Notation by work

Hilbert 12: contracting ambient torsion

All notation · Glossary · Source coverage

196 symbol spellings in 133 mathematical passages across 3 selected source files. Subscripts and superscripts have their own entries alongside their base symbols.

The same symbol can have several meanings here. A passage classified as a definition may define another part of its expression. Read the linked context before assigning a meaning.

All 196 entries are available below.

+Use in source · 31 passages

Search aliases: +

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 15
ω^2+ω+1=0, β^3=7, ε=2-β, ρ=ε^-1=β^2+2β+4.
Exact mathematical passage
```
\omega^2+\omega+1=0,\quad \beta^3=7,\quad
\varepsilon=2-\beta,\quad \rho=\varepsilon^{-1}=\beta^2+2\beta+4.
```

=Use in source · 77 passages

Search aliases: =

Link to this entry ·

Use in source · H12_Ambient_Torsion_README.md, line 5
This research package continues the pure cubic experiment K= Q(β), β^3=7.
Exact mathematical passage
```
K=\mathbb Q(\beta)
```

AUse in source · 3 passages

Search aliases: A

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 155
A_0=4k+5ℓ+j, B_0=4ℓ+5j, C_0=4j,
Exact mathematical passage
```
A_0=4k+5\ell+j,\qquad B_0=4\ell+5j,\qquad C_0=4j,
```

aUse in source · 9 passages

Search aliases: a

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 45
q_a(z)=fracθ_0(z;a)^5θ_0(5z;5a).
Exact mathematical passage
```
q_a(z)=\frac{\theta_0(z;a)^5}{\theta_0(5z;5a)}.
```

A_0Use in source · 3 passages

Search aliases: A_0

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 155
A_0=4k+5ℓ+j, B_0=4ℓ+5j, C_0=4j,
Exact mathematical passage
```
A_0=4k+5\ell+j,\qquad B_0=4\ell+5j,\qquad C_0=4j,
```

a_0Use in source · 1 passage

Search aliases: a_0

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 385
The covector a_0=(1,0,0)^T, corresponding to the oriented complementary pairing with w, has an orbit of length **21** under this matrix. Its first three iterates are
Exact mathematical passage
```
a_0=(1,0,0)^T
```

a_xUse in source · 1 passage

Search aliases: a_x

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 208
Let d_x=(a_x,b_x,c_x)^T be the integer carry vector from Section 3. At stage m, the carry is d_x,m=R^m-1d_x∈ Z^3. Set W_x,m=ρ^m-1z_T x. By the arithmetic transport formula,
Exact mathematical passage
```
d_x=(a_x,b_x,c_x)^T
```

a_{x,m}Use in source · 2 passages

Search aliases: a_x,m

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 210
ρ^m z_x=W_x,m+a_x,mτ+b_x,mσ+c_x,m.
Exact mathematical passage
```
\rho^m z_x=W_{x,m}+a_{x,m}\tau+b_{x,m}\sigma+c_{x,m}.
```

BUse in source · 3 passages

Search aliases: B

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 155
A_0=4k+5ℓ+j, B_0=4ℓ+5j, C_0=4j,
Exact mathematical passage
```
A_0=4k+5\ell+j,\qquad B_0=4\ell+5j,\qquad C_0=4j,
```

bUse in source · 6 passages

Search aliases: b

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 169
ρ z_k,ℓ,j= z_k',ℓ',j'+aτ+bσ+c
Exact mathematical passage
```
\boxed{
\rho z_{k,\ell,j}=
 z_{k',\ell',j'}+a\tau+b\sigma+c
}
```

B_0Use in source · 3 passages

Search aliases: B_0

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 155
A_0=4k+5ℓ+j, B_0=4ℓ+5j, C_0=4j,
Exact mathematical passage
```
A_0=4k+5\ell+j,\qquad B_0=4\ell+5j,\qquad C_0=4j,
```

b_xUse in source · 1 passage

Search aliases: b_x

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 208
Let d_x=(a_x,b_x,c_x)^T be the integer carry vector from Section 3. At stage m, the carry is d_x,m=R^m-1d_x∈ Z^3. Set W_x,m=ρ^m-1z_T x. By the arithmetic transport formula,
Exact mathematical passage
```
d_x=(a_x,b_x,c_x)^T
```

b_{x,m}Use in source · 2 passages

Search aliases: b_x,m

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 210
ρ^m z_x=W_x,m+a_x,mτ+b_x,mσ+c_x,m.
Exact mathematical passage
```
\rho^m z_x=W_{x,m}+a_{x,m}\tau+b_{x,m}\sigma+c_{x,m}.
```

\bmodUse in source · 2 passages

Search aliases: bmod

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 167
Write (k',ℓ',j')=(A_0,B_0,C_0)bmod7. Then
Exact mathematical passage
```
(k',\ell',j')=(A_0,B_0,C_0)\bmod7
```

CUse in source · 9 passages

Search aliases: C

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 155
A_0=4k+5ℓ+j, B_0=4ℓ+5j, C_0=4j,
Exact mathematical passage
```
A_0=4k+5\ell+j,\qquad B_0=4\ell+5j,\qquad C_0=4j,
```

cUse in source · 4 passages

Search aliases: c

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 169
ρ z_k,ℓ,j= z_k',ℓ',j'+aτ+bσ+c
Exact mathematical passage
```
\boxed{
\rho z_{k,\ell,j}=
 z_{k',\ell',j'}+a\tau+b\sigma+c
}
```

C_0Use in source · 3 passages

Search aliases: C_0

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 155
A_0=4k+5ℓ+j, B_0=4ℓ+5j, C_0=4j,
Exact mathematical passage
```
A_0=4k+5\ell+j,\qquad B_0=4\ell+5j,\qquad C_0=4j,
```

C_mUse in source · 1 passage

Search aliases: C_m

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 364
f_m(x)=C_m(x)f_m-1(Tx), x∈ V.
Exact mathematical passage
```
f_m(x)=C_m(x)f_{m-1}(Tx),\qquad x\in V.
```

C_{\mathrm{pol}}Use in source · 1 passage

Search aliases: C_pol mathrm

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 348
Im(C_polU_Γ^3)=0
Exact mathematical passage
```
\operatorname{Im}(C_{\mathrm{pol}}U_\Gamma^3)=0
```

C_{T^{-m}x}^{(m)}Use in source · 1 passage

Search aliases: C_T^-mx^(m)

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 246
F(z_x)=ε^2 ∏_m=1^∞ (C_T^-mx^(m))^-1.
Exact mathematical passage
```
\boxed{
F(z_x)=\varepsilon^2
\prod_{m=1}^{\infty}
\left(C_{T^{-m}x}^{(m)}\right)^{-1}.
}
```

c_xUse in source · 1 passage

Search aliases: c_x

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 208
Let d_x=(a_x,b_x,c_x)^T be the integer carry vector from Section 3. At stage m, the carry is d_x,m=R^m-1d_x∈ Z^3. Set W_x,m=ρ^m-1z_T x. By the arithmetic transport formula,
Exact mathematical passage
```
d_x=(a_x,b_x,c_x)^T
```

c_{x,m}Use in source · 1 passage

Search aliases: c_x,m

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 210
ρ^m z_x=W_x,m+a_x,mτ+b_x,mσ+c_x,m.
Exact mathematical passage
```
\rho^m z_x=W_{x,m}+a_{x,m}\tau+b_{x,m}\sigma+c_{x,m}.
```

C_x^{(m)}Use in source · 3 passages

Search aliases: C_x^(m)

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 227
C_x^(m)= P(q_t^-1;b_x,m,W_x,m,σ) P(q_σ;a_x,m,W_x,m+b_x,mσ,τ).
Exact mathematical passage
```
\boxed{
C_x^{(m)}=
\mathcal P(q_t^{-1};b_{x,m},W_{x,m},\sigma)\,
\mathcal P(q_\sigma;a_{x,m},W_{x,m}+b_{x,m}\sigma,\tau).
}
```

dUse in source · 2 passages

Search aliases: d

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 208
Let d_x=(a_x,b_x,c_x)^T be the integer carry vector from Section 3. At stage m, the carry is d_x,m=R^m-1d_x∈ Z^3. Set W_x,m=ρ^m-1z_T x. By the arithmetic transport formula,
Exact mathematical passage
```
d_x=(a_x,b_x,c_x)^T
```

d_xUse in source · 2 passages

Search aliases: d_x

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 208
Let d_x=(a_x,b_x,c_x)^T be the integer carry vector from Section 3. At stage m, the carry is d_x,m=R^m-1d_x∈ Z^3. Set W_x,m=ρ^m-1z_T x. By the arithmetic transport formula,
Exact mathematical passage
```
d_x=(a_x,b_x,c_x)^T
```

d_{x,m}Use in source · 1 passage

Search aliases: d_x,m

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 208
Let d_x=(a_x,b_x,c_x)^T be the integer carry vector from Section 3. At stage m, the carry is d_x,m=R^m-1d_x∈ Z^3. Set W_x,m=ρ^m-1z_T x. By the arithmetic transport formula,
Exact mathematical passage
```
d_{x,m}=R^{m-1}d_x\in\mathbb Z^3
```

\detUse in source · 1 passage

Search aliases: det

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 124
Since δ,η,ξ are the images of u/7,v/7,w/7, and det[u v w]=1, the 343 labels represent all of tfrac17Λ/Λ. Here this is torsion of the rank-three lattice quotient, not a claim that the nondiscrete rank-three quotient is an ordinary elliptic curve.
Exact mathematical passage
```
\det[u\ v\ w]=1
```

eUse in source · 3 passages

Search aliases: e

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 262
L_t,σ(z)=-5∑_substackm≥1 5nmid m fracP^m e^2π imz-Q^m e^-2π imz m(1-P^m)(1-Q^m), P=e^2π it, Q=e^2π iσ.
Exact mathematical passage
```
L_{t,\sigma}(z)=-5\sum_{\substack{m\ge1\\5\nmid m}}
\frac{P^m e^{2\pi imz}-Q^m e^{-2\pi imz}}
{m(1-P^m)(1-Q^m)},\qquad
P=e^{2\pi it},\ Q=e^{2\pi i\sigma}.
```

e^{-2\pi i nq z}Use in source · 1 passage

Search aliases: e^-2π i nq z pi

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 297
-5∑_substackq≥1 5nmid nq fracP^q e^2π i nq z-Q^q e^-2π i nq z q(1-P^q)(1-Q^q).
Exact mathematical passage
```
-5\sum_{\substack{q\ge1\\5\nmid nq}}
\frac{P^q e^{2\pi i nq z}-Q^q e^{-2\pi i nq z}}
{q(1-P^q)(1-Q^q)}.
```

e^{-2\pi imz}Use in source · 1 passage

Search aliases: e^-2π imz pi

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 262
L_t,σ(z)=-5∑_substackm≥1 5nmid m fracP^m e^2π imz-Q^m e^-2π imz m(1-P^m)(1-Q^m), P=e^2π it, Q=e^2π iσ.
Exact mathematical passage
```
L_{t,\sigma}(z)=-5\sum_{\substack{m\ge1\\5\nmid m}}
\frac{P^m e^{2\pi imz}-Q^m e^{-2\pi imz}}
{m(1-P^m)(1-Q^m)},\qquad
P=e^{2\pi it},\ Q=e^{2\pi i\sigma}.
```

e^{2\pi i nq z}Use in source · 1 passage

Search aliases: e^2π i nq z pi

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 297
-5∑_substackq≥1 5nmid nq fracP^q e^2π i nq z-Q^q e^-2π i nq z q(1-P^q)(1-Q^q).
Exact mathematical passage
```
-5\sum_{\substack{q\ge1\\5\nmid nq}}
\frac{P^q e^{2\pi i nq z}-Q^q e^{-2\pi i nq z}}
{q(1-P^q)(1-Q^q)}.
```

e^{2\pi i/7}Use in source · 1 passage

Search aliases: e^2π i/7 pi

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 330
Fix the oriented seventh-root phase ζ_7=e^2π i/7. The additive character against the top dual direction is
Exact mathematical passage
```
\zeta_7=e^{2\pi i/7}
```

e^{2\pi imz}Use in source · 1 passage

Search aliases: e^2π imz pi

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 262
L_t,σ(z)=-5∑_substackm≥1 5nmid m fracP^m e^2π imz-Q^m e^-2π imz m(1-P^m)(1-Q^m), P=e^2π it, Q=e^2π iσ.
Exact mathematical passage
```
L_{t,\sigma}(z)=-5\sum_{\substack{m\ge1\\5\nmid m}}
\frac{P^m e^{2\pi imz}-Q^m e^{-2\pi imz}}
{m(1-P^m)(1-Q^m)},\qquad
P=e^{2\pi it},\ Q=e^{2\pi i\sigma}.
```

e^{2\pi it}Use in source · 1 passage

Search aliases: e^2π it pi

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 262
L_t,σ(z)=-5∑_substackm≥1 5nmid m fracP^m e^2π imz-Q^m e^-2π imz m(1-P^m)(1-Q^m), P=e^2π it, Q=e^2π iσ.
Exact mathematical passage
```
L_{t,\sigma}(z)=-5\sum_{\substack{m\ge1\\5\nmid m}}
\frac{P^m e^{2\pi imz}-Q^m e^{-2\pi imz}}
{m(1-P^m)(1-Q^m)},\qquad
P=e^{2\pi it},\ Q=e^{2\pi i\sigma}.
```

e^{2\pi i\sigma}Use in source · 1 passage

Search aliases: e^2π iσ pi sigma

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 262
L_t,σ(z)=-5∑_substackm≥1 5nmid m fracP^m e^2π imz-Q^m e^-2π imz m(1-P^m)(1-Q^m), P=e^2π it, Q=e^2π iσ.
Exact mathematical passage
```
L_{t,\sigma}(z)=-5\sum_{\substack{m\ge1\\5\nmid m}}
\frac{P^m e^{2\pi imz}-Q^m e^{-2\pi imz}}
{m(1-P^m)(1-Q^m)},\qquad
P=e^{2\pi it},\ Q=e^{2\pi i\sigma}.
```

\mathrm{ell}Use in source · 1 passage

Search aliases: ell mathrm

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 30
F(z)=fracΓ_ell(z;τ,σ)^5 Γ_ell(5z;5τ,5σ), U_Γ=F(z_0).
Exact mathematical passage
```
F(z)=\frac{\Gamma_{\mathrm{ell}}(z;\tau,\sigma)^5}
{\Gamma_{\mathrm{ell}}(5z;5\tau,5\sigma)},\qquad U_\Gamma=F(z_0).
```

\expUse in source · 1 passage

Search aliases: exp

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 356
Let V=( Z/7)^3, ζ=exp(2π i/7), and let F be the ordinary finite Fourier transform
Exact mathematical passage
```
\zeta=\exp(2\pi i/7)
```

FUse in source · 11 passages

Search aliases: F

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 30
F(z)=fracΓ_ell(z;τ,σ)^5 Γ_ell(5z;5τ,5σ), U_Γ=F(z_0).
Exact mathematical passage
```
F(z)=\frac{\Gamma_{\mathrm{ell}}(z;\tau,\sigma)^5}
{\Gamma_{\mathrm{ell}}(5z;5\tau,5\sigma)},\qquad U_\Gamma=F(z_0).
```

\mathbb FUse in source · 1 passage

Search aliases: F mathbb

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 90
Because R^3-I≠0, the order of R in GL_3( F_7) is exactly 21. Its one-dimensional eigenline has scalar action 4, of order three. Thus the order-three torsion packet is the semisimple slice of a nonsemisimple order-21 arithmetic operator.
Exact mathematical passage
```
\mathrm{GL}_3(\mathbb F_7)
```

\mathcal FUse in source · 1 passage

Search aliases: F mathcal

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 356
Let V=( Z/7)^3, ζ=exp(2π i/7), and let F be the ordinary finite Fourier transform
Exact mathematical passage
```
\mathcal F
```

fUse in source · 5 passages

Search aliases: f

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 204
f_x^(m)=F(ρ^m z_x), m≥0.
Exact mathematical passage
```
f_x^{(m)}=F(\rho^m z_x),\qquad m\ge0.
```

\mathbb F_7Use in source · 1 passage

Search aliases: F_7 mathbb

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 90
Because R^3-I≠0, the order of R in GL_3( F_7) is exactly 21. Its one-dimensional eigenline has scalar action 4, of order three. Thus the order-three torsion packet is the semisimple slice of a nonsemisimple order-21 arithmetic operator.
Exact mathematical passage
```
\mathrm{GL}_3(\mathbb F_7)
```

f_mUse in source · 1 passage

Search aliases: f_m

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 364
f_m(x)=C_m(x)f_m-1(Tx), x∈ V.
Exact mathematical passage
```
f_m(x)=C_m(x)f_{m-1}(Tx),\qquad x\in V.
```

f_{m-1}Use in source · 1 passage

Search aliases: f_m-1

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 364
f_m(x)=C_m(x)f_m-1(Tx), x∈ V.
Exact mathematical passage
```
f_m(x)=C_m(x)f_{m-1}(Tx),\qquad x\in V.
```

f_{T x}^{(m-1)}Use in source · 1 passage

Search aliases: f_T x^(m-1)

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 237
f_x^(m)=C_x^(m) f_T x^(m-1).
Exact mathematical passage
```
\boxed{f_x^{(m)}=C_x^{(m)} f_{T x}^{(m-1)}.}
```

f_x^{(m)}Use in source · 3 passages

Search aliases: f_x^(m)

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 204
f_x^(m)=F(ρ^m z_x), m≥0.
Exact mathematical passage
```
f_x^{(m)}=F(\rho^m z_x),\qquad m\ge0.
```

GUse in source · 1 passage

Search aliases: G

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 326
G_H=0 · 0 · 1 0 · 1 · 0 1 · 0 · 0.
Exact mathematical passage
```
G_H=\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}.
```

gUse in source · 1 passage

Search aliases: g

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 216
P(g;n,w,h)= ∏_r=0^n-1g(w+rh), · n>0, 1, · n=0, ∏_r=n^-1g(w+rh)^-1, · n<0.
Exact mathematical passage
```
\mathcal P(g;n,w,h)=
\begin{cases}
\prod_{r=0}^{n-1}g(w+rh),&n>0,\\
1,&n=0,\\
\prod_{r=n}^{-1}g(w+rh)^{-1},&n<0.
\end{cases}
```

G_HUse in source · 1 passage

Search aliases: G_H

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 326
G_H=0 · 0 · 1 0 · 1 · 0 1 · 0 · 0.
Exact mathematical passage
```
G_H=\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}.
```

\mathrm{GL}Use in source · 1 passage

Search aliases: GL mathrm

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 90
Because R^3-I≠0, the order of R in GL_3( F_7) is exactly 21. Its one-dimensional eigenline has scalar action 4, of order three. Thus the order-three torsion packet is the semisimple slice of a nonsemisimple order-21 arithmetic operator.
Exact mathematical passage
```
\mathrm{GL}_3(\mathbb F_7)
```

\mathrm{GL}_3Use in source · 1 passage

Search aliases: GL_3 mathrm

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 90
Because R^3-I≠0, the order of R in GL_3( F_7) is exactly 21. Its one-dimensional eigenline has scalar action 4, of order three. Thus the order-three torsion packet is the semisimple slice of a nonsemisimple order-21 arithmetic operator.
Exact mathematical passage
```
\mathrm{GL}_3(\mathbb F_7)
```

HUse in source · 4 passages

Search aliases: H

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 94
H=4 · -4 · 15 -4 · 1 · 0 15 · 0 · 0, M^T H=HM,
Exact mathematical passage
```
H=\begin{pmatrix}4&-4&15\\-4&1&0\\15&0&0\end{pmatrix},\qquad
M^T H=HM,
```

hUse in source · 1 passage

Search aliases: h

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 216
P(g;n,w,h)= ∏_r=0^n-1g(w+rh), · n>0, 1, · n=0, ∏_r=n^-1g(w+rh)^-1, · n<0.
Exact mathematical passage
```
\mathcal P(g;n,w,h)=
\begin{cases}
\prod_{r=0}^{n-1}g(w+rh),&n>0,\\
1,&n=0,\\
\prod_{r=n}^{-1}g(w+rh)^{-1},&n<0.
\end{cases}
```

IUse in source · 4 passages

Search aliases: I

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 63
Let N=M-2Ipmod7. Then N^3=0 and N^2≠0. Define
Exact mathematical passage
```
N=M-2I\pmod7
```

iUse in source · 4 passages

Search aliases: i

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 262
L_t,σ(z)=-5∑_substackm≥1 5nmid m fracP^m e^2π imz-Q^m e^-2π imz m(1-P^m)(1-Q^m), P=e^2π it, Q=e^2π iσ.
Exact mathematical passage
```
L_{t,\sigma}(z)=-5\sum_{\substack{m\ge1\\5\nmid m}}
\frac{P^m e^{2\pi imz}-Q^m e^{-2\pi imz}}
{m(1-P^m)(1-Q^m)},\qquad
P=e^{2\pi it},\ Q=e^{2\pi i\sigma}.
```

\operatorname{Im}Use in source · 1 passage

Search aliases: Im operatorname

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 348
Im(C_polU_Γ^3)=0
Exact mathematical passage
```
\operatorname{Im}(C_{\mathrm{pol}}U_\Gamma^3)=0
```

JUse in source · 1 passage

Search aliases: J

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 346
A successful next theorem would compute a character-sensitive, correctly based analytic transport and prove that its arithmetic evaluation agrees with the oriented Gauss/Jacobi coordinate J=2+3ω. The fixed desired scalar identity remains
Exact mathematical passage
```
J=2+3\omega
```

jUse in source · 20 passages

Search aliases: j

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 119
z_k,ℓ,j=z_0+kδ+ℓη+jξ, k,ℓ,j∈{0,…,6}.
Exact mathematical passage
```
\boxed{z_{k,\ell,j}=z_0+k\delta+\ell\eta+j\xi,\qquad
k,\ell,j\in\{0,\ldots,6\}.}
```

j'Use in source · 2 passages

Search aliases: j'

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 167
Write (k',ℓ',j')=(A_0,B_0,C_0)bmod7. Then
Exact mathematical passage
```
(k',\ell',j')=(A_0,B_0,C_0)\bmod7
```

KUse in source · 1 passage

Search aliases: K

Link to this entry ·

Use in source · H12_Ambient_Torsion_README.md, line 5
This research package continues the pure cubic experiment K= Q(β), β^3=7.
Exact mathematical passage
```
K=\mathbb Q(\beta)
```

kUse in source · 17 passages

Search aliases: k

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 117
The seven-point packet has arguments z_0+kδ. The first refinement consists of z_0+kδ+ℓη, with 49 labels. The full ambient packet is
Exact mathematical passage
```
z_0+k\delta
```

k'Use in source · 2 passages

Search aliases: k'

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 167
Write (k',ℓ',j')=(A_0,B_0,C_0)bmod7. Then
Exact mathematical passage
```
(k',\ell',j')=(A_0,B_0,C_0)\bmod7
```

LUse in source · 2 passages

Search aliases: L

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 262
L_t,σ(z)=-5∑_substackm≥1 5nmid m fracP^m e^2π imz-Q^m e^-2π imz m(1-P^m)(1-Q^m), P=e^2π it, Q=e^2π iσ.
Exact mathematical passage
```
L_{t,\sigma}(z)=-5\sum_{\substack{m\ge1\\5\nmid m}}
\frac{P^m e^{2\pi imz}-Q^m e^{-2\pi imz}}
{m(1-P^m)(1-Q^m)},\qquad
P=e^{2\pi it},\ Q=e^{2\pi i\sigma}.
```

L_{t,\sigma}Use in source · 2 passages

Search aliases: L_t,σ sigma

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 262
L_t,σ(z)=-5∑_substackm≥1 5nmid m fracP^m e^2π imz-Q^m e^-2π imz m(1-P^m)(1-Q^m), P=e^2π it, Q=e^2π iσ.
Exact mathematical passage
```
L_{t,\sigma}(z)=-5\sum_{\substack{m\ge1\\5\nmid m}}
\frac{P^m e^{2\pi imz}-Q^m e^{-2\pi imz}}
{m(1-P^m)(1-Q^m)},\qquad
P=e^{2\pi it},\ Q=e^{2\pi i\sigma}.
```

\lfloorUse in source · 1 passage

Search aliases: lfloor

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 161
q_1=lfloor A_0/7rfloor, q_2=lfloor B_0/7rfloor, q_3=lfloor C_0/7rfloor.
Exact mathematical passage
```
q_1=\lfloor A_0/7\rfloor,\quad
q_2=\lfloor B_0/7\rfloor,\quad
q_3=\lfloor C_0/7\rfloor.
```

MUse in source · 3 passages

Search aliases: M

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 54
M=0 · 1 · 0 -4 · 2 · 15 -1 · 0 · 4, R=M^-1=8 · -4 · 15 1 · 0 · 0 2 · -1 · 4.
Exact mathematical passage
```
M=\begin{pmatrix}0&1&0\\-4&2&15\\-1&0&4\end{pmatrix},\qquad
R=M^{-1}=\begin{pmatrix}8&-4&15\\1&0&0\\2&-1&4\end{pmatrix}.
```

mUse in source · 15 passages

Search aliases: m

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 204
f_x^(m)=F(ρ^m z_x), m≥0.
Exact mathematical passage
```
f_x^{(m)}=F(\rho^m z_x),\qquad m\ge0.
```

M^{-1}Use in source · 1 passage

Search aliases: M^-1

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 54
M=0 · 1 · 0 -4 · 2 · 15 -1 · 0 · 4, R=M^-1=8 · -4 · 15 1 · 0 · 0 2 · -1 · 4.
Exact mathematical passage
```
M=\begin{pmatrix}0&1&0\\-4&2&15\\-1&0&4\end{pmatrix},\qquad
R=M^{-1}=\begin{pmatrix}8&-4&15\\1&0&0\\2&-1&4\end{pmatrix}.
```

M^TUse in source · 1 passage

Search aliases: M^T

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 94
H=4 · -4 · 15 -4 · 1 · 0 15 · 0 · 0, M^T H=HM,
Exact mathematical passage
```
H=\begin{pmatrix}4&-4&15\\-4&1&0\\15&0&0\end{pmatrix},\qquad
M^T H=HM,
```

\midUse in source · 2 passages

Search aliases: mid

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 279
N_nF(z)= F(nz), · 5nmid n, 1, · 5mid n.
Exact mathematical passage
```
\boxed{
\mathcal N_nF(z)=
\begin{cases}F(nz),&5\nmid n,\\1,&5\mid n.\end{cases}
}
```

NUse in source · 8 passages

Search aliases: N

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 63
Let N=M-2Ipmod7. Then N^3=0 and N^2≠0. Define
Exact mathematical passage
```
N=M-2I\pmod7
```

\mathcal NUse in source · 3 passages

Search aliases: N mathcal

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 271
N_n F(z)= ∏_u=1-n^0∏_v=0^n-1∏_j=0^n-1 F(z+fracut+vσ+jn).
Exact mathematical passage
```
\mathcal N_n F(z)=
\prod_{u=1-n}^{0}\prod_{v=0}^{n-1}\prod_{j=0}^{n-1}
F\left(z+\frac{ut+v\sigma+j}{n}\right).
```

nUse in source · 15 passages

Search aliases: n

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 214
For an integer n, define the oriented finite product
Exact mathematical passage
```
n
```

N^2Use in source · 4 passages

Search aliases: N^2

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 63
Let N=M-2Ipmod7. Then N^3=0 and N^2≠0. Define
Exact mathematical passage
```
N^2\ne0
```

N^3Use in source · 1 passage

Search aliases: N^3

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 63
Let N=M-2Ipmod7. Then N^3=0 and N^2≠0. Define
Exact mathematical passage
```
N^3=0
```

\mathcal N_7Use in source · 1 passage

Search aliases: N_7 mathcal

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 307
N_7F(1/3)=F(7/3)=F(1/3)=U_Γ.
Exact mathematical passage
```
\boxed{\mathcal N_7F(1/3)=F(7/3)=F(1/3)=U_\Gamma.}
```

\mathcal N_nUse in source · 2 passages

Search aliases: N_n mathcal

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 271
N_n F(z)= ∏_u=1-n^0∏_v=0^n-1∏_j=0^n-1 F(z+fracut+vσ+jn).
Exact mathematical passage
```
\mathcal N_n F(z)=
\prod_{u=1-n}^{0}\prod_{v=0}^{n-1}\prod_{j=0}^{n-1}
F\left(z+\frac{ut+v\sigma+j}{n}\right).
```

\nmidUse in source · 6 passages

Search aliases: nmid

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 262
L_t,σ(z)=-5∑_substackm≥1 5nmid m fracP^m e^2π imz-Q^m e^-2π imz m(1-P^m)(1-Q^m), P=e^2π it, Q=e^2π iσ.
Exact mathematical passage
```
L_{t,\sigma}(z)=-5\sum_{\substack{m\ge1\\5\nmid m}}
\frac{P^m e^{2\pi imz}-Q^m e^{-2\pi imz}}
{m(1-P^m)(1-Q^m)},\qquad
P=e^{2\pi it},\ Q=e^{2\pi i\sigma}.
```

OUse in source · 1 passage

Search aliases: O

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 242
As ρ^m z_x→0, all f_x^(m)→ F(0)=ε^2. Every multiplier is nonzero at the chosen arguments, and C_x^(m)=1+O(|ρ|^m-1).
Exact mathematical passage
```
C_x^{(m)}=1+O(|\rho|^{m-1})
```

PUse in source · 3 passages

Search aliases: P

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 262
L_t,σ(z)=-5∑_substackm≥1 5nmid m fracP^m e^2π imz-Q^m e^-2π imz m(1-P^m)(1-Q^m), P=e^2π it, Q=e^2π iσ.
Exact mathematical passage
```
L_{t,\sigma}(z)=-5\sum_{\substack{m\ge1\\5\nmid m}}
\frac{P^m e^{2\pi imz}-Q^m e^{-2\pi imz}}
{m(1-P^m)(1-Q^m)},\qquad
P=e^{2\pi it},\ Q=e^{2\pi i\sigma}.
```

\mathcal PUse in source · 2 passages

Search aliases: P mathcal

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 216
P(g;n,w,h)= ∏_r=0^n-1g(w+rh), · n>0, 1, · n=0, ∏_r=n^-1g(w+rh)^-1, · n<0.
Exact mathematical passage
```
\mathcal P(g;n,w,h)=
\begin{cases}
\prod_{r=0}^{n-1}g(w+rh),&n>0,\\
1,&n=0,\\
\prod_{r=n}^{-1}g(w+rh)^{-1},&n<0.
\end{cases}
```

P^{-(n-1)q}Use in source · 1 passage

Search aliases: P^-(n-1)q

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 288
∑_u=1-n^0P^qu =P^-(n-1)qfrac1-P^nq1-P^q, ∑_v=0^n-1Q^qv =frac1-Q^nq1-Q^q.
Exact mathematical passage
```
\sum_{u=1-n}^{0}P^{qu}
=P^{-(n-1)q}\frac{1-P^{nq}}{1-P^q},\qquad
\sum_{v=0}^{n-1}Q^{qv}
=\frac{1-Q^{nq}}{1-Q^q}.
```

P^mUse in source · 1 passage

Search aliases: P^m

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 262
L_t,σ(z)=-5∑_substackm≥1 5nmid m fracP^m e^2π imz-Q^m e^-2π imz m(1-P^m)(1-Q^m), P=e^2π it, Q=e^2π iσ.
Exact mathematical passage
```
L_{t,\sigma}(z)=-5\sum_{\substack{m\ge1\\5\nmid m}}
\frac{P^m e^{2\pi imz}-Q^m e^{-2\pi imz}}
{m(1-P^m)(1-Q^m)},\qquad
P=e^{2\pi it},\ Q=e^{2\pi i\sigma}.
```

P^{nq}Use in source · 1 passage

Search aliases: P^nq

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 288
∑_u=1-n^0P^qu =P^-(n-1)qfrac1-P^nq1-P^q, ∑_v=0^n-1Q^qv =frac1-Q^nq1-Q^q.
Exact mathematical passage
```
\sum_{u=1-n}^{0}P^{qu}
=P^{-(n-1)q}\frac{1-P^{nq}}{1-P^q},\qquad
\sum_{v=0}^{n-1}Q^{qv}
=\frac{1-Q^{nq}}{1-Q^q}.
```

P^qUse in source · 2 passages

Search aliases: P^q

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 288
∑_u=1-n^0P^qu =P^-(n-1)qfrac1-P^nq1-P^q, ∑_v=0^n-1Q^qv =frac1-Q^nq1-Q^q.
Exact mathematical passage
```
\sum_{u=1-n}^{0}P^{qu}
=P^{-(n-1)q}\frac{1-P^{nq}}{1-P^q},\qquad
\sum_{v=0}^{n-1}Q^{qv}
=\frac{1-Q^{nq}}{1-Q^q}.
```

P^{qu}Use in source · 1 passage

Search aliases: P^qu

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 288
∑_u=1-n^0P^qu =P^-(n-1)qfrac1-P^nq1-P^q, ∑_v=0^n-1Q^qv =frac1-Q^nq1-Q^q.
Exact mathematical passage
```
\sum_{u=1-n}^{0}P^{qu}
=P^{-(n-1)q}\frac{1-P^{nq}}{1-P^q},\qquad
\sum_{v=0}^{n-1}Q^{qv}
=\frac{1-Q^{nq}}{1-Q^q}.
```

\pmodUse in source · 5 passages

Search aliases: pmod

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 63
Let N=M-2Ipmod7. Then N^3=0 and N^2≠0. Define
Exact mathematical passage
```
N=M-2I\pmod7
```

\mathrm{pol}Use in source · 1 passage

Search aliases: pol mathrm

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 348
Im(C_polU_Γ^3)=0
Exact mathematical passage
```
\operatorname{Im}(C_{\mathrm{pol}}U_\Gamma^3)=0
```

QUse in source · 3 passages

Search aliases: Q

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 262
L_t,σ(z)=-5∑_substackm≥1 5nmid m fracP^m e^2π imz-Q^m e^-2π imz m(1-P^m)(1-Q^m), P=e^2π it, Q=e^2π iσ.
Exact mathematical passage
```
L_{t,\sigma}(z)=-5\sum_{\substack{m\ge1\\5\nmid m}}
\frac{P^m e^{2\pi imz}-Q^m e^{-2\pi imz}}
{m(1-P^m)(1-Q^m)},\qquad
P=e^{2\pi it},\ Q=e^{2\pi i\sigma}.
```

\mathbb QUse in source · 1 passage

Search aliases: Q mathbb

Link to this entry ·

Use in source · H12_Ambient_Torsion_README.md, line 5
This research package continues the pure cubic experiment K= Q(β), β^3=7.
Exact mathematical passage
```
K=\mathbb Q(\beta)
```

qUse in source · 10 passages

Search aliases: q

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 37
F(z+1)=F(z), F(z+τ)=q_σ(z)F(z), F(z+σ)=q_t(z)^-1F(z),
Exact mathematical passage
```
F(z+1)=F(z),\qquad
F(z+\tau)=q_\sigma(z)F(z),\qquad
F(z+\sigma)=q_t(z)^{-1}F(z),
```

Q^mUse in source · 1 passage

Search aliases: Q^m

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 262
L_t,σ(z)=-5∑_substackm≥1 5nmid m fracP^m e^2π imz-Q^m e^-2π imz m(1-P^m)(1-Q^m), P=e^2π it, Q=e^2π iσ.
Exact mathematical passage
```
L_{t,\sigma}(z)=-5\sum_{\substack{m\ge1\\5\nmid m}}
\frac{P^m e^{2\pi imz}-Q^m e^{-2\pi imz}}
{m(1-P^m)(1-Q^m)},\qquad
P=e^{2\pi it},\ Q=e^{2\pi i\sigma}.
```

Q^{nq}Use in source · 1 passage

Search aliases: Q^nq

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 288
∑_u=1-n^0P^qu =P^-(n-1)qfrac1-P^nq1-P^q, ∑_v=0^n-1Q^qv =frac1-Q^nq1-Q^q.
Exact mathematical passage
```
\sum_{u=1-n}^{0}P^{qu}
=P^{-(n-1)q}\frac{1-P^{nq}}{1-P^q},\qquad
\sum_{v=0}^{n-1}Q^{qv}
=\frac{1-Q^{nq}}{1-Q^q}.
```

Q^qUse in source · 2 passages

Search aliases: Q^q

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 288
∑_u=1-n^0P^qu =P^-(n-1)qfrac1-P^nq1-P^q, ∑_v=0^n-1Q^qv =frac1-Q^nq1-Q^q.
Exact mathematical passage
```
\sum_{u=1-n}^{0}P^{qu}
=P^{-(n-1)q}\frac{1-P^{nq}}{1-P^q},\qquad
\sum_{v=0}^{n-1}Q^{qv}
=\frac{1-Q^{nq}}{1-Q^q}.
```

Q^{qv}Use in source · 1 passage

Search aliases: Q^qv

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 288
∑_u=1-n^0P^qu =P^-(n-1)qfrac1-P^nq1-P^q, ∑_v=0^n-1Q^qv =frac1-Q^nq1-Q^q.
Exact mathematical passage
```
\sum_{u=1-n}^{0}P^{qu}
=P^{-(n-1)q}\frac{1-P^{nq}}{1-P^q},\qquad
\sum_{v=0}^{n-1}Q^{qv}
=\frac{1-Q^{nq}}{1-Q^q}.
```

q_1Use in source · 2 passages

Search aliases: q_1

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 161
q_1=lfloor A_0/7rfloor, q_2=lfloor B_0/7rfloor, q_3=lfloor C_0/7rfloor.
Exact mathematical passage
```
q_1=\lfloor A_0/7\rfloor,\quad
q_2=\lfloor B_0/7\rfloor,\quad
q_3=\lfloor C_0/7\rfloor.
```

q_2Use in source · 2 passages

Search aliases: q_2

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 161
q_1=lfloor A_0/7rfloor, q_2=lfloor B_0/7rfloor, q_3=lfloor C_0/7rfloor.
Exact mathematical passage
```
q_1=\lfloor A_0/7\rfloor,\quad
q_2=\lfloor B_0/7\rfloor,\quad
q_3=\lfloor C_0/7\rfloor.
```

q_3Use in source · 2 passages

Search aliases: q_3

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 161
q_1=lfloor A_0/7rfloor, q_2=lfloor B_0/7rfloor, q_3=lfloor C_0/7rfloor.
Exact mathematical passage
```
q_1=\lfloor A_0/7\rfloor,\quad
q_2=\lfloor B_0/7\rfloor,\quad
q_3=\lfloor C_0/7\rfloor.
```

q_aUse in source · 1 passage

Search aliases: q_a

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 45
q_a(z)=fracθ_0(z;a)^5θ_0(5z;5a).
Exact mathematical passage
```
q_a(z)=\frac{\theta_0(z;a)^5}{\theta_0(5z;5a)}.
```

q_tUse in source · 1 passage

Search aliases: q_t

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 37
F(z+1)=F(z), F(z+τ)=q_σ(z)F(z), F(z+σ)=q_t(z)^-1F(z),
Exact mathematical passage
```
F(z+1)=F(z),\qquad
F(z+\tau)=q_\sigma(z)F(z),\qquad
F(z+\sigma)=q_t(z)^{-1}F(z),
```

q_t^{-1}Use in source · 1 passage

Search aliases: q_t^-1

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 227
C_x^(m)= P(q_t^-1;b_x,m,W_x,m,σ) P(q_σ;a_x,m,W_x,m+b_x,mσ,τ).
Exact mathematical passage
```
\boxed{
C_x^{(m)}=
\mathcal P(q_t^{-1};b_{x,m},W_{x,m},\sigma)\,
\mathcal P(q_\sigma;a_{x,m},W_{x,m}+b_{x,m}\sigma,\tau).
}
```

q_\sigmaUse in source · 2 passages

Search aliases: q_σ sigma

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 37
F(z+1)=F(z), F(z+τ)=q_σ(z)F(z), F(z+σ)=q_t(z)^-1F(z),
Exact mathematical passage
```
F(z+1)=F(z),\qquad
F(z+\tau)=q_\sigma(z)F(z),\qquad
F(z+\sigma)=q_t(z)^{-1}F(z),
```

RUse in source · 7 passages

Search aliases: R

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 54
M=0 · 1 · 0 -4 · 2 · 15 -1 · 0 · 4, R=M^-1=8 · -4 · 15 1 · 0 · 0 2 · -1 · 4.
Exact mathematical passage
```
M=\begin{pmatrix}0&1&0\\-4&2&15\\-1&0&4\end{pmatrix},\qquad
R=M^{-1}=\begin{pmatrix}8&-4&15\\1&0&0\\2&-1&4\end{pmatrix}.
```

rUse in source · 1 passage

Search aliases: r

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 216
P(g;n,w,h)= ∏_r=0^n-1g(w+rh), · n>0, 1, · n=0, ∏_r=n^-1g(w+rh)^-1, · n<0.
Exact mathematical passage
```
\mathcal P(g;n,w,h)=
\begin{cases}
\prod_{r=0}^{n-1}g(w+rh),&n>0,\\
1,&n=0,\\
\prod_{r=n}^{-1}g(w+rh)^{-1},&n<0.
\end{cases}
```

R^{21}Use in source · 1 passage

Search aliases: R^21

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 85
R^3-I=2N+5N^2pmod7, R^7=4Ipmod7, R^21=Ipmod7.
Exact mathematical passage
```
R^3-I=2N+5N^2\pmod7,\qquad
R^7=4I\pmod7,\qquad R^{21}=I\pmod7.
```

R^3Use in source · 2 passages

Search aliases: R^3

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 85
R^3-I=2N+5N^2pmod7, R^7=4Ipmod7, R^21=Ipmod7.
Exact mathematical passage
```
R^3-I=2N+5N^2\pmod7,\qquad
R^7=4I\pmod7,\qquad R^{21}=I\pmod7.
```

R^7Use in source · 1 passage

Search aliases: R^7

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 85
R^3-I=2N+5N^2pmod7, R^7=4Ipmod7, R^21=Ipmod7.
Exact mathematical passage
```
R^3-I=2N+5N^2\pmod7,\qquad
R^7=4I\pmod7,\qquad R^{21}=I\pmod7.
```

R^{m-1}Use in source · 1 passage

Search aliases: R^m-1

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 208
Let d_x=(a_x,b_x,c_x)^T be the integer carry vector from Section 3. At stage m, the carry is d_x,m=R^m-1d_x∈ Z^3. Set W_x,m=ρ^m-1z_T x. By the arithmetic transport formula,
Exact mathematical passage
```
d_{x,m}=R^{m-1}d_x\in\mathbb Z^3
```

\rfloorUse in source · 1 passage

Search aliases: rfloor

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 161
q_1=lfloor A_0/7rfloor, q_2=lfloor B_0/7rfloor, q_3=lfloor C_0/7rfloor.
Exact mathematical passage
```
q_1=\lfloor A_0/7\rfloor,\quad
q_2=\lfloor B_0/7\rfloor,\quad
q_3=\lfloor C_0/7\rfloor.
```

\squareUse in source · 1 passage

Search aliases: square

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 303
When 5nmid n, the condition 5nmid nq is exactly 5nmid q, giving L_t,σ(nz). When 5mid n, no terms remain and the logarithm is zero. Exponentiating proves the equality in the convergence region. Meromorphic continuation gives the stated identities. square
Exact mathematical passage
```
\square
```

TUse in source · 14 passages

Search aliases: T

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 65
u=(1,2,4)^T, v=(0,1,2)^T, w=(0,0,1)^T.
Exact mathematical passage
```
u=(1,2,4)^T,\quad v=(0,1,2)^T,\quad w=(0,0,1)^T.
```

tUse in source · 8 passages

Search aliases: t

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 22
τ=fracβ^2+2β15, σ=-fracβ+215, t=-τ, z_0=frac13.
Exact mathematical passage
```
\tau=\frac{\beta^2+2\beta}{15},\qquad
\sigma=-\frac{\beta+2}{15},\qquad t=-\tau,
\quad z_0=\frac13.
```

T^{-m}Use in source · 2 passages

Search aliases: T^-m

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 246
F(z_x)=ε^2 ∏_m=1^∞ (C_T^-mx^(m))^-1.
Exact mathematical passage
```
\boxed{
F(z_x)=\varepsilon^2
\prod_{m=1}^{\infty}
\left(C_{T^{-m}x}^{(m)}\right)^{-1}.
}
```

T^{-T}Use in source · 2 passages

Search aliases: T^-T

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 370
widehat f_m(a)=frac1343∑_b∈ V widehat C_m(a-b) widehat f_m-1(T^-Tb).
Exact mathematical passage
```
\boxed{
\widehat f_m(a)=\frac1{343}\sum_{b\in V}
\widehat C_m(a-b)\,\widehat f_{m-1}(T^{-T}b).
}
```

UUse in source · 5 passages

Search aliases: U

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 30
F(z)=fracΓ_ell(z;τ,σ)^5 Γ_ell(5z;5τ,5σ), U_Γ=F(z_0).
Exact mathematical passage
```
F(z)=\frac{\Gamma_{\mathrm{ell}}(z;\tau,\sigma)^5}
{\Gamma_{\mathrm{ell}}(5z;5\tau,5\sigma)},\qquad U_\Gamma=F(z_0).
```

uUse in source · 15 passages

Search aliases: u

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 65
u=(1,2,4)^T, v=(0,1,2)^T, w=(0,0,1)^T.
Exact mathematical passage
```
u=(1,2,4)^T,\quad v=(0,1,2)^T,\quad w=(0,0,1)^T.
```

U_\GammaUse in source · 4 passages

Search aliases: U_Γ Gamma

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 30
F(z)=fracΓ_ell(z;τ,σ)^5 Γ_ell(5z;5τ,5σ), U_Γ=F(z_0).
Exact mathematical passage
```
F(z)=\frac{\Gamma_{\mathrm{ell}}(z;\tau,\sigma)^5}
{\Gamma_{\mathrm{ell}}(5z;5\tau,5\sigma)},\qquad U_\Gamma=F(z_0).
```

U_\Gamma^3Use in source · 1 passage

Search aliases: U_Γ^3 Gamma

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 348
Im(C_polU_Γ^3)=0
Exact mathematical passage
```
\operatorname{Im}(C_{\mathrm{pol}}U_\Gamma^3)=0
```

VUse in source · 5 passages

Search aliases: V

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 202
Let x=(k,ℓ,j)∈ V=( Z/7)^3 and z_x=z_k,ℓ,j. Define
Exact mathematical passage
```
x=(k,\ell,j)\in V=(\mathbb Z/7)^3
```

vUse in source · 14 passages

Search aliases: v

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 65
u=(1,2,4)^T, v=(0,1,2)^T, w=(0,0,1)^T.
Exact mathematical passage
```
u=(1,2,4)^T,\quad v=(0,1,2)^T,\quad w=(0,0,1)^T.
```

WUse in source · 3 passages

Search aliases: W

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 208
Let d_x=(a_x,b_x,c_x)^T be the integer carry vector from Section 3. At stage m, the carry is d_x,m=R^m-1d_x∈ Z^3. Set W_x,m=ρ^m-1z_T x. By the arithmetic transport formula,
Exact mathematical passage
```
W_{x,m}=\rho^{m-1}z_{T x}
```

wUse in source · 16 passages

Search aliases: w

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 65
u=(1,2,4)^T, v=(0,1,2)^T, w=(0,0,1)^T.
Exact mathematical passage
```
u=(1,2,4)^T,\quad v=(0,1,2)^T,\quad w=(0,0,1)^T.
```

W_{x,m}Use in source · 3 passages

Search aliases: W_x,m

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 208
Let d_x=(a_x,b_x,c_x)^T be the integer carry vector from Section 3. At stage m, the carry is d_x,m=R^m-1d_x∈ Z^3. Set W_x,m=ρ^m-1z_T x. By the arithmetic transport formula,
Exact mathematical passage
```
W_{x,m}=\rho^{m-1}z_{T x}
```

\widehat CUse in source · 1 passage

Search aliases: widehat C widehat

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 370
widehat f_m(a)=frac1343∑_b∈ V widehat C_m(a-b) widehat f_m-1(T^-Tb).
Exact mathematical passage
```
\boxed{
\widehat f_m(a)=\frac1{343}\sum_{b\in V}
\widehat C_m(a-b)\,\widehat f_{m-1}(T^{-T}b).
}
```

\widehat C_mUse in source · 1 passage

Search aliases: widehat C_m widehat

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 370
widehat f_m(a)=frac1343∑_b∈ V widehat C_m(a-b) widehat f_m-1(T^-Tb).
Exact mathematical passage
```
\boxed{
\widehat f_m(a)=\frac1{343}\sum_{b\in V}
\widehat C_m(a-b)\,\widehat f_{m-1}(T^{-T}b).
}
```

\widehat fUse in source · 2 passages

Search aliases: widehat f widehat

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 358
widehat f(a)=∑_x∈ Vζ^a· xf(x), a∈ V.
Exact mathematical passage
```
\widehat f(a)=\sum_{x\in V}\zeta^{a\cdot x}f(x),\qquad a\in V.
```

\widehat f_mUse in source · 1 passage

Search aliases: widehat f_m widehat

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 370
widehat f_m(a)=frac1343∑_b∈ V widehat C_m(a-b) widehat f_m-1(T^-Tb).
Exact mathematical passage
```
\boxed{
\widehat f_m(a)=\frac1{343}\sum_{b\in V}
\widehat C_m(a-b)\,\widehat f_{m-1}(T^{-T}b).
}
```

\widehat f_{m-1}Use in source · 1 passage

Search aliases: widehat f_m-1 widehat

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 370
widehat f_m(a)=frac1343∑_b∈ V widehat C_m(a-b) widehat f_m-1(T^-Tb).
Exact mathematical passage
```
\boxed{
\widehat f_m(a)=\frac1{343}\sum_{b\in V}
\widehat C_m(a-b)\,\widehat f_{m-1}(T^{-T}b).
}
```

xUse in source · 15 passages

Search aliases: x

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 202
Let x=(k,ℓ,j)∈ V=( Z/7)^3 and z_x=z_k,ℓ,j. Define
Exact mathematical passage
```
x=(k,\ell,j)\in V=(\mathbb Z/7)^3
```

\mathbb ZUse in source · 6 passages

Search aliases: Z mathbb

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 50
The previously established zero-anchor normalization is F(0)=ε^2≠0. The divisor support is contained in Λ_5= Zτ+ Zσ+tfrac15 Z. These analytic facts are inherited inputs, not freshly formalized in Lean here.
Exact mathematical passage
```
\Lambda_5=\mathbb Z\tau+\mathbb Z\sigma+\tfrac15\mathbb Z
```

zUse in source · 23 passages

Search aliases: z

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 22
τ=fracβ^2+2β15, σ=-fracβ+215, t=-τ, z_0=frac13.
Exact mathematical passage
```
\tau=\frac{\beta^2+2\beta}{15},\qquad
\sigma=-\frac{\beta+2}{15},\qquad t=-\tau,
\quad z_0=\frac13.
```

\mathbb Z^3Use in source · 1 passage

Search aliases: Z^3 mathbb

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 208
Let d_x=(a_x,b_x,c_x)^T be the integer carry vector from Section 3. At stage m, the carry is d_x,m=R^m-1d_x∈ Z^3. Set W_x,m=ρ^m-1z_T x. By the arithmetic transport formula,
Exact mathematical passage
```
d_{x,m}=R^{m-1}d_x\in\mathbb Z^3
```

z_0Use in source · 7 passages

Search aliases: z_0

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 22
τ=fracβ^2+2β15, σ=-fracβ+215, t=-τ, z_0=frac13.
Exact mathematical passage
```
\tau=\frac{\beta^2+2\beta}{15},\qquad
\sigma=-\frac{\beta+2}{15},\qquad t=-\tau,
\quad z_0=\frac13.
```

z_{k',\ell',j'}Use in source · 1 passage

Search aliases: z_k',ℓ',j' ell

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 169
ρ z_k,ℓ,j= z_k',ℓ',j'+aτ+bσ+c
Exact mathematical passage
```
\boxed{
\rho z_{k,\ell,j}=
 z_{k',\ell',j'}+a\tau+b\sigma+c
}
```

z_{k,\ell,j}Use in source · 4 passages

Search aliases: z_k,ℓ,j ell

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 119
z_k,ℓ,j=z_0+kδ+ℓη+jξ, k,ℓ,j∈{0,…,6}.
Exact mathematical passage
```
\boxed{z_{k,\ell,j}=z_0+k\delta+\ell\eta+j\xi,\qquad
k,\ell,j\in\{0,\ldots,6\}.}
```

z_{T x}Use in source · 1 passage

Search aliases: z_T x

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 208
Let d_x=(a_x,b_x,c_x)^T be the integer carry vector from Section 3. At stage m, the carry is d_x,m=R^m-1d_x∈ Z^3. Set W_x,m=ρ^m-1z_T x. By the arithmetic transport formula,
Exact mathematical passage
```
W_{x,m}=\rho^{m-1}z_{T x}
```

z_xUse in source · 5 passages

Search aliases: z_x

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 202
Let x=(k,ℓ,j)∈ V=( Z/7)^3 and z_x=z_k,ℓ,j. Define
Exact mathematical passage
```
z_x=z_{k,\ell,j}
```

|Use in source · 2 passages

Search aliases: |

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 20
Choose the complex embedding β=7^1/3ω^2, so |ρ|<1. Put
Exact mathematical passage
```
|\rho|<1
```

|^{m-1}Use in source · 1 passage

Search aliases: |^m-1

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 242
As ρ^m z_x→0, all f_x^(m)→ F(0)=ε^2. Every multiplier is nonzero at the chosen arguments, and C_x^(m)=1+O(|ρ|^m-1).
Exact mathematical passage
```
C_x^{(m)}=1+O(|\rho|^{m-1})
```

\cdotUse in source · 4 passages

Search aliases: · cdot

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 145
343=1+2·3+16·21.
Exact mathematical passage
```
\boxed{343=1+2\cdot3+16\cdot21.}
```

\betaUse in source · 5 passages

Search aliases: β beta

Link to this entry ·

Use in source · H12_Ambient_Torsion_README.md, line 5
This research package continues the pure cubic experiment K= Q(β), β^3=7.
Exact mathematical passage
```
K=\mathbb Q(\beta)
```

\beta^2Use in source · 2 passages

Search aliases: β^2 beta

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 15
ω^2+ω+1=0, β^3=7, ε=2-β, ρ=ε^-1=β^2+2β+4.
Exact mathematical passage
```
\omega^2+\omega+1=0,\quad \beta^3=7,\quad
\varepsilon=2-\beta,\quad \rho=\varepsilon^{-1}=\beta^2+2\beta+4.
```

\beta^3Use in source · 2 passages

Search aliases: β^3 beta

Link to this entry ·

Use in source · H12_Ambient_Torsion_README.md, line 5
This research package continues the pure cubic experiment K= Q(β), β^3=7.
Exact mathematical passage
```
\beta^3=7
```

\GammaUse in source · 5 passages

Search aliases: Γ Gamma

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 30
F(z)=fracΓ_ell(z;τ,σ)^5 Γ_ell(5z;5τ,5σ), U_Γ=F(z_0).
Exact mathematical passage
```
F(z)=\frac{\Gamma_{\mathrm{ell}}(z;\tau,\sigma)^5}
{\Gamma_{\mathrm{ell}}(5z;5\tau,5\sigma)},\qquad U_\Gamma=F(z_0).
```

\Gamma_{\mathrm{ell}}Use in source · 1 passage

Search aliases: Γ_ell Gamma mathrm

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 30
F(z)=fracΓ_ell(z;τ,σ)^5 Γ_ell(5z;5τ,5σ), U_Γ=F(z_0).
Exact mathematical passage
```
F(z)=\frac{\Gamma_{\mathrm{ell}}(z;\tau,\sigma)^5}
{\Gamma_{\mathrm{ell}}(5z;5\tau,5\sigma)},\qquad U_\Gamma=F(z_0).
```

\deltaUse in source · 7 passages

Search aliases: δ delta

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 111
δ=fracτ+2σ+47, η=fracσ+27, ξ=frac17.
Exact mathematical passage
```
\delta=\frac{\tau+2\sigma+4}{7},\qquad
\eta=\frac{\sigma+2}{7},\qquad
\xi=\frac17.
```

\varepsilonUse in source · 6 passages

Search aliases: ε varepsilon

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 15
ω^2+ω+1=0, β^3=7, ε=2-β, ρ=ε^-1=β^2+2β+4.
Exact mathematical passage
```
\omega^2+\omega+1=0,\quad \beta^3=7,\quad
\varepsilon=2-\beta,\quad \rho=\varepsilon^{-1}=\beta^2+2\beta+4.
```

\varepsilon^{-1}Use in source · 1 passage

Search aliases: ε^-1 varepsilon

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 15
ω^2+ω+1=0, β^3=7, ε=2-β, ρ=ε^-1=β^2+2β+4.
Exact mathematical passage
```
\omega^2+\omega+1=0,\quad \beta^3=7,\quad
\varepsilon=2-\beta,\quad \rho=\varepsilon^{-1}=\beta^2+2\beta+4.
```

\varepsilon^2Use in source · 3 passages

Search aliases: ε^2 varepsilon

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 49
The previously established zero-anchor normalization is F(0)=ε^2≠0. The divisor support is contained in Λ_5= Zτ+ Zσ+tfrac15 Z. These analytic facts are inherited inputs, not freshly formalized in Lean here.
Exact mathematical passage
```
F(0)=\varepsilon^2\ne0
```

\zetaUse in source · 5 passages

Search aliases: ζ zeta

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 330
Fix the oriented seventh-root phase ζ_7=e^2π i/7. The additive character against the top dual direction is
Exact mathematical passage
```
\zeta_7=e^{2\pi i/7}
```

\zeta^{a\cdot x}Use in source · 1 passage

Search aliases: ζ^a· x zeta cdot

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 358
widehat f(a)=∑_x∈ Vζ^a· xf(x), a∈ V.
Exact mathematical passage
```
\widehat f(a)=\sum_{x\in V}\zeta^{a\cdot x}f(x),\qquad a\in V.
```

\zeta_7Use in source · 1 passage

Search aliases: ζ_7 zeta

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 330
Fix the oriented seventh-root phase ζ_7=e^2π i/7. The additive character against the top dual direction is
Exact mathematical passage
```
\zeta_7=e^{2\pi i/7}
```

\zeta_7^{3(4k+5\ell+j)}Use in source · 1 passage

Search aliases: ζ_7^3(4k+5ℓ+j) zeta ell

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 338
ψ_w(T(k,ℓ,j))=ζ_7^3(4k+5ℓ+j).
Exact mathematical passage
```
\boxed{\psi_w(T(k,\ell,j))=\zeta_7^{3(4k+5\ell+j)}.}
```

\zeta_7^{3k}Use in source · 1 passage

Search aliases: ζ_7^3k zeta

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 332
ψ_w(k,ℓ,j)=ζ_7^3k.
Exact mathematical passage
```
\boxed{\psi_w(k,\ell,j)=\zeta_7^{3k}.}
```

\etaUse in source · 6 passages

Search aliases: η eta

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 111
δ=fracτ+2σ+47, η=fracσ+27, ξ=frac17.
Exact mathematical passage
```
\delta=\frac{\tau+2\sigma+4}{7},\qquad
\eta=\frac{\sigma+2}{7},\qquad
\xi=\frac17.
```

\thetaUse in source · 1 passage

Search aliases: θ theta

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 45
q_a(z)=fracθ_0(z;a)^5θ_0(5z;5a).
Exact mathematical passage
```
q_a(z)=\frac{\theta_0(z;a)^5}{\theta_0(5z;5a)}.
```

\theta_0Use in source · 1 passage

Search aliases: θ_0 theta

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 45
q_a(z)=fracθ_0(z;a)^5θ_0(5z;5a).
Exact mathematical passage
```
q_a(z)=\frac{\theta_0(z;a)^5}{\theta_0(5z;5a)}.
```

\LambdaUse in source · 6 passages

Search aliases: Λ Lambda

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 50
The previously established zero-anchor normalization is F(0)=ε^2≠0. The divisor support is contained in Λ_5= Zτ+ Zσ+tfrac15 Z. These analytic facts are inherited inputs, not freshly formalized in Lean here.
Exact mathematical passage
```
\Lambda_5=\mathbb Z\tau+\mathbb Z\sigma+\tfrac15\mathbb Z
```

\Lambda_5Use in source · 4 passages

Search aliases: Λ_5 Lambda

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 50
The previously established zero-anchor normalization is F(0)=ε^2≠0. The divisor support is contained in Λ_5= Zτ+ Zσ+tfrac15 Z. These analytic facts are inherited inputs, not freshly formalized in Lean here.
Exact mathematical passage
```
\Lambda_5=\mathbb Z\tau+\mathbb Z\sigma+\tfrac15\mathbb Z
```

\xiUse in source · 5 passages

Search aliases: ξ xi

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 111
δ=fracτ+2σ+47, η=fracσ+27, ξ=frac17.
Exact mathematical passage
```
\delta=\frac{\tau+2\sigma+4}{7},\qquad
\eta=\frac{\sigma+2}{7},\qquad
\xi=\frac17.
```

\piUse in source · 4 passages

Search aliases: π pi

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 262
L_t,σ(z)=-5∑_substackm≥1 5nmid m fracP^m e^2π imz-Q^m e^-2π imz m(1-P^m)(1-Q^m), P=e^2π it, Q=e^2π iσ.
Exact mathematical passage
```
L_{t,\sigma}(z)=-5\sum_{\substack{m\ge1\\5\nmid m}}
\frac{P^m e^{2\pi imz}-Q^m e^{-2\pi imz}}
{m(1-P^m)(1-Q^m)},\qquad
P=e^{2\pi it},\ Q=e^{2\pi i\sigma}.
```

\rhoUse in source · 12 passages

Search aliases: ρ rho

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 15
ω^2+ω+1=0, β^3=7, ε=2-β, ρ=ε^-1=β^2+2β+4.
Exact mathematical passage
```
\omega^2+\omega+1=0,\quad \beta^3=7,\quad
\varepsilon=2-\beta,\quad \rho=\varepsilon^{-1}=\beta^2+2\beta+4.
```

\rho^mUse in source · 3 passages

Search aliases: ρ^m rho

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 204
f_x^(m)=F(ρ^m z_x), m≥0.
Exact mathematical passage
```
f_x^{(m)}=F(\rho^m z_x),\qquad m\ge0.
```

\rho^{m-1}Use in source · 1 passage

Search aliases: ρ^m-1 rho

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 208
Let d_x=(a_x,b_x,c_x)^T be the integer carry vector from Section 3. At stage m, the carry is d_x,m=R^m-1d_x∈ Z^3. Set W_x,m=ρ^m-1z_T x. By the arithmetic transport formula,
Exact mathematical passage
```
W_{x,m}=\rho^{m-1}z_{T x}
```

\sigmaUse in source · 18 passages

Search aliases: σ sigma

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 22
τ=fracβ^2+2β15, σ=-fracβ+215, t=-τ, z_0=frac13.
Exact mathematical passage
```
\tau=\frac{\beta^2+2\beta}{15},\qquad
\sigma=-\frac{\beta+2}{15},\qquad t=-\tau,
\quad z_0=\frac13.
```

\tauUse in source · 13 passages

Search aliases: τ tau

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 22
τ=fracβ^2+2β15, σ=-fracβ+215, t=-τ, z_0=frac13.
Exact mathematical passage
```
\tau=\frac{\beta^2+2\beta}{15},\qquad
\sigma=-\frac{\beta+2}{15},\qquad t=-\tau,
\quad z_0=\frac13.
```

\psiUse in source · 2 passages

Search aliases: ψ psi

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 332
ψ_w(k,ℓ,j)=ζ_7^3k.
Exact mathematical passage
```
\boxed{\psi_w(k,\ell,j)=\zeta_7^{3k}.}
```

\psi_wUse in source · 2 passages

Search aliases: ψ_w psi

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 332
ψ_w(k,ℓ,j)=ζ_7^3k.
Exact mathematical passage
```
\boxed{\psi_w(k,\ell,j)=\zeta_7^{3k}.}
```

\omegaUse in source · 3 passages

Search aliases: ω omega

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 15
ω^2+ω+1=0, β^3=7, ε=2-β, ρ=ε^-1=β^2+2β+4.
Exact mathematical passage
```
\omega^2+\omega+1=0,\quad \beta^3=7,\quad
\varepsilon=2-\beta,\quad \rho=\varepsilon^{-1}=\beta^2+2\beta+4.
```

\omega^2Use in source · 2 passages

Search aliases: ω^2 omega

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 15
ω^2+ω+1=0, β^3=7, ε=2-β, ρ=ε^-1=β^2+2β+4.
Exact mathematical passage
```
\omega^2+\omega+1=0,\quad \beta^3=7,\quad
\varepsilon=2-\beta,\quad \rho=\varepsilon^{-1}=\beta^2+2\beta+4.
```

\ldotsUse in source · 1 passage

Search aliases: … ldots

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 119
z_k,ℓ,j=z_0+kδ+ℓη+jξ, k,ℓ,j∈{0,…,6}.
Exact mathematical passage
```
\boxed{z_{k,\ell,j}=z_0+k\delta+\ell\eta+j\xi,\qquad
k,\ell,j\in\{0,\ldots,6\}.}
```

\ellUse in source · 16 passages

Search aliases: ℓ ell

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 117
The seven-point packet has arguments z_0+kδ. The first refinement consists of z_0+kδ+ℓη, with 49 labels. The full ambient packet is
Exact mathematical passage
```
z_0+k\delta+\ell\eta
```

\ell'Use in source · 2 passages

Search aliases: ℓ' ell

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 167
Write (k',ℓ',j')=(A_0,B_0,C_0)bmod7. Then
Exact mathematical passage
```
(k',\ell',j')=(A_0,B_0,C_0)\bmod7
```

\toUse in source · 2 passages

Search aliases: → to

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 241
As ρ^m z_x→0, all f_x^(m)→ F(0)=ε^2. Every multiplier is nonzero at the chosen arguments, and C_x^(m)=1+O(|ρ|^m-1).
Exact mathematical passage
```
\rho^m z_x\to0
```

\inUse in source · 6 passages

Search aliases: ∈ in

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 119
z_k,ℓ,j=z_0+kδ+ℓη+jξ, k,ℓ,j∈{0,…,6}.
Exact mathematical passage
```
\boxed{z_{k,\ell,j}=z_0+k\delta+\ell\eta+j\xi,\qquad
k,\ell,j\in\{0,\ldots,6\}.}
```

\prodUse in source · 4 passages

Search aliases: ∏ prod

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 216
P(g;n,w,h)= ∏_r=0^n-1g(w+rh), · n>0, 1, · n=0, ∏_r=n^-1g(w+rh)^-1, · n<0.
Exact mathematical passage
```
\mathcal P(g;n,w,h)=
\begin{cases}
\prod_{r=0}^{n-1}g(w+rh),&n>0,\\
1,&n=0,\\
\prod_{r=n}^{-1}g(w+rh)^{-1},&n<0.
\end{cases}
```

\prod_{(u,v,j)\ne(0,0,0)}Use in source · 1 passage

Search aliases: ∏_(u,v,j)≠(0,0,0) prod ne

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 313
∏_(u,v,j)≠(0,0,0) F(frac13+fracut+vσ+j7)=1.
Exact mathematical passage
```
\boxed{
\prod_{(u,v,j)\ne(0,0,0)}
F\left(\frac13+\frac{ut+v\sigma+j}{7}\right)=1.
}
```

\prod_{j=0}^{n-1}Use in source · 1 passage

Search aliases: ∏_j=0^n-1 prod

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 271
N_n F(z)= ∏_u=1-n^0∏_v=0^n-1∏_j=0^n-1 F(z+fracut+vσ+jn).
Exact mathematical passage
```
\mathcal N_n F(z)=
\prod_{u=1-n}^{0}\prod_{v=0}^{n-1}\prod_{j=0}^{n-1}
F\left(z+\frac{ut+v\sigma+j}{n}\right).
```

\prod_{m=1}^{\infty}Use in source · 1 passage

Search aliases: ∏_m=1^∞ prod infty

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 246
F(z_x)=ε^2 ∏_m=1^∞ (C_T^-mx^(m))^-1.
Exact mathematical passage
```
\boxed{
F(z_x)=\varepsilon^2
\prod_{m=1}^{\infty}
\left(C_{T^{-m}x}^{(m)}\right)^{-1}.
}
```

\prod_{r=0}^{n-1}Use in source · 1 passage

Search aliases: ∏_r=0^n-1 prod

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 216
P(g;n,w,h)= ∏_r=0^n-1g(w+rh), · n>0, 1, · n=0, ∏_r=n^-1g(w+rh)^-1, · n<0.
Exact mathematical passage
```
\mathcal P(g;n,w,h)=
\begin{cases}
\prod_{r=0}^{n-1}g(w+rh),&n>0,\\
1,&n=0,\\
\prod_{r=n}^{-1}g(w+rh)^{-1},&n<0.
\end{cases}
```

\prod_{r=n}^{-1}Use in source · 1 passage

Search aliases: ∏_r=n^-1 prod

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 216
P(g;n,w,h)= ∏_r=0^n-1g(w+rh), · n>0, 1, · n=0, ∏_r=n^-1g(w+rh)^-1, · n<0.
Exact mathematical passage
```
\mathcal P(g;n,w,h)=
\begin{cases}
\prod_{r=0}^{n-1}g(w+rh),&n>0,\\
1,&n=0,\\
\prod_{r=n}^{-1}g(w+rh)^{-1},&n<0.
\end{cases}
```

\prod_{u=1-n}^{0}Use in source · 1 passage

Search aliases: ∏_u=1-n^0 prod

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 271
N_n F(z)= ∏_u=1-n^0∏_v=0^n-1∏_j=0^n-1 F(z+fracut+vσ+jn).
Exact mathematical passage
```
\mathcal N_n F(z)=
\prod_{u=1-n}^{0}\prod_{v=0}^{n-1}\prod_{j=0}^{n-1}
F\left(z+\frac{ut+v\sigma+j}{n}\right).
```

\prod_{v=0}^{n-1}Use in source · 1 passage

Search aliases: ∏_v=0^n-1 prod

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 271
N_n F(z)= ∏_u=1-n^0∏_v=0^n-1∏_j=0^n-1 F(z+fracut+vσ+jn).
Exact mathematical passage
```
\mathcal N_n F(z)=
\prod_{u=1-n}^{0}\prod_{v=0}^{n-1}\prod_{j=0}^{n-1}
F\left(z+\frac{ut+v\sigma+j}{n}\right).
```

\sumUse in source · 5 passages

Search aliases: ∑ sum

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 262
L_t,σ(z)=-5∑_substackm≥1 5nmid m fracP^m e^2π imz-Q^m e^-2π imz m(1-P^m)(1-Q^m), P=e^2π it, Q=e^2π iσ.
Exact mathematical passage
```
L_{t,\sigma}(z)=-5\sum_{\substack{m\ge1\\5\nmid m}}
\frac{P^m e^{2\pi imz}-Q^m e^{-2\pi imz}}
{m(1-P^m)(1-Q^m)},\qquad
P=e^{2\pi it},\ Q=e^{2\pi i\sigma}.
```

\sum_{b\in V}Use in source · 1 passage

Search aliases: ∑_b∈ V sum in

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 370
widehat f_m(a)=frac1343∑_b∈ V widehat C_m(a-b) widehat f_m-1(T^-Tb).
Exact mathematical passage
```
\boxed{
\widehat f_m(a)=\frac1{343}\sum_{b\in V}
\widehat C_m(a-b)\,\widehat f_{m-1}(T^{-T}b).
}
```

\sum_{\substack{m\ge1\\5\nmid m}}Use in source · 1 passage

Search aliases: ∑_substackm≥1 5nmid m sum substack ge nmid

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 262
L_t,σ(z)=-5∑_substackm≥1 5nmid m fracP^m e^2π imz-Q^m e^-2π imz m(1-P^m)(1-Q^m), P=e^2π it, Q=e^2π iσ.
Exact mathematical passage
```
L_{t,\sigma}(z)=-5\sum_{\substack{m\ge1\\5\nmid m}}
\frac{P^m e^{2\pi imz}-Q^m e^{-2\pi imz}}
{m(1-P^m)(1-Q^m)},\qquad
P=e^{2\pi it},\ Q=e^{2\pi i\sigma}.
```

\sum_{\substack{q\ge1\\5\nmid nq}}Use in source · 1 passage

Search aliases: ∑_substackq≥1 5nmid nq sum substack ge nmid

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 297
-5∑_substackq≥1 5nmid nq fracP^q e^2π i nq z-Q^q e^-2π i nq z q(1-P^q)(1-Q^q).
Exact mathematical passage
```
-5\sum_{\substack{q\ge1\\5\nmid nq}}
\frac{P^q e^{2\pi i nq z}-Q^q e^{-2\pi i nq z}}
{q(1-P^q)(1-Q^q)}.
```

\sum_{u=1-n}^{0}Use in source · 1 passage

Search aliases: ∑_u=1-n^0 sum

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 288
∑_u=1-n^0P^qu =P^-(n-1)qfrac1-P^nq1-P^q, ∑_v=0^n-1Q^qv =frac1-Q^nq1-Q^q.
Exact mathematical passage
```
\sum_{u=1-n}^{0}P^{qu}
=P^{-(n-1)q}\frac{1-P^{nq}}{1-P^q},\qquad
\sum_{v=0}^{n-1}Q^{qv}
=\frac{1-Q^{nq}}{1-Q^q}.
```

\sum_{v=0}^{n-1}Use in source · 1 passage

Search aliases: ∑_v=0^n-1 sum

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 288
∑_u=1-n^0P^qu =P^-(n-1)qfrac1-P^nq1-P^q, ∑_v=0^n-1Q^qv =frac1-Q^nq1-Q^q.
Exact mathematical passage
```
\sum_{u=1-n}^{0}P^{qu}
=P^{-(n-1)q}\frac{1-P^{nq}}{1-P^q},\qquad
\sum_{v=0}^{n-1}Q^{qv}
=\frac{1-Q^{nq}}{1-Q^q}.
```

\sum_{x\in V}Use in source · 1 passage

Search aliases: ∑_x∈ V sum in

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 358
widehat f(a)=∑_x∈ Vζ^a· xf(x), a∈ V.
Exact mathematical passage
```
\widehat f(a)=\sum_{x\in V}\zeta^{a\cdot x}f(x),\qquad a\in V.
```

\inftyUse in source · 1 passage

Search aliases: ∞ infty

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 246
F(z_x)=ε^2 ∏_m=1^∞ (C_T^-mx^(m))^-1.
Exact mathematical passage
```
\boxed{
F(z_x)=\varepsilon^2
\prod_{m=1}^{\infty}
\left(C_{T^{-m}x}^{(m)}\right)^{-1}.
}
```

\neUse in source · 4 passages

Search aliases: ≠ ne

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 49
The previously established zero-anchor normalization is F(0)=ε^2≠0. The divisor support is contained in Λ_5= Zτ+ Zσ+tfrac15 Z. These analytic facts are inherited inputs, not freshly formalized in Lean here.
Exact mathematical passage
```
F(0)=\varepsilon^2\ne0
```

\geUse in source · 4 passages

Search aliases: ≥ ge

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 204
f_x^(m)=F(ρ^m z_x), m≥0.
Exact mathematical passage
```
f_x^{(m)}=F(\rho^m z_x),\qquad m\ge0.
```

\langleUse in source · 2 passages

Search aliases: ⟨ langle

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 105
The essential duality is ⟨ u,w⟩_H=1 while ⟨ u,u⟩_H=⟨ u,v⟩_H=0 modulo seven. A stable linear subspace containing w also contains v=Nw and u=N^2w. Thus both the full unit action and this nontrivial socle-to-top pairing require the complete three-dimensional module.
Exact mathematical passage
```
\langle u,w\rangle_H=1
```

\rangleUse in source · 2 passages

Search aliases: ⟩ rangle

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 105
The essential duality is ⟨ u,w⟩_H=1 while ⟨ u,u⟩_H=⟨ u,v⟩_H=0 modulo seven. A stable linear subspace containing w also contains v=Nw and u=N^2w. Thus both the full unit action and this nontrivial socle-to-top pairing require the complete three-dimensional module.
Exact mathematical passage
```
\langle u,w\rangle_H=1
```

\rangle_HUse in source · 2 passages

Search aliases: ⟩_H rangle

Link to this entry ·

Use in source · H12_Contracting_Ambient_Torsion_Research_Note.md, line 105
The essential duality is ⟨ u,w⟩_H=1 while ⟨ u,u⟩_H=⟨ u,v⟩_H=0 modulo seven. A stable linear subspace containing w also contains v=Nw and u=N^2w. Thus both the full unit action and this nontrivial socle-to-top pairing require the complete three-dimensional module.
Exact mathematical passage
```
\langle u,w\rangle_H=1
```

Source coverage

Automatic extraction covers TeX and Markdown math delimiters and common equation environments in these selected files. It does not interpret diagrams, images, PDF text, program identifiers, or historical archives. Occurrences are navigation aids, not a proof review.

Download all entries, passages and source checksums (JSON) · Collection coverage

research/hilbert12-contracting-ambient-torsion/H12_Ambient_Torsion_README.md · 2 mathematical passages
research/hilbert12-contracting-ambient-torsion/H12_Contracting_Ambient_Torsion_Research_Note.md · 131 mathematical passages
research/hilbert12-contracting-ambient-torsion/H12_Full_Ambient_Torsion_Verification_Log.txt · 0 mathematical passages