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Paper X: Universal Residual Seeds and Conditional Occupied-System Completion
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723 symbol spellings in 2060 mathematical passages across 37 selected source files. Subscripts and superscripts have their own entries alongside their base symbols.
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=Definition passage · 424 passages
Search aliases: =
Definition passage: Residual seed system · Read passage · Paper_X_Block27_Amended.tex, line 163
[Residual seed system] For bar x∈ M(k), define the residual fiber M_bar x(A) = {x∈ M(A):red_A(x)=bar x}. The point bar x is the residual root. The family of all nonempty fibers is the residual special fiber, and the formal deformation problems above them form the residual forest.
Exact mathematical passage
\mathcal M_{\bar x}(A) = \{x\in\mathcal M(A):\red_A(x)=\bar x\}.
ADefinition passage · 68 passages
Search aliases: A
Definition passage: Residual seed system · Read passage · Paper_X_Block27_Amended.tex, line 163
[Residual seed system] For bar x∈ M(k), define the residual fiber M_bar x(A) = {x∈ M(A):red_A(x)=bar x}. The point bar x is the residual root. The family of all nonempty fibers is the residual special fiber, and the formal deformation problems above them form the residual forest.
Exact mathematical passage
\mathcal M_{\bar x}(A) = \{x\in\mathcal M(A):\red_A(x)=\bar x\}.
aDefinition passage · 91 passages
Search aliases: a
Definition passage: Visibility weight · Read passage · Paper_X_Block27_Amended.tex, line 476
[Visibility weight] For two integral packet coordinates a,b∈ O, define their separation depth w(a,b)=min{m≥1:anot≡ bpmodϖ^m}, when such an m exists. A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.
Exact mathematical passage
a,b\in\Ocal
bDefinition passage · 30 passages
Search aliases: b
Definition passage: Visibility weight · Read passage · Paper_X_Block27_Amended.tex, line 476
[Visibility weight] For two integral packet coordinates a,b∈ O, define their separation depth w(a,b)=min{m≥1:anot≡ bpmodϖ^m}, when such an m exists. A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.
Exact mathematical passage
a,b\in\Ocal
\bar xDefinition passage · 26 passages
Search aliases: bar x bar
Definition passage: Residual seed system · Read passage · Paper_X_Block27_Amended.tex, line 162
[Residual seed system] For bar x∈ M(k), define the residual fiber M_bar x(A) = {x∈ M(A):red_A(x)=bar x}. The point bar x is the residual root. The family of all nonempty fibers is the residual special fiber, and the formal deformation problems above them form the residual forest.
Exact mathematical passage
\bar x\in\mathcal M(k)
\bar\rhoDefinition passage · 11 passages
Search aliases: barρ bar rho
Definition passage: Admissible residual transport · Read passage · Paper_X_Block27_Amended.tex, line 653
[Admissible residual transport] An admissible transport from barρ to barρ' consists of T:barρ⟼barρ' together with natural maps T_A:Def_barρ(A)⟶Def_barρ'(A) for every coefficient algebra A, compatible with base change and reduction. When the transport is reversible, it is an isomorphism in the residual transport groupoid. One-way operations give a directed category.
Exact mathematical passage
\bar\rho
\bar\rho'Definition passage · 3 passages
Search aliases: barρ' bar rho
Definition passage: Admissible residual transport · Read passage · Paper_X_Block27_Amended.tex, line 653
[Admissible residual transport] An admissible transport from barρ to barρ' consists of T:barρ⟼barρ' together with natural maps T_A:Def_barρ(A)⟶Def_barρ'(A) for every coefficient algebra A, compatible with base change and reduction. When the transport is reversible, it is an isomorphism in the residual transport groupoid. One-way operations give a directed category.
Exact mathematical passage
\bar\rho'
\ClDefinition passage · 6 passages
Search aliases: Cl operatorname
Definition passage: Directed seed rank · Read passage · Paper_X_Block27_Amended.tex, line 668
[Directed seed rank] For a directed transport graph G, define srank_G = min{|S|:Cl_G(S)=V(G)}, where Cl_G(S) is the set of vertices reachable from S.
Exact mathematical passage
\srank_G = \min\{|S|:\Cl_G(S)=V(G)\},
\Cl_GDefinition passage · 2 passages
Search aliases: Cl_G Cl operatorname
Definition passage: Directed seed rank · Read passage · Paper_X_Block27_Amended.tex, line 668
[Directed seed rank] For a directed transport graph G, define srank_G = min{|S|:Cl_G(S)=V(G)}, where Cl_G(S) is the set of vertices reachable from S.
Exact mathematical passage
\srank_G = \min\{|S|:\Cl_G(S)=V(G)\},
\CompatDefinition passage · 10 passages
Search aliases: Compat C mathfrak
Definition passage: Common arithmetic spine · Read passage · Paper_X_Block27_Amended.tex, line 763
[Common arithmetic spine] Let M be a number field. A common spine C consists of: item characteristic polynomials P_v(X)∈ M[X] at almost all places; item coefficient-prime independent weight and determinant data; item prescribed local Weil--Deligne or inertial packet labels at the ramified places. For each finite place λ of M, let X_λ,m( C) be the finite set of admissible objects modulo ϖ_λ^m realizing the reduction of this spine.
Exact mathematical passage
\Compat
\DefDefinition passage · 2 passages
Search aliases: Def operatorname
Definition passage: Admissible residual transport · Read passage · Paper_X_Block27_Amended.tex, line 658
[Admissible residual transport] An admissible transport from barρ to barρ' consists of T:barρ⟼barρ' together with natural maps T_A:Def_barρ(A)⟶Def_barρ'(A) for every coefficient algebra A, compatible with base change and reduction. When the transport is reversible, it is an isomorphism in the residual transport groupoid. One-way operations give a directed category.
Exact mathematical passage
\mathcal T_A:\Def_{\bar\rho}(A)\longrightarrow\Def_{\bar\rho'}(A)
\Def_{\bar\rho}Definition passage · 2 passages
Search aliases: Def_barρ Def bar rho operatorname bar rho
Definition passage: Admissible residual transport · Read passage · Paper_X_Block27_Amended.tex, line 658
[Admissible residual transport] An admissible transport from barρ to barρ' consists of T:barρ⟼barρ' together with natural maps T_A:Def_barρ(A)⟶Def_barρ'(A) for every coefficient algebra A, compatible with base change and reduction. When the transport is reversible, it is an isomorphism in the residual transport groupoid. One-way operations give a directed category.
Exact mathematical passage
\mathcal T_A:\Def_{\bar\rho}(A)\longrightarrow\Def_{\bar\rho'}(A)
\Def_{\bar\rho'}Definition passage · 1 passage
Search aliases: Def_barρ' Def bar rho operatorname bar rho
Definition passage: Admissible residual transport · Read passage · Paper_X_Block27_Amended.tex, line 658
[Admissible residual transport] An admissible transport from barρ to barρ' consists of T:barρ⟼barρ' together with natural maps T_A:Def_barρ(A)⟶Def_barρ'(A) for every coefficient algebra A, compatible with base change and reduction. When the transport is reversible, it is an isomorphism in the residual transport groupoid. One-way operations give a directed category.
Exact mathematical passage
\mathcal T_A:\Def_{\bar\rho}(A)\longrightarrow\Def_{\bar\rho'}(A)
\dimDefinition passage · 7 passages
Search aliases: dim
Definition passage: Branching dimension · Read passage · Paper_X_Block27_Amended.tex, line 613
[Branching dimension] When the limit exists, define the finite-level counting exponent dim_seed(bar x) = lim_m→∞ fraclog_q#X_m(bar x)m. A limsup may be used in general.
Exact mathematical passage
\dim_{\mathrm{seed}}(\bar x) = \lim_{m\to\infty} \frac{\log_q\#X_m(\bar x)}{m}.
\dim_{\mathrm{seed}}Definition passage · 2 passages
Search aliases: dim_seed dim mathrm
Definition passage: Branching dimension · Read passage · Paper_X_Block27_Amended.tex, line 613
[Branching dimension] When the limit exists, define the finite-level counting exponent dim_seed(bar x) = lim_m→∞ fraclog_q#X_m(bar x)m. A limsup may be used in general.
Exact mathematical passage
\dim_{\mathrm{seed}}(\bar x) = \lim_{m\to\infty} \frac{\log_q\#X_m(\bar x)}{m}.
GDefinition passage · 49 passages
Search aliases: G
Definition passage: Directed seed rank · Read passage · Paper_X_Block27_Amended.tex, line 667
[Directed seed rank] For a directed transport graph G, define srank_G = min{|S|:Cl_G(S)=V(G)}, where Cl_G(S) is the set of vertices reachable from S.
Exact mathematical passage
G
kDefinition passage · 59 passages
Search aliases: k
Definition passage: Residual seed system · Read passage · Paper_X_Block27_Amended.tex, line 162
[Residual seed system] For bar x∈ M(k), define the residual fiber M_bar x(A) = {x∈ M(A):red_A(x)=bar x}. The point bar x is the residual root. The family of all nonempty fibers is the residual special fiber, and the formal deformation problems above them form the residual forest.
Exact mathematical passage
\bar x\in\mathcal M(k)
\limDefinition passage · 2 passages
Search aliases: lim
Definition passage: Branching dimension · Read passage · Paper_X_Block27_Amended.tex, line 613
[Branching dimension] When the limit exists, define the finite-level counting exponent dim_seed(bar x) = lim_m→∞ fraclog_q#X_m(bar x)m. A limsup may be used in general.
Exact mathematical passage
\dim_{\mathrm{seed}}(\bar x) = \lim_{m\to\infty} \frac{\log_q\#X_m(\bar x)}{m}.
\lim_{m\to\infty}Definition passage · 1 passage
Search aliases: lim_m→∞ lim to infty
Definition passage: Branching dimension · Read passage · Paper_X_Block27_Amended.tex, line 613
[Branching dimension] When the limit exists, define the finite-level counting exponent dim_seed(bar x) = lim_m→∞ fraclog_q#X_m(bar x)m. A limsup may be used in general.
Exact mathematical passage
\dim_{\mathrm{seed}}(\bar x) = \lim_{m\to\infty} \frac{\log_q\#X_m(\bar x)}{m}.
\logDefinition passage · 5 passages
Search aliases: log
Definition passage: Branching dimension · Read passage · Paper_X_Block27_Amended.tex, line 613
[Branching dimension] When the limit exists, define the finite-level counting exponent dim_seed(bar x) = lim_m→∞ fraclog_q#X_m(bar x)m. A limsup may be used in general.
Exact mathematical passage
\dim_{\mathrm{seed}}(\bar x) = \lim_{m\to\infty} \frac{\log_q\#X_m(\bar x)}{m}.
\log_qDefinition passage · 1 passage
Search aliases: log_q log
Definition passage: Branching dimension · Read passage · Paper_X_Block27_Amended.tex, line 613
[Branching dimension] When the limit exists, define the finite-level counting exponent dim_seed(bar x) = lim_m→∞ fraclog_q#X_m(bar x)m. A limsup may be used in general.
Exact mathematical passage
\dim_{\mathrm{seed}}(\bar x) = \lim_{m\to\infty} \frac{\log_q\#X_m(\bar x)}{m}.
MDefinition passage · 23 passages
Search aliases: M
Definition passage: Common arithmetic spine · Read passage · Paper_X_Block27_Amended.tex, line 763
[Common arithmetic spine] Let M be a number field. A common spine C consists of: item characteristic polynomials P_v(X)∈ M[X] at almost all places; item coefficient-prime independent weight and determinant data; item prescribed local Weil--Deligne or inertial packet labels at the ramified places. For each finite place λ of M, let X_λ,m( C) be the finite set of admissible objects modulo ϖ_λ^m realizing the reduction of this spine.
Exact mathematical passage
M
\mathcal MDefinition passage · 12 passages
Search aliases: M mathcal
Definition passage: Residual seed system · Read passage · Paper_X_Block27_Amended.tex, line 162
[Residual seed system] For bar x∈ M(k), define the residual fiber M_bar x(A) = {x∈ M(A):red_A(x)=bar x}. The point bar x is the residual root. The family of all nonempty fibers is the residual special fiber, and the formal deformation problems above them form the residual forest.
Exact mathematical passage
\bar x\in\mathcal M(k)
mDefinition passage · 177 passages
Search aliases: m
Definition passage: Visibility weight · Read passage · Paper_X_Block27_Amended.tex, line 477
[Visibility weight] For two integral packet coordinates a,b∈ O, define their separation depth w(a,b)=min{m≥1:anot≡ bpmodϖ^m}, when such an m exists. A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.
Exact mathematical passage
w(a,b)=\min\{m\ge1:a\not\equiv b\pmod{\varpi^m}\},
\mathcal M_{\bar x}Definition passage · 6 passages
Search aliases: M_bar x mathcal bar
Definition passage: Residual seed system · Read passage · Paper_X_Block27_Amended.tex, line 163
[Residual seed system] For bar x∈ M(k), define the residual fiber M_bar x(A) = {x∈ M(A):red_A(x)=bar x}. The point bar x is the residual root. The family of all nonempty fibers is the residual special fiber, and the formal deformation problems above them form the residual forest.
Exact mathematical passage
\mathcal M_{\bar x}(A) = \{x\in\mathcal M(A):\red_A(x)=\bar x\}.
\minDefinition passage · 2 passages
Search aliases: min
Definition passage: Visibility weight · Read passage · Paper_X_Block27_Amended.tex, line 477
[Visibility weight] For two integral packet coordinates a,b∈ O, define their separation depth w(a,b)=min{m≥1:anot≡ bpmodϖ^m}, when such an m exists. A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.
Exact mathematical passage
w(a,b)=\min\{m\ge1:a\not\equiv b\pmod{\varpi^m}\},
\notDefinition passage · 8 passages
Search aliases: not
Definition passage: Visibility weight · Read passage · Paper_X_Block27_Amended.tex, line 477
[Visibility weight] For two integral packet coordinates a,b∈ O, define their separation depth w(a,b)=min{m≥1:anot≡ bpmodϖ^m}, when such an m exists. A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.
Exact mathematical passage
w(a,b)=\min\{m\ge1:a\not\equiv b\pmod{\varpi^m}\},
\OcalDefinition passage · 29 passages
Search aliases: Ocal O mathcal
Definition passage: Visibility weight · Read passage · Paper_X_Block27_Amended.tex, line 476
[Visibility weight] For two integral packet coordinates a,b∈ O, define their separation depth w(a,b)=min{m≥1:anot≡ bpmodϖ^m}, when such an m exists. A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.
Exact mathematical passage
a,b\in\Ocal
PDefinition passage · 41 passages
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Definition passage: Common arithmetic spine · Read passage · Paper_X_Block27_Amended.tex, line 765
[Common arithmetic spine] Let M be a number field. A common spine C consists of: item characteristic polynomials P_v(X)∈ M[X] at almost all places; item coefficient-prime independent weight and determinant data; item prescribed local Weil--Deligne or inertial packet labels at the ramified places. For each finite place λ of M, let X_λ,m( C) be the finite set of admissible objects modulo ϖ_λ^m realizing the reduction of this spine.
Exact mathematical passage
P_v(X)\in M[X]
P_vDefinition passage · 2 passages
Search aliases: P_v
Definition passage: Common arithmetic spine · Read passage · Paper_X_Block27_Amended.tex, line 765
[Common arithmetic spine] Let M be a number field. A common spine C consists of: item characteristic polynomials P_v(X)∈ M[X] at almost all places; item coefficient-prime independent weight and determinant data; item prescribed local Weil--Deligne or inertial packet labels at the ramified places. For each finite place λ of M, let X_λ,m( C) be the finite set of admissible objects modulo ϖ_λ^m realizing the reduction of this spine.
Exact mathematical passage
P_v(X)\in M[X]
\pmodDefinition passage · 40 passages
Search aliases: pmod
Definition passage: Visibility weight · Read passage · Paper_X_Block27_Amended.tex, line 477
[Visibility weight] For two integral packet coordinates a,b∈ O, define their separation depth w(a,b)=min{m≥1:anot≡ bpmodϖ^m}, when such an m exists. A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.
Exact mathematical passage
w(a,b)=\min\{m\ge1:a\not\equiv b\pmod{\varpi^m}\},
qDefinition passage · 54 passages
Search aliases: q
Definition passage: Branching dimension · Read passage · Paper_X_Block27_Amended.tex, line 613
[Branching dimension] When the limit exists, define the finite-level counting exponent dim_seed(bar x) = lim_m→∞ fraclog_q#X_m(bar x)m. A limsup may be used in general.
Exact mathematical passage
\dim_{\mathrm{seed}}(\bar x) = \lim_{m\to\infty} \frac{\log_q\#X_m(\bar x)}{m}.
\redDefinition passage · 4 passages
Search aliases: red operatorname
Definition passage: Residual seed system · Read passage · Paper_X_Block27_Amended.tex, line 163
[Residual seed system] For bar x∈ M(k), define the residual fiber M_bar x(A) = {x∈ M(A):red_A(x)=bar x}. The point bar x is the residual root. The family of all nonempty fibers is the residual special fiber, and the formal deformation problems above them form the residual forest.
Exact mathematical passage
\mathcal M_{\bar x}(A) = \{x\in\mathcal M(A):\red_A(x)=\bar x\}.
\red_ADefinition passage · 3 passages
Search aliases: red_A red operatorname
Definition passage: Residual seed system · Read passage · Paper_X_Block27_Amended.tex, line 163
[Residual seed system] For bar x∈ M(k), define the residual fiber M_bar x(A) = {x∈ M(A):red_A(x)=bar x}. The point bar x is the residual root. The family of all nonempty fibers is the residual special fiber, and the formal deformation problems above them form the residual forest.
Exact mathematical passage
\mathcal M_{\bar x}(A) = \{x\in\mathcal M(A):\red_A(x)=\bar x\}.
SDefinition passage · 29 passages
Search aliases: S
Definition passage: Directed seed rank · Read passage · Paper_X_Block27_Amended.tex, line 668
[Directed seed rank] For a directed transport graph G, define srank_G = min{|S|:Cl_G(S)=V(G)}, where Cl_G(S) is the set of vertices reachable from S.
Exact mathematical passage
\srank_G = \min\{|S|:\Cl_G(S)=V(G)\},
\mathrm{seed}Definition passage · 2 passages
Search aliases: seed mathrm
Definition passage: Branching dimension · Read passage · Paper_X_Block27_Amended.tex, line 613
[Branching dimension] When the limit exists, define the finite-level counting exponent dim_seed(bar x) = lim_m→∞ fraclog_q#X_m(bar x)m. A limsup may be used in general.
Exact mathematical passage
\dim_{\mathrm{seed}}(\bar x) = \lim_{m\to\infty} \frac{\log_q\#X_m(\bar x)}{m}.
\srankDefinition passage · 2 passages
Search aliases: srank operatorname
Definition passage: Directed seed rank · Read passage · Paper_X_Block27_Amended.tex, line 668
[Directed seed rank] For a directed transport graph G, define srank_G = min{|S|:Cl_G(S)=V(G)}, where Cl_G(S) is the set of vertices reachable from S.
Exact mathematical passage
\srank_G = \min\{|S|:\Cl_G(S)=V(G)\},
\srank_GDefinition passage · 2 passages
Search aliases: srank_G srank operatorname
Definition passage: Directed seed rank · Read passage · Paper_X_Block27_Amended.tex, line 668
[Directed seed rank] For a directed transport graph G, define srank_G = min{|S|:Cl_G(S)=V(G)}, where Cl_G(S) is the set of vertices reachable from S.
Exact mathematical passage
\srank_G = \min\{|S|:\Cl_G(S)=V(G)\},
TDefinition passage · 51 passages
Search aliases: T
Definition passage: Seed zeta function · Read passage · Paper_X_Block27_Amended.tex, line 532
[Seed zeta function] For a residual root bar x with finite levels, define Z_bar x(T) = ∑_m≥1#X_m(bar x)T^m. When a weighted chart is specified, write Z_ w(T).
Exact mathematical passage
\boxed{ Z_{\bar x}(T) = \sum_{m\ge1}\#X_m(\bar x)T^m. }
\mathcal TDefinition passage · 2 passages
Search aliases: T mathcal
Definition passage: Admissible residual transport · Read passage · Paper_X_Block27_Amended.tex, line 654
[Admissible residual transport] An admissible transport from barρ to barρ' consists of T:barρ⟼barρ' together with natural maps T_A:Def_barρ(A)⟶Def_barρ'(A) for every coefficient algebra A, compatible with base change and reduction. When the transport is reversible, it is an isomorphism in the residual transport groupoid. One-way operations give a directed category.
Exact mathematical passage
\mathcal T:\bar\rho\longmapsto\bar\rho'
T^mDefinition passage · 4 passages
Search aliases: T^m
Definition passage: Seed zeta function · Read passage · Paper_X_Block27_Amended.tex, line 532
[Seed zeta function] For a residual root bar x with finite levels, define Z_bar x(T) = ∑_m≥1#X_m(bar x)T^m. When a weighted chart is specified, write Z_ w(T).
Exact mathematical passage
\boxed{ Z_{\bar x}(T) = \sum_{m\ge1}\#X_m(\bar x)T^m. }
\mathcal T_ADefinition passage · 1 passage
Search aliases: T_A mathcal
Definition passage: Admissible residual transport · Read passage · Paper_X_Block27_Amended.tex, line 658
[Admissible residual transport] An admissible transport from barρ to barρ' consists of T:barρ⟼barρ' together with natural maps T_A:Def_barρ(A)⟶Def_barρ'(A) for every coefficient algebra A, compatible with base change and reduction. When the transport is reversible, it is an isomorphism in the residual transport groupoid. One-way operations give a directed category.
Exact mathematical passage
\mathcal T_A:\Def_{\bar\rho}(A)\longrightarrow\Def_{\bar\rho'}(A)
VDefinition passage · 11 passages
Search aliases: V
Definition passage: Directed seed rank · Read passage · Paper_X_Block27_Amended.tex, line 668
[Directed seed rank] For a directed transport graph G, define srank_G = min{|S|:Cl_G(S)=V(G)}, where Cl_G(S) is the set of vertices reachable from S.
Exact mathematical passage
\srank_G = \min\{|S|:\Cl_G(S)=V(G)\},
vDefinition passage · 34 passages
Search aliases: v
Definition passage: Common arithmetic spine · Read passage · Paper_X_Block27_Amended.tex, line 765
[Common arithmetic spine] Let M be a number field. A common spine C consists of: item characteristic polynomials P_v(X)∈ M[X] at almost all places; item coefficient-prime independent weight and determinant data; item prescribed local Weil--Deligne or inertial packet labels at the ramified places. For each finite place λ of M, let X_λ,m( C) be the finite set of admissible objects modulo ϖ_λ^m realizing the reduction of this spine.
Exact mathematical passage
P_v(X)\in M[X]
\bm wDefinition passage · 4 passages
Search aliases: w bm
Definition passage: Seed zeta function · Read passage · Paper_X_Block27_Amended.tex, line 539
[Seed zeta function] For a residual root bar x with finite levels, define Z_bar x(T) = ∑_m≥1#X_m(bar x)T^m. When a weighted chart is specified, write Z_ w(T).
Exact mathematical passage
Z_{\bm w}(T)
wDefinition passage · 38 passages
Search aliases: w
Definition passage: Visibility weight · Read passage · Paper_X_Block27_Amended.tex, line 477
[Visibility weight] For two integral packet coordinates a,b∈ O, define their separation depth w(a,b)=min{m≥1:anot≡ bpmodϖ^m}, when such an m exists. A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.
Exact mathematical passage
w(a,b)=\min\{m\ge1:a\not\equiv b\pmod{\varpi^m}\},
XDefinition passage · 61 passages
Search aliases: X
Definition passage: Seed zeta function · Read passage · Paper_X_Block27_Amended.tex, line 532
[Seed zeta function] For a residual root bar x with finite levels, define Z_bar x(T) = ∑_m≥1#X_m(bar x)T^m. When a weighted chart is specified, write Z_ w(T).
Exact mathematical passage
\boxed{ Z_{\bar x}(T) = \sum_{m\ge1}\#X_m(\bar x)T^m. }
xDefinition passage · 109 passages
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Definition passage: Residual seed system · Read passage · Paper_X_Block27_Amended.tex, line 163
[Residual seed system] For bar x∈ M(k), define the residual fiber M_bar x(A) = {x∈ M(A):red_A(x)=bar x}. The point bar x is the residual root. The family of all nonempty fibers is the residual special fiber, and the formal deformation problems above them form the residual forest.
Exact mathematical passage
\mathcal M_{\bar x}(A) = \{x\in\mathcal M(A):\red_A(x)=\bar x\}.
X_mDefinition passage · 16 passages
Search aliases: X_m
Definition passage: Seed zeta function · Read passage · Paper_X_Block27_Amended.tex, line 532
[Seed zeta function] For a residual root bar x with finite levels, define Z_bar x(T) = ∑_m≥1#X_m(bar x)T^m. When a weighted chart is specified, write Z_ w(T).
Exact mathematical passage
\boxed{ Z_{\bar x}(T) = \sum_{m\ge1}\#X_m(\bar x)T^m. }
X_{\lambda,m}Definition passage · 5 passages
Search aliases: X_λ,m lambda
Definition passage: Common arithmetic spine · Read passage · Paper_X_Block27_Amended.tex, line 770
ic spine] Let M be a number field. A common spine C consists of: item characteristic polynomials P_v(X)∈ M[X] at almost all places; item coefficient-prime independent weight and determinant data; item prescribed local Weil--Deligne or inertial packet labels at the ramified places. For each finite place λ of M, let X_λ,m( C) be the finite set of admissible objects modulo ϖ_λ^m realizing the reduction of this spine.
Exact mathematical passage
X_{\lambda,m}(\Compat)
ZDefinition passage · 18 passages
Search aliases: Z
Definition passage: Seed zeta function · Read passage · Paper_X_Block27_Amended.tex, line 532
[Seed zeta function] For a residual root bar x with finite levels, define Z_bar x(T) = ∑_m≥1#X_m(bar x)T^m. When a weighted chart is specified, write Z_ w(T).
Exact mathematical passage
\boxed{ Z_{\bar x}(T) = \sum_{m\ge1}\#X_m(\bar x)T^m. }
Z_{\bm w}Definition passage · 2 passages
Search aliases: Z_ w bm
Definition passage: Seed zeta function · Read passage · Paper_X_Block27_Amended.tex, line 539
[Seed zeta function] For a residual root bar x with finite levels, define Z_bar x(T) = ∑_m≥1#X_m(bar x)T^m. When a weighted chart is specified, write Z_ w(T).
Exact mathematical passage
Z_{\bm w}(T)
Z_{\bar x}Definition passage · 1 passage
Search aliases: Z_bar x bar
Definition passage: Seed zeta function · Read passage · Paper_X_Block27_Amended.tex, line 532
[Seed zeta function] For a residual root bar x with finite levels, define Z_bar x(T) = ∑_m≥1#X_m(bar x)T^m. When a weighted chart is specified, write Z_ w(T).
Exact mathematical passage
\boxed{ Z_{\bar x}(T) = \sum_{m\ge1}\#X_m(\bar x)T^m. }
|Definition passage · 18 passages
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Definition passage: Directed seed rank · Read passage · Paper_X_Block27_Amended.tex, line 668
[Directed seed rank] For a directed transport graph G, define srank_G = min{|S|:Cl_G(S)=V(G)}, where Cl_G(S) is the set of vertices reachable from S.
Exact mathematical passage
\srank_G = \min\{|S|:\Cl_G(S)=V(G)\},
\lambdaDefinition passage · 40 passages
Search aliases: λ lambda
Definition passage: Common arithmetic spine · Read passage · Paper_X_Block27_Amended.tex, line 769
efinition[Common arithmetic spine] Let M be a number field. A common spine C consists of: item characteristic polynomials P_v(X)∈ M[X] at almost all places; item coefficient-prime independent weight and determinant data; item prescribed local Weil--Deligne or inertial packet labels at the ramified places. For each finite place λ of M, let X_λ,m( C) be the finite set of admissible objects modulo ϖ_λ^m realizing the reduction of this spine.
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\lambda
\varpiDefinition passage · 30 passages
Search aliases: ϖ varpi
Definition passage: Visibility weight · Read passage · Paper_X_Block27_Amended.tex, line 477
[Visibility weight] For two integral packet coordinates a,b∈ O, define their separation depth w(a,b)=min{m≥1:anot≡ bpmodϖ^m}, when such an m exists. A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.
Exact mathematical passage
w(a,b)=\min\{m\ge1:a\not\equiv b\pmod{\varpi^m}\},
\varpi^mDefinition passage · 14 passages
Search aliases: ϖ^m varpi
Definition passage: Visibility weight · Read passage · Paper_X_Block27_Amended.tex, line 477
[Visibility weight] For two integral packet coordinates a,b∈ O, define their separation depth w(a,b)=min{m≥1:anot≡ bpmodϖ^m}, when such an m exists. A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.
Exact mathematical passage
w(a,b)=\min\{m\ge1:a\not\equiv b\pmod{\varpi^m}\},
\varpi_\lambda^mDefinition passage · 1 passage
Search aliases: ϖ_λ^m varpi lambda
Definition passage: Common arithmetic spine · Read passage · Paper_X_Block27_Amended.tex, line 773
at consists of: item characteristic polynomials P_v(X)∈ M[X] at almost all places; item coefficient-prime independent weight and determinant data; item prescribed local Weil--Deligne or inertial packet labels at the ramified places. For each finite place λ of M, let X_λ,m( C) be the finite set of admissible objects modulo ϖ_λ^m realizing the reduction of this spine.
Exact mathematical passage
\varpi_\lambda^m
\toDefinition passage · 39 passages
Search aliases: → to
Definition passage: Branching dimension · Read passage · Paper_X_Block27_Amended.tex, line 613
[Branching dimension] When the limit exists, define the finite-level counting exponent dim_seed(bar x) = lim_m→∞ fraclog_q#X_m(bar x)m. A limsup may be used in general.
Exact mathematical passage
\dim_{\mathrm{seed}}(\bar x) = \lim_{m\to\infty} \frac{\log_q\#X_m(\bar x)}{m}.
\inDefinition passage · 90 passages
Search aliases: ∈ in
Definition passage: Residual seed system · Read passage · Paper_X_Block27_Amended.tex, line 162
[Residual seed system] For bar x∈ M(k), define the residual fiber M_bar x(A) = {x∈ M(A):red_A(x)=bar x}. The point bar x is the residual root. The family of all nonempty fibers is the residual special fiber, and the formal deformation problems above them form the residual forest.
Exact mathematical passage
\bar x\in\mathcal M(k)
\sumDefinition passage · 30 passages
Search aliases: ∑ sum
Definition passage: Seed zeta function · Read passage · Paper_X_Block27_Amended.tex, line 532
[Seed zeta function] For a residual root bar x with finite levels, define Z_bar x(T) = ∑_m≥1#X_m(bar x)T^m. When a weighted chart is specified, write Z_ w(T).
Exact mathematical passage
\boxed{ Z_{\bar x}(T) = \sum_{m\ge1}\#X_m(\bar x)T^m. }
\sum_{m\ge1}Definition passage · 3 passages
Search aliases: ∑_m≥1 sum ge
Definition passage: Seed zeta function · Read passage · Paper_X_Block27_Amended.tex, line 532
[Seed zeta function] For a residual root bar x with finite levels, define Z_bar x(T) = ∑_m≥1#X_m(bar x)T^m. When a weighted chart is specified, write Z_ w(T).
Exact mathematical passage
\boxed{ Z_{\bar x}(T) = \sum_{m\ge1}\#X_m(\bar x)T^m. }
\inftyDefinition passage · 5 passages
Search aliases: ∞ infty
Definition passage: Branching dimension · Read passage · Paper_X_Block27_Amended.tex, line 613
[Branching dimension] When the limit exists, define the finite-level counting exponent dim_seed(bar x) = lim_m→∞ fraclog_q#X_m(bar x)m. A limsup may be used in general.
Exact mathematical passage
\dim_{\mathrm{seed}}(\bar x) = \lim_{m\to\infty} \frac{\log_q\#X_m(\bar x)}{m}.
\equivDefinition passage · 31 passages
Search aliases: ≡ equiv
Definition passage: Visibility weight · Read passage · Paper_X_Block27_Amended.tex, line 477
[Visibility weight] For two integral packet coordinates a,b∈ O, define their separation depth w(a,b)=min{m≥1:anot≡ bpmodϖ^m}, when such an m exists. A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.
Exact mathematical passage
w(a,b)=\min\{m\ge1:a\not\equiv b\pmod{\varpi^m}\},
\geDefinition passage · 30 passages
Search aliases: ≥ ge
Definition passage: Visibility weight · Read passage · Paper_X_Block27_Amended.tex, line 477
[Visibility weight] For two integral packet coordinates a,b∈ O, define their separation depth w(a,b)=min{m≥1:anot≡ bpmodϖ^m}, when such an m exists. A dyadic packet decomposition is weighted by these first visibility depths and by the valuations of any denominators required by its rational projectors.
Exact mathematical passage
w(a,b)=\min\{m\ge1:a\not\equiv b\pmod{\varpi^m}\},
\longrightarrowDefinition passage · 12 passages
Search aliases: ⟶ longrightarrow
Definition passage: Admissible residual transport · Read passage · Paper_X_Block27_Amended.tex, line 658
[Admissible residual transport] An admissible transport from barρ to barρ' consists of T:barρ⟼barρ' together with natural maps T_A:Def_barρ(A)⟶Def_barρ'(A) for every coefficient algebra A, compatible with base change and reduction. When the transport is reversible, it is an isomorphism in the residual transport groupoid. One-way operations give a directed category.
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\mathcal T_A:\Def_{\bar\rho}(A)\longrightarrow\Def_{\bar\rho'}(A)
\longmapstoDefinition passage · 3 passages
Search aliases: ⟼ longmapsto
Definition passage: Admissible residual transport · Read passage · Paper_X_Block27_Amended.tex, line 654
[Admissible residual transport] An admissible transport from barρ to barρ' consists of T:barρ⟼barρ' together with natural maps T_A:Def_barρ(A)⟶Def_barρ'(A) for every coefficient algebra A, compatible with base change and reduction. When the transport is reversible, it is an isomorphism in the residual transport groupoid. One-way operations give a directed category.
Exact mathematical passage
\mathcal T:\bar\rho\longmapsto\bar\rho'
\mathcal AIntroduction passage · 7 passages
Search aliases: A mathcal
Introduction passage · Read passage · Paper_X_Block27_Amended.tex, line 1258
General finite-field Frobenius comparison (Block 27). Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a nonzero maximal ideal above (p), and the induced residue map is injective.
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\mathcal A
A^2Introduction passage · 6 passages
Search aliases: A^2
Introduction passage · Read passage · Paper_X_Block27_Amended.tex, line 2103
To prove this, apply the preceding averaging correction with both representations equal to ρ. This produces an exact equivariant matrix A with the prescribed reduction at depth k, so that A^2-A∈(π)^kM_n(R). Averaging alone need not preserve idempotence. Use the integral polynomial f(A)=3A^2-2A^3. The identities f(A)-A=(1-2A)(A^2-A), f(A)^2-f(A)=(A^2-A)^2(4A^2-4A-3) show that successive corrections preserve the retained reduction and improve the idempotence defect. In these formulas scalar consta…
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A^2-A\in(\pi)^kM_n(R)
A^3Introduction passage · 1 passage
Search aliases: A^3
Introduction passage · Read passage · Paper_X_Block27_Amended.tex, line 2105
To prove this, apply the preceding averaging correction with both representations equal to ρ. This produces an exact equivariant matrix A with the prescribed reduction at depth k, so that A^2-A∈(π)^kM_n(R). Averaging alone need not preserve idempotence. Use the integral polynomial f(A)=3A^2-2A^3. The identities f(A)-A=(1-2A)(A^2-A), f(A)^2-f(A)=(A^2-A)^2(4A^2-4A-3) show that successive corrections preserve the retained reduction and improve the idempotence defect. In these formulas scalar consta…
Exact mathematical passage
f(A)=3A^2-2A^3.
\absnIntroduction passage · 8 passages
Search aliases: absn
Introduction passage · Read passage · residue_continuation_body.tex, line 23
Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding counting block. Its norm function scales by t^d under a positive real dilation t. The full-paper ledger remains 17 checked claims and 3 partial claims.
Exact mathematical passage
\Nm(x)=\absn{n(x)}
\bmodIntroduction passage · 20 passages
Search aliases: bmod
Introduction passage · Read passage · residue_continuation_body.tex, line 118
· =#μ(K) #{I≠0:[I]=C, N I≤ X, P( N I N J)}. The same norm-preserving equivalence carries arbitrary complex weights. In particular, write G_J,χ(Y) for the sum of χ( N(x)bmod m) over these generators with N(x)≤ Y, and T_C,χ(X) for the corresponding sum over ideals in class C. For multiplicative χ preserving 0 and 1, the exact identity is G_J,χ(X N J) =#μ(K) χ( N Jbmod m) T_C,χ(X). Both signs are included before applying this full roots-of-unity factor. A root of unity can interchange the sign port…
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\chi(\Nm(x)\bmod m)
CIntroduction passage · 70 passages
Search aliases: C
Introduction passage · Read passage · residue_continuation_body.tex, line 108
Let [J]=C^-1 in Cl( O), and let G(J) denote the actual fundamental-cone generator set. For every predicate P: R→Prop and every real X, the norm-preserving correspondence gives · #{x∈ G(J): N(x)≤ X N J, P( N(x))}
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[J]=C^{-1}
cIntroduction passage · 23 passages
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Introduction passage · Read passage · conditional_completion.tex, line 15
Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality of numerical traces. The selected family ⊕_iρ_i,λ^⊕ m_i has a split equivariant inclusion in the carrier. The finite trace sample uniquely determines…
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c_i
C^{-1}Introduction passage · 5 passages
Search aliases: C^-1
Introduction passage · Read passage · residue_continuation_body.tex, line 108
Let [J]=C^-1 in Cl( O), and let G(J) denote the actual fundamental-cone generator set. For every predicate P: R→Prop and every real X, the norm-preserving correspondence gives · #{x∈ G(J): N(x)≤ X N J, P( N(x))}
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[J]=C^{-1}
c_iIntroduction passage · 10 passages
Search aliases: c_i
Introduction passage · Read passage · conditional_completion.tex, line 15
Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality of numerical traces. The selected family ⊕_iρ_i,λ^⊕ m_i has a split equivariant inclusion in the carrier. The finite trace sample uniquely determines…
Exact mathematical passage
c_i
\mathrm{car}Introduction passage · 1 passage
Search aliases: car mathrm
Introduction passage · Read passage · conditional_completion.tex, line 16
Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality of numerical traces. The selected family ⊕_iρ_i,λ^⊕ m_i has a split equivariant inclusion in the carrier. The finite trace sample uniquely determines…
Exact mathematical passage
V_{\mathrm{car},\lambda}\simeq\bigoplus_i\rho_{i,\lambda}^{\oplus c_i}, \qquad U_{0,\lambda}\simeq\bigoplus_i\rho_{i,\lambda}^{\oplus m_i}.
DIntroduction passage · 33 passages
Search aliases: D
Introduction passage · Read passage · residue_continuation_body.tex, line 24
Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding counting block. Its norm function scales by t^d under a positive real dilation t. The full-paper ledger remains 17 checked claims and 3 partial claims.
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D
dIntroduction passage · 79 passages
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Introduction passage · Read passage · residue_continuation_body.tex, line 20
Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding counting block. Its norm function scales by t^d under a positive real dilation t. The full-paper ledger remains 17 checked claims and 3 partial claims.
Exact mathematical passage
d\geq1
EIntroduction passage · 55 passages
Search aliases: E
Introduction passage · Read passage · Paper_X_Block27_Amended.tex, line 2117
qquad f(A)^2-f(A)=(A^2-A)^2(4A^2-4A-3) show that successive corrections preserve the retained reduction and improve the idempotence defect. In these formulas scalar constants denote scalar matrices. The Lean proof establishes at least one additional ideal digit at each step, constructs the resulting compatible family of finite-level matrices, and uses completeness and separatedness to recover E. Every matrix commuting with A commutes with E. There is no division by 2, so the correction also work…
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E
fIntroduction passage · 19 passages
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Introduction passage · Read passage · Paper_X_Block27_Amended.tex, line 2105
To prove this, apply the preceding averaging correction with both representations equal to ρ. This produces an exact equivariant matrix A with the prescribed reduction at depth k, so that A^2-A∈(π)^kM_n(R). Averaging alone need not preserve idempotence. Use the integral polynomial f(A)=3A^2-2A^3. The identities f(A)-A=(1-2A)(A^2-A), f(A)^2-f(A)=(A^2-A)^2(4A^2-4A-3) show that successive corrections preserve the retained reduction and improve the idempotence defect. In these formulas scalar consta…
Exact mathematical passage
f(A)=3A^2-2A^3.
\mathcal GIntroduction passage · 2 passages
Search aliases: G mathcal
Introduction passage · Read passage · residue_continuation_body.tex, line 108
Let [J]=C^-1 in Cl( O), and let G(J) denote the actual fundamental-cone generator set. For every predicate P: R→Prop and every real X, the norm-preserving correspondence gives · #{x∈ G(J): N(x)≤ X N J, P( N(x))}
Exact mathematical passage
\mathcal G(J)
G_{J,\chi}Introduction passage · 3 passages
Search aliases: G_J,χ chi
Introduction passage · Read passage · residue_continuation_body.tex, line 117
· =#μ(K) #{I≠0:[I]=C, N I≤ X, P( N I N J)}. The same norm-preserving equivalence carries arbitrary complex weights. In particular, write G_J,χ(Y) for the sum of χ( N(x)bmod m) over these generators with N(x)≤ Y, and T_C,χ(X) for the corresponding sum over ideals in class C. For multiplicative χ preserving 0 and 1, the exact identity is G_J,χ(X N J) =#μ(K) χ( N Jbmod m) T_C,χ(X). Both signs are included before applying this full roots-of-unity factor. A root of unity can interchange the sign port…
Exact mathematical passage
G_{J,\chi}(Y)
G_{\Q}Introduction passage · 10 passages
Search aliases: G_Q G_ Q Q mathbf
Introduction passage · Read passage · Paper_X_Block27_Amended.tex, line 2402
Block 21 completes this restricted arithmetic interpretation and its conditional completion. The packet is defined by one fixed quotient; it is not the collection of all quadratic characters of G_ Q. No local ramification criterion, Hodge--Tate weight, Weil--Deligne attachment, or cross-coefficient local invariant was established by Block 21. The full manuscript still requires its remaining packet identifications and arithmetic comparisons. At the Block 21 checkpoint, the ledger recorded fifteen…
Exact mathematical passage
G_{\Q}
IIntroduction passage · 59 passages
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Introduction passage · Read passage · residue_continuation_body.tex, line 21
Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding counting block. Its norm function scales by t^d under a positive real dilation t. The full-paper ledger remains 17 checked claims and 3 partial claims.
Exact mathematical passage
\Nm I=\#(\OO/I)
iIntroduction passage · 111 passages
Search aliases: i
Introduction passage · Read passage · conditional_completion.tex, line 15
Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality of numerical traces. The selected family ⊕_iρ_i,λ^⊕ m_i has a split equivariant inclusion in the carrier. The finite trace sample uniquely determines…
Exact mathematical passage
\rho_i
JIntroduction passage · 42 passages
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Introduction passage · Read passage · residue_continuation_body.tex, line 24
Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding counting block. Its norm function scales by t^d under a positive real dilation t. The full-paper ledger remains 17 checked claims and 3 partial claims.
Exact mathematical passage
L(J)=\iota(J)
jIntroduction passage · 47 passages
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Introduction passage · Read passage · Paper_X_Block27_Amended.tex, line 1258
General finite-field Frobenius comparison (Block 27). Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a nonzero maximal ideal above (p), and the induced residue map is injective.
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j:L\hookrightarrow\overline{\Q}
KIntroduction passage · 45 passages
Search aliases: K
Introduction passage · Read passage · residue_continuation_body.tex, line 20
Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding counting block. Its norm function scales by t^d under a positive real dilation t. The full-paper ledger remains 17 checked claims and 3 partial claims.
Exact mathematical passage
K
LIntroduction passage · 45 passages
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Introduction passage · Read passage · Paper_X_Block27_Amended.tex, line 1257
General finite-field Frobenius comparison (Block 27). Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a nonzero maximal ideal above (p), and the induced residue map is injective.
Exact mathematical passage
L/\Q
m_iIntroduction passage · 19 passages
Search aliases: m_i
Introduction passage · Read passage · conditional_completion.tex, line 15
Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality of numerical traces. The selected family ⊕_iρ_i,λ^⊕ m_i has a split equivariant inclusion in the carrier. The finite trace sample uniquely determines…
Exact mathematical passage
m_i
M_nIntroduction passage · 3 passages
Search aliases: M_n
Introduction passage · Read passage · Paper_X_Block27_Amended.tex, line 2103
To prove this, apply the preceding averaging correction with both representations equal to ρ. This produces an exact equivariant matrix A with the prescribed reduction at depth k, so that A^2-A∈(π)^kM_n(R). Averaging alone need not preserve idempotence. Use the integral polynomial f(A)=3A^2-2A^3. The identities f(A)-A=(1-2A)(A^2-A), f(A)^2-f(A)=(A^2-A)^2(4A^2-4A-3) show that successive corrections preserve the retained reduction and improve the idempotence defect. In these formulas scalar consta…
Exact mathematical passage
A^2-A\in(\pi)^kM_n(R)
nIntroduction passage · 61 passages
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Introduction passage · Read passage · Paper_X_Block27_Amended.tex, line 2103
To prove this, apply the preceding averaging correction with both representations equal to ρ. This produces an exact equivariant matrix A with the prescribed reduction at depth k, so that A^2-A∈(π)^kM_n(R). Averaging alone need not preserve idempotence. Use the integral polynomial f(A)=3A^2-2A^3. The identities f(A)-A=(1-2A)(A^2-A), f(A)^2-f(A)=(A^2-A)^2(4A^2-4A-3) show that successive corrections preserve the retained reduction and improve the idempotence defect. In these formulas scalar consta…
Exact mathematical passage
A^2-A\in(\pi)^kM_n(R)
\NmIntroduction passage · 17 passages
Search aliases: Nm N operatorname
Introduction passage · Read passage · residue_continuation_body.tex, line 21
Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding counting block. Its norm function scales by t^d under a positive real dilation t. The full-paper ledger remains 17 checked claims and 3 partial claims.
Exact mathematical passage
\Nm I=\#(\OO/I)
\Nm_{K/\mathbb Q}Introduction passage · 1 passage
Search aliases: Nm_K/ Q N_K/ Q Nm mathbb operatorname mathbb
Introduction passage · Read passage · residue_continuation_body.tex, line 22
Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding counting block. Its norm function scales by t^d under a positive real dilation t. The full-paper ledger remains 17 checked claims and 3 partial claims.
Exact mathematical passage
n(x)=\Nm_{K/\mathbb Q}(x)\in\ZZ
\mathcal OIntroduction passage · 7 passages
Search aliases: O mathcal
Introduction passage · Read passage · Paper_X_Block27_Amended.tex, line 1260
General finite-field Frobenius comparison (Block 27). Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a nonzero maximal ideal above (p), and the induced residue map is injective.
Exact mathematical passage
\mathcal O_L\to\mathcal A
\mathcal O_LIntroduction passage · 7 passages
Search aliases: O_L mathcal
Introduction passage · Read passage · Paper_X_Block27_Amended.tex, line 1260
General finite-field Frobenius comparison (Block 27). Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a nonzero maximal ideal above (p), and the induced residue map is injective.
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\mathcal O_L\to\mathcal A
\OOIntroduction passage · 16 passages
Search aliases: OO OO O mathcal
Introduction passage · Read passage · residue_continuation_body.tex, line 21
Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding counting block. Its norm function scales by t^d under a positive real dilation t. The full-paper ledger remains 17 checked claims and 3 partial claims.
Exact mathematical passage
\OO=\OO_K
\OO_KIntroduction passage · 1 passage
Search aliases: OO_K O_K OO mathcal
Introduction passage · Read passage · residue_continuation_body.tex, line 21
Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding counting block. Its norm function scales by t^d under a positive real dilation t. The full-paper ledger remains 17 checked claims and 3 partial claims.
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\OO=\OO_K
\overline{\Q}Introduction passage · 7 passages
Search aliases: overlineQ overline Q overline mathbf
Introduction passage · Read passage · Paper_X_Block27_Amended.tex, line 1258
General finite-field Frobenius comparison (Block 27). Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a nonzero maximal ideal above (p), and the induced residue map is injective.
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j:L\hookrightarrow\overline{\Q}
\mathfrak PIntroduction passage · 5 passages
Search aliases: P mathfrak
Introduction passage · Read passage · Paper_X_Block27_Amended.tex, line 1262
arison (Block 27). Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a nonzero maximal ideal above (p), and the induced residue map is injective.
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\mathfrak P\subset\mathcal O_L
pIntroduction passage · 107 passages
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Introduction passage · Read passage · Paper_X_Block27_Amended.tex, line 1261
te-field Frobenius comparison (Block 27). Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a nonzero maximal ideal above (p), and the induced residue map is injective.
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(p)
\mathrm{Prop}Introduction passage · 1 passage
Search aliases: Prop mathrm
Introduction passage · Read passage · residue_continuation_body.tex, line 109
Let [J]=C^-1 in Cl( O), and let G(J) denote the actual fundamental-cone generator set. For every predicate P: R→Prop and every real X, the norm-preserving correspondence gives · #{x∈ G(J): N(x)≤ X N J, P( N(x))}
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P:\RR\to\mathrm{Prop}
\QIntroduction passage · 27 passages
Search aliases: Q mathbf
Introduction passage · Read passage · Paper_X_Block27_Amended.tex, line 1257
General finite-field Frobenius comparison (Block 27). Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a nonzero maximal ideal above (p), and the induced residue map is injective.
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L/\Q
\mathbb QIntroduction passage · 10 passages
Search aliases: Q mathbb
Introduction passage · Read passage · residue_continuation_body.tex, line 22
Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding counting block. Its norm function scales by t^d under a positive real dilation t. The full-paper ledger remains 17 checked claims and 3 partial claims.
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n(x)=\Nm_{K/\mathbb Q}(x)\in\ZZ
\mathfrak QIntroduction passage · 9 passages
Search aliases: Q mathfrak
Introduction passage · Read passage · Paper_X_Block27_Amended.tex, line 1261
General finite-field Frobenius comparison (Block 27). Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a nonzero maximal ideal above (p), and the induced residue map is injective.
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\mathfrak Q\subset\mathcal A
RIntroduction passage · 49 passages
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Introduction passage · Read passage · Paper_X_Block27_Amended.tex, line 2103
To prove this, apply the preceding averaging correction with both representations equal to ρ. This produces an exact equivariant matrix A with the prescribed reduction at depth k, so that A^2-A∈(π)^kM_n(R). Averaging alone need not preserve idempotence. Use the integral polynomial f(A)=3A^2-2A^3. The identities f(A)-A=(1-2A)(A^2-A), f(A)^2-f(A)=(A^2-A)^2(4A^2-4A-3) show that successive corrections preserve the retained reduction and improve the idempotence defect. In these formulas scalar consta…
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A^2-A\in(\pi)^kM_n(R)
\RRIntroduction passage · 1 passage
Search aliases: RR R mathbb
Introduction passage · Read passage · residue_continuation_body.tex, line 109
Let [J]=C^-1 in Cl( O), and let G(J) denote the actual fundamental-cone generator set. For every predicate P: R→Prop and every real X, the norm-preserving correspondence gives · #{x∈ G(J): N(x)≤ X N J, P( N(x))}
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P:\RR\to\mathrm{Prop}
tIntroduction passage · 34 passages
Search aliases: t
Introduction passage · Read passage · residue_continuation_body.tex, line 26
egree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding counting block. Its norm function scales by t^d under a positive real dilation t. The full-paper ledger remains 17 checked claims and 3 partial claims.
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t^d
t^dIntroduction passage · 5 passages
Search aliases: t^d
Introduction passage · Read passage · residue_continuation_body.tex, line 26
egree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding counting block. Its norm function scales by t^d under a positive real dilation t. The full-paper ledger remains 17 checked claims and 3 partial claims.
Exact mathematical passage
t^d
T_{C,\chi}Introduction passage · 3 passages
Search aliases: T_C,χ chi
Introduction passage · Read passage · residue_continuation_body.tex, line 119
· =#μ(K) #{I≠0:[I]=C, N I≤ X, P( N I N J)}. The same norm-preserving equivalence carries arbitrary complex weights. In particular, write G_J,χ(Y) for the sum of χ( N(x)bmod m) over these generators with N(x)≤ Y, and T_C,χ(X) for the corresponding sum over ideals in class C. For multiplicative χ preserving 0 and 1, the exact identity is G_J,χ(X N J) =#μ(K) χ( N Jbmod m) T_C,χ(X). Both signs are included before applying this full roots-of-unity factor. A root of unity can interchange the sign port…
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T_{C,\chi}(X)
UIntroduction passage · 31 passages
Search aliases: U
Introduction passage · Read passage · conditional_completion.tex, line 16
Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality of numerical traces. The selected family ⊕_iρ_i,λ^⊕ m_i has a split equivariant inclusion in the carrier. The finite trace sample uniquely determines…
Exact mathematical passage
V_{\mathrm{car},\lambda}\simeq\bigoplus_i\rho_{i,\lambda}^{\oplus c_i}, \qquad U_{0,\lambda}\simeq\bigoplus_i\rho_{i,\lambda}^{\oplus m_i}.
U_{0,\lambda}Introduction passage · 1 passage
Search aliases: U_0,λ lambda
Introduction passage · Read passage · conditional_completion.tex, line 16
Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality of numerical traces. The selected family ⊕_iρ_i,λ^⊕ m_i has a split equivariant inclusion in the carrier. The finite trace sample uniquely determines…
Exact mathematical passage
V_{\mathrm{car},\lambda}\simeq\bigoplus_i\rho_{i,\lambda}^{\oplus c_i}, \qquad U_{0,\lambda}\simeq\bigoplus_i\rho_{i,\lambda}^{\oplus m_i}.
V_{\mathrm{car},\lambda}Introduction passage · 1 passage
Search aliases: V_car,λ mathrm lambda
Introduction passage · Read passage · conditional_completion.tex, line 16
Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality of numerical traces. The selected family ⊕_iρ_i,λ^⊕ m_i has a split equivariant inclusion in the carrier. The finite trace sample uniquely determines…
Exact mathematical passage
V_{\mathrm{car},\lambda}\simeq\bigoplus_i\rho_{i,\lambda}^{\oplus c_i}, \qquad U_{0,\lambda}\simeq\bigoplus_i\rho_{i,\lambda}^{\oplus m_i}.
YIntroduction passage · 6 passages
Search aliases: Y
Introduction passage · Read passage · residue_continuation_body.tex, line 117
· =#μ(K) #{I≠0:[I]=C, N I≤ X, P( N I N J)}. The same norm-preserving equivalence carries arbitrary complex weights. In particular, write G_J,χ(Y) for the sum of χ( N(x)bmod m) over these generators with N(x)≤ Y, and T_C,χ(X) for the corresponding sum over ideals in class C. For multiplicative χ preserving 0 and 1, the exact identity is G_J,χ(X N J) =#μ(K) χ( N Jbmod m) T_C,χ(X). Both signs are included before applying this full roots-of-unity factor. A root of unity can interchange the sign port…
Exact mathematical passage
G_{J,\chi}(Y)
\mathbb ZIntroduction passage · 18 passages
Search aliases: Z mathbb
Introduction passage · Read passage · Paper_X_Block27_Amended.tex, line 1259
General finite-field Frobenius comparison (Block 27). Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a nonzero maximal ideal above (p), and the induced residue map is injective.
Exact mathematical passage
\mathbb Z
\ZZIntroduction passage · 25 passages
Search aliases: ZZ ZZ Z mathbb
Introduction passage · Read passage · residue_continuation_body.tex, line 22
Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding counting block. Its norm function scales by t^d under a positive real dilation t. The full-paper ledger remains 17 checked claims and 3 partial claims.
Exact mathematical passage
n(x)=\Nm_{K/\mathbb Q}(x)\in\ZZ
\iotaIntroduction passage · 4 passages
Search aliases: ι iota
Introduction passage · Read passage · residue_continuation_body.tex, line 23
Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding counting block. Its norm function scales by t^d under a positive real dilation t. The full-paper ledger remains 17 checked claims and 3 partial claims.
Exact mathematical passage
\iota
\muIntroduction passage · 6 passages
Search aliases: μ mu
Introduction passage · Read passage · residue_continuation_body.tex, line 111
Let [J]=C^-1 in Cl( O), and let G(J) denote the actual fundamental-cone generator set. For every predicate P: R→Prop and every real X, the norm-preserving correspondence gives · #{x∈ G(J): N(x)≤ X N J, P( N(x))} · =#μ(K) #{I≠0:[I]=C, N I≤ X, P( N I N J)}. The same norm-preserving equivalence carries arbitrary complex weights. In particular, write G_J,χ(Y) for the sum of χ( N(x)bmod m) over these generators with N(x)≤ Y, and T_C,χ(X) for the corresponding sum over ideals in class C. For multiplic…
Exact mathematical passage
&\#\{x\in\mathcal G(J):\Nm(x)\leq X\Nm J,\ P(\Nm(x))\}\\ &\qquad=\#\mu(K)\, \#\{I\ne0:[I]=C,\ \Nm I\leq X,\ P(\Nm I\,\Nm J)\}.
\piIntroduction passage · 29 passages
Search aliases: π pi
Introduction passage · Read passage · Paper_X_Block27_Amended.tex, line 2103
To prove this, apply the preceding averaging correction with both representations equal to ρ. This produces an exact equivariant matrix A with the prescribed reduction at depth k, so that A^2-A∈(π)^kM_n(R). Averaging alone need not preserve idempotence. Use the integral polynomial f(A)=3A^2-2A^3. The identities f(A)-A=(1-2A)(A^2-A), f(A)^2-f(A)=(A^2-A)^2(4A^2-4A-3) show that successive corrections preserve the retained reduction and improve the idempotence defect. In these formulas scalar consta…
Exact mathematical passage
A^2-A\in(\pi)^kM_n(R)
\rhoIntroduction passage · 33 passages
Search aliases: ρ rho
Introduction passage · Read passage · Paper_X_Block27_Amended.tex, line 2101
To prove this, apply the preceding averaging correction with both representations equal to ρ. This produces an exact equivariant matrix A with the prescribed reduction at depth k, so that A^2-A∈(π)^kM_n(R). Averaging alone need not preserve idempotence. Use the integral polynomial f(A)=3A^2-2A^3. The identities f(A)-A=(1-2A)(A^2-A), f(A)^2-f(A)=(A^2-A)^2(4A^2-4A-3) show that successive corrections preserve the retained reduction and improve the idempotence defect. In these formulas scalar consta…
Exact mathematical passage
\rho
\rho_iIntroduction passage · 3 passages
Search aliases: ρ_i rho
Introduction passage · Read passage · conditional_completion.tex, line 15
Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality of numerical traces. The selected family ⊕_iρ_i,λ^⊕ m_i has a split equivariant inclusion in the carrier. The finite trace sample uniquely determines…
Exact mathematical passage
\rho_i
\rho_{i,\lambda}^{\oplus c_i}Introduction passage · 1 passage
Search aliases: ρ_i,λ^⊕ c_i rho lambda oplus
Introduction passage · Read passage · conditional_completion.tex, line 16
Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality of numerical traces. The selected family ⊕_iρ_i,λ^⊕ m_i has a split equivariant inclusion in the carrier. The finite trace sample uniquely determines…
Exact mathematical passage
V_{\mathrm{car},\lambda}\simeq\bigoplus_i\rho_{i,\lambda}^{\oplus c_i}, \qquad U_{0,\lambda}\simeq\bigoplus_i\rho_{i,\lambda}^{\oplus m_i}.
\rho_{i,\lambda}^{\oplus m_i}Introduction passage · 2 passages
Search aliases: ρ_i,λ^⊕ m_i rho lambda oplus
Introduction passage · Read passage · conditional_completion.tex, line 16
Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality of numerical traces. The selected family ⊕_iρ_i,λ^⊕ m_i has a split equivariant inclusion in the carrier. The finite trace sample uniquely determines…
Exact mathematical passage
V_{\mathrm{car},\lambda}\simeq\bigoplus_i\rho_{i,\lambda}^{\oplus c_i}, \qquad U_{0,\lambda}\simeq\bigoplus_i\rho_{i,\lambda}^{\oplus m_i}.
\chiIntroduction passage · 43 passages
Search aliases: χ chi
Introduction passage · Read passage · residue_continuation_body.tex, line 117
· =#μ(K) #{I≠0:[I]=C, N I≤ X, P( N I N J)}. The same norm-preserving equivalence carries arbitrary complex weights. In particular, write G_J,χ(Y) for the sum of χ( N(x)bmod m) over these generators with N(x)≤ Y, and T_C,χ(X) for the corresponding sum over ideals in class C. For multiplicative χ preserving 0 and 1, the exact identity is G_J,χ(X N J) =#μ(K) χ( N Jbmod m) T_C,χ(X). Both signs are included before applying this full roots-of-unity factor. A root of unity can interchange the sign port…
Exact mathematical passage
G_{J,\chi}(Y)
\hookrightarrowIntroduction passage · 6 passages
Search aliases: ↪ hookrightarrow
Introduction passage · Read passage · Paper_X_Block27_Amended.tex, line 1258
General finite-field Frobenius comparison (Block 27). Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a nonzero maximal ideal above (p), and the induced residue map is injective.
Exact mathematical passage
j:L\hookrightarrow\overline{\Q}
\simeqIntroduction passage · 9 passages
Search aliases: ≃ simeq
Introduction passage · Read passage · conditional_completion.tex, line 16
Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality of numerical traces. The selected family ⊕_iρ_i,λ^⊕ m_i has a split equivariant inclusion in the carrier. The finite trace sample uniquely determines…
Exact mathematical passage
V_{\mathrm{car},\lambda}\simeq\bigoplus_i\rho_{i,\lambda}^{\oplus c_i}, \qquad U_{0,\lambda}\simeq\bigoplus_i\rho_{i,\lambda}^{\oplus m_i}.
\neIntroduction passage · 39 passages
Search aliases: ≠ ne
Introduction passage · Read passage · residue_continuation_body.tex, line 111
Let [J]=C^-1 in Cl( O), and let G(J) denote the actual fundamental-cone generator set. For every predicate P: R→Prop and every real X, the norm-preserving correspondence gives · #{x∈ G(J): N(x)≤ X N J, P( N(x))} · =#μ(K) #{I≠0:[I]=C, N I≤ X, P( N I N J)}. The same norm-preserving equivalence carries arbitrary complex weights. In particular, write G_J,χ(Y) for the sum of χ( N(x)bmod m) over these generators with N(x)≤ Y, and T_C,χ(X) for the corresponding sum over ideals in class C. For multiplic…
Exact mathematical passage
&\#\{x\in\mathcal G(J):\Nm(x)\leq X\Nm J,\ P(\Nm(x))\}\\ &\qquad=\#\mu(K)\, \#\{I\ne0:[I]=C,\ \Nm I\leq X,\ P(\Nm I\,\Nm J)\}.
\leqIntroduction passage · 9 passages
Search aliases: ≤ leq
Introduction passage · Read passage · conditional_completion.tex, line 15
Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality of numerical traces. The selected family ⊕_iρ_i,λ^⊕ m_i has a split equivariant inclusion in the carrier. The finite trace sample uniquely determines…
Exact mathematical passage
m_i\leq c_i
\geqIntroduction passage · 9 passages
Search aliases: ≥ geq
Introduction passage · Read passage · residue_continuation_body.tex, line 20
Scope and notation. Let K be a number field of degree d≥1, with ring of integers O= O_K. Write N I=#( O/I) for the absolute norm of a nonzero integral ideal and n(x)= N_K/ Q(x)∈ Z for x∈ O. Thus N(x)=absnn(x). Let ι be the mixed embedding in its Euclidean coordinates, L(J)=ι(J), and D the actual unit-norm fundamental cone used in the preceding counting block. Its norm function scales by t^d under a positive real dilation t. The full-paper ledger remains 17 checked claims and 3 partial claims.
Exact mathematical passage
d\geq1
\subsetIntroduction passage · 5 passages
Search aliases: ⊂ subset
Introduction passage · Read passage · Paper_X_Block27_Amended.tex, line 1261
General finite-field Frobenius comparison (Block 27). Let L/ Q now be any finite normal number field, with a chosen embedding j:L↪overline Q. Write A for the integral closure of Z in overline Q. The embedding restricts to an injective map O_L→ A which intertwines the actual Galois actions. If Q⊂ A is maximal above (p), its contraction P⊂ O_L is a nonzero maximal ideal above (p), and the induced residue map is injective.
Exact mathematical passage
\mathfrak Q\subset\mathcal A
\bigoplusIntroduction passage · 10 passages
Search aliases: ⊕ bigoplus
Introduction passage · Read passage · conditional_completion.tex, line 16
Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality of numerical traces. The selected family ⊕_iρ_i,λ^⊕ m_i has a split equivariant inclusion in the carrier. The finite trace sample uniquely determines…
Exact mathematical passage
V_{\mathrm{car},\lambda}\simeq\bigoplus_i\rho_{i,\lambda}^{\oplus c_i}, \qquad U_{0,\lambda}\simeq\bigoplus_i\rho_{i,\lambda}^{\oplus m_i}.
\oplusIntroduction passage · 9 passages
Search aliases: ⊕ oplus
Introduction passage · Read passage · conditional_completion.tex, line 16
Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality of numerical traces. The selected family ⊕_iρ_i,λ^⊕ m_i has a split equivariant inclusion in the carrier. The finite trace sample uniquely determines…
Exact mathematical passage
V_{\mathrm{car},\lambda}\simeq\bigoplus_i\rho_{i,\lambda}^{\oplus c_i}, \qquad U_{0,\lambda}\simeq\bigoplus_i\rho_{i,\lambda}^{\oplus m_i}.
\bigoplus_iIntroduction passage · 4 passages
Search aliases: ⊕_i bigoplus
Introduction passage · Read passage · conditional_completion.tex, line 16
Write ρ_i for the supplied constituents, c_i for the carrier multiplicities and m_i for the selected multiplicities, with m_i≤ c_i. At a distinguished coefficient place λ, the input presentations have the form V_car,λ≃⊕_iρ_i,λ^⊕ c_i, U_0,λ≃⊕_iρ_i,λ^⊕ m_i. These are actual equivariant presentations. They are not inferred from an unexplained equality of numerical traces. The selected family ⊕_iρ_i,λ^⊕ m_i has a split equivariant inclusion in the carrier. The finite trace sample uniquely determines…
Exact mathematical passage
V_{\mathrm{car},\lambda}\simeq\bigoplus_i\rho_{i,\lambda}^{\oplus c_i}, \qquad U_{0,\lambda}\simeq\bigoplus_i\rho_{i,\lambda}^{\oplus m_i}.
*Use in source · 3 passages
Search aliases: *
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 712
Why the finite hypothesis cannot be dropped. Take vertices {*}sqcup{v_n:n∈ N}, with * isolated and edges v_n+1→ v_n. Every strongly connected component is a singleton, and {*} is the unique source component. No finite seed set covers all vertices: it can reach only indices no larger than the largest chosen index. Both the unique-source statement and the failure of every finite cover are checked in Lean. Hence one source component in an infinite condensation does not imply that one seed reaches a…
Exact mathematical passage
\{*\}\sqcup\{v_n:n\in\mathbf N\}
+Use in source · 87 passages
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Use in source · Read passage · Paper_X_Block27_Amended.tex, line 141
residual root + positive-depth deformation coordinates.
Exact mathematical passage
\boxed{ \text{residual root} \quad+\quad \text{positive-depth deformation coordinates}. }
\bm aUse in source · 1 passage
Search aliases: a bm
Use in source · pages.tex, line 272
For target (0,1,1), every pair of conditions is occupied: (0,1,0) witnesses the first two, (0,0,1) the first and third, and (1,1,1) the second and third. But the triple itself is absent. Multiplying all three selectors gives the zero matrix. E_ a=∏_i=1^3 I-T_i, · a_i=0, T_i, · a_i=1. Tr(E_ a)=rank(E_ a). The checks compute all eight joint projectors, their pairwise orthogonality, and their sum. The sum is the identity even though two labels are absent: those two projectors are simply zero. takea…
Exact mathematical passage
E_{\bm a}=\prod_{i=1}^{3} \begin{cases}I-T_i,&a_i=0,\\T_i,&a_i=1.\end{cases} \qquad \Tr(E_{\bm a})=\rank(E_{\bm a}).
a^{-1}Use in source · 2 passages
Search aliases: a^-1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1287
The choice of absolute prime and Frobenius lift is also controlled. Let ( Q_u,γ_u) and ( Q_v,γ_v) be any two actual arithmetic witnesses at the same rational prime. Transitivity on primes of A above (p) gives a∈ G_ Q with Q_v=a Q_u, γ_v^-1aγ_u a^-1∈ I_ Q_v. Consequently, any representation trivial on I_ Q_v has the same trace on the two witnesses. For representations factoring through L, triviality on this actual absolute inertia is derived whenever p∉ S_L. The restrictions of all absolute witne…
Exact mathematical passage
\mathfrak Q_v=a\mathfrak Q_u,\qquad \gamma_v^{-1}a\gamma_u a^{-1}\in I_{\mathfrak Q_v}.
a^2Use in source · 3 passages
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Use in source · 22_norm_table.tex, line 3
[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round] path[use as bounding box] (0,-.6) rectangle (14,4.8); node[font=small,align=center,font=small] at (3,4.7) N(a+bu)=a^2-2b^2pmod5; path[draw=black!30,fill=white] (1.45,3.38) rectangle (2.07,4); node[font=small,align=center,font=small] at (1.76,3.69) 0; path[draw=black!30,fill=black!8] (2.07,3.38) rectangle (2.69,4); node[font=small,align=center,font=small] at (2.38,3.69) 3; path[draw=black!30,fill=black!8] (2.69,3.38) rectangle (3.31,4); node[…
Exact mathematical passage
N(a+bu)=a^2-2b^2\pmod5
a^dUse in source · 3 passages
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Use in source · pages.tex, line 356
vpage24A nontrivial character may still be exceptionalTHE EXACT SCALAR TESTMain text: §9; scalar/Mellin checkpoint: scalar-power and nonscalar-witness boundaries. A scalar a∈FF_5^× acts on a degree-d algebra with determinant a^d. Therefore the scalar cancellation route tests χ(a^d), not merely χ(a). vfig24_scalar_exception headA failed sufficient test is not an impossibility theorem Let χ_4(2)=i generate the character group of FF_5^×, and write χ_k=χ_4^k. Since 2 generates the unit group, ∃ a:χ_…
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a^d
a^jUse in source · 1 passage
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Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1578
Finite-exponential reconstruction and a cubic inverse (Block 13). Let K be a characteristic-zero field and let (r,N) be supplied group data with nilpotent N and r(g)N=ν(g)Nr(g). For a supplied scalar map satisfying t(gh)=t(g)+ν(g)t(h), the formalization constructs ρ(g)=exp(t(g)N)r(g), exp(aN)=∑_j=0^d-1fraca^jj!N^j (N^d=0). Lean proves the representation laws, actual finite-copy and split-map compatibility, and joint continuity from continuity of N, r and t with the stated continuous vector-space…
Exact mathematical passage
\rho(g)=\exp(t(g)N)r(g),\qquad \exp(aN)=\sum_{j=0}^{d-1}\frac{a^j}{j!}N^j\quad(N^d=0).
A_CUse in source · 3 passages
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Use in source · Read passage · residue_continuation_body.tex, line 138
Ordinary class characters: an unconditional power saving. The preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl( O)→ C, define the actual finite ideal sum S_w(X)=∑_substackI≠0
Exact mathematical passage
A_C(X)=c_KX+O_{K,C}(X^{1-1/d}),\qquad c_K=\frac{2^{r_1}(2\pi)^{r_2}R_K}{\#\mu(K)\sqrt{\absn{\Delta_K}}}.
a_iUse in source · 6 passages
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Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1622
Let E be a field of characteristic zero and let T_1,…,T_N be commuting operators on a finite-dimensional E-vector space A, each diagonalizable over E. Thus the selected spectra split over the coefficient field. Let a_i∈ E be a selected target scalar for T_i, possibly absent from its spectrum. Define e_i = ∏_μ∈Spec(T_i)∖{a_i} fracT_i-μ Ia_i-μ, e_T=∏_i=1^Ne_i, with e_i=0 if a_i is absent.
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a_i\in E
a_{i,1}Use in source · 1 passage
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Use in source · Read passage · Paper_X_Block27_Amended.tex, line 304
[Smooth formal branch count] If R_bar x≅ O[[t_1,…,t_d]] as complete local O-algebras, then #X_m(bar x)=q^d(m-1). Fix a uniformizer and a section s:k→ O of reduction, with s(0)=0. Every O-valued branch is encoded by d positive-depth digit streams t_i=s(a_i,1)ϖ+s(a_i,2)ϖ^2+⋯, a_i,j∈ k,
Exact mathematical passage
t_i=s(a_{i,1})\varpi+s(a_{i,2})\varpi^2+\cdots, \qquad a_{i,j}\in k,
a_{i,2}Use in source · 1 passage
Search aliases: a_i,2
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 304
[Smooth formal branch count] If R_bar x≅ O[[t_1,…,t_d]] as complete local O-algebras, then #X_m(bar x)=q^d(m-1). Fix a uniformizer and a section s:k→ O of reduction, with s(0)=0. Every O-valued branch is encoded by d positive-depth digit streams t_i=s(a_i,1)ϖ+s(a_i,2)ϖ^2+⋯, a_i,j∈ k,
Exact mathematical passage
t_i=s(a_{i,1})\varpi+s(a_{i,2})\varpi^2+\cdots, \qquad a_{i,j}\in k,
a_{i,j}Use in source · 3 passages
Search aliases: a_i,j
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 304
[Smooth formal branch count] If R_bar x≅ O[[t_1,…,t_d]] as complete local O-algebras, then #X_m(bar x)=q^d(m-1). Fix a uniformizer and a section s:k→ O of reduction, with s(0)=0. Every O-valued branch is encoded by d positive-depth digit streams t_i=s(a_i,1)ϖ+s(a_i,2)ϖ^2+⋯, a_i,j∈ k,
Exact mathematical passage
t_i=s(a_{i,1})\varpi+s(a_{i,2})\varpi^2+\cdots, \qquad a_{i,j}\in k,
a_kUse in source · 7 passages
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Use in source · 20_closedness.tex, line 10
scriptsize] at (5.96,4.45) 3^2; node[font=small,align=center,font=scriptsize] at (7.19,4.45) 3^3; node[font=small,align=center,font=scriptsize] at (8.42,4.45) 3^4; node[font=small,align=center,font=scriptsize] at (9.65,4.45) 3^5; node[font=small,align=center,font=scriptsize] at (10.88,4.45) 3^6; node[font=small,align=center,anchor=east] at (1.95,3.85) a_k=3^0; path[draw=black!25,fill=black!18] (2.93,3.6) rectangle (4.08,4.08); node[font=small,align=center,font=scriptsize] at (3.5,3.85) 1; path[d…
Exact mathematical passage
a_k=3^{0}
a_mUse in source · 3 passages
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Use in source · Read passage · Paper_X_Block27_Amended.tex, line 557
[Weighted seed zeta function] Let 1≤ w_1≤⋯≤ w_d, put W=∑_iw_i, and let a_m=q^∑_imax(m-w_i,0). Then Z_ w(T) = ∑_m=1^w_da_mT^m + q^-Wfrac(q^dT)^w_d+11-q^dT. In particular, every weighted smooth chart has a rational seed zeta function.
Exact mathematical passage
a_m=q^{\sum_i\max(m-w_i,0)}.
a_{m-1}Use in source · 3 passages
Search aliases: a_m-1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1975
The residual module partitions coordinates modulo p, verifies compatibility x_m+1≡ x_mpmodp^m, and reconstructs the digit a_m-1 by a_m-1=lfloorfracx_mp^m-1rfloorpmod p.
Exact mathematical passage
a_{m-1}
A_pUse in source · 1 passage
Search aliases: A_p
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1796
For each supplied required packet p, let A_p be its finite-dimensional carrier over a characteristic-zero field E_p, and let e_p be the actual joint spectral projector for its finite list of split commuting conditions. No order on E_p is assumed. The source notation Tr(e_p)>0 means that the unique natural number n_p with Tr(e_p)=(n_p:E_p) is positive. The checked projector formula identifies n_p with the dimension of the actual joint eigenspace.
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A_p
A_RUse in source · 2 passages
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Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2335
Let R be any commutative ring and B any commutative R-algebra. The coordinate ring A_R=R[X]/(X^2-1) represents precisely this specified packet: Hom_R-alg(A_R,B) ≃{x∈ B:x^2=1} ≃Hom(C_2,B^×) ≃ P_q(B). The first map evaluates the coordinate. Its inverse is the quotient universal property. A root is a unit with itself as inverse and determines a character by its value on the generator of C_2. Finally pull back along q. All inverse identities and naturality under coefficient homomorphisms are proved.…
Exact mathematical passage
A_R=R[X]/(X^2-1)
A_TUse in source · 3 passages
Search aliases: A_T
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1635
[Exact finite occupancy trace] The operator e_T is the projector onto the joint eigenspace A_T = {v:T_iv=a_iv for every i}. Moreover, Tr(e_T)=rank(e_T)=dim A_T∈ Z_≥0. The trace is identified with its unique nonnegative integer preimage under Z↪ E. Thus Tr(e_T)>0 refers to that integer, not to an ordering of an arbitrary field E. The finite packet is occupied exactly when this integer is positive. In positive characteristic the field-valued trace can vanish for a nonzero projector; the rank test,…
Exact mathematical passage
A_T = \{v:T_iv=a_iv\text{ for every }i\}.
\adUse in source · 3 passages
Search aliases: ad operatorname
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 339
[Tangent and obstruction spaces] For a framed representation problem, first-order lifts are described by Z^1(G,adbarρ). For an unframed Schur problem, tangent directions are described by H^1(G,adbarρ). Relations in a presentation of a deformation ring map into H^2(G,adbarρ), which contains the obstruction classes.
Exact mathematical passage
Z^1(G,\ad\bar\rho)
\operatorname{Aut}Use in source · 2 passages
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Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2478
The inertia theorem uses actual residue actions. For a valuation subring A of an extension field L/K, let D_A be its decomposition subgroup in Aut_K(L) and put I_A=ker(D_A⟶Aut(k_A)), where k_A is the residue field. If 2 is a unit in A, every member of I_A fixes every y∈ L with y^2=-1. Indeed y^-1=-y, so y belongs to A and is a unit there. An automorphism sends y to y or -y. Their residues are distinct because 2 and y are units, whereas inertia fixes residues. The second possibility is excluded.
Exact mathematical passage
\operatorname{Aut}_K(L)
\operatorname{Aut}_KUse in source · 1 passage
Search aliases: Aut_K operatorname
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2478
The inertia theorem uses actual residue actions. For a valuation subring A of an extension field L/K, let D_A be its decomposition subgroup in Aut_K(L) and put I_A=ker(D_A⟶Aut(k_A)), where k_A is the residue field. If 2 is a unit in A, every member of I_A fixes every y∈ L with y^2=-1. Indeed y^-1=-y, so y belongs to A and is a unit there. An automorphism sends y to y or -y. Their residues are distinct because 2 and y are units, whereas inertia fixes residues. The second possibility is excluded.
Exact mathematical passage
\operatorname{Aut}_K(L)
BUse in source · 32 passages
Search aliases: B
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 377
For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet are cochain maps, so they induce maps on these genuine quotients.
Exact mathematical passage
Z^r=\ker d_r,\qquad B^r=\operatorname{im}d_{r-1}\subseteq Z^r, \qquad H^r=Z^r/B^r,
\bm bUse in source · 4 passages
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Use in source · pages.tex, line 27
C m= b · A finite trace system for carrier multiplicities. · Integrality, positivity, or available capacity automatically.
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C\bm m=\bm b
\boldsymbol bUse in source · 4 passages
Search aliases: b boldsymbol
Use in source · 15_separator.tex, line 36
6.55,3.45) 2; path[draw=black!30,fill=white] (6.15,2.25) rectangle (6.95,3.05); node[font=small,align=center,font=small] at (6.55,2.65) 0; path[draw=black!30,fill=black!8] (6.15,1.45) rectangle (6.95,2.25); node[font=small,align=center,font=small] at (6.55,1.85) 1; node[font=small,align=center] at (7.45,2.6) =; node[font=small,align=center,font=small] at (8.45,4.55) b; path[draw=black!30,fill=black!8] (8.05,3.05) rectangle (8.85,3.85); node[font=small,align=center,font=small] at (8.45,3.45) 4; p…
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\boldsymbol b
b^2Use in source · 3 passages
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Use in source · 22_norm_table.tex, line 3
[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round] path[use as bounding box] (0,-.6) rectangle (14,4.8); node[font=small,align=center,font=small] at (3,4.7) N(a+bu)=a^2-2b^2pmod5; path[draw=black!30,fill=white] (1.45,3.38) rectangle (2.07,4); node[font=small,align=center,font=small] at (1.76,3.69) 0; path[draw=black!30,fill=black!8] (2.07,3.38) rectangle (2.69,4); node[font=small,align=center,font=small] at (2.38,3.69) 3; path[draw=black!30,fill=black!8] (2.69,3.38) rectangle (3.31,4); node[…
Exact mathematical passage
N(a+bu)=a^2-2b^2\pmod5
B^rUse in source · 1 passage
Search aliases: B^r
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 377
For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet are cochain maps, so they induce maps on these genuine quotients.
Exact mathematical passage
Z^r=\ker d_r,\qquad B^r=\operatorname{im}d_{r-1}\subseteq Z^r, \qquad H^r=Z^r/B^r,
B^\timesUse in source · 2 passages
Search aliases: B^× times
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2325
hmetic completion (Block 21). Fix the continuous quotient q:G_ Q⟶ C_2 obtained by the action on a chosen square root i of -1 in the algebraic closure. Its surjectivity and the equivalence q(σ)=1 Longleftrightarrow σ(i)=i are proved using the existing actual quadratic Galois construction. For a commutative ring B, define the arithmetic packet P_q(B)= {χ:G_ Q⟶ B^×midker q⊆kerχ}, where the maps are group homomorphisms. Unique descent through the actual surjective map q is derived, rather than suppl…
Exact mathematical passage
\mathcal P_q(B)= \{\chi:G_{\Q}\longrightarrow B^\times\mid\ker q\subseteq\ker\chi\},
B_0Use in source · 1 passage
Search aliases: B_0
Use in source · Read passage · residue_continuation_body.tex, line 237
The assembly identifies the residue sum with the sum over actual generators, recombines the sign portions, and uses Y=t^d to obtain absnG_J,χ(Y)≤ B_0(Y^1-1/d+1). Equation~[source reference] then permits division by the nonzero factor #μ(K)χ( N Jbmod m). The final theorem supplies its finite means and counting estimates internally. For d=1 the exponent is zero and the conclusion is a bounded sum, so the argument does not require d>1.
Exact mathematical passage
\absn{G_{J,\chi}(Y)}\leq B_0(Y^{1-1/d}+1)
B_j^rUse in source · 1 passage
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Use in source · Read passage · Paper_X_Block27_Amended.tex, line 381
For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet are cochain maps, so they induce maps on these genuine quotients.
Exact mathematical passage
Z_j^r,B_j^r,H_j^r
B_wUse in source · 2 passages
Search aliases: B_w
Use in source · Read passage · residue_continuation_body.tex, line 150
Nm I≤ Xw([I]). Exact grouping by ideal class gives S_w(X)=∑_C w(C)A_C(X). If ∑_Cw(C)=0, the common leading coefficient cancels, so there is B_w≥0 such that absnS_w(X)≤ B_wX^1-1/d (X≥1). For every nontrivial character χ:Cl( O)→ C^×, the finite character sum is zero. Thus the displayed estimate applies without an additional arithmetic cancellation hypothesis. This conclusion concerns ordinary class characters; it does not identify them with the further norm-residue characters needed for Frobenius …
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B_w\geq0
\bar EUse in source · 7 passages
Search aliases: bar E bar
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2087
Lifting a summand inside a supplied finite-group carrier (Block 18). Let R be a commutative integral domain complete and separated for the (π)-adic filtration. Let Γ be a finite group, and let ρ:Γ→GL_n(R) be an actual integral representation. Assume that a=|Γ|≠0 in R and amidπ^e. For k>0, suppose bar E∈ M_n(R/(π)^e+k), bar E^2=bar E, barρ(g)bar E=bar Ebarρ(g) (g∈Γ). There is an actual integral matrix E satisfying E^2=E, ρ(g)E=Eρ(g), Ebmod(π)^k=bar Ebmod(π)^k. No integral summand or coherent fami…
Exact mathematical passage
\bar E\in M_n\bigl(R/(\pi)^{e+k}\bigr),\qquad \bar E^2=\bar E,\qquad \bar\rho(g)\bar E=\bar E\bar\rho(g)\quad(g\in\Gamma).
\bar E^2Use in source · 1 passage
Search aliases: bar E^2 bar
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2087
Lifting a summand inside a supplied finite-group carrier (Block 18). Let R be a commutative integral domain complete and separated for the (π)-adic filtration. Let Γ be a finite group, and let ρ:Γ→GL_n(R) be an actual integral representation. Assume that a=|Γ|≠0 in R and amidπ^e. For k>0, suppose bar E∈ M_n(R/(π)^e+k), bar E^2=bar E, barρ(g)bar E=bar Ebarρ(g) (g∈Γ). There is an actual integral matrix E satisfying E^2=E, ρ(g)E=Eρ(g), Ebmod(π)^k=bar Ebmod(π)^k. No integral summand or coherent fami…
Exact mathematical passage
\bar E\in M_n\bigl(R/(\pi)^{e+k}\bigr),\qquad \bar E^2=\bar E,\qquad \bar\rho(g)\bar E=\bar E\bar\rho(g)\quad(g\in\Gamma).
\bar E_iUse in source · 2 passages
Search aliases: bar E_i bar
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2161
Simultaneous projector-family lifting and rigidity (Block 19). Keep the complete domain, supplied finite-group carrier and group-order hypotheses of Block 18. Let J be any finite label set and k>0. Suppose (bar E_i)_i∈ J is a complete orthogonal family of equivariant idempotents modulo (π)^e+k. There exists an integral equivariant family (E_i)_i∈ J satisfying E_i^2=E_i, E_iE_j=0 (i≠ j), ∑_i∈ JE_i=1, E_ibmod(π)^k=bar E_ibmod(π)^k. The loss is at most e, independent of |J| and the ambient matrix r…
Exact mathematical passage
(\bar E_i)_{i\in J}
\bigsqcupUse in source · 2 passages
Search aliases: bigsqcup
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 199
[Residual partition] For every coefficient algebra A, M(A) = bigsqcup_bar x∈ M(k) M_bar x(A).
Exact mathematical passage
\boxed{ \mathcal M(A) = \bigsqcup_{\bar x\in\mathcal M(k)} \mathcal M_{\bar x}(A). }
\bigsqcup_{\bar x}Use in source · 1 passage
Search aliases: bigsqcup_bar x bigsqcup bar
Use in source · pages.tex, line 41
For example, the addresses 1,4,7,10,…,25 may be deeply connected by higher-digit changes. They remain disjoint from the addresses reducing to zero. In a genuine deformation problem, the same conclusion applies to operations whose reduction is the identity, not to every operation that happens to be called a deformation. red(x+3h)=red(x), M(A)=bigsqcup_bar x M_bar x(A). takeawayVertical completeness can fill one entire residual fiber without reaching a single point in a different fiber. captiontex…
Exact mathematical passage
\operatorname{red}(x+3h)=\operatorname{red}(x), \qquad \mathcal M(A)=\bigsqcup_{\bar x}\mathcal M_{\bar x}(A).
\bigsqcup_{\bar x\in\mathcal M(k)}Use in source · 1 passage
Search aliases: bigsqcup_bar x∈ M(k) bigsqcup bar in mathcal
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 199
[Residual partition] For every coefficient algebra A, M(A) = bigsqcup_bar x∈ M(k) M_bar x(A).
Exact mathematical passage
\boxed{ \mathcal M(A) = \bigsqcup_{\bar x\in\mathcal M(k)} \mathcal M_{\bar x}(A). }
\bulletUse in source · 8 passages
Search aliases: bullet
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 357
[Cohomology packet decomposition] Under these hypotheses, each packet is a subcomplex and H^r(C^bullet) ≅ ⊕_jH^r(P_jC^bullet). In particular, H^1=⊕_jH^1_j, H^2=⊕_jH^2_j, so tangent and obstruction spaces reconstruct exactly from their packet components.
Exact mathematical passage
\boxed{ H^r(C^\bullet) \cong \bigoplus_jH^r(P_jC^\bullet). }
\bm cUse in source · 1 passage
Search aliases: c bm
Use in source · pages.tex, line 235
vpage16A solution is not yet a summandINTEGRAL, NONNEGATIVE, WITHIN CAPACITYMain text: Theorem 9.2 and its multiplicity-recovery example. Keep the same separating matrix C and suppose the available carrier has capacities c=(2,1,1). A recovered vector describes a genuine sub-sum of that carrier only if each entry is an integer and 0≤ m_i≤ c_i. vfig16_multiplicities headThree independent rejection tests The trace vector (1,0,0) gives (1/6,1/6,1/3), so it fails integrality. The vector (2,-2,-1) giv…
Exact mathematical passage
\bm c=(2,1,1)
C^0Use in source · 2 passages
Search aliases: C^0
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 343
Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.
Exact mathematical passage
C^0\xrightarrow{d_0}C^1\xrightarrow{d_1}C^2\xrightarrow{}\cdots
C^1Use in source · 3 passages
Search aliases: C^1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 343
Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.
Exact mathematical passage
C^0\xrightarrow{d_0}C^1\xrightarrow{d_1}C^2\xrightarrow{}\cdots
C^2Use in source · 2 passages
Search aliases: C^2
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 343
Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.
Exact mathematical passage
C^0\xrightarrow{d_0}C^1\xrightarrow{d_1}C^2\xrightarrow{}\cdots
C^\bulletUse in source · 6 passages
Search aliases: C^bullet bullet
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 357
[Cohomology packet decomposition] Under these hypotheses, each packet is a subcomplex and H^r(C^bullet) ≅ ⊕_jH^r(P_jC^bullet). In particular, H^1=⊕_jH^1_j, H^2=⊕_jH^2_j, so tangent and obstruction spaces reconstruct exactly from their packet components.
Exact mathematical passage
\boxed{ H^r(C^\bullet) \cong \bigoplus_jH^r(P_jC^\bullet). }
C^iUse in source · 2 passages
Search aliases: C^i
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2002
Modular Gaussian elimination computes the ranks of the differentials. For C^i-1xrightarrowd_i-1C^ixrightarrowd_iC^i+1, the finite dimension is evaluated as dim H^i = dim C^i-rank d_i-rank d_i-1. Packet subcomplexes are computed separately and checked against the total.
Exact mathematical passage
C^{i-1}\xrightarrow{d_{i-1}}C^i\xrightarrow{d_i}C^{i+1},
C^{i+1}Use in source · 1 passage
Search aliases: C^i+1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2002
Modular Gaussian elimination computes the ranks of the differentials. For C^i-1xrightarrowd_i-1C^ixrightarrowd_iC^i+1, the finite dimension is evaluated as dim H^i = dim C^i-rank d_i-rank d_i-1. Packet subcomplexes are computed separately and checked against the total.
Exact mathematical passage
C^{i-1}\xrightarrow{d_{i-1}}C^i\xrightarrow{d_i}C^{i+1},
C^{i-1}Use in source · 1 passage
Search aliases: C^i-1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2002
Modular Gaussian elimination computes the ranks of the differentials. For C^i-1xrightarrowd_i-1C^ixrightarrowd_iC^i+1, the finite dimension is evaluated as dim H^i = dim C^i-rank d_i-rank d_i-1. Packet subcomplexes are computed separately and checked against the total.
Exact mathematical passage
C^{i-1}\xrightarrow{d_{i-1}}C^i\xrightarrow{d_i}C^{i+1},
C^rUse in source · 1 passage
Search aliases: C^r
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 346
Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.
Exact mathematical passage
P_j^r:C^r\to C^r
C_2Use in source · 8 passages
Search aliases: C_2
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1328
Finite deletion can destroy density. In the actual discrete group C_2={1,a}, set s(0)=a and s(m+1)=1. The original samples are surjective and satisfy quotient coverage. Removing index zero leaves only the identity, so both density and quotient coverage fail. The actual rational sign and trivial characters agree on the remaining samples but differ globally. This checked example explains why ordinary coverage alone cannot justify deleting the bad places in the system comparison.
Exact mathematical passage
C_2=\{1,a\}
C_j^\bulletUse in source · 5 passages
Search aliases: C_j^bullet bullet
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 382
For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet are cochain maps, so they induce maps on these genuine quotients.
Exact mathematical passage
C_j^\bullet
C_j^rUse in source · 1 passage
Search aliases: C_j^r
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 375
For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet are cochain maps, so they induce maps on these genuine quotients.
Exact mathematical passage
C_j^r=\operatorname{im}P_j^r
c_KUse in source · 1 passage
Search aliases: c_K
Use in source · Read passage · residue_continuation_body.tex, line 138
Ordinary class characters: an unconditional power saving. The preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl( O)→ C, define the actual finite ideal sum S_w(X)=∑_substackI≠0
Exact mathematical passage
A_C(X)=c_KX+O_{K,C}(X^{1-1/d}),\qquad c_K=\frac{2^{r_1}(2\pi)^{r_2}R_K}{\#\mu(K)\sqrt{\absn{\Delta_K}}}.
C_mUse in source · 4 passages
Search aliases: C_m
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2365
There is also a completion theorem without chosen compatible characters. Suppose that R/I^m is finite for every m≥1, and that C_m⊆ P_q(R/I^m) are allowed subsets such that every reduction of an allowed member is allowed. Then · ∃χ∈ P_q(R), χ is continuous and χbmod I^m∈ C_m for all m≥1
Exact mathematical passage
C_m\subseteq\mathcal P_q(R/I^m)
\CCUse in source · 5 passages
Search aliases: CC C mathbb
Use in source · Read passage · residue_continuation_body.tex, line 49
The estimate is proved from the actual boundary charts and the fixed-lattice discrepancy theorem. The index factor comes from the covolume identity covol(L(M))=q covol(L(J)). For a complex weight w:J/M→ C, summing the estimates gives the leading term fracvol(D)t^dq covol(L(J))∑_a∈ J/Mw(a), with error at most B∑_aabsnw(a)(t^d-1+1). If the finite sum of weights is zero, only that error remains. Writing Y=t^d gives an error O(Y^1-1/d+1) for the actual norm threshold. These assertions include d=1.
Exact mathematical passage
w:J/M\to\CC
\CC^{\times}Use in source · 1 passage
Search aliases: CC^× C^× CC times mathbb times
Use in source · Read passage · residue_continuation_body.tex, line 154
Nm I≤ Xw([I]). Exact grouping by ideal class gives S_w(X)=∑_C w(C)A_C(X). If ∑_Cw(C)=0, the common leading coefficient cancels, so there is B_w≥0 such that absnS_w(X)≤ B_wX^1-1/d (X≥1). For every nontrivial character χ:Cl( O)→ C^×, the finite character sum is zero. Thus the displayed estimate applies without an additional arithmetic cancellation hypothesis. This conclusion concerns ordinary class characters; it does not identify them with the further norm-residue characters needed for Frobenius …
Exact mathematical passage
\chi:\Cl(\OO)\to\CC^{\times}
\operatorname{char}Use in source · 1 passage
Search aliases: char operatorname
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 155
Let O be a complete discrete valuation ring with uniformizer ϖ, maximal ideal (ϖ), residue field k= O/(ϖ), |k|=q, and fraction field K. When a characteristic-zero conclusion is intended, assume additionally that charK=0; the general counting statements do not require that restriction. Let M be a moduli functor on complete local Noetherian O-algebras with residue field k, equipped with a well-defined reduction map red_A: M(A)⟶ M(k). For a Galois representation this means either that a framed inte…
Exact mathematical passage
\operatorname{char}K=0
\covolUse in source · 3 passages
Search aliases: covol operatorname
Use in source · Read passage · residue_continuation_body.tex, line 38
There is a constant B≥0, depending on K,M,J, such that, for every a∈ J/M and every real t≥1, |#{x∈ J:xbmod M=a, ι(x)∈ tD} -fracvol(D)q covol(L(J))t^d| ≤ B(t^d-1+1). The same constant works for all residues a.
Exact mathematical passage
\left|\#\{x\in J:x\bmod M=a,\ \iota(x)\in tD\} -\frac{\vol(D)}{q\,\covol(L(J))}t^d\right| \leq B(t^{d-1}+1).
D^{-1}Use in source · 1 passage
Search aliases: D^-1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1408
ing every remaining coordinate with zero. Define r:U_c→ U_m by restriction to the selected copies. Lean proves rj=1_U_m, jρ_m(g)=ρ_c(g)j, rρ_c(g)=ρ_m(g)r. The image of j is exactly the subspace whose unused coordinates vanish. Consequently jr is an equivariant idempotent. A supplied equivariant isomorphism D:Wxrightarrow∼U_c gives the split maps D^-1j and rD inside the original carrier. The decomposition D is an input; these statements do not construct a decomposition of an unknown arithmetic ca…
Exact mathematical passage
D^{-1}j
d^2Use in source · 1 passage
Search aliases: d^2
Use in source · pages.tex, line 24
C^bullet,d · Cochains and a differential with d^2=0. · Its arithmetic identification with a Galois problem.
Exact mathematical passage
d^2=0
D^\bulletUse in source · 1 passage
Search aliases: D^bullet bullet
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 417
Naturality. If a cochain map F:C^bullet→ D^bullet intertwines the supplied projectors, it restricts to maps F_j:C_j^bullet→ D_j^bullet. The direct-sum decomposition commutes with H^r(F) and the family H^r(F_j). This is checked using representatives in the actual quotients, independently of any dimension computation.
Exact mathematical passage
F:C^\bullet\to D^\bullet
D_+Use in source · 1 passage
Search aliases: D_+
Use in source · Read passage · residue_continuation_body.tex, line 84
The relation to the geometric sign is also proved. If σ ranges over real embeddings and τ over one embedding from each complex pair, then n(x)=∏_σσ(x)∏_τabsnτ(x)^2. For x≠0, the second factor is positive. Thus the sign of n(x) is exactly the sign of the real-coordinate product. Cutting D into its positive and negative portions D_+,D_- adds no boundary outside the existing frontier, and both portions retain the proved boundary atlas. They are disjoint and their union is D.
Exact mathematical passage
D_+,D_-
D_-Use in source · 1 passage
Search aliases: D_-
Use in source · Read passage · residue_continuation_body.tex, line 84
The relation to the geometric sign is also proved. If σ ranges over real embeddings and τ over one embedding from each complex pair, then n(x)=∏_σσ(x)∏_τabsnτ(x)^2. For x≠0, the second factor is positive. Thus the sign of n(x) is exactly the sign of the real-coordinate product. Cutting D into its positive and negative portions D_+,D_- adds no boundary outside the existing frontier, and both portions retain the proved boundary atlas. They are disjoint and their union is D.
Exact mathematical passage
D_+,D_-
d_0Use in source · 8 passages
Search aliases: d_0
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 343
Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.
Exact mathematical passage
C^0\xrightarrow{d_0}C^1\xrightarrow{d_1}C^2\xrightarrow{}\cdots
d_1Use in source · 7 passages
Search aliases: d_1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 343
Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.
Exact mathematical passage
C^0\xrightarrow{d_0}C^1\xrightarrow{d_1}C^2\xrightarrow{}\cdots
D_AUse in source · 4 passages
Search aliases: D_A
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2477
The inertia theorem uses actual residue actions. For a valuation subring A of an extension field L/K, let D_A be its decomposition subgroup in Aut_K(L) and put I_A=ker(D_A⟶Aut(k_A)), where k_A is the residue field. If 2 is a unit in A, every member of I_A fixes every y∈ L with y^2=-1. Indeed y^-1=-y, so y belongs to A and is a unit there. An automorphism sends y to y or -y. Their residues are distinct because 2 and y are units, whereas inertia fixes residues. The second possibility is excluded.
Exact mathematical passage
D_A
D_iUse in source · 1 passage
Search aliases: D_i
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1512
Writing Q_D(T) for the reversed local characteristic polynomial, Lean proves Q_D(T)=det(1-T r(φ)mid E(D,H)), Q_D(0)=1, Q_⊕_iD_i^⊕ m_i(T)=∏_iQ_D_i(T)^m_i. The abstract formulas do not need characteristic zero. Normality and finite dimension appear explicitly where used. Arithmetic identification of G,H,ν and φ remains separate. In the geometric-Frobenius convention, ν(φ) is the inverse residue cardinality; it is this unit, rather than the residue cardinality, that enters the earlier special-chain…
Exact mathematical passage
Q_D(T)=\det(1-T r(\varphi)\mid E(D,H)),\qquad Q_D(0)=1, \qquad Q_{\bigoplus_iD_i^{\oplus m_i}}(T)=\prod_iQ_{D_i}(T)^{m_i}.
d_iUse in source · 2 passages
Search aliases: d_i
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2002
Modular Gaussian elimination computes the ranks of the differentials. For C^i-1xrightarrowd_i-1C^ixrightarrowd_iC^i+1, the finite dimension is evaluated as dim H^i = dim C^i-rank d_i-rank d_i-1. Packet subcomplexes are computed separately and checked against the total.
Exact mathematical passage
C^{i-1}\xrightarrow{d_{i-1}}C^i\xrightarrow{d_i}C^{i+1},
d_{i-1}Use in source · 2 passages
Search aliases: d_i-1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2002
Modular Gaussian elimination computes the ranks of the differentials. For C^i-1xrightarrowd_i-1C^ixrightarrowd_iC^i+1, the finite dimension is evaluated as dim H^i = dim C^i-rank d_i-rank d_i-1. Packet subcomplexes are computed separately and checked against the total.
Exact mathematical passage
C^{i-1}\xrightarrow{d_{i-1}}C^i\xrightarrow{d_i}C^{i+1},
D_i^{\oplus m_i}Use in source · 1 passage
Search aliases: D_i^⊕ m_i oplus
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1512
Writing Q_D(T) for the reversed local characteristic polynomial, Lean proves Q_D(T)=det(1-T r(φ)mid E(D,H)), Q_D(0)=1, Q_⊕_iD_i^⊕ m_i(T)=∏_iQ_D_i(T)^m_i. The abstract formulas do not need characteristic zero. Normality and finite dimension appear explicitly where used. Arithmetic identification of G,H,ν and φ remains separate. In the geometric-Frobenius convention, ν(φ) is the inverse residue cardinality; it is this unit, rather than the residue cardinality, that enters the earlier special-chain…
Exact mathematical passage
Q_D(T)=\det(1-T r(\varphi)\mid E(D,H)),\qquad Q_D(0)=1, \qquad Q_{\bigoplus_iD_i^{\oplus m_i}}(T)=\prod_iQ_{D_i}(T)^{m_i}.
D_j^\bulletUse in source · 1 passage
Search aliases: D_j^bullet bullet
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 418
Naturality. If a cochain map F:C^bullet→ D^bullet intertwines the supplied projectors, it restricts to maps F_j:C_j^bullet→ D_j^bullet. The direct-sum decomposition commutes with H^r(F) and the family H^r(F_j). This is checked using representatives in the actual quotients, independently of any dimension computation.
Exact mathematical passage
F_j:C_j^\bullet\to D_j^\bullet
d_rUse in source · 3 passages
Search aliases: d_r
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 347
Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.
Exact mathematical passage
\sum_jP_j^r=I, \qquad P_i^rP_j^r=\delta_{ij}P_j^r, \qquad d_rP_j^r=P_j^{r+1}d_r.
d_{r-1}Use in source · 1 passage
Search aliases: d_r-1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 377
For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet are cochain maps, so they induce maps on these genuine quotients.
Exact mathematical passage
Z^r=\ker d_r,\qquad B^r=\operatorname{im}d_{r-1}\subseteq Z^r, \qquad H^r=Z^r/B^r,
d_sUse in source · 3 passages
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Use in source · Read passage · Paper_X_Block27_Amended.tex, line 949
Actual local comparisons from fixed-depth certificates (Block 15). Let R be an I-adically complete separated commutative ring with finite coefficient quotients and an injective map into a field K. Fix a finite matrix rank. For finitely many local group maps ϕ_s:H_s→ G and integral template representations τ_s, fix detecting depths d_s. At precision m retain matrices P_s,m as part of the admissible object and require ρ_m(ϕ_s(h))P_s,m=P_s,mτ_s,m(h). For m≥ d_s require their determinants to remain …
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d_s
\ddUse in source · 2 passages
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Use in source · pages.tex, line 407
IN ENDPOINT MODELMain text: §9; retained scalar/Mellin continuation: summatory measurability and first-order Mellin control. Use the explicit envelope A(x)=x^1/2 for x≥1. It is not an ideal-counting function. It is a model showing why a power saving can make a Mellin expression regular at s=1. vfig27_mellin headCompute the endpoint rather than just naming it For real s>1/2, I(s)=s∫_1^∞ A(x)x^-s-1 dd x =fracss-1/2. At s=1+h this gives I(1+h)-I(1)=-frachh+1/2, |I(1+h)-I(1)|≤4|h| (|h|≤1/4). The der…
Exact mathematical passage
I(s)=s\int_1^\infty A(x)x^{-s-1}\,\dd x =\frac{s}{s-1/2}.
\detUse in source · 12 passages
Search aliases: det
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1021
R with uniformizer π, fraction field K and matrix rank n. If a fixed template τ has full K-linear action span, finite generation and denominator clearing give a fixed c such that π^c M_n(R)⊆span_R{τ(g):g∈ G}. Normalize a field comparison to an integral P with a unit entry. The displayed span constructs an integral B with BP=π^cI, so det Pmidπ^cn, det Pnot≡0pmodπ^cn+1. One c works for all integral representations in the fixed field-isomorphism class. Jacobson density derives full span from actual…
Exact mathematical passage
\det P\mid\pi^{cn},\qquad \det P\not\equiv0\pmod{\pi^{cn+1}}.
\diagUse in source · 5 passages
Search aliases: diag
Use in source · pages.tex, line 5
A successful answer to one question does not supply the next. vfig01_three_questions headThree models to keep in view The scalar equation x^2+1=0 has the root 1 modulo two, but no root modulo four. Over powers of five, the residues 2,7,57,182 form the beginning of a compatible branch. Neither fact constructs an automorphic carrier. In a supplied finite carrier, an idempotent such as diag(1,0) detects an occupied subspace; it does not explain where that carrier came from.
Exact mathematical passage
\diag(1,0)
\operatorname{diag}Use in source · 11 passages
Search aliases: diag operatorname
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 930
An actual Z_3 example applies the class-level completion theorem for the cyclic group of order two and polynomial X^2-1, producing a nonidentity generator. Another checked example has compatible classes represented by diag(1,-1) modulo 3 and diag(-1,1) modulo 9: these chosen matrices disagree under reduction, and an actual swap of basis repairs them. Arbitrary independent representatives therefore do not replace the framing argument.
Exact mathematical passage
\operatorname{diag}(1,-1)
\operatorname{disc}Use in source · 1 passage
Search aliases: disc operatorname
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1273
The exceptional primes are controlled by the actual field discriminant: S_L={p rational prime:pmiddisc(L)}. This set is finite. For p∉ S_L, every prime of O_L above (p) is unramified. Lean proves that the actual inertia cardinality is the ramification index. It follows that inertia is trivial and the arithmetic Frobenius at a fixed finite prime is unique as a field automorphism. Neither trivial inertia nor uniqueness is assumed in place of the arithmetic unramifiedness condition.
Exact mathematical passage
S_L=\{p\text{ rational prime}:p\mid\operatorname{disc}(L)\}.
\downarrowUse in source · 1 passage
Search aliases: downarrow
Use in source · Read passage · residue_continuation_body.tex, line 174
rability, and hence that of S_w, is proved from this fact. For zero-sum w, define F_w(h)=(1+h)∫_1^∞x^-2-hS_w(x) dx (h≥0). Since 1-1/d<1, the logarithmic first moment M_w=∫_1^∞(1+log x) absnx^-2S_w(x) dx is finite. Lean proves the explicit estimate absnF_w(h)-F_w(0)≤ hM_w for every h≥0, and therefore F_w(h)=F_w(0)+O(h) as hdownarrow0. The function here is the displayed integral of the actual ideal sum. Its identification with the intended arithmetic Euler product is a separate step.
Exact mathematical passage
h\downarrow0
eUse in source · 70 passages
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Use in source · Read passage · Paper_X_Block27_Amended.tex, line 582
[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.
Exact mathematical passage
e\ge1
E^2Use in source · 1 passage
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Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2093
d for the (π)-adic filtration. Let Γ be a finite group, and let ρ:Γ→GL_n(R) be an actual integral representation. Assume that a=|Γ|≠0 in R and amidπ^e. For k>0, suppose bar E∈ M_n(R/(π)^e+k), bar E^2=bar E, barρ(g)bar E=bar Ebarρ(g) (g∈Γ). There is an actual integral matrix E satisfying E^2=E, ρ(g)E=Eρ(g), Ebmod(π)^k=bar Ebmod(π)^k. No integral summand or coherent family of quotient projectors is assumed. No local-ring assumption or finiteness of the residue field is needed.
Exact mathematical passage
E^2=E,\qquad \rho(g)E=E\rho(g),\qquad E\bmod(\pi)^k=\bar E\bmod(\pi)^k.
E_{\bm a}Use in source · 1 passage
Search aliases: E_ a bm
Use in source · pages.tex, line 272
For target (0,1,1), every pair of conditions is occupied: (0,1,0) witnesses the first two, (0,0,1) the first and third, and (1,1,1) the second and third. But the triple itself is absent. Multiplying all three selectors gives the zero matrix. E_ a=∏_i=1^3 I-T_i, · a_i=0, T_i, · a_i=1. Tr(E_ a)=rank(E_ a). The checks compute all eight joint projectors, their pairwise orthogonality, and their sum. The sum is the identity even though two labels are absent: those two projectors are simply zero. takea…
Exact mathematical passage
E_{\bm a}=\prod_{i=1}^{3} \begin{cases}I-T_i,&a_i=0,\\T_i,&a_i=1.\end{cases} \qquad \Tr(E_{\bm a})=\rank(E_{\bm a}).
e_0Use in source · 13 passages
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Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1467
The exact special chain has basis e_0,…,e_n-1 and operators Ne_0=0, Ne_j=e_j-1 (j>0), Fe_j=aq^-je_j, a,q∈ K^×. Lean proves N^n=0, and for n>0 also N^n-1≠0 and ker N=Ke_0. Consequently its kernel polynomial is X-a, whereas its whole-space polynomial is ∏_j=0^n-1(X-aq^-j). The zero-length chain is included as a zero-dimensional space; the assertion ker N=Ke_0 is made only for positive length.
Exact mathematical passage
e_0,\ldots,e_{n-1}
e_{01}Use in source · 1 passage
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Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2212
For the continuous Z_2 carrier ρ(s)=diag(1,-1,1), conjugate the three coordinate projectors by 1+2e_01 and reduce modulo 4. They form a complete equivariant family. The general theorem constructs three nonzero integral summands retaining this family modulo 2. No exact equivariant family retains the whole prescribed family modulo 4. Separate checked examples show that independent equivariant idempotent lifts can lose orthogonality, and that distinct complete families with the same reduction can b…
Exact mathematical passage
1+2e_{01}
e_1Use in source · 7 passages
Search aliases: e_1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1524
An explicit action of C_2× Z has ν(φ)=1/3, F=diag(2,6,18) and Ne_1=e_0, Ne_2=e_1, Ne_0=0. Trivial inertia gives local dimension one and denominator 1-2T; scalar sign inertia gives dimension zero and denominator one. The entire F and N are equal in these two examples, and N^2≠0. Thus inertia cannot be omitted. Conversely, unramified chains of lengths one and three have equal Euler denominators while only the first has N=0. Euler factors therefore do not identify complete monodromy data.
Exact mathematical passage
Ne_1=e_0
e_2Use in source · 1 passage
Search aliases: e_2
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1524
An explicit action of C_2× Z has ν(φ)=1/3, F=diag(2,6,18) and Ne_1=e_0, Ne_2=e_1, Ne_0=0. Trivial inertia gives local dimension one and denominator 1-2T; scalar sign inertia gives dimension zero and denominator one. The entire F and N are equal in these two examples, and N^2≠0. Thus inertia cannot be omitted. Conversely, unramified chains of lengths one and three have equal Euler denominators while only the first has N=0. Euler factors therefore do not identify complete monodromy data.
Exact mathematical passage
Ne_2=e_1
e_3Use in source · 1 passage
Search aliases: e_3
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1609
0 · 0 · 1, ρ(d)=diag(1,3,9). Reversing the factor order or dropping the quadratic term gives checked failures. For the length-four chain, the quadratic logarithm has erroneous e_3-to-e_0 coefficient -1/3, so its cubic restriction is substantive. A zero-parameter sample also fails to recover N.
Exact mathematical passage
e_3
E_iUse in source · 8 passages
Search aliases: E_i
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2163
Simultaneous projector-family lifting and rigidity (Block 19). Keep the complete domain, supplied finite-group carrier and group-order hypotheses of Block 18. Let J be any finite label set and k>0. Suppose (bar E_i)_i∈ J is a complete orthogonal family of equivariant idempotents modulo (π)^e+k. There exists an integral equivariant family (E_i)_i∈ J satisfying E_i^2=E_i, E_iE_j=0 (i≠ j), ∑_i∈ JE_i=1, E_ibmod(π)^k=bar E_ibmod(π)^k. The loss is at most e, independent of |J| and the ambient matrix r…
Exact mathematical passage
(E_i)_{i\in J}
e_iUse in source · 3 passages
Search aliases: e_i
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1623
Let E be a field of characteristic zero and let T_1,…,T_N be commuting operators on a finite-dimensional E-vector space A, each diagonalizable over E. Thus the selected spectra split over the coefficient field. Let a_i∈ E be a selected target scalar for T_i, possibly absent from its spectrum. Define e_i = ∏_μ∈Spec(T_i)∖{a_i} fracT_i-μ Ia_i-μ, e_T=∏_i=1^Ne_i, with e_i=0 if a_i is absent.
Exact mathematical passage
e_i = \prod_{\mu\in\Spec(T_i)\setminus\{a_i\}} \frac{T_i-\mu I}{a_i-\mu}, \qquad e_T=\prod_{i=1}^{N}e_i,
E_i^2Use in source · 1 passage
Search aliases: E_i^2
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2164
Simultaneous projector-family lifting and rigidity (Block 19). Keep the complete domain, supplied finite-group carrier and group-order hypotheses of Block 18. Let J be any finite label set and k>0. Suppose (bar E_i)_i∈ J is a complete orthogonal family of equivariant idempotents modulo (π)^e+k. There exists an integral equivariant family (E_i)_i∈ J satisfying E_i^2=E_i, E_iE_j=0 (i≠ j), ∑_i∈ JE_i=1, E_ibmod(π)^k=bar E_ibmod(π)^k. The loss is at most e, independent of |J| and the ambient matrix r…
Exact mathematical passage
E_i^2=E_i,\qquad E_iE_j=0\ (i\ne j),\qquad \sum_{i\in J}E_i=1,\qquad E_i\bmod(\pi)^k=\bar E_i\bmod(\pi)^k.
E_jUse in source · 1 passage
Search aliases: E_j
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2164
Simultaneous projector-family lifting and rigidity (Block 19). Keep the complete domain, supplied finite-group carrier and group-order hypotheses of Block 18. Let J be any finite label set and k>0. Suppose (bar E_i)_i∈ J is a complete orthogonal family of equivariant idempotents modulo (π)^e+k. There exists an integral equivariant family (E_i)_i∈ J satisfying E_i^2=E_i, E_iE_j=0 (i≠ j), ∑_i∈ JE_i=1, E_ibmod(π)^k=bar E_ibmod(π)^k. The loss is at most e, independent of |J| and the ambient matrix r…
Exact mathematical passage
E_i^2=E_i,\qquad E_iE_j=0\ (i\ne j),\qquad \sum_{i\in J}E_i=1,\qquad E_i\bmod(\pi)^k=\bar E_i\bmod(\pi)^k.
e_jUse in source · 1 passage
Search aliases: e_j
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1468
The exact special chain has basis e_0,…,e_n-1 and operators Ne_0=0, Ne_j=e_j-1 (j>0), Fe_j=aq^-je_j, a,q∈ K^×. Lean proves N^n=0, and for n>0 also N^n-1≠0 and ker N=Ke_0. Consequently its kernel polynomial is X-a, whereas its whole-space polynomial is ∏_j=0^n-1(X-aq^-j). The zero-length chain is included as a zero-dimensional space; the assertion ker N=Ke_0 is made only for positive length.
Exact mathematical passage
Ne_0=0,\qquad Ne_j=e_{j-1}\ (j>0),\qquad Fe_j=aq^{-j}e_j, \qquad a,q\in K^\times.
e_{j-1}Use in source · 1 passage
Search aliases: e_j-1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1468
The exact special chain has basis e_0,…,e_n-1 and operators Ne_0=0, Ne_j=e_j-1 (j>0), Fe_j=aq^-je_j, a,q∈ K^×. Lean proves N^n=0, and for n>0 also N^n-1≠0 and ker N=Ke_0. Consequently its kernel polynomial is X-a, whereas its whole-space polynomial is ∏_j=0^n-1(X-aq^-j). The zero-length chain is included as a zero-dimensional space; the assertion ker N=Ke_0 is made only for positive length.
Exact mathematical passage
Ne_0=0,\qquad Ne_j=e_{j-1}\ (j>0),\qquad Fe_j=aq^{-j}e_j, \qquad a,q\in K^\times.
e_{n-1}Use in source · 1 passage
Search aliases: e_n-1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1467
The exact special chain has basis e_0,…,e_n-1 and operators Ne_0=0, Ne_j=e_j-1 (j>0), Fe_j=aq^-je_j, a,q∈ K^×. Lean proves N^n=0, and for n>0 also N^n-1≠0 and ker N=Ke_0. Consequently its kernel polynomial is X-a, whereas its whole-space polynomial is ∏_j=0^n-1(X-aq^-j). The zero-length chain is included as a zero-dimensional space; the assertion ker N=Ke_0 is made only for positive length.
Exact mathematical passage
e_0,\ldots,e_{n-1}
E_pUse in source · 3 passages
Search aliases: E_p
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1797
For each supplied required packet p, let A_p be its finite-dimensional carrier over a characteristic-zero field E_p, and let e_p be the actual joint spectral projector for its finite list of split commuting conditions. No order on E_p is assumed. The source notation Tr(e_p)>0 means that the unique natural number n_p with Tr(e_p)=(n_p:E_p) is positive. The checked projector formula identifies n_p with the dimension of the actual joint eigenspace.
Exact mathematical passage
E_p
e_pUse in source · 4 passages
Search aliases: e_p
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1797
For each supplied required packet p, let A_p be its finite-dimensional carrier over a characteristic-zero field E_p, and let e_p be the actual joint spectral projector for its finite list of split commuting conditions. No order on E_p is assumed. The source notation Tr(e_p)>0 means that the unique natural number n_p with Tr(e_p)=(n_p:E_p) is positive. The checked projector formula identifies n_p with the dimension of the actual joint eigenspace.
Exact mathematical passage
e_p
e_TUse in source · 7 passages
Search aliases: e_T
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1623
Let E be a field of characteristic zero and let T_1,…,T_N be commuting operators on a finite-dimensional E-vector space A, each diagonalizable over E. Thus the selected spectra split over the coefficient field. Let a_i∈ E be a selected target scalar for T_i, possibly absent from its spectrum. Define e_i = ∏_μ∈Spec(T_i)∖{a_i} fracT_i-μ Ia_i-μ, e_T=∏_i=1^Ne_i, with e_i=0 if a_i is absent.
Exact mathematical passage
e_i = \prod_{\mu\in\Spec(T_i)\setminus\{a_i\}} \frac{T_i-\mu I}{a_i-\mu}, \qquad e_T=\prod_{i=1}^{N}e_i,
e_{T,a}Use in source · 1 passage
Search aliases: e_T,a
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1681
Naturality and finite hypotheses. If F T_i=U_i F and the source operators split, then F e_T,a=e_U,aF. This follows first on source eigenvectors and then on their span. In particular, a change of basis conjugates the constructed projector and preserves its trace. Adding conditions decreases the actual joint eigenspace and cannot increase its dimension. These statements do not assume an orthonormal basis or a positive definite inner product.
Exact mathematical passage
F e_{T,a}=e_{U,a}F
e_{U,a}Use in source · 1 passage
Search aliases: e_U,a
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1681
Naturality and finite hypotheses. If F T_i=U_i F and the source operators split, then F e_T,a=e_U,aF. This follows first on source eigenvectors and then on their span. In particular, a change of basis conjugates the constructed projector and preserves its trace. Adding conditions decreases the actual joint eigenspace and cannot increase its dimension. These statements do not assume an orthonormal basis or a positive definite inner product.
Exact mathematical passage
F e_{T,a}=e_{U,a}F
E_\lambdaUse in source · 1 passage
Search aliases: E_λ lambda
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1059
ion] Let mathscr W_1,…,mathscr W_r be pairwise nonisomorphic strictly compatible systems whose members are absolutely irreducible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficient-field embedding σ_λ:E↪ E_λ therefore sends C to an invertible matrix. Its entries agree with the ordinary representation traces Tr(W_i,λ(Frob_v_j)) when the selected places …
Exact mathematical passage
\sigma_\lambda:E\hookrightarrow E_\lambda
\operatorname{End}Use in source · 1 passage
Search aliases: End operatorname
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1389
Explicit summands inside a supplied decomposition. Let K be a field, let G be a monoid, and let ρ_i:G→End_K(W_i) be actual representations on finite-dimensional spaces, indexed by a finite set I. For natural numbers m_i≤ c_i, set U_m=∏_i∈ I∏_k<m_iW_i, U_c=∏_i∈ I∏_k<c_iW_i. These finite products realize the corresponding finite direct sums. The representation on U_m acts componentwise. Define j:U_m→ U_c by retaining the first m_i copies and filling every remaining coordinate with zero. Define r:U…
Exact mathematical passage
\rho_i:G\to\operatorname{End}_K(W_i)
\operatorname{End}_KUse in source · 1 passage
Search aliases: End_K operatorname
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1389
Explicit summands inside a supplied decomposition. Let K be a field, let G be a monoid, and let ρ_i:G→End_K(W_i) be actual representations on finite-dimensional spaces, indexed by a finite set I. For natural numbers m_i≤ c_i, set U_m=∏_i∈ I∏_k<m_iW_i, U_c=∏_i∈ I∏_k<c_iW_i. These finite products realize the corresponding finite direct sums. The representation on U_m acts componentwise. Define j:U_m→ U_c by retaining the first m_i copies and filling every remaining coordinate with zero. Define r:U…
Exact mathematical passage
\rho_i:G\to\operatorname{End}_K(W_i)
\expUse in source · 8 passages
Search aliases: exp
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1578
Finite-exponential reconstruction and a cubic inverse (Block 13). Let K be a characteristic-zero field and let (r,N) be supplied group data with nilpotent N and r(g)N=ν(g)Nr(g). For a supplied scalar map satisfying t(gh)=t(g)+ν(g)t(h), the formalization constructs ρ(g)=exp(t(g)N)r(g), exp(aN)=∑_j=0^d-1fraca^jj!N^j (N^d=0). Lean proves the representation laws, actual finite-copy and split-map compatibility, and joint continuity from continuity of N, r and t with the stated continuous vector-space…
Exact mathematical passage
\rho(g)=\exp(t(g)N)r(g),\qquad \exp(aN)=\sum_{j=0}^{d-1}\frac{a^j}{j!}N^j\quad(N^d=0).
FUse in source · 37 passages
Search aliases: F
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 172
For a Galois problem the principal example is barρ:G_F,S⟶ GL_n(k), with a deformation functor Def_barρ cut out by a determinant and selected local conditions.
Exact mathematical passage
\bar\rho:G_{F,S}\longrightarrow \GL_n(k),
\FUse in source · 1 passage
Search aliases: F mathbf
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 433
[Finite packet complex] The exact modular test uses two invariant subcomplexes over F_5. Their cohomology dimensions are (0,0,0), (1,2,1), and the reconstructed total is dim(H^0,H^1,H^2)=(1,2,1). The packet-sum residual is zero.
Exact mathematical passage
\F_5
\mathbb FUse in source · 9 passages
Search aliases: F mathbb
Use in source · 01_three_questions.tex, line 6
[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round] path[use as bounding box] (0,-.6) rectangle (14,4.8); draw[black!25] (4.55,0) -- (4.55,4.45); draw[black!25] (9.25,0) -- (9.25,4.45); node[font=small,align=center] at (2.1,4.35) A residual root; node[font=small,align=center] at (2.1,3.55) x^2+1=0 over F_2; path[draw,fill=black!12] (2.1,2.55) circle (0.24); node[font=small,align=center] at (2.1,2.55) 1; node[font=small,align=center] at (2.1,1.45) one point mod 2; node[font=small,align=center,…
Exact mathematical passage
\mathbb F_2
\mathbf FUse in source · 2 passages
Search aliases: F mathbf
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 448
New checked realization of the finite pattern. The Block 04 implementation separately uses F_5^2xrightarrowd_0 F_5^4xrightarrowd_1 F_5^2, d_0(a,b)=(a,0,0,0), d_1(a,b,c,d)=(b,0). The first packet is the exact complex with dimensions (1,2,1), and the second packet has zero differentials and dimensions (1,2,1). The actual middle cohomology is linearly isomorphic to F_5^2, which is checked in Lean. Independent exhaustive coset tests check all three cohomology degrees after changes of basis. This is …
Exact mathematical passage
\mathbf F_5^2\xrightarrow{d_0}\mathbf F_5^4\xrightarrow{d_1}\mathbf F_5^2, \quad d_0(a,b)=(a,0,0,0),\quad d_1(a,b,c,d)=(b,0).
f_0Use in source · 4 passages
Search aliases: f_0
Use in source · 10_cohomology.tex, line 43
,align=center,font=small] at (6.135,2.265) 0; path[draw=black!30,fill=white] (6.48,1.92) rectangle (7.17,2.61); node[font=small,align=center,font=small] at (6.825,2.265) 0; path[draw=black!30,fill=white] (7.17,1.92) rectangle (7.86,2.61); node[font=small,align=center,font=small] at (7.515,2.265) 0; node[font=small,align=center,anchor=east,font=scriptsize] at (4.95,2.955) f_0; node[font=small,align=center,anchor=east,font=scriptsize] at (4.95,2.265) f_1; node[font=small,align=center,font=scriptsi…
Exact mathematical passage
f_0
f_1Use in source · 5 passages
Search aliases: f_1
Use in source · 10_cohomology.tex, line 44
48,1.92) rectangle (7.17,2.61); node[font=small,align=center,font=small] at (6.825,2.265) 0; path[draw=black!30,fill=white] (7.17,1.92) rectangle (7.86,2.61); node[font=small,align=center,font=small] at (7.515,2.265) 0; node[font=small,align=center,anchor=east,font=scriptsize] at (4.95,2.955) f_0; node[font=small,align=center,anchor=east,font=scriptsize] at (4.95,2.265) f_1; node[font=small,align=center,font=scriptsize] at (5.445,3.53) u_0; node[font=small,align=center,font=scriptsize] at (6.135…
Exact mathematical passage
f_1
\mathbb F_2Use in source · 1 passage
Search aliases: F_2 mathbb
Use in source · 01_three_questions.tex, line 6
[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round] path[use as bounding box] (0,-.6) rectangle (14,4.8); draw[black!25] (4.55,0) -- (4.55,4.45); draw[black!25] (9.25,0) -- (9.25,4.45); node[font=small,align=center] at (2.1,4.35) A residual root; node[font=small,align=center] at (2.1,3.55) x^2+1=0 over F_2; path[draw,fill=black!12] (2.1,2.55) circle (0.24); node[font=small,align=center] at (2.1,2.55) 1; node[font=small,align=center] at (2.1,1.45) one point mod 2; node[font=small,align=center,…
Exact mathematical passage
\mathbb F_2
\mathbb F_{25}Use in source · 1 passage
Search aliases: F_25 mathbb
Use in source · 22_norm_table.tex, line 64
gn=center,font=scriptsize] at (1.76,4.23) b=0; node[font=small,align=center,font=scriptsize] at (2.38,4.23) b=1; node[font=small,align=center,font=scriptsize] at (3,4.23) b=2; node[font=small,align=center,font=scriptsize] at (3.62,4.23) b=3; node[font=small,align=center,font=scriptsize] at (4.24,4.23) b=4; node[font=small,align=center] at (9.6,4.25) u^2=2 in F_25; node[font=small,align=center] at (9.6,3.2) M_a+bu=a · 2b
Exact mathematical passage
\mathbb F_{25}
\mathbb F_3Use in source · 2 passages
Search aliases: F_3 mathbb
Use in source · 29_coefficients.tex, line 3
[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round] path[use as bounding box] (0,-.6) rectangle (14,4.8); node[font=small,align=center] at (2.2,4.3) Scalar over F_3; node[font=small,align=center] at (2.2,3.25) x=0,1,2; node[font=small,align=center] at (2.2,2.35) x^2+1=1,2,2; node[font=small,align=center] at (2.2,1.25) no solution; draw[black!25] (4.45,0.3) -- (4.45,4.45); node[font=small,align=center] at (7,4.3) Matrix over F_3; path[draw=black!30,fill=white] (6.2,2.65) rectangle (7,3.45); no…
Exact mathematical passage
\mathbb F_3
\F_5Use in source · 1 passage
Search aliases: F_5 F mathbf
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 433
[Finite packet complex] The exact modular test uses two invariant subcomplexes over F_5. Their cohomology dimensions are (0,0,0), (1,2,1), and the reconstructed total is dim(H^0,H^1,H^2)=(1,2,1). The packet-sum residual is zero.
Exact mathematical passage
\F_5
\mathbb F_5Use in source · 2 passages
Search aliases: F_5 mathbb
Use in source · 14_spine.tex, line 24
35,0.67) rectangle (7.8,1.53); node[font=small,align=center,font=small] at (7.075,1.1) 2; path[draw=black!30,fill=black!8] (7.8,0.67) rectangle (9.25,1.53); node[font=small,align=center,font=small] at (8.525,1.1) 6; node[font=small,align=center,anchor=east,font=scriptsize] at (4.75,2.82) Z; node[font=small,align=center,anchor=east,font=scriptsize] at (4.75,1.96) F_5; node[font=small,align=center,anchor=east,font=scriptsize] at (4.75,1.1) F_7; node[font=small,align=center,font=scriptsize] at (5.6…
Exact mathematical passage
\mathbb F_5
\mathbf F_5^2Use in source · 2 passages
Search aliases: F_5^2 mathbf
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 448
New checked realization of the finite pattern. The Block 04 implementation separately uses F_5^2xrightarrowd_0 F_5^4xrightarrowd_1 F_5^2, d_0(a,b)=(a,0,0,0), d_1(a,b,c,d)=(b,0). The first packet is the exact complex with dimensions (1,2,1), and the second packet has zero differentials and dimensions (1,2,1). The actual middle cohomology is linearly isomorphic to F_5^2, which is checked in Lean. Independent exhaustive coset tests check all three cohomology degrees after changes of basis. This is …
Exact mathematical passage
\mathbf F_5^2\xrightarrow{d_0}\mathbf F_5^4\xrightarrow{d_1}\mathbf F_5^2, \quad d_0(a,b)=(a,0,0,0),\quad d_1(a,b,c,d)=(b,0).
\mathbf F_5^4Use in source · 1 passage
Search aliases: F_5^4 mathbf
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 448
New checked realization of the finite pattern. The Block 04 implementation separately uses F_5^2xrightarrowd_0 F_5^4xrightarrowd_1 F_5^2, d_0(a,b)=(a,0,0,0), d_1(a,b,c,d)=(b,0). The first packet is the exact complex with dimensions (1,2,1), and the second packet has zero differentials and dimensions (1,2,1). The actual middle cohomology is linearly isomorphic to F_5^2, which is checked in Lean. Independent exhaustive coset tests check all three cohomology degrees after changes of basis. This is …
Exact mathematical passage
\mathbf F_5^2\xrightarrow{d_0}\mathbf F_5^4\xrightarrow{d_1}\mathbf F_5^2, \quad d_0(a,b)=(a,0,0,0),\quad d_1(a,b,c,d)=(b,0).
\mathbb F_7Use in source · 1 passage
Search aliases: F_7 mathbb
Use in source · 14_spine.tex, line 25
2; path[draw=black!30,fill=black!8] (7.8,0.67) rectangle (9.25,1.53); node[font=small,align=center,font=small] at (8.525,1.1) 6; node[font=small,align=center,anchor=east,font=scriptsize] at (4.75,2.82) Z; node[font=small,align=center,anchor=east,font=scriptsize] at (4.75,1.96) F_5; node[font=small,align=center,anchor=east,font=scriptsize] at (4.75,1.1) F_7; node[font=small,align=center,font=scriptsize] at (5.625,3.48) T^2; node[font=small,align=center,font=scriptsize] at (7.075,3.48) T; node[fon…
Exact mathematical passage
\mathbb F_7
\mathbb F_9Use in source · 1 passage
Search aliases: F_9 mathbb
Use in source · 29_coefficients.tex, line 19
!30,fill=black!8] (6.2,1.85) rectangle (7,2.65); node[font=small,align=center,font=small] at (6.6,2.25) 1; path[draw=black!30,fill=white] (7,1.85) rectangle (7.8,2.65); node[font=small,align=center,font=small] at (7.4,2.25) 0; node[font=small,align=center] at (7,0.85) A^2+I=0; draw[black!25] (9.3,0.3) -- (9.3,4.45); node[font=small,align=center] at (11.7,4.3) Scalar over F_9; node[font=small,align=center] at (11.7,3.25) u^2=-1; node[font=small,align=center] at (11.7,2.2) roots u,-u; node[font=sm…
Exact mathematical passage
\mathbb F_9
F_iUse in source · 9 passages
Search aliases: F_i
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1447
Algebraic monodromy data on the selected copies. Fix a field K and a nonzero scalar q∈ K^×. On each constituent W_i, supply an automorphism F_i and a nilpotent endomorphism N_i with F_iN_iF_i^-1=qN_i. The scalar q is common to this finite family. The componentwise operators F_m,N_m on U_m satisfy the same relation; F_m is invertible and N_m is nilpotent. The split maps constructed above intertwine both operators. The proof first identifies the independently defined kernel through an equivariant …
Exact mathematical passage
F_i
F_i^{-1}Use in source · 1 passage
Search aliases: F_i^-1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1448
Algebraic monodromy data on the selected copies. Fix a field K and a nonzero scalar q∈ K^×. On each constituent W_i, supply an automorphism F_i and a nilpotent endomorphism N_i with F_iN_iF_i^-1=qN_i. The scalar q is common to this finite family. The componentwise operators F_m,N_m on U_m satisfy the same relation; F_m is invertible and N_m is nilpotent. The split maps constructed above intertwine both operators. The proof first identifies the independently defined kernel through an equivariant …
Exact mathematical passage
F_iN_iF_i^{-1}=qN_i
F_jUse in source · 2 passages
Search aliases: F_j
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 418
Naturality. If a cochain map F:C^bullet→ D^bullet intertwines the supplied projectors, it restricts to maps F_j:C_j^bullet→ D_j^bullet. The direct-sum decomposition commutes with H^r(F) and the family H^r(F_j). This is checked using representatives in the actual quotients, independently of any dimension computation.
Exact mathematical passage
F_j:C_j^\bullet\to D_j^\bullet
F_mUse in source · 3 passages
Search aliases: F_m
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1449
Algebraic monodromy data on the selected copies. Fix a field K and a nonzero scalar q∈ K^×. On each constituent W_i, supply an automorphism F_i and a nilpotent endomorphism N_i with F_iN_iF_i^-1=qN_i. The scalar q is common to this finite family. The componentwise operators F_m,N_m on U_m satisfy the same relation; F_m is invertible and N_m is nilpotent. The split maps constructed above intertwine both operators. The proof first identifies the independently defined kernel through an equivariant …
Exact mathematical passage
F_m,N_m
\mathbb F_pUse in source · 1 passage
Search aliases: F_p mathbb
Use in source · 28_frobenius_boundary.tex, line 20
nter,font=scriptsize] at (4.85,4) 19; node[font=small,align=center,font=scriptsize] at (4.85,3.3) 0; node[font=small,align=center,font=scriptsize] at (5.45,4) 23; node[font=small,align=center,font=scriptsize] at (5.45,3.3) 0; node[font=small,align=center] at (3.4,4.65) odd primes p; node[font=small,align=center,font=scriptsize] at (3.4,2.7) roots of x^2+1 in F_p; node[font=small,align=center,font=footnotesize] at (3.4,1.5) two factorization types
Exact mathematical passage
\mathbb F_p
F_wUse in source · 3 passages
Search aliases: F_w
Use in source · Read passage · residue_continuation_body.tex, line 164
An actual Mellin endpoint. The counting functions are constant on the integer norm steps. Their measurability, and hence that of S_w, is proved from this fact. For zero-sum w, define F_w(h)=(1+h)∫_1^∞x^-2-hS_w(x) dx (h≥0). Since 1-1/d<1, the logarithmic first moment M_w=∫_1^∞(1+log x) absnx^-2S_w(x) dx is finite. Lean proves the explicit estimate absnF_w(h)-F_w(0)≤ hM_w for every h≥0, and therefore F_w(h)=F_w(0)+O(h) as hdownarrow0. The function here is the displayed integral of the actual ideal…
Exact mathematical passage
F_w(h)=(1+h)\int_1^{\infty}x^{-2-h}S_w(x)\,dx \qquad(h\geq0).
F_{\chi_i}Use in source · 2 passages
Search aliases: F_χ_i chi
Use in source · Read passage · residue_continuation_body.tex, line 181
For a finite family of nontrivial ordinary class characters, suppose functions p,Z satisfy h p(h)→ P≠0, h Z(h)→ Q≠0, and, for all sufficiently small h>0, Z(h)=(p(h)∏_i F_χ_i(h))^e, e∈ N. The checked real-axis criterion then gives e=1 and F_χ_i(0)≠0 for every i. The pole limits and product identity remain hypotheses of this application.
Exact mathematical passage
Z(h)=\left(p(h)\prod_i F_{\chi_i}(h)\right)^e, \qquad e\in\NN.
\FFUse in source · 15 passages
Search aliases: FF
Use in source · pages.tex, line 141
vpage10See cycles, boundaries, and classesA TWO-PACKET COCHAIN COMPLEXMain text: Theorem 4.2 and the finite-complex example. Over FF_5, use bases (e_0,e_1), (u_0,u_1,u_2,u_3), and (f_0,f_1) in degrees zero, one, and two. The only nonzero differential values are d_0e_0=u_0 and d_1u_1=f_0. Consequently d_1d_0=0. vfig10_cohomology headThe quotient is the important step In degree one, the cycles are Span(u_0,u_2,u_3), but u_0 is a boundary. Thus the cohomology is represented by u_2,u_3, not by every…
Exact mathematical passage
\FF_5
\FF_{25}Use in source · 1 passage
Search aliases: FF_25 FF
Use in source · pages.tex, line 322
vpage22A norm is a determinant you can computeTHE QUOTIENT ALGEBRA FF_25Main text: §9; retained scalar/Mellin continuation: actual quotient determinant norms. Use A=FF_5[u]/(u^2-2). Since two is not a square modulo five, this quotient is a field with 25 elements. Every element is uniquely a+bu, with a,b∈FF_5. vfig22_norm_table headCoordinates, multiplication, and norm agree In the basis (1,u), multiplication by a+bu has matrix M_a,b=mata · 2b
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\FF_{25}
\FF_3Use in source · 6 passages
Search aliases: FF_3 FF
Use in source · pages.tex, line 435
vpage29Changing coefficients changes the problemSCALAR ABSENCE VERSUS REPRESENTATION EXISTENCEMain text: Example 8.4 and the declared coefficient categories in §§8--9. Over FF_3, no scalar solves x^2+1=0. But the matrix A=mat0 · -1
Exact mathematical passage
\FF_3
\FF_5Use in source · 4 passages
Search aliases: FF_5 FF
Use in source · pages.tex, line 141
vpage10See cycles, boundaries, and classesA TWO-PACKET COCHAIN COMPLEXMain text: Theorem 4.2 and the finite-complex example. Over FF_5, use bases (e_0,e_1), (u_0,u_1,u_2,u_3), and (f_0,f_1) in degrees zero, one, and two. The only nonzero differential values are d_0e_0=u_0 and d_1u_1=f_0. Consequently d_1d_0=0. vfig10_cohomology headThe quotient is the important step In degree one, the cycles are Span(u_0,u_2,u_3), but u_0 is a boundary. Thus the cohomology is represented by u_2,u_3, not by every…
Exact mathematical passage
\FF_5
\FF_5^5Use in source · 1 passage
Search aliases: FF_5^5 FF
Use in source · pages.tex, line 284
theses do real workTRACE AND SEMISIMPLICITYMain text: Theorem 10.1 and the checked false controls following it. The finite occupancy theorem uses a split semisimple commuting family in characteristic zero. Two tiny matrices show why neither the coefficient condition nor semisimplicity should disappear from the statement. vfig19_hypotheses headTrace zero need not mean an empty image On FF_5^5, the identity is an idempotent of rank five, but its field-valued trace is 5=0 in FF_5. Rank and localiza…
Exact mathematical passage
\FF_5^5
\FF_5^\timesUse in source · 2 passages
Search aliases: FF_5^× FF times
Use in source · pages.tex, line 356
vpage24A nontrivial character may still be exceptionalTHE EXACT SCALAR TESTMain text: §9; scalar/Mellin checkpoint: scalar-power and nonscalar-witness boundaries. A scalar a∈FF_5^× acts on a degree-d algebra with determinant a^d. Therefore the scalar cancellation route tests χ(a^d), not merely χ(a). vfig24_scalar_exception headA failed sufficient test is not an impossibility theorem Let χ_4(2)=i generate the character group of FF_5^×, and write χ_k=χ_4^k. Since 2 generates the unit group, ∃ a:χ_…
Exact mathematical passage
a\in\FF_5^\times
\FF_9Use in source · 2 passages
Search aliases: FF_9 FF
Use in source · pages.tex, line 440
1 · 0∈ M_2(FF_3) satisfies A^2+I=0. And over FF_9=FF_3[u]/(u^2+1), the scalars u and -u solve the same polynomial. vfig29_coefficients headThree outputs with different types The failed scalar problem over FF_3, the successful matrix problem over FF_3, and the successful scalar problem over FF_9 are not contradictory answers. They have different allowed objects. The matrix is multiplication by u in the FF_3-basis (1,u) of the extension field.
Exact mathematical passage
\FF_9=\FF_3[u]/(u^2+1)
\operatorname{Frob}Use in source · 1 passage
Search aliases: Frob operatorname
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1062
cible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficient-field embedding σ_λ:E↪ E_λ therefore sends C to an invertible matrix. Its entries agree with the ordinary representation traces Tr(W_i,λ(Frob_v_j)) when the selected places satisfy the unramified compatibility convention for that coefficient prime. In particular, places above the residue characte…
Exact mathematical passage
\operatorname{Tr}(W_{i,\lambda}(\operatorname{Frob}_{v_j}))
\operatorname{Frob}_{v_j}Use in source · 1 passage
Search aliases: Frob_v_j operatorname
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1062
cible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficient-field embedding σ_λ:E↪ E_λ therefore sends C to an invertible matrix. Its entries agree with the ordinary representation traces Tr(W_i,λ(Frob_v_j)) when the selected places satisfy the unramified compatibility convention for that coefficient prime. In particular, places above the residue characte…
Exact mathematical passage
\operatorname{Tr}(W_{i,\lambda}(\operatorname{Frob}_{v_j}))
gUse in source · 24 passages
Search aliases: g
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 484
[The two sign branches] The characters χ_+(g)=1 and χ_-(g)=-1 have the same reduction modulo 2, but 1not≡-1pmod4. Their first separation depth is therefore 2. One mod-two root contains two branches that become distinct at the next level.
Exact mathematical passage
\chi_+(g)=1
g^{-1}Use in source · 2 passages
Search aliases: g^-1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1001
Derived comparison bounds (Block 16). For a finite group Γ whose order is a unit in a commutative local coefficient ring R, any prescribed equivariant residual isomorphism lifts to an integral isomorphism. Lift its matrix to A and define P=|Γ|^-1∑_g∈Γρ(g)Aτ(g^-1). Reindexing proves equivariance, and reduction recovers the prescribed matrix. Its determinant is a unit. Complete local coefficients with finite quotients, occupied compatible finite conditions and this residual type therefore give act…
Exact mathematical passage
P=|\Gamma|^{-1}\sum_{g\in\Gamma}\rho(g)A\tau(g^{-1}).
G_{\mathbb Q}Use in source · 3 passages
Search aliases: G_ Q mathbb
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 968
The formalization specializes to the actual maps G_ Q_p→ G_ Q induced by the field embeddings and choices of algebraic closures. A nontrivial continuous quadratic character of G_ Q, constructed from its action on i^2=-1, gives the integral 2-adic actions τ(σ)=diag(1,-1), ρ(σ)=1 · -1
Exact mathematical passage
G_{\mathbb Q_p}\to G_{\mathbb Q}
G_{\mathbb Q_p}Use in source · 2 passages
Search aliases: G_ Q_p mathbb
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 968
The formalization specializes to the actual maps G_ Q_p→ G_ Q induced by the field embeddings and choices of algebraic closures. A nontrivial continuous quadratic character of G_ Q, constructed from its action on i^2=-1, gives the integral 2-adic actions τ(σ)=diag(1,-1), ρ(σ)=1 · -1
Exact mathematical passage
G_{\mathbb Q_p}\to G_{\mathbb Q}
G_{F,S}Use in source · 2 passages
Search aliases: G_F,S
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 172
For a Galois problem the principal example is barρ:G_F,S⟶ GL_n(k), with a deformation functor Def_barρ cut out by a determinant and selected local conditions.
Exact mathematical passage
\bar\rho:G_{F,S}\longrightarrow \GL_n(k),
G_KUse in source · 1 passage
Search aliases: G_K
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2413
Actual quadratic local conditions and completion (Block 22). Continue with the fixed packet P_q above. For a characteristic-zero field K, let j_K:G_K⟶ G_ Q be the actual field-induced homomorphism between absolute Galois groups. It includes a chosen embedding of algebraic closures. Let i_K be the image of the chosen square root i under that embedding. Lean proves compatibility with the Galois actions and hence (q∘ j_K)(σ)=1 Longleftrightarrow σ(i_K)=i_K. The fixed-field theorem for the algebraic…
Exact mathematical passage
j_K:G_K\longrightarrow G_{\Q}
G_{\Q_p}Use in source · 2 passages
Search aliases: G_Q_p G_ Q_p Q mathbf
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2498
This inertia is the actual residue-action kernel inside D_A. No equality D_A=G_ Q_p is established in this block. Nor is ramification at two proved: the nontrivial full local character there is a separate result. At three the Z_2-valued sign character is nontrivial on the full local Galois group but trivial on every inertia subgroup covered by the theorem. Thus local triviality and the stated inertia triviality are distinguished by checked arithmetic examples.
Exact mathematical passage
D_A=G_{\Q_p}
\operatorname{Gal}Use in source · 3 passages
Search aliases: Gal operatorname
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1134
Arithmetic coverage for a fixed cyclotomic extension (Block 25). Let n>0 and let L= Q(ζ_n) be a cyclotomic number field. The canonical isomorphism Gal(L/ Q)≃( Z/n Z)^× defines σ_p by σ_p(ζ_n)=ζ_n^p when pnmid n. For every automorphism σ, finite set B⊂ N and bound b, Lean proves that there is a rational prime p>b, with p∉ B and pnmid n, such that σ_p=σ. The prime-existence step uses the proved Dirichlet theorem in the pinned Mathlib dependency.
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\operatorname{Gal}(L/\Q)\simeq(\mathbb Z/n\mathbb Z)^\times
\gcdUse in source · 4 passages
Search aliases: gcd
Use in source · Read passage · residue_continuation_body.tex, line 99
The exact ideal filter. Suppose IJ=(x), n(x) has the indicated sign, and gcd( N J,m)=1. For c∈ Z/m Z, N I≡ cpmod m Longleftrightarrow overline n_ε([x])=c( N Jbmod m). This gives an actual finite residue target, which is allowed to be empty. The product identity before cancellation does not require coprimality.
Exact mathematical passage
\gcd(\Nm J,m)=1
\GLUse in source · 1 passage
Search aliases: GL operatorname
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 172
For a Galois problem the principal example is barρ:G_F,S⟶ GL_n(k), with a deformation functor Def_barρ cut out by a determinant and selected local conditions.
Exact mathematical passage
\bar\rho:G_{F,S}\longrightarrow \GL_n(k),
\mathrm{GL}Use in source · 3 passages
Search aliases: GL mathrm
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2085
Lifting a summand inside a supplied finite-group carrier (Block 18). Let R be a commutative integral domain complete and separated for the (π)-adic filtration. Let Γ be a finite group, and let ρ:Γ→GL_n(R) be an actual integral representation. Assume that a=|Γ|≠0 in R and amidπ^e. For k>0, suppose bar E∈ M_n(R/(π)^e+k), bar E^2=bar E, barρ(g)bar E=bar Ebarρ(g) (g∈Γ). There is an actual integral matrix E satisfying E^2=E, ρ(g)E=Eρ(g), Ebmod(π)^k=bar Ebmod(π)^k. No integral summand or coherent fami…
Exact mathematical passage
\rho:\Gamma\to\mathrm{GL}_n(R)
\operatorname{GL}Use in source · 1 passage
Search aliases: GL operatorname
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 868
Checked representation-level completion and its arithmetic boundary. The eighth formalization block works with actual matrix representations of a fixed group, not only with underlying sets. For a complete separated ring R and ideal I, compatible maps ρ_m:G→GL_d(R/I^m) reconstruct a unique integral representation for that fixed compatible family. Multiplication and inverses are verified after every reduction. If all finite representations are continuous for the discrete quotient topology, the int…
Exact mathematical passage
\rho_m:G\to\operatorname{GL}_d(R/I^m)
\mathrm{GL}_2Use in source · 2 passages
Search aliases: GL_2 mathrm
Use in source · references.tex, line 65
bibitemKisin2010 Mark Kisin. newblock Deformations of G_ Q_p and GL_2( Q_p) representations. newblock In Laurent Berger, Christophe Breuil, and Pierre Colmez, editors, em Repr esentations p-adiques de Groupes p-adiques II: Repr esentations de GL_2( Q_p) et (φ,Γ)-modules, volume 330 of em Ast erisque, pages 511--528. Soci et e Math ematique de France, Paris, 2010.
Exact mathematical passage
\mathrm{GL}_2(\mathbb Q_p)
\operatorname{GL}_dUse in source · 1 passage
Search aliases: GL_d operatorname
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 868
Checked representation-level completion and its arithmetic boundary. The eighth formalization block works with actual matrix representations of a fixed group, not only with underlying sets. For a complete separated ring R and ideal I, compatible maps ρ_m:G→GL_d(R/I^m) reconstruct a unique integral representation for that fixed compatible family. Multiplication and inverses are verified after every reduction. If all finite representations are continuous for the discrete quotient topology, the int…
Exact mathematical passage
\rho_m:G\to\operatorname{GL}_d(R/I^m)
\GL_nUse in source · 1 passage
Search aliases: GL_n GL operatorname
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 172
For a Galois problem the principal example is barρ:G_F,S⟶ GL_n(k), with a deformation functor Def_barρ cut out by a determinant and selected local conditions.
Exact mathematical passage
\bar\rho:G_{F,S}\longrightarrow \GL_n(k),
\mathrm{GL}_nUse in source · 1 passage
Search aliases: GL_n mathrm
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2085
Lifting a summand inside a supplied finite-group carrier (Block 18). Let R be a commutative integral domain complete and separated for the (π)-adic filtration. Let Γ be a finite group, and let ρ:Γ→GL_n(R) be an actual integral representation. Assume that a=|Γ|≠0 in R and amidπ^e. For k>0, suppose bar E∈ M_n(R/(π)^e+k), bar E^2=bar E, barρ(g)bar E=bar Ebarρ(g) (g∈Γ). There is an actual integral matrix E satisfying E^2=E, ρ(g)E=Eρ(g), Ebmod(π)^k=bar Ebmod(π)^k. No integral summand or coherent fami…
Exact mathematical passage
\rho:\Gamma\to\mathrm{GL}_n(R)
HUse in source · 29 passages
Search aliases: H
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 339
[Tangent and obstruction spaces] For a framed representation problem, first-order lifts are described by Z^1(G,adbarρ). For an unframed Schur problem, tangent directions are described by H^1(G,adbarρ). Relations in a presentation of a deformation ring map into H^2(G,adbarρ), which contains the obstruction classes.
Exact mathematical passage
H^1(G,\ad\bar\rho)
hUse in source · 20 passages
Search aliases: h
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 240
[Exact residual partition] The finite implementation partitions the integers 0,…,8 into the three classes modulo 3: {0,3,6}, {1,4,7}, {2,5,8}. Every tested correction of the form x↦ x+p h preserves the residual class exactly.
Exact mathematical passage
x\mapsto x+p h
H^0Use in source · 3 passages
Search aliases: H^0
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 440
[Finite packet complex] The exact modular test uses two invariant subcomplexes over F_5. Their cohomology dimensions are (0,0,0), (1,2,1), and the reconstructed total is dim(H^0,H^1,H^2)=(1,2,1). The packet-sum residual is zero.
Exact mathematical passage
\dim(H^0,H^1,H^2)=(1,2,1).
H^1Use in source · 5 passages
Search aliases: H^1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 339
[Tangent and obstruction spaces] For a framed representation problem, first-order lifts are described by Z^1(G,adbarρ). For an unframed Schur problem, tangent directions are described by H^1(G,adbarρ). Relations in a presentation of a deformation ring map into H^2(G,adbarρ), which contains the obstruction classes.
Exact mathematical passage
H^1(G,\ad\bar\rho)
H^1_jUse in source · 1 passage
Search aliases: H^1_j
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 365
[Cohomology packet decomposition] Under these hypotheses, each packet is a subcomplex and H^r(C^bullet) ≅ ⊕_jH^r(P_jC^bullet). In particular, H^1=⊕_jH^1_j, H^2=⊕_jH^2_j, so tangent and obstruction spaces reconstruct exactly from their packet components.
Exact mathematical passage
H^1=\bigoplus_jH^1_j, \qquad H^2=\bigoplus_jH^2_j,
H^2Use in source · 6 passages
Search aliases: H^2
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 339
[Tangent and obstruction spaces] For a framed representation problem, first-order lifts are described by Z^1(G,adbarρ). For an unframed Schur problem, tangent directions are described by H^1(G,adbarρ). Relations in a presentation of a deformation ring map into H^2(G,adbarρ), which contains the obstruction classes.
Exact mathematical passage
H^2(G,\ad\bar\rho)
H^2_jUse in source · 1 passage
Search aliases: H^2_j
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 365
[Cohomology packet decomposition] Under these hypotheses, each packet is a subcomplex and H^r(C^bullet) ≅ ⊕_jH^r(P_jC^bullet). In particular, H^1=⊕_jH^1_j, H^2=⊕_jH^2_j, so tangent and obstruction spaces reconstruct exactly from their packet components.
Exact mathematical passage
H^1=\bigoplus_jH^1_j, \qquad H^2=\bigoplus_jH^2_j,
H^iUse in source · 1 passage
Search aliases: H^i
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2006
Modular Gaussian elimination computes the ranks of the differentials. For C^i-1xrightarrowd_i-1C^ixrightarrowd_iC^i+1, the finite dimension is evaluated as dim H^i = dim C^i-rank d_i-rank d_i-1. Packet subcomplexes are computed separately and checked against the total.
Exact mathematical passage
\dim H^i = \dim C^i-\rank d_i-\rank d_{i-1}.
H^rUse in source · 5 passages
Search aliases: H^r
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 357
[Cohomology packet decomposition] Under these hypotheses, each packet is a subcomplex and H^r(C^bullet) ≅ ⊕_jH^r(P_jC^bullet). In particular, H^1=⊕_jH^1_j, H^2=⊕_jH^2_j, so tangent and obstruction spaces reconstruct exactly from their packet components.
Exact mathematical passage
\boxed{ H^r(C^\bullet) \cong \bigoplus_jH^r(P_jC^\bullet). }
H_j^rUse in source · 1 passage
Search aliases: H_j^r
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 381
For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet are cochain maps, so they induce maps on these genuine quotients.
Exact mathematical passage
Z_j^r,B_j^r,H_j^r
H_sUse in source · 1 passage
Search aliases: H_s
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 948
Actual local comparisons from fixed-depth certificates (Block 15). Let R be an I-adically complete separated commutative ring with finite coefficient quotients and an injective map into a field K. Fix a finite matrix rank. For finitely many local group maps ϕ_s:H_s→ G and integral template representations τ_s, fix detecting depths d_s. At precision m retain matrices P_s,m as part of the admissible object and require ρ_m(ϕ_s(h))P_s,m=P_s,mτ_s,m(h). For m≥ d_s require their determinants to remain …
Exact mathematical passage
\phi_s:H_s\to G
\HomUse in source · 1 passage
Search aliases: Hom operatorname
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2337
Let R be any commutative ring and B any commutative R-algebra. The coordinate ring A_R=R[X]/(X^2-1) represents precisely this specified packet: Hom_R-alg(A_R,B) ≃{x∈ B:x^2=1} ≃Hom(C_2,B^×) ≃ P_q(B). The first map evaluates the coordinate. Its inverse is the quotient universal property. A root is a unit with itself as inverse and determines a character by its value on the generator of C_2. Finally pull back along q. All inverse identities and naturality under coefficient homomorphisms are proved.…
Exact mathematical passage
\Hom_{R\text{-alg}}(A_R,B) \simeq\{x\in B:x^2=1\} \simeq\Hom(C_2,B^\times) \simeq\mathcal P_q(B).
\Hom_{R\text{-alg}}Use in source · 1 passage
Search aliases: Hom_R-alg Hom text operatorname text
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2337
Let R be any commutative ring and B any commutative R-algebra. The coordinate ring A_R=R[X]/(X^2-1) represents precisely this specified packet: Hom_R-alg(A_R,B) ≃{x∈ B:x^2=1} ≃Hom(C_2,B^×) ≃ P_q(B). The first map evaluates the coordinate. Its inverse is the quotient universal property. A root is a unit with itself as inverse and determines a character by its value on the generator of C_2. Finally pull back along q. All inverse identities and naturality under coefficient homomorphisms are proved.…
Exact mathematical passage
\Hom_{R\text{-alg}}(A_R,B) \simeq\{x\in B:x^2=1\} \simeq\Hom(C_2,B^\times) \simeq\mathcal P_q(B).
i^2Use in source · 1 passage
Search aliases: i^2
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 970
The formalization specializes to the actual maps G_ Q_p→ G_ Q induced by the field embeddings and choices of algebraic closures. A nontrivial continuous quadratic character of G_ Q, constructed from its action on i^2=-1, gives the integral 2-adic actions τ(σ)=diag(1,-1), ρ(σ)=1 · -1
Exact mathematical passage
i^2=-1
I^{d_s}Use in source · 1 passage
Search aliases: I^d_s
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 955
e map into a field K. Fix a finite matrix rank. For finitely many local group maps ϕ_s:H_s→ G and integral template representations τ_s, fix detecting depths d_s. At precision m retain matrices P_s,m as part of the admissible object and require ρ_m(ϕ_s(h))P_s,m=P_s,mτ_s,m(h). For m≥ d_s require their determinants to remain nonzero after reduction to R/I^d_s. Earlier levels omit that test. Finite occupied augmented levels give a compatible branch of both representations and comparisons. The integ…
Exact mathematical passage
R/I^{d_s}
I^kUse in source · 2 passages
Search aliases: I^k
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 908
Completion from conjugacy classes (Block 14). Over an I-adically complete separated commutative coefficient ring R, for a fixed finite rank, the formalization now takes finite-level objects to be actual matrix representations modulo conjugation by an invertible matrix. Assume the coefficient quotients R/I^k are finite. Conjugation commutes with coefficient reduction, and each conjugacy class has finitely many frames over a finite coefficient ring. Thus finite allowed class levels give finite ful…
Exact mathematical passage
R/I^k
i^kUse in source · 1 passage
Search aliases: i^k
Use in source · 24_scalar_exception.tex, line 61
,align=center,font=scriptsize] at (8.5,1.05) no; path[draw=black!30,fill=black!16] (9.3,0.75) rectangle (10.7,1.33); node[font=small,align=center,font=scriptsize] at (10,1.05) yes; path[draw=black!30,fill=black!16] (10.8,0.75) rectangle (12.2,1.33); node[font=small,align=center,font=scriptsize] at (11.5,1.05) yes; node[font=small,align=center,font=footnotesize] at (7,0.05) For χ_k(2)=i^k, a witness exists exactly when kdnot≡0pmod4.;
Exact mathematical passage
\chi_k(2)=i^k
I^mUse in source · 9 passages
Search aliases: I^m
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 868
Checked representation-level completion and its arithmetic boundary. The eighth formalization block works with actual matrix representations of a fixed group, not only with underlying sets. For a complete separated ring R and ideal I, compatible maps ρ_m:G→GL_d(R/I^m) reconstruct a unique integral representation for that fixed compatible family. Multiplication and inverses are verified after every reduction. If all finite representations are continuous for the discrete quotient topology, the int…
Exact mathematical passage
\rho_m:G\to\operatorname{GL}_d(R/I^m)
I_{\mathfrak Q_v}Use in source · 2 passages
Search aliases: I_ Q_v mathfrak
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1287
The choice of absolute prime and Frobenius lift is also controlled. Let ( Q_u,γ_u) and ( Q_v,γ_v) be any two actual arithmetic witnesses at the same rational prime. Transitivity on primes of A above (p) gives a∈ G_ Q with Q_v=a Q_u, γ_v^-1aγ_u a^-1∈ I_ Q_v. Consequently, any representation trivial on I_ Q_v has the same trace on the two witnesses. For representations factoring through L, triviality on this actual absolute inertia is derived whenever p∉ S_L. The restrictions of all absolute witne…
Exact mathematical passage
\mathfrak Q_v=a\mathfrak Q_u,\qquad \gamma_v^{-1}a\gamma_u a^{-1}\in I_{\mathfrak Q_v}.
I_2Use in source · 1 passage
Search aliases: I_2
Use in source · 19_hypotheses.tex, line 67
.7); node[font=small,align=center,font=small] at (8,2.3) 0; path[draw=black!30,fill=white] (8.4,1.9) rectangle (9.2,2.7); node[font=small,align=center,font=small] at (8.8,2.3) 0; node[font=small,align=center] at (11.6,2.95) only eigenvalue 0; node[font=small,align=center] at (11.6,2.1) dimker N=1; node[font=small,align=center] at (10.4,0.8) sole-eigenvalue product =I_2; node[font=small,align=center,font=footnotesize] at (7,-0.05) Keep characteristic zero for trace detection, and split semisimpli…
Exact mathematical passage
=I_2
I_5Use in source · 1 passage
Search aliases: I_5
Use in source · 19_hypotheses.tex, line 3
[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round] path[use as bounding box] (0,-.6) rectangle (14,4.8); node[font=small,align=center,font=small] at (2.7,4.65) I_5 over F_5; path[draw=black!30,fill=black!8] (1.5,3.47) rectangle (1.98,3.95); node[font=small,align=center,font=small] at (1.74,3.71) 1; path[draw=black!30,fill=white] (1.98,3.47) rectangle (2.46,3.95); node[font=small,align=center,font=small] at (2.22,3.71) 0; path[draw=black!30,fill=white] (2.46,3.47) rectangle (2.94,3.95); node[…
Exact mathematical passage
I_5
I_AUse in source · 2 passages
Search aliases: I_A
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2479
The inertia theorem uses actual residue actions. For a valuation subring A of an extension field L/K, let D_A be its decomposition subgroup in Aut_K(L) and put I_A=ker(D_A⟶Aut(k_A)), where k_A is the residue field. If 2 is a unit in A, every member of I_A fixes every y∈ L with y^2=-1. Indeed y^-1=-y, so y belongs to A and is a unit there. An automorphism sends y to y or -y. Their residues are distinct because 2 and y are units, whereas inertia fixes residues. The second possibility is excluded.
Exact mathematical passage
I_A=\ker\bigl(D_A\longrightarrow\operatorname{Aut}(k_A)\bigr),
i_jUse in source · 5 passages
Search aliases: i_j
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 382
For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet are cochain maps, so they induce maps on these genuine quotients.
Exact mathematical passage
i_j:C_j^\bullet\to C^\bullet
i_KUse in source · 2 passages
Search aliases: i_K
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2417
Actual quadratic local conditions and completion (Block 22). Continue with the fixed packet P_q above. For a characteristic-zero field K, let j_K:G_K⟶ G_ Q be the actual field-induced homomorphism between absolute Galois groups. It includes a chosen embedding of algebraic closures. Let i_K be the image of the chosen square root i under that embedding. Lean proves compatibility with the Galois actions and hence (q∘ j_K)(σ)=1 Longleftrightarrow σ(i_K)=i_K. The fixed-field theorem for the algebraic…
Exact mathematical passage
i_K
\operatorname{im}Use in source · 7 passages
Search aliases: im operatorname
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 375
For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet are cochain maps, so they induce maps on these genuine quotients.
Exact mathematical passage
C_j^r=\operatorname{im}P_j^r
j_KUse in source · 4 passages
Search aliases: j_K
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2413
Actual quadratic local conditions and completion (Block 22). Continue with the fixed packet P_q above. For a characteristic-zero field K, let j_K:G_K⟶ G_ Q be the actual field-induced homomorphism between absolute Galois groups. It includes a chosen embedding of algebraic closures. Let i_K be the image of the chosen square root i under that embedding. Lean proves compatibility with the Galois actions and hence (q∘ j_K)(σ)=1 Longleftrightarrow σ(i_K)=i_K. The fixed-field theorem for the algebraic…
Exact mathematical passage
j_K:G_K\longrightarrow G_{\Q}
K^\timesUse in source · 3 passages
Search aliases: K^× times
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1446
Algebraic monodromy data on the selected copies. Fix a field K and a nonzero scalar q∈ K^×. On each constituent W_i, supply an automorphism F_i and a nilpotent endomorphism N_i with F_iN_iF_i^-1=qN_i. The scalar q is common to this finite family. The componentwise operators F_m,N_m on U_m satisfy the same relation; F_m is invertible and N_m is nilpotent. The split maps constructed above intertwine both operators. The proof first identifies the independently defined kernel through an equivariant …
Exact mathematical passage
q\in K^\times
k_AUse in source · 2 passages
Search aliases: k_A
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2479
The inertia theorem uses actual residue actions. For a valuation subring A of an extension field L/K, let D_A be its decomposition subgroup in Aut_K(L) and put I_A=ker(D_A⟶Aut(k_A)), where k_A is the residue field. If 2 is a unit in A, every member of I_A fixes every y∈ L with y^2=-1. Indeed y^-1=-y, so y belongs to A and is a unit there. An automorphism sends y to y or -y. Their residues are distinct because 2 and y are units, whereas inertia fixes residues. The second possibility is excluded.
Exact mathematical passage
I_A=\ker\bigl(D_A\longrightarrow\operatorname{Aut}(k_A)\bigr),
\kerUse in source · 16 passages
Search aliases: ker
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 377
For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet are cochain maps, so they induce maps on these genuine quotients.
Exact mathematical passage
Z^r=\ker d_r,\qquad B^r=\operatorname{im}d_{r-1}\subseteq Z^r, \qquad H^r=Z^r/B^r,
L^2Use in source · 3 passages
Search aliases: L^2
Use in source · 26_boundary_count.tex, line 291
at (3,4.4) [0,12)^2∩ Z^2; node[font=small,align=center] at (9.8,3.75) four full 5×5 cells; node[font=small,align=center] at (9.8,2.85) 100 bulk points: sum 0; node[font=small,align=center] at (9.8,1.95) 44 boundary points; node[font=small,align=center] at (9.8,1.05) actual total S_12=1; node[font=small,align=center,font=footnotesize] at (7,-0.1) |S_L|≤ L^2-(L-(Lbmod5))^2 for this square-lattice model.;
Exact mathematical passage
|S_L|\le L^2-(L-(L\bmod5))^2
L_dUse in source · 1 passage
Search aliases: L_d
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2232
General finite logarithmic recovery (Block 20). Let A be a rational algebra, without any commutativity assumption on A. For a unipotent element U and a positive integer d satisfying (U-1)^d=0, put L_d(U)=∑_j=1^d-1frac(-1)^j+1j(U-1)^j. The value is independent of the chosen annihilating exponent. The finite logarithm and the nilpotent exponential satisfy L(exp N)=N, exp(L(U))=U. Both maps preserve every annihilating exponent, and hence preserve the exact nilpotency class of N and exp(N)-1, or of …
Exact mathematical passage
L_d(U)=\sum_{j=1}^{d-1}\frac{(-1)^{j+1}}{j}(U-1)^j.
\landUse in source · 1 passage
Search aliases: land
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1806
An admissible realization here includes continuous coefficient-field members and membership of every finite reduction in its designated admissible class. Under the completion hypotheses and these supplied carrier hypotheses, Lean proves · ∃ ρ, (x_p)_p: ρ is an admissible realization of the full chosen spine, 0≠ x_p∈im(e_p) for every p · Longleftrightarrow (∀λ, X_λ,1( C)≠∅) land (∀λ, ∀ m≥2, X_λ,m( C)≠∅) land (∀ p, n_p>0). The packet index may be infinite. Each x_p belongs to its own supplied carr…
Exact mathematical passage
\begin{aligned} &\exists\,\rho,\ (x_p)_p:\quad \rho\text{ is an admissible realization of the full chosen spine},\quad 0\ne x_p\in\operatorname{im}(e_p)\text{ for every }p\\ &\quad\Longleftrightarrow\quad \bigl(\forall\lambda,\ X_{\lambda,1}(\Compat)\ne\varnothing\bigr) \ \land\ \bigl(\forall\lambda,\ \forall m\ge2, X_{\lambda,m}(\Compat)\ne\varnothing\bigr) \ \land\ \bigl(\forall p,\ n_p>0\bigr). \end{aligned}
\lceilUse in source · 5 passages
Search aliases: lceil
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 587
[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.
Exact mathematical passage
\boxed{ q^{m-\lceil m/e\rceil} }
\lfloorUse in source · 1 passage
Search aliases: lfloor
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1976
The residual module partitions coordinates modulo p, verifies compatibility x_m+1≡ x_mpmodp^m, and reconstructs the digit a_m-1 by a_m-1=lfloorfracx_mp^m-1rfloorpmod p.
Exact mathematical passage
a_{m-1}=\left\lfloor\frac{x_m}{p^{m-1}}\right\rfloor\pmod p.
\lim_mUse in source · 1 passage
Search aliases: lim_m lim
Use in source · pages.tex, line 107
headCount vertices and count branches separately There are three roots modulo nine: 0,3,6. But the only roots modulo 27 are 0,9,18, and all three reduce to zero modulo nine. The other two vertices at the previous level have no children. Counting them as permanent branches would be incorrect. #X_m=1,3,3,9,9,27,…, Z(T)=fracT+3T^21-3T^2. The count-growth exponent is lim_mlog_3(#X_m)/m=1/2. Nevertheless, a compatible branch reconstructs x∈ZZ_3 with x^2=0. Since ZZ_3 has no nonzero nilpotent, the onl…
Exact mathematical passage
\lim_m\log_3(\#X_m)/m=1/2
\operatorname{LocallyTrivial}Use in source · 4 passages
Search aliases: LocallyTrivial operatorname
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2438
For every commutative coefficient ring B and χ∈ P_q(B), define χ_K=χ∘ j_K. The local character is continuous and commutes with coefficient change. The precise local admissibility condition is LocallyTrivial_K(χ) Longleftrightarrow χ_K=1. Descent through the fixed quadratic quotient proves LocallyTrivial_K(χ) Longleftrightarrow (∃ x∈ K, x^2=-1) lor χ=1. No reducedness, characteristic-zero or domain hypothesis is imposed on B. In particular, the formula applies to the finite coefficient quotients …
Exact mathematical passage
\operatorname{LocallyTrivial}_K(\chi) \quad\Longleftrightarrow\quad \chi_K=1.
\operatorname{LocallyTrivial}_KUse in source · 3 passages
Search aliases: LocallyTrivial_K operatorname
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2438
For every commutative coefficient ring B and χ∈ P_q(B), define χ_K=χ∘ j_K. The local character is continuous and commutes with coefficient change. The precise local admissibility condition is LocallyTrivial_K(χ) Longleftrightarrow χ_K=1. Descent through the fixed quadratic quotient proves LocallyTrivial_K(χ) Longleftrightarrow (∃ x∈ K, x^2=-1) lor χ=1. No reducedness, characteristic-zero or domain hypothesis is imposed on B. In particular, the formula applies to the finite coefficient quotients …
Exact mathematical passage
\operatorname{LocallyTrivial}_K(\chi) \quad\Longleftrightarrow\quad \chi_K=1.
\operatorname{LocallyTrivial}_{\Q_p}Use in source · 1 passage
Search aliases: LocallyTrivial_Q_p LocallyTrivial_ Q_p operatorname Q operatorname mathbf
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2464
l. Reduction modulo three rules out a root in Q_3, and reduction modulo four rules out a root in Q_2. At an odd prime, any residue root of X^2+1 has nonzero derivative and lifts by the pinned Mathlib Hensel theorem. The finite-field square criterion then gives the stated congruence. An explicit Hensel application at the residue root 2 modulo 5 supplies a root in Z_5. Consequently LocallyTrivial_ Q_p(χ) Longleftrightarrow p≡1pmod4 lor χ=1. The arithmetic prime p is independent of the coefficient …
Exact mathematical passage
\boxed{ \operatorname{LocallyTrivial}_{\Q_p}(\chi) \quad\Longleftrightarrow\quad p\equiv1\pmod4\ \lor\ \chi=1. }
\log_2Use in source · 1 passage
Search aliases: log_2 log
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1590
Block 13 also established an extracted inverse on the explicit class N^3=0: log_2 U=(U-I)-tfrac12(U-I)^2, a^-1log_2(exp(aN))=N (a≠0). Both exponential/logarithmic inverse identities are checked on the cubic nilpotent and unipotent loci. If a supplied common sample u has r_D(u)=r_E(u)=I and t(u)≠0, an intertwiner of the reconstructed cubic actions recovers both original group equivariance and monodromy equivariance. The general nilpotence exponential theorem is not restricted to this cubic class,…
Exact mathematical passage
\log_2 U=(U-I)-\tfrac12(U-I)^2,\qquad a^{-1}\log_2(\exp(aN))=N\quad(a\ne0).
\log_3Use in source · 1 passage
Search aliases: log_3 log
Use in source · pages.tex, line 107
headCount vertices and count branches separately There are three roots modulo nine: 0,3,6. But the only roots modulo 27 are 0,9,18, and all three reduce to zero modulo nine. The other two vertices at the previous level have no children. Counting them as permanent branches would be incorrect. #X_m=1,3,3,9,9,27,…, Z(T)=fracT+3T^21-3T^2. The count-growth exponent is lim_mlog_3(#X_m)/m=1/2. Nevertheless, a compatible branch reconstructs x∈ZZ_3 with x^2=0. Since ZZ_3 has no nonzero nilpotent, the onl…
Exact mathematical passage
\lim_m\log_3(\#X_m)/m=1/2
\LongleftrightarrowUse in source · 15 passages
Search aliases: Longleftrightarrow
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 407
Scope of the splitting. The algebraic proof works over any commutative ring and does not divide by 2, so it applies in characteristic two when the stated integral chain projectors exist. It does not construct those projectors. The identification of a particular Galois tangent or obstruction problem with this cohomology remains the preceding classical input. The splitting detects vanishing: H^r(C^bullet)=0 Longleftrightarrow H^r(C_j^bullet)=0 for every j. It does not force either side to vanish o…
Exact mathematical passage
H^r(C^\bullet)=0\quad\Longleftrightarrow\quad H^r(C_j^\bullet)=0\text{ for every }j.
\LongrightarrowUse in source · 1 passage
Search aliases: Longrightarrow
Use in source · pages.tex, line 132
In the paper, finite nonempty levels and actual total reduction maps provide compactness even when some vertices die and some transition maps are not onto. One can retain only vertices with descendants at arbitrarily large depths. Finiteness makes that surviving part nonempty and permits compatible choices. X_m≠∅ for all m notLongrightarrow varprojlim_m X_m≠∅ without the compactness hypotheses. takeawayArbitrarily long finite paths are not automatically one infinite path. The finite-level hypoth…
Exact mathematical passage
X_m\ne\varnothing\ \text{for all }m \quad\not\Longrightarrow\quad \varprojlim_m X_m\ne\varnothing \quad\text{without the compactness hypotheses.}
\lorUse in source · 3 passages
Search aliases: lor
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2443
For every commutative coefficient ring B and χ∈ P_q(B), define χ_K=χ∘ j_K. The local character is continuous and commutes with coefficient change. The precise local admissibility condition is LocallyTrivial_K(χ) Longleftrightarrow χ_K=1. Descent through the fixed quadratic quotient proves LocallyTrivial_K(χ) Longleftrightarrow (∃ x∈ K, x^2=-1) lor χ=1. No reducedness, characteristic-zero or domain hypothesis is imposed on B. In particular, the formula applies to the finite coefficient quotients …
Exact mathematical passage
\operatorname{LocallyTrivial}_K(\chi) \quad\Longleftrightarrow\quad \bigl(\exists x\in K,\ x^2=-1\bigr)\ \lor\ \chi=1.
\lvertUse in source · 1 passage
Search aliases: lvert
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2457
The p-adic square-root criterion is proved for every prime: (∃ x∈ Q_p, x^2=-1) Longleftrightarrow p≡1pmod4. A root satisfies lvert xrvert_p^2=1 and is therefore integral. Reduction modulo three rules out a root in Q_3, and reduction modulo four rules out a root in Q_2. At an odd prime, any residue root of X^2+1 has nonzero derivative and lifts by the pinned Mathlib Hensel theorem. The finite-field square criterion then gives the stated congruence. An explicit Hensel application at the residue ro…
Exact mathematical passage
\lvert x\rvert_p^2=1
\bm mUse in source · 10 passages
Search aliases: m bm
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1365
Use the same multiplicities in every member of the compatible carrier. Direct sums preserve strict compatibility, and m_i≤ c_i makes V_λ a summand of W_λ. At the separating places, the target trace vector is C m. Full column rank makes m the unique solution.
Exact mathematical passage
C\bm m
\boldsymbol mUse in source · 1 passage
Search aliases: m boldsymbol
Use in source · 15_separator.tex, line 28
tsize] at (1.95,2.65) (12); node[font=small,align=center,anchor=east,font=scriptsize] at (1.95,1.85) (123); node[font=small,align=center,font=scriptsize] at (2.5,4.08) trivial; node[font=small,align=center,font=scriptsize] at (3.3,4.08) sign; node[font=small,align=center,font=scriptsize] at (4.1,4.08) standard; node[font=small,align=center,font=small] at (6.55,4.55) m; path[draw=black!30,fill=black!8] (6.15,3.05) rectangle (6.95,3.85); node[font=small,align=center,font=small] at (6.55,3.45) 2; p…
Exact mathematical passage
\boldsymbol m
m^dUse in source · 1 passage
Search aliases: m^d
Use in source · Read passage · residue_continuation_body.tex, line 73
The residue norm and its sign. For m∈ N, reducing the multiplication matrices of O modulo m and taking their determinants constructs a monoid homomorphism overline n: O/m O⟶ Z/m Z, overline n([x])=n(x)bmod m. The construction is proved independent of the integral basis. For m≠0, its source is finite of cardinality m^d. Consequently x-y∈ m O implies n(x)≡ n(y)pmod m. No surjectivity of overline n is asserted.
Exact mathematical passage
m^d
m_1Use in source · 3 passages
Search aliases: m_1
Use in source · pages.tex, line 260
More generally, the kernel of C modulo 2^n has exactly two elements: (0,0,0), (2^n-1,2^n-1,0). To see this, the second equation gives m_1=m_2; the third gives m_3=2m_1; the first then gives 6m_1=0. Three is a unit modulo 2^n, leaving precisely the displayed possibilities.
Exact mathematical passage
m_1=m_2
M_2Use in source · 1 passage
Search aliases: M_2
Use in source · pages.tex, line 436
vpage29Changing coefficients changes the problemSCALAR ABSENCE VERSUS REPRESENTATION EXISTENCEMain text: Example 8.4 and the declared coefficient categories in §§8--9. Over FF_3, no scalar solves x^2+1=0. But the matrix A=mat0 · -1 1 · 0∈ M_2(FF_3) satisfies A^2+I=0. And over FF_9=FF_3[u]/(u^2+1), the scalars u and -u solve the same polynomial. vfig29_coefficients headThree outputs with different types The failed scalar problem over FF_3, the successful matrix problem over FF_3, and the successf…
Exact mathematical passage
A=\mat{0&-1\\1&0}\in M_2(\FF_3) \quad\text{satisfies}\quad A^2+I=0.
m_2Use in source · 1 passage
Search aliases: m_2
Use in source · pages.tex, line 260
More generally, the kernel of C modulo 2^n has exactly two elements: (0,0,0), (2^n-1,2^n-1,0). To see this, the second equation gives m_1=m_2; the third gives m_3=2m_1; the first then gives 6m_1=0. Three is a unit modulo 2^n, leaving precisely the displayed possibilities.
Exact mathematical passage
m_1=m_2
m_3Use in source · 1 passage
Search aliases: m_3
Use in source · pages.tex, line 260
More generally, the kernel of C modulo 2^n has exactly two elements: (0,0,0), (2^n-1,2^n-1,0). To see this, the second equation gives m_1=m_2; the third gives m_3=2m_1; the first then gives 6m_1=0. Three is a unit modulo 2^n, leaving precisely the displayed possibilities.
Exact mathematical passage
m_3=2m_1
M_{a+bu}Use in source · 1 passage
Search aliases: M_a+bu
Use in source · 22_norm_table.tex, line 65
mall,align=center,font=scriptsize] at (2.38,4.23) b=1; node[font=small,align=center,font=scriptsize] at (3,4.23) b=2; node[font=small,align=center,font=scriptsize] at (3.62,4.23) b=3; node[font=small,align=center,font=scriptsize] at (4.24,4.23) b=4; node[font=small,align=center] at (9.6,4.25) u^2=2 in F_25; node[font=small,align=center] at (9.6,3.2) M_a+bu=a · 2b b · a; node[font=small,align=center] at (9.6,2) #N^-1(0)=1; node[font=small,align=center] at (9.6,1.05) #N^-1(r)=6 for r≠0; node[font=…
Exact mathematical passage
M_{a+bu}=\begin{pmatrix}a&2b\\b&a\end{pmatrix}
M_{a,b}Use in source · 1 passage
Search aliases: M_a,b
Use in source · pages.tex, line 327
GEBRA FF_25Main text: §9; retained scalar/Mellin continuation: actual quotient determinant norms. Use A=FF_5[u]/(u^2-2). Since two is not a square modulo five, this quotient is a field with 25 elements. Every element is uniquely a+bu, with a,b∈FF_5. vfig22_norm_table headCoordinates, multiplication, and norm agree In the basis (1,u), multiplication by a+bu has matrix M_a,b=mata · 2b b · a, N(a+bu)=det M_a,b=a^2-2b^2pmod5. The table shows every norm value. Only the zero element has norm zero. Eac…
Exact mathematical passage
M_{a,b}=\mat{a&2b\\b&a}, \qquad N(a+bu)=\det M_{a,b}=a^2-2b^2\pmod5.
M_rUse in source · 1 passage
Search aliases: M_r
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1056
[Finite Frobenius separation] Let mathscr W_1,…,mathscr W_r be pairwise nonisomorphic strictly compatible systems whose members are absolutely irreducible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficient-field embedding σ_λ:E↪ E_λ therefore sends C to an invertible matrix. Its entries agree with the ordinary representation traces Tr(W_i,λ(Frob_v_j)) …
Exact mathematical passage
C=(t_i(v_j))_{j,i}\in M_r(E),\qquad \det C\ne0.
M_wUse in source · 2 passages
Search aliases: M_w
Use in source · Read passage · residue_continuation_body.tex, line 169
An actual Mellin endpoint. The counting functions are constant on the integer norm steps. Their measurability, and hence that of S_w, is proved from this fact. For zero-sum w, define F_w(h)=(1+h)∫_1^∞x^-2-hS_w(x) dx (h≥0). Since 1-1/d<1, the logarithmic first moment M_w=∫_1^∞(1+log x) absnx^-2S_w(x) dx is finite. Lean proves the explicit estimate absnF_w(h)-F_w(0)≤ hM_w for every h≥0, and therefore F_w(h)=F_w(0)+O(h) as hdownarrow0. The function here is the displayed integral of the actual ideal…
Exact mathematical passage
M_w=\int_1^{\infty}(1+\log x)\,\absn{x^{-2}S_w(x)}\,dx
\matUse in source · 3 passages
Search aliases: mat
Use in source · pages.tex, line 223
of S_3: trivial, sign, and standard. On the identity, a transposition, and a three-cycle, their values form the matrix below. Rows are samples; columns are constituent types. vfig15_separator headWhat full rank buys Since det C=6, these samples distinguish every rational multiplicity vector. A target with two trivial constituents, no sign constituent, and one standard constituent has m=mat2 0 1, b=C m=mat4 2 1. Solving C m= b recovers the unique rational vector. The standard character is not int…
Exact mathematical passage
\bm m=\mat{2\\0\\1},\qquad \bm b=C\bm m=\mat{4\\2\\1}.
\mathscrUse in source · 5 passages
Search aliases: mathscr
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1052
[Finite Frobenius separation] Let mathscr W_1,…,mathscr W_r be pairwise nonisomorphic strictly compatible systems whose members are absolutely irreducible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficient-field embedding σ_λ:E↪ E_λ therefore sends C to an invertible matrix. Its entries agree with the ordinary representation traces Tr(W_i,λ(Frob_v_j)) …
Exact mathematical passage
\mathscr W_1,\ldots,\mathscr W_r
\maxUse in source · 4 passages
Search aliases: max
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 505
[Weighted smooth-chart count] For weights w=(w_1,…,w_d), N_ w(m) = q^∑_i=1^dmax(m-w_i,0).
Exact mathematical passage
\boxed{ N_{\bm w}(m) = q^{\sum_{i=1}^d\max(m-w_i,0)}. }
\midUse in source · 7 passages
Search aliases: mid
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 316
For the digit claim, subtract the chosen representative of the residue and divide the remainder by the uniformizer. Iterating gives unique digits. Conversely, the partial sums of any digit stream form a compatible Cauchy family and reconstruct uniquely by completeness and separatedness. More precisely, for each parameter there is a unique stream (a_i,j)_j≥1 such that ϖ^n+1mid t_i-∑_j=1^ns(a_i,j)ϖ^j (n≥0). The section is a set-theoretic section of reduction, not an asserted embedding of residue f…
Exact mathematical passage
\varpi^{n+1}\mid t_i-\sum_{j=1}^{n}s(a_{i,j})\varpi^j\qquad(n\ge0).
NUse in source · 79 passages
Search aliases: N
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 311
A local map sends each parameter into the image of (ϖ) in O/ϖ^m. Conversely, every tuple in that image is nilpotent and defines a coefficient-preserving evaluation homomorphism. This homomorphism is local. To prove uniqueness without assuming continuity of an arbitrary competing homomorphism, observe in one variable that a homomorphism killing X^N only sees the finite polynomial truncation modulo X^N. Induction over the finite number of variables gives uniqueness in all dimensions. The actual lo…
Exact mathematical passage
X^N
\mathbb NUse in source · 3 passages
Search aliases: N mathbb
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1138
Arithmetic coverage for a fixed cyclotomic extension (Block 25). Let n>0 and let L= Q(ζ_n) be a cyclotomic number field. The canonical isomorphism Gal(L/ Q)≃( Z/n Z)^× defines σ_p by σ_p(ζ_n)=ζ_n^p when pnmid n. For every automorphism σ, finite set B⊂ N and bound b, Lean proves that there is a rational prime p>b, with p∉ B and pnmid n, such that σ_p=σ. The prime-existence step uses the proved Dirichlet theorem in the pinned Mathlib dependency.
Exact mathematical passage
B\subset\mathbb N
\mathbf NUse in source · 1 passage
Search aliases: N mathbf
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 712
Why the finite hypothesis cannot be dropped. Take vertices {*}sqcup{v_n:n∈ N}, with * isolated and edges v_n+1→ v_n. Every strongly connected component is a singleton, and {*} is the unique source component. No finite seed set covers all vertices: it can reach only indices no larger than the largest chosen index. Both the unique-source statement and the failure of every finite cover are checked in Lean. Hence one source component in an infinite condensation does not imply that one seed reaches a…
Exact mathematical passage
\{*\}\sqcup\{v_n:n\in\mathbf N\}
N^{-1}Use in source · 2 passages
Search aliases: N^-1
Use in source · 22_norm_table.tex, line 66
b · a; node[font=small,align=center] at (9.6,2) #N^-1(0)=1; node[font=small,align=center] at (9.6,1.05) #N^-1(r)=6 for r≠0; node[font=small,align=center,font=footnotesize] at (7,-0.1) All 25 elements are shown. Every nonzero element is a unit.;
Exact mathematical passage
\#N^{-1}(0)=1
N^2Use in source · 2 passages
Search aliases: N^2
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1527
An explicit action of C_2× Z has ν(φ)=1/3, F=diag(2,6,18) and Ne_1=e_0, Ne_2=e_1, Ne_0=0. Trivial inertia gives local dimension one and denominator 1-2T; scalar sign inertia gives dimension zero and denominator one. The entire F and N are equal in these two examples, and N^2≠0. Thus inertia cannot be omitted. Conversely, unramified chains of lengths one and three have equal Euler denominators while only the first has N=0. Euler factors therefore do not identify complete monodromy data.
Exact mathematical passage
N^2\ne0
N^3Use in source · 1 passage
Search aliases: N^3
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1589
Block 13 also established an extracted inverse on the explicit class N^3=0: log_2 U=(U-I)-tfrac12(U-I)^2, a^-1log_2(exp(aN))=N (a≠0). Both exponential/logarithmic inverse identities are checked on the cubic nilpotent and unipotent loci. If a supplied common sample u has r_D(u)=r_E(u)=I and t(u)≠0, an intertwiner of the reconstructed cubic actions recovers both original group equivariance and monodromy equivariance. The general nilpotence exponential theorem is not restricted to this cubic class,…
Exact mathematical passage
N^3=0
N^5Use in source · 1 passage
Search aliases: N^5
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2292
Actual affine representations with monodromy chains of every finite length invoke these theorems. In particular a length-six example has N^5≠0 and N^6=0 and is recovered from its actual reconstructed action. The length-four example still defeats the old quadratic logarithm, while the full finite logarithm recovers it.
Exact mathematical passage
N^5\ne0
N^6Use in source · 1 passage
Search aliases: N^6
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2292
Actual affine representations with monodromy chains of every finite length invoke these theorems. In particular a length-six example has N^5≠0 and N^6=0 and is recovered from its actual reconstructed action. The length-four example still defeats the old quadratic logarithm, while the full finite logarithm recovers it.
Exact mathematical passage
N^6=0
N^dUse in source · 2 passages
Search aliases: N^d
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1578
Finite-exponential reconstruction and a cubic inverse (Block 13). Let K be a characteristic-zero field and let (r,N) be supplied group data with nilpotent N and r(g)N=ν(g)Nr(g). For a supplied scalar map satisfying t(gh)=t(g)+ν(g)t(h), the formalization constructs ρ(g)=exp(t(g)N)r(g), exp(aN)=∑_j=0^d-1fraca^jj!N^j (N^d=0). Lean proves the representation laws, actual finite-copy and split-map compatibility, and joint continuity from continuity of N, r and t with the stated continuous vector-space…
Exact mathematical passage
\rho(g)=\exp(t(g)N)r(g),\qquad \exp(aN)=\sum_{j=0}^{d-1}\frac{a^j}{j!}N^j\quad(N^d=0).
N^jUse in source · 1 passage
Search aliases: N^j
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1578
Finite-exponential reconstruction and a cubic inverse (Block 13). Let K be a characteristic-zero field and let (r,N) be supplied group data with nilpotent N and r(g)N=ν(g)Nr(g). For a supplied scalar map satisfying t(gh)=t(g)+ν(g)t(h), the formalization constructs ρ(g)=exp(t(g)N)r(g), exp(aN)=∑_j=0^d-1fraca^jj!N^j (N^d=0). Lean proves the representation laws, actual finite-copy and split-map compatibility, and joint continuity from continuity of N, r and t with the stated continuous vector-space…
Exact mathematical passage
\rho(g)=\exp(t(g)N)r(g),\qquad \exp(aN)=\sum_{j=0}^{d-1}\frac{a^j}{j!}N^j\quad(N^d=0).
N^nUse in source · 1 passage
Search aliases: N^n
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1472
The exact special chain has basis e_0,…,e_n-1 and operators Ne_0=0, Ne_j=e_j-1 (j>0), Fe_j=aq^-je_j, a,q∈ K^×. Lean proves N^n=0, and for n>0 also N^n-1≠0 and ker N=Ke_0. Consequently its kernel polynomial is X-a, whereas its whole-space polynomial is ∏_j=0^n-1(X-aq^-j). The zero-length chain is included as a zero-dimensional space; the assertion ker N=Ke_0 is made only for positive length.
Exact mathematical passage
N^n=0
N^{n-1}Use in source · 1 passage
Search aliases: N^n-1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1472
The exact special chain has basis e_0,…,e_n-1 and operators Ne_0=0, Ne_j=e_j-1 (j>0), Fe_j=aq^-je_j, a,q∈ K^×. Lean proves N^n=0, and for n>0 also N^n-1≠0 and ker N=Ke_0. Consequently its kernel polynomial is X-a, whereas its whole-space polynomial is ∏_j=0^n-1(X-aq^-j). The zero-length chain is included as a zero-dimensional space; the assertion ker N=Ke_0 is made only for positive length.
Exact mathematical passage
N^{n-1}\ne0
N_{\mathbb Q/\mathbb Q}Use in source · 2 passages
Search aliases: N_ Q/ Q mathbb mathbb
Use in source · 21_signed_norm.tex, line 26
ode[font=small,align=center,font=scriptsize] at (6.75,2.2) integer generators; path[draw,fill=black!15] (3.409091,2.75) circle (0.12); draw[->] (3.409091,2.55) -- (6.55,1.75); path[draw,fill=black!15] (8.181818,2.75) circle (0.12); draw[->] (8.181818,2.55) -- (6.55,1.75); node[font=small,align=center] at (6.55,1.45) same residue 2pmod5; node[font=small,align=center] at (3,4.35) N_ Q/ Q(-3)=-3; node[font=small,align=center] at (10.5,4.35) N_ Q/ Q(2)=2; node[font=small,align=center] at (3,0.45) N(…
Exact mathematical passage
N_{\mathbb Q/\mathbb Q}(-3)=-3
N_{\bm w}Use in source · 1 passage
Search aliases: N_ w bm
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 505
[Weighted smooth-chart count] For weights w=(w_1,…,w_d), N_ w(m) = q^∑_i=1^dmax(m-w_i,0).
Exact mathematical passage
\boxed{ N_{\bm w}(m) = q^{\sum_{i=1}^d\max(m-w_i,0)}. }
N_DUse in source · 1 passage
Search aliases: N_D
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2269
For supplied covariance data and the twisted cocycle of Block 13, let ρ_D(g)=exp(t(g)N_D)r_D(g). If a common element u satisfies r_D(u)=r_E(u)=1 and t(u)≠0, every intertwiner of ρ_D and ρ_E recovers both the original r-intertwiner and the N-intertwiner. There is no cubic nilpotence restriction. This recovers equivariant projectors for both original components. Reconstructed-action isomorphisms induce actual isomorphisms of the inertia-fixed monodromy kernels. Their local characteristic polynomia…
Exact mathematical passage
\rho_D(g)=\exp(t(g)N_D)r_D(g)
N_iUse in source · 6 passages
Search aliases: N_i
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1447
Algebraic monodromy data on the selected copies. Fix a field K and a nonzero scalar q∈ K^×. On each constituent W_i, supply an automorphism F_i and a nilpotent endomorphism N_i with F_iN_iF_i^-1=qN_i. The scalar q is common to this finite family. The componentwise operators F_m,N_m on U_m satisfy the same relation; F_m is invertible and N_m is nilpotent. The split maps constructed above intertwine both operators. The proof first identifies the independently defined kernel through an equivariant …
Exact mathematical passage
N_i
N_{K/\QQ}Use in source · 1 passage
Search aliases: N_K/QQ QQ
Use in source · pages.tex, line 308
ALSMain text: §9, arithmetic residue-counting continuation; signed-norm comparison. Even in the number field QQ, congruent generators need not have congruent absolute norms. The integers 2 and -3 are congruent modulo five. Their signed field norms are congruent, but their absolute values are not. vfig21_signed_norm headThe smallest counterexample is already decisive For K=QQ, N_K/QQ(2)=2, N_K/QQ(-3)=-3≡2pmod5, whereas N((2))=2, N((-3))=3not≡2pmod5. Here N((a)) is the positive index of the ideal …
Exact mathematical passage
N_{K/\QQ}(2)=2,\qquad N_{K/\QQ}(-3)=-3\equiv2\pmod5,
N_mUse in source · 5 passages
Search aliases: N_m
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1449
Algebraic monodromy data on the selected copies. Fix a field K and a nonzero scalar q∈ K^×. On each constituent W_i, supply an automorphism F_i and a nilpotent endomorphism N_i with F_iN_iF_i^-1=qN_i. The scalar q is common to this finite family. The componentwise operators F_m,N_m on U_m satisfy the same relation; F_m is invertible and N_m is nilpotent. The split maps constructed above intertwine both operators. The proof first identifies the independently defined kernel through an equivariant …
Exact mathematical passage
F_m,N_m
N_{m+1}Use in source · 1 passage
Search aliases: N_m+1
Use in source · pages.tex, line 191
t must work at every depthONE REVERSIBLE EXAMPLEMain text: Definition 7.1 and the reduction-compatibility requirement. For the two roots of x^2+1=0 modulo powers of five, negation is an actual reversible transport. It preserves the equation, is its own inverse, and commutes with reduction. vfig13_transport headOne successful input is not a natural transformation At every depth m, N_m(x)=-xpmod5^m, r_m+1,mN_m+1=N_mr_m+1,m. The two compatible branches beginning at 2 and 3 are exchanged. Because 2n…
Exact mathematical passage
N_m(x)=-x\pmod{5^m},\qquad r_{m+1,m}N_{m+1}=N_mr_{m+1,m}.
n_pUse in source · 4 passages
Search aliases: n_p
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1800
For each supplied required packet p, let A_p be its finite-dimensional carrier over a characteristic-zero field E_p, and let e_p be the actual joint spectral projector for its finite list of split commuting conditions. No order on E_p is assumed. The source notation Tr(e_p)>0 means that the unique natural number n_p with Tr(e_p)=(n_p:E_p) is positive. The checked projector formula identifies n_p with the dimension of the actual joint eigenspace.
Exact mathematical passage
n_p
\nmidUse in source · 5 passages
Search aliases: nmid
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 777
The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.
Exact mathematical passage
\boxed{ \Sys_{\mathrm{sys}} = \Sys_2^{\mathrm{wt}} \times_{\Compat} \prod_{\lambda\nmid2}'\Sys_\lambda. }
\NNUse in source · 2 passages
Search aliases: NN N mathbb
Use in source · Read passage · residue_continuation_body.tex, line 66
The residue norm and its sign. For m∈ N, reducing the multiplication matrices of O modulo m and taking their determinants constructs a monoid homomorphism overline n: O/m O⟶ Z/m Z, overline n([x])=n(x)bmod m. The construction is proved independent of the integral basis. For m≠0, its source is finite of cardinality m^d. Consequently x-y∈ m O implies n(x)≡ n(y)pmod m. No surjectivity of overline n is asserted.
Exact mathematical passage
m\in\NN
OUse in source · 4 passages
Search aliases: O
Use in source · Read passage · residue_continuation_body.tex, line 56
attice discrepancy theorem. The index factor comes from the covolume identity covol(L(M))=q covol(L(J)). For a complex weight w:J/M→ C, summing the estimates gives the leading term fracvol(D)t^dq covol(L(J))∑_a∈ J/Mw(a), with error at most B∑_aabsnw(a)(t^d-1+1). If the finite sum of weights is zero, only that error remains. Writing Y=t^d gives an error O(Y^1-1/d+1) for the actual norm threshold. These assertions include d=1.
Exact mathematical passage
O(Y^{1-1/d}+1)
O_{K,C}Use in source · 1 passage
Search aliases: O_K,C
Use in source · Read passage · residue_continuation_body.tex, line 138
Ordinary class characters: an unconditional power saving. The preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl( O)→ C, define the actual finite ideal sum S_w(X)=∑_substackI≠0
Exact mathematical passage
A_C(X)=c_KX+O_{K,C}(X^{1-1/d}),\qquad c_K=\frac{2^{r_1}(2\pi)^{r_2}R_K}{\#\mu(K)\sqrt{\absn{\Delta_K}}}.
O_{K,J,m,\chi}Use in source · 1 passage
Search aliases: O_K,J,m,χ chi
Use in source · Read passage · residue_continuation_body.tex, line 225
The assembled ideal norm-character estimate. Let C∈Cl( O), let J≠0 be an integral ideal with [J]=C^-1, and let m>0. Let χ: Z/m Z→ C preserve multiplication, 0, and 1. Assume a residue-ring unit satisfying [source reference], and assume χ( N Jbmod m)≠0. Then T_C,χ(X) =∑_substackI≠0, [I]=C N I≤ X χ( N Ibmod m) =O_K,J,m,χ(X^1-1/d). More precisely, some B≥0 bounds its absolute value by BX^1-1/d for every real X≥1.
Exact mathematical passage
T_{C,\chi}(X) =\sum_{\substack{I\ne0,\ [I]=C\\\Nm I\leq X}} \chi(\Nm I\bmod m) =O_{K,J,m,\chi}(X^{1-1/d}).
\Ocal_\lambdaUse in source · 2 passages
Search aliases: Ocal_λ O_λ Ocal lambda mathcal lambda
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 885
All coefficients of the characteristic polynomial commute with reduction. Consequently, prescribed integral characteristic polynomials at every precision hold exactly in the reconstructed representation. For a common coefficient field E, local integral polynomials lie in O_λ[X] and are compared with P_v∈ E[X] only after mapping both into the local coefficient field. No field embedding E→ O_λ is assumed. The allowed-place predicate is explicit and may depend on λ. Trivial action on a supplied sub…
Exact mathematical passage
\Ocal_\lambda[X]
\overline aUse in source · 1 passage
Search aliases: overline a overline
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1191
The arithmetic witnesses use the integral closure A of Z in overline Q. For each rational prime p, Lean constructs a maximal ideal Q of A above (p) and an actual element γ_p∈ G_ Q such that γ_p∈Stab_G_ Q( Q), overlineγ_p(a)=overline a^ p (a∈ A). The existence argument applies Mathlib's proved profinite stabilizer-surjectivity theorem to this actual invariant ring. The residue p-power map is proved bijective by taking p-th roots in overline Q and showing that they remain integral when their p-th …
Exact mathematical passage
\gamma_p\in\operatorname{Stab}_{G_{\Q}}(\mathfrak Q),\qquad \overline{\gamma_p(a)}=\overline a^{\,p} \quad(a\in\mathcal A).
\overline a^{\,p}Use in source · 1 passage
Search aliases: overline a^ p overline
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1191
The arithmetic witnesses use the integral closure A of Z in overline Q. For each rational prime p, Lean constructs a maximal ideal Q of A above (p) and an actual element γ_p∈ G_ Q such that γ_p∈Stab_G_ Q( Q), overlineγ_p(a)=overline a^ p (a∈ A). The existence argument applies Mathlib's proved profinite stabilizer-surjectivity theorem to this actual invariant ring. The residue p-power map is proved bijective by taking p-th roots in overline Q and showing that they remain integral when their p-th …
Exact mathematical passage
\gamma_p\in\operatorname{Stab}_{G_{\Q}}(\mathfrak Q),\qquad \overline{\gamma_p(a)}=\overline a^{\,p} \quad(a\in\mathcal A).
\overline nUse in source · 8 passages
Search aliases: overline n overline
Use in source · Read passage · residue_continuation_body.tex, line 68
The residue norm and its sign. For m∈ N, reducing the multiplication matrices of O modulo m and taking their determinants constructs a monoid homomorphism overline n: O/m O⟶ Z/m Z, overline n([x])=n(x)bmod m. The construction is proved independent of the integral basis. For m≠0, its source is finite of cardinality m^d. Consequently x-y∈ m O implies n(x)≡ n(y)pmod m. No surjectivity of overline n is asserted.
Exact mathematical passage
\overline n:\OO/m\OO\longrightarrow\ZZ/m\ZZ, \qquad \overline n([x])=n(x)\bmod m.
\overline n_JUse in source · 2 passages
Search aliases: overline n_J overline
Use in source · Read passage · residue_continuation_body.tex, line 202
An actual unit action on ideal residues. Fix m>0 and a nonzero integral ideal J. Multiplication by u∈( O/m O)^× gives a proved permutation of J/mJ. Its inverse follows from (ab-1)x∈ mJ when ab-1∈ m O and x∈ J. No surjectivity of J/mJ→ O/m O or coprimality condition on J is used. The well-defined residue norm on this quotient satisfies overline n_J(u· a)=overline n(u) overline n_J(a). For a multiplicative χ: Z/m Z→ C preserving 1, an exhibited unit with u∈( O/m O)^×, χ(overline n(u))≠1 therefore …
Exact mathematical passage
\overline n_J(u\cdot a)=\overline n(u)\,\overline n_J(a).
\overline n_{\varepsilon}Use in source · 3 passages
Search aliases: overline n_ε overline varepsilon
Use in source · Read passage · residue_continuation_body.tex, line 89
For ε∈{1,-1} define overline n_ε(a)=εoverline n(a). On the corresponding sign portion, N(x)bmod m=overline n_ε([x]). The sign cannot be dropped: -1≡2pmod3, whereas N(-1)=1 and N(2)=2 for principal ideals of Z.
Exact mathematical passage
\overline n_{\varepsilon}(a)=\varepsilon\overline n(a)
\overline yUse in source · 1 passage
Search aliases: overline y overline
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1213
When pnmid n, the comparison with the finite cyclotomic sample is proved: q_L(γ_p)=σ_p, σ_p(ζ_n)=ζ_n^p. Reduction distinguishes n-th roots of unity at such a prime. Indeed, if x^n=y^n=1 and their reductions agree, the factorization of x^n-y^n gives (x-y)∑_i=0^n-1x^i y^n-1-i=0. The second factor reduces to noverline y^ n-1≠0, so the domain property gives x=y. This proves both Frobenius recognition on the cyclotomic generator and triviality of its inertia action. Consequently, every representation…
Exact mathematical passage
n\overline y^{\,n-1}\ne0
\overline y^{\,n-1}Use in source · 1 passage
Search aliases: overline y^ n-1 overline
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1213
When pnmid n, the comparison with the finite cyclotomic sample is proved: q_L(γ_p)=σ_p, σ_p(ζ_n)=ζ_n^p. Reduction distinguishes n-th roots of unity at such a prime. Indeed, if x^n=y^n=1 and their reductions agree, the factorization of x^n-y^n gives (x-y)∑_i=0^n-1x^i y^n-1-i=0. The second factor reduces to noverline y^ n-1≠0, so the domain property gives x=y. This proves both Frobenius recognition on the cyclotomic generator and triviality of its inertia action. Consequently, every representation…
Exact mathematical passage
n\overline y^{\,n-1}\ne0
\overline{\Q_p}Use in source · 1 passage
Search aliases: overlineQ_p overline Q_p overline Q overline mathbf
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2489
For every prime p, Lean constructs a valuation subring A⊆overline Q_p containing the integral image of Z_p such that the induced map Z_p→ A is local. For odd p, two is a unit. The preceding argument applies to every valuation subring containing that integral image, and hence every member of P_q(B) is trivial on its inertia subgroup after actual local restriction. The existence theorem supplies a locally dominating choice, so this assertion is not vacuous.
Exact mathematical passage
A\subseteq\overline{\Q_p}
\overline{x}Use in source · 1 passage
Search aliases: overlinex overline
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1147
The arithmetic attachment is checked in this finite cyclotomic case. There is an actual maximal prime P of O_L above (p), its ramification index is one, and σ_p belongs to its decomposition group. For every integral x∈ O_L, overlineσ_p(x)=overlinex^ p in O_L/ P. The residue identity is proved on all integral elements using O_L= Z[ζ_n]. Existence of P is obtained by lying over, rather than assumed. This specifies arithmetic Frobenius, with p-power residue action. The action and unramifiedness sta…
Exact mathematical passage
\overline{\sigma_p(x)}=\overline{x}^{\,p} \quad\text{in }\mathcal O_L/\mathfrak P.
\overline{x}^{\,p}Use in source · 1 passage
Search aliases: overlinex^ p overline
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1147
The arithmetic attachment is checked in this finite cyclotomic case. There is an actual maximal prime P of O_L above (p), its ramification index is one, and σ_p belongs to its decomposition group. For every integral x∈ O_L, overlineσ_p(x)=overlinex^ p in O_L/ P. The residue identity is proved on all integral elements using O_L= Z[ζ_n]. Existence of P is obtained by lying over, rather than assumed. This specifies arithmetic Frobenius, with p-power residue action. The action and unramifiedness sta…
Exact mathematical passage
\overline{\sigma_p(x)}=\overline{x}^{\,p} \quad\text{in }\mathcal O_L/\mathfrak P.
\overline{\gamma_p(a)}Use in source · 1 passage
Search aliases: overlineγ_p(a) overline gamma
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1191
The arithmetic witnesses use the integral closure A of Z in overline Q. For each rational prime p, Lean constructs a maximal ideal Q of A above (p) and an actual element γ_p∈ G_ Q such that γ_p∈Stab_G_ Q( Q), overlineγ_p(a)=overline a^ p (a∈ A). The existence argument applies Mathlib's proved profinite stabilizer-surjectivity theorem to this actual invariant ring. The residue p-power map is proved bijective by taking p-th roots in overline Q and showing that they remain integral when their p-th …
Exact mathematical passage
\gamma_p\in\operatorname{Stab}_{G_{\Q}}(\mathfrak Q),\qquad \overline{\gamma_p(a)}=\overline a^{\,p} \quad(a\in\mathcal A).
\overline{\sigma_p(x)}Use in source · 1 passage
Search aliases: overlineσ_p(x) overline sigma
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1147
The arithmetic attachment is checked in this finite cyclotomic case. There is an actual maximal prime P of O_L above (p), its ramification index is one, and σ_p belongs to its decomposition group. For every integral x∈ O_L, overlineσ_p(x)=overlinex^ p in O_L/ P. The residue identity is proved on all integral elements using O_L= Z[ζ_n]. Existence of P is obtained by lying over, rather than assumed. This specifies arithmetic Frobenius, with p-power residue action. The action and unramifiedness sta…
Exact mathematical passage
\overline{\sigma_p(x)}=\overline{x}^{\,p} \quad\text{in }\mathcal O_L/\mathfrak P.
\mathcal PUse in source · 12 passages
Search aliases: P mathcal
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2325
hmetic completion (Block 21). Fix the continuous quotient q:G_ Q⟶ C_2 obtained by the action on a chosen square root i of -1 in the algebraic closure. Its surjectivity and the equivalence q(σ)=1 Longleftrightarrow σ(i)=i are proved using the existing actual quadratic Galois construction. For a commutative ring B, define the arithmetic packet P_q(B)= {χ:G_ Q⟶ B^×midker q⊆kerχ}, where the maps are group homomorphisms. Unique descent through the actual surjective map q is derived, rather than suppl…
Exact mathematical passage
\mathcal P_q(B)= \{\chi:G_{\Q}\longrightarrow B^\times\mid\ker q\subseteq\ker\chi\},
P^0Use in source · 3 passages
Search aliases: P^0
Use in source · pages.tex, line 154
vpage11An arbitrary cut need not descendTHE CHAIN-MAP CONDITIONMain text: Theorem 4.2 and the scope of the cohomology splitting. Retain the previous complex. The valid A-packet projectors are P^0=diag(1,0), P^1=diag(1,1,0,0), and P^2=diag(1,0). They satisfy d_rP^r=P^r+1d_r. vfig11_chain_defect headA boundary can cross an incompatible cut Replace the degree-one projector by Q^1=diag(0,1,0,0) while keeping P^0. Then d_0P^0e_0=u_0, Q^1d_0e_0=0, (d_0P^0-Q^1d_0)e_0=u_0≠0. The selected degree-zero com…
Exact mathematical passage
P^0=\diag(1,0)
P^1Use in source · 1 passage
Search aliases: P^1
Use in source · pages.tex, line 154
vpage11An arbitrary cut need not descendTHE CHAIN-MAP CONDITIONMain text: Theorem 4.2 and the scope of the cohomology splitting. Retain the previous complex. The valid A-packet projectors are P^0=diag(1,0), P^1=diag(1,1,0,0), and P^2=diag(1,0). They satisfy d_rP^r=P^r+1d_r. vfig11_chain_defect headA boundary can cross an incompatible cut Replace the degree-one projector by Q^1=diag(0,1,0,0) while keeping P^0. Then d_0P^0e_0=u_0, Q^1d_0e_0=0, (d_0P^0-Q^1d_0)e_0=u_0≠0. The selected degree-zero com…
Exact mathematical passage
P^1=\diag(1,1,0,0)
P^2Use in source · 1 passage
Search aliases: P^2
Use in source · pages.tex, line 154
vpage11An arbitrary cut need not descendTHE CHAIN-MAP CONDITIONMain text: Theorem 4.2 and the scope of the cohomology splitting. Retain the previous complex. The valid A-packet projectors are P^0=diag(1,0), P^1=diag(1,1,0,0), and P^2=diag(1,0). They satisfy d_rP^r=P^r+1d_r. vfig11_chain_defect headA boundary can cross an incompatible cut Replace the degree-one projector by Q^1=diag(0,1,0,0) while keeping P^0. Then d_0P^0e_0=u_0, Q^1d_0e_0=0, (d_0P^0-Q^1d_0)e_0=u_0≠0. The selected degree-zero com…
Exact mathematical passage
P^2=\diag(1,0)
p^mUse in source · 6 passages
Search aliases: p^m
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1972
The residual module partitions coordinates modulo p, verifies compatibility x_m+1≡ x_mpmodp^m, and reconstructs the digit a_m-1 by a_m-1=lfloorfracx_mp^m-1rfloorpmod p.
Exact mathematical passage
x_{m+1}\equiv x_m\pmod{p^m},
p^{m-1}Use in source · 2 passages
Search aliases: p^m-1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1976
The residual module partitions coordinates modulo p, verifies compatibility x_m+1≡ x_mpmodp^m, and reconstructs the digit a_m-1 by a_m-1=lfloorfracx_mp^m-1rfloorpmod p.
Exact mathematical passage
a_{m-1}=\left\lfloor\frac{x_m}{p^{m-1}}\right\rfloor\pmod p.
P^rUse in source · 1 passage
Search aliases: P^r
Use in source · pages.tex, line 154
vpage11An arbitrary cut need not descendTHE CHAIN-MAP CONDITIONMain text: Theorem 4.2 and the scope of the cohomology splitting. Retain the previous complex. The valid A-packet projectors are P^0=diag(1,0), P^1=diag(1,1,0,0), and P^2=diag(1,0). They satisfy d_rP^r=P^r+1d_r. vfig11_chain_defect headA boundary can cross an incompatible cut Replace the degree-one projector by Q^1=diag(0,1,0,0) while keeping P^0. Then d_0P^0e_0=u_0, Q^1d_0e_0=0, (d_0P^0-Q^1d_0)e_0=u_0≠0. The selected degree-zero com…
Exact mathematical passage
d_rP^r=P^{r+1}d_r
P^{r+1}Use in source · 1 passage
Search aliases: P^r+1
Use in source · pages.tex, line 154
vpage11An arbitrary cut need not descendTHE CHAIN-MAP CONDITIONMain text: Theorem 4.2 and the scope of the cohomology splitting. Retain the previous complex. The valid A-packet projectors are P^0=diag(1,0), P^1=diag(1,1,0,0), and P^2=diag(1,0). They satisfy d_rP^r=P^r+1d_r. vfig11_chain_defect headA boundary can cross an incompatible cut Replace the degree-one projector by Q^1=diag(0,1,0,0) while keeping P^0. Then d_0P^0e_0=u_0, Q^1d_0e_0=0, (d_0P^0-Q^1d_0)e_0=u_0≠0. The selected degree-zero com…
Exact mathematical passage
d_rP^r=P^{r+1}d_r
P_{22}^2Use in source · 1 passage
Search aliases: P_22^2
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 984
0 · 1, every integral intertwiner with ρ_1 has determinant aP_22^2. Taking a=π^n for a nonzero π∈ I rules out every fixed detecting depth, even though all the field-valued approximants are isomorphic and their integral limit is trivial. This does not refute Claim 8.2, which already assumes inverse-limit closure of its chosen local conditions.
Exact mathematical passage
aP_{22}^2
P_A^0Use in source · 1 passage
Search aliases: P_A^0
Use in source · 11_chain_defect.tex, line 69
node[font=small,align=center,font=small] at (5.915,1.865) 0; path[draw=black!30,fill=white] (6.22,1.56) rectangle (6.83,2.17); node[font=small,align=center,font=small] at (6.525,1.865) 0; path[draw=black!30,fill=white] (6.83,1.56) rectangle (7.44,2.17); node[font=small,align=center,font=small] at (7.135,1.865) 0; node[font=small,align=center,font=small] at (10.16,4.7) d_0P_A^0-Q^1d_0; path[draw=black!30,fill=black!8] (9.55,3.39) rectangle (10.16,4); node[font=small,align=center,font=small] at (9…
Exact mathematical passage
d_0P_A^0-Q^1d_0
P_A^1Use in source · 1 passage
Search aliases: P_A^1
Use in source · 11_chain_defect.tex, line 3
[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round] path[use as bounding box] (0,-.6) rectangle (14,4.8); node[font=small,align=center,font=small] at (2.47,4.7) valid P_A^1; path[draw=black!30,fill=black!8] (1.25,3.39) rectangle (1.86,4); node[font=small,align=center,font=small] at (1.555,3.695) 1; path[draw=black!30,fill=white] (1.86,3.39) rectangle (2.47,4); node[font=small,align=center,font=small] at (2.165,3.695) 0; path[draw=black!30,fill=white] (2.47,3.39) rectangle (3.08,4); node[font=…
Exact mathematical passage
P_A^1
p_iUse in source · 1 passage
Search aliases: p_i
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 387
For i≠ j, orthogonality gives H(p_i)H(i_j)=0. Idempotence gives H(p_j)H(i_j)=I, and completeness gives ∑_j H(i_j)H(p_j)=I. Thus the maps [z]⟼([P_j^r z])_j, ([z_j])_j⟼[∑_j i_j z_j] are inverse linear maps. The finite product equals the direct sum, proving the assertion. In the formalization, both sides are also identified with Mathlib's standard homology objects of the genuine graded complexes.
Exact mathematical passage
H(p_i)H(i_j)=0
p_{i,a}Use in source · 2 passages
Search aliases: p_i,a
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1652
For each operator, its finite set of actual eigenvalues is used without multiplicity. The polynomial p_i,a(X)=∏_μ∈Spec(T_i)∖{a} (a-μ)^-1(X-μ) is used when a occurs, and the zero polynomial otherwise. For an eigenvector T_i v=bv, evaluation gives p_i,a(T_i)v=v · b=a,
Exact mathematical passage
p_{i,a}(X)=\prod_{\mu\in\Spec(T_i)\setminus\{a\}} (a-\mu)^{-1}(X-\mu)
p_{i,a_i}Use in source · 1 passage
Search aliases: p_i,a_i
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1661
0 · b≠ a. Because the eigenspaces span A, this proves that e_i=p_i,a_i(T_i) maps all of A into the selected eigenspace and fixes that eigenspace pointwise. It therefore proves the individual projector and image assertions, including an absent target.
Exact mathematical passage
e_i=p_{i,a_i}(T_i)
P_{i,g}Use in source · 1 passage
Search aliases: P_i,g
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1422
(XI-ρ_m(g)) =∏_i∈ Idet(XI-ρ_i(g))^m_i, Tr(ρ_m(g)) =∑_i∈ I(m_i:K)Tr(ρ_i(g)). Constituents may have different dimensions. Zero multiplicities, zero-dimensional constituents, and the empty index set are included. If at a permitted sample g the constituent polynomials are the images of P_i,g∈ E[X] under a specified map E→ K, the summand polynomial is the image of ∏_iP_i,g^m_i. Applying this independently at each coefficient field preserves a supplied common polynomial family. This does not establish…
Exact mathematical passage
P_{i,g}\in E[X]
P_{i,g}^{m_i}Use in source · 1 passage
Search aliases: P_i,g^m_i
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1424
eratornameTr(ρ_m(g)) =∑_i∈ I(m_i:K)Tr(ρ_i(g)). Constituents may have different dimensions. Zero multiplicities, zero-dimensional constituents, and the empty index set are included. If at a permitted sample g the constituent polynomials are the images of P_i,g∈ E[X] under a specified map E→ K, the summand polynomial is the image of ∏_iP_i,g^m_i. Applying this independently at each coefficient field preserves a supplied common polynomial family. This does not establish the Hodge--Tate or Weil--Del…
Exact mathematical passage
\prod_iP_{i,g}^{m_i}
P_i^rUse in source · 1 passage
Search aliases: P_i^r
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 347
Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.
Exact mathematical passage
\sum_jP_j^r=I, \qquad P_i^rP_j^r=\delta_{ij}P_j^r, \qquad d_rP_j^r=P_j^{r+1}d_r.
P_jUse in source · 1 passage
Search aliases: P_j
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 357
[Cohomology packet decomposition] Under these hypotheses, each packet is a subcomplex and H^r(C^bullet) ≅ ⊕_jH^r(P_jC^bullet). In particular, H^1=⊕_jH^1_j, H^2=⊕_jH^2_j, so tangent and obstruction spaces reconstruct exactly from their packet components.
Exact mathematical passage
\boxed{ H^r(C^\bullet) \cong \bigoplus_jH^r(P_jC^\bullet). }
p_jUse in source · 6 passages
Search aliases: p_j
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 383
h packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet are cochain maps, so they induce maps on these genuine quotients.
Exact mathematical passage
p_j:C^\bullet\to C_j^\bullet
P_j^rUse in source · 4 passages
Search aliases: P_j^r
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 346
Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.
Exact mathematical passage
P_j^r:C^r\to C^r
P_j^{r+1}Use in source · 1 passage
Search aliases: P_j^r+1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 347
Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.
Exact mathematical passage
\sum_jP_j^r=I, \qquad P_i^rP_j^r=\delta_{ij}P_j^r, \qquad d_rP_j^r=P_j^{r+1}d_r.
\mathcal P_qUse in source · 12 passages
Search aliases: P_q mathcal
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2325
hmetic completion (Block 21). Fix the continuous quotient q:G_ Q⟶ C_2 obtained by the action on a chosen square root i of -1 in the algebraic closure. Its surjectivity and the equivalence q(σ)=1 Longleftrightarrow σ(i)=i are proved using the existing actual quadratic Galois construction. For a commutative ring B, define the arithmetic packet P_q(B)= {χ:G_ Q⟶ B^×midker q⊆kerχ}, where the maps are group homomorphisms. Unique descent through the actual surjective map q is derived, rather than suppl…
Exact mathematical passage
\mathcal P_q(B)= \{\chi:G_{\Q}\longrightarrow B^\times\mid\ker q\subseteq\ker\chi\},
p_sUse in source · 2 passages
Search aliases: p_s
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2521
Suppose now that R is I-adically complete and that every positive quotient R/I^m is finite. Let C_m⊆ P_q(R/I^m) be allowed subsets preserved by reduction, and let (p_s)_s∈ S be any indexed family of arithmetic primes. The index set S need not be finite. Lean proves the simultaneous arithmetic completion criterion · ∃χ∈ P_q(R), χ continuous, χbmod I^m∈ C_m (m≥1), χ_ Q_p_s=1 (s∈ S)
Exact mathematical passage
(p_s)_{s\in S}
P_{s,m}Use in source · 2 passages
Search aliases: P_s,m
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 949
agraphActual local comparisons from fixed-depth certificates (Block 15). Let R be an I-adically complete separated commutative ring with finite coefficient quotients and an injective map into a field K. Fix a finite matrix rank. For finitely many local group maps ϕ_s:H_s→ G and integral template representations τ_s, fix detecting depths d_s. At precision m retain matrices P_s,m as part of the admissible object and require ρ_m(ϕ_s(h))P_s,m=P_s,mτ_s,m(h). For m≥ d_s require their determinants to r…
Exact mathematical passage
P_{s,m}
QUse in source · 11 passages
Search aliases: Q
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1511
Writing Q_D(T) for the reversed local characteristic polynomial, Lean proves Q_D(T)=det(1-T r(φ)mid E(D,H)), Q_D(0)=1, Q_⊕_iD_i^⊕ m_i(T)=∏_iQ_D_i(T)^m_i. The abstract formulas do not need characteristic zero. Normality and finite dimension appear explicitly where used. Arithmetic identification of G,H,ν and φ remains separate. In the geometric-Frobenius convention, ν(φ) is the inverse residue cardinality; it is this unit, rather than the residue cardinality, that enters the earlier special-chain…
Exact mathematical passage
Q_D(T)
\mathbb
QUse in source · 1 passage
Search aliases: Q mathbb
Use in source · references.tex, line 65
bibitemKisin2010 Mark Kisin. newblock Deformations of G_ Q_p and GL_2( Q_p) representations. newblock In Laurent Berger, Christophe Breuil, and Pierre Colmez, editors, em Repr esentations p-adiques de Groupes p-adiques II: Repr esentations de GL_2( Q_p) et (φ,Γ)-modules, volume 330 of em Ast erisque, pages 511--528. Soci et e Math ematique de France, Paris, 2010.
Exact mathematical passage
\mathrm{GL}_2(\mathbb Q_p)
\mathbf QUse in source · 1 passage
Search aliases: Q mathbf
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1439
For the three-character example below, the construction produces an actual three-dimensional summand inside the carrier of dimension seven. At the sample (0,1) its characteristic polynomial is (X-1)^2(X+1). A separate checked counterexample uses the coordinate inclusion Q→ Q^2, its coordinate retraction, and the swap involution: the split law holds, but compression sends the swap to zero and therefore fails to preserve its square. Thus selecting whole constituent copies, with the proved equivari…
Exact mathematical passage
\mathbf Q\to\mathbf Q^2
q^{-j}Use in source · 2 passages
Search aliases: q^-j
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1468
The exact special chain has basis e_0,…,e_n-1 and operators Ne_0=0, Ne_j=e_j-1 (j>0), Fe_j=aq^-je_j, a,q∈ K^×. Lean proves N^n=0, and for n>0 also N^n-1≠0 and ker N=Ke_0. Consequently its kernel polynomial is X-a, whereas its whole-space polynomial is ∏_j=0^n-1(X-aq^-j). The zero-length chain is included as a zero-dimensional space; the assertion ker N=Ke_0 is made only for positive length.
Exact mathematical passage
Ne_0=0,\qquad Ne_j=e_{j-1}\ (j>0),\qquad Fe_j=aq^{-j}e_j, \qquad a,q\in K^\times.
q^{-W}Use in source · 1 passage
Search aliases: q^-W
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 561
[Weighted seed zeta function] Let 1≤ w_1≤⋯≤ w_d, put W=∑_iw_i, and let a_m=q^∑_imax(m-w_i,0). Then Z_ w(T) = ∑_m=1^w_da_mT^m + q^-Wfrac(q^dT)^w_d+11-q^dT. In particular, every weighted smooth chart has a rational seed zeta function.
Exact mathematical passage
\boxed{ Z_{\bm w}(T) = \sum_{m=1}^{w_d}a_mT^m + q^{-W}\frac{(q^dT)^{w_d+1}}{1-q^dT}. }
Q^1Use in source · 4 passages
Search aliases: Q^1
Use in source · 11_chain_defect.tex, line 36
font=small,align=center,font=small] at (2.165,1.865) 0; path[draw=black!30,fill=white] (2.47,1.56) rectangle (3.08,2.17); node[font=small,align=center,font=small] at (2.775,1.865) 0; path[draw=black!30,fill=white] (3.08,1.56) rectangle (3.69,2.17); node[font=small,align=center,font=small] at (3.385,1.865) 0; node[font=small,align=center,font=small] at (6.22,4.7) invalid Q^1; path[draw=black!30,fill=white] (5,3.39) rectangle (5.61,4); node[font=small,align=center,font=small] at (5.305,3.695) 0; p…
Exact mathematical passage
Q^1
\mathbf Q^2Use in source · 1 passage
Search aliases: Q^2 mathbf
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1439
For the three-character example below, the construction produces an actual three-dimensional summand inside the carrier of dimension seven. At the sample (0,1) its characteristic polynomial is (X-1)^2(X+1). A separate checked counterexample uses the coordinate inclusion Q→ Q^2, its coordinate retraction, and the swap involution: the split law holds, but compression sends the swap to zero and therefore fails to preserve its square. Thus selecting whole constituent copies, with the proved equivari…
Exact mathematical passage
\mathbf Q\to\mathbf Q^2
\Q^4Use in source · 1 passage
Search aliases: Q^4 Q mathbf
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1688
A specified finite realization. On Q^4, take the diagonal operators with diagonal lists (1,1,1,0), (2,2,0,0), (3,0,0,0), (0,4,4,4) and target scalars (1,2,3,4). All four targets occur individually. Imposing these conditions successively gives actual projector ranks 3,2,1,0. The last packet is empty despite individual feasibility. The formal code checks these operators, projector formulas, traces and dimensions. This is a newly specified realization, not a claim to have recovered the original com…
Exact mathematical passage
\Q^4
q^dUse in source · 2 passages
Search aliases: q^d
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 544
[Smooth seed zeta function] For a smooth formal component of dimension d, Z_d(T)=fracT1-q^dT.
Exact mathematical passage
\boxed{ Z_d(T)=\frac{T}{1-q^dT}. }
q^{d(m-1)}Use in source · 3 passages
Search aliases: q^d(m-1)
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 298
[Smooth formal branch count] If R_bar x≅ O[[t_1,…,t_d]] as complete local O-algebras, then #X_m(bar x)=q^d(m-1). Fix a uniformizer and a section s:k→ O of reduction, with s(0)=0. Every O-valued branch is encoded by d positive-depth digit streams t_i=s(a_i,1)ϖ+s(a_i,2)ϖ^2+⋯, a_i,j∈ k,
Exact mathematical passage
\boxed{ \#X_m(\bar x)=q^{d(m-1)}. }
q^{dm-W}Use in source · 1 passage
Search aliases: q^dm-W
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 575
The finite sum records the depths before all coordinates are active. For m≥ w_d+1, a_m=q^dm-W, so the remaining terms form one geometric series.
Exact mathematical passage
a_m=q^{dm-W},
q^{e-1}Use in source · 1 passage
Search aliases: q^e-1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 593
[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.
Exact mathematical passage
\boxed{ Z_e(T) = \frac{\sum_{r=1}^{e}q^{r-1}T^r} {1-q^{e-1}T^e}. }
q^{m-\lceil m/e\rceil}Use in source · 2 passages
Search aliases: q^m-lceil m/erceil lceil rceil
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 587
[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.
Exact mathematical passage
\boxed{ q^{m-\lceil m/e\rceil} }
q^{m-w_i}Use in source · 1 passage
Search aliases: q^m-w_i
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 515
The image of ϖ^w_i O in O/ϖ^m has one element when m≤ w_i, and q^m-w_i elements when m>w_i. Multiplying the independent coordinate counts gives the formula.
Exact mathematical passage
q^{m-w_i}
q^{r-1}Use in source · 1 passage
Search aliases: q^r-1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 593
[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.
Exact mathematical passage
\boxed{ Z_e(T) = \frac{\sum_{r=1}^{e}q^{r-1}T^r} {1-q^{e-1}T^e}. }
q^{\sum_{i=1}^d\max(m-w_i,0)}Use in source · 1 passage
Search aliases: q^∑_i=1^dmax(m-w_i,0) sum max
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 505
[Weighted smooth-chart count] For weights w=(w_1,…,w_d), N_ w(m) = q^∑_i=1^dmax(m-w_i,0).
Exact mathematical passage
\boxed{ N_{\bm w}(m) = q^{\sum_{i=1}^d\max(m-w_i,0)}. }
q^{\sum_i\max(m-w_i,0)}Use in source · 1 passage
Search aliases: q^∑_imax(m-w_i,0) sum max
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 557
[Weighted seed zeta function] Let 1≤ w_1≤⋯≤ w_d, put W=∑_iw_i, and let a_m=q^∑_imax(m-w_i,0). Then Z_ w(T) = ∑_m=1^w_da_mT^m + q^-Wfrac(q^dT)^w_d+11-q^dT. In particular, every weighted smooth chart has a rational seed zeta function.
Exact mathematical passage
a_m=q^{\sum_i\max(m-w_i,0)}.
\Q_2Use in source · 1 passage
Search aliases: Q_2 Q mathbf
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2459
The p-adic square-root criterion is proved for every prime: (∃ x∈ Q_p, x^2=-1) Longleftrightarrow p≡1pmod4. A root satisfies lvert xrvert_p^2=1 and is therefore integral. Reduction modulo three rules out a root in Q_3, and reduction modulo four rules out a root in Q_2. At an odd prime, any residue root of X^2+1 has nonzero derivative and lifts by the pinned Mathlib Hensel theorem. The finite-field square criterion then gives the stated congruence. An explicit Hensel application at the residue ro…
Exact mathematical passage
\Q_2
\mathbb Q_2Use in source · 1 passage
Search aliases: Q_2 mathbb
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 978
0 · 2 at a nonidentity quadratic element. The determinant 2 is detected modulo 4. The representations are conjugate over Q_2 but not integrally conjugate. A separate Z_3 example invokes simultaneous completion and reconstructs a comparison which is an integral nonunit and a field unit.
Exact mathematical passage
\mathbb Q_2
\Q_3Use in source · 1 passage
Search aliases: Q_3 Q mathbf
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2458
The p-adic square-root criterion is proved for every prime: (∃ x∈ Q_p, x^2=-1) Longleftrightarrow p≡1pmod4. A root satisfies lvert xrvert_p^2=1 and is therefore integral. Reduction modulo three rules out a root in Q_3, and reduction modulo four rules out a root in Q_2. At an odd prime, any residue root of X^2+1 has nonzero derivative and lifts by the pinned Mathlib Hensel theorem. The finite-field square criterion then gives the stated congruence. An explicit Hensel application at the residue ro…
Exact mathematical passage
\Q_3
Q_DUse in source · 2 passages
Search aliases: Q_D
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1511
Writing Q_D(T) for the reversed local characteristic polynomial, Lean proves Q_D(T)=det(1-T r(φ)mid E(D,H)), Q_D(0)=1, Q_⊕_iD_i^⊕ m_i(T)=∏_iQ_D_i(T)^m_i. The abstract formulas do not need characteristic zero. Normality and finite dimension appear explicitly where used. Arithmetic identification of G,H,ν and φ remains separate. In the geometric-Frobenius convention, ν(φ) is the inverse residue cardinality; it is this unit, rather than the residue cardinality, that enters the earlier special-chain…
Exact mathematical passage
Q_D(T)
Q_{D_i}Use in source · 1 passage
Search aliases: Q_D_i
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1512
Writing Q_D(T) for the reversed local characteristic polynomial, Lean proves Q_D(T)=det(1-T r(φ)mid E(D,H)), Q_D(0)=1, Q_⊕_iD_i^⊕ m_i(T)=∏_iQ_D_i(T)^m_i. The abstract formulas do not need characteristic zero. Normality and finite dimension appear explicitly where used. Arithmetic identification of G,H,ν and φ remains separate. In the geometric-Frobenius convention, ν(φ) is the inverse residue cardinality; it is this unit, rather than the residue cardinality, that enters the earlier special-chain…
Exact mathematical passage
Q_D(T)=\det(1-T r(\varphi)\mid E(D,H)),\qquad Q_D(0)=1, \qquad Q_{\bigoplus_iD_i^{\oplus m_i}}(T)=\prod_iQ_{D_i}(T)^{m_i}.
\mathfrak Q_jUse in source · 1 passage
Search aliases: Q_j mathfrak
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1230
For a compatible family ρ_i,λ of representations of the fixed finite group, set V_i,λ=ρ_i,λ∘ q_L. Suppose the actual absolute-Galois members are irreducible and the systems are pairwise componentwise nonisomorphic. The finite prime separator now yields distinct permitted rational primes p_j, actual prime ideals Q_j and arithmetic Frobenius witnesses γ_j as above, outside any prescribed finite set, such that the common algebraic trace matrix has nonzero determinant. Every coefficient embedding pr…
Exact mathematical passage
\mathfrak Q_j
q_LUse in source · 5 passages
Search aliases: q_L
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1175
Absolute Galois restriction and arithmetic witnesses (Block 26). Fix a cyclotomic number field L= Q(ζ_n) with n>0 and an embedding j:L↪overline Q. The new restriction map q_L:G_ Q⟶Gal(L/ Q), j(q_L(γ)x)=γ(j(x)), is an actual continuous surjective homomorphism with open kernel. Surjectivity is derived by extending automorphisms to the algebraic closure. Pullback along a surjective homomorphism preserves and reflects equivariant linear isomorphisms, invariant subspaces, and irreducibility. These as…
Exact mathematical passage
q_L:G_{\Q}\longrightarrow\operatorname{Gal}(L/\Q),\qquad j(q_L(\gamma)x)=\gamma(j(x)),
\Q_pUse in source · 4 passages
Search aliases: Q_p Q mathbf
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2453
The p-adic square-root criterion is proved for every prime: (∃ x∈ Q_p, x^2=-1) Longleftrightarrow p≡1pmod4. A root satisfies lvert xrvert_p^2=1 and is therefore integral. Reduction modulo three rules out a root in Q_3, and reduction modulo four rules out a root in Q_2. At an odd prime, any residue root of X^2+1 has nonzero derivative and lifts by the pinned Mathlib Hensel theorem. The finite-field square criterion then gives the stated congruence. An explicit Hensel application at the residue ro…
Exact mathematical passage
\bigl(\exists x\in\Q_p,\ x^2=-1\bigr) \quad\Longleftrightarrow\quad p\equiv1\pmod4.
\mathbb
Q_pUse in source · 1 passage
Search aliases: Q_p mathbb
Use in source · references.tex, line 65
bibitemKisin2010 Mark Kisin. newblock Deformations of G_ Q_p and GL_2( Q_p) representations. newblock In Laurent Berger, Christophe Breuil, and Pierre Colmez, editors, em Repr esentations p-adiques de Groupes p-adiques II: Repr esentations de GL_2( Q_p) et (φ,Γ)-modules, volume 330 of em Ast erisque, pages 511--528. Soci et e Math ematique de France, Paris, 2010.
Exact mathematical passage
\mathrm{GL}_2(\mathbb Q_p)
\mathbb Q_pUse in source · 3 passages
Search aliases: Q_p mathbb
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 968
The formalization specializes to the actual maps G_ Q_p→ G_ Q induced by the field embeddings and choices of algebraic closures. A nontrivial continuous quadratic character of G_ Q, constructed from its action on i^2=-1, gives the integral 2-adic actions τ(σ)=diag(1,-1), ρ(σ)=1 · -1
Exact mathematical passage
G_{\mathbb Q_p}\to G_{\mathbb Q}
\Q_{p_s}Use in source · 1 passage
Search aliases: Q_p_s Q mathbf
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2524
Suppose now that R is I-adically complete and that every positive quotient R/I^m is finite. Let C_m⊆ P_q(R/I^m) be allowed subsets preserved by reduction, and let (p_s)_s∈ S be any indexed family of arithmetic primes. The index set S need not be finite. Lean proves the simultaneous arithmetic completion criterion · ∃χ∈ P_q(R), χ continuous, χbmod I^m∈ C_m (m≥1), χ_ Q_p_s=1 (s∈ S) · Longleftrightarrow ∀ m≥1, ∃ψ∈ C_m, ∀ s∈ S, p_s≡1pmod4 lor ψ=1. The constructed character also satisfies every odd-p…
Exact mathematical passage
\begin{split} &\exists\chi\in\mathcal P_q(R),\quad \chi\text{ continuous},\quad \chi\bmod I^m\in C_m\ (m\ge1),\quad \chi_{\Q_{p_s}}=1\ (s\in S)\\ &\quad\Longleftrightarrow\quad \forall m\ge1,\ \exists\psi\in C_m,\quad \forall s\in S,\ p_s\equiv1\pmod4\ \lor\ \psi=1. \end{split}
\mathfrak Q_uUse in source · 2 passages
Search aliases: Q_u mathfrak
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1284
The choice of absolute prime and Frobenius lift is also controlled. Let ( Q_u,γ_u) and ( Q_v,γ_v) be any two actual arithmetic witnesses at the same rational prime. Transitivity on primes of A above (p) gives a∈ G_ Q with Q_v=a Q_u, γ_v^-1aγ_u a^-1∈ I_ Q_v. Consequently, any representation trivial on I_ Q_v has the same trace on the two witnesses. For representations factoring through L, triviality on this actual absolute inertia is derived whenever p∉ S_L. The restrictions of all absolute witne…
Exact mathematical passage
(\mathfrak Q_u,\gamma_u)
\mathfrak Q_vUse in source · 3 passages
Search aliases: Q_v mathfrak
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1284
The choice of absolute prime and Frobenius lift is also controlled. Let ( Q_u,γ_u) and ( Q_v,γ_v) be any two actual arithmetic witnesses at the same rational prime. Transitivity on primes of A above (p) gives a∈ G_ Q with Q_v=a Q_u, γ_v^-1aγ_u a^-1∈ I_ Q_v. Consequently, any representation trivial on I_ Q_v has the same trace on the two witnesses. For representations factoring through L, triviality on this actual absolute inertia is derived whenever p∉ S_L. The restrictions of all absolute witne…
Exact mathematical passage
(\mathfrak Q_v,\gamma_v)
Q_{\bigoplus_iD_i^{\oplus m_i}}Use in source · 1 passage
Search aliases: Q_⊕_iD_i^⊕ m_i bigoplus oplus
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1512
Writing Q_D(T) for the reversed local characteristic polynomial, Lean proves Q_D(T)=det(1-T r(φ)mid E(D,H)), Q_D(0)=1, Q_⊕_iD_i^⊕ m_i(T)=∏_iQ_D_i(T)^m_i. The abstract formulas do not need characteristic zero. Normality and finite dimension appear explicitly where used. Arithmetic identification of G,H,ν and φ remains separate. In the geometric-Frobenius convention, ν(φ) is the inverse residue cardinality; it is this unit, rather than the residue cardinality, that enters the earlier special-chain…
Exact mathematical passage
Q_D(T)=\det(1-T r(\varphi)\mid E(D,H)),\qquad Q_D(0)=1, \qquad Q_{\bigoplus_iD_i^{\oplus m_i}}(T)=\prod_iQ_{D_i}(T)^{m_i}.
\QQUse in source · 3 passages
Search aliases: QQ
Use in source · pages.tex, line 304
vpage21The sign of a norm is arithmetic dataELEMENTS VERSUS IDEALSMain text: §9, arithmetic residue-counting continuation; signed-norm comparison. Even in the number field QQ, congruent generators need not have congruent absolute norms. The integers 2 and -3 are congruent modulo five. Their signed field norms are congruent, but their absolute values are not. vfig21_signed_norm headThe smallest counterexample is already decisive For K=QQ, N_K/QQ(2)=2, N_K/QQ(-3)=-3≡2pmod5, whereas N((2))=2, N((-3…
Exact mathematical passage
\QQ
rUse in source · 66 passages
Search aliases: r
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 250
Fix bar x∈ M(k). For m≥1, set X_m(bar x)= M_bar x( O/ϖ^m). Reduction gives transition maps r_m+1,m:X_m+1(bar x)⟶ X_m(bar x). A compatible branch is a sequence (x_m)_m≥1 satisfying r_m+1,m(x_m+1)=x_m.
Exact mathematical passage
r_{m+1,m}:X_{m+1}(\bar x)\longrightarrow X_m(\bar x).
r^2Use in source · 1 passage
Search aliases: r^2
Use in source · pages.tex, line 394
For a general positive integer L, write L=5q+r, 0≤ r<5. The complete tiled square has side L-r, so |S_L|≤ L^2-(L-r)^2=2Lr-r^2≤8L. The area is X=L^2, making this a square-root-in-area bound for this square-box model. The proof is the exact tile decomposition; checking many L values is only a regression check of the implementation.
Exact mathematical passage
|S_L|\le L^2-(L-r)^2=2Lr-r^2\le8L.
R^nUse in source · 3 passages
Search aliases: R^n
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2123
The actual submodules U=imE and W=ker E are invariant, and addition gives an equivariant isomorphism U⊕ W≃ R^n. The inclusion i:U↪ R^n and retraction r(v)=Ev are constructed as linear maps and satisfy ri=1_U and ir=E. Their equivariance is proved. Coordinatewise reduction maps U onto im(bar Ebmod(π)^k) and maps W onto ker(bar Ebmod(π)^k). Both equalities are surjectivity statements onto the actual submodules over the finite-level quotient, not merely containment. The same projector defines a sum…
Exact mathematical passage
U\oplus W\simeq R^n.
r_1Use in source · 2 passages
Search aliases: r_1
Use in source · Read passage · residue_continuation_body.tex, line 138
Ordinary class characters: an unconditional power saving. The preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl( O)→ C, define the actual finite ideal sum S_w(X)=∑_substackI≠0
Exact mathematical passage
A_C(X)=c_KX+O_{K,C}(X^{1-1/d}),\qquad c_K=\frac{2^{r_1}(2\pi)^{r_2}R_K}{\#\mu(K)\sqrt{\absn{\Delta_K}}}.
r_2Use in source · 2 passages
Search aliases: r_2
Use in source · Read passage · residue_continuation_body.tex, line 138
Ordinary class characters: an unconditional power saving. The preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl( O)→ C, define the actual finite ideal sum S_w(X)=∑_substackI≠0
Exact mathematical passage
A_C(X)=c_KX+O_{K,C}(X^{1-1/d}),\qquad c_K=\frac{2^{r_1}(2\pi)^{r_2}R_K}{\#\mu(K)\sqrt{\absn{\Delta_K}}}.
R_{\bar x}Use in source · 7 passages
Search aliases: R_bar x bar
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 259
[Represented complete deformation problems] When a deformation functor is represented by a complete local ring R_bar x, its O-valued points are continuous local maps R_bar x→ O. Completeness identifies such maps with compatible maps to every quotient O/ϖ^m.
Exact mathematical passage
R_{\bar x}
r_DUse in source · 4 passages
Search aliases: r_D
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1596
Block 13 also established an extracted inverse on the explicit class N^3=0: log_2 U=(U-I)-tfrac12(U-I)^2, a^-1log_2(exp(aN))=N (a≠0). Both exponential/logarithmic inverse identities are checked on the cubic nilpotent and unipotent loci. If a supplied common sample u has r_D(u)=r_E(u)=I and t(u)≠0, an intertwiner of the reconstructed cubic actions recovers both original group equivariance and monodromy equivariance. The general nilpotence exponential theorem is not restricted to this cubic class,…
Exact mathematical passage
r_D(u)=r_E(u)=I
r_EUse in source · 3 passages
Search aliases: r_E
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1596
Block 13 also established an extracted inverse on the explicit class N^3=0: log_2 U=(U-I)-tfrac12(U-I)^2, a^-1log_2(exp(aN))=N (a≠0). Both exponential/logarithmic inverse identities are checked on the cubic nilpotent and unipotent loci. If a supplied common sample u has r_D(u)=r_E(u)=I and t(u)≠0, an intertwiner of the reconstructed cubic actions recovers both original group equivariance and monodromy equivariance. The general nilpotence exponential theorem is not restricted to this cubic class,…
Exact mathematical passage
r_D(u)=r_E(u)=I
R_KUse in source · 2 passages
Search aliases: R_K
Use in source · Read passage · residue_continuation_body.tex, line 138
Ordinary class characters: an unconditional power saving. The preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl( O)→ C, define the actual finite ideal sum S_w(X)=∑_substackI≠0
Exact mathematical passage
A_C(X)=c_KX+O_{K,C}(X^{1-1/d}),\qquad c_K=\frac{2^{r_1}(2\pi)^{r_2}R_K}{\#\mu(K)\sqrt{\absn{\Delta_K}}}.
r_{m+1,m}Use in source · 3 passages
Search aliases: r_m+1,m
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 250
Fix bar x∈ M(k). For m≥1, set X_m(bar x)= M_bar x( O/ϖ^m). Reduction gives transition maps r_m+1,m:X_m+1(bar x)⟶ X_m(bar x). A compatible branch is a sequence (x_m)_m≥1 satisfying r_m+1,m(x_m+1)=x_m.
Exact mathematical passage
r_{m+1,m}:X_{m+1}(\bar x)\longrightarrow X_m(\bar x).
\operatorname{rank}Use in source · 1 passage
Search aliases: rank operatorname
Use in source · 01_three_questions.tex, line 38
.15); node[font=small,align=center,font=small] at (12.045,2.735) 0; path[draw=black!30,fill=white] (10.8,1.49) rectangle (11.63,2.32); node[font=small,align=center,font=small] at (11.215,1.905) 0; path[draw=black!30,fill=white] (11.63,1.49) rectangle (12.46,2.32); node[font=small,align=center,font=small] at (12.045,1.905) 0; node[font=small,align=center] at (11.65,1.05) rankE=1; node[font=small,align=center,font=footnotesize] at (11.65,0.4) positivity needs a carrier;
Exact mathematical passage
\operatorname{rank}E=1
\rankUse in source · 3 passages
Search aliases: rank operatorname
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1641
[Exact finite occupancy trace] The operator e_T is the projector onto the joint eigenspace A_T = {v:T_iv=a_iv for every i}. Moreover, Tr(e_T)=rank(e_T)=dim A_T∈ Z_≥0. The trace is identified with its unique nonnegative integer preimage under Z↪ E. Thus Tr(e_T)>0 refers to that integer, not to an ordering of an arbitrary field E. The finite packet is occupied exactly when this integer is positive. In positive characteristic the field-valued trace can vanish for a nonzero projector; the rank test,…
Exact mathematical passage
\boxed{ \Tr(e_T)=\rank(e_T)=\dim A_T\in\Z_{\ge0}. }
\rceilUse in source · 5 passages
Search aliases: rceil
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 587
[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.
Exact mathematical passage
\boxed{ q^{m-\lceil m/e\rceil} }
\operatorname{red}Use in source · 1 passage
Search aliases: red operatorname
Use in source · pages.tex, line 41
For example, the addresses 1,4,7,10,…,25 may be deeply connected by higher-digit changes. They remain disjoint from the addresses reducing to zero. In a genuine deformation problem, the same conclusion applies to operations whose reduction is the identity, not to every operation that happens to be called a deformation. red(x+3h)=red(x), M(A)=bigsqcup_bar x M_bar x(A). takeawayVertical completeness can fill one entire residual fiber without reaching a single point in a different fiber. captiontex…
Exact mathematical passage
\operatorname{red}(x+3h)=\operatorname{red}(x), \qquad \mathcal M(A)=\bigsqcup_{\bar x}\mathcal M_{\bar x}(A).
\rfloorUse in source · 1 passage
Search aliases: rfloor
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1976
The residual module partitions coordinates modulo p, verifies compatibility x_m+1≡ x_mpmodp^m, and reconstructs the digit a_m-1 by a_m-1=lfloorfracx_mp^m-1rfloorpmod p.
Exact mathematical passage
a_{m-1}=\left\lfloor\frac{x_m}{p^{m-1}}\right\rfloor\pmod p.
\rvertUse in source · 1 passage
Search aliases: rvert
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2457
The p-adic square-root criterion is proved for every prime: (∃ x∈ Q_p, x^2=-1) Longleftrightarrow p≡1pmod4. A root satisfies lvert xrvert_p^2=1 and is therefore integral. Reduction modulo three rules out a root in Q_3, and reduction modulo four rules out a root in Q_2. At an odd prime, any residue root of X^2+1 has nonzero derivative and lifts by the pinned Mathlib Hensel theorem. The finite-field square criterion then gives the stated congruence. An explicit Hensel application at the residue ro…
Exact mathematical passage
\lvert x\rvert_p^2=1
\rvert_p^2Use in source · 1 passage
Search aliases: rvert_p^2 rvert
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2457
The p-adic square-root criterion is proved for every prime: (∃ x∈ Q_p, x^2=-1) Longleftrightarrow p≡1pmod4. A root satisfies lvert xrvert_p^2=1 and is therefore integral. Reduction modulo three rules out a root in Q_3, and reduction modulo four rules out a root in Q_2. At an odd prime, any residue root of X^2+1 has nonzero derivative and lifts by the pinned Mathlib Hensel theorem. The finite-field square criterion then gives the stated congruence. An explicit Hensel application at the residue ro…
Exact mathematical passage
\lvert x\rvert_p^2=1
sUse in source · 25 passages
Search aliases: s
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 303
[Smooth formal branch count] If R_bar x≅ O[[t_1,…,t_d]] as complete local O-algebras, then #X_m(bar x)=q^d(m-1). Fix a uniformizer and a section s:k→ O of reduction, with s(0)=0. Every O-valued branch is encoded by d positive-depth digit streams t_i=s(a_i,1)ϖ+s(a_i,2)ϖ^2+⋯, a_i,j∈ k,
Exact mathematical passage
s:k\to\Ocal
S_{12}Use in source · 1 passage
Search aliases: S_12
Use in source · 26_boundary_count.tex, line 290
width=.8pt] (2.7,1.85) rectangle (4.2,3.35); node[font=small,align=center] at (3,4.4) [0,12)^2∩ Z^2; node[font=small,align=center] at (9.8,3.75) four full 5×5 cells; node[font=small,align=center] at (9.8,2.85) 100 bulk points: sum 0; node[font=small,align=center] at (9.8,1.95) 44 boundary points; node[font=small,align=center] at (9.8,1.05) actual total S_12=1; node[font=small,align=center,font=footnotesize] at (7,-0.1) |S_L|≤ L^2-(L-(Lbmod5))^2 for this square-lattice model.;
Exact mathematical passage
S_{12}=1
S_3Use in source · 3 passages
Search aliases: S_3
Use in source · 15_separator.tex, line 3
[x=1cm,y=1cm,>=Stealth,line cap=round,line join=round] path[use as bounding box] (0,-.6) rectangle (14,4.8); node[font=small,align=center,font=small] at (3.3,4.55) C: actual S_3 character table; path[draw=black!30,fill=black!8] (2.1,3.05) rectangle (2.9,3.85); node[font=small,align=center,font=small] at (2.5,3.45) 1; path[draw=black!30,fill=black!8] (2.9,3.05) rectangle (3.7,3.85); node[font=small,align=center,font=small] at (3.3,3.45) 1; path[draw=black!30,fill=black!8] (3.7,3.05) rectangle (4.…
Exact mathematical passage
S_3
S_LUse in source · 7 passages
Search aliases: S_L
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1273
The exceptional primes are controlled by the actual field discriminant: S_L={p rational prime:pmiddisc(L)}. This set is finite. For p∉ S_L, every prime of O_L above (p) is unramified. Lean proves that the actual inertia cardinality is the ramification index. It follows that inertia is trivial and the arithmetic Frobenius at a fixed finite prime is unique as a field automorphism. Neither trivial inertia nor uniqueness is assumed in place of the arithmetic unramifiedness condition.
Exact mathematical passage
S_L=\{p\text{ rational prime}:p\mid\operatorname{disc}(L)\}.
S_wUse in source · 6 passages
Search aliases: S_w
Use in source · Read passage · residue_continuation_body.tex, line 145
e preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl( O)→ C, define the actual finite ideal sum S_w(X)=∑_substackI≠0 N I≤ Xw([I]). Exact grouping by ideal class gives S_w(X)=∑_C w(C)A_C(X). If ∑_Cw(C)=0, the common leading coefficient cancels, so there is B_w≥0 s…
Exact mathematical passage
S_w(X)=\sum_{\substack{I\ne0\\\Nm I\leq X}}w([I]).
\mathrm{sign}Use in source · 1 passage
Search aliases: sign mathrm
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1042
0 · -1) and τ(g)=diag(1,-1). It preserves the residual comparison and has determinant one. Pullback through the quadratic character of the actual G_ Q gives continuous arithmetic examples. In contrast, the identity and diag(1,sign) over Z/4 agree modulo two but are not integrally conjugate. Their group order is not a unit. These restricted results do not supply general arithmetic packet attachment, all finite-level existence statements, or cross-coefficient compatibility.
Exact mathematical passage
\operatorname{diag}(1,\mathrm{sign})
\SpanUse in source · 1 passage
Search aliases: Span
Use in source · pages.tex, line 144
ies, and classesA TWO-PACKET COCHAIN COMPLEXMain text: Theorem 4.2 and the finite-complex example. Over FF_5, use bases (e_0,e_1), (u_0,u_1,u_2,u_3), and (f_0,f_1) in degrees zero, one, and two. The only nonzero differential values are d_0e_0=u_0 and d_1u_1=f_0. Consequently d_1d_0=0. vfig10_cohomology headThe quotient is the important step In degree one, the cycles are Span(u_0,u_2,u_3), but u_0 is a boundary. Thus the cohomology is represented by u_2,u_3, not by every vector killed by d_1. In …
Exact mathematical passage
\Span(u_0,u_2,u_3)
\operatorname{span}Use in source · 1 passage
Search aliases: span operatorname
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1016
For a second sector let R be a DVR with uniformizer π, fraction field K and matrix rank n. If a fixed template τ has full K-linear action span, finite generation and denominator clearing give a fixed c such that π^c M_n(R)⊆span_R{τ(g):g∈ G}. Normalize a field comparison to an integral P with a unit entry. The displayed span constructs an integral B with BP=π^cI, so det Pmidπ^cn, det Pnot≡0pmodπ^cn+1. One c works for all integral representations in the fixed field-isomorphism class. Jacobson dens…
Exact mathematical passage
\pi^c M_n(R)\subseteq\operatorname{span}_R\{\tau(g):g\in G\}.
\operatorname{span}_RUse in source · 1 passage
Search aliases: span_R operatorname
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1016
For a second sector let R be a DVR with uniformizer π, fraction field K and matrix rank n. If a fixed template τ has full K-linear action span, finite generation and denominator clearing give a fixed c such that π^c M_n(R)⊆span_R{τ(g):g∈ G}. Normalize a field comparison to an integral P with a unit entry. The displayed span constructs an integral B with BP=π^cI, so det Pmidπ^cn, det Pnot≡0pmodπ^cn+1. One c works for all integral representations in the fixed field-isomorphism class. Jacobson dens…
Exact mathematical passage
\pi^c M_n(R)\subseteq\operatorname{span}_R\{\tau(g):g\in G\}.
\SpecUse in source · 2 passages
Search aliases: Spec operatorname
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1623
Let E be a field of characteristic zero and let T_1,…,T_N be commuting operators on a finite-dimensional E-vector space A, each diagonalizable over E. Thus the selected spectra split over the coefficient field. Let a_i∈ E be a selected target scalar for T_i, possibly absent from its spectrum. Define e_i = ∏_μ∈Spec(T_i)∖{a_i} fracT_i-μ Ia_i-μ, e_T=∏_i=1^Ne_i, with e_i=0 if a_i is absent.
Exact mathematical passage
e_i = \prod_{\mu\in\Spec(T_i)\setminus\{a_i\}} \frac{T_i-\mu I}{a_i-\mu}, \qquad e_T=\prod_{i=1}^{N}e_i,
\sqcupUse in source · 1 passage
Search aliases: sqcup
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 712
Why the finite hypothesis cannot be dropped. Take vertices {*}sqcup{v_n:n∈ N}, with * isolated and edges v_n+1→ v_n. Every strongly connected component is a singleton, and {*} is the unique source component. No finite seed set covers all vertices: it can reach only indices no larger than the largest chosen index. Both the unique-source statement and the failure of every finite cover are checked in Lean. Hence one source component in an infinite condensation does not imply that one seed reaches a…
Exact mathematical passage
\{*\}\sqcup\{v_n:n\in\mathbf N\}
\operatorname{srank}Use in source · 1 passage
Search aliases: srank operatorname
Use in source · 12_source_scc.tex, line 29
ill=white] (7,3.8) circle (0.22); node[font=small,align=center] at (7,3.8) C; path[draw,fill=white] (11.7,2.4) circle (0.22); node[font=small,align=center] at (11.7,2.4) F; path[draw,fill=white] (7,0.65) circle (0.22); node[font=small,align=center] at (7,0.65) G; node[font=small,align=center] at (11.1,4.35) three source components; node[font=small,align=center] at (11.5,0.7) srank=3; node[font=small,align=center,font=footnotesize] at (7,-0.1) A minimum set meets each dashed source component.;
Exact mathematical passage
\operatorname{srank}=3
\operatorname{Stab}Use in source · 1 passage
Search aliases: Stab operatorname
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1191
The arithmetic witnesses use the integral closure A of Z in overline Q. For each rational prime p, Lean constructs a maximal ideal Q of A above (p) and an actual element γ_p∈ G_ Q such that γ_p∈Stab_G_ Q( Q), overlineγ_p(a)=overline a^ p (a∈ A). The existence argument applies Mathlib's proved profinite stabilizer-surjectivity theorem to this actual invariant ring. The residue p-power map is proved bijective by taking p-th roots in overline Q and showing that they remain integral when their p-th …
Exact mathematical passage
\gamma_p\in\operatorname{Stab}_{G_{\Q}}(\mathfrak Q),\qquad \overline{\gamma_p(a)}=\overline a^{\,p} \quad(a\in\mathcal A).
\operatorname{Stab}_{G_{\Q}}Use in source · 1 passage
Search aliases: Stab_G_Q Stab_G_ Q operatorname Q operatorname mathbf
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1191
The arithmetic witnesses use the integral closure A of Z in overline Q. For each rational prime p, Lean constructs a maximal ideal Q of A above (p) and an actual element γ_p∈ G_ Q such that γ_p∈Stab_G_ Q( Q), overlineγ_p(a)=overline a^ p (a∈ A). The existence argument applies Mathlib's proved profinite stabilizer-surjectivity theorem to this actual invariant ring. The residue p-power map is proved bijective by taking p-th roots in overline Q and showing that they remain integral when their p-th …
Exact mathematical passage
\gamma_p\in\operatorname{Stab}_{G_{\Q}}(\mathfrak Q),\qquad \overline{\gamma_p(a)}=\overline a^{\,p} \quad(a\in\mathcal A).
\SysUse in source · 1 passage
Search aliases: Sys mathscr S mathscr
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 777
The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.
Exact mathematical passage
\boxed{ \Sys_{\mathrm{sys}} = \Sys_2^{\mathrm{wt}} \times_{\Compat} \prod_{\lambda\nmid2}'\Sys_\lambda. }
\mathrm{sys}Use in source · 1 passage
Search aliases: sys mathrm
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 777
The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.
Exact mathematical passage
\boxed{ \Sys_{\mathrm{sys}} = \Sys_2^{\mathrm{wt}} \times_{\Compat} \prod_{\lambda\nmid2}'\Sys_\lambda. }
\Sys_2^{\mathrm{wt}}Use in source · 1 passage
Search aliases: Sys_2^wt mathscr S_2^wt Sys mathrm mathscr mathrm
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 777
The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.
Exact mathematical passage
\boxed{ \Sys_{\mathrm{sys}} = \Sys_2^{\mathrm{wt}} \times_{\Compat} \prod_{\lambda\nmid2}'\Sys_\lambda. }
\Sys_{\mathrm{sys}}Use in source · 1 passage
Search aliases: Sys_sys mathscr S_sys Sys mathrm mathscr mathrm
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 777
The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.
Exact mathematical passage
\boxed{ \Sys_{\mathrm{sys}} = \Sys_2^{\mathrm{wt}} \times_{\Compat} \prod_{\lambda\nmid2}'\Sys_\lambda. }
\Sys_\lambdaUse in source · 1 passage
Search aliases: Sys_λ mathscr S_λ Sys lambda mathscr lambda
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 777
The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.
Exact mathematical passage
\boxed{ \Sys_{\mathrm{sys}} = \Sys_2^{\mathrm{wt}} \times_{\Compat} \prod_{\lambda\nmid2}'\Sys_\lambda. }
T^2Use in source · 9 passages
Search aliases: T^2
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1862
For weights (1,2,4), the counts are 1,2,8,32,256,2048,…. The first coordinate is active immediately, the second one level later, and the third at depth five. The exact rational function generated by the code is Z_(1,2,4)(T) = fracT(1-6T-8T^2-32T^3)1-8T. The negative signs in this presentation arise from moving the denominator to 1-8T; every power-series coefficient is positive.
Exact mathematical passage
Z_{(1,2,4)}(T) = \frac{T(1-6T-8T^2-32T^3)}{1-8T}.
T^3Use in source · 2 passages
Search aliases: T^3
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1862
For weights (1,2,4), the counts are 1,2,8,32,256,2048,…. The first coordinate is active immediately, the second one level later, and the third at depth five. The exact rational function generated by the code is Z_(1,2,4)(T) = fracT(1-6T-8T^2-32T^3)1-8T. The negative signs in this presentation arise from moving the denominator to 1-8T; every power-series coefficient is positive.
Exact mathematical passage
Z_{(1,2,4)}(T) = \frac{T(1-6T-8T^2-32T^3)}{1-8T}.
t^{d-1}Use in source · 2 passages
Search aliases: t^d-1
Use in source · Read passage · residue_continuation_body.tex, line 38
There is a constant B≥0, depending on K,M,J, such that, for every a∈ J/M and every real t≥1, |#{x∈ J:xbmod M=a, ι(x)∈ tD} -fracvol(D)q covol(L(J))t^d| ≤ B(t^d-1+1). The same constant works for all residues a.
Exact mathematical passage
\left|\#\{x\in J:x\bmod M=a,\ \iota(x)\in tD\} -\frac{\vol(D)}{q\,\covol(L(J))}t^d\right| \leq B(t^{d-1}+1).
T^eUse in source · 1 passage
Search aliases: T^e
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 593
[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.
Exact mathematical passage
\boxed{ Z_e(T) = \frac{\sum_{r=1}^{e}q^{r-1}T^r} {1-q^{e-1}T^e}. }
T^rUse in source · 1 passage
Search aliases: T^r
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 593
[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.
Exact mathematical passage
\boxed{ Z_e(T) = \frac{\sum_{r=1}^{e}q^{r-1}T^r} {1-q^{e-1}T^e}. }
T_1Use in source · 5 passages
Search aliases: T_1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1622
Let E be a field of characteristic zero and let T_1,…,T_N be commuting operators on a finite-dimensional E-vector space A, each diagonalizable over E. Thus the selected spectra split over the coefficient field. Let a_i∈ E be a selected target scalar for T_i, possibly absent from its spectrum. Define e_i = ∏_μ∈Spec(T_i)∖{a_i} fracT_i-μ Ia_i-μ, e_T=∏_i=1^Ne_i, with e_i=0 if a_i is absent.
Exact mathematical passage
T_1,\ldots,T_N
t_1Use in source · 4 passages
Search aliases: t_1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 294
[Smooth formal branch count] If R_bar x≅ O[[t_1,…,t_d]] as complete local O-algebras, then #X_m(bar x)=q^d(m-1). Fix a uniformizer and a section s:k→ O of reduction, with s(0)=0. Every O-valued branch is encoded by d positive-depth digit streams t_i=s(a_i,1)ϖ+s(a_i,2)ϖ^2+⋯, a_i,j∈ k,
Exact mathematical passage
R_{\bar x}\cong\Ocal[[t_1,\ldots,t_d]]\quad\text{as complete local }\Ocal\text{-algebras},
T_2Use in source · 3 passages
Search aliases: T_2
Use in source · 18_occupancy.tex, line 25
h[draw=black!35,fill=black!18] (6.06,2.63) rectangle (7.46,3.12); path[draw=black!35,fill=white] (7.59,2.63) rectangle (8.99,3.12); path[draw=black!35,fill=white] (9.12,2.63) rectangle (10.52,3.12); path[draw=black!35,fill=white] (10.65,2.63) rectangle (12.05,3.12); node[font=small,align=center] at (13,2.89) 3; node[font=small,align=center,anchor=east,font=scriptsize] at (2.65,2.13) T_1=T_2=0; path[draw=black!35,fill=black!18] (3,1.87) rectangle (4.4,2.36); path[draw=black!35,fill=black!18] (4.5…
Exact mathematical passage
T_1=T_2=0
t_2Use in source · 3 passages
Search aliases: t_2
Use in source · 05_smooth_grid.tex, line 173
anchor=east] at (1.04,1.835) 9; node[font=small,align=center,font=scriptsize] at (3.915,0.12) 18; node[font=small,align=center,font=scriptsize,anchor=east] at (1.04,3.065) 18; node[font=small,align=center,font=scriptsize] at (4.735,0.12) 24; node[font=small,align=center,font=scriptsize,anchor=east] at (1.04,3.885) 24; node[font=small,align=center] at (3.1,4.45) (t_1,t_2)∈(3 Z/27 Z)^2; node[font=small,align=center] at (9.55,3.6) 9×9=81 points; node[font=small,align=center] at (9.55,2.55) (t_1,t_2…
Exact mathematical passage
(t_1,t_2)\in(3\mathbb Z/27\mathbb Z)^2
T_3Use in source · 2 passages
Search aliases: T_3
Use in source · 18_occupancy.tex, line 33
path[draw=black!35,fill=white] (6.06,1.87) rectangle (7.46,2.36); path[draw=black!35,fill=white] (7.59,1.87) rectangle (8.99,2.36); path[draw=black!35,fill=white] (9.12,1.87) rectangle (10.52,2.36); path[draw=black!35,fill=white] (10.65,1.87) rectangle (12.05,2.36); node[font=small,align=center] at (13,2.13) 2; node[font=small,align=center,anchor=east,font=scriptsize] at (2.65,1.37) T_1=T_2=T_3=0; path[draw=black!35,fill=black!18] (3,1.11) rectangle (4.4,1.6); path[draw=black!35,fill=white] (4.5…
Exact mathematical passage
T_1=T_2=T_3=0
t_dUse in source · 1 passage
Search aliases: t_d
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 294
[Smooth formal branch count] If R_bar x≅ O[[t_1,…,t_d]] as complete local O-algebras, then #X_m(bar x)=q^d(m-1). Fix a uniformizer and a section s:k→ O of reduction, with s(0)=0. Every O-valued branch is encoded by d positive-depth digit streams t_i=s(a_i,1)ϖ+s(a_i,2)ϖ^2+⋯, a_i,j∈ k,
Exact mathematical passage
R_{\bar x}\cong\Ocal[[t_1,\ldots,t_d]]\quad\text{as complete local }\Ocal\text{-algebras},
T_iUse in source · 11 passages
Search aliases: T_i
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1622
Let E be a field of characteristic zero and let T_1,…,T_N be commuting operators on a finite-dimensional E-vector space A, each diagonalizable over E. Thus the selected spectra split over the coefficient field. Let a_i∈ E be a selected target scalar for T_i, possibly absent from its spectrum. Define e_i = ∏_μ∈Spec(T_i)∖{a_i} fracT_i-μ Ia_i-μ, e_T=∏_i=1^Ne_i, with e_i=0 if a_i is absent.
Exact mathematical passage
T_i
t_iUse in source · 7 passages
Search aliases: t_i
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 304
[Smooth formal branch count] If R_bar x≅ O[[t_1,…,t_d]] as complete local O-algebras, then #X_m(bar x)=q^d(m-1). Fix a uniformizer and a section s:k→ O of reduction, with s(0)=0. Every O-valued branch is encoded by d positive-depth digit streams t_i=s(a_i,1)ϖ+s(a_i,2)ϖ^2+⋯, a_i,j∈ k,
Exact mathematical passage
t_i=s(a_{i,1})\varpi+s(a_{i,2})\varpi^2+\cdots, \qquad a_{i,j}\in k,
t_jUse in source · 1 passage
Search aliases: t_j
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1115
Suppose two members are isomorphic at λ_0. Injectivity of the common coefficient embedding gives t_i(v)=t_j(v) at every place good for λ_0. At another coefficient prime λ, this equality compares actual member traces on the intersection of the two good-place sets. Conjugacy density of that intersection and continuity extend the equality to the whole group; character rigidity then gives an actual isomorphism at λ. Thus the nonisomorphism of systems supplies the required nonisomorphism of members. …
Exact mathematical passage
t_i(v)=t_j(v)
T_NUse in source · 1 passage
Search aliases: T_N
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1622
Let E be a field of characteristic zero and let T_1,…,T_N be commuting operators on a finite-dimensional E-vector space A, each diagonalizable over E. Thus the selected spectra split over the coefficient field. Let a_i∈ E be a selected target scalar for T_i, possibly absent from its spectrum. Define e_i = ∏_μ∈Spec(T_i)∖{a_i} fracT_i-μ Ia_i-μ, e_T=∏_i=1^Ne_i, with e_i=0 if a_i is absent.
Exact mathematical passage
T_1,\ldots,T_N
\TrUse in source · 6 passages
Search aliases: Tr operatorname
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1641
[Exact finite occupancy trace] The operator e_T is the projector onto the joint eigenspace A_T = {v:T_iv=a_iv for every i}. Moreover, Tr(e_T)=rank(e_T)=dim A_T∈ Z_≥0. The trace is identified with its unique nonnegative integer preimage under Z↪ E. Thus Tr(e_T)>0 refers to that integer, not to an ordering of an arbitrary field E. The finite packet is occupied exactly when this integer is positive. In positive characteristic the field-valued trace can vanish for a nonzero projector; the rank test,…
Exact mathematical passage
\boxed{ \Tr(e_T)=\rank(e_T)=\dim A_T\in\Z_{\ge0}. }
\operatorname{Tr}Use in source · 2 passages
Search aliases: Tr operatorname
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1062
cible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficient-field embedding σ_λ:E↪ E_λ therefore sends C to an invertible matrix. Its entries agree with the ordinary representation traces Tr(W_i,λ(Frob_v_j)) when the selected places satisfy the unramified compatibility convention for that coefficient prime. In particular, places above the residue characte…
Exact mathematical passage
\operatorname{Tr}(W_{i,\lambda}(\operatorname{Frob}_{v_j}))
\operatorname{tr}Use in source · 1 passage
Search aliases: tr operatorname
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1234
le and the systems are pairwise componentwise nonisomorphic. The finite prime separator now yields distinct permitted rational primes p_j, actual prime ideals Q_j and arithmetic Frobenius witnesses γ_j as above, outside any prescribed finite set, such that the common algebraic trace matrix has nonzero determinant. Every coefficient embedding preserves this nonvanishing, and ι_λ(t_i(p_j))= tr(V_i,λ(γ_j)) whenever p_j is good for that coefficient member. The operator and trace values agree for eve…
Exact mathematical passage
\iota_\lambda(t_i(p_j))= \operatorname{tr}(V_{i,\lambda}(\gamma_j))
uUse in source · 57 passages
Search aliases: u
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 664
For a finite collection of roots, record a directed edge u→ v whenever occupancy or the relevant property is known to propagate from u to v.
Exact mathematical passage
u\to v
U^{-1}Use in source · 1 passage
Search aliases: U^-1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2203
For rigidity alone, let R be a commutative local ring, let π be a nonunit, and let E,F be complete equivariant families on the same carrier with E_i≡ F_ipmod(π)^k for every label and k>0. Then U=∑_i∈ JF_iE_i, U≡1pmod(π)^k, UE_i=F_iU. Reduction reflects invertibility in this setting, so U is an integral unit and UE_iU^-1=F_i. It commutes with the carrier and induces actual equivariant isomorphisms imE_i≃imF_i. This rigidity theorem requires neither a finite acting group, a domain, nor completenes…
Exact mathematical passage
UE_iU^{-1}=F_i
u^2Use in source · 4 passages
Search aliases: u^2
Use in source · 22_norm_table.tex, line 64
=small,align=center,font=scriptsize] at (1.76,4.23) b=0; node[font=small,align=center,font=scriptsize] at (2.38,4.23) b=1; node[font=small,align=center,font=scriptsize] at (3,4.23) b=2; node[font=small,align=center,font=scriptsize] at (3.62,4.23) b=3; node[font=small,align=center,font=scriptsize] at (4.24,4.23) b=4; node[font=small,align=center] at (9.6,4.25) u^2=2 in F_25; node[font=small,align=center] at (9.6,3.2) M_a+bu=a · 2b
Exact mathematical passage
u^2=2
u^{ab}Use in source · 2 passages
Search aliases: u^ab
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2280
For actual finite original inertia images the common kernel sample is derived. Suppose u∈ H satisfies ν(u)=1 and t(u)≠0 and both r_D(H) and r_E(H) are finite. Their sampled actions have positive finite orders a,b. At u^ab both original actions are trivial, and t(u^ab)=(ab)t(u)≠0 by the cocycle identity and characteristic zero. Thus the same full intertwiner equivalence holds using the actual finite-image hypotheses, without a separately supplied common trivial-action sample. The initial nonzero …
Exact mathematical passage
u^{ab}
u_0Use in source · 9 passages
Search aliases: u_0
Use in source · 10_cohomology.tex, line 20
nt=small,align=center,font=small] at (2.39,2.3) 0; path[draw=black!30,fill=white] (1.4,1.31) rectangle (2.06,1.97); node[font=small,align=center,font=small] at (1.73,1.64) 0; path[draw=black!30,fill=white] (2.06,1.31) rectangle (2.72,1.97); node[font=small,align=center,font=small] at (2.39,1.64) 0; node[font=small,align=center,anchor=east,font=scriptsize] at (1.25,3.62) u_0; node[font=small,align=center,anchor=east,font=scriptsize] at (1.25,2.96) u_1; node[font=small,align=center,anchor=east,fon…
Exact mathematical passage
u_0
u_1Use in source · 5 passages
Search aliases: u_1
Use in source · 10_cohomology.tex, line 21
te] (1.4,1.31) rectangle (2.06,1.97); node[font=small,align=center,font=small] at (1.73,1.64) 0; path[draw=black!30,fill=white] (2.06,1.31) rectangle (2.72,1.97); node[font=small,align=center,font=small] at (2.39,1.64) 0; node[font=small,align=center,anchor=east,font=scriptsize] at (1.25,3.62) u_0; node[font=small,align=center,anchor=east,font=scriptsize] at (1.25,2.96) u_1; node[font=small,align=center,anchor=east,font=scriptsize] at (1.25,2.3) u_2; node[font=small,align=center,anchor=east,font…
Exact mathematical passage
u_1
u_2Use in source · 7 passages
Search aliases: u_2
Use in source · 10_cohomology.tex, line 22
(1.73,1.64) 0; path[draw=black!30,fill=white] (2.06,1.31) rectangle (2.72,1.97); node[font=small,align=center,font=small] at (2.39,1.64) 0; node[font=small,align=center,anchor=east,font=scriptsize] at (1.25,3.62) u_0; node[font=small,align=center,anchor=east,font=scriptsize] at (1.25,2.96) u_1; node[font=small,align=center,anchor=east,font=scriptsize] at (1.25,2.3) u_2; node[font=small,align=center,anchor=east,font=scriptsize] at (1.25,1.64) u_3; node[font=small,align=center,font=scriptsize] at …
Exact mathematical passage
u_2
u_3Use in source · 7 passages
Search aliases: u_3
Use in source · 10_cohomology.tex, line 23
; node[font=small,align=center,font=small] at (2.39,1.64) 0; node[font=small,align=center,anchor=east,font=scriptsize] at (1.25,3.62) u_0; node[font=small,align=center,anchor=east,font=scriptsize] at (1.25,2.96) u_1; node[font=small,align=center,anchor=east,font=scriptsize] at (1.25,2.3) u_2; node[font=small,align=center,anchor=east,font=scriptsize] at (1.25,1.64) u_3; node[font=small,align=center,font=scriptsize] at (1.73,4.18) e_0; node[font=small,align=center,font=scriptsize] at (2.39,4.18) e…
Exact mathematical passage
u_3
U_cUse in source · 4 passages
Search aliases: U_c
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1392
Explicit summands inside a supplied decomposition. Let K be a field, let G be a monoid, and let ρ_i:G→End_K(W_i) be actual representations on finite-dimensional spaces, indexed by a finite set I. For natural numbers m_i≤ c_i, set U_m=∏_i∈ I∏_k<m_iW_i, U_c=∏_i∈ I∏_k<c_iW_i. These finite products realize the corresponding finite direct sums. The representation on U_m acts componentwise. Define j:U_m→ U_c by retaining the first m_i copies and filling every remaining coordinate with zero. Define r:U…
Exact mathematical passage
U_m=\prod_{i\in I}\prod_{k<m_i}W_i, \qquad U_c=\prod_{i\in I}\prod_{k<c_i}W_i.
U_iUse in source · 4 passages
Search aliases: U_i
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1680
Naturality and finite hypotheses. If F T_i=U_i F and the source operators split, then F e_T,a=e_U,aF. This follows first on source eigenvectors and then on their span. In particular, a change of basis conjugates the constructed projector and preserves its trace. Adding conditions decreases the actual joint eigenspace and cannot increase its dimension. These statements do not assume an orthonormal basis or a positive definite inner product.
Exact mathematical passage
F T_i=U_i F
U_mUse in source · 6 passages
Search aliases: U_m
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1392
Explicit summands inside a supplied decomposition. Let K be a field, let G be a monoid, and let ρ_i:G→End_K(W_i) be actual representations on finite-dimensional spaces, indexed by a finite set I. For natural numbers m_i≤ c_i, set U_m=∏_i∈ I∏_k<m_iW_i, U_c=∏_i∈ I∏_k<c_iW_i. These finite products realize the corresponding finite direct sums. The representation on U_m acts componentwise. Define j:U_m→ U_c by retaining the first m_i copies and filling every remaining coordinate with zero. Define r:U…
Exact mathematical passage
U_m=\prod_{i\in I}\prod_{k<m_i}W_i, \qquad U_c=\prod_{i\in I}\prod_{k<c_i}W_i.
V^HUse in source · 1 passage
Search aliases: V^H
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1496
Checked group, inertia and monodromy finite sums (Block 12). Let K be a field, G a group, and ν:G→ K^× a character. The new datum consists of an actual representation r and nilpotent N with r(g)N=ν(g)Nr(g) for every g. For a supplied subgroup H, define E(D,H)=ker N∩ V^H. This is the intersection of independently defined kernel and actual invariant subspaces. For normal H, the restricted whole-group representation and its inverse automorphisms are constructed. A change from φ to φ h, with h∈ H, l…
Exact mathematical passage
E(D,H)=\ker N\cap V^H.
v_1Use in source · 1 passage
Search aliases: v_1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1055
[Finite Frobenius separation] Let mathscr W_1,…,mathscr W_r be pairwise nonisomorphic strictly compatible systems whose members are absolutely irreducible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficient-field embedding σ_λ:E↪ E_λ therefore sends C to an invertible matrix. Its entries agree with the ordinary representation traces Tr(W_i,λ(Frob_v_j)) …
Exact mathematical passage
v_1,\ldots,v_r
v_2Use in source · 2 passages
Search aliases: v_2
Use in source · 17_precision.tex, line 21
mall,align=center,font=small] at (9.37,2) 8; draw[->,line width=.8pt] (4.35,2.8) -- (7.65,2.8); node[font=small,align=center] at (6,3.3) multiply by C; node[font=small,align=center] at (3.2,1.15) nonzero modulo 8; node[font=small,align=center] at (10,1.15) zero modulo 8; node[font=small,align=center,font=footnotesize] at (7,0.25) Modulo 16, the traces separate them. v_2(det C)=v_2(6)=1.;
Exact mathematical passage
v_2(\det C)=v_2(6)=1
v_3Use in source · 1 passage
Search aliases: v_3
Use in source · pages.tex, line 99
vpage7Many approximate roots, one limitSINGULAR POWER SEEDSMain text: Theorem 6.4 and Corollary 6.6. Now impose x^2=0 modulo 3^m. The condition is exactly 2v_3(x)≥ m, so the roots are the multiples of 3^lceil m/2rceil in that quotient. Their number is 3^m-lceil m/2rceil. vfig07_singular_forest headCount vertices and count branches separately There are three roots modulo nine: 0,3,6. But the only roots modulo 27 are 0,9,18, and all three reduce to zero modulo nine. The other two vertices at the p…
Exact mathematical passage
2v_3(x)\ge m
V_iUse in source · 1 passage
Search aliases: V_i
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1533
The remaining arithmetic comparison is WD(⊕_i V_i^⊕ m_i)≅ ⊕_iWD(V_i)^⊕ m_i for the actual attached functor, intertwining the Weil action and monodromy, together with the intended coefficient and Hodge--Tate comparisons. These are not supplied by the abstract datum. Claim 9.2 remains partial at its full arithmetic scope.
Exact mathematical passage
\operatorname{WD}(\bigoplus_i V_i^{\oplus m_i})\cong \bigoplus_i\operatorname{WD}(V_i)^{\oplus m_i}
v_iUse in source · 1 passage
Search aliases: v_i
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2185
The actual invariant images U_i=imE_i give an equivariant linear isomorphism ⊕_i∈ JU_i≃ R^n, (v_i)_i⟼∑_i v_i, v⟼(E_iv)_i. Lean represents the finite direct sum by the finite product of the image submodules. The existing coordinatewise reduction theorem maps every U_i onto the corresponding reduced projector image. No chosen bases or unproved freeness of the images are used.
Exact mathematical passage
\bigoplus_{i\in J}U_i\simeq R^n, \qquad (v_i)_i\longmapsto\sum_i v_i, \qquad v\longmapsto(E_iv)_i.
V_{i,\lambda}Use in source · 2 passages
Search aliases: V_i,λ lambda
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1227
For a compatible family ρ_i,λ of representations of the fixed finite group, set V_i,λ=ρ_i,λ∘ q_L. Suppose the actual absolute-Galois members are irreducible and the systems are pairwise componentwise nonisomorphic. The finite prime separator now yields distinct permitted rational primes p_j, actual prime ideals Q_j and arithmetic Frobenius witnesses γ_j as above, outside any prescribed finite set, such that the common algebraic trace matrix has nonzero determinant. Every coefficient embedding pr…
Exact mathematical passage
V_{i,\lambda}=\rho_{i,\lambda}\circ q_L
V_i^{\oplus m_i}Use in source · 1 passage
Search aliases: V_i^⊕ m_i oplus
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1533
The remaining arithmetic comparison is WD(⊕_i V_i^⊕ m_i)≅ ⊕_iWD(V_i)^⊕ m_i for the actual attached functor, intertwining the Weil action and monodromy, together with the intended coefficient and Hodge--Tate comparisons. These are not supplied by the abstract datum. Claim 9.2 remains partial at its full arithmetic scope.
Exact mathematical passage
\operatorname{WD}(\bigoplus_i V_i^{\oplus m_i})\cong \bigoplus_i\operatorname{WD}(V_i)^{\oplus m_i}
v_jUse in source · 2 passages
Search aliases: v_j
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1056
[Finite Frobenius separation] Let mathscr W_1,…,mathscr W_r be pairwise nonisomorphic strictly compatible systems whose members are absolutely irreducible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficient-field embedding σ_λ:E↪ E_λ therefore sends C to an invertible matrix. Its entries agree with the ordinary representation traces Tr(W_i,λ(Frob_v_j)) …
Exact mathematical passage
C=(t_i(v_j))_{j,i}\in M_r(E),\qquad \det C\ne0.
v_nUse in source · 2 passages
Search aliases: v_n
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 712
Why the finite hypothesis cannot be dropped. Take vertices {*}sqcup{v_n:n∈ N}, with * isolated and edges v_n+1→ v_n. Every strongly connected component is a singleton, and {*} is the unique source component. No finite seed set covers all vertices: it can reach only indices no larger than the largest chosen index. Both the unique-source statement and the failure of every finite cover are checked in Lean. Hence one source component in an infinite condensation does not imply that one seed reaches a…
Exact mathematical passage
\{*\}\sqcup\{v_n:n\in\mathbf N\}
v_{n+1}Use in source · 1 passage
Search aliases: v_n+1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 713
Why the finite hypothesis cannot be dropped. Take vertices {*}sqcup{v_n:n∈ N}, with * isolated and edges v_n+1→ v_n. Every strongly connected component is a singleton, and {*} is the unique source component. No finite seed set covers all vertices: it can reach only indices no larger than the largest chosen index. Both the unique-source statement and the failure of every finite cover are checked in Lean. Hence one source component in an infinite condensation does not imply that one seed reaches a…
Exact mathematical passage
v_{n+1}\to v_n
v_rUse in source · 1 passage
Search aliases: v_r
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1055
[Finite Frobenius separation] Let mathscr W_1,…,mathscr W_r be pairwise nonisomorphic strictly compatible systems whose members are absolutely irreducible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficient-field embedding σ_λ:E↪ E_λ therefore sends C to an invertible matrix. Its entries agree with the ordinary representation traces Tr(W_i,λ(Frob_v_j)) …
Exact mathematical passage
v_1,\ldots,v_r
V_\lambdaUse in source · 2 passages
Search aliases: V_λ lambda
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1353
arrier summand completion] Suppose a semisimple strictly compatible carrier decomposes as mathscr W ≅ ⊕_i=1^r mathscr W_i^⊕ c_i, with the mathscr W_i as above. At one coefficient prime λ_0, suppose a semisimple summand has the form V_λ_0 ≅ ⊕_i=1^r W_i,λ_0^⊕ m_i, 0≤ m_i≤ c_i. Then V_λ = ⊕_i=1^r W_i,λ^⊕ m_i defines a strictly compatible summand at every coefficient prime. The vector (m_i) is uniquely recoverable from finitely many Frobenius traces supplied by the separation theorem.
Exact mathematical passage
\boxed{ V_\lambda = \bigoplus_{i=1}^{r} W_{i,\lambda}^{\oplus m_i} }
V_{\lambda_0}Use in source · 1 passage
Search aliases: V_λ_0 lambda
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1344
[Compatible-carrier summand completion] Suppose a semisimple strictly compatible carrier decomposes as mathscr W ≅ ⊕_i=1^r mathscr W_i^⊕ c_i, with the mathscr W_i as above. At one coefficient prime λ_0, suppose a semisimple summand has the form V_λ_0 ≅ ⊕_i=1^r W_i,λ_0^⊕ m_i, 0≤ m_i≤ c_i. Then V_λ = ⊕_i=1^r W_i,λ^⊕ m_i defines a strictly compatible summand at every coefficient prime. The vector (m_i) is uniquely recoverable from finitely many Frobenius traces supplied by the separation theorem.
Exact mathematical passage
V_{\lambda_0} \cong \bigoplus_{i=1}^{r} W_{i,\lambda_0}^{\oplus m_i}, \qquad 0\le m_i\le c_i.
v_\piUse in source · 1 passage
Search aliases: v_π pi
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2049
For a DVR, e=v_π(|Γ|) is derived from the actual group order. The single precision e+1, independent of rank, detects integral isomorphism. It makes reduction injective on integral isomorphism classes, and finite coefficient quotients give finitely many integral classes in fixed rank. This is a formalization of classical finite-group rigidity (Maranda, 1953, Theorem~2), with an explicit proof of the comparison's precision loss.
Exact mathematical passage
e=v_\pi(|\Gamma|)
v_\varpiUse in source · 1 passage
Search aliases: v_ϖ varpi
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 605
The congruence is equivalent to e v_ϖ(x)≥ m, so x must lie in ϖ^lceil m/erceil O/ϖ^m, which has the stated cardinality. Write m=ek+r, 1≤ r≤ e. The exponent becomes (e-1)k+r-1. Summing over k≥0 and 1≤ r≤ e gives the rational expression.
Exact mathematical passage
e\,v_\varpi(x)\ge m,
\varprojlimUse in source · 5 passages
Search aliases: varprojlim
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 264
[Formal inverse-limit reconstruction] Assume M_bar x is represented by a complete local O-algebra R_bar x. Then reduction induces a natural bijection M_bar x( O) ≅ varprojlim_m M_bar x( O/ϖ^m).
Exact mathematical passage
\boxed{ \mathcal M_{\bar x}(\Ocal) \cong \varprojlim_m \mathcal M_{\bar x}(\Ocal/\varpi^m). }
\varprojlim_mUse in source · 4 passages
Search aliases: varprojlim_m varprojlim
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 264
[Formal inverse-limit reconstruction] Assume M_bar x is represented by a complete local O-algebra R_bar x. Then reduction induces a natural bijection M_bar x( O) ≅ varprojlim_m M_bar x( O/ϖ^m).
Exact mathematical passage
\boxed{ \mathcal M_{\bar x}(\Ocal) \cong \varprojlim_m \mathcal M_{\bar x}(\Ocal/\varpi^m). }
\varprojlim_{m\ge0}Use in source · 1 passage
Search aliases: varprojlim_m≥0 varprojlim ge
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2355
Let I⊆ R and suppose that R is I-adically complete. Using the checked ring-map reconstruction on the quadratic coordinate ring, Lean proves P_q(R)≃varprojlim_m≥0 P_q(R/I^m). The zero-precision quotient is included only for the formal indexing. The inverse constructs an actual arithmetic character, with exactly the prescribed reductions. It is unique, and its continuity is derived from its factorization through the same finite quotient.
Exact mathematical passage
\mathcal P_q(R)\simeq\varprojlim_{m\ge0}\mathcal P_q(R/I^m).
\volUse in source · 2 passages
Search aliases: vol operatorname
Use in source · Read passage · residue_continuation_body.tex, line 38
There is a constant B≥0, depending on K,M,J, such that, for every a∈ J/M and every real t≥1, |#{x∈ J:xbmod M=a, ι(x)∈ tD} -fracvol(D)q covol(L(J))t^d| ≤ B(t^d-1+1). The same constant works for all residues a.
Exact mathematical passage
\left|\#\{x\in J:x\bmod M=a,\ \iota(x)\in tD\} -\frac{\vol(D)}{q\,\covol(L(J))}t^d\right| \leq B(t^{d-1}+1).
WUse in source · 21 passages
Search aliases: W
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 556
[Weighted seed zeta function] Let 1≤ w_1≤⋯≤ w_d, put W=∑_iw_i, and let a_m=q^∑_imax(m-w_i,0). Then Z_ w(T) = ∑_m=1^w_da_mT^m + q^-Wfrac(q^dT)^w_d+11-q^dT. In particular, every weighted smooth chart has a rational seed zeta function.
Exact mathematical passage
W=\sum_iw_i
W_1Use in source · 1 passage
Search aliases: W_1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1052
[Finite Frobenius separation] Let mathscr W_1,…,mathscr W_r be pairwise nonisomorphic strictly compatible systems whose members are absolutely irreducible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficient-field embedding σ_λ:E↪ E_λ therefore sends C to an invertible matrix. Its entries agree with the ordinary representation traces Tr(W_i,λ(Frob_v_j)) …
Exact mathematical passage
\mathscr W_1,\ldots,\mathscr W_r
w_1Use in source · 3 passages
Search aliases: w_1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 504
[Weighted smooth-chart count] For weights w=(w_1,…,w_d), N_ w(m) = q^∑_i=1^dmax(m-w_i,0).
Exact mathematical passage
\bm w=(w_1,\ldots,w_d)
w_2Use in source · 1 passage
Search aliases: w_2
Use in source · 06_weights.tex, line 25
=black!14] (8,3.2) rectangle (9.3,3.87); node[font=small,align=center] at (8.65,3.55) 4; path[draw=black!35,fill=black!14] (9.35,3.2) rectangle (10.65,3.87); node[font=small,align=center] at (10,3.55) 5; path[draw=black!35,fill=black!14] (10.7,3.2) rectangle (12,3.87); node[font=small,align=center] at (11.35,3.55) 6; node[font=small,align=center,anchor=east] at (1.75,2.66) w_2=2; path[draw=black!35,fill=white] (2.6,2.3) rectangle (3.9,2.97); node[font=small,align=center] at (3.25,2.65) 0; path[d…
Exact mathematical passage
w_2=2
w_3Use in source · 1 passage
Search aliases: w_3
Use in source · 06_weights.tex, line 40
=black!14] (8,2.3) rectangle (9.3,2.97); node[font=small,align=center] at (8.65,2.65) 3; path[draw=black!35,fill=black!14] (9.35,2.3) rectangle (10.65,2.97); node[font=small,align=center] at (10,2.65) 4; path[draw=black!35,fill=black!14] (10.7,2.3) rectangle (12,2.97); node[font=small,align=center] at (11.35,2.65) 5; node[font=small,align=center,anchor=east] at (1.75,1.76) w_3=4; path[draw=black!35,fill=white] (2.6,1.4) rectangle (3.9,2.07); node[font=small,align=center] at (3.25,1.75) 0; path[d…
Exact mathematical passage
w_3=4
w_dUse in source · 4 passages
Search aliases: w_d
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 504
[Weighted smooth-chart count] For weights w=(w_1,…,w_d), N_ w(m) = q^∑_i=1^dmax(m-w_i,0).
Exact mathematical passage
\bm w=(w_1,\ldots,w_d)
W_iUse in source · 5 passages
Search aliases: W_i
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1104
System isomorphism and good-place overlaps (Block 25). The system equivalence used here is componentwise representation isomorphism: mathscr W_i≃mathscr W_j Longleftrightarrow ∀λ, W_i,λ≃ W_j,λ. For finite-dimensional irreducible representations of any monoid over a characteristic-zero field, the new character-rigidity theorem proves that equality of characters is equivalent to an actual equivariant linear isomorphism. Neither a finite group nor an algebraically closed coefficient field is needed…
Exact mathematical passage
\mathscr W_i\simeq\mathscr W_j \quad\Longleftrightarrow\quad \forall\lambda,\ W_{i,\lambda}\simeq W_{j,\lambda}.
w_iUse in source · 15 passages
Search aliases: w_i
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 497
A simple weighted formal model already captures delayed visibility. Let w_i≥1 and constrain t_i∈(ϖ^w_i O+ϖ^m O)/ϖ^m O. Then coordinate i is invisible until m>w_i.
Exact mathematical passage
w_i\ge1
W_{i,\lambda}Use in source · 2 passages
Search aliases: W_i,λ lambda
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1062
cible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficient-field embedding σ_λ:E↪ E_λ therefore sends C to an invertible matrix. Its entries agree with the ordinary representation traces Tr(W_i,λ(Frob_v_j)) when the selected places satisfy the unramified compatibility convention for that coefficient prime. In particular, places above the residue characte…
Exact mathematical passage
\operatorname{Tr}(W_{i,\lambda}(\operatorname{Frob}_{v_j}))
W_{i,\lambda}^{\oplus m_i}Use in source · 1 passage
Search aliases: W_i,λ^⊕ m_i lambda oplus
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1353
arrier summand completion] Suppose a semisimple strictly compatible carrier decomposes as mathscr W ≅ ⊕_i=1^r mathscr W_i^⊕ c_i, with the mathscr W_i as above. At one coefficient prime λ_0, suppose a semisimple summand has the form V_λ_0 ≅ ⊕_i=1^r W_i,λ_0^⊕ m_i, 0≤ m_i≤ c_i. Then V_λ = ⊕_i=1^r W_i,λ^⊕ m_i defines a strictly compatible summand at every coefficient prime. The vector (m_i) is uniquely recoverable from finitely many Frobenius traces supplied by the separation theorem.
Exact mathematical passage
\boxed{ V_\lambda = \bigoplus_{i=1}^{r} W_{i,\lambda}^{\oplus m_i} }
W_{i,\lambda_0}^{\oplus m_i}Use in source · 1 passage
Search aliases: W_i,λ_0^⊕ m_i lambda oplus
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1344
[Compatible-carrier summand completion] Suppose a semisimple strictly compatible carrier decomposes as mathscr W ≅ ⊕_i=1^r mathscr W_i^⊕ c_i, with the mathscr W_i as above. At one coefficient prime λ_0, suppose a semisimple summand has the form V_λ_0 ≅ ⊕_i=1^r W_i,λ_0^⊕ m_i, 0≤ m_i≤ c_i. Then V_λ = ⊕_i=1^r W_i,λ^⊕ m_i defines a strictly compatible summand at every coefficient prime. The vector (m_i) is uniquely recoverable from finitely many Frobenius traces supplied by the separation theorem.
Exact mathematical passage
V_{\lambda_0} \cong \bigoplus_{i=1}^{r} W_{i,\lambda_0}^{\oplus m_i}, \qquad 0\le m_i\le c_i.
W_i^{\oplus c_i}Use in source · 1 passage
Search aliases: W_i^⊕ c_i oplus
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1337
[Compatible-carrier summand completion] Suppose a semisimple strictly compatible carrier decomposes as mathscr W ≅ ⊕_i=1^r mathscr W_i^⊕ c_i, with the mathscr W_i as above. At one coefficient prime λ_0, suppose a semisimple summand has the form V_λ_0 ≅ ⊕_i=1^r W_i,λ_0^⊕ m_i, 0≤ m_i≤ c_i. Then V_λ = ⊕_i=1^r W_i,λ^⊕ m_i defines a strictly compatible summand at every coefficient prime. The vector (m_i) is uniquely recoverable from finitely many Frobenius traces supplied by the separation theorem.
Exact mathematical passage
\mathscr W \cong \bigoplus_{i=1}^{r} \mathscr W_i^{\oplus c_i},
W_jUse in source · 1 passage
Search aliases: W_j
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1104
System isomorphism and good-place overlaps (Block 25). The system equivalence used here is componentwise representation isomorphism: mathscr W_i≃mathscr W_j Longleftrightarrow ∀λ, W_i,λ≃ W_j,λ. For finite-dimensional irreducible representations of any monoid over a characteristic-zero field, the new character-rigidity theorem proves that equality of characters is equivalent to an actual equivariant linear isomorphism. Neither a finite group nor an algebraically closed coefficient field is needed…
Exact mathematical passage
\mathscr W_i\simeq\mathscr W_j \quad\Longleftrightarrow\quad \forall\lambda,\ W_{i,\lambda}\simeq W_{j,\lambda}.
W_{j,\lambda}Use in source · 1 passage
Search aliases: W_j,λ lambda
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1104
System isomorphism and good-place overlaps (Block 25). The system equivalence used here is componentwise representation isomorphism: mathscr W_i≃mathscr W_j Longleftrightarrow ∀λ, W_i,λ≃ W_j,λ. For finite-dimensional irreducible representations of any monoid over a characteristic-zero field, the new character-rigidity theorem proves that equality of characters is equivalent to an actual equivariant linear isomorphism. Neither a finite group nor an algebraically closed coefficient field is needed…
Exact mathematical passage
\mathscr W_i\simeq\mathscr W_j \quad\Longleftrightarrow\quad \forall\lambda,\ W_{i,\lambda}\simeq W_{j,\lambda}.
W_rUse in source · 1 passage
Search aliases: W_r
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1052
[Finite Frobenius separation] Let mathscr W_1,…,mathscr W_r be pairwise nonisomorphic strictly compatible systems whose members are absolutely irreducible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficient-field embedding σ_λ:E↪ E_λ therefore sends C to an invertible matrix. Its entries agree with the ordinary representation traces Tr(W_i,λ(Frob_v_j)) …
Exact mathematical passage
\mathscr W_1,\ldots,\mathscr W_r
W_\lambdaUse in source · 1 passage
Search aliases: W_λ lambda
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1365
Use the same multiplicities in every member of the compatible carrier. Direct sums preserve strict compatibility, and m_i≤ c_i makes V_λ a summand of W_λ. At the separating places, the target trace vector is C m. Full column rank makes m the unique solution.
Exact mathematical passage
W_\lambda
\operatorname{WD}Use in source · 1 passage
Search aliases: WD operatorname
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1533
The remaining arithmetic comparison is WD(⊕_i V_i^⊕ m_i)≅ ⊕_iWD(V_i)^⊕ m_i for the actual attached functor, intertwining the Weil action and monodromy, together with the intended coefficient and Hodge--Tate comparisons. These are not supplied by the abstract datum. Claim 9.2 remains partial at its full arithmetic scope.
Exact mathematical passage
\operatorname{WD}(\bigoplus_i V_i^{\oplus m_i})\cong \bigoplus_i\operatorname{WD}(V_i)^{\oplus m_i}
\mathrm{wt}Use in source · 1 passage
Search aliases: wt mathrm
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 777
The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.
Exact mathematical passage
\boxed{ \Sys_{\mathrm{sys}} = \Sys_2^{\mathrm{wt}} \times_{\Compat} \prod_{\lambda\nmid2}'\Sys_\lambda. }
x^{-2}Use in source · 1 passage
Search aliases: x^-2
Use in source · Read passage · residue_continuation_body.tex, line 169
An actual Mellin endpoint. The counting functions are constant on the integer norm steps. Their measurability, and hence that of S_w, is proved from this fact. For zero-sum w, define F_w(h)=(1+h)∫_1^∞x^-2-hS_w(x) dx (h≥0). Since 1-1/d<1, the logarithmic first moment M_w=∫_1^∞(1+log x) absnx^-2S_w(x) dx is finite. Lean proves the explicit estimate absnF_w(h)-F_w(0)≤ hM_w for every h≥0, and therefore F_w(h)=F_w(0)+O(h) as hdownarrow0. The function here is the displayed integral of the actual ideal…
Exact mathematical passage
M_w=\int_1^{\infty}(1+\log x)\,\absn{x^{-2}S_w(x)}\,dx
x^{-2-h}Use in source · 1 passage
Search aliases: x^-2-h
Use in source · Read passage · residue_continuation_body.tex, line 164
An actual Mellin endpoint. The counting functions are constant on the integer norm steps. Their measurability, and hence that of S_w, is proved from this fact. For zero-sum w, define F_w(h)=(1+h)∫_1^∞x^-2-hS_w(x) dx (h≥0). Since 1-1/d<1, the logarithmic first moment M_w=∫_1^∞(1+log x) absnx^-2S_w(x) dx is finite. Lean proves the explicit estimate absnF_w(h)-F_w(0)≤ hM_w for every h≥0, and therefore F_w(h)=F_w(0)+O(h) as hdownarrow0. The function here is the displayed integral of the actual ideal…
Exact mathematical passage
F_w(h)=(1+h)\int_1^{\infty}x^{-2-h}S_w(x)\,dx \qquad(h\geq0).
x^{-3/2}Use in source · 1 passage
Search aliases: x^-3/2
Use in source · pages.tex, line 417
why a power saving can make a Mellin expression regular at s=1. vfig27_mellin headCompute the endpoint rather than just naming it For real s>1/2, I(s)=s∫_1^∞ A(x)x^-s-1 dd x =fracss-1/2. At s=1+h this gives I(1+h)-I(1)=-frachh+1/2, |I(1+h)-I(1)|≤4|h| (|h|≤1/4). The derivative at one is -2. The logarithmic moment is finite as well: ∫_1^∞ x^-3/2log x dd x=4. These are explicit calculations; the drawn curve does not establish differentiability or convergence.
Exact mathematical passage
\int_1^\infty x^{-3/2}\log x\,\dd x=4
x^{-s-1}Use in source · 1 passage
Search aliases: x^-s-1
Use in source · pages.tex, line 407
IN ENDPOINT MODELMain text: §9; retained scalar/Mellin continuation: summatory measurability and first-order Mellin control. Use the explicit envelope A(x)=x^1/2 for x≥1. It is not an ideal-counting function. It is a model showing why a power saving can make a Mellin expression regular at s=1. vfig27_mellin headCompute the endpoint rather than just naming it For real s>1/2, I(s)=s∫_1^∞ A(x)x^-s-1 dd x =fracss-1/2. At s=1+h this gives I(1+h)-I(1)=-frachh+1/2, |I(1+h)-I(1)|≤4|h| (|h|≤1/4). The der…
Exact mathematical passage
I(s)=s\int_1^\infty A(x)x^{-s-1}\,\dd x =\frac{s}{s-1/2}.
X^{1-1/d}Use in source · 5 passages
Search aliases: X^1-1/d
Use in source · Read passage · residue_continuation_body.tex, line 138
Ordinary class characters: an unconditional power saving. The preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl( O)→ C, define the actual finite ideal sum S_w(X)=∑_substackI≠0
Exact mathematical passage
A_C(X)=c_KX+O_{K,C}(X^{1-1/d}),\qquad c_K=\frac{2^{r_1}(2\pi)^{r_2}R_K}{\#\mu(K)\sqrt{\absn{\Delta_K}}}.
x^{1/2}Use in source · 1 passage
Search aliases: x^1/2
Use in source · pages.tex, line 403
vpage27What a power-saving bound buys analyticallyA MELLIN ENDPOINT MODELMain text: §9; retained scalar/Mellin continuation: summatory measurability and first-order Mellin control. Use the explicit envelope A(x)=x^1/2 for x≥1. It is not an ideal-counting function. It is a model showing why a power saving can make a Mellin expression regular at s=1. vfig27_mellin headCompute the endpoint rather than just naming it For real s>1/2, I(s)=s∫_1^∞ A(x)x^-s-1 dd x =fracss-1/2. At s=1+h this gives I(1+h)…
Exact mathematical passage
A(x)=x^{1/2}
X^2Use in source · 7 passages
Search aliases: X^2
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 928
An actual Z_3 example applies the class-level completion theorem for the cyclic group of order two and polynomial X^2-1, producing a nonidentity generator. Another checked example has compatible classes represented by diag(1,-1) modulo 3 and diag(-1,1) modulo 9: these chosen matrices disagree under reduction, and an actual swap of basis repairs them. Arbitrary independent representatives therefore do not replace the framing argument.
Exact mathematical passage
X^2-1
x^2Use in source · 20 passages
Search aliases: x^2
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 629
[Fractional singular dimension] For x^e=0, dim_seed=1-frac1e. In particular, x^2=0 has branching dimension 1/2.
Exact mathematical passage
x^2=0
x^eUse in source · 6 passages
Search aliases: x^e
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 583
[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.
Exact mathematical passage
x^e=0
x^iUse in source · 1 passage
Search aliases: x^i
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1210
When pnmid n, the comparison with the finite cyclotomic sample is proved: q_L(γ_p)=σ_p, σ_p(ζ_n)=ζ_n^p. Reduction distinguishes n-th roots of unity at such a prime. Indeed, if x^n=y^n=1 and their reductions agree, the factorization of x^n-y^n gives (x-y)∑_i=0^n-1x^i y^n-1-i=0. The second factor reduces to noverline y^ n-1≠0, so the domain property gives x=y. This proves both Frobenius recognition on the cyclotomic generator and triviality of its inertia action. Consequently, every representation…
Exact mathematical passage
(x-y)\sum_{i=0}^{n-1}x^i y^{n-1-i}=0.
X^NUse in source · 2 passages
Search aliases: X^N
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 311
A local map sends each parameter into the image of (ϖ) in O/ϖ^m. Conversely, every tuple in that image is nilpotent and defines a coefficient-preserving evaluation homomorphism. This homomorphism is local. To prove uniqueness without assuming continuity of an arbitrary competing homomorphism, observe in one variable that a homomorphism killing X^N only sees the finite polynomial truncation modulo X^N. Induction over the finite number of variables gives uniqueness in all dimensions. The actual lo…
Exact mathematical passage
X^N
x^nUse in source · 2 passages
Search aliases: x^n
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1209
When pnmid n, the comparison with the finite cyclotomic sample is proved: q_L(γ_p)=σ_p, σ_p(ζ_n)=ζ_n^p. Reduction distinguishes n-th roots of unity at such a prime. Indeed, if x^n=y^n=1 and their reductions agree, the factorization of x^n-y^n gives (x-y)∑_i=0^n-1x^i y^n-1-i=0. The second factor reduces to noverline y^ n-1≠0, so the domain property gives x=y. This proves both Frobenius recognition on the cyclotomic generator and triviality of its inertia action. Consequently, every representation…
Exact mathematical passage
x^n=y^n=1
x_0Use in source · 1 passage
Search aliases: x_0
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1980
For canonical representatives 0≤ x_m<p^m, set x_0=0. Compatibility also gives a_m-1=(x_m-x_m-1)/p^m-1∈{0,…,p-1}. The floor in the first formula is essential. For example, the compatible base-five residues 4,19 give the second digit 3, not the rational number 19/5. The executable companion already uses integer floor division. This edition corrects the displayed algorithm, not the implementation.
Exact mathematical passage
x_0=0
X_1Use in source · 1 passage
Search aliases: X_1
Use in source · 09_infinite_tail.tex, line 13
all,align=center,font=scriptsize] at (7.15,4.5) 6; node[font=small,align=center,font=scriptsize] at (8.12,4.5) 7; node[font=small,align=center,font=scriptsize] at (9.09,4.5) 8; node[font=small,align=center,font=scriptsize] at (10.06,4.5) 9; node[font=small,align=center,font=scriptsize] at (11.03,4.5) 10; node[font=small,align=center,anchor=east] at (1.25,3.93) X_1; path[draw=black!25,fill=black!16] (1.85,3.68) rectangle (2.77,4.16); path[fill=black] (2.3,3.93) circle (0.055); path[draw=black!25,…
Exact mathematical passage
X_1
x_{1}Use in source · 1 passage
Search aliases: x_1
Use in source · 04_digits.tex, line 9
nt=scriptsize] at (5.35,4.45) 5^1; node[font=small,align=center,font=scriptsize] at (6.5,4.45) 5^2; node[font=small,align=center,font=scriptsize] at (7.65,4.45) 5^3; node[font=small,align=center,font=scriptsize] at (8.8,4.45) 5^4; node[font=small,align=center,font=scriptsize] at (9.95,4.45) 5^5; node[font=small,align=center,anchor=east] at (2.4,3.85) x_1=4; path[draw=black!25,fill=black!12] (3.65,3.58) rectangle (4.75,4.06); node[font=small,align=center,font=scriptsize] at (4.2,3.83) 4; path[dra…
Exact mathematical passage
x_{1}=4
X_2Use in source · 1 passage
Search aliases: X_2
Use in source · 09_infinite_tail.tex, line 35
ath[fill=black] (9.09,3.93) circle (0.055); path[draw=black!25,fill=black!16] (9.61,3.68) rectangle (10.53,4.16); path[fill=black] (10.06,3.93) circle (0.055); path[draw=black!25,fill=black!16] (10.58,3.68) rectangle (11.5,4.16); path[fill=black] (11.03,3.93) circle (0.055); node[font=small,align=center] at (12.1,3.93) ⋯; node[font=small,align=center,anchor=east] at (1.25,3.32) X_2; path[draw=black!25,fill=white] (1.85,3.07) rectangle (2.77,3.55); path[draw=black!25,fill=black!16] (2.82,3.07) re…
Exact mathematical passage
X_2
x_{2}Use in source · 1 passage
Search aliases: x_2
Use in source · 04_digits.tex, line 22
r,font=scriptsize] at (7.65,3.83) textemdash; path[draw=black!25,fill=white] (8.25,3.58) rectangle (9.35,4.06); node[font=small,align=center,font=scriptsize] at (8.8,3.83) textemdash; path[draw=black!25,fill=white] (9.4,3.58) rectangle (10.5,4.06); node[font=small,align=center,font=scriptsize] at (9.95,3.83) textemdash; node[font=small,align=center,anchor=east] at (2.4,3.23) x_2=19; path[draw=black!25,fill=black!12] (3.65,2.96) rectangle (4.75,3.44); node[font=small,align=center,font=scriptsize]…
Exact mathematical passage
x_{2}=19
X_3Use in source · 1 passage
Search aliases: X_3
Use in source · 09_infinite_tail.tex, line 56
ath[fill=black] (9.09,3.32) circle (0.055); path[draw=black!25,fill=black!16] (9.61,3.07) rectangle (10.53,3.55); path[fill=black] (10.06,3.32) circle (0.055); path[draw=black!25,fill=black!16] (10.58,3.07) rectangle (11.5,3.55); path[fill=black] (11.03,3.32) circle (0.055); node[font=small,align=center] at (12.1,3.32) ⋯; node[font=small,align=center,anchor=east] at (1.25,2.71) X_3; path[draw=black!25,fill=white] (1.85,2.46) rectangle (2.77,2.94); path[draw=black!25,fill=white] (2.82,2.46) recta…
Exact mathematical passage
X_3
x_{3}Use in source · 1 passage
Search aliases: x_3
Use in source · 04_digits.tex, line 35
r,font=scriptsize] at (7.65,3.21) textemdash; path[draw=black!25,fill=white] (8.25,2.96) rectangle (9.35,3.44); node[font=small,align=center,font=scriptsize] at (8.8,3.21) textemdash; path[draw=black!25,fill=white] (9.4,2.96) rectangle (10.5,3.44); node[font=small,align=center,font=scriptsize] at (9.95,3.21) textemdash; node[font=small,align=center,anchor=east] at (2.4,2.61) x_3=94; path[draw=black!25,fill=black!12] (3.65,2.34) rectangle (4.75,2.82); node[font=small,align=center,font=scriptsize]…
Exact mathematical passage
x_{3}=94
X_4Use in source · 1 passage
Search aliases: X_4
Use in source · 09_infinite_tail.tex, line 76
path[fill=black] (9.09,2.71) circle (0.055); path[draw=black!25,fill=black!16] (9.61,2.46) rectangle (10.53,2.94); path[fill=black] (10.06,2.71) circle (0.055); path[draw=black!25,fill=black!16] (10.58,2.46) rectangle (11.5,2.94); path[fill=black] (11.03,2.71) circle (0.055); node[font=small,align=center] at (12.1,2.71) ⋯; node[font=small,align=center,anchor=east] at (1.25,2.1) X_4; path[draw=black!25,fill=white] (1.85,1.85) rectangle (2.77,2.33); path[draw=black!25,fill=white] (2.82,1.85) recta…
Exact mathematical passage
X_4
x_{4}Use in source · 1 passage
Search aliases: x_4
Use in source · 04_digits.tex, line 48
r,font=scriptsize] at (7.65,2.59) textemdash; path[draw=black!25,fill=white] (8.25,2.34) rectangle (9.35,2.82); node[font=small,align=center,font=scriptsize] at (8.8,2.59) textemdash; path[draw=black!25,fill=white] (9.4,2.34) rectangle (10.5,2.82); node[font=small,align=center,font=scriptsize] at (9.95,2.59) textemdash; node[font=small,align=center,anchor=east] at (2.4,1.99) x_4=469; path[draw=black!25,fill=black!12] (3.65,1.72) rectangle (4.75,2.2); node[font=small,align=center,font=scriptsize]…
Exact mathematical passage
x_{4}=469
X_5Use in source · 1 passage
Search aliases: X_5
Use in source · 09_infinite_tail.tex, line 95
; path[fill=black] (9.09,2.1) circle (0.055); path[draw=black!25,fill=black!16] (9.61,1.85) rectangle (10.53,2.33); path[fill=black] (10.06,2.1) circle (0.055); path[draw=black!25,fill=black!16] (10.58,1.85) rectangle (11.5,2.33); path[fill=black] (11.03,2.1) circle (0.055); node[font=small,align=center] at (12.1,2.1) ⋯; node[font=small,align=center,anchor=east] at (1.25,1.49) X_5; path[draw=black!25,fill=white] (1.85,1.24) rectangle (2.77,1.72); path[draw=black!25,fill=white] (2.82,1.24) rectan…
Exact mathematical passage
X_5
x_{5}Use in source · 1 passage
Search aliases: x_5
Use in source · 04_digits.tex, line 61
lign=center,font=scriptsize] at (7.65,1.97) 3; path[draw=black!25,fill=white] (8.25,1.72) rectangle (9.35,2.2); node[font=small,align=center,font=scriptsize] at (8.8,1.97) textemdash; path[draw=black!25,fill=white] (9.4,1.72) rectangle (10.5,2.2); node[font=small,align=center,font=scriptsize] at (9.95,1.97) textemdash; node[font=small,align=center,anchor=east] at (2.4,1.37) x_5=2344; path[draw=black!25,fill=black!12] (3.65,1.1) rectangle (4.75,1.58); node[font=small,align=center,font=scriptsize]…
Exact mathematical passage
x_{5}=2344
X_6Use in source · 1 passage
Search aliases: X_6
Use in source · 09_infinite_tail.tex, line 113
ath[fill=black] (9.09,1.49) circle (0.055); path[draw=black!25,fill=black!16] (9.61,1.24) rectangle (10.53,1.72); path[fill=black] (10.06,1.49) circle (0.055); path[draw=black!25,fill=black!16] (10.58,1.24) rectangle (11.5,1.72); path[fill=black] (11.03,1.49) circle (0.055); node[font=small,align=center] at (12.1,1.49) ⋯; node[font=small,align=center,anchor=east] at (1.25,0.88) X_6; path[draw=black!25,fill=white] (1.85,0.63) rectangle (2.77,1.11); path[draw=black!25,fill=white] (2.82,0.63) recta…
Exact mathematical passage
X_6
x_{6}Use in source · 1 passage
Search aliases: x_6
Use in source · 04_digits.tex, line 74
all,align=center,font=scriptsize] at (7.65,1.35) 3; path[draw=black!25,fill=black!12] (8.25,1.1) rectangle (9.35,1.58); node[font=small,align=center,font=scriptsize] at (8.8,1.35) 3; path[draw=black!25,fill=white] (9.4,1.1) rectangle (10.5,1.58); node[font=small,align=center,font=scriptsize] at (9.95,1.35) textemdash; node[font=small,align=center,anchor=east] at (2.4,0.75) x_6=11719; path[draw=black!25,fill=black!12] (3.65,0.48) rectangle (4.75,0.96); node[font=small,align=center,font=scriptsize…
Exact mathematical passage
x_{6}=11719
x_kUse in source · 3 passages
Search aliases: x_k
Use in source · pages.tex, line 292
vpage20A condition must survive the limitCLOSEDNESS IS NOT DECORATIONMain text: Theorems 8.2 and 11.1; the inverse-limit closure hypotheses. In ZZ_3, every term of the sequence x_k=3^k is nonzero. Yet x_k→0 in the three-adic topology. The property ``nonzero'' is not closed, so a limit of objects satisfying it can lose that property. vfig20_closedness headPolynomial conditions behave differently For a continuous polynomial f, if a convergent sequence satisfies f(x_k)=0 for every k, then continuit…
Exact mathematical passage
x_k=3^k
x_mUse in source · 12 passages
Search aliases: x_m
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 253
Fix bar x∈ M(k). For m≥1, set X_m(bar x)= M_bar x( O/ϖ^m). Reduction gives transition maps r_m+1,m:X_m+1(bar x)⟶ X_m(bar x). A compatible branch is a sequence (x_m)_m≥1 satisfying r_m+1,m(x_m+1)=x_m.
Exact mathematical passage
(x_m)_{m\ge1}
X_{m+1}Use in source · 2 passages
Search aliases: X_m+1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 250
Fix bar x∈ M(k). For m≥1, set X_m(bar x)= M_bar x( O/ϖ^m). Reduction gives transition maps r_m+1,m:X_m+1(bar x)⟶ X_m(bar x). A compatible branch is a sequence (x_m)_m≥1 satisfying r_m+1,m(x_m+1)=x_m.
Exact mathematical passage
r_{m+1,m}:X_{m+1}(\bar x)\longrightarrow X_m(\bar x).
x_{m+1}Use in source · 5 passages
Search aliases: x_m+1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 254
Fix bar x∈ M(k). For m≥1, set X_m(bar x)= M_bar x( O/ϖ^m). Reduction gives transition maps r_m+1,m:X_m+1(bar x)⟶ X_m(bar x). A compatible branch is a sequence (x_m)_m≥1 satisfying r_m+1,m(x_m+1)=x_m.
Exact mathematical passage
r_{m+1,m}(x_{m+1})=x_m.
x_{m-1}Use in source · 1 passage
Search aliases: x_m-1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1981
For canonical representatives 0≤ x_m<p^m, set x_0=0. Compatibility also gives a_m-1=(x_m-x_m-1)/p^m-1∈{0,…,p-1}. The floor in the first formula is essential. For example, the compatible base-five residues 4,19 give the second digit 3, not the rational number 19/5. The executable companion already uses integer floor division. This edition corrects the displayed algorithm, not the implementation.
Exact mathematical passage
a_{m-1}=(x_m-x_{m-1})/p^{m-1}\in\{0,\ldots,p-1\}
x_pUse in source · 2 passages
Search aliases: x_p
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1806
An admissible realization here includes continuous coefficient-field members and membership of every finite reduction in its designated admissible class. Under the completion hypotheses and these supplied carrier hypotheses, Lean proves · ∃ ρ, (x_p)_p: ρ is an admissible realization of the full chosen spine, 0≠ x_p∈im(e_p) for every p · Longleftrightarrow (∀λ, X_λ,1( C)≠∅) land (∀λ, ∀ m≥2, X_λ,m( C)≠∅) land (∀ p, n_p>0). The packet index may be infinite. Each x_p belongs to its own supplied carr…
Exact mathematical passage
\begin{aligned} &\exists\,\rho,\ (x_p)_p:\quad \rho\text{ is an admissible realization of the full chosen spine},\quad 0\ne x_p\in\operatorname{im}(e_p)\text{ for every }p\\ &\quad\Longleftrightarrow\quad \bigl(\forall\lambda,\ X_{\lambda,1}(\Compat)\ne\varnothing\bigr) \ \land\ \bigl(\forall\lambda,\ \forall m\ge2, X_{\lambda,m}(\Compat)\ne\varnothing\bigr) \ \land\ \bigl(\forall p,\ n_p>0\bigr). \end{aligned}
X_{\lambda,1}Use in source · 3 passages
Search aliases: X_λ,1 lambda
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 809
[Residual companion obstruction] If X_λ,1( C)=∅ for one coefficient prime λ, no compatible completion with spine C exists. Deeper perturbations at other primes cannot create the missing residual companion.
Exact mathematical passage
X_{\lambda,1}(\Compat)=\varnothing
\xrightarrowUse in source · 4 passages
Search aliases: xrightarrow
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 343
Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.
Exact mathematical passage
C^0\xrightarrow{d_0}C^1\xrightarrow{d_1}C^2\xrightarrow{}\cdots
yUse in source · 16 passages
Search aliases: y
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 214
[No crossing under positive-depth perturbation] Suppose x,y∈ M( O) are related by a deformation operation that is congruent to the identity modulo ϖ. Then red(x)=red(y). Every finite composition remains in one residual fiber. If an infinite perturbation stream converges in the ϖ-adic topology, its limit remains in the same fiber. The existence of that limit is an explicit hypothesis, not a consequence of positive depth alone.
Exact mathematical passage
x,y\in\mathcal M(\Ocal)
y^{-1}Use in source · 1 passage
Search aliases: y^-1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2483
The inertia theorem uses actual residue actions. For a valuation subring A of an extension field L/K, let D_A be its decomposition subgroup in Aut_K(L) and put I_A=ker(D_A⟶Aut(k_A)), where k_A is the residue field. If 2 is a unit in A, every member of I_A fixes every y∈ L with y^2=-1. Indeed y^-1=-y, so y belongs to A and is a unit there. An automorphism sends y to y or -y. Their residues are distinct because 2 and y are units, whereas inertia fixes residues. The second possibility is excluded.
Exact mathematical passage
y^{-1}=-y
Y^{1-1/d}Use in source · 2 passages
Search aliases: Y^1-1/d
Use in source · Read passage · residue_continuation_body.tex, line 56
attice discrepancy theorem. The index factor comes from the covolume identity covol(L(M))=q covol(L(J)). For a complex weight w:J/M→ C, summing the estimates gives the leading term fracvol(D)t^dq covol(L(J))∑_a∈ J/Mw(a), with error at most B∑_aabsnw(a)(t^d-1+1). If the finite sum of weights is zero, only that error remains. Writing Y=t^d gives an error O(Y^1-1/d+1) for the actual norm threshold. These assertions include d=1.
Exact mathematical passage
O(Y^{1-1/d}+1)
y^2Use in source · 1 passage
Search aliases: y^2
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2483
The inertia theorem uses actual residue actions. For a valuation subring A of an extension field L/K, let D_A be its decomposition subgroup in Aut_K(L) and put I_A=ker(D_A⟶Aut(k_A)), where k_A is the residue field. If 2 is a unit in A, every member of I_A fixes every y∈ L with y^2=-1. Indeed y^-1=-y, so y belongs to A and is a unit there. An automorphism sends y to y or -y. Their residues are distinct because 2 and y are units, whereas inertia fixes residues. The second possibility is excluded.
Exact mathematical passage
y^2=-1
y^nUse in source · 2 passages
Search aliases: y^n
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1209
When pnmid n, the comparison with the finite cyclotomic sample is proved: q_L(γ_p)=σ_p, σ_p(ζ_n)=ζ_n^p. Reduction distinguishes n-th roots of unity at such a prime. Indeed, if x^n=y^n=1 and their reductions agree, the factorization of x^n-y^n gives (x-y)∑_i=0^n-1x^i y^n-1-i=0. The second factor reduces to noverline y^ n-1≠0, so the domain property gives x=y. This proves both Frobenius recognition on the cyclotomic generator and triviality of its inertia action. Consequently, every representation…
Exact mathematical passage
x^n=y^n=1
y^{n-1-i}Use in source · 1 passage
Search aliases: y^n-1-i
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1210
When pnmid n, the comparison with the finite cyclotomic sample is proved: q_L(γ_p)=σ_p, σ_p(ζ_n)=ζ_n^p. Reduction distinguishes n-th roots of unity at such a prime. Indeed, if x^n=y^n=1 and their reductions agree, the factorization of x^n-y^n gives (x-y)∑_i=0^n-1x^i y^n-1-i=0. The second factor reduces to noverline y^ n-1≠0, so the domain property gives x=y. This proves both Frobenius recognition on the cyclotomic generator and triviality of its inertia action. Consequently, every representation…
Exact mathematical passage
(x-y)\sum_{i=0}^{n-1}x^i y^{n-1-i}=0.
\ZUse in source · 11 passages
Search aliases: Z mathbf
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1641
[Exact finite occupancy trace] The operator e_T is the projector onto the joint eigenspace A_T = {v:T_iv=a_iv for every i}. Moreover, Tr(e_T)=rank(e_T)=dim A_T∈ Z_≥0. The trace is identified with its unique nonnegative integer preimage under Z↪ E. Thus Tr(e_T)>0 refers to that integer, not to an ordering of an arbitrary field E. The finite packet is occupied exactly when this integer is positive. In positive characteristic the field-valued trace can vanish for a nonzero projector; the rank test,…
Exact mathematical passage
\boxed{ \Tr(e_T)=\rank(e_T)=\dim A_T\in\Z_{\ge0}. }
\mathbf ZUse in source · 1 passage
Search aliases: Z mathbf
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1554
[Multiplicity recovery] The original companion matrix was not retrieved or rerun. A new exact realization uses the trivial character and two independent sign characters of ( Z/2 Z)^2, evaluated at (0,0),(1,0),(0,1): C=1 · 1 · 1
Exact mathematical passage
(\mathbf Z/2\mathbf Z)^2
zUse in source · 3 passages
Search aliases: z
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 390
For i≠ j, orthogonality gives H(p_i)H(i_j)=0. Idempotence gives H(p_j)H(i_j)=I, and completeness gives ∑_j H(i_j)H(p_j)=I. Thus the maps [z]⟼([P_j^r z])_j, ([z_j])_j⟼[∑_j i_j z_j] are inverse linear maps. The finite product equals the direct sum, proving the assertion. In the formalization, both sides are also identified with Mathlib's standard homology objects of the genuine graded complexes.
Exact mathematical passage
[z]\longmapsto([P_j^r z])_j, \qquad ([z_j])_j\longmapsto\left[\sum_j i_j z_j\right]
Z^1Use in source · 1 passage
Search aliases: Z^1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 339
[Tangent and obstruction spaces] For a framed representation problem, first-order lifts are described by Z^1(G,adbarρ). For an unframed Schur problem, tangent directions are described by H^1(G,adbarρ). Relations in a presentation of a deformation ring map into H^2(G,adbarρ), which contains the obstruction classes.
Exact mathematical passage
Z^1(G,\ad\bar\rho)
\mathbb Z^2Use in source · 1 passage
Search aliases: Z^2 mathbb
Use in source · 26_boundary_count.tex, line 286
raw=black!22,fill=black!13] (4.5,3.65) rectangle (4.8,3.95); path[draw,fill=black] (4.65,3.8) circle (0.035); path[draw,line width=.8pt] (1.2,0.35) rectangle (2.7,1.85); path[draw,line width=.8pt] (1.2,1.85) rectangle (2.7,3.35); path[draw,line width=.8pt] (2.7,0.35) rectangle (4.2,1.85); path[draw,line width=.8pt] (2.7,1.85) rectangle (4.2,3.35); node[font=small,align=center] at (3,4.4) [0,12)^2∩ Z^2; node[font=small,align=center] at (9.8,3.75) four full 5×5 cells; node[font=small,align=center]…
Exact mathematical passage
[0,12)^2\cap\mathbb Z^2
Z^rUse in source · 1 passage
Search aliases: Z^r
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 377
For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet are cochain maps, so they induce maps on these genuine quotients.
Exact mathematical passage
Z^r=\ker d_r,\qquad B^r=\operatorname{im}d_{r-1}\subseteq Z^r, \qquad H^r=Z^r/B^r,
Z_{(1,2,4)}Use in source · 1 passage
Search aliases: Z_(1,2,4)
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1862
For weights (1,2,4), the counts are 1,2,8,32,256,2048,…. The first coordinate is active immediately, the second one level later, and the third at depth five. The exact rational function generated by the code is Z_(1,2,4)(T) = fracT(1-6T-8T^2-32T^3)1-8T. The negative signs in this presentation arise from moving the denominator to 1-8T; every power-series coefficient is positive.
Exact mathematical passage
Z_{(1,2,4)}(T) = \frac{T(1-6T-8T^2-32T^3)}{1-8T}.
\Z_2Use in source · 4 passages
Search aliases: Z_2 Z mathbf
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2383
For R= Z_2, Lean derives finiteness of all positive maximal-ideal quotients and applies this completion theorem. Since Z_2 is a domain, its only roots of X^2-1 are 1 and -1. Thus the integral packet has exactly two distinct characters: the trivial character and the sign character describing the action on i. They have the same reduction modulo two and distinct reductions modulo four.
Exact mathematical passage
R=\Z_2
\mathbb Z_2Use in source · 4 passages
Search aliases: Z_2 mathbb
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1252
The boundaries are formalized as well. Pullback along a nonsurjective homomorphism can make inequivalent irreducible representations isomorphic. Samples can cover one quotient while missing an open quotient of the full group. Finally, in Z_2, the distinct square roots 1 and -1 have the same residue, so the condition that the root order be a unit cannot be discarded.
Exact mathematical passage
\mathbb Z_2
\mathbb Z_3Use in source · 3 passages
Search aliases: Z_3 mathbb
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 927
An actual Z_3 example applies the class-level completion theorem for the cyclic group of order two and polynomial X^2-1, producing a nonidentity generator. Another checked example has compatible classes represented by diag(1,-1) modulo 3 and diag(-1,1) modulo 9: these chosen matrices disagree under reduction, and an actual swap of basis repairs them. Arbitrary independent representatives therefore do not replace the framing argument.
Exact mathematical passage
\mathbb Z_3
\Z_5Use in source · 1 passage
Search aliases: Z_5 Z mathbf
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2463
d is therefore integral. Reduction modulo three rules out a root in Q_3, and reduction modulo four rules out a root in Q_2. At an odd prime, any residue root of X^2+1 has nonzero derivative and lifts by the pinned Mathlib Hensel theorem. The finite-field square criterion then gives the stated congruence. An explicit Hensel application at the residue root 2 modulo 5 supplies a root in Z_5. Consequently LocallyTrivial_ Q_p(χ) Longleftrightarrow p≡1pmod4 lor χ=1. The arithmetic prime p is independe…
Exact mathematical passage
\Z_5
\mathbb Z_5Use in source · 1 passage
Search aliases: Z_5 mathbb
Use in source · 01_three_questions.tex, line 12
athbb F_2; path[draw,fill=black!12] (2.1,2.55) circle (0.24); node[font=small,align=center] at (2.1,2.55) 1; node[font=small,align=center] at (2.1,1.45) one point mod 2; node[font=small,align=center,font=footnotesize] at (2.1,0.55) no point mod 4; node[font=small,align=center] at (6.9,4.35) A coherent branch; node[font=small,align=center] at (6.9,3.55) x^2+1=0 over Z_5; path[draw,fill=white] (5.3,2.5) circle (0.29); node[font=small,align=center,font=scriptsize] at (5.3,2.5) 2; node[font=small,al…
Exact mathematical passage
\mathbb Z_5
Z_dUse in source · 1 passage
Search aliases: Z_d
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 544
[Smooth seed zeta function] For a smooth formal component of dimension d, Z_d(T)=fracT1-q^dT.
Exact mathematical passage
\boxed{ Z_d(T)=\frac{T}{1-q^dT}. }
Z_eUse in source · 1 passage
Search aliases: Z_e
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 593
[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.
Exact mathematical passage
\boxed{ Z_e(T) = \frac{\sum_{r=1}^{e}q^{r-1}T^r} {1-q^{e-1}T^e}. }
z_jUse in source · 1 passage
Search aliases: z_j
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 390
For i≠ j, orthogonality gives H(p_i)H(i_j)=0. Idempotence gives H(p_j)H(i_j)=I, and completeness gives ∑_j H(i_j)H(p_j)=I. Thus the maps [z]⟼([P_j^r z])_j, ([z_j])_j⟼[∑_j i_j z_j] are inverse linear maps. The finite product equals the direct sum, proving the assertion. In the formalization, both sides are also identified with Mathlib's standard homology objects of the genuine graded complexes.
Exact mathematical passage
[z]\longmapsto([P_j^r z])_j, \qquad ([z_j])_j\longmapsto\left[\sum_j i_j z_j\right]
Z_j^rUse in source · 1 passage
Search aliases: Z_j^r
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 381
For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet are cochain maps, so they induce maps on these genuine quotients.
Exact mathematical passage
Z_j^r,B_j^r,H_j^r
\Z_pUse in source · 2 passages
Search aliases: Z_p Z mathbf
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2489
For every prime p, Lean constructs a valuation subring A⊆overline Q_p containing the integral image of Z_p such that the induced map Z_p→ A is local. For odd p, two is a unit. The preceding argument applies to every valuation subring containing that integral image, and hence every member of P_q(B) is trivial on its inertia subgroup after actual local restriction. The existence theorem supplies a locally dominating choice, so this assertion is not vacuous.
Exact mathematical passage
\Z_p
\Z_{\ge0}Use in source · 1 passage
Search aliases: Z_≥0 Z ge mathbf ge
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1641
[Exact finite occupancy trace] The operator e_T is the projector onto the joint eigenspace A_T = {v:T_iv=a_iv for every i}. Moreover, Tr(e_T)=rank(e_T)=dim A_T∈ Z_≥0. The trace is identified with its unique nonnegative integer preimage under Z↪ E. Thus Tr(e_T)>0 refers to that integer, not to an ordering of an arbitrary field E. The finite packet is occupied exactly when this integer is positive. In positive characteristic the field-valued trace can vanish for a nonzero projector; the rank test,…
Exact mathematical passage
\boxed{ \Tr(e_T)=\rank(e_T)=\dim A_T\in\Z_{\ge0}. }
\ZZ^2Use in source · 1 passage
Search aliases: ZZ^2 ZZ
Use in source · pages.tex, line 388
vpage26Cancellation leaves a boundary problemA FULL LATTICE-BOX CALCULATIONMain text: §9; retained scalar/Mellin continuation: additive residue counts and norm-character bounds. Extend the function f(a,b)=χ(a^2-2b^2bmod5) periodically to ZZ^2. Every complete 5×5 residue tile has sum zero by the finite-field calculation. Only incomplete tiles can contribute to a box sum. vfig26_boundary_count headSeparate bulk cancellation from boundary size For 0≤ a,b<12, four complete tiles fill a 10×10 square …
Exact mathematical passage
\ZZ^2
\ZZ_2Use in source · 4 passages
Search aliases: ZZ_2 ZZ
Use in source · pages.tex, line 82
vpage6Weights delay visibilityWEIGHTED DYADIC CHARTSMain text: Proposition 5.3, Theorem 6.3, and §12.2. Over ZZ_2, require three coordinates to lie in 2ZZ_2, 2^2ZZ_2, and 2^4ZZ_2. At depth m, coordinate i has max(m-w_i,0) free binary digits. A zero in the table means only one allowed residue: zero. vfig06_weights headThe two different thresholds A coordinate of weight w_i first becomes nontrivial when m=w_i+1. By contrast, the first separation depth of two distinct fixed values a,b is v_2(a-b)+1…
Exact mathematical passage
\ZZ_2
\ZZ_3Use in source · 5 passages
Search aliases: ZZ_3 ZZ
Use in source · pages.tex, line 68
vpage5A smooth chart has independent choicesACTUAL FINITE POINTSMain text: Proposition 3.4 and Theorem 6.2. Take R=ZZ_3[[t_1,t_2]] and coefficient-preserving local maps to ZZ/27ZZ. Locality forces both parameters into the maximal ideal. Each therefore has nine possible values, 0,3,6,…,24. vfig05_smooth_grid headWhy the exponent is d(m-1) At depth m, the image of the maximal ideal contains 3^m-1 elements. Two independent parameters give 3^2(m-1) points. The grid displays all 81 points at depth th…
Exact mathematical passage
R=\ZZ_3[[t_1,t_2]]
\ZZ_5Use in source · 2 passages
Search aliases: ZZ_5 ZZ
Use in source · pages.tex, line 59
iv1pmod5^m, not by taking decimal approximations. At the first six depths they are 4,19,94,469,2344,11719. vfig04_digits headCompatibility is stronger than independent solvability The residues satisfy both x_m+1≡ x_mpmod5^m, 4x_m-1=3·5^m. The latter identity shows that the residual error tends to zero in the five-adic norm. The stabilized digit sequence is frac14=4+3·5+3·5^2+3·5^3+⋯ in ZZ_5. This series is not convergent in the ordinary real norm. The topology is part of the statement. To view t…
Exact mathematical passage
\frac14=4+3\cdot5+3\cdot5^2+3\cdot5^3+\cdots\quad\text{in }\ZZ_5.
|^{-1}Use in source · 1 passage
Search aliases: |^-1
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1001
Derived comparison bounds (Block 16). For a finite group Γ whose order is a unit in a commutative local coefficient ring R, any prescribed equivariant residual isomorphism lifts to an integral isomorphism. Lift its matrix to A and define P=|Γ|^-1∑_g∈Γρ(g)Aτ(g^-1). Reindexing proves equivariance, and reduction recovers the prescribed matrix. Its determinant is a unit. Complete local coefficients with finite quotients, occupied compatible finite conditions and this residual type therefore give act…
Exact mathematical passage
P=|\Gamma|^{-1}\sum_{g\in\Gamma}\rho(g)A\tau(g^{-1}).
|_3Use in source · 1 passage
Search aliases: |_3
Use in source · pages.tex, line 301
requires a proof that the chosen condition is preserved by the limit under construction. The word ``admissible'' does not itself provide that proof. takeawayCompatibility reconstructs an object. Closedness is what carries the declared equations or conditions to that object. captiontextThe finite diagram displays the retreat of the first nonzero digit. The convergence assertion follows from |3^k|_3=3^-k, not from extrapolating a finite numerical plot.
Exact mathematical passage
|3^k|_3=3^{-k}
|_{\ker N}Use in source · 1 passage
Search aliases: |_ker N ker
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1488
These statements concern supplied algebraic pairs. No Weil-group or inertia action, Hodge--Tate object, arithmetic comparison, or realization of the pair from a Galois representation is constructed here. The polynomial convention is det(XI-F|_ker N); no identification with an inertia-dependent local Euler factor is asserted. Claim 9.2 remains partial at its full arithmetic scope.
Exact mathematical passage
\det(XI-F|_{\ker N})
|_{\ker N_i}Use in source · 2 passages
Search aliases: |_ker N_i ker
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1457
,N_m on U_m satisfy the same relation; F_m is invertible and N_m is nilpotent. The split maps constructed above intertwine both operators. The proof first identifies the independently defined kernel through an equivariant linear isomorphism ker N_m≃∏_i∈ I∏_k<m_iker N_i. The relation, with q≠0, proves that both F_i and its inverse preserve ker N_i. Thus F_i|_ker N_i is an actual automorphism, and Lean proves det(XI-F_m|_ker N_m) =∏_i∈ Idet(XI-F_i|_ker N_i)^m_i. The corresponding kernel dimensions…
Exact mathematical passage
F_i|_{\ker N_i}
|_{\ker N_m}Use in source · 1 passage
Search aliases: |_ker N_m ker
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1458
and N_m is nilpotent. The split maps constructed above intertwine both operators. The proof first identifies the independently defined kernel through an equivariant linear isomorphism ker N_m≃∏_i∈ I∏_k<m_iker N_i. The relation, with q≠0, proves that both F_i and its inverse preserve ker N_i. Thus F_i|_ker N_i is an actual automorphism, and Lean proves det(XI-F_m|_ker N_m) =∏_i∈ Idet(XI-F_i|_ker N_i)^m_i. The corresponding kernel dimensions and traces are additive with multiplicity. Common consti…
Exact mathematical passage
\det(XI-F_m|_{\ker N_m}) =\prod_{i\in I}\det(XI-F_i|_{\ker N_i})^{m_i}.
\cdotUse in source · 4 passages
Search aliases: · cdot
Use in source · Read passage · residue_continuation_body.tex, line 202
An actual unit action on ideal residues. Fix m>0 and a nonzero integral ideal J. Multiplication by u∈( O/m O)^× gives a proved permutation of J/mJ. Its inverse follows from (ab-1)x∈ mJ when ab-1∈ m O and x∈ J. No surjectivity of J/mJ→ O/m O or coprimality condition on J is used. The well-defined residue norm on this quotient satisfies overline n_J(u· a)=overline n(u) overline n_J(a). For a multiplicative χ: Z/m Z→ C preserving 1, an exhibited unit with u∈( O/m O)^×, χ(overline n(u))≠1 therefore …
Exact mathematical passage
\overline n_J(u\cdot a)=\overline n(u)\,\overline n_J(a).
\timesUse in source · 17 passages
Search aliases: × times
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 777
The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.
Exact mathematical passage
\boxed{ \Sys_{\mathrm{sys}} = \Sys_2^{\mathrm{wt}} \times_{\Compat} \prod_{\lambda\nmid2}'\Sys_\lambda. }
\times_{\Compat}Use in source · 1 passage
Search aliases: ×_Compat ×_ C times Compat times mathfrak
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 777
The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.
Exact mathematical passage
\boxed{ \Sys_{\mathrm{sys}} = \Sys_2^{\mathrm{wt}} \times_{\Compat} \prod_{\lambda\nmid2}'\Sys_\lambda. }
\GammaUse in source · 14 passages
Search aliases: Γ Gamma
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 998
Derived comparison bounds (Block 16). For a finite group Γ whose order is a unit in a commutative local coefficient ring R, any prescribed equivariant residual isomorphism lifts to an integral isomorphism. Lift its matrix to A and define P=|Γ|^-1∑_g∈Γρ(g)Aτ(g^-1). Reindexing proves equivariance, and reduction recovers the prescribed matrix. Its determinant is a unit. Complete local coefficients with finite quotients, occupied compatible finite conditions and this residual type therefore give act…
Exact mathematical passage
\Gamma
\gammaUse in source · 11 passages
Search aliases: γ gamma
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1175
Absolute Galois restriction and arithmetic witnesses (Block 26). Fix a cyclotomic number field L= Q(ζ_n) with n>0 and an embedding j:L↪overline Q. The new restriction map q_L:G_ Q⟶Gal(L/ Q), j(q_L(γ)x)=γ(j(x)), is an actual continuous surjective homomorphism with open kernel. Surjectivity is derived by extending automorphisms to the algebraic closure. Pullback along a surjective homomorphism preserves and reflects equivariant linear isomorphisms, invariant subspaces, and irreducibility. These as…
Exact mathematical passage
q_L:G_{\Q}\longrightarrow\operatorname{Gal}(L/\Q),\qquad j(q_L(\gamma)x)=\gamma(j(x)),
\gamma_jUse in source · 2 passages
Search aliases: γ_j gamma
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1231
compatible family ρ_i,λ of representations of the fixed finite group, set V_i,λ=ρ_i,λ∘ q_L. Suppose the actual absolute-Galois members are irreducible and the systems are pairwise componentwise nonisomorphic. The finite prime separator now yields distinct permitted rational primes p_j, actual prime ideals Q_j and arithmetic Frobenius witnesses γ_j as above, outside any prescribed finite set, such that the common algebraic trace matrix has nonzero determinant. Every coefficient embedding preserve…
Exact mathematical passage
\gamma_j
\gamma_pUse in source · 5 passages
Search aliases: γ_p gamma
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1190
The arithmetic witnesses use the integral closure A of Z in overline Q. For each rational prime p, Lean constructs a maximal ideal Q of A above (p) and an actual element γ_p∈ G_ Q such that γ_p∈Stab_G_ Q( Q), overlineγ_p(a)=overline a^ p (a∈ A). The existence argument applies Mathlib's proved profinite stabilizer-surjectivity theorem to this actual invariant ring. The residue p-power map is proved bijective by taking p-th roots in overline Q and showing that they remain integral when their p-th …
Exact mathematical passage
\gamma_p\in G_{\Q}
\gamma_uUse in source · 2 passages
Search aliases: γ_u gamma
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1284
The choice of absolute prime and Frobenius lift is also controlled. Let ( Q_u,γ_u) and ( Q_v,γ_v) be any two actual arithmetic witnesses at the same rational prime. Transitivity on primes of A above (p) gives a∈ G_ Q with Q_v=a Q_u, γ_v^-1aγ_u a^-1∈ I_ Q_v. Consequently, any representation trivial on I_ Q_v has the same trace on the two witnesses. For representations factoring through L, triviality on this actual absolute inertia is derived whenever p∉ S_L. The restrictions of all absolute witne…
Exact mathematical passage
(\mathfrak Q_u,\gamma_u)
\gamma_vUse in source · 1 passage
Search aliases: γ_v gamma
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1284
The choice of absolute prime and Frobenius lift is also controlled. Let ( Q_u,γ_u) and ( Q_v,γ_v) be any two actual arithmetic witnesses at the same rational prime. Transitivity on primes of A above (p) gives a∈ G_ Q with Q_v=a Q_u, γ_v^-1aγ_u a^-1∈ I_ Q_v. Consequently, any representation trivial on I_ Q_v has the same trace on the two witnesses. For representations factoring through L, triviality on this actual absolute inertia is derived whenever p∉ S_L. The restrictions of all absolute witne…
Exact mathematical passage
(\mathfrak Q_v,\gamma_v)
\gamma_v^{-1}Use in source · 1 passage
Search aliases: γ_v^-1 gamma
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1287
The choice of absolute prime and Frobenius lift is also controlled. Let ( Q_u,γ_u) and ( Q_v,γ_v) be any two actual arithmetic witnesses at the same rational prime. Transitivity on primes of A above (p) gives a∈ G_ Q with Q_v=a Q_u, γ_v^-1aγ_u a^-1∈ I_ Q_v. Consequently, any representation trivial on I_ Q_v has the same trace on the two witnesses. For representations factoring through L, triviality on this actual absolute inertia is derived whenever p∉ S_L. The restrictions of all absolute witne…
Exact mathematical passage
\mathfrak Q_v=a\mathfrak Q_u,\qquad \gamma_v^{-1}a\gamma_u a^{-1}\in I_{\mathfrak Q_v}.
\DeltaUse in source · 2 passages
Search aliases: Δ Delta
Use in source · Read passage · residue_continuation_body.tex, line 138
Ordinary class characters: an unconditional power saving. The preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl( O)→ C, define the actual finite ideal sum S_w(X)=∑_substackI≠0
Exact mathematical passage
A_C(X)=c_KX+O_{K,C}(X^{1-1/d}),\qquad c_K=\frac{2^{r_1}(2\pi)^{r_2}R_K}{\#\mu(K)\sqrt{\absn{\Delta_K}}}.
\deltaUse in source · 8 passages
Search aliases: δ delta
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 347
Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.
Exact mathematical passage
\sum_jP_j^r=I, \qquad P_i^rP_j^r=\delta_{ij}P_j^r, \qquad d_rP_j^r=P_j^{r+1}d_r.
\delta_{ij}Use in source · 1 passage
Search aliases: δ_ij delta
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 347
Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.
Exact mathematical passage
\sum_jP_j^r=I, \qquad P_i^rP_j^r=\delta_{ij}P_j^r, \qquad d_rP_j^r=P_j^{r+1}d_r.
\Delta_KUse in source · 2 passages
Search aliases: Δ_K Delta
Use in source · Read passage · residue_continuation_body.tex, line 138
Ordinary class characters: an unconditional power saving. The preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl( O)→ C, define the actual finite ideal sum S_w(X)=∑_substackI≠0
Exact mathematical passage
A_C(X)=c_KX+O_{K,C}(X^{1-1/d}),\qquad c_K=\frac{2^{r_1}(2\pi)^{r_2}R_K}{\#\mu(K)\sqrt{\absn{\Delta_K}}}.
\varepsilonUse in source · 4 passages
Search aliases: ε varepsilon
Use in source · Read passage · residue_continuation_body.tex, line 88
For ε∈{1,-1} define overline n_ε(a)=εoverline n(a). On the corresponding sign portion, N(x)bmod m=overline n_ε([x]). The sign cannot be dropped: -1≡2pmod3, whereas N(-1)=1 and N(2)=2 for principal ideals of Z.
Exact mathematical passage
\varepsilon\in\{1,-1\}
\zetaUse in source · 6 passages
Search aliases: ζ zeta
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1132
Arithmetic coverage for a fixed cyclotomic extension (Block 25). Let n>0 and let L= Q(ζ_n) be a cyclotomic number field. The canonical isomorphism Gal(L/ Q)≃( Z/n Z)^× defines σ_p by σ_p(ζ_n)=ζ_n^p when pnmid n. For every automorphism σ, finite set B⊂ N and bound b, Lean proves that there is a rational prime p>b, with p∉ B and pnmid n, such that σ_p=σ. The prime-existence step uses the proved Dirichlet theorem in the pinned Mathlib dependency.
Exact mathematical passage
L=\Q(\zeta_n)
\zeta_4Use in source · 1 passage
Search aliases: ζ_4 zeta
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1165
ite set, with nonzero determinant. Every coefficient embedding preserves its nonvanishing; actual member trace comparisons retain that member's good-prime condition. No coverage or character-continuity hypothesis is added in this cyclotomic specialization: the prime coverage is proved, and the finite extension's Krull topology is discrete. A concrete conductor-four instance uses the actual field Q(ζ_4).
Exact mathematical passage
\Q(\zeta_4)
\zeta_nUse in source · 5 passages
Search aliases: ζ_n zeta
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1132
Arithmetic coverage for a fixed cyclotomic extension (Block 25). Let n>0 and let L= Q(ζ_n) be a cyclotomic number field. The canonical isomorphism Gal(L/ Q)≃( Z/n Z)^× defines σ_p by σ_p(ζ_n)=ζ_n^p when pnmid n. For every automorphism σ, finite set B⊂ N and bound b, Lean proves that there is a rational prime p>b, with p∉ B and pnmid n, such that σ_p=σ. The prime-existence step uses the proved Dirichlet theorem in the pinned Mathlib dependency.
Exact mathematical passage
L=\Q(\zeta_n)
\zeta_n^pUse in source · 2 passages
Search aliases: ζ_n^p zeta
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1137
Arithmetic coverage for a fixed cyclotomic extension (Block 25). Let n>0 and let L= Q(ζ_n) be a cyclotomic number field. The canonical isomorphism Gal(L/ Q)≃( Z/n Z)^× defines σ_p by σ_p(ζ_n)=ζ_n^p when pnmid n. For every automorphism σ, finite set B⊂ N and bound b, Lean proves that there is a rational prime p>b, with p∉ B and pnmid n, such that σ_p=σ. The prime-existence step uses the proved Dirichlet theorem in the pinned Mathlib dependency.
Exact mathematical passage
\sigma_p(\zeta_n)=\zeta_n^p
\etaUse in source · 1 passage
Search aliases: η eta
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 426
[Odd and dyadic nonlinear equations] At odd residue characteristic, a differential graded Lie model may be written in Maurer--Cartan form dη+frac12[η,η]=0, and a cyclic grading converts this into exact packet convolutions. At residue characteristic two, division by 2 is not integral. A quadratic or divided-square refinement must be retained. This is one reason the dyadic residual forest is weighted rather than obtained by formal substitution into the odd-prime formulas.
Exact mathematical passage
d\eta+\frac12[\eta,\eta]=0,
\iota_\lambdaUse in source · 1 passage
Search aliases: ι_λ iota lambda
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1234
le and the systems are pairwise componentwise nonisomorphic. The finite prime separator now yields distinct permitted rational primes p_j, actual prime ideals Q_j and arithmetic Frobenius witnesses γ_j as above, outside any prescribed finite set, such that the common algebraic trace matrix has nonzero determinant. Every coefficient embedding preserves this nonvanishing, and ι_λ(t_i(p_j))= tr(V_i,λ(γ_j)) whenever p_j is good for that coefficient member. The operator and trace values agree for eve…
Exact mathematical passage
\iota_\lambda(t_i(p_j))= \operatorname{tr}(V_{i,\lambda}(\gamma_j))
\lambda_0Use in source · 4 passages
Search aliases: λ_0 lambda
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1114
Suppose two members are isomorphic at λ_0. Injectivity of the common coefficient embedding gives t_i(v)=t_j(v) at every place good for λ_0. At another coefficient prime λ, this equality compares actual member traces on the intersection of the two good-place sets. Conjugacy density of that intersection and continuity extend the equality to the whole group; character rigidity then gives an actual isomorphism at λ. Thus the nonisomorphism of systems supplies the required nonisomorphism of members. …
Exact mathematical passage
\lambda_0
\nuUse in source · 8 passages
Search aliases: ν nu
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1493
Checked group, inertia and monodromy finite sums (Block 12). Let K be a field, G a group, and ν:G→ K^× a character. The new datum consists of an actual representation r and nilpotent N with r(g)N=ν(g)Nr(g) for every g. For a supplied subgroup H, define E(D,H)=ker N∩ V^H. This is the intersection of independently defined kernel and actual invariant subspaces. For normal H, the restricted whole-group representation and its inverse automorphisms are constructed. A change from φ to φ h, with h∈ H, l…
Exact mathematical passage
\nu:G\to K^\times
\varpi^2Use in source · 1 passage
Search aliases: ϖ^2 varpi
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 304
[Smooth formal branch count] If R_bar x≅ O[[t_1,…,t_d]] as complete local O-algebras, then #X_m(bar x)=q^d(m-1). Fix a uniformizer and a section s:k→ O of reduction, with s(0)=0. Every O-valued branch is encoded by d positive-depth digit streams t_i=s(a_i,1)ϖ+s(a_i,2)ϖ^2+⋯, a_i,j∈ k,
Exact mathematical passage
t_i=s(a_{i,1})\varpi+s(a_{i,2})\varpi^2+\cdots, \qquad a_{i,j}\in k,
\pi^cUse in source · 2 passages
Search aliases: π^c pi
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1016
For a second sector let R be a DVR with uniformizer π, fraction field K and matrix rank n. If a fixed template τ has full K-linear action span, finite generation and denominator clearing give a fixed c such that π^c M_n(R)⊆span_R{τ(g):g∈ G}. Normalize a field comparison to an integral P with a unit entry. The displayed span constructs an integral B with BP=π^cI, so det Pmidπ^cn, det Pnot≡0pmodπ^cn+1. One c works for all integral representations in the fixed field-isomorphism class. Jacobson dens…
Exact mathematical passage
\pi^c M_n(R)\subseteq\operatorname{span}_R\{\tau(g):g\in G\}.
\pi^{cn}Use in source · 1 passage
Search aliases: π^cn pi
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1021
R with uniformizer π, fraction field K and matrix rank n. If a fixed template τ has full K-linear action span, finite generation and denominator clearing give a fixed c such that π^c M_n(R)⊆span_R{τ(g):g∈ G}. Normalize a field comparison to an integral P with a unit entry. The displayed span constructs an integral B with BP=π^cI, so det Pmidπ^cn, det Pnot≡0pmodπ^cn+1. One c works for all integral representations in the fixed field-isomorphism class. Jacobson density derives full span from actual…
Exact mathematical passage
\det P\mid\pi^{cn},\qquad \det P\not\equiv0\pmod{\pi^{cn+1}}.
\pi^{cn+1}Use in source · 1 passage
Search aliases: π^cn+1 pi
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1021
R with uniformizer π, fraction field K and matrix rank n. If a fixed template τ has full K-linear action span, finite generation and denominator clearing give a fixed c such that π^c M_n(R)⊆span_R{τ(g):g∈ G}. Normalize a field comparison to an integral P with a unit entry. The displayed span constructs an integral B with BP=π^cI, so det Pmidπ^cn, det Pnot≡0pmodπ^cn+1. One c works for all integral representations in the fixed field-isomorphism class. Jacobson density derives full span from actual…
Exact mathematical passage
\det P\mid\pi^{cn},\qquad \det P\not\equiv0\pmod{\pi^{cn+1}}.
\pi^eUse in source · 3 passages
Search aliases: π^e pi
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2037
Finite-group comparison with precision loss (Block 17). Let R be a commutative domain, Γ a finite group, and ρ,τ actual integral matrix representations. Suppose a=|Γ|≠0 in R and amidπ^e. An equivariant matrix modulo (π)^e+k lifts to an integral equivariant matrix with the prescribed reduction modulo (π)^k. Indeed, lift its entries to A and form S=∑_g∈Γρ(g)Aτ(g^-1). The finite equations imply S-aA=π^e+kE with E integral. Choose b with π^e=ab and put P=A+bπ^kE. Then aP=S, so cancellation of a≠0 pr…
Exact mathematical passage
a\mid\pi^e
\pi^{e+k}Use in source · 1 passage
Search aliases: π^e+k pi
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2043
a commutative domain, Γ a finite group, and ρ,τ actual integral matrix representations. Suppose a=|Γ|≠0 in R and amidπ^e. An equivariant matrix modulo (π)^e+k lifts to an integral equivariant matrix with the prescribed reduction modulo (π)^k. Indeed, lift its entries to A and form S=∑_g∈Γρ(g)Aτ(g^-1). The finite equations imply S-aA=π^e+kE with E integral. Choose b with π^e=ab and put P=A+bπ^kE. Then aP=S, so cancellation of a≠0 proves equivariance. Over a local ring, if π is a nonunit, k>0 and …
Exact mathematical passage
S-aA=\pi^{e+k}E
\varpi^{em}Use in source · 1 passage
Search aliases: ϖ^em varpi
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 632
Counting exponent versus compatible branches. The preceding exponent counts all finite-level solutions, including nodes which cannot lift indefinitely. It is not the cardinality or dimension of the space of infinite branches. In fact every compatible tower of solutions to x^e=0, for e≥1, is the zero tower. For each m, a solution at depth em lies in ϖ^m O/ϖ^em; its reduction at depth m is therefore zero. Compatibility proves the assertion at every depth. This observation and the real logarithmic …
Exact mathematical passage
\varpi^m\Ocal/\varpi^{em}
\varpi^jUse in source · 1 passage
Search aliases: ϖ^j varpi
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 316
For the digit claim, subtract the chosen representative of the residue and divide the remainder by the uniformizer. Iterating gives unique digits. Conversely, the partial sums of any digit stream form a compatible Cauchy family and reconstruct uniquely by completeness and separatedness. More precisely, for each parameter there is a unique stream (a_i,j)_j≥1 such that ϖ^n+1mid t_i-∑_j=1^ns(a_i,j)ϖ^j (n≥0). The section is a set-theoretic section of reduction, not an asserted embedding of residue f…
Exact mathematical passage
\varpi^{n+1}\mid t_i-\sum_{j=1}^{n}s(a_{i,j})\varpi^j\qquad(n\ge0).
\pi^kUse in source · 1 passage
Search aliases: π^k pi
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2044
tegral matrix representations. Suppose a=|Γ|≠0 in R and amidπ^e. An equivariant matrix modulo (π)^e+k lifts to an integral equivariant matrix with the prescribed reduction modulo (π)^k. Indeed, lift its entries to A and form S=∑_g∈Γρ(g)Aτ(g^-1). The finite equations imply S-aA=π^e+kE with E integral. Choose b with π^e=ab and put P=A+bπ^kE. Then aP=S, so cancellation of a≠0 proves equivariance. Over a local ring, if π is a nonunit, k>0 and the supplied comparison is invertible, P is an integral u…
Exact mathematical passage
P=A+b\pi^kE
\varpi^{\lceil m/e\rceil}Use in source · 1 passage
Search aliases: ϖ^lceil m/erceil varpi lceil rceil
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 608
The congruence is equivalent to e v_ϖ(x)≥ m, so x must lie in ϖ^lceil m/erceil O/ϖ^m, which has the stated cardinality. Write m=ek+r, 1≤ r≤ e. The exponent becomes (e-1)k+r-1. Summing over k≥0 and 1≤ r≤ e gives the rational expression.
Exact mathematical passage
\varpi^{\lceil m/e\rceil}\Ocal/\varpi^m
\pi^nUse in source · 2 passages
Search aliases: π^n pi
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 984
0 · 1, every integral intertwiner with ρ_1 has determinant aP_22^2. Taking a=π^n for a nonzero π∈ I rules out every fixed detecting depth, even though all the field-valued approximants are isomorphic and their integral limit is trivial. This does not refute Claim 8.2, which already assumes inverse-limit closure of its chosen local conditions.
Exact mathematical passage
a=\pi^n
\varpi^{n+1}Use in source · 1 passage
Search aliases: ϖ^n+1 varpi
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 316
For the digit claim, subtract the chosen representative of the residue and divide the remainder by the uniformizer. Iterating gives unique digits. Conversely, the partial sums of any digit stream form a compatible Cauchy family and reconstruct uniquely by completeness and separatedness. More precisely, for each parameter there is a unique stream (a_i,j)_j≥1 such that ϖ^n+1mid t_i-∑_j=1^ns(a_i,j)ϖ^j (n≥0). The section is a set-theoretic section of reduction, not an asserted embedding of residue f…
Exact mathematical passage
\varpi^{n+1}\mid t_i-\sum_{j=1}^{n}s(a_{i,j})\varpi^j\qquad(n\ge0).
\varpi^{w_i}Use in source · 3 passages
Search aliases: ϖ^w_i varpi
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 498
A simple weighted formal model already captures delayed visibility. Let w_i≥1 and constrain t_i∈(ϖ^w_i O+ϖ^m O)/ϖ^m O. Then coordinate i is invisible until m>w_i.
Exact mathematical passage
t_i\in(\varpi^{w_i}\Ocal+\varpi^m\Ocal)/\varpi^m\Ocal.
\rho_1Use in source · 1 passage
Search aliases: ρ_1 rho
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 984
0 · 1, every integral intertwiner with ρ_1 has determinant aP_22^2. Taking a=π^n for a nonzero π∈ I rules out every fixed detecting depth, even though all the field-valued approximants are isomorphic and their integral limit is trivial. This does not refute Claim 8.2, which already assumes inverse-limit closure of its chosen local conditions.
Exact mathematical passage
\rho_1
\rho_aUse in source · 1 passage
Search aliases: ρ_a rho
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 983
The bound is substantive: for the unipotent actions ρ_a(g)=1 · ag 0 · 1, every integral intertwiner with ρ_1 has determinant aP_22^2. Taking a=π^n for a nonzero π∈ I rules out every fixed detecting depth, even though all the field-valued approximants are isomorphic and their integral limit is trivial. This does not refute Claim 8.2, which already assumes inverse-limit closure of its chosen local conditions.
Exact mathematical passage
\rho_a(g)=\begin{pmatrix}1&ag\\0&1\end{pmatrix}
\rho_cUse in source · 1 passage
Search aliases: ρ_c rho
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1400
≤ c_i, set U_m=∏_i∈ I∏_k<m_iW_i, U_c=∏_i∈ I∏_k<c_iW_i. These finite products realize the corresponding finite direct sums. The representation on U_m acts componentwise. Define j:U_m→ U_c by retaining the first m_i copies and filling every remaining coordinate with zero. Define r:U_c→ U_m by restriction to the selected copies. Lean proves rj=1_U_m, jρ_m(g)=ρ_c(g)j, rρ_c(g)=ρ_m(g)r. The image of j is exactly the subspace whose unused coordinates vanish. Consequently jr is an equivariant idempotent…
Exact mathematical passage
rj=1_{U_m},\qquad j\rho_m(g)=\rho_c(g)j,\qquad r\rho_c(g)=\rho_m(g)r.
\rho_DUse in source · 2 passages
Search aliases: ρ_D rho
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2269
For supplied covariance data and the twisted cocycle of Block 13, let ρ_D(g)=exp(t(g)N_D)r_D(g). If a common element u satisfies r_D(u)=r_E(u)=1 and t(u)≠0, every intertwiner of ρ_D and ρ_E recovers both the original r-intertwiner and the N-intertwiner. There is no cubic nilpotence restriction. This recovers equivariant projectors for both original components. Reconstructed-action isomorphisms induce actual isomorphisms of the inertia-fixed monodromy kernels. Their local characteristic polynomia…
Exact mathematical passage
\rho_D(g)=\exp(t(g)N_D)r_D(g)
\rho_EUse in source · 1 passage
Search aliases: ρ_E rho
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2271
For supplied covariance data and the twisted cocycle of Block 13, let ρ_D(g)=exp(t(g)N_D)r_D(g). If a common element u satisfies r_D(u)=r_E(u)=1 and t(u)≠0, every intertwiner of ρ_D and ρ_E recovers both the original r-intertwiner and the N-intertwiner. There is no cubic nilpotence restriction. This recovers equivariant projectors for both original components. Reconstructed-action isomorphisms induce actual isomorphisms of the inertia-fixed monodromy kernels. Their local characteristic polynomia…
Exact mathematical passage
\rho_E
\rho_{i,\lambda}Use in source · 2 passages
Search aliases: ρ_i,λ rho lambda
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1226
For a compatible family ρ_i,λ of representations of the fixed finite group, set V_i,λ=ρ_i,λ∘ q_L. Suppose the actual absolute-Galois members are irreducible and the systems are pairwise componentwise nonisomorphic. The finite prime separator now yields distinct permitted rational primes p_j, actual prime ideals Q_j and arithmetic Frobenius witnesses γ_j as above, outside any prescribed finite set, such that the common algebraic trace matrix has nonzero determinant. Every coefficient embedding pr…
Exact mathematical passage
\rho_{i,\lambda}
\rho_mUse in source · 4 passages
Search aliases: ρ_m rho
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 868
Checked representation-level completion and its arithmetic boundary. The eighth formalization block works with actual matrix representations of a fixed group, not only with underlying sets. For a complete separated ring R and ideal I, compatible maps ρ_m:G→GL_d(R/I^m) reconstruct a unique integral representation for that fixed compatible family. Multiplication and inverses are verified after every reduction. If all finite representations are continuous for the discrete quotient topology, the int…
Exact mathematical passage
\rho_m:G\to\operatorname{GL}_d(R/I^m)
\rho_nUse in source · 1 passage
Search aliases: ρ_n rho
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1779
Local properties need their own limit argument. Over a domain, for a nonzero π∈ I, the representations of the additive integer group given by ρ_n(g)=1 · π^n g 0 · 1 are all nontrivial and converge pointwise in the ideal-adic topology to the trivial representation. The convergence and the failure of limit preservation for nontriviality are formally checked. Thus not every property of approximating integral representations can be asserted for a limit. This example is not claimed to disprove stabil…
Exact mathematical passage
\rho_n(g)=\begin{pmatrix}1&\pi^n g\\0&1\end{pmatrix}
\rho_\lambdaUse in source · 2 passages
Search aliases: ρ_λ rho lambda
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 797
pine C over a number field M is fixed; item for every finite place λ of M and every m≥1, the companion level X_λ,m( C) is finite and nonempty; item the reduction forests are finitely branching; item the selected local conditions are closed under inverse limits. Then for every λ there is an infinite compatible branch ρ_λ∈varprojlim_mX_λ,m( C), and the collection {ρ_λ}_λ realizes the prescribed strictly compatible spine.
Exact mathematical passage
\rho_\lambda\in\varprojlim_mX_{\lambda,m}(\Compat),
\sigmaUse in source · 15 passages
Search aliases: σ sigma
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 972
The formalization specializes to the actual maps G_ Q_p→ G_ Q induced by the field embeddings and choices of algebraic closures. A nontrivial continuous quadratic character of G_ Q, constructed from its action on i^2=-1, gives the integral 2-adic actions τ(σ)=diag(1,-1), ρ(σ)=1 · -1 0 · -1, P=1 · 1 0 · 2 at a nonidentity quadratic element. The determinant 2 is detected modulo 4. The representations are conjugate over Q_2 but not integrally conjugate. A separate Z_3 example invokes simultaneous c…
Exact mathematical passage
\tau(\sigma)=\operatorname{diag}(1,-1),\quad \rho(\sigma)=\begin{pmatrix}1&-1\\0&-1\end{pmatrix},\quad P=\begin{pmatrix}1&1\\0&2\end{pmatrix}
\sigma_pUse in source · 6 passages
Search aliases: σ_p sigma
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1137
Arithmetic coverage for a fixed cyclotomic extension (Block 25). Let n>0 and let L= Q(ζ_n) be a cyclotomic number field. The canonical isomorphism Gal(L/ Q)≃( Z/n Z)^× defines σ_p by σ_p(ζ_n)=ζ_n^p when pnmid n. For every automorphism σ, finite set B⊂ N and bound b, Lean proves that there is a rational prime p>b, with p∉ B and pnmid n, such that σ_p=σ. The prime-existence step uses the proved Dirichlet theorem in the pinned Mathlib dependency.
Exact mathematical passage
\sigma_p
\sigma_\lambdaUse in source · 1 passage
Search aliases: σ_λ sigma lambda
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1059
ion] Let mathscr W_1,…,mathscr W_r be pairwise nonisomorphic strictly compatible systems whose members are absolutely irreducible, with common algebraic unramified trace values t_i(v) in a number field E. There are r distinct permitted unramified places v_1,…,v_r for which C=(t_i(v_j))_j,i∈ M_r(E), det C≠0. Every coefficient-field embedding σ_λ:E↪ E_λ therefore sends C to an invertible matrix. Its entries agree with the ordinary representation traces Tr(W_i,λ(Frob_v_j)) when the selected places …
Exact mathematical passage
\sigma_\lambda:E\hookrightarrow E_\lambda
\tauUse in source · 12 passages
Search aliases: τ tau
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 949
Actual local comparisons from fixed-depth certificates (Block 15). Let R be an I-adically complete separated commutative ring with finite coefficient quotients and an injective map into a field K. Fix a finite matrix rank. For finitely many local group maps ϕ_s:H_s→ G and integral template representations τ_s, fix detecting depths d_s. At precision m retain matrices P_s,m as part of the admissible object and require ρ_m(ϕ_s(h))P_s,m=P_s,mτ_s,m(h). For m≥ d_s require their determinants to remain …
Exact mathematical passage
\tau_s
\tau_sUse in source · 1 passage
Search aliases: τ_s tau
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 949
Actual local comparisons from fixed-depth certificates (Block 15). Let R be an I-adically complete separated commutative ring with finite coefficient quotients and an injective map into a field K. Fix a finite matrix rank. For finitely many local group maps ϕ_s:H_s→ G and integral template representations τ_s, fix detecting depths d_s. At precision m retain matrices P_s,m as part of the admissible object and require ρ_m(ϕ_s(h))P_s,m=P_s,mτ_s,m(h). For m≥ d_s require their determinants to remain …
Exact mathematical passage
\tau_s
\tau_{s,m}Use in source · 1 passage
Search aliases: τ_s,m tau
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 951
cates (Block 15). Let R be an I-adically complete separated commutative ring with finite coefficient quotients and an injective map into a field K. Fix a finite matrix rank. For finitely many local group maps ϕ_s:H_s→ G and integral template representations τ_s, fix detecting depths d_s. At precision m retain matrices P_s,m as part of the admissible object and require ρ_m(ϕ_s(h))P_s,m=P_s,mτ_s,m(h). For m≥ d_s require their determinants to remain nonzero after reduction to R/I^d_s. Earlier level…
Exact mathematical passage
\rho_m(\phi_s(h))P_{s,m}=P_{s,m}\tau_{s,m}(h).
\phiUse in source · 2 passages
Search aliases: ϕ phi
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 948
Actual local comparisons from fixed-depth certificates (Block 15). Let R be an I-adically complete separated commutative ring with finite coefficient quotients and an injective map into a field K. Fix a finite matrix rank. For finitely many local group maps ϕ_s:H_s→ G and integral template representations τ_s, fix detecting depths d_s. At precision m retain matrices P_s,m as part of the admissible object and require ρ_m(ϕ_s(h))P_s,m=P_s,mτ_s,m(h). For m≥ d_s require their determinants to remain …
Exact mathematical passage
\phi_s:H_s\to G
\varphiUse in source · 7 passages
Search aliases: φ varphi
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1501
character. The new datum consists of an actual representation r and nilpotent N with r(g)N=ν(g)Nr(g) for every g. For a supplied subgroup H, define E(D,H)=ker N∩ V^H. This is the intersection of independently defined kernel and actual invariant subspaces. For normal H, the restricted whole-group representation and its inverse automorphisms are constructed. A change from φ to φ h, with h∈ H, leaves the restricted operator itself unchanged.
Exact mathematical passage
\varphi
\phi_sUse in source · 2 passages
Search aliases: ϕ_s phi
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 948
Actual local comparisons from fixed-depth certificates (Block 15). Let R be an I-adically complete separated commutative ring with finite coefficient quotients and an injective map into a field K. Fix a finite matrix rank. For finitely many local group maps ϕ_s:H_s→ G and integral template representations τ_s, fix detecting depths d_s. At precision m retain matrices P_s,m as part of the admissible object and require ρ_m(ϕ_s(h))P_s,m=P_s,mτ_s,m(h). For m≥ d_s require their determinants to remain …
Exact mathematical passage
\phi_s:H_s\to G
\chi_+Use in source · 1 passage
Search aliases: χ_+ chi
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 484
[The two sign branches] The characters χ_+(g)=1 and χ_-(g)=-1 have the same reduction modulo 2, but 1not≡-1pmod4. Their first separation depth is therefore 2. One mod-two root contains two branches that become distinct at the next level.
Exact mathematical passage
\chi_+(g)=1
\chi_-Use in source · 1 passage
Search aliases: χ_- chi
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 484
[The two sign branches] The characters χ_+(g)=1 and χ_-(g)=-1 have the same reduction modulo 2, but 1not≡-1pmod4. Their first separation depth is therefore 2. One mod-two root contains two branches that become distinct at the next level.
Exact mathematical passage
\chi_-(g)=-1
\chi_4Use in source · 1 passage
Search aliases: χ_4 chi
Use in source · pages.tex, line 359
ter may still be exceptionalTHE EXACT SCALAR TESTMain text: §9; scalar/Mellin checkpoint: scalar-power and nonscalar-witness boundaries. A scalar a∈FF_5^× acts on a degree-d algebra with determinant a^d. Therefore the scalar cancellation route tests χ(a^d), not merely χ(a). vfig24_scalar_exception headA failed sufficient test is not an impossibility theorem Let χ_4(2)=i generate the character group of FF_5^×, and write χ_k=χ_4^k. Since 2 generates the unit group, ∃ a:χ_k(a^d)≠1 Longleftrightarro…
Exact mathematical passage
\chi_4(2)=i
\chi_4^kUse in source · 1 passage
Search aliases: χ_4^k chi
Use in source · pages.tex, line 359
ar/Mellin checkpoint: scalar-power and nonscalar-witness boundaries. A scalar a∈FF_5^× acts on a degree-d algebra with determinant a^d. Therefore the scalar cancellation route tests χ(a^d), not merely χ(a). vfig24_scalar_exception headA failed sufficient test is not an impossibility theorem Let χ_4(2)=i generate the character group of FF_5^×, and write χ_k=χ_4^k. Since 2 generates the unit group, ∃ a:χ_k(a^d)≠1 Longleftrightarrow kdnot≡0pmod4. For the quadratic character k=2 and degree d=2, ever…
Exact mathematical passage
\chi_k=\chi_4^k
\chi_iUse in source · 2 passages
Search aliases: χ_i chi
Use in source · Read passage · residue_continuation_body.tex, line 181
For a finite family of nontrivial ordinary class characters, suppose functions p,Z satisfy h p(h)→ P≠0, h Z(h)→ Q≠0, and, for all sufficiently small h>0, Z(h)=(p(h)∏_i F_χ_i(h))^e, e∈ N. The checked real-axis criterion then gives e=1 and F_χ_i(0)≠0 for every i. The pole limits and product identity remain hypotheses of this application.
Exact mathematical passage
Z(h)=\left(p(h)\prod_i F_{\chi_i}(h)\right)^e, \qquad e\in\NN.
\chi_KUse in source · 2 passages
Search aliases: χ_K chi
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2435
For every commutative coefficient ring B and χ∈ P_q(B), define χ_K=χ∘ j_K. The local character is continuous and commutes with coefficient change. The precise local admissibility condition is LocallyTrivial_K(χ) Longleftrightarrow χ_K=1. Descent through the fixed quadratic quotient proves LocallyTrivial_K(χ) Longleftrightarrow (∃ x∈ K, x^2=-1) lor χ=1. No reducedness, characteristic-zero or domain hypothesis is imposed on B. In particular, the formula applies to the finite coefficient quotients …
Exact mathematical passage
\chi_K=\chi\circ j_K
\chi_kUse in source · 3 passages
Search aliases: χ_k chi
Use in source · 24_scalar_exception.tex, line 61
,align=center,font=scriptsize] at (8.5,1.05) no; path[draw=black!30,fill=black!16] (9.3,0.75) rectangle (10.7,1.33); node[font=small,align=center,font=scriptsize] at (10,1.05) yes; path[draw=black!30,fill=black!16] (10.8,0.75) rectangle (12.2,1.33); node[font=small,align=center,font=scriptsize] at (11.5,1.05) yes; node[font=small,align=center,font=footnotesize] at (7,0.05) For χ_k(2)=i^k, a witness exists exactly when kdnot≡0pmod4.;
Exact mathematical passage
\chi_k(2)=i^k
\chi_{\Q_{p_s}}Use in source · 1 passage
Search aliases: χ_Q_p_s χ_ Q_p_s chi Q chi mathbf
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2524
Suppose now that R is I-adically complete and that every positive quotient R/I^m is finite. Let C_m⊆ P_q(R/I^m) be allowed subsets preserved by reduction, and let (p_s)_s∈ S be any indexed family of arithmetic primes. The index set S need not be finite. Lean proves the simultaneous arithmetic completion criterion · ∃χ∈ P_q(R), χ continuous, χbmod I^m∈ C_m (m≥1), χ_ Q_p_s=1 (s∈ S) · Longleftrightarrow ∀ m≥1, ∃ψ∈ C_m, ∀ s∈ S, p_s≡1pmod4 lor ψ=1. The constructed character also satisfies every odd-p…
Exact mathematical passage
\begin{split} &\exists\chi\in\mathcal P_q(R),\quad \chi\text{ continuous},\quad \chi\bmod I^m\in C_m\ (m\ge1),\quad \chi_{\Q_{p_s}}=1\ (s\in S)\\ &\quad\Longleftrightarrow\quad \forall m\ge1,\ \exists\psi\in C_m,\quad \forall s\in S,\ p_s\equiv1\pmod4\ \lor\ \psi=1. \end{split}
\psiUse in source · 1 passage
Search aliases: ψ psi
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2524
Suppose now that R is I-adically complete and that every positive quotient R/I^m is finite. Let C_m⊆ P_q(R/I^m) be allowed subsets preserved by reduction, and let (p_s)_s∈ S be any indexed family of arithmetic primes. The index set S need not be finite. Lean proves the simultaneous arithmetic completion criterion · ∃χ∈ P_q(R), χ continuous, χbmod I^m∈ C_m (m≥1), χ_ Q_p_s=1 (s∈ S) · Longleftrightarrow ∀ m≥1, ∃ψ∈ C_m, ∀ s∈ S, p_s≡1pmod4 lor ψ=1. The constructed character also satisfies every odd-p…
Exact mathematical passage
\begin{split} &\exists\chi\in\mathcal P_q(R),\quad \chi\text{ continuous},\quad \chi\bmod I^m\in C_m\ (m\ge1),\quad \chi_{\Q_{p_s}}=1\ (s\in S)\\ &\quad\Longleftrightarrow\quad \forall m\ge1,\ \exists\psi\in C_m,\quad \forall s\in S,\ p_s\equiv1\pmod4\ \lor\ \psi=1. \end{split}
\ldotsUse in source · 16 passages
Search aliases: … ldots
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 222
ation preserves the residual root, and induction proves the finite statement. For a convergent stream, convergence at the first finite precision makes its reduction eventually equal to that of its limit. Every term already has the initial reduction, so the limit has that reduction too. Completeness applies to Cauchy streams; it does not make every root-preserving stream Cauchy. For example, 0,3,0,3,… has constant reduction modulo 3 but no limit modulo 9.
Exact mathematical passage
0,3,0,3,\ldots
\ellUse in source · 2 passages
Search aliases: ℓ ell
Use in source · references.tex, line 75
bibitemLarsenPink1992 Michael Larsen and Richard Pink. newblock On ℓ-independence of algebraic monodromy groups in compatible systems of representations. newblock em Inventiones Mathematicae, 107(3):603--636, 1992.
Exact mathematical passage
\ell
\mapstoUse in source · 4 passages
Search aliases: ↦ mapsto
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 240
[Exact residual partition] The finite implementation partitions the integers 0,…,8 into the three classes modulo 3: {0,3,6}, {1,4,7}, {2,5,8}. Every tested correction of the form x↦ x+p h preserves the residual class exactly.
Exact mathematical passage
x\mapsto x+p h
\forallUse in source · 3 passages
Search aliases: ∀ forall
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1104
System isomorphism and good-place overlaps (Block 25). The system equivalence used here is componentwise representation isomorphism: mathscr W_i≃mathscr W_j Longleftrightarrow ∀λ, W_i,λ≃ W_j,λ. For finite-dimensional irreducible representations of any monoid over a characteristic-zero field, the new character-rigidity theorem proves that equality of characters is equivalent to an actual equivariant linear isomorphism. Neither a finite group nor an algebraically closed coefficient field is needed…
Exact mathematical passage
\mathscr W_i\simeq\mathscr W_j \quad\Longleftrightarrow\quad \forall\lambda,\ W_{i,\lambda}\simeq W_{j,\lambda}.
\existsUse in source · 7 passages
Search aliases: ∃ exists
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1806
An admissible realization here includes continuous coefficient-field members and membership of every finite reduction in its designated admissible class. Under the completion hypotheses and these supplied carrier hypotheses, Lean proves · ∃ ρ, (x_p)_p: ρ is an admissible realization of the full chosen spine, 0≠ x_p∈im(e_p) for every p · Longleftrightarrow (∀λ, X_λ,1( C)≠∅) land (∀λ, ∀ m≥2, X_λ,m( C)≠∅) land (∀ p, n_p>0). The packet index may be infinite. Each x_p belongs to its own supplied carr…
Exact mathematical passage
\begin{aligned} &\exists\,\rho,\ (x_p)_p:\quad \rho\text{ is an admissible realization of the full chosen spine},\quad 0\ne x_p\in\operatorname{im}(e_p)\text{ for every }p\\ &\quad\Longleftrightarrow\quad \bigl(\forall\lambda,\ X_{\lambda,1}(\Compat)\ne\varnothing\bigr) \ \land\ \bigl(\forall\lambda,\ \forall m\ge2, X_{\lambda,m}(\Compat)\ne\varnothing\bigr) \ \land\ \bigl(\forall p,\ n_p>0\bigr). \end{aligned}
\varnothingUse in source · 5 passages
Search aliases: ∅ varnothing
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 809
[Residual companion obstruction] If X_λ,1( C)=∅ for one coefficient prime λ, no compatible completion with spine C exists. Deeper perturbations at other primes cannot create the missing residual companion.
Exact mathematical passage
X_{\lambda,1}(\Compat)=\varnothing
\notinUse in source · 4 passages
Search aliases: ∉ notin
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1139
ixed cyclotomic extension (Block 25). Let n>0 and let L= Q(ζ_n) be a cyclotomic number field. The canonical isomorphism Gal(L/ Q)≃( Z/n Z)^× defines σ_p by σ_p(ζ_n)=ζ_n^p when pnmid n. For every automorphism σ, finite set B⊂ N and bound b, Lean proves that there is a rational prime p>b, with p∉ B and pnmid n, such that σ_p=σ. The prime-existence step uses the proved Dirichlet theorem in the pinned Mathlib dependency.
Exact mathematical passage
p\notin B
\prodUse in source · 13 passages
Search aliases: ∏ prod
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 777
The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.
Exact mathematical passage
\boxed{ \Sys_{\mathrm{sys}} = \Sys_2^{\mathrm{wt}} \times_{\Compat} \prod_{\lambda\nmid2}'\Sys_\lambda. }
\prod_iUse in source · 3 passages
Search aliases: ∏_i prod
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1424
eratornameTr(ρ_m(g)) =∑_i∈ I(m_i:K)Tr(ρ_i(g)). Constituents may have different dimensions. Zero multiplicities, zero-dimensional constituents, and the empty index set are included. If at a permitted sample g the constituent polynomials are the images of P_i,g∈ E[X] under a specified map E→ K, the summand polynomial is the image of ∏_iP_i,g^m_i. Applying this independently at each coefficient field preserves a supplied common polynomial family. This does not establish the Hodge--Tate or Weil--Del…
Exact mathematical passage
\prod_iP_{i,g}^{m_i}
\prod_{i=1}^{3}Use in source · 1 passage
Search aliases: ∏_i=1^3 prod
Use in source · pages.tex, line 272
For target (0,1,1), every pair of conditions is occupied: (0,1,0) witnesses the first two, (0,0,1) the first and third, and (1,1,1) the second and third. But the triple itself is absent. Multiplying all three selectors gives the zero matrix. E_ a=∏_i=1^3 I-T_i, · a_i=0, T_i, · a_i=1. Tr(E_ a)=rank(E_ a). The checks compute all eight joint projectors, their pairwise orthogonality, and their sum. The sum is the identity even though two labels are absent: those two projectors are simply zero. takea…
Exact mathematical passage
E_{\bm a}=\prod_{i=1}^{3} \begin{cases}I-T_i,&a_i=0,\\T_i,&a_i=1.\end{cases} \qquad \Tr(E_{\bm a})=\rank(E_{\bm a}).
\prod_{i=1}^{N}Use in source · 1 passage
Search aliases: ∏_i=1^N prod
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1623
Let E be a field of characteristic zero and let T_1,…,T_N be commuting operators on a finite-dimensional E-vector space A, each diagonalizable over E. Thus the selected spectra split over the coefficient field. Let a_i∈ E be a selected target scalar for T_i, possibly absent from its spectrum. Define e_i = ∏_μ∈Spec(T_i)∖{a_i} fracT_i-μ Ia_i-μ, e_T=∏_i=1^Ne_i, with e_i=0 if a_i is absent.
Exact mathematical passage
e_i = \prod_{\mu\in\Spec(T_i)\setminus\{a_i\}} \frac{T_i-\mu I}{a_i-\mu}, \qquad e_T=\prod_{i=1}^{N}e_i,
\prod_{i\in I}Use in source · 4 passages
Search aliases: ∏_i∈ I prod in
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1392
Explicit summands inside a supplied decomposition. Let K be a field, let G be a monoid, and let ρ_i:G→End_K(W_i) be actual representations on finite-dimensional spaces, indexed by a finite set I. For natural numbers m_i≤ c_i, set U_m=∏_i∈ I∏_k<m_iW_i, U_c=∏_i∈ I∏_k<c_iW_i. These finite products realize the corresponding finite direct sums. The representation on U_m acts componentwise. Define j:U_m→ U_c by retaining the first m_i copies and filling every remaining coordinate with zero. Define r:U…
Exact mathematical passage
U_m=\prod_{i\in I}\prod_{k<m_i}W_i, \qquad U_c=\prod_{i\in I}\prod_{k<c_i}W_i.
\prod_{j=0}^{n-1}Use in source · 1 passage
Search aliases: ∏_j=0^n-1 prod
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1474
The exact special chain has basis e_0,…,e_n-1 and operators Ne_0=0, Ne_j=e_j-1 (j>0), Fe_j=aq^-je_j, a,q∈ K^×. Lean proves N^n=0, and for n>0 also N^n-1≠0 and ker N=Ke_0. Consequently its kernel polynomial is X-a, whereas its whole-space polynomial is ∏_j=0^n-1(X-aq^-j). The zero-length chain is included as a zero-dimensional space; the assertion ker N=Ke_0 is made only for positive length.
Exact mathematical passage
\prod_{j=0}^{n-1}(X-aq^{-j})
\prod_{k<c_i}Use in source · 1 passage
Search aliases: ∏_k<c_i prod
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1392
Explicit summands inside a supplied decomposition. Let K be a field, let G be a monoid, and let ρ_i:G→End_K(W_i) be actual representations on finite-dimensional spaces, indexed by a finite set I. For natural numbers m_i≤ c_i, set U_m=∏_i∈ I∏_k<m_iW_i, U_c=∏_i∈ I∏_k<c_iW_i. These finite products realize the corresponding finite direct sums. The representation on U_m acts componentwise. Define j:U_m→ U_c by retaining the first m_i copies and filling every remaining coordinate with zero. Define r:U…
Exact mathematical passage
U_m=\prod_{i\in I}\prod_{k<m_i}W_i, \qquad U_c=\prod_{i\in I}\prod_{k<c_i}W_i.
\prod_{k<m_i}Use in source · 2 passages
Search aliases: ∏_k<m_i prod
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1392
Explicit summands inside a supplied decomposition. Let K be a field, let G be a monoid, and let ρ_i:G→End_K(W_i) be actual representations on finite-dimensional spaces, indexed by a finite set I. For natural numbers m_i≤ c_i, set U_m=∏_i∈ I∏_k<m_iW_i, U_c=∏_i∈ I∏_k<c_iW_i. These finite products realize the corresponding finite direct sums. The representation on U_m acts componentwise. Define j:U_m→ U_c by retaining the first m_i copies and filling every remaining coordinate with zero. Define r:U…
Exact mathematical passage
U_m=\prod_{i\in I}\prod_{k<m_i}W_i, \qquad U_c=\prod_{i\in I}\prod_{k<c_i}W_i.
\prod_{\lambda\nmid2}'Use in source · 1 passage
Search aliases: ∏_λnmid2' prod lambda nmid
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 777
The system seed is the fibered object mathscr S_sys = mathscr S_2^wt ×_ C ∏_λnmid2'mathscr S_λ. The fiber product notation states a compatibility requirement. It does not prove that every fiber is nonempty.
Exact mathematical passage
\boxed{ \Sys_{\mathrm{sys}} = \Sys_2^{\mathrm{wt}} \times_{\Compat} \prod_{\lambda\nmid2}'\Sys_\lambda. }
\prod_{\mu\in\Spec(T_i)\setminus\{a_i\}}Use in source · 1 passage
Search aliases: ∏_μ∈Spec(T_i)∖{a_i} prod mu in Spec setminus prod mu in operatorname setminus
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1623
Let E be a field of characteristic zero and let T_1,…,T_N be commuting operators on a finite-dimensional E-vector space A, each diagonalizable over E. Thus the selected spectra split over the coefficient field. Let a_i∈ E be a selected target scalar for T_i, possibly absent from its spectrum. Define e_i = ∏_μ∈Spec(T_i)∖{a_i} fracT_i-μ Ia_i-μ, e_T=∏_i=1^Ne_i, with e_i=0 if a_i is absent.
Exact mathematical passage
e_i = \prod_{\mu\in\Spec(T_i)\setminus\{a_i\}} \frac{T_i-\mu I}{a_i-\mu}, \qquad e_T=\prod_{i=1}^{N}e_i,
\prod_{\mu\in\Spec(T_i)\setminus\{a\}}Use in source · 1 passage
Search aliases: ∏_μ∈Spec(T_i)∖{a} prod mu in Spec setminus prod mu in operatorname setminus
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1652
For each operator, its finite set of actual eigenvalues is used without multiplicity. The polynomial p_i,a(X)=∏_μ∈Spec(T_i)∖{a} (a-μ)^-1(X-μ) is used when a occurs, and the zero polynomial otherwise. For an eigenvector T_i v=bv, evaluation gives p_i,a(T_i)v=v · b=a,
Exact mathematical passage
p_{i,a}(X)=\prod_{\mu\in\Spec(T_i)\setminus\{a\}} (a-\mu)^{-1}(X-\mu)
\prod_{\sigma}Use in source · 1 passage
Search aliases: ∏_σ prod sigma
Use in source · Read passage · residue_continuation_body.tex, line 79
The relation to the geometric sign is also proved. If σ ranges over real embeddings and τ over one embedding from each complex pair, then n(x)=∏_σσ(x)∏_τabsnτ(x)^2. For x≠0, the second factor is positive. Thus the sign of n(x) is exactly the sign of the real-coordinate product. Cutting D into its positive and negative portions D_+,D_- adds no boundary outside the existing frontier, and both portions retain the proved boundary atlas. They are disjoint and their union is D.
Exact mathematical passage
n(x)=\prod_{\sigma}\sigma(x)\prod_{\tau}\absn{\tau(x)}^2.
\prod_{\tau}Use in source · 1 passage
Search aliases: ∏_τ prod tau
Use in source · Read passage · residue_continuation_body.tex, line 79
The relation to the geometric sign is also proved. If σ ranges over real embeddings and τ over one embedding from each complex pair, then n(x)=∏_σσ(x)∏_τabsnτ(x)^2. For x≠0, the second factor is positive. Thus the sign of n(x) is exactly the sign of the real-coordinate product. Cutting D into its positive and negative portions D_+,D_- adds no boundary outside the existing frontier, and both portions retain the proved boundary atlas. They are disjoint and their union is D.
Exact mathematical passage
n(x)=\prod_{\sigma}\sigma(x)\prod_{\tau}\absn{\tau(x)}^2.
\sum_aUse in source · 1 passage
Search aliases: ∑_a sum
Use in source · Read passage · residue_continuation_body.tex, line 54
The estimate is proved from the actual boundary charts and the fixed-lattice discrepancy theorem. The index factor comes from the covolume identity covol(L(M))=q covol(L(J)). For a complex weight w:J/M→ C, summing the estimates gives the leading term fracvol(D)t^dq covol(L(J))∑_a∈ J/Mw(a), with error at most B∑_aabsnw(a)(t^d-1+1). If the finite sum of weights is zero, only that error remains. Writing Y=t^d gives an error O(Y^1-1/d+1) for the actual norm threshold. These assertions include d=1.
Exact mathematical passage
B\sum_a\absn{w(a)}(t^{d-1}+1)
\sum_{a\in J/M}Use in source · 1 passage
Search aliases: ∑_a∈ J/M sum in
Use in source · Read passage · residue_continuation_body.tex, line 51
The estimate is proved from the actual boundary charts and the fixed-lattice discrepancy theorem. The index factor comes from the covolume identity covol(L(M))=q covol(L(J)). For a complex weight w:J/M→ C, summing the estimates gives the leading term fracvol(D)t^dq covol(L(J))∑_a∈ J/Mw(a), with error at most B∑_aabsnw(a)(t^d-1+1). If the finite sum of weights is zero, only that error remains. Writing Y=t^d gives an error O(Y^1-1/d+1) for the actual norm threshold. These assertions include d=1.
Exact mathematical passage
\frac{\vol(D)t^d}{q\,\covol(L(J))}\sum_{a\in J/M}w(a),
\sum_{a\in J/mJ}Use in source · 1 passage
Search aliases: ∑_a∈ J/mJ sum in
Use in source · Read passage · residue_continuation_body.tex, line 211
ty of J/mJ→ O/m O or coprimality condition on J is used. The well-defined residue norm on this quotient satisfies overline n_J(u· a)=overline n(u) overline n_J(a). For a multiplicative χ: Z/m Z→ C preserving 1, an exhibited unit with u∈( O/m O)^×, χ(overline n(u))≠1 therefore forces ∑_a∈ J/mJχ(s overline n_J(a))=0 (s∈ Z/m Z). Taking s=1,-1 gives the two finite means needed by the sign-corrected norm weights. Their vanishing follows from the actual permutation.
Exact mathematical passage
\sum_{a\in J/mJ}\chi\bigl(s\,\overline n_J(a)\bigr)=0 \qquad(s\in\ZZ/m\ZZ).
\sum_CUse in source · 2 passages
Search aliases: ∑_C sum
Use in source · Read passage · residue_continuation_body.tex, line 148
Nm I≤ Xw([I]). Exact grouping by ideal class gives S_w(X)=∑_C w(C)A_C(X). If ∑_Cw(C)=0, the common leading coefficient cancels, so there is B_w≥0 such that absnS_w(X)≤ B_wX^1-1/d (X≥1). For every nontrivial character χ:Cl( O)→ C^×, the finite character sum is zero. Thus the displayed estimate applies without an additional arithmetic cancellation hypothesis. This conclusion concerns ordinary class characters; it does not identify them with the further norm-residue characters needed for Frobenius …
Exact mathematical passage
S_w(X)=\sum_C w(C)A_C(X)
\sum_{g\in\Gamma}Use in source · 2 passages
Search aliases: ∑_g∈Γ sum in Gamma
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1001
Derived comparison bounds (Block 16). For a finite group Γ whose order is a unit in a commutative local coefficient ring R, any prescribed equivariant residual isomorphism lifts to an integral isomorphism. Lift its matrix to A and define P=|Γ|^-1∑_g∈Γρ(g)Aτ(g^-1). Reindexing proves equivariance, and reduction recovers the prescribed matrix. Its determinant is a unit. Complete local coefficients with finite quotients, occupied compatible finite conditions and this residual type therefore give act…
Exact mathematical passage
P=|\Gamma|^{-1}\sum_{g\in\Gamma}\rho(g)A\tau(g^{-1}).
\sum_iUse in source · 3 passages
Search aliases: ∑_i sum
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 556
[Weighted seed zeta function] Let 1≤ w_1≤⋯≤ w_d, put W=∑_iw_i, and let a_m=q^∑_imax(m-w_i,0). Then Z_ w(T) = ∑_m=1^w_da_mT^m + q^-Wfrac(q^dT)^w_d+11-q^dT. In particular, every weighted smooth chart has a rational seed zeta function.
Exact mathematical passage
W=\sum_iw_i
\sum_{i=0}^{n-1}Use in source · 1 passage
Search aliases: ∑_i=0^n-1 sum
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1210
When pnmid n, the comparison with the finite cyclotomic sample is proved: q_L(γ_p)=σ_p, σ_p(ζ_n)=ζ_n^p. Reduction distinguishes n-th roots of unity at such a prime. Indeed, if x^n=y^n=1 and their reductions agree, the factorization of x^n-y^n gives (x-y)∑_i=0^n-1x^i y^n-1-i=0. The second factor reduces to noverline y^ n-1≠0, so the domain property gives x=y. This proves both Frobenius recognition on the cyclotomic generator and triviality of its inertia action. Consequently, every representation…
Exact mathematical passage
(x-y)\sum_{i=0}^{n-1}x^i y^{n-1-i}=0.
\sum_{i=1}^dUse in source · 1 passage
Search aliases: ∑_i=1^d sum
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 505
[Weighted smooth-chart count] For weights w=(w_1,…,w_d), N_ w(m) = q^∑_i=1^dmax(m-w_i,0).
Exact mathematical passage
\boxed{ N_{\bm w}(m) = q^{\sum_{i=1}^d\max(m-w_i,0)}. }
\sum_{i\in I}Use in source · 1 passage
Search aliases: ∑_i∈ I sum in
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1413
The same formalization proves, for each g∈ G, det(XI-ρ_m(g)) =∏_i∈ Idet(XI-ρ_i(g))^m_i, Tr(ρ_m(g)) =∑_i∈ I(m_i:K)Tr(ρ_i(g)). Constituents may have different dimensions. Zero multiplicities, zero-dimensional constituents, and the empty index set are included. If at a permitted sample g the constituent polynomials are the images of P_i,g∈ E[X] under a specified map E→ K, the summand polynomial is the image of ∏_iP_i,g^m_i. Applying this independently at each coefficient field preserves a supplied …
Exact mathematical passage
\det\bigl(XI-\rho_m(g)\bigr) =\prod_{i\in I}\det\bigl(XI-\rho_i(g)\bigr)^{m_i}, \qquad \operatorname{Tr}\bigl(\rho_m(g)\bigr) =\sum_{i\in I}(m_i:K)\operatorname{Tr}\bigl(\rho_i(g)\bigr).
\sum_{i\in J}Use in source · 2 passages
Search aliases: ∑_i∈ J sum in
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2164
Simultaneous projector-family lifting and rigidity (Block 19). Keep the complete domain, supplied finite-group carrier and group-order hypotheses of Block 18. Let J be any finite label set and k>0. Suppose (bar E_i)_i∈ J is a complete orthogonal family of equivariant idempotents modulo (π)^e+k. There exists an integral equivariant family (E_i)_i∈ J satisfying E_i^2=E_i, E_iE_j=0 (i≠ j), ∑_i∈ JE_i=1, E_ibmod(π)^k=bar E_ibmod(π)^k. The loss is at most e, independent of |J| and the ambient matrix r…
Exact mathematical passage
E_i^2=E_i,\qquad E_iE_j=0\ (i\ne j),\qquad \sum_{i\in J}E_i=1,\qquad E_i\bmod(\pi)^k=\bar E_i\bmod(\pi)^k.
\sum_jUse in source · 3 passages
Search aliases: ∑_j sum
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 347
Let C^0xrightarrowd_0C^1xrightarrowd_1C^2xrightarrow⋯ be a cochain complex over a field. Suppose a finite family of projectors P_j^r:C^r→ C^r satisfies ∑_jP_j^r=I, P_i^rP_j^r=δ_ijP_j^r, d_rP_j^r=P_j^r+1d_r.
Exact mathematical passage
\sum_jP_j^r=I, \qquad P_i^rP_j^r=\delta_{ij}P_j^r, \qquad d_rP_j^r=P_j^{r+1}d_r.
\sum_{j=0}^{d-1}Use in source · 1 passage
Search aliases: ∑_j=0^d-1 sum
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1578
Finite-exponential reconstruction and a cubic inverse (Block 13). Let K be a characteristic-zero field and let (r,N) be supplied group data with nilpotent N and r(g)N=ν(g)Nr(g). For a supplied scalar map satisfying t(gh)=t(g)+ν(g)t(h), the formalization constructs ρ(g)=exp(t(g)N)r(g), exp(aN)=∑_j=0^d-1fraca^jj!N^j (N^d=0). Lean proves the representation laws, actual finite-copy and split-map compatibility, and joint continuity from continuity of N, r and t with the stated continuous vector-space…
Exact mathematical passage
\rho(g)=\exp(t(g)N)r(g),\qquad \exp(aN)=\sum_{j=0}^{d-1}\frac{a^j}{j!}N^j\quad(N^d=0).
\sum_{j=1}^{d-1}Use in source · 2 passages
Search aliases: ∑_j=1^d-1 sum
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2232
General finite logarithmic recovery (Block 20). Let A be a rational algebra, without any commutativity assumption on A. For a unipotent element U and a positive integer d satisfying (U-1)^d=0, put L_d(U)=∑_j=1^d-1frac(-1)^j+1j(U-1)^j. The value is independent of the chosen annihilating exponent. The finite logarithm and the nilpotent exponential satisfy L(exp N)=N, exp(L(U))=U. Both maps preserve every annihilating exponent, and hence preserve the exact nilpotency class of N and exp(N)-1, or of …
Exact mathematical passage
L_d(U)=\sum_{j=1}^{d-1}\frac{(-1)^{j+1}}{j}(U-1)^j.
\sum_{j=1}^{n}Use in source · 1 passage
Search aliases: ∑_j=1^n sum
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 316
For the digit claim, subtract the chosen representative of the residue and divide the remainder by the uniformizer. Iterating gives unique digits. Conversely, the partial sums of any digit stream form a compatible Cauchy family and reconstruct uniquely by completeness and separatedness. More precisely, for each parameter there is a unique stream (a_i,j)_j≥1 such that ϖ^n+1mid t_i-∑_j=1^ns(a_i,j)ϖ^j (n≥0). The section is a set-theoretic section of reduction, not an asserted embedding of residue f…
Exact mathematical passage
\varpi^{n+1}\mid t_i-\sum_{j=1}^{n}s(a_{i,j})\varpi^j\qquad(n\ge0).
\sum_{m=1}^{w_d}Use in source · 1 passage
Search aliases: ∑_m=1^w_d sum
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 561
[Weighted seed zeta function] Let 1≤ w_1≤⋯≤ w_d, put W=∑_iw_i, and let a_m=q^∑_imax(m-w_i,0). Then Z_ w(T) = ∑_m=1^w_da_mT^m + q^-Wfrac(q^dT)^w_d+11-q^dT. In particular, every weighted smooth chart has a rational seed zeta function.
Exact mathematical passage
\boxed{ Z_{\bm w}(T) = \sum_{m=1}^{w_d}a_mT^m + q^{-W}\frac{(q^dT)^{w_d+1}}{1-q^dT}. }
\sum_{r=1}^{e}Use in source · 1 passage
Search aliases: ∑_r=1^e sum
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 593
[Singular power seed] For integers e≥1 and m≥1, in O/ϖ^m the equation x^e=0 has exactly q^m-lceil m/erceil solutions. Its seed zeta function is Z_e(T) = frac∑_r=1^eq^r-1T^r 1-q^e-1T^e.
Exact mathematical passage
\boxed{ Z_e(T) = \frac{\sum_{r=1}^{e}q^{r-1}T^r} {1-q^{e-1}T^e}. }
\sum_{\substack{I\ne0\\\Nm I\leq X}}Use in source · 1 passage
Search aliases: ∑_substackI≠0 Nm I≤ X ∑_substackI≠0 N I≤ X sum substack ne Nm leq sum substack ne operatorname leq
Use in source · Read passage · residue_continuation_body.tex, line 145
e preceding block proved, for every ordinary ideal class C, A_C(X)=c_KX+O_K,C(X^1-1/d), c_K=frac2^r_1(2π)^r_2R_K#μ(K)sqrtabsnΔ_K. Here A_C counts actual nonzero integral ideals, r_1,r_2 count the real and complex places, R_K is the regulator, and Δ_K is the discriminant. For a weight w:Cl( O)→ C, define the actual finite ideal sum S_w(X)=∑_substackI≠0 N I≤ Xw([I]). Exact grouping by ideal class gives S_w(X)=∑_C w(C)A_C(X). If ∑_Cw(C)=0, the common leading coefficient cancels, so there is B_w≥0 s…
Exact mathematical passage
S_w(X)=\sum_{\substack{I\ne0\\\Nm I\leq X}}w([I]).
\sum_{\substack{I\ne0,\ [I]=C\\\Nm I\leq X}}Use in source · 1 passage
Search aliases: ∑_substackI≠0, [I]=C Nm I≤ X ∑_substackI≠0, [I]=C N I≤ X sum substack ne Nm leq sum substack ne operatorname leq
Use in source · Read passage · residue_continuation_body.tex, line 225
The assembled ideal norm-character estimate. Let C∈Cl( O), let J≠0 be an integral ideal with [J]=C^-1, and let m>0. Let χ: Z/m Z→ C preserve multiplication, 0, and 1. Assume a residue-ring unit satisfying [source reference], and assume χ( N Jbmod m)≠0. Then T_C,χ(X) =∑_substackI≠0, [I]=C N I≤ X χ( N Ibmod m) =O_K,J,m,χ(X^1-1/d). More precisely, some B≥0 bounds its absolute value by BX^1-1/d for every real X≥1.
Exact mathematical passage
T_{C,\chi}(X) =\sum_{\substack{I\ne0,\ [I]=C\\\Nm I\leq X}} \chi(\Nm I\bmod m) =O_{K,J,m,\chi}(X^{1-1/d}).
\sum_{z\in A}Use in source · 1 passage
Search aliases: ∑_z∈ A sum in
Use in source · pages.tex, line 343
tained scalar/Mellin continuation: norm-residue lattice mean cancellation. Let χ be the quadratic character modulo five, extended by χ(0)=0. In the field on the preceding page, choose v=2u. Its norm is N(v)=2, so χ(N(v))=-1. vfig23_mean_cancellation headThe whole sum is paired with its negative Multiplication by v permutes the 25 elements. Norm multiplicativity gives ∑_z∈ Aχ(N(z)) =∑_z∈ Aχ(N(vz)) =-∑_z∈ Aχ(N(z)). The sum is an integer, so it is zero. The origin contributes zero. The other 24 ele…
Exact mathematical passage
\sum_{z\in A}\chi(N(z)) =\sum_{z\in A}\chi(N(vz)) =-\sum_{z\in A}\chi(N(z)).
\setminusUse in source · 2 passages
Search aliases: ∖ setminus
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1623
Let E be a field of characteristic zero and let T_1,…,T_N be commuting operators on a finite-dimensional E-vector space A, each diagonalizable over E. Thus the selected spectra split over the coefficient field. Let a_i∈ E be a selected target scalar for T_i, possibly absent from its spectrum. Define e_i = ∏_μ∈Spec(T_i)∖{a_i} fracT_i-μ Ia_i-μ, e_T=∏_i=1^Ne_i, with e_i=0 if a_i is absent.
Exact mathematical passage
e_i = \prod_{\mu\in\Spec(T_i)\setminus\{a_i\}} \frac{T_i-\mu I}{a_i-\mu}, \qquad e_T=\prod_{i=1}^{N}e_i,
\circUse in source · 4 passages
Search aliases: ∘ circ
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1227
For a compatible family ρ_i,λ of representations of the fixed finite group, set V_i,λ=ρ_i,λ∘ q_L. Suppose the actual absolute-Galois members are irreducible and the systems are pairwise componentwise nonisomorphic. The finite prime separator now yields distinct permitted rational primes p_j, actual prime ideals Q_j and arithmetic Frobenius witnesses γ_j as above, outside any prescribed finite set, such that the common algebraic trace matrix has nonzero determinant. Every coefficient embedding pr…
Exact mathematical passage
V_{i,\lambda}=\rho_{i,\lambda}\circ q_L
\capUse in source · 3 passages
Search aliases: ∩ cap
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1496
Checked group, inertia and monodromy finite sums (Block 12). Let K be a field, G a group, and ν:G→ K^× a character. The new datum consists of an actual representation r and nilpotent N with r(g)N=ν(g)Nr(g) for every g. For a supplied subgroup H, define E(D,H)=ker N∩ V^H. This is the intersection of independently defined kernel and actual invariant subspaces. For normal H, the restricted whole-group representation and its inverse automorphisms are constructed. A change from φ to φ h, with h∈ H, l…
Exact mathematical passage
E(D,H)=\ker N\cap V^H.
\cupUse in source · 1 passage
Search aliases: ∪ cup
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1309
The remaining prime-existence obligation now has a precise formulation. For every finite Galois subfield L, every finite set B of rational primes, and every σ∈Gal(L/ Q), require a prime p∉ B∪ S_L and an actual absolute arithmetic witness γ_p such that q_L(γ_p)∼σ. Lean proves that this assertion is equivalent to the cofinite quotient coverage required by the general separator, for any fixed choice of actual Frobenius witnesses. The equivalence derives independence from the witness choices; it doe…
Exact mathematical passage
p\notin B\cup S_L
\intUse in source · 4 passages
Search aliases: ∫ int
Use in source · Read passage · residue_continuation_body.tex, line 164
An actual Mellin endpoint. The counting functions are constant on the integer norm steps. Their measurability, and hence that of S_w, is proved from this fact. For zero-sum w, define F_w(h)=(1+h)∫_1^∞x^-2-hS_w(x) dx (h≥0). Since 1-1/d<1, the logarithmic first moment M_w=∫_1^∞(1+log x) absnx^-2S_w(x) dx is finite. Lean proves the explicit estimate absnF_w(h)-F_w(0)≤ hM_w for every h≥0, and therefore F_w(h)=F_w(0)+O(h) as hdownarrow0. The function here is the displayed integral of the actual ideal…
Exact mathematical passage
F_w(h)=(1+h)\int_1^{\infty}x^{-2-h}S_w(x)\,dx \qquad(h\geq0).
\int_1^\inftyUse in source · 2 passages
Search aliases: ∫_1^∞ int infty
Use in source · pages.tex, line 407
IN ENDPOINT MODELMain text: §9; retained scalar/Mellin continuation: summatory measurability and first-order Mellin control. Use the explicit envelope A(x)=x^1/2 for x≥1. It is not an ideal-counting function. It is a model showing why a power saving can make a Mellin expression regular at s=1. vfig27_mellin headCompute the endpoint rather than just naming it For real s>1/2, I(s)=s∫_1^∞ A(x)x^-s-1 dd x =fracss-1/2. At s=1+h this gives I(1+h)-I(1)=-frachh+1/2, |I(1+h)-I(1)|≤4|h| (|h|≤1/4). The der…
Exact mathematical passage
I(s)=s\int_1^\infty A(x)x^{-s-1}\,\dd x =\frac{s}{s-1/2}.
\int_1^{\infty}Use in source · 2 passages
Search aliases: ∫_1^∞ int infty
Use in source · Read passage · residue_continuation_body.tex, line 164
An actual Mellin endpoint. The counting functions are constant on the integer norm steps. Their measurability, and hence that of S_w, is proved from this fact. For zero-sum w, define F_w(h)=(1+h)∫_1^∞x^-2-hS_w(x) dx (h≥0). Since 1-1/d<1, the logarithmic first moment M_w=∫_1^∞(1+log x) absnx^-2S_w(x) dx is finite. Lean proves the explicit estimate absnF_w(h)-F_w(0)≤ hM_w for every h≥0, and therefore F_w(h)=F_w(0)+O(h) as hdownarrow0. The function here is the displayed integral of the actual ideal…
Exact mathematical passage
F_w(h)=(1+h)\int_1^{\infty}x^{-2-h}S_w(x)\,dx \qquad(h\geq0).
\simUse in source · 2 passages
Search aliases: ∼ sim
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1311
The remaining prime-existence obligation now has a precise formulation. For every finite Galois subfield L, every finite set B of rational primes, and every σ∈Gal(L/ Q), require a prime p∉ B∪ S_L and an actual absolute arithmetic witness γ_p such that q_L(γ_p)∼σ. Lean proves that this assertion is equivalent to the cofinite quotient coverage required by the general separator, for any fixed choice of actual Frobenius witnesses. The equivalence derives independence from the witness choices; it doe…
Exact mathematical passage
q_L(\gamma_p)\sim\sigma.
\congUse in source · 7 passages
Search aliases: ≅ cong
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 264
[Formal inverse-limit reconstruction] Assume M_bar x is represented by a complete local O-algebra R_bar x. Then reduction induces a natural bijection M_bar x( O) ≅ varprojlim_m M_bar x( O/ϖ^m).
Exact mathematical passage
\boxed{ \mathcal M_{\bar x}(\Ocal) \cong \varprojlim_m \mathcal M_{\bar x}(\Ocal/\varpi^m). }
\leUse in source · 16 passages
Search aliases: ≤ le
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 515
The image of ϖ^w_i O in O/ϖ^m has one element when m≤ w_i, and q^m-w_i elements when m>w_i. Multiplying the independent coordinate counts gives the formula.
Exact mathematical passage
m\le w_i
\subseteqUse in source · 8 passages
Search aliases: ⊆ subseteq
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 377
For each packet, use the actual image submodules C_j^r=imP_j^r. Commutation with the differential makes these a subcomplex. Write Z^r=ker d_r, B^r=imd_r-1⊆ Z^r, H^r=Z^r/B^r, and define Z_j^r,B_j^r,H_j^r using the restricted differentials of C_j^bullet. The inclusion i_j:C_j^bullet→ C^bullet and the image-valued projection p_j:C^bullet→ C_j^bullet are cochain maps, so they induce maps on these genuine quotients.
Exact mathematical passage
Z^r=\ker d_r,\qquad B^r=\operatorname{im}d_{r-1}\subseteq Z^r, \qquad H^r=Z^r/B^r,
\bigoplus_{i=1}^{r}Use in source · 3 passages
Search aliases: ⊕_i=1^r bigoplus
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 1337
[Compatible-carrier summand completion] Suppose a semisimple strictly compatible carrier decomposes as mathscr W ≅ ⊕_i=1^r mathscr W_i^⊕ c_i, with the mathscr W_i as above. At one coefficient prime λ_0, suppose a semisimple summand has the form V_λ_0 ≅ ⊕_i=1^r W_i,λ_0^⊕ m_i, 0≤ m_i≤ c_i. Then V_λ = ⊕_i=1^r W_i,λ^⊕ m_i defines a strictly compatible summand at every coefficient prime. The vector (m_i) is uniquely recoverable from finitely many Frobenius traces supplied by the separation theorem.
Exact mathematical passage
\mathscr W \cong \bigoplus_{i=1}^{r} \mathscr W_i^{\oplus c_i},
\bigoplus_{i\in J}Use in source · 1 passage
Search aliases: ⊕_i∈ J bigoplus in
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 2185
The actual invariant images U_i=imE_i give an equivariant linear isomorphism ⊕_i∈ JU_i≃ R^n, (v_i)_i⟼∑_i v_i, v⟼(E_iv)_i. Lean represents the finite direct sum by the finite product of the image submodules. The existing coordinatewise reduction theorem maps every U_i onto the corresponding reduced projector image. No chosen bases or unproved freeness of the images are used.
Exact mathematical passage
\bigoplus_{i\in J}U_i\simeq R^n, \qquad (v_i)_i\longmapsto\sum_i v_i, \qquad v\longmapsto(E_iv)_i.
\bigoplus_jUse in source · 2 passages
Search aliases: ⊕_j bigoplus
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 357
[Cohomology packet decomposition] Under these hypotheses, each packet is a subcomplex and H^r(C^bullet) ≅ ⊕_jH^r(P_jC^bullet). In particular, H^1=⊕_jH^1_j, H^2=⊕_jH^2_j, so tangent and obstruction spaces reconstruct exactly from their packet components.
Exact mathematical passage
\boxed{ H^r(C^\bullet) \cong \bigoplus_jH^r(P_jC^\bullet). }
\cdotsUse in source · 10 passages
Search aliases: ⋯ cdots
Use in source · Read passage · Paper_X_Block27_Amended.tex, line 304
[Smooth formal branch count] If R_bar x≅ O[[t_1,…,t_d]] as complete local O-algebras, then #X_m(bar x)=q^d(m-1). Fix a uniformizer and a section s:k→ O of reduction, with s(0)=0. Every O-valued branch is encoded by d positive-depth digit streams t_i=s(a_i,1)ϖ+s(a_i,2)ϖ^2+⋯, a_i,j∈ k,
Exact mathematical passage
t_i=s(a_{i,1})\varpi+s(a_{i,2})\varpi^2+\cdots, \qquad a_{i,j}\in k,
\langleUse in source · 3 passages
Search aliases: ⟨ langle
Use in source · 10_cohomology.tex, line 50
node[font=small,align=center,font=scriptsize] at (5.445,3.53) u_0; node[font=small,align=center,font=scriptsize] at (6.135,3.53) u_1; node[font=small,align=center,font=scriptsize] at (6.825,3.53) u_2; node[font=small,align=center,font=scriptsize] at (7.515,3.53) u_3; node[font=small,align=center] at (11,3.7) d_1d_0=0; node[font=small,align=center] at (11,2.65) H^0=⟨[e_1]⟩; node[font=small,align=center] at (11,1.85) H^1=⟨[u_2],[u_3]⟩; node[font=small,align=center] at (11,1.05) H^2=⟨[f_1]⟩; node[f…
Exact mathematical passage
H^0=\langle[e_1]\rangle
\rangleUse in source · 3 passages
Search aliases: ⟩ rangle
Use in source · 10_cohomology.tex, line 50
node[font=small,align=center,font=scriptsize] at (5.445,3.53) u_0; node[font=small,align=center,font=scriptsize] at (6.135,3.53) u_1; node[font=small,align=center,font=scriptsize] at (6.825,3.53) u_2; node[font=small,align=center,font=scriptsize] at (7.515,3.53) u_3; node[font=small,align=center] at (11,3.7) d_1d_0=0; node[font=small,align=center] at (11,2.65) H^0=⟨[e_1]⟩; node[font=small,align=center] at (11,1.85) H^1=⟨[u_2],[u_3]⟩; node[font=small,align=center] at (11,1.05) H^2=⟨[f_1]⟩; node[f…
Exact mathematical passage
H^0=\langle[e_1]\rangle
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