=norma^(m)^2=∑_n≥1fracm^4(m^2+n^2)^2, B_m=normv^(m)^2=∑_n≥1fracn^2(m^2+n^2)^2. Then frac m4≤ A_m≤frac4m3, frac14(m+1)≤ B_m≤frac2m. For the sums A_m,K,B_m,K through any integer K≥ m, 0≤ A_m-A_m,K≤fracm^43K^3, 0≤ B_m-B_m,K≤frac1K. For n>m, one has (a_n^(m))^2≤ m^4/n^4, so the correction sequence is square-summable. The diagonal operator is bounded with norm one, and w=(1/n)∈ℓ^2, hence the residual sequence is also in ℓ^2. The operator is self-…
Exact mathematical passage
\label{eq:soft-tails} 0\le A_m-A_{m,K}\le\frac{m^4}{3K^3},\qquad 0\le B_m-B_{m,K}\le\frac1K.
\K^{m+p}Use in source · 1 passage
Search aliases: K^m+p K
Use in source · 06_boundaries.tex, line 99
m has zero left side and nonzero right side, assign arbitrary values to its free variables and solve for its pivot variables. This supplies a solution y_0. For any vector y, Ty=b Longleftrightarrow T(y-y_0)=0. Translation by y_0 is therefore a bijection from ker T to the answer set, with inverse y↦ y-y_0. Rank--nullity applied to the actual linear map T:K^n→K^m+p gives the ambiguity dimension. Dependent prescription rows do not alter this argument.
Exact mathematical passage
T:\K^n\to\K^{m+p}
\K^{m\times n}Use in source · 1 passage
Search aliases: K^m× n K times
Use in source · 06_boundaries.tex, line 72
For a finite linear system over a field, the same question admits independently checkable positive and negative certificates. Let A∈K^m× n encode compatibility, P∈K^p× n the protected values, and x∈K^n the raw candidate. Let the columns of N be a basis of ker P. The correction equation is ANz=Ax, y=x-Nz. It is equivalent to the stacked system Ay=0, Py=Px.
Exact mathematical passage
A\in\K^{m\times n}
\K^nUse in source · 2 passages
Search aliases: K^n K
Use in source · 06_boundaries.tex, line 72
For a finite linear system over a field, the same question admits independently checkable positive and negative certificates. Let A∈K^m× n encode compatibility, P∈K^p× n the protected values, and x∈K^n the raw candidate. Let the columns of N be a basis of ker P. The correction equation is ANz=Ax, y=x-Nz. It is equivalent to the stacked system Ay=0, Py=Px.
Exact mathematical passage
x\in\K^n
\K^{p\times n}Use in source · 1 passage
Search aliases: K^p× n K times
Use in source · 06_boundaries.tex, line 72
For a finite linear system over a field, the same question admits independently checkable positive and negative certificates. Let A∈K^m× n encode compatibility, P∈K^p× n the protected values, and x∈K^n the raw candidate. Let the columns of N be a basis of ker P. The correction equation is ANz=Ax, y=x-Nz. It is equivalent to the stacked system Ay=0, Py=Px.
Exact mathematical passage
P\in\K^{p\times n}
K^\perpUse in source · 11 passages
Search aliases: K^⊥ perp
Use in source · 05_geometry.tex, line 132
ntitative exactness] The following are equivalent: item ran D is closed. item There is C≥0 with normx-qx≤ CnormDx for all x. item There is C≥0 such that every x has some exact y with normx-y≤ CnormDx. When the range is closed, the least admissible constant is the operator norm of the inverse of D:K^⊥→ran D. In the zero operator case that inverse has zero dimensional domain and its norm is zero. Let K=ker D and K^⊥ carry the induced Hilbert norm. Define D_⊥ u=Du, but regard its target as ran D wi…
Exact mathematical passage
D:K^\perp\to\ran D
K_{\mathfrak H}Use in source · 1 passage
Search aliases: K_ H mathfrak
Use in source · 10_grammar.tex, line 57
Monotonicity under grammar extension. [Admissible expansion] Suppose every valid description in G translates to a valid description in H with the same semantic group. Then every G-realizable group is H-realizable. If every translated description has length at most f(lvert drvert), then K_ H(G)≤ f(K_ G(G)) whenever K_ G(G) is finite. Translate a shortest valid G description of G. Its semantic output remains G, and its length has the stated bound. An obstruction to a small grammar can therefore di…
Exact mathematical passage
K_{\mathfrak H}(G)\le f(K_{\mathfrak G}(G))
k_0Use in source · 2 passages
Search aliases: k_0
Use in source · 05_geometry.tex, line 200
Let k_0=P^M_ker Bh_0 and h_*=h_0-k_0. Then Bh_*=Bh_0=Dx and h_*⊥ker B in M. If a different initial correction is chosen, it differs from h_0 by an element of ker B. Its orthogonal-complement component is therefore again h_*. This proves that the construction is independent of the feasibility witness.
Exact mathematical passage
k_0=P^M_{\ker B}h_0
k_iUse in source · 1 passage
Search aliases: k_i
Use in source · 03a_observation.tex, line 117
For surjectivity, let F be finitely observed. Choose one pair (k_n,a_n) at each output level, satisfying F(x)_n=a_n(x_k_n) for every coherent input. Define ϕ(n)=max_0≤ i≤ nk_i, f_n=a_nρ^X_k_n,ϕ(n). The maximum is of a nonempty finite set. It gives a monotone schedule with k_n≤ϕ(n). For every point x, coherence in X gives f_n(x_ϕ(n)) =a_n(ρ^X_k_n,ϕ(n)x_ϕ(n)) =a_n(x_k_n)=F(x)_n. tag*
Exact mathematical passage
\phi(n)=\max_{0\le i\le n}k_i, \qquad f_n=a_n\rho^X_{k_n,\phi(n)}.
k_nUse in source · 11 passages
Search aliases: k_n
Use in source · 03a_observation.tex, line 116
For surjectivity, let F be finitely observed. Choose one pair (k_n,a_n) at each output level, satisfying F(x)_n=a_n(x_k_n) for every coherent input. Define ϕ(n)=max_0≤ i≤ nk_i, f_n=a_nρ^X_k_n,ϕ(n). The maximum is of a nonempty finite set. It gives a monotone schedule with k_n≤ϕ(n). For every point x, coherence in X gives f_n(x_ϕ(n)) =a_n(ρ^X_k_n,ϕ(n)x_ϕ(n)) =a_n(x_k_n)=F(x)_n. tag*
Exact mathematical passage
(k_n,a_n)
\mathcal LUse in source · 1 passage
Search aliases: L mathcal
Use in source · 03_towers.tex, line 92
chapBanach repair and constructed split coordinates An actual complete candidate space. Let X_n,Y_n be Banach spaces over a complete nontrivially normed field, with continuous linear bonding maps and uniform bounds normρ^X_n≤ B_X, normρ^Y_n≤ B_Y. Use the spaces of bounded operator families E=ℓ^∞_n L(X_n,Y_n), R=ℓ^∞_n L(X_n+1,Y_n). The source formalization derives the required completeness from the operator spaces and their bounded product. The same differential satisfies norm∂ f_∞≤(B_X+B_Y)norm …
Exact mathematical passage
E=\ell^\infty_n\mathcal L(X_n,Y_n),\qquad R=\ell^\infty_n\mathcal L(X_{n+1},Y_n).
lUse in source · 2 passages
Search aliases: l
Use in source · 05h_regularized_repair.tex, line 197
[3pt] q=0 · No · Residual vanishes, correction norms diverge, no exact protected answer. [3pt] q>0 · Yes · Strong convergence to the least-squares correction of minimum norm, not an exact answer. [3pt] q>0 · No · A positive residual floor and diverging correction norms occur together. The last case is not hypothetical. Let Tu=(u_n/n) on ℓ^2, define Su=(Tu,0) into ℓ^2⊕R, and take r=((1/n),1). Then b=((1/n),0) is in the closure but not the range, while q=1. The same escaping corrections as before …
Exact mathematical passage
3pt] $q=0$ & No & Residual vanishes, correction norms diverge, no exact protected answer.\\[3pt] $q>0$ & Yes & Strong convergence to the least-squares correction of minimum norm, not an exact answer.\\[3pt] $q>0$ & No & A positive residual floor and diverging correction norms occur together.\\\bottomrule \end{tabularx} \end{center} The last case is not hypothetical. Let $Tu=(u_n/n)$ on $\ell^2$, define $Su=(Tu,0)$ into $\ell^2\oplus\R$, and take $r=((1/n),1)$. Then $b=((1/n),0)$ is in the closure but not the range, while $q=1$. The same escaping corrections as before coexist with an irreducible orthogonal residual. Positive floor and unbounded correction cost are independent features. \subsection{Noise is controlled at a fixed parameter} \fixedplate{34_noise_filter}{For one singular value $\sigma$, the response to residual-data noise is $\sigma/(\sigma^2+\alpha)$. Its maximum occurs at $\sigma=\sqrt\alpha$ and equals $1/(2\sqrt\alpha)$. The plot evaluates this formula and the universal horizontal bounds at three explicit positive parameters.}{fig:noise-filter} \begin{theorem}[Data sensitivity and a convergent parameter choice]\label{WR09} For two residual data vectors $r,\widetilde r$ with $\norm{r-\widetilde r}\le\delta$ and the same $T,\alpha$, \[ \norm{a_\alpha(r)-a_\alpha(\widetilde r)} \le\frac{\delta}{2\sqrt\alpha},\qquad \norm{T(a_\alpha(r)-a_\alpha(\widetilde r))}\le\delta.
L^3Use in source · 2 passages
Search aliases: L^3
Use in source · 05i_diagonal_regularization.tex, line 99
The tails require a bound independent of sufficiently fine discretization. Since N≥ mL/2, the inequalities f(x)≤ x^-4 and g(x)≤ x^-2 give frac1m∑_n>Nf(n/m) ≤ m^3∑_n>Nn^-4≤fracm^33N^3 ≤frac83L^3, frac1m∑_n>Ng(n/m) ≤ m∑_n>Nn^-2≤frac mN≤frac2L. The integral tails are at most 1/(3L^3) and 1/L. Given an error tolerance, choose L first to control both discrete and integral tails, then choose m to control the compact-interval sums. This justifies passing from finite-interval Riemann convergence to the …
Exact mathematical passage
\frac1m\sum_{n>N}f(n/m) \le m^3\sum_{n>N}n^{-4}\le\frac{m^3}{3N^3} \le\frac8{3L^3},
L_0Use in source · 1 passage
Search aliases: L_0
Use in source · 03_towers.tex, line 325
A whole cascade. At scale j, let C_j:R_j→ R_j+1 and J_j:R_j+1→ R_j be bounded linear maps with C_jJ_j=I. For x∈ R_0, define x_0=x, x_j+1=C_jx_j, L_0=I, L_N=J_0⋯ J_N-1, h_j=x_j-J_jx_j+1, Q_Nx=L_Nx_N, S_N=I-Q_N. The map Q_N restores the surviving coarse remainder to the original space. It is this restored remainder, not just x_N, that controls reconstruction.
Exact mathematical passage
x_0=x,\quad x_{j+1}=C_jx_j,\quad L_0=I,\quad L_N=J_0\cdots J_{N-1},
L_{0,N}Use in source · 2 passages
Search aliases: L_0,N
Use in source · 03_towers.tex, line 442
Now Q_Nr=L_0,Nx_N, so the support assertion applies to the restored remainder. Since S_Nr=r-Q_Nr, it gives the reconstruction equation. Finally, if two residuals have identical details at every scale, the finite telescope makes their S_N reconstructions equal for every N. At any fixed coordinate choose N with k+1<2^N. The reconstruction equation then makes their original coordinates equal. Hence the residuals are equal. sourceKS28 The strict inequality matters: the next coordinate, k=2^N-1, can …
Exact mathematical passage
Q_Nr=L_{0,N}x_N
l_bUse in source · 1 passage
Search aliases: l_b
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
l_cUse in source · 1 passage
Search aliases: l_c
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
l_dUse in source · 1 passage
Search aliases: l_d
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
l_gUse in source · 1 passage
Search aliases: l_g
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
l_iUse in source · 1 passage
Search aliases: l_i
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
L_jUse in source · 5 passages
Search aliases: L_j
Use in source · 03_towers.tex, line 337
[Finite reconstruction and nested remainders] For every N, x=Q_Nx+∑_j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of C_jJ_j=I gives D_NL_N=I_R_N. This cancellation takes place between adjacent maps with matching domains. No identification of different residual spaces is needed…
Exact mathematical passage
x=Q_Nx+\sum_{j<N}L_jh_j, \qquad S_Nx=\sum_{j<N}L_jh_j.
L_{j+1}Use in source · 1 passage
Search aliases: L_j+1
Use in source · 03_towers.tex, line 357
Since L_j+1=L_jJ_j and x_j+1=C_jx_j, L_jh_j=L_j(x_j-J_jx_j+1)=Q_jx-Q_j+1x. Summing from j=0 to N-1 leaves x-Q_Nx because Q_0=I. This proves both finite reconstruction identities, including N=0, when the detail sum is empty.
Exact mathematical passage
L_{j+1}=L_jJ_j
L_{j+1,N}Use in source · 3 passages
Search aliases: L_j+1,N
Use in source · 03_towers.tex, line 431
Set L_j,0=I_R_j and L_j,N+1=J_jL_j+1,N. We prove, simultaneously for all starting scales j, that k+1<2^N Longrightarrow (L_j,Nz)_k=0 (z∈ R_j+N). For N=0, the inequality k+1<1 has no solution. Suppose the assertion holds at depth N. At depth N+1, an even coordinate vanishes by the displayed lift formula. For an odd coordinate k=2m+1, the hypothesis gives 2(m+1)<2^N+1, m+1<2^N. The induction hypothesis at starting scale j+1 makes (L_j+1,Nz)_m zero. Composing that zero operator with s^(j)_2m gives …
Exact mathematical passage
L_{j,N+1}=J_jL_{j+1,N}
L_{j,0}Use in source · 1 passage
Search aliases: L_j,0
Use in source · 03_towers.tex, line 431
Set L_j,0=I_R_j and L_j,N+1=J_jL_j+1,N. We prove, simultaneously for all starting scales j, that k+1<2^N Longrightarrow (L_j,Nz)_k=0 (z∈ R_j+N). For N=0, the inequality k+1<1 has no solution. Suppose the assertion holds at depth N. At depth N+1, an even coordinate vanishes by the displayed lift formula. For an odd coordinate k=2m+1, the hypothesis gives 2(m+1)<2^N+1, m+1<2^N. The induction hypothesis at starting scale j+1 makes (L_j+1,Nz)_m zero. Composing that zero operator with s^(j)_2m gives …
Exact mathematical passage
L_{j,0}=I_{R_j}
L_{j,N}Use in source · 2 passages
Search aliases: L_j,N
Use in source · 03_towers.tex, line 432
Set L_j,0=I_R_j and L_j,N+1=J_jL_j+1,N. We prove, simultaneously for all starting scales j, that k+1<2^N Longrightarrow (L_j,Nz)_k=0 (z∈ R_j+N). For N=0, the inequality k+1<1 has no solution. Suppose the assertion holds at depth N. At depth N+1, an even coordinate vanishes by the displayed lift formula. For an odd coordinate k=2m+1, the hypothesis gives 2(m+1)<2^N+1, m+1<2^N. The induction hypothesis at starting scale j+1 makes (L_j+1,Nz)_m zero. Composing that zero operator with s^(j)_2m gives …
Exact mathematical passage
k+1<2^N\quad\Longrightarrow\quad (L_{j,N}z)_k=0 \qquad(z\in R_{j+N}).
L_{j,N+1}Use in source · 1 passage
Search aliases: L_j,N+1
Use in source · 03_towers.tex, line 431
Set L_j,0=I_R_j and L_j,N+1=J_jL_j+1,N. We prove, simultaneously for all starting scales j, that k+1<2^N Longrightarrow (L_j,Nz)_k=0 (z∈ R_j+N). For N=0, the inequality k+1<1 has no solution. Suppose the assertion holds at depth N. At depth N+1, an even coordinate vanishes by the displayed lift formula. For an odd coordinate k=2m+1, the hypothesis gives 2(m+1)<2^N+1, m+1<2^N. The induction hypothesis at starting scale j+1 makes (L_j+1,Nz)_m zero. Composing that zero operator with s^(j)_2m gives …
Exact mathematical passage
L_{j,N+1}=J_jL_{j+1,N}
l_lUse in source · 1 passage
Search aliases: l_l
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
L_MUse in source · 3 passages
Search aliases: L_M
Use in source · 03_towers.tex, line 368
Idempotence follows from D_NL_N=I: Q_N^2=L_N(D_NL_N)D_N=Q_N. For M≥ N, put C_N,M=C_M-1⋯ C_N and J_N,M=J_N⋯ J_M-1, using identities for M=N. Then D_M=C_N,MD_N, L_M=L_NJ_N,M, C_N,MJ_N,M=I. The identities D_NL_M=J_N,M and D_ML_N=C_N,M now give Q_NQ_M=L_NJ_N,MD_M=Q_M, Q_MQ_N=L_MC_N,MD_N=Q_M. Both orders have been checked, rather than inferred from one range inclusion.
Exact mathematical passage
D_M=C_{N,M}D_N,\quad L_M=L_NJ_{N,M},\quad C_{N,M}J_{N,M}=I.
l_mUse in source · 1 passage
Search aliases: l_m
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
L_NUse in source · 9 passages
Search aliases: L_N
Use in source · 03_towers.tex, line 325
A whole cascade. At scale j, let C_j:R_j→ R_j+1 and J_j:R_j+1→ R_j be bounded linear maps with C_jJ_j=I. For x∈ R_0, define x_0=x, x_j+1=C_jx_j, L_0=I, L_N=J_0⋯ J_N-1, h_j=x_j-J_jx_j+1, Q_Nx=L_Nx_N, S_N=I-Q_N. The map Q_N restores the surviving coarse remainder to the original space. It is this restored remainder, not just x_N, that controls reconstruction.
Exact mathematical passage
x_0=x,\quad x_{j+1}=C_jx_j,\quad L_0=I,\quad L_N=J_0\cdots J_{N-1},
l_nUse in source · 1 passage
Search aliases: l_n
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
l_tUse in source · 1 passage
Search aliases: l_t
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
l_wUse in source · 1 passage
Search aliases: l_w
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
\lceilUse in source · 4 passages
Search aliases: lceil
Use in source · 05i_diagonal_regularization.tex, line 155
[An explicit noisy attainable repair] For the attainable datum b=(1/n^2), suppose normwidetilde b-b≤δ with 0<δ≤1. Choose m=lceilδ^-2/3rceil, α=m^-2. Then the noisy regularized correction satisfies norma_α(widetilde b)-a^† ≤sqrtfrac2m+fracδ m2 ≤(sqrt2+1)δ^1/3. The exact-data bias has coordinates n/(m^2+n^2), so its squared norm is precisely B_m. By Proposition~[source reference], norma_m^-2(b)-a^†≤sqrt2/m. Theorem~[source reference] at the same parameter gives norma_m^-2(widetilde b)-a_m^-2(b) ≤f…
Exact mathematical passage
m=\lceil\delta^{-2/3}\rceil,\qquad\alpha=m^{-2}.
\lfloorUse in source · 1 passage
Search aliases: lfloor
Use in source · 05i_diagonal_regularization.tex, line 91
)=fracπ4. The hard prefix correction at budget M=sqrt N instead has sqrt N normw-T1_N⟶1. Step 1: control the infinite Riemann sums uniformly. Define f(x)=frac1(1+x^2)^2, g(x)=fracx^2(1+x^2)^2. Then A_m/m=m^-1∑_n≥1f(n/m) and mB_m=m^-1∑_n≥1g(n/m). Fix L≥2 and put N=lfloor mLrfloor. On [0,L], both functions are uniformly continuous. If ω_f(h) is a modulus of continuity there, comparison on the N intervals of length 1/m gives |frac1m∑_n=1^N f(n/m)-∫_0^N/mf(x) dx| ≤ Lω_f(1/m). The missing interval [N…
Exact mathematical passage
N=\lfloor mL\rfloor
\limUse in source · 9 passages
Search aliases: lim
Use in source · 03_towers.tex, line 147
Continuity of d implies d(P_Af)=lim_m r_m=0. Protection holds before taking a limit: pA=0 gives pf^(m+1)=pf^(m), hence pf^(m)=pf for every m. Continuity of p gives p(P_Af)=pf. This explains why continuity of the observable belongs to this analytic statement.
Exact mathematical passage
d(P_Af)=\lim_m r_m=0
\lim_kUse in source · 3 passages
Search aliases: lim_k lim
Use in source · 05f_bounded_realization.tex, line 37
For each finite J, Bessel's inequality and passage to the limit give ∑_j=1^Jc_j^2 =lim_k∑_j=1^J⟨ a_n_k,e_j⟩^2≤ M^2. The nondecreasing partial sums are therefore bounded. Orthogonality makes the partial vector sums ∑_j≤ Jc_je_j Cauchy, so completeness defines a∈ H_0 with norm a≤ M.
Exact mathematical passage
\sum_{j=1}^Jc_j^2 =\lim_k\sum_{j=1}^J\langle a_{n_k},e_j\rangle^2\le M^2.
\lim_mUse in source · 1 passage
Search aliases: lim_m lim
Use in source · 03_towers.tex, line 147
Continuity of d implies d(P_Af)=lim_m r_m=0. Protection holds before taking a limit: pA=0 gives pf^(m+1)=pf^(m), hence pf^(m)=pf for every m. Continuity of p gives p(P_Af)=pf. This explains why continuity of the observable belongs to this analytic statement.
Exact mathematical passage
d(P_Af)=\lim_m r_m=0
\lim_{M\to\infty}Use in source · 2 passages
Search aliases: lim_M→∞ lim to infty
Use in source · 05g_budget_certificates.tex, line 15
sidual minimum exists [Attainment and shape of the budget profile] For a bounded linear map between real Hilbert spaces, the infimum defining β(M) is attained for every M≥0. The profile is nonincreasing and convex. For 0≤ M_1≤ M_2, 0≤β(M_1)-β(M_2)≤norm T(M_2-M_1). Writing C=overlineran T, b=P_Cr, and q=normr-b, one has lim_M→∞β(M)=q. The equality β(M)=q at a finite budget is equivalent to the existence of a with Ta=b and norm a≤ M. In particular, β(M)=0 Longleftrightarrow ∃ a∈ H: Ta=r, norm a≤ M…
Exact mathematical passage
\lim_{M\to\infty}\beta(M)=q.
\lim_nUse in source · 3 passages
Search aliases: lim_n lim
Use in source · 05_geometry.tex, line 147
Now assume condition 2. To prove closedness, take an arbitrary sequence r_n∈ran D with r_n→ r in R. Choose x_n with Dx_n=r_n and put u_n=x_n-qx_n. Then u_n∈ K^⊥ and Du_n=r_n. Applying the estimate to u_n-u_m gives normu_n-u_m≤ Cnormr_n-r_m. Thus (u_n) is Cauchy in the complete space K^⊥. Let its limit be u. Continuity gives Du=lim_n r_n=r, so r∈ran D. In a normed space, a set containing the limits of all its convergent sequences is closed. This proves condition 1 without presupposing that the in…
Exact mathematical passage
Du=\lim_n r_n=r
\liminfUse in source · 4 passages
Search aliases: liminf
Use in source · 05f_bounded_realization.tex, line 27
The compactness needed for the existence argument. [Bounded sequences have weakly convergent subsequences] Every bounded sequence (a_n) in a real Hilbert space has a subsequence converging weakly to some a. Moreover, norm a≤liminf_knorma_n_k. Choose M≥0 with norma_n≤ M. If M=0, every term is zero and the conclusion is immediate. Otherwise let H_0 be the closed linear span of the countably many vectors a_n. Finite rational linear combinations of those vectors form a countable dense set, so H_0 is…
Exact mathematical passage
\norm a\le\liminf_k\norm{a_{n_k}}.
\liminf_kUse in source · 4 passages
Search aliases: liminf_k liminf
Use in source · 05f_bounded_realization.tex, line 27
The compactness needed for the existence argument. [Bounded sequences have weakly convergent subsequences] Every bounded sequence (a_n) in a real Hilbert space has a subsequence converging weakly to some a. Moreover, norm a≤liminf_knorma_n_k. Choose M≥0 with norma_n≤ M. If M=0, every term is zero and the conclusion is immediate. Otherwise let H_0 be the closed linear span of the countably many vectors a_n. Finite rational linear combinations of those vectors form a countable dense set, so H_0 is…
Exact mathematical passage
\norm a\le\liminf_k\norm{a_{n_k}}.
\linewidthUse in source · 1 passage
Search aliases: linewidth
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
\logUse in source · 1 passage
Search aliases: log
Use in source · 09_workshop.tex, line 51
Solution. The sampled value is η_ϕ(k)=2^-k. Recovering the original index gives k=log_2(n+1) on the sampled stages, so the value is still 1/(n+1). The schedule preserves vanishing but changes the meaning of the horizontal axis. No faster original process has been proved. Proposition~[source reference] is a convergence equivalence under a monotone cofinal schedule, not a rate-invariance theorem.
Exact mathematical passage
k=\log_2(n+1)
\log_2Use in source · 1 passage
Search aliases: log_2 log
Use in source · 09_workshop.tex, line 51
Solution. The sampled value is η_ϕ(k)=2^-k. Recovering the original index gives k=log_2(n+1) on the sampled stages, so the value is still 1/(n+1). The schedule preserves vanishing but changes the meaning of the horizontal axis. No faster original process has been proved. Proposition~[source reference] is a convergence equivalence under a monotone cofinal schedule, not a rate-invariance theorem.
Exact mathematical passage
k=\log_2(n+1)
m^{-1}Use in source · 5 passages
Search aliases: m^-1
Use in source · 05i_diagonal_regularization.tex, line 91
infinite problem, lim_M→∞Mβ_T,w(M)=fracπ4. The hard prefix correction at budget M=sqrt N instead has sqrt N normw-T1_N⟶1. Step 1: control the infinite Riemann sums uniformly. Define f(x)=frac1(1+x^2)^2, g(x)=fracx^2(1+x^2)^2. Then A_m/m=m^-1∑_n≥1f(n/m) and mB_m=m^-1∑_n≥1g(n/m). Fix L≥2 and put N=lfloor mLrfloor. On [0,L], both functions are uniformly continuous. If ω_f(h) is a modulus of continuity there, comparison on the N intervals of length 1/m gives |frac1m∑_n=1^N f(n/m)-∫_0^N/mf(x) dx| ≤ L…
Exact mathematical passage
A_m/m=m^{-1}\sum_{n\ge1}f(n/m)
m^{-1/2}Use in source · 1 passage
Search aliases: m^-1/2
Use in source · 05i_diagonal_regularization.tex, line 63
Thus the correction norm grows on the order of sqrt m, the residual norm decays on the order of m^-1/2, and the penalty α A_m=A_m/m^2 tends to zero. The smooth correction avoids the hard discontinuity at a prefix boundary, but it still approaches the forbidden constant sequence coordinate by coordinate.
Exact mathematical passage
m^{-1/2}
m^{-2}Use in source · 8 passages
Search aliases: m^-2
Use in source · 05i_diagonal_regularization.tex, line 5
The entire regularized family is explicit. Use Tu=(u_n/n) and r=w=(1/n) as in Section~[source reference]. Choose α=m^-2 for an integer m≥1. The normal equation in each coordinate gives a^(m)_n=fracm^2m^2+n^2, v^(m)_n=r_n-(Ta^(m))_n=fracnm^2+n^2. The repaired ambient candidate is (-a^(m),1). Its protected coordinate is one for every m.
Exact mathematical passage
\alpha=m^{-2}
M^2Use in source · 1 passage
Search aliases: M^2
Use in source · 05f_bounded_realization.tex, line 37
For each finite J, Bessel's inequality and passage to the limit give ∑_j=1^Jc_j^2 =lim_k∑_j=1^J⟨ a_n_k,e_j⟩^2≤ M^2. The nondecreasing partial sums are therefore bounded. Orthogonality makes the partial vector sums ∑_j≤ Jc_je_j Cauchy, so completeness defines a∈ H_0 with norm a≤ M.
Exact mathematical passage
\sum_{j=1}^Jc_j^2 =\lim_k\sum_{j=1}^J\langle a_{n_k},e_j\rangle^2\le M^2.
m^2Use in source · 11 passages
Search aliases: m^2
Use in source · 05i_diagonal_regularization.tex, line 6
The entire regularized family is explicit. Use Tu=(u_n/n) and r=w=(1/n) as in Section~[source reference]. Choose α=m^-2 for an integer m≥1. The normal equation in each coordinate gives a^(m)_n=fracm^2m^2+n^2, v^(m)_n=r_n-(Ta^(m))_n=fracnm^2+n^2. The repaired ambient candidate is (-a^(m),1). Its protected coordinate is one for every m.
Exact mathematical passage
a^{(m)}_n=\frac{m^2}{m^2+n^2},\qquad v^{(m)}_n=r_n-(Ta^{(m)})_n=\frac{n}{m^2+n^2}.
m^3Use in source · 4 passages
Search aliases: m^3
Use in source · 05i_diagonal_regularization.tex, line 38
For 1≤ n≤ m, m^2/(m^2+n^2)≥1/2. There are m such coordinates, so A_m≥ m/4. For the upper bound, the first m squares are at most one, and the decreasing function x^-4 gives ∑_n>mfrac1n^4≤∫_m^∞ x^-4 dx=frac13m^3. Thus A_m≤ m+m^4/(3m^3)=4m/3.
Exact mathematical passage
\sum_{n>m}\frac1{n^4}\le\int_m^\infty x^{-4}\,dx=\frac1{3m^3}.
m^4Use in source · 6 passages
Search aliases: m^4
Use in source · 05i_diagonal_regularization.tex, line 14
[Energy, residual, and certified tails] The displayed a^(m) belongs to ℓ^2 and is the unique regularized correction at α=m^-2. Set A_m=norma^(m)^2=∑_n≥1fracm^4(m^2+n^2)^2, B_m=normv^(m)^2=∑_n≥1fracn^2(m^2+n^2)^2. Then frac m4≤ A_m≤frac4m3, frac14(m+1)≤ B_m≤frac2m. For the sums A_m,K,B_m,K through any integer K≥ m, 0≤ A_m-A_m,K≤fracm^43K^3, 0≤ B_m-B_m,K≤frac1K. For n>m, one has (a_n^(m))^2≤ m^4/n^4, so the correction sequence is square-summable. The diagonal operator is bounded with norm one, and…
Exact mathematical passage
A_m=\norm{a^{(m)}}^2=\sum_{n\ge1}\frac{m^4}{(m^2+n^2)^2},\qquad B_m=\norm{v^{(m)}}^2=\sum_{n\ge1}\frac{n^2}{(m^2+n^2)^2}.
M_{21}Use in source · 1 passage
Search aliases: M_21
Use in source · 03b_repair_transport.tex, line 76
For two consecutive compatible changes (S_1,C_1), (S_2,C_2), with compatible choices of sections at the three ends, their mismatches satisfy M_21=S_2M_1+M_2C_1. If dx_j→0 in norm for a fixed change and fixed sections, the repair discrepancy tends to zero. First DS=Cd sends ker d into ker D, and DM=DSA-DBC=CdA-C=C-C=0. Thus both terms in the proposed exact coordinate have the correct codomain. Given u∈ker d, compute the residual and exact coordinates of S(u+Ar) separately: DS(u+Ar)=Cr, S(u+Ar)-BC…
Exact mathematical passage
M_{21}=S_2M_1+M_2C_1.
m_aUse in source · 1 passage
Search aliases: m_a
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
M_cUse in source · 1 passage
Search aliases: M_c
Use in source · 03b_repair_transport.tex, line 125
sectionA two-coordinate model in which the routes differ Use E=F=R^2, write candidates as (u,r), and take d(u,r)=D(u,r)=r, A(r)=B(r)=(0,r). The exact space is the horizontal axis and both repairs are P(u,r)=(u,0). For a real parameter c, set S_c(u,r)=(u+cr,r), C=I. The defect square commutes because the second coordinate is unchanged. The mismatch is M_c(r)=(cr,0). Hence PS_c(u,r)=(u+cr,0), S_cP(u,r)=(u,0). With c=2 and input (0,1), one route gives (2,0) and the other gives (0,0). Both have zero…
Exact mathematical passage
M_c(r)=(cr,0)
M_{c_1}Use in source · 1 passage
Search aliases: M_c_1
Use in source · 03b_repair_transport.tex, line 133
Two shears compose by adding their parameters. In the mismatch law this appears as M_c_2+c_1=S_c_2M_c_1+M_c_2, and S_c_2 fixes the exact horizontal axis. The identity is therefore a simple instance of the same composition rule, not an unrelated coordinate trick.
Exact mathematical passage
M_{c_2+c_1}=S_{c_2}M_{c_1}+M_{c_2}
M_{c_2}Use in source · 1 passage
Search aliases: M_c_2
Use in source · 03b_repair_transport.tex, line 133
Two shears compose by adding their parameters. In the mismatch law this appears as M_c_2+c_1=S_c_2M_c_1+M_c_2, and S_c_2 fixes the exact horizontal axis. The identity is therefore a simple instance of the same composition rule, not an unrelated coordinate trick.
Exact mathematical passage
M_{c_2+c_1}=S_{c_2}M_{c_1}+M_{c_2}
M_{c_2+c_1}Use in source · 1 passage
Search aliases: M_c_2+c_1
Use in source · 03b_repair_transport.tex, line 133
Two shears compose by adding their parameters. In the mismatch law this appears as M_c_2+c_1=S_c_2M_c_1+M_c_2, and S_c_2 fixes the exact horizontal axis. The identity is therefore a simple instance of the same composition rule, not an unrelated coordinate trick.
Exact mathematical passage
M_{c_2+c_1}=S_{c_2}M_{c_1}+M_{c_2}
M_iUse in source · 1 passage
Search aliases: M_i
Use in source · 01_contracts.tex, line 65
[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If M_j≠0 and j is not reachable, P_j annihilates the entire hull but not M. An element of the direct sum has finite support. Applying one primitive oper…
Exact mathematical passage
M=\bigoplus_{i\in I}M_i
m_iUse in source · 1 passage
Search aliases: m_i
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
M_jUse in source · 2 passages
Search aliases: M_j
Use in source · 01_contracts.tex, line 66
[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If M_j≠0 and j is not reachable, P_j annihilates the entire hull but not M. An element of the direct sum has finite support. Applying one primitive oper…
Exact mathematical passage
\Hull_{\mathcal O}(S)\subseteq \bigoplus_{j\in\operatorname{Reach}_E(F)}M_j.
M_mUse in source · 9 passages
Search aliases: M_m
Use in source · 05i_diagonal_regularization.tex, line 118
Step 3: move from regularization parameters to arbitrary budgets. Set M_m=sqrtA_m. Every positive coordinate m^2/(m^2+n^2) increases strictly with m, so A_m, and hence M_m, increases strictly. Since A_m/m→ c=π/4>0, these budgets tend to infinity and (fracM_m+1M_m)^2 =fracA_m+1/(m+1)A_m/mfracm+1m⟶1. Proposition~[source reference] identifies β(M_m)=sqrtB_m. Thus M_mβ(M_m)=sqrt(A_m/m)(mB_m)⟶ c. For every sufficiently large M, choose m with M_m≤ M≤ M_m+1. Monotonicity of β gives M_mβ(M_m+1)≤ Mβ(M)≤ …
Exact mathematical passage
M_m=\sqrt{A_m}
M_{m+1}Use in source · 5 passages
Search aliases: M_m+1
Use in source · 05i_diagonal_regularization.tex, line 119
Step 3: move from regularization parameters to arbitrary budgets. Set M_m=sqrtA_m. Every positive coordinate m^2/(m^2+n^2) increases strictly with m, so A_m, and hence M_m, increases strictly. Since A_m/m→ c=π/4>0, these budgets tend to infinity and (fracM_m+1M_m)^2 =fracA_m+1/(m+1)A_m/mfracm+1m⟶1. Proposition~[source reference] identifies β(M_m)=sqrtB_m. Thus M_mβ(M_m)=sqrt(A_m/m)(mB_m)⟶ c. For every sufficiently large M, choose m with M_m≤ M≤ M_m+1. Monotonicity of β gives M_mβ(M_m+1)≤ Mβ(M)≤ …
Exact mathematical passage
\left(\frac{M_{m+1}}{M_m}\right)^2 =\frac{A_{m+1}/(m+1)}{A_m/m}\frac{m+1}{m}\longrightarrow1.
m_NUse in source · 3 passages
Search aliases: m_N
Use in source · 02_ambient.tex, line 7
Finite stages and two connecting systems. For N≥1, define A_N=∏_i<NC_p_i, m_N=∏_i<Np_i. The Chinese remainder map sends kbmod m_N to (kbmod p_i)_i<N. It is an isomorphism, so A_N is cyclic and its all-one vector is a generator.
Exact mathematical passage
A_N=\prod_{i<N}C_{p_i},\qquad m_N=\prod_{i<N}p_i.
m_N^2Use in source · 4 passages
Search aliases: m_N^2
Use in source · 05f_bounded_realization.tex, line 129
Suppose these energies are bounded, and let L=lim_n m_n^2. Given ϵ>0, choose N with L-m_N^2<ϵ^2. For m≥ n≥ N, norma_m-a_n^2=m_m^2-m_n^2≤ L-m_N^2<ϵ^2. Thus (a_n) is Cauchy and has a norm limit a_*. For each fixed n, all terms from that index onward belong to the closed set A_n, so a_*∈ A_n. This places a_* in their entire intersection.
Exact mathematical passage
L-m_N^2<\epsilon^2
m_rUse in source · 1 passage
Search aliases: m_r
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
m_tUse in source · 1 passage
Search aliases: m_t
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
M_vUse in source · 2 passages
Search aliases: M_v
Use in source · 01_views.tex, line 139
Correspondences must also be coherent. Suppose natural maps form a span FxleftarrowpMxrightarrowqG, and fix a∈Γ(F), b∈Γ(G). At each view define the ordinary fiber B_a,b(v)={m∈ M_v:p_v(m)=a_v, q_v(m)=b_v}. Naturality and coherence provide its restrictions. A global bridge is an element of B(a,b)={z∈Γ(M):Γ(p)z=a, Γ(q)z=b}.
Exact mathematical passage
B_{a,b}(v)=\{m\in M_v:p_v(m)=a_v,\ q_v(m)=b_v\}.
M_\alphaUse in source · 5 passages
Search aliases: M_α alpha
Use in source · 05h_regularized_repair.tex, line 90
Parameter monotonicity and the fixed-budget problem. [Regularized answers lie on the budget frontier] If 0<α<γ, then norma_α≥norma_γ, normr-Ta_α≤normr-Ta_γ. At M_α=norma_α, β(M_α)=normr-Ta_α. When v_α=r-Ta_α≠0, its normalization is an optimizing witness in Theorem~[source reference] at that budget. Let A_α=norma_α^2 and R_α=normr-Ta_α^2. Comparing the two admissible minimizers in each other's objectives gives R_α+α A_α≤ R_γ+α A_γ, R_γ+γ A_γ≤ R_α+γ A_α. Add these inequalities and cancel the resid…
Exact mathematical passage
M_\alpha=\norm{a_\alpha}
\MatchUse in source · 1 passage
Search aliases: Match
Use in source · 01_views.tex, line 102
[Strict gluing] Under [source reference], restriction induces an equivalence Γ(F)≃Match(F,(U_i)). Every fixed matching family has a unique global glue. Step 1: every coordinate can be selected. Apply the cover condition to the identity arrow of v. It gives an index containing v. Choose one such index i(v) and put x_v=(s_i(v))_v. The choice is an existence construction. It is not an algorithm selecting patches in a preferred order. If v∈ U_j, matching on the overlap U_i(v)∩ U_j gives x_v=(s_j)_v.…
Exact mathematical passage
\Gamma(F)\simeq\Match(F,(U_i)).
\maxUse in source · 15 passages
Search aliases: max
Use in source · 03a_observation.tex, line 78
ctions, two reindexed maps out of X are eventually equal exactly when they induce the same realized function. Surjective adjacent bonding maps are a sufficient condition for surjective point projections. Realized equality follows from eventual equality by the preceding theorem, so only its converse remains. Suppose |f|=|g|. Fix the output level n, and let k=max{ϕ(n),ψ(n)}. To prove equality of two maps defined on X_k, fix an arbitrary element a∈ X_k. The assumed surjectivity of ev_k supplies a c…
Exact mathematical passage
k=\max\{\phi(n),\psi(n)\}
\max_{0\le i\le n}Use in source · 1 passage
Search aliases: max_0≤ i≤ n max le le
Use in source · 03a_observation.tex, line 117
For surjectivity, let F be finitely observed. Choose one pair (k_n,a_n) at each output level, satisfying F(x)_n=a_n(x_k_n) for every coherent input. Define ϕ(n)=max_0≤ i≤ nk_i, f_n=a_nρ^X_k_n,ϕ(n). The maximum is of a nonempty finite set. It gives a monotone schedule with k_n≤ϕ(n). For every point x, coherence in X gives f_n(x_ϕ(n)) =a_n(ρ^X_k_n,ϕ(n)x_ϕ(n)) =a_n(x_k_n)=F(x)_n. tag*
Exact mathematical passage
\phi(n)=\max_{0\le i\le n}k_i, \qquad f_n=a_n\rho^X_{k_n,\phi(n)}.
\max_iUse in source · 2 passages
Search aliases: max_i max
Use in source · 05_geometry.tex, line 82
The maps are linear. On the finite vertex product, normR_pz_∞≤(max_inormT_i^-1)norm z, normx_o≤norm x_∞. These bounds also hold on the subspaces K and ker d, proving continuity of both directions. No completeness is needed for these finite bounds.
Exact mathematical passage
\norm{R_pz}_\infty\le\left(\max_i\norm{T_i^{-1}}\right)\norm z, \qquad \norm{x_o}\le\norm x_\infty.
\max_{\norm z\le1}Use in source · 1 passage
Search aliases: max_norm z≤1 max norm le
Use in source · 05g_budget_certificates.tex, line 67
A dual witness can rule out every correction in the budget. [A sharp budget certificate] For every M≥0, β_T,r(M)= max_norm z≤1(⟨ r,z⟩-MnormT^*z). Thus any single z with norm z≤1 and ⟨ r,z⟩-MnormT^*z>ε certifies that no correction of norm at most M has residual at most ε. For every norm a≤ M and norm z≤1, the adjoint identity and Cauchy--Schwarz give ⟨ r,z⟩-MnormT^*z · =⟨ r-Ta,z⟩+⟨ a,T^*z⟩-MnormT^*z
Exact mathematical passage
\label{eq:budget-dual} \beta_{T,r}(M)= \max_{\norm z\le1}\bigl(\langle r,z\rangle-M\norm{T^*z}\bigr).
\midruleUse in source · 1 passage
Search aliases: midrule
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
n^{-1}Use in source · 1 passage
Search aliases: n^-1
Use in source · 05_geometry.tex, line 159
A small residual far from exactness. On ℓ^2(N), index the standard basis from one and define De_n=n^-1e_n. The map is bounded and injective, so its exact space is zero. Yet norme_n=1, normDe_n=1/n⟶0. No finite C can bound dist(e_n,ker D) by CnormDe_n. The range is not closed. For example finite truncations of (1/n)_n≥1 belong to the range and converge in ℓ^2, but a preimage of the limit would be the constant sequence, which does not belong to ℓ^2.
Exact mathematical passage
De_n=n^{-1}e_n
n^{-2}Use in source · 8 passages
Search aliases: n^-2
Use in source · 05i_diagonal_regularization.tex, line 59
Finally, for K≥ m, both omitted tails consist of nonnegative terms. Bound the first by m^4∑_n>Kn^-4 and the second by ∑_n>Kn^-2. The same upper integral comparisons give the two certified remainders in [source reference]. No finite truncation is substituted for an equality of infinite energies. writtenWR10
Exact mathematical passage
\sum_{n>K}n^{-2}
n^{-3}Use in source · 1 passage
Search aliases: n^-3
Use in source · 09_workshop.tex, line 91
Solution. The noise term is at most 1/(2n), but the solver term is at most one. Thus the displayed tolerance does not provide the required guarantee. The scalar counterexample T=0, r=0, a_n=1 meets it and does not approach the exact optimizer. Replacing it by n^-3 makes the solver term at most 1/n, so the triangle inequality and the assumed vanishing bias prove convergence. This is a sufficient worst-case rule, not a necessary condition for every particular operator.
Exact mathematical passage
n^{-3}
n^{-4}Use in source · 2 passages
Search aliases: n^-4
Use in source · 05i_diagonal_regularization.tex, line 59
Finally, for K≥ m, both omitted tails consist of nonnegative terms. Bound the first by m^4∑_n>Kn^-4 and the second by ∑_n>Kn^-2. The same upper integral comparisons give the two certified remainders in [source reference]. No finite truncation is substituted for an equality of infinite energies. writtenWR10
Exact mathematical passage
m^4\sum_{n>K}n^{-4}
N^2Use in source · 1 passage
Search aliases: N^2
Use in source · 05e_escape.tex, line 86
Escape is not an accident of this particular algorithm. Suppose an arbitrary protected candidate (u,1) satisfies normd(u,1)≤ε. On its first N coordinates, ∑_n=1^N|u_n+1|^2 ≤ N^2∑_n=1^Nfrac|u_n+1|^2n^2 ≤ N^2ε^2. The reverse triangle inequality in R^N gives norm u≥sqrt N-Nε (N≥1). If a sequence of protected candidates has residual tending to zero, then for each fixed N, its eventual correction norms are at least, for example, sqrt N/2. Taking N arbitrarily large shows that these norms must tend to…
Exact mathematical passage
\sum_{n=1}^{N}|u_n+1|^2 \le N^2\sum_{n=1}^{N}\frac{|u_n+1|^2}{n^2} \le N^2\varepsilon^2.
\N^2Use in source · 1 passage
Search aliases: N^2 N mathbb
Use in source · 10_grammar.tex, line 17
Repeated diagonal coding. Fix a computable pairing bijection ⟨ j,k⟩: N^2→ N. For a freely chosen set C⊆ N, define a new selector A_C by 1_A_C(2⟨ j,k⟩) · =1- 1_B_j(2⟨ j,k⟩),
Exact mathematical passage
\langle j,k\rangle:\N^2\to\N
n^2Use in source · 24 passages
Search aliases: n^2
Use in source · 03b_repair_transport.tex, line 186
Unbounded gaps cause a different issue. With ϕ(n)=n^2, the bounded residual family r_k=1 has block sums 2n+1. These are unbounded. Therefore block accumulation cannot be a map from every bounded residual family to a bounded sampled residual family in this example. The constant residual is not claimed to be the differential of a bounded candidate on the identity tower. It is a witness about the proposed operator on the entire residual space.
Exact mathematical passage
\phi(n)=n^2
N^3Use in source · 1 passage
Search aliases: N^3
Use in source · 05i_diagonal_regularization.tex, line 99
The tails require a bound independent of sufficiently fine discretization. Since N≥ mL/2, the inequalities f(x)≤ x^-4 and g(x)≤ x^-2 give frac1m∑_n>Nf(n/m) ≤ m^3∑_n>Nn^-4≤fracm^33N^3 ≤frac83L^3, frac1m∑_n>Ng(n/m) ≤ m∑_n>Nn^-2≤frac mN≤frac2L. The integral tails are at most 1/(3L^3) and 1/L. Given an error tolerance, choose L first to control both discrete and integral tails, then choose m to control the compact-interval sums. This justifies passing from finite-interval Riemann convergence to the …
Exact mathematical passage
\frac1m\sum_{n>N}f(n/m) \le m^3\sum_{n>N}n^{-4}\le\frac{m^3}{3N^3} \le\frac8{3L^3},
n^4Use in source · 2 passages
Search aliases: n^4
Use in source · 05i_diagonal_regularization.tex, line 30
fracn^2(m^2+n^2)^2. Then frac m4≤ A_m≤frac4m3, frac14(m+1)≤ B_m≤frac2m. For the sums A_m,K,B_m,K through any integer K≥ m, 0≤ A_m-A_m,K≤fracm^43K^3, 0≤ B_m-B_m,K≤frac1K. For n>m, one has (a_n^(m))^2≤ m^4/n^4, so the correction sequence is square-summable. The diagonal operator is bounded with norm one, and w=(1/n)∈ℓ^2, hence the residual sequence is also in ℓ^2. The operator is self-adjoint, and the coordinate equality frac1nfracnm^2+n^2 =m^-2fracm^2m^2+n^2 proves T^*v^(m)=α a^(m) as an equality…
Exact mathematical passage
(a_n^{(m)})^2\le m^4/n^4
\N_{>0}Use in source · 2 passages
Search aliases: N_>0 N
Use in source · 05e_escape.tex, line 7
The complete spaces and the preservation rule. Index sequences by the positive integers. Set E=ℓ^2(N_>0;R)⊕R, R=ℓ^2(N_>0;R), norm(u,t)_E^2=∑_n≥1|u_n|^2+|t|^2. Let w=(1/n)_n≥1. It belongs to ℓ^2, because ∑_n≥1frac1n^2 ≤1+∑_n≥2frac1n(n-1)=2. Define the defect and protected observable by d(u,t)_n=fracu_n+tn, p(u,t)=t. The diagonal map T:u↦(u_n/n)_n has operator norm one. Since d(u,t)=Tu+tw, normd(u,t)≤norm u+|t|norm w ≤sqrt1+norm w^2 norm(u,t)_E. Thus both d and p are continuous linear maps between…
Exact mathematical passage
E=\ell^2(\N_{>0};\R)\oplus\R,\qquad R=\ell^2(\N_{>0};\R),\qquad \norm{(u,t)}_E^2=\sum_{n\ge1}|u_n|^2+|t|^2.
n_cUse in source · 1 passage
Search aliases: n_c
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
n_eUse in source · 1 passage
Search aliases: n_e
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
n_iUse in source · 1 passage
Search aliases: n_i
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
n_kUse in source · 15 passages
Search aliases: n_k
Use in source · 05f_bounded_realization.tex, line 27
The compactness needed for the existence argument. [Bounded sequences have weakly convergent subsequences] Every bounded sequence (a_n) in a real Hilbert space has a subsequence converging weakly to some a. Moreover, norm a≤liminf_knorma_n_k. Choose M≥0 with norma_n≤ M. If M=0, every term is zero and the conclusion is immediate. Otherwise let H_0 be the closed linear span of the countably many vectors a_n. Finite rational linear combinations of those vectors form a countable dense set, so H_0 is…
Exact mathematical passage
\norm a\le\liminf_k\norm{a_{n_k}}.
N_nUse in source · 3 passages
Search aliases: N_n
Use in source · 03_towers.tex, line 225
Protected information beyond the seed. When the target grows, its new kernel directions are independent information. Define N_n(f)=(I-s_nρ^Y_n)f_n+1, (p_Lf)_0=f_0, (p_Lf)_n+1=N_n(f). Each N_n(f) lands in kerρ^Y_n. The observable p_L retains both the seed and every new output component, relative to the chosen sections.
Exact mathematical passage
N_n(f)=(I-s_n\rho^Y_n)f_{n+1},\qquad (p_Lf)_0=f_0,\quad(p_Lf)_{n+1}=N_n(f).
n_oUse in source · 1 passage
Search aliases: n_o
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
N_pUse in source · 9 passages
Search aliases: N_p
Use in source · 05_geometry.tex, line 246
Every factor in the bound has a meaning. Define the route gain A_p=max_iγ(overlinep_i), using reverse routes so that the error is measured in each destination view. The path telescope gives normx-R_p(x_o)_∞≤ A_pnormdx_∞. Let N_p be the residual normalization (N_pr)_ij=T_ir_ij. The exact identity H=N_pdR_p implies norm H(x_o)_∞ ≤normN_p(norm d A_p+1)normdx_∞.
Exact mathematical passage
N_p
N_p^{-1}Use in source · 1 passage
Search aliases: N_p^-1
Use in source · 05_geometry.tex, line 285
dist(w,ker d)≤ Cnormdw_∞ for every vertex family w. Fix z∈ K^⊥ and take w=R_pz. Every exact family e has e_o∈ K, and root evaluation has norm at most one. Therefore norm z=dist(z,K)≤normz-e_o≤normR_pz-e_∞. Taking the infimum over e∈ker d is legitimate without assuming that this global nearest point is attained. It gives norm z≤ CnormdR_pz_∞ ≤ CnormN_p^-1norm Hz_∞. Here N_p is a continuous linear equivalence on the finite edge product because every component T_i is one. For a general root vector,…
Exact mathematical passage
\norm z\le C\norm{dR_pz}_\infty \le C\norm{N_p^{-1}}\norm{\mathcal Hz}_\infty.
\neqfinUse in source · 5 passages
Search aliases: neqfin ≠^* ne
Use in source · 07_classification.tex, line 80
[Torsion-image obstruction] If A≠^* B, every homomorphism G_A→ G_B has image contained in D. Consequently neither group embeds in the other, and neither is a quotient of the other. A homomorphism preserves torsion and induces a map Q→ Q on the quotients, necessarily multiplication by a rational r. If r=0, its image lies in D.
Exact mathematical passage
A\neqfin B
\newlineUse in source · 1 passage
Search aliases: newline
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
OUse in source · 1 passage
Search aliases: O
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
o_aUse in source · 1 passage
Search aliases: o_a
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
o_fUse in source · 1 passage
Search aliases: o_f
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
o_iUse in source · 1 passage
Search aliases: o_i
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
o_pUse in source · 1 passage
Search aliases: o_p
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
\mathfrak o_WUse in source · 1 passage
Search aliases: o_W mathfrak
Use in source · 02a_prescription.tex, line 34
[Functorial forgetting of protected views] The homomorphism π_W,U is well defined and surjective. For U⊆ W⊆ Z, it satisfies π_U,U=I, π_W,Uπ_Z,W=π_Z,U. For every s∈Γ_W(A), π_W,U o_W(s)= o_U(s|_U). In particular an obstructed restriction remains obstructed when more compatible prescribed data are retained. A successful extension on the larger prescribed domain gives a successful extension of its restriction. If r-r'∈ B_W, then r-r'∈ B_U, proving well-definedness. Every class modulo B_U has a repre…
Exact mathematical passage
\label{eq:prescribed-naturality} \pi_{W,U}\mathfrak o_W(s)=\mathfrak o_U(s|_U).
o_zUse in source · 1 passage
Search aliases: o_z
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
\one_{A_C}Use in source · 1 passage
Search aliases: one_A_C 1_A_C one mathbf
Use in source · 10_grammar.tex, line 18
Repeated diagonal coding. Fix a computable pairing bijection ⟨ j,k⟩: N^2→ N. For a freely chosen set C⊆ N, define a new selector A_C by 1_A_C(2⟨ j,k⟩) · =1- 1_B_j(2⟨ j,k⟩), 1_A_C(2⟨ j,k⟩+1) · = 1_C(j). Every natural number has exactly one of these forms, so the rules define a total subset. The even positions defeat grammar outputs. The odd positions retain an independent continuum of choices, with each chosen bit repeated infinitely often.
Exact mathematical passage
\one_{A_C}(2\langle j,k\rangle) &=1-\one_{B_j}(2\langle j,k\rangle),\label{eq:diageven}\\ \one_{A_C}(2\langle j,k\rangle+1) &=\one_C(j).\label{eq:diagodd}
\one_{B_j}Use in source · 1 passage
Search aliases: one_B_j 1_B_j one mathbf
Use in source · 10_grammar.tex, line 18
Repeated diagonal coding. Fix a computable pairing bijection ⟨ j,k⟩: N^2→ N. For a freely chosen set C⊆ N, define a new selector A_C by 1_A_C(2⟨ j,k⟩) · =1- 1_B_j(2⟨ j,k⟩), 1_A_C(2⟨ j,k⟩+1) · = 1_C(j). Every natural number has exactly one of these forms, so the rules define a total subset. The even positions defeat grammar outputs. The odd positions retain an independent continuum of choices, with each chosen bit repeated infinitely often.
Exact mathematical passage
\one_{A_C}(2\langle j,k\rangle) &=1-\one_{B_j}(2\langle j,k\rangle),\label{eq:diageven}\\ \one_{A_C}(2\langle j,k\rangle+1) &=\one_C(j).\label{eq:diagodd}
\one_CUse in source · 1 passage
Search aliases: one_C 1_C one mathbf
Use in source · 10_grammar.tex, line 18
Repeated diagonal coding. Fix a computable pairing bijection ⟨ j,k⟩: N^2→ N. For a freely chosen set C⊆ N, define a new selector A_C by 1_A_C(2⟨ j,k⟩) · =1- 1_B_j(2⟨ j,k⟩), 1_A_C(2⟨ j,k⟩+1) · = 1_C(j). Every natural number has exactly one of these forms, so the rules define a total subset. The even positions defeat grammar outputs. The odd positions retain an independent continuum of choices, with each chosen bit repeated infinitely often.
Exact mathematical passage
\one_{A_C}(2\langle j,k\rangle) &=1-\one_{B_j}(2\langle j,k\rangle),\label{eq:diageven}\\ \one_{A_C}(2\langle j,k\rangle+1) &=\one_C(j).\label{eq:diagodd}
\one_{S_A}Use in source · 2 passages
Search aliases: one_S_A 1_S_A one mathbf
Use in source · 05_construction.tex, line 99
[One cocycle value reads one coordinate] For every prime p, c_A(1/p,(p-1)/p)=- 1_S_A(p)e_p. The support bound leaves only the coordinate p. At that coordinate both fractional lifts are zero, and the lift of their sum, which is one, is 1_S_A(p). Dropping the cocycle is therefore not a harmless simplification. The direct product operation on Q× D removes exactly these defects and gives a different, split extension.
Exact mathematical passage
c_A(1/p,(p-1)/p)=-\one_{S_A}(p)e_p.
\overline BUse in source · 4 passages
Search aliases: overline B overline
Use in source · 05d_floors.tex, line 18
[The floor detects closure] For every b∈ B, 0≤ν_B(Ω)≤normΩ-b. Moreover, ν_B(Ω)=0 Longleftrightarrow Ω∈overline B. If B is closed, this is equivalent to Ω∈ B. The set of candidate distances is nonempty because 0∈ B, bounded below by zero, and contains the finite value normΩ. Its infimum is therefore a finite nonnegative real, and the defining lower-bound property gives ν_B(Ω)≤normΩ-b for every b∈ B.
Exact mathematical passage
\nu_B(\Omega)=0\quad\Longleftrightarrow\quad\Omega\in\overline B.
\overline{\mathcal F}Use in source · 3 passages
Search aliases: overline F overline mathcal
Use in source · 03_towers.tex, line 392
[Density criterion for norm recovery] Suppose sup_NnormQ_N≤ K<∞. Then the norm recoverable vectors are exactly overline F. In particular all vectors are norm recoverable if and only if the finite core is dense. The nested identities first imply that the kernels increase: if Q_My=0 and N≥ M, then Q_Ny=Q_NQ_My=0. Thus the finite core really is an increasing union of linear subspaces.
Exact mathematical passage
\overline{\mathcal F}
\overline LUse in source · 1 passage
Search aliases: overline L overline
Use in source · 02_obstructions.tex, line 197
Equivalently, if r-r'∈ d(ker p), then L(r-r')=0, so L(r)=L(r'). Hence overline L([r])=L(r) is a well-defined homomorphism on R/d(ker p). Its value on [dx] is nonzero. This supplies the quotient interpretation of the same contradiction, without assuming that every nonzero quotient class can be detected by a map into one fixed preferred coefficient group. sourceKS15 This is often the most economical negative proof. One need not classify every possible correction. It is enough to find an invariant …
Exact mathematical passage
\overline L([r])=L(r)
\overline MUse in source · 1 passage
Search aliases: overline M overline
Use in source · 07_classification.tex, line 56
All homomorphisms are diagonal. The product P is also a commutative ring under coordinatewise multiplication. Multiplication by any λ∈ P preserves D, so it induces an additive endomorphism overline M_λ of V. Because division in V is unique, this induced endomorphism is Q-linear.
Exact mathematical passage
\overline M_\lambda
\overline M_\lambdaUse in source · 1 passage
Search aliases: overline M_λ overline lambda
Use in source · 07_classification.tex, line 56
All homomorphisms are diagonal. The product P is also a commutative ring under coordinatewise multiplication. Multiplication by any λ∈ P preserves D, so it induces an additive endomorphism overline M_λ of V. Because division in V is unique, this induced endomorphism is Q-linear.
Exact mathematical passage
\overline M_\lambda
\overline RUse in source · 2 passages
Search aliases: overline R overline
Use in source · 03_towers.tex, line 306
chapMultiscale residuals and the limits of reconstruction A split coarsener retains its missing detail. Let C:R→overline R and J:overline R→ R be linear maps with CJ=I. Put H=I-JC. Then r=JCr+Hr, CH=0, H^2=H. The two coordinates (Cr,Hr) retain all information. The coarse coordinate alone generally does not.
Exact mathematical passage
C:R\to\overline R
\overline{d(\ker p)}Use in source · 1 passage
Search aliases: overlined(ker p) overline ker
Use in source · 05d_floors.tex, line 73
The bridge back to exactification. Proposition~[source reference] and Proposition~[source reference] fit together closely. If corrections produce a norm-convergent, protected path with vanishing residual, the actual quotient class vanishes. If only the corrected residuals tend to zero, all that follows is membership in overlined(ker p).
Exact mathematical passage
\overline{d(\ker p)}
\overline{p_i}Use in source · 1 passage
Search aliases: overlinep_i overline
Use in source · 05_geometry.tex, line 239
Every factor in the bound has a meaning. Define the route gain A_p=max_iγ(overlinep_i), using reverse routes so that the error is measured in each destination view. The path telescope gives normx-R_p(x_o)_∞≤ A_pnormdx_∞. Let N_p be the residual normalization (N_pr)_ij=T_ir_ij. The exact identity H=N_pdR_p implies norm H(x_o)_∞ ≤normN_p(norm d A_p+1)normdx_∞.
Exact mathematical passage
A_p=\max_i\gamma(\overline{p_i}),
\overline{\ran T}Use in source · 5 passages
Search aliases: overlineran T overline ran
Use in source · 05f_bounded_realization.tex, line 88
An immediate contrapositive is useful. If r∈overlineran T∖ran T, then every sequence with Ta_n→ r satisfies norma_n→∞. Otherwise infinitely many terms would lie in one bounded ball, contradicting the theorem along that subsequence. This recovers the escape mechanism of Section~[source reference] without using the diagonal formula.
Exact mathematical passage
r\in\overline{\ran T}\setminus\ran T
\overline{S(2nB_X)}Use in source · 1 passage
Search aliases: overlineS(2nB_X) overline
Use in source · 12_proof_tools.tex, line 122
mma~[source reference], at least one of the closed sets on the right has nonempty interior. Thus there are n≥1, y_0∈ Y, and ρ>0 such that B(y_0,ρ)⊆overlineS(nB_X). Here B(y_0,ρ) is an open ball. If normh<ρ, both y_0+h and y_0 belong to this closure. Choose approximating image sequences for both. Subtracting their preimages shows h∈overlineS(2nB_X). Scaling by 2n gives a constant c=ρ/(2n)>0 with B(0,c)⊆overlineS(B_X). This only gives approximate preimages. We next turn them into an actual preimag…
Exact mathematical passage
h\in\overline{S(2nB_X)}
\overline{S(B_X)}Use in source · 1 passage
Search aliases: overlineS(B_X) overline
Use in source · 12_proof_tools.tex, line 123
ere are n≥1, y_0∈ Y, and ρ>0 such that B(y_0,ρ)⊆overlineS(nB_X). Here B(y_0,ρ) is an open ball. If normh<ρ, both y_0+h and y_0 belong to this closure. Choose approximating image sequences for both. Subtracting their preimages shows h∈overlineS(2nB_X). Scaling by 2n gives a constant c=ρ/(2n)>0 with B(0,c)⊆overlineS(B_X). This only gives approximate preimages. We next turn them into an actual preimage with a summable budget.
Exact mathematical passage
\label{eq:inverse-closure-ball} B(0,c)\subseteq\overline{S(B_X)}.
\overline{\langle u\rangle}Use in source · 1 passage
Search aliases: overline⟨ u⟩ overline langle rangle
Use in source · 02_ambient.tex, line 73
A single topological generator after completion. [Topological cyclicity] Let u=(1,1,1,…)∈ P. Then overline⟨ u⟩=P. In particular, trank(P)=1. A basic open set prescribes a tuple on finitely many distinct prime coordinates. The Chinese remainder theorem supplies an integer k with the specified residues. The element ku belongs to the given open set. Therefore the cyclic subgroup is dense. The product is nontrivial, so zero generators do not suffice. The generator u is absent from D. The topological…
Exact mathematical passage
\overline{\langle u\rangle}=P.
\mathcal PUse in source · 5 passages
Search aliases: P mathcal
Use in source · 03_towers.tex, line 346
[Finite reconstruction and nested remainders] For every N, x=Q_Nx+∑_j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of C_jJ_j=I gives D_NL_N=I_R_N. This cancellation takes place between adjacent maps with matching domains. No identification of different residual spaces is needed…
Exact mathematical passage
\mathcal P_\infty=\{x:Q_Nx=x\text{ for every }N\}.
P^2Use in source · 2 passages
Search aliases: P^2
Use in source · 03_towers.tex, line 214
Set P=I-A∂. Then ∂ P=∂-∂ A∂=0, P^2=I-2A∂+A(∂ A)∂=I-A∂=P. The expression 2A∂ abbreviates addition of the same operator twice and uses no division by two. Every exact f is fixed because ∂ f=0. Since (Ar)_0=0, the seed of Pf is f_0. Finally f-Pf=A∂ f, and the bound on A gives the asserted residual estimate. The denominators above are justified exactly by the strict propagation gap. sourceKS20 The surjectivity is onto all bounded residual families. It is stronger than the assertion that some exact o…
Exact mathematical passage
\partial P=\partial-\partial A\partial=0,\qquad P^2=I-2A\partial+A(\partial A)\partial=I-A\partial=P.
p^2Use in source · 1 passage
Search aliases: p^2
Use in source · 04_symmetry.tex, line 171
For H=S_3, these elements are the identity and the three transpositions. There are exactly two orbit classes. The identity cocycle has automorphism group S_3, which is noncommutative. A nonidentity involution gives a coherent cocycle whose regular twisted section equation x=px has no solution. The only nontrivial cocycle relation is p^2=1. At a single view the gauge action is p↦ apa^-1. All three transpositions are conjugate and none is conjugate to the identity. The stabilizer is therefore the …
Exact mathematical passage
p^2=1
P^M_{\ker B}Use in source · 3 passages
Search aliases: P^M_ker B ker
Use in source · 05_geometry.tex, line 200
Let k_0=P^M_ker Bh_0 and h_*=h_0-k_0. Then Bh_*=Bh_0=Dx and h_*⊥ker B in M. If a different initial correction is chosen, it differs from h_0 by an element of ker B. Its orthogonal-complement component is therefore again h_*. This proves that the construction is independent of the feasibility witness.
Exact mathematical passage
k_0=P^M_{\ker B}h_0
p_0Use in source · 1 passage
Search aliases: p_0
Use in source · 00_notation.tex, line 5
Notation and principal results. addcontentslinetocchapterNotation and principal results markbothNotation and principal results All groups are written additively unless a finite target group is explicitly allowed to be nonabelian. The natural numbers start at zero. Primes are indexed by p_0=2, p_1=3, p_2=5, p_3=7, …. @p0.23textwidthp0.71textwidth@ Symbol · Meaning
Exact mathematical passage
p_0=2,\quad p_1=3,\quad p_2=5,\quad p_3=7,\quad\ldots.
p_1Use in source · 1 passage
Search aliases: p_1
Use in source · 00_notation.tex, line 5
Notation and principal results. addcontentslinetocchapterNotation and principal results markbothNotation and principal results All groups are written additively unless a finite target group is explicitly allowed to be nonabelian. The natural numbers start at zero. Primes are indexed by p_0=2, p_1=3, p_2=5, p_3=7, …. @p0.23textwidthp0.71textwidth@ Symbol · Meaning
Exact mathematical passage
p_0=2,\quad p_1=3,\quad p_2=5,\quad p_3=7,\quad\ldots.
p_2Use in source · 1 passage
Search aliases: p_2
Use in source · 00_notation.tex, line 5
Notation and principal results. addcontentslinetocchapterNotation and principal results markbothNotation and principal results All groups are written additively unless a finite target group is explicitly allowed to be nonabelian. The natural numbers start at zero. Primes are indexed by p_0=2, p_1=3, p_2=5, p_3=7, …. @p0.23textwidthp0.71textwidth@ Symbol · Meaning
Exact mathematical passage
p_0=2,\quad p_1=3,\quad p_2=5,\quad p_3=7,\quad\ldots.
p_{2n}Use in source · 4 passages
Search aliases: p_2n
Use in source · 06_roots.tex, line 116
The information retained by a marked element. For the distinguished element u_A=s_A(1), the coordinate criterion simplifies to pmid_G_Au_A Longleftrightarrow p∉ S_A. Consequently, for each n, p_2nmid_G_Au_A · Longleftrightarrow n∈ A, p_2n+1mid_G_Au_A · Longleftrightarrow n∉ A. Exactly one root equation is solvable. This binary pattern is the selector written in intrinsic group language.
Exact mathematical passage
p_{2n}\mid_{G_A}u_A&\quad\Longleftrightarrow\quad n\in A,\label{eq:evenroot}\\ p_{2n+1}\mid_{G_A}u_A&\quad\Longleftrightarrow\quad n\notin A.\label{eq:oddroot}
p_{2n+1}Use in source · 3 passages
Search aliases: p_2n+1
Use in source · 06_roots.tex, line 116
The information retained by a marked element. For the distinguished element u_A=s_A(1), the coordinate criterion simplifies to pmid_G_Au_A Longleftrightarrow p∉ S_A. Consequently, for each n, p_2nmid_G_Au_A · Longleftrightarrow n∈ A, p_2n+1mid_G_Au_A · Longleftrightarrow n∉ A. Exactly one root equation is solvable. This binary pattern is the selector written in intrinsic group language.
Exact mathematical passage
p_{2n}\mid_{G_A}u_A&\quad\Longleftrightarrow\quad n\in A,\label{eq:evenroot}\\ p_{2n+1}\mid_{G_A}u_A&\quad\Longleftrightarrow\quad n\notin A.\label{eq:oddroot}
p_{2n+a_n}Use in source · 1 passage
Search aliases: p_2n+a_n
Use in source · 05_construction.tex, line 7
Choosing a rational line. Let A⊆ N, write a_n= 1_A(n), and define S_A={p_2n+a_n:n∈ N}. Exactly one prime from each pair is selected. Both S_A and its complement are infinite. Let u_A∈ P be its indicator: (u_A)_p=1, · p∈ S_A,
Exact mathematical passage
S_A=\{p_{2n+a_n}:n\in\N\}.
p_3Use in source · 1 passage
Search aliases: p_3
Use in source · 00_notation.tex, line 5
Notation and principal results. addcontentslinetocchapterNotation and principal results markbothNotation and principal results All groups are written additively unless a finite target group is explicitly allowed to be nonabelian. The natural numbers start at zero. Primes are indexed by p_0=2, p_1=3, p_2=5, p_3=7, …. @p0.23textwidthp0.71textwidth@ Symbol · Meaning
Exact mathematical passage
p_0=2,\quad p_1=3,\quad p_2=5,\quad p_3=7,\quad\ldots.
P_aUse in source · 4 passages
Search aliases: P_a
Use in source · 09_workshop.tex, line 11
2. A residual chart can change without changing the defect. Let d:R^2→R be d(u,r)=r. For a∈R, let A_a(r)=(ar,r). Compute the exact repair P_a, compare it with P_b, and decide when they commute.
Exact mathematical passage
P_a
p_AUse in source · 3 passages
Search aliases: p_A
Use in source · 05b_transport_cost.tex, line 82
Preservation changes the admissible arrows. One response to collapse is to restrict arrows to those satisfying a declared preservation contract. For objects equipped with maps p_A:A→ S, a morphism can be required to satisfy p_Bf=p_A. Identities and composition preserve this equation. Unless p_A=0, the zero arrow is not admissible. In other applications the arrows might be required to be isomorphisms or to retain a particular observable. These are different choices of category, not automatic cons…
Exact mathematical passage
p_A:A\to S
P_A^2Use in source · 2 passages
Search aliases: P_A^2
Use in source · 03_towers.tex, line 158
n allows passage to the norm limit and proves linearity of P_A. The estimate with m=0 gives normP_Af≤norm f+fracnorm A1-qnormdf ≤(1+fracnorm Anorm d1-q)norm f, so the linear map is bounded. If df=0, every iterate equals f, hence P_Af=f. Its image is contained in ker d and it fixes that kernel, proving surjectivity onto its stated image and P_A^2=P_A. For q=0 the residual vanishes after the first update and the same inequalities remain valid. sourceKS19 This theorem constructs the dynamics from A…
Exact mathematical passage
P_A^2=P_A
P_BUse in source · 10 passages
Search aliases: P_B
Use in source · 03b_repair_transport.tex, line 19
[The split defect chart] The maps χ_A:E≃_L K× R, χ_A(x)=(P_Ax,dx), χ_A^-1(u,r)=u+Ar are inverse continuous linear equivalences. In these coordinates, P_A sends (u,r) to (u,0). If B is another right inverse, then A-B takes values in K, and P_Bx-P_Ax=(A-B)dx. The projections commute if and only if they are equal, and they are equal if and only if A=B. The identity dA=I gives dP_A=d-dAd=0, so the proposed first chart coordinate actually belongs to K. If u∈ K, then du=0 and P_Au=u. Therefore, for (u…
Exact mathematical passage
P_Bx-P_Ax=(A-B)dx.
P_bUse in source · 4 passages
Search aliases: P_b
Use in source · 09_workshop.tex, line 11
2. A residual chart can change without changing the defect. Let d:R^2→R be d(u,r)=r. For a∈R, let A_a(r)=(ar,r). Compute the exact repair P_a, compare it with P_b, and decide when they commute.
Exact mathematical passage
P_b
p_BUse in source · 1 passage
Search aliases: p_B
Use in source · 05b_transport_cost.tex, line 83
Preservation changes the admissible arrows. One response to collapse is to restrict arrows to those satisfying a declared preservation contract. For objects equipped with maps p_A:A→ S, a morphism can be required to satisfy p_Bf=p_A. Identities and composition preserve this equation. Unless p_A=0, the zero arrow is not admissible. In other applications the arrows might be required to be isomorphisms or to retain a particular observable. These are different choices of category, not automatic cons…
Exact mathematical passage
p_Bf=p_A.
p_bUse in source · 1 passage
Search aliases: p_b
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
p_{b_j}Use in source · 1 passage
Search aliases: p_b_j
Use in source · 03_spectrum.tex, line 56
Conversely, suppose B is infinite and enumerable in X. List its distinct elements as b_0,b_1,…. To compute b_j, wait for the (j+1)st distinct enumerated value. This procedure terminates for every j. Use canonical finite tuples with the jth coordinate in F_p_b_j. Coordinatewise addition, normalization, and equality are computable in X. If B is finite, D_B is a finite group and has a computable copy. This last existence statement does not require a uniform way to decide when a finite enumeration h…
Exact mathematical passage
\F_{p_{b_j}}
P_CUse in source · 3 passages
Search aliases: P_C
Use in source · 05g_budget_certificates.tex, line 14
ubsectionAt every finite budget the residual minimum exists [Attainment and shape of the budget profile] For a bounded linear map between real Hilbert spaces, the infimum defining β(M) is attained for every M≥0. The profile is nonincreasing and convex. For 0≤ M_1≤ M_2, 0≤β(M_1)-β(M_2)≤norm T(M_2-M_1). Writing C=overlineran T, b=P_Cr, and q=normr-b, one has lim_M→∞β(M)=q. The equality β(M)=q at a finite budget is equivalent to the existence of a with Ta=b and norm a≤ M. In particular, β(M)=0 Long…
Exact mathematical passage
b=P_Cr
p_cUse in source · 1 passage
Search aliases: p_c
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
P_{D',C'}Use in source · 1 passage
Search aliases: P_D',C'
Use in source · 05_geometry.tex, line 636
ally exact subspace. Let G,G' be the resulting joint residual gains. If normD-D'≤ε_D, normC-C'≤ε_C, then the maximum norm on the target product gives norm(D,C)-(D',C')≤max(ε_D,ε_C). Applying Theorem~[source reference] to the joint operators, and using the actual two-stage identity of Theorem~[source reference], gives normP_D,C-P_D',C' ≤max(G,G')max(ε_D,ε_C). This is a comparison of the original repair maps, not a replacement definition that happens to have a similar kernel. Both the local equati…
Exact mathematical passage
\label{eq:descentmotion} \norm{P_{D,C}-P_{D',C'}} \le\max(G,G')\max(\varepsilon_D,\varepsilon_C).
P_{D,C}Use in source · 1 passage
Search aliases: P_D,C
Use in source · 05_geometry.tex, line 636
ally exact subspace. Let G,G' be the resulting joint residual gains. If normD-D'≤ε_D, normC-C'≤ε_C, then the maximum norm on the target product gives norm(D,C)-(D',C')≤max(ε_D,ε_C). Applying Theorem~[source reference] to the joint operators, and using the actual two-stage identity of Theorem~[source reference], gives normP_D,C-P_D',C' ≤max(G,G')max(ε_D,ε_C). This is a comparison of the original repair maps, not a replacement definition that happens to have a similar kernel. Both the local equati…
Exact mathematical passage
\label{eq:descentmotion} \norm{P_{D,C}-P_{D',C'}} \le\max(G,G')\max(\varepsilon_D,\varepsilon_C).
P_HUse in source · 1 passage
Search aliases: P_H
Use in source · 05j_inexact_certificates.tex, line 20
The adjoint in g is the restricted adjoint on H, not the unrestricted ambient adjoint. Equivalently, T^*v=P_Hd^*v. Thus the correction certificate must measure the part of the normal equation that belongs to the permitted correction space. A numerical program that forgets this projection can incorrectly reject an exact protected optimizer.
Exact mathematical passage
T^*v=P_Hd^*v.
P_{H_0}Use in source · 1 passage
Search aliases: P_H_0
Use in source · 05f_bounded_realization.tex, line 49
To verify weak convergence against h∈ H_0, choose a finite basis combination h_J close to h. Then |⟨ a_n_k-a,h⟩| ≤ |⟨ a_n_k-a,h_J⟩| +2Mnormh-h_J. For a prescribed tolerance, choose h_J first to control the second term and then k to control the first by finitely many coordinate limits. For a general h in the ambient space, write h=P_H_0h+h_⊥. The orthogonal component pairs to zero with every a_n_k and with a, reducing to the preceding case. This proves weak convergence against every test vector.
Exact mathematical passage
h=P_{H_0}h+h_\perp
P_iUse in source · 2 passages
Search aliases: P_i
Use in source · 01_contracts.tex, line 65
[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If M_j≠0 and j is not reachable, P_j annihilates the entire hull but not M. An element of the direct sum has finite support. Applying one primitive oper…
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P_i
p_iUse in source · 11 passages
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Use in source · 05_geometry.tex, line 4
default. In a declared linear model, the compatibility equations become a bounded operator. Closed range, holonomy, and conditioning then determine what small residuals actually guarantee. Graph views, holonomy, and reconstruction. Actual transports on a finite graph. Let a finite connected simple graph have vertex set V, root o, and a chosen route p_i:o→ i for every vertex, with the route at o empty. Use a common real normed fiber H. An inverse consistent connection assigns continuous linear eq…
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p_i:o\to i
P_jUse in source · 2 passages
Search aliases: P_j
Use in source · 01_contracts.tex, line 65
[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If M_j≠0 and j is not reachable, P_j annihilates the entire hull but not M. An element of the direct sum has finite support. Applying one primitive oper…
Exact mathematical passage
P_jTP_i
P_KUse in source · 3 passages
Search aliases: P_K
Use in source · 05_geometry.tex, line 104
chapClosed range is the stability boundary Projection exists before a residual estimate. Let E be a real Hilbert space and R a real Banach space. Let D:E→ R be continuous linear. The kernel K=ker D is closed, so its orthogonal projection q=P_K exists~[source reference].
Exact mathematical passage
q=P_K
P_{\ker D}Use in source · 5 passages
Search aliases: P_ker D ker
Use in source · 05_geometry.tex, line 217
ready exact under permitted correction and the least correction is zero, so the optimal gain is zero. sourceKS56 The relevant stability boundary is therefore D(ker p), not automatically D(E). Restricting permitted changes can turn a well-conditioned unconstrained problem into an obstructed or poorly conditioned protected one. The nearest protected answer is generally not P_ker Dx.
Exact mathematical passage
P_{\ker D}x
P_{\ker D'}Use in source · 1 passage
Search aliases: P_ker D' ker
Use in source · 05_geometry.tex, line 420
chapControlled motion of exact spaces A two-sided condition number theorem. Let D,D':E→ R be continuous linear maps from a real Hilbert space to the same normed target. Write q=P_ker D, q'=P_ker D'. Suppose norm(I-q)x≤ CnormDx, norm(I-q')x≤ C'normD'x, C,C'≥0, and normD-D'≤ε.
Exact mathematical passage
q'=P_{\ker D'}
p_L^2Use in source · 1 passage
Search aliases: p_L^2
Use in source · 03_towers.tex, line 249
g p_Lf. Put B_n=I-s_nρ^Y_n. Since ρ^Y_ns_n=I, direct multiplication gives B_n^2=B_n, ρ^Y_nB_n=0, B_ns_n=0. Moreover, B_ny=y whenever ρ^Y_ny=0. Thus B_n really projects onto the target kernel. Its norm is at most 1+SB_Y. Coordinate zero of p_L is the identity, and every later coordinate applies B_n, so p_L^2=p_L and normp_L≤1+SB_Y. In particular the formula defines a map on the bounded product, not just on arbitrary coordinate families.
Exact mathematical passage
p_L^2=p_L
P_MUse in source · 1 passage
Search aliases: P_M
Use in source · 05_geometry.tex, line 342
), write ℓ=P_Lx and let Px be the projection of ℓ onto ker B inside L. Then Px∈ M, u=x-ℓ∈ L^⊥, and v=ℓ-Px∈ L is orthogonal to M. Since M⊆ L, the vector u is also orthogonal to M. Their sum x-Px=u+v is therefore orthogonal to M. A vector in M with that error is the unique orthogonal projection of x, so P=P_M.
Exact mathematical passage
P=P_M
p_nUse in source · 1 passage
Search aliases: p_n
Use in source · 03_spectrum.tex, line 84
A purely finite-state boundary. Consider a restricted grammar in which a finite automaton reads the unary word 1^n and selects the factor C_p_n exactly when it accepts.
Exact mathematical passage
C_{p_n}
p_oUse in source · 1 passage
Search aliases: p_o
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
P_{\rm anchor}Use in source · 1 passage
Search aliases: P_rm anchor rm
Use in source · 06_cases.tex, line 129
Preserving only the feasible projection of the anchor gives a different answer. Orthogonally project (2,1) to (2,0), then reconstruct along the tree. The result is P_rm anchorx=((2,0),(2,0),(2,0)). It is the unique exact family preserving the first component of the original anchor. It does not preserve its second component, and makes no claim to do so.
Exact mathematical passage
P_{\rm anchor}x=((2,0),(2,0),(2,0)).
P_{\rm nearest}Use in source · 1 passage
Search aliases: P_rm nearest rm
Use in source · 06_cases.tex, line 135
For global nearest-point repair, declare the Euclidean product norm on (R^2)^3. The closest member of [source reference] uses the mean of the three first coordinates, namely three: P_rm nearestx=((3,0),(3,0),(3,0)). To verify this without appealing only to a general theorem, write the squared error at parameter a: (2-a)^2+(4-a)^2+(3-a)^2+1^2+(-1)^2+2^2 =3(a-3)^2+8. It is minimized uniquely at a=3. The anchor-based answer has squared error eleven. The globally nearest answer has squared error eig…
Exact mathematical passage
P_{\rm nearest}x=((3,0),(3,0),(3,0)).
P_{T(\{a:\norm a\le M\})}Use in source · 1 passage
Search aliases: P_T({a:norm a≤ M}) norm le
Use in source · 05g_budget_certificates.tex, line 62
The nearest reachable image y_M=P_T({a:norm a≤ M})r is unique, so the best residual vector r-y_M is unique as well. Corrections mapping to y_M can differ by vectors in ker T. Among those corrections there is a unique least norm correction. These two uniqueness statements should not be confused with injectivity of T.
Exact mathematical passage
y_M=P_{T(\{a:\norm a\le M\})}r
p_vUse in source · 3 passages
Search aliases: p_v
Use in source · 01_views.tex, line 139
Correspondences must also be coherent. Suppose natural maps form a span FxleftarrowpMxrightarrowqG, and fix a∈Γ(F), b∈Γ(G). At each view define the ordinary fiber B_a,b(v)={m∈ M_v:p_v(m)=a_v, q_v(m)=b_v}. Naturality and coherence provide its restrictions. A global bridge is an element of B(a,b)={z∈Γ(M):Γ(p)z=a, Γ(q)z=b}.
Exact mathematical passage
B_{a,b}(v)=\{m\in M_v:p_v(m)=a_v,\ q_v(m)=b_v\}.
p_WUse in source · 1 passage
Search aliases: p_W
Use in source · 02a_prescription.tex, line 17
Forgetting a prescription gives a quotient map. Let U⊆ W. Then ker p_W⊆ker p_U, B_W⊆ B_U, H_W⊆ H_U. A correction invisible on all of W is certainly invisible on U. The direction of the quotient map is consequently π_W,U:Q_W⟶ Q_U, [r]_B_W⟼[r]_B_U. The larger prescribed domain appears in the source. More protection means a smaller subgroup is divided out, so more obstruction information can remain.
Exact mathematical passage
\ker p_W\subseteq\ker p_U,\qquad B_W\subseteq B_U, \qquad H_W\subseteq H_U.
p_wUse in source · 1 passage
Search aliases: p_w
Use in source · 01_views.tex, line 156
a coordinate preserving equivalence Γ(B_a,b)≃ B(a,b). Under a transition cover, matching local bridge sections glue to a unique global bridge with both prescribed projections unchanged. We must first check that the displayed fibers really form a view system. Let m∈ B_a,b(w) and f:v→ w. Naturality of p and coherence of a give p_v(f_M^*m)=f_F^*p_w(m)=f_F^*a_w=a_v. The same calculation with q,b gives q_v(f_M^*m)=b_v. Therefore f_M^*m belongs to B_a,b(v). Identity and composition of these restricted…
Exact mathematical passage
p_v(f_M^*m)=f_F^*p_w(m)=f_F^*a_w=a_v.
\mathcal P_\inftyUse in source · 2 passages
Search aliases: P_∞ mathcal infty
Use in source · 03_towers.tex, line 346
[Finite reconstruction and nested remainders] For every N, x=Q_Nx+∑_j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of C_jJ_j=I gives D_NL_N=I_R_N. This cancellation takes place between adjacent maps with matching domains. No identification of different residual spaces is needed…
Exact mathematical passage
\mathcal P_\infty=\{x:Q_Nx=x\text{ for every }N\}.
\pathUse in source · 1 passage
Search aliases: path
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
\operatorname{Point}Use in source · 4 passages
Search aliases: Point operatorname
Use in source · 03a_observation.tex, line 16
The realized map is |f|:Point(X)→Point(Y), |f|(x)_n=f_n(x_ϕ(n)). Equation~[source reference] proves that this is a coherent output. A second transport g_n:Y_ψ(n)→ Z_n composes with f using schedule ϕ∘ψ and coordinate map g_n f_ψ(n). The order of the schedules is fixed by the types, not by the order in which their letters are written in prose.
Exact mathematical passage
|f|:\operatorname{Point}(X)\to\operatorname{Point}(Y),\qquad |f|(x)_n=f_n(x_{\phi(n)}).
q'Use in source · 23 passages
Search aliases: q'
Use in source · 05_geometry.tex, line 420
chapControlled motion of exact spaces A two-sided condition number theorem. Let D,D':E→ R be continuous linear maps from a real Hilbert space to the same normed target. Write q=P_ker D, q'=P_ker D'. Suppose norm(I-q)x≤ CnormDx, norm(I-q')x≤ C'normD'x, C,C'≥0, and normD-D'≤ε.
Exact mathematical passage
q'=P_{\ker D'}
q^{\mathsf T}Use in source · 7 passages
Search aliases: q^ T mathsf
Use in source · 05k_stopping_rules.tex, line 13
[The invariant norm of a coordinate certificate] The actual normal residual is g=BG^-1q, norm g^2=q^ TG^-1q. For any invertible change of basis B'=BC, the same proposed correction has c'=C^-1c, G'=C^ TGC, q'=C^ Tq, and (q')^ T(G')^-1q'=q^ TG^-1q. Because B has independent columns, for every nonzero coefficient vector c, c^ TGc=c^ TB^ TBc=normBc^2>0. Thus G is positive definite and invertible. The matrix P=BG^-1B^ T is symmetric and obeys P^2=BG^-1(B^ TB)G^-1B^ T=P, PB=B. Its range is precisely t…
Exact mathematical passage
\label{eq:covector-norm} g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
q^2Use in source · 11 passages
Search aliases: q^2
Use in source · 05_geometry.tex, line 108
[Exact orthogonal repair] For every x∈ E, D(qx)=0, q^2=q, normqx≤norm x, and qx is the unique closest point to x in K. Boundedness of D makes K=ker D closed. Lemma~[source reference] constructs q=P_K and gives the orthogonal decomposition x=qx+(x-qx). The first term belongs to K, so Dqx=0. Projection fixes every point of K, giving q(qx)=qx.
Exact mathematical passage
D(qx)=0,\quad q^2=q,\quad\norm{qx}\le\norm x,
q^jUse in source · 3 passages
Search aliases: q^j
Use in source · 03_towers.tex, line 134
nd normf^(m)-P_Af ≤fracnorm A1-qnormdf^(m). It fixes every exact candidate and satisfies normP_A≤1+fracnorm Anorm d1-q. Put r_m=df^(m). Additivity of d gives r_m+1=r_m-dAr_m. Each r_m is the residual of an actual candidate, so the stated hypothesis applies at every iteration. Induction yields, for j≥0, normr_m+j≤ q^jnormr_m, normf^(m+j+1)-f^(m+j) ≤norm A q^jnormr_m. For n>m, the finite triangle inequality therefore gives normf^(n)-f^(m) ≤norm A normr_m∑_j=0^n-m-1q^j ≤fracnorm A1-qnormr_m. Since …
Exact mathematical passage
\norm{r_{m+j}}\le q^j\norm{r_m},\qquad \norm{f^{(m+j+1)}-f^{(m+j)}} \le\norm A\,q^j\norm{r_m}.
q^mUse in source · 1 passage
Search aliases: q^m
Use in source · 03_towers.tex, line 145
the residual of an actual candidate, so the stated hypothesis applies at every iteration. Induction yields, for j≥0, normr_m+j≤ q^jnormr_m, normf^(m+j+1)-f^(m+j) ≤norm A q^jnormr_m. For n>m, the finite triangle inequality therefore gives normf^(n)-f^(m) ≤norm A normr_m∑_j=0^n-m-1q^j ≤fracnorm A1-qnormr_m. Since normr_m≤ q^mnormdf tends to zero, this is a Cauchy estimate. Completeness of E supplies a limit, denoted P_Af. Passing n→∞ in the same norm inequality proves the required tail estimate.
Exact mathematical passage
\norm{r_m}\le q^m\norm{df}
q^NUse in source · 2 passages
Search aliases: q^N
Use in source · 03_towers.tex, line 478
[The recovery domain is c_0] For this shift cascade, S_Nx→ x in ℓ^∞ Longleftrightarrow x_n→0. For x_n=q^n, 0≤ q<1, the error is exactly q^N. The error norm is sup_n≥ N|x_n|. It tends to zero precisely when the sequence tends to zero. For a geometric sequence its largest tail value is the first one, q^N. sourceKS29 As a second example, take scalar spaces with C=tfrac12I, J=2I. The coarse value x_N=2^-Nx tends to zero, but restoration gives Q_Nx=x. Every detail is zero and the entire input is pers…
Exact mathematical passage
q^N
q^nUse in source · 1 passage
Search aliases: q^n
Use in source · 03_towers.tex, line 478
[The recovery domain is c_0] For this shift cascade, S_Nx→ x in ℓ^∞ Longleftrightarrow x_n→0. For x_n=q^n, 0≤ q<1, the error is exactly q^N. The error norm is sup_n≥ N|x_n|. It tends to zero precisely when the sequence tends to zero. For a geometric sequence its largest tail value is the first one, q^N. sourceKS29 As a second example, take scalar spaces with C=tfrac12I, J=2I. The coarse value x_N=2^-Nx tends to zero, but restoration gives Q_Nx=x. Every detail is zero and the entire input is pers…
Exact mathematical passage
x_n=q^n
Q_0Use in source · 2 passages
Search aliases: Q_0
Use in source · 03_towers.tex, line 361
Since L_j+1=L_jJ_j and x_j+1=C_jx_j, L_jh_j=L_j(x_j-J_jx_j+1)=Q_jx-Q_j+1x. Summing from j=0 to N-1 leaves x-Q_Nx because Q_0=I. This proves both finite reconstruction identities, including N=0, when the detail sum is empty.
Exact mathematical passage
Q_0=I
q_0Use in source · 3 passages
Search aliases: q_0
Use in source · 05_geometry.tex, line 584
[Exact motion of the line] For every real t, norm(q_t-q_0)x^2=fract^21+t^2norm x^2, normq_t-q_0=frac|t|sqrt1+t^2. In the small perturbation interface, kernel transport sends (0,b) to (-tb,b). Orthogonal projection instead sends it to (-tb/(1+t^2),b/(1+t^2)). The vector v_t=(-t,1) spans ker D_t and has squared norm 1+t^2>0. Projection onto this line is x⟼frac⟨ x,v_t⟩1+t^2v_t, which gives the matrix in [source reference]. In particular it is the orthogonal projector, not merely a map whose image i…
Exact mathematical passage
\norm{(q_t-q_0)x}^2=\frac{t^2}{1+t^2}\norm x^2, \qquad \norm{q_t-q_0}=\frac{|t|}{\sqrt{1+t^2}}.
Q_{2k}Use in source · 1 passage
Search aliases: Q_2k
Use in source · 05c_iteration.tex, line 112
Residual bursts can be compatible with convergence. Set q_2k=2, q_2k+1=1/8. Then Q_2k=4^-k, Q_2k+1=2·4^-k. For f_n=(s,Q_n), the residual doubles at the start of every pair of steps. Choose C_2k=1 and C_2k+1=7/8. These give equality in the movement bound, and ∑_n=0^∞C_nQ_n =∑_k=0^∞(1+frac74)4^-k =frac113. Thus f_n→(s,0) despite infinitely many increases. Starting at an even index 2k, the remaining movement budget is frac1134^-k. At an odd index 2k+1, it is frac83 4^-k. These bounds are larger tha…
Exact mathematical passage
Q_{2k}=4^{-k},\qquad Q_{2k+1}=2\cdot4^{-k}.
q_{2k}Use in source · 1 passage
Search aliases: q_2k
Use in source · 05c_iteration.tex, line 111
Residual bursts can be compatible with convergence. Set q_2k=2, q_2k+1=1/8. Then Q_2k=4^-k, Q_2k+1=2·4^-k. For f_n=(s,Q_n), the residual doubles at the start of every pair of steps. Choose C_2k=1 and C_2k+1=7/8. These give equality in the movement bound, and ∑_n=0^∞C_nQ_n =∑_k=0^∞(1+frac74)4^-k =frac113. Thus f_n→(s,0) despite infinitely many increases. Starting at an even index 2k, the remaining movement budget is frac1134^-k. At an odd index 2k+1, it is frac83 4^-k. These bounds are larger tha…
Exact mathematical passage
q_{2k}=2
Q_{2k+1}Use in source · 1 passage
Search aliases: Q_2k+1
Use in source · 05c_iteration.tex, line 112
Residual bursts can be compatible with convergence. Set q_2k=2, q_2k+1=1/8. Then Q_2k=4^-k, Q_2k+1=2·4^-k. For f_n=(s,Q_n), the residual doubles at the start of every pair of steps. Choose C_2k=1 and C_2k+1=7/8. These give equality in the movement bound, and ∑_n=0^∞C_nQ_n =∑_k=0^∞(1+frac74)4^-k =frac113. Thus f_n→(s,0) despite infinitely many increases. Starting at an even index 2k, the remaining movement budget is frac1134^-k. At an odd index 2k+1, it is frac83 4^-k. These bounds are larger tha…
Exact mathematical passage
Q_{2k}=4^{-k},\qquad Q_{2k+1}=2\cdot4^{-k}.
q_{2k+1}Use in source · 1 passage
Search aliases: q_2k+1
Use in source · 05c_iteration.tex, line 111
Residual bursts can be compatible with convergence. Set q_2k=2, q_2k+1=1/8. Then Q_2k=4^-k, Q_2k+1=2·4^-k. For f_n=(s,Q_n), the residual doubles at the start of every pair of steps. Choose C_2k=1 and C_2k+1=7/8. These give equality in the movement bound, and ∑_n=0^∞C_nQ_n =∑_k=0^∞(1+frac74)4^-k =frac113. Thus f_n→(s,0) despite infinitely many increases. Starting at an even index 2k, the remaining movement budget is frac1134^-k. At an odd index 2k+1, it is frac83 4^-k. These bounds are larger tha…
Exact mathematical passage
q_{2k+1}=1/8
q_fUse in source · 1 passage
Search aliases: q_f
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
q_iUse in source · 1 passage
Search aliases: q_i
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
Q_jUse in source · 2 passages
Search aliases: Q_j
Use in source · 03_towers.tex, line 358
Since L_j+1=L_jJ_j and x_j+1=C_jx_j, L_jh_j=L_j(x_j-J_jx_j+1)=Q_jx-Q_j+1x. Summing from j=0 to N-1 leaves x-Q_Nx because Q_0=I. This proves both finite reconstruction identities, including N=0, when the detail sum is empty.
Exact mathematical passage
L_jh_j=L_j(x_j-J_jx_{j+1})=Q_jx-Q_{j+1}x.
Q_{j+1}Use in source · 2 passages
Search aliases: Q_j+1
Use in source · 03_towers.tex, line 358
Since L_j+1=L_jJ_j and x_j+1=C_jx_j, L_jh_j=L_j(x_j-J_jx_j+1)=Q_jx-Q_j+1x. Summing from j=0 to N-1 leaves x-Q_Nx because Q_0=I. This proves both finite reconstruction identities, including N=0, when the detail sum is empty.
Exact mathematical passage
L_jh_j=L_j(x_j-J_jx_{j+1})=Q_jx-Q_{j+1}x.
Q_kUse in source · 2 passages
Search aliases: Q_k
Use in source · 05c_iteration.tex, line 59
[Variable-rate exactification] If Q_n→0 and ∑_n C_nQ_n<∞, then (f_n) has an exact limit and dist(f_n,x)≤ω(f_0)∑_k=n^∞C_kQ_k. For constant q_n=q∈[0,1) and constant C_n=C, the sharper restarted estimate is dist(f_n,x)≤fracC1-q ω(f_n). The estimate ω(f_n)≤ Q_nω(f_0) follows by induction. At n=0 it is equality. If it holds at n, nonnegativity of q_n permits multiplication and gives ω(f_n+1)≤ q_nω(f_n) ≤ q_nQ_nω(f_0)=Q_n+1ω(f_0). Because Q_n→0 and ω(f_0) is a fixed finite number, the residual tends t…
Exact mathematical passage
\label{eq:variable-tail} \dist(f_n,x)\le\omega(f_0)\sum_{k=n}^{\infty}C_kQ_k.
Q_MUse in source · 5 passages
Search aliases: Q_M
Use in source · 03_towers.tex, line 342
[Finite reconstruction and nested remainders] For every N, x=Q_Nx+∑_j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of C_jJ_j=I gives D_NL_N=I_R_N. This cancellation takes place between adjacent maps with matching domains. No identification of different residual spaces is needed…
Exact mathematical passage
Q_NQ_M=Q_MQ_N=Q_M.
q_nUse in source · 11 passages
Search aliases: q_n
Use in source · 05c_iteration.tex, line 47
Variable rates and a movement budget. Let q_n,C_n≥0, and define the prefix product Q_0=1, Q_n+1=q_nQ_n. Assume ω(f_n+1)≤ q_nω(f_n), dist(f_n,f_n+1)≤ C_nω(f_n).
Exact mathematical passage
q_n,C_n\ge0
Q_{n+1}Use in source · 2 passages
Search aliases: Q_n+1
Use in source · 05c_iteration.tex, line 48
Variable rates and a movement budget. Let q_n,C_n≥0, and define the prefix product Q_0=1, Q_n+1=q_nQ_n. Assume ω(f_n+1)≤ q_nω(f_n), dist(f_n,f_n+1)≤ C_nω(f_n).
Exact mathematical passage
Q_0=1,\qquad Q_{n+1}=q_nQ_n.
Q_N^2Use in source · 2 passages
Search aliases: Q_N^2
Use in source · 03_towers.tex, line 364
Idempotence follows from D_NL_N=I: Q_N^2=L_N(D_NL_N)D_N=Q_N. For M≥ N, put C_N,M=C_M-1⋯ C_N and J_N,M=J_N⋯ J_M-1, using identities for M=N. Then D_M=C_N,MD_N, L_M=L_NJ_N,M, C_N,MJ_N,M=I. The identities D_NL_M=J_N,M and D_ML_N=C_N,M now give Q_NQ_M=L_NJ_N,MD_M=Q_M, Q_MQ_N=L_MC_N,MD_N=Q_M. Both orders have been checked, rather than inferred from one range inclusion.
Exact mathematical passage
Q_N^2=L_N(D_NL_N)D_N=Q_N.
q_sUse in source · 1 passage
Search aliases: q_s
Use in source · 05_geometry.tex, line 580
-t · 1. The common right inverse r↦(r,0) supplies gain at most one for every t, and normD_s-D_t=|s-t|. The general theorem gives normq_s-q_t≤|s-t|.
Exact mathematical passage
\norm{q_s-q_t}\le|s-t|
q_tUse in source · 4 passages
Search aliases: q_t
Use in source · 05_geometry.tex, line 576
An all-real worked model. Let D_t(x,y)=x+ty. Its kernel is the line {(-tb,b):b∈R}, and its orthogonal projector is q_t=frac11+t^2 t^2 · -t -t · 1. The common right inverse r↦(r,0) supplies gain at most one for every t, and normD_s-D_t=|s-t|. The general theorem gives normq_s-q_t≤|s-t|.
Exact mathematical passage
\label{eq:lineprojector} q_t=\frac1{1+t^2} \begin{pmatrix}t^2&-t\\-t&1\end{pmatrix}.
q_v^{-1}Use in source · 3 passages
Search aliases: q_v^-1
Use in source · 04_symmetry.tex, line 83
The relative obstruction without commutativity. Prescribe a coherent partial section s of T_c on U. The based gauge group is G_U={a∈∏_vG_v:a_u=1 for u∈ U}. Choose a raw family q equal to s on U. Identity elements can be used elsewhere. Normalize the labels by c^q_f=q_v^-1c_f f^*(q_w). Its labels on arrows internal to U are one. Define the pointed orbit obstruction o_U(c,s)=[c^q]∈ Z^1(V;G)/ G_U, whose neutral point is the class of the trivial cocycle.
Exact mathematical passage
c^q_f=q_v^{-1}c_f f^*(q_w).
Q_WUse in source · 1 passage
Search aliases: Q_W
Use in source · 02a_prescription.tex, line 22
Forgetting a prescription gives a quotient map. Let U⊆ W. Then ker p_W⊆ker p_U, B_W⊆ B_U, H_W⊆ H_U. A correction invisible on all of W is certainly invisible on U. The direction of the quotient map is consequently π_W,U:Q_W⟶ Q_U, [r]_B_W⟼[r]_B_U. The larger prescribed domain appears in the source. More protection means a smaller subgroup is divided out, so more obstruction information can remain.
Exact mathematical passage
\pi_{W,U}:Q_W\longrightarrow Q_U,\qquad [r]_{B_W}\longmapsto[r]_{B_U}.
q_wUse in source · 6 passages
Search aliases: q_w
Use in source · 04_symmetry.tex, line 83
The relative obstruction without commutativity. Prescribe a coherent partial section s of T_c on U. The based gauge group is G_U={a∈∏_vG_v:a_u=1 for u∈ U}. Choose a raw family q equal to s on U. Identity elements can be used elsewhere. Normalize the labels by c^q_f=q_v^-1c_f f^*(q_w). Its labels on arrows internal to U are one. Define the pointed orbit obstruction o_U(c,s)=[c^q]∈ Z^1(V;G)/ G_U, whose neutral point is the class of the trivial cocycle.
Exact mathematical passage
c^q_f=q_v^{-1}c_f f^*(q_w).
q_zUse in source · 1 passage
Search aliases: q_z
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
\RUse in source · 46 passages
Search aliases: R
Use in source · 03_towers.tex, line 276
A growing example and the gap boundary. Use X_n=Y_n=R^n+1 with maximum norms. Let π_n delete the last coordinate, take ρ^X_n=tfrac12π_n, ρ^Y_n=π_n, and let s_n append zero. Then S=B_Y=1, B_X=1/2, and K=2.
Exact mathematical passage
X_n=Y_n=\R^{n+1}
R'Use in source · 10 passages
Search aliases: R'
Use in source · 04_symmetry.tex, line 409
Weakly commuting squares are enough. Suppose E: C→ C' and K: D→ D' are equivalences, with functors R,R' and a natural isomorphism σ:KRxRightarrow∼R'E. The square is not required to commute literally.
Exact mathematical passage
R,R'
r'Use in source · 11 passages
Search aliases: r'
Use in source · 02_obstructions.tex, line 197
Equivalently, if r-r'∈ d(ker p), then L(r-r')=0, so L(r)=L(r'). Hence overline L([r])=L(r) is a well-defined homomorphism on R/d(ker p). Its value on [dx] is nonzero. This supplies the quotient interpretation of the same contradiction, without assuming that every nonzero quotient class can be detected by a map into one fixed preferred coefficient group. sourceKS15 This is often the most economical negative proof. One need not classify every possible correction. It is enough to find an invariant …
Exact mathematical passage
r-r'\in d(\ker p)
r^{-1}Use in source · 2 passages
Search aliases: r^-1
Use in source · 04a_boundary_lab.tex, line 7
Use a constant group H as coefficients, with identity pullback homomorphisms. A cocycle on the duplicated category is specified by its label r∈ H on the arrow 0→1. Its reverse label is r^-1, and identity labels are one. A gauge (a_0,a_1) carries r to r'=a_0ra_1^-1. Every r is isomorphic to the trivial cocycle, for example with gauge (1,r). The automorphisms of r are the pairs (a_0,r^-1a_0r), a_0∈ H. Thus the unframed groupoid has one isomorphism class but retains a copy of H as the automorphism …
Exact mathematical passage
r^{-1}
\R^2Use in source · 12 passages
Search aliases: R^2 R
Use in source · 03b_repair_transport.tex, line 117
A two-coordinate model in which the routes differ. Use E=F=R^2, write candidates as (u,r), and take d(u,r)=D(u,r)=r, A(r)=B(r)=(0,r). The exact space is the horizontal axis and both repairs are P(u,r)=(u,0). For a real parameter c, set S_c(u,r)=(u+cr,r), C=I. The defect square commutes because the second coordinate is unchanged. The mismatch is M_c(r)=(cr,0). Hence PS_c(u,r)=(u+cr,0), S_cP(u,r)=(u,0). With c=2 and input (0,1), one route gives (2,0) and the other gives (0,0). Both have zero resid…
Exact mathematical passage
E=F=\R^2
\R^3Use in source · 3 passages
Search aliases: R^3 R
Use in source · 05c_iteration.tex, line 197
Three hypotheses that cannot simply be dropped. Summable movement. In maximum-norm R^3, let f_n=(s,H_n,(n+1)^-1), H_n=∑_j=1^nfrac1j. The first coordinate is protected and the residual is the last coordinate. It tends to zero with the same ratios as the polynomial example. The movement equals 1/(n+1), however, and the unprotected coordinate diverges. Here C_n=1, so ∑ C_nQ_n is the divergent harmonic series. Residual decay and semantic preservation hold, but exactification of this path fails.
Exact mathematical passage
\R^3
\R^4Use in source · 1 passage
Search aliases: R^4 R
Use in source · 06_cases.tex, line 77
Here the general protected construction is completely explicit. With candidate space E=R^4, defect d=ℓ, and protected observable p(r)=(r_0,r_1,r_2), the allowed direction is the last coordinate axis. The restricted operator d|_ker p has gain one. Without protection, the least gain for distance to kerℓ is 1/2, since normℓ=2. The different constants measure different correction spaces.
Exact mathematical passage
E=\R^4
\R^NUse in source · 3 passages
Search aliases: R^N R
Use in source · 05e_escape.tex, line 91
Escape is not an accident of this particular algorithm. Suppose an arbitrary protected candidate (u,1) satisfies normd(u,1)≤ε. On its first N coordinates, ∑_n=1^N|u_n+1|^2 ≤ N^2∑_n=1^Nfrac|u_n+1|^2n^2 ≤ N^2ε^2. The reverse triangle inequality in R^N gives norm u≥sqrt N-Nε (N≥1). If a sequence of protected candidates has residual tending to zero, then for each fixed N, its eventual correction norms are at least, for example, sqrt N/2. Taking N arbitrarily large shows that these norms must tend to…
Exact mathematical passage
\R^N
\R^{n+1}Use in source · 1 passage
Search aliases: R^n+1 R
Use in source · 03_towers.tex, line 276
A growing example and the gap boundary. Use X_n=Y_n=R^n+1 with maximum norms. Let π_n delete the last coordinate, take ρ^X_n=tfrac12π_n, ρ^Y_n=π_n, and let s_n append zero. Then S=B_Y=1, B_X=1/2, and K=2.
Exact mathematical passage
X_n=Y_n=\R^{n+1}
\R^{\N_{>0}}Use in source · 1 passage
Search aliases: R^N_>0 R N
Use in source · 05e_escape.tex, line 115
A coherent inverse-limit point still exists in a larger space. At each finite level, the protected exact equations have the unique solution (-1,…,-1;1)∈R^N⊕R. Truncation maps the solution at level N+1 to the one at level N. These finite solution sets are singletons, so their inverse limit is also a singleton. Its coordinate description is (-1,-1,…;1)∈R^N_>0×R. The coherent point is not missing. It lies outside the intended ℓ^2-based semantic space.
Exact mathematical passage
(-1,-1,\ldots;1)\in\R^{\N_{>0}}\times\R.
R_0Use in source · 10 passages
Search aliases: R_0
Use in source · 03_towers.tex, line 324
A whole cascade. At scale j, let C_j:R_j→ R_j+1 and J_j:R_j+1→ R_j be bounded linear maps with C_jJ_j=I. For x∈ R_0, define x_0=x, x_j+1=C_jx_j, L_0=I, L_N=J_0⋯ J_N-1, h_j=x_j-J_jx_j+1, Q_Nx=L_Nx_N, S_N=I-Q_N. The map Q_N restores the surviving coarse remainder to the original space. It is this restored remainder, not just x_N, that controls reconstruction.
Exact mathematical passage
x\in R_0
r_0Use in source · 10 passages
Search aliases: r_0
Use in source · 06_cases.tex, line 4
nd its numerical repair. The following calculations instantiate the preceding principles. They do not introduce a second, silently enlarged formal theory. A closed signal loop. Prescribed differences as affine view transitions. Consider four scalar views arranged around a cycle. Their values are x_0,x_1,x_2,x_3, and the prescribed differences are r_0,r_1,r_2,r_3. Indices are taken modulo four. The compatibility equations are x_i+1-x_i=r_i. Work over Q or R. These equations can be presented as a …
Exact mathematical passage
r_0,r_1,r_2,r_3
r_{01}Use in source · 3 passages
Search aliases: r_01
Use in source · 06_cases.tex, line 99
0 · -1. For one chosen orientation of each edge, the residuals are r_01=x_1-x_0, r_12=x_2-x_1, r_20=Jx_0-x_2. The reverse transports are inverses, so this defines an inverse-consistent isometric connection. Exactness first forces x_1=x_0 and x_2=x_0, and the remaining edge then forces Jx_0=x_0. Hence every exact family is ((a,0),(a,0),(a,0)), a∈R. The loop is not the identity on the whole fiber. Its fixed space is the first coordinate axis. Every anchor on that axis extends uniquely, and every a…
Exact mathematical passage
r_{01}=x_1-x_0,\qquad r_{12}=x_2-x_1,\qquad r_{20}=Jx_0-x_2.
R_1Use in source · 6 passages
Search aliases: R_1
Use in source · 05_geometry.tex, line 543
An equivalence between the moving kernels. [Explicit kernel transport] Under the hypotheses of Theorem~[source reference], J:ker D≃_Lker D', Jx=x-A'D'x, with inverse y↦ y-ADy. Moreover normJx-x≤fracaε1-aεnorm x. Let R_0=I-AD and R_1=I-A'D'. From DA=D'A'=I we obtain DR_0=0 and D'R_1=0. Each R_i fixes its corresponding kernel. Thus R_1 restricts to a map from ker D to ker D', while R_0 restricts in the other direction.
Exact mathematical passage
R_1=I-A'D'
r_1Use in source · 9 passages
Search aliases: r_1
Use in source · 06_cases.tex, line 4
nd its numerical repair. The following calculations instantiate the preceding principles. They do not introduce a second, silently enlarged formal theory. A closed signal loop. Prescribed differences as affine view transitions. Consider four scalar views arranged around a cycle. Their values are x_0,x_1,x_2,x_3, and the prescribed differences are r_0,r_1,r_2,r_3. Indices are taken modulo four. The compatibility equations are x_i+1-x_i=r_i. Work over Q or R. These equations can be presented as a …
Exact mathematical passage
r_0,r_1,r_2,r_3
r_{12}Use in source · 3 passages
Search aliases: r_12
Use in source · 06_cases.tex, line 99
0 · -1. For one chosen orientation of each edge, the residuals are r_01=x_1-x_0, r_12=x_2-x_1, r_20=Jx_0-x_2. The reverse transports are inverses, so this defines an inverse-consistent isometric connection. Exactness first forces x_1=x_0 and x_2=x_0, and the remaining edge then forces Jx_0=x_0. Hence every exact family is ((a,0),(a,0),(a,0)), a∈R. The loop is not the identity on the whole fiber. Its fixed space is the first coordinate axis. Every anchor on that axis extends uniquely, and every a…
Exact mathematical passage
r_{01}=x_1-x_0,\qquad r_{12}=x_2-x_1,\qquad r_{20}=Jx_0-x_2.
r_2Use in source · 5 passages
Search aliases: r_2
Use in source · 06_cases.tex, line 4
nd its numerical repair. The following calculations instantiate the preceding principles. They do not introduce a second, silently enlarged formal theory. A closed signal loop. Prescribed differences as affine view transitions. Consider four scalar views arranged around a cycle. Their values are x_0,x_1,x_2,x_3, and the prescribed differences are r_0,r_1,r_2,r_3. Indices are taken modulo four. The compatibility equations are x_i+1-x_i=r_i. Work over Q or R. These equations can be presented as a …
Exact mathematical passage
r_0,r_1,r_2,r_3
r_{20}Use in source · 3 passages
Search aliases: r_20
Use in source · 06_cases.tex, line 99
0 · -1. For one chosen orientation of each edge, the residuals are r_01=x_1-x_0, r_12=x_2-x_1, r_20=Jx_0-x_2. The reverse transports are inverses, so this defines an inverse-consistent isometric connection. Exactness first forces x_1=x_0 and x_2=x_0, and the remaining edge then forces Jx_0=x_0. Hence every exact family is ((a,0),(a,0),(a,0)), a∈R. The loop is not the identity on the whole fiber. Its fixed space is the first coordinate axis. Every anchor on that axis extends uniquely, and every a…
Exact mathematical passage
r_{01}=x_1-x_0,\qquad r_{12}=x_2-x_1,\qquad r_{20}=Jx_0-x_2.
r_{2n}Use in source · 1 passage
Search aliases: r_2n
Use in source · 03_towers.tex, line 313
The actual dyadic tower coarsener has the typed formula (Cr)_n=ρ^Y_2nr_2n+1+r_2nρ^X_2n+1. It is obtained by telescoping two adjacent compatibility squares. With target sections, a lift is (Jc)_2n=0, (Jc)_2n+1=s_2nc_n. The section identity gives CJ=I, so the missing detail is explicitly Hr, not an unspecified reconstruction error. The normed construction imposes the bounds needed to make these maps bounded on its operator family spaces. sourceKS25
Exact mathematical passage
\label{eq:dyadic} (Cr)_n=\rho^Y_{2n}r_{2n+1}+r_{2n}\rho^X_{2n+1}.
r_{2n+1}Use in source · 1 passage
Search aliases: r_2n+1
Use in source · 03_towers.tex, line 313
The actual dyadic tower coarsener has the typed formula (Cr)_n=ρ^Y_2nr_2n+1+r_2nρ^X_2n+1. It is obtained by telescoping two adjacent compatibility squares. With target sections, a lift is (Jc)_2n=0, (Jc)_2n+1=s_2nc_n. The section identity gives CJ=I, so the missing detail is explicitly Hr, not an unspecified reconstruction error. The normed construction imposes the bounds needed to make these maps bounded on its operator family spaces. sourceKS25
Exact mathematical passage
\label{eq:dyadic} (Cr)_n=\rho^Y_{2n}r_{2n+1}+r_{2n}\rho^X_{2n+1}.
r_3Use in source · 2 passages
Search aliases: r_3
Use in source · 06_cases.tex, line 4
nd its numerical repair. The following calculations instantiate the preceding principles. They do not introduce a second, silently enlarged formal theory. A closed signal loop. Prescribed differences as affine view transitions. Consider four scalar views arranged around a cycle. Their values are x_0,x_1,x_2,x_3, and the prescribed differences are r_0,r_1,r_2,r_3. Indices are taken modulo four. The compatibility equations are x_i+1-x_i=r_i. Work over Q or R. These equations can be presented as a …
Exact mathematical passage
r_0,r_1,r_2,r_3
r_bUse in source · 1 passage
Search aliases: r_b
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
r_cUse in source · 1 passage
Search aliases: r_c
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
R_iUse in source · 1 passage
Search aliases: R_i
Use in source · 05_geometry.tex, line 543
tween the moving kernels [Explicit kernel transport] Under the hypotheses of Theorem~[source reference], J:ker D≃_Lker D', Jx=x-A'D'x, with inverse y↦ y-ADy. Moreover normJx-x≤fracaε1-aεnorm x. Let R_0=I-AD and R_1=I-A'D'. From DA=D'A'=I we obtain DR_0=0 and D'R_1=0. Each R_i fixes its corresponding kernel. Thus R_1 restricts to a map from ker D to ker D', while R_0 restricts in the other direction.
Exact mathematical passage
R_i
r_iUse in source · 9 passages
Search aliases: r_i
Use in source · 04_symmetry.tex, line 194
a_ik(v) · =a_jk(v)a_ij(v). The triple equation implies a_ii=1 and a_ji=a_ij^-1. A morphism of descent data is a family of local gauges r_i satisfying r_j(v)a_ij(v)=a'_ij(v)r_i(v). These objects and arrows form a groupoid.
Exact mathematical passage
r_i
r_{ij}Use in source · 1 passage
Search aliases: r_ij
Use in source · 05_geometry.tex, line 246
Every factor in the bound has a meaning. Define the route gain A_p=max_iγ(overlinep_i), using reverse routes so that the error is measured in each destination view. The path telescope gives normx-R_p(x_o)_∞≤ A_pnormdx_∞. Let N_p be the residual normalization (N_pr)_ij=T_ir_ij. The exact identity H=N_pdR_p implies norm H(x_o)_∞ ≤normN_p(norm d A_p+1)normdx_∞.
Exact mathematical passage
(N_pr)_{ij}=T_ir_{ij}
r_jUse in source · 4 passages
Search aliases: r_j
Use in source · 04_symmetry.tex, line 195
a_ik(v) · =a_jk(v)a_ij(v). The triple equation implies a_ii=1 and a_ji=a_ij^-1. A morphism of descent data is a family of local gauges r_i satisfying r_j(v)a_ij(v)=a'_ij(v)r_i(v). These objects and arrows form a groupoid.
Exact mathematical passage
r_j(v)a_{ij}(v)=a'_{ij}(v)r_i(v).
R_{j+1}Use in source · 2 passages
Search aliases: R_j+1
Use in source · 03_towers.tex, line 324
A whole cascade. At scale j, let C_j:R_j→ R_j+1 and J_j:R_j+1→ R_j be bounded linear maps with C_jJ_j=I. For x∈ R_0, define x_0=x, x_j+1=C_jx_j, L_0=I, L_N=J_0⋯ J_N-1, h_j=x_j-J_jx_j+1, Q_Nx=L_Nx_N, S_N=I-Q_N. The map Q_N restores the surviving coarse remainder to the original space. It is this restored remainder, not just x_N, that controls reconstruction.
Exact mathematical passage
C_j:R_j\to R_{j+1}
r_{j+1}Use in source · 1 passage
Search aliases: r_j+1
Use in source · 12_proof_tools.tex, line 137
r a nonzero residual r∈ Y, set λ=2normr/c. The vector r/λ has norm c/2, so [source reference] supplies z∈ B_X with normSz-r/λ≤ c/4. Then u=λ z satisfies normu≤ 2normr/c, normr-Su≤normr/2. If the residual is zero, choose u=0. Start from r_0=y, repeat this construction, and define r_j+1=r_j-Su_j. Induction yields normr_j≤2^-jnormy, ∑_j≥0normu_j≤frac4cnormy. The series of preimages converges in X by completeness, to a vector x with the displayed norm bound. Continuity of S, together with the finite…
Exact mathematical passage
r_{j+1}=r_j-Su_j
R_{j+N}Use in source · 1 passage
Search aliases: R_j+N
Use in source · 03_towers.tex, line 432
Set L_j,0=I_R_j and L_j,N+1=J_jL_j+1,N. We prove, simultaneously for all starting scales j, that k+1<2^N Longrightarrow (L_j,Nz)_k=0 (z∈ R_j+N). For N=0, the inequality k+1<1 has no solution. Suppose the assertion holds at depth N. At depth N+1, an even coordinate vanishes by the displayed lift formula. For an odd coordinate k=2m+1, the hypothesis gives 2(m+1)<2^N+1, m+1<2^N. The induction hypothesis at starting scale j+1 makes (L_j+1,Nz)_m zero. Composing that zero operator with s^(j)_2m gives …
Exact mathematical passage
k+1<2^N\quad\Longrightarrow\quad (L_{j,N}z)_k=0 \qquad(z\in R_{j+N}).
r_mUse in source · 8 passages
Search aliases: r_m
Use in source · 03_towers.tex, line 129
belKS19 Starting from f^(0)=f, define f^(m+1)=f^(m)-A(df^(m)). There is a bounded linear projection P_A:E→ker d with f^(m)⟶ P_Af, p(P_Af)=p(f), and normf^(m)-P_Af ≤fracnorm A1-qnormdf^(m). It fixes every exact candidate and satisfies normP_A≤1+fracnorm Anorm d1-q. Put r_m=df^(m). Additivity of d gives r_m+1=r_m-dAr_m. Each r_m is the residual of an actual candidate, so the stated hypothesis applies at every iteration. Induction yields, for j≥0, normr_m+j≤ q^jnormr_m, normf^(m+j+1)-f^(m+j) ≤norm …
Exact mathematical passage
r_m=df^{(m)}
r_{m+1}Use in source · 1 passage
Search aliases: r_m+1
Use in source · 03_towers.tex, line 130
fine f^(m+1)=f^(m)-A(df^(m)). There is a bounded linear projection P_A:E→ker d with f^(m)⟶ P_Af, p(P_Af)=p(f), and normf^(m)-P_Af ≤fracnorm A1-qnormdf^(m). It fixes every exact candidate and satisfies normP_A≤1+fracnorm Anorm d1-q. Put r_m=df^(m). Additivity of d gives r_m+1=r_m-dAr_m. Each r_m is the residual of an actual candidate, so the stated hypothesis applies at every iteration. Induction yields, for j≥0, normr_m+j≤ q^jnormr_m, normf^(m+j+1)-f^(m+j) ≤norm A q^jnormr_m. For n>m, the finite…
Exact mathematical passage
r_{m+1}=r_m-dAr_m.
r_{m+j}Use in source · 1 passage
Search aliases: r_m+j
Use in source · 03_towers.tex, line 134
nd normf^(m)-P_Af ≤fracnorm A1-qnormdf^(m). It fixes every exact candidate and satisfies normP_A≤1+fracnorm Anorm d1-q. Put r_m=df^(m). Additivity of d gives r_m+1=r_m-dAr_m. Each r_m is the residual of an actual candidate, so the stated hypothesis applies at every iteration. Induction yields, for j≥0, normr_m+j≤ q^jnormr_m, normf^(m+j+1)-f^(m+j) ≤norm A q^jnormr_m. For n>m, the finite triangle inequality therefore gives normf^(n)-f^(m) ≤norm A normr_m∑_j=0^n-m-1q^j ≤fracnorm A1-qnormr_m. Since …
Exact mathematical passage
\norm{r_{m+j}}\le q^j\norm{r_m},\qquad \norm{f^{(m+j+1)}-f^{(m+j)}} \le\norm A\,q^j\norm{r_m}.
R_NUse in source · 2 passages
Search aliases: R_N
Use in source · 03_towers.tex, line 351
egintheorem[Finite reconstruction and nested remainders] For every N, x=Q_Nx+∑_j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of C_jJ_j=I gives D_NL_N=I_R_N. This cancellation takes place between adjacent maps with matching domains. No identification of different residual space…
Exact mathematical passage
D_N=C_{N-1}\cdots C_0:R_0\to R_N
r_NUse in source · 1 passage
Search aliases: r_N
Use in source · 12_proof_tools.tex, line 143
residual is zero, choose u=0. Start from r_0=y, repeat this construction, and define r_j+1=r_j-Su_j. Induction yields normr_j≤2^-jnormy, ∑_j≥0normu_j≤frac4cnormy. The series of preimages converges in X by completeness, to a vector x with the displayed norm bound. Continuity of S, together with the finite telescoping equation S(∑_j=0^N-1u_j)=y-r_N, proves Sx=y. Thus we may take C=4/c.
Exact mathematical passage
S\left(\sum_{j=0}^{N-1}u_j\right)=y-r_N,
r_nUse in source · 15 passages
Search aliases: r_n
Use in source · 03_towers.tex, line 173
A right inverse from target splittings. Suppose each target bonding map has a continuous linear section s_n:Y_n→ Y_n+1, ρ^Y_ns_n=I, norms_n≤ S. Assume the propagation gap SB_X<1, K=fracS1-SB_X. For r∈ R, recursively set (Ar)_0=0, (Ar)_n+1=s_n(r_n+(Ar)_nρ^X_n).
Exact mathematical passage
\label{eq:greenrecurrence} (Ar)_0=0,\qquad (Ar)_{n+1}=s_n\bigl(r_n+(Ar)_n\rho^X_n\bigr).
r_oUse in source · 1 passage
Search aliases: r_o
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
r_rUse in source · 1 passage
Search aliases: r_r
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
r_UUse in source · 9 passages
Search aliases: r_U
Use in source · 02_obstructions.tex, line 116
The exact sequence behind the criterion. The additive structures make restriction a homomorphism r_U:Γ(A)→Γ_U(A). Using the zero extension, or lift independence, o_U is also a homomorphism.
Exact mathematical passage
r_U:\Gamma(A)\to\Gamma_U(A)
r_vUse in source · 2 passages
Search aliases: r_v
Use in source · 04_symmetry.tex, line 245
Step 4: retain every morphism. Let c,d be global cocycles. Their canonical restrictions have identity overlap gauges. A morphism between those restrictions is consequently a family r_i satisfying r_j(v)=r_i(v) on each overlap. Assemble r_v from any containing patch. For an arrow choose a patch containing both endpoints. Its gauge equation proves d_f=r_vc_f[f^*(r_w)]^-1. Thus r is a global gauge. Its restriction is the original family. Conversely, assembling the restrictions of a global gauge pre…
Exact mathematical passage
r_v
r_wUse in source · 1 passage
Search aliases: r_w
Use in source · 04_symmetry.tex, line 246
Step 4: retain every morphism. Let c,d be global cocycles. Their canonical restrictions have identity overlap gauges. A morphism between those restrictions is consequently a family r_i satisfying r_j(v)=r_i(v) on each overlap. Assemble r_v from any containing patch. For an arrow choose a patch containing both endpoints. Its gauge equation proves d_f=r_vc_f[f^*(r_w)]^-1. Thus r is a global gauge. Its restriction is the original family. Conversely, assembling the restrictions of a global gauge pre…
Exact mathematical passage
d_f=r_vc_f[f^*(r_w)]^{-1}.
R_\alphaUse in source · 3 passages
Search aliases: R_α alpha
Use in source · 05h_regularized_repair.tex, line 97
et frontier] If 0<α<γ, then norma_α≥norma_γ, normr-Ta_α≤normr-Ta_γ. At M_α=norma_α, β(M_α)=normr-Ta_α. When v_α=r-Ta_α≠0, its normalization is an optimizing witness in Theorem~[source reference] at that budget. Let A_α=norma_α^2 and R_α=normr-Ta_α^2. Comparing the two admissible minimizers in each other's objectives gives R_α+α A_α≤ R_γ+α A_γ, R_γ+γ A_γ≤ R_α+γ A_α. Add these inequalities and cancel the residual terms. The result is (γ-α)(A_α-A_γ)≥0. Since γ>α, it follows that A_α≥ A_γ. Return to…
Exact mathematical passage
R_\alpha=\norm{r-Ta_\alpha}^2
R_\gammaUse in source · 2 passages
Search aliases: R_γ gamma
Use in source · 05h_regularized_repair.tex, line 98
a_α≤normr-Ta_γ. At M_α=norma_α, β(M_α)=normr-Ta_α. When v_α=r-Ta_α≠0, its normalization is an optimizing witness in Theorem~[source reference] at that budget. Let A_α=norma_α^2 and R_α=normr-Ta_α^2. Comparing the two admissible minimizers in each other's objectives gives R_α+α A_α≤ R_γ+α A_γ, R_γ+γ A_γ≤ R_α+γ A_α. Add these inequalities and cancel the residual terms. The result is (γ-α)(A_α-A_γ)≥0. Since γ>α, it follows that A_α≥ A_γ. Return to the first comparison to obtain R_α-R_γ≤α(A_γ-A_α)≤0…
Exact mathematical passage
R_\alpha+\alpha A_\alpha\le R_\gamma+\alpha A_\gamma, \qquad R_\gamma+\gamma A_\gamma\le R_\alpha+\gamma A_\alpha.
r_\inftyUse in source · 4 passages
Search aliases: r_∞ infty
Use in source · 03_towers.tex, line 194
orm A≤ K. The operator P=I-A∂ is an exact projection and (Pf)_0=f_0, normf-Pf_∞≤ Knorm∂ f_∞. Let θ=SB_X<1. The number K=S/(1-θ) is nonnegative and satisfies S+SB_XK=K. Fix r∈ R and define its coordinates recursively as in [source reference]. The initial coordinate has norm zero. If norm(Ar)_n≤ Knorm r_∞, submultiplicativity of operator norms gives norm(Ar)_n+1 · ≤norms_n(normr_n+norm(Ar)_nnormρ^X_n)
Exact mathematical passage
\norm{(Ar)_n}\le K\norm r_\infty
\R_{\ge0}^{\infty}Use in source · 1 passage
Search aliases: R_≥0^∞ R ge infty
Use in source · 05b_transport_cost.tex, line 40
stances from A to C and from C to A is bounded by d_W^sym(A,B)+d_W^sym(B,C). Taking their maximum proves the symmetric triangle law. Zero diagonal and symmetry give reflexivity and symmetry of the zero-distance relation. The triangle inequality and nonnegativity give transitivity. sourceKS78 The formal source proves these laws directly in R_≥0^∞ and constructs the zero-distance setoid. It does not identify zero distance with categorical isomorphism. Passing to the quotient can separate zero-dist…
Exact mathematical passage
\R_{\ge0}^{\infty}
\ranUse in source · 39 passages
Search aliases: ran
Use in source · 05_geometry.tex, line 120
ion gives norm x^2=normqx^2+normx-qx^2, hence normqx≤norm x. For any y∈ K, the vector qx-y lies in K, whereas x-qx lies in its orthogonal complement. Consequently normx-y^2=normx-qx^2+normqx-y^2≥normx-qx^2. Equality holds only when normqx-y=0, or y=qx. This proves nearest-point uniqueness without any closedness assumption on ran D. Closedness of the kernel and of the range serve different purposes. sourceKS42 No closed range assumption on D is needed for this projection. Such an assumption enter…
Exact mathematical passage
\ran D
\operatorname{rank}Use in source · 1 passage
Search aliases: rank operatorname
Use in source · 06_boundaries.tex, line 88
is y with Ay=0 and Py=Px. The full answer set is y+H, where H=ker A∩ker P. item There are row vectors λ∈K^1× m and μ∈K^1× p with λ A=μ P, λ Ax=1. The negative witness annihilates every admissible correction defect, since λ AN=μ PN=0. The ambiguity dimension in the positive case is dim H=n-rankA P. Set T=A
Exact mathematical passage
\dim H=n-\operatorname{rank}\begin{pmatrix}A\\P\end{pmatrix}.
\rceilUse in source · 4 passages
Search aliases: rceil
Use in source · 05i_diagonal_regularization.tex, line 155
[An explicit noisy attainable repair] For the attainable datum b=(1/n^2), suppose normwidetilde b-b≤δ with 0<δ≤1. Choose m=lceilδ^-2/3rceil, α=m^-2. Then the noisy regularized correction satisfies norma_α(widetilde b)-a^† ≤sqrtfrac2m+fracδ m2 ≤(sqrt2+1)δ^1/3. The exact-data bias has coordinates n/(m^2+n^2), so its squared norm is precisely B_m. By Proposition~[source reference], norma_m^-2(b)-a^†≤sqrt2/m. Theorem~[source reference] at the same parameter gives norma_m^-2(widetilde b)-a_m^-2(b) ≤f…
Exact mathematical passage
m=\lceil\delta^{-2/3}\rceil,\qquad\alpha=m^{-2}.
\operatorname{Reach}Use in source · 1 passage
Search aliases: Reach operatorname
Use in source · 01_contracts.tex, line 66
[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If M_j≠0 and j is not reachable, P_j annihilates the entire hull but not M. An element of the direct sum has finite support. Applying one primitive oper…
Exact mathematical passage
\Hull_{\mathcal O}(S)\subseteq \bigoplus_{j\in\operatorname{Reach}_E(F)}M_j.
\operatorname{Reach}_EUse in source · 1 passage
Search aliases: Reach_E operatorname
Use in source · 01_contracts.tex, line 66
[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If M_j≠0 and j is not reachable, P_j annihilates the entire hull but not M. An element of the direct sum has finite support. Applying one primitive oper…
Exact mathematical passage
\Hull_{\mathcal O}(S)\subseteq \bigoplus_{j\in\operatorname{Reach}_E(F)}M_j.
\rfloorUse in source · 1 passage
Search aliases: rfloor
Use in source · 05i_diagonal_regularization.tex, line 91
)=fracπ4. The hard prefix correction at budget M=sqrt N instead has sqrt N normw-T1_N⟶1. Step 1: control the infinite Riemann sums uniformly. Define f(x)=frac1(1+x^2)^2, g(x)=fracx^2(1+x^2)^2. Then A_m/m=m^-1∑_n≥1f(n/m) and mB_m=m^-1∑_n≥1g(n/m). Fix L≥2 and put N=lfloor mLrfloor. On [0,L], both functions are uniformly continuous. If ω_f(h) is a modulus of continuity there, comparison on the N intervals of length 1/m gives |frac1m∑_n=1^N f(n/m)-∫_0^N/mf(x) dx| ≤ Lω_f(1/m). The missing interval [N…
Exact mathematical passage
N=\lfloor mL\rfloor
\rightharpoonupUse in source · 1 passage
Search aliases: rightharpoonup
Use in source · 05h_regularized_repair.tex, line 175
Step 4: strengthen weak convergence to norm convergence. Take an arbitrary sequence α_ndownarrow0, or more generally positive α_n→0. The bound supplies a weakly convergent subsequence a_α_n_krightharpoonup a. Since its images converge in norm to b, the adjoint pairing argument of Theorem~[source reference] yields Ta=b. Weak lower semicontinuity gives norm a≤liminf_knorma_α_n_k ≤norma^†. Minimality and uniqueness of a^† force a=a^†. Moreover the same lower bound, together with the uniform upper b…
Exact mathematical passage
a_{\alpha_{n_k}}\rightharpoonup a
\mathcal SUse in source · 5 passages
Search aliases: S mathcal
Use in source · 03_towers.tex, line 233
[Split coordinate theorem] The observable is a bounded projection with p_LA=0 and p_LP=p_L. It separates exact candidates. If S_L={s∈ E:p_Ls=s}, then Φ:E≃_L S_L× R, Φ(f)=(p_Lf,∂ f), Φ^-1(s,r)=Ps+Ar. Repair becomes (s,r)↦(s,0). It is the unique exact repair preserving p_Lf. Put B_n=I-s_nρ^Y_n. Since ρ^Y_ns_n=I, direct multiplication gives B_n^2=B_n, ρ^Y_nB_n=0, B_ns_n=0. Moreover, B_ny=y whenever ρ^Y_ny=0. Thus B_n really projects onto the target kernel. Its norm is at most 1+SB_Y. Coordinate zer…
Exact mathematical passage
\mathcal S_L=\{s\in E:p_Ls=s\},
S'Use in source · 2 passages
Search aliases: S'
Use in source · 05k_stopping_rules.tex, line 53
For the basis change B'=BC, the same ambient vector is represented by c'=C^-1c. Also S'=SC and G'=C^ TGC. Substitute these formulas into the new coordinate normal equation: q' · =((S')^ TS'+α G')c'-(S')^ Tr
Exact mathematical passage
S'=SC
s^2Use in source · 1 passage
Search aliases: s^2
Use in source · 05k_stopping_rules.tex, line 71
In the previous example, B=(1,1)^ T, G=2, and the proposal c=1/4 gives q=-1/4. Thus q^ TG^-1q=1/32, as required. Replacing B by sB changes q to sq and G to s^2G. The Euclidean norm of the coordinate covector can be made arbitrarily small or large by changing the nonzero scalar s. The actual correction, normal residual, and certified error do not change.
Exact mathematical passage
s^2G
s_{2n}Use in source · 1 passage
Search aliases: s_2n
Use in source · 03_towers.tex, line 317
The actual dyadic tower coarsener has the typed formula (Cr)_n=ρ^Y_2nr_2n+1+r_2nρ^X_2n+1. It is obtained by telescoping two adjacent compatibility squares. With target sections, a lift is (Jc)_2n=0, (Jc)_2n+1=s_2nc_n. The section identity gives CJ=I, so the missing detail is explicitly Hr, not an unspecified reconstruction error. The normed construction imposes the bounds needed to make these maps bounded on its operator family spaces. sourceKS25
Exact mathematical passage
(Jc)_{2n}=0,\qquad (Jc)_{2n+1}=s_{2n}c_n.
S_3Use in source · 6 passages
Search aliases: S_3
Use in source · 04_symmetry.tex, line 165
A small noncommutative model. [A C_2 loop with S_3 coefficients] Take one view with a loop σ satisfying σ^2=1, and constant coefficients in a group H. A cocycle is determined by an element p∈ H whose square is the identity. Gauge equivalence is conjugacy, and its automorphisms form the centralizer of p.
Exact mathematical passage
S_3
s_aUse in source · 1 passage
Search aliases: s_a
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
S_BUse in source · 3 passages
Search aliases: S_B
Use in source · 07_classification.tex, line 31
ghtarrow A=^* B. When A=^* B, the two groups are equal as subgroups of the fixed product P. Suppose f:G_A→ G_B is an isomorphism. The image of u_A has infinite order. An isomorphism preserves the existence of a pth root for every prime p. Combining the exact signature [source reference] with Lemma~[source reference] for f(u_A) gives S_A=^* S_B. Since each set chooses exactly one prime from each pair, the corresponding sets of bits differ finitely.
Exact mathematical passage
S_A\eqfin S_B
s_BUse in source · 5 passages
Search aliases: s_B
Use in source · 07_classification.tex, line 33
Conversely, suppose A=^* B. For every rational r, the vectors s_A(r) and s_B(r) differ only on the finitely many prime pairs where the selectors differ. Thus their difference belongs to D. The descriptions G_A=s_A( Q)+D and G_B=s_B( Q)+D define the same subset of P. The normal forms need not be identical. Their coordinate change is (r,d)⟼ (r,d+s_A(r)-s_B(r)). It fixes the represented ambient element. The correction is finite exactly because the selectors differ only finitely.
Exact mathematical passage
s_B(r)
s_bUse in source · 1 passage
Search aliases: s_b
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
S_cUse in source · 2 passages
Search aliases: S_c
Use in source · 03b_repair_transport.tex, line 122
A two-coordinate model in which the routes differ. Use E=F=R^2, write candidates as (u,r), and take d(u,r)=D(u,r)=r, A(r)=B(r)=(0,r). The exact space is the horizontal axis and both repairs are P(u,r)=(u,0). For a real parameter c, set S_c(u,r)=(u+cr,r), C=I. The defect square commutes because the second coordinate is unchanged. The mismatch is M_c(r)=(cr,0). Hence PS_c(u,r)=(u+cr,0), S_cP(u,r)=(u,0). With c=2 and input (0,1), one route gives (2,0) and the other gives (0,0). Both have zero resid…
Exact mathematical passage
S_c(u,r)=(u+cr,r),\qquad C=I.
s_cUse in source · 1 passage
Search aliases: s_c
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
S_{c_2}Use in source · 2 passages
Search aliases: S_c_2
Use in source · 03b_repair_transport.tex, line 133
Two shears compose by adding their parameters. In the mismatch law this appears as M_c_2+c_1=S_c_2M_c_1+M_c_2, and S_c_2 fixes the exact horizontal axis. The identity is therefore a simple instance of the same composition rule, not an unrelated coordinate trick.
Exact mathematical passage
M_{c_2+c_1}=S_{c_2}M_{c_1}+M_{c_2}
s_eUse in source · 1 passage
Search aliases: s_e
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
s_gUse in source · 1 passage
Search aliases: s_g
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
s_{HLw}Use in source · 2 passages
Search aliases: s_HLw
Use in source · 04_symmetry.tex, line 378
For the relative claim, pullback clearly sends an extension of s to an extension of H^*s on U'=H^-1U. Conversely suppose y extends H^*s. If w∈ U, the isomorphism w≅ HLw and isomorphism closure imply HLw∈ U. Hence Lw∈ U' and y_Lw=s_HLw. Both endpoints of t_w lie in U, so partial coherence gives (Ry)_w=F(t_w)s_HLw=s_w. Recovery therefore preserves the required boundary. The two inverse identities remain valid on the subsets of extensions. Bijections preserve nonemptiness and identify any two eleme…
Exact mathematical passage
y_{Lw}=s_{HLw}
s_iUse in source · 1 passage
Search aliases: s_i
Use in source · 01_views.tex, line 93
Let (U_i)_i∈ I be full patches. A matching family consists of s_i∈Γ_U_i(F) with equal values on every overlap. The relevant covering condition is for every f:v→ w, some U_i contains both v and w. Identities imply that this condition also covers every object.
Exact mathematical passage
s_i\in\Gamma_{U_i}(F)
s_{i(v)}Use in source · 2 passages
Search aliases: s_i(v)
Use in source · 01_views.tex, line 109
rem[Strict gluing] Under [source reference], restriction induces an equivalence Γ(F)≃Match(F,(U_i)). Every fixed matching family has a unique global glue. Step 1: every coordinate can be selected. Apply the cover condition to the identity arrow of v. It gives an index containing v. Choose one such index i(v) and put x_v=(s_i(v))_v. The choice is an existence construction. It is not an algorithm selecting patches in a preferred order. If v∈ U_j, matching on the overlap U_i(v)∩ U_j gives x_v=(s_j)…
Exact mathematical passage
x_v=(s_{i(v)})_v.
s_jUse in source · 4 passages
Search aliases: s_j
Use in source · 01_views.tex, line 113
Step 1: every coordinate can be selected. Apply the cover condition to the identity arrow of v. It gives an index containing v. Choose one such index i(v) and put x_v=(s_i(v))_v. The choice is an existence construction. It is not an algorithm selecting patches in a preferred order. If v∈ U_j, matching on the overlap U_i(v)∩ U_j gives x_v=(s_j)_v. tag* In particular the value is independent of the selected index.
Exact mathematical passage
x_v=(s_j)_v. \tag{*}
\mathcal S_LUse in source · 3 passages
Search aliases: S_L mathcal
Use in source · 03_towers.tex, line 233
[Split coordinate theorem] The observable is a bounded projection with p_LA=0 and p_LP=p_L. It separates exact candidates. If S_L={s∈ E:p_Ls=s}, then Φ:E≃_L S_L× R, Φ(f)=(p_Lf,∂ f), Φ^-1(s,r)=Ps+Ar. Repair becomes (s,r)↦(s,0). It is the unique exact repair preserving p_Lf. Put B_n=I-s_nρ^Y_n. Since ρ^Y_ns_n=I, direct multiplication gives B_n^2=B_n, ρ^Y_nB_n=0, B_ns_n=0. Moreover, B_ny=y whenever ρ^Y_ny=0. Thus B_n really projects onto the target kernel. Its norm is at most 1+SB_Y. Coordinate zer…
Exact mathematical passage
\mathcal S_L=\{s\in E:p_Ls=s\},
s_n'Use in source · 1 passage
Search aliases: s_n'
Use in source · 03_towers.tex, line 293
What is canonical. For fixed splitting and protected observable, the repair above is unique. Changing the splitting can change both the observable and the repaired answer. In the growing example, replace s_n(y)=(y,0) by s_n'(y)=(y,y_0). At the first positive level, repair of the identity becomes 1/2 · 0
Exact mathematical passage
s_n'(y)=(y,y_0)
s_{n+1}Use in source · 1 passage
Search aliases: s_n+1
Use in source · 06_boundaries.tex, line 55
For surjectivity, fix y∈ Y_n and choose m≥max{s_n,s_n+1,n+1}. Since y∈ I_n,m, there is w∈ X_m with ρ_n,m(w)=y. Put z=ρ_n+1,m(w). Stabilization at level n+1 gives z∈ Y_n+1, and composition gives ρ_nz=y.
Exact mathematical passage
m\ge\max\{s_n,s_{n+1},n+1\}.
s_sUse in source · 1 passage
Search aliases: s_s
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
s_uUse in source · 1 passage
Search aliases: s_u
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
s_vUse in source · 1 passage
Search aliases: s_v
Use in source · 04_symmetry.tex, line 119
is neutral. Equality of orbit classes means there exists an actual based gauge b with b· c^q=1. At an arrow f:v→ w, this equation is b_vq_v^-1c_ff^*(q_w)[f^*(b_w)]^-1=1. Multiply on the left by q_vb_v^-1. Homomorphic preservation of products then gives c_ff^*(q_wb_w^-1)=q_vb_v^-1. Therefore x_v=q_vb_v^-1 is a twisted coherent family. On U, b_v=1 and q_v=s_v, so it extends the prescribed section.
Exact mathematical passage
q_v=s_v
s_wUse in source · 1 passage
Search aliases: s_w
Use in source · 04_symmetry.tex, line 379
For the relative claim, pullback clearly sends an extension of s to an extension of H^*s on U'=H^-1U. Conversely suppose y extends H^*s. If w∈ U, the isomorphism w≅ HLw and isomorphism closure imply HLw∈ U. Hence Lw∈ U' and y_Lw=s_HLw. Both endpoints of t_w lie in U, so partial coherence gives (Ry)_w=F(t_w)s_HLw=s_w. Recovery therefore preserves the required boundary. The two inverse identities remain valid on the subsets of extensions. Bijections preserve nonemptiness and identify any two eleme…
Exact mathematical passage
(Ry)_w=F(t_w)s_{HLw}=s_w.
S_\phiUse in source · 4 passages
Search aliases: S_ϕ phi
Use in source · 03b_repair_transport.tex, line 137
Actual tower resampling gives a defect map. Return to Banach towers with contractive source and target bonds. Let ϕ be a monotone schedule. Candidate sampling is (S_ϕ f)_n=f_ϕ(n). The sampled bonding map is the actual composite restriction between the selected levels. The residual map C_ϕ is not pointwise sampling of a scalar error score. It accumulates transported fine defects across each omitted block.
Exact mathematical passage
(S_\phi f)_n=f_{\phi(n)}.
\secUse in source · 2 passages
Search aliases: sec
Use in source · 05i_diagonal_regularization.tex, line 110
Step 2: evaluate the integrals. The substitution x=tanθ gives dx=sec^2θ dθ and 1+x^2=sec^2θ, so ∫_0^∞ f(x) dx=∫_0^π/2cos^2θ dθ=fracπ4, ∫_0^∞ g(x) dx=∫_0^π/2sin^2θ dθ=fracπ4. Step 1 now proves the two stated asymptotics.
Exact mathematical passage
dx=\sec^2\theta\,d\theta
\sec^2Use in source · 2 passages
Search aliases: sec^2 sec
Use in source · 05i_diagonal_regularization.tex, line 110
Step 2: evaluate the integrals. The substitution x=tanθ gives dx=sec^2θ dθ and 1+x^2=sec^2θ, so ∫_0^∞ f(x) dx=∫_0^π/2cos^2θ dθ=fracπ4, ∫_0^∞ g(x) dx=∫_0^π/2sin^2θ dθ=fracπ4. Step 1 now proves the two stated asymptotics.
Exact mathematical passage
dx=\sec^2\theta\,d\theta
\operatorname{Set}Use in source · 1 passage
Search aliases: Set operatorname
Use in source · 04_symmetry.tex, line 275
chapA genuine stack on the composition site The preceding theorem can be placed in a standard site, rather than called stack-like. We use the usual effective-descent meaning of a stack~[source reference]. The base of this site is the inclusion poset of full patches P(V)=(Set(ObV),⊆). This patch category is different from the original category of views.
Exact mathematical passage
\mathcal P(\V)=(\operatorname{Set}(\Ob\V),\subseteq).
\sinUse in source · 1 passage
Search aliases: sin
Use in source · 05i_diagonal_regularization.tex, line 111
Step 2: evaluate the integrals. The substitution x=tanθ gives dx=sec^2θ dθ and 1+x^2=sec^2θ, so ∫_0^∞ f(x) dx=∫_0^π/2cos^2θ dθ=fracπ4, ∫_0^∞ g(x) dx=∫_0^π/2sin^2θ dθ=fracπ4. Step 1 now proves the two stated asymptotics.
Exact mathematical passage
\int_0^\infty f(x)\,dx=\int_0^{\pi/2}\cos^2\theta\,d\theta=\frac\pi4, \qquad \int_0^\infty g(x)\,dx=\int_0^{\pi/2}\sin^2\theta\,d\theta=\frac\pi4.
\sin^2Use in source · 1 passage
Search aliases: sin^2 sin
Use in source · 05i_diagonal_regularization.tex, line 111
Step 2: evaluate the integrals. The substitution x=tanθ gives dx=sec^2θ dθ and 1+x^2=sec^2θ, so ∫_0^∞ f(x) dx=∫_0^π/2cos^2θ dθ=fracπ4, ∫_0^∞ g(x) dx=∫_0^π/2sin^2θ dθ=fracπ4. Step 1 now proves the two stated asymptotics.
Exact mathematical passage
\int_0^\infty f(x)\,dx=\int_0^{\pi/2}\cos^2\theta\,d\theta=\frac\pi4, \qquad \int_0^\infty g(x)\,dx=\int_0^{\pi/2}\sin^2\theta\,d\theta=\frac\pi4.
\operatorname{span}Use in source · 1 passage
Search aliases: span operatorname
Use in source · 06_cases.tex, line 25
In matrix form, the difference operator is D= -1 · 1 · 0 · 0 0 · -1 · 1 · 0 0 · 0 · -1 · 1 1 · 0 · 0 · -1 , ker D=span{(1,1,1,1)}. The explicit construction proves ran D=kerℓ, ℓ(r)=∑_i r_i. Thus ℓ is a complete feasibility test for this prescribed-difference problem. It annihilates every actual difference family, but takes a nonzero value on an incompatible one. This is the finite obstruction-certificate pattern of Proposition~[source reference] and Proposition~[source reference] in a fully visi…
Exact mathematical passage
D=\begin{pmatrix} -1&1&0&0\\ 0&-1&1&0\\ 0&0&-1&1\\ 1&0&0&-1 \end{pmatrix},\qquad \ker D=\operatorname{span}\{(1,1,1,1)\}.
\srank_{\End(G_A)}Use in source · 1 passage
Search aliases: srank_End(G_A) srank End operatorname operatorname
Use in source · 07_classification.tex, line 123
Infinite seed rank before completion. [Seed-rank separation] For every A, srank_End(G_A)(G_A)=aleph_0. No finite family topologically generates G_A, even after arbitrary endomorphisms are admitted. Yet its completion has ordinary topological generating rank one. Each element s_A(r)+d has only finitely many nonzero coordinates outside S_A. A finite family therefore omits infinitely many unselected prime coordinates. By Theorem~[source reference], every endomorphism preserves a zero coordinate as …
Exact mathematical passage
\srank_{\End(G_A)}(G_A)=\aleph_0.
\StabUse in source · 1 passage
Search aliases: Stab
Use in source · 04_symmetry.tex, line 73
Automorphisms and ambiguity are not the same question. The stabilizer of a cocycle is the group of families with a_v=c_f f^*(a_w)c_f^-1. If one gauge a:c→ d is known, all gauges from c to d are uniquely of the form ah, with h∈Stab(c). Indeed a^-1b stabilizes c for every other gauge b. This is the ordinary Hom torsor principle in a groupoid.
Exact mathematical passage
h\in\Stab(c)
\SubUse in source · 4 passages
Search aliases: Sub operatorname
Use in source · 03_spectrum.tex, line 14
[Subgroup and isomorphism classification] The map B↦ D_B is an order isomorphism from P( N) onto Sub(D). Every subgroup is fully invariant under all endomorphisms of D, and D_B≅ D_C Longleftrightarrow B=C. Let H≤ D and let x∈ H. Its support is a finite set F of primes. For p∈ F, choose by the Chinese remainder theorem an integer k that is one modulo p and zero modulo every other prime in F. Then kx is supported only at p, where it is nonzero. It generates the entire C_p coordinate. Thus H contai…
Exact mathematical passage
\Sub(D)
\subsectionUse in source · 2 passages
Search aliases: subsection
Use in source · 05h_regularized_repair.tex, line 197
[3pt] q=0 · No · Residual vanishes, correction norms diverge, no exact protected answer. [3pt] q>0 · Yes · Strong convergence to the least-squares correction of minimum norm, not an exact answer. [3pt] q>0 · No · A positive residual floor and diverging correction norms occur together. The last case is not hypothetical. Let Tu=(u_n/n) on ℓ^2, define Su=(Tu,0) into ℓ^2⊕R, and take r=((1/n),1). Then b=((1/n),0) is in the closure but not the range, while q=1. The same escaping corrections as before …
Exact mathematical passage
3pt] $q=0$ & No & Residual vanishes, correction norms diverge, no exact protected answer.\\[3pt] $q>0$ & Yes & Strong convergence to the least-squares correction of minimum norm, not an exact answer.\\[3pt] $q>0$ & No & A positive residual floor and diverging correction norms occur together.\\\bottomrule \end{tabularx} \end{center} The last case is not hypothetical. Let $Tu=(u_n/n)$ on $\ell^2$, define $Su=(Tu,0)$ into $\ell^2\oplus\R$, and take $r=((1/n),1)$. Then $b=((1/n),0)$ is in the closure but not the range, while $q=1$. The same escaping corrections as before coexist with an irreducible orthogonal residual. Positive floor and unbounded correction cost are independent features. \subsection{Noise is controlled at a fixed parameter} \fixedplate{34_noise_filter}{For one singular value $\sigma$, the response to residual-data noise is $\sigma/(\sigma^2+\alpha)$. Its maximum occurs at $\sigma=\sqrt\alpha$ and equals $1/(2\sqrt\alpha)$. The plot evaluates this formula and the universal horizontal bounds at three explicit positive parameters.}{fig:noise-filter} \begin{theorem}[Data sensitivity and a convergent parameter choice]\label{WR09} For two residual data vectors $r,\widetilde r$ with $\norm{r-\widetilde r}\le\delta$ and the same $T,\alpha$, \[ \norm{a_\alpha(r)-a_\alpha(\widetilde r)} \le\frac{\delta}{2\sqrt\alpha},\qquad \norm{T(a_\alpha(r)-a_\alpha(\widetilde r))}\le\delta.
\sup_kUse in source · 1 passage
Search aliases: sup_k sup
Use in source · 03b_repair_transport.tex, line 166
uad normC_ϕ≤ m, ∂_ϕ S_ϕ=C_ϕ∂. Exact candidates remain exact under every monotone common resampling, even without a gap bound. If A and B are bounded right inverses for the fine and sampled differentials, then normP_BS_ϕ f-S_ϕ P_Af ≤(norm A+mnorm B)norm∂ f. For a bounded candidate family, sup_nnormf_ϕ(n)≤sup_knormf_k, proving the sampling bound. For a bounded residual family, each summand in [source reference] is a map from X_b to Y_a and has norm at most norm r_∞, because every composite bond is…
Exact mathematical passage
\sup_n\norm{f_{\phi(n)}}\le\sup_k\norm{f_k}
\sup_NUse in source · 1 passage
Search aliases: sup_N sup
Use in source · 03_towers.tex, line 392
[Density criterion for norm recovery] Suppose sup_NnormQ_N≤ K<∞. Then the norm recoverable vectors are exactly overline F. In particular all vectors are norm recoverable if and only if the finite core is dense. The nested identities first imply that the kernels increase: if Q_My=0 and N≥ M, then Q_Ny=Q_NQ_My=0. Thus the finite core really is an increasing union of linear subspaces.
Exact mathematical passage
\sup_N\norm{Q_N}\le K<\infty
\sup_{\norm{z}\le1}Use in source · 1 passage
Search aliases: sup_normz≤1 sup norm le
Use in source · 12_proof_tools.tex, line 82
linear map between real Hilbert spaces. For fixed z∈ R, the functional h↦⟨ Th,z⟩ is bounded by normTnormznormh. The lemma therefore defines a unique vector T^*z∈ H with ⟨ Th,z⟩=⟨ h,T^*z⟩. Uniqueness of representatives proves that T^* is linear. The same bound proves normT^*z≤normTnormz. Conversely, normTh=sup_normz≤1|⟨ Th,z⟩| ≤normT^*normh, where the first equality follows by testing z=Th/normTh when Th≠0. Hence normT^*=normT.
Exact mathematical passage
\norm{Th}=\sup_{\norm{z}\le1}|\langle Th,z\rangle| \le\norm{T^*}\norm{h},
\sup_{n\ge N}Use in source · 1 passage
Search aliases: sup_n≥ N sup ge
Use in source · 03_towers.tex, line 481
[The recovery domain is c_0] For this shift cascade, S_Nx→ x in ℓ^∞ Longleftrightarrow x_n→0. For x_n=q^n, 0≤ q<1, the error is exactly q^N. The error norm is sup_n≥ N|x_n|. It tends to zero precisely when the sequence tends to zero. For a geometric sequence its largest tail value is the first one, q^N. sourceKS29 As a second example, take scalar spaces with C=tfrac12I, J=2I. The coarse value x_N=2^-Nx tends to zero, but restoration gives Q_Nx=x. Every detail is zero and the entire input is pers…
Exact mathematical passage
\sup_{n\ge N}|x_n|
\mathcal TUse in source · 4 passages
Search aliases: T mathcal
Use in source · 04_symmetry.tex, line 297
For transitivity, suppose R covers W, and that for every U∈ R the pullback of a sieve T to U covers U. Fix a composable pair q in W. Its support S=|q| belongs to R. Since patches are full, q is also a composable pair in S. The covering condition for the pullback of T to S therefore puts the identity inclusion S⊆ S in that pullback. By the definition of pullback, this says S∈ T. It holds for every q, so T covers W.
Exact mathematical passage
\mathcal T
T^0Use in source · 1 passage
Search aliases: T^0
Use in source · 12_proof_tools.tex, line 167
Submultiplicativity gives normT^j≤θ^j, with the convention T^0=I. The finite partial sums are therefore operator-norm Cauchy, since for M>N, ∑_j=N+1^MT^j ≤∑_j=N+1^Mθ^j≤fracθ^N+11-θ. Their limit Z has norm at most ∑_j≥0θ^j=1/(1-θ). The operator identities for finite sums are (I-T)∑_j=0^NT^j=I-T^N+1, (∑_j=0^NT^j)(I-T)=I-T^N+1. Multiplication by a fixed bounded operator is continuous in operator norm, and normT^N+1≤θ^N+1→0. Passing to the limit proves both inverse identities. When W is the zero spa…
Exact mathematical passage
T^0=I
t^3Use in source · 1 passage
Search aliases: t^3
Use in source · 05_geometry.tex, line 602
-t · -t^2. The two off-diagonal entries of its square are zero, since -t^3+t^3=0. Both diagonal entries are (t^4+t^2)/(1+t^2)^2=t^2/(1+t^2). Thus B_t^ TB_t=B_t^2=fract^21+t^2I. Pairing this identity with an arbitrary vector proves the all-vector norm formula. It bounds the operator norm above by |t|/sqrt1+t^2, and evaluating any unit vector gives equality. The argument includes t=0, where both sides vanish.
Exact mathematical passage
-t^3+t^3=0
t^4Use in source · 1 passage
Search aliases: t^4
Use in source · 05_geometry.tex, line 602
-t · -t^2. The two off-diagonal entries of its square are zero, since -t^3+t^3=0. Both diagonal entries are (t^4+t^2)/(1+t^2)^2=t^2/(1+t^2). Thus B_t^ TB_t=B_t^2=fract^21+t^2I. Pairing this identity with an arbitrary vector proves the all-vector norm formula. It bounds the operator norm above by |t|/sqrt1+t^2, and evaluating any unit vector gives equality. The argument includes t=0, where both sides vanish.
Exact mathematical passage
(t^4+t^2)/(1+t^2)^2=t^2/(1+t^2)
T^jUse in source · 4 passages
Search aliases: T^j
Use in source · 12_proof_tools.tex, line 156
Neumann inversion with both inverse identities. [A strict perturbation radius] Let W be a real Banach space and T:W→ W bounded linear, with normT≤θ<1 and θ≥0. Then Z=∑_j=0^∞T^j exists in operator norm and satisfies (I-T)Z=Z(I-T)=I, normZ≤frac11-θ. First recall why the space of bounded operators on W is complete. An operator-norm Cauchy sequence A_n makes A_nx Cauchy for each x, and hence defines Ax=lim_nA_nx. Pointwise limits preserve linearity. The sequence of operator norms is bounded, so the …
Exact mathematical passage
Z=\sum_{j=0}^{\infty}T^j
T^nUse in source · 1 passage
Search aliases: T^n
Use in source · 05_geometry.tex, line 503
The perturbed map is surjective, with D'A'=I, normA'≤fraca1-aε. Its exact projection satisfies normx-q'x≤fraca1-aεnormD'x, normq-q'≤fracaε1-aε. The map T=(D-D')A acts on the Banach target R and satisfies norm T≤ aε<1. The operator series Z=∑_n=0^∞ T^n converges in operator norm and obeys norm Z≤(1-aε)^-1. Multiplying its finite partial sums on either side by I-T gives I-T^N+1. Passing to the operator-norm limit proves (I-T)Z=Z(I-T)=I. Appendix~[source reference] supplies the full series argument…
Exact mathematical passage
Z=\sum_{n=0}^\infty T^n
T^{N+1}Use in source · 3 passages
Search aliases: T^N+1
Use in source · 05_geometry.tex, line 506
1-aεnormD'x, normq-q'≤fracaε1-aε. The map T=(D-D')A acts on the Banach target R and satisfies norm T≤ aε<1. The operator series Z=∑_n=0^∞ T^n converges in operator norm and obeys norm Z≤(1-aε)^-1. Multiplying its finite partial sums on either side by I-T gives I-T^N+1. Passing to the operator-norm limit proves (I-T)Z=Z(I-T)=I. Appendix~[source reference] supplies the full series argument, including completeness of the operator space.
Exact mathematical passage
I-T^{N+1}
t_aUse in source · 1 passage
Search aliases: t_a
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
t_bUse in source · 1 passage
Search aliases: t_b
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
t_dUse in source · 1 passage
Search aliases: t_d
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
t_eUse in source · 1 passage
Search aliases: t_e
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
t_{Hv}Use in source · 2 passages
Search aliases: t_Hv
Use in source · 04_symmetry.tex, line 372
If x∈Γ(F), coherence along t_w gives (RH^*x)_w=F(t_w)x_HLw=x_w. For the other composite, the triangle identity says t_Hv=H(η_v). Coherence of y along η_v:v→ LHv yields (H^*Ry)_v=F(t_Hv)y_LHv=F(Hη_v)y_LHv=y_v. These coordinate equations prove both inverse identities.
Exact mathematical passage
t_{Hv}=H(\eta_v)
T_i^{-1}Use in source · 5 passages
Search aliases: T_i^-1
Use in source · 05_geometry.tex, line 54
Holonomy is the remaining exactness condition. Write T_i=T_p_i. Reconstruction from a proposed root value is (R_pz)_i=T_i^-1z. For each directed edge define the root loop operator K_ij=T_iτ_ijT_j^-1, K=⋂_(i,j)∈vec Eker(K_ij-I). Each K_ij is the transport around an actual rooted loop. No flatness assumption has been made.
Exact mathematical passage
(R_pz)_i=T_i^{-1}z.
T_j^{-1}Use in source · 2 passages
Search aliases: T_j^-1
Use in source · 05_geometry.tex, line 58
Holonomy is the remaining exactness condition. Write T_i=T_p_i. Reconstruction from a proposed root value is (R_pz)_i=T_i^-1z. For each directed edge define the root loop operator K_ij=T_iτ_ijT_j^-1, K=⋂_(i,j)∈vec Eker(K_ij-I). Each K_ij is the transport around an actual rooted loop. No flatness assumption has been made.
Exact mathematical passage
K_{ij}=T_i\tau_{ij}T_j^{-1},\qquad K=\bigcap_{(i,j)\in\vec E}\ker(K_{ij}-I).
t_mUse in source · 1 passage
Search aliases: t_m
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
T_NUse in source · 1 passage
Search aliases: T_N
Use in source · 05e_escape.tex, line 101
Every finite operator is invertible, but the gains diverge. On R^N, let T_N=diag(1,1/2,…,1/N). It is invertible and normT_N^-1=N in the Euclidean norm. The upper bound follows by multiplying each coordinate by an integer at most N, and equality is attained at the last coordinate vector. Full finite rank therefore supplies no uniform gain as N→∞.
Exact mathematical passage
T_N=\operatorname{diag}(1,1/2,\ldots,1/N).
T_N^{-1}Use in source · 1 passage
Search aliases: T_N^-1
Use in source · 05e_escape.tex, line 104
Every finite operator is invertible, but the gains diverge. On R^N, let T_N=diag(1,1/2,…,1/N). It is invertible and normT_N^-1=N in the Euclidean norm. The upper bound follows by multiplying each coordinate by an integer at most N, and equality is attained at the last coordinate vector. Full finite rank therefore supplies no uniform gain as N→∞.
Exact mathematical passage
\norm{T_N^{-1}}=N
t_oUse in source · 1 passage
Search aliases: t_o
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
t_pUse in source · 1 passage
Search aliases: t_p
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
T_{p_i}Use in source · 1 passage
Search aliases: T_p_i
Use in source · 05_geometry.tex, line 53
Holonomy is the remaining exactness condition. Write T_i=T_p_i. Reconstruction from a proposed root value is (R_pz)_i=T_i^-1z. For each directed edge define the root loop operator K_ij=T_iτ_ijT_j^-1, K=⋂_(i,j)∈vec Eker(K_ij-I). Each K_ij is the transport around an actual rooted loop. No flatness assumption has been made.
Exact mathematical passage
T_i=T_{p_i}
t_rUse in source · 1 passage
Search aliases: t_r
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
t_v^{-1}Use in source · 2 passages
Search aliases: t_v^-1
Use in source · 04_symmetry.tex, line 98
If t is another raw lift, define a_v=t_v^-1q_v. On U both lifts equal s, so a_v=1 and a∈ G_U. Its action is (a· c^q)_f · =t_v^-1q_v q_v^-1c_ff^*(q_w) [f^*(t_w^-1q_w)]^-1
Exact mathematical passage
a_v=t_v^{-1}q_v
t_wUse in source · 9 passages
Search aliases: t_w
Use in source · 04_symmetry.tex, line 99
If t is another raw lift, define a_v=t_v^-1q_v. On U both lifts equal s, so a_v=1 and a∈ G_U. Its action is (a· c^q)_f · =t_v^-1q_v q_v^-1c_ff^*(q_w) [f^*(t_w^-1q_w)]^-1 · =t_v^-1c_ff^*(q_w) [f^*(q_w)]^-1f^*(t_w) =t_v^-1c_ff^*(t_w)=c_f^t. Thus changing the raw lift leaves the based orbit unchanged.
Exact mathematical passage
\begin{aligned} (a\cdot c^q)_f &=t_v^{-1}q_v\,q_v^{-1}c_ff^*(q_w) [f^*(t_w^{-1}q_w)]^{-1}\\ &=t_v^{-1}c_ff^*(q_w) [f^*(q_w)]^{-1}f^*(t_w) =t_v^{-1}c_ff^*(t_w)=c_f^t. \end{aligned}
t_{w'}Use in source · 2 passages
Search aliases: t_w'
Use in source · 04_symmetry.tex, line 354
ce and uniqueness of the prescribed extension are preserved and reflected. Write the unit of the equivalence as η:I_V⇒ LH, and take the unit and counit with their triangle identities. Set t_w=ε_w^-1:w→ HLw. Given y∈Γ(H^*F), define (Ry)_w=F(t_w)y_Lw. For an arrow f:w→ w', naturality of the inverse counit states HL(f) t_w=t_w'f. Contravariance of F and coherence of y therefore give F(f)(Ry)_w' · =F(f)F(t_w')y_Lw'
Exact mathematical passage
HL(f)\,t_w=t_{w'}f.
t_w^{-1}Use in source · 1 passage
Search aliases: t_w^-1
Use in source · 04_symmetry.tex, line 99
If t is another raw lift, define a_v=t_v^-1q_v. On U both lifts equal s, so a_v=1 and a∈ G_U. Its action is (a· c^q)_f · =t_v^-1q_v q_v^-1c_ff^*(q_w) [f^*(t_w^-1q_w)]^-1 · =t_v^-1c_ff^*(q_w) [f^*(q_w)]^-1f^*(t_w) =t_v^-1c_ff^*(t_w)=c_f^t. Thus changing the raw lift leaves the based orbit unchanged.
Exact mathematical passage
\begin{aligned} (a\cdot c^q)_f &=t_v^{-1}q_v\,q_v^{-1}c_ff^*(q_w) [f^*(t_w^{-1}q_w)]^{-1}\\ &=t_v^{-1}c_ff^*(q_w) [f^*(q_w)]^{-1}f^*(t_w) =t_v^{-1}c_ff^*(t_w)=c_f^t. \end{aligned}
\tanUse in source · 1 passage
Search aliases: tan
Use in source · 05i_diagonal_regularization.tex, line 110
Step 2: evaluate the integrals. The substitution x=tanθ gives dx=sec^2θ dθ and 1+x^2=sec^2θ, so ∫_0^∞ f(x) dx=∫_0^π/2cos^2θ dθ=fracπ4, ∫_0^∞ g(x) dx=∫_0^π/2sin^2θ dθ=fracπ4. Step 1 now proves the two stated asymptotics.
Exact mathematical passage
x=\tan\theta
\topruleUse in source · 1 passage
Search aliases: toprule
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
\TorUse in source · 4 passages
Search aliases: Tor operatorname
Use in source · 00_notation.tex, line 33
The main statements have the following form: Tor(G_A) · =D, · G_A/D · ≅ Q, G_A/mG_A · ≅ C_rad(m), · widehatG_A · ≅ P, G_A≅ G_B · iff A=^* B, · · G_A has an X-computable copy · iff A≤_T X. · · The extension by D does not split. Distinct eventual selector classes do not embed in one another. All of these conclusions are proved within the book, with the exact hypotheses retained.
Exact mathematical passage
\Tor(G_A)&=D,&G_A/D&\cong\Q,\\ G_A/mG_A&\cong C_{\rad(m)},&\widehat{G_A}&\cong P,\\ G_A\cong G_B&\iff A\eqfin B,&&\\ G_A\text{ has an }X\text{-computable copy}&\iff A\le_T X.&&
U'Use in source · 4 passages
Search aliases: U'
Use in source · 04_symmetry.tex, line 342
[Equivalence of section spaces] Pullback induces Γ(F)≃Γ(H^*F). If U⊆ObW is closed under isomorphism and U'=H^-1U, it also induces Ext_U^F(s)≃Ext_U'^H^*F(H^*s). Existence and uniqueness of the prescribed extension are preserved and reflected. Write the unit of the equivalence as η:I_V⇒ LH, and take the unit and counit with their triangle identities. Set t_w=ε_w^-1:w→ HLw. Given y∈Γ(H^*F), define (Ry)_w=F(t_w)y_Lw. For an arrow f:w→ w', naturality of the inverse counit states HL(f) t_w=t_w'f. Cont…
Exact mathematical passage
U'=H^{-1}U
u'Use in source · 3 passages
Search aliases: u'
Use in source · 12_proof_tools.tex, line 72
For any h∈ H, the vector h-ℓ(h)v/ℓ(v) belongs to L. Pairing it with v therefore gives ⟨ h,v⟩=fracℓ(h)ℓ(v)normv^2. Consequently u=ℓ(v)v/normv^2 represents ℓ. If u' also represents it, pairing the difference of the two formulas with h=u-u' gives normu-u'^2=0, hence uniqueness.
Exact mathematical passage
u'
U'_xUse in source · 1 passage
Search aliases: U'_x
Use in source · 04b_phase.tex, line 67
Changing lifts to U'_x=(-1)^a(x)U_x changes the phase to c'(x,y)=c(x,y)+a(x)+a(y)-a(x+y). No such change makes c' vanish identically. The antisymmetric part is unchanged by every scalar change of lift. In this model it is κ(x,y)=c(x,y)-c(y,x) =x_2y_1-y_2x_1. This is a directly observable obstruction certificate for the model. Its nonzero value at (e_1,e_2) is enough, without a search over gauges.
Exact mathematical passage
U'_x=(-1)^{a(x)}U_x
u^2Use in source · 3 passages
Search aliases: u^2
Use in source · 05_geometry.tex, line 335
[The two-stage repair is the joint projection] The operator P is the orthogonal projection onto M, equivalently onto the kernel of Δ=(D,C). If u=x-P_Lx and v=P_Lx-Px, then u⊥ v, normx-Px^2=norm u^2+norm v^2. The subspace L=ker D is closed, hence Hilbert with the inherited inner product. The restriction B=C|_L is continuous, so its kernel is a closed subspace of L. Under the inclusion into E, this kernel is exactly M=L∩ker C. Thus both projections used to define P exist.
Exact mathematical passage
u\perp v,\qquad\norm{x-Px}^2=\norm u^2+\norm v^2.
U_{(p,q)}Use in source · 1 passage
Search aliases: U_(p,q)
Use in source · 04b_phase.tex, line 45
The phase is visible in actual matrices. Over R, define X=0 · 1 1 · 0, Z=1 · 0 0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do not commute.
Exact mathematical passage
X=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad Z=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\qquad U_{(p,q)}=X^pZ^q\quad(p,q\in\{0,1\}).
u_BUse in source · 4 passages
Search aliases: u_B
Use in source · 07_classification.tex, line 45
[Marked classification] An isomorphism (G_A,u_A)≅(G_B,u_B) exists if and only if A=B. A marked isomorphism preserves the exact two-prime root test [source reference]--[source reference] at every index. Therefore every bit agrees. The converse is the identity map.
Exact mathematical passage
(G_A,u_A)\cong(G_B,u_B)
U_{e_1}Use in source · 2 passages
Search aliases: U_e_1
Use in source · 04b_phase.tex, line 54
0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do not commute.
Exact mathematical passage
U_{e_1}U_{e_2}=XZ
U_{e_2}Use in source · 2 passages
Search aliases: U_e_2
Use in source · 04b_phase.tex, line 54
0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do not commute.
Exact mathematical passage
U_{e_1}U_{e_2}=XZ
U_iUse in source · 18 passages
Search aliases: U_i
Use in source · 01_views.tex, line 93
Let (U_i)_i∈ I be full patches. A matching family consists of s_i∈Γ_U_i(F) with equal values on every overlap. The relevant covering condition is for every f:v→ w, some U_i contains both v and w. Identities imply that this condition also covers every object.
Exact mathematical passage
(U_i)_{i\in I}
U_{i(v)}Use in source · 1 passage
Search aliases: U_i(v)
Use in source · 01_views.tex, line 112
has a unique global glue. Step 1: every coordinate can be selected. Apply the cover condition to the identity arrow of v. It gives an index containing v. Choose one such index i(v) and put x_v=(s_i(v))_v. The choice is an existence construction. It is not an algorithm selecting patches in a preferred order. If v∈ U_j, matching on the overlap U_i(v)∩ U_j gives x_v=(s_j)_v. tag* In particular the value is independent of the selected index.
Exact mathematical passage
U_{i(v)}\cap U_j
U_jUse in source · 9 passages
Search aliases: U_j
Use in source · 01_views.tex, line 112
_i)). Every fixed matching family has a unique global glue. Step 1: every coordinate can be selected. Apply the cover condition to the identity arrow of v. It gives an index containing v. Choose one such index i(v) and put x_v=(s_i(v))_v. The choice is an existence construction. It is not an algorithm selecting patches in a preferred order. If v∈ U_j, matching on the overlap U_i(v)∩ U_j gives x_v=(s_j)_v. tag* In particular the value is independent of the selected index.
Exact mathematical passage
v\in U_j
u_jUse in source · 3 passages
Search aliases: u_j
Use in source · 12_proof_tools.tex, line 137
r a nonzero residual r∈ Y, set λ=2normr/c. The vector r/λ has norm c/2, so [source reference] supplies z∈ B_X with normSz-r/λ≤ c/4. Then u=λ z satisfies normu≤ 2normr/c, normr-Su≤normr/2. If the residual is zero, choose u=0. Start from r_0=y, repeat this construction, and define r_j+1=r_j-Su_j. Induction yields normr_j≤2^-jnormy, ∑_j≥0normu_j≤frac4cnormy. The series of preimages converges in X by completeness, to a vector x with the displayed norm bound. Continuity of S, together with the finite…
Exact mathematical passage
r_{j+1}=r_j-Su_j
U_mUse in source · 1 passage
Search aliases: U_m
Use in source · 05_geometry.tex, line 39
τ_i_0i_1⋯τ_i_k-1i_k for 1≤ k≤ m. The transport convention makes every U_k a map from the fiber at i_k to that at i_0. Since U_k+1=U_kτ_i_ki_k+1, the k-th transported defect is U_k(dx)_i_ki_k+1 =U_k+1x_i_k+1-U_kx_i_k. Summing over k=0,…,m-1 cancels each interior value once with each sign. The remaining terms are U_mx_i_m-U_0x_i_0=T_qx_i_m-x_i_0, proving the identity with its actual ordered products. For an empty walk both sides are zero.
Exact mathematical passage
U_mx_{i_m}-U_0x_{i_0}=T_qx_{i_m}-x_{i_0}
u_mUse in source · 2 passages
Search aliases: u_m
Use in source · 05_geometry.tex, line 143
Now assume condition 2. To prove closedness, take an arbitrary sequence r_n∈ran D with r_n→ r in R. Choose x_n with Dx_n=r_n and put u_n=x_n-qx_n. Then u_n∈ K^⊥ and Du_n=r_n. Applying the estimate to u_n-u_m gives normu_n-u_m≤ Cnormr_n-r_m. Thus (u_n) is Cauchy in the complete space K^⊥. Let its limit be u. Continuity gives Du=lim_n r_n=r, so r∈ran D. In a normed space, a set containing the limits of all its convergent sequences is closed. This proves condition 1 without presupposing that the in…
Exact mathematical passage
u_n-u_m
u_nUse in source · 18 passages
Search aliases: u_n
Use in source · 05_geometry.tex, line 143
Now assume condition 2. To prove closedness, take an arbitrary sequence r_n∈ran D with r_n→ r in R. Choose x_n with Dx_n=r_n and put u_n=x_n-qx_n. Then u_n∈ K^⊥ and Du_n=r_n. Applying the estimate to u_n-u_m gives normu_n-u_m≤ Cnormr_n-r_m. Thus (u_n) is Cauchy in the complete space K^⊥. Let its limit be u. Continuity gives Du=lim_n r_n=r, so r∈ran D. In a normed space, a set containing the limits of all its convergent sequences is closed. This proves condition 1 without presupposing that the in…
Exact mathematical passage
u_n=x_n-qx_n
U_tUse in source · 2 passages
Search aliases: U_t
Use in source · 05_geometry.tex, line 314
Small holonomy can still be poorly conditioned. In R^2, the rational rotation U_t=frac11+t^2 1-t^2 · -2t 2t · 1-t^2 satisfies norm(U_t-I)z^2=frac4t^21+t^2norm z^2. For t≠0 its fixed space is zero and its residual gain is sqrt1+t^2/(2|t|), which diverges as t→0. At t=0, the entire fiber is fixed. Therefore small loop discrepancy does not supply a uniform stable realization estimate. The fixed directions and their conditioning must be analyzed together.
Exact mathematical passage
U_t=\frac1{1+t^2} \begin{pmatrix}1-t^2&-2t\\2t&1-t^2\end{pmatrix}
U_xUse in source · 5 passages
Search aliases: U_x
Use in source · 04b_phase.tex, line 51
0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do not commute.
Exact mathematical passage
\label{eq:phase-matrices} U_xU_y=(-1)^{c(x,y)}U_{x+y}.
U_{x+y}Use in source · 1 passage
Search aliases: U_x+y
Use in source · 04b_phase.tex, line 51
0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do not commute.
Exact mathematical passage
\label{eq:phase-matrices} U_xU_y=(-1)^{c(x,y)}U_{x+y}.
U_{x+y+z}Use in source · 2 passages
Search aliases: U_x+y+z
Use in source · 04b_phase.tex, line 57
Associativity of three lifted matrices requires the same total sign on both parenthesizations. Equation~[source reference] is exactly that requirement: (U_xU_y)U_z =(-1)^c(x,y)+c(x+y,z)U_x+y+z, U_x(U_yU_z) =(-1)^c(y,z)+c(x,y+z)U_x+y+z. Thus a nonzero phase is not automatically a failed associativity law. It can be the data that make the composition law explicit.
Exact mathematical passage
(U_xU_y)U_z =(-1)^{c(x,y)+c(x+y,z)}U_{x+y+z},
U_yUse in source · 3 passages
Search aliases: U_y
Use in source · 04b_phase.tex, line 51
0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do not commute.
Exact mathematical passage
\label{eq:phase-matrices} U_xU_y=(-1)^{c(x,y)}U_{x+y}.
U_zUse in source · 2 passages
Search aliases: U_z
Use in source · 04b_phase.tex, line 57
Associativity of three lifted matrices requires the same total sign on both parenthesizations. Equation~[source reference] is exactly that requirement: (U_xU_y)U_z =(-1)^c(x,y)+c(x+y,z)U_x+y+z, U_x(U_yU_z) =(-1)^c(y,z)+c(x,y+z)U_x+y+z. Thus a nonzero phase is not automatically a failed associativity law. It can be the data that make the composition law explicit.
Exact mathematical passage
(U_xU_y)U_z =(-1)^{c(x,y)+c(x+y,z)}U_{x+y+z},
v^{\mathsf T}Use in source · 1 passage
Search aliases: v^ T mathsf
Use in source · 06_cases.tex, line 158
-1 · 0 · -1. The ordinary cycle block satisfies A^ TA=3I-11^ T. It is zero on the constant line and equals 3I on its orthogonal complement. The reflected block satisfies B^ TB=I+vv^ T, v=(1,-1,1)^ T. Since norm v^2=3, this matrix has eigenvalue four in the direction of v and eigenvalue one on its orthogonal complement. In particular it is positive definite.
Exact mathematical passage
B^{\mathsf T}B=I+vv^{\mathsf T},\qquad v=(1,-1,1)^{\mathsf T}.
v^{(m)}Use in source · 2 passages
Search aliases: v^(m)
Use in source · 05i_diagonal_regularization.tex, line 14
[Energy, residual, and certified tails] The displayed a^(m) belongs to ℓ^2 and is the unique regularized correction at α=m^-2. Set A_m=norma^(m)^2=∑_n≥1fracm^4(m^2+n^2)^2, B_m=normv^(m)^2=∑_n≥1fracn^2(m^2+n^2)^2. Then frac m4≤ A_m≤frac4m3, frac14(m+1)≤ B_m≤frac2m. For the sums A_m,K,B_m,K through any integer K≥ m, 0≤ A_m-A_m,K≤fracm^43K^3, 0≤ B_m-B_m,K≤frac1K. For n>m, one has (a_n^(m))^2≤ m^4/n^4, so the correction sequence is square-summable. The diagonal operator is bounded with norm one, and…
Exact mathematical passage
A_m=\norm{a^{(m)}}^2=\sum_{n\ge1}\frac{m^4}{(m^2+n^2)^2},\qquad B_m=\norm{v^{(m)}}^2=\sum_{n\ge1}\frac{n^2}{(m^2+n^2)^2}.
v^{(m)}_nUse in source · 1 passage
Search aliases: v^(m)_n
Use in source · 05i_diagonal_regularization.tex, line 6
The entire regularized family is explicit. Use Tu=(u_n/n) and r=w=(1/n) as in Section~[source reference]. Choose α=m^-2 for an integer m≥1. The normal equation in each coordinate gives a^(m)_n=fracm^2m^2+n^2, v^(m)_n=r_n-(Ta^(m))_n=fracnm^2+n^2. The repaired ambient candidate is (-a^(m),1). Its protected coordinate is one for every m.
Exact mathematical passage
a^{(m)}_n=\frac{m^2}{m^2+n^2},\qquad v^{(m)}_n=r_n-(Ta^{(m)})_n=\frac{n}{m^2+n^2}.
v^{-1}Use in source · 1 passage
Search aliases: v^-1
Use in source · 04_symmetry.tex, line 148
Conversely, if x is a twisted extension of the same prescribed data, define h_v=(x_0)_v^-1x_v. On U its value is one. For every arrow, h_v · =[c_f f^*((x_0)_w)]^-1[c_f f^*(x_w)]
Exact mathematical passage
h_v=(x_0)_v^{-1}x_v
v^2Use in source · 7 passages
Search aliases: v^2
Use in source · 05_geometry.tex, line 335
[The two-stage repair is the joint projection] The operator P is the orthogonal projection onto M, equivalently onto the kernel of Δ=(D,C). If u=x-P_Lx and v=P_Lx-Px, then u⊥ v, normx-Px^2=norm u^2+norm v^2. The subspace L=ker D is closed, hence Hilbert with the inherited inner product. The restriction B=C|_L is continuous, so its kernel is a closed subspace of L. Under the inclusion into E, this kernel is exactly M=L∩ker C. Thus both projections used to define P exist.
Exact mathematical passage
u\perp v,\qquad\norm{x-Px}^2=\norm u^2+\norm v^2.
v^\perpUse in source · 1 passage
Search aliases: v^⊥ perp
Use in source · 06_cases.tex, line 163
These two identities show that the least nonnegative gain for the full defect-to-distance estimate in this Euclidean model is one. The lower bound is attained, for example, on a candidate with zero first components and second-component vector (1,1,0), which lies in v^⊥. The squared input distance and squared residual are both two.
Exact mathematical passage
v^\perp
v_nUse in source · 1 passage
Search aliases: v_n
Use in source · 05j_inexact_certificates.tex, line 76
Why a vanishing normal residual is not enough. Let T=0 on R, let r=0, and take α_n=n^-2 for n≥1. The exact optimizer is zero for every such n. The fixed proposal a_n=1 instead has v_n=0, g_n=α_n⟶0, J_α_n(a_n)=α_n⟶0, absa_n-a_α_n=1. Thus the fit residual, the normal residual, and even the absolute objective value can all tend to zero without convergence to the selected optimizer. The certificate correctly reports absg_n/α_n=1.
Exact mathematical passage
v_n=0,\qquad g_n=\alpha_n\longrightarrow0,\qquad J_{\alpha_n}(a_n)=\alpha_n\longrightarrow0, \qquad\abs{a_n-a_{\alpha_n}}=1.
v_tUse in source · 2 passages
Search aliases: v_t
Use in source · 05_geometry.tex, line 591
[Exact motion of the line] For every real t, norm(q_t-q_0)x^2=fract^21+t^2norm x^2, normq_t-q_0=frac|t|sqrt1+t^2. In the small perturbation interface, kernel transport sends (0,b) to (-tb,b). Orthogonal projection instead sends it to (-tb/(1+t^2),b/(1+t^2)). The vector v_t=(-t,1) spans ker D_t and has squared norm 1+t^2>0. Projection onto this line is x⟼frac⟨ x,v_t⟩1+t^2v_t, which gives the matrix in [source reference]. In particular it is the orthogonal projector, not merely a map whose image i…
Exact mathematical passage
v_t=(-t,1)
v_\alphaUse in source · 1 passage
Search aliases: v_α alpha
Use in source · 05h_regularized_repair.tex, line 94
Parameter monotonicity and the fixed-budget problem. [Regularized answers lie on the budget frontier] If 0<α<γ, then norma_α≥norma_γ, normr-Ta_α≤normr-Ta_γ. At M_α=norma_α, β(M_α)=normr-Ta_α. When v_α=r-Ta_α≠0, its normalization is an optimizing witness in Theorem~[source reference] at that budget. Let A_α=norma_α^2 and R_α=normr-Ta_α^2. Comparing the two admissible minimizers in each other's objectives gives R_α+α A_α≤ R_γ+α A_γ, R_γ+γ A_γ≤ R_α+γ A_α. Add these inequalities and cancel the resid…
Exact mathematical passage
v_\alpha=r-Ta_\alpha\ne0
\varinjlimUse in source · 1 passage
Search aliases: varinjlim
Use in source · 02_ambient.tex, line 25
Zero extension gives injections A_N→ A_N+1. Forgetting the last coordinate gives surjections A_N+1→ A_N. These are different diagrams. Their limits are D=varinjlim_N A_N=⊕_p C_p, P=varprojlim_N A_N=∏_p C_p. The first group contains only finite-support vectors. The second contains arbitrary compatible coordinate assignments.
Exact mathematical passage
D=\varinjlim_N A_N=\bigoplus_p C_p, \qquad P=\varprojlim_N A_N=\prod_p C_p.
\varinjlim_NUse in source · 1 passage
Search aliases: varinjlim_N varinjlim
Use in source · 02_ambient.tex, line 25
Zero extension gives injections A_N→ A_N+1. Forgetting the last coordinate gives surjections A_N+1→ A_N. These are different diagrams. Their limits are D=varinjlim_N A_N=⊕_p C_p, P=varprojlim_N A_N=∏_p C_p. The first group contains only finite-support vectors. The second contains arbitrary compatible coordinate assignments.
Exact mathematical passage
D=\varinjlim_N A_N=\bigoplus_p C_p, \qquad P=\varprojlim_N A_N=\prod_p C_p.
\varprojlimUse in source · 3 passages
Search aliases: varprojlim
Use in source · 01_contracts.tex, line 83
The distinction is easiest to express in terms of an embedding G⟶ varprojlim G/N. Residual finiteness makes this map injective. The inverse limit includes all compatible tuples. The image of the original G can nevertheless be a proper dense subgroup. Forgetting that image can forget the object even while retaining enough coordinates to distinguish every element once the object is supplied.
Exact mathematical passage
G\longrightarrow \varprojlim G/N.
\varprojlim_NUse in source · 2 passages
Search aliases: varprojlim_N varprojlim
Use in source · 02_ambient.tex, line 25
Zero extension gives injections A_N→ A_N+1. Forgetting the last coordinate gives surjections A_N+1→ A_N. These are different diagrams. Their limits are D=varinjlim_N A_N=⊕_p C_p, P=varprojlim_N A_N=∏_p C_p. The first group contains only finite-support vectors. The second contains arbitrary compatible coordinate assignments.
Exact mathematical passage
D=\varinjlim_N A_N=\bigoplus_p C_p, \qquad P=\varprojlim_N A_N=\prod_p C_p.
\vec EUse in source · 2 passages
Search aliases: vec E vec
Use in source · 05_geometry.tex, line 7
The candidate and residual spaces are finite products with maximum norms: E=∏_i∈ VH, R=∏_(i,j)∈vec EH, (dx)_ij=τ_ijx_j-x_i. Both orientations of an edge are retained. If M bounds the norms of all measured edge maps, then norm d≤ M+1. The empty edge case has the usual zero residual.
Exact mathematical passage
E=\prod_{i\in V}H,\qquad R=\prod_{(i,j)\in\vec E}H,\qquad (dx)_{ij}=\tau_{ij}x_j-x_i.
\WUse in source · 4 passages
Search aliases: W
Use in source · 04_symmetry.tex, line 339
chapEquivalent presentations and identified boundaries Transporting the same question. Let H:V→W be part of a category equivalence, with inverse L and counit ε:HL⇒id_W. Pull a view system F on W back along H. Sampling a global section gives (H^*x)_v=x_Hv.
Exact mathematical passage
H:\V\to\W
w'Use in source · 3 passages
Search aliases: w'
Use in source · 04_symmetry.tex, line 353
[ Ext_U^F(s)≃Ext_U'^H^*F(H^*s). Existence and uniqueness of the prescribed extension are preserved and reflected. Write the unit of the equivalence as η:I_V⇒ LH, and take the unit and counit with their triangle identities. Set t_w=ε_w^-1:w→ HLw. Given y∈Γ(H^*F), define (Ry)_w=F(t_w)y_Lw. For an arrow f:w→ w', naturality of the inverse counit states HL(f) t_w=t_w'f. Contravariance of F and coherence of y therefore give F(f)(Ry)_w' · =F(f)F(t_w')y_Lw'
Exact mathematical passage
f:w\to w'
w^{-1}Use in source · 1 passage
Search aliases: w^-1
Use in source · 04_symmetry.tex, line 149
Conversely, if x is a twisted extension of the same prescribed data, define h_v=(x_0)_v^-1x_v. On U its value is one. For every arrow, h_v · =[c_f f^*((x_0)_w)]^-1[c_f f^*(x_w)] · =f^*((x_0)_w)^-1c_f^-1c_f f^*(x_w) · =f^*((x_0)_w^-1x_w)=f^*(h_w). The cancellation uses inverse multiplication in its actual order. No commuting factors have been moved past one another. Hence h∈ H_U. Pointwise group cancellation shows that h↦ x_0h and x↦ x_0^-1x are inverse maps.
Exact mathematical passage
\begin{aligned} h_v &=[c_f f^*((x_0)_w)]^{-1}[c_f f^*(x_w)]\\ &=f^*((x_0)_w)^{-1}c_f^{-1}c_f f^*(x_w)\\ &=f^*((x_0)_w^{-1}x_w)=f^*(h_w). \end{aligned}
w^2Use in source · 1 passage
Search aliases: w^2
Use in source · 05e_escape.tex, line 22
usR, R=ℓ^2(N_>0;R), norm(u,t)_E^2=∑_n≥1|u_n|^2+|t|^2. Let w=(1/n)_n≥1. It belongs to ℓ^2, because ∑_n≥1frac1n^2 ≤1+∑_n≥2frac1n(n-1)=2. Define the defect and protected observable by d(u,t)_n=fracu_n+tn, p(u,t)=t. The diagonal map T:u↦(u_n/n)_n has operator norm one. Since d(u,t)=Tu+tw, normd(u,t)≤norm u+|t|norm w ≤sqrt1+norm w^2 norm(u,t)_E. Thus both d and p are continuous linear maps between the declared complete spaces. No failure of completeness or continuity is built into the model.
Exact mathematical passage
\norm{d(u,t)}\le\norm u+|t|\norm w \le\sqrt{1+\norm w^2}\,\norm{(u,t)}_E.
W_AUse in source · 1 passage
Search aliases: W_A
Use in source · 05_construction.tex, line 15
0, · p∉ S_A. Because u_A has infinite support, q(u_A)≠0. We can therefore define a rational line and its preimage by W_A= Q q(u_A), G_A=q^-1(W_A). This definition produces an actual group for every set A. It does not claim an effective method for obtaining the selector bits of an arbitrary A.
Exact mathematical passage
W_A=\Q q(u_A),\qquad G_A=q^{-1}(W_A).
w_nUse in source · 1 passage
Search aliases: w_n
Use in source · 05e_escape.tex, line 42
The obstruction class is nonzero while its floor is zero. The admissible correction space is ker p=ℓ^2⊕{0}, with correction image B=d(ker p)=ran T ={r∈ℓ^2:(nr_n)_n≥1∈ℓ^2}. It contains every finitely supported sequence, so it is dense in R. But w∉ B, because (nw_n)_n=(1,1,…) is not square summable. Therefore [w]≠0 in R/B, ν_B(w)=0. This is the middle row of the preceding classification. The example is a worked Hilbert-space derivation without a separate Lean instance. Its finite-coordinate calcul…
Exact mathematical passage
(nw_n)_n=(1,1,\ldots)
\widehat DUse in source · 2 passages
Search aliases: widehat D widehat
Use in source · 02_ambient.tex, line 92
The complete finite-quotient system. [Profinite completion of the direct sum] The canonical embedding of D into P identifies P with widehat D. Let f:D→ F be a homomorphism to a finite group. The image of a prime-order factor C_p is trivial unless p divides lvert Frvert. Thus f vanishes on all but finitely many prime components and factors through a finite coordinate product. Kernels of finite coordinate projections are therefore cofinal among the finite-index normal subgroups of D. Taking their …
Exact mathematical passage
\widehat D
\widehat fUse in source · 2 passages
Search aliases: widehat f widehat
Use in source · 09_reconstruction.tex, line 68
[Reflection within the paired-prime family] For a homomorphism f:G_A→ G_B, the induced completion map widehat f is an isomorphism if and only if f is an isomorphism. One implication follows from functoriality. Conversely, suppose widehat f is an isomorphism. By Theorem~[source reference], its coordinate action is multiplication by a diagonal sequence λ. Each λ_p must be nonzero, so f is injective. If A≠^* B, Theorem~[source reference] would force its image to be torsion, contradicting injectivit…
Exact mathematical passage
\widehat f
\widehat rUse in source · 2 passages
Search aliases: widehat r widehat
Use in source · 06_cases.tex, line 50
To repair the residual, now regard the vector of edge data as a candidate in its own right. Its defect is ℓ(r). The exact subspace is the zero-sum hyperplane. With the ordinary Euclidean norm and no protected edge coordinates, the nearest compatible residual is widehat r=r-fracℓ(r)41 =(frac34,-frac14,-frac14,-frac14). The correction is perpendicular to the hyperplane, so the projection theorem proves it is the unique nearest vector there. Its squared size is 1/4. This is a repair of the edge dat…
Exact mathematical passage
\widehat r=r-\frac{\ell(r)}4\mathbf1 =\left(\frac34,-\frac14,-\frac14,-\frac14\right).
\widehat xUse in source · 2 passages
Search aliases: widehat x widehat
Use in source · 06_cases.tex, line 57
Anchor the repaired signal at zero. Its primitive is widehat x=(0,frac34,frac12,frac14). The next partial sum is zero, so it closes. If instead the desired representative has mean zero, subtract the mean 3/8: widehat x_rm mean =(-frac38,frac38,frac18,-frac18). Both primitives realize exactly widehat r. They differ by an element of ker D. The anchor condition and the mean condition are two different selections from the same affine solution fiber.
Exact mathematical passage
\label{eq:cycle-anchor-primitive} \widehat x=\left(0,\frac34,\frac12,\frac14\right).
\widehat x_{\rm mean}Use in source · 1 passage
Search aliases: widehat x_rm mean widehat rm
Use in source · 06_cases.tex, line 61
Anchor the repaired signal at zero. Its primitive is widehat x=(0,frac34,frac12,frac14). The next partial sum is zero, so it closes. If instead the desired representative has mean zero, subtract the mean 3/8: widehat x_rm mean =(-frac38,frac38,frac18,-frac18). Both primitives realize exactly widehat r. They differ by an element of ker D. The anchor condition and the mean condition are two different selections from the same affine solution fiber.
Exact mathematical passage
\widehat x_{\rm mean} =\left(-\frac38,\frac38,\frac18,-\frac18\right).
\widehat{G_A}Use in source · 2 passages
Search aliases: widehatG_A widehat
Use in source · 00_notation.tex, line 33
The main statements have the following form: Tor(G_A) · =D, · G_A/D · ≅ Q, G_A/mG_A · ≅ C_rad(m), · widehatG_A · ≅ P, G_A≅ G_B · iff A=^* B, · · G_A has an X-computable copy · iff A≤_T X. · · The extension by D does not split. Distinct eventual selector classes do not embed in one another. All of these conclusions are proved within the book, with the exact hypotheses retained.
Exact mathematical passage
\Tor(G_A)&=D,&G_A/D&\cong\Q,\\ G_A/mG_A&\cong C_{\rad(m)},&\widehat{G_A}&\cong P,\\ G_A\cong G_B&\iff A\eqfin B,&&\\ G_A\text{ has an }X\text{-computable copy}&\iff A\le_T X.&&
\widehat{H_W}Use in source · 1 passage
Search aliases: widehatH_W widehat
Use in source · 04_purity.tex, line 61
em[One finite-quotient system for every preimage] For every rational subspace W and every m≥1, the coordinate map induces an isomorphism H_W/mH_W ≅ ∏_pmid mC_p. For mmid n, the quotient transition map is the coordinate restriction from primes dividing n to primes dividing m. These quotients are cofinal among all finite quotients, and widehatH_W≅ P. The coordinate map onto the displayed finite product is surjective already on D. Its kernel is mH_W by Theorem~[source reference]. The transition ass…
Exact mathematical passage
\widehat{H_W}\cong P.
\widetilde aUse in source · 3 passages
Search aliases: widetilde a widetilde
Use in source · 05k_stopping_rules.tex, line 106
arepsilon_nα_n→0. If the proposed corrections a_n∈ H satisfy the displayed data and normal-residual bounds, then a_n→ a^† in norm. At every index, norma_n-a^† ≤fracε_nα_n +fracδ_n2sqrtα_n +norma_α_n(r)-a^†. At index n, distinguish two exact minimizers: let widetilde a_n=a_α_n(widetilde r_n) solve the measured problem and let a_n^0=a_α_n(r) solve the intended-data problem. All three corrections, including the proposal a_n, belong to the same space H.
Exact mathematical passage
\widetilde a_n=a_{\alpha_n}(\widetilde r_n)
\widetilde a_nUse in source · 3 passages
Search aliases: widetilde a_n widetilde
Use in source · 05k_stopping_rules.tex, line 106
arepsilon_nα_n→0. If the proposed corrections a_n∈ H satisfy the displayed data and normal-residual bounds, then a_n→ a^† in norm. At every index, norma_n-a^† ≤fracε_nα_n +fracδ_n2sqrtα_n +norma_α_n(r)-a^†. At index n, distinguish two exact minimizers: let widetilde a_n=a_α_n(widetilde r_n) solve the measured problem and let a_n^0=a_α_n(r) solve the intended-data problem. All three corrections, including the proposal a_n, belong to the same space H.
Exact mathematical passage
\widetilde a_n=a_{\alpha_n}(\widetilde r_n)
\widetilde bUse in source · 3 passages
Search aliases: widetilde b widetilde
Use in source · 05i_diagonal_regularization.tex, line 154
[An explicit noisy attainable repair] For the attainable datum b=(1/n^2), suppose normwidetilde b-b≤δ with 0<δ≤1. Choose m=lceilδ^-2/3rceil, α=m^-2. Then the noisy regularized correction satisfies norma_α(widetilde b)-a^† ≤sqrtfrac2m+fracδ m2 ≤(sqrt2+1)δ^1/3. The exact-data bias has coordinates n/(m^2+n^2), so its squared norm is precisely B_m. By Proposition~[source reference], norma_m^-2(b)-a^†≤sqrt2/m. Theorem~[source reference] at the same parameter gives norma_m^-2(widetilde b)-a_m^-2(b) ≤f…
Exact mathematical passage
\norm{\widetilde b-b}\le\delta
\widetilde cUse in source · 4 passages
Search aliases: widetilde c widetilde
Use in source · 04_symmetry.tex, line 215
Step 2: obtain literal matching labels. Define widetilde c_i(f)=b_i(v)^-1c_i(f)f^*(b_i(w)) (f:v→ w in U_i). This is a gauge transform and hence a cocycle on U_i. If the arrow lies also in U_j, substitute the original overlap equation and (*) at its two endpoints: widetilde c_j(f) · =b_i(v)^-1a_ij(v)^-1 a_ij(v)c_i(f)[f^*a_ij(w)]^-1 f^*(a_ij(w)b_i(w))
Exact mathematical passage
\widetilde c_i(f)=b_i(v)^{-1}c_i(f)f^*(b_i(w)) \quad(f:v\to w\text{ in }U_i).
\widetilde c_iUse in source · 4 passages
Search aliases: widetilde c_i widetilde
Use in source · 04_symmetry.tex, line 215
Step 2: obtain literal matching labels. Define widetilde c_i(f)=b_i(v)^-1c_i(f)f^*(b_i(w)) (f:v→ w in U_i). This is a gauge transform and hence a cocycle on U_i. If the arrow lies also in U_j, substitute the original overlap equation and (*) at its two endpoints: widetilde c_j(f) · =b_i(v)^-1a_ij(v)^-1 a_ij(v)c_i(f)[f^*a_ij(w)]^-1 f^*(a_ij(w)b_i(w))
Exact mathematical passage
\widetilde c_i(f)=b_i(v)^{-1}c_i(f)f^*(b_i(w)) \quad(f:v\to w\text{ in }U_i).
\widetilde c_jUse in source · 1 passage
Search aliases: widetilde c_j widetilde
Use in source · 04_symmetry.tex, line 220
Step 2: obtain literal matching labels. Define widetilde c_i(f)=b_i(v)^-1c_i(f)f^*(b_i(w)) (f:v→ w in U_i). This is a gauge transform and hence a cocycle on U_i. If the arrow lies also in U_j, substitute the original overlap equation and (*) at its two endpoints: widetilde c_j(f) · =b_i(v)^-1a_ij(v)^-1 a_ij(v)c_i(f)[f^*a_ij(w)]^-1 f^*(a_ij(w)b_i(w)) · =b_i(v)^-1c_i(f)f^*(b_i(w)) =widetilde c_i(f). Every cancellation is between neighboring inverse factors. Commutativity is not required.
Exact mathematical passage
\begin{aligned} \widetilde c_j(f) &=b_i(v)^{-1}a_{ij}(v)^{-1} a_{ij}(v)c_i(f)[f^*a_{ij}(w)]^{-1} f^*(a_{ij}(w)b_i(w))\\ &=b_i(v)^{-1}c_i(f)f^*(b_i(w)) =\widetilde c_i(f). \end{aligned}
\widetilde DUse in source · 19 passages
Search aliases: widetilde D widetilde
Use in source · 05a_certificates.tex, line 20
[Measurement and operator uncertainty] With the reference certificate above, normx-qx≤ C(norm y+η) if normDx-y≤η. More generally, let widetilde D:E→ R be the bounded linear operator actually used in the measurement. If normD-widetilde D≤ε, normwidetilde D x-y≤η, then normx-qx≤ C(norm y+η+εnorm x). The projection in this conclusion is still the projection for the reference operator D. The first assertion follows from normDx≤norm y+η. For the second, decompose Dx=(D-widetilde D)x+(widetilde D x-y)…
Exact mathematical passage
\widetilde D:E\to R
\widetilde D_tUse in source · 1 passage
Search aliases: widetilde D_t widetilde
Use in source · 05a_certificates.tex, line 90
plate22_gain_budgetFor the kernel-preserving family widetilde D_t(a,b)=(1-t)a, the exact residual gain is 1/(1-t). It deteriorates as the strict budget is exhausted. At the excluded endpoint the kernel changes, so the finite-radius estimate does not extend by substituting t=1.fig:gain-budget
Exact mathematical passage
\widetilde D_t(a,b)=(1-t)a
\widetilde GUse in source · 2 passages
Search aliases: widetilde G widetilde
Use in source · 04b_phase.tex, line 81
Retaining the phase restores an ordinary multiplication law. Instead of trying to discard the phase, include it in the state. On the eight-element set widetilde G= F_2× G, put (a,x)·(b,y)=(a+b+c(x,y),x+y). The first coordinate of the left-associated product of three elements is a+b+d+c(x,y)+c(x+y,z). The right-associated coordinate is a+b+d+c(y,z)+c(x,y+z). Equation~[source reference] proves equality. Normalization gives identity (0,0), and (a,x)^-1=(a+c(x,x),x). Both inverse products are the id…
Exact mathematical passage
\widetilde G=\mathbf F_2\times G
\widetilde rUse in source · 17 passages
Search aliases: widetilde r widetilde
Use in source · 05g_budget_certificates.tex, line 133
How measurement and model uncertainty affect the budget. [A robust budget profile] Suppose the bounded linear map widetilde T:H→ Y and the datum widetilde r∈ Y satisfy normT-widetilde T≤η, normr-widetilde r≤δ, η,δ≥0. For every M≥0, absβ_T,r(M)-β_widetilde T,widetilde r(M) ≤δ+η M. In particular, for norm z≤1, β_T,r(M)≥ ⟨widetilde r,z⟩-Mnormwidetilde T^*z -δ-η M. Fix a feasible correction a. The two residual vectors have difference (r-Ta)-(widetilde r-widetilde Ta) =(r-widetilde r)-(T-widetilde T)…
Exact mathematical passage
\widetilde r\in Y
\widetilde r_nUse in source · 1 passage
Search aliases: widetilde r_n widetilde
Use in source · 05k_stopping_rules.tex, line 106
arepsilon_nα_n→0. If the proposed corrections a_n∈ H satisfy the displayed data and normal-residual bounds, then a_n→ a^† in norm. At every index, norma_n-a^† ≤fracε_nα_n +fracδ_n2sqrtα_n +norma_α_n(r)-a^†. At index n, distinguish two exact minimizers: let widetilde a_n=a_α_n(widetilde r_n) solve the measured problem and let a_n^0=a_α_n(r) solve the intended-data problem. All three corrections, including the proposal a_n, belong to the same space H.
Exact mathematical passage
\widetilde a_n=a_{\alpha_n}(\widetilde r_n)
\widetilde s'Use in source · 3 passages
Search aliases: widetilde s' widetilde
Use in source · 02_obstructions.tex, line 64
The proof is independent of the selected raw lift. Indeed, if widetilde s' is another lift, k=widetilde s'-widetilde s lies in ker p_U, and δwidetilde s'-δwidetilde s=δ k∈ B_U. Thus the two quotient classes agree before either feasibility direction is invoked. A raw lift always exists in this additive model by using zero outside U. Constructing a raw lift does not solve the coherence equations. sourceKS09 The quotient is constructed before the solvability test. The theorem therefore does not def…
Exact mathematical passage
\widetilde s'
\widetilde TUse in source · 9 passages
Search aliases: widetilde T widetilde
Use in source · 05g_budget_certificates.tex, line 133
How measurement and model uncertainty affect the budget. [A robust budget profile] Suppose the bounded linear map widetilde T:H→ Y and the datum widetilde r∈ Y satisfy normT-widetilde T≤η, normr-widetilde r≤δ, η,δ≥0. For every M≥0, absβ_T,r(M)-β_widetilde T,widetilde r(M) ≤δ+η M. In particular, for norm z≤1, β_T,r(M)≥ ⟨widetilde r,z⟩-Mnormwidetilde T^*z -δ-η M. Fix a feasible correction a. The two residual vectors have difference (r-Ta)-(widetilde r-widetilde Ta) =(r-widetilde r)-(T-widetilde T)…
Exact mathematical passage
\widetilde T:H\to Y
\widetilde tUse in source · 2 passages
Search aliases: widetilde t widetilde
Use in source · 02_obstructions.tex, line 127
Γ(A)xrightarrowr_UΓ_U(A) xrightarrow o_UQ_U is exact at Γ_U(A). Thus ker o_U=im r_U, and restriction is onto precisely when o_U is the zero map. First verify that o_U is a homomorphism, since exactness is being asserted in an additive setting. Given s,t∈Γ_U(A), choose raw lifts widetilde s,widetilde t. Their sum lifts s+t. Lift independence and additivity imply o_U(s+t) =[δ(widetilde s+widetilde t)] =[δwidetilde s]+[δwidetilde t] = o_U(s)+ o_U(t). The zero lift gives o_U(0)=0. Restriction r_U is…
Exact mathematical passage
\widetilde s,\widetilde t
\widetilde T^*Use in source · 2 passages
Search aliases: widetilde T^* widetilde
Use in source · 05g_budget_certificates.tex, line 144
ct the budget [A robust budget profile] Suppose the bounded linear map widetilde T:H→ Y and the datum widetilde r∈ Y satisfy normT-widetilde T≤η, normr-widetilde r≤δ, η,δ≥0. For every M≥0, absβ_T,r(M)-β_widetilde T,widetilde r(M) ≤δ+η M. In particular, for norm z≤1, β_T,r(M)≥ ⟨widetilde r,z⟩-Mnormwidetilde T^*z -δ-η M. Fix a feasible correction a. The two residual vectors have difference (r-Ta)-(widetilde r-widetilde Ta) =(r-widetilde r)-(T-widetilde T)a, whose norm is at most c=δ+η M. The rever…
Exact mathematical passage
\beta_{T,r}(M)\ge \langle\widetilde r,z\rangle-M\norm{\widetilde T^*z} -\delta-\eta M.
\widetilde xUse in source · 1 passage
Search aliases: widetilde x widetilde
Use in source · 06_cases.tex, line 71
Protecting three edges forces a different repair. Suppose the first three values (r_0,r_1,r_2)=(1,0,0) are required to remain fixed. The admissible corrections then affect only the final edge. Compatibility forces widetilde r=(1,0,0,-1), widetilde x=(0,1,1,1) for anchor zero. This is the unique repair preserving those three edges. Its squared correction size is one rather than 1/4. The larger error is not a failure of nearest repair: the feasible set has changed.
Exact mathematical passage
\widetilde r=(1,0,0,-1),\qquad \widetilde x=(0,1,1,1)
x^{-1}Use in source · 1 passage
Search aliases: x^-1
Use in source · 04_symmetry.tex, line 171
erent cocycle whose regular twisted section equation x=px has no solution. The only nontrivial cocycle relation is p^2=1. At a single view the gauge action is p↦ apa^-1. All three transpositions are conjugate and none is conjugate to the identity. The stabilizer is therefore the centralizer. Finally x=px implies p=1 after multiplication by x^-1. sourceKS33 The example makes an important boundary visible. A cocycle can exist and glue as an object even when it admits no global trivializing section…
Exact mathematical passage
x^{-1}
x^{-2}Use in source · 2 passages
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Use in source · 05i_diagonal_regularization.tex, line 54
For the lower bound, n>m implies m^2+n^2≤2n^2, hence fracn^2(m^2+n^2)^2≥frac14n^2. The other direction of the decreasing-function integral comparison gives ∑_n>mfrac1n^2≥∫_m+1^∞ x^-2 dx=frac1m+1. This proves B_m≥1/(4(m+1)).
Exact mathematical passage
\sum_{n>m}\frac1{n^2}\ge\int_{m+1}^\infty x^{-2}\,dx=\frac1{m+1}.
x^{-4}Use in source · 3 passages
Search aliases: x^-4
Use in source · 05i_diagonal_regularization.tex, line 37
For 1≤ n≤ m, m^2/(m^2+n^2)≥1/2. There are m such coordinates, so A_m≥ m/4. For the upper bound, the first m squares are at most one, and the decreasing function x^-4 gives ∑_n>mfrac1n^4≤∫_m^∞ x^-4 dx=frac13m^3. Thus A_m≤ m+m^4/(3m^3)=4m/3.
Exact mathematical passage
x^{-4}
X^2Use in source · 1 passage
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Use in source · 04b_phase.tex, line 50
0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do not commute.
Exact mathematical passage
X^2=Z^2=I
x^2Use in source · 5 passages
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Use in source · 05_geometry.tex, line 116
Pythagoras applied to that decomposition gives norm x^2=normqx^2+normx-qx^2, hence normqx≤norm x. For any y∈ K, the vector qx-y lies in K, whereas x-qx lies in its orthogonal complement. Consequently normx-y^2=normx-qx^2+normqx-y^2≥normx-qx^2. Equality holds only when normqx-y=0, or y=qx. This proves nearest-point uniqueness without any closedness assumption on ran D. Closedness of the kernel and of the range serve different purposes. sourceKS42 No closed range assumption on D is needed for this…
Exact mathematical passage
\norm x^2=\norm{qx}^2+\norm{x-qx}^2
X^pUse in source · 1 passage
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Use in source · 04b_phase.tex, line 45
The phase is visible in actual matrices. Over R, define X=0 · 1 1 · 0, Z=1 · 0 0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do not commute.
Exact mathematical passage
X=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad Z=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\qquad U_{(p,q)}=X^pZ^q\quad(p,q\in\{0,1\}).
X^{y_1}Use in source · 1 passage
Search aliases: X^y_1
Use in source · 04b_phase.tex, line 50
0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do not commute.
Exact mathematical passage
X^{y_1}
X_0Use in source · 1 passage
Search aliases: X_0
Use in source · 03_towers.tex, line 5
ned detailResolution gives a concrete family of views. Its compatibility differential connects exact maps, protected information, and residual coordinates. Recovering every finite coordinate is then separated from convergence in a chosen norm. Resolution towers and exact transport. The sequential model. A tower in a category C is a sequence ⋯xrightarrowρ_2X_2xrightarrowρ_1X_1 xrightarrowρ_0X_0. For sets, a point is a sequence x_n∈ X_n with ρ_n(x_n+1)=x_n. Composite restrictions turn the tower in…
Exact mathematical passage
\cdots\xrightarrow{\rho_2}X_2\xrightarrow{\rho_1}X_1 \xrightarrow{\rho_0}X_0.
x_0Use in source · 38 passages
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Use in source · 02_obstructions.tex, line 83
[Classification of a nonempty extension fiber] If x_0∈Ext_U(s), translation gives an equivalence H_U≃Ext_U(s), h⟼ x_0+h. Consequently the extension is unique exactly when o_U(s)=0 and H_U=0. Define Θ:H_U→Ext_U(s) by Θ(h)=x_0+h. It lands in the extension set because δ(x_0+h)=δ x_0+δ h=0, p_U(x_0+h)=s+0=s. Define Ψ:Ext_U(s)→ H_U by Ψ(x)=x-x_0. Its differential is zero, and its restriction is s-s=0, so the claimed target is correct. The formulas give ΨΘ(h)=x_0+h-x_0=h, ΘΨ(x)=x_0+(x-x_0)=x. This pro…
Exact mathematical passage
x_0\in\Ext_U(s)
x_0^{-1}Use in source · 1 passage
Search aliases: x_0^-1
Use in source · 04_symmetry.tex, line 157
· =f^*((x_0)_w^-1x_w)=f^*(h_w). The cancellation uses inverse multiplication in its actual order. No commuting factors have been moved past one another. Hence h∈ H_U. Pointwise group cancellation shows that h↦ x_0h and x↦ x_0^-1x are inverse maps.
Exact mathematical passage
x\mapsto x_0^{-1}x
X_1Use in source · 1 passage
Search aliases: X_1
Use in source · 03_towers.tex, line 5
ned detailResolution gives a concrete family of views. Its compatibility differential connects exact maps, protected information, and residual coordinates. Recovering every finite coordinate is then separated from convergence in a chosen norm. Resolution towers and exact transport. The sequential model. A tower in a category C is a sequence ⋯xrightarrowρ_2X_2xrightarrowρ_1X_1 xrightarrowρ_0X_0. For sets, a point is a sequence x_n∈ X_n with ρ_n(x_n+1)=x_n. Composite restrictions turn the tower in…
Exact mathematical passage
\cdots\xrightarrow{\rho_2}X_2\xrightarrow{\rho_1}X_1 \xrightarrow{\rho_0}X_0.
x_1Use in source · 15 passages
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Use in source · 02_obstructions.tex, line 104
The choice of x_0 only supplies an origin for the parameters. If x_1=x_0+h_1 is used instead, the parameter of x changes from x-x_0 to x-x_1=(x-x_0)-h_1. The affine set of answers itself has not changed. sourceKS10 An extension fiber is therefore an affine copy, or torsor, of the invisible coherent group. Selecting one solution gives coordinates on the fiber. Without a selected solution, the affine structure does not designate an origin.
Exact mathematical passage
x_1=x_0+h_1
X_2Use in source · 1 passage
Search aliases: X_2
Use in source · 03_towers.tex, line 5
ned detailResolution gives a concrete family of views. Its compatibility differential connects exact maps, protected information, and residual coordinates. Recovering every finite coordinate is then separated from convergence in a chosen norm. Resolution towers and exact transport. The sequential model. A tower in a category C is a sequence ⋯xrightarrowρ_2X_2xrightarrowρ_1X_1 xrightarrowρ_0X_0. For sets, a point is a sequence x_n∈ X_n with ρ_n(x_n+1)=x_n. Composite restrictions turn the tower in…
Exact mathematical passage
\cdots\xrightarrow{\rho_2}X_2\xrightarrow{\rho_1}X_1 \xrightarrow{\rho_0}X_0.
x_2Use in source · 18 passages
Search aliases: x_2
Use in source · 02a_prescription.tex, line 55
An initially free parameter can become an obstruction Consider the chain of views 0→1→2, with fiber Z^2 at each view. Write the maps in the direction in which data are restricted: ρ_0,1(a,b)=(a,0), ρ_1,2(a,b)=(a,2b), ρ_0,2(a,b)=(a,0). The composite identity ρ_0,1ρ_1,2=ρ_0,2 is literal. A global family is exactly x_0=(k,0), x_1=(k,2n), x_2=(k,n), k,n∈Z. Indeed the first coordinate must agree at every view. The second coordinate at view zero is forced to vanish, and the second coordinate at view o…
Exact mathematical passage
\label{eq:branch-family} x_0=(k,0),\qquad x_1=(k,2n),\qquad x_2=(k,n), \qquad k,n\in\Z.
x_{2n+1}Use in source · 1 passage
Search aliases: x_2n+1
Use in source · 03_spectrum.tex, line 79
For the relator description, introduce a commuting generator x_i for each prime index, impose p_i x_i=0, and impose x_2n+1=0 whenever program n is observed to halt on input n. The resulting quotient retains exactly the prime factors indexed by B_K. All relators are effectively enumerable. Deciding whether an odd generator has become trivial is not supplied by this enumeration. The conclusion is not absolute finite indescribability. The displayed definition is finite. What fails is the stronger c…
Exact mathematical passage
x_{2n+1}=0
x_3Use in source · 7 passages
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Use in source · 05_geometry.tex, line 219
For D(x_1,x_2,x_3)=x_1-x_2, p(x)=(x_1,x_3), and x=(2,0,5), the ordinary orthogonal answer is (1,1,5). The protected answer is (2,2,5). The respective squared errors are two and four. Both have zero residual, but only the latter preserves the declared data. The theorem is formalized in codeProtectedHilbertRepair.lean, using the actual operator on ker p. The displayed three-coordinate values are an illustrative calculation, not a separate universal algorithm.
Exact mathematical passage
D(x_1,x_2,x_3)=x_1-x_2
X_bUse in source · 2 passages
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Use in source · 03b_repair_transport.tex, line 147
For a block from a to b, its value on a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a,b. The dyadic coarsener in Section~[source reference] is the case b=a+2.
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X_b
x_cUse in source · 1 passage
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Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
x_{HLw}Use in source · 1 passage
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Use in source · 04_symmetry.tex, line 369
If x∈Γ(F), coherence along t_w gives (RH^*x)_w=F(t_w)x_HLw=x_w. For the other composite, the triangle identity says t_Hv=H(η_v). Coherence of y along η_v:v→ LHv yields (H^*Ry)_v=F(t_Hv)y_LHv=F(Hη_v)y_LHv=y_v. These coordinate equations prove both inverse identities.
Exact mathematical passage
(RH^*x)_w=F(t_w)x_{HLw}=x_w.
x_{Hv}Use in source · 1 passage
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Use in source · 04_symmetry.tex, line 339
chapEquivalent presentations and identified boundaries Transporting the same question. Let H:V→W be part of a category equivalence, with inverse L and counit ε:HL⇒id_W. Pull a view system F on W back along H. Sampling a global section gives (H^*x)_v=x_Hv.
Exact mathematical passage
(H^*x)_v=x_{Hv}
x_iUse in source · 10 passages
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Use in source · 03a_observation.tex, line 148
Bitwise complementation is represented at level n by complementing the first n bits. The left shift F(x)_i=x_i+1 uses the schedule ϕ(n)=n+1. A nearest-neighbor difference over F_2, G(x)_i=x_i+x_i+1, also uses n+1 input bits for n output bits. Applying G twice uses n+2 bits, and the middle terms cancel: G(G(x))_i=x_i+x_i+2. The compatible finite maps make each statement meaningful on the entire coherent sequence, not merely on one chosen truncation. These are worked instances of the scheduled for…
Exact mathematical passage
G(x)_i=x_i+x_{i+1},
x_{i+1}Use in source · 3 passages
Search aliases: x_i+1
Use in source · 03a_observation.tex, line 147
Bitwise complementation is represented at level n by complementing the first n bits. The left shift F(x)_i=x_i+1 uses the schedule ϕ(n)=n+1. A nearest-neighbor difference over F_2, G(x)_i=x_i+x_i+1, also uses n+1 input bits for n output bits. Applying G twice uses n+2 bits, and the middle terms cancel: G(G(x))_i=x_i+x_i+2. The compatible finite maps make each statement meaningful on the entire coherent sequence, not merely on one chosen truncation. These are worked instances of the scheduled for…
Exact mathematical passage
F(x)_i=x_{i+1}
x_{i+2}Use in source · 1 passage
Search aliases: x_i+2
Use in source · 03a_observation.tex, line 152
Bitwise complementation is represented at level n by complementing the first n bits. The left shift F(x)_i=x_i+1 uses the schedule ϕ(n)=n+1. A nearest-neighbor difference over F_2, G(x)_i=x_i+x_i+1, also uses n+1 input bits for n output bits. Applying G twice uses n+2 bits, and the middle terms cancel: G(G(x))_i=x_i+x_i+2. The compatible finite maps make each statement meaningful on the entire coherent sequence, not merely on one chosen truncation. These are worked instances of the scheduled for…
Exact mathematical passage
G(G(x))_i=x_i+x_{i+2}.
x_jUse in source · 8 passages
Search aliases: x_j
Use in source · 03_towers.tex, line 325
A whole cascade. At scale j, let C_j:R_j→ R_j+1 and J_j:R_j+1→ R_j be bounded linear maps with C_jJ_j=I. For x∈ R_0, define x_0=x, x_j+1=C_jx_j, L_0=I, L_N=J_0⋯ J_N-1, h_j=x_j-J_jx_j+1, Q_Nx=L_Nx_N, S_N=I-Q_N. The map Q_N restores the surviving coarse remainder to the original space. It is this restored remainder, not just x_N, that controls reconstruction.
Exact mathematical passage
x_0=x,\quad x_{j+1}=C_jx_j,\quad L_0=I,\quad L_N=J_0\cdots J_{N-1},
x_{j+1}Use in source · 4 passages
Search aliases: x_j+1
Use in source · 03_towers.tex, line 325
A whole cascade. At scale j, let C_j:R_j→ R_j+1 and J_j:R_j+1→ R_j be bounded linear maps with C_jJ_j=I. For x∈ R_0, define x_0=x, x_j+1=C_jx_j, L_0=I, L_N=J_0⋯ J_N-1, h_j=x_j-J_jx_j+1, Q_Nx=L_Nx_N, S_N=I-Q_N. The map Q_N restores the surviving coarse remainder to the original space. It is this restored remainder, not just x_N, that controls reconstruction.
Exact mathematical passage
x_0=x,\quad x_{j+1}=C_jx_j,\quad L_0=I,\quad L_N=J_0\cdots J_{N-1},
X_kUse in source · 11 passages
Search aliases: X_k
Use in source · 03a_observation.tex, line 24
Equality is witnessed at finite stages. Two different schedules can represent the same transformation. To compare (ϕ,f) and (ψ,g), require that for every output level n there be a finite level k≥ϕ(n),ψ(n) with f_nρ^X_ϕ(n),k=g_nρ^X_ψ(n),k as functions on all of X_k. Call this eventual equality. The witnessing k may depend on n. Equality is required on the whole finite-level fiber, not only on states that happen to extend globally.
Exact mathematical passage
\label{eq:eventual-equality} f_n\rho^X_{\phi(n),k}=g_n\rho^X_{\psi(n),k} \quad\text{as functions on all of }X_k.
x_kUse in source · 6 passages
Search aliases: x_k
Use in source · 03_towers.tex, line 63
For each k, unboundedness gives an index n with k≤ϕ(n). Set x_k=ρ_k,ϕ(n)y_n. If m is another permissible index, choose ℓ≥ n,m. Monotonicity permits both composite restrictions, and sampled coherence gives ρ_k,ϕ(n)y_n =ρ_k,ϕ(n)ρ_ϕ(n),ϕ(ℓ)y_ℓ =ρ_k,ϕ(ℓ)y_ℓ =ρ_k,ϕ(m)y_m. Thus x_k is independent of the choice, including when the schedule repeats a level.
Exact mathematical passage
x_k=\rho_{k,\phi(n)}y_n
x_{k+1}Use in source · 1 passage
Search aliases: x_k+1
Use in source · 03_towers.tex, line 73
For the adjacent equation at k, choose one n with k+1≤ϕ(n). Then ρ_k,k+1x_k+1 =ρ_k,k+1ρ_k+1,ϕ(n)y_n =ρ_k,ϕ(n)y_n=x_k. The recovered family is therefore a point of the original tower.
Exact mathematical passage
\rho_{k,k+1}x_{k+1} =\rho_{k,k+1}\rho_{k+1,\phi(n)}y_n =\rho_{k,\phi(n)}y_n=x_k.
x_{k_n}Use in source · 2 passages
Search aliases: x_k_n
Use in source · 03a_observation.tex, line 116
For surjectivity, let F be finitely observed. Choose one pair (k_n,a_n) at each output level, satisfying F(x)_n=a_n(x_k_n) for every coherent input. Define ϕ(n)=max_0≤ i≤ nk_i, f_n=a_nρ^X_k_n,ϕ(n). The maximum is of a nonempty finite set. It gives a monotone schedule with k_n≤ϕ(n). For every point x, coherence in X gives f_n(x_ϕ(n)) =a_n(ρ^X_k_n,ϕ(n)x_ϕ(n)) =a_n(x_k_n)=F(x)_n. tag*
Exact mathematical passage
F(x)_n=a_n(x_{k_n})
x_mUse in source · 1 passage
Search aliases: x_m
Use in source · 06_boundaries.tex, line 62
Finally, fix a level n. Every global point satisfies x_n=ρ_n,m(x_m) for each m≥ n, hence x_n∈ Y_n. Conversely, start with any y∈ Y_n, lift recursively above level n through the surjections just proved, and define the lower coordinates by x_i=ρ_i,n(y). The composite restriction laws verify coherence across and below level n. The resulting point evaluates to y. These two inclusions prove the asserted equality of evaluation images, not merely nonemptiness of the inverse limit. sourceKS58 This is th…
Exact mathematical passage
x_n=\rho_{n,m}(x_m)
x_{m+1}Use in source · 1 passage
Search aliases: x_m+1
Use in source · 06_boundaries.tex, line 27
Suppose instead that x=(x_n) is a global point. Equality along each inclusion says that the underlying natural numbers of x_n+1 and x_n are equal. Induction therefore makes every underlying coordinate equal to the number m underlying x_0. Membership of x_m+1 in X_m+1 now gives m≥ m+1, a contradiction. Thus no global point exists.
Exact mathematical passage
x_{m+1}
x_NUse in source · 7 passages
Search aliases: x_N
Use in source · 03_towers.tex, line 329
A whole cascade. At scale j, let C_j:R_j→ R_j+1 and J_j:R_j+1→ R_j be bounded linear maps with C_jJ_j=I. For x∈ R_0, define x_0=x, x_j+1=C_jx_j, L_0=I, L_N=J_0⋯ J_N-1, h_j=x_j-J_jx_j+1, Q_Nx=L_Nx_N, S_N=I-Q_N. The map Q_N restores the surviving coarse remainder to the original space. It is this restored remainder, not just x_N, that controls reconstruction.
Exact mathematical passage
h_j=x_j-J_jx_{j+1},\qquad Q_Nx=L_Nx_N, \qquad S_N=I-Q_N.
x_nUse in source · 21 passages
Search aliases: x_n
Use in source · 03_towers.tex, line 9
information, and residual coordinates. Recovering every finite coordinate is then separated from convergence in a chosen norm. Resolution towers and exact transport. The sequential model. A tower in a category C is a sequence ⋯xrightarrowρ_2X_2xrightarrowρ_1X_1 xrightarrowρ_0X_0. For sets, a point is a sequence x_n∈ X_n with ρ_n(x_n+1)=x_n. Composite restrictions turn the tower into a view system on N, with an arrow i→ j when i≤ j. Its section space is the same coherent point type, with the same…
Exact mathematical passage
x_n\in X_n
x_{n+1}Use in source · 5 passages
Search aliases: x_n+1
Use in source · 03_towers.tex, line 9
sidual coordinates. Recovering every finite coordinate is then separated from convergence in a chosen norm. Resolution towers and exact transport. The sequential model. A tower in a category C is a sequence ⋯xrightarrowρ_2X_2xrightarrowρ_1X_1 xrightarrowρ_0X_0. For sets, a point is a sequence x_n∈ X_n with ρ_n(x_n+1)=x_n. Composite restrictions turn the tower into a view system on N, with an arrow i→ j when i≤ j. Its section space is the same coherent point type, with the same coordinates.
Exact mathematical passage
\rho_n(x_{n+1})=x_n
x_oUse in source · 16 passages
Search aliases: x_o
Use in source · 05_geometry.tex, line 79
Conversely, let x∈ker d. The path telescope along p_i has all zero summands, hence T_ix_i=x_o for every vertex. Applying T_i^-1 gives x_i=T_i^-1x_o, so x=R_p(x_o). The first part then places x_o in K. We have proved both inverse equations and uniqueness of an exact extension of a root value.
Exact mathematical passage
T_ix_i=x_o
x_pUse in source · 3 passages
Search aliases: x_p
Use in source · 02_ambient.tex, line 45
Conversely, suppose mx=0 for a positive integer m. Whenever pnmid m, multiplication by m is invertible in C_p, and hence x_p=0. The support of x is contained in the finite set of prime divisors of m. Thus x∈ D. The all-one vector has infinite support, so it belongs to P∖ D.
Exact mathematical passage
x_p=0
x_zUse in source · 1 passage
Search aliases: x_z
Use in source · 02_obstructions.tex, line 21
[Differential identities] The map δ:E→ R is additive, Γ(A)=kerδ, and (δ x)_1_v=0, (δ x)_g∘ f=f^*((δ x)_g)+(δ x)_f. Additivity follows coordinate by coordinate. Vanishing is exactly the defining coherence equation. For composition, add and subtract f^*x_w: f^*g^*x_z-x_v =f^*(g^*x_z-x_w)+(f^*x_w-x_v). The identity equation uses (1_v)^*=id. sourceKS08 This exact formula is more informative than a scalar error score. It specifies how defects transform and combine. Norms can be added later, but they …
Exact mathematical passage
f^*g^*x_z-x_v =f^*(g^*x_z-x_w)+(f^*x_w-x_v).
X_\thetaUse in source · 6 passages
Search aliases: X_θ theta
Use in source · 03a_observation.tex, line 162
Cofinal changes also preserve the morphism calculus. A monotone unbounded schedule θ gives the sampled resolution X_θ. There is a transport from X to X_θ that reads the selected level. For the reverse direction choose, for each n, a level j(n) with n≤θ(j(n)), and restrict the available value down to X_n. Taking a least such index gives a monotone schedule.
Exact mathematical passage
X_\theta
x_{\phi(i)}Use in source · 1 passage
Search aliases: x_ϕ(i) phi
Use in source · 03a_observation.tex, line 129
The remaining obligation is naturality on all finite states. Fix i≤ j and u∈ X_ϕ(j). By surjectivity of that point projection, choose x with x_ϕ(j)=u. Equation (*) and output coherence yield ρ^Y_i,jf_j(u) · =ρ^Y_i,jF(x)_j=F(x)_i, f_iρ^X_ϕ(i),ϕ(j)(u) · =f_i(x_ϕ(i))=F(x)_i. The state u was arbitrary. Hence the two finite-level functions are equal, which proves scheduled naturality. The resulting transport realizes F by (*).
Exact mathematical passage
\begin{aligned} \rho^Y_{i,j}f_j(u)&=\rho^Y_{i,j}F(x)_j=F(x)_i,\\ f_i\rho^X_{\phi(i),\phi(j)}(u)&=f_i(x_{\phi(i)})=F(x)_i. \end{aligned}
X_{\phi(j)}Use in source · 1 passage
Search aliases: X_ϕ(j) phi
Use in source · 03a_observation.tex, line 128
The remaining obligation is naturality on all finite states. Fix i≤ j and u∈ X_ϕ(j). By surjectivity of that point projection, choose x with x_ϕ(j)=u. Equation (*) and output coherence yield ρ^Y_i,jf_j(u) · =ρ^Y_i,jF(x)_j=F(x)_i,
Exact mathematical passage
u\in X_{\phi(j)}
x_{\phi(j)}Use in source · 1 passage
Search aliases: x_ϕ(j) phi
Use in source · 03a_observation.tex, line 128
The remaining obligation is naturality on all finite states. Fix i≤ j and u∈ X_ϕ(j). By surjectivity of that point projection, choose x with x_ϕ(j)=u. Equation (*) and output coherence yield ρ^Y_i,jf_j(u) · =ρ^Y_i,jF(x)_j=F(x)_i,
Exact mathematical passage
x_{\phi(j)}=u
x_{\phi(n)}Use in source · 6 passages
Search aliases: x_ϕ(n) phi
Use in source · 03_towers.tex, line 80
Sampling a full point and recovering its kth coordinate gives ρ_k,ϕ(n)x_ϕ(n)=x_k by its original coherence. Recovering a sampled point and evaluating at k=ϕ(n) allows the index n itself, giving ρ_ϕ(n),ϕ(n)y_n=y_n. These are both inverse identities. No surjectivity of the bonding maps or of the point evaluations was used. sourceKS18 This theorem is about exact coherent points. Raw residual size is a different quantity. On identity scalar towers, f_n=(-1)^n has nonzero fine differences, but its ev…
Exact mathematical passage
\rho_{k,\phi(n)}x_{\phi(n)}=x_k
x_{\phi(\psi(n))}Use in source · 1 passage
Search aliases: x_ϕ(ψ(n)) phi psi
Use in source · 03a_observation.tex, line 57
Finally, evaluate a witness equation at the kth coordinate of a coherent point x. Its two restricted arguments become x_ϕ(n) and x_ψ(n), proving equality of the realized outputs. Thus realization is well defined on the quotient. For a composite, direct evaluation gives |g∘ f|(x)_n=g_n(f_ψ(n)(x_ϕ(ψ(n)))) =(|g|(|f|(x)))_n. The identity law is equally coordinatewise. Joint determination of output points proves the functor laws. sourceKS61 This is an actual quotient category in the formal source. No…
Exact mathematical passage
|g\circ f|(x)_n=g_n\bigl(f_{\psi(n)}(x_{\phi(\psi(n))})\bigr) =(|g|(|f|(x)))_n.
x_{\psi(n)}Use in source · 2 passages
Search aliases: x_ψ(n) psi
Use in source · 03a_observation.tex, line 56
Finally, evaluate a witness equation at the kth coordinate of a coherent point x. Its two restricted arguments become x_ϕ(n) and x_ψ(n), proving equality of the realized outputs. Thus realization is well defined on the quotient. For a composite, direct evaluation gives |g∘ f|(x)_n=g_n(f_ψ(n)(x_ϕ(ψ(n)))) =(|g|(|f|(x)))_n. The identity law is equally coordinatewise. Joint determination of output points proves the functor laws. sourceKS61 This is an actual quotient category in the formal source. No…
Exact mathematical passage
x_{\psi(n)}
x_\inftyUse in source · 2 passages
Search aliases: x_∞ infty
Use in source · 05_geometry.tex, line 82
The maps are linear. On the finite vertex product, normR_pz_∞≤(max_inormT_i^-1)norm z, normx_o≤norm x_∞. These bounds also hold on the subspaces K and ker d, proving continuity of both directions. No completeness is needed for these finite bounds.
Exact mathematical passage
\norm{R_pz}_\infty\le\left(\max_i\norm{T_i^{-1}}\right)\norm z, \qquad \norm{x_o}\le\norm x_\infty.
\xleftarrowUse in source · 1 passage
Search aliases: xleftarrow
Use in source · 01_views.tex, line 138
Correspondences must also be coherent. Suppose natural maps form a span FxleftarrowpMxrightarrowqG, and fix a∈Γ(F), b∈Γ(G). At each view define the ordinary fiber B_a,b(v)={m∈ M_v:p_v(m)=a_v, q_v(m)=b_v}. Naturality and coherence provide its restrictions. A global bridge is an element of B(a,b)={z∈Γ(M):Γ(p)z=a, Γ(q)z=b}.
Exact mathematical passage
F\xleftarrow{p}M\xrightarrow{q}G
\xRightarrowUse in source · 1 passage
Search aliases: xRightarrow
Use in source · 04_symmetry.tex, line 410
Weakly commuting squares are enough. Suppose E: C→ C' and K: D→ D' are equivalences, with functors R,R' and a natural isomorphism σ:KRxRightarrow∼R'E. The square is not required to commute literally.
Exact mathematical passage
\sigma:KR\xRightarrow{\sim}R'E.
y_*Use in source · 7 passages
Search aliases: y_*
Use in source · 05_geometry.tex, line 180
[Nearest feasible protected repair] Every feasible x has a unique closest exact answer y_* preserving p(x). Its correction h_*=x-y_* is the unique vector in the orthogonal complement of ker B inside M satisfying Bh_*=Dx. For every other feasible answer y, normx-y^2=normx-y_*^2+normy_*-y^2. Moreover, the following conditions are equivalent: ran B is closed Longleftrightarrow ∃ C≥0 ∀ feasible x: normx-y_*≤ CnormDx. When these conditions hold, the least nonnegative constant is the transverse invers…
Exact mathematical passage
y_*
Y_0Use in source · 1 passage
Search aliases: Y_0
Use in source · 06_boundaries.tex, line 60
Choose x_0∈ Y_0 and recursively choose x_n+1∈ Y_n+1 above x_n. The preceding surjectivity justifies each step. Adjacent coherence implies every composite coherence equation, so this gives a point. This is an existence construction using countably many choices, not an asserted effective procedure for locating the stabilization indices.
Exact mathematical passage
x_0\in Y_0
y_1Use in source · 6 passages
Search aliases: y_1
Use in source · 04b_phase.tex, line 8
A two-bit phase law. Write F_2=Z/2Z additively, and let G= F_2^2, x=(x_1,x_2), c(x,y)=x_2y_1. All entries and operations in this subsection lie in F_2. In particular minus and plus agree, although retaining a minus sign in the coboundary formula helps explain its origin.
Exact mathematical passage
G=\mathbf F_2^2,\qquad x=(x_1,x_2),\qquad c(x,y)=x_2y_1.
y_2Use in source · 2 passages
Search aliases: y_2
Use in source · 04b_phase.tex, line 26
on c(x,y)+c(x+y,z)=c(y,z)+c(x,y+z). There is no function a:G→ F_2 such that c(x,y)=a(x)+a(y)-a(x+y) for every x,y∈ G. Normalization follows from multiplication by zero. Expanding the two sides of [source reference] gives, respectively, x_2y_1+(x_2+y_2)z_1, y_2z_1+x_2(y_1+z_1), which are equal by distributivity.
Exact mathematical passage
x_2y_1+(x_2+y_2)z_1, \qquad y_2z_1+x_2(y_1+z_1),
Y_aUse in source · 2 passages
Search aliases: Y_a
Use in source · 03b_repair_transport.tex, line 147
For a block from a to b, its value on a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a,b. The dyadic coarsener in Section~[source reference] is the case b=a+2.
Exact mathematical passage
Y_a
y_iUse in source · 2 passages
Search aliases: y_i
Use in source · 05_geometry.tex, line 300
Flatness does not remove amplification. Consider a scalar chain with n+1 vertices, where n≥1, and coordinates e_i(z)=2^iz. Set τ_ij=e_ie_j^-1. This connection is flat, but reverse transport along one edge multiplies norms by two. Let x_i=8·2^i-1, y_i=7·2^i. The family y is exact and preserves x_o=7. On a forward edge the residual of x is 1/2, and on its reverse it is -1. Thus normdx_∞=1, normx-Px_∞=2^n-1. For n≥2, a route length bound of n fails. This is amplification with trivial holonomy, dist…
Exact mathematical passage
x_i=8\cdot2^i-1,\qquad y_i=7\cdot2^i.
y_{LHv}Use in source · 1 passage
Search aliases: y_LHv
Use in source · 04_symmetry.tex, line 373
If x∈Γ(F), coherence along t_w gives (RH^*x)_w=F(t_w)x_HLw=x_w. For the other composite, the triangle identity says t_Hv=H(η_v). Coherence of y along η_v:v→ LHv yields (H^*Ry)_v=F(t_Hv)y_LHv=F(Hη_v)y_LHv=y_v. These coordinate equations prove both inverse identities.
Exact mathematical passage
(H^*Ry)_v=F(t_{Hv})y_{LHv}=F(H\eta_v)y_{LHv}=y_v.
y_{Lw}Use in source · 3 passages
Search aliases: y_Lw
Use in source · 04_symmetry.tex, line 350
sm and U'=H^-1U, it also induces Ext_U^F(s)≃Ext_U'^H^*F(H^*s). Existence and uniqueness of the prescribed extension are preserved and reflected. Write the unit of the equivalence as η:I_V⇒ LH, and take the unit and counit with their triangle identities. Set t_w=ε_w^-1:w→ HLw. Given y∈Γ(H^*F), define (Ry)_w=F(t_w)y_Lw. For an arrow f:w→ w', naturality of the inverse counit states HL(f) t_w=t_w'f. Contravariance of F and coherence of y therefore give F(f)(Ry)_w' · =F(f)F(t_w')y_Lw'
Exact mathematical passage
(Ry)_w=F(t_w)y_{Lw}.
y_{Lw'}Use in source · 1 passage
Search aliases: y_Lw'
Use in source · 04_symmetry.tex, line 358
sition Write the unit of the equivalence as η:I_V⇒ LH, and take the unit and counit with their triangle identities. Set t_w=ε_w^-1:w→ HLw. Given y∈Γ(H^*F), define (Ry)_w=F(t_w)y_Lw. For an arrow f:w→ w', naturality of the inverse counit states HL(f) t_w=t_w'f. Contravariance of F and coherence of y therefore give F(f)(Ry)_w' · =F(f)F(t_w')y_Lw' · =F(t_w)F(HL(f))y_Lw' =F(t_w)y_Lw=(Ry)_w. Thus recovery gives an actual coherent section.
Exact mathematical passage
\begin{aligned} F(f)(Ry)_{w'} &=F(f)F(t_{w'})y_{Lw'}\\ &=F(t_w)F(HL(f))y_{Lw'} =F(t_w)y_{Lw}=(Ry)_w. \end{aligned}
y_MUse in source · 3 passages
Search aliases: y_M
Use in source · 05g_budget_certificates.tex, line 62
The nearest reachable image y_M=P_T({a:norm a≤ M})r is unique, so the best residual vector r-y_M is unique as well. Corrections mapping to y_M can differ by vectors in ker T. Among those corrections there is a unique least norm correction. These two uniqueness statements should not be confused with injectivity of T.
Exact mathematical passage
y_M=P_{T(\{a:\norm a\le M\})}r
y_NUse in source · 4 passages
Search aliases: y_N
Use in source · 03_towers.tex, line 397
Suppose Q_Nx→0. Put y_N=S_Nx=x-Q_Nx. Idempotence gives Q_Ny_N=Q_Nx-Q_N^2x=0, so y_N∈ F. Since x-y_N=Q_Nx→0, these are finite-core approximants converging to x. Hence x∈overline F.
Exact mathematical passage
y_N=S_Nx=x-Q_Nx
y_pUse in source · 3 passages
Search aliases: y_p
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
y_tUse in source · 1 passage
Search aliases: y_t
Use in source · 11_written_extensions.tex, line 9
[4pt] [source reference] · Nested affine minimum corrections · Verified [4pt] [source reference] · Attainment and shape of the budget profile · Verified [4pt] [source reference] · Sharp budget certificate · Verified [4pt] [source reference] · Robust budget comparison · Verified [4pt] [source reference] · Unique regularized correction · Verified [4pt] [source reference] · Parameter and budget comparison · Verified [4pt] [source reference] · Regularization limit alternative · Verified [4pt] [sourc…
Exact mathematical passage
4pt] \ref{WR02} & Nested affine minimum corrections & Verified\\[4pt] \ref{WR03} & Attainment and shape of the budget profile & Verified\\[4pt] \ref{WR04} & Sharp budget certificate & Verified\\[4pt] \ref{WR05} & Robust budget comparison & Verified\\[4pt] \ref{WR06} & Unique regularized correction & Verified\\[4pt] \ref{WR07} & Parameter and budget comparison & Verified\\[4pt] \ref{WR08} & Regularization limit alternative & Verified\\[4pt] \ref{WR09} & Data sensitivity and convergent parameter choice & Verified\\[4pt] \ref{WR10} & Infinite diagonal family and certified tails & Verified\\[4pt] \ref{WR11} & Sharp diagonal budget asymptotic & Verified\\[4pt] \ref{WR12} & Rounded noisy diagonal rate & Verified\\[4pt] \ref{WR13} & Normal-residual solver certificate & Verified\\[4pt] \ref{WR14} & Regularized primal and dual gap & Verified\\[4pt] \ref{WR15} & Coordinate certificate and arbitrary basis change & Verified\\[4pt] \ref{WR16} & Consistent inexact stopping rule & Verified\\ \bottomrule\end{longtable} \subsection{Bounded realization and budget geometry}\label{app:wr-source-map} The formal bounded-realization theorem concerns bounded linear maps between real Hilbert spaces. It gives an exact preimage within the same norm bound as the approximating corrections. Its proof uses compactness of a dual ball and the Riesz isometric equivalence. The printed proof uses the weak-subsequence lemma, Lemma~\ref{lem:weak-subsequence}, whose particular proof route is not separately formalized here. Neither route requires separability or closed range. Neither asserts strong convergence of the original approximating sequence. The nested-constraint theorem constructs the minimum-norm points of nonempty closed affine sets. It proves the exact energy identity, norm monotonicity, and the equivalence between a nonempty intersection and bounded minimum norms. Under that bound, the corrections converge strongly to the unique minimum-norm point of the intersection. If the intersection is empty, their norms tend to infinity. The affine assumption is retained: for general convex sets the corresponding energy statement is an inequality. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BoundedApproximateRealization} & \path{exists_solution_of_bounded_residual_tendsto_zero}\newline\path{mem_range_iff_bounded_approximation}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{NestedMinimumCorrections} & \path{nested_affine_energy}\newline\path{nested_affine_intersection_iff_bounded}\newline\path{nested_affine_strong_realization}\newline\path{nested_affine_escape}\\[5pt] \path{ProtectedBoundedRealization} & \path{exists_exact_repair_of_bounded_approximation}\\ \bottomrule\end{longtable} All namespaces in this appendix have the prefix \path{KomorebiSpace}. The protected application uses the actual subtype $\ker p$ with its inherited Hilbert norm. It constructs a repaired candidate preserving the declared observable and the original correction budget. The budget-profile theorems concern bounded operators between real complete Hilbert spaces. The budget is nonnegative and may be zero. Attainment of a residual minimum over a bounded correction ball does not require the entire operator range to be closed. The sharp support formula supplies an actual maximizing witness, expressed formally by \code{IsGreatest}. It does not infer attainment from the mere existence of a supremum. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{BudgetProfile} & \path{exists_budget_minimizer}\newline\path{beta_eq_iInf}\newline\path{beta_tendsto_floor}\newline\path{beta_eq_floor_iff}\newline\path{beta_eq_zero_iff}\\[5pt] \path{BudgetProfileGeometry} & \path{beta_antitone}\newline\path{beta_zero}\newline\path{beta_convex_combination}\newline\path{beta_profile_bounds}\\[5pt] \path{BudgetDuality} & \path{dualValue_le_beta}\newline\path{exists_attaining_dual_witness}\newline\path{sharp_budget_duality}\newline\path{strict_exclusion_certificate}\\[5pt] \path{BudgetRobustness} & \path{candidate_residual_le}\newline\path{beta_abs_sub_le}\newline\path{nominal_dual_lower_certificate}\newline\path{excludes_budget_of_nominal_dual_certificate}\\ \bottomrule\end{longtable} The robustness comparison uses the same correction space for both problems. Its bound $\delta+\eta M$ controls data and operator uncertainty at budget $M$. A changing protected kernel is a different problem and is not covered by that estimate. \subsection{Regularization, noise, and stopping} For positive parameter $\alpha$, the regularized normal operator is constructed as a continuous linear equivalence. Its inverse defines the correction. The formal development proves uniqueness, the energy identity, parameter monotonicity, budget optimality, and the two data-sensitivity estimates. The fixed-space protected application uses the restricted operator and its actual adjoint. The limit theory distinguishes projection onto the closure of the range from membership in the range itself. Fitted values converge to the projected datum and residual norms converge to the irreducible floor. If that projected datum is attained, the corrections converge in norm to its unique minimum-norm preimage. Otherwise their norms tend to infinity. The data and stopping theorems use this limit theorem together with their explicit parameter, noise, and solver-tolerance assumptions. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{exists_unique_minimizer}\newline\path{energy_identity}\newline\path{residual_best_at_budget}\newline\path{parameter_norm_antitone}\newline\path{parameter_residual_monotone}\newline\path{data_noise_bound}\newline\path{data_fitted_bound}\\[5pt] \path{RegularizationLimits} & \path{fitted_tendsto_projectedDatum}\newline\path{residual_norm_tendsto_floor}\newline\path{weighted_penalty_tendsto_zero}\newline\path{norm_tendsto_atTop_of_not_mem_range}\\[5pt] \path{RegularizationStrongLimit} & \path{exists_unique_minimum_and_limit}\\[5pt] \path{RegularizationNoiseLimits} & \path{data_noise_coefficient_attained}\newline\path{noisy_parameter_choice}\newline\path{inexact_parameter_choice}\newline\path{stopping_rule_sequence}\\ \bottomrule\end{longtable} The stopping theorem compares proposals, exact regularized corrections, and the intended minimum correction in one fixed Hilbert model. It includes the pointwise three-term bound and the convergence conclusion. Protection is exact at every index. The explicit diagonal schedule at the end of Section~\ref{sec:stopping-rules} combines this solver bound with the formalized noisy rate in Corollary~\ref{WR12}. The declaration \path{DiagonalRegularization.diagonal_inexact_rounded_rate} proves this specialization with the literal ceiling, the data-noise hypothesis, and the actual measured-data normal-residual bound. The exact-data bias estimate is derived rather than assumed. \subsection{Certificates and their coordinate metric} The solver certificate controls correction error, fitted error, and objective excess through the actual Hilbert normal residual. The primal and dual identity supplies objective bracketing and a correction-distance bound for an arbitrary dual witness. These are universal statements over the declared Hilbert spaces, with $\alpha>0$ wherever coercivity is used. \begin{longtable}{@{}p{.38\linewidth}p{.55\linewidth}@{}} \toprule Namespace & Representative declarations\\\midrule\endhead \path{RegularizedHilbertRepair} & \path{correction_error_bound}\newline\path{fitted_error_bound}\newline\path{objective_gap_bound}\\[5pt] \path{RegularizationCertificates} & \path{primal_dual_gap}\newline\path{dual_maximizer}\newline\path{gap_controls_correction}\newline\path{residual_witness_gap}\\[5pt] \path{CoordinateCovector} & \path{gram_isUnit}\newline\path{protected_coordinate_certificate}\newline\path{computed_normalResidual_changedBasis}\newline\path{computed_certificate_changedBasis}\newline\path{changedBasis_coordinates}\\ \bottomrule\end{longtable} The coordinate certificate uses a basis of the entire permitted correction space, with its inherited Hilbert metric. The proof derives invertibility of the Gram matrix from that basis, identifies the coordinate residual with the pairings of the Hilbert normal residual, and proves \[ g=BG^{-1}q,\qquad \norm g^2=q^{\mathsf T}G^{-1}q.
y_vUse in source · 5 passages
Search aliases: y_v
Use in source · 01_views.tex, line 34
), there is a unique map c:T→Γ(F) satisfying ev_v c=c_v for every view. We first verify that the evaluation maps themselves form a compatible family. If x∈Γ(F) and f:v→ w, the defining equation for x gives f^*(ev_w x)=f^*(x_w)=x_v=ev_v x. If x,y∈Γ(F) have equal evaluations, then x_v=y_v for every v. Equality of functions into a dependent product gives equality of their underlying families. Membership in Γ(F) is a property of a family, not a further choice of coordinate. Thus x=y.
Exact mathematical passage
x_v=y_v
y_wUse in source · 1 passage
Search aliases: y_w
Use in source · 04_symmetry.tex, line 56
If d=a· c and x is a section of T_c, put y_v=a_vx_v. Then d_ff^*(y_w) =a_vc_f[f^*(a_w)]^-1f^*(a_w)f^*(x_w) =a_vx_v=y_v. The inverse map is multiplication by a_v^-1 at each view, proving an equivalence of the actual section sets.
Exact mathematical passage
d_ff^*(y_w) =a_vc_f[f^*(a_w)]^{-1}f^*(a_w)f^*(x_w) =a_vx_v=y_v.
Y_\etaUse in source · 1 passage
Search aliases: Y_η eta
Use in source · 03a_observation.tex, line 166
[Cofinal invariance of eventual transports] Sampling and recovery are inverse in the eventual transport category. Consequently, for monotone unbounded schedules θ and η, there is a bijection EventualHom(X,Y) ≃ EventualHom(X_θ,Y_η). On point spaces it agrees with the previously constructed cofinal realization equivalences. Let s:X→ X_θ be sampling, with schedule θ and identity level maps. For recovery choose the least j(n) with n≤θ(j(n)). Unboundedness makes this set nonempty. Increasing n only s…
Exact mathematical passage
\operatorname{EventualHom}(X,Y) \simeq \operatorname{EventualHom}(X_\theta,Y_\eta).
Y_{\psi(n)}Use in source · 1 passage
Search aliases: Y_ψ(n) psi
Use in source · 03a_observation.tex, line 20
The realized map is |f|:Point(X)→Point(Y), |f|(x)_n=f_n(x_ϕ(n)). Equation~[source reference] proves that this is a coherent output. A second transport g_n:Y_ψ(n)→ Z_n composes with f using schedule ϕ∘ψ and coordinate map g_n f_ψ(n). The order of the schedules is fixed by the types, not by the order in which their letters are written in prose.
Exact mathematical passage
g_n:Y_{\psi(n)}\to Z_n
y_\ellUse in source · 1 passage
Search aliases: y_ℓ ell
Use in source · 03_towers.tex, line 64
For each k, unboundedness gives an index n with k≤ϕ(n). Set x_k=ρ_k,ϕ(n)y_n. If m is another permissible index, choose ℓ≥ n,m. Monotonicity permits both composite restrictions, and sampled coherence gives ρ_k,ϕ(n)y_n =ρ_k,ϕ(n)ρ_ϕ(n),ϕ(ℓ)y_ℓ =ρ_k,ϕ(ℓ)y_ℓ =ρ_k,ϕ(m)y_m. Thus x_k is independent of the choice, including when the schedule repeats a level.
Exact mathematical passage
\rho_{k,\phi(n)}y_n =\rho_{k,\phi(n)}\rho_{\phi(n),\phi(\ell)}y_\ell =\rho_{k,\phi(\ell)}y_\ell =\rho_{k,\phi(m)}y_m.
ZUse in source · 41 passages
Search aliases: Z
Use in source · 02a_prescription.tex, line 29
[Functorial forgetting of protected views] The homomorphism π_W,U is well defined and surjective. For U⊆ W⊆ Z, it satisfies π_U,U=I, π_W,Uπ_Z,W=π_Z,U. For every s∈Γ_W(A), π_W,U o_W(s)= o_U(s|_U). In particular an obstructed restriction remains obstructed when more compatible prescribed data are retained. A successful extension on the larger prescribed domain gives a successful extension of its restriction. If r-r'∈ B_W, then r-r'∈ B_U, proving well-definedness. Every class modulo B_U has a repre…
Exact mathematical passage
U\subseteq W\subseteq Z
\ZUse in source · 14 passages
Search aliases: Z Z mathbb
Use in source · 02_obstructions.tex, line 110
[Parity and an independent ambiguity] For two views with f^*(m)=2m on Z, a prescribed value k at the coarse view extends exactly when k is even. The extension is (k,k/2) and is unique. In particular 1 is obstructed and 2 extends nontrivially. The parity statement is an instance of Theorem~[source reference]. If a separate, disconnected integer view is added and left unprescribed, every extendable seed has infinitely many completions. This last variant is a direct application of Theorem~[source r…
Exact mathematical passage
\Z
Z^1Use in source · 4 passages
Search aliases: Z^1
Use in source · 04_symmetry.tex, line 66
to e, then (ba)· c=b·(a· c)=e, giving composition. The identity label is 1 and the inverse label is a^-1. Associativity and the two inverse identities now follow from the corresponding multiplication laws in the product group. The label ba is therefore forced by the order of application, not an arbitrary notational convention. sourceKS30 The action groupoid Z^1(V;G)//∏_vG_v retains comparison witnesses. Its orbit set Z^1(V;G)/∏_vG_v retains only isomorphism classes. There is no reason for the or…
Exact mathematical passage
Z^1(\V;G)//\prod_vG_v
Z^2Use in source · 1 passage
Search aliases: Z^2
Use in source · 04b_phase.tex, line 50
0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do not commute.
Exact mathematical passage
X^2=Z^2=I
\Z^2Use in source · 1 passage
Search aliases: Z^2 Z
Use in source · 02a_prescription.tex, line 48
An initially free parameter can become an obstruction. Consider the chain of views 0→1→2, with fiber Z^2 at each view. Write the maps in the direction in which data are restricted: ρ_0,1(a,b)=(a,0), ρ_1,2(a,b)=(a,2b), ρ_0,2(a,b)=(a,0). The composite identity ρ_0,1ρ_1,2=ρ_0,2 is literal. A global family is exactly x_0=(k,0), x_1=(k,2n), x_2=(k,n), k,n∈Z. Indeed the first coordinate must agree at every view. The second coordinate at view zero is forced to vanish, and the second coordinate at view …
Exact mathematical passage
\Z^2
z^2Use in source · 4 passages
Search aliases: z^2
Use in source · 05_geometry.tex, line 319
2t · 1-t^2 satisfies norm(U_t-I)z^2=frac4t^21+t^2norm z^2. For t≠0 its fixed space is zero and its residual gain is sqrt1+t^2/(2|t|), which diverges as t→0. At t=0, the entire fiber is fixed. Therefore small loop discrepancy does not supply a uniform stable realization estimate. The fixed directions and their conditioning must be analyzed together.
Exact mathematical passage
\norm{(U_t-I)z}^2=\frac{4t^2}{1+t^2}\norm z^2.
Z^qUse in source · 1 passage
Search aliases: Z^q
Use in source · 04b_phase.tex, line 45
The phase is visible in actual matrices. Over R, define X=0 · 1 1 · 0, Z=1 · 0 0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do not commute.
Exact mathematical passage
X=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad Z=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\qquad U_{(p,q)}=X^pZ^q\quad(p,q\in\{0,1\}).
Z^{x_2}Use in source · 1 passage
Search aliases: Z^x_2
Use in source · 04b_phase.tex, line 50
0 · -1, U_(p,q)=X^pZ^q (p,q∈{0,1}). Direct multiplication gives X^2=Z^2=I and ZX=-XZ. Moving Z^x_2 past X^y_1 therefore yields U_xU_y=(-1)^c(x,y)U_x+y. For example U_e_1U_e_2=XZ, while U_e_2U_e_1=-XZ. The two-bit labels add commutatively, but their chosen matrix lifts do not commute.
Exact mathematical passage
Z^{x_2}
z_1Use in source · 1 passage
Search aliases: z_1
Use in source · 04b_phase.tex, line 26
on c(x,y)+c(x+y,z)=c(y,z)+c(x,y+z). There is no function a:G→ F_2 such that c(x,y)=a(x)+a(y)-a(x+y) for every x,y∈ G. Normalization follows from multiplication by zero. Expanding the two sides of [source reference] gives, respectively, x_2y_1+(x_2+y_2)z_1, y_2z_1+x_2(y_1+z_1), which are equal by distributivity.
Exact mathematical passage
x_2y_1+(x_2+y_2)z_1, \qquad y_2z_1+x_2(y_1+z_1),
Z_nUse in source · 1 passage
Search aliases: Z_n
Use in source · 03a_observation.tex, line 20
The realized map is |f|:Point(X)→Point(Y), |f|(x)_n=f_n(x_ϕ(n)). Equation~[source reference] proves that this is a coherent output. A second transport g_n:Y_ψ(n)→ Z_n composes with f using schedule ϕ∘ψ and coordinate map g_n f_ψ(n). The order of the schedules is fixed by the types, not by the order in which their letters are written in prose.
Exact mathematical passage
g_n:Y_{\psi(n)}\to Z_n
z_vUse in source · 3 passages
Search aliases: z_v
Use in source · 01_views.tex, line 161
Given u∈Γ(B_a,b), forget each fiber-membership condition and call the underlying family z_v∈ M_v. Coherence of u is precisely coherence of z, so z∈Γ(M). The equations p_vz_v=a_v hold at every coordinate. Joint determination in Γ(F) yields Γ(p)z=a, and the equations for q similarly yield Γ(q)z=b. Thus z is a global bridge.
Exact mathematical passage
z_v\in M_v
z_\alphaUse in source · 4 passages
Search aliases: z_α alpha
Use in source · 05j_inexact_certificates.tex, line 101
[Exact regularized primal and dual gap] For every a∈ H and z∈ Y, J_α(a)- D_α(z) =normTa-r+z^2+ fracnormT^*z-α a^2α. Consequently D_α(z)≤ J_α(a) for every pair. At a=a_α and z=z_α=r-Ta_α, equality holds, so the dual maximum is attained and equals the primal minimum. Moreover, αnorma-a_α^2 ≤ J_α(a)- D_α(z). For the easily computed choice z=r-Ta, J_α(a)- D_α(r-Ta)=fracnorm g^2α. For arbitrary a,z, expand the first square as normTa-r+z^2 =normTa-r^2+norm z^2+2⟨ Ta,z⟩-2⟨ r,z⟩, and the scaled second s…
Exact mathematical passage
z=z_\alpha=r-Ta_\alpha
|^2Use in source · 2 passages
Search aliases: |^2
Use in source · 05e_escape.tex, line 7
The complete spaces and the preservation rule. Index sequences by the positive integers. Set E=ℓ^2(N_>0;R)⊕R, R=ℓ^2(N_>0;R), norm(u,t)_E^2=∑_n≥1|u_n|^2+|t|^2. Let w=(1/n)_n≥1. It belongs to ℓ^2, because ∑_n≥1frac1n^2 ≤1+∑_n≥2frac1n(n-1)=2. Define the defect and protected observable by d(u,t)_n=fracu_n+tn, p(u,t)=t. The diagonal map T:u↦(u_n/n)_n has operator norm one. Since d(u,t)=Tu+tw, normd(u,t)≤norm u+|t|norm w ≤sqrt1+norm w^2 norm(u,t)_E. Thus both d and p are continuous linear maps between…
Exact mathematical passage
E=\ell^2(\N_{>0};\R)\oplus\R,\qquad R=\ell^2(\N_{>0};\R),\qquad \norm{(u,t)}_E^2=\sum_{n\ge1}|u_n|^2+|t|^2.
|_{\ker D}Use in source · 1 passage
Search aliases: |_ker D ker
Use in source · 05_geometry.tex, line 379
[Closed range for the joint defect] Assume ran D is closed. Then ran(D,C) is closed Longleftrightarrow ran(C|_ker D) is closed. Under either condition, a finite joint residual-to-repair bound exists. Suppose the range of the joint map is closed. To test closedness of ran B, take a sequence Bk_n→ y with k_n∈ker D. The joint images are (Dk_n,Ck_n)=(0,Bk_n)⟶(0,y). Closedness supplies x∈ E with (Dx,Cx)=(0,y). The first coordinate gives x∈ker D, and the second gives Bx=y. Hence the matching range is …
Exact mathematical passage
\ran(D,C)\text{ is closed}\quad\Longleftrightarrow\quad \ran(C|_{\ker D})\text{ is closed}.
|_{\ker p}Use in source · 1 passage
Search aliases: |_ker p ker
Use in source · 06_cases.tex, line 77
Here the general protected construction is completely explicit. With candidate space E=R^4, defect d=ℓ, and protected observable p(r)=(r_0,r_1,r_2), the allowed direction is the last coordinate axis. The restricted operator d|_ker p has gain one. Without protection, the least gain for distance to kerℓ is 1/2, since normℓ=2. The different constants measure different correction spaces.
Exact mathematical passage
d|_{\ker p}
|_MUse in source · 1 passage
Search aliases: |_M
Use in source · 05_geometry.tex, line 174
Stable repair with protected data. The preceding orthogonal projection does not generally preserve a separately declared observable. To join the analytic theory to Section~[source reference], let p:E→ S be continuous linear into a normed space. Define M=ker p, B=D|_M:M→ R, H=M∩ker D. The closed subspace M inherits its Hilbert structure from E. A candidate x is feasible for protected repair when an exact y with p(y)=p(x) exists. The algebraic criterion is still Dx∈ran B.
Exact mathematical passage
M=\ker p,\qquad B=D|_M:M\to R,\qquad H=M\cap\ker D.
|_UUse in source · 9 passages
Search aliases: |_U
Use in source · 01_views.tex, line 88
chapPartial data, gluing, and correspondence fibers Prescribed views. For U⊆ObV, use the full subcategory on U. A partial section s∈Γ_U(F) satisfies all transition equations whose endpoints lie in U. Define Ext_U(s)={x∈Γ(F):x|_U=s}. This fiber can be empty, a singleton, or much larger. A partial section is already coherent on its prescribed domain. A raw assignment that fails an internal transition equation is not a partial section of this kind.
Exact mathematical passage
\label{eq:extension} \Ext_U(s)=\{x\in\Gamma(F):x|_U=s\}.
|_{U_j}Use in source · 1 passage
Search aliases: |_U_j
Use in source · 01_views.tex, line 124
Step 3: assembly and restriction are inverse. Formula (*) proves x|_U_j=s_j for every j. Conversely, start with y∈Γ(F) and assemble its restrictions. The resulting coordinate at v is y_v, regardless of which containing patch is chosen. Joint determination gives the original y. These two identities are exactly the inverse identities for the displayed equivalence.
Exact mathematical passage
x|_{U_j}=s_j
|_WUse in source · 1 passage
Search aliases: |_W
Use in source · 04_symmetry.tex, line 306
The actual patch pseudofunctor. On a patch W, use K_G(W)=Z^1(V|_W;G|_W)//∏_v∈ WG_v. Restriction keeps the existing cocycle values and gauge values on the smaller patch. It first gives a strict category valued functor on the opposite patch poset. Its standard pseudofunctor form has identity comparison isomorphisms, with the required coherence laws.
Exact mathematical passage
\mathfrak K_G(W)=Z^1(\V|_W;G|_W)//\prod_{v\in W}G_v.
\pmUse in source · 1 passage
Search aliases: ± pm
Use in source · 04b_phase.tex, line 97
Projection (a,x)↦ x is a homomorphism onto G. Its kernel is {(0,0),(1,0)}, which commutes with every element. The map (a,x)⟼(-1)^aU_x is a homomorphism by [source reference]. Its eight matrices are distinct: they are ± I,± X,± Z,± XZ. It is therefore faithful. The phase extension is not an assertion that G itself has acquired an ordinary representation with these same lifts. The represented group has changed.
Exact mathematical passage
\pm I,\pm X,\pm Z,\pm XZ
\timesUse in source · 17 passages
Search aliases: × times
Use in source · 03_towers.tex, line 237
[Split coordinate theorem] The observable is a bounded projection with p_LA=0 and p_LP=p_L. It separates exact candidates. If S_L={s∈ E:p_Ls=s}, then Φ:E≃_L S_L× R, Φ(f)=(p_Lf,∂ f), Φ^-1(s,r)=Ps+Ar. Repair becomes (s,r)↦(s,0). It is the unique exact repair preserving p_Lf. Put B_n=I-s_nρ^Y_n. Since ρ^Y_ns_n=I, direct multiplication gives B_n^2=B_n, ρ^Y_nB_n=0, B_ns_n=0. Moreover, B_ny=y whenever ρ^Y_ny=0. Thus B_n really projects onto the target kernel. Its norm is at most 1+SB_Y. Coordinate zer…
Exact mathematical passage
\Phi:E\simeq_L\mathcal S_L\times R, \qquad \Phi(f)=(p_Lf,\partial f), \qquad \Phi^{-1}(s,r)=Ps+Ar.
\alpha^{-1}Use in source · 7 passages
Search aliases: α^-1 alpha
Use in source · 05h_regularized_repair.tex, line 34
To prove closedness of its range, suppose Bh_n→ y. Applying the lower bound to h_n-h_m gives normh_n-h_m≤α^-1normBh_n-Bh_m. Thus h_n is Cauchy. Completeness of H supplies a limit h, and continuity gives Bh=y. If y is orthogonal to the range of B, then for every h, 0=⟨ Bh,y⟩=⟨ h,By⟩. Taking h=By gives By=0, and injectivity gives y=0. A closed subspace with zero orthogonal complement is the whole Hilbert space, so ran B=H.
Exact mathematical passage
\norm{h_n-h_m}\le\alpha^{-1}\norm{Bh_n-Bh_m}.
\alpha^{-1/2}Use in source · 1 passage
Search aliases: α^-1/2 alpha
Use in source · 05j_inexact_certificates.tex, line 72
fixedplate37_solver_amplificationA fixed normal-equation tolerance has different consequences at different regularization parameters. The curves are universal certified upper bounds for a declared normal-residual norm, not observed errors of a particular experiment. The correction estimate grows as α^-1 while the fitted-data estimate grows as α^-1/2.fig:solver-amplification
Exact mathematical passage
\alpha^{-1/2}
\alpha^2Use in source · 1 passage
Search aliases: α^2 alpha
Use in source · 05j_inexact_certificates.tex, line 48
To obtain the sharper fitted-data bound, retain the quadratic dependence on norm e instead of substituting the first estimate. Multiplying [source reference] by α and rearranging gives αnormTe^2 · ≤αnorm gnorm e-α^2norm e^2 · =fracnorm g^24 -(αnorm e-fracnorm g2)^2 ≤fracnorm g^24. Division by α and square roots give the factor 1/(2sqrtα).
Exact mathematical passage
\begin{aligned} \alpha\norm{Te}^2 &\le\alpha\norm g\norm e-\alpha^2\norm e^2\\ &=\frac{\norm g^2}{4} -\left(\alpha\norm e-\frac{\norm g}{2}\right)^2 \le\frac{\norm g^2}{4}. \end{aligned}
\alpha_nUse in source · 16 passages
Search aliases: α_n alpha
Use in source · 05h_regularized_repair.tex, line 175
Step 4: strengthen weak convergence to norm convergence. Take an arbitrary sequence α_ndownarrow0, or more generally positive α_n→0. The bound supplies a weakly convergent subsequence a_α_n_krightharpoonup a. Since its images converge in norm to b, the adjoint pairing argument of Theorem~[source reference] yields Ta=b. Weak lower semicontinuity gives norm a≤liminf_knorma_α_n_k ≤norma^†. Minimality and uniqueness of a^† force a=a^†. Moreover the same lower bound, together with the uniform upper b…
Exact mathematical passage
\alpha_n\downarrow0
\alpha_{n_k}Use in source · 3 passages
Search aliases: α_n_k alpha
Use in source · 05h_regularized_repair.tex, line 175
Step 4: strengthen weak convergence to norm convergence. Take an arbitrary sequence α_ndownarrow0, or more generally positive α_n→0. The bound supplies a weakly convergent subsequence a_α_n_krightharpoonup a. Since its images converge in norm to b, the adjoint pairing argument of Theorem~[source reference] yields Ta=b. Weak lower semicontinuity gives norm a≤liminf_knorma_α_n_k ≤norma^†. Minimality and uniqueness of a^† force a=a^†. Moreover the same lower bound, together with the uniform upper b…
Exact mathematical passage
a_{\alpha_{n_k}}\rightharpoonup a
\alpha_vUse in source · 3 passages
Search aliases: α_v alpha
Use in source · 01_views.tex, line 54
Maps of realizations. Let α:F→ G consist of maps α_v:F_v→ G_v with α_v f_F^*=f_G^*α_w. The induced map on realizations is Γ(α)(x)_v=α_v(x_v). The naturality square is exactly what is needed to show that the image family is coherent.
Exact mathematical passage
\alpha_v:F_v\to G_v
\alpha_wUse in source · 1 passage
Search aliases: α_w alpha
Use in source · 01_views.tex, line 55
Maps of realizations. Let α:F→ G consist of maps α_v:F_v→ G_v with α_v f_F^*=f_G^*α_w. The induced map on realizations is Γ(α)(x)_v=α_v(x_v). The naturality square is exactly what is needed to show that the image family is coherent.
Exact mathematical passage
\alpha_v f_F^*=f_G^*\alpha_w.
\beta_nUse in source · 2 passages
Search aliases: β_n beta
Use in source · 05c_iteration.tex, line 150
When two repair paths select the same point. [Path independence requires a comparison] Let f_n→ x and g_n→ y in a metric space. If dist(f_n,g_n)≤β_n, β_n→0, then x=y. In contrast, exactness and preservation of a nonzero observable alone do not force a unique answer. For each n, the triangle inequality gives dist(x,y)≤dist(x,f_n)+dist(f_n,g_n)+dist(g_n,y) ≤dist(x,f_n)+β_n+dist(g_n,y). Given any ε>0, choose n large enough that the three terms on the right are each below ε/3. Thus dist(x,y)≤ε for e…
Exact mathematical passage
\dist(f_n,g_n)\le\beta_n,\qquad\beta_n\to0,
\beta_{T,w}Use in source · 1 passage
Search aliases: β_T,w beta
Use in source · 05i_diagonal_regularization.tex, line 77
[A sharp budget asymptotic for the diagonal datum] As m→∞, fracA_mm⟶fracπ4, mB_m⟶fracπ4. For the exact budget profile of this infinite problem, lim_M→∞Mβ_T,w(M)=fracπ4. The hard prefix correction at budget M=sqrt N instead has sqrt N normw-T1_N⟶1. Step 1: control the infinite Riemann sums uniformly. Define f(x)=frac1(1+x^2)^2, g(x)=fracx^2(1+x^2)^2. Then A_m/m=m^-1∑_n≥1f(n/m) and mB_m=m^-1∑_n≥1g(n/m). Fix L≥2 and put N=lfloor mLrfloor. On [0,L], both functions are uniformly continuous. If ω_f(h)…
Exact mathematical passage
\label{eq:sharp-budget} \lim_{M\to\infty}M\beta_{T,w}(M)=\frac\pi4.
\beta_{\widetilde T,\widetilde r}Use in source · 3 passages
Search aliases: β_widetilde T,widetilde r beta widetilde widetilde
Use in source · 05g_budget_certificates.tex, line 139
How measurement and model uncertainty affect the budget. [A robust budget profile] Suppose the bounded linear map widetilde T:H→ Y and the datum widetilde r∈ Y satisfy normT-widetilde T≤η, normr-widetilde r≤δ, η,δ≥0. For every M≥0, absβ_T,r(M)-β_widetilde T,widetilde r(M) ≤δ+η M. In particular, for norm z≤1, β_T,r(M)≥ ⟨widetilde r,z⟩-Mnormwidetilde T^*z -δ-η M. Fix a feasible correction a. The two residual vectors have difference (r-Ta)-(widetilde r-widetilde Ta) =(r-widetilde r)-(T-widetilde T)…
Exact mathematical passage
\abs{\beta_{T,r}(M)-\beta_{\widetilde T,\widetilde r}(M)} \le\delta+\eta M.
\beta_xUse in source · 3 passages
Search aliases: β_x beta
Use in source · 04_symmetry.tex, line 423
[Invariance of the marked realization problem] The map (x,ϕ)⟼(Ex,K(ϕ)σ_x^-1), a⟼ E(a) defines an equivalence E_b(R)≃ E_Kb(R'). For an object (x,ϕ) put β_x=K(ϕ)σ_x^-1:R'Ex→ Kb. This is an isomorphism of the required type. If a:(x,ϕ)→(y,ψ), naturality of σ says R'E(a)σ_x=σ_yKR(a), σ_y^-1R'E(a)=KR(a)σ_x^-1. Consequently β_yR'E(a) =K(ψ)KR(a)σ_x^-1 =K(ψ R(a))σ_x^-1 =K(ϕ)σ_x^-1=β_x. So E(a) is a marked arrow. Identity and composition are preserved because the underlying arrow map is that of the functo…
Exact mathematical passage
\beta_x=K(\phi)\sigma_x^{-1}:R'Ex\to Kb
\beta_yUse in source · 1 passage
Search aliases: β_y beta
Use in source · 04_symmetry.tex, line 429
), a⟼ E(a) defines an equivalence E_b(R)≃ E_Kb(R'). For an object (x,ϕ) put β_x=K(ϕ)σ_x^-1:R'Ex→ Kb. This is an isomorphism of the required type. If a:(x,ϕ)→(y,ψ), naturality of σ says R'E(a)σ_x=σ_yKR(a), σ_y^-1R'E(a)=KR(a)σ_x^-1. Consequently β_yR'E(a) =K(ψ)KR(a)σ_x^-1 =K(ψ R(a))σ_x^-1 =K(ϕ)σ_x^-1=β_x. So E(a) is a marked arrow. Identity and composition are preserved because the underlying arrow map is that of the functor E.
Exact mathematical passage
\beta_yR'E(a) =K(\psi)KR(a)\sigma_x^{-1} =K(\psi R(a))\sigma_x^{-1} =K(\phi)\sigma_x^{-1}=\beta_x.
\Gamma_{U_i}Use in source · 1 passage
Search aliases: Γ_U_i Gamma
Use in source · 01_views.tex, line 93
Let (U_i)_i∈ I be full patches. A matching family consists of s_i∈Γ_U_i(F) with equal values on every overlap. The relevant covering condition is for every f:v→ w, some U_i contains both v and w. Identities imply that this condition also covers every object.
Exact mathematical passage
s_i\in\Gamma_{U_i}(F)
\Gamma_WUse in source · 3 passages
Search aliases: Γ_W Gamma
Use in source · 02_obstructions.tex, line 145
Extension in stages. For U⊆ W⊆ObV, an extension first to W and then globally must retain the intermediate choice. There is an explicit equivalence Ext_U(s)≃ coprod_substackt∈Γ_W(A) t|_U=sExt_W(t). It sends a global extension to its restriction to W, together with itself as an extension of that restriction. The inverse forgets the intermediate coordinate. These operations are inverse because the required intermediate coordinate of a global section is its actual restriction. A coherent intermediat…
Exact mathematical passage
\label{eq:stages} \Ext_U(s)\simeq \coprod_{\substack{t\in\Gamma_W(A)\\t|_U=s}}\Ext_W(t).
\DeltaUse in source · 1 passage
Search aliases: Δ Delta
Use in source · 05_geometry.tex, line 334
[The two-stage repair is the joint projection] The operator P is the orthogonal projection onto M, equivalently onto the kernel of Δ=(D,C). If u=x-P_Lx and v=P_Lx-Px, then u⊥ v, normx-Px^2=norm u^2+norm v^2. The subspace L=ker D is closed, hence Hilbert with the inherited inner product. The restriction B=C|_L is continuous, so its kernel is a closed subspace of L. Under the inclusion into E, this kernel is exactly M=L∩ker C. Thus both projections used to define P exist.
Exact mathematical passage
\Delta=(D,C)
\delta^{-2/3}Use in source · 5 passages
Search aliases: δ^-2/3 delta
Use in source · 05i_diagonal_regularization.tex, line 155
[An explicit noisy attainable repair] For the attainable datum b=(1/n^2), suppose normwidetilde b-b≤δ with 0<δ≤1. Choose m=lceilδ^-2/3rceil, α=m^-2. Then the noisy regularized correction satisfies norma_α(widetilde b)-a^† ≤sqrtfrac2m+fracδ m2 ≤(sqrt2+1)δ^1/3. The exact-data bias has coordinates n/(m^2+n^2), so its squared norm is precisely B_m. By Proposition~[source reference], norma_m^-2(b)-a^†≤sqrt2/m. Theorem~[source reference] at the same parameter gives norma_m^-2(widetilde b)-a_m^-2(b) ≤f…
Exact mathematical passage
m=\lceil\delta^{-2/3}\rceil,\qquad\alpha=m^{-2}.
\delta^{1/3}Use in source · 6 passages
Search aliases: δ^1/3 delta
Use in source · 05i_diagonal_regularization.tex, line 159
[An explicit noisy attainable repair] For the attainable datum b=(1/n^2), suppose normwidetilde b-b≤δ with 0<δ≤1. Choose m=lceilδ^-2/3rceil, α=m^-2. Then the noisy regularized correction satisfies norma_α(widetilde b)-a^† ≤sqrtfrac2m+fracδ m2 ≤(sqrt2+1)δ^1/3. The exact-data bias has coordinates n/(m^2+n^2), so its squared norm is precisely B_m. By Proposition~[source reference], norma_m^-2(b)-a^†≤sqrt2/m. Theorem~[source reference] at the same parameter gives norma_m^-2(widetilde b)-a_m^-2(b) ≤f…
Exact mathematical passage
\norm{a_\alpha(\widetilde b)-a^\dagger} \le\sqrt{\frac2m}+\frac{\delta m}{2} \le(\sqrt2+1)\delta^{1/3}.
\delta^2Use in source · 1 passage
Search aliases: δ^2 delta
Use in source · 05h_regularized_repair.tex, line 229
e=widetilde r-r. Subtract the two normal equations to obtain (T^*T+α I)h=T^*e. Pair with h and use the adjoint identity: normTh^2+αnorm h^2 =⟨ e,Th⟩≤δnormTh. Put t=normTh≥0. If t>0, dropping the nonnegative penalty and dividing by t gives t≤δ. If t=0, that inequality is already true. Completing the scalar square gives αnorm h^2≤δ t-t^2 =fracδ^24-(t-δ/2)^2 ≤fracδ^24. Because α>0, taking square roots proves the correction bound. This also explains why the coefficient is 1/(2sqrtα) rather than the …
Exact mathematical passage
\alpha\norm h^2\le\delta t-t^2 =\frac{\delta^2}{4}-(t-\delta/2)^2 \le\frac{\delta^2}{4}.
\delta^{4/3}Use in source · 1 passage
Search aliases: δ^4/3 delta
Use in source · 05k_stopping_rules.tex, line 136
lceilδ^-2/3rceil and α=m^-2 as before. If an inexact solver satisfies ε≤καδ^1/3 (κ≥0), then the same argument gives norma-a^†≤(sqrt2+1+κ)δ^1/3. Thus the inner solve can retain the proven outer accuracy if its normal residual decreases on the scale αδ^1/3. Since α is on the scale δ^4/3, the corresponding normal-residual scale is δ^5/3, with the stated ceiling factors retained in the exact rule. This is a sufficient rule for this problem, not a universal optimal stopping law.
Exact mathematical passage
\delta^{4/3}
\delta^{5/3}Use in source · 1 passage
Search aliases: δ^5/3 delta
Use in source · 05k_stopping_rules.tex, line 136
an inexact solver satisfies ε≤καδ^1/3 (κ≥0), then the same argument gives norma-a^†≤(sqrt2+1+κ)δ^1/3. Thus the inner solve can retain the proven outer accuracy if its normal residual decreases on the scale αδ^1/3. Since α is on the scale δ^4/3, the corresponding normal-residual scale is δ^5/3, with the stated ceiling factors retained in the exact rule. This is a sufficient rule for this problem, not a universal optimal stopping law.
Exact mathematical passage
\delta^{5/3}
\delta_nUse in source · 3 passages
Search aliases: δ_n delta
Use in source · 05k_stopping_rules.tex, line 92
[A consistent inexact stopping rule] Assume the projected intended datum belongs to the actual range of T, with minimum-norm preimage a^†. Along a sequence of positive parameters and measured problems, suppose α_n→0, fracδ_nsqrtα_n→0, fracε_nα_n→0. If the proposed corrections a_n∈ H satisfy the displayed data and normal-residual bounds, then a_n→ a^† in norm. At every index, norma_n-a^† ≤fracε_nα_n +fracδ_n2sqrtα_n +norma_α_n(r)-a^†. At index n, distinguish two exact minimizers: let widetilde a_…
Exact mathematical passage
\alpha_n\to0,\qquad \frac{\delta_n}{\sqrt{\alpha_n}}\to0,\qquad \frac{\varepsilon_n}{\alpha_n}\to0.
\epsilonUse in source · 17 passages
Search aliases: ϵ epsilon
Use in source · 03_towers.tex, line 399
Conversely, fix x∈overline F and ϵ>0. If K=0, the common operator bound gives normQ_Nx=0 for every N, so convergence is immediate. Suppose K>0. Choose y∈ F satisfying normx-y<ϵ/(K+1). Some M has Q_My=0. For every N≥ M, nesting and the common operator bound give normQ_Nx=normQ_N(x-y)≤ Knormx-y <ϵfracKK+1≤ϵ. The approximation y is chosen before the depth N, giving precisely the quantifier order in norm convergence. This proves equality of the two sets. Applying it to every x∈ R_0 gives the density…
Exact mathematical passage
\epsilon>0
\varepsilonUse in source · 79 passages
Search aliases: ε varepsilon
Use in source · 04_symmetry.tex, line 339
chapEquivalent presentations and identified boundaries Transporting the same question. Let H:V→W be part of a category equivalence, with inverse L and counit ε:HL⇒id_W. Pull a view system F on W back along H. Sampling a global section gives (H^*x)_v=x_Hv.
Exact mathematical passage
\varepsilon:HL\Rightarrow\id_\W
\epsilon^2Use in source · 3 passages
Search aliases: ϵ^2 epsilon
Use in source · 05f_bounded_realization.tex, line 129
Suppose these energies are bounded, and let L=lim_n m_n^2. Given ϵ>0, choose N with L-m_N^2<ϵ^2. For m≥ n≥ N, norma_m-a_n^2=m_m^2-m_n^2≤ L-m_N^2<ϵ^2. Thus (a_n) is Cauchy and has a norm limit a_*. For each fixed n, all terms from that index onward belong to the closed set A_n, so a_*∈ A_n. This places a_* in their entire intersection.
Exact mathematical passage
L-m_N^2<\epsilon^2
\varepsilon^2Use in source · 2 passages
Search aliases: ε^2 varepsilon
Use in source · 05_geometry.tex, line 461
Now write (q-q')x=q(I-q')x-(I-q)q'x. The two displayed terms lie in ker D and (ker D)^⊥, respectively, so they are orthogonal. The input components (I-q')x and q'x are also orthogonal. Applying the dual estimate to the first and the original cross estimate to the second gives norm(q-q')x^2 · =normq(I-q')x^2+norm(I-q)q'x^2 · ≤ (C'ε)^2norm(I-q')x^2 +(Cε)^2normq'x^2 · ≤ε^2max(C,C')^2 (norm(I-q')x^2+normq'x^2) · =ε^2max(C,C')^2norm x^2. The bound normD-D'≤ε implies ε≥0, so taking square roots yields…
Exact mathematical passage
\begin{aligned} \norm{(q-q')x}^2 &=\norm{q(I-q')x}^2+\norm{(I-q)q'x}^2\\ &\le (C'\varepsilon)^2\norm{(I-q')x}^2 +(C\varepsilon)^2\norm{q'x}^2\\ &\le\varepsilon^2\max(C,C')^2 \bigl(\norm{(I-q')x}^2+\norm{q'x}^2\bigr)\\ &=\varepsilon^2\max(C,C')^2\norm x^2. \end{aligned}
\varepsilon_CUse in source · 3 passages
Search aliases: ε_C varepsilon
Use in source · 05_geometry.tex, line 627
Motion of the actual descent repair. Compare pairs (D,C) and (D',C') on the same raw Hilbert space. Suppose each pair has closed local range and closed overlap range on its own locally exact subspace. Let G,G' be the resulting joint residual gains. If normD-D'≤ε_D, normC-C'≤ε_C, then the maximum norm on the target product gives norm(D,C)-(D',C')≤max(ε_D,ε_C). Applying Theorem~[source reference] to the joint operators, and using the actual two-stage identity of Theorem~[source reference], gives n…
Exact mathematical passage
\norm{D-D'}\le\varepsilon_D,\qquad \norm{C-C'}\le\varepsilon_C,
\varepsilon_DUse in source · 3 passages
Search aliases: ε_D varepsilon
Use in source · 05_geometry.tex, line 627
Motion of the actual descent repair. Compare pairs (D,C) and (D',C') on the same raw Hilbert space. Suppose each pair has closed local range and closed overlap range on its own locally exact subspace. Let G,G' be the resulting joint residual gains. If normD-D'≤ε_D, normC-C'≤ε_C, then the maximum norm on the target product gives norm(D,C)-(D',C')≤max(ε_D,ε_C). Applying Theorem~[source reference] to the joint operators, and using the actual two-stage identity of Theorem~[source reference], gives n…
Exact mathematical passage
\norm{D-D'}\le\varepsilon_D,\qquad \norm{C-C'}\le\varepsilon_C,
\varepsilon_nUse in source · 3 passages
Search aliases: ε_n varepsilon
Use in source · 05k_stopping_rules.tex, line 92
[A consistent inexact stopping rule] Assume the projected intended datum belongs to the actual range of T, with minimum-norm preimage a^†. Along a sequence of positive parameters and measured problems, suppose α_n→0, fracδ_nsqrtα_n→0, fracε_nα_n→0. If the proposed corrections a_n∈ H satisfy the displayed data and normal-residual bounds, then a_n→ a^† in norm. At every index, norma_n-a^† ≤fracε_nα_n +fracδ_n2sqrtα_n +norma_α_n(r)-a^†. At index n, distinguish two exact minimizers: let widetilde a_…
Exact mathematical passage
\alpha_n\to0,\qquad \frac{\delta_n}{\sqrt{\alpha_n}}\to0,\qquad \frac{\varepsilon_n}{\alpha_n}\to0.
\varepsilon_w^{-1}Use in source · 1 passage
Search aliases: ε_w^-1 varepsilon
Use in source · 04_symmetry.tex, line 349
simeqΓ(H^*F). If U⊆ObW is closed under isomorphism and U'=H^-1U, it also induces Ext_U^F(s)≃Ext_U'^H^*F(H^*s). Existence and uniqueness of the prescribed extension are preserved and reflected. Write the unit of the equivalence as η:I_V⇒ LH, and take the unit and counit with their triangle identities. Set t_w=ε_w^-1:w→ HLw. Given y∈Γ(H^*F), define (Ry)_w=F(t_w)y_Lw. For an arrow f:w→ w', naturality of the inverse counit states HL(f) t_w=t_w'f. Contravariance of F and coherence of y therefore give…
Exact mathematical passage
t_w=\varepsilon_w^{-1}:w\to HLw
\etaUse in source · 39 passages
Search aliases: η eta
Use in source · 03a_observation.tex, line 165
[Cofinal invariance of eventual transports] Sampling and recovery are inverse in the eventual transport category. Consequently, for monotone unbounded schedules θ and η, there is a bijection EventualHom(X,Y) ≃ EventualHom(X_θ,Y_η). On point spaces it agrees with the previously constructed cofinal realization equivalences. Let s:X→ X_θ be sampling, with schedule θ and identity level maps. For recovery choose the least j(n) with n≤θ(j(n)). Unboundedness makes this set nonempty. Increasing n only s…
Exact mathematical passage
\eta
\eta_{2n}Use in source · 1 passage
Search aliases: η_2n eta
Use in source · 05c_iteration.tex, line 191
For failure without monotonicity, take η_2n=0, η_2n+1=1, and ϕ(n)=2n. All sampled values vanish, but infinitely many original values equal one. sourceKS75 Even in the decreasing case, the numerical rate changes. The profile η_n=1/(n+1), sampled at ϕ(n)=2^n-1, becomes 2^-n. This is exponential decay in the number of observations, not a proof of a faster correction process at the original resolution scale.
Exact mathematical passage
\eta_{2n}=0
\eta_{2n+1}Use in source · 1 passage
Search aliases: η_2n+1 eta
Use in source · 05c_iteration.tex, line 191
For failure without monotonicity, take η_2n=0, η_2n+1=1, and ϕ(n)=2n. All sampled values vanish, but infinitely many original values equal one. sourceKS75 Even in the decreasing case, the numerical rate changes. The profile η_n=1/(n+1), sampled at ϕ(n)=2^n-1, becomes 2^-n. This is exponential decay in the number of observations, not a proof of a faster correction process at the original resolution scale.
Exact mathematical passage
\eta_{2n+1}=1
\eta_nUse in source · 5 passages
Search aliases: η_n eta
Use in source · 05c_iteration.tex, line 179
Changing the sampling schedule does not preserve a rate. A residual profile η_n indexed by resolution is not an iteration trajectory. Nevertheless a useful convergence property is invariant under a cofinal schedule when the profile is monotone.
Exact mathematical passage
\eta_n
\eta_vUse in source · 3 passages
Search aliases: η_v eta
Use in source · 04_symmetry.tex, line 372
If x∈Γ(F), coherence along t_w gives (RH^*x)_w=F(t_w)x_HLw=x_w. For the other composite, the triangle identity says t_Hv=H(η_v). Coherence of y along η_v:v→ LHv yields (H^*Ry)_v=F(t_Hv)y_LHv=F(Hη_v)y_LHv=y_v. These coordinate equations prove both inverse identities.
Exact mathematical passage
t_{Hv}=H(\eta_v)
\eta_{\phi(k)}Use in source · 1 passage
Search aliases: η_ϕ(k) eta phi
Use in source · 09_workshop.tex, line 51
Solution. The sampled value is η_ϕ(k)=2^-k. Recovering the original index gives k=log_2(n+1) on the sampled stages, so the value is still 1/(n+1). The schedule preserves vanishing but changes the meaning of the horizontal axis. No faster original process has been proved. Proposition~[source reference] is a convergence equivalence under a monotone cofinal schedule, not a rate-invariance theorem.
Exact mathematical passage
\eta_{\phi(k)}=2^{-k}
\eta_{\phi(n)}Use in source · 1 passage
Search aliases: η_ϕ(n) eta phi
Use in source · 05c_iteration.tex, line 183
[Cofinal sampling of a decreasing profile] Suppose η_n≥0 is nonincreasing and ϕ:N→N is monotone and unbounded. Then η_n→0 Longleftrightarrow η_ϕ(n)→0. Without the monotonicity of η, the reverse implication can fail even for a cofinal schedule. The schedule tends to infinity, so convergence of the full profile implies convergence of the sampled one. Conversely, choose a sampled value below any prescribed ε>0. Every later index of the original profile has no larger value, by monotonicity. This pro…
Exact mathematical passage
\eta_n\to0\quad\Longleftrightarrow\quad\eta_{\phi(n)}\to0.
\ThetaUse in source · 3 passages
Search aliases: Θ Theta
Use in source · 02_obstructions.tex, line 90
[Classification of a nonempty extension fiber] If x_0∈Ext_U(s), translation gives an equivalence H_U≃Ext_U(s), h⟼ x_0+h. Consequently the extension is unique exactly when o_U(s)=0 and H_U=0. Define Θ:H_U→Ext_U(s) by Θ(h)=x_0+h. It lands in the extension set because δ(x_0+h)=δ x_0+δ h=0, p_U(x_0+h)=s+0=s. Define Ψ:Ext_U(s)→ H_U by Ψ(x)=x-x_0. Its differential is zero, and its restriction is s-s=0, so the claimed target is correct. The formulas give ΨΘ(h)=x_0+h-x_0=h, ΘΨ(x)=x_0+(x-x_0)=x. This pro…
Exact mathematical passage
\Theta:H_U\to\Ext_U(s)
\thetaUse in source · 33 passages
Search aliases: θ theta
Use in source · 03_towers.tex, line 190
[Split tower correction] The recurrence defines a continuous linear map A:R→ E with ∂ A=I_R, norm A≤ K. The operator P=I-A∂ is an exact projection and (Pf)_0=f_0, normf-Pf_∞≤ Knorm∂ f_∞. Let θ=SB_X<1. The number K=S/(1-θ) is nonnegative and satisfies S+SB_XK=K. Fix r∈ R and define its coordinates recursively as in [source reference]. The initial coordinate has norm zero. If norm(Ar)_n≤ Knorm r_∞, submultiplicativity of operator norms gives norm(Ar)_n+1 · ≤norms_n(normr_n+norm(Ar)_nnormρ^X_n)
Exact mathematical passage
\theta=SB_X<1
\theta^jUse in source · 3 passages
Search aliases: θ^j theta
Use in source · 12_proof_tools.tex, line 167
Submultiplicativity gives normT^j≤θ^j, with the convention T^0=I. The finite partial sums are therefore operator-norm Cauchy, since for M>N, ∑_j=N+1^MT^j ≤∑_j=N+1^Mθ^j≤fracθ^N+11-θ. Their limit Z has norm at most ∑_j≥0θ^j=1/(1-θ). The operator identities for finite sums are (I-T)∑_j=0^NT^j=I-T^N+1, (∑_j=0^NT^j)(I-T)=I-T^N+1. Multiplication by a fixed bounded operator is continuous in operator norm, and normT^N+1≤θ^N+1→0. Passing to the limit proves both inverse identities. When W is the zero spa…
Exact mathematical passage
\norm{T^j}\le\theta^j
\theta^{N+1}Use in source · 2 passages
Search aliases: θ^N+1 theta
Use in source · 12_proof_tools.tex, line 168
Submultiplicativity gives normT^j≤θ^j, with the convention T^0=I. The finite partial sums are therefore operator-norm Cauchy, since for M>N, ∑_j=N+1^MT^j ≤∑_j=N+1^Mθ^j≤fracθ^N+11-θ. Their limit Z has norm at most ∑_j≥0θ^j=1/(1-θ). The operator identities for finite sums are (I-T)∑_j=0^NT^j=I-T^N+1, (∑_j=0^NT^j)(I-T)=I-T^N+1. Multiplication by a fixed bounded operator is continuous in operator norm, and normT^N+1≤θ^N+1→0. Passing to the limit proves both inverse identities. When W is the zero spa…
Exact mathematical passage
\left\|\sum_{j=N+1}^{M}T^j\right\| \le\sum_{j=N+1}^{M}\theta^j\le\frac{\theta^{N+1}}{1-\theta}.
\lambda_pUse in source · 6 passages
Search aliases: λ_p lambda
Use in source · 07_classification.tex, line 61
[Complete homomorphism description] Every homomorphism f:G_A→ G_B has the form f(g)_p=λ_p g_p (p prime) for a unique λ∈ P. Conversely, coordinate multiplication by λ defines such a homomorphism exactly when q(λ u_A)∈ Q q(u_B). Equivalently, there is a rational r for which λ u_A-s_B(r) has finite support. For a homomorphism f, the composite G_A→ G_B→ C_p kills pG_A. The quotient G_A/pG_A is canonically C_p. Every endomorphism of this cyclic additive group is multiplication by a unique scalar λ_p.…
Exact mathematical passage
f(g)_p=\lambda_p g_p\qquad(p\text{ prime})
\lambda_p^{-1}Use in source · 1 passage
Search aliases: λ_p^-1 lambda
Use in source · 07_classification.tex, line 113
[Automorphisms] An endomorphism with diagonal λ∈ E_A is an automorphism if and only if every coordinate λ_p is nonzero. In that case its rational quotient scalar is nonzero, and its inverse has diagonal λ_p^-1. An automorphism restricts to an automorphism of each torsion factor, so every scalar is nonzero. Conversely, if every scalar is nonzero, the eventual rational scalar cannot be zero because S_A is infinite. Outside finitely many selected primes, the inverse diagonal is reduction of its rec…
Exact mathematical passage
\lambda_p^{-1}
\nu_2Use in source · 2 passages
Search aliases: ν_2 nu
Use in source · 03_towers.tex, line 452
[The dyadic support pattern] For the same lifts, 2^Nnmid(k+1) Longrightarrow (Q_Nr)_k=0. Thus coordinate k is certainly recovered once N>ν_2(k+1), where ν_2(m) is the largest power-of-two exponent dividing the positive integer m. The only possible unrecovered coordinates at depth N are 2^N-1, 2·2^N-1,quad3·2^N-1, …. Prove the assertion for L_j,N by induction, uniformly in j. At depth zero every positive integer is divisible by one, so there is nothing to prove. At depth N+1, even coordinates van…
Exact mathematical passage
N>\nu_2(k+1)
\nu_{B_n}Use in source · 2 passages
Search aliases: ν_B_n nu
Use in source · 05d_floors.tex, line 55
[Uniformly positive floors obstruct asymptotic repair] Suppose c>0 and eventually c≤ν_B_n(Ω_n). For any choices b_n∈ B_n, it is impossible that normΩ_n-b_n⟶0. Every allowed correction satisfies normΩ_n-b_n≥ν_B_n(Ω_n). The corrected norms are therefore eventually at least c. They cannot converge to zero. sourceKS77 The quantifier covers all admissible sequences, not just the output of one repair routine. On the other hand, the hypothesis is an eventually uniform positive lower bound. Having a non…
Exact mathematical passage
c\le\nu_{B_n}(\Omega_n).
\pi_mUse in source · 1 passage
Search aliases: π_m pi
Use in source · 06_roots.tex, line 58
The quotient formula. [All integer quotients] For m≥1, the coordinate map π_m:G_A⟶∏_pmid mC_p is surjective with kernel mG_A. Thus G_A/mG_A≅∏_pmid mC_p≅ C_rad(m). For mmid n, the transition map is the corresponding coordinate projection. A vector supported on the prime divisors of m realizes any tuple in the target, proving surjectivity. The root criterion identifies the kernel. The Chinese remainder theorem identifies the finite product with the squarefree cyclic group. The transition assertion…
Exact mathematical passage
\pi_m:G_A\longrightarrow\prod_{p\mid m}C_p
\pi_nUse in source · 3 passages
Search aliases: π_n pi
Use in source · 03_towers.tex, line 276
A growing example and the gap boundary. Use X_n=Y_n=R^n+1 with maximum norms. Let π_n delete the last coordinate, take ρ^X_n=tfrac12π_n, ρ^Y_n=π_n, and let s_n append zero. Then S=B_Y=1, B_X=1/2, and K=2.
Exact mathematical passage
\pi_n
\pi_UUse in source · 2 passages
Search aliases: π_U pi
Use in source · 02_obstructions.tex, line 49
[Extension criterion] For every coherent partial section, o_U(s)=0 Longleftrightarrow Ext_U(s)≠∅. Let π_U:R→ R/B_U be the quotient homomorphism. By the definition of a quotient by a subgroup, π_U(r)=0 Longleftrightarrow r∈ B_U. Here B_U is the image of ker p_U under δ. Consequently membership of δwidetilde s means the existence of a specific h∈ E satisfying both p_Uh=0 and δ h=δwidetilde s. The subgroup need not be closed in any topology. No topology has been introduced.
Exact mathematical passage
\pi_U:R\to R/B_U
\pi_{U,U}Use in source · 1 passage
Search aliases: π_U,U pi
Use in source · 02a_prescription.tex, line 30
[Functorial forgetting of protected views] The homomorphism π_W,U is well defined and surjective. For U⊆ W⊆ Z, it satisfies π_U,U=I, π_W,Uπ_Z,W=π_Z,U. For every s∈Γ_W(A), π_W,U o_W(s)= o_U(s|_U). In particular an obstructed restriction remains obstructed when more compatible prescribed data are retained. A successful extension on the larger prescribed domain gives a successful extension of its restriction. If r-r'∈ B_W, then r-r'∈ B_U, proving well-definedness. Every class modulo B_U has a repre…
Exact mathematical passage
\pi_{U,U}=I,\qquad \pi_{W,U}\pi_{Z,W}=\pi_{Z,U}.
\pi_{Z,U}Use in source · 1 passage
Search aliases: π_Z,U pi
Use in source · 02a_prescription.tex, line 30
[Functorial forgetting of protected views] The homomorphism π_W,U is well defined and surjective. For U⊆ W⊆ Z, it satisfies π_U,U=I, π_W,Uπ_Z,W=π_Z,U. For every s∈Γ_W(A), π_W,U o_W(s)= o_U(s|_U). In particular an obstructed restriction remains obstructed when more compatible prescribed data are retained. A successful extension on the larger prescribed domain gives a successful extension of its restriction. If r-r'∈ B_W, then r-r'∈ B_U, proving well-definedness. Every class modulo B_U has a repre…
Exact mathematical passage
\pi_{U,U}=I,\qquad \pi_{W,U}\pi_{Z,W}=\pi_{Z,U}.
\pi_{Z,W}Use in source · 1 passage
Search aliases: π_Z,W pi
Use in source · 02a_prescription.tex, line 30
[Functorial forgetting of protected views] The homomorphism π_W,U is well defined and surjective. For U⊆ W⊆ Z, it satisfies π_U,U=I, π_W,Uπ_Z,W=π_Z,U. For every s∈Γ_W(A), π_W,U o_W(s)= o_U(s|_U). In particular an obstructed restriction remains obstructed when more compatible prescribed data are retained. A successful extension on the larger prescribed domain gives a successful extension of its restriction. If r-r'∈ B_W, then r-r'∈ B_U, proving well-definedness. Every class modulo B_U has a repre…
Exact mathematical passage
\pi_{U,U}=I,\qquad \pi_{W,U}\pi_{Z,W}=\pi_{Z,U}.
\rho^X_{2n+1}Use in source · 1 passage
Search aliases: ρ^X_2n+1 rho
Use in source · 03_towers.tex, line 313
The actual dyadic tower coarsener has the typed formula (Cr)_n=ρ^Y_2nr_2n+1+r_2nρ^X_2n+1. It is obtained by telescoping two adjacent compatibility squares. With target sections, a lift is (Jc)_2n=0, (Jc)_2n+1=s_2nc_n. The section identity gives CJ=I, so the missing detail is explicitly Hr, not an unspecified reconstruction error. The normed construction imposes the bounds needed to make these maps bounded on its operator family spaces. sourceKS25
Exact mathematical passage
\label{eq:dyadic} (Cr)_n=\rho^Y_{2n}r_{2n+1}+r_{2n}\rho^X_{2n+1}.
\rho^X_{a,b}Use in source · 2 passages
Search aliases: ρ^X_a,b rho
Use in source · 03b_repair_transport.tex, line 148
a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a,b. The dyadic coarsener in Section~[source reference] is the case b=a+2.
Exact mathematical passage
C_{a,b}(\partial f)=\rho^Y_{a,b}f_b-f_a\rho^X_{a,b}.
\rho^X_kUse in source · 1 passage
Search aliases: ρ^X_k rho
Use in source · 03b_repair_transport.tex, line 169
To verify the commuting square, insert r_k=ρ^Y_kf_k+1-f_kρ^X_k. The k-th term becomes ρ^Y_a,k+1f_k+1ρ^X_k+1,b -ρ^Y_a,kf_kρ^X_k,b. Successive internal terms cancel in the finite sum, leaving ρ^Y_a,bf_b-f_aρ^X_a,b. This is exactly the sampled differential. No infinite series interchange is involved.
Exact mathematical passage
r_k=\rho^Y_kf_{k+1}-f_k\rho^X_k
\rho^X_{k+1,b}Use in source · 2 passages
Search aliases: ρ^X_k+1,b rho
Use in source · 03b_repair_transport.tex, line 143
For a block from a to b, its value on a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a,b. The dyadic coarsener in Section~[source reference] is the case b=a+2.
Exact mathematical passage
\label{eq:block-defect-sum} (C_{a,b}r)=\sum_{k=a}^{b-1} \rho^Y_{a,k}\,r_k\,\rho^X_{k+1,b}.
\rho^X_{k,b}Use in source · 1 passage
Search aliases: ρ^X_k,b rho
Use in source · 03b_repair_transport.tex, line 170
To verify the commuting square, insert r_k=ρ^Y_kf_k+1-f_kρ^X_k. The k-th term becomes ρ^Y_a,k+1f_k+1ρ^X_k+1,b -ρ^Y_a,kf_kρ^X_k,b. Successive internal terms cancel in the finite sum, leaving ρ^Y_a,bf_b-f_aρ^X_a,b. This is exactly the sampled differential. No infinite series interchange is involved.
Exact mathematical passage
\rho^Y_{a,k+1}f_{k+1}\rho^X_{k+1,b} -\rho^Y_{a,k}f_k\rho^X_{k,b}.
\rho^X_{k,\ell}Use in source · 1 passage
Search aliases: ρ^X_k,ℓ rho ell
Use in source · 03a_observation.tex, line 34
ality is an equivalence relation and is preserved by composition. Its quotient therefore gives a category of resolutions and eventual transports. Realization descends to a functor into sets, preserving identities and composition. A finite-stage equality remains true after moving to a later stage. Indeed, if it holds on X_k and ℓ≥ k, precompose both sides with ρ^X_k,ℓ and use the composite-restriction laws. Reflexivity uses k=ϕ(n), symmetry reverses the equality, and transitivity moves two witnes…
Exact mathematical passage
\rho^X_{k,\ell}
\rho^X_{k_n,\phi(n)}Use in source · 2 passages
Search aliases: ρ^X_k_n,ϕ(n) rho phi
Use in source · 03a_observation.tex, line 117
For surjectivity, let F be finitely observed. Choose one pair (k_n,a_n) at each output level, satisfying F(x)_n=a_n(x_k_n) for every coherent input. Define ϕ(n)=max_0≤ i≤ nk_i, f_n=a_nρ^X_k_n,ϕ(n). The maximum is of a nonempty finite set. It gives a monotone schedule with k_n≤ϕ(n). For every point x, coherence in X gives f_n(x_ϕ(n)) =a_n(ρ^X_k_n,ϕ(n)x_ϕ(n)) =a_n(x_k_n)=F(x)_n. tag*
Exact mathematical passage
\phi(n)=\max_{0\le i\le n}k_i, \qquad f_n=a_n\rho^X_{k_n,\phi(n)}.
\rho^X_nUse in source · 11 passages
Search aliases: ρ^X_n rho
Use in source · 03_towers.tex, line 12
A candidate transport from X to Y consists of maps f_n:X_n→ Y_n. It is exact when ρ^Y_n f_n+1=f_nρ^X_n for all n.
Exact mathematical passage
\label{eq:exacttower} \rho^Y_n f_{n+1}=f_n\rho^X_n\quad\text{for all }n.
\rho^X_{n,\theta(j(n))}Use in source · 1 passage
Search aliases: ρ^X_n,θ(j(n)) rho theta
Use in source · 03a_observation.tex, line 174
with the previously constructed cofinal realization equivalences. Let s:X→ X_θ be sampling, with schedule θ and identity level maps. For recovery choose the least j(n) with n≤θ(j(n)). Unboundedness makes this set nonempty. Increasing n only strengthens the inequality, so j is monotone. The level map of r:X_θ→ X is ρ^X_n,θ(j(n)). Composite restriction proves its naturality.
Exact mathematical passage
\rho^X_{n,\theta(j(n))}
\rho^X_{\theta(n),\theta(j(\theta(n)))}Use in source · 1 passage
Search aliases: ρ^X_θ(n),θ(j(θ(n))) rho theta theta theta
Use in source · 03a_observation.tex, line 178
For sr:X_θ→ X_θ, the schedule at output n is j(θ(n)), and the level map is ρ^X_θ(n),θ(j(θ(n))). Compare it with the identity at a sampled index k≥max{n,j(θ(n))}. Restricting either route from level k of the sampled system gives the same original restriction ρ^X_θ(n),θ(k). This proves eventual equality even when θ has repeated values. Hence the two classes are inverse isomorphisms, without requiring them to be literal inverse schedules.
Exact mathematical passage
\rho^X_{\theta(n),\theta(j(\theta(n)))}
\rho^X_{\theta(n),\theta(k)}Use in source · 1 passage
Search aliases: ρ^X_θ(n),θ(k) rho theta theta
Use in source · 03a_observation.tex, line 182
For sr:X_θ→ X_θ, the schedule at output n is j(θ(n)), and the level map is ρ^X_θ(n),θ(j(θ(n))). Compare it with the identity at a sampled index k≥max{n,j(θ(n))}. Restricting either route from level k of the sampled system gives the same original restriction ρ^X_θ(n),θ(k). This proves eventual equality even when θ has repeated values. Hence the two classes are inverse isomorphisms, without requiring them to be literal inverse schedules.
Exact mathematical passage
\rho^X_{\theta(n),\theta(k)}
\rho^X_{\phi'(j),k}Use in source · 2 passages
Search aliases: ρ^X_ϕ'(j),k rho phi
Use in source · 03a_observation.tex, line 41
We check compatibility with composition with all indices displayed. Write f,f' with schedules ϕ,ϕ' and g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applying the naturality equations and then the two witness equations gives g_nf_ψ(n)ρ^X_ϕ(ψ(n)),k · =g_nρ^Y_ψ(n),jf_jρ^X_ϕ(j),k
Exact mathematical passage
f_j\rho^X_{\phi(j),k}=f'_j\rho^X_{\phi'(j),k}.
\rho^X_{\phi'(\psi'(n)),k}Use in source · 1 passage
Search aliases: ρ^X_ϕ'(ψ'(n)),k rho phi psi
Use in source · 03a_observation.tex, line 45
nd g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applying the naturality equations and then the two witness equations gives g_nf_ψ(n)ρ^X_ϕ(ψ(n)),k · =g_nρ^Y_ψ(n),jf_jρ^X_ϕ(j),k · =g'_nρ^Y_ψ'(n),jf_jρ^X_ϕ(j),k · =g'_nρ^Y_ψ'(n),jf'_jρ^X_ϕ'(j),k · =g'_nf'_ψ'(n)ρ^X_ϕ'(ψ'(n)),k. This is the required equality o…
Exact mathematical passage
\begin{aligned} g_nf_{\psi(n)}\rho^X_{\phi(\psi(n)),k} &=g_n\rho^Y_{\psi(n),j}f_j\rho^X_{\phi(j),k}\\ &=g'_n\rho^Y_{\psi'(n),j}f_j\rho^X_{\phi(j),k}\\ &=g'_n\rho^Y_{\psi'(n),j}f'_j\rho^X_{\phi'(j),k}\\ &=g'_nf'_{\psi'(n)}\rho^X_{\phi'(\psi'(n)),k}. \end{aligned}
\rho^X_{\phi(j),k}Use in source · 2 passages
Search aliases: ρ^X_ϕ(j),k rho phi
Use in source · 03a_observation.tex, line 41
We check compatibility with composition with all indices displayed. Write f,f' with schedules ϕ,ϕ' and g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applying the naturality equations and then the two witness equations gives g_nf_ψ(n)ρ^X_ϕ(ψ(n)),k · =g_nρ^Y_ψ(n),jf_jρ^X_ϕ(j),k
Exact mathematical passage
f_j\rho^X_{\phi(j),k}=f'_j\rho^X_{\phi'(j),k}.
\rho^X_{\phi(n),k}Use in source · 2 passages
Search aliases: ρ^X_ϕ(n),k rho phi
Use in source · 03a_observation.tex, line 24
Equality is witnessed at finite stages. Two different schedules can represent the same transformation. To compare (ϕ,f) and (ψ,g), require that for every output level n there be a finite level k≥ϕ(n),ψ(n) with f_nρ^X_ϕ(n),k=g_nρ^X_ψ(n),k as functions on all of X_k. Call this eventual equality. The witnessing k may depend on n. Equality is required on the whole finite-level fiber, not only on states that happen to extend globally.
Exact mathematical passage
\label{eq:eventual-equality} f_n\rho^X_{\phi(n),k}=g_n\rho^X_{\psi(n),k} \quad\text{as functions on all of }X_k.
\rho^X_{\phi(\psi(n)),k}Use in source · 1 passage
Search aliases: ρ^X_ϕ(ψ(n)),k rho phi psi
Use in source · 03a_observation.tex, line 45
nd g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applying the naturality equations and then the two witness equations gives g_nf_ψ(n)ρ^X_ϕ(ψ(n)),k · =g_nρ^Y_ψ(n),jf_jρ^X_ϕ(j),k · =g'_nρ^Y_ψ'(n),jf_jρ^X_ϕ(j),k · =g'_nρ^Y_ψ'(n),jf'_jρ^X_ϕ'(j),k · =g'_nf'_ψ'(n)ρ^X_ϕ'(ψ'(n)),k. This is the required equality o…
Exact mathematical passage
\begin{aligned} g_nf_{\psi(n)}\rho^X_{\phi(\psi(n)),k} &=g_n\rho^Y_{\psi(n),j}f_j\rho^X_{\phi(j),k}\\ &=g'_n\rho^Y_{\psi'(n),j}f_j\rho^X_{\phi(j),k}\\ &=g'_n\rho^Y_{\psi'(n),j}f'_j\rho^X_{\phi'(j),k}\\ &=g'_nf'_{\psi'(n)}\rho^X_{\phi'(\psi'(n)),k}. \end{aligned}
\rho^X_{\psi(n),k}Use in source · 2 passages
Search aliases: ρ^X_ψ(n),k rho psi
Use in source · 03a_observation.tex, line 24
Equality is witnessed at finite stages. Two different schedules can represent the same transformation. To compare (ϕ,f) and (ψ,g), require that for every output level n there be a finite level k≥ϕ(n),ψ(n) with f_nρ^X_ϕ(n),k=g_nρ^X_ψ(n),k as functions on all of X_k. Call this eventual equality. The witnessing k may depend on n. Equality is required on the whole finite-level fiber, not only on states that happen to extend globally.
Exact mathematical passage
\label{eq:eventual-equality} f_n\rho^X_{\phi(n),k}=g_n\rho^X_{\psi(n),k} \quad\text{as functions on all of }X_k.
\rho^Y_{2n}Use in source · 1 passage
Search aliases: ρ^Y_2n rho
Use in source · 03_towers.tex, line 313
The actual dyadic tower coarsener has the typed formula (Cr)_n=ρ^Y_2nr_2n+1+r_2nρ^X_2n+1. It is obtained by telescoping two adjacent compatibility squares. With target sections, a lift is (Jc)_2n=0, (Jc)_2n+1=s_2nc_n. The section identity gives CJ=I, so the missing detail is explicitly Hr, not an unspecified reconstruction error. The normed construction imposes the bounds needed to make these maps bounded on its operator family spaces. sourceKS25
Exact mathematical passage
\label{eq:dyadic} (Cr)_n=\rho^Y_{2n}r_{2n+1}+r_{2n}\rho^X_{2n+1}.
\rho^Y_{a,b}Use in source · 2 passages
Search aliases: ρ^Y_a,b rho
Use in source · 03b_repair_transport.tex, line 148
a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a,b. The dyadic coarsener in Section~[source reference] is the case b=a+2.
Exact mathematical passage
C_{a,b}(\partial f)=\rho^Y_{a,b}f_b-f_a\rho^X_{a,b}.
\rho^Y_{a,k}Use in source · 2 passages
Search aliases: ρ^Y_a,k rho
Use in source · 03b_repair_transport.tex, line 143
For a block from a to b, its value on a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a,b. The dyadic coarsener in Section~[source reference] is the case b=a+2.
Exact mathematical passage
\label{eq:block-defect-sum} (C_{a,b}r)=\sum_{k=a}^{b-1} \rho^Y_{a,k}\,r_k\,\rho^X_{k+1,b}.
\rho^Y_{a,k+1}Use in source · 1 passage
Search aliases: ρ^Y_a,k+1 rho
Use in source · 03b_repair_transport.tex, line 170
To verify the commuting square, insert r_k=ρ^Y_kf_k+1-f_kρ^X_k. The k-th term becomes ρ^Y_a,k+1f_k+1ρ^X_k+1,b -ρ^Y_a,kf_kρ^X_k,b. Successive internal terms cancel in the finite sum, leaving ρ^Y_a,bf_b-f_aρ^X_a,b. This is exactly the sampled differential. No infinite series interchange is involved.
Exact mathematical passage
\rho^Y_{a,k+1}f_{k+1}\rho^X_{k+1,b} -\rho^Y_{a,k}f_k\rho^X_{k,b}.
\rho^Y_kUse in source · 1 passage
Search aliases: ρ^Y_k rho
Use in source · 03b_repair_transport.tex, line 169
To verify the commuting square, insert r_k=ρ^Y_kf_k+1-f_kρ^X_k. The k-th term becomes ρ^Y_a,k+1f_k+1ρ^X_k+1,b -ρ^Y_a,kf_kρ^X_k,b. Successive internal terms cancel in the finite sum, leaving ρ^Y_a,bf_b-f_aρ^X_a,b. This is exactly the sampled differential. No infinite series interchange is involved.
Exact mathematical passage
r_k=\rho^Y_kf_{k+1}-f_k\rho^X_k
\rho^Y_nUse in source · 19 passages
Search aliases: ρ^Y_n rho
Use in source · 03_towers.tex, line 12
A candidate transport from X to Y consists of maps f_n:X_n→ Y_n. It is exact when ρ^Y_n f_n+1=f_nρ^X_n for all n.
Exact mathematical passage
\label{eq:exacttower} \rho^Y_n f_{n+1}=f_n\rho^X_n\quad\text{for all }n.
\rho^Y_{\psi'(n),j}Use in source · 2 passages
Search aliases: ρ^Y_ψ'(n),j rho psi
Use in source · 03a_observation.tex, line 37
We check compatibility with composition with all indices displayed. Write f,f' with schedules ϕ,ϕ' and g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applying the naturality equations and then the two witness equations gives g_nf_ψ(n)ρ^X_ϕ(ψ(n)),k · =g_nρ^Y_ψ(n),jf_jρ^X_ϕ(j),k
Exact mathematical passage
g_n\rho^Y_{\psi(n),j}=g'_n\rho^Y_{\psi'(n),j}.
\rho^Y_{\psi(n),j}Use in source · 2 passages
Search aliases: ρ^Y_ψ(n),j rho psi
Use in source · 03a_observation.tex, line 37
We check compatibility with composition with all indices displayed. Write f,f' with schedules ϕ,ϕ' and g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applying the naturality equations and then the two witness equations gives g_nf_ψ(n)ρ^X_ϕ(ψ(n)),k · =g_nρ^Y_ψ(n),jf_jρ^X_ϕ(j),k
Exact mathematical passage
g_n\rho^Y_{\psi(n),j}=g'_n\rho^Y_{\psi'(n),j}.
\rho^Z_nUse in source · 2 passages
Search aliases: ρ^Z_n rho
Use in source · 03_towers.tex, line 20
[The exact tower category] Identity maps and composites of exact transports are exact. An exact transport induces a map of coherent point spaces by x↦(f_nx_n)_n, preserving identities and composition. For two exact transports, substitute ρ^Z_ng_n+1=g_nρ^Y_n and ρ^Y_nf_n+1=f_nρ^X_n into the composite square. For a coherent point, ρ^Y_n f_n+1x_n+1 =f_nρ^X_nx_n+1=f_nx_n. Identity, associativity, and composition on points are checked coordinatewise. sourceKS16 This is a category on a fixed schedule.…
Exact mathematical passage
\rho^Z_ng_{n+1}=g_n\rho^Y_n
\rho_0Use in source · 1 passage
Search aliases: ρ_0 rho
Use in source · 03_towers.tex, line 5
ned detailResolution gives a concrete family of views. Its compatibility differential connects exact maps, protected information, and residual coordinates. Recovering every finite coordinate is then separated from convergence in a chosen norm. Resolution towers and exact transport. The sequential model. A tower in a category C is a sequence ⋯xrightarrowρ_2X_2xrightarrowρ_1X_1 xrightarrowρ_0X_0. For sets, a point is a sequence x_n∈ X_n with ρ_n(x_n+1)=x_n. Composite restrictions turn the tower in…
Exact mathematical passage
\cdots\xrightarrow{\rho_2}X_2\xrightarrow{\rho_1}X_1 \xrightarrow{\rho_0}X_0.
\rho_{0,1}Use in source · 3 passages
Search aliases: ρ_0,1 rho
Use in source · 02a_prescription.tex, line 49
An initially free parameter can become an obstruction. Consider the chain of views 0→1→2, with fiber Z^2 at each view. Write the maps in the direction in which data are restricted: ρ_0,1(a,b)=(a,0), ρ_1,2(a,b)=(a,2b), ρ_0,2(a,b)=(a,0). The composite identity ρ_0,1ρ_1,2=ρ_0,2 is literal. A global family is exactly x_0=(k,0), x_1=(k,2n), x_2=(k,n), k,n∈Z. Indeed the first coordinate must agree at every view. The second coordinate at view zero is forced to vanish, and the second coordinate at view …
Exact mathematical passage
\rho_{0,1}(a,b)=(a,0),\qquad \rho_{1,2}(a,b)=(a,2b),\qquad \rho_{0,2}(a,b)=(a,0).
\rho_{0,2}Use in source · 2 passages
Search aliases: ρ_0,2 rho
Use in source · 02a_prescription.tex, line 49
An initially free parameter can become an obstruction. Consider the chain of views 0→1→2, with fiber Z^2 at each view. Write the maps in the direction in which data are restricted: ρ_0,1(a,b)=(a,0), ρ_1,2(a,b)=(a,2b), ρ_0,2(a,b)=(a,0). The composite identity ρ_0,1ρ_1,2=ρ_0,2 is literal. A global family is exactly x_0=(k,0), x_1=(k,2n), x_2=(k,n), k,n∈Z. Indeed the first coordinate must agree at every view. The second coordinate at view zero is forced to vanish, and the second coordinate at view …
Exact mathematical passage
\rho_{0,1}(a,b)=(a,0),\qquad \rho_{1,2}(a,b)=(a,2b),\qquad \rho_{0,2}(a,b)=(a,0).
\rho_1Use in source · 1 passage
Search aliases: ρ_1 rho
Use in source · 03_towers.tex, line 5
ned detailResolution gives a concrete family of views. Its compatibility differential connects exact maps, protected information, and residual coordinates. Recovering every finite coordinate is then separated from convergence in a chosen norm. Resolution towers and exact transport. The sequential model. A tower in a category C is a sequence ⋯xrightarrowρ_2X_2xrightarrowρ_1X_1 xrightarrowρ_0X_0. For sets, a point is a sequence x_n∈ X_n with ρ_n(x_n+1)=x_n. Composite restrictions turn the tower in…
Exact mathematical passage
\cdots\xrightarrow{\rho_2}X_2\xrightarrow{\rho_1}X_1 \xrightarrow{\rho_0}X_0.
\rho_{1,2}Use in source · 2 passages
Search aliases: ρ_1,2 rho
Use in source · 02a_prescription.tex, line 49
An initially free parameter can become an obstruction. Consider the chain of views 0→1→2, with fiber Z^2 at each view. Write the maps in the direction in which data are restricted: ρ_0,1(a,b)=(a,0), ρ_1,2(a,b)=(a,2b), ρ_0,2(a,b)=(a,0). The composite identity ρ_0,1ρ_1,2=ρ_0,2 is literal. A global family is exactly x_0=(k,0), x_1=(k,2n), x_2=(k,n), k,n∈Z. Indeed the first coordinate must agree at every view. The second coordinate at view zero is forced to vanish, and the second coordinate at view …
Exact mathematical passage
\rho_{0,1}(a,b)=(a,0),\qquad \rho_{1,2}(a,b)=(a,2b),\qquad \rho_{0,2}(a,b)=(a,0).
\rho_2Use in source · 1 passage
Search aliases: ρ_2 rho
Use in source · 03_towers.tex, line 5
ned detailResolution gives a concrete family of views. Its compatibility differential connects exact maps, protected information, and residual coordinates. Recovering every finite coordinate is then separated from convergence in a chosen norm. Resolution towers and exact transport. The sequential model. A tower in a category C is a sequence ⋯xrightarrowρ_2X_2xrightarrowρ_1X_1 xrightarrowρ_0X_0. For sets, a point is a sequence x_n∈ X_n with ρ_n(x_n+1)=x_n. Composite restrictions turn the tower in…
Exact mathematical passage
\cdots\xrightarrow{\rho_2}X_2\xrightarrow{\rho_1}X_1 \xrightarrow{\rho_0}X_0.
\rho_AUse in source · 2 passages
Search aliases: ρ_A rho
Use in source · 05_construction.tex, line 110
Structure of the extension. By Chapter~[source reference], the group is pure in P, reduced, residually finite, and dense. Its torsion is D, and the rational coordinate gives the exact sequence 0⟶ D⟶ G_Axrightarrowρ_A Q⟶0, ρ_A(s_A(r)+d)=r. It does not split.
Exact mathematical passage
\label{eq:extension} 0\longrightarrow D\longrightarrow G_A\xrightarrow{\rho_A}\Q\longrightarrow0, \qquad \rho_A(s_A(r)+d)=r.
\rho_{i,n}Use in source · 1 passage
Search aliases: ρ_i,n rho
Use in source · 06_boundaries.tex, line 62
Finally, fix a level n. Every global point satisfies x_n=ρ_n,m(x_m) for each m≥ n, hence x_n∈ Y_n. Conversely, start with any y∈ Y_n, lift recursively above level n through the surjections just proved, and define the lower coordinates by x_i=ρ_i,n(y). The composite restriction laws verify coherence across and below level n. The resulting point evaluates to y. These two inclusions prove the asserted equality of evaluation images, not merely nonemptiness of the inverse limit. sourceKS58 This is th…
Exact mathematical passage
x_i=\rho_{i,n}(y)
\rho_{k+1,\phi(n)}Use in source · 1 passage
Search aliases: ρ_k+1,ϕ(n) rho phi
Use in source · 03_towers.tex, line 73
For the adjacent equation at k, choose one n with k+1≤ϕ(n). Then ρ_k,k+1x_k+1 =ρ_k,k+1ρ_k+1,ϕ(n)y_n =ρ_k,ϕ(n)y_n=x_k. The recovered family is therefore a point of the original tower.
Exact mathematical passage
\rho_{k,k+1}x_{k+1} =\rho_{k,k+1}\rho_{k+1,\phi(n)}y_n =\rho_{k,\phi(n)}y_n=x_k.
\rho_{k,k+1}Use in source · 1 passage
Search aliases: ρ_k,k+1 rho
Use in source · 03_towers.tex, line 73
For the adjacent equation at k, choose one n with k+1≤ϕ(n). Then ρ_k,k+1x_k+1 =ρ_k,k+1ρ_k+1,ϕ(n)y_n =ρ_k,ϕ(n)y_n=x_k. The recovered family is therefore a point of the original tower.
Exact mathematical passage
\rho_{k,k+1}x_{k+1} =\rho_{k,k+1}\rho_{k+1,\phi(n)}y_n =\rho_{k,\phi(n)}y_n=x_k.
\rho_{k,\phi(m)}Use in source · 1 passage
Search aliases: ρ_k,ϕ(m) rho phi
Use in source · 03_towers.tex, line 64
For each k, unboundedness gives an index n with k≤ϕ(n). Set x_k=ρ_k,ϕ(n)y_n. If m is another permissible index, choose ℓ≥ n,m. Monotonicity permits both composite restrictions, and sampled coherence gives ρ_k,ϕ(n)y_n =ρ_k,ϕ(n)ρ_ϕ(n),ϕ(ℓ)y_ℓ =ρ_k,ϕ(ℓ)y_ℓ =ρ_k,ϕ(m)y_m. Thus x_k is independent of the choice, including when the schedule repeats a level.
Exact mathematical passage
\rho_{k,\phi(n)}y_n =\rho_{k,\phi(n)}\rho_{\phi(n),\phi(\ell)}y_\ell =\rho_{k,\phi(\ell)}y_\ell =\rho_{k,\phi(m)}y_m.
\rho_{k,\phi(n)}Use in source · 4 passages
Search aliases: ρ_k,ϕ(n) rho phi
Use in source · 03_towers.tex, line 63
For each k, unboundedness gives an index n with k≤ϕ(n). Set x_k=ρ_k,ϕ(n)y_n. If m is another permissible index, choose ℓ≥ n,m. Monotonicity permits both composite restrictions, and sampled coherence gives ρ_k,ϕ(n)y_n =ρ_k,ϕ(n)ρ_ϕ(n),ϕ(ℓ)y_ℓ =ρ_k,ϕ(ℓ)y_ℓ =ρ_k,ϕ(m)y_m. Thus x_k is independent of the choice, including when the schedule repeats a level.
Exact mathematical passage
x_k=\rho_{k,\phi(n)}y_n
\rho_{k,\phi(\ell)}Use in source · 1 passage
Search aliases: ρ_k,ϕ(ℓ) rho phi ell
Use in source · 03_towers.tex, line 64
For each k, unboundedness gives an index n with k≤ϕ(n). Set x_k=ρ_k,ϕ(n)y_n. If m is another permissible index, choose ℓ≥ n,m. Monotonicity permits both composite restrictions, and sampled coherence gives ρ_k,ϕ(n)y_n =ρ_k,ϕ(n)ρ_ϕ(n),ϕ(ℓ)y_ℓ =ρ_k,ϕ(ℓ)y_ℓ =ρ_k,ϕ(m)y_m. Thus x_k is independent of the choice, including when the schedule repeats a level.
Exact mathematical passage
\rho_{k,\phi(n)}y_n =\rho_{k,\phi(n)}\rho_{\phi(n),\phi(\ell)}y_\ell =\rho_{k,\phi(\ell)}y_\ell =\rho_{k,\phi(m)}y_m.
\rho_{\phi(n),\phi(n)}Use in source · 1 passage
Search aliases: ρ_ϕ(n),ϕ(n) rho phi phi
Use in source · 03_towers.tex, line 80
Sampling a full point and recovering its kth coordinate gives ρ_k,ϕ(n)x_ϕ(n)=x_k by its original coherence. Recovering a sampled point and evaluating at k=ϕ(n) allows the index n itself, giving ρ_ϕ(n),ϕ(n)y_n=y_n. These are both inverse identities. No surjectivity of the bonding maps or of the point evaluations was used. sourceKS18 This theorem is about exact coherent points. Raw residual size is a different quantity. On identity scalar towers, f_n=(-1)^n has nonzero fine differences, but its ev…
Exact mathematical passage
\rho_{\phi(n),\phi(n)}y_n=y_n
\rho_{\phi(n),\phi(\ell)}Use in source · 1 passage
Search aliases: ρ_ϕ(n),ϕ(ℓ) rho phi phi ell
Use in source · 03_towers.tex, line 64
For each k, unboundedness gives an index n with k≤ϕ(n). Set x_k=ρ_k,ϕ(n)y_n. If m is another permissible index, choose ℓ≥ n,m. Monotonicity permits both composite restrictions, and sampled coherence gives ρ_k,ϕ(n)y_n =ρ_k,ϕ(n)ρ_ϕ(n),ϕ(ℓ)y_ℓ =ρ_k,ϕ(ℓ)y_ℓ =ρ_k,ϕ(m)y_m. Thus x_k is independent of the choice, including when the schedule repeats a level.
Exact mathematical passage
\rho_{k,\phi(n)}y_n =\rho_{k,\phi(n)}\rho_{\phi(n),\phi(\ell)}y_\ell =\rho_{k,\phi(\ell)}y_\ell =\rho_{k,\phi(m)}y_m.
\sigmaUse in source · 14 passages
Search aliases: σ sigma
Use in source · 04_symmetry.tex, line 166
A small noncommutative model. [A C_2 loop with S_3 coefficients] Take one view with a loop σ satisfying σ^2=1, and constant coefficients in a group H. A cocycle is determined by an element p∈ H whose square is the identity. Gauge equivalence is conjugacy, and its automorphisms form the centralizer of p.
Exact mathematical passage
\sigma
\sigma^2Use in source · 2 passages
Search aliases: σ^2 sigma
Use in source · 04_symmetry.tex, line 166
A small noncommutative model. [A C_2 loop with S_3 coefficients] Take one view with a loop σ satisfying σ^2=1, and constant coefficients in a group H. A cocycle is determined by an element p∈ H whose square is the identity. Gauge equivalence is conjugacy, and its automorphisms form the centralizer of p.
Exact mathematical passage
\sigma^2=1
\sigma_pUse in source · 1 passage
Search aliases: σ_p sigma
Use in source · 03_spectrum.tex, line 10
The entire subgroup lattice. For B⊆ N, set D_B=⊕_i∈ BC_p_i≤ D. A prime spectrum records which prime orders occur. Define σ_p(M)=1 when M contains a nonzero element annihilated by p, and set it to zero otherwise. Since p is prime, a nonzero such element has order exactly p.
Exact mathematical passage
\sigma_p(M)=1
\sigma_{p_i}Use in source · 1 passage
Search aliases: σ_p_i sigma
Use in source · 03_spectrum.tex, line 22
Let B be the set of indices of these factors. Every element of H lies in D_B, and every finite sum of the factors in B lies in H. Hence H=D_B. Uniqueness follows by testing membership of e_p_i. Proposition~[source reference] proves full invariance. Finally σ_p_i(D_B)=1 exactly when i∈ B, and these values are isomorphism invariants. The lattice operations have literal set-theoretic descriptions: D_B+D_C=D_B∪ C, D_B∩ D_C=D_B∩ C, D=D_B⊕ D_ N∖ B. The infinite support choice is therefore already visi…
Exact mathematical passage
\sigma_{p_i}(D_B)=1
\sigma_xUse in source · 3 passages
Search aliases: σ_x sigma
Use in source · 04_symmetry.tex, line 424
he marked realization problem] The map (x,ϕ)⟼(Ex,K(ϕ)σ_x^-1), a⟼ E(a) defines an equivalence E_b(R)≃ E_Kb(R'). For an object (x,ϕ) put β_x=K(ϕ)σ_x^-1:R'Ex→ Kb. This is an isomorphism of the required type. If a:(x,ϕ)→(y,ψ), naturality of σ says R'E(a)σ_x=σ_yKR(a), σ_y^-1R'E(a)=KR(a)σ_x^-1. Consequently β_yR'E(a) =K(ψ)KR(a)σ_x^-1 =K(ψ R(a))σ_x^-1 =K(ϕ)σ_x^-1=β_x. So E(a) is a marked arrow. Identity and composition are preserved because the underlying arrow map is that of the functor E.
Exact mathematical passage
R'E(a)\sigma_x=\sigma_yKR(a), \qquad \sigma_y^{-1}R'E(a)=KR(a)\sigma_x^{-1}.
\sigma_x^{-1}Use in source · 5 passages
Search aliases: σ_x^-1 sigma
Use in source · 04_symmetry.tex, line 417
[Invariance of the marked realization problem] The map (x,ϕ)⟼(Ex,K(ϕ)σ_x^-1), a⟼ E(a) defines an equivalence E_b(R)≃ E_Kb(R'). For an object (x,ϕ) put β_x=K(ϕ)σ_x^-1:R'Ex→ Kb. This is an isomorphism of the required type. If a:(x,ϕ)→(y,ψ), naturality of σ says R'E(a)σ_x=σ_yKR(a), σ_y^-1R'E(a)=KR(a)σ_x^-1. Consequently β_yR'E(a) =K(ψ)KR(a)σ_x^-1 =K(ψ R(a))σ_x^-1 =K(ϕ)σ_x^-1=β_x. So E(a) is a marked arrow. Identity and composition are preserved because the underlying arrow map is that of the functo…
Exact mathematical passage
(x,\phi)\longmapsto(Ex,K(\phi)\sigma_x^{-1}),\qquad a\longmapsto E(a)
\sigma_yUse in source · 1 passage
Search aliases: σ_y sigma
Use in source · 04_symmetry.tex, line 424
he marked realization problem] The map (x,ϕ)⟼(Ex,K(ϕ)σ_x^-1), a⟼ E(a) defines an equivalence E_b(R)≃ E_Kb(R'). For an object (x,ϕ) put β_x=K(ϕ)σ_x^-1:R'Ex→ Kb. This is an isomorphism of the required type. If a:(x,ϕ)→(y,ψ), naturality of σ says R'E(a)σ_x=σ_yKR(a), σ_y^-1R'E(a)=KR(a)σ_x^-1. Consequently β_yR'E(a) =K(ψ)KR(a)σ_x^-1 =K(ψ R(a))σ_x^-1 =K(ϕ)σ_x^-1=β_x. So E(a) is a marked arrow. Identity and composition are preserved because the underlying arrow map is that of the functor E.
Exact mathematical passage
R'E(a)\sigma_x=\sigma_yKR(a), \qquad \sigma_y^{-1}R'E(a)=KR(a)\sigma_x^{-1}.
\sigma_y^{-1}Use in source · 1 passage
Search aliases: σ_y^-1 sigma
Use in source · 04_symmetry.tex, line 424
he marked realization problem] The map (x,ϕ)⟼(Ex,K(ϕ)σ_x^-1), a⟼ E(a) defines an equivalence E_b(R)≃ E_Kb(R'). For an object (x,ϕ) put β_x=K(ϕ)σ_x^-1:R'Ex→ Kb. This is an isomorphism of the required type. If a:(x,ϕ)→(y,ψ), naturality of σ says R'E(a)σ_x=σ_yKR(a), σ_y^-1R'E(a)=KR(a)σ_x^-1. Consequently β_yR'E(a) =K(ψ)KR(a)σ_x^-1 =K(ψ R(a))σ_x^-1 =K(ϕ)σ_x^-1=β_x. So E(a) is a marked arrow. Identity and composition are preserved because the underlying arrow map is that of the functor E.
Exact mathematical passage
R'E(a)\sigma_x=\sigma_yKR(a), \qquad \sigma_y^{-1}R'E(a)=KR(a)\sigma_x^{-1}.
\tau_{i_ji_{j+1}}Use in source · 1 passage
Search aliases: τ_i_ji_j+1 tau
Use in source · 05_geometry.tex, line 42
The triangle inequality and the operator norm bound now give normT_qx_i_m-x_i_0 ≤∑_k=0^m-1normU_knorm(dx)_i_ki_k+1 ≤(∑_k=0^m-1∏_j=0^k-1normτ_i_ji_j+1)normdx_∞. The bracket is γ(q), with the empty product equal to one. Repeated vertices or edges do not affect the cancellation. Reordering the edge maps would, in general, invalidate it. sourceKS40 For isometric maps, γ(q)≤|q|. For arbitrary bounded equivalences, route length cannot replace the amplification gain. Reversing a route can have a differ…
Exact mathematical passage
\norm{T_qx_{i_m}-x_{i_0}} \le\sum_{k=0}^{m-1}\norm{U_k}\norm{(dx)_{i_ki_{k+1}}} \le\left(\sum_{k=0}^{m-1}\prod_{j=0}^{k-1}\norm{\tau_{i_ji_{j+1}}}\right)\norm{dx}_\infty.
\tau_{i_{m-1}i_m}Use in source · 1 passage
Search aliases: τ_i_m-1i_m tau
Use in source · 05_geometry.tex, line 15
Distinct walks are retained as arrows of the view category. The product along a walk q=(i_0,…,i_m) is T_q=τ_i_0i_1⋯τ_i_m-1i_m. Concatenation gives the restriction composition law. An exact vertex family defines a coherent walk-view section, and a coherent walk-view section satisfies every edge equation. Thus this graph construction returns to the original ambient realization type rather than creating a disconnected numerical model.
Exact mathematical passage
T_q=\tau_{i_0i_1}\cdots\tau_{i_{m-1}i_m}.
\tau_{ij}Use in source · 6 passages
Search aliases: τ_ij tau
Use in source · 05_geometry.tex, line 4
lly guarantee. Graph views, holonomy, and reconstruction. Actual transports on a finite graph. Let a finite connected simple graph have vertex set V, root o, and a chosen route p_i:o→ i for every vertex, with the route at o empty. Use a common real normed fiber H. An inverse consistent connection assigns continuous linear equivalences τ_ij, with τ_ji=τ_ij^-1. Our convention is that τ_ij carries the value at j into the coordinates at i.
Exact mathematical passage
\tau_{ij}
\tau_{ij}^{-1}Use in source · 1 passage
Search aliases: τ_ij^-1 tau
Use in source · 05_geometry.tex, line 4
tionGraph views, holonomy, and reconstruction Actual transports on a finite graph. Let a finite connected simple graph have vertex set V, root o, and a chosen route p_i:o→ i for every vertex, with the route at o empty. Use a common real normed fiber H. An inverse consistent connection assigns continuous linear equivalences τ_ij, with τ_ji=τ_ij^-1. Our convention is that τ_ij carries the value at j into the coordinates at i.
Exact mathematical passage
\tau_{ji}=\tau_{ij}^{-1}
\tau_{ji}Use in source · 1 passage
Search aliases: τ_ji tau
Use in source · 05_geometry.tex, line 4
tionGraph views, holonomy, and reconstruction Actual transports on a finite graph. Let a finite connected simple graph have vertex set V, root o, and a chosen route p_i:o→ i for every vertex, with the route at o empty. Use a common real normed fiber H. An inverse consistent connection assigns continuous linear equivalences τ_ij, with τ_ji=τ_ij^-1. Our convention is that τ_ij carries the value at j into the coordinates at i.
Exact mathematical passage
\tau_{ji}=\tau_{ij}^{-1}
\PhiUse in source · 1 passage
Search aliases: Φ Phi
Use in source · 03_towers.tex, line 237
[Split coordinate theorem] The observable is a bounded projection with p_LA=0 and p_LP=p_L. It separates exact candidates. If S_L={s∈ E:p_Ls=s}, then Φ:E≃_L S_L× R, Φ(f)=(p_Lf,∂ f), Φ^-1(s,r)=Ps+Ar. Repair becomes (s,r)↦(s,0). It is the unique exact repair preserving p_Lf. Put B_n=I-s_nρ^Y_n. Since ρ^Y_ns_n=I, direct multiplication gives B_n^2=B_n, ρ^Y_nB_n=0, B_ns_n=0. Moreover, B_ny=y whenever ρ^Y_ny=0. Thus B_n really projects onto the target kernel. Its norm is at most 1+SB_Y. Coordinate zer…
Exact mathematical passage
\Phi:E\simeq_L\mathcal S_L\times R, \qquad \Phi(f)=(p_Lf,\partial f), \qquad \Phi^{-1}(s,r)=Ps+Ar.
\phi'Use in source · 4 passages
Search aliases: ϕ' phi
Use in source · 03a_observation.tex, line 36
We check compatibility with composition with all indices displayed. Write f,f' with schedules ϕ,ϕ' and g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applying the naturality equations and then the two witness equations gives g_nf_ψ(n)ρ^X_ϕ(ψ(n)),k · =g_nρ^Y_ψ(n),jf_jρ^X_ϕ(j),k
Exact mathematical passage
\phi,\phi'
\Phi^{-1}Use in source · 1 passage
Search aliases: Φ^-1 Phi
Use in source · 03_towers.tex, line 237
[Split coordinate theorem] The observable is a bounded projection with p_LA=0 and p_LP=p_L. It separates exact candidates. If S_L={s∈ E:p_Ls=s}, then Φ:E≃_L S_L× R, Φ(f)=(p_Lf,∂ f), Φ^-1(s,r)=Ps+Ar. Repair becomes (s,r)↦(s,0). It is the unique exact repair preserving p_Lf. Put B_n=I-s_nρ^Y_n. Since ρ^Y_ns_n=I, direct multiplication gives B_n^2=B_n, ρ^Y_nB_n=0, B_ns_n=0. Moreover, B_ny=y whenever ρ^Y_ny=0. Thus B_n really projects onto the target kernel. Its norm is at most 1+SB_Y. Coordinate zer…
Exact mathematical passage
\Phi:E\simeq_L\mathcal S_L\times R, \qquad \Phi(f)=(p_Lf,\partial f), \qquad \Phi^{-1}(s,r)=Ps+Ar.
\chi_A^{-1}Use in source · 2 passages
Search aliases: χ_A^-1 chi
Use in source · 03b_repair_transport.tex, line 13
[The split defect chart] The maps χ_A:E≃_L K× R, χ_A(x)=(P_Ax,dx), χ_A^-1(u,r)=u+Ar are inverse continuous linear equivalences. In these coordinates, P_A sends (u,r) to (u,0). If B is another right inverse, then A-B takes values in K, and P_Bx-P_Ax=(A-B)dx. The projections commute if and only if they are equal, and they are equal if and only if A=B. The identity dA=I gives dP_A=d-dAd=0, so the proposed first chart coordinate actually belongs to K. If u∈ K, then du=0 and P_Au=u. Therefore, for (u…
Exact mathematical passage
\chi_A:E\simeq_L K\times R,\qquad \chi_A(x)=(P_Ax,dx),\qquad \chi_A^{-1}(u,r)=u+Ar
\chi_BUse in source · 1 passage
Search aliases: χ_B chi
Use in source · 03b_repair_transport.tex, line 61
[Triangular transport and repair mismatch] In split defect coordinates, the change of representation is χ_BSχ_A^-1(u,r)=(Su+Mr,Cr). The two repair routes differ by P_BSx-SP_Ax=M(dx). Consequently they agree on exact inputs, and normP_BSx-SP_Ax≤norm Mnormdx, norm M≤norm Snorm A+norm Bnorm C. They agree on all inputs if and only if SA=BC.
Exact mathematical passage
\label{eq:triangular-chart} \chi_BS\chi_A^{-1}(u,r)=(Su+Mr,Cr).
\PsiUse in source · 3 passages
Search aliases: Ψ Psi
Use in source · 02_obstructions.tex, line 95
10 If x_0∈Ext_U(s), translation gives an equivalence H_U≃Ext_U(s), h⟼ x_0+h. Consequently the extension is unique exactly when o_U(s)=0 and H_U=0. Define Θ:H_U→Ext_U(s) by Θ(h)=x_0+h. It lands in the extension set because δ(x_0+h)=δ x_0+δ h=0, p_U(x_0+h)=s+0=s. Define Ψ:Ext_U(s)→ H_U by Ψ(x)=x-x_0. Its differential is zero, and its restriction is s-s=0, so the claimed target is correct. The formulas give ΨΘ(h)=x_0+h-x_0=h, ΘΨ(x)=x_0+(x-x_0)=x. This proves the equivalence rather than only a compa…
Exact mathematical passage
\Psi:\Ext_U(s)\to H_U
\psi'Use in source · 4 passages
Search aliases: ψ' psi
Use in source · 03a_observation.tex, line 36
We check compatibility with composition with all indices displayed. Write f,f' with schedules ϕ,ϕ' and g,g' with schedules ψ,ψ'. Fix an output level n. Choose j≥ψ(n),ψ'(n) with g_nρ^Y_ψ(n),j=g'_nρ^Y_ψ'(n),j. Next choose k≥ϕ(j),ϕ'(j) with f_jρ^X_ϕ(j),k=f'_jρ^X_ϕ'(j),k. Monotonicity implies that k is also at least both composite input levels. Applying the naturality equations and then the two witness equations gives g_nf_ψ(n)ρ^X_ϕ(ψ(n)),k · =g_nρ^Y_ψ(n),jf_jρ^X_ϕ(j),k
Exact mathematical passage
\psi,\psi'
\psi^{-1}Use in source · 2 passages
Search aliases: ψ^-1 psi
Use in source · 04_symmetry.tex, line 455
Replacing b by an isomorphic boundary b' also gives an equivalence, by composing the identification with b→ b'. For R=id, the marked realization groupoid has center (b,1_b), and between (x,ϕ) and (y,ψ) there is exactly one compatible arrow ψ^-1ϕ. This is a useful example of contractible realization data, not a claim that the original groupoid has no symmetries.
Exact mathematical passage
\psi^{-1}\phi
\OmegaUse in source · 21 passages
Search aliases: Ω Omega
Use in source · 05d_floors.tex, line 6
Distance to the actual correction image. Let R be a normed additive group and B≤ R an additive subgroup. In the protected linear model, B=d(ker p). For a residual Ω∈ R, define ν_B(Ω)=inf_b∈ BnormΩ-b. The subgroup contains zero, so this is a finite nonnegative real number. It is an infimum, not an assertion that a nearest admissible residual exists.
Exact mathematical passage
\Omega\in R
\omega_fUse in source · 2 passages
Search aliases: ω_f omega
Use in source · 05i_diagonal_regularization.tex, line 91
stead has sqrt N normw-T1_N⟶1. Step 1: control the infinite Riemann sums uniformly. Define f(x)=frac1(1+x^2)^2, g(x)=fracx^2(1+x^2)^2. Then A_m/m=m^-1∑_n≥1f(n/m) and mB_m=m^-1∑_n≥1g(n/m). Fix L≥2 and put N=lfloor mLrfloor. On [0,L], both functions are uniformly continuous. If ω_f(h) is a modulus of continuity there, comparison on the N intervals of length 1/m gives |frac1m∑_n=1^N f(n/m)-∫_0^N/mf(x) dx| ≤ Lω_f(1/m). The missing interval [N/m,L] has length below 1/m and a bounded integrand, so it …
Exact mathematical passage
\omega_f(h)
\Omega_nUse in source · 5 passages
Search aliases: Ω_n Omega
Use in source · 05d_floors.tex, line 51
A persistent floor blocks every correction sequence. The correction image can vary with the resolution or the admissibility rule. Let B_n≤ R and let Ω_n∈ R be given.
Exact mathematical passage
\Omega_n\in R
\daggerUse in source · 22 passages
Search aliases: † dagger
Use in source · 05h_regularized_repair.tex, line 147
[The regularization limit alternative] As αdownarrow0, Ta_α⟶ b, normr-Ta_α⟶ q, αnorma_α^2⟶0. If b∈ran T, let a^† be its unique minimum norm preimage. Then a_α⟶ a^† in norm. If b∉ran T, then norma_α→∞. Step 1: identify the unavoidable component. Since r-b⊥overlineran T, the adjoint identity gives T^*(r-b)=0. Thus a_α(r)=a_α(b). For every correction the orthogonal decomposition is normr-Ta^2=q^2+normb-Ta^2. In particular both the residual square and the minimum objective are at least q^2.
Exact mathematical passage
a^\dagger
\ell^\inftyUse in source · 2 passages
Search aliases: ℓ^∞ ell infty
Use in source · 03_towers.tex, line 470
Two boundary examples. On ℓ^∞(N,R), let C remove the first coordinate and let J prepend zero. Then Q_Nx is the tail of x, placed at its original coordinates with zeros before N. Every fixed coordinate is eventually zero in the remainder, and normQ_N≤1. Nevertheless the constant sequence 1 satisfies normQ_N1=1 for every N.
Exact mathematical passage
\ell^\infty(\N,\R)
\ell^\infty_nUse in source · 1 passage
Search aliases: ℓ^∞_n ell infty
Use in source · 03_towers.tex, line 92
chapBanach repair and constructed split coordinates An actual complete candidate space. Let X_n,Y_n be Banach spaces over a complete nontrivially normed field, with continuous linear bonding maps and uniform bounds normρ^X_n≤ B_X, normρ^Y_n≤ B_Y. Use the spaces of bounded operator families E=ℓ^∞_n L(X_n,Y_n), R=ℓ^∞_n L(X_n+1,Y_n). The source formalization derives the required completeness from the operator spaces and their bounded product. The same differential satisfies norm∂ f_∞≤(B_X+B_Y)norm …
Exact mathematical passage
E=\ell^\infty_n\mathcal L(X_n,Y_n),\qquad R=\ell^\infty_n\mathcal L(X_{n+1},Y_n).
\RightarrowUse in source · 2 passages
Search aliases: ⇒ Rightarrow
Use in source · 04_symmetry.tex, line 339
chapEquivalent presentations and identified boundaries Transporting the same question. Let H:V→W be part of a category equivalence, with inverse L and counit ε:HL⇒id_W. Pull a view system F on W back along H. Sampling a global section gives (H^*x)_v=x_Hv.
Exact mathematical passage
\varepsilon:HL\Rightarrow\id_\W
\forallUse in source · 4 passages
Search aliases: ∀ forall
Use in source · 03a_observation.tex, line 68
When global behavior determines the finite presentation. Say that X has surjective point projections when ∀ k ev_k:Point(X)→ X_k is onto. This means that every finite input state really occurs in some coherent global input. It is stronger than nonemptiness of the global space.
Exact mathematical passage
\forall k\quad \operatorname{ev}_k:\operatorname{Point}(X)\to X_k \quad\text{is onto}.
\partial_\phiUse in source · 1 passage
Search aliases: ∂_ϕ partial phi
Use in source · 03b_repair_transport.tex, line 155
[Bounded-gap resampling and selected repair] Suppose ϕ(n+1)-ϕ(n)≤ m for a fixed natural number m. Then the actual sampling maps are bounded and satisfy normS_ϕ≤1, normC_ϕ≤ m, ∂_ϕ S_ϕ=C_ϕ∂. Exact candidates remain exact under every monotone common resampling, even without a gap bound. If A and B are bounded right inverses for the fine and sampled differentials, then normP_BS_ϕ f-S_ϕ P_Af ≤(norm A+mnorm B)norm∂ f. For a bounded candidate family, sup_nnormf_ϕ(n)≤sup_knormf_k, proving the sampling b…
Exact mathematical passage
\norm{S_\phi}\le1,\qquad \norm{C_\phi}\le m, \qquad \partial_\phi S_\phi=C_\phi\partial.
\existsUse in source · 8 passages
Search aliases: ∃ exists
Use in source · 02_obstructions.tex, line 157
[Semantics preserving correction] For x∈ E, the following conditions are equivalent: · [dx]=0 in R/d(ker p), · ∃ h∈ E: p(h)=0, d(h)=d(x), · ∃ y∈ E: d(y)=0, p(y)=p(x). The class of dx is unchanged when an admissible correction is subtracted from x. The image d(ker p) is a subgroup of R. The defining property of the quotient says [dx]=0 Longleftrightarrow dx∈ d(ker p) Longleftrightarrow ∃ h∈ E: ph=0, dh=dx. This is the first equivalence with its witness made explicit.
Exact mathematical passage
\begin{aligned} &[dx]=0\text{ in }R/d(\ker p),\\ &\exists h\in E:\ p(h)=0,\quad d(h)=d(x),\\ &\exists y\in E:\ d(y)=0,\quad p(y)=p(x). \end{aligned}
\prod_{(i,j)\in\vec E}Use in source · 1 passage
Search aliases: ∏_(i,j)∈vec E prod in vec
Use in source · 05_geometry.tex, line 7
The candidate and residual spaces are finite products with maximum norms: E=∏_i∈ VH, R=∏_(i,j)∈vec EH, (dx)_ij=τ_ijx_j-x_i. Both orientations of an edge are retained. If M bounds the norms of all measured edge maps, then norm d≤ M+1. The empty edge case has the usual zero residual.
Exact mathematical passage
E=\prod_{i\in V}H,\qquad R=\prod_{(i,j)\in\vec E}H,\qquad (dx)_{ij}=\tau_{ij}x_j-x_i.
\prod_{f:v\to w}Use in source · 1 passage
Search aliases: ∏_f:v→ w prod to
Use in source · 02_obstructions.tex, line 6
The compatibility differential. Assume each fiber A_v is an abelian group and each restriction is additive. Set E=∏_v A_v, R=∏_f:v→ wA_v, (δ x)_f=f^*(x_w)-x_v. The group R retains an entry for every arrow. It is not silently identified with a cohomology group.
Exact mathematical passage
E=\prod_v A_v,\qquad R=\prod_{f:v\to w}A_v, \qquad (\delta x)_f=f^*(x_w)-x_v.
\prod_{i<N}Use in source · 1 passage
Search aliases: ∏_i<N prod
Use in source · 02_ambient.tex, line 7
Finite stages and two connecting systems. For N≥1, define A_N=∏_i<NC_p_i, m_N=∏_i<Np_i. The Chinese remainder map sends kbmod m_N to (kbmod p_i)_i<N. It is an isomorphism, so A_N is cyclic and its all-one vector is a generator.
Exact mathematical passage
A_N=\prod_{i<N}C_{p_i},\qquad m_N=\prod_{i<N}p_i.
\prod_{i\in V}Use in source · 1 passage
Search aliases: ∏_i∈ V prod in
Use in source · 05_geometry.tex, line 7
The candidate and residual spaces are finite products with maximum norms: E=∏_i∈ VH, R=∏_(i,j)∈vec EH, (dx)_ij=τ_ijx_j-x_i. Both orientations of an edge are retained. If M bounds the norms of all measured edge maps, then norm d≤ M+1. The empty edge case has the usual zero residual.
Exact mathematical passage
E=\prod_{i\in V}H,\qquad R=\prod_{(i,j)\in\vec E}H,\qquad (dx)_{ij}=\tau_{ij}x_j-x_i.
\prod_{j<N}Use in source · 1 passage
Search aliases: ∏_j<N prod
Use in source · 03_towers.tex, line 408
nce. This proves equality of the two sets. Applying it to every x∈ R_0 gives the density criterion. sourceKS27 There are other sufficient conditions. If the transported details are absolutely summable in the Banach space R_0, their sum exists and the remaining boundary lies in P_∞. If that persistent subspace is zero, the sum equals x. Alternatively, normx-S_Nx≤(∏_j<NnormJ_j)normx_N gives a direct sufficient decay test. A decreasing coarse norm by itself is not enough when restoration gains grow…
Exact mathematical passage
\norm{x-S_Nx}\le\left(\prod_{j<N}\norm{J_j}\right)\norm{x_N}
\prod_{j=0}^{k-1}Use in source · 1 passage
Search aliases: ∏_j=0^k-1 prod
Use in source · 05_geometry.tex, line 42
The triangle inequality and the operator norm bound now give normT_qx_i_m-x_i_0 ≤∑_k=0^m-1normU_knorm(dx)_i_ki_k+1 ≤(∑_k=0^m-1∏_j=0^k-1normτ_i_ji_j+1)normdx_∞. The bracket is γ(q), with the empty product equal to one. Repeated vertices or edges do not affect the cancellation. Reordering the edge maps would, in general, invalidate it. sourceKS40 For isometric maps, γ(q)≤|q|. For arbitrary bounded equivalences, route length cannot replace the amplification gain. Reversing a route can have a differ…
Exact mathematical passage
\norm{T_qx_{i_m}-x_{i_0}} \le\sum_{k=0}^{m-1}\norm{U_k}\norm{(dx)_{i_ki_{k+1}}} \le\left(\sum_{k=0}^{m-1}\prod_{j=0}^{k-1}\norm{\tau_{i_ji_{j+1}}}\right)\norm{dx}_\infty.
\prod_{p\mid m}Use in source · 4 passages
Search aliases: ∏_pmid m prod mid
Use in source · 04_purity.tex, line 57
[One finite-quotient system for every preimage] For every rational subspace W and every m≥1, the coordinate map induces an isomorphism H_W/mH_W ≅ ∏_pmid mC_p. For mmid n, the quotient transition map is the coordinate restriction from primes dividing n to primes dividing m. These quotients are cofinal among all finite quotients, and widehatH_W≅ P. The coordinate map onto the displayed finite product is surjective already on D. Its kernel is mH_W by Theorem~[source reference]. The transition asser…
Exact mathematical passage
H_W/mH_W\ \cong\ \prod_{p\mid m}C_p.
\prod_{u\in U}Use in source · 1 passage
Search aliases: ∏_u∈ U prod in
Use in source · 02a_prescription.tex, line 6
Three maps that must not be confused. Keep the additive view system of Section~[source reference]. For a prescribed domain U, there are three distinct maps: p_U:E⟶∏_u∈ UA_u, r_U:Γ(A)⟶Γ_U(A), o_U:Γ_U(A)⟶ Q_U. The first restricts a raw family, whether or not it is coherent. The second restricts a global coherent family. The third detects whether an already coherent partial family has a global completion. Their domains are different. In particular, surjectivity of the raw coordinate projection does…
Exact mathematical passage
p_U:E\longrightarrow\prod_{u\in U}A_u,\qquad r_U:\Gamma(A)\longrightarrow\Gamma_U(A),\qquad \mathfrak o_U:\Gamma_U(A)\longrightarrow Q_U.
\prod_vUse in source · 5 passages
Search aliases: ∏_v prod
Use in source · 02_obstructions.tex, line 6
The compatibility differential. Assume each fiber A_v is an abelian group and each restriction is additive. Set E=∏_v A_v, R=∏_f:v→ wA_v, (δ x)_f=f^*(x_w)-x_v. The group R retains an entry for every arrow. It is not silently identified with a cohomology group.
Exact mathematical passage
E=\prod_v A_v,\qquad R=\prod_{f:v\to w}A_v, \qquad (\delta x)_f=f^*(x_w)-x_v.
\prod_{v\in W}Use in source · 1 passage
Search aliases: ∏_v∈ W prod in
Use in source · 04_symmetry.tex, line 306
The actual patch pseudofunctor. On a patch W, use K_G(W)=Z^1(V|_W;G|_W)//∏_v∈ WG_v. Restriction keeps the existing cocycle values and gauge values on the smaller patch. It first gives a strict category valued functor on the opposite patch poset. Its standard pseudofunctor form has identity comparison isomorphisms, with the required coherence laws.
Exact mathematical passage
\mathfrak K_G(W)=Z^1(\V|_W;G|_W)//\prod_{v\in W}G_v.
\sum_iUse in source · 2 passages
Search aliases: ∑_i sum
Use in source · 06_cases.tex, line 10
Every edge map is bijective. Around the cycle their composite is translation by -∑_i r_i. A global realization must be fixed by that translation. Thus the example makes local invertibility and global compatibility visibly different without needing a noncommutative group.
Exact mathematical passage
-\sum_i r_i
\sum_{j<N}Use in source · 1 passage
Search aliases: ∑_j<N sum
Use in source · 03_towers.tex, line 337
[Finite reconstruction and nested remainders] For every N, x=Q_Nx+∑_j<NL_jh_j, S_Nx=∑_j<NL_jh_j. Each Q_N is a projection, and for N≤ M, Q_NQ_M=Q_MQ_N=Q_M. The vectors with all details zero are exactly the persistent subspace P_∞={x:Q_Nx=x for every N}. Let D_0=I and D_N=C_N-1⋯ C_0:R_0→ R_N. Thus x_N=D_Nx, Q_N=L_ND_N, and repeated use of C_jJ_j=I gives D_NL_N=I_R_N. This cancellation takes place between adjacent maps with matching domains. No identification of different residual spaces is needed…
Exact mathematical passage
x=Q_Nx+\sum_{j<N}L_jh_j, \qquad S_Nx=\sum_{j<N}L_jh_j.
\sum_{j=0}^{N}Use in source · 1 passage
Search aliases: ∑_j=0^N sum
Use in source · 12_proof_tools.tex, line 173
Submultiplicativity gives normT^j≤θ^j, with the convention T^0=I. The finite partial sums are therefore operator-norm Cauchy, since for M>N, ∑_j=N+1^MT^j ≤∑_j=N+1^Mθ^j≤fracθ^N+11-θ. Their limit Z has norm at most ∑_j≥0θ^j=1/(1-θ). The operator identities for finite sums are (I-T)∑_j=0^NT^j=I-T^N+1, (∑_j=0^NT^j)(I-T)=I-T^N+1. Multiplication by a fixed bounded operator is continuous in operator norm, and normT^N+1≤θ^N+1→0. Passing to the limit proves both inverse identities. When W is the zero spa…
Exact mathematical passage
(I-T)\sum_{j=0}^{N}T^j=I-T^{N+1},\qquad \left(\sum_{j=0}^{N}T^j\right)(I-T)=I-T^{N+1}.
\sum_{j=0}^{N-1}Use in source · 1 passage
Search aliases: ∑_j=0^N-1 sum
Use in source · 12_proof_tools.tex, line 143
residual is zero, choose u=0. Start from r_0=y, repeat this construction, and define r_j+1=r_j-Su_j. Induction yields normr_j≤2^-jnormy, ∑_j≥0normu_j≤frac4cnormy. The series of preimages converges in X by completeness, to a vector x with the displayed norm bound. Continuity of S, together with the finite telescoping equation S(∑_j=0^N-1u_j)=y-r_N, proves Sx=y. Thus we may take C=4/c.
Exact mathematical passage
S\left(\sum_{j=0}^{N-1}u_j\right)=y-r_N,
\sum_{j=0}^{n-m-1}Use in source · 1 passage
Search aliases: ∑_j=0^n-m-1 sum
Use in source · 03_towers.tex, line 140
Put r_m=df^(m). Additivity of d gives r_m+1=r_m-dAr_m. Each r_m is the residual of an actual candidate, so the stated hypothesis applies at every iteration. Induction yields, for j≥0, normr_m+j≤ q^jnormr_m, normf^(m+j+1)-f^(m+j) ≤norm A q^jnormr_m. For n>m, the finite triangle inequality therefore gives normf^(n)-f^(m) ≤norm A normr_m∑_j=0^n-m-1q^j ≤fracnorm A1-qnormr_m. Since normr_m≤ q^mnormdf tends to zero, this is a Cauchy estimate. Completeness of E supplies a limit, denoted P_Af. Passing n…
Exact mathematical passage
\norm{f^{(n)}-f^{(m)}} \le\norm A\,\norm{r_m}\sum_{j=0}^{n-m-1}q^j \le\frac{\norm A}{1-q}\norm{r_m}.
\sum_{j=0}^{\infty}Use in source · 1 passage
Search aliases: ∑_j=0^∞ sum infty
Use in source · 12_proof_tools.tex, line 156
Neumann inversion with both inverse identities. [A strict perturbation radius] Let W be a real Banach space and T:W→ W bounded linear, with normT≤θ<1 and θ≥0. Then Z=∑_j=0^∞T^j exists in operator norm and satisfies (I-T)Z=Z(I-T)=I, normZ≤frac11-θ. First recall why the space of bounded operators on W is complete. An operator-norm Cauchy sequence A_n makes A_nx Cauchy for each x, and hence defines Ax=lim_nA_nx. Pointwise limits preserve linearity. The sequence of operator norms is bounded, so the …
Exact mathematical passage
Z=\sum_{j=0}^{\infty}T^j
\sum_{j=1}^JUse in source · 1 passage
Search aliases: ∑_j=1^J sum
Use in source · 05f_bounded_realization.tex, line 37
For each finite J, Bessel's inequality and passage to the limit give ∑_j=1^Jc_j^2 =lim_k∑_j=1^J⟨ a_n_k,e_j⟩^2≤ M^2. The nondecreasing partial sums are therefore bounded. Orthogonality makes the partial vector sums ∑_j≤ Jc_je_j Cauchy, so completeness defines a∈ H_0 with norm a≤ M.
Exact mathematical passage
\sum_{j=1}^Jc_j^2 =\lim_k\sum_{j=1}^J\langle a_{n_k},e_j\rangle^2\le M^2.
\sum_{j=1}^{n}Use in source · 2 passages
Search aliases: ∑_j=1^n sum
Use in source · 05c_iteration.tex, line 198
Three hypotheses that cannot simply be dropped. Summable movement. In maximum-norm R^3, let f_n=(s,H_n,(n+1)^-1), H_n=∑_j=1^nfrac1j. The first coordinate is protected and the residual is the last coordinate. It tends to zero with the same ratios as the polynomial example. The movement equals 1/(n+1), however, and the unprotected coordinate diverges. Here C_n=1, so ∑ C_nQ_n is the divergent harmonic series. Residual decay and semantic preservation hold, but exactification of this path fails.
Exact mathematical passage
f_n=(s,H_n,(n+1)^{-1}),\qquad H_n=\sum_{j=1}^{n}\frac1j.
\sum_{j=N+1}^{M}Use in source · 1 passage
Search aliases: ∑_j=N+1^M sum
Use in source · 12_proof_tools.tex, line 168
Submultiplicativity gives normT^j≤θ^j, with the convention T^0=I. The finite partial sums are therefore operator-norm Cauchy, since for M>N, ∑_j=N+1^MT^j ≤∑_j=N+1^Mθ^j≤fracθ^N+11-θ. Their limit Z has norm at most ∑_j≥0θ^j=1/(1-θ). The operator identities for finite sums are (I-T)∑_j=0^NT^j=I-T^N+1, (∑_j=0^NT^j)(I-T)=I-T^N+1. Multiplication by a fixed bounded operator is continuous in operator norm, and normT^N+1≤θ^N+1→0. Passing to the limit proves both inverse identities. When W is the zero spa…
Exact mathematical passage
\left\|\sum_{j=N+1}^{M}T^j\right\| \le\sum_{j=N+1}^{M}\theta^j\le\frac{\theta^{N+1}}{1-\theta}.
\sum_{j\le J}Use in source · 1 passage
Search aliases: ∑_j≤ J sum le
Use in source · 05f_bounded_realization.tex, line 41
For each finite J, Bessel's inequality and passage to the limit give ∑_j=1^Jc_j^2 =lim_k∑_j=1^J⟨ a_n_k,e_j⟩^2≤ M^2. The nondecreasing partial sums are therefore bounded. Orthogonality makes the partial vector sums ∑_j≤ Jc_je_j Cauchy, so completeness defines a∈ H_0 with norm a≤ M.
Exact mathematical passage
\sum_{j\le J}c_je_j
\sum_{j\ge0}Use in source · 2 passages
Search aliases: ∑_j≥0 sum ge
Use in source · 12_proof_tools.tex, line 138
λ=2normr/c. The vector r/λ has norm c/2, so [source reference] supplies z∈ B_X with normSz-r/λ≤ c/4. Then u=λ z satisfies normu≤ 2normr/c, normr-Su≤normr/2. If the residual is zero, choose u=0. Start from r_0=y, repeat this construction, and define r_j+1=r_j-Su_j. Induction yields normr_j≤2^-jnormy, ∑_j≥0normu_j≤frac4cnormy. The series of preimages converges in X by completeness, to a vector x with the displayed norm bound. Continuity of S, together with the finite telescoping equation S(∑_j=0^N…
Exact mathematical passage
\norm{r_j}\le2^{-j}\norm{y},\qquad \sum_{j\ge0}\norm{u_j}\le\frac{4}{c}\norm{y}.
\sum_{k=0}^{\infty}Use in source · 1 passage
Search aliases: ∑_k=0^∞ sum infty
Use in source · 05c_iteration.tex, line 116
Residual bursts can be compatible with convergence. Set q_2k=2, q_2k+1=1/8. Then Q_2k=4^-k, Q_2k+1=2·4^-k. For f_n=(s,Q_n), the residual doubles at the start of every pair of steps. Choose C_2k=1 and C_2k+1=7/8. These give equality in the movement bound, and ∑_n=0^∞C_nQ_n =∑_k=0^∞(1+frac74)4^-k =frac113. Thus f_n→(s,0) despite infinitely many increases. Starting at an even index 2k, the remaining movement budget is frac1134^-k. At an odd index 2k+1, it is frac83 4^-k. These bounds are larger tha…
Exact mathematical passage
\sum_{n=0}^{\infty}C_nQ_n =\sum_{k=0}^{\infty}\left(1+\frac74\right)4^{-k} =\frac{11}{3}.
\sum_{k=a}^{b-1}Use in source · 1 passage
Search aliases: ∑_k=a^b-1 sum
Use in source · 03b_repair_transport.tex, line 143
For a block from a to b, its value on a residual family r is (C_a,br)=∑_k=a^b-1 ρ^Y_a,k r_k ρ^X_k+1,b. For the schedule ϕ, set (C_ϕ r)_n=C_ϕ(n),ϕ(n+1)r. Every summand maps X_b into Y_a. These surrounding restrictions are necessary when the spaces change with the level. For an actual differential, the sum telescopes to C_a,b(∂ f)=ρ^Y_a,bf_b-f_aρ^X_a,b. The dyadic coarsener in Section~[source reference] is the case b=a+2.
Exact mathematical passage
\label{eq:block-defect-sum} (C_{a,b}r)=\sum_{k=a}^{b-1} \rho^Y_{a,k}\,r_k\,\rho^X_{k+1,b}.
\sum_{k=n}^{m-1}Use in source · 1 passage
Search aliases: ∑_k=n^m-1 sum
Use in source · 05c_iteration.tex, line 29
∑_n=0^∞b_n<∞, and dist(f_n,f_n+1)≤ b_n, ω(f_n)⟶0. Then there is a limit x∈ M satisfying f_n⟶ x, ω(x)=0, dist(f_n,x)≤∑_k=n^∞b_k. For integers m>n, apply the triangle inequality successively along the finite chain f_n,f_n+1,…,f_m. It gives dist(f_n,f_m)≤∑_k=n^m-1dist(f_k,f_k+1) ≤∑_k=n^m-1b_k≤∑_k=n^∞b_k. Given ε>0, convergence of the nonnegative series supplies N for which the final tail is below ε whenever n≥ N. Interchanging m,n when necessary proves the Cauchy condition for all pairs of indices …
Exact mathematical passage
\dist(f_n,f_m)\le\sum_{k=n}^{m-1}\dist(f_k,f_{k+1}) \le\sum_{k=n}^{m-1}b_k\le\sum_{k=n}^{\infty}b_k.
\sum_{k=n}^{\infty}Use in source · 4 passages
Search aliases: ∑_k=n^∞ sum infty
Use in source · 05c_iteration.tex, line 22
[Summable movement gives exactification] Suppose b_n≥0, ∑_n=0^∞b_n<∞, and dist(f_n,f_n+1)≤ b_n, ω(f_n)⟶0. Then there is a limit x∈ M satisfying f_n⟶ x, ω(x)=0, dist(f_n,x)≤∑_k=n^∞b_k. For integers m>n, apply the triangle inequality successively along the finite chain f_n,f_n+1,…,f_m. It gives dist(f_n,f_m)≤∑_k=n^m-1dist(f_k,f_k+1) ≤∑_k=n^m-1b_k≤∑_k=n^∞b_k. Given ε>0, convergence of the nonnegative series supplies N for which the final tail is below ε whenever n≥ N. Interchanging m,n when necessa…
Exact mathematical passage
f_n\longrightarrow x,\qquad\omega(x)=0,\qquad \dist(f_n,x)\le\sum_{k=n}^{\infty}b_k.
\sum_nUse in source · 1 passage
Search aliases: ∑_n sum
Use in source · 05c_iteration.tex, line 58
[Variable-rate exactification] If Q_n→0 and ∑_n C_nQ_n<∞, then (f_n) has an exact limit and dist(f_n,x)≤ω(f_0)∑_k=n^∞C_kQ_k. For constant q_n=q∈[0,1) and constant C_n=C, the sharper restarted estimate is dist(f_n,x)≤fracC1-q ω(f_n). The estimate ω(f_n)≤ Q_nω(f_0) follows by induction. At n=0 it is equality. If it holds at n, nonnegativity of q_n permits multiplication and gives ω(f_n+1)≤ q_nω(f_n) ≤ q_nQ_nω(f_0)=Q_n+1ω(f_0). Because Q_n→0 and ω(f_0) is a fixed finite number, the residual tends t…
Exact mathematical passage
\sum_n C_nQ_n<\infty
\sum_{n=0}^\inftyUse in source · 1 passage
Search aliases: ∑_n=0^∞ sum infty
Use in source · 05_geometry.tex, line 503
The perturbed map is surjective, with D'A'=I, normA'≤fraca1-aε. Its exact projection satisfies normx-q'x≤fraca1-aεnormD'x, normq-q'≤fracaε1-aε. The map T=(D-D')A acts on the Banach target R and satisfies norm T≤ aε<1. The operator series Z=∑_n=0^∞ T^n converges in operator norm and obeys norm Z≤(1-aε)^-1. Multiplying its finite partial sums on either side by I-T gives I-T^N+1. Passing to the operator-norm limit proves (I-T)Z=Z(I-T)=I. Appendix~[source reference] supplies the full series argument…
Exact mathematical passage
Z=\sum_{n=0}^\infty T^n
\sum_{n=0}^{\infty}Use in source · 2 passages
Search aliases: ∑_n=0^∞ sum infty
Use in source · 05c_iteration.tex, line 17
[Summable movement gives exactification] Suppose b_n≥0, ∑_n=0^∞b_n<∞, and dist(f_n,f_n+1)≤ b_n, ω(f_n)⟶0. Then there is a limit x∈ M satisfying f_n⟶ x, ω(x)=0, dist(f_n,x)≤∑_k=n^∞b_k. For integers m>n, apply the triangle inequality successively along the finite chain f_n,f_n+1,…,f_m. It gives dist(f_n,f_m)≤∑_k=n^m-1dist(f_k,f_k+1) ≤∑_k=n^m-1b_k≤∑_k=n^∞b_k. Given ε>0, convergence of the nonnegative series supplies N for which the final tail is below ε whenever n≥ N. Interchanging m,n when necessa…
Exact mathematical passage
\sum_{n=0}^{\infty}b_n<\infty
\sum_{n=1}^mUse in source · 1 passage
Search aliases: ∑_n=1^m sum
Use in source · 05i_diagonal_regularization.tex, line 44
For n≤ m, the residual square is at most n^2/m^4. The sum-of-squares estimate ∑_n=1^m n^2=fracm(m+1)(2m+1)6≤ m^3 holds for m≥1, so this part contributes at most 1/m. For n>m, each residual square is at most 1/n^2, whose tail is at most 1/m. This proves B_m≤2/m.
Exact mathematical passage
\sum_{n=1}^m n^2=\frac{m(m+1)(2m+1)}6\le m^3
\sum_{n=1}^NUse in source · 1 passage
Search aliases: ∑_n=1^N sum
Use in source · 05i_diagonal_regularization.tex, line 92
1: control the infinite Riemann sums uniformly. Define f(x)=frac1(1+x^2)^2, g(x)=fracx^2(1+x^2)^2. Then A_m/m=m^-1∑_n≥1f(n/m) and mB_m=m^-1∑_n≥1g(n/m). Fix L≥2 and put N=lfloor mLrfloor. On [0,L], both functions are uniformly continuous. If ω_f(h) is a modulus of continuity there, comparison on the N intervals of length 1/m gives |frac1m∑_n=1^N f(n/m)-∫_0^N/mf(x) dx| ≤ Lω_f(1/m). The missing interval [N/m,L] has length below 1/m and a bounded integrand, so it contributes a term tending to zero. …
Exact mathematical passage
\left|\frac1m\sum_{n=1}^N f(n/m)-\int_0^{N/m}f(x)\,dx\right| \le L\omega_f(1/m).
\sum_{n=1}^{N}Use in source · 2 passages
Search aliases: ∑_n=1^N sum
Use in source · 05e_escape.tex, line 86
Escape is not an accident of this particular algorithm. Suppose an arbitrary protected candidate (u,1) satisfies normd(u,1)≤ε. On its first N coordinates, ∑_n=1^N|u_n+1|^2 ≤ N^2∑_n=1^Nfrac|u_n+1|^2n^2 ≤ N^2ε^2. The reverse triangle inequality in R^N gives norm u≥sqrt N-Nε (N≥1). If a sequence of protected candidates has residual tending to zero, then for each fixed N, its eventual correction norms are at least, for example, sqrt N/2. Taking N arbitrarily large shows that these norms must tend to…
Exact mathematical passage
\sum_{n=1}^{N}|u_n+1|^2 \le N^2\sum_{n=1}^{N}\frac{|u_n+1|^2}{n^2} \le N^2\varepsilon^2.
\sum_{n>K}Use in source · 2 passages
Search aliases: ∑_n>K sum
Use in source · 05i_diagonal_regularization.tex, line 59
Finally, for K≥ m, both omitted tails consist of nonnegative terms. Bound the first by m^4∑_n>Kn^-4 and the second by ∑_n>Kn^-2. The same upper integral comparisons give the two certified remainders in [source reference]. No finite truncation is substituted for an equality of infinite energies. writtenWR10
Exact mathematical passage
m^4\sum_{n>K}n^{-4}
\sum_{n>m}Use in source · 2 passages
Search aliases: ∑_n>m sum
Use in source · 05i_diagonal_regularization.tex, line 38
For 1≤ n≤ m, m^2/(m^2+n^2)≥1/2. There are m such coordinates, so A_m≥ m/4. For the upper bound, the first m squares are at most one, and the decreasing function x^-4 gives ∑_n>mfrac1n^4≤∫_m^∞ x^-4 dx=frac13m^3. Thus A_m≤ m+m^4/(3m^3)=4m/3.
Exact mathematical passage
\sum_{n>m}\frac1{n^4}\le\int_m^\infty x^{-4}\,dx=\frac1{3m^3}.
\sum_{n>N}Use in source · 6 passages
Search aliases: ∑_n>N sum
Use in source · 05e_escape.tex, line 59
1/n · n>N, norma_N^2=N, normf_N^2=N+1. For every N≥1, telescoping comparisons give frac1N+1≤normd(f_N)^2 =∑_n>Nfrac1n^2≤frac1N. Indeed 1/[n(n+1)]≤1/n^2≤1/[n(n-1)] for n≥2. Summing the two telescoping bounds proves [source reference] without numerically evaluating an infinite series.
Exact mathematical passage
\label{eq:escape-tail} \frac1{N+1}\le\norm{d(f_N)}^2 =\sum_{n>N}\frac1{n^2}\le\frac1N.
\sum_{n\ge1}Use in source · 5 passages
Search aliases: ∑_n≥1 sum ge
Use in source · 05e_escape.tex, line 7
The complete spaces and the preservation rule. Index sequences by the positive integers. Set E=ℓ^2(N_>0;R)⊕R, R=ℓ^2(N_>0;R), norm(u,t)_E^2=∑_n≥1|u_n|^2+|t|^2. Let w=(1/n)_n≥1. It belongs to ℓ^2, because ∑_n≥1frac1n^2 ≤1+∑_n≥2frac1n(n-1)=2. Define the defect and protected observable by d(u,t)_n=fracu_n+tn, p(u,t)=t. The diagonal map T:u↦(u_n/n)_n has operator norm one. Since d(u,t)=Tu+tw, normd(u,t)≤norm u+|t|norm w ≤sqrt1+norm w^2 norm(u,t)_E. Thus both d and p are continuous linear maps between…
Exact mathematical passage
E=\ell^2(\N_{>0};\R)\oplus\R,\qquad R=\ell^2(\N_{>0};\R),\qquad \norm{(u,t)}_E^2=\sum_{n\ge1}|u_n|^2+|t|^2.
\sum_{n\ge2}Use in source · 1 passage
Search aliases: ∑_n≥2 sum ge
Use in source · 05e_escape.tex, line 13
The complete spaces and the preservation rule. Index sequences by the positive integers. Set E=ℓ^2(N_>0;R)⊕R, R=ℓ^2(N_>0;R), norm(u,t)_E^2=∑_n≥1|u_n|^2+|t|^2. Let w=(1/n)_n≥1. It belongs to ℓ^2, because ∑_n≥1frac1n^2 ≤1+∑_n≥2frac1n(n-1)=2. Define the defect and protected observable by d(u,t)_n=fracu_n+tn, p(u,t)=t. The diagonal map T:u↦(u_n/n)_n has operator norm one. Since d(u,t)=Tu+tw, normd(u,t)≤norm u+|t|norm w ≤sqrt1+norm w^2 norm(u,t)_E. Thus both d and p are continuous linear maps between…
Exact mathematical passage
\sum_{n\ge1}\frac1{n^2} \le1+\sum_{n\ge2}\frac1{n(n-1)}=2.
\sum_{p\mid m}Use in source · 1 passage
Search aliases: ∑_pmid m sum mid
Use in source · 11_atlas.tex, line 62
Finite quotient representatives. For any target g and positive multiplier m, define the finite vector d_m(g)=∑_pmid mg_p e_p. Then g-d_m(g) has zero coordinates at every prime dividing m. The root theorem provides a witness that it belongs to mG_A. Thus d_m(g) is an exact finite representative of the coset of g.
Exact mathematical passage
d_m(g)=\sum_{p\mid m}g_p e_p.
\setminusUse in source · 5 passages
Search aliases: ∖ setminus
Use in source · 05d_floors.tex, line 40
Ω∈overline B∖ B · Nonzero · Arbitrarily small corrected residuals exist, but no exact cancellation exists.
Exact mathematical passage
\Omega\in\overline B\setminus B
\cupUse in source · 2 passages
Search aliases: ∪ cup
Use in source · 03_spectrum.tex, line 25
e factors. Every element of H lies in D_B, and every finite sum of the factors in B lies in H. Hence H=D_B. Uniqueness follows by testing membership of e_p_i. Proposition~[source reference] proves full invariance. Finally σ_p_i(D_B)=1 exactly when i∈ B, and these values are isomorphism invariants. The lattice operations have literal set-theoretic descriptions: D_B+D_C=D_B∪ C, D_B∩ D_C=D_B∩ C, D=D_B⊕ D_ N∖ B. The infinite support choice is therefore already visible in an elementary algebraic latt…
Exact mathematical passage
D_B+D_C=D_{B\cup C},\qquad D_B\cap D_C=D_{B\cap C}, \qquad D=D_B\oplus D_{\N\setminus B}.
\intUse in source · 5 passages
Search aliases: ∫ int
Use in source · 01_views.tex, line 83
The category of elements ∫_V F and Γ(F) also have different roles. The former organizes individual appearances and their transitions. The latter selects coherent families across all views. The category of elements serves here as explanatory notation, while the Lean formalization directly represents families and their coherence.
Exact mathematical passage
\int_\V F
\int_0^{N/m}Use in source · 1 passage
Search aliases: ∫_0^N/m int
Use in source · 05i_diagonal_regularization.tex, line 92
1: control the infinite Riemann sums uniformly. Define f(x)=frac1(1+x^2)^2, g(x)=fracx^2(1+x^2)^2. Then A_m/m=m^-1∑_n≥1f(n/m) and mB_m=m^-1∑_n≥1g(n/m). Fix L≥2 and put N=lfloor mLrfloor. On [0,L], both functions are uniformly continuous. If ω_f(h) is a modulus of continuity there, comparison on the N intervals of length 1/m gives |frac1m∑_n=1^N f(n/m)-∫_0^N/mf(x) dx| ≤ Lω_f(1/m). The missing interval [N/m,L] has length below 1/m and a bounded integrand, so it contributes a term tending to zero. …
Exact mathematical passage
\left|\frac1m\sum_{n=1}^N f(n/m)-\int_0^{N/m}f(x)\,dx\right| \le L\omega_f(1/m).
\int_0^{\pi/2}Use in source · 1 passage
Search aliases: ∫_0^π/2 int pi
Use in source · 05i_diagonal_regularization.tex, line 111
Step 2: evaluate the integrals. The substitution x=tanθ gives dx=sec^2θ dθ and 1+x^2=sec^2θ, so ∫_0^∞ f(x) dx=∫_0^π/2cos^2θ dθ=fracπ4, ∫_0^∞ g(x) dx=∫_0^π/2sin^2θ dθ=fracπ4. Step 1 now proves the two stated asymptotics.
Exact mathematical passage
\int_0^\infty f(x)\,dx=\int_0^{\pi/2}\cos^2\theta\,d\theta=\frac\pi4, \qquad \int_0^\infty g(x)\,dx=\int_0^{\pi/2}\sin^2\theta\,d\theta=\frac\pi4.
\int_0^\inftyUse in source · 1 passage
Search aliases: ∫_0^∞ int infty
Use in source · 05i_diagonal_regularization.tex, line 111
Step 2: evaluate the integrals. The substitution x=tanθ gives dx=sec^2θ dθ and 1+x^2=sec^2θ, so ∫_0^∞ f(x) dx=∫_0^π/2cos^2θ dθ=fracπ4, ∫_0^∞ g(x) dx=∫_0^π/2sin^2θ dθ=fracπ4. Step 1 now proves the two stated asymptotics.
Exact mathematical passage
\int_0^\infty f(x)\,dx=\int_0^{\pi/2}\cos^2\theta\,d\theta=\frac\pi4, \qquad \int_0^\infty g(x)\,dx=\int_0^{\pi/2}\sin^2\theta\,d\theta=\frac\pi4.
\int_{m+1}^\inftyUse in source · 1 passage
Search aliases: ∫_m+1^∞ int infty
Use in source · 05i_diagonal_regularization.tex, line 54
For the lower bound, n>m implies m^2+n^2≤2n^2, hence fracn^2(m^2+n^2)^2≥frac14n^2. The other direction of the decreasing-function integral comparison gives ∑_n>mfrac1n^2≥∫_m+1^∞ x^-2 dx=frac1m+1. This proves B_m≥1/(4(m+1)).
Exact mathematical passage
\sum_{n>m}\frac1{n^2}\ge\int_{m+1}^\infty x^{-2}\,dx=\frac1{m+1}.
\int_m^\inftyUse in source · 1 passage
Search aliases: ∫_m^∞ int infty
Use in source · 05i_diagonal_regularization.tex, line 38
For 1≤ n≤ m, m^2/(m^2+n^2)≥1/2. There are m such coordinates, so A_m≥ m/4. For the upper bound, the first m squares are at most one, and the decreasing function x^-4 gives ∑_n>mfrac1n^4≤∫_m^∞ x^-4 dx=frac13m^3. Thus A_m≤ m+m^4/(3m^3)=4m/3.
Exact mathematical passage
\sum_{n>m}\frac1{n^4}\le\int_m^\infty x^{-4}\,dx=\frac1{3m^3}.
\int_\VUse in source · 1 passage
Search aliases: ∫_V int V
Use in source · 01_views.tex, line 83
The category of elements ∫_V F and Γ(F) also have different roles. The former organizes individual appearances and their transitions. The latter selects coherent families across all views. The category of elements serves here as explanatory notation, while the Lean formalization directly represents families and their coherence.
Exact mathematical passage
\int_\V F
\neUse in source · 32 passages
Search aliases: ≠ ne
Use in source · 02_obstructions.tex, line 44
[Extension criterion] For every coherent partial section, o_U(s)=0 Longleftrightarrow Ext_U(s)≠∅. Let π_U:R→ R/B_U be the quotient homomorphism. By the definition of a quotient by a subgroup, π_U(r)=0 Longleftrightarrow r∈ B_U. Here B_U is the image of ker p_U under δ. Consequently membership of δwidetilde s means the existence of a specific h∈ E satisfying both p_Uh=0 and δ h=δwidetilde s. The subgroup need not be closed in any topology. No topology has been introduced.
Exact mathematical passage
\mathfrak o_U(s)=0\quad\Longleftrightarrow\quad\Ext_U(s)\ne\varnothing.
\subsetUse in source · 1 passage
Search aliases: ⊂ subset
Use in source · 05f_bounded_realization.tex, line 61
[Bounded approximate feasibility] For a bounded linear map T:H→ Y between real Hilbert spaces and r∈ Y, the following are equivalent: · r∈ran T, · there is a bounded sequence (a_n)⊂ H with normTa_n-r⟶0. If all norma_n≤ M, an exact solution exists with norm at most M. If Ta=r, the constant sequence a_n=a is bounded and has zero residual. For the converse, let norma_n≤ M and Ta_n→ r in norm. By the preceding lemma there is a subsequence a_n_k converging weakly to a∈ H with norm a≤ M.
Exact mathematical passage
\begin{split} &r\in\ran T,\\ &\text{there is a bounded sequence }(a_n)\subset H \text{ with }\norm{Ta_n-r}\longrightarrow0. \end{split}
\supseteqUse in source · 1 passage
Search aliases: ⊇ supseteq
Use in source · 05f_bounded_realization.tex, line 98
Suppose now that A_1⊇ A_2⊇⋯ are nonempty closed affine subsets of a real Hilbert correction space H. A typical A_n consists of corrections satisfying the first n bounded linear observation equations. The sets are affine because the equations are linear, and closed because the observations are continuous.
Exact mathematical passage
A_1\supseteq A_2\supseteq\cdots
\oplusUse in source · 8 passages
Search aliases: ⊕ oplus
Use in source · 05e_escape.tex, line 7
The complete spaces and the preservation rule. Index sequences by the positive integers. Set E=ℓ^2(N_>0;R)⊕R, R=ℓ^2(N_>0;R), norm(u,t)_E^2=∑_n≥1|u_n|^2+|t|^2. Let w=(1/n)_n≥1. It belongs to ℓ^2, because ∑_n≥1frac1n^2 ≤1+∑_n≥2frac1n(n-1)=2. Define the defect and protected observable by d(u,t)_n=fracu_n+tn, p(u,t)=t. The diagonal map T:u↦(u_n/n)_n has operator norm one. Since d(u,t)=Tu+tw, normd(u,t)≤norm u+|t|norm w ≤sqrt1+norm w^2 norm(u,t)_E. Thus both d and p are continuous linear maps between…
Exact mathematical passage
E=\ell^2(\N_{>0};\R)\oplus\R,\qquad R=\ell^2(\N_{>0};\R),\qquad \norm{(u,t)}_E^2=\sum_{n\ge1}|u_n|^2+|t|^2.
\bigoplus_{i\in B}Use in source · 1 passage
Search aliases: ⊕_i∈ B bigoplus in
Use in source · 03_spectrum.tex, line 7
The entire subgroup lattice. For B⊆ N, set D_B=⊕_i∈ BC_p_i≤ D. A prime spectrum records which prime orders occur. Define σ_p(M)=1 when M contains a nonzero element annihilated by p, and set it to zero otherwise. Since p is prime, a nonzero such element has order exactly p.
Exact mathematical passage
D_B=\bigoplus_{i\in B}C_{p_i}\le D.
\bigoplus_{i\in I}Use in source · 1 passage
Search aliases: ⊕_i∈ I bigoplus in
Use in source · 01_contracts.tex, line 65
[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If M_j≠0 and j is not reachable, P_j annihilates the entire hull but not M. An element of the direct sum has finite support. Applying one primitive oper…
Exact mathematical passage
M=\bigoplus_{i\in I}M_i
\bigoplus_{j\in\operatorname{Reach}_E(F)}Use in source · 1 passage
Search aliases: ⊕_j∈Reach_E(F) bigoplus in operatorname
Use in source · 01_contracts.tex, line 66
[Reachability obstruction] Let M=⊕_i∈ IM_i be a direct sum of abelian groups with coordinate projections P_i. Suppose every primitive operator T is an endomorphism and its transfer block P_jTP_i vanishes unless i→ j is an allowed edge of a directed graph E. If F is the union of the supports of a seed family S, then Hull_ O(S)⊆ ⊕_j∈Reach_E(F)M_j. If M_j≠0 and j is not reachable, P_j annihilates the entire hull but not M. An element of the direct sum has finite support. Applying one primitive oper…
Exact mathematical passage
\Hull_{\mathcal O}(S)\subseteq \bigoplus_{j\in\operatorname{Reach}_E(F)}M_j.
\bigoplus_{p\mid m}Use in source · 1 passage
Search aliases: ⊕_pmid m bigoplus mid
Use in source · 06_roots.tex, line 43
[All roots and their number] If my=g is solvable, its solution set is the coset y_0+G_A[m], G_A[m]=⊕_pmid mC_p. It has exactly rad(m) elements. Two solutions differ by an element annihilated by m, and adding such an element to a solution gives another. The ambient product has a nonzero m-torsion coordinate only when pmid m. All those finite coordinate factors lie in D⊆ G_A. The kernel is therefore the displayed finite direct sum, whose cardinal is rad(m). Repeated prime powers do not enlarge the…
Exact mathematical passage
y_0+G_A[m],\qquad G_A[m]=\bigoplus_{p\mid m}C_p.
\perp_MUse in source · 1 passage
Search aliases: ⊥_M perp
Use in source · 05_geometry.tex, line 208
If ran B is closed, Theorem~[source reference] applied on M gives a bounded inverse of B on (ker B)^⊥_M. The vector h_* is exactly that inverse applied to Dx, so its norm obeys the required bound.
Exact mathematical passage
(\ker B)^{\perp_M}
\bigcap_{(i,j)\in\vec E}Use in source · 1 passage
Search aliases: ⋂_(i,j)∈vec E bigcap in vec
Use in source · 05_geometry.tex, line 58
Holonomy is the remaining exactness condition. Write T_i=T_p_i. Reconstruction from a proposed root value is (R_pz)_i=T_i^-1z. For each directed edge define the root loop operator K_ij=T_iτ_ijT_j^-1, K=⋂_(i,j)∈vec Eker(K_ij-I). Each K_ij is the transport around an actual rooted loop. No flatness assumption has been made.
Exact mathematical passage
K_{ij}=T_i\tau_{ij}T_j^{-1},\qquad K=\bigcap_{(i,j)\in\vec E}\ker(K_{ij}-I).
\bigcup_NUse in source · 1 passage
Search aliases: ⋃_N bigcup
Use in source · 03_towers.tex, line 386
Convergence has its own theorem. Call x norm recoverable when Q_Nx→0. This is equivalent to S_Nx→ x, but it does not follow merely from vanishing persistent subspace. Let F=⋃_Nker Q_N be the finite core.
Exact mathematical passage
\mathcal F=\bigcup_N\ker Q_N
\rangle^2Use in source · 1 passage
Search aliases: ⟩^2 rangle
Use in source · 05f_bounded_realization.tex, line 37
For each finite J, Bessel's inequality and passage to the limit give ∑_j=1^Jc_j^2 =lim_k∑_j=1^J⟨ a_n_k,e_j⟩^2≤ M^2. The nondecreasing partial sums are therefore bounded. Orthogonality makes the partial vector sums ∑_j≤ Jc_je_j Cauchy, so completeness defines a∈ H_0 with norm a≤ M.
Exact mathematical passage
\sum_{j=1}^Jc_j^2 =\lim_k\sum_{j=1}^J\langle a_{n_k},e_j\rangle^2\le M^2.
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