detailed_3.tex
Prime-Contact Coherence Calculus
detailed_3.tex
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1 \PartChapter{Detailed Theory: Kummer, Fermat, Polylogarithmic, and Coherence Objects}{Kummer Globalization and the Fermat Character Packet}
2
3 \section{The contact modulus}
4 Let
5 \[
6 c=\prod_{i=1}^{r}p_i^{e_i},
7 \qquad
8 R_c=\prod_i p_i,
9 \qquad
10 M=\frac{c}{R_c}
11 =\prod_i p_i^{e_i-1}.
12 \]
13 The filtered output contact packet is the cyclic group
14 \[
15 \Z/M\Z.
16 \]
17 The first global question is whether this packet can be realized by one arithmetic class without losing most of its order.
18
19 \section{The global Kummer map}
20 Kummer theory gives
21 \[
22 H^1(\Q,\mu_M)
23 \cong
24 \Q^\times/(\Q^\times)^M.
25 \]
26 Define
27 \[
28 \boxed{
29 \kappa_c:
30 \Z/M\Z
31 \longrightarrow
32 H^1(\Q,\mu_M),
33 \qquad
34 k\longmapsto[c^k].
35 }
36 \]
37 The valuation vector of $c$ is
38 \[
39 (e_1,\ldots,e_r)
40 \in
41 (\Z/M\Z)^r.
42 \]
43
44 \begin{theorem}[Exact Kummer kernel]
45 Let
46 \[
47 g_c=\gcd(M,e_1,\ldots,e_r).
48 \]
49 Then
50 \[
51 \boxed{
52 |\ker\kappa_c|=g_c,
53 \qquad
54 |\im\kappa_c|=M/g_c.
55 }
56 \]
57 \end{theorem}
58
59 \begin{proof}
60 The class $[c]^k$ is trivial exactly when every prime valuation $ke_i$ is divisible by $M$. The least positive such $k$ is
61 \[
62 \operatorname{lcm}_i
63 \frac{M}{\gcd(M,e_i)}
64 =
65 \frac{M}{\gcd(M,e_1,\ldots,e_r)}.
66 \]
67 This is the order of the image. The kernel size follows.
68 \end{proof}
69
70 Because
71 \[
72 g_c\le\min_i e_i\le\log_2 c,
73 \]
74 we obtain the universal near-injectivity estimate
75 \[
76 \boxed{
77 |\ker\kappa_c|\le\log_2c.
78 }
79 \]
80 This closes the global-occupancy problem for the contact packet up to logarithmic loss.
81
82 \section{Balanced Kummer class}
83 The class of $c$ does not remember the additive relation. A better coordinate is
84 \[
85 \beta=-\frac ba=1-\frac ca.
86 \]
87 Let
88 \[
89 \nu_q=v_q(b)-v_q(a)
90 \]
91 for input primes $q\mid ab$. The order of $[\beta]$ in
92 \[
93 \Q^\times/(\Q^\times)^M
94 \]
95 is determined by the signed valuation vector, with one possible dyadic sign correction when $M$ is even. If
96 \[
97 g_{\mathrm{val}}
98 =
99 \gcd(M,\{\nu_q\}),
100 \qquad
101 m_0=M/g_{\mathrm{val}},
102 \]
103 then the order is $m_0$ unless $M$ is even and $m_0$ is odd, in which case it is $2m_0$.
104
105 The balanced class retains the input-output relation and usually has full contact order. Its kernel remains logarithmically bounded.
106
107 \section{The Kummer--Fermat lift}
108 Choose algebraic numbers $u$ and $v$ satisfying
109 \[
110 u^M=\frac ac,
111 \qquad
112 v^M=\frac bc.
113 \]
114 Then
115 \[
116 \boxed{u^M+v^M=1.}
117 \]
118 Thus
119 \[
120 P_M=[u:v:1]
121 \]
122 lies on the Fermat curve
123 \[
124 F_M:X^M+Y^M=Z^M.
125 \]
126 The construction is canonical up to the action of $\mu_M^2$.
127
128 \begin{theorem}[Exact height compression]
129 \[
130 \boxed{
131 h_{\mathcal O(1)}(P_M)=\frac{\log c}{M}.
132 }
133 \]
134 \end{theorem}
135
136 \begin{proof}
137 The coordinatewise $M$th-power map sends $P_M$ to $[a:b:c]$. Absolute logarithmic projective height multiplies by $M$ under this map. Since the primitive positive triple has height $\log c$, the result follows.
138 \end{proof}
139
140 The contact modulus divides the displayed coordinate height by the full powerful part.
141
142 \section{Fermat character polyphase}
143 The group $\mu_M^2$ acts by
144 \[
145 (\xi,\eta)\cdot(X,Y)=(\xi X,\eta Y).
146 \]
147 Let $\sigma$ and $\tau$ denote the two coordinate rotations. For
148 \[
149 (r,s)\in(\Z/M\Z)^2,
150 \]
151 define
152 \[
153 P_{r,s}
154 =
155 \frac1{M^2}
156 \sum_{i,j=0}^{M-1}
157 \zeta_M^{-ri-sj}
158 \sigma^i\tau^j.
159 \]
160 Character orthogonality gives
161 \[
162 P_{r,s}P_{r',s'}
163 =
164 \delta_{r,r'}\delta_{s,s'}P_{r,s}
165 \]
166 and
167 \[
168 \boxed{
169 \sum_{r,s}P_{r,s}=I.
170 }
171 \]
172 This is an exact two-dimensional character filter bank.
173
174 \section{Line-bundle cost of the full Fermat model}
175 For $M\ge4$, adjunction gives
176 \[
177 K_{F_M}\cong\mathcal O_{F_M}(M-3).
178 \]
179 For compatible Weil-height normalizations,
180 \[
181 h_{K_{F_M}}(P_M)
182 =
183 (M-3)h_{\mathcal O(1)}(P_M)+O(1)
184 =
185 \frac{M-3}{M}\log c+O(1).
186 \]
187 If the metric on $K_{F_M}$ is defined by the induced tensor-power metric on $\mathcal O(1)^{\otimes(M-3)}$, the displayed equality holds without the $O(1)$ term. Thus the full plane-curve polarization reintroduces a factor of order $M$ into the height accounting.
188
189 \begin{proposition}[Line-bundle cost of the full plane model]
190 The Kummer lift reduces the $\mathcal O(1)$ coordinate height by $M$, while the canonical line bundle of the complete Fermat curve has degree factor $M-3$. This rules out a height saving obtained merely by retaining the full plane-curve carrier with its induced polarization. It does not rule out a lower-complexity quotient equipped with a different arithmetic metric.
191 \end{proposition}
192
193 A successful route must therefore select and control a much smaller quotient of the character bank. Fermat character decompositions and Jacobi-sum factors provide the established ambient framework for such quotients~\cite{AokiFermat,OtsuboYamazaki,ColemanMcCallum}.
194
195 \chapter{The Polylogarithmic Three-Point Path Packet}
196
197 \section{Root-distribution compression}
198 Let $u^M=z$ and let $\zeta_M$ be a primitive $M$th root of unity. For $n\ge1$,
199 \[
200 \boxed{
201 \sum_{j=0}^{M-1}
202 \operatorname{Li}_n(\zeta_M^ju)
203 =
204 M^{1-n}\operatorname{Li}_n(u^M).
205 }
206 \]
207 Inside the disk of convergence, this follows by summing the defining series and using the root-of-unity filter. Analytic continuation extends the identity with compatible branches.
208
209 The inverse synthesis gain at weight $n$ is $M^{n-1}$. Through weights one, two, and three, the total gain is
210 \[
211 \boxed{1\cdot M\cdot M^2=M^3.}
212 \]
213 The analytic branch count becomes fixed, but the integral transition matrix retains a large determinant.
214
215 \section{Three-point functional closure}
216 Set
217 \[
218 z=\frac ac,
219 \qquad
220 1-z=\frac bc,
221 \qquad
222 \beta=1-\frac1z=-\frac ba.
223 \]
224 The Rogers dilogarithm
225 \[
226 L_2(z)
227 =
228 \operatorname{Li}_2(z)
229 +
230 \frac12\log z\log(1-z)
231 \]
232 satisfies
233 \[
234 \boxed{
235 L_2(z)+L_2(1-z)=\frac{\pi^2}{6}.
236 }
237 \]
238 At weight three,
239 \[
240 \boxed{
241 \begin{aligned}
242 &\operatorname{Li}_3(z)
243 +\operatorname{Li}_3(1-z)
244 +\operatorname{Li}_3(\beta)\\
245 &\qquad=
246 \zeta(3)
247 +\frac16\log^3z
248 +\frac{\pi^2}{6}\log z
249 -\frac12\log^2z\log(1-z).
250 \end{aligned}
251 }
252 \]
253 The balanced Kummer coordinate is therefore exactly the third argument in the depth-three functional closure.
254
255 A fixed path packet is
256 \[
257 \Pi_{\le3}(z)
258 =
259 \left(
260 \log z,
261 \log(1-z),
262 L_2(z),
263 \operatorname{Li}_3(z),
264 \operatorname{Li}_3(1-z)
265 \right).
266 \]
267 Its coordinate count is independent of $M$.
268
269 \section{Archimedean envelope}
270 For $0<z<1$,
271 \[
272 |\log z|,
273 |\log(1-z)|
274 \le\log c.
275 \]
276 The positive dilogarithm and trilogarithm values on $(0,1)$ are bounded by $\zeta(2)$ and $\zeta(3)$. The three-point identity gives
277 \[
278 \left|\operatorname{Li}_3\left(-\frac ba\right)\right|
279 \le
280 3\zeta(3)
281 +
282 \frac{\pi^2}{6}\log c
283 +
284 \frac23(\log c)^3.
285 \]
286 Thus the complex packet has only cubic logarithmic growth. The surrounding motivic and regulator language is informed by the polylogarithmic complexes of Goncharov, the regulator framework discussed by Zagier, and Borel's rank theorem~\cite{Goncharov,ZagierPolylog,Borel}.
287
288 This is a genuine nonlinear compression of the displayed complex data. It is not yet an integral compression. A fixed-rank integer lattice carrying a large torsion quotient must pay through covolume, denominator, algebraic degree, or a short vector.
289
290 \section{Fixed-rank integral uncertainty}
291 Let $A\in M_d(\Z)$ be nonsingular and suppose $D\mid\det A$. If every entry satisfies $|A_{ij}|\le B$, then Hadamard's inequality gives
292 \[
293 D\le|\det A|\le d^{d/2}B^d.
294 \]
295 Hence
296 \[
297 \boxed{
298 B\ge\frac{D^{1/d}}{\sqrt d}.
299 }
300 \]
301 A small complex period packet does not imply a small integral realization.
302
303 \section{Exact finite-place notch}
304 At an output prime $p^e\parallel c$,
305 \[
306 \beta=1-\frac ca.
307 \]
308 Since $p\nmid a$,
309 \[
310 v_p(\beta-1)=e.
311 \]
312 For odd $p$,
313 \[
314 \boxed{
315 v_p(\log_p\beta)=e.
316 }
317 \]
318 The corresponding weighted dyadic identity holds on the principal branch when $e\ge2$.
319
320 Writing
321 \[
322 \beta=-\prod_{q\mid ab}q^{\nu_q},
323 \]
324 we obtain the cross-prime logarithmic filter relation
325 \[
326 \boxed{
327 \log_p\beta
328 =
329 \sum_{q\mid ab}
330 \nu_q\log_p\langle q\rangle_p.
331 }
332 \]
333 The same depth-three path therefore has a small complex envelope and exact deep finite notches.
334
335 \section{Weight-three motivation and limitation}
336 Weight two provides strong functional closure but no nonzero pure rational regulator direction. Weight three is the first plausible rational regulator layer. This motivates the depth-three path packet. It does not prove that weight three suffices uniformly. A complete proof may require adaptive higher weights or a different motivic quotient.
337
338 \section{Conditional motivic transfer}
339 Let $K_\beta$ be the cyclic group generated by the balanced Kummer class. Suppose there exists an integral finite object $\mathcal H_3(a,b,c)$ and a canonical map
340 \[
341 \Theta_3:K_\beta\longrightarrow\mathcal H_3(a,b,c)
342 \]
343 satisfying
344 \[
345 |\ker\Theta_3|\le C_\eps R^\eps
346 \]
347 and
348 \[
349 |\mathcal H_3(a,b,c)|\le C'_\eps R_aR_bR^\eps.
350 \]
351 Then the strong $abc$ conjecture follows after absorbing the logarithmic Kummer kernel.
352
353 The missing object must combine finite Kummer depth, the three-point additive relation, an integral motivic or syntomic lattice, and an archimedean period norm.
354
355 \chapter{Cross-Prime Logarithmic Coherence}
356
357 \section{The signed input exponent signal}
358 Let
359 \[
360 S_{ab}=\{q:q\mid ab\}.
361 \]
362 Define
363 \[
364 \nu_q=v_q(b)-v_q(a).
365 \]
366 Then
367 \[
368 -\frac ba=-\prod_{q\mid ab}q^{\nu_q}.
369 \]
370 For every output prime $p\mid c$, the same input exponent packet produces the deep principal unit
371 \[
372 1-\frac ca.
373 \]
374 The output persistence is therefore a simultaneous cross-prime cancellation phenomenon.
375
376 \section{Odd logarithmic rows}
377 For odd $p^e\parallel c$, define the finite-depth row coefficient
378 \[
379 \ell_{p,e}(q)
380 =
381 \frac1p\log_p\langle q\rangle_p
382 \pmod{p^{e-1}}.
383 \]
384 The balanced relation gives
385 \[
386 \boxed{
387 \sum_{q\mid ab}
388 \nu_q\ell_{p,e}(q)
389 \equiv0
390 \pmod{p^{e-1}}.
391 }
392 \]
393 Define the truncated valuation on $\Z/p^{e-1}\Z$ by
394 \[
395 \bar v_p(0)=e-1,
396 \qquad
397 \bar v_p(x)=\min\{e-1,v_p(\widetilde x)\}
398 \]
399 for any integer lift $\widetilde x$ of a nonzero residue. Then
400 \[
401 \boxed{
402 \bar v_p(\ell_{p,e}(q))
403 =
404 \min\{e-1,v_p(q^{p-1}-1)-1\}.
405 }
406 \]
407 Exceptional entrywise depth is a generalized Wieferich event.
408
409 \section{Weighted dyadic row}
410 For an odd input prime $q$, define
411 \[
412 \ell_{2,e}(q)
413 =
414 \frac14
415 \log_2\left((-1)^{(q-1)/2}q\right)
416 \pmod{2^{e-2}}.
417 \]
418 With the analogous truncated valuation on $\Z/2^{e-2}\Z$,
419 \[
420 \bar v_2(\ell_{2,e}(q))
421 =
422 \min\{e-2,v_2(q^2-1)-3\}.
423 \]
424 A separate sign packet supplies the missing first dyadic layer. This is the weighted dyadic branch in its final logarithmic form.
425
426 \section{Sequential factorization of the powerful part}
427 The factorization is defined prime by prime. For an odd repeated output prime $p^e\parallel c$, set
428 \[
429 d_p=e-1.
430 \]
431 For a repeated dyadic prime $2^e\parallel c$, remove one sign layer and set
432 \[
433 s_2=1,
434 \qquad
435 d_2=e-2.
436 \]
437 If $2\nmid M$, take $s_2=0$. Put $\epsilon_2=2^{s_2}$. Let $K_{\mathrm{Kum}}$ be the kernel size of the balanced Kummer class and put
438 \[
439 r_p=\min\{d_p,v_p(K_{\mathrm{Kum}})\}.
440 \]
441 The balanced order formula implies
442 \[
443 K_{\mathrm{Kum}}\mid M/\epsilon_2,
444 \]
445 so $K_{\mathrm{Kum}}=\prod_{p\mid c}p^{r_p}$. Write $K=K_{\mathrm{Kum}}$. After dividing the input exponent vector by $K$, let $a_{p,q}$ denote the resulting odd or weighted-dyadic logarithmic row coefficients. Define
446 \[
447 w_p
448 =
449 \min\left
450 \{
451 d_p-r_p,
452 \min_{q\mid ab}\bar v_p(a_{p,q})
453 \right\},
454 \qquad
455 \chi_p=d_p-r_p-w_p.
456 \]
457 The normalized global factors are
458 \[
459 K=\prod_{p\mid c}p^{r_p},
460 \qquad
461 W=\prod_{p\mid c}p^{w_p},
462 \qquad
463 C=\prod_{p\mid c}p^{\chi_p}.
464 \]
465 The balanced Kummer order formula guarantees that the dyadic sign layer is removed before the principal $2$-adic Kummer depth, so these exponents partition the contact depth. Hence
466 \[
467 \boxed{
468 M=\epsilon_2KWC.
469 }
470 \]
471 The order of removal matters. The local logarithmic defect is measured only after the Kummer preload has been divided out. The resulting row is then divided by its common logarithmic depth, leaving a primitive row modulo $p^{\chi_p}$.
472
473 \section{The coherence character}
474 After normalization, every output prime produces a primitive row
475 \[
476 g_p
477 \in
478 (\Z/p^{\chi_p}\Z)^{S_{ab}}
479 \]
480 satisfying
481 \[
482 g_p\cdot\nu_0
483 \equiv0
484 \pmod{p^{\chi_p}},
485 \]
486 where $\nu_0$ is the Kummer-normalized input exponent vector. The Chinese remainder theorem combines the local rows into one primitive row
487 \[
488 g\in(\Z/C\Z)^{S_{ab}}.
489 \]
490 Define the coherence character
491 \[
492 \boxed{
493 \chi_{\mathrm{coh}}(x)
494 =
495 \exp\left(\frac{2\pi i}{C}g\cdot x\right).
496 }
497 \]
498 Its kernel is the adelic coherence lattice
499 \[
500 \boxed{
501 \Lambda_{\mathrm{coh}}
502 =
503 \{x\in\Z^{S_{ab}}:g\cdot x\equiv0\pmod C\}.
504 }
505 \]
506 Since $g$ is primitive modulo $C$,
507 \[
508 [\Z^{S_{ab}}:\Lambda_{\mathrm{coh}}]=C
509 \]
510 and
511 \[
512 \Z^{S_{ab}}/\Lambda_{\mathrm{coh}}\cong\Z/C\Z.
513 \]
514 The normalized exponent vector lies in the lattice.
515
516 \section{Worked lattice for \texorpdfstring{$3+125=128$}{3+125=128}}
517 Here
518 \[
519 M=64
520 \]
521 and
522 \[
523 (\epsilon_2,K,W,C)=(2,1,1,32).
524 \]
525 The input vector is
526 \[
527 \nu_0=(-1,3)
528 \]
529 on the prime coordinates $(3,5)$. The residual row is
530 \[
531 (29,31)\pmod{32}.
532 \]
533 Thus
534 \[
535 \boxed{
536 \Lambda_{\mathrm{coh}}
537 =
538 \{(x_3,x_5)\in\Z^2:29x_3+31x_5\equiv0\pmod{32}\}.
539 }
540 \]
541 Indeed,
542 \[
543 29(-1)+31(3)=64\equiv0\pmod{32}.
544 \]
545 The five residual dyadic contact layers are exactly the index of this lattice.
546
547 \section{Generalized Wieferich versus collective coherence}
548 There are two qualitatively different sources of large output depth.
549
550 \begin{enumerate}[leftmargin=1.5em]
551 \item An entrywise defect can make every coefficient in one row divisible by a large power of $p$. This is the factor $W$.
552 \item Ordinary coefficients can cancel coherently against the specific exponent vector. This is the factor $C$.
553 \end{enumerate}
554
555 Finite data shows that $W$ is usually one, while $C$ is often large. The typical difficult triple is therefore not explained by one exceptional Wieferich edge. It is explained by coordinated cancellation among several ordinary edges.
556
557 \section{Why abstract lattice geometry is insufficient}
558 Let $v\in\Z^s$ be primitive, with $s\ge2$. For every $C\ge2$, there exists a primitive row $g\pmod C$ satisfying
559 \[
560 g\cdot v\equiv0\pmod C.
561 \]
562 Thus a fixed short vector can lie in primitive congruence lattices of arbitrarily large index.
563
564 \begin{theorem}[Arbitrary-index annihilator]
565 Abstract lattice geometry cannot bound the coherence index from the length of the known relation vector alone.
566 \end{theorem}
567
568 The proof completes $v$ to a unimodular basis and chooses a primitive row annihilating its first basis coordinate modulo $C$.
569
570 The missing theorem must use the arithmetic origin of $g$: its entries are simultaneous $p$-adic logarithms of the actual input primes, and the output primes are exactly the divisors of $a+b$.
571
572 \section{Why positive divisibility is circular}
573 If positive integers $A$ and $B$ satisfy $c\mid A+B$, then
574 \[
575 A+B\ge c,
576 \]
577 so
578 \[
579 \max(A,B)\ge c/2.
580 \]
581 A direct search for a small positive $S$-unit relation divisible by $c$ cannot produce an $abc$ saving. Signed cancellation, a determinant, a regulator, or a functional equation is necessary.
582
583 \section{Exact reformulation of strong \texorpdfstring{$abc$}{abc}}
584 Since
585 \[
586 c=R_c\epsilon_2KWC
587 \]
588 and $\epsilon_2\le2$ while $K\le\log_2c$, the all-$\eps$ strong $abc$ statement is equivalent, after the usual absorption of bounded and logarithmic factors into an arbitrarily small power, to the target
589 \[
590 \boxed{
591 WC
592 \le
593 C_\eps R_aR_bR^\eps.
594 }
595 \]
596 In the generic finite examples $W=1$, leaving the coherence conductor bound
597 \[
598 C\le C_\eps R_aR_bR^\eps.
599 \]
600
601 \section{Adelic coherence regulator object}
602 Package the surviving data as
603 \[
604 \boxed{
605 \mathfrak R_{a,b,c}
606 =
607 \left(
608 U_{ab},
609 \beta^{1/K},
610 \{\chi_p\}_{p\mid c},
611 \chi_{\mathrm{coh}},
612 \Lambda_{\mathrm{coh}},
613 \norm{\cdot}_\infty
614 \right).
615 }
616 \]
617 Here $U_{ab}$ is the input $S$-unit group, the local characters are the normalized logarithmic rows, and the archimedean norm records input height.
618
619 The open theorem is an adelic slope or regulator inequality bounding the finite conductor $\log C$ by the input radical and an arbitrarily small global leakage term. General simultaneous linear-form theory contains the right kinds of local and archimedean quantities, but known general estimates are much too coarse for the $R^\eps$ scale.
620
621 \chapter{The General Prime-Contact Coherence Functor}
622
623 \section{Sparse additive relations}
624 Let
625 \[
626 \epsilon_1n_1+\cdots+\epsilon_rn_r=0,
627 \qquad
628 \epsilon_i\in\{\pm1\},
629 \]
630 be primitive. For every term define
631 \[
632 R_i=\rad(n_i),
633 \qquad
634 M_i=n_i/R_i.
635 \]
636 The prime-contact coherence calculus assigns:
637
638 \begin{enumerate}[leftmargin=1.5em]
639 \item filtered contact packets for every repeated prime power
640 \item local Witt coordinates and dyadic normalization
641 \item Kummer classes for balanced multiplicative ratios
642 \item cross-prime logarithmic filter matrices
643 \item coherence characters and kernel lattices
644 \item optional elliptic, hypergeometric, Fermat, polylogarithmic, or automorphic lifts
645 \end{enumerate}
646
647 \section{Canonical operations}
648 The construction is compatible with several basic operations.
649
650 \subsection{Permutation}
651 Reordering the terms permutes sector labels and leaves the underlying contact and Kummer data unchanged.
652
653 \subsection{Sign reversal}
654 Multiplying the relation by $-1$ reverses the signed exponent packet but leaves the kernel lattice invariant.
655
656 \subsection{Primitive reduction}
657 A common gcd must be removed before forming the contact object. Otherwise one introduces a spurious prime channel shared by every term.
658
659 \subsection{Primary decomposition}
660 Every filtered contact packet and coherence quotient decomposes canonically into its prime-primary factors.
661
662 \subsection{Polyphase refinement}
663 Valuation words may be refined to any modulus. Exact synthesis norms must accompany every refinement.
664
665 \subsection{Kummer quotient}
666 Common powers in a balanced multiplicative relation should be divided out before logarithmic normalization.
667
668 \subsection{Functorial gluing}
669 If two sparse relations share a term, the corresponding contact and Kummer objects should glue across that term. A complete gluing theorem remains open.
670
671 \section{Prime-contact data structure}
672 The reusable object attached to a sparse relation is
673 \[
674 \boxed{
675 \mathfrak A(\mathbf n,\boldsymbol\epsilon)
676 =
677 \left(
678 \mathbb W_{\mathbf n},
679 \kappa_{\mathbf n},
680 \mathcal L_{\mathbf n},
681 \Lambda_{\mathrm{coh}},
682 \mathcal R_\infty,
683 \mathcal M_{\mathrm{glob}}
684 \right).
685 }
686 \]
687 The first four components are finite and local-global arithmetic data. The last two are optional archimedean and global-motivic realizations. Keeping them separate prevents an unproved global comparison from being built into the definition.
688
689 \section{Why the calculus extends beyond \texorpdfstring{$abc$}{abc}}
690 The essential input is not the particular equation $a+b=c$. It is a sparse additive relation with repeated prime depth. This occurs naturally in generalized Fermat equations, perfect-power gaps, recurrence identities, and Bernoulli congruence problems. The later application chapters develop these directions at a programmatic level.
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