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Hilbert 12: ambient dual slice
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161 symbol spellings in 126 mathematical passages across 1 selected source files. Subscripts and superscripts have their own entries alongside their base symbols.
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+Use in source · 25 passages
Search aliases: +
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 13
Put ω^2+ω+1=0, J=2+3ω, bar J=-1-3ω. Then J+bar J=1, Jbar J=7. With
Exact mathematical passage
\omega^2+\omega+1=0
=Use in source · 78 passages
Search aliases: =
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 5
**Author:** Komorebi P. **Research status:** New exact organization and checks of finite algebra. The analytic comparison is open. **Working case:** K= Q(β), β^3=7, with the selected complex embedding.
Exact mathematical passage
K=\mathbb Q(\beta)
AUse in source · 9 passages
Search aliases: A
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 44
On A=E_ω^3, E_ω= C/ Z[ω], use the product principal Weil pairing composed with the integral symmetric isogeny represented by H. For seventh-torsion coordinates X=(a+ω b)/7, Y=(c+ω d)/7, write
Exact mathematical passage
A=E_\omega^3
aUse in source · 7 passages
Search aliases: a
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 44
On A=E_ω^3, E_ω= C/ Z[ω], use the product principal Weil pairing composed with the integral symmetric isogeny represented by H. For seventh-torsion coordinates X=(a+ω b)/7, Y=(c+ω d)/7, write
Exact mathematical passage
X=(a+\omega b)/7
A'Use in source · 1 passage
Search aliases: A'
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 129
D_χ(z)=∑_k=1^6χ(k)fracf'_k(z)f_k(z) =fracA'(z)A(z)+ωfracB'(z)B(z)
Exact mathematical passage
D_\chi(z)=\sum_{k=1}^6\chi(k)\frac{f'_k(z)}{f_k(z)} =\frac{A'(z)}{A(z)}+\omega\frac{B'(z)}{B(z)}
a^{\mathsf T}Use in source · 1 passage
Search aliases: a^ T mathsf
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 46
e_7,H(X,Y)=ζ_7^a^ THd-b^ THc.
Exact mathematical passage
e_{7,H}(X,Y)=\zeta_7^{a^{\mathsf T}Hd-b^{\mathsf T}Hc}.
A^\veeUse in source · 2 passages
Search aliases: A^vee vee
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 50
The pairing is perfect since 7nmiddet H. Its source is the canonical pairing of A[7] with A^vee[7]. The symmetric map H:A→ A^vee need not define an ample polarization: it is indefinite. The exact matrix identity gives contragredient covariance
Exact mathematical passage
A^\vee[7]
a_1Use in source · 1 passage
Search aliases: a_1
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 115
(a_1,…,a_6)=(1,0,-1,-1,0,1), (b_1,…,b_6)=(0,1,-1,-1,1,0).
Exact mathematical passage
(a_1,\ldots,a_6)=(1,0,-1,-1,0,1),\quad (b_1,\ldots,b_6)=(0,1,-1,-1,1,0).
a_6Use in source · 1 passage
Search aliases: a_6
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 115
(a_1,…,a_6)=(1,0,-1,-1,0,1), (b_1,…,b_6)=(0,1,-1,-1,1,0).
Exact mathematical passage
(a_1,\ldots,a_6)=(1,0,-1,-1,0,1),\quad (b_1,\ldots,b_6)=(0,1,-1,-1,1,0).
a_kUse in source · 1 passage
Search aliases: a_k
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 113
The cubic character χ(k)=a_k+ω b_k has integer coefficient vectors
Exact mathematical passage
\chi(k)=a_k+\omega b_k
\mathrm{alg}Use in source · 1 passage
Search aliases: alg mathrm
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 167
The analytic-to-arithmetic equality U_Γ=U_alg is still unproved. In the later polarized presentation the remaining statement is Im(C_polU_Γ^3)=0, given the established modulus and independent branch conditions. The two-coordinate packet identifies where a primitive character can live, but no identification with the analytic gamma packet has been proved. In particular, no gerbe trivialization, Eisenstein deck descent, normalized scalar comparison, or new Lean compilation is supplied here.
Exact mathematical passage
U_\Gamma=U_{\mathrm{alg}}
BUse in source · 4 passages
Search aliases: B
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 122
A(z)=fracf_1f_6f_3f_4, B(z)=fracf_2f_5f_3f_4.
Exact mathematical passage
A(z)=\frac{f_1f_6}{f_3f_4},\qquad B(z)=\frac{f_2f_5}{f_3f_4}.
bUse in source · 8 passages
Search aliases: b
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 19
and basis b_0=β^2+2β, b_1=-β-2, b_2=15, multiplication by ϵ=2-β has matrix
Exact mathematical passage
b_0=\beta^2+2\beta
B'Use in source · 1 passage
Search aliases: B'
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 129
D_χ(z)=∑_k=1^6χ(k)fracf'_k(z)f_k(z) =fracA'(z)A(z)+ωfracB'(z)B(z)
Exact mathematical passage
D_\chi(z)=\sum_{k=1}^6\chi(k)\frac{f'_k(z)}{f_k(z)} =\frac{A'(z)}{A(z)}+\omega\frac{B'(z)}{B(z)}
b^{\mathsf T}Use in source · 1 passage
Search aliases: b^ T mathsf
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 46
e_7,H(X,Y)=ζ_7^a^ THd-b^ THc.
Exact mathematical passage
e_{7,H}(X,Y)=\zeta_7^{a^{\mathsf T}Hd-b^{\mathsf T}Hc}.
b_0Use in source · 1 passage
Search aliases: b_0
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 19
and basis b_0=β^2+2β, b_1=-β-2, b_2=15, multiplication by ϵ=2-β has matrix
Exact mathematical passage
b_0=\beta^2+2\beta
b_1Use in source · 2 passages
Search aliases: b_1
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 19
and basis b_0=β^2+2β, b_1=-β-2, b_2=15, multiplication by ϵ=2-β has matrix
Exact mathematical passage
b_1=-\beta-2
b_2Use in source · 1 passage
Search aliases: b_2
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 19
and basis b_0=β^2+2β, b_1=-β-2, b_2=15, multiplication by ϵ=2-β has matrix
Exact mathematical passage
b_2=15
b_6Use in source · 1 passage
Search aliases: b_6
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 115
(a_1,…,a_6)=(1,0,-1,-1,0,1), (b_1,…,b_6)=(0,1,-1,-1,1,0).
Exact mathematical passage
(a_1,\ldots,a_6)=(1,0,-1,-1,0,1),\quad (b_1,\ldots,b_6)=(0,1,-1,-1,1,0).
b_iUse in source · 1 passage
Search aliases: b_i
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 25
The coefficient functional λ(x)=[β^2]x, applied after reduction via β^3=7, is equivalently Tr_K/ Q(x/(3β^2)). The Gram matrix H_ij=λ(b_i b_j) is
Exact mathematical passage
H_{ij}=\lambda(b_i b_j)
b_jUse in source · 1 passage
Search aliases: b_j
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 25
The coefficient functional λ(x)=[β^2]x, applied after reduction via β^3=7, is equivalently Tr_K/ Q(x/(3β^2)). The Gram matrix H_ij=λ(b_i b_j) is
Exact mathematical passage
H_{ij}=\lambda(b_i b_j)
b_kUse in source · 1 passage
Search aliases: b_k
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 113
The cubic character χ(k)=a_k+ω b_k has integer coefficient vectors
Exact mathematical passage
\chi(k)=a_k+\omega b_k
\bar JUse in source · 9 passages
Search aliases: bar J bar
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 13
Put ω^2+ω+1=0, J=2+3ω, bar J=-1-3ω. Then J+bar J=1, Jbar J=7. With
Exact mathematical passage
\bar J=-1-3\omega
\bmodUse in source · 1 passage
Search aliases: bmod
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 134
is unambiguous and meromorphic. Under formal unit relabeling f_k↦ f_2kbmod7 the functions transform exactly as
Exact mathematical passage
f_k\mapsto f_{2k\bmod7}
CUse in source · 1 passage
Search aliases: C
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 167
The analytic-to-arithmetic equality U_Γ=U_alg is still unproved. In the later polarized presentation the remaining statement is Im(C_polU_Γ^3)=0, given the established modulus and independent branch conditions. The two-coordinate packet identifies where a primitive character can live, but no identification with the analytic gamma packet has been proved. In particular, no gerbe trivialization, Eisenstein deck descent, normalized scalar comparison, or new Lean compilation is supplied here.
Exact mathematical passage
\operatorname{Im}(C_{\mathrm{pol}}U_\Gamma^3)=0
\mathbb CUse in source · 2 passages
Search aliases: C mathbb
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 44
On A=E_ω^3, E_ω= C/ Z[ω], use the product principal Weil pairing composed with the integral symmetric isogeny represented by H. For seventh-torsion coordinates X=(a+ω b)/7, Y=(c+ω d)/7, write
Exact mathematical passage
E_\omega=\mathbb C/\mathbb Z[\omega]
cUse in source · 2 passages
Search aliases: c
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 44
On A=E_ω^3, E_ω= C/ Z[ω], use the product principal Weil pairing composed with the integral symmetric isogeny represented by H. For seventh-torsion coordinates X=(a+ω b)/7, Y=(c+ω d)/7, write
Exact mathematical passage
Y=(c+\omega d)/7
\mathbb C^3Use in source · 1 passage
Search aliases: C^3 mathbb
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 99
Thus the exact principal-seven shifts z+kδ in the earlier gamma-divisor paper are the projections of k(P+P'). This holds **on the universal cover**; π does not automatically descend from A= C^3/ Z[ω]^3 to the original triptic quotient because of the extra Eisenstein periods. The missing deck multipliers cannot be discarded.
Exact mathematical passage
A=\mathbb C^3/\mathbb Z[\omega]^3
C_{\mathrm{pol}}Use in source · 1 passage
Search aliases: C_pol mathrm
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 167
The analytic-to-arithmetic equality U_Γ=U_alg is still unproved. In the later polarized presentation the remaining statement is Im(C_polU_Γ^3)=0, given the established modulus and independent branch conditions. The two-coordinate packet identifies where a primitive character can live, but no identification with the analytic gamma packet has been proved. In particular, no gerbe trivialization, Eisenstein deck descent, normalized scalar comparison, or new Lean compilation is supplied here.
Exact mathematical passage
\operatorname{Im}(C_{\mathrm{pol}}U_\Gamma^3)=0
DUse in source · 4 passages
Search aliases: D
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 129
D_χ(z)=∑_k=1^6χ(k)fracf'_k(z)f_k(z) =fracA'(z)A(z)+ωfracB'(z)B(z)
Exact mathematical passage
D_\chi(z)=\sum_{k=1}^6\chi(k)\frac{f'_k(z)}{f_k(z)} =\frac{A'(z)}{A(z)}+\omega\frac{B'(z)}{B(z)}
dUse in source · 2 passages
Search aliases: d
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 44
On A=E_ω^3, E_ω= C/ Z[ω], use the product principal Weil pairing composed with the integral symmetric isogeny represented by H. For seventh-torsion coordinates X=(a+ω b)/7, Y=(c+ω d)/7, write
Exact mathematical passage
Y=(c+\omega d)/7
D_\chiUse in source · 4 passages
Search aliases: D_χ chi
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 129
D_χ(z)=∑_k=1^6χ(k)fracf'_k(z)f_k(z) =fracA'(z)A(z)+ωfracB'(z)B(z)
Exact mathematical passage
D_\chi(z)=\sum_{k=1}^6\chi(k)\frac{f'_k(z)}{f_k(z)} =\frac{A'(z)}{A(z)}+\omega\frac{B'(z)}{B(z)}
\detUse in source · 3 passages
Search aliases: det
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 21
M=0 · 1 · 0 -4 · 2 · 15 -1 · 0 · 4, det M=1.
Exact mathematical passage
M=\begin{pmatrix}0&1&0\\-4&2&15\\-1&0&4\end{pmatrix},\quad \det M=1.
EUse in source · 2 passages
Search aliases: E
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 44
On A=E_ω^3, E_ω= C/ Z[ω], use the product principal Weil pairing composed with the integral symmetric isogeny represented by H. For seventh-torsion coordinates X=(a+ω b)/7, Y=(c+ω d)/7, write
Exact mathematical passage
A=E_\omega^3
eUse in source · 5 passages
Search aliases: e
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 46
e_7,H(X,Y)=ζ_7^a^ THd-b^ THc.
Exact mathematical passage
e_{7,H}(X,Y)=\zeta_7^{a^{\mathsf T}Hd-b^{\mathsf T}Hc}.
e_{7,H}Use in source · 5 passages
Search aliases: e_7,H
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 46
e_7,H(X,Y)=ζ_7^a^ THd-b^ THc.
Exact mathematical passage
e_{7,H}(X,Y)=\zeta_7^{a^{\mathsf T}Hd-b^{\mathsf T}Hc}.
E_\omegaUse in source · 1 passage
Search aliases: E_ω omega
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 44
On A=E_ω^3, E_ω= C/ Z[ω], use the product principal Weil pairing composed with the integral symmetric isogeny represented by H. For seventh-torsion coordinates X=(a+ω b)/7, Y=(c+ω d)/7, write
Exact mathematical passage
E_\omega=\mathbb C/\mathbb Z[\omega]
E_\omega^3Use in source · 1 passage
Search aliases: E_ω^3 omega
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 44
On A=E_ω^3, E_ω= C/ Z[ω], use the product principal Weil pairing composed with the integral symmetric isogeny represented by H. For seventh-torsion coordinates X=(a+ω b)/7, Y=(c+ω d)/7, write
Exact mathematical passage
A=E_\omega^3
\mathrm{ell}Use in source · 1 passage
Search aliases: ell mathrm
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 111
Let F(z)=Γ_ell(z;τ,σ)^5/Γ_ell(5z;5τ,5σ) and f_k(z)=F(z+kδ). The inherited principal-seven product is P_7(z)=∏_k=0^6 f_k(z), the trivial character sector.
Exact mathematical passage
F(z)=\Gamma_{\mathrm{ell}}(z;\tau,\sigma)^5/\Gamma_{\mathrm{ell}}(5z;5\tau,5\sigma)
FUse in source · 5 passages
Search aliases: F
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 111
Let F(z)=Γ_ell(z;τ,σ)^5/Γ_ell(5z;5τ,5σ) and f_k(z)=F(z+kδ). The inherited principal-seven product is P_7(z)=∏_k=0^6 f_k(z), the trivial character sector.
Exact mathematical passage
F(z)=\Gamma_{\mathrm{ell}}(z;\tau,\sigma)^5/\Gamma_{\mathrm{ell}}(5z;5\tau,5\sigma)
\mathbb FUse in source · 3 passages
Search aliases: F mathbb
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 31
Also (M-2I)^3=-7I. Thus over F_7, N=M-2I has one nilpotent block of length three. Set u=(1,2,4)^ T, v=(0,1,2)^ T, w=(0,0,1)^ T. Then Nu=0, Nv=u, Nw=v modulo seven, and
Exact mathematical passage
\mathbb F_7
fUse in source · 8 passages
Search aliases: f
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 78
Equivalently, the primitive multiplicative-character projector applied to the additive function f(k)=e_7,H(kP,Q)=ζ_7^3k produces G_3/6:
Exact mathematical passage
f(k)=e_{7,H}(kP,Q)=\zeta_7^{3k}
f'_kUse in source · 1 passage
Search aliases: f'_k
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 129
D_χ(z)=∑_k=1^6χ(k)fracf'_k(z)f_k(z) =fracA'(z)A(z)+ωfracB'(z)B(z)
Exact mathematical passage
D_\chi(z)=\sum_{k=1}^6\chi(k)\frac{f'_k(z)}{f_k(z)} =\frac{A'(z)}{A(z)}+\omega\frac{B'(z)}{B(z)}
f_1Use in source · 1 passage
Search aliases: f_1
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 122
A(z)=fracf_1f_6f_3f_4, B(z)=fracf_2f_5f_3f_4.
Exact mathematical passage
A(z)=\frac{f_1f_6}{f_3f_4},\qquad B(z)=\frac{f_2f_5}{f_3f_4}.
f_2Use in source · 2 passages
Search aliases: f_2
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 122
A(z)=fracf_1f_6f_3f_4, B(z)=fracf_2f_5f_3f_4.
Exact mathematical passage
A(z)=\frac{f_1f_6}{f_3f_4},\qquad B(z)=\frac{f_2f_5}{f_3f_4}.
f_{2k\bmod7}Use in source · 1 passage
Search aliases: f_2kbmod7 bmod
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 134
is unambiguous and meromorphic. Under formal unit relabeling f_k↦ f_2kbmod7 the functions transform exactly as
Exact mathematical passage
f_k\mapsto f_{2k\bmod7}
f_3Use in source · 1 passage
Search aliases: f_3
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 122
A(z)=fracf_1f_6f_3f_4, B(z)=fracf_2f_5f_3f_4.
Exact mathematical passage
A(z)=\frac{f_1f_6}{f_3f_4},\qquad B(z)=\frac{f_2f_5}{f_3f_4}.
f_4Use in source · 1 passage
Search aliases: f_4
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 122
A(z)=fracf_1f_6f_3f_4, B(z)=fracf_2f_5f_3f_4.
Exact mathematical passage
A(z)=\frac{f_1f_6}{f_3f_4},\qquad B(z)=\frac{f_2f_5}{f_3f_4}.
f_5Use in source · 1 passage
Search aliases: f_5
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 122
A(z)=fracf_1f_6f_3f_4, B(z)=fracf_2f_5f_3f_4.
Exact mathematical passage
A(z)=\frac{f_1f_6}{f_3f_4},\qquad B(z)=\frac{f_2f_5}{f_3f_4}.
f_6Use in source · 1 passage
Search aliases: f_6
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 122
A(z)=fracf_1f_6f_3f_4, B(z)=fracf_2f_5f_3f_4.
Exact mathematical passage
A(z)=\frac{f_1f_6}{f_3f_4},\qquad B(z)=\frac{f_2f_5}{f_3f_4}.
\mathbb F_7Use in source · 2 passages
Search aliases: F_7 mathbb
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 31
Also (M-2I)^3=-7I. Thus over F_7, N=M-2I has one nilpotent block of length three. Set u=(1,2,4)^ T, v=(0,1,2)^ T, w=(0,0,1)^ T. Then Nu=0, Nv=u, Nw=v modulo seven, and
Exact mathematical passage
\mathbb F_7
\mathbb F_7^\timesUse in source · 1 passage
Search aliases: F_7^× mathbb times
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 64
The nontrivial phase remains unchanged for Q=(w+su+tv)/bar J, since u pairs trivially with u,v modulo seven. Other nonzero top-quotient choices produce phases ζ_7^3r, r∈ F_7^×.
Exact mathematical passage
r\in\mathbb F_7^\times
f_kUse in source · 4 passages
Search aliases: f_k
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 111
Let F(z)=Γ_ell(z;τ,σ)^5/Γ_ell(5z;5τ,5σ) and f_k(z)=F(z+kδ). The inherited principal-seven product is P_7(z)=∏_k=0^6 f_k(z), the trivial character sector.
Exact mathematical passage
f_k(z)=F(z+k\delta)
GUse in source · 5 passages
Search aliases: G
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 70
Let χ(1),…,χ(6)=(1,ω,ω^2,ω^2,ω,1) and G_r=∑_k=1^6χ(k)ζ_7^rk. Then for every r≠0,
Exact mathematical passage
G_r=\sum_{k=1}^{6}\chi(k)\zeta_7^{rk}
G_1Use in source · 1 passage
Search aliases: G_1
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 72
G_r=χ(r)^-1G_1, G_r^3=7J.
Exact mathematical passage
G_r=\chi(r)^{-1}G_1,\qquad G_r^3=7J.
G_3Use in source · 2 passages
Search aliases: G_3
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 78
Equivalently, the primitive multiplicative-character projector applied to the additive function f(k)=e_7,H(kP,Q)=ζ_7^3k produces G_3/6:
Exact mathematical passage
G_3/6
G_rUse in source · 3 passages
Search aliases: G_r
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 70
Let χ(1),…,χ(6)=(1,ω,ω^2,ω^2,ω,1) and G_r=∑_k=1^6χ(k)ζ_7^rk. Then for every r≠0,
Exact mathematical passage
G_r=\sum_{k=1}^{6}\chi(k)\zeta_7^{rk}
G_r^3Use in source · 1 passage
Search aliases: G_r^3
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 72
G_r=χ(r)^-1G_1, G_r^3=7J.
Exact mathematical passage
G_r=\chi(r)^{-1}G_1,\qquad G_r^3=7J.
HUse in source · 15 passages
Search aliases: H
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 25
The coefficient functional λ(x)=[β^2]x, applied after reduction via β^3=7, is equivalently Tr_K/ Q(x/(3β^2)). The Gram matrix H_ij=λ(b_i b_j) is
Exact mathematical passage
H_{ij}=\lambda(b_i b_j)
H_+Use in source · 2 passages
Search aliases: H_+
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 66
There cannot be a positive definite real symmetric matrix H_+ with M^ TH_+=H_+M: that would make M similar to a real symmetric matrix, but χ_M(X)=X^3-6X^2+12X-1 has negative discriminant -1323 and two nonreal roots. Hence the Poincare duality description is safer than requiring an invariant ample polarization. Positive-definite lifts congruent to H modulo seven still give the same finite pairing, but do not preserve the global self-adjoint identity.
Exact mathematical passage
H_+
H_{ij}Use in source · 1 passage
Search aliases: H_ij
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 25
The coefficient functional λ(x)=[β^2]x, applied after reduction via β^3=7, is equivalently Tr_K/ Q(x/(3β^2)). The Gram matrix H_ij=λ(b_i b_j) is
Exact mathematical passage
H_{ij}=\lambda(b_i b_j)
IUse in source · 4 passages
Search aliases: I
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 31
Also (M-2I)^3=-7I. Thus over F_7, N=M-2I has one nilpotent block of length three. Set u=(1,2,4)^ T, v=(0,1,2)^ T, w=(0,0,1)^ T. Then Nu=0, Nv=u, Nw=v modulo seven, and
Exact mathematical passage
(M-2I)^3=-7I
iUse in source · 1 passage
Search aliases: i
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 25
The coefficient functional λ(x)=[β^2]x, applied after reduction via β^3=7, is equivalently Tr_K/ Q(x/(3β^2)). The Gram matrix H_ij=λ(b_i b_j) is
Exact mathematical passage
H_{ij}=\lambda(b_i b_j)
\operatorname{Im}Use in source · 1 passage
Search aliases: Im operatorname
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 167
The analytic-to-arithmetic equality U_Γ=U_alg is still unproved. In the later polarized presentation the remaining statement is Im(C_polU_Γ^3)=0, given the established modulus and independent branch conditions. The two-coordinate packet identifies where a primitive character can live, but no identification with the analytic gamma packet has been proved. In particular, no gerbe trivialization, Eisenstein deck descent, normalized scalar comparison, or new Lean compilation is supplied here.
Exact mathematical passage
\operatorname{Im}(C_{\mathrm{pol}}U_\Gamma^3)=0
JUse in source · 8 passages
Search aliases: J
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 13
Put ω^2+ω+1=0, J=2+3ω, bar J=-1-3ω. Then J+bar J=1, Jbar J=7. With
Exact mathematical passage
J=2+3\omega
jUse in source · 2 passages
Search aliases: j
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 25
The coefficient functional λ(x)=[β^2]x, applied after reduction via β^3=7, is equivalently Tr_K/ Q(x/(3β^2)). The Gram matrix H_ij=λ(b_i b_j) is
Exact mathematical passage
H_{ij}=\lambda(b_i b_j)
KUse in source · 2 passages
Search aliases: K
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 5
**Author:** Komorebi P. **Research status:** New exact organization and checks of finite algebra. The analytic comparison is open. **Working case:** K= Q(β), β^3=7, with the selected complex embedding.
Exact mathematical passage
K=\mathbb Q(\beta)
kUse in source · 14 passages
Search aliases: k
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 70
Let χ(1),…,χ(6)=(1,ω,ω^2,ω^2,ω,1) and G_r=∑_k=1^6χ(k)ζ_7^rk. Then for every r≠0,
Exact mathematical passage
G_r=\sum_{k=1}^{6}\chi(k)\zeta_7^{rk}
\kerUse in source · 2 passages
Search aliases: ker
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 38
Indeed u represents β^2+56≡β^2, v represents 28-β≡-β, and w represents 15≡1 in F_7[β]/(β^3). The codifferent pairing extracts the coefficient of β^2. It is perfect, and pairs the socle against the top quotient. In particular, u^ TH vanishes on ker N^2=span(u,v) and is nonzero on w.
Exact mathematical passage
\ker N^2=\operatorname{span}(u,v)
LUse in source · 1 passage
Search aliases: L
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 15
L= Z(β^2+2β)+ Z(-β-2)+15 Z
Exact mathematical passage
L=\mathbb Z(\beta^2+2\beta)+\mathbb Z(-\beta-2)+15\mathbb Z
MUse in source · 10 passages
Search aliases: M
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 21
M=0 · 1 · 0 -4 · 2 · 15 -1 · 0 · 4, det M=1.
Exact mathematical passage
M=\begin{pmatrix}0&1&0\\-4&2&15\\-1&0&4\end{pmatrix},\quad \det M=1.
M^\mathsfUse in source · 1 passage
Search aliases: M^ mathsf
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 66
There cannot be a positive definite real symmetric matrix H_+ with M^ TH_+=H_+M: that would make M similar to a real symmetric matrix, but χ_M(X)=X^3-6X^2+12X-1 has negative discriminant -1323 and two nonreal roots. Hence the Poincare duality description is safer than requiring an invariant ample polarization. Positive-definite lifts congruent to H modulo seven still give the same finite pairing, but do not preserve the global self-adjoint identity.
Exact mathematical passage
M^\mathsf TH_+=H_+M
M^{\mathsf T}Use in source · 1 passage
Search aliases: M^ T mathsf
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 27
H=4 · -4 · 15 -4 · 1 · 0 15 · 0 · 0, det H=-225, M^ TH=HM.
Exact mathematical passage
H=\begin{pmatrix}4&-4&15\\-4&1&0\\15&0&0\end{pmatrix},\qquad \det H=-225,\qquad M^{\mathsf T}H=HM.
M^{-1}Use in source · 1 passage
Search aliases: M^-1
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 52
e_7,H(MX,M^-1Y)=e_7,H(X,Y).
Exact mathematical passage
e_{7,H}(MX,M^{-1}Y)=e_{7,H}(X,Y).
NUse in source · 5 passages
Search aliases: N
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 31
Also (M-2I)^3=-7I. Thus over F_7, N=M-2I has one nilpotent block of length three. Set u=(1,2,4)^ T, v=(0,1,2)^ T, w=(0,0,1)^ T. Then Nu=0, Nv=u, Nw=v modulo seven, and
Exact mathematical passage
N=M-2I
N^2Use in source · 1 passage
Search aliases: N^2
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 38
Indeed u represents β^2+56≡β^2, v represents 28-β≡-β, and w represents 15≡1 in F_7[β]/(β^3). The codifferent pairing extracts the coefficient of β^2. It is perfect, and pairs the socle against the top quotient. In particular, u^ TH vanishes on ker N^2=span(u,v) and is nonzero on w.
Exact mathematical passage
\ker N^2=\operatorname{span}(u,v)
\nmidUse in source · 1 passage
Search aliases: nmid
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 50
The pairing is perfect since 7nmiddet H. Its source is the canonical pairing of A[7] with A^vee[7]. The symmetric map H:A→ A^vee need not define an ample polarization: it is indefinite. The exact matrix identity gives contragredient covariance
Exact mathematical passage
7\nmid\det H
\operatorname{ord}Use in source · 1 passage
Search aliases: ord operatorname
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 143
There is a concrete obstruction to replacing the pair (A,B) by a single character-power product. The prior divisor formula gives ord_τ F=4, and δ has order seven modulo Λ_5= Zτ+ Zσ+(1/5) Z. At z=τ-2δ, exactly the term f_2 has a zero, hence
Exact mathematical passage
\operatorname{ord}_\tau F=4
\operatorname{ord}_\tauUse in source · 1 passage
Search aliases: ord_τ operatorname tau
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 143
There is a concrete obstruction to replacing the pair (A,B) by a single character-power product. The prior divisor formula gives ord_τ F=4, and δ has order seven modulo Λ_5= Zτ+ Zσ+(1/5) Z. At z=τ-2δ, exactly the term f_2 has a zero, hence
Exact mathematical passage
\operatorname{ord}_\tau F=4
PUse in source · 8 passages
Search aliases: P
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 56
Define P=u/J, P'=u/bar J, and Q=w/bar J. The coefficient expansions 1/J=(-1-3ω)/7 and 1/bar J=(2+3ω)/7 yield
Exact mathematical passage
P=u/J
pUse in source · 2 passages
Search aliases: p
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 86
In the original analytic period row p=(τ,σ,1), let
Exact mathematical passage
p=(\tau,\sigma,1)
P'Use in source · 4 passages
Search aliases: P'
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 56
Define P=u/J, P'=u/bar J, and Q=w/bar J. The coefficient expansions 1/J=(-1-3ω)/7 and 1/bar J=(2+3ω)/7 yield
Exact mathematical passage
P'=u/\bar J
P_7Use in source · 1 passage
Search aliases: P_7
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 111
Let F(z)=Γ_ell(z;τ,σ)^5/Γ_ell(5z;5τ,5σ) and f_k(z)=F(z+kδ). The inherited principal-seven product is P_7(z)=∏_k=0^6 f_k(z), the trivial character sector.
Exact mathematical passage
P_7(z)=\prod_{k=0}^6 f_k(z)
\pmodUse in source · 3 passages
Search aliases: pmod
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 33
[u v w]^ TH[u v w]≡ 0 · 0 · 1 0 · 1 · 0 1 · 0 · 0pmod7.
Exact mathematical passage
[u\ v\ w]^{\mathsf T}H[u\ v\ w]\equiv \begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}\pmod7.
\mathrm{pol}Use in source · 1 passage
Search aliases: pol mathrm
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 167
The analytic-to-arithmetic equality U_Γ=U_alg is still unproved. In the later polarized presentation the remaining statement is Im(C_polU_Γ^3)=0, given the established modulus and independent branch conditions. The two-coordinate packet identifies where a primitive character can live, but no identification with the analytic gamma packet has been proved. In particular, no gerbe trivialization, Eisenstein deck descent, normalized scalar comparison, or new Lean compilation is supplied here.
Exact mathematical passage
\operatorname{Im}(C_{\mathrm{pol}}U_\Gamma^3)=0
QUse in source · 4 passages
Search aliases: Q
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 56
Define P=u/J, P'=u/bar J, and Q=w/bar J. The coefficient expansions 1/J=(-1-3ω)/7 and 1/bar J=(2+3ω)/7 yield
Exact mathematical passage
Q=w/\bar J
\mathbb QUse in source · 2 passages
Search aliases: Q mathbb
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 5
**Author:** Komorebi P. **Research status:** New exact organization and checks of finite algebra. The analytic comparison is open. **Working case:** K= Q(β), β^3=7, with the selected complex embedding.
Exact mathematical passage
K=\mathbb Q(\beta)
qUse in source · 3 passages
Search aliases: q
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 155
The inherited shift law is F(z+σ)/F(z)=q_t(z)^-1, where t=-τ and q_t(z)=θ_0(z;t)^5/θ_0(5z;5t). Therefore for k=1,…,6,
Exact mathematical passage
F(z+\sigma)/F(z)=q_t(z)^{-1}
q_tUse in source · 3 passages
Search aliases: q_t
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 155
The inherited shift law is F(z+σ)/F(z)=q_t(z)^-1, where t=-τ and q_t(z)=θ_0(z;t)^5/θ_0(5z;5t). Therefore for k=1,…,6,
Exact mathematical passage
F(z+\sigma)/F(z)=q_t(z)^{-1}
rUse in source · 6 passages
Search aliases: r
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 64
The nontrivial phase remains unchanged for Q=(w+su+tv)/bar J, since u pairs trivially with u,v modulo seven. Other nonzero top-quotient choices produce phases ζ_7^3r, r∈ F_7^×.
Exact mathematical passage
\zeta_7^{3r}
\operatorname{Res}Use in source · 1 passage
Search aliases: Res operatorname
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 145
Res_z=τ-2δD_χ(z)=4χ(2)=4ω.
Exact mathematical passage
\operatorname{Res}_{z=\tau-2\delta}D_\chi(z)=4\chi(2)=4\omega.
\operatorname{Res}_{z=\tau-2\delta}Use in source · 1 passage
Search aliases: Res_z=τ-2δ operatorname tau delta
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 145
Res_z=τ-2δD_χ(z)=4χ(2)=4ω.
Exact mathematical passage
\operatorname{Res}_{z=\tau-2\delta}D_\chi(z)=4\chi(2)=4\omega.
sUse in source · 1 passage
Search aliases: s
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 64
The nontrivial phase remains unchanged for Q=(w+su+tv)/bar J, since u pairs trivially with u,v modulo seven. Other nonzero top-quotient choices produce phases ζ_7^3r, r∈ F_7^×.
Exact mathematical passage
Q=(w+su+tv)/\bar J
\operatorname{span}Use in source · 1 passage
Search aliases: span operatorname
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 38
Indeed u represents β^2+56≡β^2, v represents 28-β≡-β, and w represents 15≡1 in F_7[β]/(β^3). The codifferent pairing extracts the coefficient of β^2. It is perfect, and pairs the socle against the top quotient. In particular, u^ TH vanishes on ker N^2=span(u,v) and is nonzero on w.
Exact mathematical passage
\ker N^2=\operatorname{span}(u,v)
TUse in source · 3 passages
Search aliases: T
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 66
There cannot be a positive definite real symmetric matrix H_+ with M^ TH_+=H_+M: that would make M similar to a real symmetric matrix, but χ_M(X)=X^3-6X^2+12X-1 has negative discriminant -1323 and two nonreal roots. Hence the Poincare duality description is safer than requiring an invariant ample polarization. Positive-definite lifts congruent to H modulo seven still give the same finite pairing, but do not preserve the global self-adjoint identity.
Exact mathematical passage
M^\mathsf TH_+=H_+M
\mathsf TUse in source · 7 passages
Search aliases: T mathsf
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 27
H=4 · -4 · 15 -4 · 1 · 0 15 · 0 · 0, det H=-225, M^ TH=HM.
Exact mathematical passage
H=\begin{pmatrix}4&-4&15\\-4&1&0\\15&0&0\end{pmatrix},\qquad \det H=-225,\qquad M^{\mathsf T}H=HM.
tUse in source · 5 passages
Search aliases: t
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 64
The nontrivial phase remains unchanged for Q=(w+su+tv)/bar J, since u pairs trivially with u,v modulo seven. Other nonzero top-quotient choices produce phases ζ_7^3r, r∈ F_7^×.
Exact mathematical passage
Q=(w+su+tv)/\bar J
T^3Use in source · 1 passage
Search aliases: T^3
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 136
(A,B)⟼(B/A,1/A), T=-1 · 1 -1 · 0, T^3=I.
Exact mathematical passage
(A,B)\longmapsto(B/A,1/A),\qquad T=\begin{pmatrix}-1&1\\-1&0\end{pmatrix},\quad T^3=I.
\operatorname{Tr}Use in source · 1 passage
Search aliases: Tr operatorname
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 25
The coefficient functional λ(x)=[β^2]x, applied after reduction via β^3=7, is equivalently Tr_K/ Q(x/(3β^2)). The Gram matrix H_ij=λ(b_i b_j) is
Exact mathematical passage
\operatorname{Tr}_{K/\mathbb Q}(x/(3\beta^2))
\operatorname{Tr}_{K/\mathbb Q}Use in source · 1 passage
Search aliases: Tr_K/ Q operatorname mathbb
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 25
The coefficient functional λ(x)=[β^2]x, applied after reduction via β^3=7, is equivalently Tr_K/ Q(x/(3β^2)). The Gram matrix H_ij=λ(b_i b_j) is
Exact mathematical passage
\operatorname{Tr}_{K/\mathbb Q}(x/(3\beta^2))
UUse in source · 2 passages
Search aliases: U
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 167
The analytic-to-arithmetic equality U_Γ=U_alg is still unproved. In the later polarized presentation the remaining statement is Im(C_polU_Γ^3)=0, given the established modulus and independent branch conditions. The two-coordinate packet identifies where a primitive character can live, but no identification with the analytic gamma packet has been proved. In particular, no gerbe trivialization, Eisenstein deck descent, normalized scalar comparison, or new Lean compilation is supplied here.
Exact mathematical passage
U_\Gamma=U_{\mathrm{alg}}
uUse in source · 15 passages
Search aliases: u
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 31
Also (M-2I)^3=-7I. Thus over F_7, N=M-2I has one nilpotent block of length three. Set u=(1,2,4)^ T, v=(0,1,2)^ T, w=(0,0,1)^ T. Then Nu=0, Nv=u, Nw=v modulo seven, and
Exact mathematical passage
u=(1,2,4)^{\mathsf T}
u^{\mathsf T}Use in source · 1 passage
Search aliases: u^ T mathsf
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 38
Indeed u represents β^2+56≡β^2, v represents 28-β≡-β, and w represents 15≡1 in F_7[β]/(β^3). The codifferent pairing extracts the coefficient of β^2. It is perfect, and pairs the socle against the top quotient. In particular, u^ TH vanishes on ker N^2=span(u,v) and is nonzero on w.
Exact mathematical passage
u^{\mathsf T}H
U_{\mathrm{alg}}Use in source · 1 passage
Search aliases: U_alg mathrm
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 167
The analytic-to-arithmetic equality U_Γ=U_alg is still unproved. In the later polarized presentation the remaining statement is Im(C_polU_Γ^3)=0, given the established modulus and independent branch conditions. The two-coordinate packet identifies where a primitive character can live, but no identification with the analytic gamma packet has been proved. In particular, no gerbe trivialization, Eisenstein deck descent, normalized scalar comparison, or new Lean compilation is supplied here.
Exact mathematical passage
U_\Gamma=U_{\mathrm{alg}}
U_\GammaUse in source · 1 passage
Search aliases: U_Γ Gamma
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 167
The analytic-to-arithmetic equality U_Γ=U_alg is still unproved. In the later polarized presentation the remaining statement is Im(C_polU_Γ^3)=0, given the established modulus and independent branch conditions. The two-coordinate packet identifies where a primitive character can live, but no identification with the analytic gamma packet has been proved. In particular, no gerbe trivialization, Eisenstein deck descent, normalized scalar comparison, or new Lean compilation is supplied here.
Exact mathematical passage
U_\Gamma=U_{\mathrm{alg}}
U_\Gamma^3Use in source · 1 passage
Search aliases: U_Γ^3 Gamma
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 167
The analytic-to-arithmetic equality U_Γ=U_alg is still unproved. In the later polarized presentation the remaining statement is Im(C_polU_Γ^3)=0, given the established modulus and independent branch conditions. The two-coordinate packet identifies where a primitive character can live, but no identification with the analytic gamma packet has been proved. In particular, no gerbe trivialization, Eisenstein deck descent, normalized scalar comparison, or new Lean compilation is supplied here.
Exact mathematical passage
\operatorname{Im}(C_{\mathrm{pol}}U_\Gamma^3)=0
vUse in source · 9 passages
Search aliases: v
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 31
Also (M-2I)^3=-7I. Thus over F_7, N=M-2I has one nilpotent block of length three. Set u=(1,2,4)^ T, v=(0,1,2)^ T, w=(0,0,1)^ T. Then Nu=0, Nv=u, Nw=v modulo seven, and
Exact mathematical passage
v=(0,1,2)^{\mathsf T}
\veeUse in source · 2 passages
Search aliases: vee
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 50
The pairing is perfect since 7nmiddet H. Its source is the canonical pairing of A[7] with A^vee[7]. The symmetric map H:A→ A^vee need not define an ample polarization: it is indefinite. The exact matrix identity gives contragredient covariance
Exact mathematical passage
A^\vee[7]
wUse in source · 8 passages
Search aliases: w
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 31
Also (M-2I)^3=-7I. Thus over F_7, N=M-2I has one nilpotent block of length three. Set u=(1,2,4)^ T, v=(0,1,2)^ T, w=(0,0,1)^ T. Then Nu=0, Nv=u, Nw=v modulo seven, and
Exact mathematical passage
w=(0,0,1)^{\mathsf T}
XUse in source · 4 passages
Search aliases: X
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 44
On A=E_ω^3, E_ω= C/ Z[ω], use the product principal Weil pairing composed with the integral symmetric isogeny represented by H. For seventh-torsion coordinates X=(a+ω b)/7, Y=(c+ω d)/7, write
Exact mathematical passage
X=(a+\omega b)/7
xUse in source · 2 passages
Search aliases: x
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 25
The coefficient functional λ(x)=[β^2]x, applied after reduction via β^3=7, is equivalently Tr_K/ Q(x/(3β^2)). The Gram matrix H_ij=λ(b_i b_j) is
Exact mathematical passage
\lambda(x)=[\beta^2]x
X^2Use in source · 1 passage
Search aliases: X^2
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 66
There cannot be a positive definite real symmetric matrix H_+ with M^ TH_+=H_+M: that would make M similar to a real symmetric matrix, but χ_M(X)=X^3-6X^2+12X-1 has negative discriminant -1323 and two nonreal roots. Hence the Poincare duality description is safer than requiring an invariant ample polarization. Positive-definite lifts congruent to H modulo seven still give the same finite pairing, but do not preserve the global self-adjoint identity.
Exact mathematical passage
\chi_M(X)=X^3-6X^2+12X-1
X^3Use in source · 1 passage
Search aliases: X^3
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 66
There cannot be a positive definite real symmetric matrix H_+ with M^ TH_+=H_+M: that would make M similar to a real symmetric matrix, but χ_M(X)=X^3-6X^2+12X-1 has negative discriminant -1323 and two nonreal roots. Hence the Poincare duality description is safer than requiring an invariant ample polarization. Positive-definite lifts congruent to H modulo seven still give the same finite pairing, but do not preserve the global self-adjoint identity.
Exact mathematical passage
\chi_M(X)=X^3-6X^2+12X-1
YUse in source · 5 passages
Search aliases: Y
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 44
On A=E_ω^3, E_ω= C/ Z[ω], use the product principal Weil pairing composed with the integral symmetric isogeny represented by H. For seventh-torsion coordinates X=(a+ω b)/7, Y=(c+ω d)/7, write
Exact mathematical passage
Y=(c+\omega d)/7
\mathbb ZUse in source · 6 passages
Search aliases: Z mathbb
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 15
L= Z(β^2+2β)+ Z(-β-2)+15 Z
Exact mathematical passage
L=\mathbb Z(\beta^2+2\beta)+\mathbb Z(-\beta-2)+15\mathbb Z
zUse in source · 13 passages
Search aliases: z
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 76
The first identity follows by index substitution and the second by the classical Gauss and Jacobi identities. All six cubes were also checked by exact coefficient convolution in Z[ω,z]/(ω^2+ω+1,z^7-1). Thus (G_r/β)^3=J, independent of the chosen nonzero dual direction. The individual cube-root branch is not fixed by this statement.
Exact mathematical passage
\mathbb Z[\omega,z]/(\omega^2+\omega+1,z^7-1)
z^7Use in source · 1 passage
Search aliases: z^7
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 76
The first identity follows by index substitution and the second by the classical Gauss and Jacobi identities. All six cubes were also checked by exact coefficient convolution in Z[ω,z]/(ω^2+ω+1,z^7-1). Thus (G_r/β)^3=J, independent of the chosen nonzero dual direction. The individual cube-root branch is not fixed by this statement.
Exact mathematical passage
\mathbb Z[\omega,z]/(\omega^2+\omega+1,z^7-1)
\timesUse in source · 1 passage
Search aliases: × times
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 64
The nontrivial phase remains unchanged for Q=(w+su+tv)/bar J, since u pairs trivially with u,v modulo seven. Other nonzero top-quotient choices produce phases ζ_7^3r, r∈ F_7^×.
Exact mathematical passage
r\in\mathbb F_7^\times
\betaUse in source · 17 passages
Search aliases: β beta
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 5
**Author:** Komorebi P. **Research status:** New exact organization and checks of finite algebra. The analytic comparison is open. **Working case:** K= Q(β), β^3=7, with the selected complex embedding.
Exact mathematical passage
K=\mathbb Q(\beta)
\beta^2Use in source · 7 passages
Search aliases: β^2 beta
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 15
L= Z(β^2+2β)+ Z(-β-2)+15 Z
Exact mathematical passage
L=\mathbb Z(\beta^2+2\beta)+\mathbb Z(-\beta-2)+15\mathbb Z
\beta^3Use in source · 3 passages
Search aliases: β^3 beta
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 5
**Author:** Komorebi P. **Research status:** New exact organization and checks of finite algebra. The analytic comparison is open. **Working case:** K= Q(β), β^3=7, with the selected complex embedding.
Exact mathematical passage
\beta^3=7
\GammaUse in source · 3 passages
Search aliases: Γ Gamma
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 111
Let F(z)=Γ_ell(z;τ,σ)^5/Γ_ell(5z;5τ,5σ) and f_k(z)=F(z+kδ). The inherited principal-seven product is P_7(z)=∏_k=0^6 f_k(z), the trivial character sector.
Exact mathematical passage
F(z)=\Gamma_{\mathrm{ell}}(z;\tau,\sigma)^5/\Gamma_{\mathrm{ell}}(5z;5\tau,5\sigma)
\Gamma_{\mathrm{ell}}Use in source · 1 passage
Search aliases: Γ_ell Gamma mathrm
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 111
Let F(z)=Γ_ell(z;τ,σ)^5/Γ_ell(5z;5τ,5σ) and f_k(z)=F(z+kδ). The inherited principal-seven product is P_7(z)=∏_k=0^6 f_k(z), the trivial character sector.
Exact mathematical passage
F(z)=\Gamma_{\mathrm{ell}}(z;\tau,\sigma)^5/\Gamma_{\mathrm{ell}}(5z;5\tau,5\sigma)
\deltaUse in source · 11 passages
Search aliases: δ delta
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 88
τ=fracβ^2+2β15, σ=-fracβ+215, δ=fracτ+2σ+47=fracβ^2+56105.
Exact mathematical passage
\tau=\frac{\beta^2+2\beta}{15},\quad\sigma=-\frac{\beta+2}{15},\quad \delta=\frac{\tau+2\sigma+4}{7}=\frac{\beta^2+56}{105}.
\epsilonUse in source · 4 passages
Search aliases: ϵ epsilon
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 19
and basis b_0=β^2+2β, b_1=-β-2, b_2=15, multiplication by ϵ=2-β has matrix
Exact mathematical passage
\epsilon=2-\beta
\zetaUse in source · 5 passages
Search aliases: ζ zeta
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 46
e_7,H(X,Y)=ζ_7^a^ THd-b^ THc.
Exact mathematical passage
e_{7,H}(X,Y)=\zeta_7^{a^{\mathsf T}Hd-b^{\mathsf T}Hc}.
\zeta_7^3Use in source · 1 passage
Search aliases: ζ_7^3 zeta
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 58
e_7,H(P,Q)=ζ_7^3, e_7,H(P,P')=1.
Exact mathematical passage
e_{7,H}(P,Q)=\zeta_7^3,\qquad e_{7,H}(P,P')=1.
\zeta_7^{3k}Use in source · 1 passage
Search aliases: ζ_7^3k zeta
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 78
Equivalently, the primitive multiplicative-character projector applied to the additive function f(k)=e_7,H(kP,Q)=ζ_7^3k produces G_3/6:
Exact mathematical passage
f(k)=e_{7,H}(kP,Q)=\zeta_7^{3k}
\zeta_7^{3r}Use in source · 1 passage
Search aliases: ζ_7^3r zeta
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 64
The nontrivial phase remains unchanged for Q=(w+su+tv)/bar J, since u pairs trivially with u,v modulo seven. Other nonzero top-quotient choices produce phases ζ_7^3r, r∈ F_7^×.
Exact mathematical passage
\zeta_7^{3r}
\zeta_7^{a^{\mathsf T}Hd-b^{\mathsf T}Hc}Use in source · 1 passage
Search aliases: ζ_7^a^ THd-b^ THc zeta mathsf mathsf
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 46
e_7,H(X,Y)=ζ_7^a^ THd-b^ THc.
Exact mathematical passage
e_{7,H}(X,Y)=\zeta_7^{a^{\mathsf T}Hd-b^{\mathsf T}Hc}.
\zeta_7^{rk}Use in source · 1 passage
Search aliases: ζ_7^rk zeta
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 70
Let χ(1),…,χ(6)=(1,ω,ω^2,ω^2,ω,1) and G_r=∑_k=1^6χ(k)ζ_7^rk. Then for every r≠0,
Exact mathematical passage
G_r=\sum_{k=1}^{6}\chi(k)\zeta_7^{rk}
\thetaUse in source · 1 passage
Search aliases: θ theta
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 155
The inherited shift law is F(z+σ)/F(z)=q_t(z)^-1, where t=-τ and q_t(z)=θ_0(z;t)^5/θ_0(5z;5t). Therefore for k=1,…,6,
Exact mathematical passage
q_t(z)=\theta_0(z;t)^5/\theta_0(5z;5t)
\theta_0Use in source · 1 passage
Search aliases: θ_0 theta
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 155
The inherited shift law is F(z+σ)/F(z)=q_t(z)^-1, where t=-τ and q_t(z)=θ_0(z;t)^5/θ_0(5z;5t). Therefore for k=1,…,6,
Exact mathematical passage
q_t(z)=\theta_0(z;t)^5/\theta_0(5z;5t)
\LambdaUse in source · 1 passage
Search aliases: Λ Lambda
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 143
There is a concrete obstruction to replacing the pair (A,B) by a single character-power product. The prior divisor formula gives ord_τ F=4, and δ has order seven modulo Λ_5= Zτ+ Zσ+(1/5) Z. At z=τ-2δ, exactly the term f_2 has a zero, hence
Exact mathematical passage
\Lambda_5=\mathbb Z\tau+\mathbb Z\sigma+(1/5)\mathbb Z
\lambdaUse in source · 2 passages
Search aliases: λ lambda
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 25
The coefficient functional λ(x)=[β^2]x, applied after reduction via β^3=7, is equivalently Tr_K/ Q(x/(3β^2)). The Gram matrix H_ij=λ(b_i b_j) is
Exact mathematical passage
\lambda(x)=[\beta^2]x
\Lambda_5Use in source · 1 passage
Search aliases: Λ_5 Lambda
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 143
There is a concrete obstruction to replacing the pair (A,B) by a single character-power product. The prior divisor formula gives ord_τ F=4, and δ has order seven modulo Λ_5= Zτ+ Zσ+(1/5) Z. At z=τ-2δ, exactly the term f_2 has a zero, hence
Exact mathematical passage
\Lambda_5=\mathbb Z\tau+\mathbb Z\sigma+(1/5)\mathbb Z
\PiUse in source · 1 passage
Search aliases: Π Pi
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 80
(Π_χ^-1f)(1)=frac16∑_k≠0χ(k)f(k)=fracG_36.
Exact mathematical passage
(\Pi_{\chi^{-1}}f)(1)=\frac16\sum_{k\ne0}\chi(k)f(k)=\frac{G_3}{6}.
\piUse in source · 5 passages
Search aliases: π pi
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 93
The period projection π(v)=pv on the universal cover satisfies π(u/7)=δ, π(w)=1, and
Exact mathematical passage
\pi(v)=pv
\Pi_{\chi^{-1}}Use in source · 1 passage
Search aliases: Π_χ^-1 Pi chi
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 80
(Π_χ^-1f)(1)=frac16∑_k≠0χ(k)f(k)=fracG_36.
Exact mathematical passage
(\Pi_{\chi^{-1}}f)(1)=\frac16\sum_{k\ne0}\chi(k)f(k)=\frac{G_3}{6}.
\sigmaUse in source · 8 passages
Search aliases: σ sigma
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 86
In the original analytic period row p=(τ,σ,1), let
Exact mathematical passage
p=(\tau,\sigma,1)
\tauUse in source · 8 passages
Search aliases: τ tau
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 86
In the original analytic period row p=(τ,σ,1), let
Exact mathematical passage
p=(\tau,\sigma,1)
\chiUse in source · 10 passages
Search aliases: χ chi
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 66
There cannot be a positive definite real symmetric matrix H_+ with M^ TH_+=H_+M: that would make M similar to a real symmetric matrix, but χ_M(X)=X^3-6X^2+12X-1 has negative discriminant -1323 and two nonreal roots. Hence the Poincare duality description is safer than requiring an invariant ample polarization. Positive-definite lifts congruent to H modulo seven still give the same finite pairing, but do not preserve the global self-adjoint identity.
Exact mathematical passage
\chi_M(X)=X^3-6X^2+12X-1
\chi^{-1}Use in source · 1 passage
Search aliases: χ^-1 chi
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 80
(Π_χ^-1f)(1)=frac16∑_k≠0χ(k)f(k)=fracG_36.
Exact mathematical passage
(\Pi_{\chi^{-1}}f)(1)=\frac16\sum_{k\ne0}\chi(k)f(k)=\frac{G_3}{6}.
\chi_MUse in source · 1 passage
Search aliases: χ_M chi
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 66
There cannot be a positive definite real symmetric matrix H_+ with M^ TH_+=H_+M: that would make M similar to a real symmetric matrix, but χ_M(X)=X^3-6X^2+12X-1 has negative discriminant -1323 and two nonreal roots. Hence the Poincare duality description is safer than requiring an invariant ample polarization. Positive-definite lifts congruent to H modulo seven still give the same finite pairing, but do not preserve the global self-adjoint identity.
Exact mathematical passage
\chi_M(X)=X^3-6X^2+12X-1
\omegaUse in source · 18 passages
Search aliases: ω omega
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 13
Put ω^2+ω+1=0, J=2+3ω, bar J=-1-3ω. Then J+bar J=1, Jbar J=7. With
Exact mathematical passage
\omega^2+\omega+1=0
\omega^2Use in source · 5 passages
Search aliases: ω^2 omega
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 13
Put ω^2+ω+1=0, J=2+3ω, bar J=-1-3ω. Then J+bar J=1, Jbar J=7. With
Exact mathematical passage
\omega^2+\omega+1=0
\ldotsUse in source · 3 passages
Search aliases: … ldots
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 70
Let χ(1),…,χ(6)=(1,ω,ω^2,ω^2,ω,1) and G_r=∑_k=1^6χ(k)ζ_7^rk. Then for every r≠0,
Exact mathematical passage
\chi(1),\ldots,\chi(6)=(1,\omega,\omega^2,\omega^2,\omega,1)
\toUse in source · 1 passage
Search aliases: → to
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 50
The pairing is perfect since 7nmiddet H. Its source is the canonical pairing of A[7] with A^vee[7]. The symmetric map H:A→ A^vee need not define an ample polarization: it is indefinite. The exact matrix identity gives contragredient covariance
Exact mathematical passage
H:A\to A^\vee
\mapstoUse in source · 4 passages
Search aliases: ↦ mapsto
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 107
Hence unit multiplication acts on the seven cover labels by k↦2kpmod7, with explicit period shifts. The inverse unit acts by k↦4k. These are precisely the finite orbits present in the earlier divisor-transport calculations.
Exact mathematical passage
k\mapsto2k\pmod7
\inUse in source · 1 passage
Search aliases: ∈ in
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 64
The nontrivial phase remains unchanged for Q=(w+su+tv)/bar J, since u pairs trivially with u,v modulo seven. Other nonzero top-quotient choices produce phases ζ_7^3r, r∈ F_7^×.
Exact mathematical passage
r\in\mathbb F_7^\times
\notinUse in source · 1 passage
Search aliases: ∉ notin
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 149
A logarithmic derivative of a single-valued meromorphic function has an integer residue. Since 4ω∉ Z, **no single meromorphic scalar** has logarithmic derivative D_χ. The two-component meromorphic realization avoids this obstruction and carries an integral order-three monomial action.
Exact mathematical passage
4\omega\notin\mathbb Z
\prodUse in source · 2 passages
Search aliases: ∏ prod
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 111
Let F(z)=Γ_ell(z;τ,σ)^5/Γ_ell(5z;5τ,5σ) and f_k(z)=F(z+kδ). The inherited principal-seven product is P_7(z)=∏_k=0^6 f_k(z), the trivial character sector.
Exact mathematical passage
P_7(z)=\prod_{k=0}^6 f_k(z)
\prod_{j=0}^{8k-1}Use in source · 1 passage
Search aliases: ∏_j=0^8k-1 prod
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 157
F(ϵ(z+kδ))= F(ϵ z+2kδ) ∏_j=0^8k-1q_t(ϵ z+2kδ+jσ)^-1.
Exact mathematical passage
F(\epsilon(z+k\delta))= F(\epsilon z+2k\delta) \prod_{j=0}^{8k-1}q_t(\epsilon z+2k\delta+j\sigma)^{-1}.
\prod_{k=0}^6Use in source · 1 passage
Search aliases: ∏_k=0^6 prod
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 111
Let F(z)=Γ_ell(z;τ,σ)^5/Γ_ell(5z;5τ,5σ) and f_k(z)=F(z+kδ). The inherited principal-seven product is P_7(z)=∏_k=0^6 f_k(z), the trivial character sector.
Exact mathematical passage
P_7(z)=\prod_{k=0}^6 f_k(z)
\sumUse in source · 3 passages
Search aliases: ∑ sum
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 70
Let χ(1),…,χ(6)=(1,ω,ω^2,ω^2,ω,1) and G_r=∑_k=1^6χ(k)ζ_7^rk. Then for every r≠0,
Exact mathematical passage
G_r=\sum_{k=1}^{6}\chi(k)\zeta_7^{rk}
\sum_{k=1}^6Use in source · 1 passage
Search aliases: ∑_k=1^6 sum
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 129
D_χ(z)=∑_k=1^6χ(k)fracf'_k(z)f_k(z) =fracA'(z)A(z)+ωfracB'(z)B(z)
Exact mathematical passage
D_\chi(z)=\sum_{k=1}^6\chi(k)\frac{f'_k(z)}{f_k(z)} =\frac{A'(z)}{A(z)}+\omega\frac{B'(z)}{B(z)}
\sum_{k=1}^{6}Use in source · 1 passage
Search aliases: ∑_k=1^6 sum
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 70
Let χ(1),…,χ(6)=(1,ω,ω^2,ω^2,ω,1) and G_r=∑_k=1^6χ(k)ζ_7^rk. Then for every r≠0,
Exact mathematical passage
G_r=\sum_{k=1}^{6}\chi(k)\zeta_7^{rk}
\sum_{k\ne0}Use in source · 1 passage
Search aliases: ∑_k≠0 sum ne
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 80
(Π_χ^-1f)(1)=frac16∑_k≠0χ(k)f(k)=fracG_36.
Exact mathematical passage
(\Pi_{\chi^{-1}}f)(1)=\frac16\sum_{k\ne0}\chi(k)f(k)=\frac{G_3}{6}.
\neUse in source · 2 passages
Search aliases: ≠ ne
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 70
Let χ(1),…,χ(6)=(1,ω,ω^2,ω^2,ω,1) and G_r=∑_k=1^6χ(k)ζ_7^rk. Then for every r≠0,
Exact mathematical passage
r\ne0
\equivUse in source · 4 passages
Search aliases: ≡ equiv
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 33
[u v w]^ TH[u v w]≡ 0 · 0 · 1 0 · 1 · 0 1 · 0 · 0pmod7.
Exact mathematical passage
[u\ v\ w]^{\mathsf T}H[u\ v\ w]\equiv \begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}\pmod7.
\longmapstoUse in source · 1 passage
Search aliases: ⟼ longmapsto
Use in source · H12_Ambient_Dual_Slice_Research_Note.md, line 136
(A,B)⟼(B/A,1/A), T=-1 · 1 -1 · 0, T^3=I.
Exact mathematical passage
(A,B)\longmapsto(B/A,1/A),\qquad T=\begin{pmatrix}-1&1\\-1&0\end{pmatrix},\quad T^3=I.
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