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777 symbol spellings in 1409 mathematical passages across 29 selected source files. Subscripts and superscripts have their own entries alongside their base symbols.
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+Introduction passage · 172 passages
Search aliases: +
Introduction passage · project_09.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).
=Introduction passage · 461 passages
Search aliases: =
Introduction passage · project_09.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).
cIntroduction passage · 287 passages
Search aliases: c
Introduction passage · project_09.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).
dIntroduction passage · 40 passages
Search aliases: d
Introduction passage · project_11.tex, line 74
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. Divide the input exponent vector by K.…
Exact mathematical passage
d_p=e-1.
d_2Introduction passage · 1 passage
Search aliases: d_2
Introduction passage · project_11.tex, line 78
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. Divide the input exponent vector by K.…
Exact mathematical passage
s_2=1, \qquad d_2=e-2.
d_pIntroduction passage · 4 passages
Search aliases: d_p
Introduction passage · project_11.tex, line 74
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. Divide the input exponent vector by K.…
Exact mathematical passage
d_p=e-1.
eIntroduction passage · 122 passages
Search aliases: e
Introduction passage · project_11.tex, line 73
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. Divide the input exponent vector by K.…
Exact mathematical passage
p^e\parallel c
FIntroduction passage · 20 passages
Search aliases: F
Introduction passage · project_09.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 · 4 passages
Search aliases: F_M
Introduction passage · project_09.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 · 46 passages
Search aliases: h
Introduction passage · project_09.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 · project_09.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 · 1 passage
Search aliases: h_K_F_M
Introduction passage · project_09.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 · 16 passages
Search aliases: I
Introduction passage · project_09.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 · 60 passages
Search aliases: i
Introduction passage · project_09.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 · 45 passages
Search aliases: j
Introduction passage · project_09.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.
KIntroduction passage · 55 passages
Search aliases: K
Introduction passage · project_09.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_{F_M}Introduction passage · 3 passages
Search aliases: K_F_M
Introduction passage · project_09.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 · project_11.tex, line 83
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. Divide the input exponent vector by K.…
Exact mathematical passage
K_{\mathrm{Kum}}
\mathrm{Kum}Introduction passage · 5 passages
Search aliases: Kum mathrm
Introduction passage · project_11.tex, line 83
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. Divide the input exponent vector by K.…
Exact mathematical passage
K_{\mathrm{Kum}}
\logIntroduction passage · 59 passages
Search aliases: log
Introduction passage · project_09.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).
MIntroduction passage · 76 passages
Search aliases: M
Introduction passage · project_09.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
M^2Introduction passage · 7 passages
Search aliases: M^2
Introduction passage · project_09.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.
\minIntroduction passage · 18 passages
Search aliases: min
Introduction passage · project_11.tex, line 84
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. Divide the input exponent vector by K. Define …
Exact mathematical passage
r_p=\min\{d_p,v_p(K_{\mathrm{Kum}})\}.
\nmidIntroduction passage · 10 passages
Search aliases: nmid
Introduction passage · project_11.tex, line 83
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. Divide the input exponent vector by K.…
Exact mathematical passage
2\nmid M
OIntroduction passage · 4 passages
Search aliases: O
Introduction passage · project_09.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 · 6 passages
Search aliases: O mathcal
Introduction passage · project_09.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 · 1 passage
Search aliases: O_F_M mathcal
Introduction passage · project_09.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 · 35 passages
Search aliases: P
Introduction passage · project_09.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.
pIntroduction passage · 217 passages
Search aliases: p
Introduction passage · project_11.tex, line 73
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. Divide the input exponent vector by K.…
Exact mathematical passage
p^e\parallel c
p^eIntroduction passage · 20 passages
Search aliases: p^e
Introduction passage · project_11.tex, line 73
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. Divide the input exponent vector by K.…
Exact mathematical passage
p^e\parallel c
P_MIntroduction passage · 4 passages
Search aliases: P_M
Introduction passage · project_09.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 · project_09.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 · project_09.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 · 14 passages
Search aliases: parallel
Introduction passage · project_11.tex, line 73
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. Divide the input exponent vector by K.…
Exact mathematical passage
p^e\parallel c
rIntroduction passage · 178 passages
Search aliases: r
Introduction passage · project_09.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 · project_09.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 · 5 passages
Search aliases: r_p
Introduction passage · project_11.tex, line 84
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. Divide the input exponent vector by K. Define …
Exact mathematical passage
r_p=\min\{d_p,v_p(K_{\mathrm{Kum}})\}.
sIntroduction passage · 62 passages
Search aliases: s
Introduction passage · project_09.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 · project_09.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 · project_11.tex, line 78
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. Divide the input exponent vector by K.…
Exact mathematical passage
s_2=1, \qquad d_2=e-2.
vIntroduction passage · 63 passages
Search aliases: v
Introduction passage · project_11.tex, line 84
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. Divide the input exponent vector by K. Define …
Exact mathematical passage
r_p=\min\{d_p,v_p(K_{\mathrm{Kum}})\}.
v_pIntroduction passage · 39 passages
Search aliases: v_p
Introduction passage · project_11.tex, line 84
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. Divide the input exponent vector by K. Define …
Exact mathematical passage
r_p=\min\{d_p,v_p(K_{\mathrm{Kum}})\}.
XIntroduction passage · 21 passages
Search aliases: X
Introduction passage · project_09.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 · 3 passages
Search aliases: Y
Introduction passage · project_09.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 · 71 passages
Search aliases: Z
Introduction passage · project_09.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,
\cdotIntroduction passage · 23 passages
Search aliases: · cdot
Introduction passage · project_09.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 · 16 passages
Search aliases: δ delta
Introduction passage · project_09.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 · project_09.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 · project_09.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}
\epsilonIntroduction passage · 7 passages
Search aliases: ϵ epsilon
Introduction passage · project_11.tex, line 83
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. Divide the input exponent vector by K.…
Exact mathematical passage
\epsilon_2=2^{s_2}
\epsilon_2Introduction passage · 7 passages
Search aliases: ϵ_2 epsilon
Introduction passage · project_11.tex, line 83
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. Divide the input exponent vector by K.…
Exact mathematical passage
\epsilon_2=2^{s_2}
\zetaIntroduction passage · 15 passages
Search aliases: ζ zeta
Introduction passage · project_09.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 · project_09.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 · 23 passages
Search aliases: η eta
Introduction passage · project_09.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 · 11 passages
Search aliases: μ mu
Introduction passage · project_09.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 · 2 passages
Search aliases: μ_M^2 mu
Introduction passage · project_09.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 · 6 passages
Search aliases: ξ xi
Introduction passage · project_09.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 · 7 passages
Search aliases: σ sigma
Introduction passage · project_09.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
\sigma
\sigma^iIntroduction passage · 1 passage
Search aliases: σ^i sigma
Introduction passage · project_09.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.
\tauIntroduction passage · 2 passages
Search aliases: τ tau
Introduction passage · project_09.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 · project_09.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.
\inIntroduction passage · 41 passages
Search aliases: ∈ in
Introduction passage · project_09.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,
\sumIntroduction passage · 61 passages
Search aliases: ∑ sum
Introduction passage · project_09.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 · project_09.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 · project_09.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 · 15 passages
Search aliases: ≅ cong
Introduction passage · project_09.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).
\geIntroduction passage · 83 passages
Search aliases: ≥ ge
Introduction passage · project_09.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 · project_09.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 · 5 passages
Search aliases: *
Use in source · project_03.tex, line 150
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
\mathbf1Use in source · 1 passage
Search aliases: 1 mathbf
Use in source · project_05.tex, line 10
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 · 1 passage
Search aliases: 1_0≤ r<e_p mathbf le
Use in source · project_05.tex, line 10
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}.
AUse in source · 56 passages
Search aliases: A
Use in source · 02_polynomial.tex, line 9
[Mason--Stothers in characteristic zero] Let k be an algebraically closed field of characteristic zero. Suppose A,B,C∈ k[X] are nonzero, pairwise coprime, satisfy A+B=C, and are not all constant. Then max{deg A,deg B,deg C}≤ n_0(ABC)-1. This is the polynomial abc theorem. Snyder gives a short primary proof in a form that also makes the derivative hypothesis explicit~[source reference]. The proof below spells out the cancellation that will motivate the arithmetic Wronskian.
Exact mathematical passage
A,B,C\in k[X]
aUse in source · 262 passages
Search aliases: a
Use in source · 00_reading.tex, line 3
How to read this piece. addcontentslinetocsectionHow to read this piece The equation a+b=c joins two kinds of structure. Multiplication describes prime factors and their repeated powers. Addition forces those three factorizations to fit one exact relation. The abc conjecture says that this compatibility limits how large the sum can be compared with the distinct primes that appear.
Exact mathematical passage
a+b=c
A'Use in source · 3 passages
Search aliases: A'
Use in source · 02_polynomial.tex, line 17
Set W=A'B-AB'. It is nonzero. Indeed, if it vanished, then (A/B)'=0, and characteristic zero would force the rational function A/B to be constant. Coprimality would then force both A and B to be constant, and so also C.
Exact mathematical passage
W=A'B-AB'
a^2Use in source · 3 passages
Search aliases: a^2
Use in source · 25_truncation.tex, line 49
Rank two exposes the loss in one line. The first two truncated rank-two expressions are H_2,2(t)=t^3/8, H_2,3(t)=fract^38+frac4t^427+fract^5216 =fract^3(27+32t+t^2)216. Clearing the displayed denominator and the projective denominator in the second gives W(a,c)=a^3(27c^2+32ac+a^2). For every p>3 dividing c, this is congruent to a^5 modulo p. Hence it is a unit at that output prime.
Exact mathematical passage
W(a,c)=a^3(27c^2+32ac+a^2).
a^3Use in source · 1 passage
Search aliases: a^3
Use in source · 25_truncation.tex, line 49
Rank two exposes the loss in one line. The first two truncated rank-two expressions are H_2,2(t)=t^3/8, H_2,3(t)=fract^38+frac4t^427+fract^5216 =fract^3(27+32t+t^2)216. Clearing the displayed denominator and the projective denominator in the second gives W(a,c)=a^3(27c^2+32ac+a^2). For every p>3 dividing c, this is congruent to a^5 modulo p. Hence it is a unit at that output prime.
Exact mathematical passage
W(a,c)=a^3(27c^2+32ac+a^2).
a^5Use in source · 1 passage
Search aliases: a^5
Use in source · 25_truncation.tex, line 52
Rank two exposes the loss in one line The first two truncated rank-two expressions are H_2,2(t)=t^3/8, H_2,3(t)=fract^38+frac4t^427+fract^5216 =fract^3(27+32t+t^2)216. Clearing the displayed denominator and the projective denominator in the second gives W(a,c)=a^3(27c^2+32ac+a^2). For every p>3 dividing c, this is congruent to a^5 modulo p. Hence it is a unit at that output prime.
Exact mathematical passage
a^5
a^dUse in source · 1 passage
Search aliases: a^d
Use in source · 25_truncation.tex, line 35
mid c and gcd(a,c)=1, then pnmid W_r,K(a,c). Expand W as a homogeneous polynomial in a,c. Modulo p, every term below the top degree in a retains a positive power of c and vanishes. The unique top subset is (K-r+1,…,K). Its indices, nonzero differences and L_K are all p-units, so its scaled coefficient is a unit. The remaining factor a^d is a unit because pnmid a. The surviving term is therefore nonzero modulo p. If an output prime above the cutoff has repeated depth, this witness has no positive…
Exact mathematical passage
a^d
a_0Use in source · 9 passages
Search aliases: a_0
Use in source · project_07.tex, line 86
Complete-intersection control case. Let F∈Z[X] be monic and squarefree over Q, with an integer simple root a_0. Put T=Z[X]/(F) and let λ: T→Z evaluate at a_0. The ring is finite free over Z and reduced over Q. Then the congruence quotient Z/λ(AnnI) and cotangent module both have order |F'(a_0)|. To see this directly, write F=(X-a_0)G. The kernel I is generated by X-a_0. Modulo I^2, the relation is G(a_0)(X-a_0)=0, so I/I^2≅Z/F'(a_0)Z. The annihilator of I is generated by G modulo F, whose evalua…
Exact mathematical passage
a_0
a_1Use in source · 1 passage
Search aliases: a_1
Use in source · project_05.tex, line 152
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
A_{ij}Use in source · 1 passage
Search aliases: A_ij
Use in source · project_10.tex, line 107
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_jUse in source · 2 passages
Search aliases: a_j
Use in source · 20_signed_filter.tex, line 77
[Exact cyclic Prony reconstruction] The distribution on a cyclic channel with r occupied residues is determined by the 2r exact consecutive samples m(0),…,m(2r-1). Write A(z)=∏_x∈ E(z-α_x)=z^r+∑_j=0^r-1a_jz^j. For every k, ∑_j=0^r-1a_jm(k+j)=-m(k+r). The coefficient matrix for 0≤ k<r is the Hankel matrix H=(m(i+j))_0≤ i,j<r. With V=(α_x^i)_i,x and D=diag(N(x)/M), one has H=VDV^ T. The square Vandermonde V is invertible and every diagonal entry of D is nonzero. Thus H is invertible, and the 2r sa…
Exact mathematical passage
A(z)=\prod_{x\in E}(z-\alpha_x)=z^r+\sum_{j=0}^{r-1}a_jz^j.
A_NUse in source · 4 passages
Search aliases: A_N
Use in source · project_02.tex, line 35
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
A_N=\Z[T]/(T^N-1).
a_nUse in source · 1 passage
Search aliases: a_n
Use in source · 01_foundations.tex, line 104
Why the epsilon cannot be removed. The uniform statement c≤ KR for one absolute constant K is false. There is an elementary family that proves this without using abc. [Failure of the zero-epsilon estimate] The ratios c/rad(abc) are unbounded among primitive positive additive triples. For n≥0, take a_n=1, b_n=4^3^n-1, c_n=4^3^n. These are primitive, b_n is odd, and rad(c_n)=2. We claim v_3(b_n)=n+1. At n=0 this is v_3(3)=1. If x=1+3^ku with k≥1 and 3nmid u, then x^3-1=3^k+1u(1+3^ku+3^2k-1u^2). Th…
Exact mathematical passage
a_n=1,\quad b_n=4^{3^n}-1,\quad c_n=4^{3^n}.
a_{p,q}Use in source · 2 passages
Search aliases: a_p,q
Use in source · project_11.tex, line 91
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. Divide the input exponent vector by K. Define a_p,q as the original odd or weighted dyadic coefficient, now reduced modulo p^d_p-r_p. Dividing the relation vector reduces the needed relation modulus. It does not divide the coefficient row by K. Define w_p = min { d_p-r_p…
Exact mathematical passage
a_{p,q}
A_rUse in source · 4 passages
Search aliases: A_r
Use in source · 26_global_bridge.tex, line 6
[Conditional integral witness criterion] Let R be an unbounded set of positive integer ranks, and put s_r=r(r+1)/2. Suppose that for each r∈ R there are constants C_r>0, A_r≥0 and E_r≥0, independent of the primitive triple, such that every primitive positive triple admits a nonzero integer W_r with Q^s_rmid W_r, |W_r|≤ C_r(ab)^s_rR^E_r(log(3c))^A_r. If E_r/s_r→0 along R, then Conjecture~[source reference] follows. Nonzero integer divisibility gives Q^s_r≤|W_r|. Divide by (ab)^s_r and use Q/(ab)=…
Exact mathematical passage
A_r\ge0
a_rUse in source · 13 passages
Search aliases: a_r
Use in source · 24_padic.tex, line 69
grows only logarithmically with each index, whereas its numerator power of t gives linearly growing valuation. Thus the terms tend to zero and the finite Cauchy--Binet expansions pass to the convergent local limit. [Exact good-prime valuation] If p>3r-1 and v_p(t)=e≥1, then v_p( H_r(t))=e r(r+1)/2. Write a_r=2r-1 and s_r=r(r+1)/2. A term indexed by k_1<⋯<k_r has valuation e∑_i k_i+2∑_i<jv_p(k_j-k_i)-a_r∑_i v_p(k_i). For every h≥1, the hypothesis gives p^h-r>a_rh. At h=1 this is p-r>a_r. Successi…
Exact mathematical passage
a_r=2r-1
A_{r,K}Use in source · 2 passages
Search aliases: A_r,K
Use in source · 25_truncation.tex, line 12
Let L_K=lcm(1,…,K), s_r=r(r+1)/2, d=rK-r(r-1)/2. Then A_r,K(t):=L_K^r^2 H_r,K(t)∈Z[t], ord_0A_r,K=s_r, deg A_r,K=d. To see integrality, scale row i by L_K^i+1 and column j by L_K^j. Each entry becomes L_K^i+j+1∑_k=1^Kfract^kk^i+j+1∈Z[t]. The total determinant scaling exponent is r^2. The unique smallest subset is (1,…,r) and the unique largest is (K-r+1,…,K), proving the order and degree.
Exact mathematical passage
A_{r,K}(t):=L_K^{r^2}\mathcal H_{r,K}(t)\in\Z[t], \qquad \ord_0A_{r,K}=s_r,\qquad \deg A_{r,K}=d.
A_\varepsilonUse in source · 4 passages
Search aliases: A_ε varepsilon
Use in source · project_06.tex, line 92
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 · project_06.tex, line 103
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 · project_06.tex, line 98
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 · project_05.tex, line 107
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)
\mathrm{act}Use in source · 4 passages
Search aliases: act mathrm
Use in source · project_05.tex, line 23
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 · project_01.tex, line 109
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 · project_08.tex, line 112
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)}. }
\operatorname{Ann}Use in source · 1 passage
Search aliases: Ann operatorname
Use in source · project_07.tex, line 90
Complete-intersection control case. Let F∈Z[X] be monic and squarefree over Q, with an integer simple root a_0. Put T=Z[X]/(F) and let λ: T→Z evaluate at a_0. The ring is finite free over Z and reduced over Q. Then the congruence quotient Z/λ(AnnI) and cotangent module both have order |F'(a_0)|. To see this directly, write F=(X-a_0)G. The kernel I is generated by X-a_0. Modulo I^2, the relation is G(a_0)(X-a_0)=0, so I/I^2≅Z/F'(a_0)Z. The annihilator of I is generated by G modulo F, whose evalua…
Exact mathematical passage
\Z/\lambda(\operatorname{Ann}I)
BUse in source · 49 passages
Search aliases: B
Use in source · 02_polynomial.tex, line 9
[Mason--Stothers in characteristic zero] Let k be an algebraically closed field of characteristic zero. Suppose A,B,C∈ k[X] are nonzero, pairwise coprime, satisfy A+B=C, and are not all constant. Then max{deg A,deg B,deg C}≤ n_0(ABC)-1. This is the polynomial abc theorem. Snyder gives a short primary proof in a form that also makes the derivative hypothesis explicit~[source reference]. The proof below spells out the cancellation that will motivate the arithmetic Wronskian.
Exact mathematical passage
A,B,C\in k[X]
bUse in source · 220 passages
Search aliases: b
Use in source · 00_reading.tex, line 3
How to read this piece. addcontentslinetocsectionHow to read this piece The equation a+b=c joins two kinds of structure. Multiplication describes prime factors and their repeated powers. Addition forces those three factorizations to fit one exact relation. The abc conjecture says that this compatibility limits how large the sum can be compared with the distinct primes that appear.
Exact mathematical passage
a+b=c
B'Use in source · 2 passages
Search aliases: B'
Use in source · 02_polynomial.tex, line 17
Set W=A'B-AB'. It is nonzero. Indeed, if it vanished, then (A/B)'=0, and characteristic zero would force the rational function A/B to be constant. Coprimality would then force both A and B to be constant, and so also C.
Exact mathematical passage
W=A'B-AB'
b^2Use in source · 2 passages
Search aliases: b^2
Use in source · project_05.tex, line 86
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 · project_10.tex, line 108
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_1Use in source · 1 passage
Search aliases: B_1
Use in source · project_06.tex, line 31
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_2Use in source · 1 passage
Search aliases: B_2
Use in source · 24_padic.tex, line 109
For h_i=0, the same bound holds because its left side is nonpositive. The numerator Vandermonde only improves valuation. Since e≥1, every term in the convergent expansion has valuation at least es_r+∑_i[(k_i-i)-a_rh_i]≥ es_r-C_r,p. The ultrametric inequality gives the same bound for the sum. Multiplication by B_r restores the missing C_r,p. For rank two, B_2=2^7·3^3=3456 is a valid budget. For instance, at p=2 the two component maxima, in index order i=1,2, are three and four. At p=3 they are on…
Exact mathematical passage
B_2=2^7\cdot3^3=3456
B_dUse in source · 5 passages
Search aliases: B_d
Use in source · project_06.tex, line 10
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. }
b_hUse in source · 4 passages
Search aliases: b_h
Use in source · 24_padic.tex, line 17
Let u_1<⋯<u_r be the first r positive integers not divisible by p. Then u_j=j+lfloorfracj-1p-1rfloor, S_p,r=∑_j=1^r u_j. For h≥1, put s_h=(p-1)p^h-1 and write r=q_hs_h+b_h, 0≤ b_h<s_h. Define the finite sum E_p,r=∑_h≥1(s_hbinomq_h2+b_hq_h). The terms are zero once s_h≥ r.
Exact mathematical passage
r=q_hs_h+b_h,\qquad 0\le b_h<s_h.
b_nUse in source · 6 passages
Search aliases: b_n
Use in source · 01_foundations.tex, line 104
Why the epsilon cannot be removed. The uniform statement c≤ KR for one absolute constant K is false. There is an elementary family that proves this without using abc. [Failure of the zero-epsilon estimate] The ratios c/rad(abc) are unbounded among primitive positive additive triples. For n≥0, take a_n=1, b_n=4^3^n-1, c_n=4^3^n. These are primitive, b_n is odd, and rad(c_n)=2. We claim v_3(b_n)=n+1. At n=0 this is v_3(3)=1. If x=1+3^ku with k≥1 and 3nmid u, then x^3-1=3^k+1u(1+3^ku+3^2k-1u^2). Th…
Exact mathematical passage
a_n=1,\quad b_n=4^{3^n}-1,\quad c_n=4^{3^n}.
B_rUse in source · 3 passages
Search aliases: B_r
Use in source · 24_padic.tex, line 86
ining primes Put a_r=2r-1. For each prime p and 1≤ i≤ r, define C_r,p,i=max (0,sup_h≥1{ a_rh-(p^hlceilfrac ip^hrceil-i)}). This is a finite integer maximum because p^h eventually grows faster than h. Put C_r,p=∑_i=1^rC_r,p,i. The good-prime inequality just proved implies C_r,p=0 for p>3r-1. Hence B_r=∏_p≤3r-1p^C_r,p is a fixed positive integer depending only on the rank.
Exact mathematical passage
B_r=\prod_{p\le3r-1}p^{C_{r,p}}
\bar vUse in source · 4 passages
Search aliases: bar v bar
Use in source · project_11.tex, line 40
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)\}
\bar v_2Use in source · 1 passage
Search aliases: bar v_2 bar
Use in source · project_11.tex, line 65
Weighted dyadic row. For 2^eparallel c with e≥2 and 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}. At e=1, the output contact depth is zero and this row is omitted. 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\}.
\bar v_pUse in source · 3 passages
Search aliases: bar v_p bar
Use in source · project_11.tex, line 40
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)\}
\bigsqcupUse in source · 1 passage
Search aliases: bigsqcup
Use in source · project_04.tex, line 6
Obstruction cosets. Exclude (1,1,2) throughout this section. 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 · project_04.tex, line 6
Obstruction cosets. Exclude (1,1,2) throughout this section. 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).
CUse in source · 77 passages
Search aliases: C
Use in source · 01_foundations.tex, line 130
A known unconditional bound. Stewart and Yu proved an effective bound of a different scale~[source reference]: there is an effectively computable absolute C>0 such that every primitive positive triple satisfies c<exp (CR^1/3(log R)^3). It controls the height in terms of the radical unconditionally. Its exponential dependence is much larger than the power R^1+eps required by abc. It therefore provides a benchmark, without closing the gap this research addresses.
Exact mathematical passage
C>0
\CUse in source · 3 passages
Search aliases: C
Use in source · project_02.tex, line 56
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 · project_08.tex, line 149
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'Use in source · 1 passage
Search aliases: C'
Use in source · 02_polynomial.tex, line 22
Since C=A+B, W=A'C-AC'. At a root of C of multiplicity e, this expression is divisible by (X-α)^e-1. The three root sets are disjoint. Therefore fracABCrad(ABC)mid W. Taking degrees gives deg A+deg B+deg C-n_0(ABC)≤deg W≤deg A+deg B-1. Hence deg C≤ n_0(ABC)-1. To bound A, apply the same argument to the relation C+(-B)=A. To bound B, use C+(-A)=B. Coprimality and the radical root count are unchanged by these permutations and signs.
Exact mathematical passage
W=A'C-AC'.
C'_\epsUse in source · 3 passages
Search aliases: C'_eps eps
Use in source · project_08.tex, line 172
[Conditional filtered-transfer criterion] Suppose every primitive positive triple admits a canonical map H^0(Θ_f):Z/(c/R_c)Z⟶ H^0( C_f) satisfies, for every eps>0, with constants independent of the triple, |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'_{\eta}Use in source · 1 passage
Search aliases: C'_η eta
Use in source · 26_global_bridge.tex, line 65
If for every η>0 one proves WC≤ C_η R_aR_bR^η, then c≤ C'_ηR^1+ηlog(3c). The same absorption argument gives abc. Conversely WC≤ M_c, so the all-eps output-sector abc inequality immediately supplies this bound. The uniform combined WC estimate is therefore the exact endpoint. A bound on C alone suffices only under independent control of W, for example in a stated class where W=1.
Exact mathematical passage
c\le C'_{\eta}R^{1+\eta}\log(3c)
c^{1-\delta}Use in source · 1 passage
Search aliases: c^1-δ delta
Use in source · 26_global_bridge.tex, line 28
one rank with η_r<eps. Next choose 0<δ<fraceps-η_r1+eps. For the fixed exponent a_r, there is a constant L_r,δ such that (log(3c))^a_r≤ L_r,δc^δ for all c≥2. The elementary fact follows because a fixed logarithmic power divided by any positive power of c tends to zero, and the remainder is bounded on a finite initial interval. Thus c^1-δ≤ D_rL_r,δR^1+η_r. Raise to 1/(1-δ). The chosen inequality for δ gives frac1+η_r1-δ<1+eps. Absorb the fixed constants into K_eps and use R≥2 to obtain the desire…
Exact mathematical passage
c^{1-\delta}\le D_rL_{r,\delta}R^{1+\eta_r}.
c^2Use in source · 2 passages
Search aliases: c^2
Use in source · 25_truncation.tex, line 49
Rank two exposes the loss in one line. The first two truncated rank-two expressions are H_2,2(t)=t^3/8, H_2,3(t)=fract^38+frac4t^427+fract^5216 =fract^3(27+32t+t^2)216. Clearing the displayed denominator and the projective denominator in the second gives W(a,c)=a^3(27c^2+32ac+a^2). For every p>3 dividing c, this is congruent to a^5 modulo p. Hence it is a unit at that output prime.
Exact mathematical passage
W(a,c)=a^3(27c^2+32ac+a^2).
c^{3/2}Use in source · 1 passage
Search aliases: c^3/2
Use in source · project_05.tex, line 182
lier of this chosen parametrization and differential. The marked Frey curve need not be the optimal representative of its isogeny class. 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_ϕ^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 ene…
Exact mathematical passage
y=c^{3/2}Y
c^dUse in source · 1 passage
Search aliases: c^d
Use in source · 25_truncation.tex, line 23
For a reduced rational argument t=a/c, an obvious integral scalar is W_r,K(a,c)=c^dA_r,K(a/c). It is nonzero when 0<a<c, since the truncated positive expansion is nonzero. Thus it is a genuine nonzero arithmetic candidate. The question is whether it retains the required output-prime contact after clearing denominators.
Exact mathematical passage
W_{r,K}(a,c)=c^dA_{r,K}(a/c).
c^kUse in source · 1 passage
Search aliases: c^k
Use in source · project_09.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 · project_01.tex, line 247
ept for (1,1,2), 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. The exceptional triple (1,1,2) has zero input derivatives and is omitted from Θ. It is absorbed into the constant in the conjecture. A subpolynomial estimate Θ(a,b,c)=c^o(1) would imply the strong abc inequality. Later sections 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^\deltaUse in source · 1 passage
Search aliases: c^δ delta
Use in source · 26_global_bridge.tex, line 27
nd use Q/(ab)=c/R: (c/R)^s_r≤ C_rR^E_r(log(3c))^A_r. Taking positive roots yields c≤ D_rR^1+η_r(log(3c))^a_r, D_r=C_r^1/s_r, η_r=E_r/s_r, a_r=A_r/s_r. Fix the target eps>0. Choose one rank with η_r<eps. Next choose 0<δ<fraceps-η_r1+eps. For the fixed exponent a_r, there is a constant L_r,δ such that (log(3c))^a_r≤ L_r,δc^δ for all c≥2. The elementary fact follows because a fixed logarithmic power divided by any positive power of c tends to zero, and the remainder is bounded on a finite initial i…
Exact mathematical passage
(\log(3c))^{a_r}\le L_{r,\delta}c^\delta
c_1Use in source · 1 passage
Search aliases: c_1
Use in source · project_05.tex, line 152
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_4Use in source · 3 passages
Search aliases: c_4
Use in source · 03_consequences.tex, line 30
The precise equivalent statement uses more than the discriminant alone. For an elliptic curve over Q, let N be its conductor and let c_4,c_6 be the invariants of a global minimal integral Weierstrass equation. The modified Szpiro assertion is max{|c_4|^3,c_6^2}≤ C_eps N^6+eps for every eps>0, with a constant uniform in the elliptic curve. Oesterl e identifies this form as equivalent to the full abc conjecture~[source reference].
Exact mathematical passage
c_4,c_6
c_6Use in source · 1 passage
Search aliases: c_6
Use in source · 03_consequences.tex, line 30
The precise equivalent statement uses more than the discriminant alone. For an elliptic curve over Q, let N be its conductor and let c_4,c_6 be the invariants of a global minimal integral Weierstrass equation. The modified Szpiro assertion is max{|c_4|^3,c_6^2}≤ C_eps N^6+eps for every eps>0, with a constant uniform in the elliptic curve. Oesterl e identifies this form as equivalent to the full abc conjecture~[source reference].
Exact mathematical passage
c_4,c_6
c_6^2Use in source · 1 passage
Search aliases: c_6^2
Use in source · 03_consequences.tex, line 31
The precise equivalent statement uses more than the discriminant alone. For an elliptic curve over Q, let N be its conductor and let c_4,c_6 be the invariants of a global minimal integral Weierstrass equation. The modified Szpiro assertion is max{|c_4|^3,c_6^2}≤ C_eps N^6+eps for every eps>0, with a constant uniform in the elliptic curve. Oesterl e identifies this form as equivalent to the full abc conjecture~[source reference].
Exact mathematical passage
\max\{|c_4|^3,c_6^2\}\le C_\eps N^{6+\eps}
C_{e,N}Use in source · 2 passages
Search aliases: C_e,N
Use in source · project_02.tex, line 38
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_\epsUse in source · 7 passages
Search aliases: C_eps eps
Use in source · 03_consequences.tex, line 31
The precise equivalent statement uses more than the discriminant alone. For an elliptic curve over Q, let N be its conductor and let c_4,c_6 be the invariants of a global minimal integral Weierstrass equation. The modified Szpiro assertion is max{|c_4|^3,c_6^2}≤ C_eps N^6+eps for every eps>0, with a constant uniform in the elliptic curve. Oesterl e identifies this form as equivalent to the full abc conjecture~[source reference].
Exact mathematical passage
\max\{|c_4|^3,c_6^2\}\le C_\eps N^{6+\eps}
\mathbf C_fUse in source · 3 passages
Search aliases: C_f mathbf
Use in source · project_08.tex, line 149
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_jUse in source · 1 passage
Search aliases: c_j
Use in source · 20_signed_filter.tex, line 91
Polynomial target filters and exact count certificates. For a polynomial P(z)=∑_j=0^dc_jz^j, define the collision quadratic form E_d(P)=frac nM^2∑_x∈ EN(x)^2|P(α_x)|^2. It can be written as a positive semidefinite Gram form in the coefficients of P. Let λ_d(h)=min{ E_d(P):deg P≤ d, P(α_h)=1}. For every such filter, the target term is present with value (n/M^2)N(h)^2, so N(h)≤frac Msqrt nsqrtλ_d(h). If the right-hand side is rigorously less than one, the integer count is zero. An approximate deci…
Exact mathematical passage
P(z)=\sum_{j=0}^dc_jz^j
c_{m,1}Use in source · 1 passage
Search aliases: c_m,1
Use in source · 22_jet_filter.tex, line 43
Small coefficient norm has a limited meaning. The coefficients of P_m alternate in sign. Its absolute-coefficient polynomial evaluated at one gives ∑_s|c_m,s|=∏_j=1^m(j+1)=(m+1)!. Also c_m,1=(-1)^m, so the integer coefficient vector is primitive. After the displayed normalization, its coefficient ℓ^1 norm is one.
Exact mathematical passage
c_{m,1}=(-1)^m
c_{m,s}Use in source · 3 passages
Search aliases: c_m,s
Use in source · 22_jet_filter.tex, line 5
The universal mixed-weight jet filter. One polynomial chooses the cancelling coefficients. Let m≥0 be an integer. Define P_m(X)=X∏_j=1^m(jX-1)=∑_s=1^m+1c_m,sX^s, and the formal series Φ_m(t)=frac1(m+1)!∑_s=1^m+1c_m,sLi_s(t). Here Li_s(t)=∑_k≥1t^k/k^s is first interpreted in Q[[t]]. It also defines the usual analytic polylogarithm on |t|<1~[source reference]. The label s is its weight. The contact order will follow from its coefficients, rather than from that label.
Exact mathematical passage
P_m(X)=X\prod_{j=1}^m(jX-1)=\sum_{s=1}^{m+1}c_{m,s}X^s,
c_nUse in source · 3 passages
Search aliases: c_n
Use in source · 01_foundations.tex, line 104
Why the epsilon cannot be removed. The uniform statement c≤ KR for one absolute constant K is false. There is an elementary family that proves this without using abc. [Failure of the zero-epsilon estimate] The ratios c/rad(abc) are unbounded among primitive positive additive triples. For n≥0, take a_n=1, b_n=4^3^n-1, c_n=4^3^n. These are primitive, b_n is odd, and rad(c_n)=2. We claim v_3(b_n)=n+1. At n=0 this is v_3(3)=1. If x=1+3^ku with k≥1 and 3nmid u, then x^3-1=3^k+1u(1+3^ku+3^2k-1u^2). Th…
Exact mathematical passage
a_n=1,\quad b_n=4^{3^n}-1,\quad c_n=4^{3^n}.
C_rUse in source · 4 passages
Search aliases: C_r
Use in source · 23_hankel.tex, line 88
Since |log(b/a)|≤log c and log(c/a)≤log c, every entry has a bound by a constant times (1+log c)^j. A determinant term has total weight ∑_i=0^r-1(i+π(i)+1)=r^2. Therefore | H_r(c/a)|≤ C_r(1+log c)^r^2 for each of those consistent branches. The constant may depend on r. No nonvanishing assertion at every argument in this third chart follows from the positive expansion on 0<t<1.
Exact mathematical passage
|\mathcal H_r(c/a)|\le C_r(1+\log c)^{r^2}
C_{r,p}Use in source · 5 passages
Search aliases: C_r,p
Use in source · 24_padic.tex, line 85
An explicit fixed correction at the remaining primes. Put a_r=2r-1. For each prime p and 1≤ i≤ r, define C_r,p,i=max (0,sup_h≥1{ a_rh-(p^hlceilfrac ip^hrceil-i)}). This is a finite integer maximum because p^h eventually grows faster than h. Put C_r,p=∑_i=1^rC_r,p,i. The good-prime inequality just proved implies C_r,p=0 for p>3r-1. Hence B_r=∏_p≤3r-1p^C_r,p is a fixed positive integer depending only on the rank.
Exact mathematical passage
C_{r,p}=\sum_{i=1}^rC_{r,p,i}
C_{r,p,i}Use in source · 3 passages
Search aliases: C_r,p,i
Use in source · 24_padic.tex, line 81
An explicit fixed correction at the remaining primes. Put a_r=2r-1. For each prime p and 1≤ i≤ r, define C_r,p,i=max (0,sup_h≥1{ a_rh-(p^hlceilfrac ip^hrceil-i)}). This is a finite integer maximum because p^h eventually grows faster than h. Put C_r,p=∑_i=1^rC_r,p,i. The good-prime inequality just proved implies C_r,p=0 for p>3r-1. Hence B_r=∏_p≤3r-1p^C_r,p is a fixed positive integer depending only on the rank.
Exact mathematical passage
C_{r,p,i}=\max\!\left(0,\sup_{h\ge1}\left\{ a_rh-\left(p^h\left\lceil\frac i{p^h}\right\rceil-i\right)\right\}\right).
C_r^{1/s_r}Use in source · 1 passage
Search aliases: C_r^1/s_r
Use in source · 26_global_bridge.tex, line 19
admits a nonzero integer W_r with Q^s_rmid W_r, |W_r|≤ C_r(ab)^s_rR^E_r(log(3c))^A_r. If E_r/s_r→0 along R, then Conjecture~[source reference] follows. Nonzero integer divisibility gives Q^s_r≤|W_r|. Divide by (ab)^s_r and use Q/(ab)=c/R: (c/R)^s_r≤ C_rR^E_r(log(3c))^A_r. Taking positive roots yields c≤ D_rR^1+η_r(log(3c))^a_r, D_r=C_r^1/s_r, η_r=E_r/s_r, a_r=A_r/s_r. Fix the target eps>0. Choose one rank with η_r<eps. Next choose 0<δ<fraceps-η_r1+eps. For the fixed exponent a_r, there is a cons…
Exact mathematical passage
c\le D_rR^{1+\eta_r}(\log(3c))^{a_r}, \quad D_r=C_r^{1/s_r},\quad \eta_r=E_r/s_r,\quad a_r=A_r/s_r.
C_{\mathrm{red}}Use in source · 2 passages
Search aliases: C_red mathrm
Use in source · project_08.tex, line 21
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). Place the two-term contact complex in degrees -1,0 and define 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 · project_07.tex, line 12
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).
C_\etaUse in source · 1 passage
Search aliases: C_η eta
Use in source · 26_global_bridge.tex, line 62
If for every η>0 one proves WC≤ C_η R_aR_bR^η, then c≤ C'_ηR^1+ηlog(3c). The same absorption argument gives abc. Conversely WC≤ M_c, so the all-eps output-sector abc inequality immediately supplies this bound. The uniform combined WC estimate is therefore the exact endpoint. A bound on C alone suffices only under independent control of W, for example in a stated class where W=1.
Exact mathematical passage
WC\le C_\eta R_aR_bR^\eta,
c_\phiUse in source · 2 passages
Search aliases: c_ϕ phi
Use in source · project_05.tex, line 160
Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be a nonconstant modular parametrization, and let ϕ_E^*ω_E = 2π i c_ϕ f_E(z) dz. Here c_ϕ is the differential multiplier of this chosen parametrization and differential. The marked Frey curve need not be the optimal representative of its isogeny class. 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_ϕ^2 normf_E_Pet^2. For the Frey model, scaling x=cX, …
Exact mathematical passage
\phi_E^*\omega_E = 2\pi i\,c_\phi f_E(z)\,dz.
c_\phi^2Use in source · 1 passage
Search aliases: c_ϕ^2 phi
Use in source · project_05.tex, line 174
parametrization, and let ϕ_E^*ω_E = 2π i c_ϕ f_E(z) dz. Here c_ϕ is the differential multiplier of this chosen parametrization and differential. The marked Frey curve need not be the optimal representative of its isogeny class. 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_ϕ^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 shap…
Exact mathematical passage
\boxed{ \deg(\phi_E)A(E) = 4\pi^2 c_\phi^2 \norm{f_E}_{\mathrm{Pet}}^2. }
\mathrm{coh}Use in source · 6 passages
Search aliases: coh mathrm
Use in source · project_11.tex, line 137
er normalization, every output prime with nontrivial residual depth produces a primitive row g_p ∈ (Z/p^χ_pZ)^S_ab satisfying g_p·ν_0 ≡0 pmodp^χ_p, where ν_0 is the Kummer-normalized input exponent vector. The Chinese remainder theorem combines the local rows into one primitive row g∈(Z/CZ)^S_ab. Define the coherence character χ_coh(x) = exp(frac2π iCg· x). Its kernel is the adelic coherence lattice Λ_coh = {x∈Z^S_ab:g· x≡0pmod C}. Since g is primitive modulo C, [Z^S_ab:Λ_coh]=C and Z^S_ab/Λ_coh…
Exact mathematical passage
\boxed{ \chi_{\mathrm{coh}}(x) = \exp\left(\frac{2\pi i}{C}g\cdot x\right). }
\cokerUse in source · 1 passage
Search aliases: coker
Use in source · project_08.tex, line 100
Exact cyclic transducer. Let m,η be positive integers and put 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 · project_03.tex, line 46
[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. }
\cosUse in source · 1 passage
Search aliases: cos
Use in source · 20_signed_filter.tex, line 15
Let N(x) count assignments producing x∈ G. Repeated contributions remain separate assignments, so M=∑_x∈ GN(x)=3^s, μ(x)=N(x)/M. For a character χ∈widehat G, define widehatμ(χ)=∑_xμ(x)χ(x) =∏_j=1^sfrac1+χ(γ_j)+χ(-γ_j)3. On a cyclic channel G=Z/nZ, this becomes m(k)=∏_j=1^sfrac1+2cos(2π kγ_j/n)3. This is the discrete cosine or COS expression used in the project. It is an exact characteristic function. Individual cosine factors may be negative. No positivity of all Fourier samples is assumed.
Exact mathematical passage
m(k)=\prod_{j=1}^s\frac{1+2\cos(2\pi k\gamma_j/n)}3.
DUse in source · 16 passages
Search aliases: D
Use in source · 02_polynomial.tex, line 42
Where the integer argument stops. An ordinary integer has no distinguished derivative with both an exact Leibniz rule and an automatic relation D(a)+D(b)=D(c). Arithmetic derivatives provide a family of Leibniz maps. Additivity at one triple then imposes a lattice constraint on the prime-coordinate values.
Exact mathematical passage
D(a)+D(b)=D(c)
\mathrm dUse in source · 15 passages
Search aliases: d mathrm
Use in source · project_01.tex, line 21
e 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. First define the global free module over all primes, and restrict it to Ω_S for S-supported integers. For such 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 l…
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 · project_10.tex, line 112
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 · project_10.tex, line 108
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^{m+2}Use in source · 2 passages
Search aliases: D^m+2
Use in source · 22_jet_filter.tex, line 70
An exact differential identity. Let D=t dd/dd t be the Euler operator. Termwise formal differentiation is valid. Applying D^m+2 multiplies the coefficient at t^k by k^m+2, so D^m+2Φ_m=(-1)^m∑_k≥1binomkm+1t^k =(-1)^mfract^m+1(1-t)^m+2. The final equality is the elementary generating function for the binomial coefficients.
Exact mathematical passage
D^{m+2}
D_fUse in source · 1 passage
Search aliases: D_f
Use in source · project_07.tex, line 98
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). Let E/Q_q have split multiplicative reduction with Tate parameter q_E, and let ℓ≠ q. Scalarity of the full local Galois representation on E[ℓ^n] forces n≤min{v_ℓ(q-1),v_ℓ(v_q(q_E))}. In a Tate basis, the cyclotomic graded diagonal and the inertia Kummer extension respectively force ℓ^nmid(q-1) and ℓ^nmid v_q(q_E)~[source r…
Exact mathematical passage
D_f(\ell,q).
D_rUse in source · 2 passages
Search aliases: D_r
Use in source · 26_global_bridge.tex, line 19
admits a nonzero integer W_r with Q^s_rmid W_r, |W_r|≤ C_r(ab)^s_rR^E_r(log(3c))^A_r. If E_r/s_r→0 along R, then Conjecture~[source reference] follows. Nonzero integer divisibility gives Q^s_r≤|W_r|. Divide by (ab)^s_r and use Q/(ab)=c/R: (c/R)^s_r≤ C_rR^E_r(log(3c))^A_r. Taking positive roots yields c≤ D_rR^1+η_r(log(3c))^a_r, D_r=C_r^1/s_r, η_r=E_r/s_r, a_r=A_r/s_r. Fix the target eps>0. Choose one rank with η_r<eps. Next choose 0<δ<fraceps-η_r1+eps. For the fixed exponent a_r, there is a cons…
Exact mathematical passage
c\le D_rR^{1+\eta_r}(\log(3c))^{a_r}, \quad D_r=C_r^{1/s_r},\quad \eta_r=E_r/s_r,\quad a_r=A_r/s_r.
\mathrm d_xUse in source · 13 passages
Search aliases: d_x mathrm
Use in source · project_01.tex, line 46
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 · 3 passages
Search aliases: D_y
Use in source · project_02.tex, line 132
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
D_y(n)=n\sum_{p\mid n}\frac{y_p}{p}.
\ddUse in source · 2 passages
Search aliases: dd
Use in source · 22_jet_filter.tex, line 70
An exact differential identity. Let D=t dd/dd t be the Euler operator. Termwise formal differentiation is valid. Applying D^m+2 multiplies the coefficient at t^k by k^m+2, so D^m+2Φ_m=(-1)^m∑_k≥1binomkm+1t^k =(-1)^mfract^m+1(1-t)^m+2. The final equality is the elementary generating function for the binomial coefficients.
Exact mathematical passage
D=t\,\dd/\dd t
\degUse in source · 7 passages
Search aliases: deg
Use in source · 02_polynomial.tex, line 4
The polynomial theorem and the missing integer analogue. Roots play the role of distinct primes. For a nonzero polynomial F over an algebraically closed field, let rad(F) be the product of its distinct monic linear factors. Its degree counts distinct roots. Write n_0(F)=degrad(F).
Exact mathematical passage
n_0(F)=\deg\operatorname{rad}(F)
\detUse in source · 5 passages
Search aliases: det
Use in source · 23_hankel.tex, line 5
A nonzero Hankel contact determinant. Independent moment directions replace one filter. For an integer r≥1, define H_r(t)=det(Li_i+j+1(t))_0≤ i,j<r. It uses weights one through 2r-1. The expression is a formal series over Q and an analytic function for |t|<1. No algebraicity at a rational argument is assumed.
Exact mathematical passage
\mathcal H_r(t)=\det\bigl(\Li_{i+j+1}(t)\bigr)_{0\le i,j<r}.
\operatorname{diag}Use in source · 4 passages
Search aliases: diag operatorname
Use in source · 20_signed_filter.tex, line 84
on a cyclic channel with r occupied residues is determined by the 2r exact consecutive samples m(0),…,m(2r-1). Write A(z)=∏_x∈ E(z-α_x)=z^r+∑_j=0^r-1a_jz^j. For every k, ∑_j=0^r-1a_jm(k+j)=-m(k+r). The coefficient matrix for 0≤ k<r is the Hankel matrix H=(m(i+j))_0≤ i,j<r. With V=(α_x^i)_i,x and D=diag(N(x)/M), one has H=VDV^ T. The square Vandermonde V is invertible and every diagonal entry of D is nonzero. Thus H is invertible, and the 2r samples determine the monic annihilator A.
Exact mathematical passage
D=\operatorname{diag}(N(x)/M)
\operatorname{div}Use in source · 2 passages
Search aliases: div operatorname
Use in source · project_08.tex, line 13
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). Place the two-term contact complex in degrees -1,0 and define 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).
EUse in source · 34 passages
Search aliases: E
Use in source · 03_consequences.tex, line 25
Elliptic curves and the modified Szpiro comparison. The marked Frey curve E:y^2=x(x-a)(x+b) translates the additive triple into an elliptic curve. Its discriminant sees repeated depth, while its conductor sees prime support, with local model corrections at the small primes. This makes elliptic height estimates a natural companion to abc.
Exact mathematical passage
E:y^2=x(x-a)(x+b)
\mathcal EUse in source · 2 passages
Search aliases: E mathcal
Use in source · 20_signed_filter.tex, line 92
Polynomial target filters and exact count certificates. For a polynomial P(z)=∑_j=0^dc_jz^j, define the collision quadratic form E_d(P)=frac nM^2∑_x∈ EN(x)^2|P(α_x)|^2. It can be written as a positive semidefinite Gram form in the coefficients of P. Let λ_d(h)=min{ E_d(P):deg P≤ d, P(α_h)=1}. For every such filter, the target term is present with value (n/M^2)N(h)^2, so N(h)≤frac Msqrt nsqrtλ_d(h). If the right-hand side is rigorously less than one, the integer count is zero. An approximate deci…
Exact mathematical passage
\mathcal E_d(P)=\frac n{M^2}\sum_{x\in E}N(x)^2|P(\alpha_x)|^2.
e^{1/4}Use in source · 4 passages
Search aliases: e^1/4
Use in source · 23_hankel.tex, line 55
Real positivity and a bound uniform in rank. [Positive real determinant] For every r≥1 and every real 0<t<1, 0< H_r(t)≤ e^1/4[-log(1-t)]. The Cauchy--Binet expansion consists of positive terms and contains the nonzero term (1,…,r), so positivity follows. Equivalently, the real moment matrix is positive definite: a nonzero polynomial of degree below r cannot vanish at all the distinct nodes k^-1.
Exact mathematical passage
0<\mathcal H_r(t)\le e^{1/4}[-\log(1-t)].
e^{2\pi ix/n}Use in source · 1 passage
Search aliases: e^2π ix/n pi
Use in source · 20_signed_filter.tex, line 62
Exact innovation rank in a cyclic channel. Let G=Z/nZ, α_x=e^2π ix/n and m(k)=∑_x∈ EfracN(x)Mα_x^k. Form the shift matrix S_d=(m(k+j))_0≤ k<n, 0≤ j≤ d. It factors as a Fourier-column matrix times a Vandermonde matrix on the distinct occupied nodes. The Fourier columns are independent, and all weights N(x)/M are nonzero. Therefore rankS_d=min(r,d+1). Before degree r, a small singular value is a conditioning phenomenon, not an exact reduction in the number of occupied residues.
Exact mathematical passage
\alpha_x=e^{2\pi ix/n}
e_1Use in source · 3 passages
Search aliases: e_1
Use in source · project_09.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_{a,b,c}Use in source · 1 passage
Search aliases: E_a,b,c
Use in source · project_05.tex, line 80
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_dUse in source · 2 passages
Search aliases: E_d mathcal
Use in source · 20_signed_filter.tex, line 92
Polynomial target filters and exact count certificates. For a polynomial P(z)=∑_j=0^dc_jz^j, define the collision quadratic form E_d(P)=frac nM^2∑_x∈ EN(x)^2|P(α_x)|^2. It can be written as a positive semidefinite Gram form in the coefficients of P. Let λ_d(h)=min{ E_d(P):deg P≤ d, P(α_h)=1}. For every such filter, the target term is present with value (n/M^2)N(h)^2, so N(h)≤frac Msqrt nsqrtλ_d(h). If the right-hand side is rigorously less than one, the integer count is zero. An approximate deci…
Exact mathematical passage
\mathcal E_d(P)=\frac n{M^2}\sum_{x\in E}N(x)^2|P(\alpha_x)|^2.
e_fUse in source · 2 passages
Search aliases: e_f
Use in source · project_06.tex, line 118
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 · 4 passages
Search aliases: e_i
Use in source · project_09.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}.
e_pUse in source · 6 passages
Search aliases: e_p
Use in source · 01_foundations.tex, line 4
Radical support and the exact conjecture. The multiplicative quantity that ignores repetition. Every positive integer has a unique prime factorization n=∏_pmid np^e_p, e_p≥1. Its radical is rad(n)=∏_pmid np, rad(1)=1. The radical retains which primes occur and discards how many times they occur. Thus rad(72)=rad(2^3·3^2)=6, whereas 72/6=12 records the repeated-power contribution. If n is squarefree, then rad(n)=n.
Exact mathematical passage
n=\prod_{p\mid n}p^{e_p},\qquad e_p\ge1.
E_{p,r}Use in source · 5 passages
Search aliases: E_p,r
Use in source · 24_padic.tex, line 21
Let u_1<⋯<u_r be the first r positive integers not divisible by p. Then u_j=j+lfloorfracj-1p-1rfloor, S_p,r=∑_j=1^r u_j. For h≥1, put s_h=(p-1)p^h-1 and write r=q_hs_h+b_h, 0≤ b_h<s_h. Define the finite sum E_p,r=∑_h≥1(s_hbinomq_h2+b_hq_h). The terms are zero once s_h≥ r.
Exact mathematical passage
E_{p,r}=\sum_{h\ge1}\left(s_h\binom{q_h}{2}+b_hq_h\right).
E_rUse in source · 5 passages
Search aliases: E_r
Use in source · 26_global_bridge.tex, line 6
[Conditional integral witness criterion] Let R be an unbounded set of positive integer ranks, and put s_r=r(r+1)/2. Suppose that for each r∈ R there are constants C_r>0, A_r≥0 and E_r≥0, independent of the primitive triple, such that every primitive positive triple admits a nonzero integer W_r with Q^s_rmid W_r, |W_r|≤ C_r(ab)^s_rR^E_r(log(3c))^A_r. If E_r/s_r→0 along R, then Conjecture~[source reference] follows. Nonzero integer divisibility gives Q^s_r≤|W_r|. Divide by (ab)^s_r and use Q/(ab)=…
Exact mathematical passage
E_r\ge0
e_rUse in source · 3 passages
Search aliases: e_r
Use in source · project_09.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 · project_07.tex, line 5
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
\epsUse in source · 56 passages
Search aliases: eps
Use in source · 01_foundations.tex, line 30
[Masser--Oesterl e abc] For every real eps>0 there is a constant K_eps>0, depending only on eps, such that every positive integer triple with a+b=c and gcd(a,b)=1 satisfies c≤ K_eps rad(abc)^1+eps. The order of the quantifiers is the statement. First choose eps. Then choose a constant that works for every primitive triple. The constant may become large as eps approaches zero. It may not depend on the particular triple.
Exact mathematical passage
\eps>0
\expUse in source · 2 passages
Search aliases: exp
Use in source · 01_foundations.tex, line 131
A known unconditional bound. Stewart and Yu proved an effective bound of a different scale~[source reference]: there is an effectively computable absolute C>0 such that every primitive positive triple satisfies c<exp (CR^1/3(log R)^3). It controls the height in terms of the radical unconditionally. Its exponential dependence is much larger than the power R^1+eps required by abc. It therefore provides a benchmark, without closing the gap this research addresses.
Exact mathematical passage
c<\exp\!\left(CR^{1/3}(\log R)^3\right).
\FUse in source · 5 passages
Search aliases: F
Use in source · project_08.tex, line 70
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 for j≥2. The j=1 sequence has zero left term and is split. These carry extensions are precisely what the split persistence packet forgot.
Exact mathematical passage
\F_p
fUse in source · 14 passages
Search aliases: f
Use in source · project_02.tex, line 12
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. Set V_0(T)=0 for an absent prime. 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 · 4 passages
Search aliases: F'
Use in source · project_07.tex, line 91
Complete-intersection control case. Let F∈Z[X] be monic and squarefree over Q, with an integer simple root a_0. Put T=Z[X]/(F) and let λ: T→Z evaluate at a_0. The ring is finite free over Z and reduced over Q. Then the congruence quotient Z/λ(AnnI) and cotangent module both have order |F'(a_0)|. To see this directly, write F=(X-a_0)G. The kernel I is generated by X-a_0. Modulo I^2, the relation is G(a_0)(X-a_0)=0, so I/I^2≅Z/F'(a_0)Z. The annihilator of I is generated by G modulo F, whose evalua…
Exact mathematical passage
|F'(a_0)|.
F_{a,b,c}Use in source · 3 passages
Search aliases: F_a,b,c
Use in source · project_03.tex, line 176
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 · 3 passages
Search aliases: f_E
Use in source · project_05.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
f_E
F_{i+j+1,p}Use in source · 1 passage
Search aliases: F_i+j+1,p
Use in source · 24_padic.tex, line 5
Exact local determinant valuation and denominator budget. Depletion gives an integral local matrix. Fix a prime p and an integer r≥1. Define F_s,p(t)=∑_pnmid kfract^kk^s, H_r,p(t)=det(F_i+j+1,p(t))_0≤ i,j<r. These series converge at every t∈Q_p with v_p(t)>0. They are integral power series. The Cauchy--Binet expansion from Section~[source reference] now ranges only over increasing p-unit indices.
Exact mathematical passage
F_{s,p}(t)=\sum_{p\nmid k}\frac{t^k}{k^s},\qquad \mathcal H_{r,p}(t)=\det(F_{i+j+1,p}(t))_{0\le i,j<r}.
\F_pUse in source · 5 passages
Search aliases: F_p F
Use in source · project_08.tex, line 70
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 for j≥2. The j=1 sequence has zero left term and is split. These carry extensions are precisely what the split persistence packet forgot.
Exact mathematical passage
\F_p
F_{s,p}Use in source · 1 passage
Search aliases: F_s,p
Use in source · 24_padic.tex, line 5
Exact local determinant valuation and denominator budget. Depletion gives an integral local matrix. Fix a prime p and an integer r≥1. Define F_s,p(t)=∑_pnmid kfract^kk^s, H_r,p(t)=det(F_i+j+1,p(t))_0≤ i,j<r. These series converge at every t∈Q_p with v_p(t)>0. They are integral power series. The Cauchy--Binet expansion from Section~[source reference] now ranges only over increasing p-unit indices.
Exact mathematical passage
F_{s,p}(t)=\sum_{p\nmid k}\frac{t^k}{k^s},\qquad \mathcal H_{r,p}(t)=\det(F_{i+j+1,p}(t))_{0\le i,j<r}.
GUse in source · 12 passages
Search aliases: G
Use in source · 20_signed_filter.tex, line 3
Signed arithmetic COS and the sparse spectral leaf. What the finite assignment signal records. The spectral branch of the project asks which prime assignments realize a prescribed residue. Its exact object is a finite distribution. Fix a finite abelian group G and s eligible variable support primes. At the jth prime choose one of three permitted contributions, represented by 0,γ_j,-γ_j in additive group notation. The sum of the chosen contributions is the assignment residue.
Exact mathematical passage
G
gUse in source · 24 passages
Search aliases: g
Use in source · project_01.tex, line 253
Three-sector compression. For the empty unit sector, set g_1=0, t_1=0, and ρ_1(0)=0. Any nonzero target in that sector is infeasible. Within a nonempty 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…
Exact mathematical passage
g_1=0
g_1Use in source · 1 passage
Search aliases: g_1
Use in source · project_01.tex, line 253
Three-sector compression. For the empty unit sector, set g_1=0, t_1=0, and ρ_1(0)=0. Any nonzero target in that sector is infeasible. Within a nonempty 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…
Exact mathematical passage
g_1=0
g_aUse in source · 4 passages
Search aliases: g_a
Use in source · project_01.tex, line 260
Three-sector compression. For the empty unit sector, set g_1=0, t_1=0, and ρ_1(0)=0. Any nonzero target in that sector is infeasible. Within a nonempty 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…
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 · project_01.tex, line 263
Three-sector compression. For the empty unit sector, set g_1=0, t_1=0, and ρ_1(0)=0. Any nonzero target in that sector is infeasible. Within a nonempty 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…
Exact mathematical passage
g_b
g_cUse in source · 5 passages
Search aliases: g_c
Use in source · project_01.tex, line 263
Three-sector compression. For the empty unit sector, set g_1=0, t_1=0, and ρ_1(0)=0. Any nonzero target in that sector is infeasible. Within a nonempty 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…
Exact mathematical passage
g_c
g_pUse in source · 2 passages
Search aliases: g_p
Use in source · project_11.tex, line 121
The coherence character. Set χ_p=0 for absent or squarefree output channels. When the residual modulus is one, use the trivial row and character. After normalization, every output prime with nontrivial residual depth produces a primitive row g_p ∈ (Z/p^χ_pZ)^S_ab satisfying g_p·ν_0 ≡0 pmodp^χ_p, where ν_0 is the Kummer-normalized input exponent vector. The Chinese remainder theorem combines the local rows into one primitive row g∈(Z/CZ)^S_ab. Define the coherence character χ_coh(x) = exp(frac2π …
Exact mathematical passage
g_p \in (\Z/p^{\chi_p}\Z)^{S_{ab}}
g_{\mathrm{val}}Use in source · 1 passage
Search aliases: g_val mathrm
Use in source · project_09.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 · 14 passages
Search aliases: gcd
Use in source · 01_foundations.tex, line 14
For a positive primitive additive triple a+b=c, gcd(a,b)=1, all three pairs are coprime. For example, a divisor of a and c also divides c-a=b, hence is one. Therefore the prime supports are disjoint and R=rad(abc)=R_aR_bR_c, R_u=rad(u). Put M_u=u/R_u. The exact decomposition is abc=R M_aM_bM_c. The conjecture compares c with R, rather than with abc itself. Since abc already contains c, an estimate against the full product would lose the intended content.
Exact mathematical passage
a+b=c,\qquad \gcd(a,b)=1,
\gcd_{p\mid u}Use in source · 1 passage
Search aliases: gcd_pmid u gcd mid
Use in source · project_02.tex, line 242
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,
\ggUse in source · 1 passage
Search aliases: gg
Use in source · 20_signed_filter.tex, line 54
If the assignment model has M=3^s with s≤ω(R), choose δ=eps/2 in both estimates. Then fracφ(Q)Mgg_eps R^1+eps. Consequently a sufficiently large hypothetical exponent-gap exception, represented in this joint unit-group model, lies on the sparse side of the energy barrier. Uniform spectral mixing cannot be the universal finishing argument for this regime.
Exact mathematical passage
\frac{\varphi(Q)}M\gg_\eps R^{1+\eps}.
\gg_\epsUse in source · 1 passage
Search aliases: gg_eps gg eps
Use in source · 20_signed_filter.tex, line 54
If the assignment model has M=3^s with s≤ω(R), choose δ=eps/2 in both estimates. Then fracφ(Q)Mgg_eps R^1+eps. Consequently a sufficiently large hypothetical exponent-gap exception, represented in this joint unit-group model, lies on the sparse side of the energy barrier. Uniform spectral mixing cannot be the universal finishing argument for this regime.
Exact mathematical passage
\frac{\varphi(Q)}M\gg_\eps R^{1+\eps}.
\mathrm{glob}Use in source · 1 passage
Search aliases: glob mathrm
Use in source · project_08.tex, line 149
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.
\grUse in source · 1 passage
Search aliases: gr
Use in source · project_08.tex, line 71
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 for j≥2. The j=1 sequence has zero left term and is split. These carry extensions are precisely what the split persistence packet forgot.
Exact mathematical passage
\gr\Jcal_{p,e}\cong(\F_p)^{e-1}.
HUse in source · 22 passages
Search aliases: H
Use in source · 20_signed_filter.tex, line 84
lic Prony reconstruction] The distribution on a cyclic channel with r occupied residues is determined by the 2r exact consecutive samples m(0),…,m(2r-1). Write A(z)=∏_x∈ E(z-α_x)=z^r+∑_j=0^r-1a_jz^j. For every k, ∑_j=0^r-1a_jm(k+j)=-m(k+r). The coefficient matrix for 0≤ k<r is the Hankel matrix H=(m(i+j))_0≤ i,j<r. With V=(α_x^i)_i,x and D=diag(N(x)/M), one has H=VDV^ T. The square Vandermonde V is invertible and every diagonal entry of D is nonzero. Thus H is invertible, and the 2r samples dete…
Exact mathematical passage
H=(m(i+j))_{0\le i,j<r}
\mathcal HUse in source · 21 passages
Search aliases: H mathcal
Use in source · 23_hankel.tex, line 5
A nonzero Hankel contact determinant. Independent moment directions replace one filter. For an integer r≥1, define H_r(t)=det(Li_i+j+1(t))_0≤ i,j<r. It uses weights one through 2r-1. The expression is a formal series over Q and an analytic function for |t|<1. No algebraicity at a rational argument is assumed.
Exact mathematical passage
\mathcal H_r(t)=\det\bigl(\Li_{i+j+1}(t)\bigr)_{0\le i,j<r}.
H^0Use in source · 4 passages
Search aliases: H^0
Use in source · project_08.tex, line 44
[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 · 2 passages
Search aliases: H^1
Use in source · project_09.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.
\mathcal H_1Use in source · 1 passage
Search aliases: H_1 mathcal
Use in source · 23_hankel.tex, line 11
For the first two ranks, H_1=Li_1, H_2=Li_1Li_3-Li_2^2. The rank-two determinant compares genuinely different moment directions. Its order-three contact is not inferred from the presence of weight three.
Exact mathematical passage
\mathcal H_1=\Li_1,\qquad \mathcal H_2=\Li_1\Li_3-\Li_2^2.
\mathcal H_2Use in source · 2 passages
Search aliases: H_2 mathcal
Use in source · 23_hankel.tex, line 11
For the first two ranks, H_1=Li_1, H_2=Li_1Li_3-Li_2^2. The rank-two determinant compares genuinely different moment directions. Its order-three contact is not inferred from the presence of weight three.
Exact mathematical passage
\mathcal H_1=\Li_1,\qquad \mathcal H_2=\Li_1\Li_3-\Li_2^2.
\mathcal H_{2,2}Use in source · 1 passage
Search aliases: H_2,2 mathcal
Use in source · 25_truncation.tex, line 41
Rank two exposes the loss in one line. The first two truncated rank-two expressions are H_2,2(t)=t^3/8, H_2,3(t)=fract^38+frac4t^427+fract^5216 =fract^3(27+32t+t^2)216. Clearing the displayed denominator and the projective denominator in the second gives W(a,c)=a^3(27c^2+32ac+a^2). For every p>3 dividing c, this is congruent to a^5 modulo p. Hence it is a unit at that output prime.
Exact mathematical passage
\mathcal H_{2,2}(t)=t^3/8,
\mathcal H_{2,3}Use in source · 1 passage
Search aliases: H_2,3 mathcal
Use in source · 25_truncation.tex, line 44
Rank two exposes the loss in one line. The first two truncated rank-two expressions are H_2,2(t)=t^3/8, H_2,3(t)=fract^38+frac4t^427+fract^5216 =fract^3(27+32t+t^2)216. Clearing the displayed denominator and the projective denominator in the second gives W(a,c)=a^3(27c^2+32ac+a^2). For every p>3 dividing c, this is congruent to a^5 modulo p. Hence it is a unit at that output prime.
Exact mathematical passage
\mathcal H_{2,3}(t)=\frac{t^3}{8}+\frac{4t^4}{27}+\frac{t^5}{216} =\frac{t^3(27+32t+t^2)}{216}.
\mathcal H_{2,p}Use in source · 1 passage
Search aliases: H_2,p mathcal
Use in source · 24_padic.tex, line 54
The expansion converges p-adically. A unique term of minimum valuation cannot be cancelled by the remaining terms. Its valuation is therefore the valuation of the whole determinant. For rank two, v_p( H_2,p(t))= 3e, · p≠2, 4e+2, · p=2. At two the first unit indices are 1,3, of sum four, and their difference contributes one valuation before squaring. For p>r in arbitrary rank, the first r indices are 1,…,r and all their differences are units. Thus the general formula reduces to es_r.
Exact mathematical passage
v_p(\mathcal H_{2,p}(t))= \begin{cases}3e,&p\ne2,\\4e+2,&p=2.\end{cases}
\mathcal H_3Use in source · 3 passages
Search aliases: H_3 mathcal
Use in source · project_10.tex, line 155
Conditional motivic transfer. Let K_β be the cyclic group generated by the balanced Kummer class. Suppose that for every primitive positive triple there exists an integral finite object H_3(a,b,c) and a canonical map Θ_3:K_β⟶ H_3(a,b,c) such that, for every eps>0, constants independent of the triple give |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_iUse in source · 9 passages
Search aliases: h_i
Use in source · 24_padic.tex, line 75
If h_i=v_p(k_i)>0, then k_i-i≥ p^h_i-r>a_rh_i. If h_i=0, then k_i-i≥0 and there is no denominator loss. Thus every subset other than (1,…,r) has valuation strictly above es_r. The first subset has a unit coefficient because its indices and differences are smaller than p. Its term has valuation es_r, and a unique minimum again proves equality. The strict prime threshold is retained. A good-prime equality is stronger than the all-prime lower bound below.
Exact mathematical passage
h_i=v_p(k_i)>0
\mathcal H_rUse in source · 9 passages
Search aliases: H_r mathcal
Use in source · 23_hankel.tex, line 5
A nonzero Hankel contact determinant. Independent moment directions replace one filter. For an integer r≥1, define H_r(t)=det(Li_i+j+1(t))_0≤ i,j<r. It uses weights one through 2r-1. The expression is a formal series over Q and an analytic function for |t|<1. No algebraicity at a rational argument is assumed.
Exact mathematical passage
\mathcal H_r(t)=\det\bigl(\Li_{i+j+1}(t)\bigr)_{0\le i,j<r}.
\mathcal H_{r,K}Use in source · 2 passages
Search aliases: H_r,K mathcal
Use in source · 25_truncation.tex, line 4
A rational truncation that fails the output contact test. A genuine algebraic candidate can be tested exactly. Fix integers r≥1 and K≥ r. Replace each polylogarithm by its finite series through index K, obtaining a rational polynomial H_r,K(t). Its Cauchy--Binet expansion has the same positive subset coefficients, restricted to 1≤ k_i≤ K.
Exact mathematical passage
\mathcal H_{r,K}(t)
\mathcal H_{r,p}Use in source · 2 passages
Search aliases: H_r,p mathcal
Use in source · 24_padic.tex, line 5
Exact local determinant valuation and denominator budget. Depletion gives an integral local matrix. Fix a prime p and an integer r≥1. Define F_s,p(t)=∑_pnmid kfract^kk^s, H_r,p(t)=det(F_i+j+1,p(t))_0≤ i,j<r. These series converge at every t∈Q_p with v_p(t)>0. They are integral power series. The Cauchy--Binet expansion from Section~[source reference] now ranges only over increasing p-unit indices.
Exact mathematical passage
F_{s,p}(t)=\sum_{p\nmid k}\frac{t^k}{k^s},\qquad \mathcal H_{r,p}(t)=\det(F_{i+j+1,p}(t))_{0\le i,j<r}.
I^2Use in source · 2 passages
Search aliases: I^2
Use in source · project_07.tex, line 94
efree over Q, with an integer simple root a_0. Put T=Z[X]/(F) and let λ: T→Z evaluate at a_0. The ring is finite free over Z and reduced over Q. Then the congruence quotient Z/λ(AnnI) and cotangent module both have order |F'(a_0)|. To see this directly, write F=(X-a_0)G. The kernel I is generated by X-a_0. Modulo I^2, the relation is G(a_0)(X-a_0)=0, so I/I^2≅Z/F'(a_0)Z. The annihilator of I is generated by G modulo F, whose evaluation is G(a_0)=F'(a_0). Thus the congruence ideal is F'(a_0)Z and…
Exact mathematical passage
I^2
I_aUse in source · 1 passage
Search aliases: I_a
Use in source · project_04.tex, line 118
[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 · project_04.tex, line 118
[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 · project_04.tex, line 118
[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 · 6 passages
Search aliases: I_N
Use in source · project_02.tex, line 53
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
I_N=\ker\bigl(A_N\xrightarrow{\,T\mapsto1\,}\Z\bigr)
\imUse in source · 2 passages
Search aliases: im
Use in source · project_08.tex, line 100
Exact cyclic transducer. Let m,η be positive integers and put 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. }
\infUse in source · 1 passage
Search aliases: inf
Use in source · project_03.tex, line 112
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 · project_03.tex, line 112
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 · 4 passages
Search aliases: J
Use in source · project_03.tex, line 165
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}).
\mathbf JUse in source · 4 passages
Search aliases: J mathbf
Use in source · project_08.tex, line 27
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). Place the two-term contact complex in degrees -1,0 and define 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_aUse in source · 2 passages
Search aliases: J_a
Use in source · project_03.tex, line 176
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 · 2 passages
Search aliases: J_b
Use in source · project_03.tex, line 176
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 · 2 passages
Search aliases: J_c
Use in source · project_03.tex, line 176
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 · 4 passages
Search aliases: J_c mathbf
Use in source · project_08.tex, line 27
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). Place the two-term contact complex in degrees -1,0 and define 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 · project_03.tex, line 165
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}).
\JcalUse in source · 3 passages
Search aliases: Jcal
Use in source · project_08.tex, line 59
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 for j≥2. The j=1 sequence has zero left term and is split. 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.
\Jcal_{p,e}Use in source · 3 passages
Search aliases: Jcal_p,e Jcal
Use in source · project_08.tex, line 59
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 for j≥2. The j=1 sequence has zero left term and is split. 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.
kUse in source · 90 passages
Search aliases: k
Use in source · 01_foundations.tex, line 111
his without using abc. [Failure of the zero-epsilon estimate] The ratios c/rad(abc) are unbounded among primitive positive additive triples. For n≥0, take a_n=1, b_n=4^3^n-1, c_n=4^3^n. These are primitive, b_n is odd, and rad(c_n)=2. We claim v_3(b_n)=n+1. At n=0 this is v_3(3)=1. If x=1+3^ku with k≥1 and 3nmid u, then x^3-1=3^k+1u(1+3^ku+3^2k-1u^2). The bracket is a unit modulo three, so cubing increases the valuation by exactly one. This proves the claim by induction.
Exact mathematical passage
x=1+3^ku
k^{-(2k-1)}Use in source · 1 passage
Search aliases: k^-(2k-1)
Use in source · 23_hankel.tex, line 40
[Exact contact and first coefficient] Put s_r=r(r+1)/2. Then ord_0 H_r=s_r, [t^s_r] H_r=∏_k=1^r k^-(2k-1). The unique increasing r-tuple of positive indices with least sum is (1,…,r), of sum s_r. Its coefficient is nonzero. Its Vandermonde numerator is ∏_j=1^r-1(j!)^2. Dividing this by ∏_k=1^r k^2r-1 leaves exponent 2k-1 in the denominator at each k, giving the formula. For r=2 the first coefficient is 1/8. For r=3 it is 1/(2^3 3^5). The contact orders are 1,3,6,10,…, the triangular numbers.
Exact mathematical passage
\ord_0\mathcal H_r=s_r,\qquad [t^{s_r}]\mathcal H_r=\prod_{k=1}^r k^{-(2k-1)}.
k^{-(r-1)}Use in source · 1 passage
Search aliases: k^-(r-1)
Use in source · 23_hankel.tex, line 25
[Cauchy--Binet expansion] H_r(t)=∑_1≤ k_1<⋯<k_r t^k_1+⋯+k_r frac∏_i<j(k_j-k_i)^2∏_i=1^r k_i^2r-1. Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and including the factors t^k_i/k_i gives the stated coefficient. In detail, ∏_i<j(k_j^-1-k_i^-1)^2 =frac∏_i<j(k_j-k_i)^2∏_i k_i^2(r-1). Passing to the formal …
Exact mathematical passage
v_k=(1,k^{-1},\ldots,k^{-(r-1)})^{\mathsf T}
k^{-1}Use in source · 2 passages
Search aliases: k^-1
Use in source · 23_hankel.tex, line 25
[Cauchy--Binet expansion] H_r(t)=∑_1≤ k_1<⋯<k_r t^k_1+⋯+k_r frac∏_i<j(k_j-k_i)^2∏_i=1^r k_i^2r-1. Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and including the factors t^k_i/k_i gives the stated coefficient. In detail, ∏_i<j(k_j^-1-k_i^-1)^2 =frac∏_i<j(k_j-k_i)^2∏_i k_i^2(r-1). Passing to the formal …
Exact mathematical passage
v_k=(1,k^{-1},\ldots,k^{-(r-1)})^{\mathsf T}
k^2Use in source · 2 passages
Search aliases: k^2
Use in source · 23_hankel.tex, line 68
Hadamard's inequality for the positive matrix gives H_r(t)≤∏_i=0^r-1Li_2i+1(t) ≤[-log(1-t)]∏_i=1^r-1ζ(2i+1). For the higher weights, t^k≤1 bounds the series by its zeta value. Since all summands are positive, ∑_i≥1(ζ(2i+1)-1) =∑_k≥2frac1k(k^2-1)=frac14. The last sum telescopes using frac1k(k^2-1)=frac12(frac1k(k-1)-frac1k(k+1)). Finally log(1+x)≤ x for x≥0 bounds the zeta product by e^1/4, uniformly in its length. At t=a/c and t=b/c, the bounds become e^1/4log(c/b) and e^1/4log(c/a). Thus these …
Exact mathematical passage
\sum_{i\ge1}(\zeta(2i+1)-1) =\sum_{k\ge2}\frac1{k(k^2-1)}=\frac14.
k^{2r-1}Use in source · 1 passage
Search aliases: k^2r-1
Use in source · 23_hankel.tex, line 46
contact and first coefficient] Put s_r=r(r+1)/2. Then ord_0 H_r=s_r, [t^s_r] H_r=∏_k=1^r k^-(2k-1). The unique increasing r-tuple of positive indices with least sum is (1,…,r), of sum s_r. Its coefficient is nonzero. Its Vandermonde numerator is ∏_j=1^r-1(j!)^2. Dividing this by ∏_k=1^r k^2r-1 leaves exponent 2k-1 in the denominator at each k, giving the formula. For r=2 the first coefficient is 1/8. For r=3 it is 1/(2^3 3^5). The contact orders are 1,3,6,10,…, the triangular numbers.
Exact mathematical passage
\prod_{k=1}^r k^{2r-1}
k^{i+j+1}Use in source · 1 passage
Search aliases: k^i+j+1
Use in source · 25_truncation.tex, line 17
Let L_K=lcm(1,…,K), s_r=r(r+1)/2, d=rK-r(r-1)/2. Then A_r,K(t):=L_K^r^2 H_r,K(t)∈Z[t], ord_0A_r,K=s_r, deg A_r,K=d. To see integrality, scale row i by L_K^i+1 and column j by L_K^j. Each entry becomes L_K^i+j+1∑_k=1^Kfract^kk^i+j+1∈Z[t]. The total determinant scaling exponent is r^2. The unique smallest subset is (1,…,r) and the unique largest is (K-r+1,…,K), proving the order and degree.
Exact mathematical passage
L_K^{i+j+1}\sum_{k=1}^K\frac{t^k}{k^{i+j+1}}\in\Z[t].
k^{m+1}Use in source · 1 passage
Search aliases: k^m+1
Use in source · 22_jet_filter.tex, line 26
[Exact jet cancellation] Φ_m(t)=(-1)^m∑_k≥1binomkm+1fract^kk^m+2. Thus ord_0Φ_m=m+1, [t^m+1]Φ_m=frac(-1)^m(m+1)^m+2. The coefficient of t^k is P_m(1/k)/(m+1)!. Direct substitution gives P_m(1/k)=frac1k∏_j=1^m(frac jk-1) =(-1)^mfrac(k-1)⋯(k-m)k^m+1. Divide by (m+1)! and multiply numerator and denominator by k to get the asserted binomial coefficient. It vanishes for 1≤ k≤ m and first becomes one at k=m+1. The coefficient there is nonzero. For m=0, this is Φ_0=Li_1. For m=1, Φ_1=(Li_2-Li_1)/2=-t^2…
Exact mathematical passage
P_m(1/k)=\frac1k\prod_{j=1}^m\left(\frac jk-1\right) =(-1)^m\frac{(k-1)\cdots(k-m)}{k^{m+1}}.
k^{m+2}Use in source · 3 passages
Search aliases: k^m+2
Use in source · 22_jet_filter.tex, line 15
[Exact jet cancellation] Φ_m(t)=(-1)^m∑_k≥1binomkm+1fract^kk^m+2. Thus ord_0Φ_m=m+1, [t^m+1]Φ_m=frac(-1)^m(m+1)^m+2. The coefficient of t^k is P_m(1/k)/(m+1)!. Direct substitution gives P_m(1/k)=frac1k∏_j=1^m(frac jk-1) =(-1)^mfrac(k-1)⋯(k-m)k^m+1. Divide by (m+1)! and multiply numerator and denominator by k to get the asserted binomial coefficient. It vanishes for 1≤ k≤ m and first becomes one at k=m+1. The coefficient there is nonzero. For m=0, this is Φ_0=Li_1. For m=1, Φ_1=(Li_2-Li_1)/2=-t^2…
Exact mathematical passage
\Phi_m(t)=(-1)^m\sum_{k\ge1}\binom{k}{m+1}\frac{t^k}{k^{m+2}}.
k^sUse in source · 2 passages
Search aliases: k^s
Use in source · 22_jet_filter.tex, line 12
The universal mixed-weight jet filter. One polynomial chooses the cancelling coefficients. Let m≥0 be an integer. Define P_m(X)=X∏_j=1^m(jX-1)=∑_s=1^m+1c_m,sX^s, and the formal series Φ_m(t)=frac1(m+1)!∑_s=1^m+1c_m,sLi_s(t). Here Li_s(t)=∑_k≥1t^k/k^s is first interpreted in Q[[t]]. It also defines the usual analytic polylogarithm on |t|<1~[source reference]. The label s is its weight. The contact order will follow from its coefficients, rather than from that label.
Exact mathematical passage
\Li_s(t)=\sum_{k\ge1}t^k/k^s
k_*Use in source · 4 passages
Search aliases: k_*
Use in source · project_03.tex, line 150
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
K_1Use in source · 3 passages
Search aliases: K_1
Use in source · 03_consequences.tex, line 14
al large-exponent Fermat statement] Assume Conjecture~[source reference]. There is an integer n_0 such that, for every integer n>n_0, the equation x^n+y^n=z^n has no positive integer solution. Divide a putative solution by its common gcd. The resulting x,y,z are pairwise coprime. Apply abc to a=x^n, b=y^n, c=z^n with the fixed exponent eps=1. Then z^n≤ K_1rad(xyz)^2≤ K_1(xyz)^2<K_1z^6, because x,y<z. As z≥2, this implies 2^n-6<K_1 whenever n>6. Any fixed integer larger than 6+log_2max(1,K_1) exc…
Exact mathematical passage
z^n\le K_1\rad(xyz)^2\le K_1(xyz)^2<K_1z^6,
k_1Use in source · 3 passages
Search aliases: k_1
Use in source · 23_hankel.tex, line 18
[Cauchy--Binet expansion] H_r(t)=∑_1≤ k_1<⋯<k_r t^k_1+⋯+k_r frac∏_i<j(k_j-k_i)^2∏_i=1^r k_i^2r-1. Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and including the factors t^k_i/k_i gives the stated coefficient. In detail, ∏_i<j(k_j^-1-k_i^-1)^2 =frac∏_i<j(k_j-k_i)^2∏_i k_i^2(r-1). Passing to the formal …
Exact mathematical passage
\mathcal H_r(t)=\sum_{1\le k_1<\cdots<k_r} t^{k_1+\cdots+k_r} \frac{\prod_{i<j}(k_j-k_i)^2}{\prod_{i=1}^r k_i^{2r-1}}.
K_3Use in source · 1 passage
Search aliases: K_3
Use in source · project_10.tex, line 152
Weight-three motivation and limitation. Over Q, Borel's rank theorem gives rankK_3(Q)=0 and rankK_5(Q)=1, the rational weights two and three. This makes weight three a plausible rational regulator layer. It does not assert the absence of all weight-two periods or of regulator directions over arbitrary number fields. This motivates the depth-three path packet. It does not prove that weight three suffices uniformly. A complete proof may require adaptive higher weights or a different motivic quotie…
Exact mathematical passage
\operatorname{rank}K_3(\Q)=0
K_5Use in source · 1 passage
Search aliases: K_5
Use in source · project_10.tex, line 152
Weight-three motivation and limitation. Over Q, Borel's rank theorem gives rankK_3(Q)=0 and rankK_5(Q)=1, the rational weights two and three. This makes weight three a plausible rational regulator layer. It does not assert the absence of all weight-two periods or of regulator directions over arbitrary number fields. This motivates the depth-three path packet. It does not prove that weight three suffices uniformly. A complete proof may require adaptive higher weights or a different motivic quotie…
Exact mathematical passage
\operatorname{rank}K_5(\Q)=1
K_\epsUse in source · 8 passages
Search aliases: K_eps eps
Use in source · 01_foundations.tex, line 30
[Masser--Oesterl e abc] For every real eps>0 there is a constant K_eps>0, depending only on eps, such that every positive integer triple with a+b=c and gcd(a,b)=1 satisfies c≤ K_eps rad(abc)^1+eps. The order of the quantifiers is the statement. First choose eps. Then choose a constant that works for every primitive triple. The constant may become large as eps approaches zero. It may not depend on the particular triple.
Exact mathematical passage
K_\eps>0
K_{\eps/2}Use in source · 2 passages
Search aliases: K_eps/2 eps
Use in source · 01_foundations.tex, line 88
[Finiteness formulation] Conjecture~[source reference] is equivalent to the assertion that, for every eps>0, there are only finitely many primitive positive triples with q(a,b,c)>1+eps. Assume abc, using its bound with eps/2. A triple of quality above 1+eps satisfies R^1+eps<c≤ K_eps/2R^1+eps/2. Hence R^eps/2<K_eps/2. The radical is bounded. The same conjectural inequality then bounds c, leaving finitely many positive triples.
Exact mathematical passage
R^{1+\eps}<c\le K_{\eps/2}R^{1+\eps/2}.
k_iUse in source · 13 passages
Search aliases: k_i
Use in source · 23_hankel.tex, line 18
[Cauchy--Binet expansion] H_r(t)=∑_1≤ k_1<⋯<k_r t^k_1+⋯+k_r frac∏_i<j(k_j-k_i)^2∏_i=1^r k_i^2r-1. Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and including the factors t^k_i/k_i gives the stated coefficient. In detail, ∏_i<j(k_j^-1-k_i^-1)^2 =frac∏_i<j(k_j-k_i)^2∏_i k_i^2(r-1). Passing to the formal …
Exact mathematical passage
\mathcal H_r(t)=\sum_{1\le k_1<\cdots<k_r} t^{k_1+\cdots+k_r} \frac{\prod_{i<j}(k_j-k_i)^2}{\prod_{i=1}^r k_i^{2r-1}}.
k_i^{-1}Use in source · 2 passages
Search aliases: k_i^-1
Use in source · 23_hankel.tex, line 29
k_1<⋯<k_r t^k_1+⋯+k_r frac∏_i<j(k_j-k_i)^2∏_i=1^r k_i^2r-1. Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and including the factors t^k_i/k_i gives the stated coefficient. In detail, ∏_i<j(k_j^-1-k_i^-1)^2 =frac∏_i<j(k_j-k_i)^2∏_i k_i^2(r-1). Passing to the formal infinite sum is coefficientwise legiti…
Exact mathematical passage
k_i^{-1}
k_i^{2(r-1)}Use in source · 1 passage
Search aliases: k_i^2(r-1)
Use in source · 23_hankel.tex, line 30
orem Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and including the factors t^k_i/k_i gives the stated coefficient. In detail, ∏_i<j(k_j^-1-k_i^-1)^2 =frac∏_i<j(k_j-k_i)^2∏_i k_i^2(r-1). Passing to the formal infinite sum is coefficientwise legitimate. At a fixed power of t, only finitely many index s…
Exact mathematical passage
\prod_{i<j}(k_j^{-1}-k_i^{-1})^2 =\frac{\prod_{i<j}(k_j-k_i)^2}{\prod_i k_i^{2(r-1)}}.
k_i^{2r-1}Use in source · 1 passage
Search aliases: k_i^2r-1
Use in source · 23_hankel.tex, line 18
[Cauchy--Binet expansion] H_r(t)=∑_1≤ k_1<⋯<k_r t^k_1+⋯+k_r frac∏_i<j(k_j-k_i)^2∏_i=1^r k_i^2r-1. Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and including the factors t^k_i/k_i gives the stated coefficient. In detail, ∏_i<j(k_j^-1-k_i^-1)^2 =frac∏_i<j(k_j-k_i)^2∏_i k_i^2(r-1). Passing to the formal …
Exact mathematical passage
\mathcal H_r(t)=\sum_{1\le k_1<\cdots<k_r} t^{k_1+\cdots+k_r} \frac{\prod_{i<j}(k_j-k_i)^2}{\prod_{i=1}^r k_i^{2r-1}}.
k_jUse in source · 4 passages
Search aliases: k_j
Use in source · 23_hankel.tex, line 18
[Cauchy--Binet expansion] H_r(t)=∑_1≤ k_1<⋯<k_r t^k_1+⋯+k_r frac∏_i<j(k_j-k_i)^2∏_i=1^r k_i^2r-1. Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and including the factors t^k_i/k_i gives the stated coefficient. In detail, ∏_i<j(k_j^-1-k_i^-1)^2 =frac∏_i<j(k_j-k_i)^2∏_i k_i^2(r-1). Passing to the formal …
Exact mathematical passage
\mathcal H_r(t)=\sum_{1\le k_1<\cdots<k_r} t^{k_1+\cdots+k_r} \frac{\prod_{i<j}(k_j-k_i)^2}{\prod_{i=1}^r k_i^{2r-1}}.
k_j^{-1}Use in source · 1 passage
Search aliases: k_j^-1
Use in source · 23_hankel.tex, line 30
orem Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and including the factors t^k_i/k_i gives the stated coefficient. In detail, ∏_i<j(k_j^-1-k_i^-1)^2 =frac∏_i<j(k_j-k_i)^2∏_i k_i^2(r-1). Passing to the formal infinite sum is coefficientwise legitimate. At a fixed power of t, only finitely many index s…
Exact mathematical passage
\prod_{i<j}(k_j^{-1}-k_i^{-1})^2 =\frac{\prod_{i<j}(k_j-k_i)^2}{\prod_i k_i^{2(r-1)}}.
k_rUse in source · 3 passages
Search aliases: k_r
Use in source · 23_hankel.tex, line 18
[Cauchy--Binet expansion] H_r(t)=∑_1≤ k_1<⋯<k_r t^k_1+⋯+k_r frac∏_i<j(k_j-k_i)^2∏_i=1^r k_i^2r-1. Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and including the factors t^k_i/k_i gives the stated coefficient. In detail, ∏_i<j(k_j^-1-k_i^-1)^2 =frac∏_i<j(k_j-k_i)^2∏_i k_i^2(r-1). Passing to the formal …
Exact mathematical passage
\mathcal H_r(t)=\sum_{1\le k_1<\cdots<k_r} t^{k_1+\cdots+k_r} \frac{\prod_{i<j}(k_j-k_i)^2}{\prod_{i=1}^r k_i^{2r-1}}.
K_\betaUse in source · 2 passages
Search aliases: K_β beta
Use in source · project_10.tex, line 155
Conditional motivic transfer. Let K_β be the cyclic group generated by the balanced Kummer class. Suppose that for every primitive positive triple there exists an integral finite object H_3(a,b,c) and a canonical map Θ_3:K_β⟶ H_3(a,b,c) such that, for every eps>0, constants independent of the triple give |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 · 1 passage
Search aliases: Ker
Use in source · project_02.tex, line 177
[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 · project_01.tex, line 76
[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. }
LUse in source · 23 passages
Search aliases: L
Use in source · 25_truncation.tex, line 7
Let L_K=lcm(1,…,K), s_r=r(r+1)/2, d=rK-r(r-1)/2. Then A_r,K(t):=L_K^r^2 H_r,K(t)∈Z[t], ord_0A_r,K=s_r, deg A_r,K=d. To see integrality, scale row i by L_K^i+1 and column j by L_K^j. Each entry becomes L_K^i+j+1∑_k=1^Kfract^kk^i+j+1∈Z[t]. The total determinant scaling exponent is r^2. The unique smallest subset is (1,…,r) and the unique largest is (K-r+1,…,K), proving the order and degree.
Exact mathematical passage
L_K=\operatorname{lcm}(1,\ldots,K),\quad s_r=r(r+1)/2, \quad d=rK-r(r-1)/2.
\mathbf LUse in source · 1 passage
Search aliases: L mathbf
Use in source · project_08.tex, line 149
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_{\mathbb T_f}Use in source · 1 passage
Search aliases: L_ T_f mathbf mathbb
Use in source · project_08.tex, line 149
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_0Use in source · 4 passages
Search aliases: L_0
Use in source · project_03.tex, line 74
The exact lattice extension. Throughout the remaining height and exact-sequence constructions, exclude (1,1,2). 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 …
Exact mathematical passage
L_0=L\cap\ker(\mathsf w).
L_2Use in source · 3 passages
Search aliases: L_2
Use in source · project_10.tex, line 32
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)
L_KUse in source · 2 passages
Search aliases: L_K
Use in source · 25_truncation.tex, line 7
Let L_K=lcm(1,…,K), s_r=r(r+1)/2, d=rK-r(r-1)/2. Then A_r,K(t):=L_K^r^2 H_r,K(t)∈Z[t], ord_0A_r,K=s_r, deg A_r,K=d. To see integrality, scale row i by L_K^i+1 and column j by L_K^j. Each entry becomes L_K^i+j+1∑_k=1^Kfract^kk^i+j+1∈Z[t]. The total determinant scaling exponent is r^2. The unique smallest subset is (1,…,r) and the unique largest is (K-r+1,…,K), proving the order and degree.
Exact mathematical passage
L_K=\operatorname{lcm}(1,\ldots,K),\quad s_r=r(r+1)/2, \quad d=rK-r(r-1)/2.
L_K^{i+1}Use in source · 1 passage
Search aliases: L_K^i+1
Use in source · 25_truncation.tex, line 16
Let L_K=lcm(1,…,K), s_r=r(r+1)/2, d=rK-r(r-1)/2. Then A_r,K(t):=L_K^r^2 H_r,K(t)∈Z[t], ord_0A_r,K=s_r, deg A_r,K=d. To see integrality, scale row i by L_K^i+1 and column j by L_K^j. Each entry becomes L_K^i+j+1∑_k=1^Kfract^kk^i+j+1∈Z[t]. The total determinant scaling exponent is r^2. The unique smallest subset is (1,…,r) and the unique largest is (K-r+1,…,K), proving the order and degree.
Exact mathematical passage
L_K^{i+1}
L_K^{i+j+1}Use in source · 1 passage
Search aliases: L_K^i+j+1
Use in source · 25_truncation.tex, line 17
Let L_K=lcm(1,…,K), s_r=r(r+1)/2, d=rK-r(r-1)/2. Then A_r,K(t):=L_K^r^2 H_r,K(t)∈Z[t], ord_0A_r,K=s_r, deg A_r,K=d. To see integrality, scale row i by L_K^i+1 and column j by L_K^j. Each entry becomes L_K^i+j+1∑_k=1^Kfract^kk^i+j+1∈Z[t]. The total determinant scaling exponent is r^2. The unique smallest subset is (1,…,r) and the unique largest is (K-r+1,…,K), proving the order and degree.
Exact mathematical passage
L_K^{i+j+1}\sum_{k=1}^K\frac{t^k}{k^{i+j+1}}\in\Z[t].
L_K^jUse in source · 1 passage
Search aliases: L_K^j
Use in source · 25_truncation.tex, line 16
Let L_K=lcm(1,…,K), s_r=r(r+1)/2, d=rK-r(r-1)/2. Then A_r,K(t):=L_K^r^2 H_r,K(t)∈Z[t], ord_0A_r,K=s_r, deg A_r,K=d. To see integrality, scale row i by L_K^i+1 and column j by L_K^j. Each entry becomes L_K^i+j+1∑_k=1^Kfract^kk^i+j+1∈Z[t]. The total determinant scaling exponent is r^2. The unique smallest subset is (1,…,r) and the unique largest is (K-r+1,…,K), proving the order and degree.
Exact mathematical passage
L_K^j
L_K^{r^2}Use in source · 1 passage
Search aliases: L_K^r^2
Use in source · 25_truncation.tex, line 12
Let L_K=lcm(1,…,K), s_r=r(r+1)/2, d=rK-r(r-1)/2. Then A_r,K(t):=L_K^r^2 H_r,K(t)∈Z[t], ord_0A_r,K=s_r, deg A_r,K=d. To see integrality, scale row i by L_K^i+1 and column j by L_K^j. Each entry becomes L_K^i+j+1∑_k=1^Kfract^kk^i+j+1∈Z[t]. The total determinant scaling exponent is r^2. The unique smallest subset is (1,…,r) and the unique largest is (K-r+1,…,K), proving the order and degree.
Exact mathematical passage
A_{r,K}(t):=L_K^{r^2}\mathcal H_{r,K}(t)\in\Z[t], \qquad \ord_0A_{r,K}=s_r,\qquad \deg A_{r,K}=d.
\mathbf L_p^{\mathrm{st}}Use in source · 1 passage
Search aliases: L_p^st mathbf mathrm
Use in source · project_08.tex, line 149
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_{r,\delta}Use in source · 3 passages
Search aliases: L_r,δ delta
Use in source · 26_global_bridge.tex, line 27
Divide by (ab)^s_r and use Q/(ab)=c/R: (c/R)^s_r≤ C_rR^E_r(log(3c))^A_r. Taking positive roots yields c≤ D_rR^1+η_r(log(3c))^a_r, D_r=C_r^1/s_r, η_r=E_r/s_r, a_r=A_r/s_r. Fix the target eps>0. Choose one rank with η_r<eps. Next choose 0<δ<fraceps-η_r1+eps. For the fixed exponent a_r, there is a constant L_r,δ such that (log(3c))^a_r≤ L_r,δc^δ for all c≥2. The elementary fact follows because a fixed logarithmic power divided by any positive power of c tends to zero, and the remainder is bounded o…
Exact mathematical passage
L_{r,\delta}
\mathbf L_{R^{\mathrm{glob}}}Use in source · 1 passage
Search aliases: L_R^glob mathbf mathrm
Use in source · project_08.tex, line 149
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.
\lceilUse in source · 3 passages
Search aliases: lceil
Use in source · 24_padic.tex, line 81
An explicit fixed correction at the remaining primes. Put a_r=2r-1. For each prime p and 1≤ i≤ r, define C_r,p,i=max (0,sup_h≥1{ a_rh-(p^hlceilfrac ip^hrceil-i)}). This is a finite integer maximum because p^h eventually grows faster than h. Put C_r,p=∑_i=1^rC_r,p,i. The good-prime inequality just proved implies C_r,p=0 for p>3r-1. Hence B_r=∏_p≤3r-1p^C_r,p is a fixed positive integer depending only on the rank.
Exact mathematical passage
C_{r,p,i}=\max\!\left(0,\sup_{h\ge1}\left\{ a_rh-\left(p^h\left\lceil\frac i{p^h}\right\rceil-i\right)\right\}\right).
\lcmUse in source · 2 passages
Search aliases: lcm
Use in source · project_07.tex, line 72
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. }
\operatorname{lcm}Use in source · 2 passages
Search aliases: lcm operatorname
Use in source · 25_truncation.tex, line 7
Let L_K=lcm(1,…,K), s_r=r(r+1)/2, d=rK-r(r-1)/2. Then A_r,K(t):=L_K^r^2 H_r,K(t)∈Z[t], ord_0A_r,K=s_r, deg A_r,K=d. To see integrality, scale row i by L_K^i+1 and column j by L_K^j. Each entry becomes L_K^i+j+1∑_k=1^Kfract^kk^i+j+1∈Z[t]. The total determinant scaling exponent is r^2. The unique smallest subset is (1,…,r) and the unique largest is (K-r+1,…,K), proving the order and degree.
Exact mathematical passage
L_K=\operatorname{lcm}(1,\ldots,K),\quad s_r=r(r+1)/2, \quad d=rK-r(r-1)/2.
\operatorname{lcm}_iUse in source · 1 passage
Search aliases: lcm_i operatorname
Use in source · project_09.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 · 24_padic.tex, line 12
Let u_1<⋯<u_r be the first r positive integers not divisible by p. Then u_j=j+lfloorfracj-1p-1rfloor, S_p,r=∑_j=1^r u_j. For h≥1, put s_h=(p-1)p^h-1 and write r=q_hs_h+b_h, 0≤ b_h<s_h. Define the finite sum E_p,r=∑_h≥1(s_hbinomq_h2+b_hq_h). The terms are zero once s_h≥ r.
Exact mathematical passage
u_j=j+\left\lfloor\frac{j-1}{p-1}\right\rfloor, \qquad S_{p,r}=\sum_{j=1}^r u_j.
\LiUse in source · 14 passages
Search aliases: Li
Use in source · 22_jet_filter.tex, line 9
The universal mixed-weight jet filter. One polynomial chooses the cancelling coefficients. Let m≥0 be an integer. Define P_m(X)=X∏_j=1^m(jX-1)=∑_s=1^m+1c_m,sX^s, and the formal series Φ_m(t)=frac1(m+1)!∑_s=1^m+1c_m,sLi_s(t). Here Li_s(t)=∑_k≥1t^k/k^s is first interpreted in Q[[t]]. It also defines the usual analytic polylogarithm on |t|<1~[source reference]. The label s is its weight. The contact order will follow from its coefficients, rather than from that label.
Exact mathematical passage
\Phi_m(t)=\frac1{(m+1)!}\sum_{s=1}^{m+1}c_{m,s}\Li_s(t).
\operatorname{Li}Use in source · 5 passages
Search aliases: Li operatorname
Use in source · project_10.tex, line 6
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). }
\Li_1Use in source · 6 passages
Search aliases: Li_1 Li
Use in source · 22_jet_filter.tex, line 32
ent of t^k is P_m(1/k)/(m+1)!. Direct substitution gives P_m(1/k)=frac1k∏_j=1^m(frac jk-1) =(-1)^mfrac(k-1)⋯(k-m)k^m+1. Divide by (m+1)! and multiply numerator and denominator by k to get the asserted binomial coefficient. It vanishes for 1≤ k≤ m and first becomes one at k=m+1. The coefficient there is nonzero. For m=0, this is Φ_0=Li_1. For m=1, Φ_1=(Li_2-Li_1)/2=-t^2/8-t^3/9-⋯. The sign is part of the exact formula. Changing the weight alone would not produce the cancellation of the linear coe…
Exact mathematical passage
\Phi_0=\Li_1
\Li_2Use in source · 4 passages
Search aliases: Li_2 Li
Use in source · 22_jet_filter.tex, line 33
+1)!. Direct substitution gives P_m(1/k)=frac1k∏_j=1^m(frac jk-1) =(-1)^mfrac(k-1)⋯(k-m)k^m+1. Divide by (m+1)! and multiply numerator and denominator by k to get the asserted binomial coefficient. It vanishes for 1≤ k≤ m and first becomes one at k=m+1. The coefficient there is nonzero. For m=0, this is Φ_0=Li_1. For m=1, Φ_1=(Li_2-Li_1)/2=-t^2/8-t^3/9-⋯. The sign is part of the exact formula. Changing the weight alone would not produce the cancellation of the linear coefficient.
Exact mathematical passage
\Phi_1=(\Li_2-\Li_1)/2=-t^2/8-t^3/9-\cdots.
\operatorname{Li}_2Use in source · 1 passage
Search aliases: Li_2 operatorname
Use in source · project_10.tex, line 32
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)
\Li_2^2Use in source · 1 passage
Search aliases: Li_2^2 Li
Use in source · 23_hankel.tex, line 11
For the first two ranks, H_1=Li_1, H_2=Li_1Li_3-Li_2^2. The rank-two determinant compares genuinely different moment directions. Its order-three contact is not inferred from the presence of weight three.
Exact mathematical passage
\mathcal H_1=\Li_1,\qquad \mathcal H_2=\Li_1\Li_3-\Li_2^2.
\Li_{2i+1}Use in source · 1 passage
Search aliases: Li_2i+1 Li
Use in source · 23_hankel.tex, line 63
Hadamard's inequality for the positive matrix gives H_r(t)≤∏_i=0^r-1Li_2i+1(t) ≤[-log(1-t)]∏_i=1^r-1ζ(2i+1). For the higher weights, t^k≤1 bounds the series by its zeta value. Since all summands are positive, ∑_i≥1(ζ(2i+1)-1) =∑_k≥2frac1k(k^2-1)=frac14. The last sum telescopes using frac1k(k^2-1)=frac12(frac1k(k-1)-frac1k(k+1)). Finally log(1+x)≤ x for x≥0 bounds the zeta product by e^1/4, uniformly in its length. At t=a/c and t=b/c, the bounds become e^1/4log(c/b) and e^1/4log(c/a). Thus these …
Exact mathematical passage
\mathcal H_r(t)\le\prod_{i=0}^{r-1}\Li_{2i+1}(t) \le[-\log(1-t)]\prod_{i=1}^{r-1}\zeta(2i+1).
\Li_3Use in source · 3 passages
Search aliases: Li_3 Li
Use in source · 23_hankel.tex, line 11
For the first two ranks, H_1=Li_1, H_2=Li_1Li_3-Li_2^2. The rank-two determinant compares genuinely different moment directions. Its order-three contact is not inferred from the presence of weight three.
Exact mathematical passage
\mathcal H_1=\Li_1,\qquad \mathcal H_2=\Li_1\Li_3-\Li_2^2.
\operatorname{Li}_3Use in source · 3 passages
Search aliases: Li_3 operatorname
Use in source · project_10.tex, line 46
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). For 0<z<1, all three displayed arguments use their real branches. The dilogarithm complement identity proves the Rogers relation. The auxiliary identity Li_2(1-1/z)+Li_2(1-z)=-tfrac12log^2z follows by differentiat…
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} }
\Li_{i+j+1}Use in source · 1 passage
Search aliases: Li_i+j+1 Li
Use in source · 23_hankel.tex, line 5
A nonzero Hankel contact determinant. Independent moment directions replace one filter. For an integer r≥1, define H_r(t)=det(Li_i+j+1(t))_0≤ i,j<r. It uses weights one through 2r-1. The expression is a formal series over Q and an analytic function for |t|<1. No algebraicity at a rational argument is assumed.
Exact mathematical passage
\mathcal H_r(t)=\det\bigl(\Li_{i+j+1}(t)\bigr)_{0\le i,j<r}.
\Li_jUse in source · 2 passages
Search aliases: Li_j Li
Use in source · 23_hankel.tex, line 81
The third complex chart has a branch boundary. For fixed rank and a consistent upper or lower boundary branch at x=c/a>1, the classical integer-weight inversion formula expresses each Li_j(x) as a fixed degree-j polynomial in a chosen log(-x) plus a sign times Li_j(1/x)~[source reference]. The latter is bounded by ζ(j) for j≥2. For j=1, use Li_1(x)=-log(1-x) and x-1=b/a.
Exact mathematical passage
\Li_j(x)
\operatorname{Li}_nUse in source · 1 passage
Search aliases: Li_n operatorname
Use in source · project_10.tex, line 6
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). }
\Li_sUse in source · 2 passages
Search aliases: Li_s Li
Use in source · 22_jet_filter.tex, line 9
The universal mixed-weight jet filter. One polynomial chooses the cancelling coefficients. Let m≥0 be an integer. Define P_m(X)=X∏_j=1^m(jX-1)=∑_s=1^m+1c_m,sX^s, and the formal series Φ_m(t)=frac1(m+1)!∑_s=1^m+1c_m,sLi_s(t). Here Li_s(t)=∑_k≥1t^k/k^s is first interpreted in Q[[t]]. It also defines the usual analytic polylogarithm on |t|<1~[source reference]. The label s is its weight. The contact order will follow from its coefficients, rather than from that label.
Exact mathematical passage
\Phi_m(t)=\frac1{(m+1)!}\sum_{s=1}^{m+1}c_{m,s}\Li_s(t).
\llUse in source · 2 passages
Search aliases: ll
Use in source · 20_signed_filter.tex, line 47
The hypothetical exception regime is sparse. For a primitive triple put Q=abc/R. Suppose c>R^1+eps for a fixed eps>0. Since a,b are positive with sum c, one has ab≥ c-1. Thus, for large c, Q=fracabcR≥fracc(c-1)R>frac12R^1+2eps. The unit group modulo Q has order φ(Q). For every δ>0, frac Qφ(Q)ll_δ R^δ, 3^ω(R)ll_δ R^δ. Both elementary bounds follow by separating a finite set of small primes. For all sufficiently large primes, (1-1/p)^-1≤ p^δ and 3≤ p^δ. Every prime dividing Q also divides R.
Exact mathematical passage
\frac Q{\varphi(Q)}\ll_\delta R^\delta, \qquad 3^{\omega(R)}\ll_\delta R^\delta.
\ll_\deltaUse in source · 1 passage
Search aliases: ll_δ ll delta
Use in source · 20_signed_filter.tex, line 47
The hypothetical exception regime is sparse. For a primitive triple put Q=abc/R. Suppose c>R^1+eps for a fixed eps>0. Since a,b are positive with sum c, one has ab≥ c-1. Thus, for large c, Q=fracabcR≥fracc(c-1)R>frac12R^1+2eps. The unit group modulo Q has order φ(Q). For every δ>0, frac Qφ(Q)ll_δ R^δ, 3^ω(R)ll_δ R^δ. Both elementary bounds follow by separating a finite set of small primes. For all sufficiently large primes, (1-1/p)^-1≤ p^δ and 3≤ p^δ. Every prime dividing Q also divides R.
Exact mathematical passage
\frac Q{\varphi(Q)}\ll_\delta R^\delta, \qquad 3^{\omega(R)}\ll_\delta R^\delta.
\log^2Use in source · 3 passages
Search aliases: log^2 log
Use in source · project_10.tex, line 46
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). For 0<z<1, all three displayed arguments use their real branches. The dilogarithm complement identity proves the Rogers relation. The auxiliary identity Li_2(1-1/z)+Li_2(1-z)=-tfrac12log^2z follows by differentiat…
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 · project_10.tex, line 46
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). For 0<z<1, all three displayed arguments use their real branches. The dilogarithm complement identity proves the Rogers relation. The auxiliary identity Li_2(1-1/z)+Li_2(1-z)=-tfrac12log^2z follows by differentiat…
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 · 03_consequences.tex, line 17
^n+y^n=z^n has no positive integer solution. Divide a putative solution by its common gcd. The resulting x,y,z are pairwise coprime. Apply abc to a=x^n, b=y^n, c=z^n with the fixed exponent eps=1. Then z^n≤ K_1rad(xyz)^2≤ K_1(xyz)^2<K_1z^6, because x,y<z. As z≥2, this implies 2^n-6<K_1 whenever n>6. Any fixed integer larger than 6+log_2max(1,K_1) excludes all such exponents. The theorem does not give Fermat's Last Theorem for every n≥3. It supplies only a sufficiently-large-exponent implication,…
Exact mathematical passage
6+\log_2\max(1,K_1)
\log_pUse in source · 3 passages
Search aliases: log_p log
Use in source · project_10.tex, line 129
Exact finite-place notch. At an output prime p^eparallel c, β=1-frac ca. Since pnmid a, v_p(β-1)=e. For odd p, v_p(log_pβ)=e. The corresponding weighted dyadic identity holds on the principal branch when e≥2.
Exact mathematical passage
\boxed{ v_p(\log_p\beta)=e. }
\mathbf mUse in source · 1 passage
Search aliases: m mathbf
Use in source · project_07.tex, line 58
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}\}.
\mathsf MUse in source · 4 passages
Search aliases: M mathsf
Use in source · project_03.tex, line 22
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 · 59 passages
Search aliases: m
Use in source · 20_signed_filter.tex, line 15
Let N(x) count assignments producing x∈ G. Repeated contributions remain separate assignments, so M=∑_x∈ GN(x)=3^s, μ(x)=N(x)/M. For a character χ∈widehat G, define widehatμ(χ)=∑_xμ(x)χ(x) =∏_j=1^sfrac1+χ(γ_j)+χ(-γ_j)3. On a cyclic channel G=Z/nZ, this becomes m(k)=∏_j=1^sfrac1+2cos(2π kγ_j/n)3. This is the discrete cosine or COS expression used in the project. It is an exact characteristic function. Individual cosine factors may be negative. No positivity of all Fourier samples is assumed.
Exact mathematical passage
m(k)=\prod_{j=1}^s\frac{1+2\cos(2\pi k\gamma_j/n)}3.
M^{1-n}Use in source · 1 passage
Search aliases: M^1-n
Use in source · project_10.tex, line 6
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 · project_10.tex, line 17
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 displayed coordinate rescaling has determinant M^3. It has not been identified here with an actual integral motivic or period-lattice map. That identification would require the missing arithmetic realization.
Exact mathematical passage
\boxed{1\cdot M\cdot M^2=M^3.}
M^{n-1}Use in source · 1 passage
Search aliases: M^n-1
Use in source · project_10.tex, line 16
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 displayed coordinate rescaling has determinant M^3. It has not been identified here with an actual integral motivic or period-lattice map. That identification would require the missing arithmetic realization.
Exact mathematical passage
M^{n-1}
m_0Use in source · 4 passages
Search aliases: m_0
Use in source · project_09.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 · project_07.tex, line 72
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. }
M_aUse in source · 1 passage
Search aliases: M_a
Use in source · 01_foundations.tex, line 22
For a positive primitive additive triple a+b=c, gcd(a,b)=1, all three pairs are coprime. For example, a divisor of a and c also divides c-a=b, hence is one. Therefore the prime supports are disjoint and R=rad(abc)=R_aR_bR_c, R_u=rad(u). Put M_u=u/R_u. The exact decomposition is abc=R M_aM_bM_c. The conjecture compares c with R, rather than with abc itself. Since abc already contains c, an estimate against the full product would lose the intended content.
Exact mathematical passage
abc=R M_aM_bM_c.
\mathsf M_{a,b,c}Use in source · 4 passages
Search aliases: M_a,b,c mathsf
Use in source · project_03.tex, line 22
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_bUse in source · 1 passage
Search aliases: M_b
Use in source · 01_foundations.tex, line 22
For a positive primitive additive triple a+b=c, gcd(a,b)=1, all three pairs are coprime. For example, a divisor of a and c also divides c-a=b, hence is one. Therefore the prime supports are disjoint and R=rad(abc)=R_aR_bR_c, R_u=rad(u). Put M_u=u/R_u. The exact decomposition is abc=R M_aM_bM_c. The conjecture compares c with R, rather than with abc itself. Since abc already contains c, an estimate against the full product would lose the intended content.
Exact mathematical passage
abc=R M_aM_bM_c.
M_cUse in source · 13 passages
Search aliases: M_c
Use in source · 01_foundations.tex, line 22
For a positive primitive additive triple a+b=c, gcd(a,b)=1, all three pairs are coprime. For example, a divisor of a and c also divides c-a=b, hence is one. Therefore the prime supports are disjoint and R=rad(abc)=R_aR_bR_c, R_u=rad(u). Put M_u=u/R_u. The exact decomposition is abc=R M_aM_bM_c. The conjecture compares c with R, rather than with abc itself. Since abc already contains c, an estimate against the full product would lose the intended content.
Exact mathematical passage
abc=R M_aM_bM_c.
M_dUse in source · 1 passage
Search aliases: M_d
Use in source · project_10.tex, line 107
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)
m_iUse in source · 3 passages
Search aliases: m_i
Use in source · project_07.tex, line 58
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 · project_07.tex, line 72
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. }
M_uUse in source · 1 passage
Search aliases: M_u
Use in source · 01_foundations.tex, line 21
For a positive primitive additive triple a+b=c, gcd(a,b)=1, all three pairs are coprime. For example, a divisor of a and c also divides c-a=b, hence is one. Therefore the prime supports are disjoint and R=rad(abc)=R_aR_bR_c, R_u=rad(u). Put M_u=u/R_u. The exact decomposition is abc=R M_aM_bM_c. The conjecture compares c with R, rather than with abc itself. Since abc already contains c, an estimate against the full product would lose the intended content.
Exact mathematical passage
M_u=u/R_u
\mathscrUse in source · 4 passages
Search aliases: mathscr
Use in source · project_03.tex, line 165
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 · 02_polynomial.tex, line 10
[Mason--Stothers in characteristic zero] Let k be an algebraically closed field of characteristic zero. Suppose A,B,C∈ k[X] are nonzero, pairwise coprime, satisfy A+B=C, and are not all constant. Then max{deg A,deg B,deg C}≤ n_0(ABC)-1. This is the polynomial abc theorem. Snyder gives a short primary proof in a form that also makes the derivative hypothesis explicit~[source reference]. The proof below spells out the cancellation that will motivate the arithmetic Wronskian.
Exact mathematical passage
\max\{\deg A,\deg B,\deg C\}\le n_0(ABC)-1.
\midUse in source · 59 passages
Search aliases: mid
Use in source · 01_foundations.tex, line 4
Radical support and the exact conjecture. The multiplicative quantity that ignores repetition. Every positive integer has a unique prime factorization n=∏_pmid np^e_p, e_p≥1. Its radical is rad(n)=∏_pmid np, rad(1)=1. The radical retains which primes occur and discards how many times they occur. Thus rad(72)=rad(2^3·3^2)=6, whereas 72/6=12 records the repeated-power contribution. If n is squarefree, then rad(n)=n.
Exact mathematical passage
n=\prod_{p\mid n}p^{e_p},\qquad e_p\ge1.
\min_iUse in source · 1 passage
Search aliases: min_i min
Use in source · project_09.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 · project_03.tex, line 112
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.
\min_{q\mid ab}Use in source · 1 passage
Search aliases: min_qmid ab min mid
Use in source · project_11.tex, line 92
r formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. Divide the input exponent vector by K. Define a_p,q as the original odd or weighted dyadic coefficient, now reduced modulo p^d_p-r_p. Dividing the relation vector reduces the needed relation modulus. It does not divide the coefficient row by K. 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 ord…
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.
NUse in source · 34 passages
Search aliases: N
Use in source · 03_consequences.tex, line 30
The precise equivalent statement uses more than the discriminant alone. For an elliptic curve over Q, let N be its conductor and let c_4,c_6 be the invariants of a global minimal integral Weierstrass equation. The modified Szpiro assertion is max{|c_4|^3,c_6^2}≤ C_eps N^6+eps for every eps>0, with a constant uniform in the elliptic curve. Oesterl e identifies this form as equivalent to the full abc conjecture~[source reference].
Exact mathematical passage
N
nUse in source · 100 passages
Search aliases: n
Use in source · 01_foundations.tex, line 4
Radical support and the exact conjecture. The multiplicative quantity that ignores repetition. Every positive integer has a unique prime factorization n=∏_pmid np^e_p, e_p≥1. Its radical is rad(n)=∏_pmid np, rad(1)=1. The radical retains which primes occur and discards how many times they occur. Thus rad(72)=rad(2^3·3^2)=6, whereas 72/6=12 records the repeated-power contribution. If n is squarefree, then rad(n)=n.
Exact mathematical passage
n=\prod_{p\mid n}p^{e_p},\qquad e_p\ge1.
N^{6+\eps}Use in source · 1 passage
Search aliases: N^6+eps eps
Use in source · 03_consequences.tex, line 31
The precise equivalent statement uses more than the discriminant alone. For an elliptic curve over Q, let N be its conductor and let c_4,c_6 be the invariants of a global minimal integral Weierstrass equation. The modified Szpiro assertion is max{|c_4|^3,c_6^2}≤ C_eps N^6+eps for every eps>0, with a constant uniform in the elliptic curve. Oesterl e identifies this form as equivalent to the full abc conjecture~[source reference].
Exact mathematical passage
\max\{|c_4|^3,c_6^2\}\le C_\eps N^{6+\eps}
n_0Use in source · 7 passages
Search aliases: n_0
Use in source · 02_polynomial.tex, line 4
The polynomial theorem and the missing integer analogue. Roots play the role of distinct primes. For a nonzero polynomial F over an algebraically closed field, let rad(F) be the product of its distinct monic linear factors. Its degree counts distinct roots. Write n_0(F)=degrad(F).
Exact mathematical passage
n_0(F)=\deg\operatorname{rad}(F)
n_1Use in source · 1 passage
Search aliases: n_1
Use in source · 24_padic.tex, line 32
The residue-class energy minimum. For r distinct p-unit integers k_1,…,k_r, the valuation of their Vandermonde product is ∑_i<jv_p(k_j-k_i) =∑_h≥1#{i<j:k_i≡ k_jpmodp^h}. There are s_h unit residue classes modulo p^h. If their occupation counts are n_1,…,n_s_h, the number of coincident pairs is ∑_jbinomn_j2.
Exact mathematical passage
n_1,\ldots,n_{s_h}
N_EUse in source · 1 passage
Search aliases: N_E
Use in source · project_05.tex, line 156
Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be a nonconstant modular parametrization, and let ϕ_E^*ω_E = 2π i c_ϕ f_E(z) dz. Here c_ϕ is the differential multiplier of this chosen parametrization and differential. The marked Frey curve need not be the optimal representative of its isogeny class. 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_ϕ^2 normf_E_Pet^2. For the Frey model, scaling x=cX, …
Exact mathematical passage
\phi_E:X_0(N_E)\longrightarrow E
n_iUse in source · 2 passages
Search aliases: n_i
Use in source · 24_padic.tex, line 34
For a fixed sum r, this quantity is minimized when the counts differ by at most one. If n_i≥ n_j+2, moving one item from class i to class j lowers the pair count by n_i-n_j-1>0. At the balanced minimum, b_h classes have count q_h+1 and the others count q_h. The minimum is therefore (s_h-b_h)binomq_h2+b_hbinomq_h+12 =s_hbinomq_h2+b_hq_h. The first r positive p-units realize this minimum for every h simultaneously. Ordered unit residues repeat in blocks modulo p^h, and each full block contains eac…
Exact mathematical passage
n_i\ge n_j+2
n_jUse in source · 3 passages
Search aliases: n_j
Use in source · 24_padic.tex, line 32
The residue-class energy minimum. For r distinct p-unit integers k_1,…,k_r, the valuation of their Vandermonde product is ∑_i<jv_p(k_j-k_i) =∑_h≥1#{i<j:k_i≡ k_jpmodp^h}. There are s_h unit residue classes modulo p^h. If their occupation counts are n_1,…,n_s_h, the number of coincident pairs is ∑_jbinomn_j2.
Exact mathematical passage
\sum_j\binom{n_j}{2}
n_{s_h}Use in source · 1 passage
Search aliases: n_s_h
Use in source · 24_padic.tex, line 32
The residue-class energy minimum. For r distinct p-unit integers k_1,…,k_r, the valuation of their Vandermonde product is ∑_i<jv_p(k_j-k_i) =∑_h≥1#{i<j:k_i≡ k_jpmodp^h}. There are s_h unit residue classes modulo p^h. If their occupation counts are n_1,…,n_s_h, the number of coincident pairs is ∑_jbinomn_j2.
Exact mathematical passage
n_1,\ldots,n_{s_h}
\normUse in source · 16 passages
Search aliases: norm
Use in source · project_01.tex, line 109
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}.
oUse in source · 4 passages
Search aliases: o
Use in source · 00_reading.tex, line 24
The research has also made a substantive correction. If Q=abc/rad(abc), then the defect relevant to the conjecture is Q/(ab)=c/rad(abc). The target Q=R^o(1) is false. Its missing factor ab is restored explicitly in every final certificate.
Exact mathematical passage
Q=R^{o(1)}
\mathrm{obs}Use in source · 1 passage
Search aliases: obs mathrm
Use in source · project_01.tex, line 120
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 · 3 passages
Search aliases: ord
Use in source · 22_jet_filter.tex, line 19
[Exact jet cancellation] Φ_m(t)=(-1)^m∑_k≥1binomkm+1fract^kk^m+2. Thus ord_0Φ_m=m+1, [t^m+1]Φ_m=frac(-1)^m(m+1)^m+2. The coefficient of t^k is P_m(1/k)/(m+1)!. Direct substitution gives P_m(1/k)=frac1k∏_j=1^m(frac jk-1) =(-1)^mfrac(k-1)⋯(k-m)k^m+1. Divide by (m+1)! and multiply numerator and denominator by k to get the asserted binomial coefficient. It vanishes for 1≤ k≤ m and first becomes one at k=m+1. The coefficient there is nonzero. For m=0, this is Φ_0=Li_1. For m=1, Φ_1=(Li_2-Li_1)/2=-t^2…
Exact mathematical passage
\ord_0\Phi_m=m+1,\qquad [t^{m+1}]\Phi_m=\frac{(-1)^m}{(m+1)^{m+2}}.
\ord_0Use in source · 3 passages
Search aliases: ord_0 ord
Use in source · 22_jet_filter.tex, line 19
[Exact jet cancellation] Φ_m(t)=(-1)^m∑_k≥1binomkm+1fract^kk^m+2. Thus ord_0Φ_m=m+1, [t^m+1]Φ_m=frac(-1)^m(m+1)^m+2. The coefficient of t^k is P_m(1/k)/(m+1)!. Direct substitution gives P_m(1/k)=frac1k∏_j=1^m(frac jk-1) =(-1)^mfrac(k-1)⋯(k-m)k^m+1. Divide by (m+1)! and multiply numerator and denominator by k to get the asserted binomial coefficient. It vanishes for 1≤ k≤ m and first becomes one at k=m+1. The coefficient there is nonzero. For m=0, this is Φ_0=Li_1. For m=1, Φ_1=(Li_2-Li_1)/2=-t^2…
Exact mathematical passage
\ord_0\Phi_m=m+1,\qquad [t^{m+1}]\Phi_m=\frac{(-1)^m}{(m+1)^{m+2}}.
\overline HUse in source · 7 passages
Search aliases: overline H overline
Use in source · project_03.tex, line 112
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.
\overline{\chi(h)}Use in source · 1 passage
Search aliases: overlineχ(h) overline chi
Use in source · 20_signed_filter.tex, line 21
Character orthogonality gives the target count N(h)=frac M|G|∑_χ∈widehat Gwidehatμ(χ)overlineχ(h). The identity is finite and exact. It solves a representation question after the arithmetic channel and admissible contributions have been specified. It does not by itself bound the height of an integer triple that generated that channel.
Exact mathematical passage
N(h)=\frac M{|G|}\sum_{\chi\in\widehat G}\widehat\mu(\chi)\overline{\chi(h)}.
\overline{\omega_E}Use in source · 1 passage
Search aliases: overlineω_E overline omega
Use in source · project_05.tex, line 166
automorphic analysis--synthesis identity Let ϕ_E:X_0(N_E)⟶ E be a nonconstant modular parametrization, and let ϕ_E^*ω_E = 2π i c_ϕ f_E(z) dz. Here c_ϕ is the differential multiplier of this chosen parametrization and differential. The marked Frey curve need not be the optimal representative of its isogeny class. 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_ϕ^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2…
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 · project_08.tex, line 6
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). Place the two-term contact complex in degrees -1,0 and define 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 · project_07.tex, line 40
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 · project_07.tex, line 51
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 · project_08.tex, line 6
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). Place the two-term contact complex in degrees -1,0 and define 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 · project_07.tex, line 51
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 · project_02.tex, line 187
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^{C_{r,p}}Use in source · 1 passage
Search aliases: p^C_r,p
Use in source · 24_padic.tex, line 86
ining primes Put a_r=2r-1. For each prime p and 1≤ i≤ r, define C_r,p,i=max (0,sup_h≥1{ a_rh-(p^hlceilfrac ip^hrceil-i)}). This is a finite integer maximum because p^h eventually grows faster than h. Put C_r,p=∑_i=1^rC_r,p,i. The good-prime inequality just proved implies C_r,p=0 for p>3r-1. Hence B_r=∏_p≤3r-1p^C_r,p is a fixed positive integer depending only on the rank.
Exact mathematical passage
B_r=\prod_{p\le3r-1}p^{C_{r,p}}
p^{d_p-r_p}Use in source · 1 passage
Search aliases: p^d_p-r_p
Use in source · project_11.tex, line 91
_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. Divide the input exponent vector by K. Define a_p,q as the original odd or weighted dyadic coefficient, now reduced modulo p^d_p-r_p. Dividing the relation vector reduces the needed relation modulus. It does not divide the coefficient row by K. 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 n…
Exact mathematical passage
p^{d_p-r_p}
p^{e-1}Use in source · 13 passages
Search aliases: p^e-1
Use in source · project_01.tex, line 145
Let p^eparallel a. Every term of d_x(a) is divisible by p^e-1. Since pnmid b, the term b d_x(a) is divisible by p^e-1. Also p^emid a, so a d_x(b) is divisible by p^e, hence by p^e-1. Therefore p^e-1mid W_x(a,b). The same argument applies to primes dividing b.
Exact mathematical passage
p^{e-1}
p^{e_p}Use in source · 3 passages
Search aliases: p^e_p
Use in source · 01_foundations.tex, line 4
Radical support and the exact conjecture. The multiplicative quantity that ignores repetition. Every positive integer has a unique prime factorization n=∏_pmid np^e_p, e_p≥1. Its radical is rad(n)=∏_pmid np, rad(1)=1. The radical retains which primes occur and discards how many times they occur. Thus rad(72)=rad(2^3·3^2)=6, whereas 72/6=12 records the repeated-power contribution. If n is squarefree, then rad(n)=n.
Exact mathematical passage
n=\prod_{p\mid n}p^{e_p},\qquad e_p\ge1.
p^hUse in source · 7 passages
Search aliases: p^h
Use in source · 24_padic.tex, line 28
The residue-class energy minimum. For r distinct p-unit integers k_1,…,k_r, the valuation of their Vandermonde product is ∑_i<jv_p(k_j-k_i) =∑_h≥1#{i<j:k_i≡ k_jpmodp^h}. There are s_h unit residue classes modulo p^h. If their occupation counts are n_1,…,n_s_h, the number of coincident pairs is ∑_jbinomn_j2.
Exact mathematical passage
\sum_{i<j}v_p(k_j-k_i) =\sum_{h\ge1}\#\{i<j:k_i\equiv k_j\pmod{p^h}\}.
p^{h-1}Use in source · 1 passage
Search aliases: p^h-1
Use in source · 24_padic.tex, line 16
Let u_1<⋯<u_r be the first r positive integers not divisible by p. Then u_j=j+lfloorfracj-1p-1rfloor, S_p,r=∑_j=1^r u_j. For h≥1, put s_h=(p-1)p^h-1 and write r=q_hs_h+b_h, 0≤ b_h<s_h. Define the finite sum E_p,r=∑_h≥1(s_hbinomq_h2+b_hq_h). The terms are zero once s_h≥ r.
Exact mathematical passage
s_h=(p-1)p^{h-1}
p^{h_i}Use in source · 3 passages
Search aliases: p^h_i
Use in source · 24_padic.tex, line 75
If h_i=v_p(k_i)>0, then k_i-i≥ p^h_i-r>a_rh_i. If h_i=0, then k_i-i≥0 and there is no denominator loss. Thus every subset other than (1,…,r) has valuation strictly above es_r. The first subset has a unit coefficient because its indices and differences are smaller than p. Its term has valuation es_r, and a unique minimum again proves equality. The strict prime threshold is retained. A good-prime equality is stronger than the all-prime lower bound below.
Exact mathematical passage
k_i-i\ge p^{h_i}-r>a_rh_i
p^jUse in source · 2 passages
Search aliases: p^j
Use in source · project_08.tex, line 67
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 for j≥2. The j=1 sequence has zero left term and is split. 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 · project_08.tex, line 75
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 for j≥2. The j=1 sequence has zero left term and is split. 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^{\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 · project_08.tex, line 120
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 · project_07.tex, line 47
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 · project_08.tex, line 137
Witt realization. For e≥2, the local packet has a natural additive identification Jcal_p,e ≅ W_e-1(F_p). Division by p identifies the source with Z_p/p^e-1Z_p, and the standard additive identification W_n(F_p)≅Z/p^nZ gives the result. For e=1, take W_0=0. The Verschiebung filtration matches multiplication by successive powers of p. This is the typed local object underlying the carry chain.
Exact mathematical passage
W_n(\F_p)\cong\Z/p^n\Z
p^{r_p}Use in source · 2 passages
Search aliases: p^r_p
Use in source · project_11.tex, line 91
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. Divide the input exponent vector by K. Define a_p,q as the original odd or weighted dyadic coefficient, now reduced modulo p^d_p-r_p. Dividing the relation vector reduc…
Exact mathematical passage
K_{\mathrm{Kum}}=\prod_{p\mid c}p^{r_p}
p^{w_p}Use in source · 1 passage
Search aliases: p^w_p
Use in source · project_11.tex, line 104
put exponent vector by K. Define a_p,q as the original odd or weighted dyadic coefficient, now reduced modulo p^d_p-r_p. Dividing the relation vector reduces the needed relation modulus. It does not divide the coefficient row by K. 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-adi…
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^\deltaUse in source · 2 passages
Search aliases: p^δ delta
Use in source · 20_signed_filter.tex, line 51
are positive with sum c, one has ab≥ c-1. Thus, for large c, Q=fracabcR≥fracc(c-1)R>frac12R^1+2eps. The unit group modulo Q has order φ(Q). For every δ>0, frac Qφ(Q)ll_δ R^δ, 3^ω(R)ll_δ R^δ. Both elementary bounds follow by separating a finite set of small primes. For all sufficiently large primes, (1-1/p)^-1≤ p^δ and 3≤ p^δ. Every prime dividing Q also divides R.
Exact mathematical passage
(1-1/p)^{-1}\le p^\delta
p^{\chi_p}Use in source · 4 passages
Search aliases: p^χ_p chi
Use in source · project_11.tex, line 104
put exponent vector by K. Define a_p,q as the original odd or weighted dyadic coefficient, now reduced modulo p^d_p-r_p. Dividing the relation vector reduces the needed relation modulus. It does not divide the coefficient row by K. 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-adi…
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_1Use in source · 1 passage
Search aliases: p_1
Use in source · project_04.tex, line 64
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 · project_02.tex, line 259
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 · 4 passages
Search aliases: P_act mathrm
Use in source · project_05.tex, line 23
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 · project_01.tex, line 109
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 · project_02.tex, line 259
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 · project_02.tex, line 259
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 · project_07.tex, line 40
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 · 7 passages
Search aliases: p_i
Use in source · project_04.tex, line 68
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.
p_i^{e_i}Use in source · 1 passage
Search aliases: p_i^e_i
Use in source · project_09.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 · project_09.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 · 4 passages
Search aliases: P_m
Use in source · 22_jet_filter.tex, line 5
The universal mixed-weight jet filter. One polynomial chooses the cancelling coefficients. Let m≥0 be an integer. Define P_m(X)=X∏_j=1^m(jX-1)=∑_s=1^m+1c_m,sX^s, and the formal series Φ_m(t)=frac1(m+1)!∑_s=1^m+1c_m,sLi_s(t). Here Li_s(t)=∑_k≥1t^k/k^s is first interpreted in Q[[t]]. It also defines the usual analytic polylogarithm on |t|<1~[source reference]. The label s is its weight. The contact order will follow from its coefficients, rather than from that label.
Exact mathematical passage
P_m(X)=X\prod_{j=1}^m(jX-1)=\sum_{s=1}^{m+1}c_{m,s}X^s,
p_mUse in source · 1 passage
Search aliases: p_m
Use in source · project_04.tex, line 64
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 · project_01.tex, line 120
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 · 3 passages
Search aliases: P_u
Use in source · project_02.tex, line 221
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
r_u=\rad(u), \qquad P_u=\frac{u}{r_u}.
\mathrm{Pet}Use in source · 1 passage
Search aliases: Pet mathrm
Use in source · project_05.tex, line 174
parametrization, and let ϕ_E^*ω_E = 2π i c_ϕ f_E(z) dz. Here c_ϕ is the differential multiplier of this chosen parametrization and differential. The marked Frey curve need not be the optimal representative of its isogeny class. 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_ϕ^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 shap…
Exact mathematical passage
\boxed{ \deg(\phi_E)A(E) = 4\pi^2 c_\phi^2 \norm{f_E}_{\mathrm{Pet}}^2. }
\pmodUse in source · 14 passages
Search aliases: pmod
Use in source · 24_padic.tex, line 28
The residue-class energy minimum. For r distinct p-unit integers k_1,…,k_r, the valuation of their Vandermonde product is ∑_i<jv_p(k_j-k_i) =∑_h≥1#{i<j:k_i≡ k_jpmodp^h}. There are s_h unit residue classes modulo p^h. If their occupation counts are n_1,…,n_s_h, the number of coincident pairs is ∑_jbinomn_j2.
Exact mathematical passage
\sum_{i<j}v_p(k_j-k_i) =\sum_{h\ge1}\#\{i<j:k_i\equiv k_j\pmod{p^h}\}.
QUse in source · 35 passages
Search aliases: Q
Use in source · 00_reading.tex, line 24
The research has also made a substantive correction. If Q=abc/rad(abc), then the defect relevant to the conjecture is Q/(ab)=c/rad(abc). The target Q=R^o(1) is false. Its missing factor ab is restored explicitly in every final certificate.
Exact mathematical passage
Q=abc/\rad(abc)
\QUse in source · 15 passages
Search aliases: Q
Use in source · 03_consequences.tex, line 30
The precise equivalent statement uses more than the discriminant alone. For an elliptic curve over Q, let N be its conductor and let c_4,c_6 be the invariants of a global minimal integral Weierstrass equation. The modified Szpiro assertion is max{|c_4|^3,c_6^2}≤ C_eps N^6+eps for every eps>0, with a constant uniform in the elliptic curve. Oesterl e identifies this form as equivalent to the full abc conjecture~[source reference].
Exact mathematical passage
\Q
\mathbb QUse in source · 1 passage
Search aliases: Q mathbb
Use in source · 28_sources.tex, line 124
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, no.~4 (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
qUse in source · 45 passages
Search aliases: q
Use in source · 01_foundations.tex, line 60
Quality measures the exponent in one example. Define the quality of a primitive triple by q(a,b,c)=fraclog clog R. The denominator is positive because c≥2 and R≥2. The logarithm base does not matter. The identity c=R^q is exact, although decimal qualities are approximations.
Exact mathematical passage
q(a,b,c)=\frac{\log c}{\log R}.
q^2Use in source · 1 passage
Search aliases: q^2
Use in source · project_11.tex, line 65
Weighted dyadic row. For 2^eparallel c with e≥2 and 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}. At e=1, the output contact depth is zero and this row is omitted. 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\}.
q^{e-1}Use in source · 1 passage
Search aliases: q^e-1
Use in source · project_07.tex, line 105
ℓ(v_q(q_E))}. In a Tate basis, the cyclotomic graded diagonal and the inertia Kummer extension respectively force ℓ^nmid(q-1) and ℓ^nmid v_q(q_E)~[source reference]. This is a necessary condition, rather than a converse or a condition on Frobenius alone. 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 · 1 passage
Search aliases: q^p-1
Use in source · project_11.tex, line 46
te-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
\boxed{ \bar v_p(\ell_{p,e}(q)) = \min\{e-1,v_p(q^{p-1}-1)-1\}. }
Q^sUse in source · 1 passage
Search aliases: Q^s
Use in source · 21_corrected_target.tex, line 36
A necessary leading cost for powers of Q. A stronger elementary obstruction measures the exponent cost of any all-triple bound on the unbalanced quantity. [Necessary cost lower bound] Fix s>0. Suppose B,A≥0 and C>0 satisfy Q^s≤ C R^B(log(3c))^A for every primitive positive triple. Then B≥2s. Fix an integer d≥2 and take a=t^d, b=(t+1)^d, c=t^d+(t+1)^d for t≥1. The inputs are coprime. As rad(t^d)≤ t and rad((t+1)^d)≤ t+1, R≤ t(t+1)c, Q=fracabcR≥[t(t+1)]^d-1. The assumed inequality implies [t(t+1)]…
Exact mathematical passage
Q^s\le C R^B(\log(3c))^A
Q^{s_r}Use in source · 2 passages
Search aliases: Q^s_r
Use in source · 26_global_bridge.tex, line 7
[Conditional integral witness criterion] Let R be an unbounded set of positive integer ranks, and put s_r=r(r+1)/2. Suppose that for each r∈ R there are constants C_r>0, A_r≥0 and E_r≥0, independent of the primitive triple, such that every primitive positive triple admits a nonzero integer W_r with Q^s_rmid W_r, |W_r|≤ C_r(ab)^s_rR^E_r(log(3c))^A_r. If E_r/s_r→0 along R, then Conjecture~[source reference] follows. Nonzero integer divisibility gives Q^s_r≤|W_r|. Divide by (ab)^s_r and use Q/(ab)=…
Exact mathematical passage
Q^{s_r}\mid W_r, \qquad |W_r|\le C_r(ab)^{s_r}R^{E_r}(\log(3c))^{A_r}.
\Q^\timesUse in source · 2 passages
Search aliases: Q^× Q times
Use in source · project_09.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 · 2 passages
Search aliases: q^ν_q nu
Use in source · project_10.tex, line 137
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},
q_EUse in source · 3 passages
Search aliases: q_E
Use in source · project_07.tex, line 101
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). Let E/Q_q have split multiplicative reduction with Tate parameter q_E, and let ℓ≠ q. Scalarity of the full local Galois representation on E[ℓ^n] forces n≤min{v_ℓ(q-1),v_ℓ(v_q(q_E))}. In a Tate basis, the cyclotomic graded diagonal and the inertia Kummer extension respectively force ℓ^nmid(q-1) and ℓ^nmid v_q(q_E)~[source r…
Exact mathematical passage
q_E
q_hUse in source · 5 passages
Search aliases: q_h
Use in source · 24_padic.tex, line 17
Let u_1<⋯<u_r be the first r positive integers not divisible by p. Then u_j=j+lfloorfracj-1p-1rfloor, S_p,r=∑_j=1^r u_j. For h≥1, put s_h=(p-1)p^h-1 and write r=q_hs_h+b_h, 0≤ b_h<s_h. Define the finite sum E_p,r=∑_h≥1(s_hbinomq_h2+b_hq_h). The terms are zero once s_h≥ r.
Exact mathematical passage
r=q_hs_h+b_h,\qquad 0\le b_h<s_h.
\Q_pUse in source · 1 passage
Search aliases: Q_p Q
Use in source · 24_padic.tex, line 9
Exact local determinant valuation and denominator budget. Depletion gives an integral local matrix. Fix a prime p and an integer r≥1. Define F_s,p(t)=∑_pnmid kfract^kk^s, H_r,p(t)=det(F_i+j+1,p(t))_0≤ i,j<r. These series converge at every t∈Q_p with v_p(t)>0. They are integral power series. The Cauchy--Binet expansion from Section~[source reference] now ranges only over increasing p-unit indices.
Exact mathematical passage
t\in\Q_p
\Q_qUse in source · 1 passage
Search aliases: Q_q Q
Use in source · project_07.tex, line 101
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). Let E/Q_q have split multiplicative reduction with Tate parameter q_E, and let ℓ≠ q. Scalarity of the full local Galois representation on E[ℓ^n] forces n≤min{v_ℓ(q-1),v_ℓ(v_q(q_E))}. In a Tate basis, the cyclotomic graded diagonal and the inertia Kummer extension respectively force ℓ^nmid(q-1) and ℓ^nmid v_q(q_E)~[source r…
Exact mathematical passage
E/\Q_q
RUse in source · 101 passages
Search aliases: R
Use in source · 00_reading.tex, line 24
The research has also made a substantive correction. If Q=abc/rad(abc), then the defect relevant to the conjecture is Q/(ab)=c/rad(abc). The target Q=R^o(1) is false. Its missing factor ab is restored explicitly in every final certificate.
Exact mathematical passage
Q=R^{o(1)}
\RUse in source · 1 passage
Search aliases: R
Use in source · project_01.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
\R
\mathcal RUse in source · 3 passages
Search aliases: R mathcal
Use in source · 26_global_bridge.tex, line 6
[Conditional integral witness criterion] Let R be an unbounded set of positive integer ranks, and put s_r=r(r+1)/2. Suppose that for each r∈ R there are constants C_r>0, A_r≥0 and E_r≥0, independent of the primitive triple, such that every primitive positive triple admits a nonzero integer W_r with Q^s_rmid W_r, |W_r|≤ C_r(ab)^s_rR^E_r(log(3c))^A_r. If E_r/s_r→0 along R, then Conjecture~[source reference] follows. Nonzero integer divisibility gives Q^s_r≤|W_r|. Divide by (ab)^s_r and use Q/(ab)=…
Exact mathematical passage
\mathcal R
\mathfrak RUse in source · 1 passage
Search aliases: R mathfrak
Use in source · project_11.tex, line 249
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^{1+2\eps}Use in source · 1 passage
Search aliases: R^1+2eps eps
Use in source · 20_signed_filter.tex, line 43
The hypothetical exception regime is sparse. For a primitive triple put Q=abc/R. Suppose c>R^1+eps for a fixed eps>0. Since a,b are positive with sum c, one has ab≥ c-1. Thus, for large c, Q=fracabcR≥fracc(c-1)R>frac12R^1+2eps. The unit group modulo Q has order φ(Q). For every δ>0, frac Qφ(Q)ll_δ R^δ, 3^ω(R)ll_δ R^δ. Both elementary bounds follow by separating a finite set of small primes. For all sufficiently large primes, (1-1/p)^-1≤ p^δ and 3≤ p^δ. Every prime dividing Q also divides R.
Exact mathematical passage
Q=\frac{abc}{R}\ge\frac{c(c-1)}R>\frac12R^{1+2\eps}.
R^{1+\eps}Use in source · 6 passages
Search aliases: R^1+eps eps
Use in source · 01_foundations.tex, line 88
[Finiteness formulation] Conjecture~[source reference] is equivalent to the assertion that, for every eps>0, there are only finitely many primitive positive triples with q(a,b,c)>1+eps. Assume abc, using its bound with eps/2. A triple of quality above 1+eps satisfies R^1+eps<c≤ K_eps/2R^1+eps/2. Hence R^eps/2<K_eps/2. The radical is bounded. The same conjectural inequality then bounds c, leaving finitely many positive triples.
Exact mathematical passage
R^{1+\eps}<c\le K_{\eps/2}R^{1+\eps/2}.
R^{1+\eps/2}Use in source · 1 passage
Search aliases: R^1+eps/2 eps
Use in source · 01_foundations.tex, line 88
[Finiteness formulation] Conjecture~[source reference] is equivalent to the assertion that, for every eps>0, there are only finitely many primitive positive triples with q(a,b,c)>1+eps. Assume abc, using its bound with eps/2. A triple of quality above 1+eps satisfies R^1+eps<c≤ K_eps/2R^1+eps/2. Hence R^eps/2<K_eps/2. The radical is bounded. The same conjectural inequality then bounds c, leaving finitely many positive triples.
Exact mathematical passage
R^{1+\eps}<c\le K_{\eps/2}R^{1+\eps/2}.
R^{1+\eta}Use in source · 1 passage
Search aliases: R^1+η eta
Use in source · 26_global_bridge.tex, line 65
If for every η>0 one proves WC≤ C_η R_aR_bR^η, then c≤ C'_ηR^1+ηlog(3c). The same absorption argument gives abc. Conversely WC≤ M_c, so the all-eps output-sector abc inequality immediately supplies this bound. The uniform combined WC estimate is therefore the exact endpoint. A bound on C alone suffices only under independent control of W, for example in a stated class where W=1.
Exact mathematical passage
c\le C'_{\eta}R^{1+\eta}\log(3c)
R^{1+\eta_r}Use in source · 2 passages
Search aliases: R^1+η_r eta
Use in source · 26_global_bridge.tex, line 19
admits a nonzero integer W_r with Q^s_rmid W_r, |W_r|≤ C_r(ab)^s_rR^E_r(log(3c))^A_r. If E_r/s_r→0 along R, then Conjecture~[source reference] follows. Nonzero integer divisibility gives Q^s_r≤|W_r|. Divide by (ab)^s_r and use Q/(ab)=c/R: (c/R)^s_r≤ C_rR^E_r(log(3c))^A_r. Taking positive roots yields c≤ D_rR^1+η_r(log(3c))^a_r, D_r=C_r^1/s_r, η_r=E_r/s_r, a_r=A_r/s_r. Fix the target eps>0. Choose one rank with η_r<eps. Next choose 0<δ<fraceps-η_r1+eps. For the fixed exponent a_r, there is a cons…
Exact mathematical passage
c\le D_rR^{1+\eta_r}(\log(3c))^{a_r}, \quad D_r=C_r^{1/s_r},\quad \eta_r=E_r/s_r,\quad a_r=A_r/s_r.
R^{1/3}Use in source · 1 passage
Search aliases: R^1/3
Use in source · 01_foundations.tex, line 131
A known unconditional bound. Stewart and Yu proved an effective bound of a different scale~[source reference]: there is an effectively computable absolute C>0 such that every primitive positive triple satisfies c<exp (CR^1/3(log R)^3). It controls the height in terms of the radical unconditionally. Its exponential dependence is much larger than the power R^1+eps required by abc. It therefore provides a benchmark, without closing the gap this research addresses.
Exact mathematical passage
c<\exp\!\left(CR^{1/3}(\log R)^3\right).
r^2Use in source · 4 passages
Search aliases: r^2
Use in source · 23_hankel.tex, line 84
Since |log(b/a)|≤log c and log(c/a)≤log c, every entry has a bound by a constant times (1+log c)^j. A determinant term has total weight ∑_i=0^r-1(i+π(i)+1)=r^2. Therefore | H_r(c/a)|≤ C_r(1+log c)^r^2 for each of those consistent branches. The constant may depend on r. No nonvanishing assertion at every argument in this third chart follows from the positive expansion on 0<t<1.
Exact mathematical passage
\sum_{i=0}^{r-1}(i+\pi(i)+1)=r^2.
R^BUse in source · 1 passage
Search aliases: R^B
Use in source · 21_corrected_target.tex, line 36
A necessary leading cost for powers of Q. A stronger elementary obstruction measures the exponent cost of any all-triple bound on the unbalanced quantity. [Necessary cost lower bound] Fix s>0. Suppose B,A≥0 and C>0 satisfy Q^s≤ C R^B(log(3c))^A for every primitive positive triple. Then B≥2s. Fix an integer d≥2 and take a=t^d, b=(t+1)^d, c=t^d+(t+1)^d for t≥1. The inputs are coprime. As rad(t^d)≤ t and rad((t+1)^d)≤ t+1, R≤ t(t+1)c, Q=fracabcR≥[t(t+1)]^d-1. The assumed inequality implies [t(t+1)]…
Exact mathematical passage
Q^s\le C R^B(\log(3c))^A
R^{E_r}Use in source · 2 passages
Search aliases: R^E_r
Use in source · 26_global_bridge.tex, line 7
[Conditional integral witness criterion] Let R be an unbounded set of positive integer ranks, and put s_r=r(r+1)/2. Suppose that for each r∈ R there are constants C_r>0, A_r≥0 and E_r≥0, independent of the primitive triple, such that every primitive positive triple admits a nonzero integer W_r with Q^s_rmid W_r, |W_r|≤ C_r(ab)^s_rR^E_r(log(3c))^A_r. If E_r/s_r→0 along R, then Conjecture~[source reference] follows. Nonzero integer divisibility gives Q^s_r≤|W_r|. Divide by (ab)^s_r and use Q/(ab)=…
Exact mathematical passage
Q^{s_r}\mid W_r, \qquad |W_r|\le C_r(ab)^{s_r}R^{E_r}(\log(3c))^{A_r}.
R^\epsUse in source · 11 passages
Search aliases: R^eps eps
Use in source · 01_foundations.tex, line 48
ulations The form best adapted to this research separates the repeated depth of the output from the input prime support. [Exact normalization] For every primitive positive triple, frac cR=fracM_cR_aR_b, Q:=fracabcR, frac Qab=frac cR. Consequently Conjecture~[source reference] is exactly equivalent to each all-eps assertion M_c≤ K_eps R_aR_bR^eps, Q≤ K_eps abR^eps. Use R=R_aR_bR_c and c=R_cM_c, then cancel the indicated factors. Multiplying any displayed inequality by the positive cancelled facto…
Exact mathematical passage
M_c\le K_\eps R_aR_bR^\eps, \qquad Q\le K_\eps abR^\eps.
R^{\eps/2}Use in source · 1 passage
Search aliases: R^eps/2 eps
Use in source · 01_foundations.tex, line 91
[Finiteness formulation] Conjecture~[source reference] is equivalent to the assertion that, for every eps>0, there are only finitely many primitive positive triples with q(a,b,c)>1+eps. Assume abc, using its bound with eps/2. A triple of quality above 1+eps satisfies R^1+eps<c≤ K_eps/2R^1+eps/2. Hence R^eps/2<K_eps/2. The radical is bounded. The same conjectural inequality then bounds c, leaving finitely many positive triples.
Exact mathematical passage
R^{\eps/2}<K_{\eps/2}
R^{\mathrm{glob}}Use in source · 1 passage
Search aliases: R^glob mathrm
Use in source · project_08.tex, line 149
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^{o(1)}Use in source · 3 passages
Search aliases: R^o(1)
Use in source · 00_reading.tex, line 24
The research has also made a substantive correction. If Q=abc/rad(abc), then the defect relevant to the conjecture is Q/(ab)=c/rad(abc). The target Q=R^o(1) is false. Its missing factor ab is restored explicitly in every final certificate.
Exact mathematical passage
Q=R^{o(1)}
R^qUse in source · 1 passage
Search aliases: R^q
Use in source · 01_foundations.tex, line 63
Quality measures the exponent in one example. Define the quality of a primitive triple by q(a,b,c)=fraclog clog R. The denominator is positive because c≥2 and R≥2. The logarithm base does not matter. The identity c=R^q is exact, although decimal qualities are approximations.
Exact mathematical passage
c=R^q
R^\deltaUse in source · 1 passage
Search aliases: R^δ delta
Use in source · 20_signed_filter.tex, line 47
The hypothetical exception regime is sparse. For a primitive triple put Q=abc/R. Suppose c>R^1+eps for a fixed eps>0. Since a,b are positive with sum c, one has ab≥ c-1. Thus, for large c, Q=fracabcR≥fracc(c-1)R>frac12R^1+2eps. The unit group modulo Q has order φ(Q). For every δ>0, frac Qφ(Q)ll_δ R^δ, 3^ω(R)ll_δ R^δ. Both elementary bounds follow by separating a finite set of small primes. For all sufficiently large primes, (1-1/p)^-1≤ p^δ and 3≤ p^δ. Every prime dividing Q also divides R.
Exact mathematical passage
\frac Q{\varphi(Q)}\ll_\delta R^\delta, \qquad 3^{\omega(R)}\ll_\delta R^\delta.
R^\etaUse in source · 3 passages
Search aliases: R^η eta
Use in source · 21_corrected_target.tex, line 11
[Counterfamily for the unbalanced target] For every fixed 0≤η<1 and A≥0, no constant K makes Q≤ K R^η(log(3c))^A valid for all primitive positive triples. Take a=1, b=2^k, c=2^k+1 for integers k≥1. The triple is primitive and c is odd. Thus R=2rad(c)≤2c≤3·2^k, Q=2^k-1frac crad(c)≥2^k-1. Consequently fracQR^η(log(3c))^A ≥frac2^(1-η)k-13^η(log(3(2^k+1)))^A⟶∞. An exponential with positive exponent eventually exceeds every fixed power of k.
Exact mathematical passage
Q\le K R^\eta(\log(3c))^A
R_aUse in source · 18 passages
Search aliases: R_a
Use in source · 01_foundations.tex, line 18
For a positive primitive additive triple a+b=c, gcd(a,b)=1, all three pairs are coprime. For example, a divisor of a and c also divides c-a=b, hence is one. Therefore the prime supports are disjoint and R=rad(abc)=R_aR_bR_c, R_u=rad(u). Put M_u=u/R_u. The exact decomposition is abc=R M_aM_bM_c. The conjecture compares c with R, rather than with abc itself. Since abc already contains c, an estimate against the full product would lose the intended content.
Exact mathematical passage
R=\rad(abc)=R_aR_bR_c,\qquad R_u=\rad(u).
r_aUse in source · 7 passages
Search aliases: r_a
Use in source · project_02.tex, line 263
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 · project_11.tex, line 249
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 · 01_foundations.tex, line 18
For a positive primitive additive triple a+b=c, gcd(a,b)=1, all three pairs are coprime. For example, a divisor of a and c also divides c-a=b, hence is one. Therefore the prime supports are disjoint and R=rad(abc)=R_aR_bR_c, R_u=rad(u). Put M_u=u/R_u. The exact decomposition is abc=R M_aM_bM_c. The conjecture compares c with R, rather than with abc itself. Since abc already contains c, an estimate against the full product would lose the intended content.
Exact mathematical passage
R=\rad(abc)=R_aR_bR_c,\qquad R_u=\rad(u).
r_bUse in source · 7 passages
Search aliases: r_b
Use in source · project_02.tex, line 263
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 · 15 passages
Search aliases: R_c
Use in source · 01_foundations.tex, line 18
For a positive primitive additive triple a+b=c, gcd(a,b)=1, all three pairs are coprime. For example, a divisor of a and c also divides c-a=b, hence is one. Therefore the prime supports are disjoint and R=rad(abc)=R_aR_bR_c, R_u=rad(u). Put M_u=u/R_u. The exact decomposition is abc=R M_aM_bM_c. The conjecture compares c with R, rather than with abc itself. Since abc already contains c, an estimate against the full product would lose the intended content.
Exact mathematical passage
R=\rad(abc)=R_aR_bR_c,\qquad R_u=\rad(u).
r_cUse in source · 2 passages
Search aliases: r_c
Use in source · project_02.tex, line 273
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_nUse in source · 2 passages
Search aliases: R_n
Use in source · 01_foundations.tex, line 118
Because b_n contains 3^n+1, its powerful part contains 3^n. Thus rad(b_n)≤ fracb_n3^n, R_n=2rad(b_n)≤frac2b_n3^n. Therefore fracc_nR_n≥frac3^n c_n2b_n>frac3^n2⟶∞. The displayed lower bound does not contradict the positive-eps conjecture. It does not exhibit infinitely many triples with a fixed positive quality gap. The output grows doubly exponentially with n, while the proved defect lower bound grows exponentially. A large defect lower bound and a fixed positive exponent gap are different asser…
Exact mathematical passage
\rad(b_n)\le \frac{b_n}{3^n},\qquad R_n=2\rad(b_n)\le\frac{2b_n}{3^n}.
R_uUse in source · 2 passages
Search aliases: R_u
Use in source · 01_foundations.tex, line 18
For a positive primitive additive triple a+b=c, gcd(a,b)=1, all three pairs are coprime. For example, a divisor of a and c also divides c-a=b, hence is one. Therefore the prime supports are disjoint and R=rad(abc)=R_aR_bR_c, R_u=rad(u). Put M_u=u/R_u. The exact decomposition is abc=R M_aM_bM_c. The conjecture compares c with R, rather than with abc itself. Since abc already contains c, an estimate against the full product would lose the intended content.
Exact mathematical passage
R=\rad(abc)=R_aR_bR_c,\qquad R_u=\rad(u).
r_uUse in source · 7 passages
Search aliases: r_u
Use in source · project_02.tex, line 221
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
r_u=\rad(u), \qquad P_u=\frac{u}{r_u}.
\operatorname{rad}Use in source · 4 passages
Search aliases: rad operatorname
Use in source · 02_polynomial.tex, line 4
The polynomial theorem and the missing integer analogue. Roots play the role of distinct primes. For a nonzero polynomial F over an algebraically closed field, let rad(F) be the product of its distinct monic linear factors. Its degree counts distinct roots. Write n_0(F)=degrad(F).
Exact mathematical passage
\operatorname{rad}(F)
\radUse in source · 25 passages
Search aliases: rad
Use in source · 00_reading.tex, line 24
The research has also made a substantive correction. If Q=abc/rad(abc), then the defect relevant to the conjecture is Q/(ab)=c/rad(abc). The target Q=R^o(1) is false. Its missing factor ab is restored explicitly in every final certificate.
Exact mathematical passage
Q=abc/\rad(abc)
\operatorname{rank}Use in source · 3 passages
Search aliases: rank operatorname
Use in source · 20_signed_filter.tex, line 67
Exact innovation rank in a cyclic channel. Let G=Z/nZ, α_x=e^2π ix/n and m(k)=∑_x∈ EfracN(x)Mα_x^k. Form the shift matrix S_d=(m(k+j))_0≤ k<n, 0≤ j≤ d. It factors as a Fourier-column matrix times a Vandermonde matrix on the distinct occupied nodes. The Fourier columns are independent, and all weights N(x)/M are nonzero. Therefore rankS_d=min(r,d+1). Before degree r, a small singular value is a conditioning phenomenon, not an exact reduction in the number of occupied residues.
Exact mathematical passage
\boxed{\operatorname{rank}S_d=\min(r,d+1).}
\rceilUse in source · 3 passages
Search aliases: rceil
Use in source · 24_padic.tex, line 81
An explicit fixed correction at the remaining primes. Put a_r=2r-1. For each prime p and 1≤ i≤ r, define C_r,p,i=max (0,sup_h≥1{ a_rh-(p^hlceilfrac ip^hrceil-i)}). This is a finite integer maximum because p^h eventually grows faster than h. Put C_r,p=∑_i=1^rC_r,p,i. The good-prime inequality just proved implies C_r,p=0 for p>3r-1. Hence B_r=∏_p≤3r-1p^C_r,p is a fixed positive integer depending only on the rank.
Exact mathematical passage
C_{r,p,i}=\max\!\left(0,\sup_{h\ge1}\left\{ a_rh-\left(p^h\left\lceil\frac i{p^h}\right\rceil-i\right)\right\}\right).
\mathrm{red}Use in source · 2 passages
Search aliases: red mathrm
Use in source · project_08.tex, line 21
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). Place the two-term contact complex in degrees -1,0 and define 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 · 24_padic.tex, line 12
Let u_1<⋯<u_r be the first r positive integers not divisible by p. Then u_j=j+lfloorfracj-1p-1rfloor, S_p,r=∑_j=1^r u_j. For h≥1, put s_h=(p-1)p^h-1 and write r=q_hs_h+b_h, 0≤ b_h<s_h. Define the finite sum E_p,r=∑_h≥1(s_hbinomq_h2+b_hq_h). The terms are zero once s_h≥ r.
Exact mathematical passage
u_j=j+\left\lfloor\frac{j-1}{p-1}\right\rfloor, \qquad S_{p,r}=\sum_{j=1}^r u_j.
SUse in source · 29 passages
Search aliases: S
Use in source · 20_signed_filter.tex, line 66
Exact innovation rank in a cyclic channel. Let G=Z/nZ, α_x=e^2π ix/n and m(k)=∑_x∈ EfracN(x)Mα_x^k. Form the shift matrix S_d=(m(k+j))_0≤ k<n, 0≤ j≤ d. It factors as a Fourier-column matrix times a Vandermonde matrix on the distinct occupied nodes. The Fourier columns are independent, and all weights N(x)/M are nonzero. Therefore rankS_d=min(r,d+1). Before degree r, a small singular value is a conditioning phenomenon, not an exact reduction in the number of occupied residues.
Exact mathematical passage
S_d=(m(k+j))_{0\le k<n,\,0\le j\le d}
S_aUse in source · 1 passage
Search aliases: S_a
Use in source · project_01.tex, line 12
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. First define the global free module over all primes, and restrict it to Ω_S for S-supported integers. For such 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 ex…
Exact mathematical passage
S=S_a\sqcup S_b\sqcup S_c =\{p:p\mid abc\}.
S_{ab}Use in source · 6 passages
Search aliases: S_ab
Use in source · project_11.tex, line 5
The signed input exponent signal. Let S_ab={q:qmid ab}. Define ν_q=v_q(b)-v_q(a). Then -frac ba=-∏_qmid abq^ν_q. For every output prime pmid c, the same input exponent packet produces the deep principal unit 1-frac ca. The output persistence is therefore a simultaneous cross-prime cancellation phenomenon.
Exact mathematical passage
S_{ab}=\{q:q\mid ab\}.
S_bUse in source · 1 passage
Search aliases: S_b
Use in source · project_01.tex, line 12
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. First define the global free module over all primes, and restrict it to Ω_S for S-supported integers. For such 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 ex…
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 · project_01.tex, line 12
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. First define the global free module over all primes, and restrict it to Ω_S for S-supported integers. For such 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 ex…
Exact mathematical passage
S=S_a\sqcup S_b\sqcup S_c =\{p:p\mid abc\}.
S_dUse in source · 2 passages
Search aliases: S_d
Use in source · 20_signed_filter.tex, line 66
Exact innovation rank in a cyclic channel. Let G=Z/nZ, α_x=e^2π ix/n and m(k)=∑_x∈ EfracN(x)Mα_x^k. Form the shift matrix S_d=(m(k+j))_0≤ k<n, 0≤ j≤ d. It factors as a Fourier-column matrix times a Vandermonde matrix on the distinct occupied nodes. The Fourier columns are independent, and all weights N(x)/M are nonzero. Therefore rankS_d=min(r,d+1). Before degree r, a small singular value is a conditioning phenomenon, not an exact reduction in the number of occupied residues.
Exact mathematical passage
S_d=(m(k+j))_{0\le k<n,\,0\le j\le d}
s_hUse in source · 7 passages
Search aliases: s_h
Use in source · 24_padic.tex, line 16
Let u_1<⋯<u_r be the first r positive integers not divisible by p. Then u_j=j+lfloorfracj-1p-1rfloor, S_p,r=∑_j=1^r u_j. For h≥1, put s_h=(p-1)p^h-1 and write r=q_hs_h+b_h, 0≤ b_h<s_h. Define the finite sum E_p,r=∑_h≥1(s_hbinomq_h2+b_hq_h). The terms are zero once s_h≥ r.
Exact mathematical passage
s_h=(p-1)p^{h-1}
S_{p,r}Use in source · 3 passages
Search aliases: S_p,r
Use in source · 24_padic.tex, line 12
Let u_1<⋯<u_r be the first r positive integers not divisible by p. Then u_j=j+lfloorfracj-1p-1rfloor, S_p,r=∑_j=1^r u_j. For h≥1, put s_h=(p-1)p^h-1 and write r=q_hs_h+b_h, 0≤ b_h<s_h. Define the finite sum E_p,r=∑_h≥1(s_hbinomq_h2+b_hq_h). The terms are zero once s_h≥ r.
Exact mathematical passage
u_j=j+\left\lfloor\frac{j-1}{p-1}\right\rfloor, \qquad S_{p,r}=\sum_{j=1}^r u_j.
s_rUse in source · 19 passages
Search aliases: s_r
Use in source · 23_hankel.tex, line 39
[Exact contact and first coefficient] Put s_r=r(r+1)/2. Then ord_0 H_r=s_r, [t^s_r] H_r=∏_k=1^r k^-(2k-1). The unique increasing r-tuple of positive indices with least sum is (1,…,r), of sum s_r. Its coefficient is nonzero. Its Vandermonde numerator is ∏_j=1^r-1(j!)^2. Dividing this by ∏_k=1^r k^2r-1 leaves exponent 2k-1 in the denominator at each k, giving the formula. For r=2 the first coefficient is 1/8. For r=3 it is 1/(2^3 3^5). The contact orders are 1,3,6,10,…, the triangular numbers.
Exact mathematical passage
s_r=r(r+1)/2
\mathrm{sh}Use in source · 1 passage
Search aliases: sh mathrm
Use in source · project_02.tex, line 248
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\}.
\SNFUse in source · 2 passages
Search aliases: SNF
Use in source · project_02.tex, line 42
[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 · project_08.tex, line 6
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). Place the two-term contact complex in degrees -1,0 and define 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 · project_01.tex, line 12
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. First define the global free module over all primes, and restrict it to Ω_S for S-supported integers. For such 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 ex…
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 · project_08.tex, line 149
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.
\supUse in source · 1 passage
Search aliases: sup
Use in source · 24_padic.tex, line 81
An explicit fixed correction at the remaining primes. Put a_r=2r-1. For each prime p and 1≤ i≤ r, define C_r,p,i=max (0,sup_h≥1{ a_rh-(p^hlceilfrac ip^hrceil-i)}). This is a finite integer maximum because p^h eventually grows faster than h. Put C_r,p=∑_i=1^rC_r,p,i. The good-prime inequality just proved implies C_r,p=0 for p>3r-1. Hence B_r=∏_p≤3r-1p^C_r,p is a fixed positive integer depending only on the rank.
Exact mathematical passage
C_{r,p,i}=\max\!\left(0,\sup_{h\ge1}\left\{ a_rh-\left(p^h\left\lceil\frac i{p^h}\right\rceil-i\right)\right\}\right).
\sup_{h\ge1}Use in source · 1 passage
Search aliases: sup_h≥1 sup ge
Use in source · 24_padic.tex, line 81
An explicit fixed correction at the remaining primes. Put a_r=2r-1. For each prime p and 1≤ i≤ r, define C_r,p,i=max (0,sup_h≥1{ a_rh-(p^hlceilfrac ip^hrceil-i)}). This is a finite integer maximum because p^h eventually grows faster than h. Put C_r,p=∑_i=1^rC_r,p,i. The good-prime inequality just proved implies C_r,p=0 for p>3r-1. Hence B_r=∏_p≤3r-1p^C_r,p is a fixed positive integer depending only on the rank.
Exact mathematical passage
C_{r,p,i}=\max\!\left(0,\sup_{h\ge1}\left\{ a_rh-\left(p^h\left\lceil\frac i{p^h}\right\rceil-i\right)\right\}\right).
TUse in source · 19 passages
Search aliases: T
Use in source · project_02.tex, line 6
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. Set V_0(T)=0 for an absent prime. 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{ V_e(T)=1+T+\cdots+T^{e-1}. }
\mathbb TUse in source · 4 passages
Search aliases: T mathbb
Use in source · project_07.tex, line 58
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 · 5 passages
Search aliases: T mathbf
Use in source · project_03.tex, line 165
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}).
\mathcal TUse in source · 3 passages
Search aliases: T mathcal
Use in source · project_01.tex, line 76
[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 · 5 passages
Search aliases: T mathsf
Use in source · 20_signed_filter.tex, line 84
sidues is determined by the 2r exact consecutive samples m(0),…,m(2r-1). Write A(z)=∏_x∈ E(z-α_x)=z^r+∑_j=0^r-1a_jz^j. For every k, ∑_j=0^r-1a_jm(k+j)=-m(k+r). The coefficient matrix for 0≤ k<r is the Hankel matrix H=(m(i+j))_0≤ i,j<r. With V=(α_x^i)_i,x and D=diag(N(x)/M), one has H=VDV^ T. The square Vandermonde V is invertible and every diagonal entry of D is nonzero. Thus H is invertible, and the 2r samples determine the monic annihilator A.
Exact mathematical passage
H=VDV^{\mathsf T}
tUse in source · 102 passages
Search aliases: t
Use in source · 21_corrected_target.tex, line 42
ng cost for powers of Q A stronger elementary obstruction measures the exponent cost of any all-triple bound on the unbalanced quantity. [Necessary cost lower bound] Fix s>0. Suppose B,A≥0 and C>0 satisfy Q^s≤ C R^B(log(3c))^A for every primitive positive triple. Then B≥2s. Fix an integer d≥2 and take a=t^d, b=(t+1)^d, c=t^d+(t+1)^d for t≥1. The inputs are coprime. As rad(t^d)≤ t and rad((t+1)^d)≤ t+1, R≤ t(t+1)c, Q=fracabcR≥[t(t+1)]^d-1. The assumed inequality implies [t(t+1)]^s(d-1)≤ C[t(t+1)c…
Exact mathematical passage
a=t^d
T^2Use in source · 1 passage
Search aliases: T^2
Use in source · project_02.tex, line 165
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 · 3 passages
Search aliases: t^2
Use in source · 22_jet_filter.tex, line 33
+1)!. Direct substitution gives P_m(1/k)=frac1k∏_j=1^m(frac jk-1) =(-1)^mfrac(k-1)⋯(k-m)k^m+1. Divide by (m+1)! and multiply numerator and denominator by k to get the asserted binomial coefficient. It vanishes for 1≤ k≤ m and first becomes one at k=m+1. The coefficient there is nonzero. For m=0, this is Φ_0=Li_1. For m=1, Φ_1=(Li_2-Li_1)/2=-t^2/8-t^3/9-⋯. The sign is part of the exact formula. Changing the weight alone would not produce the cancellation of the linear coefficient.
Exact mathematical passage
\Phi_1=(\Li_2-\Li_1)/2=-t^2/8-t^3/9-\cdots.
T^3Use in source · 1 passage
Search aliases: T^3
Use in source · project_02.tex, line 187
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^3Use in source · 3 passages
Search aliases: t^3
Use in source · 22_jet_filter.tex, line 33
+1)!. Direct substitution gives P_m(1/k)=frac1k∏_j=1^m(frac jk-1) =(-1)^mfrac(k-1)⋯(k-m)k^m+1. Divide by (m+1)! and multiply numerator and denominator by k to get the asserted binomial coefficient. It vanishes for 1≤ k≤ m and first becomes one at k=m+1. The coefficient there is nonzero. For m=0, this is Φ_0=Li_1. For m=1, Φ_1=(Li_2-Li_1)/2=-t^2/8-t^3/9-⋯. The sign is part of the exact formula. Changing the weight alone would not produce the cancellation of the linear coefficient.
Exact mathematical passage
\Phi_1=(\Li_2-\Li_1)/2=-t^2/8-t^3/9-\cdots.
t^4Use in source · 1 passage
Search aliases: t^4
Use in source · 25_truncation.tex, line 44
Rank two exposes the loss in one line. The first two truncated rank-two expressions are H_2,2(t)=t^3/8, H_2,3(t)=fract^38+frac4t^427+fract^5216 =fract^3(27+32t+t^2)216. Clearing the displayed denominator and the projective denominator in the second gives W(a,c)=a^3(27c^2+32ac+a^2). For every p>3 dividing c, this is congruent to a^5 modulo p. Hence it is a unit at that output prime.
Exact mathematical passage
\mathcal H_{2,3}(t)=\frac{t^3}{8}+\frac{4t^4}{27}+\frac{t^5}{216} =\frac{t^3(27+32t+t^2)}{216}.
t^5Use in source · 1 passage
Search aliases: t^5
Use in source · 25_truncation.tex, line 44
Rank two exposes the loss in one line. The first two truncated rank-two expressions are H_2,2(t)=t^3/8, H_2,3(t)=fract^38+frac4t^427+fract^5216 =fract^3(27+32t+t^2)216. Clearing the displayed denominator and the projective denominator in the second gives W(a,c)=a^3(27c^2+32ac+a^2). For every p>3 dividing c, this is congruent to a^5 modulo p. Hence it is a unit at that output prime.
Exact mathematical passage
\mathcal H_{2,3}(t)=\frac{t^3}{8}+\frac{4t^4}{27}+\frac{t^5}{216} =\frac{t^3(27+32t+t^2)}{216}.
t^dUse in source · 3 passages
Search aliases: t^d
Use in source · 21_corrected_target.tex, line 42
ng cost for powers of Q A stronger elementary obstruction measures the exponent cost of any all-triple bound on the unbalanced quantity. [Necessary cost lower bound] Fix s>0. Suppose B,A≥0 and C>0 satisfy Q^s≤ C R^B(log(3c))^A for every primitive positive triple. Then B≥2s. Fix an integer d≥2 and take a=t^d, b=(t+1)^d, c=t^d+(t+1)^d for t≥1. The inputs are coprime. As rad(t^d)≤ t and rad((t+1)^d)≤ t+1, R≤ t(t+1)c, Q=fracabcR≥[t(t+1)]^d-1. The assumed inequality implies [t(t+1)]^s(d-1)≤ C[t(t+1)c…
Exact mathematical passage
a=t^d
T^eUse in source · 1 passage
Search aliases: T^e
Use in source · project_02.tex, line 12
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. Set V_0(T)=0 for an absent prime. 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 · 1 passage
Search aliases: T^e-1
Use in source · project_02.tex, line 6
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. Set V_0(T)=0 for an absent prime. 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{ V_e(T)=1+T+\cdots+T^{e-1}. }
t^kUse in source · 10 passages
Search aliases: t^k
Use in source · 22_jet_filter.tex, line 12
The universal mixed-weight jet filter. One polynomial chooses the cancelling coefficients. Let m≥0 be an integer. Define P_m(X)=X∏_j=1^m(jX-1)=∑_s=1^m+1c_m,sX^s, and the formal series Φ_m(t)=frac1(m+1)!∑_s=1^m+1c_m,sLi_s(t). Here Li_s(t)=∑_k≥1t^k/k^s is first interpreted in Q[[t]]. It also defines the usual analytic polylogarithm on |t|<1~[source reference]. The label s is its weight. The contact order will follow from its coefficients, rather than from that label.
Exact mathematical passage
\Li_s(t)=\sum_{k\ge1}t^k/k^s
t^{k_1+\cdots+k_r}Use in source · 1 passage
Search aliases: t^k_1+⋯+k_r cdots
Use in source · 23_hankel.tex, line 18
[Cauchy--Binet expansion] H_r(t)=∑_1≤ k_1<⋯<k_r t^k_1+⋯+k_r frac∏_i<j(k_j-k_i)^2∏_i=1^r k_i^2r-1. Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and including the factors t^k_i/k_i gives the stated coefficient. In detail, ∏_i<j(k_j^-1-k_i^-1)^2 =frac∏_i<j(k_j-k_i)^2∏_i k_i^2(r-1). Passing to the formal …
Exact mathematical passage
\mathcal H_r(t)=\sum_{1\le k_1<\cdots<k_r} t^{k_1+\cdots+k_r} \frac{\prod_{i<j}(k_j-k_i)^2}{\prod_{i=1}^r k_i^{2r-1}}.
t^{k_i}Use in source · 1 passage
Search aliases: t^k_i
Use in source · 23_hankel.tex, line 29
i<j(k_j-k_i)^2∏_i=1^r k_i^2r-1. Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and including the factors t^k_i/k_i gives the stated coefficient. In detail, ∏_i<j(k_j^-1-k_i^-1)^2 =frac∏_i<j(k_j-k_i)^2∏_i k_i^2(r-1). Passing to the formal infinite sum is coefficientwise legitimate. At a fixed power of t,…
Exact mathematical passage
t^{k_i}/k_i
t^{m+1}Use in source · 2 passages
Search aliases: t^m+1
Use in source · 22_jet_filter.tex, line 19
[Exact jet cancellation] Φ_m(t)=(-1)^m∑_k≥1binomkm+1fract^kk^m+2. Thus ord_0Φ_m=m+1, [t^m+1]Φ_m=frac(-1)^m(m+1)^m+2. The coefficient of t^k is P_m(1/k)/(m+1)!. Direct substitution gives P_m(1/k)=frac1k∏_j=1^m(frac jk-1) =(-1)^mfrac(k-1)⋯(k-m)k^m+1. Divide by (m+1)! and multiply numerator and denominator by k to get the asserted binomial coefficient. It vanishes for 1≤ k≤ m and first becomes one at k=m+1. The coefficient there is nonzero. For m=0, this is Φ_0=Li_1. For m=1, Φ_1=(Li_2-Li_1)/2=-t^2…
Exact mathematical passage
\ord_0\Phi_m=m+1,\qquad [t^{m+1}]\Phi_m=\frac{(-1)^m}{(m+1)^{m+2}}.
T^NUse in source · 1 passage
Search aliases: T^N
Use in source · project_02.tex, line 35
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
A_N=\Z[T]/(T^N-1).
t^{s_r}Use in source · 1 passage
Search aliases: t^s_r
Use in source · 23_hankel.tex, line 40
[Exact contact and first coefficient] Put s_r=r(r+1)/2. Then ord_0 H_r=s_r, [t^s_r] H_r=∏_k=1^r k^-(2k-1). The unique increasing r-tuple of positive indices with least sum is (1,…,r), of sum s_r. Its coefficient is nonzero. Its Vandermonde numerator is ∏_j=1^r-1(j!)^2. Dividing this by ∏_k=1^r k^2r-1 leaves exponent 2k-1 in the denominator at each k, giving the formula. For r=2 the first coefficient is 1/8. For r=3 it is 1/(2^3 3^5). The contact orders are 1,3,6,10,…, the triangular numbers.
Exact mathematical passage
\ord_0\mathcal H_r=s_r,\qquad [t^{s_r}]\mathcal H_r=\prod_{k=1}^r k^{-(2k-1)}.
T^{v_p(m)}Use in source · 1 passage
Search aliases: T^v_p(m)
Use in source · project_02.tex, line 22
nt 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. Set V_0(T)=0 for an absent prime. 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 · 2 passages
Search aliases: T_0
Use in source · project_05.tex, line 116
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 · 2 passages
Search aliases: T_1
Use in source · project_05.tex, line 120
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_1Use in source · 1 passage
Search aliases: t_1
Use in source · project_01.tex, line 253
Three-sector compression. For the empty unit sector, set g_1=0, t_1=0, and ρ_1(0)=0. Any nonzero target in that sector is infeasible. Within a nonempty 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…
Exact mathematical passage
t_1=0
t_aUse in source · 15 passages
Search aliases: t_a
Use in source · project_01.tex, line 264
sector is infeasible. Within a nonempty 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 · 3 passages
Search aliases: T_a,b,c mathcal
Use in source · project_01.tex, line 76
[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 · project_02.tex, line 273
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 · project_01.tex, line 264
sector is infeasible. Within a nonempty 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 · project_02.tex, line 273
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 · project_01.tex, line 264
sector is infeasible. Within a nonempty 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 · project_02.tex, line 273
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 · project_08.tex, line 149
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 · project_02.tex, line 112
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 · 2 passages
Search aliases: T_p,r
Use in source · project_02.tex, line 172
[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 · project_02.tex, line 160
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 · project_02.tex, line 187
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 · project_02.tex, line 227
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 · project_03.tex, line 46
[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 · 17 passages
Search aliases: U
Use in source · project_03.tex, line 200
[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]]
uUse in source · 40 passages
Search aliases: u
Use in source · 01_foundations.tex, line 18
For a positive primitive additive triple a+b=c, gcd(a,b)=1, all three pairs are coprime. For example, a divisor of a and c also divides c-a=b, hence is one. Therefore the prime supports are disjoint and R=rad(abc)=R_aR_bR_c, R_u=rad(u). Put M_u=u/R_u. The exact decomposition is abc=R M_aM_bM_c. The conjecture compares c with R, rather than with abc itself. Since abc already contains c, an estimate against the full product would lose the intended content.
Exact mathematical passage
R=\rad(abc)=R_aR_bR_c,\qquad R_u=\rad(u).
U^{-1}Use in source · 1 passage
Search aliases: U^-1
Use in source · project_05.tex, line 64
[Unitary no-gain principle] Let U be unitary on the shell coordinate of every active prime channel. Then normUΨ_n_2=normΨ_n_2. Transport the activation projector to UP_actU^-1. Its projected energy then also agrees exactly with the original activation energy. The same holds after exact synthesis. An arbitrary unitary transform need not preserve the energy under a fixed, untransported first-shell projection.
Exact mathematical passage
UP_{\mathrm{act}}U^{-1}
u^2Use in source · 3 passages
Search aliases: u^2
Use in source · 01_foundations.tex, line 112
re of the zero-epsilon estimate] The ratios c/rad(abc) are unbounded among primitive positive additive triples. For n≥0, take a_n=1, b_n=4^3^n-1, c_n=4^3^n. These are primitive, b_n is odd, and rad(c_n)=2. We claim v_3(b_n)=n+1. At n=0 this is v_3(3)=1. If x=1+3^ku with k≥1 and 3nmid u, then x^3-1=3^k+1u(1+3^ku+3^2k-1u^2). The bracket is a unit modulo three, so cubing increases the valuation by exactly one. This proves the claim by induction.
Exact mathematical passage
x^3-1=3^{k+1}u\left(1+3^ku+3^{2k-1}u^2\right).
u^MUse in source · 4 passages
Search aliases: u^M
Use in source · project_09.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_1Use in source · 2 passages
Search aliases: u_1
Use in source · 24_padic.tex, line 11
Let u_1<⋯<u_r be the first r positive integers not divisible by p. Then u_j=j+lfloorfracj-1p-1rfloor, S_p,r=∑_j=1^r u_j. For h≥1, put s_h=(p-1)p^h-1 and write r=q_hs_h+b_h, 0≤ b_h<s_h. Define the finite sum E_p,r=∑_h≥1(s_hbinomq_h2+b_hq_h). The terms are zero once s_h≥ r.
Exact mathematical passage
u_1<\cdots<u_r
U_2Use in source · 1 passage
Search aliases: U_2
Use in source · project_03.tex, line 200
[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 · project_11.tex, line 249
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 · project_03.tex, line 200
[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 · project_04.tex, line 92
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 ℓ_i(H)≤ u_i(H) for every i, ∑_iℓ_i(H)≤ q(t)≤∑_i u_i(H). Every co…
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).
u_jUse in source · 1 passage
Search aliases: u_j
Use in source · 24_padic.tex, line 12
Let u_1<⋯<u_r be the first r positive integers not divisible by p. Then u_j=j+lfloorfracj-1p-1rfloor, S_p,r=∑_j=1^r u_j. For h≥1, put s_h=(p-1)p^h-1 and write r=q_hs_h+b_h, 0≤ b_h<s_h. Define the finite sum E_p,r=∑_h≥1(s_hbinomq_h2+b_hq_h). The terms are zero once s_h≥ r.
Exact mathematical passage
u_j=j+\left\lfloor\frac{j-1}{p-1}\right\rfloor, \qquad S_{p,r}=\sum_{j=1}^r u_j.
u_rUse in source · 2 passages
Search aliases: u_r
Use in source · 24_padic.tex, line 11
Let u_1<⋯<u_r be the first r positive integers not divisible by p. Then u_j=j+lfloorfracj-1p-1rfloor, S_p,r=∑_j=1^r u_j. For h≥1, put s_h=(p-1)p^h-1 and write r=q_hs_h+b_h, 0≤ b_h<s_h. Define the finite sum E_p,r=∑_h≥1(s_hbinomq_h2+b_hq_h). The terms are zero once s_h≥ r.
Exact mathematical passage
u_1<\cdots<u_r
VUse in source · 16 passages
Search aliases: V
Use in source · 20_signed_filter.tex, line 84
thm:prony The distribution on a cyclic channel with r occupied residues is determined by the 2r exact consecutive samples m(0),…,m(2r-1). Write A(z)=∏_x∈ E(z-α_x)=z^r+∑_j=0^r-1a_jz^j. For every k, ∑_j=0^r-1a_jm(k+j)=-m(k+r). The coefficient matrix for 0≤ k<r is the Hankel matrix H=(m(i+j))_0≤ i,j<r. With V=(α_x^i)_i,x and D=diag(N(x)/M), one has H=VDV^ T. The square Vandermonde V is invertible and every diagonal entry of D is nonzero. Thus H is invertible, and the 2r samples determine the monic …
Exact mathematical passage
V=(\alpha_x^i)_{i,x}
V^{\mathsf T}Use in source · 1 passage
Search aliases: V^ T mathsf
Use in source · 20_signed_filter.tex, line 84
sidues is determined by the 2r exact consecutive samples m(0),…,m(2r-1). Write A(z)=∏_x∈ E(z-α_x)=z^r+∑_j=0^r-1a_jz^j. For every k, ∑_j=0^r-1a_jm(k+j)=-m(k+r). The coefficient matrix for 0≤ k<r is the Hankel matrix H=(m(i+j))_0≤ i,j<r. With V=(α_x^i)_i,x and D=diag(N(x)/M), one has H=VDV^ T. The square Vandermonde V is invertible and every diagonal entry of D is nonzero. Thus H is invertible, and the 2r samples determine the monic annihilator A.
Exact mathematical passage
H=VDV^{\mathsf T}
v^2Use in source · 1 passage
Search aliases: v^2
Use in source · project_05.tex, line 116
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 · 2 passages
Search aliases: v^M
Use in source · project_09.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_0Use in source · 1 passage
Search aliases: V_0
Use in source · project_02.tex, line 11
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. Set V_0(T)=0 for an absent prime. 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_0(T)=0
v_2Use in source · 1 passage
Search aliases: v_2
Use in source · project_11.tex, line 65
Weighted dyadic row. For 2^eparallel c with e≥2 and 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}. At e=1, the output contact depth is zero and this row is omitted. 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\}.
v_3Use in source · 2 passages
Search aliases: v_3
Use in source · 01_foundations.tex, line 108
s false. There is an elementary family that proves this without using abc. [Failure of the zero-epsilon estimate] The ratios c/rad(abc) are unbounded among primitive positive additive triples. For n≥0, take a_n=1, b_n=4^3^n-1, c_n=4^3^n. These are primitive, b_n is odd, and rad(c_n)=2. We claim v_3(b_n)=n+1. At n=0 this is v_3(3)=1. If x=1+3^ku with k≥1 and 3nmid u, then x^3-1=3^k+1u(1+3^ku+3^2k-1u^2). The bracket is a unit modulo three, so cubing increases the valuation by exactly one. This pro…
Exact mathematical passage
v_3(b_n)=n+1.
V_eUse in source · 9 passages
Search aliases: V_e
Use in source · project_02.tex, line 6
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. Set V_0(T)=0 for an absent prime. 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{ 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 · project_02.tex, line 12
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. Set V_0(T)=0 for an absent prime. 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 · project_02.tex, line 12
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. Set V_0(T)=0 for an absent prime. 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_kUse in source · 2 passages
Search aliases: v_k
Use in source · 23_hankel.tex, line 25
[Cauchy--Binet expansion] H_r(t)=∑_1≤ k_1<⋯<k_r t^k_1+⋯+k_r frac∏_i<j(k_j-k_i)^2∏_i=1^r k_i^2r-1. Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and including the factors t^k_i/k_i gives the stated coefficient. In detail, ∏_i<j(k_j^-1-k_i^-1)^2 =frac∏_i<j(k_j-k_i)^2∏_i k_i^2(r-1). Passing to the formal …
Exact mathematical passage
v_k=(1,k^{-1},\ldots,k^{-(r-1)})^{\mathsf T}
v_k^{\mathsf T}Use in source · 1 passage
Search aliases: v_k^ T mathsf
Use in source · 23_hankel.tex, line 26
[Cauchy--Binet expansion] H_r(t)=∑_1≤ k_1<⋯<k_r t^k_1+⋯+k_r frac∏_i<j(k_j-k_i)^2∏_i=1^r k_i^2r-1. Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and including the factors t^k_i/k_i gives the stated coefficient. In detail, ∏_i<j(k_j^-1-k_i^-1)^2 =frac∏_i<j(k_j-k_i)^2∏_i k_i^2(r-1). Passing to the formal …
Exact mathematical passage
\sum_{k\ge1}\frac{t^k}{k}v_kv_k^{\mathsf T}.
v_qUse in source · 4 passages
Search aliases: v_q
Use in source · project_07.tex, line 102
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). Let E/Q_q have split multiplicative reduction with Tate parameter q_E, and let ℓ≠ q. Scalarity of the full local Galois representation on E[ℓ^n] forces n≤min{v_ℓ(q-1),v_ℓ(v_q(q_E))}. In a Tate basis, the cyclotomic graded diagonal and the inertia Kummer extension respectively force ℓ^nmid(q-1) and ℓ^nmid v_q(q_E)~[source r…
Exact mathematical passage
n\le\min\{v_\ell(q-1),v_\ell(v_q(q_E))\}.
V_{v_p(n)}Use in source · 1 passage
Search aliases: V_v_p(n)
Use in source · project_02.tex, line 16
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. Set V_0(T)=0 for an absent prime. 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 · project_07.tex, line 102
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). Let E/Q_q have split multiplicative reduction with Tate parameter q_E, and let ℓ≠ q. Scalarity of the full local Galois representation on E[ℓ^n] forces n≤min{v_ℓ(q-1),v_ℓ(v_q(q_E))}. In a Tate basis, the cyclotomic graded diagonal and the inertia Kummer extension respectively force ℓ^nmid(q-1) and ℓ^nmid v_q(q_E)~[source r…
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 · project_09.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}},
WUse in source · 56 passages
Search aliases: W
Use in source · 02_polynomial.tex, line 17
Set W=A'B-AB'. It is nonzero. Indeed, if it vanished, then (A/B)'=0, and characteristic zero would force the rational function A/B to be constant. Coprimality would then force both A and B to be constant, and so also C.
Exact mathematical passage
W=A'B-AB'
\mathsf wUse in source · 8 passages
Search aliases: w mathsf
Use in source · project_03.tex, line 6
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}
wUse in source · 6 passages
Search aliases: w
Use in source · project_04.tex, line 162
Arithmetic correction and arbitrary precision. Some large high-quality triples exceed signed 64-bit range, and intermediate quantities such as ab, fracabcR, w· x can overflow even when the input terms individually fit. Every public arithmetic entry point must therefore promote integer-like inputs to arbitrary-precision integers before multiplication. Floating-point values are display-only. All logical comparisons and stopping bounds are exact rational or integer computations.
Exact mathematical passage
ab, \qquad \frac{abc}{R}, \qquad w\cdot x
W_0Use in source · 1 passage
Search aliases: W_0
Use in source · project_08.tex, line 137
Witt realization. For e≥2, the local packet has a natural additive identification Jcal_p,e ≅ W_e-1(F_p). Division by p identifies the source with Z_p/p^e-1Z_p, and the standard additive identification W_n(F_p)≅Z/p^nZ gives the result. For e=1, take W_0=0. The Verschiebung filtration matches multiplication by successive powers of p. This is the typed local object underlying the carry chain.
Exact mathematical passage
W_0=0
W_1Use in source · 1 passage
Search aliases: W_1
Use in source · project_06.tex, line 84
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_{e-1}Use in source · 1 passage
Search aliases: W_e-1
Use in source · project_08.tex, line 130
Witt realization. For e≥2, the local packet has a natural additive identification Jcal_p,e ≅ W_e-1(F_p). Division by p identifies the source with Z_p/p^e-1Z_p, and the standard additive identification W_n(F_p)≅Z/p^nZ gives the result. For e=1, take W_0=0. 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_jUse in source · 1 passage
Search aliases: W_j
Use in source · project_06.tex, line 92
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_nUse in source · 1 passage
Search aliases: W_n
Use in source · project_08.tex, line 137
Witt realization. For e≥2, the local packet has a natural additive identification Jcal_p,e ≅ W_e-1(F_p). Division by p identifies the source with Z_p/p^e-1Z_p, and the standard additive identification W_n(F_p)≅Z/p^nZ gives the result. For e=1, take W_0=0. The Verschiebung filtration matches multiplication by successive powers of p. This is the typed local object underlying the carry chain.
Exact mathematical passage
W_n(\F_p)\cong\Z/p^n\Z
\mathsf w_pUse in source · 1 passage
Search aliases: w_p mathsf
Use in source · project_03.tex, line 6
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_pUse in source · 2 passages
Search aliases: w_p
Use in source · project_11.tex, line 92
r formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. Divide the input exponent vector by K. Define a_p,q as the original odd or weighted dyadic coefficient, now reduced modulo p^d_p-r_p. Dividing the relation vector reduces the needed relation modulus. It does not divide the coefficient row by K. 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 ord…
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_rUse in source · 5 passages
Search aliases: W_r
Use in source · 26_global_bridge.tex, line 6
[Conditional integral witness criterion] Let R be an unbounded set of positive integer ranks, and put s_r=r(r+1)/2. Suppose that for each r∈ R there are constants C_r>0, A_r≥0 and E_r≥0, independent of the primitive triple, such that every primitive positive triple admits a nonzero integer W_r with Q^s_rmid W_r, |W_r|≤ C_r(ab)^s_rR^E_r(log(3c))^A_r. If E_r/s_r→0 along R, then Conjecture~[source reference] follows. Nonzero integer divisibility gives Q^s_r≤|W_r|. Divide by (ab)^s_r and use Q/(ab)=…
Exact mathematical passage
W_r
W_{r,K}Use in source · 2 passages
Search aliases: W_r,K
Use in source · 25_truncation.tex, line 23
For a reduced rational argument t=a/c, an obvious integral scalar is W_r,K(a,c)=c^dA_r,K(a/c). It is nonzero when 0<a<c, since the truncated positive expansion is nonzero. Thus it is a genuine nonzero arithmetic candidate. The question is whether it retains the required output-prime contact after clearing denominators.
Exact mathematical passage
W_{r,K}(a,c)=c^dA_{r,K}(a/c).
W_xUse in source · 12 passages
Search aliases: W_x
Use in source · project_01.tex, line 87
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,
Exact mathematical passage
\boxed{ W_x(a,b) =a\,\mathrm d_x(b)-b\,\mathrm d_x(a). }
W_yUse in source · 1 passage
Search aliases: W_y
Use in source · project_02.tex, line 263
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 · 1 passage
Search aliases: wedge
Use in source · project_05.tex, line 166
automorphic analysis--synthesis identity Let ϕ_E:X_0(N_E)⟶ E be a nonconstant modular parametrization, and let ϕ_E^*ω_E = 2π i c_ϕ f_E(z) dz. Here c_ϕ is the differential multiplier of this chosen parametrization and differential. The marked Frey curve need not be the optimal representative of its isogeny class. 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_ϕ^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2…
Exact mathematical passage
A(E) = \frac{i}{2} \int_{E(\C)} \omega_E\wedge\overline{\omega_E}.
\widehat GUse in source · 3 passages
Search aliases: widehat G widehat
Use in source · 20_signed_filter.tex, line 9
Let N(x) count assignments producing x∈ G. Repeated contributions remain separate assignments, so M=∑_x∈ GN(x)=3^s, μ(x)=N(x)/M. For a character χ∈widehat G, define widehatμ(χ)=∑_xμ(x)χ(x) =∏_j=1^sfrac1+χ(γ_j)+χ(-γ_j)3. On a cyclic channel G=Z/nZ, this becomes m(k)=∏_j=1^sfrac1+2cos(2π kγ_j/n)3. This is the discrete cosine or COS expression used in the project. It is an exact characteristic function. Individual cosine factors may be negative. No positivity of all Fourier samples is assumed.
Exact mathematical passage
\chi\in\widehat G
\widehat TUse in source · 3 passages
Search aliases: widehat T widehat
Use in source · project_02.tex, line 196
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 · project_02.tex, line 197
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 · project_02.tex, line 208
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 · project_02.tex, line 197
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 · project_02.tex, line 196
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\muUse in source · 4 passages
Search aliases: widehatμ widehat mu
Use in source · 20_signed_filter.tex, line 10
Let N(x) count assignments producing x∈ G. Repeated contributions remain separate assignments, so M=∑_x∈ GN(x)=3^s, μ(x)=N(x)/M. For a character χ∈widehat G, define widehatμ(χ)=∑_xμ(x)χ(x) =∏_j=1^sfrac1+χ(γ_j)+χ(-γ_j)3. On a cyclic channel G=Z/nZ, this becomes m(k)=∏_j=1^sfrac1+2cos(2π kγ_j/n)3. This is the discrete cosine or COS expression used in the project. It is an exact characteristic function. Individual cosine factors may be negative. No positivity of all Fourier samples is assumed.
Exact mathematical passage
\widehat\mu(\chi)=\sum_x\mu(x)\chi(x) =\prod_{j=1}^s\frac{1+\chi(\gamma_j)+\chi(-\gamma_j)}3.
\widetilde xUse in source · 2 passages
Search aliases: widetilde x widetilde
Use in source · project_11.tex, line 40
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)\}
xUse in source · 85 passages
Search aliases: x
Use in source · 01_foundations.tex, line 111
his without using abc. [Failure of the zero-epsilon estimate] The ratios c/rad(abc) are unbounded among primitive positive additive triples. For n≥0, take a_n=1, b_n=4^3^n-1, c_n=4^3^n. These are primitive, b_n is odd, and rad(c_n)=2. We claim v_3(b_n)=n+1. At n=0 this is v_3(3)=1. If x=1+3^ku with k≥1 and 3nmid u, then x^3-1=3^k+1u(1+3^ku+3^2k-1u^2). The bracket is a unit modulo three, so cubing increases the valuation by exactly one. This proves the claim by induction.
Exact mathematical passage
x=1+3^ku
x^{(a)}Use in source · 1 passage
Search aliases: x^(a)
Use in source · project_01.tex, line 278
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^3Use in source · 1 passage
Search aliases: x^3
Use in source · 01_foundations.tex, line 112
re of the zero-epsilon estimate] The ratios c/rad(abc) are unbounded among primitive positive additive triples. For n≥0, take a_n=1, b_n=4^3^n-1, c_n=4^3^n. These are primitive, b_n is odd, and rad(c_n)=2. We claim v_3(b_n)=n+1. At n=0 this is v_3(3)=1. If x=1+3^ku with k≥1 and 3nmid u, then x^3-1=3^k+1u(1+3^ku+3^2k-1u^2). The bracket is a unit modulo three, so cubing increases the valuation by exactly one. This proves the claim by induction.
Exact mathematical passage
x^3-1=3^{k+1}u\left(1+3^ku+3^{2k-1}u^2\right).
X^MUse in source · 1 passage
Search aliases: X^M
Use in source · project_09.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^nUse in source · 2 passages
Search aliases: X^n
Use in source · 02_polynomial.tex, line 37
A sharp example. Take A=X^n, B=1-X^n, C=1 for n≥1. The root sets are disjoint. There is one distinct root of A and n distinct roots of B, so n_0(ABC)=n+1. The maximal degree is n, attaining the theorem's bound. In this example W=nX^n-1, and the repeated-root divisor is exactly X^n-1 up to a nonzero scalar.
Exact mathematical passage
A=X^n
x^nUse in source · 2 passages
Search aliases: x^n
Use in source · 03_consequences.tex, line 7
ionConsequences and the scale of the missing estimate An asymptotic Fermat consequence. The following implication is elementary and illustrates how repeated powers amplify the conjecture. [Conditional large-exponent Fermat statement] Assume Conjecture~[source reference]. There is an integer n_0 such that, for every integer n>n_0, the equation x^n+y^n=z^n has no positive integer solution. Divide a putative solution by its common gcd. The resulting x,y,z are pairwise coprime. Apply abc to a=x^n, b…
Exact mathematical passage
x^n+y^n=z^n
X^{n-1}Use in source · 2 passages
Search aliases: X^n-1
Use in source · 02_polynomial.tex, line 37
A sharp example. Take A=X^n, B=1-X^n, C=1 for n≥1. The root sets are disjoint. There is one distinct root of A and n distinct roots of B, so n_0(ABC)=n+1. The maximal degree is n, attaining the theorem's bound. In this example W=nX^n-1, and the repeated-root divisor is exactly X^n-1 up to a nonzero scalar.
Exact mathematical passage
W=nX^{n-1}
X^pUse in source · 1 passage
Search aliases: X^p
Use in source · 02_polynomial.tex, line 39
The proof needs an actual nonzero derivative. In positive characteristic, X^p+(1-X^p)=1 has every derivative zero, so the characteristic-zero statement cannot simply be transferred. The correct positive-characteristic forms include a condition excluding common Frobenius powers.
Exact mathematical passage
X^p+(1-X^p)=1
X^sUse in source · 1 passage
Search aliases: X^s
Use in source · 22_jet_filter.tex, line 5
The universal mixed-weight jet filter. One polynomial chooses the cancelling coefficients. Let m≥0 be an integer. Define P_m(X)=X∏_j=1^m(jX-1)=∑_s=1^m+1c_m,sX^s, and the formal series Φ_m(t)=frac1(m+1)!∑_s=1^m+1c_m,sLi_s(t). Here Li_s(t)=∑_k≥1t^k/k^s is first interpreted in Q[[t]]. It also defines the usual analytic polylogarithm on |t|<1~[source reference]. The label s is its weight. The contact order will follow from its coefficients, rather than from that label.
Exact mathematical passage
P_m(X)=X\prod_{j=1}^m(jX-1)=\sum_{s=1}^{m+1}c_{m,s}X^s,
X_0Use in source · 1 passage
Search aliases: X_0
Use in source · project_05.tex, line 156
Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be a nonconstant modular parametrization, and let ϕ_E^*ω_E = 2π i c_ϕ f_E(z) dz. Here c_ϕ is the differential multiplier of this chosen parametrization and differential. The marked Frey curve need not be the optimal representative of its isogeny class. 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_ϕ^2 normf_E_Pet^2. For the Frey model, scaling x=cX, …
Exact mathematical passage
\phi_E:X_0(N_E)\longrightarrow E
x_0Use in source · 6 passages
Search aliases: x_0
Use in source · project_04.tex, line 5
Obstruction cosets. Exclude (1,1,2) throughout this section. 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 · project_07.tex, line 58
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_2Use in source · 1 passage
Search aliases: x_2
Use in source · project_01.tex, line 330
The minimality is elementary. At height at most five, |x_3+75x_5|≤380<448, so additivity forces x_2=0. Then x_3=-75x_5 and the same height restriction force both coordinates to vanish. Thus no transverse packet has height below six, and the displayed packet proves Θ=6.
Exact mathematical passage
x_2=0
x_3Use in source · 3 passages
Search aliases: x_3
Use in source · project_01.tex, line 330
The minimality is elementary. At height at most five, |x_3+75x_5|≤380<448, so additivity forces x_2=0. Then x_3=-75x_5 and the same height restriction force both coordinates to vanish. Thus no transverse packet has height below six, and the displayed packet proves Θ=6.
Exact mathematical passage
|x_3+75x_5|\le380<448
x_5Use in source · 3 passages
Search aliases: x_5
Use in source · project_01.tex, line 330
The minimality is elementary. At height at most five, |x_3+75x_5|≤380<448, so additivity forces x_2=0. Then x_3=-75x_5 and the same height restriction force both coordinates to vanish. Thus no transverse packet has height below six, and the displayed packet proves Θ=6.
Exact mathematical passage
|x_3+75x_5|\le380<448
x_iUse in source · 5 passages
Search aliases: x_i
Use in source · project_04.tex, line 72
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
\sum_i\frac r{p_i}x_i=t.
x_pUse in source · 2 passages
Search aliases: x_p
Use in source · project_01.tex, line 42
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 · project_07.tex, line 58
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 · 7 passages
Search aliases: xrightarrow
Use in source · project_02.tex, line 53
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
I_N=\ker\bigl(A_N\xrightarrow{\,T\mapsto1\,}\Z\bigr)
yUse in source · 16 passages
Search aliases: y
Use in source · 03_consequences.tex, line 7
ionConsequences and the scale of the missing estimate An asymptotic Fermat consequence. The following implication is elementary and illustrates how repeated powers amplify the conjecture. [Conditional large-exponent Fermat statement] Assume Conjecture~[source reference]. There is an integer n_0 such that, for every integer n>n_0, the equation x^n+y^n=z^n has no positive integer solution. Divide a putative solution by its common gcd. The resulting x,y,z are pairwise coprime. Apply abc to a=x^n, b…
Exact mathematical passage
x^n+y^n=z^n
y^2Use in source · 2 passages
Search aliases: y^2
Use in source · 03_consequences.tex, line 25
Elliptic curves and the modified Szpiro comparison. The marked Frey curve E:y^2=x(x-a)(x+b) translates the additive triple into an elliptic curve. Its discriminant sees repeated depth, while its conductor sees prime support, with local model corrections at the small primes. This makes elliptic height estimates a natural companion to abc.
Exact mathematical passage
E:y^2=x(x-a)(x+b)
Y^MUse in source · 1 passage
Search aliases: Y^M
Use in source · project_09.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 · 2 passages
Search aliases: y^n
Use in source · 03_consequences.tex, line 7
ionConsequences and the scale of the missing estimate An asymptotic Fermat consequence. The following implication is elementary and illustrates how repeated powers amplify the conjecture. [Conditional large-exponent Fermat statement] Assume Conjecture~[source reference]. There is an integer n_0 such that, for every integer n>n_0, the equation x^n+y^n=z^n has no positive integer solution. Divide a putative solution by its common gcd. The resulting x,y,z are pairwise coprime. Apply abc to a=x^n, b…
Exact mathematical passage
x^n+y^n=z^n
y_pUse in source · 2 passages
Search aliases: y_p
Use in source · project_02.tex, line 121
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 · project_02.tex, line 226
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 · project_09.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.
zUse in source · 26 passages
Search aliases: z
Use in source · 03_consequences.tex, line 7
ionConsequences and the scale of the missing estimate An asymptotic Fermat consequence. The following implication is elementary and illustrates how repeated powers amplify the conjecture. [Conditional large-exponent Fermat statement] Assume Conjecture~[source reference]. There is an integer n_0 such that, for every integer n>n_0, the equation x^n+y^n=z^n has no positive integer solution. Divide a putative solution by its common gcd. The resulting x,y,z are pairwise coprime. Apply abc to a=x^n, b…
Exact mathematical passage
x^n+y^n=z^n
\Z^2Use in source · 3 passages
Search aliases: Z^2 Z
Use in source · project_03.tex, line 22
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^6Use in source · 1 passage
Search aliases: z^6
Use in source · 03_consequences.tex, line 14
al large-exponent Fermat statement] Assume Conjecture~[source reference]. There is an integer n_0 such that, for every integer n>n_0, the equation x^n+y^n=z^n has no positive integer solution. Divide a putative solution by its common gcd. The resulting x,y,z are pairwise coprime. Apply abc to a=x^n, b=y^n, c=z^n with the fixed exponent eps=1. Then z^n≤ K_1rad(xyz)^2≤ K_1(xyz)^2<K_1z^6, because x,y<z. As z≥2, this implies 2^n-6<K_1 whenever n>6. Any fixed integer larger than 6+log_2max(1,K_1) exc…
Exact mathematical passage
z^n\le K_1\rad(xyz)^2\le K_1(xyz)^2<K_1z^6,
z^jUse in source · 2 passages
Search aliases: z^j
Use in source · 20_signed_filter.tex, line 77
[Exact cyclic Prony reconstruction] The distribution on a cyclic channel with r occupied residues is determined by the 2r exact consecutive samples m(0),…,m(2r-1). Write A(z)=∏_x∈ E(z-α_x)=z^r+∑_j=0^r-1a_jz^j. For every k, ∑_j=0^r-1a_jm(k+j)=-m(k+r). The coefficient matrix for 0≤ k<r is the Hankel matrix H=(m(i+j))_0≤ i,j<r. With V=(α_x^i)_i,x and D=diag(N(x)/M), one has H=VDV^ T. The square Vandermonde V is invertible and every diagonal entry of D is nonzero. Thus H is invertible, and the 2r sa…
Exact mathematical passage
A(z)=\prod_{x\in E}(z-\alpha_x)=z^r+\sum_{j=0}^{r-1}a_jz^j.
Z^MUse in source · 1 passage
Search aliases: Z^M
Use in source · project_09.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 · project_07.tex, line 32
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^nUse in source · 3 passages
Search aliases: z^n
Use in source · 03_consequences.tex, line 7
ionConsequences and the scale of the missing estimate An asymptotic Fermat consequence. The following implication is elementary and illustrates how repeated powers amplify the conjecture. [Conditional large-exponent Fermat statement] Assume Conjecture~[source reference]. There is an integer n_0 such that, for every integer n>n_0, the equation x^n+y^n=z^n has no positive integer solution. Divide a putative solution by its common gcd. The resulting x,y,z are pairwise coprime. Apply abc to a=x^n, b…
Exact mathematical passage
x^n+y^n=z^n
z^rUse in source · 1 passage
Search aliases: z^r
Use in source · 20_signed_filter.tex, line 77
[Exact cyclic Prony reconstruction] The distribution on a cyclic channel with r occupied residues is determined by the 2r exact consecutive samples m(0),…,m(2r-1). Write A(z)=∏_x∈ E(z-α_x)=z^r+∑_j=0^r-1a_jz^j. For every k, ∑_j=0^r-1a_jm(k+j)=-m(k+r). The coefficient matrix for 0≤ k<r is the Hankel matrix H=(m(i+j))_0≤ i,j<r. With V=(α_x^i)_i,x and D=diag(N(x)/M), one has H=VDV^ T. The square Vandermonde V is invertible and every diagonal entry of D is nonzero. Thus H is invertible, and the 2r sa…
Exact mathematical passage
A(z)=\prod_{x\in E}(z-\alpha_x)=z^r+\sum_{j=0}^{r-1}a_jz^j.
\Z^{r+1}Use in source · 1 passage
Search aliases: Z^r+1 Z
Use in source · project_07.tex, line 58
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 · project_01.tex, line 42
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 · 1 passage
Search aliases: Z^s Z
Use in source · project_11.tex, line 204
Why abstract lattice geometry is insufficient. Let v∈Z^s be primitive, with s≥2. For every C≥2, there exists a primitive row gpmod C satisfying g· v≡0pmod C. Thus a fixed short vector can lie in primitive congruence lattices of arbitrarily large index.
Exact mathematical passage
v\in\Z^s
\Z^{S_{ab}}Use in source · 3 passages
Search aliases: Z^S_ab Z
Use in source · project_11.tex, line 145
satisfying g_p·ν_0 ≡0 pmodp^χ_p, where ν_0 is the Kummer-normalized input exponent vector. The Chinese remainder theorem combines the local rows into one primitive row g∈(Z/CZ)^S_ab. Define the coherence character χ_coh(x) = exp(frac2π iCg· x). Its kernel is the adelic coherence lattice Λ_coh = {x∈Z^S_ab:g· x≡0pmod C}. Since g is primitive modulo C, [Z^S_ab:Λ_coh]=C and Z^S_ab/Λ_coh≅Z/CZ. The normalized exponent vector lies in the lattice.
Exact mathematical passage
\boxed{ \Lambda_{\mathrm{coh}} = \{x\in\Z^{S_{ab}}:g\cdot x\equiv0\pmod C\}. }
\Z_2Use in source · 1 passage
Search aliases: Z_2 Z
Use in source · project_08.tex, line 145
At odd p, the p-adic logarithm identifies (1+pZ_p)/(1+p^eZ_p) ≅ pZ_p/p^eZ_p. At 2, the additive contact packet remains uniform, while the multiplicative coordinate begins on 1+4Z_2 and therefore carries the familiar weighted dyadic shift.
Exact mathematical passage
1+4\Z_2
z_iUse in source · 4 passages
Search aliases: z_i
Use in source · project_04.tex, line 84
ix 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_i≤ lfloorfracH-ξ_ip_irf…
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 · 22_jet_filter.tex, line 49
The depleted local series. For a prime p, delete the terms whose index is divisible by p: Φ_m,p(t)=(-1)^m∑_pnmid kbinomkm+1fract^kk^m+2∈Z_p[[t]]. It converges when v_p(t)>0. [Exact local jet valuation] If v_p(t)=e≥1, then v_p(Φ_m,p(t))= (m+1)e, · pnmid m+1,
Exact mathematical passage
\Phi_{m,p}(t)=(-1)^m\sum_{p\nmid k}\binom{k}{m+1}\frac{t^k}{k^{m+2}}\in\Z_p[[t]].
|Use in source · 35 passages
Search aliases: |
Use in source · 03_consequences.tex, line 31
The precise equivalent statement uses more than the discriminant alone. For an elliptic curve over Q, let N be its conductor and let c_4,c_6 be the invariants of a global minimal integral Weierstrass equation. The modified Szpiro assertion is max{|c_4|^3,c_6^2}≤ C_eps N^6+eps for every eps>0, with a constant uniform in the elliptic curve. Oesterl e identifies this form as equivalent to the full abc conjecture~[source reference].
Exact mathematical passage
\max\{|c_4|^3,c_6^2\}\le C_\eps N^{6+\eps}
|^2Use in source · 4 passages
Search aliases: |^2
Use in source · 20_signed_filter.tex, line 28
Sparsity forces transverse energy. Let E={x:N(x)>0} and r=|E|. Always r≤ M. Parseval yields ∑_χ∈widehat G|widehatμ(χ)|^2 =frac|G|M^2∑_xN(x)^2. Cauchy--Schwarz on the occupied support gives ∑_xN(x)^2≥ M^2/r. Hence [Sparse energy barrier] ∑_χ≠1|widehatμ(χ)|^2 ≥frac|G|r-1≥frac|G|M-1. When few assignment atoms lie in a large group, substantial nontrivial Fourier energy is unavoidable. A failure of uniform mixing is therefore a structural feature of the signal, rather than evidence that the arithmeti…
Exact mathematical passage
\sum_{\chi\in\widehat G}|\widehat\mu(\chi)|^2 =\frac{|G|}{M^2}\sum_xN(x)^2.
|^3Use in source · 1 passage
Search aliases: |^3
Use in source · 03_consequences.tex, line 31
The precise equivalent statement uses more than the discriminant alone. For an elliptic curve over Q, let N be its conductor and let c_4,c_6 be the invariants of a global minimal integral Weierstrass equation. The modified Szpiro assertion is max{|c_4|^3,c_6^2}≤ C_eps N^6+eps for every eps>0, with a constant uniform in the elliptic curve. Oesterl e identifies this form as equivalent to the full abc conjecture~[source reference].
Exact mathematical passage
\max\{|c_4|^3,c_6^2\}\le C_\eps N^{6+\eps}
\pmUse in source · 2 passages
Search aliases: ± pm
Use in source · project_03.tex, line 60
Rows restricted to one sector are proportional, so same-sector 2×2 minors vanish. If p and q lie in two different nonempty sectors, the corresponding minor is ±fracabcpq = ± QfracRpq. Every nonzero minor is therefore divisible by Q. Conversely, for each prime ℓmid R, choose p=ℓ and choose q in a different nonempty sector. Since R is squarefree, R/(pq) is not divisible by ℓ. Thus the gcd of the reduced minors R/(pq) is one. The gcd of all nonzero minors is exactly Q. The determinantal-divisor des…
Exact mathematical passage
\pm\frac{abc}{pq} = \pm Q\frac{R}{pq}.
\timesUse in source · 6 passages
Search aliases: × times
Use in source · 20_signed_filter.tex, line 117
A Galois coordinate with a controlled denominator. For an occupied target define Δ_h(E)=∏_x∈ E∖{h}(α_h-α_x). This is a nonzero algebraic integer in Q(ζ_n). For every unit u modulo n, its conjugate is Δ_uh(uE). The absolute norm is a nonzero integer. Hence ∏_u∈(Z/nZ)^×|Δ_uh(uE)|≥1. At least one factor is at least one. In that coordinate, the Lagrange numerator has coefficient ℓ^1 norm at most 2^r-1, and dividing by the target denominator preserves that upper bound.
Exact mathematical passage
\prod_{u\in(\Z/n\Z)^\times}|\Delta_{uh}(uE)|\ge1.
\alphaUse in source · 18 passages
Search aliases: α alpha
Use in source · 02_polynomial.tex, line 6
The analogy is direct. Prime powers p^e become repeated roots (X-α)^e. A derivative lowers the multiplicity by one. The divisor left over after that operation is F/rad(F), just as n/rad(n) is the repeated-power part of an integer.
Exact mathematical passage
(X-\alpha)^e
\alpha_hUse in source · 3 passages
Search aliases: α_h alpha
Use in source · 20_signed_filter.tex, line 96
Polynomial target filters and exact count certificates. For a polynomial P(z)=∑_j=0^dc_jz^j, define the collision quadratic form E_d(P)=frac nM^2∑_x∈ EN(x)^2|P(α_x)|^2. It can be written as a positive semidefinite Gram form in the coefficients of P. Let λ_d(h)=min{ E_d(P):deg P≤ d, P(α_h)=1}. For every such filter, the target term is present with value (n/M^2)N(h)^2, so N(h)≤frac Msqrt nsqrtλ_d(h). If the right-hand side is rigorously less than one, the integer count is zero. An approximate deci…
Exact mathematical passage
\lambda_d(h)=\min\{\mathcal E_d(P):\deg P\le d,\ P(\alpha_h)=1\}.
\alpha_uUse in source · 2 passages
Search aliases: α_u alpha
Use in source · project_04.tex, line 151
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_xUse in source · 4 passages
Search aliases: α_x alpha
Use in source · 20_signed_filter.tex, line 62
Exact innovation rank in a cyclic channel. Let G=Z/nZ, α_x=e^2π ix/n and m(k)=∑_x∈ EfracN(x)Mα_x^k. Form the shift matrix S_d=(m(k+j))_0≤ k<n, 0≤ j≤ d. It factors as a Fourier-column matrix times a Vandermonde matrix on the distinct occupied nodes. The Fourier columns are independent, and all weights N(x)/M are nonzero. Therefore rankS_d=min(r,d+1). Before degree r, a small singular value is a conditioning phenomenon, not an exact reduction in the number of occupied residues.
Exact mathematical passage
\alpha_x=e^{2\pi ix/n}
\alpha_x^iUse in source · 1 passage
Search aliases: α_x^i alpha
Use in source · 20_signed_filter.tex, line 84
thm:prony The distribution on a cyclic channel with r occupied residues is determined by the 2r exact consecutive samples m(0),…,m(2r-1). Write A(z)=∏_x∈ E(z-α_x)=z^r+∑_j=0^r-1a_jz^j. For every k, ∑_j=0^r-1a_jm(k+j)=-m(k+r). The coefficient matrix for 0≤ k<r is the Hankel matrix H=(m(i+j))_0≤ i,j<r. With V=(α_x^i)_i,x and D=diag(N(x)/M), one has H=VDV^ T. The square Vandermonde V is invertible and every diagonal entry of D is nonzero. Thus H is invertible, and the 2r samples determine the monic …
Exact mathematical passage
V=(\alpha_x^i)_{i,x}
\alpha_x^kUse in source · 1 passage
Search aliases: α_x^k alpha
Use in source · 20_signed_filter.tex, line 63
Exact innovation rank in a cyclic channel. Let G=Z/nZ, α_x=e^2π ix/n and m(k)=∑_x∈ EfracN(x)Mα_x^k. Form the shift matrix S_d=(m(k+j))_0≤ k<n, 0≤ j≤ d. It factors as a Fourier-column matrix times a Vandermonde matrix on the distinct occupied nodes. The Fourier columns are independent, and all weights N(x)/M are nonzero. Therefore rankS_d=min(r,d+1). Before degree r, a small singular value is a conditioning phenomenon, not an exact reduction in the number of occupied residues.
Exact mathematical passage
m(k)=\sum_{x\in E}\frac{N(x)}M\alpha_x^k.
\alpha_\ellUse in source · 2 passages
Search aliases: α_ℓ alpha ell
Use in source · project_02.tex, line 92
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 · 12 passages
Search aliases: β beta
Use in source · project_09.tex, line 84
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
\beta=-\frac ba=1-\frac ca.
\beta^{1/K}Use in source · 1 passage
Search aliases: β^1/K beta
Use in source · project_11.tex, line 249
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). }
\GammaUse in source · 3 passages
Search aliases: Γ Gamma
Use in source · project_03.tex, line 150
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
\gammaUse in source · 3 passages
Search aliases: γ gamma
Use in source · 20_signed_filter.tex, line 3
ic COS and the sparse spectral leaf What the finite assignment signal records. The spectral branch of the project asks which prime assignments realize a prescribed residue. Its exact object is a finite distribution. Fix a finite abelian group G and s eligible variable support primes. At the jth prime choose one of three permitted contributions, represented by 0,γ_j,-γ_j in additive group notation. The sum of the chosen contributions is the assignment residue.
Exact mathematical passage
0,\gamma_j,-\gamma_j
\gamma_jUse in source · 3 passages
Search aliases: γ_j gamma
Use in source · 20_signed_filter.tex, line 3
ic COS and the sparse spectral leaf What the finite assignment signal records. The spectral branch of the project asks which prime assignments realize a prescribed residue. Its exact object is a finite distribution. Fix a finite abelian group G and s eligible variable support primes. At the jth prime choose one of three permitted contributions, represented by 0,γ_j,-γ_j in additive group notation. The sum of the chosen contributions is the assignment residue.
Exact mathematical passage
0,\gamma_j,-\gamma_j
\DeltaUse in source · 9 passages
Search aliases: Δ Delta
Use in source · 03_consequences.tex, line 36
This equivalence is cited as an external theorem. We do not derive its full small-prime and minimal-model accounting here. The ordinary statement involving only |Δ_min| should not be silently substituted for the max-invariant formulation. A proposed project route through the Frey curve must preserve the marked integral model and the normalization of its differential, rather than only its rational isogeny class.
Exact mathematical passage
|\Delta_{\min}|
\delta_0Use in source · 1 passage
Search aliases: δ_0 delta
Use in source · project_05.tex, line 71
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 · project_05.tex, line 71
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 · project_08.tex, line 112
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 · project_05.tex, line 71
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 · 1 passage
Search aliases: Δ_E Delta
Use in source · project_05.tex, line 86
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
\Delta_hUse in source · 1 passage
Search aliases: Δ_h Delta
Use in source · 20_signed_filter.tex, line 113
A Galois coordinate with a controlled denominator. For an occupied target define Δ_h(E)=∏_x∈ E∖{h}(α_h-α_x). This is a nonzero algebraic integer in Q(ζ_n). For every unit u modulo n, its conjugate is Δ_uh(uE). The absolute norm is a nonzero integer. Hence ∏_u∈(Z/nZ)^×|Δ_uh(uE)|≥1. At least one factor is at least one. In that coordinate, the Lagrange numerator has coefficient ℓ^1 norm at most 2^r-1, and dividing by the target denominator preserves that upper bound.
Exact mathematical passage
\Delta_h(E)=\prod_{x\in E\setminus\{h\}}(\alpha_h-\alpha_x).
\Delta_{\min}Use in source · 1 passage
Search aliases: Δ_min Delta min
Use in source · 03_consequences.tex, line 36
This equivalence is cited as an external theorem. We do not derive its full small-prime and minimal-model accounting here. The ordinary statement involving only |Δ_min| should not be silently substituted for the max-invariant formulation. A proposed project route through the Frey curve must preserve the marked integral model and the normalization of its differential, rather than only its rational isogeny class.
Exact mathematical passage
|\Delta_{\min}|
\Delta_{uh}Use in source · 2 passages
Search aliases: Δ_uh Delta
Use in source · 20_signed_filter.tex, line 116
A Galois coordinate with a controlled denominator. For an occupied target define Δ_h(E)=∏_x∈ E∖{h}(α_h-α_x). This is a nonzero algebraic integer in Q(ζ_n). For every unit u modulo n, its conjugate is Δ_uh(uE). The absolute norm is a nonzero integer. Hence ∏_u∈(Z/nZ)^×|Δ_uh(uE)|≥1. At least one factor is at least one. In that coordinate, the Lagrange numerator has coefficient ℓ^1 norm at most 2^r-1, and dividing by the target denominator preserves that upper bound.
Exact mathematical passage
\Delta_{uh}(uE)
\varepsilonUse in source · 5 passages
Search aliases: ε varepsilon
Use in source · project_06.tex, line 88
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 · project_06.tex, line 103
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'),
\varepsilon_jUse in source · 1 passage
Search aliases: ε_j varepsilon
Use in source · project_06.tex, line 92
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).
\zeta^eUse in source · 3 passages
Search aliases: ζ^e zeta
Use in source · project_02.tex, line 57
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
V_e(\zeta) = \frac{1-\zeta^e}{1-\zeta}, \qquad \zeta^N=1, \quad \zeta\neq1.
\zeta^NUse in source · 2 passages
Search aliases: ζ^N zeta
Use in source · project_02.tex, line 57
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
V_e(\zeta) = \frac{1-\zeta^e}{1-\zeta}, \qquad \zeta^N=1, \quad \zeta\neq1.
\zeta_MUse in source · 1 passage
Search aliases: ζ_M zeta
Use in source · project_10.tex, line 5
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 · 1 passage
Search aliases: ζ_M^j zeta
Use in source · project_10.tex, line 6
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). }
\zeta_nUse in source · 1 passage
Search aliases: ζ_n zeta
Use in source · 20_signed_filter.tex, line 116
A Galois coordinate with a controlled denominator. For an occupied target define Δ_h(E)=∏_x∈ E∖{h}(α_h-α_x). This is a nonzero algebraic integer in Q(ζ_n). For every unit u modulo n, its conjugate is Δ_uh(uE). The absolute norm is a nonzero integer. Hence ∏_u∈(Z/nZ)^×|Δ_uh(uE)|≥1. At least one factor is at least one. In that coordinate, the Lagrange numerator has coefficient ℓ^1 norm at most 2^r-1, and dividing by the target denominator preserves that upper bound.
Exact mathematical passage
\Q(\zeta_n)
\eta_fUse in source · 4 passages
Search aliases: η_f eta
Use in source · project_06.tex, line 119
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_rUse in source · 5 passages
Search aliases: η_r eta
Use in source · 26_global_bridge.tex, line 19
admits a nonzero integer W_r with Q^s_rmid W_r, |W_r|≤ C_r(ab)^s_rR^E_r(log(3c))^A_r. If E_r/s_r→0 along R, then Conjecture~[source reference] follows. Nonzero integer divisibility gives Q^s_r≤|W_r|. Divide by (ab)^s_r and use Q/(ab)=c/R: (c/R)^s_r≤ C_rR^E_r(log(3c))^A_r. Taking positive roots yields c≤ D_rR^1+η_r(log(3c))^a_r, D_r=C_r^1/s_r, η_r=E_r/s_r, a_r=A_r/s_r. Fix the target eps>0. Choose one rank with η_r<eps. Next choose 0<δ<fraceps-η_r1+eps. For the fixed exponent a_r, there is a cons…
Exact mathematical passage
c\le D_rR^{1+\eta_r}(\log(3c))^{a_r}, \quad D_r=C_r^{1/s_r},\quad \eta_r=E_r/s_r,\quad a_r=A_r/s_r.
\eta_UUse in source · 2 passages
Search aliases: η_U eta
Use in source · project_07.tex, line 17
[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 · 11 passages
Search aliases: Θ Theta
Use in source · 27_evidence.tex, line 28
The nine canonical implementation groups also reproduced their archived values, after two disclosed filesystem path adaptations. The mathematical code was unchanged. At 3+125=128, the weighted derivative minimum is Θ=6, while the exponent-free moving-search minimum is Φ=8. They optimize different packets. The latter has obstruction multiple one and exact master gap one. At 1+4374=4375, the moving-search minimum is 62 and its exact master gap is 248/243.
Exact mathematical passage
\Theta=6
\Theta_3Use in source · 2 passages
Search aliases: Θ_3 Theta
Use in source · project_10.tex, line 156
Conditional motivic transfer. Let K_β be the cyclic group generated by the balanced Kummer class. Suppose that for every primitive positive triple there exists an integral finite object H_3(a,b,c) and a canonical map Θ_3:K_β⟶ H_3(a,b,c) such that, for every eps>0, constants independent of the triple give |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 · project_08.tex, line 164
[Conditional filtered-transfer criterion] Suppose every primitive positive triple admits a canonical map H^0(Θ_f):Z/(c/R_c)Z⟶ H^0( C_f) satisfies, for every eps>0, with constants independent of the triple, |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)
\kappaUse in source · 3 passages
Search aliases: κ kappa
Use in source · project_09.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]. }
\kappa_cUse in source · 3 passages
Search aliases: κ_c kappa
Use in source · project_09.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]. }
\LambdaUse in source · 20 passages
Search aliases: Λ Lambda
Use in source · project_03.tex, line 6
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}
\lambdaUse in source · 15 passages
Search aliases: λ lambda
Use in source · 20_signed_filter.tex, line 96
Polynomial target filters and exact count certificates. For a polynomial P(z)=∑_j=0^dc_jz^j, define the collision quadratic form E_d(P)=frac nM^2∑_x∈ EN(x)^2|P(α_x)|^2. It can be written as a positive semidefinite Gram form in the coefficients of P. Let λ_d(h)=min{ E_d(P):deg P≤ d, P(α_h)=1}. For every such filter, the target term is present with value (n/M^2)N(h)^2, so N(h)≤frac Msqrt nsqrtλ_d(h). If the right-hand side is rigorously less than one, the integer count is zero. An approximate deci…
Exact mathematical passage
\lambda_d(h)=\min\{\mathcal E_d(P):\deg P\le d,\ P(\alpha_h)=1\}.
\lambda^{\mathsf T}Use in source · 1 passage
Search aliases: λ^ T lambda mathsf
Use in source · project_01.tex, line 109
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 · project_01.tex, line 254
Three-sector compression. For the empty unit sector, set g_1=0, t_1=0, and ρ_1(0)=0. Any nonzero target in that sector is infeasible. Within a nonempty 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…
Exact mathematical passage
\lambda^{(a)} = \left(\frac{a v_p(a)}p\right)_{p\mid a}
\Lambda_{\mathrm{coh}}Use in source · 5 passages
Search aliases: Λ_coh Lambda mathrm
Use in source · project_11.tex, line 145
satisfying g_p·ν_0 ≡0 pmodp^χ_p, where ν_0 is the Kummer-normalized input exponent vector. The Chinese remainder theorem combines the local rows into one primitive row g∈(Z/CZ)^S_ab. Define the coherence character χ_coh(x) = exp(frac2π iCg· x). Its kernel is the adelic coherence lattice Λ_coh = {x∈Z^S_ab:g· x≡0pmod C}. Since g is primitive modulo C, [Z^S_ab:Λ_coh]=C and Z^S_ab/Λ_coh≅Z/CZ. The normalized exponent vector lies in the lattice.
Exact mathematical passage
\boxed{ \Lambda_{\mathrm{coh}} = \{x\in\Z^{S_{ab}}:g\cdot x\equiv0\pmod C\}. }
\lambda_dUse in source · 2 passages
Search aliases: λ_d lambda
Use in source · 20_signed_filter.tex, line 96
Polynomial target filters and exact count certificates. For a polynomial P(z)=∑_j=0^dc_jz^j, define the collision quadratic form E_d(P)=frac nM^2∑_x∈ EN(x)^2|P(α_x)|^2. It can be written as a positive semidefinite Gram form in the coefficients of P. Let λ_d(h)=min{ E_d(P):deg P≤ d, P(α_h)=1}. For every such filter, the target term is present with value (n/M^2)N(h)^2, so N(h)≤frac Msqrt nsqrtλ_d(h). If the right-hand side is rigorously less than one, the integer count is zero. An approximate deci…
Exact mathematical passage
\lambda_d(h)=\min\{\mathcal E_d(P):\deg P\le d,\ P(\alpha_h)=1\}.
\Lambda_pUse in source · 1 passage
Search aliases: Λ_p Lambda
Use in source · project_03.tex, line 6
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 · project_01.tex, line 63
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_rUse in source · 1 passage
Search aliases: λ_r lambda
Use in source · 20_signed_filter.tex, line 105
If h is absent, a degree-r polynomial vanishes at every occupied node and takes value one at α_h, so λ_r(h)=0. If h is occupied, its degree-(r-1) Lagrange polynomial isolates exactly that node. Thus λ_r-1(h)=frac nM^2N(h)^2. At the innovation degree the count bound is exact. The remaining computational issue is finding the needed filter with controlled exact coefficients and reliable arithmetic, rather than hoping for an unexplained asymptotic gain.
Exact mathematical passage
\lambda_r(h)=0
\lambda_{r-1}Use in source · 1 passage
Search aliases: λ_r-1 lambda
Use in source · 20_signed_filter.tex, line 106
If h is absent, a degree-r polynomial vanishes at every occupied node and takes value one at α_h, so λ_r(h)=0. If h is occupied, its degree-(r-1) Lagrange polynomial isolates exactly that node. Thus λ_r-1(h)=frac nM^2N(h)^2. At the innovation degree the count bound is exact. The remaining computational issue is finding the needed filter with controlled exact coefficients and reliable arithmetic, rather than hoping for an unexplained asymptotic gain.
Exact mathematical passage
\lambda_{r-1}(h)=\frac n{M^2}N(h)^2.
\Lambda_UUse in source · 4 passages
Search aliases: Λ_U Lambda
Use in source · project_07.tex, line 6
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 · project_07.tex, line 6
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_MUse in source · 2 passages
Search aliases: μ_M mu
Use in source · project_09.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 · project_01.tex, line 94
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 · project_01.tex, line 116
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 · project_01.tex, line 120
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 · 11 passages
Search aliases: ν nu
Use in source · project_05.tex, line 71
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_0Use in source · 3 passages
Search aliases: ν_0 nu
Use in source · project_11.tex, line 127
The coherence character. Set χ_p=0 for absent or squarefree output channels. When the residual modulus is one, use the trivial row and character. After normalization, every output prime with nontrivial residual depth produces a primitive row g_p ∈ (Z/p^χ_pZ)^S_ab satisfying g_p·ν_0 ≡0 pmodp^χ_p, where ν_0 is the Kummer-normalized input exponent vector. The Chinese remainder theorem combines the local rows into one primitive row g∈(Z/CZ)^S_ab. Define the coherence character χ_coh(x) = exp(frac2π …
Exact mathematical passage
g_p\cdot\nu_0 \equiv0 \pmod{p^{\chi_p}},
\nu_{a,b,c}Use in source · 1 passage
Search aliases: ν_a,b,c nu
Use in source · project_05.tex, line 71
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 · 7 passages
Search aliases: ν_q nu
Use in source · project_09.tex, line 88
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
\nu_q=v_q(b)-v_q(a)
\xi_iUse in source · 3 passages
Search aliases: ξ_i xi
Use in source · project_04.tex, line 83
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_i≤ lfl…
Exact mathematical passage
\xi_i
\xi_pUse in source · 2 passages
Search aliases: ξ_p xi
Use in source · project_01.tex, line 17
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. First define the global free module over all primes, and restrict it to Ω_S for S-supported integers. For such 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 ex…
Exact mathematical passage
\Omega_S=\bigoplus_{p\in S}\Z\xi_p.
\PiUse in source · 1 passage
Search aliases: Π Pi
Use in source · project_10.tex, line 72
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 · 20_signed_filter.tex, line 15
Let N(x) count assignments producing x∈ G. Repeated contributions remain separate assignments, so M=∑_x∈ GN(x)=3^s, μ(x)=N(x)/M. For a character χ∈widehat G, define widehatμ(χ)=∑_xμ(x)χ(x) =∏_j=1^sfrac1+χ(γ_j)+χ(-γ_j)3. On a cyclic channel G=Z/nZ, this becomes m(k)=∏_j=1^sfrac1+2cos(2π kγ_j/n)3. This is the discrete cosine or COS expression used in the project. It is an exact characteristic function. Individual cosine factors may be negative. No positivity of all Fourier samples is assumed.
Exact mathematical passage
m(k)=\prod_{j=1}^s\frac{1+2\cos(2\pi k\gamma_j/n)}3.
\pi^2Use in source · 5 passages
Search aliases: π^2 pi
Use in source · project_05.tex, line 174
parametrization, and let ϕ_E^*ω_E = 2π i c_ϕ f_E(z) dz. Here c_ϕ is the differential multiplier of this chosen parametrization and differential. The marked Frey curve need not be the optimal representative of its isogeny class. 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_ϕ^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 shap…
Exact mathematical passage
\boxed{ \deg(\phi_E)A(E) = 4\pi^2 c_\phi^2 \norm{f_E}_{\mathrm{Pet}}^2. }
\Pi_{\le3}Use in source · 1 passage
Search aliases: Π_≤3 Pi le
Use in source · project_10.tex, line 72
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 · 4 passages
Search aliases: ρ rho
Use in source · project_01.tex, line 253
Three-sector compression. For the empty unit sector, set g_1=0, t_1=0, and ρ_1(0)=0. Any nonzero target in that sector is infeasible. Within a nonempty 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…
Exact mathematical passage
\rho_1(0)=0
\rho_1Use in source · 1 passage
Search aliases: ρ_1 rho
Use in source · project_01.tex, line 253
Three-sector compression. For the empty unit sector, set g_1=0, t_1=0, and ρ_1(0)=0. Any nonzero target in that sector is infeasible. Within a nonempty 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…
Exact mathematical passage
\rho_1(0)=0
\rho_aUse in source · 2 passages
Search aliases: ρ_a rho
Use in source · project_01.tex, line 278
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 · project_01.tex, line 287
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 · project_01.tex, line 287
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 · project_02.tex, line 248
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_aUse in source · 1 passage
Search aliases: σ_a sigma
Use in source · project_03.tex, line 128
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
\Delta(a,b,c) = \min\left\{ \sigma_a+\sigma_b, \frac cb(\sigma_a+\sigma_c), \frac ca(\sigma_b+\sigma_c) \right\}.
\sigma_{a,b,c}Use in source · 1 passage
Search aliases: σ_a,b,c sigma
Use in source · project_08.tex, line 6
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). Place the two-term contact complex in degrees -1,0 and define 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_bUse in source · 1 passage
Search aliases: σ_b sigma
Use in source · project_03.tex, line 128
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
\Delta(a,b,c) = \min\left\{ \sigma_a+\sigma_b, \frac cb(\sigma_a+\sigma_c), \frac ca(\sigma_b+\sigma_c) \right\}.
\sigma_cUse in source · 1 passage
Search aliases: σ_c sigma
Use in source · project_03.tex, line 128
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
\Delta(a,b,c) = \min\left\{ \sigma_a+\sigma_b, \frac cb(\sigma_a+\sigma_c), \frac ca(\sigma_b+\sigma_c) \right\}.
\sigma_uUse in source · 1 passage
Search aliases: σ_u sigma
Use in source · project_03.tex, line 124
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 · 18 passages
Search aliases: Φ Phi
Use in source · 22_jet_filter.tex, line 9
The universal mixed-weight jet filter. One polynomial chooses the cancelling coefficients. Let m≥0 be an integer. Define P_m(X)=X∏_j=1^m(jX-1)=∑_s=1^m+1c_m,sX^s, and the formal series Φ_m(t)=frac1(m+1)!∑_s=1^m+1c_m,sLi_s(t). Here Li_s(t)=∑_k≥1t^k/k^s is first interpreted in Q[[t]]. It also defines the usual analytic polylogarithm on |t|<1~[source reference]. The label s is its weight. The contact order will follow from its coefficients, rather than from that label.
Exact mathematical passage
\Phi_m(t)=\frac1{(m+1)!}\sum_{s=1}^{m+1}c_{m,s}\Li_s(t).
\phiUse in source · 4 passages
Search aliases: ϕ phi
Use in source · project_05.tex, line 156
Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be a nonconstant modular parametrization, and let ϕ_E^*ω_E = 2π i c_ϕ f_E(z) dz. Here c_ϕ is the differential multiplier of this chosen parametrization and differential. The marked Frey curve need not be the optimal representative of its isogeny class. 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_ϕ^2 normf_E_Pet^2. For the Frey model, scaling x=cX, …
Exact mathematical passage
\phi_E:X_0(N_E)\longrightarrow E
\varphiUse in source · 8 passages
Search aliases: φ varphi
Use in source · 20_signed_filter.tex, line 46
The hypothetical exception regime is sparse. For a primitive triple put Q=abc/R. Suppose c>R^1+eps for a fixed eps>0. Since a,b are positive with sum c, one has ab≥ c-1. Thus, for large c, Q=fracabcR≥fracc(c-1)R>frac12R^1+2eps. The unit group modulo Q has order φ(Q). For every δ>0, frac Qφ(Q)ll_δ R^δ, 3^ω(R)ll_δ R^δ. Both elementary bounds follow by separating a finite set of small primes. For all sufficiently large primes, (1-1/p)^-1≤ p^δ and 3≤ p^δ. Every prime dividing Q also divides R.
Exact mathematical passage
\varphi(Q)
\Phi_0Use in source · 1 passage
Search aliases: Φ_0 Phi
Use in source · 22_jet_filter.tex, line 32
ent of t^k is P_m(1/k)/(m+1)!. Direct substitution gives P_m(1/k)=frac1k∏_j=1^m(frac jk-1) =(-1)^mfrac(k-1)⋯(k-m)k^m+1. Divide by (m+1)! and multiply numerator and denominator by k to get the asserted binomial coefficient. It vanishes for 1≤ k≤ m and first becomes one at k=m+1. The coefficient there is nonzero. For m=0, this is Φ_0=Li_1. For m=1, Φ_1=(Li_2-Li_1)/2=-t^2/8-t^3/9-⋯. The sign is part of the exact formula. Changing the weight alone would not produce the cancellation of the linear coe…
Exact mathematical passage
\Phi_0=\Li_1
\Phi_1Use in source · 1 passage
Search aliases: Φ_1 Phi
Use in source · 22_jet_filter.tex, line 33
+1)!. Direct substitution gives P_m(1/k)=frac1k∏_j=1^m(frac jk-1) =(-1)^mfrac(k-1)⋯(k-m)k^m+1. Divide by (m+1)! and multiply numerator and denominator by k to get the asserted binomial coefficient. It vanishes for 1≤ k≤ m and first becomes one at k=m+1. The coefficient there is nonzero. For m=0, this is Φ_0=Li_1. For m=1, Φ_1=(Li_2-Li_1)/2=-t^2/8-t^3/9-⋯. The sign is part of the exact formula. Changing the weight alone would not produce the cancellation of the linear coefficient.
Exact mathematical passage
\Phi_1=(\Li_2-\Li_1)/2=-t^2/8-t^3/9-\cdots.
\phi_EUse in source · 2 passages
Search aliases: ϕ_E phi
Use in source · project_05.tex, line 156
Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be a nonconstant modular parametrization, and let ϕ_E^*ω_E = 2π i c_ϕ f_E(z) dz. Here c_ϕ is the differential multiplier of this chosen parametrization and differential. The marked Frey curve need not be the optimal representative of its isogeny class. 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_ϕ^2 normf_E_Pet^2. For the Frey model, scaling x=cX, …
Exact mathematical passage
\phi_E:X_0(N_E)\longrightarrow E
\phi_E^*Use in source · 1 passage
Search aliases: ϕ_E^* phi
Use in source · project_05.tex, line 160
Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be a nonconstant modular parametrization, and let ϕ_E^*ω_E = 2π i c_ϕ f_E(z) dz. Here c_ϕ is the differential multiplier of this chosen parametrization and differential. The marked Frey curve need not be the optimal representative of its isogeny class. 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_ϕ^2 normf_E_Pet^2. For the Frey model, scaling x=cX, …
Exact mathematical passage
\phi_E^*\omega_E = 2\pi i\,c_\phi f_E(z)\,dz.
\Phi_mUse in source · 5 passages
Search aliases: Φ_m Phi
Use in source · 22_jet_filter.tex, line 9
The universal mixed-weight jet filter. One polynomial chooses the cancelling coefficients. Let m≥0 be an integer. Define P_m(X)=X∏_j=1^m(jX-1)=∑_s=1^m+1c_m,sX^s, and the formal series Φ_m(t)=frac1(m+1)!∑_s=1^m+1c_m,sLi_s(t). Here Li_s(t)=∑_k≥1t^k/k^s is first interpreted in Q[[t]]. It also defines the usual analytic polylogarithm on |t|<1~[source reference]. The label s is its weight. The contact order will follow from its coefficients, rather than from that label.
Exact mathematical passage
\Phi_m(t)=\frac1{(m+1)!}\sum_{s=1}^{m+1}c_{m,s}\Li_s(t).
\Phi_{m,p}Use in source · 3 passages
Search aliases: Φ_m,p Phi
Use in source · 22_jet_filter.tex, line 49
The depleted local series. For a prime p, delete the terms whose index is divisible by p: Φ_m,p(t)=(-1)^m∑_pnmid kbinomkm+1fract^kk^m+2∈Z_p[[t]]. It converges when v_p(t)>0. [Exact local jet valuation] If v_p(t)=e≥1, then v_p(Φ_m,p(t))= (m+1)e, · pnmid m+1,
Exact mathematical passage
\Phi_{m,p}(t)=(-1)^m\sum_{p\nmid k}\binom{k}{m+1}\frac{t^k}{k^{m+2}}\in\Z_p[[t]].
\chiUse in source · 13 passages
Search aliases: χ chi
Use in source · 20_signed_filter.tex, line 9
Let N(x) count assignments producing x∈ G. Repeated contributions remain separate assignments, so M=∑_x∈ GN(x)=3^s, μ(x)=N(x)/M. For a character χ∈widehat G, define widehatμ(χ)=∑_xμ(x)χ(x) =∏_j=1^sfrac1+χ(γ_j)+χ(-γ_j)3. On a cyclic channel G=Z/nZ, this becomes m(k)=∏_j=1^sfrac1+2cos(2π kγ_j/n)3. This is the discrete cosine or COS expression used in the project. It is an exact characteristic function. Individual cosine factors may be negative. No positivity of all Fourier samples is assumed.
Exact mathematical passage
\chi\in\widehat G
\chi_{\mathrm{coh}}Use in source · 2 passages
Search aliases: χ_coh chi mathrm
Use in source · project_11.tex, line 137
er normalization, every output prime with nontrivial residual depth produces a primitive row g_p ∈ (Z/p^χ_pZ)^S_ab satisfying g_p·ν_0 ≡0 pmodp^χ_p, where ν_0 is the Kummer-normalized input exponent vector. The Chinese remainder theorem combines the local rows into one primitive row g∈(Z/CZ)^S_ab. Define the coherence character χ_coh(x) = exp(frac2π iCg· x). Its kernel is the adelic coherence lattice Λ_coh = {x∈Z^S_ab:g· x≡0pmod C}. Since g is primitive modulo C, [Z^S_ab:Λ_coh]=C and Z^S_ab/Λ_coh…
Exact mathematical passage
\boxed{ \chi_{\mathrm{coh}}(x) = \exp\left(\frac{2\pi i}{C}g\cdot x\right). }
\chi_pUse in source · 7 passages
Search aliases: χ_p chi
Use in source · project_11.tex, line 92
r formula implies K_Kummid M/ϵ_2, so K_Kum=∏_pmid cp^r_p. Write K=K_Kum. Divide the input exponent vector by K. Define a_p,q as the original odd or weighted dyadic coefficient, now reduced modulo p^d_p-r_p. Dividing the relation vector reduces the needed relation modulus. It does not divide the coefficient row by K. 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 ord…
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.
\PsiUse in source · 6 passages
Search aliases: Ψ Psi
Use in source · project_05.tex, line 10
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 · project_05.tex, line 38
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 · project_05.tex, line 38
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 · project_05.tex, line 38
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 · project_05.tex, line 38
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 · 4 passages
Search aliases: Ψ_n Psi
Use in source · project_05.tex, line 10
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 · 2 passages
Search aliases: Ω Omega
Use in source · project_01.tex, line 17
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. First define the global free module over all primes, and restrict it to Ω_S for S-supported integers. For such 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 ex…
Exact mathematical passage
\Omega_S=\bigoplus_{p\in S}\Z\xi_p.
\omegaUse in source · 8 passages
Search aliases: ω omega
Use in source · 20_signed_filter.tex, line 47
The hypothetical exception regime is sparse. For a primitive triple put Q=abc/R. Suppose c>R^1+eps for a fixed eps>0. Since a,b are positive with sum c, one has ab≥ c-1. Thus, for large c, Q=fracabcR≥fracc(c-1)R>frac12R^1+2eps. The unit group modulo Q has order φ(Q). For every δ>0, frac Qφ(Q)ll_δ R^δ, 3^ω(R)ll_δ R^δ. Both elementary bounds follow by separating a finite set of small primes. For all sufficiently large primes, (1-1/p)^-1≤ p^δ and 3≤ p^δ. Every prime dividing Q also divides R.
Exact mathematical passage
\frac Q{\varphi(Q)}\ll_\delta R^\delta, \qquad 3^{\omega(R)}\ll_\delta R^\delta.
\omega_EUse in source · 2 passages
Search aliases: ω_E omega
Use in source · project_05.tex, line 160
Exact automorphic analysis--synthesis identity. Let ϕ_E:X_0(N_E)⟶ E be a nonconstant modular parametrization, and let ϕ_E^*ω_E = 2π i c_ϕ f_E(z) dz. Here c_ϕ is the differential multiplier of this chosen parametrization and differential. The marked Frey curve need not be the optimal representative of its isogeny class. 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_ϕ^2 normf_E_Pet^2. For the Frey model, scaling x=cX, …
Exact mathematical passage
\phi_E^*\omega_E = 2\pi i\,c_\phi f_E(z)\,dz.
\Omega_SUse in source · 2 passages
Search aliases: Ω_S Omega
Use in source · project_01.tex, line 17
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. First define the global free module over all primes, and restrict it to Ω_S for S-supported integers. For such 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 ex…
Exact mathematical passage
\Omega_S=\bigoplus_{p\in S}\Z\xi_p.
\ldotsUse in source · 28 passages
Search aliases: … ldots
Use in source · 20_signed_filter.tex, line 73
[Exact cyclic Prony reconstruction] The distribution on a cyclic channel with r occupied residues is determined by the 2r exact consecutive samples m(0),…,m(2r-1). Write A(z)=∏_x∈ E(z-α_x)=z^r+∑_j=0^r-1a_jz^j. For every k, ∑_j=0^r-1a_jm(k+j)=-m(k+r). The coefficient matrix for 0≤ k<r is the Hankel matrix H=(m(i+j))_0≤ i,j<r. With V=(α_x^i)_i,x and D=diag(N(x)/M), one has H=VDV^ T. The square Vandermonde V is invertible and every diagonal entry of D is nonzero. Thus H is invertible, and the 2r sa…
Exact mathematical passage
m(0),\ldots,m(2r-1)
\ellUse in source · 23 passages
Search aliases: ℓ ell
Use in source · 20_signed_filter.tex, line 120
get define Δ_h(E)=∏_x∈ E∖{h}(α_h-α_x). This is a nonzero algebraic integer in Q(ζ_n). For every unit u modulo n, its conjugate is Δ_uh(uE). The absolute norm is a nonzero integer. Hence ∏_u∈(Z/nZ)^×|Δ_uh(uE)|≥1. At least one factor is at least one. In that coordinate, the Lagrange numerator has coefficient ℓ^1 norm at most 2^r-1, and dividing by the target denominator preserves that upper bound.
Exact mathematical passage
\ell^1
\ell^1Use in source · 2 passages
Search aliases: ℓ^1 ell
Use in source · 20_signed_filter.tex, line 120
get define Δ_h(E)=∏_x∈ E∖{h}(α_h-α_x). This is a nonzero algebraic integer in Q(ζ_n). For every unit u modulo n, its conjugate is Δ_uh(uE). The absolute norm is a nonzero integer. Hence ∏_u∈(Z/nZ)^×|Δ_uh(uE)|≥1. At least one factor is at least one. In that coordinate, the Lagrange numerator has coefficient ℓ^1 norm at most 2^r-1, and dividing by the target denominator preserves that upper bound.
Exact mathematical passage
\ell^1
\ell^nUse in source · 3 passages
Search aliases: ℓ^n ell
Use in source · project_07.tex, line 101
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). Let E/Q_q have split multiplicative reduction with Tate parameter q_E, and let ℓ≠ q. Scalarity of the full local Galois representation on E[ℓ^n] forces n≤min{v_ℓ(q-1),v_ℓ(v_q(q_E))}. In a Tate basis, the cyclotomic graded diagonal and the inertia Kummer extension respectively force ℓ^nmid(q-1) and ℓ^nmid v_q(q_E)~[source r…
Exact mathematical passage
E[\ell^n]
\ell^{\alpha_\ell}Use in source · 2 passages
Search aliases: ℓ^α_ℓ ell alpha ell
Use in source · project_02.tex, line 92
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 · 2 passages
Search aliases: ℓ_2,e ell
Use in source · project_11.tex, line 57
Weighted dyadic row. For 2^eparallel c with e≥2 and 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}. At e=1, the output contact depth is zero and this row is omitted. A separate sign packet supplies the missing first dyadic layer. This is the weighted dyadic branch in its final logarithmic form.
Exact mathematical passage
\ell_{2,e}(q) = \frac14 \log_2\left((-1)^{(q-1)/2}q\right) \pmod{2^{e-2}}.
\ell_iUse in source · 2 passages
Search aliases: ℓ_i ell
Use in source · project_04.tex, line 92
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 ℓ_i(H)≤ u_i(H) for every i, ∑_iℓ_i(H)≤ q(t)≤∑_i u_i(H). Every co…
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_{p,e}Use in source · 3 passages
Search aliases: ℓ_p,e ell
Use in source · project_11.tex, line 24
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
\ell_{p,e}(q) = \frac1p\log_p\langle q\rangle_p \pmod{p^{e-1}}.
\toUse in source · 5 passages
Search aliases: → to
Use in source · 21_corrected_target.tex, line 51
Fix an integer d≥2 and take a=t^d, b=(t+1)^d, c=t^d+(t+1)^d for t≥1. The inputs are coprime. As rad(t^d)≤ t and rad((t+1)^d)≤ t+1, R≤ t(t+1)c, Q=fracabcR≥[t(t+1)]^d-1. The assumed inequality implies [t(t+1)]^s(d-1)≤ C[t(t+1)c]^B(log(3c))^A. Take logarithms, divide by log t and send t to infinity. Since log c/log t→ d and loglog(3c)/log t→0, this yields 2s(d-1)≤ B(d+2). It holds for every d. Sending d to infinity proves B≥2s. No contact multiplier can make the ratio B/s tend to zero for this fals…
Exact mathematical passage
\log c/\log t\to d
\mapstoUse in source · 2 passages
Search aliases: ↦ mapsto
Use in source · project_02.tex, line 53
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
I_N=\ker\bigl(A_N\xrightarrow{\,T\mapsto1\,}\Z\bigr)
\hookrightarrowUse in source · 1 passage
Search aliases: ↪ hookrightarrow
Use in source · project_08.tex, line 35
d ∞ are div(a), div(b), div(c). For the output cusp, let C=div(c), C_red=div(R_c). Place the two-term contact complex in degrees -1,0 and define J_c = [ O(-C)⟶ O(-C_red)]. On global sections, J_c = [cZ↪ R_cZ] ≃ [Zxrightarrowc/R_cZ].
Exact mathematical passage
\mathbf J_c = [c\Z\hookrightarrow R_c\Z] \simeq [\Z\xrightarrow{c/R_c}\Z].
\partialUse in source · 2 passages
Search aliases: ∂ partial
Use in source · project_02.tex, line 16
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. Set V_0(T)=0 for an absent prime. 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 · project_02.tex, line 16
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. Set V_0(T)=0 for an absent prime. 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 · project_04.tex, line 118
[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.
\prodUse in source · 40 passages
Search aliases: ∏ prod
Use in source · 01_foundations.tex, line 4
Radical support and the exact conjecture. The multiplicative quantity that ignores repetition. Every positive integer has a unique prime factorization n=∏_pmid np^e_p, e_p≥1. Its radical is rad(n)=∏_pmid np, rad(1)=1. The radical retains which primes occur and discards how many times they occur. Thus rad(72)=rad(2^3·3^2)=6, whereas 72/6=12 records the repeated-power contribution. If n is squarefree, then rad(n)=n.
Exact mathematical passage
n=\prod_{p\mid n}p^{e_p},\qquad e_p\ge1.
\prod_{d\ge1}Use in source · 1 passage
Search aliases: ∏_d≥1 prod ge
Use in source · project_06.tex, line 22
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 · 2 passages
Search aliases: ∏_d≥2 prod ge
Use in source · project_06.tex, line 35
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 · 23_hankel.tex, line 30
orem Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and including the factors t^k_i/k_i gives the stated coefficient. In detail, ∏_i<j(k_j^-1-k_i^-1)^2 =frac∏_i<j(k_j-k_i)^2∏_i k_i^2(r-1). Passing to the formal infinite sum is coefficientwise legitimate. At a fixed power of t, only finitely many index s…
Exact mathematical passage
\prod_{i<j}(k_j^{-1}-k_i^{-1})^2 =\frac{\prod_{i<j}(k_j-k_i)^2}{\prod_i k_i^{2(r-1)}}.
\prod_{i<j}Use in source · 2 passages
Search aliases: ∏_i<j prod
Use in source · 23_hankel.tex, line 18
[Cauchy--Binet expansion] H_r(t)=∑_1≤ k_1<⋯<k_r t^k_1+⋯+k_r frac∏_i<j(k_j-k_i)^2∏_i=1^r k_i^2r-1. Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and including the factors t^k_i/k_i gives the stated coefficient. In detail, ∏_i<j(k_j^-1-k_i^-1)^2 =frac∏_i<j(k_j-k_i)^2∏_i k_i^2(r-1). Passing to the formal …
Exact mathematical passage
\mathcal H_r(t)=\sum_{1\le k_1<\cdots<k_r} t^{k_1+\cdots+k_r} \frac{\prod_{i<j}(k_j-k_i)^2}{\prod_{i=1}^r k_i^{2r-1}}.
\prod_{i=0}^{r-1}Use in source · 1 passage
Search aliases: ∏_i=0^r-1 prod
Use in source · 23_hankel.tex, line 63
Hadamard's inequality for the positive matrix gives H_r(t)≤∏_i=0^r-1Li_2i+1(t) ≤[-log(1-t)]∏_i=1^r-1ζ(2i+1). For the higher weights, t^k≤1 bounds the series by its zeta value. Since all summands are positive, ∑_i≥1(ζ(2i+1)-1) =∑_k≥2frac1k(k^2-1)=frac14. The last sum telescopes using frac1k(k^2-1)=frac12(frac1k(k-1)-frac1k(k+1)). Finally log(1+x)≤ x for x≥0 bounds the zeta product by e^1/4, uniformly in its length. At t=a/c and t=b/c, the bounds become e^1/4log(c/b) and e^1/4log(c/a). Thus these …
Exact mathematical passage
\mathcal H_r(t)\le\prod_{i=0}^{r-1}\Li_{2i+1}(t) \le[-\log(1-t)]\prod_{i=1}^{r-1}\zeta(2i+1).
\prod_{i=1}^rUse in source · 1 passage
Search aliases: ∏_i=1^r prod
Use in source · 23_hankel.tex, line 18
[Cauchy--Binet expansion] H_r(t)=∑_1≤ k_1<⋯<k_r t^k_1+⋯+k_r frac∏_i<j(k_j-k_i)^2∏_i=1^r k_i^2r-1. Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and including the factors t^k_i/k_i gives the stated coefficient. In detail, ∏_i<j(k_j^-1-k_i^-1)^2 =frac∏_i<j(k_j-k_i)^2∏_i k_i^2(r-1). Passing to the formal …
Exact mathematical passage
\mathcal H_r(t)=\sum_{1\le k_1<\cdots<k_r} t^{k_1+\cdots+k_r} \frac{\prod_{i<j}(k_j-k_i)^2}{\prod_{i=1}^r k_i^{2r-1}}.
\prod_{i=1}^{r}Use in source · 1 passage
Search aliases: ∏_i=1^r prod
Use in source · project_09.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_{i=1}^{r-1}Use in source · 1 passage
Search aliases: ∏_i=1^r-1 prod
Use in source · 23_hankel.tex, line 63
Hadamard's inequality for the positive matrix gives H_r(t)≤∏_i=0^r-1Li_2i+1(t) ≤[-log(1-t)]∏_i=1^r-1ζ(2i+1). For the higher weights, t^k≤1 bounds the series by its zeta value. Since all summands are positive, ∑_i≥1(ζ(2i+1)-1) =∑_k≥2frac1k(k^2-1)=frac14. The last sum telescopes using frac1k(k^2-1)=frac12(frac1k(k-1)-frac1k(k+1)). Finally log(1+x)≤ x for x≥0 bounds the zeta product by e^1/4, uniformly in its length. At t=a/c and t=b/c, the bounds become e^1/4log(c/b) and e^1/4log(c/a). Thus these …
Exact mathematical passage
\mathcal H_r(t)\le\prod_{i=0}^{r-1}\Li_{2i+1}(t) \le[-\log(1-t)]\prod_{i=1}^{r-1}\zeta(2i+1).
\prod_{j=1}^mUse in source · 3 passages
Search aliases: ∏_j=1^m prod
Use in source · 22_jet_filter.tex, line 5
The universal mixed-weight jet filter. One polynomial chooses the cancelling coefficients. Let m≥0 be an integer. Define P_m(X)=X∏_j=1^m(jX-1)=∑_s=1^m+1c_m,sX^s, and the formal series Φ_m(t)=frac1(m+1)!∑_s=1^m+1c_m,sLi_s(t). Here Li_s(t)=∑_k≥1t^k/k^s is first interpreted in Q[[t]]. It also defines the usual analytic polylogarithm on |t|<1~[source reference]. The label s is its weight. The contact order will follow from its coefficients, rather than from that label.
Exact mathematical passage
P_m(X)=X\prod_{j=1}^m(jX-1)=\sum_{s=1}^{m+1}c_{m,s}X^s,
\prod_{j=1}^{r}Use in source · 1 passage
Search aliases: ∏_j=1^r prod
Use in source · project_06.tex, line 92
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_{j=1}^{r-1}Use in source · 1 passage
Search aliases: ∏_j=1^r-1 prod
Use in source · 23_hankel.tex, line 46
[Exact contact and first coefficient] Put s_r=r(r+1)/2. Then ord_0 H_r=s_r, [t^s_r] H_r=∏_k=1^r k^-(2k-1). The unique increasing r-tuple of positive indices with least sum is (1,…,r), of sum s_r. Its coefficient is nonzero. Its Vandermonde numerator is ∏_j=1^r-1(j!)^2. Dividing this by ∏_k=1^r k^2r-1 leaves exponent 2k-1 in the denominator at each k, giving the formula. For r=2 the first coefficient is 1/8. For r=3 it is 1/(2^3 3^5). The contact orders are 1,3,6,10,…, the triangular numbers.
Exact mathematical passage
\prod_{j=1}^{r-1}(j!)^2
\prod_{j=1}^sUse in source · 2 passages
Search aliases: ∏_j=1^s prod
Use in source · 20_signed_filter.tex, line 10
Let N(x) count assignments producing x∈ G. Repeated contributions remain separate assignments, so M=∑_x∈ GN(x)=3^s, μ(x)=N(x)/M. For a character χ∈widehat G, define widehatμ(χ)=∑_xμ(x)χ(x) =∏_j=1^sfrac1+χ(γ_j)+χ(-γ_j)3. On a cyclic channel G=Z/nZ, this becomes m(k)=∏_j=1^sfrac1+2cos(2π kγ_j/n)3. This is the discrete cosine or COS expression used in the project. It is an exact characteristic function. Individual cosine factors may be negative. No positivity of all Fourier samples is assumed.
Exact mathematical passage
\widehat\mu(\chi)=\sum_x\mu(x)\chi(x) =\prod_{j=1}^s\frac{1+\chi(\gamma_j)+\chi(-\gamma_j)}3.
\prod_{k=1}^rUse in source · 2 passages
Search aliases: ∏_k=1^r prod
Use in source · 23_hankel.tex, line 40
[Exact contact and first coefficient] Put s_r=r(r+1)/2. Then ord_0 H_r=s_r, [t^s_r] H_r=∏_k=1^r k^-(2k-1). The unique increasing r-tuple of positive indices with least sum is (1,…,r), of sum s_r. Its coefficient is nonzero. Its Vandermonde numerator is ∏_j=1^r-1(j!)^2. Dividing this by ∏_k=1^r k^2r-1 leaves exponent 2k-1 in the denominator at each k, giving the formula. For r=2 the first coefficient is 1/8. For r=3 it is 1/(2^3 3^5). The contact orders are 1,3,6,10,…, the triangular numbers.
Exact mathematical passage
\ord_0\mathcal H_r=s_r,\qquad [t^{s_r}]\mathcal H_r=\prod_{k=1}^r k^{-(2k-1)}.
\prod_pUse in source · 2 passages
Search aliases: ∏_p prod
Use in source · project_05.tex, line 6
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^e\parallel abc}Use in source · 1 passage
Search aliases: ∏_p^eparallel abc prod parallel
Use in source · project_01.tex, line 167
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 · project_07.tex, line 47
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 · project_03.tex, line 165
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 c}Use in source · 2 passages
Search aliases: ∏_pmid c prod mid
Use in source · project_11.tex, line 91
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. Divide the input exponent vector by K. Define a_p,q as the original odd or weighted dyadic coefficient, now reduced modulo p^d_p-r_p. Dividing the relation vector reduc…
Exact mathematical passage
K_{\mathrm{Kum}}=\prod_{p\mid c}p^{r_p}
\prod_{p\mid n}Use in source · 2 passages
Search aliases: ∏_pmid n prod mid
Use in source · 01_foundations.tex, line 4
Radical support and the exact conjecture. The multiplicative quantity that ignores repetition. Every positive integer has a unique prime factorization n=∏_pmid np^e_p, e_p≥1. Its radical is rad(n)=∏_pmid np, rad(1)=1. The radical retains which primes occur and discards how many times they occur. Thus rad(72)=rad(2^3·3^2)=6, whereas 72/6=12 records the repeated-power contribution. If n is squarefree, then rad(n)=n.
Exact mathematical passage
n=\prod_{p\mid n}p^{e_p},\qquad e_p\ge1.
\prod_{p\le3r-1}Use in source · 1 passage
Search aliases: ∏_p≤3r-1 prod le
Use in source · 24_padic.tex, line 86
ining primes Put a_r=2r-1. For each prime p and 1≤ i≤ r, define C_r,p,i=max (0,sup_h≥1{ a_rh-(p^hlceilfrac ip^hrceil-i)}). This is a finite integer maximum because p^h eventually grows faster than h. Put C_r,p=∑_i=1^rC_r,p,i. The good-prime inequality just proved implies C_r,p=0 for p>3r-1. Hence B_r=∏_p≤3r-1p^C_r,p is a fixed positive integer depending only on the rank.
Exact mathematical passage
B_r=\prod_{p\le3r-1}p^{C_{r,p}}
\prod_{q\mid ab}Use in source · 2 passages
Search aliases: ∏_qmid ab prod mid
Use in source · project_10.tex, line 137
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_{r=0}^{e-1}Use in source · 3 passages
Search aliases: ∏_r=0^e-1 prod
Use in source · project_02.tex, line 160
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 · project_06.tex, line 10
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 · project_02.tex, line 67
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_{u\in(\Z/n\Z)^\times}Use in source · 1 passage
Search aliases: ∏_u∈(Z/nZ)^× prod in Z Z times
Use in source · 20_signed_filter.tex, line 117
A Galois coordinate with a controlled denominator. For an occupied target define Δ_h(E)=∏_x∈ E∖{h}(α_h-α_x). This is a nonzero algebraic integer in Q(ζ_n). For every unit u modulo n, its conjugate is Δ_uh(uE). The absolute norm is a nonzero integer. Hence ∏_u∈(Z/nZ)^×|Δ_uh(uE)|≥1. At least one factor is at least one. In that coordinate, the Lagrange numerator has coefficient ℓ^1 norm at most 2^r-1, and dividing by the target denominator preserves that upper bound.
Exact mathematical passage
\prod_{u\in(\Z/n\Z)^\times}|\Delta_{uh}(uE)|\ge1.
\prod_{x\in E}Use in source · 1 passage
Search aliases: ∏_x∈ E prod in
Use in source · 20_signed_filter.tex, line 77
[Exact cyclic Prony reconstruction] The distribution on a cyclic channel with r occupied residues is determined by the 2r exact consecutive samples m(0),…,m(2r-1). Write A(z)=∏_x∈ E(z-α_x)=z^r+∑_j=0^r-1a_jz^j. For every k, ∑_j=0^r-1a_jm(k+j)=-m(k+r). The coefficient matrix for 0≤ k<r is the Hankel matrix H=(m(i+j))_0≤ i,j<r. With V=(α_x^i)_i,x and D=diag(N(x)/M), one has H=VDV^ T. The square Vandermonde V is invertible and every diagonal entry of D is nonzero. Thus H is invertible, and the 2r sa…
Exact mathematical passage
A(z)=\prod_{x\in E}(z-\alpha_x)=z^r+\sum_{j=0}^{r-1}a_jz^j.
\prod_{x\in E\setminus\{h\}}Use in source · 1 passage
Search aliases: ∏_x∈ E∖{h} prod in setminus
Use in source · 20_signed_filter.tex, line 113
A Galois coordinate with a controlled denominator. For an occupied target define Δ_h(E)=∏_x∈ E∖{h}(α_h-α_x). This is a nonzero algebraic integer in Q(ζ_n). For every unit u modulo n, its conjugate is Δ_uh(uE). The absolute norm is a nonzero integer. Hence ∏_u∈(Z/nZ)^×|Δ_uh(uE)|≥1. At least one factor is at least one. In that coordinate, the Lagrange numerator has coefficient ℓ^1 norm at most 2^r-1, and dividing by the target denominator preserves that upper bound.
Exact mathematical passage
\Delta_h(E)=\prod_{x\in E\setminus\{h\}}(\alpha_h-\alpha_x).
\prod_{\ell\text{ odd}}Use in source · 2 passages
Search aliases: ∏_ℓ odd prod ell text
Use in source · project_02.tex, line 92
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_{1\le k_1<\cdots<k_r}Use in source · 1 passage
Search aliases: ∑_1≤ k_1<⋯<k_r sum le cdots
Use in source · 23_hankel.tex, line 18
[Cauchy--Binet expansion] H_r(t)=∑_1≤ k_1<⋯<k_r t^k_1+⋯+k_r frac∏_i<j(k_j-k_i)^2∏_i=1^r k_i^2r-1. Write v_k=(1,k^-1,…,k^-(r-1))^ T. The moment matrix is ∑_k≥1fract^kkv_kv_k^ T. For a finite sum, Cauchy--Binet expands its determinant over r-element index subsets. The determinant of the chosen columns is a Vandermonde in k_i^-1. Squaring it and including the factors t^k_i/k_i gives the stated coefficient. In detail, ∏_i<j(k_j^-1-k_i^-1)^2 =frac∏_i<j(k_j-k_i)^2∏_i k_i^2(r-1). Passing to the formal …
Exact mathematical passage
\mathcal H_r(t)=\sum_{1\le k_1<\cdots<k_r} t^{k_1+\cdots+k_r} \frac{\prod_{i<j}(k_j-k_i)^2}{\prod_{i=1}^r k_i^{2r-1}}.
\sum_{d\mid e}Use in source · 1 passage
Search aliases: ∑_dmid e sum mid
Use in source · project_06.tex, line 18
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 · project_06.tex, line 57
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_{h\ge1}Use in source · 2 passages
Search aliases: ∑_h≥1 sum ge
Use in source · 24_padic.tex, line 21
Let u_1<⋯<u_r be the first r positive integers not divisible by p. Then u_j=j+lfloorfracj-1p-1rfloor, S_p,r=∑_j=1^r u_j. For h≥1, put s_h=(p-1)p^h-1 and write r=q_hs_h+b_h, 0≤ b_h<s_h. Define the finite sum E_p,r=∑_h≥1(s_hbinomq_h2+b_hq_h). The terms are zero once s_h≥ r.
Exact mathematical passage
E_{p,r}=\sum_{h\ge1}\left(s_h\binom{q_h}{2}+b_hq_h\right).
\sum_iUse in source · 5 passages
Search aliases: ∑_i sum
Use in source · 24_padic.tex, line 70
ves linearly growing valuation. Thus the terms tend to zero and the finite Cauchy--Binet expansions pass to the convergent local limit. [Exact good-prime valuation] If p>3r-1 and v_p(t)=e≥1, then v_p( H_r(t))=e r(r+1)/2. Write a_r=2r-1 and s_r=r(r+1)/2. A term indexed by k_1<⋯<k_r has valuation e∑_i k_i+2∑_i<jv_p(k_j-k_i)-a_r∑_i v_p(k_i). For every h≥1, the hypothesis gives p^h-r>a_rh. At h=1 this is p-r>a_r. Successive increments of p^h exceed a_r, proving the inequality for all h.
Exact mathematical passage
e\sum_i k_i+2\sum_{i<j}v_p(k_j-k_i)-a_r\sum_i v_p(k_i).
\sum_{i<j}Use in source · 2 passages
Search aliases: ∑_i<j sum
Use in source · 24_padic.tex, line 28
The residue-class energy minimum. For r distinct p-unit integers k_1,…,k_r, the valuation of their Vandermonde product is ∑_i<jv_p(k_j-k_i) =∑_h≥1#{i<j:k_i≡ k_jpmodp^h}. There are s_h unit residue classes modulo p^h. If their occupation counts are n_1,…,n_s_h, the number of coincident pairs is ∑_jbinomn_j2.
Exact mathematical passage
\sum_{i<j}v_p(k_j-k_i) =\sum_{h\ge1}\#\{i<j:k_i\equiv k_j\pmod{p^h}\}.
\sum_{i=0}^{r-1}Use in source · 1 passage
Search aliases: ∑_i=0^r-1 sum
Use in source · 23_hankel.tex, line 84
Since |log(b/a)|≤log c and log(c/a)≤log c, every entry has a bound by a constant times (1+log c)^j. A determinant term has total weight ∑_i=0^r-1(i+π(i)+1)=r^2. Therefore | H_r(c/a)|≤ C_r(1+log c)^r^2 for each of those consistent branches. The constant may depend on r. No nonvanishing assertion at every argument in this third chart follows from the positive expansion on 0<t<1.
Exact mathematical passage
\sum_{i=0}^{r-1}(i+\pi(i)+1)=r^2.
\sum_{i=1}^rUse in source · 1 passage
Search aliases: ∑_i=1^r sum
Use in source · 24_padic.tex, line 85
An explicit fixed correction at the remaining primes. Put a_r=2r-1. For each prime p and 1≤ i≤ r, define C_r,p,i=max (0,sup_h≥1{ a_rh-(p^hlceilfrac ip^hrceil-i)}). This is a finite integer maximum because p^h eventually grows faster than h. Put C_r,p=∑_i=1^rC_r,p,i. The good-prime inequality just proved implies C_r,p=0 for p>3r-1. Hence B_r=∏_p≤3r-1p^C_r,p is a fixed positive integer depending only on the rank.
Exact mathematical passage
C_{r,p}=\sum_{i=1}^rC_{r,p,i}
\sum_{i\ge1}Use in source · 1 passage
Search aliases: ∑_i≥1 sum ge
Use in source · 23_hankel.tex, line 68
Hadamard's inequality for the positive matrix gives H_r(t)≤∏_i=0^r-1Li_2i+1(t) ≤[-log(1-t)]∏_i=1^r-1ζ(2i+1). For the higher weights, t^k≤1 bounds the series by its zeta value. Since all summands are positive, ∑_i≥1(ζ(2i+1)-1) =∑_k≥2frac1k(k^2-1)=frac14. The last sum telescopes using frac1k(k^2-1)=frac12(frac1k(k-1)-frac1k(k+1)). Finally log(1+x)≤ x for x≥0 bounds the zeta product by e^1/4, uniformly in its length. At t=a/c and t=b/c, the bounds become e^1/4log(c/b) and e^1/4log(c/a). Thus these …
Exact mathematical passage
\sum_{i\ge1}(\zeta(2i+1)-1) =\sum_{k\ge2}\frac1{k(k^2-1)}=\frac14.
\sum_jUse in source · 2 passages
Search aliases: ∑_j sum
Use in source · 24_padic.tex, line 32
The residue-class energy minimum. For r distinct p-unit integers k_1,…,k_r, the valuation of their Vandermonde product is ∑_i<jv_p(k_j-k_i) =∑_h≥1#{i<j:k_i≡ k_jpmodp^h}. There are s_h unit residue classes modulo p^h. If their occupation counts are n_1,…,n_s_h, the number of coincident pairs is ∑_jbinomn_j2.
Exact mathematical passage
\sum_j\binom{n_j}{2}
\sum_{j=0}^dUse in source · 1 passage
Search aliases: ∑_j=0^d sum
Use in source · 20_signed_filter.tex, line 91
Polynomial target filters and exact count certificates. For a polynomial P(z)=∑_j=0^dc_jz^j, define the collision quadratic form E_d(P)=frac nM^2∑_x∈ EN(x)^2|P(α_x)|^2. It can be written as a positive semidefinite Gram form in the coefficients of P. Let λ_d(h)=min{ E_d(P):deg P≤ d, P(α_h)=1}. For every such filter, the target term is present with value (n/M^2)N(h)^2, so N(h)≤frac Msqrt nsqrtλ_d(h). If the right-hand side is rigorously less than one, the integer count is zero. An approximate deci…
Exact mathematical passage
P(z)=\sum_{j=0}^dc_jz^j
\sum_{j=0}^{M-1}Use in source · 1 passage
Search aliases: ∑_j=0^M-1 sum
Use in source · project_10.tex, line 6
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=0}^{r-1}Use in source · 2 passages
Search aliases: ∑_j=0^r-1 sum
Use in source · 20_signed_filter.tex, line 77
[Exact cyclic Prony reconstruction] The distribution on a cyclic channel with r occupied residues is determined by the 2r exact consecutive samples m(0),…,m(2r-1). Write A(z)=∏_x∈ E(z-α_x)=z^r+∑_j=0^r-1a_jz^j. For every k, ∑_j=0^r-1a_jm(k+j)=-m(k+r). The coefficient matrix for 0≤ k<r is the Hankel matrix H=(m(i+j))_0≤ i,j<r. With V=(α_x^i)_i,x and D=diag(N(x)/M), one has H=VDV^ T. The square Vandermonde V is invertible and every diagonal entry of D is nonzero. Thus H is invertible, and the 2r sa…
Exact mathematical passage
A(z)=\prod_{x\in E}(z-\alpha_x)=z^r+\sum_{j=0}^{r-1}a_jz^j.
\sum_{j=1}^rUse in source · 1 passage
Search aliases: ∑_j=1^r sum
Use in source · 24_padic.tex, line 12
Let u_1<⋯<u_r be the first r positive integers not divisible by p. Then u_j=j+lfloorfracj-1p-1rfloor, S_p,r=∑_j=1^r u_j. For h≥1, put s_h=(p-1)p^h-1 and write r=q_hs_h+b_h, 0≤ b_h<s_h. Define the finite sum E_p,r=∑_h≥1(s_hbinomq_h2+b_hq_h). The terms are zero once s_h≥ r.
Exact mathematical passage
u_j=j+\left\lfloor\frac{j-1}{p-1}\right\rfloor, \qquad S_{p,r}=\sum_{j=1}^r u_j.
\sum_{j\neq0}Use in source · 1 passage
Search aliases: ∑_j≠0 sum neq
Use in source · project_02.tex, line 208
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=1}^KUse in source · 1 passage
Search aliases: ∑_k=1^K sum
Use in source · 25_truncation.tex, line 17
Let L_K=lcm(1,…,K), s_r=r(r+1)/2, d=rK-r(r-1)/2. Then A_r,K(t):=L_K^r^2 H_r,K(t)∈Z[t], ord_0A_r,K=s_r, deg A_r,K=d. To see integrality, scale row i by L_K^i+1 and column j by L_K^j. Each entry becomes L_K^i+j+1∑_k=1^Kfract^kk^i+j+1∈Z[t]. The total determinant scaling exponent is r^2. The unique smallest subset is (1,…,r) and the unique largest is (K-r+1,…,K), proving the order and degree.
Exact mathematical passage
L_K^{i+j+1}\sum_{k=1}^K\frac{t^k}{k^{i+j+1}}\in\Z[t].
\sum_{k\ge1}Use in source · 4 passages
Search aliases: ∑_k≥1 sum ge
Use in source · 22_jet_filter.tex, line 12
The universal mixed-weight jet filter. One polynomial chooses the cancelling coefficients. Let m≥0 be an integer. Define P_m(X)=X∏_j=1^m(jX-1)=∑_s=1^m+1c_m,sX^s, and the formal series Φ_m(t)=frac1(m+1)!∑_s=1^m+1c_m,sLi_s(t). Here Li_s(t)=∑_k≥1t^k/k^s is first interpreted in Q[[t]]. It also defines the usual analytic polylogarithm on |t|<1~[source reference]. The label s is its weight. The contact order will follow from its coefficients, rather than from that label.
Exact mathematical passage
\Li_s(t)=\sum_{k\ge1}t^k/k^s
\sum_{k\ge2}Use in source · 1 passage
Search aliases: ∑_k≥2 sum ge
Use in source · 23_hankel.tex, line 68
Hadamard's inequality for the positive matrix gives H_r(t)≤∏_i=0^r-1Li_2i+1(t) ≤[-log(1-t)]∏_i=1^r-1ζ(2i+1). For the higher weights, t^k≤1 bounds the series by its zeta value. Since all summands are positive, ∑_i≥1(ζ(2i+1)-1) =∑_k≥2frac1k(k^2-1)=frac14. The last sum telescopes using frac1k(k^2-1)=frac12(frac1k(k-1)-frac1k(k+1)). Finally log(1+x)≤ x for x≥0 bounds the zeta product by e^1/4, uniformly in its length. At t=a/c and t=b/c, the bounds become e^1/4log(c/b) and e^1/4log(c/a). Thus these …
Exact mathematical passage
\sum_{i\ge1}(\zeta(2i+1)-1) =\sum_{k\ge2}\frac1{k(k^2-1)}=\frac14.
\sum_pUse in source · 1 passage
Search aliases: ∑_p sum
Use in source · project_05.tex, line 16
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 · project_02.tex, line 112
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 · project_02.tex, line 112
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 · project_02.tex, line 112
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 · project_01.tex, line 21
e 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. First define the global free module over all primes, and restrict it to Ω_S for S-supported integers. For such 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 l…
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 · project_02.tex, line 227
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_{p\nmid k}Use in source · 2 passages
Search aliases: ∑_pnmid k sum nmid
Use in source · 22_jet_filter.tex, line 49
The depleted local series. For a prime p, delete the terms whose index is divisible by p: Φ_m,p(t)=(-1)^m∑_pnmid kbinomkm+1fract^kk^m+2∈Z_p[[t]]. It converges when v_p(t)>0. [Exact local jet valuation] If v_p(t)=e≥1, then v_p(Φ_m,p(t))= (m+1)e, · pnmid m+1,
Exact mathematical passage
\Phi_{m,p}(t)=(-1)^m\sum_{p\nmid k}\binom{k}{m+1}\frac{t^k}{k^{m+2}}\in\Z_p[[t]].
\sum_{q\mid ab}Use in source · 2 passages
Search aliases: ∑_qmid ab sum mid
Use in source · project_10.tex, line 141
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 · project_02.tex, line 165
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 · project_02.tex, line 187
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 · project_02.tex, line 227
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_sUse in source · 1 passage
Search aliases: ∑_s sum
Use in source · 22_jet_filter.tex, line 40
Small coefficient norm has a limited meaning. The coefficients of P_m alternate in sign. Its absolute-coefficient polynomial evaluated at one gives ∑_s|c_m,s|=∏_j=1^m(j+1)=(m+1)!. Also c_m,1=(-1)^m, so the integer coefficient vector is primitive. After the displayed normalization, its coefficient ℓ^1 norm is one.
Exact mathematical passage
\sum_s|c_{m,s}|=\prod_{j=1}^m(j+1)=(m+1)!.
\sum_{s=1}^{m+1}Use in source · 2 passages
Search aliases: ∑_s=1^m+1 sum
Use in source · 22_jet_filter.tex, line 5
The universal mixed-weight jet filter. One polynomial chooses the cancelling coefficients. Let m≥0 be an integer. Define P_m(X)=X∏_j=1^m(jX-1)=∑_s=1^m+1c_m,sX^s, and the formal series Φ_m(t)=frac1(m+1)!∑_s=1^m+1c_m,sLi_s(t). Here Li_s(t)=∑_k≥1t^k/k^s is first interpreted in Q[[t]]. It also defines the usual analytic polylogarithm on |t|<1~[source reference]. The label s is its weight. The contact order will follow from its coefficients, rather than from that label.
Exact mathematical passage
P_m(X)=X\prod_{j=1}^m(jX-1)=\sum_{s=1}^{m+1}c_{m,s}X^s,
\sum_xUse in source · 3 passages
Search aliases: ∑_x sum
Use in source · 20_signed_filter.tex, line 10
Let N(x) count assignments producing x∈ G. Repeated contributions remain separate assignments, so M=∑_x∈ GN(x)=3^s, μ(x)=N(x)/M. For a character χ∈widehat G, define widehatμ(χ)=∑_xμ(x)χ(x) =∏_j=1^sfrac1+χ(γ_j)+χ(-γ_j)3. On a cyclic channel G=Z/nZ, this becomes m(k)=∏_j=1^sfrac1+2cos(2π kγ_j/n)3. This is the discrete cosine or COS expression used in the project. It is an exact characteristic function. Individual cosine factors may be negative. No positivity of all Fourier samples is assumed.
Exact mathematical passage
\widehat\mu(\chi)=\sum_x\mu(x)\chi(x) =\prod_{j=1}^s\frac{1+\chi(\gamma_j)+\chi(-\gamma_j)}3.
\sum_{x\in E}Use in source · 2 passages
Search aliases: ∑_x∈ E sum in
Use in source · 20_signed_filter.tex, line 63
Exact innovation rank in a cyclic channel. Let G=Z/nZ, α_x=e^2π ix/n and m(k)=∑_x∈ EfracN(x)Mα_x^k. Form the shift matrix S_d=(m(k+j))_0≤ k<n, 0≤ j≤ d. It factors as a Fourier-column matrix times a Vandermonde matrix on the distinct occupied nodes. The Fourier columns are independent, and all weights N(x)/M are nonzero. Therefore rankS_d=min(r,d+1). Before degree r, a small singular value is a conditioning phenomenon, not an exact reduction in the number of occupied residues.
Exact mathematical passage
m(k)=\sum_{x\in E}\frac{N(x)}M\alpha_x^k.
\sum_{x\in G}Use in source · 1 passage
Search aliases: ∑_x∈ G sum in
Use in source · 20_signed_filter.tex, line 6
Let N(x) count assignments producing x∈ G. Repeated contributions remain separate assignments, so M=∑_x∈ GN(x)=3^s, μ(x)=N(x)/M. For a character χ∈widehat G, define widehatμ(χ)=∑_xμ(x)χ(x) =∏_j=1^sfrac1+χ(γ_j)+χ(-γ_j)3. On a cyclic channel G=Z/nZ, this becomes m(k)=∏_j=1^sfrac1+2cos(2π kγ_j/n)3. This is the discrete cosine or COS expression used in the project. It is an exact characteristic function. Individual cosine factors may be negative. No positivity of all Fourier samples is assumed.
Exact mathematical passage
M=\sum_{x\in G}N(x)=3^s,\qquad \mu(x)=N(x)/M.
\sum_\varepsilonUse in source · 1 passage
Search aliases: ∑_ε sum varepsilon
Use in source · project_06.tex, line 108
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. }
\sum_{\chi\in\widehat G}Use in source · 2 passages
Search aliases: ∑_χ∈widehat G sum chi in widehat
Use in source · 20_signed_filter.tex, line 21
Character orthogonality gives the target count N(h)=frac M|G|∑_χ∈widehat Gwidehatμ(χ)overlineχ(h). The identity is finite and exact. It solves a representation question after the arithmetic channel and admissible contributions have been specified. It does not by itself bound the height of an integer triple that generated that channel.
Exact mathematical passage
N(h)=\frac M{|G|}\sum_{\chi\in\widehat G}\widehat\mu(\chi)\overline{\chi(h)}.
\sum_{\chi\ne1}Use in source · 1 passage
Search aliases: ∑_χ≠1 sum chi ne
Use in source · 20_signed_filter.tex, line 34
Sparsity forces transverse energy. Let E={x:N(x)>0} and r=|E|. Always r≤ M. Parseval yields ∑_χ∈widehat G|widehatμ(χ)|^2 =frac|G|M^2∑_xN(x)^2. Cauchy--Schwarz on the occupied support gives ∑_xN(x)^2≥ M^2/r. Hence [Sparse energy barrier] ∑_χ≠1|widehatμ(χ)|^2 ≥frac|G|r-1≥frac|G|M-1. When few assignment atoms lie in a large group, substantial nontrivial Fourier energy is unavoidable. A failure of uniform mixing is therefore a structural feature of the signal, rather than evidence that the arithmeti…
Exact mathematical passage
\sum_{\chi\ne1}|\widehat\mu(\chi)|^2 \ge\frac{|G|}{r}-1\ge\frac{|G|}{M}-1.
\setminusUse in source · 1 passage
Search aliases: ∖ setminus
Use in source · 20_signed_filter.tex, line 113
A Galois coordinate with a controlled denominator. For an occupied target define Δ_h(E)=∏_x∈ E∖{h}(α_h-α_x). This is a nonzero algebraic integer in Q(ζ_n). For every unit u modulo n, its conjugate is Δ_uh(uE). The absolute norm is a nonzero integer. Hence ∏_u∈(Z/nZ)^×|Δ_uh(uE)|≥1. At least one factor is at least one. In that coordinate, the Lagrange numerator has coefficient ℓ^1 norm at most 2^r-1, and dividing by the target denominator preserves that upper bound.
Exact mathematical passage
\Delta_h(E)=\prod_{x\in E\setminus\{h\}}(\alpha_h-\alpha_x).
\inftyUse in source · 14 passages
Search aliases: ∞ infty
Use in source · 01_foundations.tex, line 123
Because b_n contains 3^n+1, its powerful part contains 3^n. Thus rad(b_n)≤ fracb_n3^n, R_n=2rad(b_n)≤frac2b_n3^n. Therefore fracc_nR_n≥frac3^n c_n2b_n>frac3^n2⟶∞. The displayed lower bound does not contradict the positive-eps conjecture. It does not exhibit infinitely many triples with a fixed positive quality gap. The output grows doubly exponentially with n, while the proved defect lower bound grows exponentially. A large defect lower bound and a fixed positive exponent gap are different asser…
Exact mathematical passage
\frac{c_n}{R_n}\ge\frac{3^n c_n}{2b_n}>\frac{3^n}{2}\longrightarrow\infty.
\capUse in source · 5 passages
Search aliases: ∩ cap
Use in source · project_01.tex, line 76
[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 · 1 passage
Search aliases: ∫ int
Use in source · project_05.tex, line 166
automorphic analysis--synthesis identity Let ϕ_E:X_0(N_E)⟶ E be a nonconstant modular parametrization, and let ϕ_E^*ω_E = 2π i c_ϕ f_E(z) dz. Here c_ϕ is the differential multiplier of this chosen parametrization and differential. The marked Frey curve need not be the optimal representative of its isogeny class. 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_ϕ^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2…
Exact mathematical passage
A(E) = \frac{i}{2} \int_{E(\C)} \omega_E\wedge\overline{\omega_E}.
\int_{E(\C)}Use in source · 1 passage
Search aliases: ∫_E(C) int C
Use in source · project_05.tex, line 166
automorphic analysis--synthesis identity Let ϕ_E:X_0(N_E)⟶ E be a nonconstant modular parametrization, and let ϕ_E^*ω_E = 2π i c_ϕ f_E(z) dz. Here c_ϕ is the differential multiplier of this chosen parametrization and differential. The marked Frey curve need not be the optimal representative of its isogeny class. 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_ϕ^2 normf_E_Pet^2. For the Frey model, scaling x=cX, y=c^3/2…
Exact mathematical passage
A(E) = \frac{i}{2} \int_{E(\C)} \omega_E\wedge\overline{\omega_E}.
\simeqUse in source · 1 passage
Search aliases: ≃ simeq
Use in source · project_08.tex, line 35
d ∞ are div(a), div(b), div(c). For the output cusp, let C=div(c), C_red=div(R_c). Place the two-term contact complex in degrees -1,0 and define J_c = [ O(-C)⟶ O(-C_red)]. On global sections, J_c = [cZ↪ R_cZ] ≃ [Zxrightarrowc/R_cZ].
Exact mathematical passage
\mathbf J_c = [c\Z\hookrightarrow R_c\Z] \simeq [\Z\xrightarrow{c/R_c}\Z].
\neUse in source · 3 passages
Search aliases: ≠ ne
Use in source · 20_signed_filter.tex, line 34
Sparsity forces transverse energy. Let E={x:N(x)>0} and r=|E|. Always r≤ M. Parseval yields ∑_χ∈widehat G|widehatμ(χ)|^2 =frac|G|M^2∑_xN(x)^2. Cauchy--Schwarz on the occupied support gives ∑_xN(x)^2≥ M^2/r. Hence [Sparse energy barrier] ∑_χ≠1|widehatμ(χ)|^2 ≥frac|G|r-1≥frac|G|M-1. When few assignment atoms lie in a large group, substantial nontrivial Fourier energy is unavoidable. A failure of uniform mixing is therefore a structural feature of the signal, rather than evidence that the arithmeti…
Exact mathematical passage
\sum_{\chi\ne1}|\widehat\mu(\chi)|^2 \ge\frac{|G|}{r}-1\ge\frac{|G|}{M}-1.
\neqUse in source · 10 passages
Search aliases: ≠ neq
Use in source · project_01.tex, line 106
[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
W_x(a,b)\neq0
\equivUse in source · 10 passages
Search aliases: ≡ equiv
Use in source · 24_padic.tex, line 28
The residue-class energy minimum. For r distinct p-unit integers k_1,…,k_r, the valuation of their Vandermonde product is ∑_i<jv_p(k_j-k_i) =∑_h≥1#{i<j:k_i≡ k_jpmodp^h}. There are s_h unit residue classes modulo p^h. If their occupation counts are n_1,…,n_s_h, the number of coincident pairs is ∑_jbinomn_j2.
Exact mathematical passage
\sum_{i<j}v_p(k_j-k_i) =\sum_{h\ge1}\#\{i<j:k_i\equiv k_j\pmod{p^h}\}.
\leUse in source · 96 passages
Search aliases: ≤ le
Use in source · 01_foundations.tex, line 31
[Masser--Oesterl e abc] For every real eps>0 there is a constant K_eps>0, depending only on eps, such that every positive integer triple with a+b=c and gcd(a,b)=1 satisfies c≤ K_eps rad(abc)^1+eps. The order of the quantifiers is the statement. First choose eps. Then choose a constant that works for every primitive triple. The constant may become large as eps approaches zero. It may not depend on the particular triple.
Exact mathematical passage
\boxed{c\le K_\eps\,\rad(abc)^{1+\eps}.}
\subsetUse in source · 1 passage
Search aliases: ⊂ subset
Use in source · project_07.tex, line 5
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\subset V
\subseteqUse in source · 2 passages
Search aliases: ⊆ subseteq
Use in source · project_06.tex, line 119
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 · 4 passages
Search aliases: ⊕ bigoplus
Use in source · project_01.tex, line 17
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. First define the global free module over all primes, and restrict it to Ω_S for S-supported integers. For such 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 ex…
Exact mathematical passage
\Omega_S=\bigoplus_{p\in S}\Z\xi_p.
\oplusUse in source · 3 passages
Search aliases: ⊕ oplus
Use in source · project_03.tex, line 78
The exact lattice extension. Throughout the remaining height and exact-sequence constructions, exclude (1,1,2). 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 …
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 · project_07.tex, line 64
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 · 1 passage
Search aliases: ⊕_p^eparallel c bigoplus parallel
Use in source · project_07.tex, line 40
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\mid c}Use in source · 1 passage
Search aliases: ⊕_pmid c bigoplus mid
Use in source · project_08.tex, line 149
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 · project_01.tex, line 17
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. First define the global free module over all primes, and restrict it to Ω_S for S-supported integers. For such 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 ex…
Exact mathematical passage
\Omega_S=\bigoplus_{p\in S}\Z\xi_p.
\perpUse in source · 3 passages
Search aliases: ⊥ perp
Use in source · project_01.tex, line 116
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 · 6 passages
Search aliases: ⋯ cdots
Use in source · 22_jet_filter.tex, line 26
[Exact jet cancellation] Φ_m(t)=(-1)^m∑_k≥1binomkm+1fract^kk^m+2. Thus ord_0Φ_m=m+1, [t^m+1]Φ_m=frac(-1)^m(m+1)^m+2. The coefficient of t^k is P_m(1/k)/(m+1)!. Direct substitution gives P_m(1/k)=frac1k∏_j=1^m(frac jk-1) =(-1)^mfrac(k-1)⋯(k-m)k^m+1. Divide by (m+1)! and multiply numerator and denominator by k to get the asserted binomial coefficient. It vanishes for 1≤ k≤ m and first becomes one at k=m+1. The coefficient there is nonzero. For m=0, this is Φ_0=Li_1. For m=1, Φ_1=(Li_2-Li_1)/2=-t^2…
Exact mathematical passage
P_m(1/k)=\frac1k\prod_{j=1}^m\left(\frac jk-1\right) =(-1)^m\frac{(k-1)\cdots(k-m)}{k^{m+1}}.
\langleUse in source · 3 passages
Search aliases: ⟨ langle
Use in source · project_07.tex, line 33
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
\langle v,v\rangle,
\rangleUse in source · 3 passages
Search aliases: ⟩ rangle
Use in source · project_07.tex, line 33
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
\langle v,v\rangle,
\rangle_pUse in source · 2 passages
Search aliases: ⟩_p rangle
Use in source · project_10.tex, line 141
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. }
\longrightarrowUse in source · 14 passages
Search aliases: ⟶ longrightarrow
Use in source · 01_foundations.tex, line 123
Because b_n contains 3^n+1, its powerful part contains 3^n. Thus rad(b_n)≤ fracb_n3^n, R_n=2rad(b_n)≤frac2b_n3^n. Therefore fracc_nR_n≥frac3^n c_n2b_n>frac3^n2⟶∞. The displayed lower bound does not contradict the positive-eps conjecture. It does not exhibit infinitely many triples with a fixed positive quality gap. The output grows doubly exponentially with n, while the proved defect lower bound grows exponentially. A large defect lower bound and a fixed positive exponent gap are different asser…
Exact mathematical passage
\frac{c_n}{R_n}\ge\frac{3^n c_n}{2b_n}>\frac{3^n}{2}\longrightarrow\infty.
\longmapstoUse in source · 3 passages
Search aliases: ⟼ longmapsto
Use in source · project_02.tex, line 172
[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.
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S:(T_{p,r})_{p,r}\longmapsto \left(\sum_rT_{p,r}\right)_p
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