Notation by work
Prime-Contact Coherence Calculus
All notation · Glossary · Source coverage
688 symbol spellings in 1371 mathematical passages across 13 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 688 entries are available below.
=Notation table · 453 passages
Search aliases: =
Notation table · tail.tex, line 444
R=rad(abc) · Activated prime support of a primitive triple
Exact mathematical passage
R=\rad(abc)
aNotation table · 245 passages
Search aliases: a
Notation table · tail.tex, line 444
R=rad(abc) · Activated prime support of a primitive triple
Exact mathematical passage
R=\rad(abc)
BNotation table · 46 passages
Search aliases: B
Notation table · tail.tex, line 453
B_d(n) · Exact-order divisor-polyphase bandwidth packet
Exact mathematical passage
B_d(n)
bNotation table · 229 passages
Search aliases: b
Notation table · tail.tex, line 444
R=rad(abc) · Activated prime support of a primitive triple
Exact mathematical passage
R=\rad(abc)
B_dNotation table · 10 passages
Search aliases: B_d
Notation table · tail.tex, line 453
B_d(n) · Exact-order divisor-polyphase bandwidth packet
Exact mathematical passage
B_d(n)
CNotation table · 120 passages
Search aliases: C
Notation table · overview.tex, line 914
C · Coherence quotient · Depth created by genuine simultaneous cancellation among the normalized cross-prime logarithmic channels
Exact mathematical passage
C
cNotation table · 268 passages
Search aliases: c
Notation table · tail.tex, line 444
R=rad(abc) · Activated prime support of a primitive triple
Exact mathematical passage
R=\rad(abc)
\mathrm{coh}Notation table · 24 passages
Search aliases: coh mathrm
Notation table · tail.tex, line 462
Λ_coh · Kernel lattice of the coherence character
Exact mathematical passage
\Lambda_{\mathrm{coh}}
dNotation table · 27 passages
Search aliases: d
Notation table · tail.tex, line 453
B_d(n) · Exact-order divisor-polyphase bandwidth packet
Exact mathematical passage
B_d(n)
HNotation table · 42 passages
Search aliases: H
Notation table · tail.tex, line 451
H(k) · Exact shortest height in obstruction coset k
Exact mathematical passage
H(k)
\JcalNotation table · 12 passages
Search aliases: Jcal
Notation table · tail.tex, line 454
Jcal_c · Filtered cusp-contact module or complex
Exact mathematical passage
\Jcal_c
\Jcal_cNotation table · 2 passages
Search aliases: Jcal_c Jcal
Notation table · tail.tex, line 454
Jcal_c · Filtered cusp-contact module or complex
Exact mathematical passage
\Jcal_c
KNotation table · 43 passages
Search aliases: K
Notation table · overview.tex, line 912
K · Kummer preload · Depth removed by a common power in the signed input valuation vector
Exact mathematical passage
K
kNotation table · 78 passages
Search aliases: k
Notation table · tail.tex, line 451
H(k) · Exact shortest height in obstruction coset k
Exact mathematical passage
H(k)
LNotation table · 30 passages
Search aliases: L
Notation table · tail.tex, line 449
L · Additive packet lattice
Exact mathematical passage
L
L_0Notation table · 9 passages
Search aliases: L_0
Notation table · tail.tex, line 450
L_0 · Wronskian-zero sublattice
Exact mathematical passage
L_0
\LcalNotation table · 6 passages
Search aliases: Lcal
Notation table · tail.tex, line 457
Lcal_P,Q · Cross-prime logarithmic filter matrix
Exact mathematical passage
\Lcal_{P,Q}
\Lcal_{P,Q}Notation table · 3 passages
Search aliases: Lcal_P,Q Lcal
Notation table · tail.tex, line 457
Lcal_P,Q · Cross-prime logarithmic filter matrix
Exact mathematical passage
\Lcal_{P,Q}
MNotation table · 95 passages
Search aliases: M
Notation table · tail.tex, line 446
M=c/R_c · Output powerful part or contact modulus
Exact mathematical passage
M=c/R_c
nNotation table · 83 passages
Search aliases: n
Notation table · tail.tex, line 453
B_d(n) · Exact-order divisor-polyphase bandwidth packet
Exact mathematical passage
B_d(n)
\overline HNotation table · 18 passages
Search aliases: overline H overline
Notation table · tail.tex, line 452
overline H · Stable real relaxation of the obstruction height
Exact mathematical passage
\overline H
PNotation table · 39 passages
Search aliases: P
Notation table · tail.tex, line 457
Lcal_P,Q · Cross-prime logarithmic filter matrix
Exact mathematical passage
\Lcal_{P,Q}
QNotation table · 22 passages
Search aliases: Q
Notation table · tail.tex, line 457
Lcal_P,Q · Cross-prime logarithmic filter matrix
Exact mathematical passage
\Lcal_{P,Q}
RNotation table · 101 passages
Search aliases: R
Notation table · tail.tex, line 444
R=rad(abc) · Activated prime support of a primitive triple
Exact mathematical passage
R=\rad(abc)
R_cNotation table · 21 passages
Search aliases: R_c
Notation table · tail.tex, line 446
M=c/R_c · Output powerful part or contact modulus
Exact mathematical passage
M=c/R_c
R_uNotation table · 1 passage
Search aliases: R_u
Notation table · tail.tex, line 445
R_u=rad(u) · Radical of one term
Exact mathematical passage
R_u=\rad(u)
\radNotation table · 28 passages
Search aliases: rad
Notation table · tail.tex, line 444
R=rad(abc) · Activated prime support of a primitive triple
Exact mathematical passage
R=\rad(abc)
uNotation table · 46 passages
Search aliases: u
Notation table · tail.tex, line 445
R_u=rad(u) · Radical of one term
Exact mathematical passage
R_u=\rad(u)
WNotation table · 75 passages
Search aliases: W
Notation table · overview.tex, line 913
W · Local logarithmic defect · Depth already present because every active logarithmic coefficient is exceptionally divisible
Exact mathematical passage
W
\WittNotation table · 4 passages
Search aliases: Witt
Notation table · tail.tex, line 455
Witt_c · Product of local truncated Witt contact crystals
Exact mathematical passage
\Witt_c
\Witt_cNotation table · 1 passage
Search aliases: Witt_c Witt
Notation table · tail.tex, line 455
Witt_c · Product of local truncated Witt contact crystals
Exact mathematical passage
\Witt_c
\epsilonNotation table · 28 passages
Search aliases: ϵ epsilon
Notation table · overview.tex, line 911
ϵ_2 · Dyadic sign layer · The separate weighted two-adic sign contribution
Exact mathematical passage
\epsilon_2
\epsilon_2Notation table · 24 passages
Search aliases: ϵ_2 epsilon
Notation table · overview.tex, line 911
ϵ_2 · Dyadic sign layer · The separate weighted two-adic sign contribution
Exact mathematical passage
\epsilon_2
\ThetaNotation table · 10 passages
Search aliases: Θ Theta
Notation table · tail.tex, line 447
Θ · Coarse shortest transverse derivative packet
Exact mathematical passage
\Theta
\kappaNotation table · 11 passages
Search aliases: κ kappa
Notation table · tail.tex, line 456
κ_c · Global Kummer map carrying the contact packet
Exact mathematical passage
\kappa_c
\kappa_cNotation table · 6 passages
Search aliases: κ_c kappa
Notation table · tail.tex, line 456
κ_c · Global Kummer map carrying the contact packet
Exact mathematical passage
\kappa_c
\LambdaNotation table · 36 passages
Search aliases: Λ Lambda
Notation table · tail.tex, line 462
Λ_coh · Kernel lattice of the coherence character
Exact mathematical passage
\Lambda_{\mathrm{coh}}
\Lambda_{\mathrm{coh}}Notation table · 15 passages
Search aliases: Λ_coh Lambda mathrm
Notation table · tail.tex, line 462
Λ_coh · Kernel lattice of the coherence character
Exact mathematical passage
\Lambda_{\mathrm{coh}}
\PhiNotation table · 13 passages
Search aliases: Φ Phi
Notation table · tail.tex, line 448
Φ · Phase-refined shortest transverse packet. The same symbol denotes the shortest nonzero detector when the context is clear
Exact mathematical passage
\Phi
\chiNotation table · 22 passages
Search aliases: χ chi
Notation table · tail.tex, line 463
χ_coh · Dual cyclic coherence character
Exact mathematical passage
\chi_{\mathrm{coh}}
\chi_{\mathrm{coh}}Notation table · 14 passages
Search aliases: χ_coh chi mathrm
Notation table · tail.tex, line 463
χ_coh · Dual cyclic coherence character
Exact mathematical passage
\chi_{\mathrm{coh}}
+Introduction passage · 137 passages
Search aliases: +
Introduction passage · appendices_more.tex, line 72
Derivation of the stable height. The real relaxation can be derived directly from aggregate sector variables. Let A, B, and C denote the aggregate derivative values on the three sectors. Additivity gives A+B=C. The normalized Wronskian is -bA+aB=kQ. For one obstruction unit k=1, the real minimization seeks to minimize H subject to |A|≤ aσ_aH, |B|≤ bσ_bH, |C|≤ cσ_cH, A+B=C. The Wronskian has the three equivalent representations -bA+aB, aC-cA, cB-bC. The triangle inequality therefore gives |W|≤ Hm…
Exact mathematical passage
A+B=C.
AIntroduction passage · 50 passages
Search aliases: A
Introduction passage · appendices_more.tex, line 71
Derivation of the stable height. The real relaxation can be derived directly from aggregate sector variables. Let A, B, and C denote the aggregate derivative values on the three sectors. Additivity gives A+B=C. The normalized Wronskian is -bA+aB=kQ. For one obstruction unit k=1, the real minimization seeks to minimize H subject to |A|≤ aσ_aH, |B|≤ bσ_bH, |C|≤ cσ_cH, A+B=C. The Wronskian has the three equivalent representations -bA+aB, aC-cA, cB-bC. The triangle inequality therefore gives |W|≤ Hm…
Exact mathematical passage
A
a_{p,q}Introduction passage · 2 passages
Search aliases: a_p,q
Introduction passage · detailed_3.tex, line 445
d d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vector by K, let a_p,q denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define w_p = min { d_p-r_p, min_qmid abbar v_p(a_p,q) }, χ_p=d_p-r_p-w_p. The normalized global factors are K=∏_pmid cp^r_p, W=∏_pmid cp^w…
Exact mathematical passage
a_{p,q}
\bar vIntroduction passage · 4 passages
Search aliases: bar v bar
Introduction passage · detailed_3.tex, line 446
be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vector by K, let a_p,q denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define w_p = min { d_p-r_p, min_qmid abbar v_p(a_p,q) }, χ_p=d_p-r_p-w_p. The normalized global factors are K=∏_pmid cp^r_p, W=∏_pmid cp^w_p, C=∏_pmid cp^χ_p. The balanced Kummer order formula guara…
Exact mathematical passage
w_p = \min\left \{ d_p-r_p, \min_{q\mid ab}\bar v_p(a_{p,q}) \right\}, \qquad \chi_p=d_p-r_p-w_p.
\bar v_pIntroduction passage · 3 passages
Search aliases: bar v_p bar
Introduction passage · detailed_3.tex, line 446
be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vector by K, let a_p,q denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define w_p = min { d_p-r_p, min_qmid abbar v_p(a_p,q) }, χ_p=d_p-r_p-w_p. The normalized global factors are K=∏_pmid cp^r_p, W=∏_pmid cp^w_p, C=∏_pmid cp^χ_p. The balanced Kummer order formula guara…
Exact mathematical passage
w_p = \min\left \{ d_p-r_p, \min_{q\mid ab}\bar v_p(a_{p,q}) \right\}, \qquad \chi_p=d_p-r_p-w_p.
d_2Introduction passage · 1 passage
Search aliases: d_2
Introduction passage · detailed_3.tex, line 432
Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vect…
Exact mathematical passage
s_2=1, \qquad d_2=e-2.
d_pIntroduction passage · 3 passages
Search aliases: d_p
Introduction passage · detailed_3.tex, line 428
Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vect…
Exact mathematical passage
d_p=e-1.
eIntroduction passage · 139 passages
Search aliases: e
Introduction passage · detailed_3.tex, line 427
Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vect…
Exact mathematical passage
p^e\parallel c
FIntroduction passage · 12 passages
Search aliases: F
Introduction passage · detailed_3.tex, line 176
Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-curve polarization reintroduces a factor of order M into the height accounting.
Exact mathematical passage
K_{F_M}\cong\mathcal O_{F_M}(M-3).
F_MIntroduction passage · 6 passages
Search aliases: F_M
Introduction passage · detailed_3.tex, line 176
Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-curve polarization reintroduces a factor of order M into the height accounting.
Exact mathematical passage
K_{F_M}\cong\mathcal O_{F_M}(M-3).
hIntroduction passage · 3 passages
Search aliases: h
Introduction passage · detailed_3.tex, line 180
Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-curve polarization reintroduces a factor of order M into the height accounting.
Exact mathematical passage
h_{K_{F_M}}(P_M) = (M-3)h_{\mathcal O(1)}(P_M)+O(1) = \frac{M-3}{M}\log c+O(1).
h_{\mathcal O(1)}Introduction passage · 2 passages
Search aliases: h_ O(1) mathcal
Introduction passage · detailed_3.tex, line 180
Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-curve polarization reintroduces a factor of order M into the height accounting.
Exact mathematical passage
h_{K_{F_M}}(P_M) = (M-3)h_{\mathcal O(1)}(P_M)+O(1) = \frac{M-3}{M}\log c+O(1).
h_{K_{F_M}}Introduction passage · 2 passages
Search aliases: h_K_F_M
Introduction passage · detailed_3.tex, line 180
Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-curve polarization reintroduces a factor of order M into the height accounting.
Exact mathematical passage
h_{K_{F_M}}(P_M) = (M-3)h_{\mathcal O(1)}(P_M)+O(1) = \frac{M-3}{M}\log c+O(1).
IIntroduction passage · 18 passages
Search aliases: I
Introduction passage · detailed_3.tex, line 167
Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.
Exact mathematical passage
\boxed{ \sum_{r,s}P_{r,s}=I. }
iIntroduction passage · 52 passages
Search aliases: i
Introduction passage · detailed_3.tex, line 152
Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.
Exact mathematical passage
P_{r,s} = \frac1{M^2} \sum_{i,j=0}^{M-1} \zeta_M^{-ri-sj} \sigma^i\tau^j.
jIntroduction passage · 14 passages
Search aliases: j
Introduction passage · detailed_3.tex, line 152
Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.
Exact mathematical passage
P_{r,s} = \frac1{M^2} \sum_{i,j=0}^{M-1} \zeta_M^{-ri-sj} \sigma^i\tau^j.
K_{F_M}Introduction passage · 5 passages
Search aliases: K_F_M
Introduction passage · detailed_3.tex, line 176
Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-curve polarization reintroduces a factor of order M into the height accounting.
Exact mathematical passage
K_{F_M}\cong\mathcal O_{F_M}(M-3).
K_{\mathrm{Kum}}Introduction passage · 5 passages
Search aliases: K_Kum mathrm
Introduction passage · detailed_3.tex, line 437
Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vect…
Exact mathematical passage
K_{\mathrm{Kum}}
\mathrm{Kum}Introduction passage · 5 passages
Search aliases: Kum mathrm
Introduction passage · detailed_3.tex, line 437
Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vect…
Exact mathematical passage
K_{\mathrm{Kum}}
\logIntroduction passage · 54 passages
Search aliases: log
Introduction passage · detailed_3.tex, line 180
Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-curve polarization reintroduces a factor of order M into the height accounting.
Exact mathematical passage
h_{K_{F_M}}(P_M) = (M-3)h_{\mathcal O(1)}(P_M)+O(1) = \frac{M-3}{M}\log c+O(1).
M^2Introduction passage · 2 passages
Search aliases: M^2
Introduction passage · detailed_3.tex, line 152
Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.
Exact mathematical passage
P_{r,s} = \frac1{M^2} \sum_{i,j=0}^{M-1} \zeta_M^{-ri-sj} \sigma^i\tau^j.
\midIntroduction passage · 64 passages
Search aliases: mid
Introduction passage · detailed_3.tex, line 442
me. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vector by K, let a_p,q denote the resulting odd or weighted-dyadic logarithmic row coefficie…
Exact mathematical passage
K_{\mathrm{Kum}}\mid M/\epsilon_2,
\minIntroduction passage · 20 passages
Search aliases: min
Introduction passage · detailed_3.tex, line 438
ial factorization of the powerful part The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vector by K,…
Exact mathematical passage
r_p=\min\{d_p,v_p(K_{\mathrm{Kum}})\}.
\min_{q\mid ab}Introduction passage · 2 passages
Search aliases: min_qmid ab min mid
Introduction passage · detailed_3.tex, line 446
be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vector by K, let a_p,q denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define w_p = min { d_p-r_p, min_qmid abbar v_p(a_p,q) }, χ_p=d_p-r_p-w_p. The normalized global factors are K=∏_pmid cp^r_p, W=∏_pmid cp^w_p, C=∏_pmid cp^χ_p. The balanced Kummer order formula guara…
Exact mathematical passage
w_p = \min\left \{ d_p-r_p, \min_{q\mid ab}\bar v_p(a_{p,q}) \right\}, \qquad \chi_p=d_p-r_p-w_p.
\nmidIntroduction passage · 3 passages
Search aliases: nmid
Introduction passage · detailed_3.tex, line 437
Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vect…
Exact mathematical passage
2\nmid M
OIntroduction passage · 6 passages
Search aliases: O
Introduction passage · detailed_3.tex, line 180
Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-curve polarization reintroduces a factor of order M into the height accounting.
Exact mathematical passage
h_{K_{F_M}}(P_M) = (M-3)h_{\mathcal O(1)}(P_M)+O(1) = \frac{M-3}{M}\log c+O(1).
\mathcal OIntroduction passage · 11 passages
Search aliases: O mathcal
Introduction passage · detailed_3.tex, line 176
Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-curve polarization reintroduces a factor of order M into the height accounting.
Exact mathematical passage
K_{F_M}\cong\mathcal O_{F_M}(M-3).
\mathcal O_{F_M}Introduction passage · 2 passages
Search aliases: O_F_M mathcal
Introduction passage · detailed_3.tex, line 176
Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-curve polarization reintroduces a factor of order M into the height accounting.
Exact mathematical passage
K_{F_M}\cong\mathcal O_{F_M}(M-3).
pIntroduction passage · 219 passages
Search aliases: p
Introduction passage · detailed_3.tex, line 427
Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vect…
Exact mathematical passage
p^e\parallel c
p^eIntroduction passage · 26 passages
Search aliases: p^e
Introduction passage · detailed_3.tex, line 427
Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vect…
Exact mathematical passage
p^e\parallel c
p^{r_p}Introduction passage · 2 passages
Search aliases: p^r_p
Introduction passage · detailed_3.tex, line 445
rallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vector by K, let a_p,q denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define w_p = min { d_p-r_p, min_qmid …
Exact mathematical passage
K_{\mathrm{Kum}}=\prod_{p\mid c}p^{r_p}
p^{w_p}Introduction passage · 3 passages
Search aliases: p^w_p
Introduction passage · detailed_3.tex, line 458
athrmKummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vector by K, let a_p,q denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define w_p = min { d_p-r_p, min_qmid abbar v_p(a_p,q) }, χ_p=d_p-r_p-w_p. The normalized global factors are K=∏_pmid cp^r_p, W=∏_pmid cp^w_p, C=∏_pmid cp^χ_p. The balanced Kummer order formula guarantees that the dyadic sign layer is removed before the principal 2-adic Kummer depth, so these exponents partitio…
Exact mathematical passage
K=\prod_{p\mid c}p^{r_p}, \qquad W=\prod_{p\mid c}p^{w_p}, \qquad C=\prod_{p\mid c}p^{\chi_p}.
p^{\chi_p}Introduction passage · 7 passages
Search aliases: p^χ_p chi
Introduction passage · detailed_3.tex, line 458
athrmKummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vector by K, let a_p,q denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define w_p = min { d_p-r_p, min_qmid abbar v_p(a_p,q) }, χ_p=d_p-r_p-w_p. The normalized global factors are K=∏_pmid cp^r_p, W=∏_pmid cp^w_p, C=∏_pmid cp^χ_p. The balanced Kummer order formula guarantees that the dyadic sign layer is removed before the principal 2-adic Kummer depth, so these exponents partitio…
Exact mathematical passage
K=\prod_{p\mid c}p^{r_p}, \qquad W=\prod_{p\mid c}p^{w_p}, \qquad C=\prod_{p\mid c}p^{\chi_p}.
P_MIntroduction passage · 6 passages
Search aliases: P_M
Introduction passage · detailed_3.tex, line 180
Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-curve polarization reintroduces a factor of order M into the height accounting.
Exact mathematical passage
h_{K_{F_M}}(P_M) = (M-3)h_{\mathcal O(1)}(P_M)+O(1) = \frac{M-3}{M}\log c+O(1).
P_{r',s'}Introduction passage · 1 passage
Search aliases: P_r',s'
Introduction passage · detailed_3.tex, line 161
Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.
Exact mathematical passage
P_{r,s}P_{r',s'} = \delta_{r,r'}\delta_{s,s'}P_{r,s}
P_{r,s}Introduction passage · 3 passages
Search aliases: P_r,s
Introduction passage · detailed_3.tex, line 152
Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.
Exact mathematical passage
P_{r,s} = \frac1{M^2} \sum_{i,j=0}^{M-1} \zeta_M^{-ri-sj} \sigma^i\tau^j.
\parallelIntroduction passage · 17 passages
Search aliases: parallel
Introduction passage · detailed_3.tex, line 427
Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vect…
Exact mathematical passage
p^e\parallel c
qIntroduction passage · 87 passages
Search aliases: q
Introduction passage · detailed_3.tex, line 445
d d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vector by K, let a_p,q denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define w_p = min { d_p-r_p, min_qmid abbar v_p(a_p,q) }, χ_p=d_p-r_p-w_p. The normalized global factors are K=∏_pmid cp^r_p, W=∏_pmid cp^w…
Exact mathematical passage
a_{p,q}
rIntroduction passage · 65 passages
Search aliases: r
Introduction passage · detailed_3.tex, line 148
Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.
Exact mathematical passage
(r,s)\in(\Z/M\Z)^2,
r'Introduction passage · 1 passage
Search aliases: r'
Introduction passage · detailed_3.tex, line 161
Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.
Exact mathematical passage
P_{r,s}P_{r',s'} = \delta_{r,r'}\delta_{s,s'}P_{r,s}
r_pIntroduction passage · 4 passages
Search aliases: r_p
Introduction passage · detailed_3.tex, line 438
ial factorization of the powerful part The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vector by K,…
Exact mathematical passage
r_p=\min\{d_p,v_p(K_{\mathrm{Kum}})\}.
sIntroduction passage · 29 passages
Search aliases: s
Introduction passage · detailed_3.tex, line 148
Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.
Exact mathematical passage
(r,s)\in(\Z/M\Z)^2,
s'Introduction passage · 1 passage
Search aliases: s'
Introduction passage · detailed_3.tex, line 161
Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.
Exact mathematical passage
P_{r,s}P_{r',s'} = \delta_{r,r'}\delta_{s,s'}P_{r,s}
s_2Introduction passage · 3 passages
Search aliases: s_2
Introduction passage · detailed_3.tex, line 432
Sequential factorization of the powerful part. The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vect…
Exact mathematical passage
s_2=1, \qquad d_2=e-2.
vIntroduction passage · 82 passages
Search aliases: v
Introduction passage · detailed_3.tex, line 438
ial factorization of the powerful part The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vector by K,…
Exact mathematical passage
r_p=\min\{d_p,v_p(K_{\mathrm{Kum}})\}.
v_pIntroduction passage · 35 passages
Search aliases: v_p
Introduction passage · detailed_3.tex, line 438
ial factorization of the powerful part The factorization is defined prime by prime. For an odd repeated output prime p^eparallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vector by K,…
Exact mathematical passage
r_p=\min\{d_p,v_p(K_{\mathrm{Kum}})\}.
wIntroduction passage · 18 passages
Search aliases: w
Introduction passage · detailed_3.tex, line 446
be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vector by K, let a_p,q denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define w_p = min { d_p-r_p, min_qmid abbar v_p(a_p,q) }, χ_p=d_p-r_p-w_p. The normalized global factors are K=∏_pmid cp^r_p, W=∏_pmid cp^w_p, C=∏_pmid cp^χ_p. The balanced Kummer order formula guara…
Exact mathematical passage
w_p = \min\left \{ d_p-r_p, \min_{q\mid ab}\bar v_p(a_{p,q}) \right\}, \qquad \chi_p=d_p-r_p-w_p.
w_pIntroduction passage · 9 passages
Search aliases: w_p
Introduction passage · detailed_3.tex, line 446
be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vector by K, let a_p,q denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define w_p = min { d_p-r_p, min_qmid abbar v_p(a_p,q) }, χ_p=d_p-r_p-w_p. The normalized global factors are K=∏_pmid cp^r_p, W=∏_pmid cp^w_p, C=∏_pmid cp^χ_p. The balanced Kummer order formula guara…
Exact mathematical passage
w_p = \min\left \{ d_p-r_p, \min_{q\mid ab}\bar v_p(a_{p,q}) \right\}, \qquad \chi_p=d_p-r_p-w_p.
XIntroduction passage · 11 passages
Search aliases: X
Introduction passage · detailed_3.tex, line 144
Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.
Exact mathematical passage
(\xi,\eta)\cdot(X,Y)=(\xi X,\eta Y).
YIntroduction passage · 7 passages
Search aliases: Y
Introduction passage · detailed_3.tex, line 144
Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.
Exact mathematical passage
(\xi,\eta)\cdot(X,Y)=(\xi X,\eta Y).
\ZIntroduction passage · 97 passages
Search aliases: Z
Introduction passage · detailed_3.tex, line 148
Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.
Exact mathematical passage
(r,s)\in(\Z/M\Z)^2,
|Introduction passage · 28 passages
Search aliases: |
Introduction passage · appendices_more.tex, line 80
Derivation of the stable height. The real relaxation can be derived directly from aggregate sector variables. Let A, B, and C denote the aggregate derivative values on the three sectors. Additivity gives A+B=C. The normalized Wronskian is -bA+aB=kQ. For one obstruction unit k=1, the real minimization seeks to minimize H subject to |A|≤ aσ_aH, |B|≤ bσ_bH, |C|≤ cσ_cH, A+B=C. The Wronskian has the three equivalent representations -bA+aB, aC-cA, cB-bC. The triangle inequality therefore gives |W|≤ Hm…
Exact mathematical passage
|A|\le a\sigma_aH, \qquad |B|\le b\sigma_bH, \qquad |C|\le c\sigma_cH, \qquad A+B=C.
\cdotIntroduction passage · 26 passages
Search aliases: · cdot
Introduction passage · detailed_3.tex, line 144
Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.
Exact mathematical passage
(\xi,\eta)\cdot(X,Y)=(\xi X,\eta Y).
\deltaIntroduction passage · 7 passages
Search aliases: δ delta
Introduction passage · detailed_3.tex, line 161
Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.
Exact mathematical passage
P_{r,s}P_{r',s'} = \delta_{r,r'}\delta_{s,s'}P_{r,s}
\delta_{r,r'}Introduction passage · 1 passage
Search aliases: δ_r,r' delta
Introduction passage · detailed_3.tex, line 161
Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.
Exact mathematical passage
P_{r,s}P_{r',s'} = \delta_{r,r'}\delta_{s,s'}P_{r,s}
\delta_{s,s'}Introduction passage · 1 passage
Search aliases: δ_s,s' delta
Introduction passage · detailed_3.tex, line 161
Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.
Exact mathematical passage
P_{r,s}P_{r',s'} = \delta_{r,r'}\delta_{s,s'}P_{r,s}
\zetaIntroduction passage · 13 passages
Search aliases: ζ zeta
Introduction passage · detailed_3.tex, line 152
Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.
Exact mathematical passage
P_{r,s} = \frac1{M^2} \sum_{i,j=0}^{M-1} \zeta_M^{-ri-sj} \sigma^i\tau^j.
\zeta_M^{-ri-sj}Introduction passage · 1 passage
Search aliases: ζ_M^-ri-sj zeta
Introduction passage · detailed_3.tex, line 152
Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.
Exact mathematical passage
P_{r,s} = \frac1{M^2} \sum_{i,j=0}^{M-1} \zeta_M^{-ri-sj} \sigma^i\tau^j.
\etaIntroduction passage · 14 passages
Search aliases: η eta
Introduction passage · detailed_3.tex, line 144
Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.
Exact mathematical passage
(\xi,\eta)\cdot(X,Y)=(\xi X,\eta Y).
\muIntroduction passage · 13 passages
Search aliases: μ mu
Introduction passage · detailed_3.tex, line 143
Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.
Exact mathematical passage
\mu_M^2
\mu_M^2Introduction passage · 3 passages
Search aliases: μ_M^2 mu
Introduction passage · detailed_3.tex, line 143
Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.
Exact mathematical passage
\mu_M^2
\xiIntroduction passage · 10 passages
Search aliases: ξ xi
Introduction passage · detailed_3.tex, line 144
Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.
Exact mathematical passage
(\xi,\eta)\cdot(X,Y)=(\xi X,\eta Y).
\sigmaIntroduction passage · 15 passages
Search aliases: σ sigma
Introduction passage · appendices_more.tex, line 80
Derivation of the stable height. The real relaxation can be derived directly from aggregate sector variables. Let A, B, and C denote the aggregate derivative values on the three sectors. Additivity gives A+B=C. The normalized Wronskian is -bA+aB=kQ. For one obstruction unit k=1, the real minimization seeks to minimize H subject to |A|≤ aσ_aH, |B|≤ bσ_bH, |C|≤ cσ_cH, A+B=C. The Wronskian has the three equivalent representations -bA+aB, aC-cA, cB-bC. The triangle inequality therefore gives |W|≤ Hm…
Exact mathematical passage
|A|\le a\sigma_aH, \qquad |B|\le b\sigma_bH, \qquad |C|\le c\sigma_cH, \qquad A+B=C.
\sigma^iIntroduction passage · 1 passage
Search aliases: σ^i sigma
Introduction passage · detailed_3.tex, line 152
Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.
Exact mathematical passage
P_{r,s} = \frac1{M^2} \sum_{i,j=0}^{M-1} \zeta_M^{-ri-sj} \sigma^i\tau^j.
\sigma_aIntroduction passage · 8 passages
Search aliases: σ_a sigma
Introduction passage · appendices_more.tex, line 80
Derivation of the stable height. The real relaxation can be derived directly from aggregate sector variables. Let A, B, and C denote the aggregate derivative values on the three sectors. Additivity gives A+B=C. The normalized Wronskian is -bA+aB=kQ. For one obstruction unit k=1, the real minimization seeks to minimize H subject to |A|≤ aσ_aH, |B|≤ bσ_bH, |C|≤ cσ_cH, A+B=C. The Wronskian has the three equivalent representations -bA+aB, aC-cA, cB-bC. The triangle inequality therefore gives |W|≤ Hm…
Exact mathematical passage
|A|\le a\sigma_aH, \qquad |B|\le b\sigma_bH, \qquad |C|\le c\sigma_cH, \qquad A+B=C.
\sigma_bIntroduction passage · 8 passages
Search aliases: σ_b sigma
Introduction passage · appendices_more.tex, line 80
Derivation of the stable height. The real relaxation can be derived directly from aggregate sector variables. Let A, B, and C denote the aggregate derivative values on the three sectors. Additivity gives A+B=C. The normalized Wronskian is -bA+aB=kQ. For one obstruction unit k=1, the real minimization seeks to minimize H subject to |A|≤ aσ_aH, |B|≤ bσ_bH, |C|≤ cσ_cH, A+B=C. The Wronskian has the three equivalent representations -bA+aB, aC-cA, cB-bC. The triangle inequality therefore gives |W|≤ Hm…
Exact mathematical passage
|A|\le a\sigma_aH, \qquad |B|\le b\sigma_bH, \qquad |C|\le c\sigma_cH, \qquad A+B=C.
\sigma_cIntroduction passage · 7 passages
Search aliases: σ_c sigma
Introduction passage · appendices_more.tex, line 80
Derivation of the stable height. The real relaxation can be derived directly from aggregate sector variables. Let A, B, and C denote the aggregate derivative values on the three sectors. Additivity gives A+B=C. The normalized Wronskian is -bA+aB=kQ. For one obstruction unit k=1, the real minimization seeks to minimize H subject to |A|≤ aσ_aH, |B|≤ bσ_bH, |C|≤ cσ_cH, A+B=C. The Wronskian has the three equivalent representations -bA+aB, aC-cA, cB-bC. The triangle inequality therefore gives |W|≤ Hm…
Exact mathematical passage
|A|\le a\sigma_aH, \qquad |B|\le b\sigma_bH, \qquad |C|\le c\sigma_cH, \qquad A+B=C.
\tauIntroduction passage · 2 passages
Search aliases: τ tau
Introduction passage · detailed_3.tex, line 147
Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.
Exact mathematical passage
\tau
\tau^jIntroduction passage · 1 passage
Search aliases: τ^j tau
Introduction passage · detailed_3.tex, line 152
Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.
Exact mathematical passage
P_{r,s} = \frac1{M^2} \sum_{i,j=0}^{M-1} \zeta_M^{-ri-sj} \sigma^i\tau^j.
\chi_pIntroduction passage · 9 passages
Search aliases: χ_p chi
Introduction passage · detailed_3.tex, line 446
be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vector by K, let a_p,q denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define w_p = min { d_p-r_p, min_qmid abbar v_p(a_p,q) }, χ_p=d_p-r_p-w_p. The normalized global factors are K=∏_pmid cp^r_p, W=∏_pmid cp^w_p, C=∏_pmid cp^χ_p. The balanced Kummer order formula guara…
Exact mathematical passage
w_p = \min\left \{ d_p-r_p, \min_{q\mid ab}\bar v_p(a_{p,q}) \right\}, \qquad \chi_p=d_p-r_p-w_p.
\inIntroduction passage · 36 passages
Search aliases: ∈ in
Introduction passage · detailed_3.tex, line 148
Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.
Exact mathematical passage
(r,s)\in(\Z/M\Z)^2,
\prodIntroduction passage · 36 passages
Search aliases: ∏ prod
Introduction passage · detailed_3.tex, line 445
rallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vector by K, let a_p,q denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define w_p = min { d_p-r_p, min_qmid …
Exact mathematical passage
K_{\mathrm{Kum}}=\prod_{p\mid c}p^{r_p}
\prod_{p\mid c}Introduction passage · 3 passages
Search aliases: ∏_pmid c prod mid
Introduction passage · detailed_3.tex, line 445
rallel c, set d_p=e-1. For a repeated dyadic prime 2^eparallel c, remove one sign layer and set s_2=1, d_2=e-2. If 2nmid M, take s_2=0. Put ϵ_2=2^s_2. Let K_Kum be the kernel size of the balanced Kummer class and put r_p=min{d_p,v_p(K_Kum)}. The balanced order formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. After dividing the input exponent vector by K, let a_p,q denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define w_p = min { d_p-r_p, min_qmid …
Exact mathematical passage
K_{\mathrm{Kum}}=\prod_{p\mid c}p^{r_p}
\sumIntroduction passage · 38 passages
Search aliases: ∑ sum
Introduction passage · detailed_3.tex, line 152
Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.
Exact mathematical passage
P_{r,s} = \frac1{M^2} \sum_{i,j=0}^{M-1} \zeta_M^{-ri-sj} \sigma^i\tau^j.
\sum_{i,j=0}^{M-1}Introduction passage · 1 passage
Search aliases: ∑_i,j=0^M-1 sum
Introduction passage · detailed_3.tex, line 152
Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.
Exact mathematical passage
P_{r,s} = \frac1{M^2} \sum_{i,j=0}^{M-1} \zeta_M^{-ri-sj} \sigma^i\tau^j.
\sum_{r,s}Introduction passage · 1 passage
Search aliases: ∑_r,s sum
Introduction passage · detailed_3.tex, line 167
Fermat character polyphase. The group μ_M^2 acts by (ξ,η)·(X,Y)=(ξ X,η Y). Let σ and τ denote the two coordinate rotations. For (r,s)∈(Z/MZ)^2, define P_r,s = frac1M^2 ∑_i,j=0^M-1 ζ_M^-ri-sj σ^iτ^j. Character orthogonality gives P_r,sP_r',s' = δ_r,r'δ_s,s'P_r,s and ∑_r,sP_r,s=I. This is an exact two-dimensional character filter bank.
Exact mathematical passage
\boxed{ \sum_{r,s}P_{r,s}=I. }
\congIntroduction passage · 24 passages
Search aliases: ≅ cong
Introduction passage · detailed_3.tex, line 176
Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-curve polarization reintroduces a factor of order M into the height accounting.
Exact mathematical passage
K_{F_M}\cong\mathcal O_{F_M}(M-3).
\leIntroduction passage · 54 passages
Search aliases: ≤ le
Introduction passage · appendices_more.tex, line 80
Derivation of the stable height. The real relaxation can be derived directly from aggregate sector variables. Let A, B, and C denote the aggregate derivative values on the three sectors. Additivity gives A+B=C. The normalized Wronskian is -bA+aB=kQ. For one obstruction unit k=1, the real minimization seeks to minimize H subject to |A|≤ aσ_aH, |B|≤ bσ_bH, |C|≤ cσ_cH, A+B=C. The Wronskian has the three equivalent representations -bA+aB, aC-cA, cB-bC. The triangle inequality therefore gives |W|≤ Hm…
Exact mathematical passage
|A|\le a\sigma_aH, \qquad |B|\le b\sigma_bH, \qquad |C|\le c\sigma_cH, \qquad A+B=C.
\geIntroduction passage · 36 passages
Search aliases: ≥ ge
Introduction passage · detailed_3.tex, line 175
Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-curve polarization reintroduces a factor of order M into the height accounting.
Exact mathematical passage
M\ge4
\otimesIntroduction passage · 2 passages
Search aliases: ⊗ otimes
Introduction passage · detailed_3.tex, line 187
Line-bundle cost of the full Fermat model. For M≥4, adjunction gives K_F_M≅ O_F_M(M-3). For compatible Weil-height normalizations, h_K_F_M(P_M) = (M-3)h_ O(1)(P_M)+O(1) = fracM-3Mlog c+O(1). If the metric on K_F_M is defined by the induced tensor-power metric on O(1)^⊗(M-3), the displayed equality holds without the O(1) term. Thus the full plane-curve polarization reintroduces a factor of order M into the height accounting.
Exact mathematical passage
\mathcal O(1)^{\otimes(M-3)}
*Use in source · 9 passages
Search aliases: *
Use in source · appendices_final.tex, line 266
Exact ratio semantics. Three ratios recur in the computation: fracΦoverline H, fracΦk_*overline H, fracH(1)Φ. They are all at least one by definition. A value greater than one is a discrete overhead, not an error. Exact rational strings should accompany decimal displays when the integers are large.
Exact mathematical passage
\frac{\Phi}{\overline H}, \qquad \frac{\Phi}{k_*\overline H}, \qquad \frac{H(1)}{\Phi}.
\mathbf1Use in source · 3 passages
Search aliases: 1 mathbf
Use in source · detailed_2.tex, line 9
Prime support as bandwidth and valuation as persistence. For n=∏_p p^e_p, define the prime-shell signal Ψ_n(p,r) = sqrtlog p 1_0≤ r<e_p. Its total energy is normΨ_n_2^2 = ∑_p e_plog p = log n. Let P_act retain only the first shell in each active prime channel. Then normP_actΨ_n_2^2 = ∑_pmid nlog p = lograd(n). Thus the radical is activated bandwidth, while the difference log n-lograd(n) is repeated persistence depth.
Exact mathematical passage
\Psi_n(p,r) = \sqrt{\log p}\,\mathbf 1_{0\le r<e_p}.
\mathbf1_{0\le r<e_p}Use in source · 3 passages
Search aliases: 1_0≤ r<e_p mathbf le
Use in source · detailed_2.tex, line 9
Prime support as bandwidth and valuation as persistence. For n=∏_p p^e_p, define the prime-shell signal Ψ_n(p,r) = sqrtlog p 1_0≤ r<e_p. Its total energy is normΨ_n_2^2 = ∑_p e_plog p = log n. Let P_act retain only the first shell in each active prime channel. Then normP_actΨ_n_2^2 = ∑_pmid nlog p = lograd(n). Thus the radical is activated bandwidth, while the difference log n-lograd(n) is repeated persistence depth.
Exact mathematical passage
\Psi_n(p,r) = \sqrt{\log p}\,\mathbf 1_{0\le r<e_p}.
\mathfrak AUse in source · 4 passages
Search aliases: A mathfrak
Use in source · detailed_3.tex, line 673
Prime-contact data structure. The reusable object attached to a sparse relation is A( n,ϵ) = ( W_ n, κ_ n, L_ n, Λ_coh, R_∞, M_glob ). The first four components are finite and local-global arithmetic data. The last two are optional archimedean and global-motivic realizations. Keeping them separate prevents an unproved global comparison from being built into the definition.
Exact mathematical passage
\boxed{ \mathfrak A(\mathbf n,\boldsymbol\epsilon) = \left( \mathbb W_{\mathbf n}, \kappa_{\mathbf n}, \mathcal L_{\mathbf n}, \Lambda_{\mathrm{coh}}, \mathcal R_\infty, \mathcal M_{\mathrm{glob}} \right). }
a^2Use in source · 3 passages
Search aliases: a^2
Use in source · detailed_2.tex, line 85
The Frey curve. Attach the marked Frey curve E_a,b,c:y^2=x(x-a)(x+b). The roots 0, a, and -b retain the three differences a, b, and c. The basic invariants are Δ_E=16a^2b^2c^2 and c_4(E)=16(a^2+ab+b^2). At every odd prime pmid abc, the curve has multiplicative reduction. The conductor records that the channel is active, while the minimal discriminant records the full depth.
Exact mathematical passage
\Delta_E=16a^2b^2c^2
A^xUse in source · 3 passages
Search aliases: A^x
Use in source · appendices.tex, line 593
Beal-type equations. A primitive counterexample to A^x+B^y=C^z, x,y,z>2, would contain three deep contact packets on disjoint prime supports. This is an especially strong setting for prime-contact filtering.
Exact mathematical passage
A^x+B^y=C^z, \qquad x,y,z>2,
a_0Use in source · 2 passages
Search aliases: a_0
Use in source · detailed_2.tex, line 397
Complete-intersection control case. Let T=Z[X]/(F) and evaluate at a simple root a_0. Then the congruence ideal and cotangent module both have order |F'(a_0)|. The defect is one. Extra multiplicity is therefore tied to non-complete-intersection structure rather than being a normalization artifact.
Exact mathematical passage
a_0
a_1Use in source · 3 passages
Search aliases: a_1
Use in source · appendices_final.tex, line 180
One-row Smith normal form. For an integer row a=(a_1,…,a_s), let d=gcd(a_1,…,a_s). There exists a unimodular matrix U∈GL_s(Z) such that aU=(d,0,…,0). Therefore the image of the row is dZ, the kernel has rank s-1, and the quotient by the kernel is infinite cyclic scaled by d.
Exact mathematical passage
a=(a_1,\ldots,a_s),
A_{ij}Use in source · 1 passage
Search aliases: A_ij
Use in source · detailed_3.tex, line 291
Fixed-rank integral uncertainty. Let A∈ M_d(Z) be nonsingular and suppose Dmiddet A. If every entry satisfies |A_ij|≤ B, then Hadamard's inequality gives D≤|det A|≤ d^d/2B^d. Hence B≥fracD^1/dsqrt d. A small complex period packet does not imply a small integral realization.
Exact mathematical passage
|A_{ij}|\le B
A_NUse in source · 8 passages
Search aliases: A_N
Use in source · appendices_final.tex, line 255
Polyphase circulant determinant. For the valuation word V_e(T)=1+T+⋯+T^e-1 in A_N=Z[T]/(T^N-1), multiplication is circulant. Its determinant is ∏_ζ^N=1V_e(ζ). Let I_N be the augmentation ideal of A_N. When gcd(e,N)=1, the nontrivial roots are permuted by ζ↦ζ^e, so multiplication by V_e has determinant one on the integral lattice I_N and is therefore an automorphism of it. On A_N/I_N≅Z, the map is multiplication by e. The exact sequence for I_N⊂ A_N then gives cokernel Z/eZ and Smith form diag(1,…
Exact mathematical passage
A_N=\Z[T]/(T^N-1)
a_sUse in source · 2 passages
Search aliases: a_s
Use in source · appendices_final.tex, line 180
One-row Smith normal form. For an integer row a=(a_1,…,a_s), let d=gcd(a_1,…,a_s). There exists a unimodular matrix U∈GL_s(Z) such that aU=(d,0,…,0). Therefore the image of the row is dZ, the kernel has rank s-1, and the quotient by the kernel is infinite cyclic scaled by d.
Exact mathematical passage
a=(a_1,\ldots,a_s),
A_\varepsilonUse in source · 5 passages
Search aliases: A_ε varepsilon
Use in source · detailed_2.tex, line 275
Atkin--Lehner Walsh packets. Let W_1,…,W_r be commuting Atkin--Lehner involutions. For a sign vector ε∈{±1}^r, define A_ε = ∏_j=1^r(I+ε_jW_j). Then A_ε^2=2^rA_ε, A_ε A_ε'=0 (ε≠ε'), and ∑_ε A_ε=2^rI. The unnormalized analysis packets are integral. The common synthesis denominator is 2^r, which is subpolynomial in a squarefree conductor. Local sign separation is therefore not the large persistence carrier.
Exact mathematical passage
A_\varepsilon = \prod_{j=1}^{r}(I+\varepsilon_jW_j).
A_{\varepsilon'}Use in source · 1 passage
Search aliases: A_ε' varepsilon
Use in source · detailed_2.tex, line 286
Atkin--Lehner Walsh packets. Let W_1,…,W_r be commuting Atkin--Lehner involutions. For a sign vector ε∈{±1}^r, define A_ε = ∏_j=1^r(I+ε_jW_j). Then A_ε^2=2^rA_ε, A_ε A_ε'=0 (ε≠ε'), and ∑_ε A_ε=2^rI. The unnormalized analysis packets are integral. The common synthesis denominator is 2^r, which is subpolynomial in a squarefree conductor. Local sign separation is therefore not the large persistence carrier.
Exact mathematical passage
A_\varepsilon A_{\varepsilon'}=0 \quad(\varepsilon\neq\varepsilon'),
A_\varepsilon^2Use in source · 1 passage
Search aliases: A_ε^2 varepsilon
Use in source · detailed_2.tex, line 281
Atkin--Lehner Walsh packets. Let W_1,…,W_r be commuting Atkin--Lehner involutions. For a sign vector ε∈{±1}^r, define A_ε = ∏_j=1^r(I+ε_jW_j). Then A_ε^2=2^rA_ε, A_ε A_ε'=0 (ε≠ε'), and ∑_ε A_ε=2^rI. The unnormalized analysis packets are integral. The common synthesis denominator is 2^r, which is subpolynomial in a squarefree conductor. Local sign separation is therefore not the large persistence carrier.
Exact mathematical passage
\boxed{ A_\varepsilon^2=2^rA_\varepsilon, }
a_\ellUse in source · 1 passage
Search aliases: a_ℓ ell
Use in source · detailed_2.tex, line 106
The automorphic eigenchannel. Modularity associates a weight-two newform f_E to the rational isogeny class of E. Its Hecke coefficients a_ℓ(E) form a global frequency signal. The conductor is the automorphic bandwidth. The newform, however, remembers the rational isogeny class rather than the marked integral representative.
Exact mathematical passage
a_\ell(E)
\absUse in source · 4 passages
Search aliases: abs
Use in source · overview.tex, line 208
Filtered contact packets. For a prime power p^eparallel n_i, define the local filtered contact packet Jcal_i,p=pZ_p/p^eZ_p. Its order is absJcal_i,p=p^e-1. Its associated graded is grJcal_i,p≅(F_p)^e-1. The distinction is essential. The filtered packet is cyclic. The associated graded has p-rank e-1. The non-split extensions between successive grades are the arithmetic carries.
Exact mathematical passage
\abs{\Jcal_{i,p}}=p^{e-1}.
\AcalUse in source · 1 passage
Search aliases: Acal
Use in source · overview.tex, line 759
The integral automorphic packet. The resulting integral automorphic object is Acal^sharp(E,ι) = ( f_E, Λ_E, ω_E, {q_p}_pmid N_E, ι, ϕ_E ). The components are: [leftmargin=1.5em] item the rational Hecke eigen-signal f_E item an integral Hecke, modular-symbol, or homology lattice Λ_E item the minimal N eron differential ω_E item local Tate-depth packets q_p item the marking ι of the three rational two-torsion differences item the modular synthesis map ϕ_E
Exact mathematical passage
\boxed{ \Acal^{\sharp}(E,\iota) = \left( f_E, \Lambda_E, \omega_E, \{q_p\}_{p\mid N_E}, \iota, \phi_E \right). }
\Acal^{\sharp}Use in source · 1 passage
Search aliases: Acal^sharp Acal sharp
Use in source · overview.tex, line 759
The integral automorphic packet. The resulting integral automorphic object is Acal^sharp(E,ι) = ( f_E, Λ_E, ω_E, {q_p}_pmid N_E, ι, ϕ_E ). The components are: [leftmargin=1.5em] item the rational Hecke eigen-signal f_E item an integral Hecke, modular-symbol, or homology lattice Λ_E item the minimal N eron differential ω_E item local Tate-depth packets q_p item the marking ι of the three rational two-torsion differences item the modular synthesis map ϕ_E
Exact mathematical passage
\boxed{ \Acal^{\sharp}(E,\iota) = \left( f_E, \Lambda_E, \omega_E, \{q_p\}_{p\mid N_E}, \iota, \phi_E \right). }
\mathrm{act}Use in source · 5 passages
Search aliases: act mathrm
Use in source · detailed_2.tex, line 22
Prime support as bandwidth and valuation as persistence. For n=∏_p p^e_p, define the prime-shell signal Ψ_n(p,r) = sqrtlog p 1_0≤ r<e_p. Its total energy is normΨ_n_2^2 = ∑_p e_plog p = log n. Let P_act retain only the first shell in each active prime channel. Then normP_actΨ_n_2^2 = ∑_pmid nlog p = lograd(n). Thus the radical is activated bandwidth, while the difference log n-lograd(n) is repeated persistence depth.
Exact mathematical passage
P_{\mathrm{act}}
\mathrm{add}Use in source · 2 passages
Search aliases: add mathrm
Use in source · detailed_1.tex, line 108
Over R, the additive projection is P_add = I- fracλλ^ Tnormλ_2^2. The obstruction direction inside the additive hyperplane is μ_⊥=P_addμ. When μ_⊥≠0, the rank-one obstruction projector is P_obs = fracμ_⊥μ_⊥^ T normμ_⊥_2^2. This is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation direction, one obstruction direction, and the remaining null packets.
Exact mathematical passage
P_{\mathrm{add}} = I- \frac{\lambda\lambda^{\mathsf T}}{\norm{\lambda}_2^2}.
\mathrm{alg}Use in source · 2 passages
Search aliases: alg mathrm
Use in source · detailed_2.tex, line 524
For the output source m=c/R_c, the corrected algebraic defect against a cyclic target of order η_f is δ_alg(c,f) = fracc/R_cgcd(c/R_c,η_f). Prime by prime, δ_alg(c,f) = ∏_p^eparallel c p^max(e-1-v_p(η_f),0). The problem is missing depth, not missing rank.
Exact mathematical passage
\boxed{ \delta_{\mathrm{alg}}(c,f) = \frac{c/R_c}{\gcd(c/R_c,\eta_f)}. }
b^2Use in source · 3 passages
Search aliases: b^2
Use in source · detailed_2.tex, line 85
The Frey curve. Attach the marked Frey curve E_a,b,c:y^2=x(x-a)(x+b). The roots 0, a, and -b retain the three differences a, b, and c. The basic invariants are Δ_E=16a^2b^2c^2 and c_4(E)=16(a^2+ab+b^2). At every odd prime pmid abc, the curve has multiplicative reduction. The conductor records that the channel is active, while the minimal discriminant records the full depth.
Exact mathematical passage
\Delta_E=16a^2b^2c^2
B^dUse in source · 1 passage
Search aliases: B^d
Use in source · detailed_3.tex, line 292
Fixed-rank integral uncertainty. Let A∈ M_d(Z) be nonsingular and suppose Dmiddet A. If every entry satisfies |A_ij|≤ B, then Hadamard's inequality gives D≤|det A|≤ d^d/2B^d. Hence B≥fracD^1/dsqrt d. A small complex period packet does not imply a small integral realization.
Exact mathematical passage
D\le|\det A|\le d^{d/2}B^d.
B^yUse in source · 3 passages
Search aliases: B^y
Use in source · appendices.tex, line 593
Beal-type equations. A primitive counterexample to A^x+B^y=C^z, x,y,z>2, would contain three deep contact packets on disjoint prime supports. This is an especially strong setting for prime-contact filtering.
Exact mathematical passage
A^x+B^y=C^z, \qquad x,y,z>2,
B_1Use in source · 1 passage
Search aliases: B_1
Use in source · detailed_2.tex, line 214
dmid e_pp. Since ∑_dmid eφ(d)=e, we have the perfect reconstruction identity n = ∏_d≥1 B_d(n)^φ(d). The first packet is B_1(n)=rad(n). The powerful part is fracnrad(n) = ∏_d≥2 B_d(n)^φ(d). This is a multiplicative polyphase decomposition indexed by exact character orders.
Exact mathematical passage
B_1(n)=\rad(n).
B_{k+1}Use in source · 1 passage
Search aliases: B_k+1
Use in source · appendices_more.tex, line 409
Bernoulli--Witt direction. The Bernoulli branch begins with the identity ∑_j=1^m-1j^k = fracB_k+1(m)-B_k+1k+1. The equation with m^k imposes strong congruences on Bernoulli values. A filtered prime-contact treatment should separate:
Exact mathematical passage
\sum_{j=1}^{m-1}j^k = \frac{B_{k+1}(m)-B_{k+1}}{k+1}.
\bar v_2Use in source · 1 passage
Search aliases: bar v_2 bar
Use in source · detailed_3.tex, line 419
Weighted dyadic row. For an odd input prime q, define ℓ_2,e(q) = frac14 log_2((-1)^(q-1)/2q) pmod2^e-2. With the analogous truncated valuation on Z/2^e-2Z, bar v_2(ℓ_2,e(q)) = min{e-2,v_2(q^2-1)-3}. A separate sign packet supplies the missing first dyadic layer. This is the weighted dyadic branch in its final logarithmic form.
Exact mathematical passage
\bar v_2(\ell_{2,e}(q)) = \min\{e-2,v_2(q^2-1)-3\}.
\bigsqcupUse in source · 1 passage
Search aliases: bigsqcup
Use in source · detailed_1.tex, line 853
Obstruction cosets. Choose a section x_0∈ L with ω(x_0)=1. Then L = bigsqcup_k∈Z (kx_0+L_0). For fixed k, the problem is a closest-vector problem to the moving target -kx_0 in the lattice L_0. The three-sector compression makes this search much more explicit.
Exact mathematical passage
L = \bigsqcup_{k\in\Z} (kx_0+L_0).
\bigsqcup_{k\in\Z}Use in source · 1 passage
Search aliases: bigsqcup_k∈Z bigsqcup in Z
Use in source · detailed_1.tex, line 853
Obstruction cosets. Choose a section x_0∈ L with ω(x_0)=1. Then L = bigsqcup_k∈Z (kx_0+L_0). For fixed k, the problem is a closest-vector problem to the moving target -kx_0 in the lattice L_0. The three-sector compression makes this search much more explicit.
Exact mathematical passage
L = \bigsqcup_{k\in\Z} (kx_0+L_0).
\bmodUse in source · 1 passage
Search aliases: bmod
Use in source · appendices_more.tex, line 338
Simultaneous Wieferich theorems. The edge weight w_p(q)=v_p(q^p-1-1)-1 measures local degeneracy. A first family of theorems could bound the number of pairs (p,q) with w_p(q)≥ k in a declared box. A more relevant target would control simultaneous rows: #{ (P,Q): rank( L_P,Qbmod p^k) is deficient }. The coherence lattice suggests studying exterior powers rather than entrywise defects alone.
Exact mathematical passage
\#\left\{ (P,Q): \operatorname{rank}(\mathcal L_{P,Q}\bmod p^k) \text{ is deficient} \right\}.
\CUse in source · 4 passages
Search aliases: C
Use in source · detailed_1.tex, line 385
Let I_N=ker(A_Nxrightarrow T↦1 Z) be the augmentation ideal. It is a free abelian group of rank N-1, and multiplication by V_e preserves it. After extension to C, the eigenvalues on I_N⊗C are V_e(ζ) = frac1-ζ^e1-ζ, ζ^N=1, ζ≠1. Because gcd(e,N)=1, the map ζ↦ζ^e permutes the nontrivial Nth roots. Therefore det(V_emid I_N) = ∏_substackζ^N=1
Exact mathematical passage
\C
\mathbf CUse in source · 3 passages
Search aliases: C mathbf
Use in source · detailed_2.tex, line 561
The filtered local-global target. The intended transducer is a filtered chain J_c ⟶ ⊕_pmid c L_p^st ⟶ L_R^glob xrightarrowR= T L_ T_f ⟶ C_f. It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archimedean size.
Exact mathematical passage
\mathbf J_c \longrightarrow \bigoplus_{p\mid c}\mathbf L_p^{\mathrm{st}} \longrightarrow \mathbf L_{R^{\mathrm{glob}}} \xrightarrow{R=\mathbb T} \mathbf L_{\mathbb T_f} \longrightarrow \mathbf C_f.
C'_\epsUse in source · 3 passages
Search aliases: C'_eps eps
Use in source · detailed_2.tex, line 584
[Conditional filtered-transfer criterion] Suppose a canonical map H^0(Θ_f):Z/(c/R_c)Z⟶ H^0( C_f) satisfies, for every eps>0, |ker H^0(Θ_f)|≤ C_eps R^eps and |H^0( C_f)|≤ C'_eps R_aR_bR^eps. Then the strong abc conjecture follows.
Exact mathematical passage
|H^0(\mathbf C_f)|\le C'_\eps R_aR_bR^\eps.
c^2Use in source · 2 passages
Search aliases: c^2
Use in source · detailed_2.tex, line 85
The Frey curve. Attach the marked Frey curve E_a,b,c:y^2=x(x-a)(x+b). The roots 0, a, and -b retain the three differences a, b, and c. The basic invariants are Δ_E=16a^2b^2c^2 and c_4(E)=16(a^2+ab+b^2). At every odd prime pmid abc, the curve has multiplicative reduction. The conductor records that the channel is active, while the minimal discriminant records the full depth.
Exact mathematical passage
\Delta_E=16a^2b^2c^2
c^{3/2}Use in source · 1 passage
Search aliases: c^3/2
Use in source · detailed_2.tex, line 181
ϕ_E:X_0(N_E)⟶ E be an optimal modular parametrization, and let ϕ_E^*ω_E = 2π i c_E f_E(z) dz. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_E^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area factor proportional to 1/c. Thus the height c can be written as a shape-corrected synthesis gain involving modular degree, Manin normalization, and Petersson energy.
Exact mathematical passage
y=c^{3/2}Y
c^kUse in source · 2 passages
Search aliases: c^k
Use in source · detailed_3.tex, line 27
The global Kummer map. Kummer theory gives H^1(Q,μ_M) ≅ Q^×/(Q^×)^M. Define κ_c: Z/MZ ⟶ H^1(Q,μ_M), k⟼[c^k]. The valuation vector of c is (e_1,…,e_r) ∈ (Z/MZ)^r.
Exact mathematical passage
\boxed{ \kappa_c: \Z/M\Z \longrightarrow H^1(\Q,\mu_M), \qquad k\longmapsto[c^k]. }
c^{o(1)}Use in source · 1 passage
Search aliases: c^o(1)
Use in source · detailed_1.tex, line 246
Define the shortest transverse packet Θ(a,b,c) = min{ normx_∞: x∈ T_a,b,c, W_x(a,b)≠0 }. Then c ≤ fracRlog2Θ(a,b,c)log c. A subpolynomial estimate Θ(a,b,c)=c^o(1) would imply the strong abc inequality. Later chapters explain why this target is too close to the conjecture to be reached by generic lattice geometry alone.
Exact mathematical passage
\Theta(a,b,c)=c^{o(1)}
C^zUse in source · 3 passages
Search aliases: C^z
Use in source · appendices.tex, line 593
Beal-type equations. A primitive counterexample to A^x+B^y=C^z, x,y,z>2, would contain three deep contact packets on disjoint prime supports. This is an especially strong setting for prime-contact filtering.
Exact mathematical passage
A^x+B^y=C^z, \qquad x,y,z>2,
C_1Use in source · 3 passages
Search aliases: C_1
Use in source · appendices_more.tex, line 294
Coherence-character composition. Suppose the two normalized coherence characters have orders C_1 and C_2. Their product character has order dividing lcm(C_1,C_2). If the shared term forces a compatibility relation, the actual order can be strictly smaller. Conversely, the combined system can expose a higher-rank determinant even when each individual quotient is cyclic.
Exact mathematical passage
C_1
c_1Use in source · 1 passage
Search aliases: c_1
Use in source · detailed_2.tex, line 151
The Legendre parameter of the parent is λ=t^2, t=v/u. The degree-two Landen transformation gives λ ⟼ frac4sqrtλ(1+sqrtλ)^2. This is exactly the child ratio a_1/c_1. The transformation preserves the automorphic channel up to the corresponding two-isogeny while lowering the integral height. It proves that the integral representative inside an isogeny class is relevant to abc.
Exact mathematical passage
a_1/c_1
C_2Use in source · 3 passages
Search aliases: C_2
Use in source · appendices_more.tex, line 294
Coherence-character composition. Suppose the two normalized coherence characters have orders C_1 and C_2. Their product character has order dividing lcm(C_1,C_2). If the shared term forces a compatibility relation, the actual order can be strictly smaller. Conversely, the combined system can expose a higher-rank determinant even when each individual quotient is cyclic.
Exact mathematical passage
C_2
c_4Use in source · 1 passage
Search aliases: c_4
Use in source · detailed_2.tex, line 89
The Frey curve. Attach the marked Frey curve E_a,b,c:y^2=x(x-a)(x+b). The roots 0, a, and -b retain the three differences a, b, and c. The basic invariants are Δ_E=16a^2b^2c^2 and c_4(E)=16(a^2+ab+b^2). At every odd prime pmid abc, the curve has multiplicative reduction. The conductor records that the channel is active, while the minimal discriminant records the full depth.
Exact mathematical passage
c_4(E)=16(a^2+ab+b^2).
c_EUse in source · 2 passages
Search aliases: c_E
Use in source · detailed_2.tex, line 159
Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be an optimal modular parametrization, and let ϕ_E^*ω_E = 2π i c_E f_E(z) dz. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_E^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area factor proportional to 1/c. Thus the height c can be written as a shape-corrected synthesis gain involving modular degree, Manin normalization, and Pete…
Exact mathematical passage
\phi_E^*\omega_E = 2\pi i\,c_E f_E(z)\,dz.
C_{e,N}Use in source · 3 passages
Search aliases: C_e,N
Use in source · detailed_1.tex, line 367
Finite polyphase rings. Fix an integer N≥2 and work in A_N=Z[T]/(T^N-1). The residue of V_e(T) records the number of valuation shells in each congruence class modulo N. Multiplication by V_e is represented by a circulant matrix C_e,N.
Exact mathematical passage
C_{e,N}
c_E^2Use in source · 2 passages
Search aliases: c_E^2
Use in source · detailed_2.tex, line 173
Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be an optimal modular parametrization, and let ϕ_E^*ω_E = 2π i c_E f_E(z) dz. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_E^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area factor proportional to 1/c. Thus the height c can be written as a shape-corrected synthesis gain involving modular degree, Manin normalization, and Pete…
Exact mathematical passage
\boxed{ \deg(\phi_E)A(E) = 4\pi^2 c_E^2 \norm{f_E}_{\mathrm{Pet}}^2. }
C_\epsUse in source · 13 passages
Search aliases: C_eps eps
Use in source · appendices.tex, line 644
Combined logarithmic--coherence conductor. For the actual rows constructed from the input primes and output-prime logarithms, prove WC≤ C_eps R_aR_bR^eps. The theorem must control both entrywise logarithmic degeneration and collective coherence and must exploit the addition law. On the subfamily W=1, it reduces to a pure bound for C. The corresponding statement is false for arbitrary primitive congruence rows.
Exact mathematical passage
WC\le C_\eps R_aR_bR^\eps.
C_fUse in source · 1 passage
Search aliases: C_f
Use in source · front_atlas.tex, line 51
Hecke filter · A_ε, e_f, C_f · local signs and integral eigenline separation · output contact depth by itself
Exact mathematical passage
C_f
\mathbf C_fUse in source · 3 passages
Search aliases: C_f mathbf
Use in source · detailed_2.tex, line 561
The filtered local-global target. The intended transducer is a filtered chain J_c ⟶ ⊕_pmid c L_p^st ⟶ L_R^glob xrightarrowR= T L_ T_f ⟶ C_f. It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archimedean size.
Exact mathematical passage
\mathbf J_c \longrightarrow \bigoplus_{p\mid c}\mathbf L_p^{\mathrm{st}} \longrightarrow \mathbf L_{R^{\mathrm{glob}}} \xrightarrow{R=\mathbb T} \mathbf L_{\mathbb T_f} \longrightarrow \mathbf C_f.
C_{\mathrm{glued}}Use in source · 1 passage
Search aliases: C_glued mathrm
Use in source · appendices_more.tex, line 306
A useful first target is: C_glued ≤ fracC_1C_2rad(C) × subpolynomial correction. The displayed formula is schematic. It expresses the hope that the shared term cancels one full support layer. No such theorem is proved.
Exact mathematical passage
\boxed{ C_{\mathrm{glued}} \le \frac{C_1C_2}{\rad(C)} \times \text{subpolynomial correction}. }
C_pUse in source · 4 passages
Search aliases: C_p
Use in source · appendices_final.tex, line 47
The local residual character is χ_p(x) = exp( frac2π ip^χ_pg_p· x ). Its kernel contains the normalized exponent vector. The local factors are therefore K_p, W_p=p^w_p, C_p=p^χ_p. At two, one sign layer ϵ_2,p is removed before the weighted logarithmic row is formed.
Exact mathematical passage
K_p, \qquad W_p=p^{w_p}, \qquad C_p=p^{\chi_p}.
C_{\mathrm{red}}Use in source · 2 passages
Search aliases: C_red mathrm
Use in source · detailed_2.tex, line 433
The three-cusp section. View a primitive triple as the integral section σ_a,b,c: SpecZ⟶ P^1_Z, σ=[a:c]. The pullbacks of the boundary divisors 0, 1, and ∞ are div(a), div(b), div(c). For the output cusp, let C=div(c), C_red=div(R_c). Define the two-term contact complex J_c = [ O(-C)⟶ O(-C_red)]. On global sections, J_c = [cZ↪ R_cZ] ≃ [Zxrightarrowc/R_cZ].
Exact mathematical passage
C=\operatorname{div}(c), \qquad C_{\mathrm{red}}=\operatorname{div}(R_c).
C_UUse in source · 3 passages
Search aliases: C_U
Use in source · detailed_2.tex, line 319
Projector denominator as a congruence exponent. Let V=U⊕ W be a rational decomposition and let Λ⊂ V be an integral lattice. Let e_U be the rational projector onto U. Define Λ_U=Λ∩ U, Λ_W=Λ∩ W, and C_U=Λ/(Λ_U+Λ_W).
Exact mathematical passage
C_U=\Lambda/(\Lambda_U+\Lambda_W).
\cokerUse in source · 1 passage
Search aliases: coker
Use in source · detailed_2.tex, line 512
Exact cyclic transducer. Let C(m)=[ZxrightarrowmZ] and C(η)=[ZxrightarrowηZ]. Set g=gcd(m,η). Multiplication by m/g in degree -1 and by η/g in degree 0 defines a chain map. On cohomology, |ker|=m/g, |im|=g, |coker|=η/g. The maximal transferable packet is the common prime-power depth.
Exact mathematical passage
\boxed{ |\ker|=m/g, \qquad |\im|=g, \qquad |\coker|=\eta/g. }
\operatorname{coker}Use in source · 1 passage
Search aliases: coker operatorname
Use in source · detailed_1.tex, line 663
[Smith obstruction theorem] Let a+b=c be primitive, with at least two of a,b,c greater than one. Then SNF( M_a,b,c) = diag(1,Q). Consequently, coker( M_a,b,c)_tors ≅ Z/QZ. The exceptional triple (1,1,2) has only one nonempty prime sector, so the matrix has rank one and is excluded from the two-row statement.
Exact mathematical passage
\boxed{ \operatorname{coker}(\mathsf M_{a,b,c})_{\mathrm{tors}} \cong \Z/Q\Z. }
DUse in source · 11 passages
Search aliases: D
Use in source · appendices.tex, line 55
The phase packet y=(-8,8,3) satisfies D_y(3)=-8, D_y(125)=200, D_y(128)=192, and -8+200=192. Its Wronskian is also 1600. The phase height is eight. The phase inequality is sharp: 128 = 30(frac13+frac15)8. This illustrates that the exponent-free phase channel can sharpen the bound even when its packet norm is larger.
Exact mathematical passage
D_y(3)=-8, \qquad D_y(125)=200, \qquad D_y(128)=192,
\mathrm dUse in source · 15 passages
Search aliases: d mathrm
Use in source · detailed_1.tex, line 20
The prime-coordinate module. Let a+b=c, gcd(a,b)=1. The three integers are pairwise coprime. Write S=S_asqcup S_bsqcup S_c ={p:pmid abc}. The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. For a positive integer n, define the universal multiplicative differential dn = n∑_pmid nfracv_p(n)p ξ_p. Every coefficient is an integer because pmid n. The map obeys the exact Leibniz law d(mn)=m dn+n dm. The verification is local. For each prime p, v_p(mn)=v_p(m)+v_p(n), so the…
Exact mathematical passage
\boxed{ \mathrm dn = n\sum_{p\mid n}\frac{v_p(n)}p\,\xi_p. }
D^{1/d}Use in source · 1 passage
Search aliases: D^1/d
Use in source · detailed_3.tex, line 296
Fixed-rank integral uncertainty. Let A∈ M_d(Z) be nonsingular and suppose Dmiddet A. If every entry satisfies |A_ij|≤ B, then Hadamard's inequality gives D≤|det A|≤ d^d/2B^d. Hence B≥fracD^1/dsqrt d. A small complex period packet does not imply a small integral realization.
Exact mathematical passage
\boxed{ B\ge\frac{D^{1/d}}{\sqrt d}. }
d^{d/2}Use in source · 1 passage
Search aliases: d^d/2
Use in source · detailed_3.tex, line 292
Fixed-rank integral uncertainty. Let A∈ M_d(Z) be nonsingular and suppose Dmiddet A. If every entry satisfies |A_ij|≤ B, then Hadamard's inequality gives D≤|det A|≤ d^d/2B^d. Hence B≥fracD^1/dsqrt d. A small complex period packet does not imply a small integral realization.
Exact mathematical passage
D\le|\det A|\le d^{d/2}B^d.
D_fUse in source · 1 passage
Search aliases: D_f
Use in source · detailed_2.tex, line 405
Two prime indices. The persistence is indexed by conductor primes qmid abc. Congruence depth is indexed by coefficient primes ℓ. The natural local signal is a matrix D_f(ℓ,q). At a split Tate prime q≠ℓ, scalarity modulo ℓ^n forces n≤min{v_ℓ(q-1),v_ℓ(v_q(q_E))}. These off-diagonal channels carry tame congruence information. They do not naturally reproduce the full base-q factor q^e-1. This mismatch motivates the later filtered contact correction and the diagonal coefficient-prime channel.
Exact mathematical passage
D_f(\ell,q).
\mathrm d_xUse in source · 13 passages
Search aliases: d_x mathrm
Use in source · detailed_1.tex, line 45
A scalar packet is an integer vector x=(x_p)_p∈ S∈Z^S. Pairing with x gives the arithmetic derivative d_x(n) = n∑_pmid nfracv_p(n)p x_p. The Leibniz rule survives scalarization.
Exact mathematical passage
\boxed{ \mathrm d_x(n) = n\sum_{p\mid n}\frac{v_p(n)}p x_p. }
D_yUse in source · 4 passages
Search aliases: D_y
Use in source · appendices.tex, line 55
The phase packet y=(-8,8,3) satisfies D_y(3)=-8, D_y(125)=200, D_y(128)=192, and -8+200=192. Its Wronskian is also 1600. The phase height is eight. The phase inequality is sharp: 128 = 30(frac13+frac15)8. This illustrates that the exponent-free phase channel can sharpen the bound even when its packet norm is larger.
Exact mathematical passage
D_y(3)=-8, \qquad D_y(125)=200, \qquad D_y(128)=192,
\degUse in source · 2 passages
Search aliases: deg
Use in source · detailed_2.tex, line 173
Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be an optimal modular parametrization, and let ϕ_E^*ω_E = 2π i c_E f_E(z) dz. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_E^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area factor proportional to 1/c. Thus the height c can be written as a shape-corrected synthesis gain involving modular degree, Manin normalization, and Pete…
Exact mathematical passage
\boxed{ \deg(\phi_E)A(E) = 4\pi^2 c_E^2 \norm{f_E}_{\mathrm{Pet}}^2. }
\detUse in source · 4 passages
Search aliases: det
Use in source · appendices_final.tex, line 169
A determinant viewpoint. When several independent exponent packets are available, replace the one-row character by an exterior-power regulator. Let L be a normalized cross-prime logarithm matrix. A nonzero r× r minor det L_I,J measures the failure of r input packets to cancel simultaneously across r output channels.
Exact mathematical passage
\det L_{I,J}
\operatorname{diag}Use in source · 7 passages
Search aliases: diag operatorname
Use in source · appendices.tex, line 76
Obstruction complex. The joint exponent-free phase--Wronskian matrix has Smith form diag(1,1600). The additive lattice maps onto 1600Z through the Wronskian. The primitive obstruction section exists algebraically. Its height remains the metric problem.
Exact mathematical passage
\operatorname{diag}(1,1600).
\operatorname{div}Use in source · 2 passages
Search aliases: div operatorname
Use in source · detailed_2.tex, line 425
The three-cusp section. View a primitive triple as the integral section σ_a,b,c: SpecZ⟶ P^1_Z, σ=[a:c]. The pullbacks of the boundary divisors 0, 1, and ∞ are div(a), div(b), div(c). For the output cusp, let C=div(c), C_red=div(R_c). Define the two-term contact complex J_c = [ O(-C)⟶ O(-C_red)]. On global sections, J_c = [cZ↪ R_cZ] ≃ [Zxrightarrowc/R_cZ].
Exact mathematical passage
\operatorname{div}(a), \qquad \operatorname{div}(b), \qquad \operatorname{div}(c).
\DownarrowUse in source · 1 passage
Search aliases: Downarrow
Use in source · front_atlas.tex, line 6
· primitive sparse relation · Downarrow · prime-packet differential and Wronskian · Downarrow · valuation-word and shell refinement · Downarrow · joint obstruction complex and exact search · Downarrow · prime-shell signal and Frey transform · Downarrow · divisor-polyphase persistence and Hecke filters · Downarrow · filtered cusp-contact and Witt packet · Downarrow · global Kummer and Fermat character packet · Downarrow · polylogarithmic path and cross-prime logarithms · Downarrow · adelic cohere…
Exact mathematical passage
\boxed{ \begin{aligned} &\text{primitive sparse relation}\\ &\quad\Downarrow\\ &\text{prime-packet differential and Wronskian}\\ &\quad\Downarrow\\ &\text{valuation-word and shell refinement}\\ &\quad\Downarrow\\ &\text{joint obstruction complex and exact search}\\ &\quad\Downarrow\\ &\text{prime-shell signal and Frey transform}\\ &\quad\Downarrow\\ &\text{divisor-polyphase persistence and Hecke filters}\\ &\quad\Downarrow\\ &\text{filtered cusp-contact and Witt packet}\\ &\quad\Downarrow\\ &\text{global Kummer and Fermat character packet}\\ &\quad\Downarrow\\ &\text{polylogarithmic path and cross-prime logarithms}\\ &\quad\Downarrow\\ &\text{adelic coherence character and kernel lattice}. \end{aligned}}
EUse in source · 25 passages
Search aliases: E
Use in source · appendices_more.tex, line 244
Consider A+B=C and C+D=E. The term C is an output in the first relation and an input in the second. Its contact packet should not be duplicated independently. A functorial theory should identify the two copies and record how depth moves across the diagram.
Exact mathematical passage
C+D=E.
\mathcal EUse in source · 1 passage
Search aliases: E mathcal
Use in source · appendices_more.tex, line 350
E_K(δ) · ={large Kummer preload}, E_W(δ) · ={large local logarithmic defect}, E_C(δ) · ={large collective coherence}, E_A(δ) · ={large automorphic congruence defect}.
Exact mathematical passage
\mathcal E_K(\delta)&=\{\text{large Kummer preload}\},\\ \mathcal E_W(\delta)&=\{\text{large local logarithmic defect}\},\\ \mathcal E_C(\delta)&=\{\text{large collective coherence}\},\\ \mathcal E_A(\delta)&=\{\text{large automorphic congruence defect}\}.
e^{2\pi i g\cdot x/C}Use in source · 1 passage
Search aliases: e^2π i g· x/C pi cdot
Use in source · appendices_final.tex, line 160
The resulting row g=(g_q)_qmid ab is primitive modulo C because every local row is primitive. Its character χ_coh(x) = e^2π i g· x/C has exact order C. The coherence lattice is its kernel.
Exact mathematical passage
\chi_{\mathrm{coh}}(x) = e^{2\pi i g\cdot x/C}
e_1Use in source · 4 passages
Search aliases: e_1
Use in source · detailed_3.tex, line 38
The global Kummer map. Kummer theory gives H^1(Q,μ_M) ≅ Q^×/(Q^×)^M. Define κ_c: Z/MZ ⟶ H^1(Q,μ_M), k⟼[c^k]. The valuation vector of c is (e_1,…,e_r) ∈ (Z/MZ)^r.
Exact mathematical passage
(e_1,\ldots,e_r) \in (\Z/M\Z)^r.
\mathcal E_AUse in source · 1 passage
Search aliases: E_A mathcal
Use in source · appendices_more.tex, line 350
E_K(δ) · ={large Kummer preload}, E_W(δ) · ={large local logarithmic defect}, E_C(δ) · ={large collective coherence}, E_A(δ) · ={large automorphic congruence defect}.
Exact mathematical passage
\mathcal E_K(\delta)&=\{\text{large Kummer preload}\},\\ \mathcal E_W(\delta)&=\{\text{large local logarithmic defect}\},\\ \mathcal E_C(\delta)&=\{\text{large collective coherence}\},\\ \mathcal E_A(\delta)&=\{\text{large automorphic congruence defect}\}.
E_{a,b,c}Use in source · 2 passages
Search aliases: E_a,b,c
Use in source · detailed_2.tex, line 79
The Frey curve. Attach the marked Frey curve E_a,b,c:y^2=x(x-a)(x+b). The roots 0, a, and -b retain the three differences a, b, and c. The basic invariants are Δ_E=16a^2b^2c^2 and c_4(E)=16(a^2+ab+b^2). At every odd prime pmid abc, the curve has multiplicative reduction. The conductor records that the channel is active, while the minimal discriminant records the full depth.
Exact mathematical passage
\boxed{ E_{a,b,c}:y^2=x(x-a)(x+b). }
\mathcal E_CUse in source · 1 passage
Search aliases: E_C mathcal
Use in source · appendices_more.tex, line 350
E_K(δ) · ={large Kummer preload}, E_W(δ) · ={large local logarithmic defect}, E_C(δ) · ={large collective coherence}, E_A(δ) · ={large automorphic congruence defect}.
Exact mathematical passage
\mathcal E_K(\delta)&=\{\text{large Kummer preload}\},\\ \mathcal E_W(\delta)&=\{\text{large local logarithmic defect}\},\\ \mathcal E_C(\delta)&=\{\text{large collective coherence}\},\\ \mathcal E_A(\delta)&=\{\text{large automorphic congruence defect}\}.
e_fUse in source · 3 passages
Search aliases: e_f
Use in source · detailed_2.tex, line 301
The same-sign separation problem. After fixing the local signs, several eigenforms can remain congruent inside the same packet. Let Λ be an integral Hecke lattice and let e_f be the rational projector onto the Frey eigenline. Define η_f = min{m≥1:m e_f(Λ)⊆Λ}. This is the integral eigenchannel separation denominator. It is an arithmetic analogue of leakage in a filter bank. The difficulty is not rational spectral isolation. It is isolation inside the integral lattice.
Exact mathematical passage
e_f
e_iUse in source · 5 passages
Search aliases: e_i
Use in source · detailed_3.tex, line 5
The contact modulus. Let c=∏_i=1^rp_i^e_i, R_c=∏_i p_i, M=fraccR_c =∏_i p_i^e_i-1. The filtered output contact packet is the cyclic group Z/MZ. The first global question is whether this packet can be realized by one arithmetic class without losing most of its order.
Exact mathematical passage
c=\prod_{i=1}^{r}p_i^{e_i}, \qquad R_c=\prod_i p_i, \qquad M=\frac{c}{R_c} =\prod_i p_i^{e_i-1}.
\mathcal E_KUse in source · 1 passage
Search aliases: E_K mathcal
Use in source · appendices_more.tex, line 350
E_K(δ) · ={large Kummer preload}, E_W(δ) · ={large local logarithmic defect}, E_C(δ) · ={large collective coherence}, E_A(δ) · ={large automorphic congruence defect}.
Exact mathematical passage
\mathcal E_K(\delta)&=\{\text{large Kummer preload}\},\\ \mathcal E_W(\delta)&=\{\text{large local logarithmic defect}\},\\ \mathcal E_C(\delta)&=\{\text{large collective coherence}\},\\ \mathcal E_A(\delta)&=\{\text{large automorphic congruence defect}\}.
e_pUse in source · 13 passages
Search aliases: e_p
Use in source · detailed_2.tex, line 5
Prime support as bandwidth and valuation as persistence. For n=∏_p p^e_p, define the prime-shell signal Ψ_n(p,r) = sqrtlog p 1_0≤ r<e_p. Its total energy is normΨ_n_2^2 = ∑_p e_plog p = log n. Let P_act retain only the first shell in each active prime channel. Then normP_actΨ_n_2^2 = ∑_pmid nlog p = lograd(n). Thus the radical is activated bandwidth, while the difference log n-lograd(n) is repeated persistence depth.
Exact mathematical passage
n=\prod_p p^{e_p},
e_rUse in source · 4 passages
Search aliases: e_r
Use in source · detailed_3.tex, line 38
The global Kummer map. Kummer theory gives H^1(Q,μ_M) ≅ Q^×/(Q^×)^M. Define κ_c: Z/MZ ⟶ H^1(Q,μ_M), k⟼[c^k]. The valuation vector of c is (e_1,…,e_r) ∈ (Z/MZ)^r.
Exact mathematical passage
(e_1,\ldots,e_r) \in (\Z/M\Z)^r.
e_UUse in source · 3 passages
Search aliases: e_U
Use in source · detailed_2.tex, line 312
Projector denominator as a congruence exponent. Let V=U⊕ W be a rational decomposition and let Λ⊂ V be an integral lattice. Let e_U be the rational projector onto U. Define Λ_U=Λ∩ U, Λ_W=Λ∩ W, and C_U=Λ/(Λ_U+Λ_W).
Exact mathematical passage
e_U
\mathcal E_WUse in source · 1 passage
Search aliases: E_W mathcal
Use in source · appendices_more.tex, line 350
E_K(δ) · ={large Kummer preload}, E_W(δ) · ={large local logarithmic defect}, E_C(δ) · ={large collective coherence}, E_A(δ) · ={large automorphic congruence defect}.
Exact mathematical passage
\mathcal E_K(\delta)&=\{\text{large Kummer preload}\},\\ \mathcal E_W(\delta)&=\{\text{large local logarithmic defect}\},\\ \mathcal E_C(\delta)&=\{\text{large collective coherence}\},\\ \mathcal E_A(\delta)&=\{\text{large automorphic congruence defect}\}.
\epsUse in source · 29 passages
Search aliases: eps
Use in source · appendices.tex, line 446
[leftmargin=1.5em] item Construct a finite source of order comparable to c/R_c. item Map it into a global target with kernel at most R^eps. item Bound the target size by R_aR_bR^eps. item Multiply by R_c and absorb logarithmic factors.
Exact mathematical passage
R^\eps
\expUse in source · 3 passages
Search aliases: exp
Use in source · appendices_final.tex, line 39
The local residual character is χ_p(x) = exp( frac2π ip^χ_pg_p· x ). Its kernel contains the normalized exponent vector. The local factors are therefore K_p, W_p=p^w_p, C_p=p^χ_p. At two, one sign layer ϵ_2,p is removed before the weighted logarithmic row is formed.
Exact mathematical passage
\chi_p(x) = \exp\left( \frac{2\pi i}{p^{\chi_p}}g_p\cdot x \right).
\FUse in source · 13 passages
Search aliases: F
Use in source · appendices.tex, line 147
Filtered contact and Witt packet. The output contact complex is [128Z↪2Z] ≃ [Zxrightarrow64Z]. Its cohomology is Jcal_c≅Z/64Z. The associated graded is (F_2)^6. The six binary layers are joined by five nontrivial carry extensions. A cyclic target of order 64 can carry the filtered source exactly, while no cyclic target can contain the split rank-six source injectively.
Exact mathematical passage
(\F_2)^6.
fUse in source · 23 passages
Search aliases: f
Use in source · detailed_1.tex, line 341
Replacing one exponent by a valuation word. The coefficient v_p(n) compresses a run of valuation shells into one scalar. For a prime power p^e, define the valuation word V_e(T)=1+T+⋯+T^e-1. It obeys the concatenation identity V_e+f(T)=V_e(T)+T^eV_f(T). The refined partial derivative ∂_p,T(n) = frac npV_v_p(n)(T) therefore satisfies the twisted product rule ∂_p,T(mn) = n∂_p,T(m) + mT^v_p(m)∂_p,T(n). At T=1, the ordinary arithmetic partial derivative is recovered.
Exact mathematical passage
V_{e+f}(T)=V_e(T)+T^eV_f(T).
F'Use in source · 1 passage
Search aliases: F'
Use in source · detailed_2.tex, line 398
Complete-intersection control case. Let T=Z[X]/(F) and evaluate at a simple root a_0. Then the congruence ideal and cotangent module both have order |F'(a_0)|. The defect is one. Extra multiplicity is therefore tied to non-complete-intersection structure rather than being a normalization artifact.
Exact mathematical passage
|F'(a_0)|.
\F_2Use in source · 2 passages
Search aliases: F_2 F
Use in source · appendices.tex, line 147
Filtered contact and Witt packet. The output contact complex is [128Z↪2Z] ≃ [Zxrightarrow64Z]. Its cohomology is Jcal_c≅Z/64Z. The associated graded is (F_2)^6. The six binary layers are joined by five nontrivial carry extensions. A cyclic target of order 64 can carry the filtered source exactly, while no cyclic target can contain the split rank-six source injectively.
Exact mathematical passage
(\F_2)^6.
F_{a,b,c}Use in source · 3 passages
Search aliases: F_a,b,c
Use in source · detailed_1.tex, line 793
The full shell deformation. Introduce one variable for every valuation shell and define mathscr J_n( T) = ∏_p^eparallel n ∏_r=0^e-1(p+T_p,r). At T=0, mathscr J_n(0)=n. For a+b=c, set F_a,b,c( T) = mathscr J_a( T) + mathscr J_b( T) - mathscr J_c( T). The shell gradient has entries frac ap, frac bp, -frac cp repeated according to valuation depth. The gcd of all entries is one.
Exact mathematical passage
F_{a,b,c}(\mathbf T) = \mathscr J_a(\mathbf T) + \mathscr J_b(\mathbf T) - \mathscr J_c(\mathbf T).
f_EUse in source · 8 passages
Search aliases: f_E
Use in source · detailed_2.tex, line 105
The automorphic eigenchannel. Modularity associates a weight-two newform f_E to the rational isogeny class of E. Its Hecke coefficients a_ℓ(E) form a global frequency signal. The conductor is the automorphic bandwidth. The newform, however, remembers the rational isogeny class rather than the marked integral representative.
Exact mathematical passage
f_E
\F_pUse in source · 11 passages
Search aliases: F_p F
Use in source · detailed_2.tex, line 482
Associated graded and carry extensions. At p^eparallel c, Jcal_p,e = pZ_p/p^eZ_p ≅ Z/p^e-1Z. Its filtration p^jZ_p/p^eZ_p has successive quotients F_p. Thus grJcal_p,e≅(F_p)^e-1. The exact sequences 0⟶ Z/p^j-1Z xrightarrow× p Z/p^jZ ⟶ F_p ⟶0 are nonsplit. These carry extensions are precisely what the split persistence packet forgot.
Exact mathematical passage
\F_p
GUse in source · 2 passages
Search aliases: G
Use in source · appendices_more.tex, line 151
Formal jet recursion. Let F( T)=0 be the shell equation and let T(t)=x^(1)t+x^(2)t^2+⋯. Suppose the gradient row Λ is primitive. Choose an integral vector v satisfying Λ· v=1. At order m, substitution gives Λ· x^(m) + G_m(x^(1),…,x^(m-1)) =0. The integral correction x^(m) = -G_m v solves the equation at that order. Induction produces an integral formal curve.
Exact mathematical passage
\Lambda\cdot x^{(m)} + G_m(x^{(1)},\ldots,x^{(m-1)}) =0.
\mathcal GUse in source · 3 passages
Search aliases: G mathcal
Use in source · overview.tex, line 334
The prime-contact coherence data structure. The reusable data attached to a primitive sparse relation is A( n,ϵ) = ( R, M, Witt_ n, κ_ n, Lcal_ n, Λ_coh, χ_coh, G_ n ). The calculus consists of exact transformations among these layers, plus the norm and determinant questions required to compare them globally.
Exact mathematical passage
\boxed{ \mathfrak A(\mathbf n,\boldsymbol\epsilon) = \left( \mathbf R, \mathbf M, \Witt_{\mathbf n}, \kappa_{\mathbf n}, \Lcal_{\mathbf n}, \Lambda_{\mathrm{coh}}, \chi_{\mathrm{coh}}, \mathcal G_{\mathbf n} \right). }
gUse in source · 30 passages
Search aliases: g
Use in source · appendices.tex, line 221
Coherence lattice. After removing the sign layer, (ϵ_2,K,W,C)=(2,1,1,32). The residual row is g=(29,31)pmod32. The lattice is Λ_coh = {(x_3,x_5):29x_3+31x_5≡0pmod32}. The input vector lies in the kernel because 29(-1)+31(3)=64. The full repeated output depth has now been transformed into one arithmetic cyclic character of order 32, after the bounded sign layer has been separated.
Exact mathematical passage
g=(29,31)\pmod{32}.
\mathcal G_{\mathbf n}Use in source · 3 passages
Search aliases: G_ n mathcal mathbf
Use in source · overview.tex, line 334
The prime-contact coherence data structure. The reusable data attached to a primitive sparse relation is A( n,ϵ) = ( R, M, Witt_ n, κ_ n, Lcal_ n, Λ_coh, χ_coh, G_ n ). The calculus consists of exact transformations among these layers, plus the norm and determinant questions required to compare them globally.
Exact mathematical passage
\boxed{ \mathfrak A(\mathbf n,\boldsymbol\epsilon) = \left( \mathbf R, \mathbf M, \Witt_{\mathbf n}, \kappa_{\mathbf n}, \Lcal_{\mathbf n}, \Lambda_{\mathrm{coh}}, \chi_{\mathrm{coh}}, \mathcal G_{\mathbf n} \right). }
g_aUse in source · 4 passages
Search aliases: g_a
Use in source · detailed_1.tex, line 259
Three-sector compression. Within the a-sector, define the integer row λ^(a) = (fraca v_p(a)p)_pmid a and let g_a=gcd{fraca v_p(a)p:pmid a}. Define g_b and g_c similarly. A one-row Smith normal form in each sector produces a unimodular coordinate system in which only one aggregate channel survives. The additive equation becomes g_at_a+g_bt_b-g_ct_c=0. The Wronskian becomes a g_bt_b-b g_at_a≠0. All remaining sector coordinates are lift gauges. They matter only because they determine the cost of re…
Exact mathematical passage
g_a=\gcd\left\{\frac{a v_p(a)}p:p\mid a\right\}.
g_bUse in source · 3 passages
Search aliases: g_b
Use in source · detailed_1.tex, line 262
Three-sector compression. Within the a-sector, define the integer row λ^(a) = (fraca v_p(a)p)_pmid a and let g_a=gcd{fraca v_p(a)p:pmid a}. Define g_b and g_c similarly. A one-row Smith normal form in each sector produces a unimodular coordinate system in which only one aggregate channel survives. The additive equation becomes g_at_a+g_bt_b-g_ct_c=0. The Wronskian becomes a g_bt_b-b g_at_a≠0. All remaining sector coordinates are lift gauges. They matter only because they determine the cost of re…
Exact mathematical passage
g_b
g_CUse in source · 1 passage
Search aliases: g_C
Use in source · appendices_more.tex, line 216
Thus Λ_C = {x:g_C· x≡0pmod C} has index C and contains the same fixed vector v. This proves that no bound on C can follow from the length or support of v alone.
Exact mathematical passage
\Lambda_C = \{x:g_C\cdot x\equiv0\pmod C\}
g_cUse in source · 5 passages
Search aliases: g_c
Use in source · detailed_1.tex, line 262
Three-sector compression. Within the a-sector, define the integer row λ^(a) = (fraca v_p(a)p)_pmid a and let g_a=gcd{fraca v_p(a)p:pmid a}. Define g_b and g_c similarly. A one-row Smith normal form in each sector produces a unimodular coordinate system in which only one aggregate channel survives. The additive equation becomes g_at_a+g_bt_b-g_ct_c=0. The Wronskian becomes a g_bt_b-b g_at_a≠0. All remaining sector coordinates are lift gauges. They matter only because they determine the cost of re…
Exact mathematical passage
g_c
G_mUse in source · 2 passages
Search aliases: G_m
Use in source · appendices_more.tex, line 151
Formal jet recursion. Let F( T)=0 be the shell equation and let T(t)=x^(1)t+x^(2)t^2+⋯. Suppose the gradient row Λ is primitive. Choose an integral vector v satisfying Λ· v=1. At order m, substitution gives Λ· x^(m) + G_m(x^(1),…,x^(m-1)) =0. The integral correction x^(m) = -G_m v solves the equation at that order. Induction produces an integral formal curve.
Exact mathematical passage
\Lambda\cdot x^{(m)} + G_m(x^{(1)},\ldots,x^{(m-1)}) =0.
g_pUse in source · 3 passages
Search aliases: g_p
Use in source · appendices_final.tex, line 39
The local residual character is χ_p(x) = exp( frac2π ip^χ_pg_p· x ). Its kernel contains the normalized exponent vector. The local factors are therefore K_p, W_p=p^w_p, C_p=p^χ_p. At two, one sign layer ϵ_2,p is removed before the weighted logarithmic row is formed.
Exact mathematical passage
\chi_p(x) = \exp\left( \frac{2\pi i}{p^{\chi_p}}g_p\cdot x \right).
g_{p,q}Use in source · 1 passage
Search aliases: g_p,q
Use in source · appendices_final.tex, line 153
CRT synthesis of the global row. The local residual moduli C_p are pairwise coprime. Let C=∏_pmid cC_p. For each input coordinate q, the Chinese remainder theorem gives a unique residue g_qpmod C whose reduction modulo C_p is the local coefficient g_p,q.
Exact mathematical passage
g_{p,q}
g_qUse in source · 2 passages
Search aliases: g_q
Use in source · appendices_final.tex, line 150
CRT synthesis of the global row. The local residual moduli C_p are pairwise coprime. Let C=∏_pmid cC_p. For each input coordinate q, the Chinese remainder theorem gives a unique residue g_qpmod C whose reduction modulo C_p is the local coefficient g_p,q.
Exact mathematical passage
g_q\pmod C
g_{\mathrm{val}}Use in source · 1 passage
Search aliases: g_val mathrm
Use in source · detailed_3.tex, line 96
Balanced Kummer class. The class of c does not remember the additive relation. A better coordinate is β=-frac ba=1-frac ca. Let ν_q=v_q(b)-v_q(a) for input primes qmid ab. The order of [β] in Q^×/(Q^×)^M is determined by the signed valuation vector, with one possible dyadic sign correction when M is even. If g_val = gcd(M,{ν_q}), m_0=M/g_val, then the order is m_0 unless M is even and m_0 is odd, in which case it is 2m_0.
Exact mathematical passage
g_{\mathrm{val}} = \gcd(M,\{\nu_q\}), \qquad m_0=M/g_{\mathrm{val}},
\gcdUse in source · 27 passages
Search aliases: gcd
Use in source · appendices.tex, line 170
Kummer and Fermat packets. The global Kummer map is κ_128:Z/64Z→ H^1(Q,μ_64). Since gcd(64,7)=1, it is injective.
Exact mathematical passage
\gcd(64,7)=1,
\gcd_{p\mid u}Use in source · 1 passage
Search aliases: gcd_pmid u gcd mid
Use in source · detailed_1.tex, line 571
Exact local lift gauges. Let r_u=rad(u), P_u=fracur_u. A shell packet y=(y_p,r) produces D_y(u) = u∑_pmid ufrac1p ∑_r=0^v_p(u)-1y_p,r =P_ut_u, where t_u= ∑_pmid u fracr_up ∑_r y_p,r. Because gcd_pmid ufracr_up=1, every integer t_u is attainable when u>1.
Exact mathematical passage
\gcd_{p\mid u}\frac{r_u}{p}=1,
\mathrm{GL}Use in source · 1 passage
Search aliases: GL mathrm
Use in source · appendices_final.tex, line 187
One-row Smith normal form. For an integer row a=(a_1,…,a_s), let d=gcd(a_1,…,a_s). There exists a unimodular matrix U∈GL_s(Z) such that aU=(d,0,…,0). Therefore the image of the row is dZ, the kernel has rank s-1, and the quotient by the kernel is infinite cyclic scaled by d.
Exact mathematical passage
U\in\mathrm{GL}_s(\Z)
\mathrm{GL}_sUse in source · 1 passage
Search aliases: GL_s mathrm
Use in source · appendices_final.tex, line 187
One-row Smith normal form. For an integer row a=(a_1,…,a_s), let d=gcd(a_1,…,a_s). There exists a unimodular matrix U∈GL_s(Z) such that aU=(d,0,…,0). Therefore the image of the row is dZ, the kernel has rank s-1, and the quotient by the kernel is infinite cyclic scaled by d.
Exact mathematical passage
U\in\mathrm{GL}_s(\Z)
\mathrm{glob}Use in source · 2 passages
Search aliases: glob mathrm
Use in source · detailed_2.tex, line 561
The filtered local-global target. The intended transducer is a filtered chain J_c ⟶ ⊕_pmid c L_p^st ⟶ L_R^glob xrightarrowR= T L_ T_f ⟶ C_f. It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archimedean size.
Exact mathematical passage
\mathbf J_c \longrightarrow \bigoplus_{p\mid c}\mathbf L_p^{\mathrm{st}} \longrightarrow \mathbf L_{R^{\mathrm{glob}}} \xrightarrow{R=\mathbb T} \mathbf L_{\mathbb T_f} \longrightarrow \mathbf C_f.
\mathrm{glued}Use in source · 1 passage
Search aliases: glued mathrm
Use in source · appendices_more.tex, line 306
A useful first target is: C_glued ≤ fracC_1C_2rad(C) × subpolynomial correction. The displayed formula is schematic. It expresses the hope that the shared term cancels one full support layer. No such theorem is proved.
Exact mathematical passage
\boxed{ C_{\mathrm{glued}} \le \frac{C_1C_2}{\rad(C)} \times \text{subpolynomial correction}. }
\grUse in source · 3 passages
Search aliases: gr
Use in source · detailed_2.tex, line 483
Associated graded and carry extensions. At p^eparallel c, Jcal_p,e = pZ_p/p^eZ_p ≅ Z/p^e-1Z. Its filtration p^jZ_p/p^eZ_p has successive quotients F_p. Thus grJcal_p,e≅(F_p)^e-1. The exact sequences 0⟶ Z/p^j-1Z xrightarrow× p Z/p^jZ ⟶ F_p ⟶0 are nonsplit. These carry extensions are precisely what the split persistence packet forgot.
Exact mathematical passage
\gr\Jcal_{p,e}\cong(\F_p)^{e-1}.
\mathbf HUse in source · 2 passages
Search aliases: H mathbf
Use in source · tail.tex, line 351
Open theorem C: filtered local-global transfer. The filtered contact construction proposes a map J_c ⟶ H_f from the cusp-contact complex into a Hecke, Selmer, or deformation complex.
Exact mathematical passage
\mathbf J_c \longrightarrow \mathbf H_f
\mathcal HUse in source · 3 passages
Search aliases: H mathcal
Use in source · detailed_3.tex, line 339
Conditional motivic transfer. Let K_β be the cyclic group generated by the balanced Kummer class. Suppose there exists an integral finite object H_3(a,b,c) and a canonical map Θ_3:K_β⟶ H_3(a,b,c) satisfying |kerΘ_3|≤ C_eps R^eps and | H_3(a,b,c)|≤ C'_eps R_aR_bR^eps. Then the strong abc conjecture follows after absorbing the logarithmic Kummer kernel.
Exact mathematical passage
\mathcal H_3(a,b,c)
H^0Use in source · 6 passages
Search aliases: H^0
Use in source · detailed_2.tex, line 456
[Filtered contact quotient] H^0( J_c) = R_cZ/cZ ≅ Z/(c/R_c)Z.
Exact mathematical passage
\boxed{ H^0(\mathbf J_c) = R_c\Z/c\Z \cong \Z/(c/R_c)\Z. }
H^1Use in source · 5 passages
Search aliases: H^1
Use in source · appendices.tex, line 166
Kummer and Fermat packets. The global Kummer map is κ_128:Z/64Z→ H^1(Q,μ_64). Since gcd(64,7)=1, it is injective.
Exact mathematical passage
\kappa_{128}:\Z/64\Z\to H^1(\Q,\mu_{64}).
\mathcal H_3Use in source · 3 passages
Search aliases: H_3 mathcal
Use in source · detailed_3.tex, line 339
Conditional motivic transfer. Let K_β be the cyclic group generated by the balanced Kummer class. Suppose there exists an integral finite object H_3(a,b,c) and a canonical map Θ_3:K_β⟶ H_3(a,b,c) satisfying |kerΘ_3|≤ C_eps R^eps and | H_3(a,b,c)|≤ C'_eps R_aR_bR^eps. Then the strong abc conjecture follows after absorbing the logarithmic Kummer kernel.
Exact mathematical passage
\mathcal H_3(a,b,c)
\mathbf H_fUse in source · 2 passages
Search aliases: H_f mathbf
Use in source · tail.tex, line 351
Open theorem C: filtered local-global transfer. The filtered contact construction proposes a map J_c ⟶ H_f from the cusp-contact complex into a Hecke, Selmer, or deformation complex.
Exact mathematical passage
\mathbf J_c \longrightarrow \mathbf H_f
I_aUse in source · 1 passage
Search aliases: I_a
Use in source · detailed_1.tex, line 966
[Moving-interval theorem] For fixed obstruction multiple k and radius H, H(k)≤ H if and only if I_a(H,k) ∩ I_b(H,k) ∩ I_c(H,k) ∩Z ≠∅.
Exact mathematical passage
I_a(H,k) \cap I_b(H,k) \cap I_c(H,k) \cap\Z \neq\varnothing.
I_bUse in source · 1 passage
Search aliases: I_b
Use in source · detailed_1.tex, line 966
[Moving-interval theorem] For fixed obstruction multiple k and radius H, H(k)≤ H if and only if I_a(H,k) ∩ I_b(H,k) ∩ I_c(H,k) ∩Z ≠∅.
Exact mathematical passage
I_a(H,k) \cap I_b(H,k) \cap I_c(H,k) \cap\Z \neq\varnothing.
I_cUse in source · 1 passage
Search aliases: I_c
Use in source · detailed_1.tex, line 966
[Moving-interval theorem] For fixed obstruction multiple k and radius H, H(k)≤ H if and only if I_a(H,k) ∩ I_b(H,k) ∩ I_c(H,k) ∩Z ≠∅.
Exact mathematical passage
I_a(H,k) \cap I_b(H,k) \cap I_c(H,k) \cap\Z \neq\varnothing.
I_NUse in source · 10 passages
Search aliases: I_N
Use in source · appendices_final.tex, line 259
Polyphase circulant determinant. For the valuation word V_e(T)=1+T+⋯+T^e-1 in A_N=Z[T]/(T^N-1), multiplication is circulant. Its determinant is ∏_ζ^N=1V_e(ζ). Let I_N be the augmentation ideal of A_N. When gcd(e,N)=1, the nontrivial roots are permuted by ζ↦ζ^e, so multiplication by V_e has determinant one on the integral lattice I_N and is therefore an automorphism of it. On A_N/I_N≅Z, the map is multiplication by e. The exact sequence for I_N⊂ A_N then gives cokernel Z/eZ and Smith form diag(1,…
Exact mathematical passage
I_N
I_XUse in source · 1 passage
Search aliases: I_X
Use in source · appendices_more.tex, line 11
Why exact synthesis prevents free coordinate splitting. Let X and Y be normed spaces, and suppose J:X⟶ Y, S:Y⟶ X satisfy SJ=I_X. Then normx = normSJx ≤ normSnormJx. Therefore normJx ≥ fracnormxnormS. An analysis map may make coordinates smaller only by increasing the synthesis norm. This elementary inequality is the common source of several no-free-refinement results.
Exact mathematical passage
SJ=I_X.
\imUse in source · 2 passages
Search aliases: im
Use in source · detailed_2.tex, line 512
Exact cyclic transducer. Let C(m)=[ZxrightarrowmZ] and C(η)=[ZxrightarrowηZ]. Set g=gcd(m,η). Multiplication by m/g in degree -1 and by η/g in degree 0 defines a chain map. On cohomology, |ker|=m/g, |im|=g, |coker|=η/g. The maximal transferable packet is the common prime-power depth.
Exact mathematical passage
\boxed{ |\ker|=m/g, \qquad |\im|=g, \qquad |\coker|=\eta/g. }
\mathrm{in}Use in source · 4 passages
Search aliases: in mathrm
Use in source · appendices_more.tex, line 250
Contact pushout. Let Jcal_C^out be the contact packet of C in the first equation and Jcal_C^in its packet in the second. Both are canonically rad(C)Z/CZ. The first gluing operation identifies them through the identity map. The total contact object is the pushout of the two relation diagrams along this common packet.
Exact mathematical passage
\Jcal_C^{\mathrm{in}}
\infUse in source · 1 passage
Search aliases: inf
Use in source · detailed_1.tex, line 729
For k∈Z, define H(k) = min{ normx_∞: x∈ L, ω(x)=k }. Three heights must be distinguished: H(1), Φ=min_k≠0H(k), overline H= inf_k≥1fracH(k)k. The shortest nonzero detector may live in a nonprimitive obstruction coset.
Exact mathematical passage
H(1), \qquad \Phi=\min_{k\neq0}H(k), \qquad \overline H= \inf_{k\ge1}\frac{H(k)}k.
\inf_{k\ge1}Use in source · 1 passage
Search aliases: inf_k≥1 inf ge
Use in source · detailed_1.tex, line 729
For k∈Z, define H(k) = min{ normx_∞: x∈ L, ω(x)=k }. Three heights must be distinguished: H(1), Φ=min_k≠0H(k), overline H= inf_k≥1fracH(k)k. The shortest nonzero detector may live in a nonprimitive obstruction coset.
Exact mathematical passage
H(1), \qquad \Phi=\min_{k\neq0}H(k), \qquad \overline H= \inf_{k\ge1}\frac{H(k)}k.
JUse in source · 8 passages
Search aliases: J
Use in source · appendices_final.tex, line 169
A determinant viewpoint. When several independent exponent packets are available, replace the one-row character by an exterior-power regulator. Let L be a normalized cross-prime logarithm matrix. A nonzero r× r minor det L_I,J measures the failure of r input packets to cancel simultaneously across r output channels.
Exact mathematical passage
\det L_{I,J}
\mathbf JUse in source · 6 passages
Search aliases: J mathbf
Use in source · detailed_2.tex, line 439
section σ_a,b,c: SpecZ⟶ P^1_Z, σ=[a:c]. The pullbacks of the boundary divisors 0, 1, and ∞ are div(a), div(b), div(c). For the output cusp, let C=div(c), C_red=div(R_c). Define the two-term contact complex J_c = [ O(-C)⟶ O(-C_red)]. On global sections, J_c = [cZ↪ R_cZ] ≃ [Zxrightarrowc/R_cZ].
Exact mathematical passage
\boxed{ \mathbf J_c = [\mathcal O(-C)\longrightarrow\mathcal O(-C_{\mathrm{red}})]. }
j^kUse in source · 1 passage
Search aliases: j^k
Use in source · appendices_more.tex, line 409
Bernoulli--Witt direction. The Bernoulli branch begins with the identity ∑_j=1^m-1j^k = fracB_k+1(m)-B_k+1k+1. The equation with m^k imposes strong congruences on Bernoulli values. A filtered prime-contact treatment should separate:
Exact mathematical passage
\sum_{j=1}^{m-1}j^k = \frac{B_{k+1}(m)-B_{k+1}}{k+1}.
J_aUse in source · 1 passage
Search aliases: J_a
Use in source · detailed_1.tex, line 793
The full shell deformation. Introduce one variable for every valuation shell and define mathscr J_n( T) = ∏_p^eparallel n ∏_r=0^e-1(p+T_p,r). At T=0, mathscr J_n(0)=n. For a+b=c, set F_a,b,c( T) = mathscr J_a( T) + mathscr J_b( T) - mathscr J_c( T). The shell gradient has entries frac ap, frac bp, -frac cp repeated according to valuation depth. The gcd of all entries is one.
Exact mathematical passage
F_{a,b,c}(\mathbf T) = \mathscr J_a(\mathbf T) + \mathscr J_b(\mathbf T) - \mathscr J_c(\mathbf T).
J_bUse in source · 1 passage
Search aliases: J_b
Use in source · detailed_1.tex, line 793
The full shell deformation. Introduce one variable for every valuation shell and define mathscr J_n( T) = ∏_p^eparallel n ∏_r=0^e-1(p+T_p,r). At T=0, mathscr J_n(0)=n. For a+b=c, set F_a,b,c( T) = mathscr J_a( T) + mathscr J_b( T) - mathscr J_c( T). The shell gradient has entries frac ap, frac bp, -frac cp repeated according to valuation depth. The gcd of all entries is one.
Exact mathematical passage
F_{a,b,c}(\mathbf T) = \mathscr J_a(\mathbf T) + \mathscr J_b(\mathbf T) - \mathscr J_c(\mathbf T).
J_cUse in source · 1 passage
Search aliases: J_c
Use in source · detailed_1.tex, line 793
The full shell deformation. Introduce one variable for every valuation shell and define mathscr J_n( T) = ∏_p^eparallel n ∏_r=0^e-1(p+T_p,r). At T=0, mathscr J_n(0)=n. For a+b=c, set F_a,b,c( T) = mathscr J_a( T) + mathscr J_b( T) - mathscr J_c( T). The shell gradient has entries frac ap, frac bp, -frac cp repeated according to valuation depth. The gcd of all entries is one.
Exact mathematical passage
F_{a,b,c}(\mathbf T) = \mathscr J_a(\mathbf T) + \mathscr J_b(\mathbf T) - \mathscr J_c(\mathbf T).
\mathbf J_cUse in source · 6 passages
Search aliases: J_c mathbf
Use in source · detailed_2.tex, line 439
section σ_a,b,c: SpecZ⟶ P^1_Z, σ=[a:c]. The pullbacks of the boundary divisors 0, 1, and ∞ are div(a), div(b), div(c). For the output cusp, let C=div(c), C_red=div(R_c). Define the two-term contact complex J_c = [ O(-C)⟶ O(-C_red)]. On global sections, J_c = [cZ↪ R_cZ] ≃ [Zxrightarrowc/R_cZ].
Exact mathematical passage
\boxed{ \mathbf J_c = [\mathcal O(-C)\longrightarrow\mathcal O(-C_{\mathrm{red}})]. }
J_nUse in source · 2 passages
Search aliases: J_n
Use in source · detailed_1.tex, line 782
The full shell deformation. Introduce one variable for every valuation shell and define mathscr J_n( T) = ∏_p^eparallel n ∏_r=0^e-1(p+T_p,r). At T=0, mathscr J_n(0)=n. For a+b=c, set F_a,b,c( T) = mathscr J_a( T) + mathscr J_b( T) - mathscr J_c( T). The shell gradient has entries frac ap, frac bp, -frac cp repeated according to valuation depth. The gcd of all entries is one.
Exact mathematical passage
\mathscr J_n(\mathbf T) = \prod_{p^e\parallel n} \prod_{r=0}^{e-1}(p+T_{p,r}).
\Jcal_C^{\mathrm{in}}Use in source · 1 passage
Search aliases: Jcal_C^in Jcal mathrm
Use in source · appendices_more.tex, line 250
Contact pushout. Let Jcal_C^out be the contact packet of C in the first equation and Jcal_C^in its packet in the second. Both are canonically rad(C)Z/CZ. The first gluing operation identifies them through the identity map. The total contact object is the pushout of the two relation diagrams along this common packet.
Exact mathematical passage
\Jcal_C^{\mathrm{in}}
\Jcal_C^{\mathrm{out}}Use in source · 1 passage
Search aliases: Jcal_C^out Jcal mathrm
Use in source · appendices_more.tex, line 250
Contact pushout. Let Jcal_C^out be the contact packet of C in the first equation and Jcal_C^in its packet in the second. Both are canonically rad(C)Z/CZ. The first gluing operation identifies them through the identity map. The total contact object is the pushout of the two relation diagrams along this common packet.
Exact mathematical passage
\Jcal_C^{\mathrm{out}}
\Jcal_{i,p}Use in source · 4 passages
Search aliases: Jcal_i,p Jcal
Use in source · overview.tex, line 204
Filtered contact packets. For a prime power p^eparallel n_i, define the local filtered contact packet Jcal_i,p=pZ_p/p^eZ_p. Its order is absJcal_i,p=p^e-1. Its associated graded is grJcal_i,p≅(F_p)^e-1. The distinction is essential. The filtered packet is cyclic. The associated graded has p-rank e-1. The non-split extensions between successive grades are the arithmetic carries.
Exact mathematical passage
\Jcal_{i,p}=p\Z_p/p^e\Z_p.
\Jcal_{n_i}Use in source · 1 passage
Search aliases: Jcal_n_i Jcal
Use in source · overview.tex, line 218
Globally, the contact packet of one term is Jcal_n_i = R_iZ/n_iZ ≅ Z/M_iZ.
Exact mathematical passage
\Jcal_{n_i} = R_i\Z/n_i\Z \cong \Z/M_i\Z.
\Jcal_{p,e}Use in source · 3 passages
Search aliases: Jcal_p,e Jcal
Use in source · detailed_2.tex, line 471
Associated graded and carry extensions. At p^eparallel c, Jcal_p,e = pZ_p/p^eZ_p ≅ Z/p^e-1Z. Its filtration p^jZ_p/p^eZ_p has successive quotients F_p. Thus grJcal_p,e≅(F_p)^e-1. The exact sequences 0⟶ Z/p^j-1Z xrightarrow× p Z/p^jZ ⟶ F_p ⟶0 are nonsplit. These carry extensions are precisely what the split persistence packet forgot.
Exact mathematical passage
\Jcal_{p,e} = p\Z_p/p^e\Z_p \cong \Z/p^{e-1}\Z.
k_*Use in source · 6 passages
Search aliases: k_*
Use in source · appendices_final.tex, line 266
Exact ratio semantics. Three ratios recur in the computation: fracΦoverline H, fracΦk_*overline H, fracH(1)Φ. They are all at least one by definition. A value greater than one is a discrete overhead, not an error. Exact rational strings should accompany decimal displays when the integers are large.
Exact mathematical passage
\frac{\Phi}{\overline H}, \qquad \frac{\Phi}{k_*\overline H}, \qquad \frac{H(1)}{\Phi}.
K_{\max}Use in source · 2 passages
Search aliases: K_max max
Use in source · appendices.tex, line 496
[leftmargin=1.5em] item Compute r_u, P_u, R, Q, and the exact stable height overline H. item Compute the primitive obstruction height H(1) by binary search on the three interval intersections. item Set K_max = lfloorfracH(1)-1overline Hrfloor. item For 2≤ k≤ K_max, skip a coset when koverline H already reaches the current best height. item Build centered CRT phase residues for the three sectors. item Compute the coarse affine line. item Binary-search the smallest H whose three admissible n-inter…
Exact mathematical passage
K_{\max} = \left\lfloor\frac{H(1)-1}{\overline H}\right\rfloor.
K_pUse in source · 2 passages
Search aliases: K_p
Use in source · appendices_final.tex, line 20
Start with the signed input exponent vector ν=(ν_q)_qmid ab, ν_q=v_q(b)-v_q(a). Let K_p be the p-primary part of the common Kummer factor. Divide the exponent vector and the local modulus by this common power. The normalized vector is primitive relative to the remaining modulus.
Exact mathematical passage
K_p
K_\betaUse in source · 2 passages
Search aliases: K_β beta
Use in source · detailed_3.tex, line 339
Conditional motivic transfer. Let K_β be the cyclic group generated by the balanced Kummer class. Suppose there exists an integral finite object H_3(a,b,c) and a canonical map Θ_3:K_β⟶ H_3(a,b,c) satisfying |kerΘ_3|≤ C_eps R^eps and | H_3(a,b,c)|≤ C'_eps R_aR_bR^eps. Then the strong abc conjecture follows after absorbing the logarithmic Kummer kernel.
Exact mathematical passage
K_\beta
\KerUse in source · 3 passages
Search aliases: Ker
Use in source · detailed_1.tex, line 506
[First-order no-free-polyphase principle] Let S:(T_p,r)_p,r⟼ (∑_rT_p,r)_p be shell synthesis. The first-order additive map and first-order Wronskian factor through S. Hence Ker S is contained in the kernels of both first-order maps.
Exact mathematical passage
\Ker S
\kerUse in source · 10 passages
Search aliases: ker
Use in source · detailed_1.tex, line 75
[6pt] -dfracc v_p(c)p, · pmid c. Then λ· x=0. The additive derivative lattice is T_a,b,c = ker(λ)∩Z^S. Since the row is nonzero, its rank is |S|-1. The problem is not existence of additive derivatives. The problem is finding an additive derivative that remains transverse to the dependence direction introduced by a and b.
Exact mathematical passage
\boxed{ \mathcal T_{a,b,c} = \ker(\lambda)\cap\Z^S. }
\mathbf LUse in source · 1 passage
Search aliases: L mathbf
Use in source · detailed_2.tex, line 561
The filtered local-global target. The intended transducer is a filtered chain J_c ⟶ ⊕_pmid c L_p^st ⟶ L_R^glob xrightarrowR= T L_ T_f ⟶ C_f. It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archimedean size.
Exact mathematical passage
\mathbf J_c \longrightarrow \bigoplus_{p\mid c}\mathbf L_p^{\mathrm{st}} \longrightarrow \mathbf L_{R^{\mathrm{glob}}} \xrightarrow{R=\mathbb T} \mathbf L_{\mathbb T_f} \longrightarrow \mathbf C_f.
\mathcal LUse in source · 2 passages
Search aliases: L mathcal
Use in source · appendices_more.tex, line 338
Simultaneous Wieferich theorems. The edge weight w_p(q)=v_p(q^p-1-1)-1 measures local degeneracy. A first family of theorems could bound the number of pairs (p,q) with w_p(q)≥ k in a declared box. A more relevant target would control simultaneous rows: #{ (P,Q): rank( L_P,Qbmod p^k) is deficient }. The coherence lattice suggests studying exterior powers rather than entrywise defects alone.
Exact mathematical passage
\#\left\{ (P,Q): \operatorname{rank}(\mathcal L_{P,Q}\bmod p^k) \text{ is deficient} \right\}.
\mathcal L_{\mathbf n}Use in source · 1 passage
Search aliases: L_ n mathcal mathbf
Use in source · detailed_3.tex, line 673
Prime-contact data structure. The reusable object attached to a sparse relation is A( n,ϵ) = ( W_ n, κ_ n, L_ n, Λ_coh, R_∞, M_glob ). The first four components are finite and local-global arithmetic data. The last two are optional archimedean and global-motivic realizations. Keeping them separate prevents an unproved global comparison from being built into the definition.
Exact mathematical passage
\boxed{ \mathfrak A(\mathbf n,\boldsymbol\epsilon) = \left( \mathbb W_{\mathbf n}, \kappa_{\mathbf n}, \mathcal L_{\mathbf n}, \Lambda_{\mathrm{coh}}, \mathcal R_\infty, \mathcal M_{\mathrm{glob}} \right). }
\mathbf L_{\mathbb T_f}Use in source · 1 passage
Search aliases: L_ T_f mathbf mathbb
Use in source · detailed_2.tex, line 561
The filtered local-global target. The intended transducer is a filtered chain J_c ⟶ ⊕_pmid c L_p^st ⟶ L_R^glob xrightarrowR= T L_ T_f ⟶ C_f. It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archimedean size.
Exact mathematical passage
\mathbf J_c \longrightarrow \bigoplus_{p\mid c}\mathbf L_p^{\mathrm{st}} \longrightarrow \mathbf L_{R^{\mathrm{glob}}} \xrightarrow{R=\mathbb T} \mathbf L_{\mathbb T_f} \longrightarrow \mathbf C_f.
L_1Use in source · 2 passages
Search aliases: L_1
Use in source · appendices_more.tex, line 282
Logarithmic matrix composition. Let L_1 and L_2 be the cross-prime logarithmic matrices of the two relations. The shared term determines a common set of prime channels. The combined system has block form L_1 · 0
Exact mathematical passage
L_1
L_2Use in source · 5 passages
Search aliases: L_2
Use in source · appendices_more.tex, line 282
Logarithmic matrix composition. Let L_1 and L_2 be the cross-prime logarithmic matrices of the two relations. The shared term determines a common set of prime channels. The combined system has block form L_1 · 0
Exact mathematical passage
L_2
L_{I,J}Use in source · 1 passage
Search aliases: L_I,J
Use in source · appendices_final.tex, line 169
A determinant viewpoint. When several independent exponent packets are available, replace the one-row character by an exterior-power regulator. Let L be a normalized cross-prime logarithm matrix. A nonzero r× r minor det L_I,J measures the failure of r input packets to cancel simultaneously across r output channels.
Exact mathematical passage
\det L_{I,J}
L_{p,q}Use in source · 2 passages
Search aliases: L_p,q
Use in source · overview.tex, line 289
Cross-prime logarithmic filters. Let Q be a set of input primes and P a set of output primes. At odd p∈ P, define L_p,q = frac1plog_p⟨ q⟩_p with the appropriate finite-depth truncation. At p=2, use the weighted principal-unit coordinate together with a separate sign bit.
Exact mathematical passage
L_{p,q} = \frac1p\log_p\langle q\rangle_p
\mathcal L_{P,Q}Use in source · 1 passage
Search aliases: L_P,Q mathcal
Use in source · appendices_more.tex, line 338
Simultaneous Wieferich theorems. The edge weight w_p(q)=v_p(q^p-1-1)-1 measures local degeneracy. A first family of theorems could bound the number of pairs (p,q) with w_p(q)≥ k in a declared box. A more relevant target would control simultaneous rows: #{ (P,Q): rank( L_P,Qbmod p^k) is deficient }. The coherence lattice suggests studying exterior powers rather than entrywise defects alone.
Exact mathematical passage
\#\left\{ (P,Q): \operatorname{rank}(\mathcal L_{P,Q}\bmod p^k) \text{ is deficient} \right\}.
\mathbf L_p^{\mathrm{st}}Use in source · 1 passage
Search aliases: L_p^st mathbf mathrm
Use in source · detailed_2.tex, line 561
The filtered local-global target. The intended transducer is a filtered chain J_c ⟶ ⊕_pmid c L_p^st ⟶ L_R^glob xrightarrowR= T L_ T_f ⟶ C_f. It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archimedean size.
Exact mathematical passage
\mathbf J_c \longrightarrow \bigoplus_{p\mid c}\mathbf L_p^{\mathrm{st}} \longrightarrow \mathbf L_{R^{\mathrm{glob}}} \xrightarrow{R=\mathbb T} \mathbf L_{\mathbb T_f} \longrightarrow \mathbf C_f.
\mathbf L_{R^{\mathrm{glob}}}Use in source · 1 passage
Search aliases: L_R^glob mathbf mathrm
Use in source · detailed_2.tex, line 561
The filtered local-global target. The intended transducer is a filtered chain J_c ⟶ ⊕_pmid c L_p^st ⟶ L_R^glob xrightarrowR= T L_ T_f ⟶ C_f. It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archimedean size.
Exact mathematical passage
\mathbf J_c \longrightarrow \bigoplus_{p\mid c}\mathbf L_p^{\mathrm{st}} \longrightarrow \mathbf L_{R^{\mathrm{glob}}} \xrightarrow{R=\mathbb T} \mathbf L_{\mathbb T_f} \longrightarrow \mathbf C_f.
\Lcal_{\mathbf n}Use in source · 3 passages
Search aliases: Lcal_ n Lcal mathbf
Use in source · overview.tex, line 334
The prime-contact coherence data structure. The reusable data attached to a primitive sparse relation is A( n,ϵ) = ( R, M, Witt_ n, κ_ n, Lcal_ n, Λ_coh, χ_coh, G_ n ). The calculus consists of exact transformations among these layers, plus the norm and determinant questions required to compare them globally.
Exact mathematical passage
\boxed{ \mathfrak A(\mathbf n,\boldsymbol\epsilon) = \left( \mathbf R, \mathbf M, \Witt_{\mathbf n}, \kappa_{\mathbf n}, \Lcal_{\mathbf n}, \Lambda_{\mathrm{coh}}, \chi_{\mathrm{coh}}, \mathcal G_{\mathbf n} \right). }
\lceilUse in source · 3 passages
Search aliases: lceil
Use in source · appendices_more.tex, line 37
For a scalar split t=y_1+⋯+y_e, the child sup norm can fall to lceilfrac|t|erceil. The synthesis map has norm e, so elceilfrac|t|erceil≥|t|. This explains why raw shell splitting cannot lower the intrinsic packet height.
Exact mathematical passage
\left\lceil\frac{|t|}{e}\right\rceil.
\lcmUse in source · 3 passages
Search aliases: lcm
Use in source · appendices_more.tex, line 295
Coherence-character composition. Suppose the two normalized coherence characters have orders C_1 and C_2. Their product character has order dividing lcm(C_1,C_2). If the shared term forces a compatibility relation, the actual order can be strictly smaller. Conversely, the combined system can expose a higher-rank determinant even when each individual quotient is cyclic.
Exact mathematical passage
\lcm(C_1,C_2).
\operatorname{lcm}Use in source · 1 passage
Search aliases: lcm operatorname
Use in source · detailed_3.tex, line 61
The class [c]^k is trivial exactly when every prime valuation ke_i is divisible by M. The least positive such k is lcm_i fracMgcd(M,e_i) = fracMgcd(M,e_1,…,e_r). This is the order of the image. The kernel size follows.
Exact mathematical passage
\operatorname{lcm}_i \frac{M}{\gcd(M,e_i)} = \frac{M}{\gcd(M,e_1,\ldots,e_r)}.
\operatorname{lcm}_iUse in source · 1 passage
Search aliases: lcm_i operatorname
Use in source · detailed_3.tex, line 61
The class [c]^k is trivial exactly when every prime valuation ke_i is divisible by M. The least positive such k is lcm_i fracMgcd(M,e_i) = fracMgcd(M,e_1,…,e_r). This is the order of the image. The kernel size follows.
Exact mathematical passage
\operatorname{lcm}_i \frac{M}{\gcd(M,e_i)} = \frac{M}{\gcd(M,e_1,\ldots,e_r)}.
\lfloorUse in source · 3 passages
Search aliases: lfloor
Use in source · appendices.tex, line 496
[leftmargin=1.5em] item Compute r_u, P_u, R, Q, and the exact stable height overline H. item Compute the primitive obstruction height H(1) by binary search on the three interval intersections. item Set K_max = lfloorfracH(1)-1overline Hrfloor. item For 2≤ k≤ K_max, skip a coset when koverline H already reaches the current best height. item Build centered CRT phase residues for the three sectors. item Compute the coarse affine line. item Binary-search the smallest H whose three admissible n-inter…
Exact mathematical passage
K_{\max} = \left\lfloor\frac{H(1)-1}{\overline H}\right\rfloor.
\operatorname{Li}Use in source · 6 passages
Search aliases: Li operatorname
Use in source · detailed_3.tex, line 199
Root-distribution compression. Let u^M=z and let ζ_M be a primitive Mth root of unity. For n≥1, ∑_j=0^M-1 Li_n(ζ_M^ju) = M^1-nLi_n(u^M). Inside the disk of convergence, this follows by summing the defining series and using the root-of-unity filter. Analytic continuation extends the identity with compatible branches.
Exact mathematical passage
\boxed{ \sum_{j=0}^{M-1} \operatorname{Li}_n(\zeta_M^ju) = M^{1-n}\operatorname{Li}_n(u^M). }
\operatorname{Li}_2Use in source · 1 passage
Search aliases: Li_2 operatorname
Use in source · detailed_3.tex, line 225
Three-point functional closure. Set z=frac ac, 1-z=frac bc, β=1-frac1z=-frac ba. The Rogers dilogarithm L_2(z) = Li_2(z) + frac12log zlog(1-z) satisfies L_2(z)+L_2(1-z)=fracπ^26. At weight three, · Li_3(z) +Li_3(1-z) +Li_3(β)
Exact mathematical passage
L_2(z) = \operatorname{Li}_2(z) + \frac12\log z\log(1-z)
\operatorname{Li}_3Use in source · 3 passages
Search aliases: Li_3 operatorname
Use in source · detailed_3.tex, line 239
Three-point functional closure. Set z=frac ac, 1-z=frac bc, β=1-frac1z=-frac ba. The Rogers dilogarithm L_2(z) = Li_2(z) + frac12log zlog(1-z) satisfies L_2(z)+L_2(1-z)=fracπ^26. At weight three, · Li_3(z) +Li_3(1-z) +Li_3(β) · = ζ(3) +frac16log^3z +fracπ^26log z -frac12log^2zlog(1-z). The balanced Kummer coordinate is therefore exactly the third argument in the depth-three functional closure.
Exact mathematical passage
\boxed{ \begin{aligned} &\operatorname{Li}_3(z) +\operatorname{Li}_3(1-z) +\operatorname{Li}_3(\beta)\\ &\qquad= \zeta(3) +\frac16\log^3z +\frac{\pi^2}{6}\log z -\frac12\log^2z\log(1-z). \end{aligned} }
\operatorname{Li}_nUse in source · 2 passages
Search aliases: Li_n operatorname
Use in source · detailed_3.tex, line 199
Root-distribution compression. Let u^M=z and let ζ_M be a primitive Mth root of unity. For n≥1, ∑_j=0^M-1 Li_n(ζ_M^ju) = M^1-nLi_n(u^M). Inside the disk of convergence, this follows by summing the defining series and using the root-of-unity filter. Analytic continuation extends the identity with compatible branches.
Exact mathematical passage
\boxed{ \sum_{j=0}^{M-1} \operatorname{Li}_n(\zeta_M^ju) = M^{1-n}\operatorname{Li}_n(u^M). }
\log^2Use in source · 1 passage
Search aliases: log^2 log
Use in source · detailed_3.tex, line 239
Three-point functional closure. Set z=frac ac, 1-z=frac bc, β=1-frac1z=-frac ba. The Rogers dilogarithm L_2(z) = Li_2(z) + frac12log zlog(1-z) satisfies L_2(z)+L_2(1-z)=fracπ^26. At weight three, · Li_3(z) +Li_3(1-z) +Li_3(β) · = ζ(3) +frac16log^3z +fracπ^26log z -frac12log^2zlog(1-z). The balanced Kummer coordinate is therefore exactly the third argument in the depth-three functional closure.
Exact mathematical passage
\boxed{ \begin{aligned} &\operatorname{Li}_3(z) +\operatorname{Li}_3(1-z) +\operatorname{Li}_3(\beta)\\ &\qquad= \zeta(3) +\frac16\log^3z +\frac{\pi^2}{6}\log z -\frac12\log^2z\log(1-z). \end{aligned} }
\log^3Use in source · 1 passage
Search aliases: log^3 log
Use in source · detailed_3.tex, line 239
Three-point functional closure. Set z=frac ac, 1-z=frac bc, β=1-frac1z=-frac ba. The Rogers dilogarithm L_2(z) = Li_2(z) + frac12log zlog(1-z) satisfies L_2(z)+L_2(1-z)=fracπ^26. At weight three, · Li_3(z) +Li_3(1-z) +Li_3(β) · = ζ(3) +frac16log^3z +fracπ^26log z -frac12log^2zlog(1-z). The balanced Kummer coordinate is therefore exactly the third argument in the depth-three functional closure.
Exact mathematical passage
\boxed{ \begin{aligned} &\operatorname{Li}_3(z) +\operatorname{Li}_3(1-z) +\operatorname{Li}_3(\beta)\\ &\qquad= \zeta(3) +\frac16\log^3z +\frac{\pi^2}{6}\log z -\frac12\log^2z\log(1-z). \end{aligned} }
\log_2Use in source · 7 passages
Search aliases: log_2 log
Use in source · appendices_final.tex, line 129
Dyadic coefficient depth. For odd q, separate the sign by writing q=(-1)^(q-1)/2⟨ q⟩_2, ⟨ q⟩_2∈1+4Z_2. Then ⟨ q⟩_2^2=q^2. The weighted coefficient is ℓ_2,e(q) = frac14log_2⟨ q⟩_2. If s=v_2(q^2-1), then v_2(ℓ_2,e(q))=s-3, truncated at e-2. The missing first layer is the separate sign bit. This is the final form of the weighted dyadic branch.
Exact mathematical passage
\ell_{2,e}(q) = \frac14\log_2\langle q\rangle_2.
\log_pUse in source · 9 passages
Search aliases: log_p log
Use in source · appendices.tex, line 417
Local logarithmic layer. For odd p, the identity v_p(log_p(1+u))=v_p(u) holds when v_p(u)>0. This yields the exact output notch depth. The coefficient formula v_p(q^p-1-1)-1 comes from the principal-unit logarithm of the input prime.
Exact mathematical passage
v_p(\log_p(1+u))=v_p(u)
\longleftrightarrowUse in source · 1 passage
Search aliases: longleftrightarrow
Use in source · overview.tex, line 700
The Frey curve as a nonlinear signal transform. Attach the marked Frey curve E_a,b,c:y^2=x(x-a)(x+b). Its discriminant is Δ_E=16a^2b^2c^2. At odd primes dividing abc, the conductor records the activated channel while the minimal discriminant records the local depth. Thus the Frey transform converts the prime-shell split into a standard automorphic split: conductor · longleftrightarrow activated bandwidth, minimal discriminant · longleftrightarrow persistence depth.
Exact mathematical passage
\begin{aligned} \text{conductor} &\longleftrightarrow \text{activated bandwidth},\\ \text{minimal discriminant} &\longleftrightarrow \text{persistence depth}. \end{aligned}
\mathbf MUse in source · 3 passages
Search aliases: M mathbf
Use in source · overview.tex, line 334
The prime-contact coherence data structure. The reusable data attached to a primitive sparse relation is A( n,ϵ) = ( R, M, Witt_ n, κ_ n, Lcal_ n, Λ_coh, χ_coh, G_ n ). The calculus consists of exact transformations among these layers, plus the norm and determinant questions required to compare them globally.
Exact mathematical passage
\boxed{ \mathfrak A(\mathbf n,\boldsymbol\epsilon) = \left( \mathbf R, \mathbf M, \Witt_{\mathbf n}, \kappa_{\mathbf n}, \Lcal_{\mathbf n}, \Lambda_{\mathrm{coh}}, \chi_{\mathrm{coh}}, \mathcal G_{\mathbf n} \right). }
\mathbf mUse in source · 1 passage
Search aliases: m mathbf
Use in source · detailed_2.tex, line 365
A toy Hecke order. Consider T( m) = {(x_0,x_1,…,x_r)∈Z^r+1:x_i≡ x_0pmodm_i}. For the target coordinate x_0, the cotangent module is p/ p^2 ≅ ⊕_i=1^rZ/m_iZ. The cyclic congruence quotient is Z/lcm(m_1,…,m_r)Z. The quotient of cotangent order by congruence order is frac∏_i m_ilcm(m_i). This is a toy Wiles-defect factor. It measures the multiplicity discarded by cyclic compression.
Exact mathematical passage
\mathbb T(\mathbf m) = \{(x_0,x_1,\ldots,x_r)\in\Z^{r+1}:x_i\equiv x_0\pmod{m_i}\}.
\mathcal MUse in source · 1 passage
Search aliases: M mathcal
Use in source · detailed_3.tex, line 673
Prime-contact data structure. The reusable object attached to a sparse relation is A( n,ϵ) = ( W_ n, κ_ n, L_ n, Λ_coh, R_∞, M_glob ). The first four components are finite and local-global arithmetic data. The last two are optional archimedean and global-motivic realizations. Keeping them separate prevents an unproved global comparison from being built into the definition.
Exact mathematical passage
\boxed{ \mathfrak A(\mathbf n,\boldsymbol\epsilon) = \left( \mathbb W_{\mathbf n}, \kappa_{\mathbf n}, \mathcal L_{\mathbf n}, \Lambda_{\mathrm{coh}}, \mathcal R_\infty, \mathcal M_{\mathrm{glob}} \right). }
\mathsf MUse in source · 5 passages
Search aliases: M mathsf
Use in source · detailed_1.tex, line 639
0, · pmid c. Consider M_a,b,c = Λ w : Z^S⟶Z^2. The first row is the exponent-free additive phase row. The second is its integer Wronskian row. Set Q=fracabcR.
Exact mathematical passage
\mathsf M_{a,b,c} = \begin{pmatrix} \Lambda\\ \mathsf w \end{pmatrix}: \Z^S\longrightarrow\Z^2.
mUse in source · 44 passages
Search aliases: m
Use in source · appendices.tex, line 609
Pillai and Hall. For x^m-y^n=k, the residual support is fixed by k. The two perfect-power packets must cancel to high precision while producing a small fixed output. This is a natural two-sector coherence problem.
Exact mathematical passage
x^m-y^n=k,
M^{1-n}Use in source · 2 passages
Search aliases: M^1-n
Use in source · detailed_3.tex, line 199
Root-distribution compression. Let u^M=z and let ζ_M be a primitive Mth root of unity. For n≥1, ∑_j=0^M-1 Li_n(ζ_M^ju) = M^1-nLi_n(u^M). Inside the disk of convergence, this follows by summing the defining series and using the root-of-unity filter. Analytic continuation extends the identity with compatible branches.
Exact mathematical passage
\boxed{ \sum_{j=0}^{M-1} \operatorname{Li}_n(\zeta_M^ju) = M^{1-n}\operatorname{Li}_n(u^M). }
M^3Use in source · 2 passages
Search aliases: M^3
Use in source · detailed_3.tex, line 210
The inverse synthesis gain at weight n is M^n-1. Through weights one, two, and three, the total gain is 1· M· M^2=M^3. The analytic branch count becomes fixed, but the integral transition matrix retains a large determinant.
Exact mathematical passage
\boxed{1\cdot M\cdot M^2=M^3.}
m^kUse in source · 3 passages
Search aliases: m^k
Use in source · appendices.tex, line 622
Bernoulli--Witt and ErdHos--Moser. Faulhaber's formula rewrites 1^k+⋯+(m-1)^k=m^k as a Bernoulli-polynomial identity. The denominator primes of Bernoulli numbers, the finite-difference divisibility, and the separate 2-adic and odd-adic lift structures suggest a different contact object: Bernoulli contact crystal + denominator prime packet + finite-difference coherence lattice. This branch should be developed independently before it is merged with the logarithmic prime-contact theory.
Exact mathematical passage
1^k+\cdots+(m-1)^k=m^k
M^{n-1}Use in source · 1 passage
Search aliases: M^n-1
Use in source · detailed_3.tex, line 209
The inverse synthesis gain at weight n is M^n-1. Through weights one, two, and three, the total gain is 1· M· M^2=M^3. The analytic branch count becomes fixed, but the integral transition matrix retains a large determinant.
Exact mathematical passage
M^{n-1}
m_0Use in source · 4 passages
Search aliases: m_0
Use in source · detailed_3.tex, line 96
Balanced Kummer class. The class of c does not remember the additive relation. A better coordinate is β=-frac ba=1-frac ca. Let ν_q=v_q(b)-v_q(a) for input primes qmid ab. The order of [β] in Q^×/(Q^×)^M is determined by the signed valuation vector, with one possible dyadic sign correction when M is even. If g_val = gcd(M,{ν_q}), m_0=M/g_val, then the order is m_0 unless M is even and m_0 is odd, in which case it is 2m_0.
Exact mathematical passage
g_{\mathrm{val}} = \gcd(M,\{\nu_q\}), \qquad m_0=M/g_{\mathrm{val}},
m_1Use in source · 1 passage
Search aliases: m_1
Use in source · detailed_2.tex, line 379
A toy Hecke order. Consider T( m) = {(x_0,x_1,…,x_r)∈Z^r+1:x_i≡ x_0pmodm_i}. For the target coordinate x_0, the cotangent module is p/ p^2 ≅ ⊕_i=1^rZ/m_iZ. The cyclic congruence quotient is Z/lcm(m_1,…,m_r)Z. The quotient of cotangent order by congruence order is frac∏_i m_ilcm(m_i). This is a toy Wiles-defect factor. It measures the multiplicity discarded by cyclic compression.
Exact mathematical passage
\boxed{ \Z/\lcm(m_1,\ldots,m_r)\Z. }
\mathsf M_{a,b,c}Use in source · 5 passages
Search aliases: M_a,b,c mathsf
Use in source · detailed_1.tex, line 639
0, · pmid c. Consider M_a,b,c = Λ w : Z^S⟶Z^2. The first row is the exponent-free additive phase row. The second is its integer Wronskian row. Set Q=fracabcR.
Exact mathematical passage
\mathsf M_{a,b,c} = \begin{pmatrix} \Lambda\\ \mathsf w \end{pmatrix}: \Z^S\longrightarrow\Z^2.
M_dUse in source · 1 passage
Search aliases: M_d
Use in source · detailed_3.tex, line 291
Fixed-rank integral uncertainty. Let A∈ M_d(Z) be nonsingular and suppose Dmiddet A. If every entry satisfies |A_ij|≤ B, then Hadamard's inequality gives D≤|det A|≤ d^d/2B^d. Hence B≥fracD^1/dsqrt d. A small complex period packet does not imply a small integral realization.
Exact mathematical passage
A\in M_d(\Z)
\mathcal M_{\mathrm{glob}}Use in source · 1 passage
Search aliases: M_glob mathcal mathrm
Use in source · detailed_3.tex, line 673
Prime-contact data structure. The reusable object attached to a sparse relation is A( n,ϵ) = ( W_ n, κ_ n, L_ n, Λ_coh, R_∞, M_glob ). The first four components are finite and local-global arithmetic data. The last two are optional archimedean and global-motivic realizations. Keeping them separate prevents an unproved global comparison from being built into the definition.
Exact mathematical passage
\boxed{ \mathfrak A(\mathbf n,\boldsymbol\epsilon) = \left( \mathbb W_{\mathbf n}, \kappa_{\mathbf n}, \mathcal L_{\mathbf n}, \Lambda_{\mathrm{coh}}, \mathcal R_\infty, \mathcal M_{\mathrm{glob}} \right). }
M_iUse in source · 4 passages
Search aliases: M_i
Use in source · detailed_3.tex, line 631
Sparse additive relations. Let ϵ_1n_1+⋯+ϵ_rn_r=0, ϵ_i∈{±1}, be primitive. For every term define R_i=rad(n_i), M_i=n_i/R_i. The prime-contact coherence calculus assigns:
Exact mathematical passage
R_i=\rad(n_i), \qquad M_i=n_i/R_i.
m_iUse in source · 3 passages
Search aliases: m_i
Use in source · detailed_2.tex, line 365
A toy Hecke order. Consider T( m) = {(x_0,x_1,…,x_r)∈Z^r+1:x_i≡ x_0pmodm_i}. For the target coordinate x_0, the cotangent module is p/ p^2 ≅ ⊕_i=1^rZ/m_iZ. The cyclic congruence quotient is Z/lcm(m_1,…,m_r)Z. The quotient of cotangent order by congruence order is frac∏_i m_ilcm(m_i). This is a toy Wiles-defect factor. It measures the multiplicity discarded by cyclic compression.
Exact mathematical passage
\mathbb T(\mathbf m) = \{(x_0,x_1,\ldots,x_r)\in\Z^{r+1}:x_i\equiv x_0\pmod{m_i}\}.
m_rUse in source · 1 passage
Search aliases: m_r
Use in source · detailed_2.tex, line 379
A toy Hecke order. Consider T( m) = {(x_0,x_1,…,x_r)∈Z^r+1:x_i≡ x_0pmodm_i}. For the target coordinate x_0, the cotangent module is p/ p^2 ≅ ⊕_i=1^rZ/m_iZ. The cyclic congruence quotient is Z/lcm(m_1,…,m_r)Z. The quotient of cotangent order by congruence order is frac∏_i m_ilcm(m_i). This is a toy Wiles-defect factor. It measures the multiplicity discarded by cyclic compression.
Exact mathematical passage
\boxed{ \Z/\lcm(m_1,\ldots,m_r)\Z. }
\mathscrUse in source · 3 passages
Search aliases: mathscr
Use in source · detailed_1.tex, line 782
The full shell deformation. Introduce one variable for every valuation shell and define mathscr J_n( T) = ∏_p^eparallel n ∏_r=0^e-1(p+T_p,r). At T=0, mathscr J_n(0)=n. For a+b=c, set F_a,b,c( T) = mathscr J_a( T) + mathscr J_b( T) - mathscr J_c( T). The shell gradient has entries frac ap, frac bp, -frac cp repeated according to valuation depth. The gcd of all entries is one.
Exact mathematical passage
\mathscr J_n(\mathbf T) = \prod_{p^e\parallel n} \prod_{r=0}^{e-1}(p+T_{p,r}).
\maxUse in source · 8 passages
Search aliases: max
Use in source · appendices.tex, line 496
[leftmargin=1.5em] item Compute r_u, P_u, R, Q, and the exact stable height overline H. item Compute the primitive obstruction height H(1) by binary search on the three interval intersections. item Set K_max = lfloorfracH(1)-1overline Hrfloor. item For 2≤ k≤ K_max, skip a coset when koverline H already reaches the current best height. item Build centered CRT phase residues for the three sectors. item Compute the coarse affine line. item Binary-search the smallest H whose three admissible n-inter…
Exact mathematical passage
K_{\max} = \left\lfloor\frac{H(1)-1}{\overline H}\right\rfloor.
\min_iUse in source · 1 passage
Search aliases: min_i min
Use in source · detailed_3.tex, line 71
Because g_c≤min_i e_i≤log_2 c, we obtain the universal near-injectivity estimate |kerκ_c|≤log_2c. This closes the global-occupancy problem for the contact packet up to logarithmic loss.
Exact mathematical passage
g_c\le\min_i e_i\le\log_2 c,
\min_{k\neq0}Use in source · 1 passage
Search aliases: min_k≠0 min neq
Use in source · detailed_1.tex, line 729
For k∈Z, define H(k) = min{ normx_∞: x∈ L, ω(x)=k }. Three heights must be distinguished: H(1), Φ=min_k≠0H(k), overline H= inf_k≥1fracH(k)k. The shortest nonzero detector may live in a nonprimitive obstruction coset.
Exact mathematical passage
H(1), \qquad \Phi=\min_{k\neq0}H(k), \qquad \overline H= \inf_{k\ge1}\frac{H(k)}k.
NUse in source · 34 passages
Search aliases: N
Use in source · appendices_final.tex, line 255
Polyphase circulant determinant. For the valuation word V_e(T)=1+T+⋯+T^e-1 in A_N=Z[T]/(T^N-1), multiplication is circulant. Its determinant is ∏_ζ^N=1V_e(ζ). Let I_N be the augmentation ideal of A_N. When gcd(e,N)=1, the nontrivial roots are permuted by ζ↦ζ^e, so multiplication by V_e has determinant one on the integral lattice I_N and is therefore an automorphism of it. On A_N/I_N≅Z, the map is multiplication by e. The exact sequence for I_N⊂ A_N then gives cokernel Z/eZ and Smith form diag(1,…
Exact mathematical passage
A_N=\Z[T]/(T^N-1)
\mathbf nUse in source · 8 passages
Search aliases: n mathbf
Use in source · detailed_3.tex, line 673
Prime-contact data structure. The reusable object attached to a sparse relation is A( n,ϵ) = ( W_ n, κ_ n, L_ n, Λ_coh, R_∞, M_glob ). The first four components are finite and local-global arithmetic data. The last two are optional archimedean and global-motivic realizations. Keeping them separate prevents an unproved global comparison from being built into the definition.
Exact mathematical passage
\boxed{ \mathfrak A(\mathbf n,\boldsymbol\epsilon) = \left( \mathbb W_{\mathbf n}, \kappa_{\mathbf n}, \mathcal L_{\mathbf n}, \Lambda_{\mathrm{coh}}, \mathcal R_\infty, \mathcal M_{\mathrm{glob}} \right). }
n_1Use in source · 2 passages
Search aliases: n_1
Use in source · detailed_3.tex, line 625
Sparse additive relations. Let ϵ_1n_1+⋯+ϵ_rn_r=0, ϵ_i∈{±1}, be primitive. For every term define R_i=rad(n_i), M_i=n_i/R_i. The prime-contact coherence calculus assigns:
Exact mathematical passage
\epsilon_1n_1+\cdots+\epsilon_rn_r=0, \qquad \epsilon_i\in\{\pm1\},
N_EUse in source · 4 passages
Search aliases: N_E
Use in source · detailed_2.tex, line 155
Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be an optimal modular parametrization, and let ϕ_E^*ω_E = 2π i c_E f_E(z) dz. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_E^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area factor proportional to 1/c. Thus the height c can be written as a shape-corrected synthesis gain involving modular degree, Manin normalization, and Pete…
Exact mathematical passage
\phi_E:X_0(N_E)\longrightarrow E
n_iUse in source · 4 passages
Search aliases: n_i
Use in source · detailed_3.tex, line 631
Sparse additive relations. Let ϵ_1n_1+⋯+ϵ_rn_r=0, ϵ_i∈{±1}, be primitive. For every term define R_i=rad(n_i), M_i=n_i/R_i. The prime-contact coherence calculus assigns:
Exact mathematical passage
R_i=\rad(n_i), \qquad M_i=n_i/R_i.
n_jUse in source · 1 passage
Search aliases: n_j
Use in source · overview.tex, line 942
A general contact functor. The construction extends to a sparse relation ∑_j=1^rϵ_jn_j=0. Choose one or more output terms whose repeated depth is to be explained. The assignment ( n,ϵ) ⟼ A( n,ϵ) is canonical under permutations, sign reversal, primitive reduction, and primary decomposition. The optional geometric lift depends on the equation, but the support, contact, Kummer, and logarithmic layers do not.
Exact mathematical passage
\sum_{j=1}^{r}\epsilon_jn_j=0.
n_rUse in source · 2 passages
Search aliases: n_r
Use in source · detailed_3.tex, line 625
Sparse additive relations. Let ϵ_1n_1+⋯+ϵ_rn_r=0, ϵ_i∈{±1}, be primitive. For every term define R_i=rad(n_i), M_i=n_i/R_i. The prime-contact coherence calculus assigns:
Exact mathematical passage
\epsilon_1n_1+\cdots+\epsilon_rn_r=0, \qquad \epsilon_i\in\{\pm1\},
\normUse in source · 23 passages
Search aliases: norm
Use in source · appendices_more.tex, line 15
Why exact synthesis prevents free coordinate splitting. Let X and Y be normed spaces, and suppose J:X⟶ Y, S:Y⟶ X satisfy SJ=I_X. Then normx = normSJx ≤ normSnormJx. Therefore normJx ≥ fracnormxnormS. An analysis map may make coordinates smaller only by increasing the synthesis norm. This elementary inequality is the common source of several no-free-refinement results.
Exact mathematical passage
\norm{x} = \norm{SJx} \le \norm{S}\norm{Jx}.
oUse in source · 1 passage
Search aliases: o
Use in source · detailed_1.tex, line 246
Define the shortest transverse packet Θ(a,b,c) = min{ normx_∞: x∈ T_a,b,c, W_x(a,b)≠0 }. Then c ≤ fracRlog2Θ(a,b,c)log c. A subpolynomial estimate Θ(a,b,c)=c^o(1) would imply the strong abc inequality. Later chapters explain why this target is too close to the conjecture to be reached by generic lattice geometry alone.
Exact mathematical passage
\Theta(a,b,c)=c^{o(1)}
\mathrm{obs}Use in source · 1 passage
Search aliases: obs mathrm
Use in source · detailed_1.tex, line 119
Over R, the additive projection is P_add = I- fracλλ^ Tnormλ_2^2. The obstruction direction inside the additive hyperplane is μ_⊥=P_addμ. When μ_⊥≠0, the rank-one obstruction projector is P_obs = fracμ_⊥μ_⊥^ T normμ_⊥_2^2. This is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation direction, one obstruction direction, and the remaining null packets.
Exact mathematical passage
P_{\mathrm{obs}} = \frac{\mu_\perp\mu_\perp^{\mathsf T}} {\norm{\mu_\perp}_2^2}.
\ordUse in source · 1 passage
Search aliases: ord
Use in source · overview.tex, line 273
[Kummer order from the valuation vector] Assume x>0. Then ord([x]) = fracMgcd(M,{ν_q}_q∈ S). For a negative rational number and even M, one additional dyadic sign layer may occur.
Exact mathematical passage
\boxed{ \ord([x]) = \frac{M}{\gcd(M,\{\nu_q\}_{q\in S})}. }
\mathrm{out}Use in source · 1 passage
Search aliases: out mathrm
Use in source · appendices_more.tex, line 250
Contact pushout. Let Jcal_C^out be the contact packet of C in the first equation and Jcal_C^in its packet in the second. Both are canonically rad(C)Z/CZ. The first gluing operation identifies them through the identity map. The total contact object is the pushout of the two relation diagrams along this common packet.
Exact mathematical passage
\Jcal_C^{\mathrm{out}}
\overline{\omega_E}Use in source · 2 passages
Search aliases: overlineω_E overline omega
Use in source · detailed_2.tex, line 165
Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be an optimal modular parametrization, and let ϕ_E^*ω_E = 2π i c_E f_E(z) dz. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_E^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area factor proportional to 1/c. Thus the height c can be written as a shape-corrected synthesis gain involving modular degree, Manin normalization, and Pete…
Exact mathematical passage
A(E) = \frac{i}{2} \int_{E(\C)} \omega_E\wedge\overline{\omega_E}.
\mathbb PUse in source · 1 passage
Search aliases: P mathbb
Use in source · detailed_2.tex, line 418
The three-cusp section. View a primitive triple as the integral section σ_a,b,c: SpecZ⟶ P^1_Z, σ=[a:c]. The pullbacks of the boundary divisors 0, 1, and ∞ are div(a), div(b), div(c). For the output cusp, let C=div(c), C_red=div(R_c). Define the two-term contact complex J_c = [ O(-C)⟶ O(-C_red)]. On global sections, J_c = [cZ↪ R_cZ] ≃ [Zxrightarrowc/R_cZ].
Exact mathematical passage
\sigma_{a,b,c}: \Spec\Z\longrightarrow\mathbb P^1_\Z, \qquad \sigma=[a:c].
\mathcal PUse in source · 2 passages
Search aliases: P mathcal
Use in source · detailed_2.tex, line 347
Why one cyclic quotient is not enough for split persistence. The split persistence module P_c ≅ ⊕_p^eparallel c (Z/pZ)^e-1 has large p-rank. The p-torsion of a cyclic target has rank at most one. Therefore every map from P_c to a cyclic group has a forced kernel of order at least ∏_p^eparallel cp^max(e-2,0). This is a true obstruction to the split source. It motivates the Hecke cotangent module p/ p^2, which can retain multiple independent congruence directions before cyclic compression.
Exact mathematical passage
\mathcal P_c \cong \bigoplus_{p^e\parallel c} (\Z/p\Z)^{e-1}
\mathfrak pUse in source · 2 passages
Search aliases: p mathfrak
Use in source · detailed_2.tex, line 358
persistence module P_c ≅ ⊕_p^eparallel c (Z/pZ)^e-1 has large p-rank. The p-torsion of a cyclic target has rank at most one. Therefore every map from P_c to a cyclic group has a forced kernel of order at least ∏_p^eparallel cp^max(e-2,0). This is a true obstruction to the split source. It motivates the Hecke cotangent module p/ p^2, which can retain multiple independent congruence directions before cyclic compression.
Exact mathematical passage
\mathfrak p/\mathfrak p^2,
\mathbb P^1_\ZUse in source · 1 passage
Search aliases: P^1_Z mathbb Z
Use in source · detailed_2.tex, line 418
The three-cusp section. View a primitive triple as the integral section σ_a,b,c: SpecZ⟶ P^1_Z, σ=[a:c]. The pullbacks of the boundary divisors 0, 1, and ∞ are div(a), div(b), div(c). For the output cusp, let C=div(c), C_red=div(R_c). Define the two-term contact complex J_c = [ O(-C)⟶ O(-C_red)]. On global sections, J_c = [cZ↪ R_cZ] ≃ [Zxrightarrowc/R_cZ].
Exact mathematical passage
\sigma_{a,b,c}: \Spec\Z\longrightarrow\mathbb P^1_\Z, \qquad \sigma=[a:c].
\mathfrak p^2Use in source · 2 passages
Search aliases: p^2 mathfrak
Use in source · detailed_2.tex, line 358
persistence module P_c ≅ ⊕_p^eparallel c (Z/pZ)^e-1 has large p-rank. The p-torsion of a cyclic target has rank at most one. Therefore every map from P_c to a cyclic group has a forced kernel of order at least ∏_p^eparallel cp^max(e-2,0). This is a true obstruction to the split source. It motivates the Hecke cotangent module p/ p^2, which can retain multiple independent congruence directions before cyclic compression.
Exact mathematical passage
\mathfrak p/\mathfrak p^2,
p^2Use in source · 1 passage
Search aliases: p^2
Use in source · detailed_1.tex, line 516
Second-order phase energy. At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^2 - frac1e∑_j≠0|widehat T_j|^2 ]. Thus nonzero polyphase energy first becomes visible at second order. Later formal-smoothness results show that this c…
Exact mathematical passage
\prod_{r=0}^{e-1} \left(1+\frac{T_r}{p}\right) = 1+ \frac1p\sum_rT_r+ \frac1{p^2}\sum_{r<s}T_rT_s+ O(T^3).
p^{e-1}Use in source · 15 passages
Search aliases: p^e-1
Use in source · appendices_final.tex, line 9
Sequential normalization. Let p^eparallel c. The local output contact packet has order p^e-1. The final factorization separates this depth into distinct mechanisms. The separation must be sequential because the same prime-power depth can otherwise be counted more than once.
Exact mathematical passage
p^{e-1}.
p^{e_p}Use in source · 5 passages
Search aliases: p^e_p
Use in source · detailed_2.tex, line 5
Prime support as bandwidth and valuation as persistence. For n=∏_p p^e_p, define the prime-shell signal Ψ_n(p,r) = sqrtlog p 1_0≤ r<e_p. Its total energy is normΨ_n_2^2 = ∑_p e_plog p = log n. Let P_act retain only the first shell in each active prime channel. Then normP_actΨ_n_2^2 = ∑_pmid nlog p = lograd(n). Thus the radical is activated bandwidth, while the difference log n-lograd(n) is repeated persistence depth.
Exact mathematical passage
n=\prod_p p^{e_p},
p^jUse in source · 3 passages
Search aliases: p^j
Use in source · detailed_2.tex, line 479
Associated graded and carry extensions. At p^eparallel c, Jcal_p,e = pZ_p/p^eZ_p ≅ Z/p^e-1Z. Its filtration p^jZ_p/p^eZ_p has successive quotients F_p. Thus grJcal_p,e≅(F_p)^e-1. The exact sequences 0⟶ Z/p^j-1Z xrightarrow× p Z/p^jZ ⟶ F_p ⟶0 are nonsplit. These carry extensions are precisely what the split persistence packet forgot.
Exact mathematical passage
p^j\Z_p/p^e\Z_p
p^{j-1}Use in source · 1 passage
Search aliases: p^j-1
Use in source · detailed_2.tex, line 487
Associated graded and carry extensions. At p^eparallel c, Jcal_p,e = pZ_p/p^eZ_p ≅ Z/p^e-1Z. Its filtration p^jZ_p/p^eZ_p has successive quotients F_p. Thus grJcal_p,e≅(F_p)^e-1. The exact sequences 0⟶ Z/p^j-1Z xrightarrow× p Z/p^jZ ⟶ F_p ⟶0 are nonsplit. These carry extensions are precisely what the split persistence packet forgot.
Exact mathematical passage
0\longrightarrow \Z/p^{j-1}\Z \xrightarrow{\times p} \Z/p^j\Z \longrightarrow \F_p \longrightarrow0
p^kUse in source · 3 passages
Search aliases: p^k
Use in source · appendices_final.tex, line 76
Why the Kummer layer comes first. Suppose every input exponent is divisible by p^k. Then the balanced rational packet is already a p^kth power in the valuation lattice. The corresponding output contact depth should not also be attributed to a local logarithm defect.
Exact mathematical passage
p^k
p^{\max(e-1-v_p(\eta_f),0)}Use in source · 1 passage
Search aliases: p^max(e-1-v_p(η_f),0) max eta
Use in source · detailed_2.tex, line 532
For the output source m=c/R_c, the corrected algebraic defect against a cyclic target of order η_f is δ_alg(c,f) = fracc/R_cgcd(c/R_c,η_f). Prime by prime, δ_alg(c,f) = ∏_p^eparallel c p^max(e-1-v_p(η_f),0). The problem is missing depth, not missing rank.
Exact mathematical passage
\delta_{\mathrm{alg}}(c,f) = \prod_{p^e\parallel c} p^{\max(e-1-v_p(\eta_f),0)}.
p^{\max(e-2,0)}Use in source · 1 passage
Search aliases: p^max(e-2,0) max
Use in source · detailed_2.tex, line 354
Why one cyclic quotient is not enough for split persistence. The split persistence module P_c ≅ ⊕_p^eparallel c (Z/pZ)^e-1 has large p-rank. The p-torsion of a cyclic target has rank at most one. Therefore every map from P_c to a cyclic group has a forced kernel of order at least ∏_p^eparallel cp^max(e-2,0). This is a true obstruction to the split source. It motivates the Hecke cotangent module p/ p^2, which can retain multiple independent congruence directions before cyclic compression.
Exact mathematical passage
\prod_{p^e\parallel c}p^{\max(e-2,0)}.
p^nUse in source · 1 passage
Search aliases: p^n
Use in source · bibliography.tex, line 86
bibitemAKN B. Antieau, A. Krause, and T. Nikolaus, ``On the K-theory of Z/p^n,'' arXiv:2405.04329, 2024.
Exact mathematical passage
\mathbb Z/p^n
p_1Use in source · 1 passage
Search aliases: p_1
Use in source · detailed_1.tex, line 911
Local CRT normal form. Fix one sector with primes p_1,…,p_m and radical r=∏_i p_i. For a nonempty sector, the local equation is ∑_ifrac rp_ix_i=t. An empty unit sector has no prime coordinates and is feasible exactly when t=0. Modulo p_i, the coordinate x_i is forced: x_i ≡ t(frac rp_i)^-1 pmodp_i. Choose the centered residue ξ_i and write x_i=ξ_i+p_i z_i. Substitution gives one quotient equation ∑_i z_i=q(t). For a box radius H, each z_i lies in the interval ℓ_i(H) = lceilfrac-H-ξ_ip_irceil ≤ z…
Exact mathematical passage
p_1,\ldots,p_m
P_aUse in source · 5 passages
Search aliases: P_a
Use in source · detailed_1.tex, line 588
The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a,
Exact mathematical passage
P_at_a+P_bt_b=P_ct_c,
P_{\mathrm{act}}Use in source · 5 passages
Search aliases: P_act mathrm
Use in source · detailed_2.tex, line 22
Prime support as bandwidth and valuation as persistence. For n=∏_p p^e_p, define the prime-shell signal Ψ_n(p,r) = sqrtlog p 1_0≤ r<e_p. Its total energy is normΨ_n_2^2 = ∑_p e_plog p = log n. Let P_act retain only the first shell in each active prime channel. Then normP_actΨ_n_2^2 = ∑_pmid nlog p = lograd(n). Thus the radical is activated bandwidth, while the difference log n-lograd(n) is repeated persistence depth.
Exact mathematical passage
P_{\mathrm{act}}
P_{\mathrm{add}}Use in source · 2 passages
Search aliases: P_add mathrm
Use in source · detailed_1.tex, line 108
Over R, the additive projection is P_add = I- fracλλ^ Tnormλ_2^2. The obstruction direction inside the additive hyperplane is μ_⊥=P_addμ. When μ_⊥≠0, the rank-one obstruction projector is P_obs = fracμ_⊥μ_⊥^ T normμ_⊥_2^2. This is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation direction, one obstruction direction, and the remaining null packets.
Exact mathematical passage
P_{\mathrm{add}} = I- \frac{\lambda\lambda^{\mathsf T}}{\norm{\lambda}_2^2}.
P_bUse in source · 5 passages
Search aliases: P_b
Use in source · detailed_1.tex, line 588
The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a,
Exact mathematical passage
P_at_a+P_bt_b=P_ct_c,
P_cUse in source · 10 passages
Search aliases: P_c
Use in source · detailed_1.tex, line 588
The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a,
Exact mathematical passage
P_at_a+P_bt_b=P_ct_c,
\mathcal P_cUse in source · 2 passages
Search aliases: P_c mathcal
Use in source · detailed_2.tex, line 347
Why one cyclic quotient is not enough for split persistence. The split persistence module P_c ≅ ⊕_p^eparallel c (Z/pZ)^e-1 has large p-rank. The p-torsion of a cyclic target has rank at most one. Therefore every map from P_c to a cyclic group has a forced kernel of order at least ∏_p^eparallel cp^max(e-2,0). This is a true obstruction to the split source. It motivates the Hecke cotangent module p/ p^2, which can retain multiple independent congruence directions before cyclic compression.
Exact mathematical passage
\mathcal P_c \cong \bigoplus_{p^e\parallel c} (\Z/p\Z)^{e-1}
p_iUse in source · 11 passages
Search aliases: p_i
Use in source · appendices_final.tex, line 208
Local CRT lift in closed form. Let r=∏_i=1^mp_i, w_i=r/p_i. The equation ∑_iw_ix_i=t has a solution because gcd(w_1,…,w_m)=1. Modulo p_i, x_i≡ t w_i^-1pmodp_i. Let ξ_i be the centered residue. Then ∑_iw_iξ_i-t is divisible by r. Set q=fract-∑_iw_iξ_ir. Writing x_i=ξ_i+p_i z_i gives ∑_i z_i=q.
Exact mathematical passage
r=\prod_{i=1}^{m}p_i, \qquad w_i=r/p_i.
p_i^{e_i}Use in source · 2 passages
Search aliases: p_i^e_i
Use in source · detailed_3.tex, line 5
The contact modulus. Let c=∏_i=1^rp_i^e_i, R_c=∏_i p_i, M=fraccR_c =∏_i p_i^e_i-1. The filtered output contact packet is the cyclic group Z/MZ. The first global question is whether this packet can be realized by one arithmetic class without losing most of its order.
Exact mathematical passage
c=\prod_{i=1}^{r}p_i^{e_i}, \qquad R_c=\prod_i p_i, \qquad M=\frac{c}{R_c} =\prod_i p_i^{e_i-1}.
p_i^{e_i-1}Use in source · 1 passage
Search aliases: p_i^e_i-1
Use in source · detailed_3.tex, line 5
The contact modulus. Let c=∏_i=1^rp_i^e_i, R_c=∏_i p_i, M=fraccR_c =∏_i p_i^e_i-1. The filtered output contact packet is the cyclic group Z/MZ. The first global question is whether this packet can be realized by one arithmetic class without losing most of its order.
Exact mathematical passage
c=\prod_{i=1}^{r}p_i^{e_i}, \qquad R_c=\prod_i p_i, \qquad M=\frac{c}{R_c} =\prod_i p_i^{e_i-1}.
p_mUse in source · 1 passage
Search aliases: p_m
Use in source · detailed_1.tex, line 911
Local CRT normal form. Fix one sector with primes p_1,…,p_m and radical r=∏_i p_i. For a nonempty sector, the local equation is ∑_ifrac rp_ix_i=t. An empty unit sector has no prime coordinates and is feasible exactly when t=0. Modulo p_i, the coordinate x_i is forced: x_i ≡ t(frac rp_i)^-1 pmodp_i. Choose the centered residue ξ_i and write x_i=ξ_i+p_i z_i. Substitution gives one quotient equation ∑_i z_i=q(t). For a box radius H, each z_i lies in the interval ℓ_i(H) = lceilfrac-H-ξ_ip_irceil ≤ z…
Exact mathematical passage
p_1,\ldots,p_m
P_{\mathrm{obs}}Use in source · 1 passage
Search aliases: P_obs mathrm
Use in source · detailed_1.tex, line 119
Over R, the additive projection is P_add = I- fracλλ^ Tnormλ_2^2. The obstruction direction inside the additive hyperplane is μ_⊥=P_addμ. When μ_⊥≠0, the rank-one obstruction projector is P_obs = fracμ_⊥μ_⊥^ T normμ_⊥_2^2. This is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation direction, one obstruction direction, and the remaining null packets.
Exact mathematical passage
P_{\mathrm{obs}} = \frac{\mu_\perp\mu_\perp^{\mathsf T}} {\norm{\mu_\perp}_2^2}.
P_uUse in source · 4 passages
Search aliases: P_u
Use in source · appendices.tex, line 493
[leftmargin=1.5em] item Compute r_u, P_u, R, Q, and the exact stable height overline H. item Compute the primitive obstruction height H(1) by binary search on the three interval intersections. item Set K_max = lfloorfracH(1)-1overline Hrfloor. item For 2≤ k≤ K_max, skip a coset when koverline H already reaches the current best height. item Build centered CRT phase residues for the three sectors. item Compute the coarse affine line. item Binary-search the smallest H whose three admissible n-inter…
Exact mathematical passage
P_u
\mathrm{Pet}Use in source · 2 passages
Search aliases: Pet mathrm
Use in source · detailed_2.tex, line 173
Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be an optimal modular parametrization, and let ϕ_E^*ω_E = 2π i c_E f_E(z) dz. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_E^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area factor proportional to 1/c. Thus the height c can be written as a shape-corrected synthesis gain involving modular degree, Manin normalization, and Pete…
Exact mathematical passage
\boxed{ \deg(\phi_E)A(E) = 4\pi^2 c_E^2 \norm{f_E}_{\mathrm{Pet}}^2. }
\mathrm{ph}Use in source · 1 passage
Search aliases: ph mathrm
Use in source · appendices.tex, line 33
Prime derivative. The additive row is λ=(1,75,-448). The phase row is λ^ph=(1,25,-64). The coarse shortest transverse packet x=(-2,6,1) satisfies -2+450=448. The Wronskian is W_x=1600=fracabcR. The coarse height is six.
Exact mathematical passage
\lambda^{\mathrm{ph}}=(1,25,-64).
\pmodUse in source · 19 passages
Search aliases: pmod
Use in source · appendices.tex, line 221
Coherence lattice. After removing the sign layer, (ϵ_2,K,W,C)=(2,1,1,32). The residual row is g=(29,31)pmod32. The lattice is Λ_coh = {(x_3,x_5):29x_3+31x_5≡0pmod32}. The input vector lies in the kernel because 29(-1)+31(3)=64. The full repeated output depth has now been transformed into one arithmetic cyclic character of order 32, after the bounded sign layer has been separated.
Exact mathematical passage
g=(29,31)\pmod{32}.
\QUse in source · 7 passages
Search aliases: Q
Use in source · appendices.tex, line 166
Kummer and Fermat packets. The global Kummer map is κ_128:Z/64Z→ H^1(Q,μ_64). Since gcd(64,7)=1, it is injective.
Exact mathematical passage
\kappa_{128}:\Z/64\Z\to H^1(\Q,\mu_{64}).
\mathbb QUse in source · 1 passage
Search aliases: Q mathbb
Use in source · bibliography.tex, line 43
bibitemBCDT C. Breuil, B. Conrad, F. Diamond, and R. Taylor, ``On the modularity of elliptic curves over Q: wild 3-adic exercises,'' Journal of the American Mathematical Society 14 (2001), 843--939. https://doi.org/10.1090/S0894-0347-01-00370-8doi:10.1090/S0894-0347-01-00370-8.
Exact mathematical passage
\mathbb Q
q^2Use in source · 3 passages
Search aliases: q^2
Use in source · appendices_final.tex, line 125
Dyadic coefficient depth. For odd q, separate the sign by writing q=(-1)^(q-1)/2⟨ q⟩_2, ⟨ q⟩_2∈1+4Z_2. Then ⟨ q⟩_2^2=q^2. The weighted coefficient is ℓ_2,e(q) = frac14log_2⟨ q⟩_2. If s=v_2(q^2-1), then v_2(ℓ_2,e(q))=s-3, truncated at e-2. The missing first layer is the separate sign bit. This is the final form of the weighted dyadic branch.
Exact mathematical passage
\langle q\rangle_2^2=q^2.
q^{e-1}Use in source · 1 passage
Search aliases: q^e-1
Use in source · detailed_2.tex, line 412
ce is indexed by conductor primes qmid abc. Congruence depth is indexed by coefficient primes ℓ. The natural local signal is a matrix D_f(ℓ,q). At a split Tate prime q≠ℓ, scalarity modulo ℓ^n forces n≤min{v_ℓ(q-1),v_ℓ(v_q(q_E))}. These off-diagonal channels carry tame congruence information. They do not naturally reproduce the full base-q factor q^e-1. This mismatch motivates the later filtered contact correction and the diagonal coefficient-prime channel.
Exact mathematical passage
q^{e-1}
q^{p-1}Use in source · 9 passages
Search aliases: q^p-1
Use in source · appendices.tex, line 421
Local logarithmic layer. For odd p, the identity v_p(log_p(1+u))=v_p(u) holds when v_p(u)>0. This yields the exact output notch depth. The coefficient formula v_p(q^p-1-1)-1 comes from the principal-unit logarithm of the input prime.
Exact mathematical passage
v_p(q^{p-1}-1)-1
\Q^\timesUse in source · 4 passages
Search aliases: Q^× Q times
Use in source · detailed_3.tex, line 21
The global Kummer map. Kummer theory gives H^1(Q,μ_M) ≅ Q^×/(Q^×)^M. Define κ_c: Z/MZ ⟶ H^1(Q,μ_M), k⟼[c^k]. The valuation vector of c is (e_1,…,e_r) ∈ (Z/MZ)^r.
Exact mathematical passage
H^1(\Q,\mu_M) \cong \Q^\times/(\Q^\times)^M.
q^{\nu_q}Use in source · 4 passages
Search aliases: q^ν_q nu
Use in source · appendices_more.tex, line 171
Exact Kummer kernel for a rational packet. Let x=±∏_q∈ Sq^ν_q and consider its class modulo Mth powers. For positive x, the order is fracMgcd(M,{ν_q}). If x<0 and M is even, the sign contributes a separate element of order two. The exact order is therefore the least common multiple of the valuation order and the sign order.
Exact mathematical passage
x=\pm\prod_{q\in S}q^{\nu_q}
q_EUse in source · 1 passage
Search aliases: q_E
Use in source · detailed_2.tex, line 409
Two prime indices. The persistence is indexed by conductor primes qmid abc. Congruence depth is indexed by coefficient primes ℓ. The natural local signal is a matrix D_f(ℓ,q). At a split Tate prime q≠ℓ, scalarity modulo ℓ^n forces n≤min{v_ℓ(q-1),v_ℓ(v_q(q_E))}. These off-diagonal channels carry tame congruence information. They do not naturally reproduce the full base-q factor q^e-1. This mismatch motivates the later filtered contact correction and the diagonal coefficient-prime channel.
Exact mathematical passage
n\le\min\{v_\ell(q-1),v_\ell(v_q(q_E))\}.
q_pUse in source · 2 passages
Search aliases: q_p
Use in source · overview.tex, line 759
The integral automorphic packet. The resulting integral automorphic object is Acal^sharp(E,ι) = ( f_E, Λ_E, ω_E, {q_p}_pmid N_E, ι, ϕ_E ). The components are: [leftmargin=1.5em] item the rational Hecke eigen-signal f_E item an integral Hecke, modular-symbol, or homology lattice Λ_E item the minimal N eron differential ω_E item local Tate-depth packets q_p item the marking ι of the three rational two-torsion differences item the modular synthesis map ϕ_E
Exact mathematical passage
\boxed{ \Acal^{\sharp}(E,\iota) = \left( f_E, \Lambda_E, \omega_E, \{q_p\}_{p\mid N_E}, \iota, \phi_E \right). }
\RUse in source · 1 passage
Search aliases: R
Use in source · detailed_1.tex, line 107
Over R, the additive projection is P_add = I- fracλλ^ Tnormλ_2^2. The obstruction direction inside the additive hyperplane is μ_⊥=P_addμ. When μ_⊥≠0, the rank-one obstruction projector is P_obs = fracμ_⊥μ_⊥^ T normμ_⊥_2^2. This is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation direction, one obstruction direction, and the remaining null packets.
Exact mathematical passage
\R
\mathbf RUse in source · 3 passages
Search aliases: R mathbf
Use in source · overview.tex, line 334
The prime-contact coherence data structure. The reusable data attached to a primitive sparse relation is A( n,ϵ) = ( R, M, Witt_ n, κ_ n, Lcal_ n, Λ_coh, χ_coh, G_ n ). The calculus consists of exact transformations among these layers, plus the norm and determinant questions required to compare them globally.
Exact mathematical passage
\boxed{ \mathfrak A(\mathbf n,\boldsymbol\epsilon) = \left( \mathbf R, \mathbf M, \Witt_{\mathbf n}, \kappa_{\mathbf n}, \Lcal_{\mathbf n}, \Lambda_{\mathrm{coh}}, \chi_{\mathrm{coh}}, \mathcal G_{\mathbf n} \right). }
\mathcal RUse in source · 1 passage
Search aliases: R mathcal
Use in source · detailed_3.tex, line 673
Prime-contact data structure. The reusable object attached to a sparse relation is A( n,ϵ) = ( W_ n, κ_ n, L_ n, Λ_coh, R_∞, M_glob ). The first four components are finite and local-global arithmetic data. The last two are optional archimedean and global-motivic realizations. Keeping them separate prevents an unproved global comparison from being built into the definition.
Exact mathematical passage
\boxed{ \mathfrak A(\mathbf n,\boldsymbol\epsilon) = \left( \mathbb W_{\mathbf n}, \kappa_{\mathbf n}, \mathcal L_{\mathbf n}, \Lambda_{\mathrm{coh}}, \mathcal R_\infty, \mathcal M_{\mathrm{glob}} \right). }
\mathfrak RUse in source · 1 passage
Search aliases: R mathfrak
Use in source · detailed_3.tex, line 603
Adelic coherence regulator object. Package the surviving data as R_a,b,c = ( U_ab, β^1/K, {χ_p}_pmid c, χ_coh, Λ_coh, norm·_∞ ). Here U_ab is the input S-unit group, the local characters are the normalized logarithmic rows, and the archimedean norm records input height.
Exact mathematical passage
\boxed{ \mathfrak R_{a,b,c} = \left( U_{ab}, \beta^{1/K}, \{\chi_p\}_{p\mid c}, \chi_{\mathrm{coh}}, \Lambda_{\mathrm{coh}}, \norm{\cdot}_\infty \right). }
R^\epsUse in source · 22 passages
Search aliases: R^eps eps
Use in source · appendices.tex, line 446
[leftmargin=1.5em] item Construct a finite source of order comparable to c/R_c. item Map it into a global target with kernel at most R^eps. item Bound the target size by R_aR_bR^eps. item Multiply by R_c and absorb logarithmic factors.
Exact mathematical passage
R^\eps
R^{\mathrm{glob}}Use in source · 1 passage
Search aliases: R^glob mathrm
Use in source · detailed_2.tex, line 561
The filtered local-global target. The intended transducer is a filtered chain J_c ⟶ ⊕_pmid c L_p^st ⟶ L_R^glob xrightarrowR= T L_ T_f ⟶ C_f. It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archimedean size.
Exact mathematical passage
\mathbf J_c \longrightarrow \bigoplus_{p\mid c}\mathbf L_p^{\mathrm{st}} \longrightarrow \mathbf L_{R^{\mathrm{glob}}} \xrightarrow{R=\mathbb T} \mathbf L_{\mathbb T_f} \longrightarrow \mathbf C_f.
R_aUse in source · 18 passages
Search aliases: R_a
Use in source · appendices.tex, line 447
[leftmargin=1.5em] item Construct a finite source of order comparable to c/R_c. item Map it into a global target with kernel at most R^eps. item Bound the target size by R_aR_bR^eps. item Multiply by R_c and absorb logarithmic factors.
Exact mathematical passage
R_aR_bR^\eps
r_aUse in source · 7 passages
Search aliases: r_a
Use in source · detailed_1.tex, line 592
The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a,
Exact mathematical passage
W_y(a,b) = P_aP_b(r_at_b-r_bt_a).
\mathfrak R_{a,b,c}Use in source · 1 passage
Search aliases: R_a,b,c mathfrak
Use in source · detailed_3.tex, line 603
Adelic coherence regulator object. Package the surviving data as R_a,b,c = ( U_ab, β^1/K, {χ_p}_pmid c, χ_coh, Λ_coh, norm·_∞ ). Here U_ab is the input S-unit group, the local characters are the normalized logarithmic rows, and the archimedean norm records input height.
Exact mathematical passage
\boxed{ \mathfrak R_{a,b,c} = \left( U_{ab}, \beta^{1/K}, \{\chi_p\}_{p\mid c}, \chi_{\mathrm{coh}}, \Lambda_{\mathrm{coh}}, \norm{\cdot}_\infty \right). }
R_bUse in source · 18 passages
Search aliases: R_b
Use in source · appendices.tex, line 447
[leftmargin=1.5em] item Construct a finite source of order comparable to c/R_c. item Map it into a global target with kernel at most R^eps. item Bound the target size by R_aR_bR^eps. item Multiply by R_c and absorb logarithmic factors.
Exact mathematical passage
R_aR_bR^\eps
r_bUse in source · 7 passages
Search aliases: r_b
Use in source · detailed_1.tex, line 592
The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a,
Exact mathematical passage
W_y(a,b) = P_aP_b(r_at_b-r_bt_a).
r_cUse in source · 2 passages
Search aliases: r_c
Use in source · detailed_1.tex, line 602
The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a, t_b(n) · =t_b^(0)+nr_b, t_c(n) · =t_c^(0)+nr_c. The unit-sector cases have no moving parameter. If a=1, then (t_a,t_b,t_c)=(0,P_c,P_b), and if b=1, then (t_a,t_b,t_c)=(-P_c,0,…
Exact mathematical passage
\begin{aligned} t_a(n)&=t_a^{(0)}+nr_a,\\ t_b(n)&=t_b^{(0)}+nr_b,\\ t_c(n)&=t_c^{(0)}+nr_c. \end{aligned}
R_iUse in source · 4 passages
Search aliases: R_i
Use in source · detailed_3.tex, line 631
Sparse additive relations. Let ϵ_1n_1+⋯+ϵ_rn_r=0, ϵ_i∈{±1}, be primitive. For every term define R_i=rad(n_i), M_i=n_i/R_i. The prime-contact coherence calculus assigns:
Exact mathematical passage
R_i=\rad(n_i), \qquad M_i=n_i/R_i.
R_{\mathrm{in}}Use in source · 3 passages
Search aliases: R_in mathrm
Use in source · overview.tex, line 930
A model target is WC ≤ C_eps R_inR^eps, where R_in is the activated input bandwidth. For abc, R_in=R_aR_b.
Exact mathematical passage
\boxed{ WC \le C_\eps R_{\mathrm{in}}R^\eps, }
r_uUse in source · 8 passages
Search aliases: r_u
Use in source · appendices.tex, line 493
[leftmargin=1.5em] item Compute r_u, P_u, R, Q, and the exact stable height overline H. item Compute the primitive obstruction height H(1) by binary search on the three interval intersections. item Set K_max = lfloorfracH(1)-1overline Hrfloor. item For 2≤ k≤ K_max, skip a coset when koverline H already reaches the current best height. item Build centered CRT phase residues for the three sectors. item Compute the coarse affine line. item Binary-search the smallest H whose three admissible n-inter…
Exact mathematical passage
r_u
\mathcal R_\inftyUse in source · 1 passage
Search aliases: R_∞ mathcal infty
Use in source · detailed_3.tex, line 673
Prime-contact data structure. The reusable object attached to a sparse relation is A( n,ϵ) = ( W_ n, κ_ n, L_ n, Λ_coh, R_∞, M_glob ). The first four components are finite and local-global arithmetic data. The last two are optional archimedean and global-motivic realizations. Keeping them separate prevents an unproved global comparison from being built into the definition.
Exact mathematical passage
\boxed{ \mathfrak A(\mathbf n,\boldsymbol\epsilon) = \left( \mathbb W_{\mathbf n}, \kappa_{\mathbf n}, \mathcal L_{\mathbf n}, \Lambda_{\mathrm{coh}}, \mathcal R_\infty, \mathcal M_{\mathrm{glob}} \right). }
\operatorname{rank}Use in source · 1 passage
Search aliases: rank operatorname
Use in source · appendices_more.tex, line 338
Simultaneous Wieferich theorems. The edge weight w_p(q)=v_p(q^p-1-1)-1 measures local degeneracy. A first family of theorems could bound the number of pairs (p,q) with w_p(q)≥ k in a declared box. A more relevant target would control simultaneous rows: #{ (P,Q): rank( L_P,Qbmod p^k) is deficient }. The coherence lattice suggests studying exterior powers rather than entrywise defects alone.
Exact mathematical passage
\#\left\{ (P,Q): \operatorname{rank}(\mathcal L_{P,Q}\bmod p^k) \text{ is deficient} \right\}.
\rceilUse in source · 3 passages
Search aliases: rceil
Use in source · appendices_more.tex, line 37
For a scalar split t=y_1+⋯+y_e, the child sup norm can fall to lceilfrac|t|erceil. The synthesis map has norm e, so elceilfrac|t|erceil≥|t|. This explains why raw shell splitting cannot lower the intrinsic packet height.
Exact mathematical passage
\left\lceil\frac{|t|}{e}\right\rceil.
\mathrm{red}Use in source · 2 passages
Search aliases: red mathrm
Use in source · detailed_2.tex, line 433
The three-cusp section. View a primitive triple as the integral section σ_a,b,c: SpecZ⟶ P^1_Z, σ=[a:c]. The pullbacks of the boundary divisors 0, 1, and ∞ are div(a), div(b), div(c). For the output cusp, let C=div(c), C_red=div(R_c). Define the two-term contact complex J_c = [ O(-C)⟶ O(-C_red)]. On global sections, J_c = [cZ↪ R_cZ] ≃ [Zxrightarrowc/R_cZ].
Exact mathematical passage
C=\operatorname{div}(c), \qquad C_{\mathrm{red}}=\operatorname{div}(R_c).
\rfloorUse in source · 3 passages
Search aliases: rfloor
Use in source · appendices.tex, line 496
[leftmargin=1.5em] item Compute r_u, P_u, R, Q, and the exact stable height overline H. item Compute the primitive obstruction height H(1) by binary search on the three interval intersections. item Set K_max = lfloorfracH(1)-1overline Hrfloor. item For 2≤ k≤ K_max, skip a coset when koverline H already reaches the current best height. item Build centered CRT phase residues for the three sectors. item Compute the coarse affine line. item Binary-search the smallest H whose three admissible n-inter…
Exact mathematical passage
K_{\max} = \left\lfloor\frac{H(1)-1}{\overline H}\right\rfloor.
SUse in source · 34 passages
Search aliases: S
Use in source · appendices_final.tex, line 409
Arithmetic endpoint. The arithmetic endpoint is the coherence character χ_coh: Z^S_ab→Z/CZ whose row is built from simultaneous logarithms of the actual input primes and whose kernel contains the normalized exponent vector forced by a+b=c.
Exact mathematical passage
\chi_{\mathrm{coh}}: \Z^{S_{ab}}\to\Z/C\Z
S_aUse in source · 1 passage
Search aliases: S_a
Use in source · detailed_1.tex, line 11
The prime-coordinate module. Let a+b=c, gcd(a,b)=1. The three integers are pairwise coprime. Write S=S_asqcup S_bsqcup S_c ={p:pmid abc}. The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. For a positive integer n, define the universal multiplicative differential dn = n∑_pmid nfracv_p(n)p ξ_p. Every coefficient is an integer because pmid n. The map obeys the exact Leibniz law d(mn)=m dn+n dm. The verification is local. For each prime p, v_p(mn)=v_p(m)+v_p(n), so the…
Exact mathematical passage
S=S_a\sqcup S_b\sqcup S_c =\{p:p\mid abc\}.
S_{ab}Use in source · 10 passages
Search aliases: S_ab
Use in source · appendices_final.tex, line 409
Arithmetic endpoint. The arithmetic endpoint is the coherence character χ_coh: Z^S_ab→Z/CZ whose row is built from simultaneous logarithms of the actual input primes and whose kernel contains the normalized exponent vector forced by a+b=c.
Exact mathematical passage
\chi_{\mathrm{coh}}: \Z^{S_{ab}}\to\Z/C\Z
S_bUse in source · 1 passage
Search aliases: S_b
Use in source · detailed_1.tex, line 11
The prime-coordinate module. Let a+b=c, gcd(a,b)=1. The three integers are pairwise coprime. Write S=S_asqcup S_bsqcup S_c ={p:pmid abc}. The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. For a positive integer n, define the universal multiplicative differential dn = n∑_pmid nfracv_p(n)p ξ_p. Every coefficient is an integer because pmid n. The map obeys the exact Leibniz law d(mn)=m dn+n dm. The verification is local. For each prime p, v_p(mn)=v_p(m)+v_p(n), so the…
Exact mathematical passage
S=S_a\sqcup S_b\sqcup S_c =\{p:p\mid abc\}.
S_cUse in source · 1 passage
Search aliases: S_c
Use in source · detailed_1.tex, line 11
The prime-coordinate module. Let a+b=c, gcd(a,b)=1. The three integers are pairwise coprime. Write S=S_asqcup S_bsqcup S_c ={p:pmid abc}. The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. For a positive integer n, define the universal multiplicative differential dn = n∑_pmid nfracv_p(n)p ξ_p. Every coefficient is an integer because pmid n. The map obeys the exact Leibniz law d(mn)=m dn+n dm. The verification is local. For each prime p, v_p(mn)=v_p(m)+v_p(n), so the…
Exact mathematical passage
S=S_a\sqcup S_b\sqcup S_c =\{p:p\mid abc\}.
\mathrm{sh}Use in source · 1 passage
Search aliases: sh mathrm
Use in source · detailed_1.tex, line 577
The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a,
Exact mathematical passage
\rho_u^{\mathrm{sh}}(t) = \min\left\{ \norm{y}_\infty: \sum_{p\mid u} \frac{r_u}{p} \sum_r y_{p,r}=t \right\}.
\sharpUse in source · 1 passage
Search aliases: sharp
Use in source · overview.tex, line 759
The integral automorphic packet. The resulting integral automorphic object is Acal^sharp(E,ι) = ( f_E, Λ_E, ω_E, {q_p}_pmid N_E, ι, ϕ_E ). The components are: [leftmargin=1.5em] item the rational Hecke eigen-signal f_E item an integral Hecke, modular-symbol, or homology lattice Λ_E item the minimal N eron differential ω_E item local Tate-depth packets q_p item the marking ι of the three rational two-torsion differences item the modular synthesis map ϕ_E
Exact mathematical passage
\boxed{ \Acal^{\sharp}(E,\iota) = \left( f_E, \Lambda_E, \omega_E, \{q_p\}_{p\mid N_E}, \iota, \phi_E \right). }
\sinUse in source · 1 passage
Search aliases: sin
Use in source · overview.tex, line 671
Its depth Fourier transform is a Dirichlet packet: widehatΨ_n,p(θ) = sqrtlog p exp(-fraci(e_p-1)θ2) fracsin(e_pθ/2)sin(θ/2). The zero phase records the exponent. Nonzero phases record the endpoint and residue geometry of the valuation run.
Exact mathematical passage
\widehat\Psi_{n,p}(\theta) = \sqrt{\log p}\, \exp\left(-\frac{i(e_p-1)\theta}{2}\right) \frac{\sin(e_p\theta/2)}{\sin(\theta/2)}.
\SNFUse in source · 3 passages
Search aliases: SNF
Use in source · detailed_1.tex, line 371
[Polyphase Smith normal form] If gcd(e,N)=1, then SNF(C_e,N) = diag(1,…,1,e).
Exact mathematical passage
\boxed{ \SNF(C_{e,N}) = \operatorname{diag}(1,\ldots,1,e). }
\SpecUse in source · 1 passage
Search aliases: Spec
Use in source · detailed_2.tex, line 418
The three-cusp section. View a primitive triple as the integral section σ_a,b,c: SpecZ⟶ P^1_Z, σ=[a:c]. The pullbacks of the boundary divisors 0, 1, and ∞ are div(a), div(b), div(c). For the output cusp, let C=div(c), C_red=div(R_c). Define the two-term contact complex J_c = [ O(-C)⟶ O(-C_red)]. On global sections, J_c = [cZ↪ R_cZ] ≃ [Zxrightarrowc/R_cZ].
Exact mathematical passage
\sigma_{a,b,c}: \Spec\Z\longrightarrow\mathbb P^1_\Z, \qquad \sigma=[a:c].
\sqcupUse in source · 1 passage
Search aliases: sqcup
Use in source · detailed_1.tex, line 11
The prime-coordinate module. Let a+b=c, gcd(a,b)=1. The three integers are pairwise coprime. Write S=S_asqcup S_bsqcup S_c ={p:pmid abc}. The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. For a positive integer n, define the universal multiplicative differential dn = n∑_pmid nfracv_p(n)p ξ_p. Every coefficient is an integer because pmid n. The map obeys the exact Leibniz law d(mn)=m dn+n dm. The verification is local. For each prime p, v_p(mn)=v_p(m)+v_p(n), so the…
Exact mathematical passage
S=S_a\sqcup S_b\sqcup S_c =\{p:p\mid abc\}.
\mathrm{st}Use in source · 1 passage
Search aliases: st mathrm
Use in source · detailed_2.tex, line 561
The filtered local-global target. The intended transducer is a filtered chain J_c ⟶ ⊕_pmid c L_p^st ⟶ L_R^glob xrightarrowR= T L_ T_f ⟶ C_f. It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archimedean size.
Exact mathematical passage
\mathbf J_c \longrightarrow \bigoplus_{p\mid c}\mathbf L_p^{\mathrm{st}} \longrightarrow \mathbf L_{R^{\mathrm{glob}}} \xrightarrow{R=\mathbb T} \mathbf L_{\mathbb T_f} \longrightarrow \mathbf C_f.
TUse in source · 26 passages
Search aliases: T
Use in source · appendices_final.tex, line 252
Polyphase circulant determinant. For the valuation word V_e(T)=1+T+⋯+T^e-1 in A_N=Z[T]/(T^N-1), multiplication is circulant. Its determinant is ∏_ζ^N=1V_e(ζ). Let I_N be the augmentation ideal of A_N. When gcd(e,N)=1, the nontrivial roots are permuted by ζ↦ζ^e, so multiplication by V_e has determinant one on the integral lattice I_N and is therefore an automorphism of it. On A_N/I_N≅Z, the map is multiplication by e. The exact sequence for I_N⊂ A_N then gives cokernel Z/eZ and Smith form diag(1,…
Exact mathematical passage
V_e(T)=1+T+\cdots+T^{e-1}
\mathbb TUse in source · 4 passages
Search aliases: T mathbb
Use in source · detailed_2.tex, line 365
A toy Hecke order. Consider T( m) = {(x_0,x_1,…,x_r)∈Z^r+1:x_i≡ x_0pmodm_i}. For the target coordinate x_0, the cotangent module is p/ p^2 ≅ ⊕_i=1^rZ/m_iZ. The cyclic congruence quotient is Z/lcm(m_1,…,m_r)Z. The quotient of cotangent order by congruence order is frac∏_i m_ilcm(m_i). This is a toy Wiles-defect factor. It measures the multiplicity discarded by cyclic compression.
Exact mathematical passage
\mathbb T(\mathbf m) = \{(x_0,x_1,\ldots,x_r)\in\Z^{r+1}:x_i\equiv x_0\pmod{m_i}\}.
\mathbf TUse in source · 7 passages
Search aliases: T mathbf
Use in source · appendices_more.tex, line 139
Formal jet recursion. Let F( T)=0 be the shell equation and let T(t)=x^(1)t+x^(2)t^2+⋯. Suppose the gradient row Λ is primitive. Choose an integral vector v satisfying Λ· v=1. At order m, substitution gives Λ· x^(m) + G_m(x^(1),…,x^(m-1)) =0. The integral correction x^(m) = -G_m v solves the equation at that order. Induction produces an integral formal curve.
Exact mathematical passage
F(\mathbf T)=0
\mathcal TUse in source · 4 passages
Search aliases: T mathcal
Use in source · detailed_1.tex, line 75
[6pt] -dfracc v_p(c)p, · pmid c. Then λ· x=0. The additive derivative lattice is T_a,b,c = ker(λ)∩Z^S. Since the row is nonzero, its rank is |S|-1. The problem is not existence of additive derivatives. The problem is finding an additive derivative that remains transverse to the dependence direction introduced by a and b.
Exact mathematical passage
\boxed{ \mathcal T_{a,b,c} = \ker(\lambda)\cap\Z^S. }
\mathsf TUse in source · 2 passages
Search aliases: T mathsf
Use in source · detailed_1.tex, line 108
Over R, the additive projection is P_add = I- fracλλ^ Tnormλ_2^2. The obstruction direction inside the additive hyperplane is μ_⊥=P_addμ. When μ_⊥≠0, the rank-one obstruction projector is P_obs = fracμ_⊥μ_⊥^ T normμ_⊥_2^2. This is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation direction, one obstruction direction, and the remaining null packets.
Exact mathematical passage
P_{\mathrm{add}} = I- \frac{\lambda\lambda^{\mathsf T}}{\norm{\lambda}_2^2}.
tUse in source · 49 passages
Search aliases: t
Use in source · appendices_final.tex, line 214
Local CRT lift in closed form. Let r=∏_i=1^mp_i, w_i=r/p_i. The equation ∑_iw_ix_i=t has a solution because gcd(w_1,…,w_m)=1. Modulo p_i, x_i≡ t w_i^-1pmodp_i. Let ξ_i be the centered residue. Then ∑_iw_iξ_i-t is divisible by r. Set q=fract-∑_iw_iξ_ir. Writing x_i=ξ_i+p_i z_i gives ∑_i z_i=q.
Exact mathematical passage
\sum_iw_ix_i=t
T^2Use in source · 1 passage
Search aliases: T^2
Use in source · detailed_1.tex, line 494
First-order no-free-refinement principle. The shell-refined factor ∏_r=0^e-1 (1+fracT_rp) has first-order expansion 1+frac1p∑_rT_r+O(T^2). Only the shell sum appears. Every internal phase in the kernel of shell synthesis is invisible to both first-order additivity and the first-order Wronskian.
Exact mathematical passage
1+\frac1p\sum_rT_r+O(T^2).
t^2Use in source · 2 passages
Search aliases: t^2
Use in source · appendices_more.tex, line 143
Formal jet recursion. Let F( T)=0 be the shell equation and let T(t)=x^(1)t+x^(2)t^2+⋯. Suppose the gradient row Λ is primitive. Choose an integral vector v satisfying Λ· v=1. At order m, substitution gives Λ· x^(m) + G_m(x^(1),…,x^(m-1)) =0. The integral correction x^(m) = -G_m v solves the equation at that order. Induction produces an integral formal curve.
Exact mathematical passage
\mathbf T(t)=x^{(1)}t+x^{(2)}t^2+\cdots.
T^3Use in source · 1 passage
Search aliases: T^3
Use in source · detailed_1.tex, line 516
Second-order phase energy. At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^2 - frac1e∑_j≠0|widehat T_j|^2 ]. Thus nonzero polyphase energy first becomes visible at second order. Later formal-smoothness results show that this c…
Exact mathematical passage
\prod_{r=0}^{e-1} \left(1+\frac{T_r}{p}\right) = 1+ \frac1p\sum_rT_r+ \frac1{p^2}\sum_{r<s}T_rT_s+ O(T^3).
T^eUse in source · 1 passage
Search aliases: T^e
Use in source · detailed_1.tex, line 341
Replacing one exponent by a valuation word. The coefficient v_p(n) compresses a run of valuation shells into one scalar. For a prime power p^e, define the valuation word V_e(T)=1+T+⋯+T^e-1. It obeys the concatenation identity V_e+f(T)=V_e(T)+T^eV_f(T). The refined partial derivative ∂_p,T(n) = frac npV_v_p(n)(T) therefore satisfies the twisted product rule ∂_p,T(mn) = n∂_p,T(m) + mT^v_p(m)∂_p,T(n). At T=1, the ordinary arithmetic partial derivative is recovered.
Exact mathematical passage
V_{e+f}(T)=V_e(T)+T^eV_f(T).
T^{e-1}Use in source · 3 passages
Search aliases: T^e-1
Use in source · appendices_final.tex, line 252
Polyphase circulant determinant. For the valuation word V_e(T)=1+T+⋯+T^e-1 in A_N=Z[T]/(T^N-1), multiplication is circulant. Its determinant is ∏_ζ^N=1V_e(ζ). Let I_N be the augmentation ideal of A_N. When gcd(e,N)=1, the nontrivial roots are permuted by ζ↦ζ^e, so multiplication by V_e has determinant one on the integral lattice I_N and is therefore an automorphism of it. On A_N/I_N≅Z, the map is multiplication by e. The exact sequence for I_N⊂ A_N then gives cokernel Z/eZ and Smith form diag(1,…
Exact mathematical passage
V_e(T)=1+T+\cdots+T^{e-1}
t^mUse in source · 1 passage
Search aliases: t^m
Use in source · appendices_more.tex, line 48
Homogeneous higher obstructions. Let O be homogeneous of degree m: O(tx)=t^m O(x). Then fracnormx | O(x)|^1/m is scale invariant. Replacing one large coordinate by several coordinates of size roughly the mth root does not improve a bound whose obstruction estimate raises those coordinates back to degree m.
Exact mathematical passage
\mathcal O(tx)=t^m\mathcal O(x).
T^NUse in source · 3 passages
Search aliases: T^N
Use in source · appendices_final.tex, line 255
Polyphase circulant determinant. For the valuation word V_e(T)=1+T+⋯+T^e-1 in A_N=Z[T]/(T^N-1), multiplication is circulant. Its determinant is ∏_ζ^N=1V_e(ζ). Let I_N be the augmentation ideal of A_N. When gcd(e,N)=1, the nontrivial roots are permuted by ζ↦ζ^e, so multiplication by V_e has determinant one on the integral lattice I_N and is therefore an automorphism of it. On A_N/I_N≅Z, the map is multiplication by e. The exact sequence for I_N⊂ A_N then gives cokernel Z/eZ and Smith form diag(1,…
Exact mathematical passage
A_N=\Z[T]/(T^N-1)
T^{v_p(m)}Use in source · 1 passage
Search aliases: T^v_p(m)
Use in source · detailed_1.tex, line 351
t by a valuation word The coefficient v_p(n) compresses a run of valuation shells into one scalar. For a prime power p^e, define the valuation word V_e(T)=1+T+⋯+T^e-1. It obeys the concatenation identity V_e+f(T)=V_e(T)+T^eV_f(T). The refined partial derivative ∂_p,T(n) = frac npV_v_p(n)(T) therefore satisfies the twisted product rule ∂_p,T(mn) = n∂_p,T(m) + mT^v_p(m)∂_p,T(n). At T=1, the ordinary arithmetic partial derivative is recovered.
Exact mathematical passage
\boxed{ \partial_{p,T}(mn) = n\partial_{p,T}(m) + mT^{v_p(m)}\partial_{p,T}(n). }
T_0Use in source · 3 passages
Search aliases: T_0
Use in source · detailed_2.tex, line 115
Dyadic Landen descent. Let u>v>0 be coprime odd integers. Consider T_0=(v^2,u^2-v^2,u^2) and T_1= ( uv, (fracu-v2)^2, (fracu+v2)^2 ). Both are primitive triples. Their radicals are equal: rad(T_0)=rad(T_1)=rad(uv(u-v)(u+v)). The second output is strictly smaller: (fracu+v2)^2<u^2. Hence the second quality is smaller.
Exact mathematical passage
T_0=(v^2,u^2-v^2,u^2)
T_1Use in source · 3 passages
Search aliases: T_1
Use in source · detailed_2.tex, line 119
Dyadic Landen descent. Let u>v>0 be coprime odd integers. Consider T_0=(v^2,u^2-v^2,u^2) and T_1= ( uv, (fracu-v2)^2, (fracu+v2)^2 ). Both are primitive triples. Their radicals are equal: rad(T_0)=rad(T_1)=rad(uv(u-v)(u+v)). The second output is strictly smaller: (fracu+v2)^2<u^2. Hence the second quality is smaller.
Exact mathematical passage
T_1= \left( uv, \left(\frac{u-v}{2}\right)^2, \left(\frac{u+v}{2}\right)^2 \right).
t_aUse in source · 15 passages
Search aliases: t_a
Use in source · detailed_1.tex, line 263
ionThree-sector compression Within the a-sector, define the integer row λ^(a) = (fraca v_p(a)p)_pmid a and let g_a=gcd{fraca v_p(a)p:pmid a}. Define g_b and g_c similarly. A one-row Smith normal form in each sector produces a unimodular coordinate system in which only one aggregate channel survives. The additive equation becomes g_at_a+g_bt_b-g_ct_c=0. The Wronskian becomes a g_bt_b-b g_at_a≠0. All remaining sector coordinates are lift gauges. They matter only because they determine the cost of …
Exact mathematical passage
\boxed{ g_at_a+g_bt_b-g_ct_c=0. }
\mathcal T_{a,b,c}Use in source · 4 passages
Search aliases: T_a,b,c mathcal
Use in source · detailed_1.tex, line 75
[6pt] -dfracc v_p(c)p, · pmid c. Then λ· x=0. The additive derivative lattice is T_a,b,c = ker(λ)∩Z^S. Since the row is nonzero, its rank is |S|-1. The problem is not existence of additive derivatives. The problem is finding an additive derivative that remains transverse to the dependence direction introduced by a and b.
Exact mathematical passage
\boxed{ \mathcal T_{a,b,c} = \ker(\lambda)\cap\Z^S. }
t_a^{(0)}Use in source · 3 passages
Search aliases: t_a^(0)
Use in source · detailed_1.tex, line 602
The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a, t_b(n) · =t_b^(0)+nr_b, t_c(n) · =t_c^(0)+nr_c. The unit-sector cases have no moving parameter. If a=1, then (t_a,t_b,t_c)=(0,P_c,P_b), and if b=1, then (t_a,t_b,t_c)=(-P_c,0,…
Exact mathematical passage
\begin{aligned} t_a(n)&=t_a^{(0)}+nr_a,\\ t_b(n)&=t_b^{(0)}+nr_b,\\ t_c(n)&=t_c^{(0)}+nr_c. \end{aligned}
t_bUse in source · 14 passages
Search aliases: t_b
Use in source · detailed_1.tex, line 263
ionThree-sector compression Within the a-sector, define the integer row λ^(a) = (fraca v_p(a)p)_pmid a and let g_a=gcd{fraca v_p(a)p:pmid a}. Define g_b and g_c similarly. A one-row Smith normal form in each sector produces a unimodular coordinate system in which only one aggregate channel survives. The additive equation becomes g_at_a+g_bt_b-g_ct_c=0. The Wronskian becomes a g_bt_b-b g_at_a≠0. All remaining sector coordinates are lift gauges. They matter only because they determine the cost of …
Exact mathematical passage
\boxed{ g_at_a+g_bt_b-g_ct_c=0. }
t_b^{(0)}Use in source · 3 passages
Search aliases: t_b^(0)
Use in source · detailed_1.tex, line 602
The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a, t_b(n) · =t_b^(0)+nr_b, t_c(n) · =t_c^(0)+nr_c. The unit-sector cases have no moving parameter. If a=1, then (t_a,t_b,t_c)=(0,P_c,P_b), and if b=1, then (t_a,t_b,t_c)=(-P_c,0,…
Exact mathematical passage
\begin{aligned} t_a(n)&=t_a^{(0)}+nr_a,\\ t_b(n)&=t_b^{(0)}+nr_b,\\ t_c(n)&=t_c^{(0)}+nr_c. \end{aligned}
t_cUse in source · 10 passages
Search aliases: t_c
Use in source · detailed_1.tex, line 263
ionThree-sector compression Within the a-sector, define the integer row λ^(a) = (fraca v_p(a)p)_pmid a and let g_a=gcd{fraca v_p(a)p:pmid a}. Define g_b and g_c similarly. A one-row Smith normal form in each sector produces a unimodular coordinate system in which only one aggregate channel survives. The additive equation becomes g_at_a+g_bt_b-g_ct_c=0. The Wronskian becomes a g_bt_b-b g_at_a≠0. All remaining sector coordinates are lift gauges. They matter only because they determine the cost of …
Exact mathematical passage
\boxed{ g_at_a+g_bt_b-g_ct_c=0. }
t_c^{(0)}Use in source · 2 passages
Search aliases: t_c^(0)
Use in source · detailed_1.tex, line 602
The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a, t_b(n) · =t_b^(0)+nr_b, t_c(n) · =t_c^(0)+nr_c. The unit-sector cases have no moving parameter. If a=1, then (t_a,t_b,t_c)=(0,P_c,P_b), and if b=1, then (t_a,t_b,t_c)=(-P_c,0,…
Exact mathematical passage
\begin{aligned} t_a(n)&=t_a^{(0)}+nr_a,\\ t_b(n)&=t_b^{(0)}+nr_b,\\ t_c(n)&=t_c^{(0)}+nr_c. \end{aligned}
\mathbb T_fUse in source · 1 passage
Search aliases: T_f mathbb
Use in source · detailed_2.tex, line 561
The filtered local-global target. The intended transducer is a filtered chain J_c ⟶ ⊕_pmid c L_p^st ⟶ L_R^glob xrightarrowR= T L_ T_f ⟶ C_f. It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archimedean size.
Exact mathematical passage
\mathbf J_c \longrightarrow \bigoplus_{p\mid c}\mathbf L_p^{\mathrm{st}} \longrightarrow \mathbf L_{R^{\mathrm{glob}}} \xrightarrow{R=\mathbb T} \mathbf L_{\mathbb T_f} \longrightarrow \mathbf C_f.
t_pUse in source · 1 passage
Search aliases: t_p
Use in source · detailed_1.tex, line 441
The coarse equation recovers the original exponent-weighted row ∑_pmid afracav_p(a)p t_p + ∑_pmid bfracbv_p(b)p t_p - ∑_pmid cfraccv_p(c)p t_p =0. The phase equation is exponent free: ∑_pmid afrac ap y_p + ∑_pmid bfrac bp y_p - ∑_pmid cfrac cp y_p =0. Define the phase derivative D_y(n)=n∑_pmid nfracy_pp. The same radical divisibility argument gives fracabcRmid aD_y(b)-bD_y(a).
Exact mathematical passage
\sum_{p\mid a}\frac{av_p(a)}p t_p + \sum_{p\mid b}\frac{bv_p(b)}p t_p - \sum_{p\mid c}\frac{cv_p(c)}p t_p =0.
T_{p,r}Use in source · 3 passages
Search aliases: T_p,r
Use in source · detailed_1.tex, line 501
[First-order no-free-polyphase principle] Let S:(T_p,r)_p,r⟼ (∑_rT_p,r)_p be shell synthesis. The first-order additive map and first-order Wronskian factor through S. Hence Ker S is contained in the kernels of both first-order maps.
Exact mathematical passage
S:(T_{p,r})_{p,r}\longmapsto \left(\sum_rT_{p,r}\right)_p
T_rUse in source · 5 passages
Search aliases: T_r
Use in source · detailed_1.tex, line 489
First-order no-free-refinement principle. The shell-refined factor ∏_r=0^e-1 (1+fracT_rp) has first-order expansion 1+frac1p∑_rT_r+O(T^2). Only the shell sum appears. Every internal phase in the kernel of shell synthesis is invisible to both first-order additivity and the first-order Wronskian.
Exact mathematical passage
\prod_{r=0}^{e-1} \left(1+\frac{T_r}{p}\right)
T_sUse in source · 3 passages
Search aliases: T_s
Use in source · detailed_1.tex, line 516
Second-order phase energy. At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^2 - frac1e∑_j≠0|widehat T_j|^2 ]. Thus nonzero polyphase energy first becomes visible at second order. Later formal-smoothness results show that this c…
Exact mathematical passage
\prod_{r=0}^{e-1} \left(1+\frac{T_r}{p}\right) = 1+ \frac1p\sum_rT_r+ \frac1{p^2}\sum_{r<s}T_rT_s+ O(T^3).
t_uUse in source · 5 passages
Search aliases: t_u
Use in source · detailed_1.tex, line 556
Exact local lift gauges. Let r_u=rad(u), P_u=fracur_u. A shell packet y=(y_p,r) produces D_y(u) = u∑_pmid ufrac1p ∑_r=0^v_p(u)-1y_p,r =P_ut_u, where t_u= ∑_pmid u fracr_up ∑_r y_p,r. Because gcd_pmid ufracr_up=1, every integer t_u is attainable when u>1.
Exact mathematical passage
D_y(u) = u\sum_{p\mid u}\frac1p \sum_{r=0}^{v_p(u)-1}y_{p,r} =P_ut_u,
\mathrm{tors}Use in source · 1 passage
Search aliases: tors mathrm
Use in source · detailed_1.tex, line 663
[Smith obstruction theorem] Let a+b=c be primitive, with at least two of a,b,c greater than one. Then SNF( M_a,b,c) = diag(1,Q). Consequently, coker( M_a,b,c)_tors ≅ Z/QZ. The exceptional triple (1,1,2) has only one nonempty prime sector, so the matrix has rank one and is excluded from the two-row statement.
Exact mathematical passage
\boxed{ \operatorname{coker}(\mathsf M_{a,b,c})_{\mathrm{tors}} \cong \Z/Q\Z. }
UUse in source · 19 passages
Search aliases: U
Use in source · appendices_final.tex, line 187
One-row Smith normal form. For an integer row a=(a_1,…,a_s), let d=gcd(a_1,…,a_s). There exists a unimodular matrix U∈GL_s(Z) such that aU=(d,0,…,0). Therefore the image of the row is dZ, the kernel has rank s-1, and the quotient by the kernel is infinite cyclic scaled by d.
Exact mathematical passage
U\in\mathrm{GL}_s(\Z)
u^2Use in source · 4 passages
Search aliases: u^2
Use in source · detailed_2.tex, line 115
Dyadic Landen descent. Let u>v>0 be coprime odd integers. Consider T_0=(v^2,u^2-v^2,u^2) and T_1= ( uv, (fracu-v2)^2, (fracu+v2)^2 ). Both are primitive triples. Their radicals are equal: rad(T_0)=rad(T_1)=rad(uv(u-v)(u+v)). The second output is strictly smaller: (fracu+v2)^2<u^2. Hence the second quality is smaller.
Exact mathematical passage
T_0=(v^2,u^2-v^2,u^2)
u^MUse in source · 7 passages
Search aliases: u^M
Use in source · detailed_3.tex, line 109
The Kummer--Fermat lift. Choose algebraic numbers u and v satisfying u^M=frac ac, v^M=frac bc. Then u^M+v^M=1. Thus P_M=[u:v:1] lies on the Fermat curve F_M:X^M+Y^M=Z^M. The construction is canonical up to the action of μ_M^2.
Exact mathematical passage
u^M=\frac ac, \qquad v^M=\frac bc.
U_2Use in source · 1 passage
Search aliases: U_2
Use in source · detailed_1.tex, line 817
[Formal affine-space theorem] Let d be the number of shell variables. After a unimodular integral change of shell coordinates, the ( T)-adic completion of Z[ T]/(F_a,b,c) at the zero section is isomorphic to a formal power-series ring Z[[U_2,…,U_d]].
Exact mathematical passage
\Z[[U_2,\ldots,U_d]]
U_{ab}Use in source · 2 passages
Search aliases: U_ab
Use in source · detailed_3.tex, line 603
Adelic coherence regulator object. Package the surviving data as R_a,b,c = ( U_ab, β^1/K, {χ_p}_pmid c, χ_coh, Λ_coh, norm·_∞ ). Here U_ab is the input S-unit group, the local characters are the normalized logarithmic rows, and the archimedean norm records input height.
Exact mathematical passage
\boxed{ \mathfrak R_{a,b,c} = \left( U_{ab}, \beta^{1/K}, \{\chi_p\}_{p\mid c}, \chi_{\mathrm{coh}}, \Lambda_{\mathrm{coh}}, \norm{\cdot}_\infty \right). }
U_dUse in source · 1 passage
Search aliases: U_d
Use in source · detailed_1.tex, line 817
[Formal affine-space theorem] Let d be the number of shell variables. After a unimodular integral change of shell coordinates, the ( T)-adic completion of Z[ T]/(F_a,b,c) at the zero section is isomorphic to a formal power-series ring Z[[U_2,…,U_d]].
Exact mathematical passage
\Z[[U_2,\ldots,U_d]]
u_iUse in source · 2 passages
Search aliases: u_i
Use in source · detailed_1.tex, line 939
p_ix_i=t. An empty unit sector has no prime coordinates and is feasible exactly when t=0. Modulo p_i, the coordinate x_i is forced: x_i ≡ t(frac rp_i)^-1 pmodp_i. Choose the centered residue ξ_i and write x_i=ξ_i+p_i z_i. Substitution gives one quotient equation ∑_i z_i=q(t). For a box radius H, each z_i lies in the interval ℓ_i(H) = lceilfrac-H-ξ_ip_irceil ≤ z_i≤ lfloorfracH-ξ_ip_irfloor =u_i(H). The local lift exists exactly when ∑_iℓ_i(H) ≤ q(t)≤ ∑_i u_i(H). This is an exact interval criterio…
Exact mathematical passage
\ell_i(H) = \left\lceil\frac{-H-\xi_i}{p_i}\right\rceil \le z_i\le \left\lfloor\frac{H-\xi_i}{p_i}\right\rfloor =u_i(H).
VUse in source · 18 passages
Search aliases: V
Use in source · appendices_final.tex, line 252
Polyphase circulant determinant. For the valuation word V_e(T)=1+T+⋯+T^e-1 in A_N=Z[T]/(T^N-1), multiplication is circulant. Its determinant is ∏_ζ^N=1V_e(ζ). Let I_N be the augmentation ideal of A_N. When gcd(e,N)=1, the nontrivial roots are permuted by ζ↦ζ^e, so multiplication by V_e has determinant one on the integral lattice I_N and is therefore an automorphism of it. On A_N/I_N≅Z, the map is multiplication by e. The exact sequence for I_N⊂ A_N then gives cokernel Z/eZ and Smith form diag(1,…
Exact mathematical passage
V_e(T)=1+T+\cdots+T^{e-1}
v^2Use in source · 2 passages
Search aliases: v^2
Use in source · detailed_2.tex, line 115
Dyadic Landen descent. Let u>v>0 be coprime odd integers. Consider T_0=(v^2,u^2-v^2,u^2) and T_1= ( uv, (fracu-v2)^2, (fracu+v2)^2 ). Both are primitive triples. Their radicals are equal: rad(T_0)=rad(T_1)=rad(uv(u-v)(u+v)). The second output is strictly smaller: (fracu+v2)^2<u^2. Hence the second quality is smaller.
Exact mathematical passage
T_0=(v^2,u^2-v^2,u^2)
v^MUse in source · 4 passages
Search aliases: v^M
Use in source · detailed_3.tex, line 109
The Kummer--Fermat lift. Choose algebraic numbers u and v satisfying u^M=frac ac, v^M=frac bc. Then u^M+v^M=1. Thus P_M=[u:v:1] lies on the Fermat curve F_M:X^M+Y^M=Z^M. The construction is canonical up to the action of μ_M^2.
Exact mathematical passage
u^M=\frac ac, \qquad v^M=\frac bc.
v_2Use in source · 6 passages
Search aliases: v_2
Use in source · appendices.tex, line 207
Polylogarithmic path. Set z=frac3128, 1-z=frac125128, β=1-frac1z=-frac1253. The depth-three functional relation closes on these three arguments. At the output prime two, v_2(β-1) = v_2(-frac1283) =7. The unscaled principal logarithm has 2-adic valuation seven. After division by four, the weighted principal coordinate occupies five filtration levels, while the separate sign packet supplies the first dyadic layer.
Exact mathematical passage
v_2(\beta-1) = v_2\left(-\frac{128}{3}\right) =7.
V_eUse in source · 15 passages
Search aliases: V_e
Use in source · appendices_final.tex, line 252
Polyphase circulant determinant. For the valuation word V_e(T)=1+T+⋯+T^e-1 in A_N=Z[T]/(T^N-1), multiplication is circulant. Its determinant is ∏_ζ^N=1V_e(ζ). Let I_N be the augmentation ideal of A_N. When gcd(e,N)=1, the nontrivial roots are permuted by ζ↦ζ^e, so multiplication by V_e has determinant one on the integral lattice I_N and is therefore an automorphism of it. On A_N/I_N≅Z, the map is multiplication by e. The exact sequence for I_N⊂ A_N then gives cokernel Z/eZ and Smith form diag(1,…
Exact mathematical passage
V_e(T)=1+T+\cdots+T^{e-1}
V_{e+f}Use in source · 1 passage
Search aliases: V_e+f
Use in source · detailed_1.tex, line 341
Replacing one exponent by a valuation word. The coefficient v_p(n) compresses a run of valuation shells into one scalar. For a prime power p^e, define the valuation word V_e(T)=1+T+⋯+T^e-1. It obeys the concatenation identity V_e+f(T)=V_e(T)+T^eV_f(T). The refined partial derivative ∂_p,T(n) = frac npV_v_p(n)(T) therefore satisfies the twisted product rule ∂_p,T(mn) = n∂_p,T(m) + mT^v_p(m)∂_p,T(n). At T=1, the ordinary arithmetic partial derivative is recovered.
Exact mathematical passage
V_{e+f}(T)=V_e(T)+T^eV_f(T).
V_fUse in source · 1 passage
Search aliases: V_f
Use in source · detailed_1.tex, line 341
Replacing one exponent by a valuation word. The coefficient v_p(n) compresses a run of valuation shells into one scalar. For a prime power p^e, define the valuation word V_e(T)=1+T+⋯+T^e-1. It obeys the concatenation identity V_e+f(T)=V_e(T)+T^eV_f(T). The refined partial derivative ∂_p,T(n) = frac npV_v_p(n)(T) therefore satisfies the twisted product rule ∂_p,T(mn) = n∂_p,T(m) + mT^v_p(m)∂_p,T(n). At T=1, the ordinary arithmetic partial derivative is recovered.
Exact mathematical passage
V_{e+f}(T)=V_e(T)+T^eV_f(T).
v_qUse in source · 4 passages
Search aliases: v_q
Use in source · appendices_final.tex, line 15
Start with the signed input exponent vector ν=(ν_q)_qmid ab, ν_q=v_q(b)-v_q(a). Let K_p be the p-primary part of the common Kummer factor. Divide the exponent vector and the local modulus by this common power. The normalized vector is primitive relative to the remaining modulus.
Exact mathematical passage
\nu=(\nu_q)_{q\mid ab}, \qquad \nu_q=v_q(b)-v_q(a).
v_sUse in source · 1 passage
Search aliases: v_s
Use in source · appendices_more.tex, line 210
Arbitrary-index congruence lattices. Let v∈Z^s be primitive. Complete v to a unimodular basis v,v_2,…,v_s. In the dual basis, choose the row that vanishes on v and takes value one on v_2. Reduce it modulo an arbitrary integer C. The resulting row is primitive modulo C and annihilates v.
Exact mathematical passage
v,v_2,\ldots,v_s.
V_{v_p(n)}Use in source · 1 passage
Search aliases: V_v_p(n)
Use in source · detailed_1.tex, line 345
Replacing one exponent by a valuation word. The coefficient v_p(n) compresses a run of valuation shells into one scalar. For a prime power p^e, define the valuation word V_e(T)=1+T+⋯+T^e-1. It obeys the concatenation identity V_e+f(T)=V_e(T)+T^eV_f(T). The refined partial derivative ∂_p,T(n) = frac npV_v_p(n)(T) therefore satisfies the twisted product rule ∂_p,T(mn) = n∂_p,T(m) + mT^v_p(m)∂_p,T(n). At T=1, the ordinary arithmetic partial derivative is recovered.
Exact mathematical passage
\partial_{p,T}(n) = \frac npV_{v_p(n)}(T)
v_\ellUse in source · 1 passage
Search aliases: v_ℓ ell
Use in source · detailed_2.tex, line 409
Two prime indices. The persistence is indexed by conductor primes qmid abc. Congruence depth is indexed by coefficient primes ℓ. The natural local signal is a matrix D_f(ℓ,q). At a split Tate prime q≠ℓ, scalarity modulo ℓ^n forces n≤min{v_ℓ(q-1),v_ℓ(v_q(q_E))}. These off-diagonal channels carry tame congruence information. They do not naturally reproduce the full base-q factor q^e-1. This mismatch motivates the later filtered contact correction and the diagonal coefficient-prime channel.
Exact mathematical passage
n\le\min\{v_\ell(q-1),v_\ell(v_q(q_E))\}.
\mathrm{val}Use in source · 1 passage
Search aliases: val mathrm
Use in source · detailed_3.tex, line 96
Balanced Kummer class. The class of c does not remember the additive relation. A better coordinate is β=-frac ba=1-frac ca. Let ν_q=v_q(b)-v_q(a) for input primes qmid ab. The order of [β] in Q^×/(Q^×)^M is determined by the signed valuation vector, with one possible dyadic sign correction when M is even. If g_val = gcd(M,{ν_q}), m_0=M/g_val, then the order is m_0 unless M is even and m_0 is odd, in which case it is 2m_0.
Exact mathematical passage
g_{\mathrm{val}} = \gcd(M,\{\nu_q\}), \qquad m_0=M/g_{\mathrm{val}},
\mathbb WUse in source · 1 passage
Search aliases: W mathbb
Use in source · detailed_3.tex, line 673
Prime-contact data structure. The reusable object attached to a sparse relation is A( n,ϵ) = ( W_ n, κ_ n, L_ n, Λ_coh, R_∞, M_glob ). The first four components are finite and local-global arithmetic data. The last two are optional archimedean and global-motivic realizations. Keeping them separate prevents an unproved global comparison from being built into the definition.
Exact mathematical passage
\boxed{ \mathfrak A(\mathbf n,\boldsymbol\epsilon) = \left( \mathbb W_{\mathbf n}, \kappa_{\mathbf n}, \mathcal L_{\mathbf n}, \Lambda_{\mathrm{coh}}, \mathcal R_\infty, \mathcal M_{\mathrm{glob}} \right). }
\mathsf wUse in source · 8 passages
Search aliases: w mathsf
Use in source · detailed_1.tex, line 623
A joint exponent-free two-row complex. The Smith obstruction is attached to the exponent-free phase, or shell-synthesis, rows rather than to the exponent-weighted derivative row. For every prime pmid abc, define Λ_p= a/p, · pmid a, b/p, · pmid b, -c/p, · pmid c, w_p= -ab/p, · pmid a, ab/p, · pmid b, 0, · pmid c. Consider M_a,b,c = Λ
Exact mathematical passage
\Lambda_p= \begin{cases} a/p,&p\mid a,\\ b/p,&p\mid b,\\ -c/p,&p\mid c, \end{cases} \qquad \mathsf w_p= \begin{cases} -ab/p,&p\mid a,\\ ab/p,&p\mid b,\\ 0,&p\mid c. \end{cases}
\mathbb W_{\mathbf n}Use in source · 1 passage
Search aliases: W_ n mathbb mathbf
Use in source · detailed_3.tex, line 673
Prime-contact data structure. The reusable object attached to a sparse relation is A( n,ϵ) = ( W_ n, κ_ n, L_ n, Λ_coh, R_∞, M_glob ). The first four components are finite and local-global arithmetic data. The last two are optional archimedean and global-motivic realizations. Keeping them separate prevents an unproved global comparison from being built into the definition.
Exact mathematical passage
\boxed{ \mathfrak A(\mathbf n,\boldsymbol\epsilon) = \left( \mathbb W_{\mathbf n}, \kappa_{\mathbf n}, \mathcal L_{\mathbf n}, \Lambda_{\mathrm{coh}}, \mathcal R_\infty, \mathcal M_{\mathrm{glob}} \right). }
W_1Use in source · 1 passage
Search aliases: W_1
Use in source · detailed_2.tex, line 267
Atkin--Lehner Walsh packets. Let W_1,…,W_r be commuting Atkin--Lehner involutions. For a sign vector ε∈{±1}^r, define A_ε = ∏_j=1^r(I+ε_jW_j). Then A_ε^2=2^rA_ε, A_ε A_ε'=0 (ε≠ε'), and ∑_ε A_ε=2^rI. The unnormalized analysis packets are integral. The common synthesis denominator is 2^r, which is subpolynomial in a squarefree conductor. Local sign separation is therefore not the large persistence carrier.
Exact mathematical passage
W_1,\ldots,W_r
w_1Use in source · 1 passage
Search aliases: w_1
Use in source · appendices_final.tex, line 218
Local CRT lift in closed form. Let r=∏_i=1^mp_i, w_i=r/p_i. The equation ∑_iw_ix_i=t has a solution because gcd(w_1,…,w_m)=1. Modulo p_i, x_i≡ t w_i^-1pmodp_i. Let ξ_i be the centered residue. Then ∑_iw_iξ_i-t is divisible by r. Set q=fract-∑_iw_iξ_ir. Writing x_i=ξ_i+p_i z_i gives ∑_i z_i=q.
Exact mathematical passage
\gcd(w_1,\ldots,w_m)=1.
W_6Use in source · 1 passage
Search aliases: W_6
Use in source · appendices.tex, line 153
The local Witt coordinate is W_6(F_2) ≅ Z/64Z. The weighted multiplicative dyadic coordinate is (1+4Z_2)/(1+2^8Z_2), which has the same order.
Exact mathematical passage
W_6(\F_2) \cong \Z/64\Z.
W_{e-1}Use in source · 4 passages
Search aliases: W_e-1
Use in source · detailed_2.tex, line 542
Witt realization. The local packet has a natural additive identification Jcal_p,e ≅ W_e-1(F_p). The Verschiebung filtration matches multiplication by successive powers of p. This is the typed local object underlying the carry chain.
Exact mathematical passage
\boxed{ \Jcal_{p,e} \cong W_{e-1}(\F_p). }
w_iUse in source · 4 passages
Search aliases: w_i
Use in source · appendices_final.tex, line 208
Local CRT lift in closed form. Let r=∏_i=1^mp_i, w_i=r/p_i. The equation ∑_iw_ix_i=t has a solution because gcd(w_1,…,w_m)=1. Modulo p_i, x_i≡ t w_i^-1pmodp_i. Let ξ_i be the centered residue. Then ∑_iw_iξ_i-t is divisible by r. Set q=fract-∑_iw_iξ_ir. Writing x_i=ξ_i+p_i z_i gives ∑_i z_i=q.
Exact mathematical passage
r=\prod_{i=1}^{m}p_i, \qquad w_i=r/p_i.
w_i^{-1}Use in source · 1 passage
Search aliases: w_i^-1
Use in source · appendices_final.tex, line 222
Local CRT lift in closed form. Let r=∏_i=1^mp_i, w_i=r/p_i. The equation ∑_iw_ix_i=t has a solution because gcd(w_1,…,w_m)=1. Modulo p_i, x_i≡ t w_i^-1pmodp_i. Let ξ_i be the centered residue. Then ∑_iw_iξ_i-t is divisible by r. Set q=fract-∑_iw_iξ_ir. Writing x_i=ξ_i+p_i z_i gives ∑_i z_i=q.
Exact mathematical passage
x_i\equiv t w_i^{-1}\pmod{p_i}.
W_jUse in source · 1 passage
Search aliases: W_j
Use in source · detailed_2.tex, line 275
Atkin--Lehner Walsh packets. Let W_1,…,W_r be commuting Atkin--Lehner involutions. For a sign vector ε∈{±1}^r, define A_ε = ∏_j=1^r(I+ε_jW_j). Then A_ε^2=2^rA_ε, A_ε A_ε'=0 (ε≠ε'), and ∑_ε A_ε=2^rI. The unnormalized analysis packets are integral. The common synthesis denominator is 2^r, which is subpolynomial in a squarefree conductor. Local sign separation is therefore not the large persistence carrier.
Exact mathematical passage
A_\varepsilon = \prod_{j=1}^{r}(I+\varepsilon_jW_j).
W_j^2Use in source · 1 passage
Search aliases: W_j^2
Use in source · appendices.tex, line 406
The Atkin--Lehner Walsh bank is a finite product of commuting involutions. Its identities are polynomial consequences of W_j^2=I.
Exact mathematical passage
W_j^2=I
w_mUse in source · 1 passage
Search aliases: w_m
Use in source · appendices_final.tex, line 218
Local CRT lift in closed form. Let r=∏_i=1^mp_i, w_i=r/p_i. The equation ∑_iw_ix_i=t has a solution because gcd(w_1,…,w_m)=1. Modulo p_i, x_i≡ t w_i^-1pmodp_i. Let ξ_i be the centered residue. Then ∑_iw_iξ_i-t is divisible by r. Set q=fract-∑_iw_iξ_ir. Writing x_i=ξ_i+p_i z_i gives ∑_i z_i=q.
Exact mathematical passage
\gcd(w_1,\ldots,w_m)=1.
W_pUse in source · 2 passages
Search aliases: W_p
Use in source · appendices.tex, line 518
[leftmargin=1.5em] item Form the signed input exponent vector ν. item Remove the common Kummer preload K. item For every output prime, construct the finite logarithmic row. item Divide each row by its common coefficient depth to obtain the local defect W_p. item Remove the dyadic sign layer separately. item Combine primitive residual rows by CRT. item Verify that the normalized exponent vector lies in the kernel. item Compute the exact quotient order C and the factorization M=ϵ_2KWC.
Exact mathematical passage
W_p
\mathsf w_pUse in source · 1 passage
Search aliases: w_p mathsf
Use in source · detailed_1.tex, line 623
A joint exponent-free two-row complex. The Smith obstruction is attached to the exponent-free phase, or shell-synthesis, rows rather than to the exponent-weighted derivative row. For every prime pmid abc, define Λ_p= a/p, · pmid a, b/p, · pmid b, -c/p, · pmid c, w_p= -ab/p, · pmid a, ab/p, · pmid b, 0, · pmid c. Consider M_a,b,c = Λ
Exact mathematical passage
\Lambda_p= \begin{cases} a/p,&p\mid a,\\ b/p,&p\mid b,\\ -c/p,&p\mid c, \end{cases} \qquad \mathsf w_p= \begin{cases} -ab/p,&p\mid a,\\ ab/p,&p\mid b,\\ 0,&p\mid c. \end{cases}
W_rUse in source · 1 passage
Search aliases: W_r
Use in source · detailed_2.tex, line 267
Atkin--Lehner Walsh packets. Let W_1,…,W_r be commuting Atkin--Lehner involutions. For a sign vector ε∈{±1}^r, define A_ε = ∏_j=1^r(I+ε_jW_j). Then A_ε^2=2^rA_ε, A_ε A_ε'=0 (ε≠ε'), and ∑_ε A_ε=2^rI. The unnormalized analysis packets are integral. The common synthesis denominator is 2^r, which is subpolynomial in a squarefree conductor. Local sign separation is therefore not the large persistence carrier.
Exact mathematical passage
W_1,\ldots,W_r
W_xUse in source · 16 passages
Search aliases: W_x
Use in source · appendices.tex, line 45
Prime derivative. The additive row is λ=(1,75,-448). The phase row is λ^ph=(1,25,-64). The coarse shortest transverse packet x=(-2,6,1) satisfies -2+450=448. The Wronskian is W_x=1600=fracabcR. The coarse height is six.
Exact mathematical passage
W_x=1600=\frac{abc}{R}.
W_yUse in source · 1 passage
Search aliases: W_y
Use in source · detailed_1.tex, line 592
The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a,
Exact mathematical passage
W_y(a,b) = P_aP_b(r_at_b-r_bt_a).
\wedgeUse in source · 2 passages
Search aliases: wedge
Use in source · detailed_2.tex, line 165
Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be an optimal modular parametrization, and let ϕ_E^*ω_E = 2π i c_E f_E(z) dz. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_E^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area factor proportional to 1/c. Thus the height c can be written as a shape-corrected synthesis gain involving modular degree, Manin normalization, and Pete…
Exact mathematical passage
A(E) = \frac{i}{2} \int_{E(\C)} \omega_E\wedge\overline{\omega_E}.
\widehat TUse in source · 3 passages
Search aliases: widehat T widehat
Use in source · detailed_1.tex, line 525
Second-order phase energy. At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^2 - frac1e∑_j≠0|widehat T_j|^2 ]. Thus nonzero polyphase energy first becomes visible at second order. Later formal-smoothness results show that this c…
Exact mathematical passage
\widehat T_j
\widehat T_{-j}Use in source · 1 passage
Search aliases: widehat T_-j widehat
Use in source · detailed_1.tex, line 526
Second-order phase energy. At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^2 - frac1e∑_j≠0|widehat T_j|^2 ]. Thus nonzero polyphase energy first becomes visible at second order. Later formal-smoothness results show that this c…
Exact mathematical passage
\sum_{r<s}T_rT_s = \frac12\left[ \widehat T_0^2 - \frac1e\sum_j \widehat T_j\widehat T_{-j} \right].
\widehat T_0Use in source · 1 passage
Search aliases: widehat T_0 widehat
Use in source · detailed_1.tex, line 537
onSecond-order phase energy At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^2 - frac1e∑_j≠0|widehat T_j|^2 ]. Thus nonzero polyphase energy first becomes visible at second order. Later formal-smoothness results show that this …
Exact mathematical passage
\sum_{r<s}T_rT_s = \frac12\left[ \left(1-\frac1e\right)|\widehat T_0|^2 - \frac1e\sum_{j\neq0}|\widehat T_j|^2 \right].
\widehat T_0^2Use in source · 1 passage
Search aliases: widehat T_0^2 widehat
Use in source · detailed_1.tex, line 526
Second-order phase energy. At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^2 - frac1e∑_j≠0|widehat T_j|^2 ]. Thus nonzero polyphase energy first becomes visible at second order. Later formal-smoothness results show that this c…
Exact mathematical passage
\sum_{r<s}T_rT_s = \frac12\left[ \widehat T_0^2 - \frac1e\sum_j \widehat T_j\widehat T_{-j} \right].
\widehat T_jUse in source · 3 passages
Search aliases: widehat T_j widehat
Use in source · detailed_1.tex, line 525
Second-order phase energy. At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^2 - frac1e∑_j≠0|widehat T_j|^2 ]. Thus nonzero polyphase energy first becomes visible at second order. Later formal-smoothness results show that this c…
Exact mathematical passage
\widehat T_j
\widehat\PsiUse in source · 1 passage
Search aliases: widehatΨ widehat Psi
Use in source · overview.tex, line 671
Its depth Fourier transform is a Dirichlet packet: widehatΨ_n,p(θ) = sqrtlog p exp(-fraci(e_p-1)θ2) fracsin(e_pθ/2)sin(θ/2). The zero phase records the exponent. Nonzero phases record the endpoint and residue geometry of the valuation run.
Exact mathematical passage
\widehat\Psi_{n,p}(\theta) = \sqrt{\log p}\, \exp\left(-\frac{i(e_p-1)\theta}{2}\right) \frac{\sin(e_p\theta/2)}{\sin(\theta/2)}.
\widehat\Psi_{n,p}Use in source · 1 passage
Search aliases: widehatΨ_n,p widehat Psi
Use in source · overview.tex, line 671
Its depth Fourier transform is a Dirichlet packet: widehatΨ_n,p(θ) = sqrtlog p exp(-fraci(e_p-1)θ2) fracsin(e_pθ/2)sin(θ/2). The zero phase records the exponent. Nonzero phases record the endpoint and residue geometry of the valuation run.
Exact mathematical passage
\widehat\Psi_{n,p}(\theta) = \sqrt{\log p}\, \exp\left(-\frac{i(e_p-1)\theta}{2}\right) \frac{\sin(e_p\theta/2)}{\sin(\theta/2)}.
\widetilde xUse in source · 2 passages
Search aliases: widetilde x widetilde
Use in source · detailed_3.tex, line 394
Odd logarithmic rows. For odd p^eparallel c, define the finite-depth row coefficient ℓ_p,e(q) = frac1plog_p⟨ q⟩_p pmodp^e-1. The balanced relation gives ∑_qmid ab ν_qℓ_p,e(q) ≡0 pmodp^e-1. Define the truncated valuation on Z/p^e-1Z by bar v_p(0)=e-1, bar v_p(x)=min{e-1,v_p(widetilde x)} for any integer lift widetilde x of a nonzero residue. Then bar v_p(ℓ_p,e(q)) = min{e-1,v_p(q^p-1-1)-1}. Exceptional entrywise depth is a generalized Wieferich event.
Exact mathematical passage
\bar v_p(0)=e-1, \qquad \bar v_p(x)=\min\{e-1,v_p(\widetilde x)\}
\Witt_{\mathbf n}Use in source · 3 passages
Search aliases: Witt_ n Witt mathbf
Use in source · overview.tex, line 334
The prime-contact coherence data structure. The reusable data attached to a primitive sparse relation is A( n,ϵ) = ( R, M, Witt_ n, κ_ n, Lcal_ n, Λ_coh, χ_coh, G_ n ). The calculus consists of exact transformations among these layers, plus the norm and determinant questions required to compare them globally.
Exact mathematical passage
\boxed{ \mathfrak A(\mathbf n,\boldsymbol\epsilon) = \left( \mathbf R, \mathbf M, \Witt_{\mathbf n}, \kappa_{\mathbf n}, \Lcal_{\mathbf n}, \Lambda_{\mathrm{coh}}, \chi_{\mathrm{coh}}, \mathcal G_{\mathbf n} \right). }
xUse in source · 101 passages
Search aliases: x
Use in source · appendices.tex, line 37
Prime derivative. The additive row is λ=(1,75,-448). The phase row is λ^ph=(1,25,-64). The coarse shortest transverse packet x=(-2,6,1) satisfies -2+450=448. The Wronskian is W_x=1600=fracabcR. The coarse height is six.
Exact mathematical passage
x=(-2,6,1)
x^{(1)}Use in source · 2 passages
Search aliases: x^(1)
Use in source · appendices_more.tex, line 143
Formal jet recursion. Let F( T)=0 be the shell equation and let T(t)=x^(1)t+x^(2)t^2+⋯. Suppose the gradient row Λ is primitive. Choose an integral vector v satisfying Λ· v=1. At order m, substitution gives Λ· x^(m) + G_m(x^(1),…,x^(m-1)) =0. The integral correction x^(m) = -G_m v solves the equation at that order. Induction produces an integral formal curve.
Exact mathematical passage
\mathbf T(t)=x^{(1)}t+x^{(2)}t^2+\cdots.
x^{(2)}Use in source · 1 passage
Search aliases: x^(2)
Use in source · appendices_more.tex, line 143
Formal jet recursion. Let F( T)=0 be the shell equation and let T(t)=x^(1)t+x^(2)t^2+⋯. Suppose the gradient row Λ is primitive. Choose an integral vector v satisfying Λ· v=1. At order m, substitution gives Λ· x^(m) + G_m(x^(1),…,x^(m-1)) =0. The integral correction x^(m) = -G_m v solves the equation at that order. Induction produces an integral formal curve.
Exact mathematical passage
\mathbf T(t)=x^{(1)}t+x^{(2)}t^2+\cdots.
x^{(a)}Use in source · 1 passage
Search aliases: x^(a)
Use in source · detailed_1.tex, line 277
For each term define ρ_a(t) = min{ normx^(a)_∞: λ^(a)· x^(a)=g_at }, with analogous functions for b and c. Then Θ(a,b,c) = min max{ρ_a(t_a),ρ_b(t_b),ρ_c(t_c)} over aggregate solutions satisfying additivity and transversality.
Exact mathematical passage
\rho_a(t) = \min\left\{ \norm{x^{(a)}}_\infty: \lambda^{(a)}\cdot x^{(a)}=g_at \right\},
x^{(m)}Use in source · 2 passages
Search aliases: x^(m)
Use in source · appendices_more.tex, line 151
Formal jet recursion. Let F( T)=0 be the shell equation and let T(t)=x^(1)t+x^(2)t^2+⋯. Suppose the gradient row Λ is primitive. Choose an integral vector v satisfying Λ· v=1. At order m, substitution gives Λ· x^(m) + G_m(x^(1),…,x^(m-1)) =0. The integral correction x^(m) = -G_m v solves the equation at that order. Induction produces an integral formal curve.
Exact mathematical passage
\Lambda\cdot x^{(m)} + G_m(x^{(1)},\ldots,x^{(m-1)}) =0.
x^{(m-1)}Use in source · 1 passage
Search aliases: x^(m-1)
Use in source · appendices_more.tex, line 151
Formal jet recursion. Let F( T)=0 be the shell equation and let T(t)=x^(1)t+x^(2)t^2+⋯. Suppose the gradient row Λ is primitive. Choose an integral vector v satisfying Λ· v=1. At order m, substitution gives Λ· x^(m) + G_m(x^(1),…,x^(m-1)) =0. The integral correction x^(m) = -G_m v solves the equation at that order. Induction produces an integral formal curve.
Exact mathematical passage
\Lambda\cdot x^{(m)} + G_m(x^{(1)},\ldots,x^{(m-1)}) =0.
X^3Use in source · 1 passage
Search aliases: X^3
Use in source · appendices_more.tex, line 393
[leftmargin=1.5em] item the 2-adic contact filtration of y^2 item the 3-adic contact filtration of x^3 item the fixed support of k item the depth-three polylogarithmic packet item the elliptic curve Y^2=X^3-k
Exact mathematical passage
Y^2=X^3-k
x^3Use in source · 5 passages
Search aliases: x^3
Use in source · appendices.tex, line 615
Hall's equation x^3-y^2=k is particularly attractive because the dyadic and triadic branches are built into the exponents. The polylogarithmic weight-three resonance may be relevant, but a rigorous connection has not yet been established.
Exact mathematical passage
x^3-y^2=k
X^{64}Use in source · 1 passage
Search aliases: X^64
Use in source · appendices.tex, line 184
For the balanced ratio β=-frac1253, the signed exponent vector on the input primes is ν=(-1,3). Its Kummer order modulo 64 is full. Choosing 64th roots gives a point on X^64+Y^64=1. The coordinate height is fraclog12864. With the tensor-power metric induced from O(1), the full plane-model canonical scale is frac6164log128. For an arbitrary compatible Weil-height normalization, this formula holds up to an additive O(1) term. It records the line-bundle cost of retaining the complete Fermat curve.
Exact mathematical passage
X^{64}+Y^{64}=1.
X^MUse in source · 1 passage
Search aliases: X^M
Use in source · detailed_3.tex, line 123
The Kummer--Fermat lift. Choose algebraic numbers u and v satisfying u^M=frac ac, v^M=frac bc. Then u^M+v^M=1. Thus P_M=[u:v:1] lies on the Fermat curve F_M:X^M+Y^M=Z^M. The construction is canonical up to the action of μ_M^2.
Exact mathematical passage
F_M:X^M+Y^M=Z^M.
x^mUse in source · 3 passages
Search aliases: x^m
Use in source · appendices.tex, line 609
Pillai and Hall. For x^m-y^n=k, the residual support is fixed by k. The two perfect-power packets must cancel to high precision while producing a small fixed output. This is a natural two-sector coherence problem.
Exact mathematical passage
x^m-y^n=k,
x^pUse in source · 2 passages
Search aliases: x^p
Use in source · appendices.tex, line 584
Generalized Fermat. For x^p+y^q=z^r, attach filtered contact packets to the three terms. The exponents multiply the local valuation depths. Balanced Kummer ratios and cross-prime logarithmic filters can eliminate many local patterns before a Frey or hypergeometric motive is attached.
Exact mathematical passage
x^p+y^q=z^r,
X_0Use in source · 2 passages
Search aliases: X_0
Use in source · detailed_2.tex, line 155
Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be an optimal modular parametrization, and let ϕ_E^*ω_E = 2π i c_E f_E(z) dz. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_E^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area factor proportional to 1/c. Thus the height c can be written as a shape-corrected synthesis gain involving modular degree, Manin normalization, and Pete…
Exact mathematical passage
\phi_E:X_0(N_E)\longrightarrow E
x_0Use in source · 6 passages
Search aliases: x_0
Use in source · detailed_1.tex, line 852
Obstruction cosets. Choose a section x_0∈ L with ω(x_0)=1. Then L = bigsqcup_k∈Z (kx_0+L_0). For fixed k, the problem is a closest-vector problem to the moving target -kx_0 in the lattice L_0. The three-sector compression makes this search much more explicit.
Exact mathematical passage
x_0\in L
x_1Use in source · 1 passage
Search aliases: x_1
Use in source · detailed_2.tex, line 365
A toy Hecke order. Consider T( m) = {(x_0,x_1,…,x_r)∈Z^r+1:x_i≡ x_0pmodm_i}. For the target coordinate x_0, the cotangent module is p/ p^2 ≅ ⊕_i=1^rZ/m_iZ. The cyclic congruence quotient is Z/lcm(m_1,…,m_r)Z. The quotient of cotangent order by congruence order is frac∏_i m_ilcm(m_i). This is a toy Wiles-defect factor. It measures the multiplicity discarded by cyclic compression.
Exact mathematical passage
\mathbb T(\mathbf m) = \{(x_0,x_1,\ldots,x_r)\in\Z^{r+1}:x_i\equiv x_0\pmod{m_i}\}.
x_3Use in source · 2 passages
Search aliases: x_3
Use in source · appendices.tex, line 225
Coherence lattice. After removing the sign layer, (ϵ_2,K,W,C)=(2,1,1,32). The residual row is g=(29,31)pmod32. The lattice is Λ_coh = {(x_3,x_5):29x_3+31x_5≡0pmod32}. The input vector lies in the kernel because 29(-1)+31(3)=64. The full repeated output depth has now been transformed into one arithmetic cyclic character of order 32, after the bounded sign layer has been separated.
Exact mathematical passage
\Lambda_{\mathrm{coh}} = \{(x_3,x_5):29x_3+31x_5\equiv0\pmod{32}\}.
x_5Use in source · 2 passages
Search aliases: x_5
Use in source · appendices.tex, line 225
Coherence lattice. After removing the sign layer, (ϵ_2,K,W,C)=(2,1,1,32). The residual row is g=(29,31)pmod32. The lattice is Λ_coh = {(x_3,x_5):29x_3+31x_5≡0pmod32}. The input vector lies in the kernel because 29(-1)+31(3)=64. The full repeated output depth has now been transformed into one arithmetic cyclic character of order 32, after the bounded sign layer has been separated.
Exact mathematical passage
\Lambda_{\mathrm{coh}} = \{(x_3,x_5):29x_3+31x_5\equiv0\pmod{32}\}.
x_iUse in source · 9 passages
Search aliases: x_i
Use in source · appendices_final.tex, line 214
Local CRT lift in closed form. Let r=∏_i=1^mp_i, w_i=r/p_i. The equation ∑_iw_ix_i=t has a solution because gcd(w_1,…,w_m)=1. Modulo p_i, x_i≡ t w_i^-1pmodp_i. Let ξ_i be the centered residue. Then ∑_iw_iξ_i-t is divisible by r. Set q=fract-∑_iw_iξ_ir. Writing x_i=ξ_i+p_i z_i gives ∑_i z_i=q.
Exact mathematical passage
\sum_iw_ix_i=t
x_pUse in source · 2 passages
Search aliases: x_p
Use in source · detailed_1.tex, line 41
A scalar packet is an integer vector x=(x_p)_p∈ S∈Z^S. Pairing with x gives the arithmetic derivative d_x(n) = n∑_pmid nfracv_p(n)p x_p. The Leibniz rule survives scalarization.
Exact mathematical passage
x=(x_p)_{p\in S}\in\Z^S.
x_rUse in source · 1 passage
Search aliases: x_r
Use in source · detailed_2.tex, line 365
A toy Hecke order. Consider T( m) = {(x_0,x_1,…,x_r)∈Z^r+1:x_i≡ x_0pmodm_i}. For the target coordinate x_0, the cotangent module is p/ p^2 ≅ ⊕_i=1^rZ/m_iZ. The cyclic congruence quotient is Z/lcm(m_1,…,m_r)Z. The quotient of cotangent order by congruence order is frac∏_i m_ilcm(m_i). This is a toy Wiles-defect factor. It measures the multiplicity discarded by cyclic compression.
Exact mathematical passage
\mathbb T(\mathbf m) = \{(x_0,x_1,\ldots,x_r)\in\Z^{r+1}:x_i\equiv x_0\pmod{m_i}\}.
\xrightarrowUse in source · 8 passages
Search aliases: xrightarrow
Use in source · appendices.tex, line 137
Filtered contact and Witt packet. The output contact complex is [128Z↪2Z] ≃ [Zxrightarrow64Z]. Its cohomology is Jcal_c≅Z/64Z. The associated graded is (F_2)^6. The six binary layers are joined by five nontrivial carry extensions. A cyclic target of order 64 can carry the filtered source exactly, while no cyclic target can contain the split rank-six source injectively.
Exact mathematical passage
[128\Z\hookrightarrow2\Z] \simeq [\Z\xrightarrow{64}\Z].
yUse in source · 32 passages
Search aliases: y
Use in source · appendices.tex, line 51
The phase packet y=(-8,8,3) satisfies D_y(3)=-8, D_y(125)=200, D_y(128)=192, and -8+200=192. Its Wronskian is also 1600. The phase height is eight. The phase inequality is sharp: 128 = 30(frac13+frac15)8. This illustrates that the exponent-free phase channel can sharpen the bound even when its packet norm is larger.
Exact mathematical passage
y=(-8,8,3)
Y^2Use in source · 1 passage
Search aliases: Y^2
Use in source · appendices_more.tex, line 393
[leftmargin=1.5em] item the 2-adic contact filtration of y^2 item the 3-adic contact filtration of x^3 item the fixed support of k item the depth-three polylogarithmic packet item the elliptic curve Y^2=X^3-k
Exact mathematical passage
Y^2=X^3-k
y^2Use in source · 8 passages
Search aliases: y^2
Use in source · appendices.tex, line 124
The marked Frey curve is y^2=x(x-3)(x+125). Its discriminant is 16·3^2·125^2·128^2. At the odd primes, the marked depths are two at 3 and six at 5. The dyadic depth requires the weighted branch.
Exact mathematical passage
y^2=x(x-3)(x+125).
Y^{64}Use in source · 1 passage
Search aliases: Y^64
Use in source · appendices.tex, line 184
For the balanced ratio β=-frac1253, the signed exponent vector on the input primes is ν=(-1,3). Its Kummer order modulo 64 is full. Choosing 64th roots gives a point on X^64+Y^64=1. The coordinate height is fraclog12864. With the tensor-power metric induced from O(1), the full plane-model canonical scale is frac6164log128. For an arbitrary compatible Weil-height normalization, this formula holds up to an additive O(1) term. It records the line-bundle cost of retaining the complete Fermat curve.
Exact mathematical passage
X^{64}+Y^{64}=1.
Y^MUse in source · 1 passage
Search aliases: Y^M
Use in source · detailed_3.tex, line 123
The Kummer--Fermat lift. Choose algebraic numbers u and v satisfying u^M=frac ac, v^M=frac bc. Then u^M+v^M=1. Thus P_M=[u:v:1] lies on the Fermat curve F_M:X^M+Y^M=Z^M. The construction is canonical up to the action of μ_M^2.
Exact mathematical passage
F_M:X^M+Y^M=Z^M.
y^nUse in source · 3 passages
Search aliases: y^n
Use in source · appendices.tex, line 609
Pillai and Hall. For x^m-y^n=k, the residual support is fixed by k. The two perfect-power packets must cancel to high precision while producing a small fixed output. This is a natural two-sector coherence problem.
Exact mathematical passage
x^m-y^n=k,
y^qUse in source · 2 passages
Search aliases: y^q
Use in source · appendices.tex, line 584
Generalized Fermat. For x^p+y^q=z^r, attach filtered contact packets to the three terms. The exponents multiply the local valuation depths. Balanced Kummer ratios and cross-prime logarithmic filters can eliminate many local patterns before a Frey or hypergeometric motive is attached.
Exact mathematical passage
x^p+y^q=z^r,
y_1Use in source · 1 passage
Search aliases: y_1
Use in source · appendices_more.tex, line 33
For a scalar split t=y_1+⋯+y_e, the child sup norm can fall to lceilfrac|t|erceil. The synthesis map has norm e, so elceilfrac|t|erceil≥|t|. This explains why raw shell splitting cannot lower the intrinsic packet height.
Exact mathematical passage
t=y_1+\cdots+y_e,
y_eUse in source · 1 passage
Search aliases: y_e
Use in source · appendices_more.tex, line 33
For a scalar split t=y_1+⋯+y_e, the child sup norm can fall to lceilfrac|t|erceil. The synthesis map has norm e, so elceilfrac|t|erceil≥|t|. This explains why raw shell splitting cannot lower the intrinsic packet height.
Exact mathematical passage
t=y_1+\cdots+y_e,
y_pUse in source · 2 passages
Search aliases: y_p
Use in source · detailed_1.tex, line 450
The coarse equation recovers the original exponent-weighted row ∑_pmid afracav_p(a)p t_p + ∑_pmid bfracbv_p(b)p t_p - ∑_pmid cfraccv_p(c)p t_p =0. The phase equation is exponent free: ∑_pmid afrac ap y_p + ∑_pmid bfrac bp y_p - ∑_pmid cfrac cp y_p =0. Define the phase derivative D_y(n)=n∑_pmid nfracy_pp. The same radical divisibility argument gives fracabcRmid aD_y(b)-bD_y(a).
Exact mathematical passage
\boxed{ \sum_{p\mid a}\frac ap y_p + \sum_{p\mid b}\frac bp y_p - \sum_{p\mid c}\frac cp y_p =0. }
y_{p,r}Use in source · 4 passages
Search aliases: y_p,r
Use in source · detailed_1.tex, line 555
Exact local lift gauges. Let r_u=rad(u), P_u=fracur_u. A shell packet y=(y_p,r) produces D_y(u) = u∑_pmid ufrac1p ∑_r=0^v_p(u)-1y_p,r =P_ut_u, where t_u= ∑_pmid u fracr_up ∑_r y_p,r. Because gcd_pmid ufracr_up=1, every integer t_u is attainable when u>1.
Exact mathematical passage
y=(y_{p,r})
ZUse in source · 1 passage
Search aliases: Z
Use in source · detailed_3.tex, line 123
The Kummer--Fermat lift. Choose algebraic numbers u and v satisfying u^M=frac ac, v^M=frac bc. Then u^M+v^M=1. Thus P_M=[u:v:1] lies on the Fermat curve F_M:X^M+Y^M=Z^M. The construction is canonical up to the action of μ_M^2.
Exact mathematical passage
F_M:X^M+Y^M=Z^M.
\mathbb ZUse in source · 1 passage
Search aliases: Z mathbb
Use in source · bibliography.tex, line 86
bibitemAKN B. Antieau, A. Krause, and T. Nikolaus, ``On the K-theory of Z/p^n,'' arXiv:2405.04329, 2024.
Exact mathematical passage
\mathbb Z/p^n
zUse in source · 25 passages
Search aliases: z
Use in source · appendices.tex, line 199
Polylogarithmic path. Set z=frac3128, 1-z=frac125128, β=1-frac1z=-frac1253. The depth-three functional relation closes on these three arguments. At the output prime two, v_2(β-1) = v_2(-frac1283) =7. The unscaled principal logarithm has 2-adic valuation seven. After division by four, the weighted principal coordinate occupies five filtration levels, while the separate sign packet supplies the first dyadic layer.
Exact mathematical passage
z=\frac3{128}, \qquad 1-z=\frac{125}{128}, \qquad \beta=1-\frac1z=-\frac{125}{3}.
\Z^2Use in source · 3 passages
Search aliases: Z^2 Z
Use in source · detailed_1.tex, line 639
0, · pmid c. Consider M_a,b,c = Λ w : Z^S⟶Z^2. The first row is the exponent-free additive phase row. The second is its integer Wronskian row. Set Q=fracabcR.
Exact mathematical passage
\mathsf M_{a,b,c} = \begin{pmatrix} \Lambda\\ \mathsf w \end{pmatrix}: \Z^S\longrightarrow\Z^2.
Z^MUse in source · 1 passage
Search aliases: Z^M
Use in source · detailed_3.tex, line 123
The Kummer--Fermat lift. Choose algebraic numbers u and v satisfying u^M=frac ac, v^M=frac bc. Then u^M+v^M=1. Thus P_M=[u:v:1] lies on the Fermat curve F_M:X^M+Y^M=Z^M. The construction is canonical up to the action of μ_M^2.
Exact mathematical passage
F_M:X^M+Y^M=Z^M.
\Z^nUse in source · 1 passage
Search aliases: Z^n Z
Use in source · detailed_2.tex, line 339
For a primitive rank-one vector v∈Z^n, the orthogonal projector denominator is ⟨ v,v⟩, and the congruence quotient is cyclic of that order.
Exact mathematical passage
v\in\Z^n
\Z^QUse in source · 3 passages
Search aliases: Z^Q Z
Use in source · overview.tex, line 304
The coherence lattice. After removing common Kummer powers and local logarithmic image defects, the surviving rows combine by the Chinese remainder theorem into a primitive cyclic character χ_coh:Z^Q⟶Z/CZ. The coherence lattice is Λ_coh = Kerχ_coh. When the row is primitive, [Z^Q:Λ_coh]=C. The normalized exponent vector of the relation lies in this kernel.
Exact mathematical passage
\chi_{\mathrm{coh}}:\Z^Q\longrightarrow\Z/C\Z.
z^rUse in source · 2 passages
Search aliases: z^r
Use in source · appendices.tex, line 584
Generalized Fermat. For x^p+y^q=z^r, attach filtered contact packets to the three terms. The exponents multiply the local valuation depths. Balanced Kummer ratios and cross-prime logarithmic filters can eliminate many local patterns before a Frey or hypergeometric motive is attached.
Exact mathematical passage
x^p+y^q=z^r,
\Z^{r+1}Use in source · 1 passage
Search aliases: Z^r+1 Z
Use in source · detailed_2.tex, line 365
A toy Hecke order. Consider T( m) = {(x_0,x_1,…,x_r)∈Z^r+1:x_i≡ x_0pmodm_i}. For the target coordinate x_0, the cotangent module is p/ p^2 ≅ ⊕_i=1^rZ/m_iZ. The cyclic congruence quotient is Z/lcm(m_1,…,m_r)Z. The quotient of cotangent order by congruence order is frac∏_i m_ilcm(m_i). This is a toy Wiles-defect factor. It measures the multiplicity discarded by cyclic compression.
Exact mathematical passage
\mathbb T(\mathbf m) = \{(x_0,x_1,\ldots,x_r)\in\Z^{r+1}:x_i\equiv x_0\pmod{m_i}\}.
\Z^SUse in source · 6 passages
Search aliases: Z^S Z
Use in source · detailed_1.tex, line 41
A scalar packet is an integer vector x=(x_p)_p∈ S∈Z^S. Pairing with x gives the arithmetic derivative d_x(n) = n∑_pmid nfracv_p(n)p x_p. The Leibniz rule survives scalarization.
Exact mathematical passage
x=(x_p)_{p\in S}\in\Z^S.
\Z^sUse in source · 3 passages
Search aliases: Z^s Z
Use in source · appendices_more.tex, line 209
Arbitrary-index congruence lattices. Let v∈Z^s be primitive. Complete v to a unimodular basis v,v_2,…,v_s. In the dual basis, choose the row that vanishes on v and takes value one on v_2. Reduce it modulo an arbitrary integer C. The resulting row is primitive modulo C and annihilates v.
Exact mathematical passage
v\in\Z^s
\Z^{S_{ab}}Use in source · 7 passages
Search aliases: Z^S_ab Z
Use in source · appendices_final.tex, line 409
Arithmetic endpoint. The arithmetic endpoint is the coherence character χ_coh: Z^S_ab→Z/CZ whose row is built from simultaneous logarithms of the actual input primes and whose kernel contains the normalized exponent vector forced by a+b=c.
Exact mathematical passage
\chi_{\mathrm{coh}}: \Z^{S_{ab}}\to\Z/C\Z
\Z_2Use in source · 4 passages
Search aliases: Z_2 Z
Use in source · appendices.tex, line 159
The local Witt coordinate is W_6(F_2) ≅ Z/64Z. The weighted multiplicative dyadic coordinate is (1+4Z_2)/(1+2^8Z_2), which has the same order.
Exact mathematical passage
(1+4\Z_2)/(1+2^8\Z_2),
z_iUse in source · 7 passages
Search aliases: z_i
Use in source · appendices_final.tex, line 233
Local CRT lift in closed form. Let r=∏_i=1^mp_i, w_i=r/p_i. The equation ∑_iw_ix_i=t has a solution because gcd(w_1,…,w_m)=1. Modulo p_i, x_i≡ t w_i^-1pmodp_i. Let ξ_i be the centered residue. Then ∑_iw_iξ_i-t is divisible by r. Set q=fract-∑_iw_iξ_ir. Writing x_i=ξ_i+p_i z_i gives ∑_i z_i=q.
Exact mathematical passage
x_i=\xi_i+p_i z_i
\Z_pUse in source · 5 passages
Search aliases: Z_p Z
Use in source · detailed_2.tex, line 471
Associated graded and carry extensions. At p^eparallel c, Jcal_p,e = pZ_p/p^eZ_p ≅ Z/p^e-1Z. Its filtration p^jZ_p/p^eZ_p has successive quotients F_p. Thus grJcal_p,e≅(F_p)^e-1. The exact sequences 0⟶ Z/p^j-1Z xrightarrow× p Z/p^jZ ⟶ F_p ⟶0 are nonsplit. These carry extensions are precisely what the split persistence packet forgot.
Exact mathematical passage
\Jcal_{p,e} = p\Z_p/p^e\Z_p \cong \Z/p^{e-1}\Z.
|^{1/m}Use in source · 1 passage
Search aliases: |^1/m
Use in source · appendices_more.tex, line 52
Homogeneous higher obstructions. Let O be homogeneous of degree m: O(tx)=t^m O(x). Then fracnormx | O(x)|^1/m is scale invariant. Replacing one large coordinate by several coordinates of size roughly the mth root does not improve a bound whose obstruction estimate raises those coordinates back to degree m.
Exact mathematical passage
\frac{\norm{x}} {|\mathcal O(x)|^{1/m}}
|^2Use in source · 1 passage
Search aliases: |^2
Use in source · detailed_1.tex, line 537
onSecond-order phase energy At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^2 - frac1e∑_j≠0|widehat T_j|^2 ]. Thus nonzero polyphase energy first becomes visible at second order. Later formal-smoothness results show that this …
Exact mathematical passage
\sum_{r<s}T_rT_s = \frac12\left[ \left(1-\frac1e\right)|\widehat T_0|^2 - \frac1e\sum_{j\neq0}|\widehat T_j|^2 \right].
\pmUse in source · 6 passages
Search aliases: ± pm
Use in source · appendices_final.tex, line 199
For the exponent-free phase obstruction matrix, every nonzero cross-sector minor is ±fracabcpq = ±fracabcRfracRpq. All such minors are divisible by abc/R. For each prime ℓmid R, choose one column with p=ℓ and one column from a different nonempty sector. The reduced minor R/(pq) is then not divisible by ℓ. Hence the gcd of the reduced minors is one and the second Smith factor is exactly abc/R.
Exact mathematical passage
\pm\frac{abc}{pq} = \pm\frac{abc}{R}\frac{R}{pq}.
\timesUse in source · 12 passages
Search aliases: × times
Use in source · appendices_final.tex, line 168
A determinant viewpoint. When several independent exponent packets are available, replace the one-row character by an exterior-power regulator. Let L be a normalized cross-prime logarithm matrix. A nonzero r× r minor det L_I,J measures the failure of r input packets to cancel simultaneously across r output channels.
Exact mathematical passage
r\times r
\alphaUse in source · 4 passages
Search aliases: α alpha
Use in source · detailed_1.tex, line 421
Dyadic and odd-adic primary lifts. Factor e=2^α∏_ℓ oddℓ^α_ℓ. The cokernel of the valuation block is Z/eZ ≅ Z/2^αZ × ∏_ℓ odd Z/ℓ^α_ℓZ. The primary factors specify the local lifting problems for the aggregate multiplicity channel. Its 2-primary component must be treated 2-adically, while the odd components admit the corresponding odd-adic lifts. The phase channels require no local denominator correction.
Exact mathematical passage
e=2^\alpha\prod_{\ell\text{ odd}}\ell^{\alpha_\ell}.
\alpha_uUse in source · 2 passages
Search aliases: α_u alpha
Use in source · detailed_1.tex, line 999
Phase and coarse separation. The local target has the form t_u(k,n)=α_uk+r_un. For a local prime pmid r_u, t_u(k,n)≡α_ukpmod p. Thus the obstruction multiple k determines the finite phase packet. The coarse parameter n changes only the quotient load. The exact search is a moving polyphase mechanism rather than a generic mixed-integer program.
Exact mathematical passage
t_u(k,n)=\alpha_uk+r_un.
\alpha_\ellUse in source · 2 passages
Search aliases: α_ℓ alpha ell
Use in source · detailed_1.tex, line 421
Dyadic and odd-adic primary lifts. Factor e=2^α∏_ℓ oddℓ^α_ℓ. The cokernel of the valuation block is Z/eZ ≅ Z/2^αZ × ∏_ℓ odd Z/ℓ^α_ℓZ. The primary factors specify the local lifting problems for the aggregate multiplicity channel. Its 2-primary component must be treated 2-adically, while the odd components admit the corresponding odd-adic lifts. The phase channels require no local denominator correction.
Exact mathematical passage
e=2^\alpha\prod_{\ell\text{ odd}}\ell^{\alpha_\ell}.
\betaUse in source · 18 passages
Search aliases: β beta
Use in source · appendices.tex, line 176
For the balanced ratio β=-frac1253, the signed exponent vector on the input primes is ν=(-1,3). Its Kummer order modulo 64 is full. Choosing 64th roots gives a point on X^64+Y^64=1. The coordinate height is fraclog12864. With the tensor-power metric induced from O(1), the full plane-model canonical scale is frac6164log128. For an arbitrary compatible Weil-height normalization, this formula holds up to an additive O(1) term. It records the line-bundle cost of retaining the complete Fermat curve.
Exact mathematical passage
\beta=-\frac{125}{3},
\beta^{1/K}Use in source · 1 passage
Search aliases: β^1/K beta
Use in source · detailed_3.tex, line 603
Adelic coherence regulator object. Package the surviving data as R_a,b,c = ( U_ab, β^1/K, {χ_p}_pmid c, χ_coh, Λ_coh, norm·_∞ ). Here U_ab is the input S-unit group, the local characters are the normalized logarithmic rows, and the archimedean norm records input height.
Exact mathematical passage
\boxed{ \mathfrak R_{a,b,c} = \left( U_{ab}, \beta^{1/K}, \{\chi_p\}_{p\mid c}, \chi_{\mathrm{coh}}, \Lambda_{\mathrm{coh}}, \norm{\cdot}_\infty \right). }
\beta_1Use in source · 1 passage
Search aliases: β_1 beta
Use in source · appendices_more.tex, line 260
Kummer fiber product. Balanced Kummer ratios from the two equations define classes β_1=-B/A and β_2=-D/C. The shared term C creates a multiplicative compatibility. The natural global object is a fiber product of the two Kummer torsors over the Kummer class of C.
Exact mathematical passage
\beta_1=-B/A
\beta_2Use in source · 1 passage
Search aliases: β_2 beta
Use in source · appendices_more.tex, line 264
Kummer fiber product. Balanced Kummer ratios from the two equations define classes β_1=-B/A and β_2=-D/C. The shared term C creates a multiplicative compatibility. The natural global object is a fiber product of the two Kummer torsors over the Kummer class of C.
Exact mathematical passage
\beta_2=-D/C.
\GammaUse in source · 3 passages
Search aliases: Γ Gamma
Use in source · detailed_1.tex, line 767
ve hyperplane. The exact continuous width can be computed from the three sector weights. Let σ_u=∑_pmid ufrac1p and Δ(a,b,c) = min{ σ_a+σ_b, frac cb(σ_a+σ_c), frac ca(σ_b+σ_c) }. Then overline H = fraccRΔ(a,b,c). The discrete master inequality becomes c≤ RΔΦ. Since Φ=overline H k_*Γ for the optimal obstruction multiple k_* and an integrality factor Γ≥1, ΔΦ = frac cR k_*Γ. This is the no-escape identity. Optimizing the same first-order invariant cannot lower the continuous floor c/R.
Exact mathematical passage
\Phi=\overline H\,k_*\Gamma
\DeltaUse in source · 11 passages
Search aliases: Δ Delta
Use in source · appendices_more.tex, line 120
Since Q=abc/R, the exact minimum is overline H = fraccRΔ(a,b,c), where Δ = min{ σ_a+σ_b, frac cb(σ_a+σ_c), frac ca(σ_b+σ_c) }. The real problem is therefore completely solved. The unresolved content lies in the integral departure from this stable width.
Exact mathematical passage
\overline H = \frac{c}{R\Delta(a,b,c)},
\delta_0Use in source · 1 passage
Search aliases: δ_0 delta
Use in source · detailed_2.tex, line 70
The three-spike texorpdfstringp-adicp-adic coherence signal. Represent the additive triple by ν_a,b,c=δ_0+δ_a+δ_c. The pairwise differences are a, b, and c. At a prime dividing exactly one of these integers, exactly one pair of spikes is p-adically close. The depth of that edge is the corresponding valuation.
Exact mathematical passage
\nu_{a,b,c}=\delta_0+\delta_a+\delta_c.
\delta_aUse in source · 1 passage
Search aliases: δ_a delta
Use in source · detailed_2.tex, line 70
The three-spike texorpdfstringp-adicp-adic coherence signal. Represent the additive triple by ν_a,b,c=δ_0+δ_a+δ_c. The pairwise differences are a, b, and c. At a prime dividing exactly one of these integers, exactly one pair of spikes is p-adically close. The depth of that edge is the corresponding valuation.
Exact mathematical passage
\nu_{a,b,c}=\delta_0+\delta_a+\delta_c.
\delta_{\mathrm{alg}}Use in source · 2 passages
Search aliases: δ_alg delta mathrm
Use in source · detailed_2.tex, line 524
For the output source m=c/R_c, the corrected algebraic defect against a cyclic target of order η_f is δ_alg(c,f) = fracc/R_cgcd(c/R_c,η_f). Prime by prime, δ_alg(c,f) = ∏_p^eparallel c p^max(e-1-v_p(η_f),0). The problem is missing depth, not missing rank.
Exact mathematical passage
\boxed{ \delta_{\mathrm{alg}}(c,f) = \frac{c/R_c}{\gcd(c/R_c,\eta_f)}. }
\delta_cUse in source · 1 passage
Search aliases: δ_c delta
Use in source · detailed_2.tex, line 70
The three-spike texorpdfstringp-adicp-adic coherence signal. Represent the additive triple by ν_a,b,c=δ_0+δ_a+δ_c. The pairwise differences are a, b, and c. At a prime dividing exactly one of these integers, exactly one pair of spikes is p-adically close. The depth of that edge is the corresponding valuation.
Exact mathematical passage
\nu_{a,b,c}=\delta_0+\delta_a+\delta_c.
\Delta_EUse in source · 3 passages
Search aliases: Δ_E Delta
Use in source · detailed_2.tex, line 85
The Frey curve. Attach the marked Frey curve E_a,b,c:y^2=x(x-a)(x+b). The roots 0, a, and -b retain the three differences a, b, and c. The basic invariants are Δ_E=16a^2b^2c^2 and c_4(E)=16(a^2+ab+b^2). At every odd prime pmid abc, the curve has multiplicative reduction. The conductor records that the channel is active, while the minimal discriminant records the full depth.
Exact mathematical passage
\Delta_E=16a^2b^2c^2
\boldsymbol\epsilonUse in source · 4 passages
Search aliases: ϵ boldsymbol epsilon
Use in source · detailed_3.tex, line 673
Prime-contact data structure. The reusable object attached to a sparse relation is A( n,ϵ) = ( W_ n, κ_ n, L_ n, Λ_coh, R_∞, M_glob ). The first four components are finite and local-global arithmetic data. The last two are optional archimedean and global-motivic realizations. Keeping them separate prevents an unproved global comparison from being built into the definition.
Exact mathematical passage
\boxed{ \mathfrak A(\mathbf n,\boldsymbol\epsilon) = \left( \mathbb W_{\mathbf n}, \kappa_{\mathbf n}, \mathcal L_{\mathbf n}, \Lambda_{\mathrm{coh}}, \mathcal R_\infty, \mathcal M_{\mathrm{glob}} \right). }
\varepsilonUse in source · 6 passages
Search aliases: ε varepsilon
Use in source · detailed_2.tex, line 271
Atkin--Lehner Walsh packets. Let W_1,…,W_r be commuting Atkin--Lehner involutions. For a sign vector ε∈{±1}^r, define A_ε = ∏_j=1^r(I+ε_jW_j). Then A_ε^2=2^rA_ε, A_ε A_ε'=0 (ε≠ε'), and ∑_ε A_ε=2^rI. The unnormalized analysis packets are integral. The common synthesis denominator is 2^r, which is subpolynomial in a squarefree conductor. Local sign separation is therefore not the large persistence carrier.
Exact mathematical passage
\varepsilon\in\{\pm1\}^r,
\varepsilon'Use in source · 1 passage
Search aliases: ε' varepsilon
Use in source · detailed_2.tex, line 286
Atkin--Lehner Walsh packets. Let W_1,…,W_r be commuting Atkin--Lehner involutions. For a sign vector ε∈{±1}^r, define A_ε = ∏_j=1^r(I+ε_jW_j). Then A_ε^2=2^rA_ε, A_ε A_ε'=0 (ε≠ε'), and ∑_ε A_ε=2^rI. The unnormalized analysis packets are integral. The common synthesis denominator is 2^r, which is subpolynomial in a squarefree conductor. Local sign separation is therefore not the large persistence carrier.
Exact mathematical passage
A_\varepsilon A_{\varepsilon'}=0 \quad(\varepsilon\neq\varepsilon'),
\epsilon_1Use in source · 2 passages
Search aliases: ϵ_1 epsilon
Use in source · detailed_3.tex, line 625
Sparse additive relations. Let ϵ_1n_1+⋯+ϵ_rn_r=0, ϵ_i∈{±1}, be primitive. For every term define R_i=rad(n_i), M_i=n_i/R_i. The prime-contact coherence calculus assigns:
Exact mathematical passage
\epsilon_1n_1+\cdots+\epsilon_rn_r=0, \qquad \epsilon_i\in\{\pm1\},
\epsilon_{2,p}Use in source · 1 passage
Search aliases: ϵ_2,p epsilon
Use in source · appendices_final.tex, line 54
The local residual character is χ_p(x) = exp( frac2π ip^χ_pg_p· x ). Its kernel contains the normalized exponent vector. The local factors are therefore K_p, W_p=p^w_p, C_p=p^χ_p. At two, one sign layer ϵ_2,p is removed before the weighted logarithmic row is formed.
Exact mathematical passage
\epsilon_{2,p}
\epsilon_iUse in source · 2 passages
Search aliases: ϵ_i epsilon
Use in source · detailed_3.tex, line 625
Sparse additive relations. Let ϵ_1n_1+⋯+ϵ_rn_r=0, ϵ_i∈{±1}, be primitive. For every term define R_i=rad(n_i), M_i=n_i/R_i. The prime-contact coherence calculus assigns:
Exact mathematical passage
\epsilon_1n_1+\cdots+\epsilon_rn_r=0, \qquad \epsilon_i\in\{\pm1\},
\epsilon_jUse in source · 1 passage
Search aliases: ϵ_j epsilon
Use in source · overview.tex, line 942
A general contact functor. The construction extends to a sparse relation ∑_j=1^rϵ_jn_j=0. Choose one or more output terms whose repeated depth is to be explained. The assignment ( n,ϵ) ⟼ A( n,ϵ) is canonical under permutations, sign reversal, primitive reduction, and primary decomposition. The optional geometric lift depends on the equation, but the support, contact, Kummer, and logarithmic layers do not.
Exact mathematical passage
\sum_{j=1}^{r}\epsilon_jn_j=0.
\varepsilon_jUse in source · 1 passage
Search aliases: ε_j varepsilon
Use in source · detailed_2.tex, line 275
Atkin--Lehner Walsh packets. Let W_1,…,W_r be commuting Atkin--Lehner involutions. For a sign vector ε∈{±1}^r, define A_ε = ∏_j=1^r(I+ε_jW_j). Then A_ε^2=2^rA_ε, A_ε A_ε'=0 (ε≠ε'), and ∑_ε A_ε=2^rI. The unnormalized analysis packets are integral. The common synthesis denominator is 2^r, which is subpolynomial in a squarefree conductor. Local sign separation is therefore not the large persistence carrier.
Exact mathematical passage
A_\varepsilon = \prod_{j=1}^{r}(I+\varepsilon_jW_j).
\epsilon_rUse in source · 2 passages
Search aliases: ϵ_r epsilon
Use in source · detailed_3.tex, line 625
Sparse additive relations. Let ϵ_1n_1+⋯+ϵ_rn_r=0, ϵ_i∈{±1}, be primitive. For every term define R_i=rad(n_i), M_i=n_i/R_i. The prime-contact coherence calculus assigns:
Exact mathematical passage
\epsilon_1n_1+\cdots+\epsilon_rn_r=0, \qquad \epsilon_i\in\{\pm1\},
\zeta^eUse in source · 4 passages
Search aliases: ζ^e zeta
Use in source · appendices_final.tex, line 259
Polyphase circulant determinant. For the valuation word V_e(T)=1+T+⋯+T^e-1 in A_N=Z[T]/(T^N-1), multiplication is circulant. Its determinant is ∏_ζ^N=1V_e(ζ). Let I_N be the augmentation ideal of A_N. When gcd(e,N)=1, the nontrivial roots are permuted by ζ↦ζ^e, so multiplication by V_e has determinant one on the integral lattice I_N and is therefore an automorphism of it. On A_N/I_N≅Z, the map is multiplication by e. The exact sequence for I_N⊂ A_N then gives cokernel Z/eZ and Smith form diag(1,…
Exact mathematical passage
\zeta\mapsto\zeta^e
\zeta^NUse in source · 3 passages
Search aliases: ζ^N zeta
Use in source · appendices_final.tex, line 256
Polyphase circulant determinant. For the valuation word V_e(T)=1+T+⋯+T^e-1 in A_N=Z[T]/(T^N-1), multiplication is circulant. Its determinant is ∏_ζ^N=1V_e(ζ). Let I_N be the augmentation ideal of A_N. When gcd(e,N)=1, the nontrivial roots are permuted by ζ↦ζ^e, so multiplication by V_e has determinant one on the integral lattice I_N and is therefore an automorphism of it. On A_N/I_N≅Z, the map is multiplication by e. The exact sequence for I_N⊂ A_N then gives cokernel Z/eZ and Smith form diag(1,…
Exact mathematical passage
\prod_{\zeta^N=1}V_e(\zeta).
\zeta_MUse in source · 1 passage
Search aliases: ζ_M zeta
Use in source · detailed_3.tex, line 198
Root-distribution compression. Let u^M=z and let ζ_M be a primitive Mth root of unity. For n≥1, ∑_j=0^M-1 Li_n(ζ_M^ju) = M^1-nLi_n(u^M). Inside the disk of convergence, this follows by summing the defining series and using the root-of-unity filter. Analytic continuation extends the identity with compatible branches.
Exact mathematical passage
\zeta_M
\zeta_M^jUse in source · 2 passages
Search aliases: ζ_M^j zeta
Use in source · detailed_3.tex, line 199
Root-distribution compression. Let u^M=z and let ζ_M be a primitive Mth root of unity. For n≥1, ∑_j=0^M-1 Li_n(ζ_M^ju) = M^1-nLi_n(u^M). Inside the disk of convergence, this follows by summing the defining series and using the root-of-unity filter. Analytic continuation extends the identity with compatible branches.
Exact mathematical passage
\boxed{ \sum_{j=0}^{M-1} \operatorname{Li}_n(\zeta_M^ju) = M^{1-n}\operatorname{Li}_n(u^M). }
\eta_fUse in source · 4 passages
Search aliases: η_f eta
Use in source · detailed_2.tex, line 302
The same-sign separation problem. After fixing the local signs, several eigenforms can remain congruent inside the same packet. Let Λ be an integral Hecke lattice and let e_f be the rational projector onto the Frey eigenline. Define η_f = min{m≥1:m e_f(Λ)⊆Λ}. This is the integral eigenchannel separation denominator. It is an arithmetic analogue of leakage in a filter bank. The difficulty is not rational spectral isolation. It is isolation inside the integral lattice.
Exact mathematical passage
\eta_f = \min\{m\ge1:m e_f(\Lambda)\subseteq\Lambda\}.
\eta_UUse in source · 2 passages
Search aliases: η_U eta
Use in source · detailed_2.tex, line 324
[Integral projector theorem] The least positive integer η_U satisfying η_U e_U(Λ)⊆Λ is the exponent of the finite quotient C_U.
Exact mathematical passage
\eta_U
\thetaUse in source · 1 passage
Search aliases: θ theta
Use in source · overview.tex, line 671
Its depth Fourier transform is a Dirichlet packet: widehatΨ_n,p(θ) = sqrtlog p exp(-fraci(e_p-1)θ2) fracsin(e_pθ/2)sin(θ/2). The zero phase records the exponent. Nonzero phases record the endpoint and residue geometry of the valuation run.
Exact mathematical passage
\widehat\Psi_{n,p}(\theta) = \sqrt{\log p}\, \exp\left(-\frac{i(e_p-1)\theta}{2}\right) \frac{\sin(e_p\theta/2)}{\sin(\theta/2)}.
\Theta_3Use in source · 2 passages
Search aliases: Θ_3 Theta
Use in source · detailed_3.tex, line 340
Conditional motivic transfer. Let K_β be the cyclic group generated by the balanced Kummer class. Suppose there exists an integral finite object H_3(a,b,c) and a canonical map Θ_3:K_β⟶ H_3(a,b,c) satisfying |kerΘ_3|≤ C_eps R^eps and | H_3(a,b,c)|≤ C'_eps R_aR_bR^eps. Then the strong abc conjecture follows after absorbing the logarithmic Kummer kernel.
Exact mathematical passage
\Theta_3:K_\beta\longrightarrow\mathcal H_3(a,b,c)
\Theta_fUse in source · 2 passages
Search aliases: Θ_f Theta
Use in source · detailed_2.tex, line 576
[Conditional filtered-transfer criterion] Suppose a canonical map H^0(Θ_f):Z/(c/R_c)Z⟶ H^0( C_f) satisfies, for every eps>0, |ker H^0(Θ_f)|≤ C_eps R^eps and |H^0( C_f)|≤ C'_eps R_aR_bR^eps. Then the strong abc conjecture follows.
Exact mathematical passage
H^0(\Theta_f):\Z/(c/R_c)\Z\longrightarrow H^0(\mathbf C_f)
\iotaUse in source · 2 passages
Search aliases: ι iota
Use in source · overview.tex, line 759
The integral automorphic packet. The resulting integral automorphic object is Acal^sharp(E,ι) = ( f_E, Λ_E, ω_E, {q_p}_pmid N_E, ι, ϕ_E ). The components are: [leftmargin=1.5em] item the rational Hecke eigen-signal f_E item an integral Hecke, modular-symbol, or homology lattice Λ_E item the minimal N eron differential ω_E item local Tate-depth packets q_p item the marking ι of the three rational two-torsion differences item the modular synthesis map ϕ_E
Exact mathematical passage
\boxed{ \Acal^{\sharp}(E,\iota) = \left( f_E, \Lambda_E, \omega_E, \{q_p\}_{p\mid N_E}, \iota, \phi_E \right). }
\kappa_{\mathbf n}Use in source · 4 passages
Search aliases: κ_ n kappa mathbf
Use in source · detailed_3.tex, line 673
Prime-contact data structure. The reusable object attached to a sparse relation is A( n,ϵ) = ( W_ n, κ_ n, L_ n, Λ_coh, R_∞, M_glob ). The first four components are finite and local-global arithmetic data. The last two are optional archimedean and global-motivic realizations. Keeping them separate prevents an unproved global comparison from being built into the definition.
Exact mathematical passage
\boxed{ \mathfrak A(\mathbf n,\boldsymbol\epsilon) = \left( \mathbb W_{\mathbf n}, \kappa_{\mathbf n}, \mathcal L_{\mathbf n}, \Lambda_{\mathrm{coh}}, \mathcal R_\infty, \mathcal M_{\mathrm{glob}} \right). }
\kappa_{128}Use in source · 1 passage
Search aliases: κ_128 kappa
Use in source · appendices.tex, line 166
Kummer and Fermat packets. The global Kummer map is κ_128:Z/64Z→ H^1(Q,μ_64). Since gcd(64,7)=1, it is injective.
Exact mathematical passage
\kappa_{128}:\Z/64\Z\to H^1(\Q,\mu_{64}).
\lambdaUse in source · 13 passages
Search aliases: λ lambda
Use in source · appendices.tex, line 29
Prime derivative. The additive row is λ=(1,75,-448). The phase row is λ^ph=(1,25,-64). The coarse shortest transverse packet x=(-2,6,1) satisfies -2+450=448. The Wronskian is W_x=1600=fracabcR. The coarse height is six.
Exact mathematical passage
\lambda=(1,75,-448).
\lambda^{\mathsf T}Use in source · 1 passage
Search aliases: λ^ T lambda mathsf
Use in source · detailed_1.tex, line 108
Over R, the additive projection is P_add = I- fracλλ^ Tnormλ_2^2. The obstruction direction inside the additive hyperplane is μ_⊥=P_addμ. When μ_⊥≠0, the rank-one obstruction projector is P_obs = fracμ_⊥μ_⊥^ T normμ_⊥_2^2. This is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation direction, one obstruction direction, and the remaining null packets.
Exact mathematical passage
P_{\mathrm{add}} = I- \frac{\lambda\lambda^{\mathsf T}}{\norm{\lambda}_2^2}.
\lambda^{(a)}Use in source · 2 passages
Search aliases: λ^(a) lambda
Use in source · detailed_1.tex, line 253
Three-sector compression. Within the a-sector, define the integer row λ^(a) = (fraca v_p(a)p)_pmid a and let g_a=gcd{fraca v_p(a)p:pmid a}. Define g_b and g_c similarly. A one-row Smith normal form in each sector produces a unimodular coordinate system in which only one aggregate channel survives. The additive equation becomes g_at_a+g_bt_b-g_ct_c=0. The Wronskian becomes a g_bt_b-b g_at_a≠0. All remaining sector coordinates are lift gauges. They matter only because they determine the cost of re…
Exact mathematical passage
\lambda^{(a)} = \left(\frac{a v_p(a)}p\right)_{p\mid a}
\lambda^{\mathrm{ph}}Use in source · 1 passage
Search aliases: λ^ph lambda mathrm
Use in source · appendices.tex, line 33
Prime derivative. The additive row is λ=(1,75,-448). The phase row is λ^ph=(1,25,-64). The coarse shortest transverse packet x=(-2,6,1) satisfies -2+450=448. The Wronskian is W_x=1600=fracabcR. The coarse height is six.
Exact mathematical passage
\lambda^{\mathrm{ph}}=(1,25,-64).
\lambda_0Use in source · 2 passages
Search aliases: λ_0 lambda
Use in source · overview.tex, line 742
Their Legendre parameters are linked by the standard dyadic Landen transformation. If λ_0=(frac vu)^2, then the complementary child parameter is λ_1 = frac4uv(u+v)^2 = frac4sqrtλ_0(1+sqrtλ_0)^2. This is the exact two-isogeny refinement law behind the family.
Exact mathematical passage
\lambda_0=\left(\frac vu\right)^2,
\lambda_1Use in source · 1 passage
Search aliases: λ_1 lambda
Use in source · overview.tex, line 746
Their Legendre parameters are linked by the standard dyadic Landen transformation. If λ_0=(frac vu)^2, then the complementary child parameter is λ_1 = frac4uv(u+v)^2 = frac4sqrtλ_0(1+sqrtλ_0)^2. This is the exact two-isogeny refinement law behind the family.
Exact mathematical passage
\lambda_1 = \frac{4uv}{(u+v)^2} = \frac{4\sqrt{\lambda_0}}{(1+\sqrt{\lambda_0})^2}.
\Lambda_CUse in source · 1 passage
Search aliases: Λ_C Lambda
Use in source · appendices_more.tex, line 216
Thus Λ_C = {x:g_C· x≡0pmod C} has index C and contains the same fixed vector v. This proves that no bound on C can follow from the length or support of v alone.
Exact mathematical passage
\Lambda_C = \{x:g_C\cdot x\equiv0\pmod C\}
\Lambda_EUse in source · 2 passages
Search aliases: Λ_E Lambda
Use in source · overview.tex, line 759
The integral automorphic packet. The resulting integral automorphic object is Acal^sharp(E,ι) = ( f_E, Λ_E, ω_E, {q_p}_pmid N_E, ι, ϕ_E ). The components are: [leftmargin=1.5em] item the rational Hecke eigen-signal f_E item an integral Hecke, modular-symbol, or homology lattice Λ_E item the minimal N eron differential ω_E item local Tate-depth packets q_p item the marking ι of the three rational two-torsion differences item the modular synthesis map ϕ_E
Exact mathematical passage
\boxed{ \Acal^{\sharp}(E,\iota) = \left( f_E, \Lambda_E, \omega_E, \{q_p\}_{p\mid N_E}, \iota, \phi_E \right). }
\Lambda_pUse in source · 1 passage
Search aliases: Λ_p Lambda
Use in source · detailed_1.tex, line 623
A joint exponent-free two-row complex. The Smith obstruction is attached to the exponent-free phase, or shell-synthesis, rows rather than to the exponent-weighted derivative row. For every prime pmid abc, define Λ_p= a/p, · pmid a, b/p, · pmid b, -c/p, · pmid c, w_p= -ab/p, · pmid a, ab/p, · pmid b, 0, · pmid c. Consider M_a,b,c = Λ
Exact mathematical passage
\Lambda_p= \begin{cases} a/p,&p\mid a,\\ b/p,&p\mid b,\\ -c/p,&p\mid c, \end{cases} \qquad \mathsf w_p= \begin{cases} -ab/p,&p\mid a,\\ ab/p,&p\mid b,\\ 0,&p\mid c. \end{cases}
\lambda_pUse in source · 1 passage
Search aliases: λ_p lambda
Use in source · detailed_1.tex, line 62
The additive row. The condition d_x(a)+ d_x(b)= d_x(c) is one integer equation. Define λ_p= dfraca v_p(a)p, · pmid a, [6pt] dfracb v_p(b)p, · pmid b, [6pt] -dfracc v_p(c)p, · pmid c. Then λ· x=0. The additive derivative lattice is T_a,b,c = ker(λ)∩Z^S. Since the row is nonzero, its rank is |S|-1. The problem is not existence of additive derivatives. The problem is finding an additive derivative that remains transverse to the dependence direction introduced by a and b.
Exact mathematical passage
\lambda_p= \begin{cases} \dfrac{a v_p(a)}p,&p\mid a,\\[6pt] \dfrac{b v_p(b)}p,&p\mid b,\\[6pt] -\dfrac{c v_p(c)}p,&p\mid c. \end{cases}
\Lambda_UUse in source · 4 passages
Search aliases: Λ_U Lambda
Use in source · detailed_2.tex, line 313
Projector denominator as a congruence exponent. Let V=U⊕ W be a rational decomposition and let Λ⊂ V be an integral lattice. Let e_U be the rational projector onto U. Define Λ_U=Λ∩ U, Λ_W=Λ∩ W, and C_U=Λ/(Λ_U+Λ_W).
Exact mathematical passage
\Lambda_U=\Lambda\cap U, \qquad \Lambda_W=\Lambda\cap W,
\Lambda_WUse in source · 3 passages
Search aliases: Λ_W Lambda
Use in source · detailed_2.tex, line 313
Projector denominator as a congruence exponent. Let V=U⊕ W be a rational decomposition and let Λ⊂ V be an integral lattice. Let e_U be the rational projector onto U. Define Λ_U=Λ∩ U, Λ_W=Λ∩ W, and C_U=Λ/(Λ_U+Λ_W).
Exact mathematical passage
\Lambda_U=\Lambda\cap U, \qquad \Lambda_W=\Lambda\cap W,
\mu_{64}Use in source · 1 passage
Search aliases: μ_64 mu
Use in source · appendices.tex, line 166
Kummer and Fermat packets. The global Kummer map is κ_128:Z/64Z→ H^1(Q,μ_64). Since gcd(64,7)=1, it is injective.
Exact mathematical passage
\kappa_{128}:\Z/64\Z\to H^1(\Q,\mu_{64}).
\mu_MUse in source · 4 passages
Search aliases: μ_M mu
Use in source · detailed_3.tex, line 21
The global Kummer map. Kummer theory gives H^1(Q,μ_M) ≅ Q^×/(Q^×)^M. Define κ_c: Z/MZ ⟶ H^1(Q,μ_M), k⟼[c^k]. The valuation vector of c is (e_1,…,e_r) ∈ (Z/MZ)^r.
Exact mathematical passage
H^1(\Q,\mu_M) \cong \Q^\times/(\Q^\times)^M.
\mu_pUse in source · 1 passage
Search aliases: μ_p mu
Use in source · detailed_1.tex, line 93
The arithmetic Wronskian. Define W_x(a,b) =a d_x(b)-b d_x(a). Writing μ_p= -dfracv_p(a)p, · pmid a, [6pt] dfracv_p(b)p, · pmid b, [6pt] 0, · pmid c, we obtain W_x(a,b)=ab μ· x. An additive packet is transverse when W_x(a,b)≠0. It is degenerate when the two input terms remain derivative-proportional.
Exact mathematical passage
\mu_p= \begin{cases} -\dfrac{v_p(a)}p,&p\mid a,\\[6pt] \dfrac{v_p(b)}p,&p\mid b,\\[6pt] 0,&p\mid c, \end{cases}
\mu_\perpUse in source · 3 passages
Search aliases: μ_⊥ mu perp
Use in source · detailed_1.tex, line 115
Over R, the additive projection is P_add = I- fracλλ^ Tnormλ_2^2. The obstruction direction inside the additive hyperplane is μ_⊥=P_addμ. When μ_⊥≠0, the rank-one obstruction projector is P_obs = fracμ_⊥μ_⊥^ T normμ_⊥_2^2. This is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation direction, one obstruction direction, and the remaining null packets.
Exact mathematical passage
\mu_\perp=P_{\mathrm{add}}\mu.
\mu_\perp^{\mathsf T}Use in source · 1 passage
Search aliases: μ_⊥^ T mu perp mathsf
Use in source · detailed_1.tex, line 119
Over R, the additive projection is P_add = I- fracλλ^ Tnormλ_2^2. The obstruction direction inside the additive hyperplane is μ_⊥=P_addμ. When μ_⊥≠0, the rank-one obstruction projector is P_obs = fracμ_⊥μ_⊥^ T normμ_⊥_2^2. This is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation direction, one obstruction direction, and the remaining null packets.
Exact mathematical passage
P_{\mathrm{obs}} = \frac{\mu_\perp\mu_\perp^{\mathsf T}} {\norm{\mu_\perp}_2^2}.
\nuUse in source · 23 passages
Search aliases: ν nu
Use in source · appendices.tex, line 180
For the balanced ratio β=-frac1253, the signed exponent vector on the input primes is ν=(-1,3). Its Kummer order modulo 64 is full. Choosing 64th roots gives a point on X^64+Y^64=1. The coordinate height is fraclog12864. With the tensor-power metric induced from O(1), the full plane-model canonical scale is frac6164log128. For an arbitrary compatible Weil-height normalization, this formula holds up to an additive O(1) term. It records the line-bundle cost of retaining the complete Fermat curve.
Exact mathematical passage
\nu=(-1,3).
\nu_0Use in source · 4 passages
Search aliases: ν_0 nu
Use in source · appendices_final.tex, line 292
ν_0 · normalized input exponent vector
Exact mathematical passage
\nu_0
\nu_{a,b,c}Use in source · 1 passage
Search aliases: ν_a,b,c nu
Use in source · detailed_2.tex, line 70
The three-spike texorpdfstringp-adicp-adic coherence signal. Represent the additive triple by ν_a,b,c=δ_0+δ_a+δ_c. The pairwise differences are a, b, and c. At a prime dividing exactly one of these integers, exactly one pair of spikes is p-adically close. The depth of that edge is the corresponding valuation.
Exact mathematical passage
\nu_{a,b,c}=\delta_0+\delta_a+\delta_c.
\nu_qUse in source · 15 passages
Search aliases: ν_q nu
Use in source · appendices_final.tex, line 15
Start with the signed input exponent vector ν=(ν_q)_qmid ab, ν_q=v_q(b)-v_q(a). Let K_p be the p-primary part of the common Kummer factor. Divide the exponent vector and the local modulus by this common power. The normalized vector is primitive relative to the remaining modulus.
Exact mathematical passage
\nu=(\nu_q)_{q\mid ab}, \qquad \nu_q=v_q(b)-v_q(a).
\xi_iUse in source · 7 passages
Search aliases: ξ_i xi
Use in source · appendices_final.tex, line 225
Local CRT lift in closed form. Let r=∏_i=1^mp_i, w_i=r/p_i. The equation ∑_iw_ix_i=t has a solution because gcd(w_1,…,w_m)=1. Modulo p_i, x_i≡ t w_i^-1pmodp_i. Let ξ_i be the centered residue. Then ∑_iw_iξ_i-t is divisible by r. Set q=fract-∑_iw_iξ_ir. Writing x_i=ξ_i+p_i z_i gives ∑_i z_i=q.
Exact mathematical passage
\xi_i
\xi_pUse in source · 2 passages
Search aliases: ξ_p xi
Use in source · detailed_1.tex, line 16
The prime-coordinate module. Let a+b=c, gcd(a,b)=1. The three integers are pairwise coprime. Write S=S_asqcup S_bsqcup S_c ={p:pmid abc}. The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. For a positive integer n, define the universal multiplicative differential dn = n∑_pmid nfracv_p(n)p ξ_p. Every coefficient is an integer because pmid n. The map obeys the exact Leibniz law d(mn)=m dn+n dm. The verification is local. For each prime p, v_p(mn)=v_p(m)+v_p(n), so the…
Exact mathematical passage
\Omega_S=\bigoplus_{p\in S}\Z\xi_p.
\PiUse in source · 2 passages
Search aliases: Π Pi
Use in source · detailed_3.tex, line 256
A fixed path packet is Π_≤3(z) = ( log z, log(1-z), L_2(z), Li_3(z), Li_3(1-z) ). Its coordinate count is independent of M.
Exact mathematical passage
\Pi_{\le3}(z) = \left( \log z, \log(1-z), L_2(z), \operatorname{Li}_3(z), \operatorname{Li}_3(1-z) \right).
\piUse in source · 10 passages
Search aliases: π pi
Use in source · appendices_final.tex, line 39
The local residual character is χ_p(x) = exp( frac2π ip^χ_pg_p· x ). Its kernel contains the normalized exponent vector. The local factors are therefore K_p, W_p=p^w_p, C_p=p^χ_p. At two, one sign layer ϵ_2,p is removed before the weighted logarithmic row is formed.
Exact mathematical passage
\chi_p(x) = \exp\left( \frac{2\pi i}{p^{\chi_p}}g_p\cdot x \right).
\pi^2Use in source · 5 passages
Search aliases: π^2 pi
Use in source · detailed_2.tex, line 173
Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be an optimal modular parametrization, and let ϕ_E^*ω_E = 2π i c_E f_E(z) dz. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_E^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area factor proportional to 1/c. Thus the height c can be written as a shape-corrected synthesis gain involving modular degree, Manin normalization, and Pete…
Exact mathematical passage
\boxed{ \deg(\phi_E)A(E) = 4\pi^2 c_E^2 \norm{f_E}_{\mathrm{Pet}}^2. }
\Pi_{\le3}Use in source · 2 passages
Search aliases: Π_≤3 Pi le
Use in source · detailed_3.tex, line 256
A fixed path packet is Π_≤3(z) = ( log z, log(1-z), L_2(z), Li_3(z), Li_3(1-z) ). Its coordinate count is independent of M.
Exact mathematical passage
\Pi_{\le3}(z) = \left( \log z, \log(1-z), L_2(z), \operatorname{Li}_3(z), \operatorname{Li}_3(1-z) \right).
\rhoUse in source · 3 passages
Search aliases: ρ rho
Use in source · detailed_1.tex, line 277
For each term define ρ_a(t) = min{ normx^(a)_∞: λ^(a)· x^(a)=g_at }, with analogous functions for b and c. Then Θ(a,b,c) = min max{ρ_a(t_a),ρ_b(t_b),ρ_c(t_c)} over aggregate solutions satisfying additivity and transversality.
Exact mathematical passage
\rho_a(t) = \min\left\{ \norm{x^{(a)}}_\infty: \lambda^{(a)}\cdot x^{(a)}=g_at \right\},
\rho_aUse in source · 2 passages
Search aliases: ρ_a rho
Use in source · detailed_1.tex, line 277
For each term define ρ_a(t) = min{ normx^(a)_∞: λ^(a)· x^(a)=g_at }, with analogous functions for b and c. Then Θ(a,b,c) = min max{ρ_a(t_a),ρ_b(t_b),ρ_c(t_c)} over aggregate solutions satisfying additivity and transversality.
Exact mathematical passage
\rho_a(t) = \min\left\{ \norm{x^{(a)}}_\infty: \lambda^{(a)}\cdot x^{(a)}=g_at \right\},
\rho_bUse in source · 1 passage
Search aliases: ρ_b rho
Use in source · detailed_1.tex, line 286
For each term define ρ_a(t) = min{ normx^(a)_∞: λ^(a)· x^(a)=g_at }, with analogous functions for b and c. Then Θ(a,b,c) = min max{ρ_a(t_a),ρ_b(t_b),ρ_c(t_c)} over aggregate solutions satisfying additivity and transversality.
Exact mathematical passage
\Theta(a,b,c) = \min \max\{\rho_a(t_a),\rho_b(t_b),\rho_c(t_c)\}
\rho_cUse in source · 1 passage
Search aliases: ρ_c rho
Use in source · detailed_1.tex, line 286
For each term define ρ_a(t) = min{ normx^(a)_∞: λ^(a)· x^(a)=g_at }, with analogous functions for b and c. Then Θ(a,b,c) = min max{ρ_a(t_a),ρ_b(t_b),ρ_c(t_c)} over aggregate solutions satisfying additivity and transversality.
Exact mathematical passage
\Theta(a,b,c) = \min \max\{\rho_a(t_a),\rho_b(t_b),\rho_c(t_c)\}
\rho_u^{\mathrm{sh}}Use in source · 1 passage
Search aliases: ρ_u^sh rho mathrm
Use in source · detailed_1.tex, line 577
The shell lift gauge is ρ_u^sh(t) = min{ normy_∞: ∑_pmid u fracr_up ∑_r y_p,r=t }. At the aggregate level, P_at_a+P_bt_b=P_ct_c, and W_y(a,b) = P_aP_b(r_at_b-r_bt_a). The radical divisor is attained sharply when r_at_b-r_bt_a=P_c. When a,b>1, all sharp aggregate solutions form a one-parameter family t_a(n) · =t_a^(0)+nr_a,
Exact mathematical passage
\rho_u^{\mathrm{sh}}(t) = \min\left\{ \norm{y}_\infty: \sum_{p\mid u} \frac{r_u}{p} \sum_r y_{p,r}=t \right\}.
\sigma_{a,b,c}Use in source · 1 passage
Search aliases: σ_a,b,c sigma
Use in source · detailed_2.tex, line 418
The three-cusp section. View a primitive triple as the integral section σ_a,b,c: SpecZ⟶ P^1_Z, σ=[a:c]. The pullbacks of the boundary divisors 0, 1, and ∞ are div(a), div(b), div(c). For the output cusp, let C=div(c), C_red=div(R_c). Define the two-term contact complex J_c = [ O(-C)⟶ O(-C_red)]. On global sections, J_c = [cZ↪ R_cZ] ≃ [Zxrightarrowc/R_cZ].
Exact mathematical passage
\sigma_{a,b,c}: \Spec\Z\longrightarrow\mathbb P^1_\Z, \qquad \sigma=[a:c].
\sigma_uUse in source · 1 passage
Search aliases: σ_u sigma
Use in source · detailed_1.tex, line 741
The stable real height. Relax the integral packet to a real vector in the additive hyperplane. The exact continuous width can be computed from the three sector weights. Let σ_u=∑_pmid ufrac1p and Δ(a,b,c) = min{ σ_a+σ_b, frac cb(σ_a+σ_c), frac ca(σ_b+σ_c) }. Then overline H = fraccRΔ(a,b,c). The discrete master inequality becomes c≤ RΔΦ. Since Φ=overline H k_*Γ for the optimal obstruction multiple k_* and an integrality factor Γ≥1, ΔΦ = frac cR k_*Γ. This is the no-escape identity. Optimizing th…
Exact mathematical passage
\sigma_u=\sum_{p\mid u}\frac1p
\phiUse in source · 8 passages
Search aliases: ϕ phi
Use in source · detailed_2.tex, line 155
Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be an optimal modular parametrization, and let ϕ_E^*ω_E = 2π i c_E f_E(z) dz. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_E^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area factor proportional to 1/c. Thus the height c can be written as a shape-corrected synthesis gain involving modular degree, Manin normalization, and Pete…
Exact mathematical passage
\phi_E:X_0(N_E)\longrightarrow E
\varphiUse in source · 7 passages
Search aliases: φ varphi
Use in source · detailed_2.tex, line 201
dmid e_pp. Since ∑_dmid eφ(d)=e, we have the perfect reconstruction identity n = ∏_d≥1 B_d(n)^φ(d). The first packet is B_1(n)=rad(n). The powerful part is fracnrad(n) = ∏_d≥2 B_d(n)^φ(d). This is a multiplicative polyphase decomposition indexed by exact character orders.
Exact mathematical passage
\sum_{d\mid e}\varphi(d)=e,
\phi_EUse in source · 6 passages
Search aliases: ϕ_E phi
Use in source · detailed_2.tex, line 155
Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be an optimal modular parametrization, and let ϕ_E^*ω_E = 2π i c_E f_E(z) dz. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_E^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area factor proportional to 1/c. Thus the height c can be written as a shape-corrected synthesis gain involving modular degree, Manin normalization, and Pete…
Exact mathematical passage
\phi_E:X_0(N_E)\longrightarrow E
\phi_E^*Use in source · 2 passages
Search aliases: ϕ_E^* phi
Use in source · detailed_2.tex, line 159
Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be an optimal modular parametrization, and let ϕ_E^*ω_E = 2π i c_E f_E(z) dz. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_E^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area factor proportional to 1/c. Thus the height c can be written as a shape-corrected synthesis gain involving modular degree, Manin normalization, and Pete…
Exact mathematical passage
\phi_E^*\omega_E = 2\pi i\,c_E f_E(z)\,dz.
\PsiUse in source · 11 passages
Search aliases: Ψ Psi
Use in source · detailed_2.tex, line 9
Prime support as bandwidth and valuation as persistence. For n=∏_p p^e_p, define the prime-shell signal Ψ_n(p,r) = sqrtlog p 1_0≤ r<e_p. Its total energy is normΨ_n_2^2 = ∑_p e_plog p = log n. Let P_act retain only the first shell in each active prime channel. Then normP_actΨ_n_2^2 = ∑_pmid nlog p = lograd(n). Thus the radical is activated bandwidth, while the difference log n-lograd(n) is repeated persistence depth.
Exact mathematical passage
\Psi_n(p,r) = \sqrt{\log p}\,\mathbf 1_{0\le r<e_p}.
\Psi_aUse in source · 1 passage
Search aliases: Ψ_a Psi
Use in source · detailed_2.tex, line 37
For a primitive triple, the direct-sum signal Ψ_a,b,c = Ψ_a⊕Ψ_b⊕Ψ_c has disjoint prime support in the three sectors. The standard quality becomes the exact energy ratio q(a,b,c) = fraclog clograd(abc) = fracnormP_cΨ_a,b,c_2^2 normP_actΨ_a,b,c_2^2. High quality means large output persistence relative to activated global bandwidth.
Exact mathematical passage
\Psi_{a,b,c} = \Psi_a\oplus\Psi_b\oplus\Psi_c
\Psi_{a,b,c}Use in source · 2 passages
Search aliases: Ψ_a,b,c Psi
Use in source · detailed_2.tex, line 37
For a primitive triple, the direct-sum signal Ψ_a,b,c = Ψ_a⊕Ψ_b⊕Ψ_c has disjoint prime support in the three sectors. The standard quality becomes the exact energy ratio q(a,b,c) = fraclog clograd(abc) = fracnormP_cΨ_a,b,c_2^2 normP_actΨ_a,b,c_2^2. High quality means large output persistence relative to activated global bandwidth.
Exact mathematical passage
\Psi_{a,b,c} = \Psi_a\oplus\Psi_b\oplus\Psi_c
\Psi_bUse in source · 1 passage
Search aliases: Ψ_b Psi
Use in source · detailed_2.tex, line 37
For a primitive triple, the direct-sum signal Ψ_a,b,c = Ψ_a⊕Ψ_b⊕Ψ_c has disjoint prime support in the three sectors. The standard quality becomes the exact energy ratio q(a,b,c) = fraclog clograd(abc) = fracnormP_cΨ_a,b,c_2^2 normP_actΨ_a,b,c_2^2. High quality means large output persistence relative to activated global bandwidth.
Exact mathematical passage
\Psi_{a,b,c} = \Psi_a\oplus\Psi_b\oplus\Psi_c
\Psi_cUse in source · 1 passage
Search aliases: Ψ_c Psi
Use in source · detailed_2.tex, line 37
For a primitive triple, the direct-sum signal Ψ_a,b,c = Ψ_a⊕Ψ_b⊕Ψ_c has disjoint prime support in the three sectors. The standard quality becomes the exact energy ratio q(a,b,c) = fraclog clograd(abc) = fracnormP_cΨ_a,b,c_2^2 normP_actΨ_a,b,c_2^2. High quality means large output persistence relative to activated global bandwidth.
Exact mathematical passage
\Psi_{a,b,c} = \Psi_a\oplus\Psi_b\oplus\Psi_c
\Psi_nUse in source · 9 passages
Search aliases: Ψ_n Psi
Use in source · detailed_2.tex, line 9
Prime support as bandwidth and valuation as persistence. For n=∏_p p^e_p, define the prime-shell signal Ψ_n(p,r) = sqrtlog p 1_0≤ r<e_p. Its total energy is normΨ_n_2^2 = ∑_p e_plog p = log n. Let P_act retain only the first shell in each active prime channel. Then normP_actΨ_n_2^2 = ∑_pmid nlog p = lograd(n). Thus the radical is activated bandwidth, while the difference log n-lograd(n) is repeated persistence depth.
Exact mathematical passage
\Psi_n(p,r) = \sqrt{\log p}\,\mathbf 1_{0\le r<e_p}.
\OmegaUse in source · 1 passage
Search aliases: Ω Omega
Use in source · detailed_1.tex, line 16
The prime-coordinate module. Let a+b=c, gcd(a,b)=1. The three integers are pairwise coprime. Write S=S_asqcup S_bsqcup S_c ={p:pmid abc}. The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. For a positive integer n, define the universal multiplicative differential dn = n∑_pmid nfracv_p(n)p ξ_p. Every coefficient is an integer because pmid n. The map obeys the exact Leibniz law d(mn)=m dn+n dm. The verification is local. For each prime p, v_p(mn)=v_p(m)+v_p(n), so the…
Exact mathematical passage
\Omega_S=\bigoplus_{p\in S}\Z\xi_p.
\omegaUse in source · 13 passages
Search aliases: ω omega
Use in source · appendices_final.tex, line 84
Odd-prime coefficient depth. Let q≠ p be prime. Write q=ω(q)⟨ q⟩_p, where ω(q) is the Teichm uller factor. Since ⟨ q⟩_p^p-1=q^p-1, we have log_p⟨ q⟩_p = frac1p-1log_p(q^p-1). If s=v_p(q^p-1-1)>0, then v_p(log_p(q^p-1))=s. Therefore v_p(frac1plog_p⟨ q⟩_p) =s-1. Truncating at the available contact depth gives v_p(ℓ_p,e(q)) = min{e-1,s-1}.
Exact mathematical passage
q=\omega(q)\langle q\rangle_p,
\omega_EUse in source · 6 passages
Search aliases: ω_E omega
Use in source · detailed_2.tex, line 159
Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be an optimal modular parametrization, and let ϕ_E^*ω_E = 2π i c_E f_E(z) dz. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_E^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area factor proportional to 1/c. Thus the height c can be written as a shape-corrected synthesis gain involving modular degree, Manin normalization, and Pete…
Exact mathematical passage
\phi_E^*\omega_E = 2\pi i\,c_E f_E(z)\,dz.
\Omega_SUse in source · 1 passage
Search aliases: Ω_S Omega
Use in source · detailed_1.tex, line 16
The prime-coordinate module. Let a+b=c, gcd(a,b)=1. The three integers are pairwise coprime. Write S=S_asqcup S_bsqcup S_c ={p:pmid abc}. The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. For a positive integer n, define the universal multiplicative differential dn = n∑_pmid nfracv_p(n)p ξ_p. Every coefficient is an integer because pmid n. The map obeys the exact Leibniz law d(mn)=m dn+n dm. The verification is local. For each prime p, v_p(mn)=v_p(m)+v_p(n), so the…
Exact mathematical passage
\Omega_S=\bigoplus_{p\in S}\Z\xi_p.
\ldotsUse in source · 20 passages
Search aliases: … ldots
Use in source · appendices_final.tex, line 180
One-row Smith normal form. For an integer row a=(a_1,…,a_s), let d=gcd(a_1,…,a_s). There exists a unimodular matrix U∈GL_s(Z) such that aU=(d,0,…,0). Therefore the image of the row is dZ, the kernel has rank s-1, and the quotient by the kernel is infinite cyclic scaled by d.
Exact mathematical passage
a=(a_1,\ldots,a_s),
\ellUse in source · 38 passages
Search aliases: ℓ ell
Use in source · appendices_final.tex, line 23
For odd p, form the logarithmic row ℓ_p,e(q) = frac1plog_p⟨ q⟩_p pmodp^e-1. Let w_p = min_qmid ab v_p(ℓ_p,e(q)) after Kummer normalization. Divide the row and the modulus by p^w_p. The remaining row is primitive modulo the residual power p^χ_p.
Exact mathematical passage
\ell_{p,e}(q) = \frac1p\log_p\langle q\rangle_p \pmod{p^{e-1}}.
\ell^{-1}Use in source · 1 passage
Search aliases: ℓ^-1 ell
Use in source · overview.tex, line 639
No-go theorem: linear Hecke averaging. In the local Tate model, the ℓ+1 isogeny branches multiply contact depth by ℓ, underbraceℓ^-1,…,ℓ^-1_ℓ branches. Uniform averaging preserves the linear mean exactly. Typical logarithmic depth falls, but rare deep branches retain the mean. Linear Hecke averaging is redistribution rather than contraction.
Exact mathematical passage
\ell, \quad \underbrace{\ell^{-1},\ldots,\ell^{-1}}_{\ell\text{ branches}}.
\ell^{2k-n}Use in source · 2 passages
Search aliases: ℓ^2k-n ell
Use in source · appendices_more.tex, line 188
The mean-preserving Hecke cascade. At a Tate curve, an ℓ-isogeny packet contains one branch multiplying depth by ℓ and ℓ branches multiplying depth by 1/ℓ. Under uniform averaging, fracℓ+ℓ(1/ℓ)ℓ+1=1. After n independent refinements, if the deep branch is chosen k times, the normalized depth is ℓ^2k-n with multiplicity binom nkℓ^n-k. Therefore ∑_k=0^n fracbinom nkℓ^n-k(ℓ+1)^n ℓ^2k-n =1. The first moment is an exact martingale. The mean logarithmic depth is -nfracℓ-1ℓ+1logℓ. Most branches become s…
Exact mathematical passage
\ell^{2k-n}
\ell^nUse in source · 1 passage
Search aliases: ℓ^n ell
Use in source · detailed_2.tex, line 408
Two prime indices. The persistence is indexed by conductor primes qmid abc. Congruence depth is indexed by coefficient primes ℓ. The natural local signal is a matrix D_f(ℓ,q). At a split Tate prime q≠ℓ, scalarity modulo ℓ^n forces n≤min{v_ℓ(q-1),v_ℓ(v_q(q_E))}. These off-diagonal channels carry tame congruence information. They do not naturally reproduce the full base-q factor q^e-1. This mismatch motivates the later filtered contact correction and the diagonal coefficient-prime channel.
Exact mathematical passage
\ell^n
\ell^{n-k}Use in source · 2 passages
Search aliases: ℓ^n-k ell
Use in source · appendices_more.tex, line 192
The mean-preserving Hecke cascade. At a Tate curve, an ℓ-isogeny packet contains one branch multiplying depth by ℓ and ℓ branches multiplying depth by 1/ℓ. Under uniform averaging, fracℓ+ℓ(1/ℓ)ℓ+1=1. After n independent refinements, if the deep branch is chosen k times, the normalized depth is ℓ^2k-n with multiplicity binom nkℓ^n-k. Therefore ∑_k=0^n fracbinom nkℓ^n-k(ℓ+1)^n ℓ^2k-n =1. The first moment is an exact martingale. The mean logarithmic depth is -nfracℓ-1ℓ+1logℓ. Most branches become s…
Exact mathematical passage
\binom nk\ell^{n-k}.
\ell^{\alpha_\ell}Use in source · 2 passages
Search aliases: ℓ^α_ℓ ell alpha ell
Use in source · detailed_1.tex, line 421
Dyadic and odd-adic primary lifts. Factor e=2^α∏_ℓ oddℓ^α_ℓ. The cokernel of the valuation block is Z/eZ ≅ Z/2^αZ × ∏_ℓ odd Z/ℓ^α_ℓZ. The primary factors specify the local lifting problems for the aggregate multiplicity channel. Its 2-primary component must be treated 2-adically, while the odd components admit the corresponding odd-adic lifts. The phase channels require no local denominator correction.
Exact mathematical passage
e=2^\alpha\prod_{\ell\text{ odd}}\ell^{\alpha_\ell}.
\ell_{2,e}Use in source · 4 passages
Search aliases: ℓ_2,e ell
Use in source · appendices_final.tex, line 129
Dyadic coefficient depth. For odd q, separate the sign by writing q=(-1)^(q-1)/2⟨ q⟩_2, ⟨ q⟩_2∈1+4Z_2. Then ⟨ q⟩_2^2=q^2. The weighted coefficient is ℓ_2,e(q) = frac14log_2⟨ q⟩_2. If s=v_2(q^2-1), then v_2(ℓ_2,e(q))=s-3, truncated at e-2. The missing first layer is the separate sign bit. This is the final form of the weighted dyadic branch.
Exact mathematical passage
\ell_{2,e}(q) = \frac14\log_2\langle q\rangle_2.
\ell_iUse in source · 2 passages
Search aliases: ℓ_i ell
Use in source · detailed_1.tex, line 939
p_ix_i=t. An empty unit sector has no prime coordinates and is feasible exactly when t=0. Modulo p_i, the coordinate x_i is forced: x_i ≡ t(frac rp_i)^-1 pmodp_i. Choose the centered residue ξ_i and write x_i=ξ_i+p_i z_i. Substitution gives one quotient equation ∑_i z_i=q(t). For a box radius H, each z_i lies in the interval ℓ_i(H) = lceilfrac-H-ξ_ip_irceil ≤ z_i≤ lfloorfracH-ξ_ip_irfloor =u_i(H). The local lift exists exactly when ∑_iℓ_i(H) ≤ q(t)≤ ∑_i u_i(H). This is an exact interval criterio…
Exact mathematical passage
\ell_i(H) = \left\lceil\frac{-H-\xi_i}{p_i}\right\rceil \le z_i\le \left\lfloor\frac{H-\xi_i}{p_i}\right\rfloor =u_i(H).
\ell_pUse in source · 1 passage
Search aliases: ℓ_p ell
Use in source · tail.tex, line 130
Simultaneous Wieferich and regulator theory. This is the closest immediate field because the cross-prime logarithmic matrix is already the main object. For finite sets of primes P and Q, define Lcal_P,Q = ( ℓ_p(q) )_p∈ P,q∈ Q. Natural theorem targets include: [leftmargin=1.5em] item a rank criterion for the matrix modulo prescribed prime powers item an exact decomposition of depth into entry defects, minor defects, and relation-specific cancellation item a density-zero or power-saving theorem fo…
Exact mathematical passage
\Lcal_{P,Q} = \left( \ell_p(q) \right)_{p\in P,q\in Q}.
\ell_{p,e}Use in source · 6 passages
Search aliases: ℓ_p,e ell
Use in source · appendices_final.tex, line 23
For odd p, form the logarithmic row ℓ_p,e(q) = frac1plog_p⟨ q⟩_p pmodp^e-1. Let w_p = min_qmid ab v_p(ℓ_p,e(q)) after Kummer normalization. Divide the row and the modulus by p^w_p. The remaining row is primitive modulo the residual power p^χ_p.
Exact mathematical passage
\ell_{p,e}(q) = \frac1p\log_p\langle q\rangle_p \pmod{p^{e-1}}.
\toUse in source · 4 passages
Search aliases: → to
Use in source · appendices.tex, line 166
Kummer and Fermat packets. The global Kummer map is κ_128:Z/64Z→ H^1(Q,μ_64). Since gcd(64,7)=1, it is injective.
Exact mathematical passage
\kappa_{128}:\Z/64\Z\to H^1(\Q,\mu_{64}).
\mapstoUse in source · 3 passages
Search aliases: ↦ mapsto
Use in source · appendices_final.tex, line 259
Polyphase circulant determinant. For the valuation word V_e(T)=1+T+⋯+T^e-1 in A_N=Z[T]/(T^N-1), multiplication is circulant. Its determinant is ∏_ζ^N=1V_e(ζ). Let I_N be the augmentation ideal of A_N. When gcd(e,N)=1, the nontrivial roots are permuted by ζ↦ζ^e, so multiplication by V_e has determinant one on the integral lattice I_N and is therefore an automorphism of it. On A_N/I_N≅Z, the map is multiplication by e. The exact sequence for I_N⊂ A_N then gives cokernel Z/eZ and Smith form diag(1,…
Exact mathematical passage
\zeta\mapsto\zeta^e
\hookrightarrowUse in source · 2 passages
Search aliases: ↪ hookrightarrow
Use in source · appendices.tex, line 137
Filtered contact and Witt packet. The output contact complex is [128Z↪2Z] ≃ [Zxrightarrow64Z]. Its cohomology is Jcal_c≅Z/64Z. The associated graded is (F_2)^6. The six binary layers are joined by five nontrivial carry extensions. A cyclic target of order 64 can carry the filtered source exactly, while no cyclic target can contain the split rank-six source injectively.
Exact mathematical passage
[128\Z\hookrightarrow2\Z] \simeq [\Z\xrightarrow{64}\Z].
\partialUse in source · 2 passages
Search aliases: ∂ partial
Use in source · detailed_1.tex, line 345
Replacing one exponent by a valuation word. The coefficient v_p(n) compresses a run of valuation shells into one scalar. For a prime power p^e, define the valuation word V_e(T)=1+T+⋯+T^e-1. It obeys the concatenation identity V_e+f(T)=V_e(T)+T^eV_f(T). The refined partial derivative ∂_p,T(n) = frac npV_v_p(n)(T) therefore satisfies the twisted product rule ∂_p,T(mn) = n∂_p,T(m) + mT^v_p(m)∂_p,T(n). At T=1, the ordinary arithmetic partial derivative is recovered.
Exact mathematical passage
\partial_{p,T}(n) = \frac npV_{v_p(n)}(T)
\partial_{p,T}Use in source · 2 passages
Search aliases: ∂_p,T partial
Use in source · detailed_1.tex, line 345
Replacing one exponent by a valuation word. The coefficient v_p(n) compresses a run of valuation shells into one scalar. For a prime power p^e, define the valuation word V_e(T)=1+T+⋯+T^e-1. It obeys the concatenation identity V_e+f(T)=V_e(T)+T^eV_f(T). The refined partial derivative ∂_p,T(n) = frac npV_v_p(n)(T) therefore satisfies the twisted product rule ∂_p,T(mn) = n∂_p,T(m) + mT^v_p(m)∂_p,T(n). At T=1, the ordinary arithmetic partial derivative is recovered.
Exact mathematical passage
\partial_{p,T}(n) = \frac npV_{v_p(n)}(T)
\varnothingUse in source · 1 passage
Search aliases: ∅ varnothing
Use in source · detailed_1.tex, line 966
[Moving-interval theorem] For fixed obstruction multiple k and radius H, H(k)≤ H if and only if I_a(H,k) ∩ I_b(H,k) ∩ I_c(H,k) ∩Z ≠∅.
Exact mathematical passage
I_a(H,k) \cap I_b(H,k) \cap I_c(H,k) \cap\Z \neq\varnothing.
\prod_{d\ge1}Use in source · 2 passages
Search aliases: ∏_d≥1 prod ge
Use in source · detailed_2.tex, line 205
dmid e_pp. Since ∑_dmid eφ(d)=e, we have the perfect reconstruction identity n = ∏_d≥1 B_d(n)^φ(d). The first packet is B_1(n)=rad(n). The powerful part is fracnrad(n) = ∏_d≥2 B_d(n)^φ(d). This is a multiplicative polyphase decomposition indexed by exact character orders.
Exact mathematical passage
\boxed{ n = \prod_{d\ge1} B_d(n)^{\varphi(d)}. }
\prod_{d\ge2}Use in source · 3 passages
Search aliases: ∏_d≥2 prod ge
Use in source · detailed_2.tex, line 218
dmid e_pp. Since ∑_dmid eφ(d)=e, we have the perfect reconstruction identity n = ∏_d≥1 B_d(n)^φ(d). The first packet is B_1(n)=rad(n). The powerful part is fracnrad(n) = ∏_d≥2 B_d(n)^φ(d). This is a multiplicative polyphase decomposition indexed by exact character orders.
Exact mathematical passage
\boxed{ \frac{n}{\rad(n)} = \prod_{d\ge2} B_d(n)^{\varphi(d)}. }
\prod_iUse in source · 4 passages
Search aliases: ∏_i prod
Use in source · detailed_1.tex, line 915
Local CRT normal form. Fix one sector with primes p_1,…,p_m and radical r=∏_i p_i. For a nonempty sector, the local equation is ∑_ifrac rp_ix_i=t. An empty unit sector has no prime coordinates and is feasible exactly when t=0. Modulo p_i, the coordinate x_i is forced: x_i ≡ t(frac rp_i)^-1 pmodp_i. Choose the centered residue ξ_i and write x_i=ξ_i+p_i z_i. Substitution gives one quotient equation ∑_i z_i=q(t). For a box radius H, each z_i lies in the interval ℓ_i(H) = lceilfrac-H-ξ_ip_irceil ≤ z…
Exact mathematical passage
r=\prod_i p_i.
\prod_{i=1}^{m}Use in source · 1 passage
Search aliases: ∏_i=1^m prod
Use in source · appendices_final.tex, line 208
Local CRT lift in closed form. Let r=∏_i=1^mp_i, w_i=r/p_i. The equation ∑_iw_ix_i=t has a solution because gcd(w_1,…,w_m)=1. Modulo p_i, x_i≡ t w_i^-1pmodp_i. Let ξ_i be the centered residue. Then ∑_iw_iξ_i-t is divisible by r. Set q=fract-∑_iw_iξ_ir. Writing x_i=ξ_i+p_i z_i gives ∑_i z_i=q.
Exact mathematical passage
r=\prod_{i=1}^{m}p_i, \qquad w_i=r/p_i.
\prod_{i=1}^{r}Use in source · 1 passage
Search aliases: ∏_i=1^r prod
Use in source · detailed_3.tex, line 5
The contact modulus. Let c=∏_i=1^rp_i^e_i, R_c=∏_i p_i, M=fraccR_c =∏_i p_i^e_i-1. The filtered output contact packet is the cyclic group Z/MZ. The first global question is whether this packet can be realized by one arithmetic class without losing most of its order.
Exact mathematical passage
c=\prod_{i=1}^{r}p_i^{e_i}, \qquad R_c=\prod_i p_i, \qquad M=\frac{c}{R_c} =\prod_i p_i^{e_i-1}.
\prod_{j=1}^{r}Use in source · 1 passage
Search aliases: ∏_j=1^r prod
Use in source · detailed_2.tex, line 275
Atkin--Lehner Walsh packets. Let W_1,…,W_r be commuting Atkin--Lehner involutions. For a sign vector ε∈{±1}^r, define A_ε = ∏_j=1^r(I+ε_jW_j). Then A_ε^2=2^rA_ε, A_ε A_ε'=0 (ε≠ε'), and ∑_ε A_ε=2^rI. The unnormalized analysis packets are integral. The common synthesis denominator is 2^r, which is subpolynomial in a squarefree conductor. Local sign separation is therefore not the large persistence carrier.
Exact mathematical passage
A_\varepsilon = \prod_{j=1}^{r}(I+\varepsilon_jW_j).
\prod_pUse in source · 5 passages
Search aliases: ∏_p prod
Use in source · detailed_2.tex, line 5
Prime support as bandwidth and valuation as persistence. For n=∏_p p^e_p, define the prime-shell signal Ψ_n(p,r) = sqrtlog p 1_0≤ r<e_p. Its total energy is normΨ_n_2^2 = ∑_p e_plog p = log n. Let P_act retain only the first shell in each active prime channel. Then normP_actΨ_n_2^2 = ∑_pmid nlog p = lograd(n). Thus the radical is activated bandwidth, while the difference log n-lograd(n) is repeated persistence depth.
Exact mathematical passage
n=\prod_p p^{e_p},
\prod_{p,r}Use in source · 1 passage
Search aliases: ∏_p,r prod
Use in source · detailed_1.tex, line 832
Gauge collapse. Within one sector, the product ∏_p,r(p+T_p,r) depends only on the aggregate product of its shell variables. Internal phase directions that preserve this product form a gauge torus. The full shell equation factors through three aggregate sector products A, B, C, with one linear relation aA+bB=cC. Thus all formal complexity collapses to a two-dimensional aggregate plane plus gauge directions. This explains why local refinements repeatedly fail to create a new global obstruction.
Exact mathematical passage
\prod_{p,r}(p+T_{p,r})
\prod_{p^e\parallel abc}Use in source · 1 passage
Search aliases: ∏_p^eparallel abc prod parallel
Use in source · detailed_1.tex, line 166
The prime supports of a, b, and c are disjoint. Multiplying all local divisibilities gives ∏_p^eparallel abcp^e-1 = fracabcR mid W_x(a,b).
Exact mathematical passage
\prod_{p^e\parallel abc}p^{e-1} = \frac{abc}{R} \mid W_x(a,b).
\prod_{p^e\parallel c}Use in source · 2 passages
Search aliases: ∏_p^eparallel c prod parallel
Use in source · detailed_2.tex, line 354
Why one cyclic quotient is not enough for split persistence. The split persistence module P_c ≅ ⊕_p^eparallel c (Z/pZ)^e-1 has large p-rank. The p-torsion of a cyclic target has rank at most one. Therefore every map from P_c to a cyclic group has a forced kernel of order at least ∏_p^eparallel cp^max(e-2,0). This is a true obstruction to the split source. It motivates the Hecke cotangent module p/ p^2, which can retain multiple independent congruence directions before cyclic compression.
Exact mathematical passage
\prod_{p^e\parallel c}p^{\max(e-2,0)}.
\prod_{p^e\parallel n}Use in source · 1 passage
Search aliases: ∏_p^eparallel n prod parallel
Use in source · detailed_1.tex, line 782
The full shell deformation. Introduce one variable for every valuation shell and define mathscr J_n( T) = ∏_p^eparallel n ∏_r=0^e-1(p+T_p,r). At T=0, mathscr J_n(0)=n. For a+b=c, set F_a,b,c( T) = mathscr J_a( T) + mathscr J_b( T) - mathscr J_c( T). The shell gradient has entries frac ap, frac bp, -frac cp repeated according to valuation depth. The gcd of all entries is one.
Exact mathematical passage
\mathscr J_n(\mathbf T) = \prod_{p^e\parallel n} \prod_{r=0}^{e-1}(p+T_{p,r}).
\prod_{p\mid n,\ d\mid e_p}Use in source · 1 passage
Search aliases: ∏_pmid n, dmid e_p prod mid mid
Use in source · overview.tex, line 480
Divisor-polyphase perfect reconstruction. For n=∏_pp^e_p, define B_d(n)=∏_pmid n, dmid e_pp. Then n=∏_d≥1B_d(n)^φ(d) and fracnrad(n) = ∏_d≥2B_d(n)^φ(d). This is an exact multiplicative polyphase decomposition of prime-power persistence.
Exact mathematical passage
B_d(n)=\prod_{p\mid n,\ d\mid e_p}p.
\prod_{q\mid ab}Use in source · 2 passages
Search aliases: ∏_qmid ab prod mid
Use in source · detailed_3.tex, line 321
Writing β=-∏_qmid abq^ν_q, we obtain the cross-prime logarithmic filter relation log_pβ = ∑_qmid ab ν_qlog_p⟨ q⟩_p. The same depth-three path therefore has a small complex envelope and exact deep finite notches.
Exact mathematical passage
\beta=-\prod_{q\mid ab}q^{\nu_q},
\prod_{q\in S}Use in source · 2 passages
Search aliases: ∏_q∈ S prod in
Use in source · appendices_more.tex, line 171
Exact Kummer kernel for a rational packet. Let x=±∏_q∈ Sq^ν_q and consider its class modulo Mth powers. For positive x, the order is fracMgcd(M,{ν_q}). If x<0 and M is even, the sign contributes a separate element of order two. The exact order is therefore the least common multiple of the valuation order and the sign order.
Exact mathematical passage
x=\pm\prod_{q\in S}q^{\nu_q}
\prod_{r=0}^{e-1}Use in source · 3 passages
Search aliases: ∏_r=0^e-1 prod
Use in source · detailed_1.tex, line 489
First-order no-free-refinement principle. The shell-refined factor ∏_r=0^e-1 (1+fracT_rp) has first-order expansion 1+frac1p∑_rT_r+O(T^2). Only the shell sum appears. Every internal phase in the kernel of shell synthesis is invisible to both first-order additivity and the first-order Wronskian.
Exact mathematical passage
\prod_{r=0}^{e-1} \left(1+\frac{T_r}{p}\right)
\prod_{\substack{p\mid n\\d\mid e_p}}Use in source · 1 passage
Search aliases: ∏_substackpmid n dmid e_p prod substack mid d mid
Use in source · detailed_2.tex, line 193
Exact-order divisor packets. Let n=∏_p p^e_p. For every d≥1, define B_d(n) = ∏_substackpmid n dmid e_pp. Since ∑_dmid eφ(d)=e, we have the perfect reconstruction identity n = ∏_d≥1 B_d(n)^φ(d). The first packet is B_1(n)=rad(n). The powerful part is fracnrad(n) = ∏_d≥2 B_d(n)^φ(d). This is a multiplicative polyphase decomposition indexed by exact character orders.
Exact mathematical passage
\boxed{ B_d(n) = \prod_{\substack{p\mid n\\d\mid e_p}}p. }
\prod_{\substack{\zeta^N=1\\\zeta\neq1}}Use in source · 1 passage
Search aliases: ∏_substackζ^N=1 ζ≠1 prod substack zeta zeta neq
Use in source · detailed_1.tex, line 396
ghtarrow T↦1 Z) be the augmentation ideal. It is a free abelian group of rank N-1, and multiplication by V_e preserves it. After extension to C, the eigenvalues on I_N⊗C are V_e(ζ) = frac1-ζ^e1-ζ, ζ^N=1, ζ≠1. Because gcd(e,N)=1, the map ζ↦ζ^e permutes the nontrivial Nth roots. Therefore det(V_emid I_N) = ∏_substackζ^N=1 ζ≠1 frac1-ζ^e1-ζ =1. Thus multiplication by V_e is an automorphism of the integral lattice I_N. On the quotient A_N/I_N≅Z, it acts as multiplication by V_e(1)=e. Applying the sna…
Exact mathematical passage
\det\bigl(V_e\mid I_N\bigr) = \prod_{\substack{\zeta^N=1\\\zeta\neq1}} \frac{1-\zeta^e}{1-\zeta} =1.
\prod_{\zeta^N=1}Use in source · 1 passage
Search aliases: ∏_ζ^N=1 prod zeta
Use in source · appendices_final.tex, line 256
Polyphase circulant determinant. For the valuation word V_e(T)=1+T+⋯+T^e-1 in A_N=Z[T]/(T^N-1), multiplication is circulant. Its determinant is ∏_ζ^N=1V_e(ζ). Let I_N be the augmentation ideal of A_N. When gcd(e,N)=1, the nontrivial roots are permuted by ζ↦ζ^e, so multiplication by V_e has determinant one on the integral lattice I_N and is therefore an automorphism of it. On A_N/I_N≅Z, the map is multiplication by e. The exact sequence for I_N⊂ A_N then gives cokernel Z/eZ and Smith form diag(1,…
Exact mathematical passage
\prod_{\zeta^N=1}V_e(\zeta).
\prod_{\ell\text{ odd}}Use in source · 2 passages
Search aliases: ∏_ℓ odd prod ell text
Use in source · detailed_1.tex, line 421
Dyadic and odd-adic primary lifts. Factor e=2^α∏_ℓ oddℓ^α_ℓ. The cokernel of the valuation block is Z/eZ ≅ Z/2^αZ × ∏_ℓ odd Z/ℓ^α_ℓZ. The primary factors specify the local lifting problems for the aggregate multiplicity channel. Its 2-primary component must be treated 2-adically, while the odd components admit the corresponding odd-adic lifts. The phase channels require no local denominator correction.
Exact mathematical passage
e=2^\alpha\prod_{\ell\text{ odd}}\ell^{\alpha_\ell}.
\sum_{d\mid e}Use in source · 1 passage
Search aliases: ∑_dmid e sum mid
Use in source · detailed_2.tex, line 201
dmid e_pp. Since ∑_dmid eφ(d)=e, we have the perfect reconstruction identity n = ∏_d≥1 B_d(n)^φ(d). The first packet is B_1(n)=rad(n). The powerful part is fracnrad(n) = ∏_d≥2 B_d(n)^φ(d). This is a multiplicative polyphase decomposition indexed by exact character orders.
Exact mathematical passage
\sum_{d\mid e}\varphi(d)=e,
\sum_{d\ge2}Use in source · 1 passage
Search aliases: ∑_d≥2 sum ge
Use in source · detailed_2.tex, line 240
Quality as input-output persistence imbalance. For a primitive triple, q-1 = frac log(c/R_c)-log(R_aR_b) log R . Using the divisor packets, q-1 = frac ∑_d≥2φ(d)log B_d(c) - log(R_aR_b) log R. A high-quality triple has more repeated-depth persistence in the output than the input radicals supply as activated bandwidth.
Exact mathematical passage
\boxed{ q-1 = \frac{ \sum_{d\ge2}\varphi(d)\log B_d(c) - \log(R_aR_b) }{\log R}. }
\sum_iUse in source · 7 passages
Search aliases: ∑_i sum
Use in source · appendices_final.tex, line 214
Local CRT lift in closed form. Let r=∏_i=1^mp_i, w_i=r/p_i. The equation ∑_iw_ix_i=t has a solution because gcd(w_1,…,w_m)=1. Modulo p_i, x_i≡ t w_i^-1pmodp_i. Let ξ_i be the centered residue. Then ∑_iw_iξ_i-t is divisible by r. Set q=fract-∑_iw_iξ_ir. Writing x_i=ξ_i+p_i z_i gives ∑_i z_i=q.
Exact mathematical passage
\sum_iw_ix_i=t
\sum_jUse in source · 1 passage
Search aliases: ∑_j sum
Use in source · detailed_1.tex, line 526
Second-order phase energy. At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^2 - frac1e∑_j≠0|widehat T_j|^2 ]. Thus nonzero polyphase energy first becomes visible at second order. Later formal-smoothness results show that this c…
Exact mathematical passage
\sum_{r<s}T_rT_s = \frac12\left[ \widehat T_0^2 - \frac1e\sum_j \widehat T_j\widehat T_{-j} \right].
\sum_{j=0}^{M-1}Use in source · 2 passages
Search aliases: ∑_j=0^M-1 sum
Use in source · detailed_3.tex, line 199
Root-distribution compression. Let u^M=z and let ζ_M be a primitive Mth root of unity. For n≥1, ∑_j=0^M-1 Li_n(ζ_M^ju) = M^1-nLi_n(u^M). Inside the disk of convergence, this follows by summing the defining series and using the root-of-unity filter. Analytic continuation extends the identity with compatible branches.
Exact mathematical passage
\boxed{ \sum_{j=0}^{M-1} \operatorname{Li}_n(\zeta_M^ju) = M^{1-n}\operatorname{Li}_n(u^M). }
\sum_{j=1}^{m-1}Use in source · 1 passage
Search aliases: ∑_j=1^m-1 sum
Use in source · appendices_more.tex, line 409
Bernoulli--Witt direction. The Bernoulli branch begins with the identity ∑_j=1^m-1j^k = fracB_k+1(m)-B_k+1k+1. The equation with m^k imposes strong congruences on Bernoulli values. A filtered prime-contact treatment should separate:
Exact mathematical passage
\sum_{j=1}^{m-1}j^k = \frac{B_{k+1}(m)-B_{k+1}}{k+1}.
\sum_{j=1}^{r}Use in source · 1 passage
Search aliases: ∑_j=1^r sum
Use in source · overview.tex, line 942
A general contact functor. The construction extends to a sparse relation ∑_j=1^rϵ_jn_j=0. Choose one or more output terms whose repeated depth is to be explained. The assignment ( n,ϵ) ⟼ A( n,ϵ) is canonical under permutations, sign reversal, primitive reduction, and primary decomposition. The optional geometric lift depends on the equation, but the support, contact, Kummer, and logarithmic layers do not.
Exact mathematical passage
\sum_{j=1}^{r}\epsilon_jn_j=0.
\sum_{j\neq0}Use in source · 1 passage
Search aliases: ∑_j≠0 sum neq
Use in source · detailed_1.tex, line 537
onSecond-order phase energy At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^2 - frac1e∑_j≠0|widehat T_j|^2 ]. Thus nonzero polyphase energy first becomes visible at second order. Later formal-smoothness results show that this …
Exact mathematical passage
\sum_{r<s}T_rT_s = \frac12\left[ \left(1-\frac1e\right)|\widehat T_0|^2 - \frac1e\sum_{j\neq0}|\widehat T_j|^2 \right].
\sum_{k=0}^nUse in source · 1 passage
Search aliases: ∑_k=0^n sum
Use in source · appendices_more.tex, line 196
Hecke cascade At a Tate curve, an ℓ-isogeny packet contains one branch multiplying depth by ℓ and ℓ branches multiplying depth by 1/ℓ. Under uniform averaging, fracℓ+ℓ(1/ℓ)ℓ+1=1. After n independent refinements, if the deep branch is chosen k times, the normalized depth is ℓ^2k-n with multiplicity binom nkℓ^n-k. Therefore ∑_k=0^n fracbinom nkℓ^n-k(ℓ+1)^n ℓ^2k-n =1. The first moment is an exact martingale. The mean logarithmic depth is -nfracℓ-1ℓ+1logℓ. Most branches become shallow, while rare br…
Exact mathematical passage
\sum_{k=0}^n \frac{\binom nk\ell^{n-k}}{(\ell+1)^n} \ell^{2k-n} =1.
\sum_pUse in source · 1 passage
Search aliases: ∑_p sum
Use in source · detailed_2.tex, line 15
Prime support as bandwidth and valuation as persistence. For n=∏_p p^e_p, define the prime-shell signal Ψ_n(p,r) = sqrtlog p 1_0≤ r<e_p. Its total energy is normΨ_n_2^2 = ∑_p e_plog p = log n. Let P_act retain only the first shell in each active prime channel. Then normP_actΨ_n_2^2 = ∑_pmid nlog p = lograd(n). Thus the radical is activated bandwidth, while the difference log n-lograd(n) is repeated persistence depth.
Exact mathematical passage
\norm{\Psi_n}_2^2 = \sum_p e_p\log p = \boxed{\log n}.
\sum_{p\mid a}Use in source · 3 passages
Search aliases: ∑_pmid a sum mid
Use in source · detailed_1.tex, line 441
The coarse equation recovers the original exponent-weighted row ∑_pmid afracav_p(a)p t_p + ∑_pmid bfracbv_p(b)p t_p - ∑_pmid cfraccv_p(c)p t_p =0. The phase equation is exponent free: ∑_pmid afrac ap y_p + ∑_pmid bfrac bp y_p - ∑_pmid cfrac cp y_p =0. Define the phase derivative D_y(n)=n∑_pmid nfracy_pp. The same radical divisibility argument gives fracabcRmid aD_y(b)-bD_y(a).
Exact mathematical passage
\sum_{p\mid a}\frac{av_p(a)}p t_p + \sum_{p\mid b}\frac{bv_p(b)}p t_p - \sum_{p\mid c}\frac{cv_p(c)}p t_p =0.
\sum_{p\mid b}Use in source · 3 passages
Search aliases: ∑_pmid b sum mid
Use in source · detailed_1.tex, line 441
The coarse equation recovers the original exponent-weighted row ∑_pmid afracav_p(a)p t_p + ∑_pmid bfracbv_p(b)p t_p - ∑_pmid cfraccv_p(c)p t_p =0. The phase equation is exponent free: ∑_pmid afrac ap y_p + ∑_pmid bfrac bp y_p - ∑_pmid cfrac cp y_p =0. Define the phase derivative D_y(n)=n∑_pmid nfracy_pp. The same radical divisibility argument gives fracabcRmid aD_y(b)-bD_y(a).
Exact mathematical passage
\sum_{p\mid a}\frac{av_p(a)}p t_p + \sum_{p\mid b}\frac{bv_p(b)}p t_p - \sum_{p\mid c}\frac{cv_p(c)}p t_p =0.
\sum_{p\mid c}Use in source · 2 passages
Search aliases: ∑_pmid c sum mid
Use in source · detailed_1.tex, line 441
The coarse equation recovers the original exponent-weighted row ∑_pmid afracav_p(a)p t_p + ∑_pmid bfracbv_p(b)p t_p - ∑_pmid cfraccv_p(c)p t_p =0. The phase equation is exponent free: ∑_pmid afrac ap y_p + ∑_pmid bfrac bp y_p - ∑_pmid cfrac cp y_p =0. Define the phase derivative D_y(n)=n∑_pmid nfracy_pp. The same radical divisibility argument gives fracabcRmid aD_y(b)-bD_y(a).
Exact mathematical passage
\sum_{p\mid a}\frac{av_p(a)}p t_p + \sum_{p\mid b}\frac{bv_p(b)}p t_p - \sum_{p\mid c}\frac{cv_p(c)}p t_p =0.
\sum_{p\mid n}Use in source · 6 passages
Search aliases: ∑_pmid n sum mid
Use in source · detailed_1.tex, line 20
The prime-coordinate module. Let a+b=c, gcd(a,b)=1. The three integers are pairwise coprime. Write S=S_asqcup S_bsqcup S_c ={p:pmid abc}. The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. For a positive integer n, define the universal multiplicative differential dn = n∑_pmid nfracv_p(n)p ξ_p. Every coefficient is an integer because pmid n. The map obeys the exact Leibniz law d(mn)=m dn+n dm. The verification is local. For each prime p, v_p(mn)=v_p(m)+v_p(n), so the…
Exact mathematical passage
\boxed{ \mathrm dn = n\sum_{p\mid n}\frac{v_p(n)}p\,\xi_p. }
\sum_{p\mid u}Use in source · 4 passages
Search aliases: ∑_pmid u sum mid
Use in source · detailed_1.tex, line 556
Exact local lift gauges. Let r_u=rad(u), P_u=fracur_u. A shell packet y=(y_p,r) produces D_y(u) = u∑_pmid ufrac1p ∑_r=0^v_p(u)-1y_p,r =P_ut_u, where t_u= ∑_pmid u fracr_up ∑_r y_p,r. Because gcd_pmid ufracr_up=1, every integer t_u is attainable when u>1.
Exact mathematical passage
D_y(u) = u\sum_{p\mid u}\frac1p \sum_{r=0}^{v_p(u)-1}y_{p,r} =P_ut_u,
\sum_{q\mid ab}Use in source · 2 passages
Search aliases: ∑_qmid ab sum mid
Use in source · detailed_3.tex, line 325
Writing β=-∏_qmid abq^ν_q, we obtain the cross-prime logarithmic filter relation log_pβ = ∑_qmid ab ν_qlog_p⟨ q⟩_p. The same depth-three path therefore has a small complex envelope and exact deep finite notches.
Exact mathematical passage
\boxed{ \log_p\beta = \sum_{q\mid ab} \nu_q\log_p\langle q\rangle_p. }
\sum_rUse in source · 5 passages
Search aliases: ∑_r sum
Use in source · detailed_1.tex, line 494
First-order no-free-refinement principle. The shell-refined factor ∏_r=0^e-1 (1+fracT_rp) has first-order expansion 1+frac1p∑_rT_r+O(T^2). Only the shell sum appears. Every internal phase in the kernel of shell synthesis is invisible to both first-order additivity and the first-order Wronskian.
Exact mathematical passage
1+\frac1p\sum_rT_r+O(T^2).
\sum_{r<s}Use in source · 3 passages
Search aliases: ∑_r<s sum
Use in source · detailed_1.tex, line 516
Second-order phase energy. At second order, ∏_r=0^e-1 (1+fracT_rp) = 1+ frac1p∑_rT_r+ frac1p^2∑_r<sT_rT_s+ O(T^3). Let widehat T_j be the discrete Fourier transform of the shell coordinates. Parseval gives ∑_r<sT_rT_s = frac12[ widehat T_0^2 - frac1e∑_j widehat T_jwidehat T_-j ]. For real coordinates, ∑_r<sT_rT_s = frac12[ (1-frac1e)|widehat T_0|^2 - frac1e∑_j≠0|widehat T_j|^2 ]. Thus nonzero polyphase energy first becomes visible at second order. Later formal-smoothness results show that this c…
Exact mathematical passage
\prod_{r=0}^{e-1} \left(1+\frac{T_r}{p}\right) = 1+ \frac1p\sum_rT_r+ \frac1{p^2}\sum_{r<s}T_rT_s+ O(T^3).
\sum_{r=0}^{v_p(u)-1}Use in source · 1 passage
Search aliases: ∑_r=0^v_p(u)-1 sum
Use in source · detailed_1.tex, line 556
Exact local lift gauges. Let r_u=rad(u), P_u=fracur_u. A shell packet y=(y_p,r) produces D_y(u) = u∑_pmid ufrac1p ∑_r=0^v_p(u)-1y_p,r =P_ut_u, where t_u= ∑_pmid u fracr_up ∑_r y_p,r. Because gcd_pmid ufracr_up=1, every integer t_u is attainable when u>1.
Exact mathematical passage
D_y(u) = u\sum_{p\mid u}\frac1p \sum_{r=0}^{v_p(u)-1}y_{p,r} =P_ut_u,
\sum_\varepsilonUse in source · 1 passage
Search aliases: ∑_ε sum varepsilon
Use in source · detailed_2.tex, line 291
Atkin--Lehner Walsh packets. Let W_1,…,W_r be commuting Atkin--Lehner involutions. For a sign vector ε∈{±1}^r, define A_ε = ∏_j=1^r(I+ε_jW_j). Then A_ε^2=2^rA_ε, A_ε A_ε'=0 (ε≠ε'), and ∑_ε A_ε=2^rI. The unnormalized analysis packets are integral. The common synthesis denominator is 2^r, which is subpolynomial in a squarefree conductor. Local sign separation is therefore not the large persistence carrier.
Exact mathematical passage
\boxed{ \sum_\varepsilon A_\varepsilon=2^rI. }
\inftyUse in source · 12 passages
Search aliases: ∞ infty
Use in source · detailed_1.tex, line 184
The master height inequality. For every positive n, | d_x(n)| · ≤ nnormx_∞ ∑_pmid nfracv_p(n)p. Since p≥2, frac1p≤frac12log2log p. Therefore ∑_pmid nfracv_p(n)p ≤ fraclog n2log2, and | d_x(n)| ≤ fracnlog n2log2normx_∞. Consequently, |W_x(a,b)| · ≤ a| d_x(b)|+b| d_x(a)|
Exact mathematical passage
\begin{aligned} |\mathrm d_x(n)| &\le n\norm{x}_\infty \sum_{p\mid n}\frac{v_p(n)}p. \end{aligned}
\capUse in source · 5 passages
Search aliases: ∩ cap
Use in source · detailed_1.tex, line 75
[6pt] -dfracc v_p(c)p, · pmid c. Then λ· x=0. The additive derivative lattice is T_a,b,c = ker(λ)∩Z^S. Since the row is nonzero, its rank is |S|-1. The problem is not existence of additive derivatives. The problem is finding an additive derivative that remains transverse to the dependence direction introduced by a and b.
Exact mathematical passage
\boxed{ \mathcal T_{a,b,c} = \ker(\lambda)\cap\Z^S. }
\intUse in source · 2 passages
Search aliases: ∫ int
Use in source · detailed_2.tex, line 165
Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be an optimal modular parametrization, and let ϕ_E^*ω_E = 2π i c_E f_E(z) dz. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_E^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area factor proportional to 1/c. Thus the height c can be written as a shape-corrected synthesis gain involving modular degree, Manin normalization, and Pete…
Exact mathematical passage
A(E) = \frac{i}{2} \int_{E(\C)} \omega_E\wedge\overline{\omega_E}.
\int_{E(\C)}Use in source · 2 passages
Search aliases: ∫_E(C) int C
Use in source · detailed_2.tex, line 165
Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be an optimal modular parametrization, and let ϕ_E^*ω_E = 2π i c_E f_E(z) dz. Define the differential area A(E) = fraci2 ∫_E(C) ω_Ewedgeoverlineω_E. Pulling the area form back gives deg(ϕ_E)A(E) = 4π^2 c_E^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2Y gives an area factor proportional to 1/c. Thus the height c can be written as a shape-corrected synthesis gain involving modular degree, Manin normalization, and Pete…
Exact mathematical passage
A(E) = \frac{i}{2} \int_{E(\C)} \omega_E\wedge\overline{\omega_E}.
\simeqUse in source · 2 passages
Search aliases: ≃ simeq
Use in source · appendices.tex, line 137
Filtered contact and Witt packet. The output contact complex is [128Z↪2Z] ≃ [Zxrightarrow64Z]. Its cohomology is Jcal_c≅Z/64Z. The associated graded is (F_2)^6. The six binary layers are joined by five nontrivial carry extensions. A cyclic target of order 64 can carry the filtered source exactly, while no cyclic target can contain the split rank-six source injectively.
Exact mathematical passage
[128\Z\hookrightarrow2\Z] \simeq [\Z\xrightarrow{64}\Z].
\neUse in source · 1 passage
Search aliases: ≠ ne
Use in source · tail.tex, line 195
Pillai and fixed perfect-power gaps. For x^m-y^n=k, k≠0 fixed, two deep contact packets collide through a residual channel whose prime support is fixed. This is more rigid than unrestricted abc because the output bandwidth is prescribed by rad(k).
Exact mathematical passage
x^m-y^n=k, \qquad k\ne0\text{ fixed},
\neqUse in source · 12 passages
Search aliases: ≠ neq
Use in source · appendices_final.tex, line 83
Odd-prime coefficient depth. Let q≠ p be prime. Write q=ω(q)⟨ q⟩_p, where ω(q) is the Teichm uller factor. Since ⟨ q⟩_p^p-1=q^p-1, we have log_p⟨ q⟩_p = frac1p-1log_p(q^p-1). If s=v_p(q^p-1-1)>0, then v_p(log_p(q^p-1))=s. Therefore v_p(frac1plog_p⟨ q⟩_p) =s-1. Truncating at the available contact depth gives v_p(ℓ_p,e(q)) = min{e-1,s-1}.
Exact mathematical passage
q\neq p
\equivUse in source · 12 passages
Search aliases: ≡ equiv
Use in source · appendices.tex, line 225
Coherence lattice. After removing the sign layer, (ϵ_2,K,W,C)=(2,1,1,32). The residual row is g=(29,31)pmod32. The lattice is Λ_coh = {(x_3,x_5):29x_3+31x_5≡0pmod32}. The input vector lies in the kernel because 29(-1)+31(3)=64. The full repeated output depth has now been transformed into one arithmetic cyclic character of order 32, after the bounded sign layer has been separated.
Exact mathematical passage
\Lambda_{\mathrm{coh}} = \{(x_3,x_5):29x_3+31x_5\equiv0\pmod{32}\}.
\subsetUse in source · 3 passages
Search aliases: ⊂ subset
Use in source · appendices_final.tex, line 259
lication is circulant. Its determinant is ∏_ζ^N=1V_e(ζ). Let I_N be the augmentation ideal of A_N. When gcd(e,N)=1, the nontrivial roots are permuted by ζ↦ζ^e, so multiplication by V_e has determinant one on the integral lattice I_N and is therefore an automorphism of it. On A_N/I_N≅Z, the map is multiplication by e. The exact sequence for I_N⊂ A_N then gives cokernel Z/eZ and Smith form diag(1,…,1,e).
Exact mathematical passage
I_N\subset A_N
\subseteqUse in source · 2 passages
Search aliases: ⊆ subseteq
Use in source · detailed_2.tex, line 302
The same-sign separation problem. After fixing the local signs, several eigenforms can remain congruent inside the same packet. Let Λ be an integral Hecke lattice and let e_f be the rational projector onto the Frey eigenline. Define η_f = min{m≥1:m e_f(Λ)⊆Λ}. This is the integral eigenchannel separation denominator. It is an arithmetic analogue of leakage in a filter bank. The difficulty is not rational spectral isolation. It is isolation inside the integral lattice.
Exact mathematical passage
\eta_f = \min\{m\ge1:m e_f(\Lambda)\subseteq\Lambda\}.
\bigoplusUse in source · 6 passages
Search aliases: ⊕ bigoplus
Use in source · detailed_1.tex, line 16
The prime-coordinate module. Let a+b=c, gcd(a,b)=1. The three integers are pairwise coprime. Write S=S_asqcup S_bsqcup S_c ={p:pmid abc}. The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. For a positive integer n, define the universal multiplicative differential dn = n∑_pmid nfracv_p(n)p ξ_p. Every coefficient is an integer because pmid n. The map obeys the exact Leibniz law d(mn)=m dn+n dm. The verification is local. For each prime p, v_p(mn)=v_p(m)+v_p(n), so the…
Exact mathematical passage
\Omega_S=\bigoplus_{p\in S}\Z\xi_p.
\oplusUse in source · 3 passages
Search aliases: ⊕ oplus
Use in source · detailed_1.tex, line 695
The exact lattice extension. Define L=ker(Λ)∩Z^S and L_0=L∩ker( w). Because Λ is primitive, choose t∈Z^S with Λ(t)=1. Then Z^S=L⊕Z t. The image of M_a,b,c is generated by (1, w(t)) together with (0, w(L)). Its index in Z^2 is therefore the index of w(L) in Z. Theorem~[source reference] shows that this index is Q, so w(L)=QZ. Hence ω(x)=frac w· xQ is an integer-valued surjection and gives the exact sequence 0⟶ L_0 ⟶ L xrightarrowω Z ⟶0. The extension splits algebraically. The difficult quantity i…
Exact mathematical passage
\Z^S=L\oplus\Z t.
\bigoplus_{i=1}^{r}Use in source · 1 passage
Search aliases: ⊕_i=1^r bigoplus
Use in source · detailed_2.tex, line 371
A toy Hecke order. Consider T( m) = {(x_0,x_1,…,x_r)∈Z^r+1:x_i≡ x_0pmodm_i}. For the target coordinate x_0, the cotangent module is p/ p^2 ≅ ⊕_i=1^rZ/m_iZ. The cyclic congruence quotient is Z/lcm(m_1,…,m_r)Z. The quotient of cotangent order by congruence order is frac∏_i m_ilcm(m_i). This is a toy Wiles-defect factor. It measures the multiplicity discarded by cyclic compression.
Exact mathematical passage
\boxed{ \mathfrak p/\mathfrak p^2 \cong \bigoplus_{i=1}^{r}\Z/m_i\Z. }
\bigoplus_{p^e\parallel c}Use in source · 2 passages
Search aliases: ⊕_p^eparallel c bigoplus parallel
Use in source · detailed_2.tex, line 347
Why one cyclic quotient is not enough for split persistence. The split persistence module P_c ≅ ⊕_p^eparallel c (Z/pZ)^e-1 has large p-rank. The p-torsion of a cyclic target has rank at most one. Therefore every map from P_c to a cyclic group has a forced kernel of order at least ∏_p^eparallel cp^max(e-2,0). This is a true obstruction to the split source. It motivates the Hecke cotangent module p/ p^2, which can retain multiple independent congruence directions before cyclic compression.
Exact mathematical passage
\mathcal P_c \cong \bigoplus_{p^e\parallel c} (\Z/p\Z)^{e-1}
\bigoplus_{p^e\parallel n}Use in source · 1 passage
Search aliases: ⊕_p^eparallel n bigoplus parallel
Use in source · overview.tex, line 236
[Exact contact quotient] For every positive integer n, rad(n)Z/nZ ≅ Z/(n/rad(n))Z. Its associated graded is gr(rad(n)Z/nZ) ≅ ⊕_p^eparallel n(F_p)^e-1.
Exact mathematical passage
\boxed{ \gr\left(\rad(n)\Z/n\Z\right) \cong \bigoplus_{p^e\parallel n}(\F_p)^{e-1}. }
\bigoplus_{p\mid c}Use in source · 1 passage
Search aliases: ⊕_pmid c bigoplus mid
Use in source · detailed_2.tex, line 561
The filtered local-global target. The intended transducer is a filtered chain J_c ⟶ ⊕_pmid c L_p^st ⟶ L_R^glob xrightarrowR= T L_ T_f ⟶ C_f. It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archimedean size.
Exact mathematical passage
\mathbf J_c \longrightarrow \bigoplus_{p\mid c}\mathbf L_p^{\mathrm{st}} \longrightarrow \mathbf L_{R^{\mathrm{glob}}} \xrightarrow{R=\mathbb T} \mathbf L_{\mathbb T_f} \longrightarrow \mathbf C_f.
\bigoplus_{p\in S}Use in source · 1 passage
Search aliases: ⊕_p∈ S bigoplus in
Use in source · detailed_1.tex, line 16
The prime-coordinate module. Let a+b=c, gcd(a,b)=1. The three integers are pairwise coprime. Write S=S_asqcup S_bsqcup S_c ={p:pmid abc}. The first object of the calculus is the free prime-packet module Ω_S=⊕_p∈ SZξ_p. For a positive integer n, define the universal multiplicative differential dn = n∑_pmid nfracv_p(n)p ξ_p. Every coefficient is an integer because pmid n. The map obeys the exact Leibniz law d(mn)=m dn+n dm. The verification is local. For each prime p, v_p(mn)=v_p(m)+v_p(n), so the…
Exact mathematical passage
\Omega_S=\bigoplus_{p\in S}\Z\xi_p.
\perpUse in source · 3 passages
Search aliases: ⊥ perp
Use in source · detailed_1.tex, line 115
Over R, the additive projection is P_add = I- fracλλ^ Tnormλ_2^2. The obstruction direction inside the additive hyperplane is μ_⊥=P_addμ. When μ_⊥≠0, the rank-one obstruction projector is P_obs = fracμ_⊥μ_⊥^ T normμ_⊥_2^2. This is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation direction, one obstruction direction, and the remaining null packets.
Exact mathematical passage
\mu_\perp=P_{\mathrm{add}}\mu.
\cdotsUse in source · 9 passages
Search aliases: ⋯ cdots
Use in source · appendices.tex, line 622
Bernoulli--Witt and ErdHos--Moser. Faulhaber's formula rewrites 1^k+⋯+(m-1)^k=m^k as a Bernoulli-polynomial identity. The denominator primes of Bernoulli numbers, the finite-difference divisibility, and the separate 2-adic and odd-adic lift structures suggest a different contact object: Bernoulli contact crystal + denominator prime packet + finite-difference coherence lattice. This branch should be developed independently before it is merged with the logarithmic prime-contact theory.
Exact mathematical passage
1^k+\cdots+(m-1)^k=m^k
\langleUse in source · 12 passages
Search aliases: ⟨ langle
Use in source · appendices_final.tex, line 23
For odd p, form the logarithmic row ℓ_p,e(q) = frac1plog_p⟨ q⟩_p pmodp^e-1. Let w_p = min_qmid ab v_p(ℓ_p,e(q)) after Kummer normalization. Divide the row and the modulus by p^w_p. The remaining row is primitive modulo the residual power p^χ_p.
Exact mathematical passage
\ell_{p,e}(q) = \frac1p\log_p\langle q\rangle_p \pmod{p^{e-1}}.
\rangleUse in source · 12 passages
Search aliases: ⟩ rangle
Use in source · appendices_final.tex, line 23
For odd p, form the logarithmic row ℓ_p,e(q) = frac1plog_p⟨ q⟩_p pmodp^e-1. Let w_p = min_qmid ab v_p(ℓ_p,e(q)) after Kummer normalization. Divide the row and the modulus by p^w_p. The remaining row is primitive modulo the residual power p^χ_p.
Exact mathematical passage
\ell_{p,e}(q) = \frac1p\log_p\langle q\rangle_p \pmod{p^{e-1}}.
\rangle_2Use in source · 2 passages
Search aliases: ⟩_2 rangle
Use in source · appendices_final.tex, line 119
Dyadic coefficient depth. For odd q, separate the sign by writing q=(-1)^(q-1)/2⟨ q⟩_2, ⟨ q⟩_2∈1+4Z_2. Then ⟨ q⟩_2^2=q^2. The weighted coefficient is ℓ_2,e(q) = frac14log_2⟨ q⟩_2. If s=v_2(q^2-1), then v_2(ℓ_2,e(q))=s-3, truncated at e-2. The missing first layer is the separate sign bit. This is the final form of the weighted dyadic branch.
Exact mathematical passage
q=(-1)^{(q-1)/2}\langle q\rangle_2, \qquad \langle q\rangle_2\in1+4\Z_2.
\rangle_2^2Use in source · 1 passage
Search aliases: ⟩_2^2 rangle
Use in source · appendices_final.tex, line 125
Dyadic coefficient depth. For odd q, separate the sign by writing q=(-1)^(q-1)/2⟨ q⟩_2, ⟨ q⟩_2∈1+4Z_2. Then ⟨ q⟩_2^2=q^2. The weighted coefficient is ℓ_2,e(q) = frac14log_2⟨ q⟩_2. If s=v_2(q^2-1), then v_2(ℓ_2,e(q))=s-3, truncated at e-2. The missing first layer is the separate sign bit. This is the final form of the weighted dyadic branch.
Exact mathematical passage
\langle q\rangle_2^2=q^2.
\rangle_pUse in source · 7 passages
Search aliases: ⟩_p rangle
Use in source · appendices_final.tex, line 23
For odd p, form the logarithmic row ℓ_p,e(q) = frac1plog_p⟨ q⟩_p pmodp^e-1. Let w_p = min_qmid ab v_p(ℓ_p,e(q)) after Kummer normalization. Divide the row and the modulus by p^w_p. The remaining row is primitive modulo the residual power p^χ_p.
Exact mathematical passage
\ell_{p,e}(q) = \frac1p\log_p\langle q\rangle_p \pmod{p^{e-1}}.
\rangle_p^{p-1}Use in source · 1 passage
Search aliases: ⟩_p^p-1 rangle
Use in source · appendices_final.tex, line 88
Odd-prime coefficient depth. Let q≠ p be prime. Write q=ω(q)⟨ q⟩_p, where ω(q) is the Teichm uller factor. Since ⟨ q⟩_p^p-1=q^p-1, we have log_p⟨ q⟩_p = frac1p-1log_p(q^p-1). If s=v_p(q^p-1-1)>0, then v_p(log_p(q^p-1))=s. Therefore v_p(frac1plog_p⟨ q⟩_p) =s-1. Truncating at the available contact depth gives v_p(ℓ_p,e(q)) = min{e-1,s-1}.
Exact mathematical passage
\langle q\rangle_p^{p-1}=q^{p-1},
\longrightarrowUse in source · 24 passages
Search aliases: ⟶ longrightarrow
Use in source · appendices_more.tex, line 5
Why exact synthesis prevents free coordinate splitting. Let X and Y be normed spaces, and suppose J:X⟶ Y, S:Y⟶ X satisfy SJ=I_X. Then normx = normSJx ≤ normSnormJx. Therefore normJx ≥ fracnormxnormS. An analysis map may make coordinates smaller only by increasing the synthesis norm. This elementary inequality is the common source of several no-free-refinement results.
Exact mathematical passage
J:X\longrightarrow Y, \qquad S:Y\longrightarrow X
\longmapstoUse in source · 5 passages
Search aliases: ⟼ longmapsto
Use in source · detailed_1.tex, line 501
[First-order no-free-polyphase principle] Let S:(T_p,r)_p,r⟼ (∑_rT_p,r)_p be shell synthesis. The first-order additive map and first-order Wronskian factor through S. Hence Ker S is contained in the kernels of both first-order maps.
Exact mathematical passage
S:(T_{p,r})_{p,r}\longmapsto \left(\sum_rT_{p,r}\right)_p
Source coverage
Automatic extraction covers TeX and Markdown math delimiters and common equation environments in these selected files. It does not interpret diagrams, images, PDF text, program identifiers, or historical archives. Occurrences are navigation aids, not a proof review.
Download all entries, passages and source checksums (JSON) · Collection coverage
- research/prime-contact-coherence/Projector_Resolved_Prime_Contact_Coherence_Calculus_Verified_Final.tex · 0 mathematical passages
- research/prime-contact-coherence/README.md · 0 mathematical passages
- research/prime-contact-coherence/chapters/appendices.tex · 140 mathematical passages
- research/prime-contact-coherence/chapters/appendices_final.tex · 120 mathematical passages
- research/prime-contact-coherence/chapters/appendices_more.tex · 134 mathematical passages
- research/prime-contact-coherence/chapters/bibliography.tex · 5 mathematical passages
- research/prime-contact-coherence/chapters/detailed_1.tex · 292 mathematical passages
- research/prime-contact-coherence/chapters/detailed_2.tex · 158 mathematical passages
- research/prime-contact-coherence/chapters/detailed_3.tex · 194 mathematical passages
- research/prime-contact-coherence/chapters/front_atlas.tex · 33 mathematical passages
- research/prime-contact-coherence/chapters/overview.tex · 200 mathematical passages
- research/prime-contact-coherence/chapters/tail.tex · 95 mathematical passages
- research/prime-contact-coherence/verification/final_pdfinfo.txt · 0 mathematical passages