Prime-Contact Coherence Calculus

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1 \PartChapter{Detailed Theory: Signal, Automorphic, and Filtered Contact Structures}{Prime-Shell Signals and the Frey Nonlinear Transform}
2 
3 \section{Prime support as bandwidth and valuation as persistence}
4 For
5 \[
6  n=\prod_p p^{e_p},
7 \]
8 define the prime-shell signal
9 \[
10  \Psi_n(p,r)
11  =
12  \sqrt{\log p}\,\mathbf 1_{0\le r<e_p}.
13 \]
14 Its total energy is
15 \[
16  \norm{\Psi_n}_2^2
17  =
18  \sum_p e_p\log p
19  =
20  \boxed{\log n}.
21 \]
22 Let $P_{\mathrm{act}}$ retain only the first shell in each active prime channel. Then
23 \[
24  \norm{P_{\mathrm{act}}\Psi_n}_2^2
25  =
26  \sum_{p\mid n}\log p
27  =
28  \boxed{\log\rad(n)}.
29 \]
30 Thus the radical is activated bandwidth, while the difference
31 \[
32  \log n-\log\rad(n)
33 \]
34 is repeated persistence depth.
35 
36 For a primitive triple, the direct-sum signal
37 \[
38  \Psi_{a,b,c}
39  =
40  \Psi_a\oplus\Psi_b\oplus\Psi_c
41 \]
42 has disjoint prime support in the three sectors. The standard quality becomes the exact energy ratio
43 \[
44  \boxed{
45  q(a,b,c)
46  =
47  \frac{\log c}{\log\rad(abc)}
48  =
49  \frac{\norm{P_c\Psi_{a,b,c}}_2^2}
50  {\norm{P_{\mathrm{act}}\Psi_{a,b,c}}_2^2}.
51  }
52 \]
53 High quality means large output persistence relative to activated global bandwidth.
54 
55 \section{Unitary transforms cannot create a radical saving}
56 Apply a unitary Fourier, wavelet, or polyphase transform in the shell variable. Parseval preserves total shell energy. Therefore the ratio between total depth energy and activated support energy is unchanged by a unitary shell analysis followed by exact synthesis.
57 
58 \begin{proposition}[Unitary no-gain principle]
59 Let $U$ be unitary on the shell coordinate of every active prime channel. Then
60 \[
61  \norm{U\Psi_n}_2=\norm{\Psi_n}_2.
62 \]
63 No inequality that depends only on these two energies becomes stronger after applying $U$.
64 \end{proposition}
65 
66 Signal theory is still useful because it reveals which nonunitary or nonlinear operations would be required. It also separates local persistence from global support before introducing a nonlinear arithmetic carrier.
67 
68 \section{The three-spike \texorpdfstring{$p$-adic}{p-adic} coherence signal}
69 Represent the additive triple by
70 \[
71  \nu_{a,b,c}=\delta_0+\delta_a+\delta_c.
72 \]
73 The pairwise differences are $a$, $b$, and $c$. At a prime dividing exactly one of these integers, exactly one pair of spikes is $p$-adically close. The depth of that edge is the corresponding valuation.
74 
75 The triple therefore determines a prime-indexed coherence triangle. The radical counts the active triangles. The valuations count how long a pair remains coupled through the $p$-adic refinement tree. This picture anticipates the later cross-prime logarithmic network.
76 
77 \section{The Frey curve}
78 Attach the marked Frey curve
79 \[
80  \boxed{
81  E_{a,b,c}:y^2=x(x-a)(x+b).
82  }
83 \]
84 The roots $0$, $a$, and $-b$ retain the three differences $a$, $b$, and $c$. The basic invariants are
85 \[
86  \Delta_E=16a^2b^2c^2
87 \]
88 and
89 \[
90  c_4(E)=16(a^2+ab+b^2).
91 \]
92 At every odd prime $p\mid abc$, the curve has multiplicative reduction. The conductor records that the channel is active, while the minimal discriminant records the full depth.
93 
94 In signal language, the Frey curve is a nonlinear encoder that converts:
95 \[
96  \begin{aligned}
97  \text{prime support}&\longrightarrow\text{conductor},\\
98  \text{valuation depth}&\longrightarrow\text{minimal discriminant depth},\\
99  \text{additive relation}&\longrightarrow\text{elliptic geometry}.
100  \end{aligned}
101 \]
102 This is precisely the kind of nonlinear transform that a unitary shell system could not provide.
103 
104 \section{The automorphic eigenchannel}
105 Modularity associates a weight-two newform $f_E$ to the rational isogeny class of $E$. Its Hecke coefficients
106 \[
107  a_\ell(E)
108 \]
109 form a global frequency signal. The conductor is the automorphic bandwidth. The newform, however, remembers the rational isogeny class rather than the marked integral representative.
110 
111 This information loss is essential. Different integral Frey models can lie in the same rational isogeny or twist channel while having different minimal discriminant depth, N\'eron differential, and modular synthesis cost. A proof cannot use the rational eigen-signal alone.
112 
113 \section{Dyadic Landen descent}
114 Let $u>v>0$ be coprime odd integers. Consider
115 \[
116  T_0=(v^2,u^2-v^2,u^2)
117 \]
118 and
119 \[
120  T_1=
121  \left(
122  uv,
123  \left(\frac{u-v}{2}\right)^2,
124  \left(\frac{u+v}{2}\right)^2
125  \right).
126 \]
127 Both are primitive triples. Their radicals are equal:
128 \[
129  \rad(T_0)=\rad(T_1)=\rad\bigl(uv(u-v)(u+v)\bigr).
130 \]
131 The second output is strictly smaller:
132 \[
133  \left(\frac{u+v}{2}\right)^2<u^2.
134 \]
135 Hence the second quality is smaller.
136 
137 The Legendre parameter of the parent is
138 \[
139  \lambda=t^2,
140  \qquad
141  t=v/u.
142 \]
143 The degree-two Landen transformation gives
144 \[
145  \boxed{
146  \lambda
147  \longmapsto
148  \frac{4\sqrt\lambda}{(1+\sqrt\lambda)^2}.
149  }
150 \]
151 This is exactly the child ratio $a_1/c_1$. The transformation preserves the automorphic channel up to the corresponding two-isogeny while lowering the integral height. It proves that the integral representative inside an isogeny class is relevant to $abc$.
152 
153 \section{Exact automorphic analysis--synthesis identity}
154 Let
155 \[
156  \phi_E:X_0(N_E)\longrightarrow E
157 \]
158 be an optimal modular parametrization, and let
159 \[
160  \phi_E^*\omega_E
161  =
162  2\pi i\,c_E f_E(z)\,dz.
163 \]
164 Define the differential area
165 \[
166  A(E)
167  =
168  \frac{i}{2}
169  \int_{E(\C)}
170  \omega_E\wedge\overline{\omega_E}.
171 \]
172 Pulling the area form back gives
173 \[
174  \boxed{
175  \deg(\phi_E)A(E)
176  =
177  4\pi^2 c_E^2
178  \norm{f_E}_{\mathrm{Pet}}^2.
179  }
180 \]
181 For the Frey model, scaling $x=cX$, $y=c^{3/2}Y$ gives an area factor proportional to $1/c$. Thus the height $c$ can be written as a shape-corrected synthesis gain involving modular degree, Manin normalization, and Petersson energy.
182 
183 The identity does not prove a conductor-scale upper bound. It identifies the exact quantities that an integral automorphic frame theorem would have to control.
184 
185 \chapter{Divisor-Polyphase Persistence and the Hecke Filter Bank}
186 
187 \section{Exact-order divisor packets}
188 Let
189 \[
190  n=\prod_p p^{e_p}.
191 \]
192 For every $d\ge1$, define
193 \[
194  \boxed{
195  B_d(n)
196  =
197  \prod_{\substack{p\mid n\\d\mid e_p}}p.
198  }
199 \]
200 Since
201 \[
202  \sum_{d\mid e}\varphi(d)=e,
203 \]
204 we have the perfect reconstruction identity
205 \[
206  \boxed{
207  n
208  =
209  \prod_{d\ge1}
210  B_d(n)^{\varphi(d)}.
211  }
212 \]
213 The first packet is
214 \[
215  B_1(n)=\rad(n).
216 \]
217 The powerful part is
218 \[
219  \boxed{
220  \frac{n}{\rad(n)}
221  =
222  \prod_{d\ge2}
223  B_d(n)^{\varphi(d)}.
224  }
225 \]
226 This is a multiplicative polyphase decomposition indexed by exact character orders.
227 
228 \section{Quality as input-output persistence imbalance}
229 For a primitive triple,
230 \[
231  q-1
232  =
233  \frac{
234  \log(c/R_c)-\log(R_aR_b)
235  }{
236  \log R
237  }.
238 \]
239 Using the divisor packets,
240 \[
241  \boxed{
242  q-1
243  =
244  \frac{
245  \sum_{d\ge2}\varphi(d)\log B_d(c)
246  -
247  \log(R_aR_b)
248  }{\log R}.
249  }
250 \]
251 A high-quality triple has more repeated-depth persistence in the output than the input radicals supply as activated bandwidth.
252 
253 Strong $abc$ is equivalent to
254 \[
255  \prod_{d\ge2}
256  B_d(c)^{\varphi(d)}
257  \le
258  C_\eps R_aR_bR^\eps.
259 \]
260 The decomposition makes the target transparent, but not easier by itself. A global channel must explain why the exact-order packets cannot all be large simultaneously.
261 
262 \section{Local Tate-depth interpretation}
263 At an odd prime $p^e\parallel c$, the Frey discriminant has local depth $2e$. The divisors of $e$ organize the levels at which prime-power coefficient systems can become insensitive to the local extension. The exact-order divisor packet therefore has a local representation-theoretic interpretation, although local disappearance does not automatically produce a global low-level eigenform.
264 
265 \section{Atkin--Lehner Walsh packets}
266 Let
267 \[
268  W_1,\ldots,W_r
269 \]
270 be commuting Atkin--Lehner involutions. For a sign vector
271 \[
272  \varepsilon\in\{\pm1\}^r,
273 \]
274 define
275 \[
276  A_\varepsilon
277  =
278  \prod_{j=1}^{r}(I+\varepsilon_jW_j).
279 \]
280 Then
281 \[
282  \boxed{
283  A_\varepsilon^2=2^rA_\varepsilon,
284  }
285 \]
286 \[
287  A_\varepsilon A_{\varepsilon'}=0
288  \quad(\varepsilon\neq\varepsilon'),
289 \]
290 and
291 \[
292  \boxed{
293  \sum_\varepsilon A_\varepsilon=2^rI.
294  }
295 \]
296 The unnormalized analysis packets are integral. The common synthesis denominator is $2^r$, which is subpolynomial in a squarefree conductor. Local sign separation is therefore not the large persistence carrier.
297 
298 The automorphic interpretation uses the standard Atkin--Lehner and modular-newform framework. The integral separation problem is related to congruence modules and modular degrees~\cite{Wiles,TaylorWiles,BCDT,Agashe,CesnaviciusManin}.
299 
300 \section{The same-sign separation problem}
301 After fixing the local signs, several eigenforms can remain congruent inside the same packet. Let $\Lambda$ be an integral Hecke lattice and let $e_f$ be the rational projector onto the Frey eigenline. Define
302 \[
303  \eta_f
304  =
305  \min\{m\ge1:m e_f(\Lambda)\subseteq\Lambda\}.
306 \]
307 This is the integral eigenchannel separation denominator. It is an arithmetic analogue of leakage in a filter bank. The difficulty is not rational spectral isolation. It is isolation inside the integral lattice.
308 
309 \chapter{Integral Projectors, Cotangent Multiplicity, and Wiles Defect}
310 
311 \section{Projector denominator as a congruence exponent}
312 Let $V=U\oplus W$ be a rational decomposition and let $\Lambda\subset V$ be an integral lattice. Let $e_U$ be the rational projector onto $U$. Define
313 \[
314  \Lambda_U=\Lambda\cap U,
315  \qquad
316  \Lambda_W=\Lambda\cap W,
317 \]
318 and
319 \[
320  C_U=\Lambda/(\Lambda_U+\Lambda_W).
321 \]
322 
323 \begin{theorem}[Integral projector theorem]
324 The least positive integer $\eta_U$ satisfying
325 \[
326  \eta_U e_U(\Lambda)\subseteq\Lambda
327 \]
328 is the exponent of the finite quotient $C_U$.
329 \end{theorem}
330 
331 \begin{proof}
332 For $x=u+w\in\Lambda$, the condition $m e_U(x)=mu\in\Lambda_U$ is equivalent to
333 \[
334  mx=mu+mw\in\Lambda_U+\Lambda_W.
335 \]
336 Thus $m$ clears every projector denominator exactly when it annihilates every class in $C_U$.
337 \end{proof}
338 
339 For a primitive rank-one vector $v\in\Z^n$, the orthogonal projector denominator is
340 \[
341  \langle v,v\rangle,
342 \]
343 and the congruence quotient is cyclic of that order.
344 
345 \section{Why one cyclic quotient is not enough for split persistence}
346 The split persistence module
347 \[
348  \mathcal P_c
349  \cong
350  \bigoplus_{p^e\parallel c}
351  (\Z/p\Z)^{e-1}
352 \]
353 has large $p$-rank. The $p$-torsion of a cyclic target has rank at most one. Therefore every map from $\mathcal P_c$ to a cyclic group has a forced kernel of order at least
354 \[
355  \prod_{p^e\parallel c}p^{\max(e-2,0)}.
356 \]
357 This is a true obstruction to the split source. It motivates the Hecke cotangent module
358 \[
359  \mathfrak p/\mathfrak p^2,
360 \]
361 which can retain multiple independent congruence directions before cyclic compression.
362 
363 \section{A toy Hecke order}
364 Consider
365 \[
366  \mathbb T(\mathbf m)
367  =
368  \{(x_0,x_1,\ldots,x_r)\in\Z^{r+1}:x_i\equiv x_0\pmod{m_i}\}.
369 \]
370 For the target coordinate $x_0$, the cotangent module is
371 \[
372  \boxed{
373  \mathfrak p/\mathfrak p^2
374  \cong
375  \bigoplus_{i=1}^{r}\Z/m_i\Z.
376  }
377 \]
378 The cyclic congruence quotient is
379 \[
380  \boxed{
381  \Z/\lcm(m_1,\ldots,m_r)\Z.
382  }
383 \]
384 The quotient of cotangent order by congruence order is
385 \[
386  \frac{\prod_i m_i}{\lcm(m_i)}.
387 \]
388 This is a toy Wiles-defect factor. It measures the multiplicity discarded by cyclic compression.
389 
390 The comparison between cotangent and congruence data, including non-complete-intersection defects and local--global formulas, is developed in the Wiles-defect literature~\cite{BKM1,BKM2}.
391 
392 \section{Complete-intersection control case}
393 Let
394 \[
395  \mathbb T=\Z[X]/(F)
396 \]
397 and evaluate at a simple root $a_0$. Then the congruence ideal and cotangent module both have order
398 \[
399  |F'(a_0)|.
400 \]
401 The defect is one. Extra multiplicity is therefore tied to non-complete-intersection structure rather than being a normalization artifact.
402 
403 \section{Two prime indices}
404 The persistence is indexed by conductor primes $q\mid abc$. Congruence depth is indexed by coefficient primes $\ell$. The natural local signal is a matrix
405 \[
406  D_f(\ell,q).
407 \]
408 At a split Tate prime $q\neq\ell$, scalarity modulo $\ell^n$ forces
409 \[
410  n\le\min\{v_\ell(q-1),v_\ell(v_q(q_E))\}.
411 \]
412 These off-diagonal channels carry tame congruence information. They do not naturally reproduce the full base-$q$ factor $q^{e-1}$. This mismatch motivates the later filtered contact correction and the diagonal coefficient-prime channel.
413 
414 \chapter{Filtered Cusp Contact and Witt Carry Structure}
415 
416 \section{The three-cusp section}
417 View a primitive triple as the integral section
418 \[
419  \sigma_{a,b,c}:
420  \Spec\Z\longrightarrow\mathbb P^1_\Z,
421  \qquad
422  \sigma=[a:c].
423 \]
424 The pullbacks of the boundary divisors $0$, $1$, and $\infty$ are
425 \[
426  \operatorname{div}(a),
427  \qquad
428  \operatorname{div}(b),
429  \qquad
430  \operatorname{div}(c).
431 \]
432 For the output cusp, let
433 \[
434  C=\operatorname{div}(c),
435  \qquad
436  C_{\mathrm{red}}=\operatorname{div}(R_c).
437 \]
438 Define the two-term contact complex
439 \[
440  \boxed{
441  \mathbf J_c
442  =
443  [\mathcal O(-C)\longrightarrow\mathcal O(-C_{\mathrm{red}})].
444  }
445 \]
446 On global sections,
447 \[
448  \mathbf J_c
449  =
450  [c\Z\hookrightarrow R_c\Z]
451  \simeq
452  [\Z\xrightarrow{c/R_c}\Z].
453 \]
454 
455 \begin{theorem}[Filtered contact quotient]
456 \[
457  \boxed{
458  H^0(\mathbf J_c)
459  =
460  R_c\Z/c\Z
461  \cong
462  \Z/(c/R_c)\Z.
463  }
464 \]
465 \end{theorem}
466 
467 The output persistence is cyclic before grading.
468 
469 \section{Associated graded and carry extensions}
470 At $p^e\parallel c$,
471 \[
472  \Jcal_{p,e}
473  =
474  p\Z_p/p^e\Z_p
475  \cong
476  \Z/p^{e-1}\Z.
477 \]
478 Its filtration
479 \[
480  p^j\Z_p/p^e\Z_p
481 \]
482 has successive quotients $\F_p$. Thus
483 \[
484  \gr\Jcal_{p,e}\cong(\F_p)^{e-1}.
485 \]
486 The exact sequences
487 \[
488  0\longrightarrow
489  \Z/p^{j-1}\Z
490  \xrightarrow{\times p}
491  \Z/p^j\Z
492  \longrightarrow
493  \F_p
494  \longrightarrow0
495 \]
496 are nonsplit. These carry extensions are precisely what the split persistence packet forgot.
497 
498 \begin{corollary}[Correction of the cyclic-rank barrier]
499 The $p$-rank obstruction applies to the associated graded source. It does not obstruct a filtered map from $\Z/p^{e-1}\Z$ to a cyclic target of matching depth.
500 \end{corollary}
501 
502 \section{Exact cyclic transducer}
503 Let
504 \[
505  C(m)=[\Z\xrightarrow{m}\Z]
506 \]
507 and
508 \[
509  C(\eta)=[\Z\xrightarrow{\eta}\Z].
510 \]
511 Set $g=\gcd(m,\eta)$. Multiplication by $m/g$ in degree $-1$ and by $\eta/g$ in degree $0$ defines a chain map. On cohomology,
512 \[
513  \boxed{
514  |\ker|=m/g,
515  \qquad
516  |\im|=g,
517  \qquad
518  |\coker|=\eta/g.
519  }
520 \]
521 The maximal transferable packet is the common prime-power depth.
522 
523 For the output source $m=c/R_c$, the corrected algebraic defect against a cyclic target of order $\eta_f$ is
524 \[
525  \boxed{
526  \delta_{\mathrm{alg}}(c,f)
527  =
528  \frac{c/R_c}{\gcd(c/R_c,\eta_f)}.
529  }
530 \]
531 Prime by prime,
532 \[
533  \delta_{\mathrm{alg}}(c,f)
534  =
535  \prod_{p^e\parallel c}
536  p^{\max(e-1-v_p(\eta_f),0)}.
537 \]
538 The problem is missing depth, not missing rank.
539 
540 \section{Witt realization}
541 The local packet has a natural additive identification
542 \[
543  \boxed{
544  \Jcal_{p,e}
545  \cong
546  W_{e-1}(\F_p).
547  }
548 \]
549 The Verschiebung filtration matches multiplication by successive powers of $p$. This is the typed local object underlying the carry chain.
550 
551 At odd $p$, the $p$-adic logarithm identifies
552 \[
553  (1+p\Z_p)/(1+p^e\Z_p)
554  \cong
555  p\Z_p/p^e\Z_p.
556 \]
557 At $2$, the additive contact packet remains uniform, while the multiplicative coordinate begins on $1+4\Z_2$ and therefore carries the familiar weighted dyadic shift.
558 
559 \section{The filtered local-global target}
560 The intended transducer is a filtered chain
561 \[
562  \mathbf J_c
563  \longrightarrow
564  \bigoplus_{p\mid c}\mathbf L_p^{\mathrm{st}}
565  \longrightarrow
566  \mathbf L_{R^{\mathrm{glob}}}
567  \xrightarrow{R=\mathbb T}
568  \mathbf L_{\mathbb T_f}
569  \longrightarrow
570  \mathbf C_f.
571 \]
572 It must preserve the Bockstein carry maps. The local contact packet exists canonically. The unproved step is its transport through global deformation and Hecke complexes with small kernel and controlled archimedean size.
573 
574 \begin{theorem}[Conditional filtered-transfer criterion]
575 Suppose a canonical map
576 \[
577  H^0(\Theta_f):\Z/(c/R_c)\Z\longrightarrow H^0(\mathbf C_f)
578 \]
579 satisfies, for every $\eps>0$,
580 \[
581  |\ker H^0(\Theta_f)|\le C_\eps R^\eps
582 \]
583 and
584 \[
585  |H^0(\mathbf C_f)|\le C'_\eps R_aR_bR^\eps.
586 \]
587 Then the strong $abc$ conjecture follows.
588 \end{theorem}
589 
590 \begin{proof}
591 The image has order
592 \[
593  \frac{c/R_c}{|\ker|}
594 \]
595 and is bounded by the target order. Therefore
596 \[
597  \frac{c/R_c}{C_\eps R^\eps}
598  \le
599  C'_\eps R_aR_bR^\eps.
600 \]
601 Multiplying by $R_c$ and replacing $\eps$ by $\eps/2$ gives the strong bound.
602 \end{proof}

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