H12_Contracting_Ambient_Torsion_Research_Note.md
Hilbert 12: contracting ambient torsion
H12_Contracting_Ambient_Torsion_Research_Note.md
1 # Contracting Arithmetic Torsion and Theta Transport in the Pure Cubic Hilbert 12 Test
2
3 **Author:** Komorebi P.
4
5 **Status:** Exact arithmetic, finite group, and analytic distribution identities. The required elliptic-gamma to class-field comparison remains open.
6
7 ## Abstract
8
9 The inverse cubic unit acting on the full seventh-torsion reduction of the original period lattice has order 21, although its distinguished eigenline has order three. The seven-point packet therefore omits the unipotent extension needed to retain the perfect dual pairing with the complementary torsion direction. An explicit 343-point ambient packet restores that data. Its action has one fixed point, two three-cycles, and sixteen twenty-one-cycles. Exact integer period carries are obtained for every point, yielding a theta-valued contracting transport for the full packet. Independently, a three-direction elliptic-gamma distribution formula is proved by roots-of-unity filtering. In the conductor-three specialization its untwisted seventh norm cancels the distinguished gamma value, identifying a precise limitation of scalar norms. The remaining mathematical problem is a normalized, character-sensitive analytic-to-arithmetic comparison.
10
11 ## 1. Fixed setting
12
13 Let
14
15 \[
16 \omega^2+\omega+1=0,\quad \beta^3=7,\quad
17 \varepsilon=2-\beta,\quad \rho=\varepsilon^{-1}=\beta^2+2\beta+4.
18 \]
19
20 Choose the complex embedding \(\beta=7^{1/3}\omega^2\), so \(|\rho|<1\). Put
21
22 \[
23 \tau=\frac{\beta^2+2\beta}{15},\qquad
24 \sigma=-\frac{\beta+2}{15},\qquad t=-\tau,
25 \quad z_0=\frac13.
26 \]
27
28 The analytic function is the fifth-smoothed elliptic gamma
29
30 \[
31 F(z)=\frac{\Gamma_{\mathrm{ell}}(z;\tau,\sigma)^5}
32 {\Gamma_{\mathrm{ell}}(5z;5\tau,5\sigma)},\qquad U_\Gamma=F(z_0).
33 \]
34
35 The inherited exact functional equations are
36
37 \[
38 F(z+1)=F(z),\qquad
39 F(z+\tau)=q_\sigma(z)F(z),\qquad
40 F(z+\sigma)=q_t(z)^{-1}F(z),
41 \]
42
43 where
44
45 \[
46 q_a(z)=\frac{\theta_0(z;a)^5}{\theta_0(5z;5a)}.
47 \]
48
49 The previously established zero-anchor normalization is \(F(0)=\varepsilon^2\ne0\). The divisor support is contained in
50 \(\Lambda_5=\mathbb Z\tau+\mathbb Z\sigma+\tfrac15\mathbb Z\). These analytic facts are inherited inputs, not freshly formalized in Lean here.
51
52 The arithmetic lattice is \(\Lambda=\mathbb Z\tau+\mathbb Z\sigma+\mathbb Z\). Multiplication by \(\varepsilon\) on its ordered basis is
53
54 \[
55 M=\begin{pmatrix}0&1&0\\-4&2&15\\-1&0&4\end{pmatrix},\qquad
56 R=M^{-1}=\begin{pmatrix}8&-4&15\\1&0&0\\2&-1&4\end{pmatrix}.
57 \]
58
59 We use \(R\) for the integral matrix and \(\rho\) for the corresponding scalar multiplication.
60
61 ## 2. Semisimple and unipotent actions at seven
62
63 Let \(N=M-2I\pmod7\). Then \(N^3=0\) and \(N^2\ne0\). Define
64
65 \[
66 u=(1,2,4)^T,\quad v=(0,1,2)^T,\quad w=(0,0,1)^T.
67 \]
68
69 Modulo seven they give the complete Jordan chain
70
71 \[
72 Nu=0,\qquad Nv=u,\qquad Nw=v.
73 \]
74
75 The matrix with columns \(u,v,w\) has determinant one. Consequently the three vectors form an integral basis.
76
77 In characteristic seven,
78
79 \[
80 R=(2I+N)^{-1}=4I+5N+N^2.
81 \]
82
83 Cubing and taking seventh powers gives
84
85 \[
86 R^3-I=2N+5N^2\pmod7,\qquad
87 R^7=4I\pmod7,\qquad R^{21}=I\pmod7.
88 \]
89
90 Because \(R^3-I\ne0\), the order of \(R\) in \(\mathrm{GL}_3(\mathbb F_7)\) 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.
91
92 For reference, the codifferent pairing from the earlier study has Gram matrix
93
94 \[
95 H=\begin{pmatrix}4&-4&15\\-4&1&0\\15&0&0\end{pmatrix},\qquad
96 M^T H=HM,
97 \]
98
99 and, in the ordered basis \(u,v,w\), its reduction is
100
101 \[
102 \begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}\pmod7.
103 \]
104
105 The essential duality is \(\langle u,w\rangle_H=1\) while \(\langle u,u\rangle_H=\langle u,v\rangle_H=0\) 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.
106
107 ## 3. Three levels of torsion packets
108
109 Put
110
111 \[
112 \delta=\frac{\tau+2\sigma+4}{7},\qquad
113 \eta=\frac{\sigma+2}{7},\qquad
114 \xi=\frac17.
115 \]
116
117 The seven-point packet has arguments \(z_0+k\delta\). The first refinement consists of \(z_0+k\delta+\ell\eta\), with 49 labels. The full ambient packet is
118
119 \[
120 \boxed{z_{k,\ell,j}=z_0+k\delta+\ell\eta+j\xi,\qquad
121 k,\ell,j\in\{0,\ldots,6\}.}
122 \]
123
124 Since \(\delta,\eta,\xi\) are the images of \(u/7,v/7,w/7\), and \(\det[u\ v\ w]=1\), the 343 labels represent all of \(\tfrac17\Lambda/\Lambda\). Here this is torsion of the rank-three lattice quotient, not a claim that the nondiscrete rank-three quotient is an ordinary elliptic curve.
125
126 The three generator transport equations are exact:
127
128 \[
129 \begin{aligned}
130 \rho z_0&=z_0+5\tau+1,\\
131 \rho\delta&=4\delta+8\tau-\sigma,\\
132 \rho\eta&=5\delta+4\eta+3\tau-2\sigma-3,\\
133 \rho\xi&=\delta+5\eta+4\xi+2\tau-\sigma-2.
134 \end{aligned}
135 \]
136
137 Hence the induced action on finite labels is
138
139 \[
140 \boxed{T(k,\ell,j)=(4k+5\ell+j,\ 4\ell+5j,\ 4j)\pmod7.}
141 \]
142
143 The complete orbit decomposition is
144
145 \[
146 \boxed{343=1+2\cdot3+16\cdot21.}
147 \]
148
149 Restricting to \(j=0\) gives \(49=1+2\cdot3+2\cdot21\). Restricting again to \(\ell=j=0\) gives \(7=1+2\cdot3\). All orbit decompositions were enumerated exactly. The quotient action, not a heuristic spectral analogy, produces these numbers.
150
151 ### The explicit 343 carry formula
152
153 For \((k,\ell,j)\), set
154
155 \[
156 A_0=4k+5\ell+j,\qquad B_0=4\ell+5j,\qquad C_0=4j,
157 \]
158
159 and let
160
161 \[
162 q_1=\lfloor A_0/7\rfloor,\quad
163 q_2=\lfloor B_0/7\rfloor,\quad
164 q_3=\lfloor C_0/7\rfloor.
165 \]
166
167 Write \((k',\ell',j')=(A_0,B_0,C_0)\bmod7\). Then
168
169 \[
170 \boxed{
171 \rho z_{k,\ell,j}=
172 z_{k',\ell',j'}+a\tau+b\sigma+c
173 }
174 \]
175
176 with the **integer** carries
177
178 \[
179 \begin{aligned}
180 a&=5+8k+3\ell+2j+q_1,\\
181 b&=-k-2\ell-j+2q_1+q_2,\\
182 c&=1-3\ell-2j+4q_1+2q_2+q_3.
183 \end{aligned}
184 \]
185
186 These identities were checked symbolically for every one of the 343 labels. They are also immediate by substituting the four generator equations above and reducing each generator modulo its seventh period.
187
188 ### Divisor avoidance
189
190 No \(z_{k,\ell,j}\) lies in \(\Lambda_5\). To see this, compare the rationally independent coefficients of \(\tau,\sigma,1\). The coefficient of \(\tau\) forces \(k=0\), that of \(\sigma\) then forces \(\ell=0\), while \(1/3+j/7\) is never in \(\tfrac15\mathbb Z\). The unit \(\rho\) preserves \(\Lambda_5\) since
191
192 \[
193 \rho\tau=8\tau+\sigma+2,\qquad
194 \rho\sigma=-4\tau-1,\qquad
195 \rho/5=3\tau+4/5,
196 \]
197
198 and the inverse \(\varepsilon\) also preserves \(\Lambda_5\). Thus every contracting orbit avoids the divisor of \(F\).
199
200 ## 4. An exact theta-valued contracting transport
201
202 Let \(x=(k,\ell,j)\in V=(\mathbb Z/7)^3\) and \(z_x=z_{k,\ell,j}\). Define
203
204 \[
205 f_x^{(m)}=F(\rho^m z_x),\qquad m\ge0.
206 \]
207
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 \(d_{x,m}=R^{m-1}d_x\in\mathbb Z^3\). Set \(W_{x,m}=\rho^{m-1}z_{T x}\). By the arithmetic transport formula,
209
210 \[
211 \rho^m z_x=W_{x,m}+a_{x,m}\tau+b_{x,m}\sigma+c_{x,m}.
212 \]
213
214 For an integer \(n\), define the oriented finite product
215
216 \[
217 \mathcal P(g;n,w,h)=
218 \begin{cases}
219 \prod_{r=0}^{n-1}g(w+rh),&n>0,\\
220 1,&n=0,\\
221 \prod_{r=n}^{-1}g(w+rh)^{-1},&n<0.
222 \end{cases}
223 \]
224
225 The multiplier is the explicit theta product
226
227 \[
228 \boxed{
229 C_x^{(m)}=
230 \mathcal P(q_t^{-1};b_{x,m},W_{x,m},\sigma)\,
231 \mathcal P(q_\sigma;a_{x,m},W_{x,m}+b_{x,m}\sigma,\tau).
232 }
233 \]
234
235 It follows solely from the gamma shift identities that
236
237 \[
238 \boxed{f_x^{(m)}=C_x^{(m)} f_{T x}^{(m-1)}.}
239 \]
240
241 As \(\rho^m z_x\to0\), all \(f_x^{(m)}\to F(0)=\varepsilon^2\). Every multiplier is nonzero at the chosen arguments, and
242 \(C_x^{(m)}=1+O(|\rho|^{m-1})\).
243
244 Thus the finite telescope has an absolutely convergent infinite limit:
245
246 \[
247 \boxed{
248 F(z_x)=\varepsilon^2
249 \prod_{m=1}^{\infty}
250 \left(C_{T^{-m}x}^{(m)}\right)^{-1}.
251 }
252 \]
253
254 This exact expression is branch-free: it uses only ordinary meromorphic theta quotients and their products. For the seven-point eigenline \(T^{-m}k=2^m k\bmod7\), recovering the previous seven-packet identity. The full 343-point form also transports all unipotent directions.
255
256 This is a normalized *analytic* realization. No conclusion that its special values belong to the cubic class field is drawn.
257
258 ## 5. A complete three-direction distribution identity
259
260 The absolutely convergent logarithm of our fifth-smoothed gamma on a suitable strip is
261
262 \[
263 L_{t,\sigma}(z)=-5\sum_{\substack{m\ge1\\5\nmid m}}
264 \frac{P^m e^{2\pi imz}-Q^m e^{-2\pi imz}}
265 {m(1-P^m)(1-Q^m)},\qquad
266 P=e^{2\pi it},\ Q=e^{2\pi i\sigma}.
267 \]
268
269 For \(n\ge1\), define the complete rank-three distribution
270
271 \[
272 \mathcal N_n F(z)=
273 \prod_{u=1-n}^{0}\prod_{v=0}^{n-1}\prod_{j=0}^{n-1}
274 F\left(z+\frac{ut+v\sigma+j}{n}\right).
275 \]
276
277 **Distribution theorem.** As meromorphic functions,
278
279 \[
280 \boxed{
281 \mathcal N_nF(z)=
282 \begin{cases}F(nz),&5\nmid n,\\1,&5\mid n.\end{cases}
283 }
284 \]
285
286 **Proof.** Work first where all logarithmic series converge normally, which includes real \(z\) in the fixed upper half-plane conventions. Summing over \(j\) gives zero unless the Fourier index is divisible by \(n\). For Fourier index \(nq\), the two other finite geometric series yield
287
288 \[
289 \sum_{u=1-n}^{0}P^{qu}
290 =P^{-(n-1)q}\frac{1-P^{nq}}{1-P^q},\qquad
291 \sum_{v=0}^{n-1}Q^{qv}
292 =\frac{1-Q^{nq}}{1-Q^q}.
293 \]
294
295 The analogous inverse sums in the negative part cancel its denominators in the same way. Thus the total logarithm is
296
297 \[
298 -5\sum_{\substack{q\ge1\\5\nmid nq}}
299 \frac{P^q e^{2\pi i nq z}-Q^q e^{-2\pi i nq z}}
300 {q(1-P^q)(1-Q^q)}.
301 \]
302
303 When \(5\nmid n\), the condition \(5\nmid nq\) is exactly \(5\nmid q\), giving \(L_{t,\sigma}(nz)\). When \(5\mid n\), no terms remain and the logarithm is zero. Exponentiating proves the equality in the convergence region. Meromorphic continuation gives the stated identities. \(\square\)
304
305 At \(n=7\) and \(z_0=1/3\), period-one invariance gives
306
307 \[
308 \boxed{\mathcal N_7F(1/3)=F(7/3)=F(1/3)=U_\Gamma.}
309 \]
310
311 The chosen shifts \((ut+v\sigma+j)/7\) represent the same full 343-element torsion module as \(k\delta+\ell\eta+j\xi\), but their selected representatives differ by integer periods, which introduces explicit theta factors. In the canonical distribution representatives the fixed, zero-shift factor is \(U_\Gamma\). Removing it gives the exact 342-factor identity
312
313 \[
314 \boxed{
315 \prod_{(u,v,j)\ne(0,0,0)}
316 F\left(\frac13+\frac{ut+v\sigma+j}{7}\right)=1.
317 }
318 \]
319
320 **Interpretation.** The untwisted complete norm cannot determine the arithmetic phase of \(U_\Gamma\), because that factor cancels. Extra finite torsion alone does not remove the need for an oriented or character-sensitive comparison.
321
322 ## 6. The dual-character target
323
324 In the integral Jordan basis \(u,v,w\), the perfect codifferent pairing modulo seven has matrix
325
326 \[
327 G_H=\begin{pmatrix}0&0&1\\0&1&0\\1&0&0\end{pmatrix}.
328 \]
329
330 Fix the oriented seventh-root phase \(\zeta_7=e^{2\pi i/7}\). The additive character against the top dual direction is
331
332 \[
333 \boxed{\psi_w(k,\ell,j)=\zeta_7^{3k}.}
334 \]
335
336 Under inverse-unit transport, its exponent changes as
337
338 \[
339 \boxed{\psi_w(T(k,\ell,j))=\zeta_7^{3(4k+5\ell+j)}.}
340 \]
341
342 So the dual character is sensitive to all levels of the Jordan chain after transport, unlike its restriction to the seven-point eigenline. This explicitly identifies the missing extension data.
343
344 One must not form arbitrary complex-character powers of the gamma values and call the result a single-valued meromorphic function. The earlier primitive-sector residue obstruction rules that out. A valid comparison should use integral monomial coordinates, a geometric theta/gerbe lift, or a coherently normalized logarithmic/cocycle construction.
345
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 \(J=2+3\omega\). The fixed desired scalar identity remains
347
348 \[
349 \operatorname{Im}(C_{\mathrm{pol}}U_\Gamma^3)=0
350 \]
351
352 with the already established modulus and separately specified branch conditions. None of the present identities establishes this equality.
353
354 ## 7. Character transport in the full Fourier space
355
356 Let \(V=(\mathbb Z/7)^3\), \(\zeta=\exp(2\pi i/7)\), and let \(\mathcal F\) be the ordinary finite Fourier transform
357
358 \[
359 \widehat f(a)=\sum_{x\in V}\zeta^{a\cdot x}f(x),\qquad a\in V.
360 \]
361
362 It uses no multivalued powers of the function values. For the actual theta-return vector, the preceding theorem gives
363
364 \[
365 f_m(x)=C_m(x)f_{m-1}(Tx),\qquad x\in V.
366 \]
367
368 Fourier inversion and a change of variables give the **exact dual transfer formula**
369
370 \[
371 \boxed{
372 \widehat f_m(a)=\frac1{343}\sum_{b\in V}
373 \widehat C_m(a-b)\,\widehat f_{m-1}(T^{-T}b).
374 }
375 \]
376
377 Thus the arithmetic operation sends each dual character to its inverse-transpose transport, while multiplication by the analytic theta multiplier produces convolution in Fourier space. This retains precisely the mode coupling that disappears if one diagonalizes only the cubic permutation and drops its theta carries.
378
379 In the basis \(u,v,w\), the dual transport matrix is
380
381 \[
382 T^{-T}=\begin{pmatrix}2&0&0\\1&2&0\\0&1&2\end{pmatrix}\pmod7.
383 \]
384
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
386
387 \[
388 (2,1,0)^T,\qquad (4,4,1)^T,\qquad(1,5,6)^T.
389 \]
390
391 A primitive dual character therefore does not close under a single order-three Fourier eigenvalue. Even before analytic theta mixing, it generates a twenty-one-term arithmetic orbit. The missing phase comparison should respect this full dual transport and its prescribed orientation.
392
393 The displayed Fourier formula is finite linear algebra and is proved independently of the unproved arithmetic evaluation. It supplies an exact, single-valued computational target, not a claim that any weighted gamma sum is already algebraic.
394
395 ## 8. Verification record and boundaries
396
397 The accompanying verifier checks 343 exact carries, the full orbit classification, the order-21 reduction, the length-three pairing, the two lower-level packets, and the arithmetic support argument. The seven-packet companion checks the rank-two monomial action and first-step theta formulas.
398
399 An independent numerical distribution test covers \(n=2,3,5,7\) with 38-digit arithmetic and finite truncation. The absolute logarithmic residual in the n=7 test was approximately \(10^{-37}\). These are regression tests and not directed interval certificates. The multiplication theorem itself is the exact Fourier-series proof above.
400
401 **Not proved:** the BCG elliptic-gamma algebraicity and reciprocity conjectures, the exact polarized phase, the Eisenstein deck descent/gerbe identification, or the general Hilbert 12 problem. No new Lean compilation was performed for this checkpoint.
402
403 ## Source foundations
404
405 - *Cubic Divisor Transport and Reconstruction*, Komorebi P.
406 - *An Ambient Cyclic Cubic and the Elliptic Gamma Fourier Obstruction*, Komorebi P.
407 - *An ambient dual slice for the pure cubic Hilbert 12 experiment*, Komorebi P.
408 - *Holomorphic descent, complementary gerbe torsion, and the oriented Gauss character*, Komorebi P.
409 - Bergeron, Charollois and García, *Elliptic Units for Complex Cubic Fields*. The algebraicity and complex reciprocity claims remain conjectural in the source material.
410
411 The results above are mathematical continuations under the inherited analytic conventions. The algebraic identities are self-contained, and every claim beyond their scope is explicitly separated from them.