Prime-Contact Coherence Calculus

Download original file

This is the original TeX manuscript source. Use the paper’s PDF for typeset equations.

1 \frontmatter
2 \begin{titlepage}
3 \thispagestyle{empty}
4 \centering
5 \vspace*{0.52in}
6 {\Large\color{Blue}\bfseries PROJECTOR-RESOLVED PRIME-CONTACT COHERENCE CALCULUS\par}
7 \vspace{0.22in}
8 {\Huge\bfseries Sparse Additive Relations, Prime-Power Contact, and Adelic Coherence\par}
9 \vspace{0.20in}
10 {\Large\color{Muted}Prime packets, valuation refinement, filtered contact, Kummer classes, polylogarithmic paths, coherence lattices, and applications in Diophantine arithmetic\par}
11 \vspace{0.50in}
12 \begin{minipage}{0.88\textwidth}
13 \large
14 This paper develops a single calculus for sparse additive relations with repeated prime-power depth. The construction begins with prime-coordinate differentials and exact obstruction lattices, passes through valuation words, filtered contact modules, Kummer globalization, and polylogarithmic path packets, and culminates in an adelic coherence character built from simultaneous $p$-adic logarithms of the actual input primes. Exact theorems, finite calculations, conditional implications, and structural limitations are distinguished throughout.
15 \end{minipage}
16 \vfill
17 {\Large\bfseries Komorebi P.\par}
18 \vspace{0.30in}
19 {\large July 2026\par}
20 \end{titlepage}
21 
22 \chapter*{Abstract}
23 \addcontentsline{toc}{chapter}{Abstract}
24 Let
25 \[
26  a+b=c,
27  \qquad
28  \gcd(a,b)=1,
29  \qquad
30  R=\rad(abc).
31 \]
32 This paper develops a prime-contact coherence calculus for the interaction between sparse addition, squarefree prime support, and repeated valuation depth. The first layer is a prime-packet differential whose additive row and arithmetic Wronskian yield the exact divisibility
33 \[
34  \frac{abc}{R}\mid W.
35 \]
36 Valuation words and polyphase refinement separate aggregate multiplicity from dyadic and odd-adic phase data. Smith reduction identifies the exact obstruction torsion, while a moving-interval search computes the first-order optimum and proves that the corresponding master inequality cannot fall below its real relaxation.
37 
38 The nonlinear layers preserve information that the first-order lattice discards. Prime-shell signals distinguish support energy from depth energy. The marked Frey curve separates conductor support from discriminant depth. Filtered cusp-contact modules retain prime-power carries, truncated Witt vectors encode their typed local structure, and Kummer theory globalizes the contact packet with at most logarithmic loss. A Fermat lift and a depth-three polylogarithmic packet preserve the additive three-point relation, while exact synthesis estimates show where geometric or integral cost reappears.
39 
40 For each output prime, simultaneous $p$-adic logarithms of the input primes define a finite filter row. After removing the dyadic sign layer, common Kummer powers, and entrywise logarithmic defects, the residual rows combine into a cyclic coherence character
41 \[
42  \chi_{\mathrm{coh}}:\Z^{S_{ab}}\longrightarrow\Z/C\Z
43 \]
44 whose kernel contains the normalized signed exponent vector. The output powerful part factors exactly as
45 \[
46  \frac{c}{\rad(c)}=\epsilon_2KWC.
47 \]
48 The remaining uniform problem is an arithmetic bound on the local logarithmic defect $W$ and coherence conductor $C$. The paper proves the exact reductions and finite identities used to reach this object. It does not prove the $abc$ conjecture or the other major conjectures discussed in the applications.
49 
50 \chapter*{Statement types}
51 \addcontentsline{toc}{chapter}{Statement types}
52 \begin{longtable}{L{0.18\textwidth}L{0.72\textwidth}}
53 \toprule
54 Label & Meaning\\
55 \midrule
56 \endfirsthead
57 \toprule
58 Label & Meaning\\
59 \midrule
60 \endhead
61 \textbf{Exact} & A theorem or identity is proved under the stated hypotheses.\\
62 \textbf{Finite verification} & An exact or high-precision calculation is reproduced on a declared finite data set. It is not an asymptotic theorem.\\
63 \textbf{Conditional} & A theorem proves that a stated estimate would imply $abc$ or another target, while the estimate itself remains open.\\
64 \textbf{Structural limitation} & A theorem proves that a proposed class of methods cannot supply the required gain without an additional ingredient.\\
65 \bottomrule
66 \end{longtable}
67 
68 \BoundaryBox{\textbf{Scope.} No major open conjecture is asserted as proved. All conjectural estimates and all uses of finite evidence are labeled explicitly.}
69 
70 \chapter*{Organization}
71 \addcontentsline{toc}{chapter}{Organization}
72 The paper is organized by mathematical dependency rather than by chronology. The opening part states the principal objects and exact implications. The next three parts give the detailed theory of prime packets, signal and automorphic structures, filtered contact, Kummer and Fermat packets, polylogarithmic paths, and the final coherence lattice. The final part contains worked examples, exact computational methods, prime-theory applications, and the unresolved arithmetic theorems.
73 
74 Three reading paths are useful.
75 \begin{enumerate}[leftmargin=1.5em]
76 \item \textbf{Structural path.} Read the opening chapter, object comparison, coherence factorization, and central open theorems.
77 \item \textbf{Proof path.} Read the detailed parts in order, beginning with prime-packet divisibility and ending with the coherence lattice.
78 \item \textbf{Applications path.} Read the general functor, computational methods, and the sections on Wieferich networks, exceptional $abc$, generalized Fermat, Hall--Pillai, and Bernoulli--Witt problems.
79 \end{enumerate}
80 
81 \tableofcontents
82 \mainmatter
83 
84 \PartChapter{Foundations and Main Results}{Architecture and Principal Results}
85 
86 \section{Layered construction}
87 The calculus is organized as a chain of exact objects. Each transition either preserves a required invariant, identifies a precise loss, or proves that a broader class of transformations cannot create the desired arithmetic gain.
88 
89 \begin{center}
90 \begin{tikzpicture}[
91  node distance=7mm and 8mm,
92  every node/.style={font=\small,align=center},
93  box/.style={draw,rounded corners,minimum width=28mm,minimum height=13mm,text width=27mm,inner sep=3mm},
94  arr/.style={-{Latex[length=2.1mm]},thick}
95 ]
96 \node[box,fill=PaleBlue] (i) {prime derivative\\and Wronskian};
97 \node[box,fill=PaleBlue,right=of i] (ii) {valuation words\\and polyphase};
98 \node[box,fill=PaleTeal,right=of ii] (iii) {obstruction complex\\and formal geometry};
99 \node[box,fill=PaleTeal,right=of iii] (iv) {exact moving search\\and real floor};
100 
101 \node[box,fill=PaleMagenta,below=of i] (viii) {filtered contact\\and carry structure};
102 \node[box,fill=PaleMagenta,right=of viii] (vii) {integral projectors\\and cotangent data};
103 \node[box,fill=PaleOrange,right=of vii] (vi) {divisor persistence\\and Hecke filters};
104 \node[box,fill=PaleOrange,right=of vi] (v) {prime-shell signal\\and Frey transform};
105 
106 \node[box,fill=PaleBlue,below=of viii] (ix) {Witt--Kummer\\and Fermat packet};
107 \node[box,fill=PaleBlue,right=of ix] (x) {polylog path\\and integral cost};
108 \node[box,fill=PaleTeal,right=of x] (xi) {adelic logarithms\\and coherence lattice};
109 \node[box,fill=PaleRed,right=of xi] (open) {open arithmetic\\conductor bound};
110 
111 \draw[arr] (i)--(ii);\draw[arr] (ii)--(iii);\draw[arr] (iii)--(iv);
112 \draw[arr] (iv.south)--(v.north);
113 \draw[arr] (v.west)--(vi.east);\draw[arr] (vi.west)--(vii.east);\draw[arr] (vii.west)--(viii.east);
114 \draw[arr] (viii.south)--(ix.north);
115 \draw[arr] (ix)--(x);\draw[arr] (x)--(xi);\draw[arr] (xi)--(open);
116 \end{tikzpicture}
117 \end{center}
118 
119 \begingroup\footnotesize\setlength{\tabcolsep}{3pt}
120 \begin{longtable}{L{0.17\textwidth}L{0.22\textwidth}L{0.29\textwidth}L{0.23\textwidth}}
121 \toprule
122 Layer & Principal object & Exact conclusion & Remaining limitation\\
123 \midrule
124 \endfirsthead
125 \toprule
126 Layer & Principal object & Exact conclusion & Remaining limitation\\
127 \midrule
128 \endhead
129 Prime differential & Additive lattice and arithmetic Wronskian & Radical divisibility, master height inequality, sector compression & Uniform small transversality is not automatic\\
130 Valuation refinement & Valuation words and phase packets & Circulant Smith form, exponent-free phases, separate dyadic and odd-adic lifts & Local liftability does not control global height\\
131 Obstruction theory & Joint integer complex & Exact torsion, formal smoothness, all-order local lifting & The surviving obstruction is metric\\
132 Exact search & Moving obstruction cosets & CRT interval search, finite stopping, complete first-order optimum & The inequality equals the real floor times overhead\\
133 Signal layer & Prime-shell signal and Frey transform & Exact energy identities, unitary no-gain, marked automorphic data & Rational spectral data forgets integral depth\\
134 Persistence layer & Divisor packets and Hecke sign filters & Perfect exponent reconstruction and integral Walsh filters & Same-sign integral separation remains arithmetic\\
135 Filtered contact & Cusp-contact and Witt modules & Carries restored, cyclic transfer law, typed local depth & A global filtered transfer is not constructed\\
136 Kummer--Fermat & Kummer class and Fermat point & Near-lossless globalization and exact character bank & Full geometric carrier is too large\\
137 Polylogarithmic path & Depth-three three-point packet & Fixed complex packet and exact distribution relations & Integral synthesis cost remains\\
138 Coherence lattice & Simultaneous logarithmic character & Exact factorization $M=\epsilon_2KWC$ and cyclic kernel lattice & Radical-scale bound for $WC$ remains open\\
139 \bottomrule
140 \end{longtable}
141 \endgroup
142 
143 \section{Structural transitions}
144 Each major transition changes the category of the object being measured.
145 \begin{enumerate}[leftmargin=1.5em]
146 \item A short derivative vector is replaced by a joint obstruction complex.
147 \item Local lift trees are replaced by a global branch carrying an archimedean metric.
148 \item First-order optimization is solved exactly and separated from the uniform arithmetic estimate.
149 \item Shell energy is enriched by a marked integral automorphic packet.
150 \item Split residue layers are replaced by filtered contact modules when carry extensions matter.
151 \item Filtered contact is globalized by a Kummer class.
152 \item The full Fermat character bank is compressed to a fixed-depth polylogarithmic path packet.
153 \item The remaining finite depth is encoded by one simultaneous logarithmic coherence character.
154 \end{enumerate}
155 
156 \ResultBox{The final structure is not selected by notation. Each surviving object is forced by an exact theorem showing which information the preceding object retained and which information it lost.}
157 
158 \section{Reusable core}
159 The construction extends to primitive sparse additive relations beyond three terms:
160 \[
161 \boxed{
162 \begin{aligned}
163 \text{prime support}
164 &\longrightarrow
165 \text{filtered contact}
166 \longrightarrow
167 \text{Kummer globalization}\\
168 &\longrightarrow
169 \text{cross-prime logarithmic coherence}
170 \longrightarrow
171 \text{equation-specific global geometry}.
172 \end{aligned}
173 }
174 \]
175 The general functorial form is developed after the three-term case.
176 
177 \chapter{Common notation and the master object}
178 
179 \section{Primitive sparse additive relations}
180 Let
181 \[
182 \epsilon_1n_1+\cdots+\epsilon_rn_r=0,
183 \qquad
184 \epsilon_i\in\{\pm1\},
185 \]
186 be primitive in the sense that the integers have no common prime divisor. The $abc$ relation is the case
187 \[
188 a+b-c=0.
189 \]
190 For each term define
191 \[
192 R_i=\rad(n_i),
193 \qquad
194 M_i=\frac{n_i}{R_i}.
195 \]
196 The radical $R_i$ records activated prime support. The powerful part $M_i$ records repeated valuation depth.
197 
198 \section{Filtered contact packets}
199 For a prime power
200 \[
201 p^e\parallel n_i,
202 \]
203 define the local filtered contact packet
204 \[
205 \Jcal_{i,p}=p\Z_p/p^e\Z_p.
206 \]
207 Its order is
208 \[
209 \abs{\Jcal_{i,p}}=p^{e-1}.
210 \]
211 Its associated graded is
212 \[
213 \gr\Jcal_{i,p}\cong(\F_p)^{e-1}.
214 \]
215 The distinction is essential. The filtered packet is cyclic. The associated graded has $p$-rank $e-1$. The non-split extensions between successive grades are the arithmetic carries.
216 
217 Globally, the contact packet of one term is
218 \[
219 \Jcal_{n_i}
220 =
221 R_i\Z/n_i\Z
222 \cong
223 \Z/M_i\Z.
224 \]
225 
226 \begin{theorem}[Exact contact quotient]
227 For every positive integer $n$,
228 \[
229 \boxed{
230 \rad(n)\Z/n\Z
231 \cong
232 \Z/(n/\rad(n))\Z.
233 }
234 \]
235 Its associated graded is
236 \[
237 \boxed{
238 \gr\left(\rad(n)\Z/n\Z\right)
239 \cong
240 \bigoplus_{p^e\parallel n}(\F_p)^{e-1}.
241 }
242 \]
243 \end{theorem}
244 
245 \begin{proof}
246 Multiplication by $\rad(n)$ identifies $\Z/(n/\rad(n))\Z$ with $\rad(n)\Z/n\Z$. The primary filtration is the product of the filtrations $p^j\Z_p/p^e\Z_p$. Each consecutive quotient is one-dimensional over $\F_p$.
247 \end{proof}
248 
249 \section{Witt coordinates}
250 The local packet has a natural $p$-typical coordinate description
251 \[
252 \Jcal_{i,p}
253 \cong
254 W_{e-1}(\F_p)
255 \]
256 as an additive filtered object. Frobenius and Verschiebung make the prime type explicit. The dyadic branch uses the same additive depth but a shifted multiplicative logarithm coordinate on principal units.
257 
258 \section{Kummer globalization}
259 Let
260 \[
261 x=\prod_{q\in S}q^{\nu_q}\in\Q^\times
262 \]
263 and fix a modulus $M$. The Kummer class of $x$ lies in
264 \[
265 H^1(\Q,\mu_M)
266 \cong
267 \Q^\times/(\Q^\times)^M.
268 \]
269 Its order is controlled exactly by the valuation vector.
270 
271 \begin{theorem}[Kummer order from the valuation vector]
272 Assume $x>0$. Then
273 \[
274 \boxed{
275 \ord([x])
276 =
277 \frac{M}{\gcd(M,\{\nu_q\}_{q\in S})}.
278 }
279 \]
280 For a negative rational number and even $M$, one additional dyadic sign layer may occur.
281 \end{theorem}
282 
283 \begin{proof}
284 The class $[x]^k$ is trivial exactly when every $k\nu_q$ is divisible by $M$. The least positive such $k$ is the least common multiple of $M/\gcd(M,\nu_q)$, which is $M/\gcd(M,\{\nu_q\})$.
285 \end{proof}
286 
287 \section{Cross-prime logarithmic filters}
288 Let $Q$ be a set of input primes and $P$ a set of output primes. At odd $p\in P$, define
289 \[
290 L_{p,q}
291 =
292 \frac1p\log_p\langle q\rangle_p
293 \]
294 with the appropriate finite-depth truncation. At $p=2$, use the weighted principal-unit coordinate together with a separate sign bit.
295 
296 The matrix
297 \[
298 \Lcal_{P,Q}=(L_{p,q})
299 \]
300 is a cross-prime signal operator. A sparse additive relation supplies a short signed valuation vector $\nu$ satisfying deep congruences in every output row.
301 
302 \section{The coherence lattice}
303 After removing common Kummer powers and local logarithmic image defects, the surviving rows combine by the Chinese remainder theorem into a primitive cyclic character
304 \[
305 \chi_{\mathrm{coh}}:\Z^Q\longrightarrow\Z/C\Z.
306 \]
307 The coherence lattice is
308 \[
309 \boxed{
310 \Lambda_{\mathrm{coh}}
311 =
312 \Ker\chi_{\mathrm{coh}}.
313 }
314 \]
315 When the row is primitive,
316 \[
317 [\Z^Q:\Lambda_{\mathrm{coh}}]=C.
318 \]
319 The normalized exponent vector of the relation lies in this kernel.
320 
321 \section{The optional global realization}
322 The same prime-contact data may be lifted further when the Diophantine equation supplies additional geometry. Possible carriers include:
323 \begin{itemize}[leftmargin=1.5em]
324 \item a Frey elliptic curve or higher-dimensional Frey variety
325 \item a hypergeometric motive
326 \item a Fermat or superelliptic cover
327 \item a polylogarithmic path torsor on a punctured projective line
328 \item a Hecke, deformation, Selmer, or syntomic complex
329 \end{itemize}
330 The finite contact and coherence layers exist before any such lift. The global realization is equation-dependent.
331 
332 \section{The prime-contact coherence data structure}
333 The reusable data attached to a primitive sparse relation is
334 \[
335 \boxed{
336 \mathfrak A(\mathbf n,\boldsymbol\epsilon)
337 =
338 \left(
339 \mathbf R,
340 \mathbf M,
341 \Witt_{\mathbf n},
342 \kappa_{\mathbf n},
343 \Lcal_{\mathbf n},
344 \Lambda_{\mathrm{coh}},
345 \chi_{\mathrm{coh}},
346 \mathcal G_{\mathbf n}
347 \right).
348 }
349 \]
350 The calculus consists of exact transformations among these layers, plus the norm and determinant questions required to compare them globally.
351 
352 \chapter{Exact Results and Structural Limitations}
353 
354 \section{Prime-packet derivative and radical divisibility}
355 For a primitive triple
356 \[
357 a+b=c,
358 \qquad
359 R=\rad(abc),
360 \]
361 the prime-packet construction attaches one coordinate to every prime dividing $abc$. An arithmetic derivative produces an additivity row, while an arithmetic Wronskian produces a nondegeneracy row. The key local divisibility theorem is
362 \[
363 \boxed{
364 \frac{abc}{R}
365 \mid
366 W_x(a,b)
367 }
368 \]
369 for every additive packet $x$. A nonzero Wronskian therefore satisfies
370 \[
371 \abs{W_x(a,b)}\ge\frac{abc}{R}.
372 \]
373 Combining this with the derivative-size bound yields a valid master inequality involving the shortest transverse packet. The theorem is exact. The missing estimate on the packet size was not proved.
374 
375 \section{Valuation-word Smith form}
376 For the valuation word
377 \[
378 V_e(T)=1+T+\cdots+T^{e-1}
379 \]
380 acting on
381 \[
382 \Z[T]/(T^N-1),
383 \]
384 the phase/coarse block has Smith form
385 \[
386 \boxed{
387 \SNF(C_{e,N})
388 =
389 \operatorname{diag}(1,\ldots,1,e)
390 }
391 \]
392 when $\gcd(e,N)=1$. Thus all nonconstant phase channels are integrally invertible and the entire multiplicity is concentrated in one aggregate channel. The proof uses the augmentation ideal: multiplication by $V_e$ has determinant one on that integral rank-$N-1$ lattice and acts as multiplication by $e$ on the quotient.
393 
394 The aggregate obstruction then factors into a weighted dyadic lift and the odd-adic lifts. This proves exact local solvability. It does not prove a small global archimedean representative.
395 
396 \section{Exact obstruction complex}
397 The exponent-free phase, or shell-synthesis, additivity and Wronskian rows define an integer matrix whose Smith form is
398 \[
399 \boxed{
400 \operatorname{diag}\left(1,\frac{abc}{R}\right).
401 }
402 \]
403 This two-row statement applies whenever at least two prime sectors are nonempty. The degenerate triple $(1,1,2)$ has rank one. On the additive phase lattice $L$, the normalized Wronskian gives an exact sequence
404 \[
405 0\longrightarrow L_0
406 \longrightarrow L
407 \longrightarrow\Z
408 \longrightarrow0.
409 \]
410 The sequence splits as an exact sequence of abelian groups, and all local lift equations are solvable. The full shell deformation is formally smooth over $\Z$. Every compatible first-order tangent extends to all formal orders.
411 
412 This proves that the obstruction is not a failure of local lifting or higher formal continuation.
413 
414 \section{Exact first-order moving search}
415 The moving-coset search solves
416 \[
417 H(k)
418 =
419 \min\{\norm{x}_\infty:x\in L,\ \omega(x)=k\}
420 \]
421 exactly for every supplied factored triple. Sector compression and the Chinese remainder theorem reduce each fixed-$k$ problem to the overlap of three integer intervals. A real lower bound supplies a finite stopping rule in $k$.
422 
423 The decisive identity is
424 \[
425 \boxed{
426 \overline H
427 =
428 \frac{c}{R\Delta}
429 }
430 \]
431 for the exact real relaxation. The master inequality
432 \[
433 c\le R\Delta\Phi
434 \]
435 therefore becomes
436 \[
437 \boxed{
438 c
439 \le
440 c\frac{\Phi}{\overline H},
441 \qquad
442 \frac{\Phi}{\overline H}\ge1.
443 }
444 \]
445 The first-order search is complete and cannot create a new $abc$ gain.
446 
447 \section{Prime-shell energy identities}
448 For
449 \[
450 n=\prod_pp^{e_p},
451 \]
452 define the shell signal
453 \[
454 \Psi_n(p,r)=\sqrt{\log p}\,\mathbf 1_{0\le r<e_p}.
455 \]
456 Then
457 \[
458 \boxed{
459 \norm{\Psi_n}_2^2=\log n.
460 }
461 \]
462 If $P_{\mathrm{act}}$ retains only the first shell at every active prime, then
463 \[
464 \boxed{
465 \norm{P_{\mathrm{act}}\Psi_n}_2^2
466 =
467 \log\rad(n).
468 }
469 \]
470 For the direct-sum signal of an $abc$ triple, the standard quality is the output-sector energy divided by activated bandwidth.
471 
472 Every unitary transform in the depth coordinate preserves this energy ratio. Fourier, wavelet, and unitary polyphase analysis can reorganize the shell signal but cannot by themselves create the radical saving.
473 
474 \section{Divisor-polyphase perfect reconstruction}
475 For
476 \[
477 n=\prod_pp^{e_p},
478 \]
479 define
480 \[
481 B_d(n)=\prod_{p\mid n,\ d\mid e_p}p.
482 \]
483 Then
484 \[
485 \boxed{
486 n=\prod_{d\ge1}B_d(n)^{\varphi(d)}
487 }
488 \]
489 and
490 \[
491 \boxed{
492 \frac{n}{\rad(n)}
493 =
494 \prod_{d\ge2}B_d(n)^{\varphi(d)}.
495 }
496 \]
497 This is an exact multiplicative polyphase decomposition of prime-power persistence.
498 
499 \section{Filtered contact correction}
500 The split module
501 \[
502 \bigoplus_{p^e\parallel c}(\F_p)^{e-1}
503 \]
504 is the associated graded of the cyclic contact module
505 \[
506 \boxed{
507 R_c\Z/c\Z
508 \cong
509 \Z/(c/R_c)\Z.
510 }
511 \]
512 The direct split-to-cyclic rank obstruction is therefore a theorem about the graded approximation, not the filtered source.
513 
514 For cyclic packets of orders $m$ and $\eta$, the largest possible image has order
515 \[
516 \gcd(m,\eta).
517 \]
518 The exact source loss is
519 \[
520 \boxed{
521 \frac{m}{\gcd(m,\eta)}.
522 }
523 \]
524 Thus the correct cyclic obstruction is prime-power depth shortfall.
525 
526 \section{Witt--Kummer globalization}
527 The local contact packet is naturally a truncated Witt packet
528 \[
529 W_{e-1}(\F_p).
530 \]
531 Globally, the Kummer map
532 \[
533 \kappa_c:\Z/M\Z\longrightarrow H^1(\Q,\mu_M),
534 \qquad
535 k\longmapsto[c^k]
536 \]
537 has exact kernel
538 \[
539 \boxed{
540 \gcd(M,e_1,\ldots,e_r)
541 }
542 \]
543 when
544 \[
545 c=\prod_ip_i^{e_i}.
546 \]
547 The kernel is at most $\log_2c$. Therefore most of the contact packet is already a global Galois class.
548 
549 \section{Fermat lift and full-plane line-bundle cost}
550 Choose
551 \[
552 u^M=\frac ac,
553 \qquad
554 v^M=\frac bc.
555 \]
556 Then
557 \[
558 u^M+v^M=1.
559 \]
560 The point $[u:v:1]$ on the Fermat curve has exact projective height
561 \[
562 \boxed{
563 \frac{\log c}{M}.
564 }
565 \]
566 However
567 \[
568 K_{F_M}\cong\mathcal O_{F_M}(M-3).
569 \]
570 For compatible Weil-height normalizations,
571 \[
572 h_{K_{F_M}}(P_M)
573 =
574 \frac{M-3}{M}\log c+O(1).
575 \]
576 Thus the full plane-curve polarization reintroduces a factor of order $M$. This observation concerns the complete Fermat model with its induced line bundle and does not exclude a lower-complexity quotient with a different arithmetic metric.
577 
578 \section{Polylogarithmic path compression}
579 The root-of-unity distribution identity
580 \[
581 \boxed{
582 \sum_{j=0}^{M-1}\operatorname{Li}_n(\zeta_M^ju)
583 =
584 M^{1-n}\operatorname{Li}_n(u^M)
585 }
586 \]
587 compresses an $M$-branch orbit to one weight-$n$ coordinate. Through weights one, two, and three, the synthesis gains multiply to
588 \[
589 M^3.
590 \]
591 The complex depth-three path packet has at most cubic logarithmic growth. A fixed-rank integral presentation carrying order $M$ must nevertheless have a large coefficient, denominator, degree, or covolume. Small complex periods are not enough.
592 
593 \section{Adelic coherence factorization}
594 For the balanced input ratio, the output contact modulus factors exactly as
595 \[
596 \boxed{
597 M=\epsilon_2KWC.
598 }
599 \]
600 Here $\epsilon_2$ is the weighted dyadic sign layer, $K$ is the common Kummer power, $W$ is the residual local logarithmic image defect, and $C$ is the genuine cross-prime coherence quotient.
601 
602 The coherence quotient is the index of an explicit lattice
603 \[
604 \Lambda_{\mathrm{coh}}
605 \subset
606 \Z^{S_{ab}}.
607 \]
608 The normalized input exponent vector lies in this lattice. The corresponding dual character has order $C$.
609 
610 \section{No-go theorem: abstract lattice geometry}
611 Let $v\in\Z^s$ be primitive with $s\ge2$. For every prescribed modulus $C$, one can choose a primitive row modulo $C$ whose kernel contains $v$. Therefore a fixed short primitive relation can lie in a primitive congruence lattice of arbitrarily large index.
612 
613 \[
614 \boxed{
615 \text{Short vector plus large lattice index is possible abstractly at every scale.}
616 }
617 \]
618 The arithmetic origin of the logarithmic row is essential.
619 
620 \section{No-go theorem: direct positive contact}
621 If positive integers $A$ and $B$ satisfy
622 \[
623 c\mid A+B,
624 \]
625 then
626 \[
627 A+B\ge c
628 \]
629 and
630 \[
631 \boxed{
632 \max(A,B)\ge\frac c2.
633 }
634 \]
635 A direct positive $S$-unit relation divisible by $c$ is already at the scale one is trying to bound.
636 
637 \section{No-go theorem: linear Hecke averaging}
638 In the local Tate model, the $\ell+1$ isogeny branches multiply contact depth by
639 \[
640 \ell,
641 \quad
642 \underbrace{\ell^{-1},\ldots,\ell^{-1}}_{\ell\text{ branches}}.
643 \]
644 Uniform averaging preserves the linear mean exactly. Typical logarithmic depth falls, but rare deep branches retain the mean. Linear Hecke averaging is redistribution rather than contraction.
645 
646 \chapter{Signal Theory and Integral Automorphic Aliasing}
647 
648 \section{Information retained by signal transforms}
649 The signal representation asks which transforms retain the squarefree support, repeated valuation depth, additive marking, and integral normalization needed by the $abc$ inequality. The conclusions are sharp.
650 
651 \begin{itemize}[leftmargin=1.5em]
652 \item The prime-shell representation retains radical support and valuation depth exactly.
653 \item Unitary transforms of shell depth preserve the decisive energy ratio and cannot create a saving.
654 \item The Frey curve is a nonlinear encoder separating conductor support from discriminant depth.
655 \item A rational automorphic eigen-signal is insufficient because integral representatives in one isogeny or twist family can have different marked heights.
656 \item The relevant automorphic packet must contain the eigen-signal, an integral lattice, local depth, differential normalization, marking, and a synthesis map.
657 \end{itemize}
658 
659 \section{Prime-shell signals}
660 For a positive integer
661 \[
662 n=\prod_pp^{e_p},
663 \]
664 the shell signal
665 \[
666 \Psi_n(p,r)=\sqrt{\log p}\,\mathbf1_{0\le r<e_p}
667 \]
668 is a rectangular pulse of length $e_p$ in each prime channel.
669 
670 Its depth Fourier transform is a Dirichlet packet:
671 \[
672 \widehat\Psi_{n,p}(\theta)
673 =
674 \sqrt{\log p}\,
675 \exp\left(-\frac{i(e_p-1)\theta}{2}\right)
676 \frac{\sin(e_p\theta/2)}{\sin(\theta/2)}.
677 \]
678 The zero phase records the exponent. Nonzero phases record the endpoint and residue geometry of the valuation run.
679 
680 This explains why polyphase refinement was natural. It also proves why a unitary depth transform is not enough. Parseval preserves the total shell energy
681 \[
682 \log n
683 \]
684 and the activated first-shell energy
685 \[
686 \log\rad(n).
687 \]
688 The transform changes coordinates, not the underlying imbalance.
689 
690 \section{The Frey curve as a nonlinear signal transform}
691 Attach the marked Frey curve
692 \[
693 E_{a,b,c}:y^2=x(x-a)(x+b).
694 \]
695 Its discriminant is
696 \[
697 \Delta_E=16a^2b^2c^2.
698 \]
699 At odd primes dividing $abc$, the conductor records the activated channel while the minimal discriminant records the local depth. Thus the Frey transform converts the prime-shell split into a standard automorphic split:
700 \[
701 \begin{aligned}
702 \text{conductor}
703 &\longleftrightarrow
704 \text{activated bandwidth},\\
705 \text{minimal discriminant}
706 &\longleftrightarrow
707 \text{persistence depth}.
708 \end{aligned}
709 \]
710 
711 The associated newform provides a global Hecke-frequency signal and a Langlands-theoretic realization of the arithmetic packet.
712 
713 \section{Information loss inside an isogeny channel}
714 Curves in the same rational isogeny class share the same rational newform. The rational Hecke signal therefore does not determine the marked integral representative that arose from the original triple.
715 
716 A useful exact family appears when $u>v>0$ are coprime odd integers. Define
717 \[
718 T_0
719 =
720 \left(v^2,u^2-v^2,u^2\right)
721 \]
722 and
723 \[
724 T_1
725 =
726 \left(
727 uv,
728 \left(\frac{u-v}{2}\right)^2,
729 \left(\frac{u+v}{2}\right)^2
730 \right).
731 \]
732 Both are primitive additive triples. They have the same radical
733 \[
734 \rad\bigl(uv(u-v)(u+v)\bigr),
735 \]
736 while the second output is smaller:
737 \[
738 \left(\frac{u+v}{2}\right)^2<u^2.
739 \]
740 
741 Their Legendre parameters are linked by the standard dyadic Landen transformation. If
742 \[
743 \lambda_0=\left(\frac vu\right)^2,
744 \]
745 then the complementary child parameter is
746 \[
747 \lambda_1
748 =
749 \frac{4uv}{(u+v)^2}
750 =
751 \frac{4\sqrt{\lambda_0}}{(1+\sqrt{\lambda_0})^2}.
752 \]
753 This is the exact two-isogeny refinement law behind the family.
754 
755 The family shows that one rational automorphic channel can contain several integral representatives with the same radical and different additive heights. Ordinary eigenvalues alone do not encode the $abc$ quantity.
756 
757 \section{The integral automorphic packet}
758 The resulting integral automorphic object is
759 \[
760 \boxed{
761 \Acal^{\sharp}(E,\iota)
762 =
763 \left(
764  f_E,
765  \Lambda_E,
766  \omega_E,
767  \{q_p\}_{p\mid N_E},
768  \iota,
769  \phi_E
770 \right).
771 }
772 \]
773 The components are:
774 \begin{itemize}[leftmargin=1.5em]
775 \item the rational Hecke eigen-signal $f_E$
776 \item an integral Hecke, modular-symbol, or homology lattice $\Lambda_E$
777 \item the minimal N\'eron differential $\omega_E$
778 \item local Tate-depth packets $q_p$
779 \item the marking $\iota$ of the three rational two-torsion differences
780 \item the modular synthesis map $\phi_E$
781 \end{itemize}
782 
783 The newform is one coordinate of the object rather than the entire object.
784 
785 \section{Exact automorphic analysis--synthesis identity}
786 Let
787 \[
788 \phi_E:X_0(N_E)\longrightarrow E
789 \]
790 be an optimal modular parametrization, and write
791 \[
792 \phi_E^*\omega_E
793 =
794 2\pi i\,c_Ef_E(z)\,dz.
795 \]
796 If
797 \[
798 A(E)
799 =
800 \frac{i}{2}
801 \int_{E(\C)}\omega_E\wedge\overline{\omega_E},
802 \]
803 then change of variables under the finite map gives
804 \[
805 \boxed{
806 \deg(\phi_E)A(E)
807 =
808 4\pi^2c_E^2\norm{f_E}_{\mathrm{Pet}}^2.
809 }
810 \]
811 This is an exact analysis--synthesis energy identity.
812 
813 For a marked Frey representative, the scaling from the normalized Legendre shape to the integral curve introduces the additive height. Schematically,
814 \[
815 \boxed{
816 \text{height}
817 =
818 \text{shape factor}
819 \times
820 \frac{\text{synthesis multiplicity}}
821 {\text{integral normalization}\times\text{spectral energy}}.
822 }
823 \]
824 The formula explains why modular degree, Manin constant, N\'eron scaling, and Petersson energy all enter the integral synthesis identity.
825 
826 \section{Integral separation problem}
827 The signal representation isolates the central integral question:
828 
829 \begin{quote}
830 Can the Frey eigenchannel be isolated and synthesized inside the integral automorphic lattice with cost almost linear in its conductor bandwidth?
831 \end{quote}
832 
833 Divisor-polyphase persistence makes the local depth explicit. Integral projector theory separates sign filtering from same-sign congruence. Filtered contact restores the carry structure that a split residue packet would lose. The resulting requirement is precise: a valid global carrier must preserve marked local depth, integral eigenline data, and the synthesis metric simultaneously.
834 
835 \chapter{The Prime-Contact Coherence Calculus}
836 
837 \section{Definition of the calculus}
838 A prime-contact coherence system attached to a primitive sparse additive relation consists of the data
839 \[
840 \boxed{
841 \mathfrak A(\mathbf n,\boldsymbol\epsilon)
842 =
843 \left(
844 \mathbf R,
845 \mathbf M,
846 \Witt_{\mathbf n},
847 \kappa_{\mathbf n},
848 \Lcal_{\mathbf n},
849 \Lambda_{\mathrm{coh}},
850 \chi_{\mathrm{coh}},
851 \mathcal G_{\mathbf n}
852 \right).
853 }
854 \]
855 Here:
856 \begin{itemize}[leftmargin=1.5em]
857 \item $\mathbf R$ records squarefree prime support
858 \item $\mathbf M$ records powerful parts
859 \item $\Witt_{\mathbf n}$ is the product of filtered Witt contact crystals
860 \item $\kappa_{\mathbf n}$ is the system of global Kummer classes
861 \item $\Lcal_{\mathbf n}$ is the cross-prime logarithmic filter matrix
862 \item $\Lambda_{\mathrm{coh}}$ is the simultaneous congruence kernel
863 \item $\chi_{\mathrm{coh}}$ is the dual cyclic coherence character
864 \item $\mathcal G_{\mathbf n}$ is an optional equation-specific geometric or automorphic realization
865 \end{itemize}
866 
867 The calculus is the collection of exact transformations among these layers, together with the norm and determinant questions required to compare them globally.
868 
869 \section{Canonical operations}
870 The following operations are exact.
871 
872 \subsection{Permutation}
873 Permuting the terms or the prime coordinates permutes the corresponding packet components and leaves all invariant factors unchanged.
874 
875 \subsection{Sign reversal}
876 Multiplying the full relation by $-1$ changes the signed exponent vector but not the contact depths, Kummer orders, or coherence indices.
877 
878 \subsection{Primitive reduction}
879 Dividing a nonprimitive relation by the common gcd removes a support layer shared by every term. The primitive relation is the correct input to the calculus because pairwise support separation is then visible.
880 
881 \subsection{Primary decomposition}
882 Every filtered contact object decomposes exactly into its $p$-primary components. The dyadic branch may require a shifted multiplicative coordinate, but the additive filtered length remains $e-1$.
883 
884 \subsection{Polyphase refinement}
885 A valuation word can be refined into residue phases. Exact synthesis must be retained when comparing norms. Phase-coordinate shrinkage without synthesis normalization is not an intrinsic gain.
886 
887 \subsection{Kummer quotient}
888 Common divisibility in the valuation vector produces an exact cyclic quotient of the contact modulus. The quotient kernel is computed by a gcd.
889 
890 \subsection{Logarithmic normalization}
891 Local image defects of the logarithmic row can be factored before measuring cross-channel coherence. This separates individual Wieferich-type degeneration from collective cancellation.
892 
893 \section{The master factorization for an output term}
894 For the three-term $abc$ relation, the final exact factorization is
895 \[
896 \boxed{
897 M=\epsilon_2KWC.
898 }
899 \]
900 The factors distinguish four mechanisms.
901 
902 \begin{longtable}{L{0.14\textwidth}L{0.28\textwidth}L{0.48\textwidth}}
903 \toprule
904 Factor & Name & Meaning\\
905 \midrule
906 \endfirsthead
907 \toprule
908 Factor & Name & Meaning\\
909 \midrule
910 \endhead
911 $\epsilon_2$ & Dyadic sign layer & The separate weighted two-adic sign contribution\\
912 $K$ & Kummer preload & Depth removed by a common power in the signed input valuation vector\\
913 $W$ & Local logarithmic defect & Depth already present because every active logarithmic coefficient is exceptionally divisible\\
914 $C$ & Coherence quotient & Depth created by genuine simultaneous cancellation among the normalized cross-prime logarithmic channels\\
915 \bottomrule
916 \end{longtable}
917 
918 In the finite data set, the Kummer and Wieferich layers were usually trivial. Most difficult triples placed their repeated depth in $C$.
919 
920 \section{The simultaneous coherence problem}
921 Let $Q$ be the input prime set and $P$ the repeated output prime set. After normalization, each output row is primitive modulo its remaining prime-power depth. The rows assemble into one cyclic map
922 \[
923 \chi_{\mathrm{coh}}:\Z^Q\longrightarrow\Z/C\Z.
924 \]
925 The exponent packet of the additive relation lies in the kernel.
926 
927 The missing theorem is not a generic short-vector theorem. It is an arithmetic conductor estimate for a row whose entries are reductions of $p$-adic logarithms of the actual input primes and whose kernel vector is forced by the additive equation.
928 
929 A model target is
930 \[
931 \boxed{
932 WC
933 \le
934 C_\eps
935 R_{\mathrm{in}}R^\eps,
936 }
937 \]
938 where $R_{\mathrm{in}}$ is the activated input bandwidth. For $abc$, $R_{\mathrm{in}}=R_aR_b$.
939 
940 \section{A general contact functor}
941 The construction extends to a sparse relation
942 \[
943 \sum_{j=1}^{r}\epsilon_jn_j=0.
944 \]
945 Choose one or more output terms whose repeated depth is to be explained. The assignment
946 \[
947 (\mathbf n,\boldsymbol\epsilon)
948 \longmapsto
949 \mathfrak A(\mathbf n,\boldsymbol\epsilon)
950 \]
951 is canonical under permutations, sign reversal, primitive reduction, and primary decomposition. The optional geometric lift depends on the equation, but the support, contact, Kummer, and logarithmic layers do not.
952 
953 A complete categorical theory would additionally specify morphisms between relations and prove compatibility under gluing. The present volume records the object-level theory and the exact transformations already used.
954 
955 \section{Why the calculus is broader than abc}
956 The construction needs only:
957 \begin{enumerate}[leftmargin=1.5em]
958 \item a sparse additive relation
959 \item factorization into prime support and depth
960 \item one or more output terms whose repeated depth is to be explained
961 \item multiplicative ratios connecting the input and output terms
962 \end{enumerate}
963 A Frey curve or automorphic representation is optional. The prime-contact and coherence layers exist for every sparse relation.
964 
965 \begin{center}
966 \resizebox{0.98\textwidth}{!}{%
967 \begin{tikzpicture}[
968  node distance=10mm and 12mm,
969  every node/.style={font=\small,align=center},
970  box/.style={draw,rounded corners,text width=29mm,minimum height=15mm,inner sep=3mm},
971  arr/.style={-{Latex[length=2.1mm]},thick}
972 ]
973 \node[box,fill=PaleBlue] (rel) {sparse additive relation};
974 \node[box,fill=PaleBlue,right=of rel] (supp) {radical support\\and powerful depth};
975 \node[box,fill=PaleTeal,right=of supp] (cont) {filtered contact\\and Witt carries};
976 \node[box,fill=PaleOrange,right=of cont] (kum) {global Kummer\\packet};
977 
978 \node[box,fill=PaleRed,below=of rel] (glob) {optional motive,\\Frey object, or regulator};
979 \node[box,fill=PaleMagenta,right=of glob] (coh) {coherence character\\and kernel lattice};
980 \node[box,fill=PaleMagenta,right=of coh] (log) {cross-prime\\logarithmic filters};
981 \draw[arr] (rel)--(supp);\draw[arr] (supp)--(cont);\draw[arr] (cont)--(kum);
982 \draw[arr] (kum.south)--(log.north);\draw[arr] (log.west)--(coh.east);\draw[arr] (coh.west)--(glob.east);
983 \end{tikzpicture}%
984 }
985 \end{center}
986 

Back to source controls

Back to Prime-Contact Coherence Calculus