appendices.tex
Prime-Contact Coherence Calculus
appendices.tex
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1 \appendix
2 \PartChapter{Worked Examples, Exact Calculations, and Applications}{One Triple Through the Entire Calculus: \texorpdfstring{$3+125=128$}{3+125=128}}
3
4 \section{Arithmetic data}
5 Consider
6 \[
7 a=3,
8 \qquad
9 b=125=5^3,
10 \qquad
11 c=128=2^7.
12 \]
13 The radical is
14 \[
15 R=2\cdot3\cdot5=30.
16 \]
17 The quality is
18 \[
19 q=\frac{\log128}{\log30}.
20 \]
21 The output contact modulus is
22 \[
23 M=\frac{128}{2}=64.
24 \]
25 This example is especially useful because each sector consists of one prime, the radical divisor is attained sharply, the output has a long dyadic filtration, and every later object can be written explicitly.
26
27 \section{Prime derivative}
28 The additive row is
29 \[
30 \lambda=(1,75,-448).
31 \]
32 The phase row is
33 \[
34 \lambda^{\mathrm{ph}}=(1,25,-64).
35 \]
36 The coarse shortest transverse packet
37 \[
38 x=(-2,6,1)
39 \]
40 satisfies
41 \[
42 -2+450=448.
43 \]
44 The Wronskian is
45 \[
46 W_x=1600=\frac{abc}{R}.
47 \]
48 The coarse height is six.
49
50 The phase packet
51 \[
52 y=(-8,8,3)
53 \]
54 satisfies
55 \[
56 D_y(3)=-8,
57 \qquad
58 D_y(125)=200,
59 \qquad
60 D_y(128)=192,
61 \]
62 and
63 \[
64 -8+200=192.
65 \]
66 Its Wronskian is also $1600$. The phase height is eight. The phase inequality is sharp:
67 \[
68 128
69 =
70 30\left(\frac13+\frac15\right)8.
71 \]
72 This illustrates that the exponent-free phase channel can sharpen the bound even when its packet norm is larger.
73
74 \section{Obstruction complex}
75 The joint exponent-free phase--Wronskian matrix has Smith form
76 \[
77 \operatorname{diag}(1,1600).
78 \]
79 The additive lattice maps onto $1600\Z$ through the Wronskian. The primitive obstruction section exists algebraically. Its height remains the metric problem.
80
81 The stable real height is computed from
82 \[
83 \sigma_a=\frac13,
84 \qquad
85 \sigma_b=\frac15,
86 \qquad
87 \sigma_c=\frac12.
88 \]
89 The smallest sector pair is
90 \[
91 \sigma_a+\sigma_b=\frac8{15}.
92 \]
93 Thus
94 \[
95 \overline H
96 =
97 \frac{128}{30(8/15)}
98 =8.
99 \]
100 The phase packet has height eight, so the integral optimum reaches the real lower barrier exactly.
101
102 \section{Moving search}
103 The primitive obstruction phase is already optimal. The certified stopping bound contains no later coset capable of improving height eight because
104 \[
105 k\overline H\ge8
106 \]
107 for every $k\ge1$. This example is therefore a clean calibration of the no-escape identity.
108
109 \section{Signal and Frey layers}
110 The prime-shell energy of $c$ is
111 \[
112 \log128=7\log2,
113 \]
114 while the activated output bandwidth is
115 \[
116 \log2.
117 \]
118 The output persistence energy is
119 \[
120 6\log2=\log64.
121 \]
122
123 The marked Frey curve is
124 \[
125 y^2=x(x-3)(x+125).
126 \]
127 Its discriminant is
128 \[
129 16\cdot3^2\cdot125^2\cdot128^2.
130 \]
131 At the odd primes, the marked depths are two at $3$ and six at $5$. The dyadic depth requires the weighted branch.
132
133 The rational automorphic channel alone does not determine the marked dyadic depth of the displayed Frey representative. Any passage to an isogenous or twisted representative must therefore carry the differential scaling and the marked local $2$-adic data explicitly.
134
135 \section{Filtered contact and Witt packet}
136 The output contact complex is
137 \[
138 [128\Z\hookrightarrow2\Z]
139 \simeq
140 [\Z\xrightarrow{64}\Z].
141 \]
142 Its cohomology is
143 \[
144 \Jcal_c\cong\Z/64\Z.
145 \]
146 The associated graded is
147 \[
148 (\F_2)^6.
149 \]
150 The six binary layers are joined by five nontrivial carry extensions. A cyclic target of order $64$ can carry the filtered source exactly, while no cyclic target can contain the split rank-six source injectively.
151
152 The local Witt coordinate is
153 \[
154 W_6(\F_2)
155 \cong
156 \Z/64\Z.
157 \]
158 The weighted multiplicative dyadic coordinate is
159 \[
160 (1+4\Z_2)/(1+2^8\Z_2),
161 \]
162 which has the same order.
163
164 \section{Kummer and Fermat packets}
165 The global Kummer map is
166 \[
167 \kappa_{128}:\Z/64\Z\to H^1(\Q,\mu_{64}).
168 \]
169 Since
170 \[
171 \gcd(64,7)=1,
172 \]
173 it is injective.
174
175 For the balanced ratio
176 \[
177 \beta=-\frac{125}{3},
178 \]
179 the signed exponent vector on the input primes is
180 \[
181 \nu=(-1,3).
182 \]
183 Its Kummer order modulo $64$ is full. Choosing $64$th roots gives a point on
184 \[
185 X^{64}+Y^{64}=1.
186 \]
187 The coordinate height is
188 \[
189 \frac{\log128}{64}.
190 \]
191 With the tensor-power metric induced from $\mathcal O(1)$, the full plane-model canonical scale is
192 \[
193 \frac{61}{64}\log128.
194 \]
195 For an arbitrary compatible Weil-height normalization, this formula holds up to an additive $O(1)$ term. It records the line-bundle cost of retaining the complete Fermat curve.
196
197 \section{Polylogarithmic path}
198 Set
199 \[
200 z=\frac3{128},
201 \qquad
202 1-z=\frac{125}{128},
203 \qquad
204 \beta=1-\frac1z=-\frac{125}{3}.
205 \]
206 The depth-three functional relation closes on these three arguments. At the output prime two,
207 \[
208 v_2(\beta-1)
209 =
210 v_2\left(-\frac{128}{3}\right)
211 =7.
212 \]
213 The unscaled principal logarithm has $2$-adic valuation seven. After division by four, the weighted principal coordinate occupies five filtration levels, while the separate sign packet supplies the first dyadic layer.
214
215 \section{Coherence lattice}
216 After removing the sign layer,
217 \[
218 (\epsilon_2,K,W,C)=(2,1,1,32).
219 \]
220 The residual row is
221 \[
222 g=(29,31)\pmod{32}.
223 \]
224 The lattice is
225 \[
226 \Lambda_{\mathrm{coh}}
227 =
228 \{(x_3,x_5):29x_3+31x_5\equiv0\pmod{32}\}.
229 \]
230 The input vector lies in the kernel because
231 \[
232 29(-1)+31(3)=64.
233 \]
234 The full repeated output depth has now been transformed into one arithmetic cyclic character of order $32$, after the bounded sign layer has been separated.
235
236 \section{What this example teaches}
237 The example is simultaneously favorable and cautionary.
238
239 \begin{enumerate}[leftmargin=1.5em]
240 \item The radical divisor is sharp.
241 \item The phase packet reaches the continuous lower barrier.
242 \item The Kummer map is injective.
243 \item The local logarithmic defect is trivial.
244 \item The remaining depth is pure coherence.
245 \end{enumerate}
246
247 So even the cleanest example does not prove the needed conductor bound. It isolates the final problem in its purest form.
248
249 \chapter{Further Calibration Examples}
250
251 \section{\texorpdfstring{$1+2400=2401$}{1+2400=2401} and dyadic Landen descent}
252 The triple is
253 \[
254 1+2400=2401=7^4.
255 \]
256 The radical is
257 \[
258 R=2\cdot3\cdot5\cdot7=210.
259 \]
260 The output contact modulus is
261 \[
262 M=7^3=343.
263 \]
264 The global Kummer kernel is one because
265 \[
266 \gcd(343,4)=1.
267 \]
268
269 The triple belongs to the Landen family with
270 \[
271 u=49,
272 \qquad
273 v=1.
274 \]
275 The dyadic child is
276 \[
277 49+576=625.
278 \]
279 Both triples have radical $210$. The child has lower output height. The corresponding Frey curves occupy the same two-isogeny packet, while their marked integral representatives have different heights and local depths.
280
281 The output persistence of the child is
282 \[
283 \frac{625}{5}=125.
284 \]
285 The modular degree of one associated minimal curve is a power of two in the stored example. This disproves the naive claim that the output persistence must divide one modular degree. The local marked depth and global eigenline separation are distinct channels.
286
287 \section{\texorpdfstring{$289+6272=6561$}{289+6272=6561}}
288 Here
289 \[
290 289=17^2,
291 \qquad
292 6272=2^7\cdot7^2,
293 \qquad
294 6561=3^8.
295 \]
296 The radical is
297 \[
298 R=2\cdot3\cdot7\cdot17=714.
299 \]
300 The output contact modulus is
301 \[
302 M=3^7=2187.
303 \]
304 The Kummer kernel is
305 \[
306 \gcd(2187,8)=1.
307 \]
308 Thus the output contact packet globalizes injectively.
309
310 The input exponent vector contains several nontrivial channels. The high output depth is not explained by a generalized Wieferich defect in one coefficient. It is a collective coherence event. This is representative of the generic high-quality behavior in the finite data set.
311
312 \section{\texorpdfstring{$1+4374=4375$}{1+4374=4375}}
313 The output is
314 \[
315 4375=5^4\cdot7.
316 \]
317 The output radical is $35$, and
318 \[
319 M=5^3=125
320 \]
321 The output contact packet has order $125$. This one-prime output is a clean setting for separating Kummer preload, local logarithmic defect, and residual coherence.
322
323 The broader point is independent of this normalization detail. A single repeated output prime creates a one-prime contact packet. The coherence problem then reduces to one logarithmic row, while the Kummer and Wieferich factors can be computed exactly.
324
325 \section{A large high-quality triple}
326 Consider the stored high-quality example
327 \[
328 24833
329 +
330 5020969537415167
331 =
332 5020969537440000.
333 \]
334 The output factorization contains
335 \[
336 2^8 3^{22}5^4.
337 \]
338 The output contact modulus is
339 \[
340 M=2^7 3^{21}5^3
341 =167365651248000.
342 \]
343 This exceeds many fixed-width intermediate scales. It is the canonical regression case for arbitrary-precision arithmetic.
344
345 The split associated graded has rank
346 \[
347 7+21+3=31.
348 \]
349 The filtered source is one cyclic group of order $M$. The global Kummer kernel is only two in the finite data set. Thus almost the entire enormous contact packet survives globalization. The unresolved arithmetic is the economical global synthesis of the remaining coherence data.
350
351 \section{A generalized Wieferich example}
352 For
353 \[
354 1+57121=57122,
355 \]
356 with
357 \[
358 57121=239^2,
359 \qquad
360 57122=2\cdot13^4,
361 \]
362 the output contact modulus is
363 \[
364 M=13^3.
365 \]
366 The local logarithm of the input prime $239$ at $13$ has exceptional depth. In the four-factor decomposition,
367 \[
368 (\epsilon_2,K,W,C)=(1,1,13^3,1).
369 \]
370 The entire repeated output depth is explained by one generalized Wieferich edge. This is a rare mechanism. Most finite examples have $W=1$ and large collective coherence factor $C$.
371
372 \chapter{Collected Theorems and Dependencies}
373
374 \section{Purpose}
375 This chapter records the dependency structure of the exact results. It does not repeat every calculation. It shows which statements rely only on elementary integer algebra, which use local analytic identities, and which invoke external arithmetic theory.
376
377 \section{Elementary integer layer}
378 The following results are elementary consequences of valuations, Smith normal form, the Chinese remainder theorem, and exact lattice algebra.
379
380 \begin{enumerate}[leftmargin=1.5em]
381 \item The universal prime-packet differential obeys Leibniz.
382 \item The additive condition is one integer row.
383 \item The arithmetic Wronskian is one additional row.
384 \item The divisor $abc/R$ divides every additive Wronskian.
385 \item Sector compression reduces each prime block to one aggregate coordinate plus lift gauges.
386 \item The joint obstruction matrix has torsion invariant $abc/R$.
387 \item The additive lattice fits into
388 \[
389 0\to L_0\to L\to\Z\to0.
390 \]
391 \item The sharp aggregate solutions form an affine integer line.
392 \item Each local lift reduces to centered CRT residues and one quotient-sum equation.
393 \item The first feasible interval intersection gives the exact coset height.
394 \item The continuous lower bound gives finite stopping in the obstruction multiple.
395 \item The filtered contact quotient is $R_c\Z/c\Z$.
396 \item The associated graded is the direct sum of residue-field layers.
397 \item The maximal image between two cyclic packets has order equal to their gcd.
398 \item The global Kummer kernel is the gcd of the modulus with the valuation vector.
399 \item The Fermat character projectors satisfy exact orthogonality and completeness.
400 \item The coherence lattice has index equal to the coherence quotient.
401 \end{enumerate}
402
403 \section{Finite Fourier and polyphase layer}
404 The valuation-word Smith theorem uses circulant eigenvalues. The determinant calculation is analytic in appearance but finite and exact. The only nontrivial invariant factor is the aggregate exponent because the nonconstant cyclotomic phases multiply to a unit.
405
406 The Atkin--Lehner Walsh bank is a finite product of commuting involutions. Its identities are polynomial consequences of $W_j^2=I$.
407
408 The Fermat projector bank and polylogarithmic distribution relation are finite-character orthogonality identities. The latter is first proved in the disk of convergence and then continued analytically.
409
410 \section{Formal deformation layer}
411 Formal smoothness of the shell deformation uses the primitive gradient. Once one coordinate has linear coefficient one, the formal implicit-function recursion is integral at every order. This proves extension existence. It does not bound coefficient growth.
412
413 The gauge-collapse statement is an algebraic factorization through aggregate sector products. It explains why internal phase curvature can be corrected formally.
414
415 \section{Local logarithmic layer}
416 For odd $p$, the identity
417 \[
418 v_p(\log_p(1+u))=v_p(u)
419 \]
420 holds when $v_p(u)>0$. This yields the exact output notch depth. The coefficient formula
421 \[
422 v_p(q^{p-1}-1)-1
423 \]
424 comes from the principal-unit logarithm of the input prime.
425
426 At $2$, the logarithm begins on $1+4\Z_2$. The separate sign bit and the shifted principal filtration must be retained. This is the source of the weighted dyadic normalization.
427
428 \section{External arithmetic layer}
429 The following contextual bridges use established external theory rather than proofs given in this paper.
430
431 \begin{enumerate}[leftmargin=1.5em]
432 \item Modularity associates a weight-two newform to a Frey curve.
433 \item Tate uniformization interprets multiplicative reduction through a local parameter.
434 \item Hecke cotangent spaces and Wiles defects compare congruence and deformation multiplicity.
435 \item Fermat Jacobians decompose through character-related factors and Jacobi sums.
436 \item Motivic, syntomic, and Deligne regulators provide candidate realizations for the depth-three path packet.
437 \end{enumerate}
438
439 No external theorem is treated as providing the missing coherence-conductor estimate.
440
441 \section{Conditional implications}
442 Every conditional $abc$ theorem in this paper has the same logical shape.
443
444 \begin{enumerate}[leftmargin=1.5em]
445 \item Construct a finite source of order comparable to $c/R_c$.
446 \item Map it into a global target with kernel at most $R^\eps$.
447 \item Bound the target size by $R_aR_bR^\eps$.
448 \item Multiply by $R_c$ and absorb logarithmic factors.
449 \end{enumerate}
450
451 The differences among the conditional theorems lie in the source and target objects. The latest endpoint uses the coherence character or an integral adelic regulator attached to it.
452
453 \chapter{Exact Computational Procedures}
454
455 \section{Exact arithmetic contract}
456 All factorizations, radicals, contact moduli, obstruction rows, CRT residues, lattice indices, and stopping bounds are computed with arbitrary-precision integers. Floating point is never used to decide feasibility, equality, divisibility, or search termination.
457
458 A public arithmetic function must promote integer-like inputs before multiplication. This prevents silent overflow when a NumPy scalar enters an otherwise exact computation.
459
460 Every display ratio should also retain an exact rational representation when possible. Decimal output is for plotting and human comparison only.
461
462 \section{Factorization and data normalization}
463 For each triple, the canonical record contains
464 \[
465 (a,b,c),
466 \quad
467 \{v_p(a)\},
468 \quad
469 \{v_p(b)\},
470 \quad
471 \{v_p(c)\},
472 \quad
473 R,
474 \quad
475 q(a,b,c).
476 \]
477 Derived objects must be computed from this record. No displayed factorization is accepted as input without verification.
478
479 The triple is normalized by:
480
481 \begin{enumerate}[leftmargin=1.5em]
482 \item checking $a+b=c$
483 \item checking $\gcd(a,b)=1$
484 \item ordering $a\le b$ when a search convention requires it
485 \item computing radicals from exact prime sets
486 \item separating dyadic and odd-adic branches
487 \end{enumerate}
488
489 \section{Exact moving-search pseudocode}
490 For one triple:
491
492 \begin{enumerate}[leftmargin=1.5em]
493 \item Compute $r_u$, $P_u$, $R$, $Q$, and the exact stable height $\overline H$.
494 \item Compute the primitive obstruction height $H(1)$ by binary search on the three interval intersections.
495 \item Set
496 \[
497 K_{\max}
498 =
499 \left\lfloor\frac{H(1)-1}{\overline H}\right\rfloor.
500 \]
501 \item For $2\le k\le K_{\max}$, skip a coset when $k\overline H$ already reaches the current best height.
502 \item Build centered CRT phase residues for the three sectors.
503 \item Compute the coarse affine line.
504 \item Binary-search the smallest $H$ whose three admissible $n$-intervals meet.
505 \item Construct one witness, verify additivity and obstruction multiple, and update the stopping bound.
506 \item Store the preceding empty radius as the no-smaller-box certificate.
507 \end{enumerate}
508
509 This is an exact algorithm after factorization. It is much faster than a generic mixed-integer formulation because the geometry has been compressed before search.
510
511 \section{Coherence-lattice construction}
512 The coherence lattice is constructed as follows.
513
514 \begin{enumerate}[leftmargin=1.5em]
515 \item Form the signed input exponent vector $\nu$.
516 \item Remove the common Kummer preload $K$.
517 \item For every output prime, construct the finite logarithmic row.
518 \item Divide each row by its common coefficient depth to obtain the local defect $W_p$.
519 \item Remove the dyadic sign layer separately.
520 \item Combine primitive residual rows by CRT.
521 \item Verify that the normalized exponent vector lies in the kernel.
522 \item Compute the exact quotient order $C$ and the factorization
523 \[
524 M=\epsilon_2KWC.
525 \]
526 \end{enumerate}
527
528 The sequential order prevents one depth layer from being counted twice.
529
530 \section{Certificate philosophy}
531 Every finite claim should be accompanied by one of four certificate types.
532
533 \begin{longtable}{p{0.22\textwidth}p{0.68\textwidth}}
534 \toprule
535 Certificate & Content\\
536 \midrule
537 Algebraic & exact identity, factorization, Smith form, or divisibility check\\
538 Feasibility & explicit witness satisfying all integer constraints\\
539 Infeasibility & empty interval, dual bound, or no-smaller-box certificate\\
540 Numerical & high-precision residual with a declared tolerance and independently computed input\\
541 \bottomrule
542 \end{longtable}
543
544 A finite data set is evidence about the declared range. It is not an asymptotic theorem.
545
546 \chapter{Prime-Theory Applications}
547
548 \section{Simultaneous Wieferich networks}
549 Define a directed prime network by the edge depth
550 \[
551 w_p(q)=v_p(q^{p-1}-1)-1
552 \]
553 for odd $p$, with the weighted dyadic analogue. An additive relation selects a finite bipartite subnetwork from input primes to output primes.
554
555 Three mechanisms can create deep output contact:
556
557 \begin{enumerate}[leftmargin=1.5em]
558 \item exceptional individual edges
559 \item degenerate minors of the logarithmic matrix
560 \item collective cancellation against the signed exponent vector
561 \end{enumerate}
562
563 A first unconditional objective is to prove density or power-saving bounds for networks exhibiting large simultaneous defects. This is closer to the exact object than a direct treatment of arbitrary prime gaps.
564
565 \section{Exceptional \texorpdfstring{$abc$}{abc} classification}
566 A structural classifier should attach to every high-quality triple the tuple
567 \[
568 (\epsilon_2,K,W,C)
569 \]
570 and additional minor data from the cross-prime logarithm matrix. A high-quality exceptional set may then be partitioned into:
571
572 \begin{enumerate}[leftmargin=1.5em]
573 \item Kummer-preloaded families
574 \item generalized Wieferich families
575 \item low-rank regulator families
576 \item collective-coherence families
577 \item automorphic-congruence families
578 \end{enumerate}
579
580 Even without a uniform $abc$ proof, a theorem showing that one or more mechanisms occur in a density-zero family would be meaningful progress.
581
582 \section{Generalized Fermat}
583 For
584 \[
585 x^p+y^q=z^r,
586 \]
587 attach filtered contact packets to the three terms. The exponents multiply the local valuation depths. Balanced Kummer ratios and cross-prime logarithmic filters can eliminate many local patterns before a Frey or hypergeometric motive is attached.
588
589 A realistic first target is a family theorem: under a nondegenerate logarithmic-minor condition, prove that no primitive solution exists outside a finite computable set. The modular or hypergeometric method would then handle the exceptional residual signatures.
590
591 \section{Beal-type equations}
592 A primitive counterexample to
593 \[
594 A^x+B^y=C^z,
595 \qquad x,y,z>2,
596 \]
597 would contain three deep contact packets on disjoint prime supports. This is an especially strong setting for prime-contact filtering.
598
599 A concrete intermediate theorem is:
600
601 \begin{quote}
602 If one repeated output prime gives a unit minor in the normalized cross-prime logarithm matrix, then no primitive Beal solution exists in the declared exponent family.
603 \end{quote}
604
605 The exact formulation depends on the rank and local coefficient prime.
606
607 \section{Pillai and Hall}
608 For
609 \[
610 x^m-y^n=k,
611 \]
612 the residual support is fixed by $k$. The two perfect-power packets must cancel to high precision while producing a small fixed output. This is a natural two-sector coherence problem.
613
614 Hall's equation
615 \[
616 x^3-y^2=k
617 \]
618 is particularly attractive because the dyadic and triadic branches are built into the exponents. The polylogarithmic weight-three resonance may be relevant, but a rigorous connection has not yet been established.
619
620 \section{Bernoulli--Witt and Erd\H{o}s--Moser}
621 Faulhaber's formula rewrites
622 \[
623 1^k+\cdots+(m-1)^k=m^k
624 \]
625 as a Bernoulli-polynomial identity. The denominator primes of Bernoulli numbers, the finite-difference divisibility, and the separate $2$-adic and odd-adic lift structures suggest a different contact object:
626 \[
627 \boxed{
628 \text{Bernoulli contact crystal}
629 +
630 \text{denominator prime packet}
631 +
632 \text{finite-difference coherence lattice}.
633 }
634 \]
635 This branch should be developed independently before it is merged with the logarithmic prime-contact theory.
636
637 \section{Why some famous prime problems are later targets}
638 Twin primes, Goldbach, and the Riemann hypothesis are dominated by prime support rather than repeated prime depth. The calculus is strongest when exponents exceed one and carry filtrations matter. These problems may eventually connect through a broader trace or spectral formalism, but they are not the most credible immediate applications.
639
640 \chapter{Unresolved Arithmetic Problems}
641
642 \section{Combined logarithmic--coherence conductor}
643 For the actual rows constructed from the input primes and output-prime logarithms, prove
644 \[
645 WC\le C_\eps R_aR_bR^\eps.
646 \]
647 The theorem must control both entrywise logarithmic degeneration and collective coherence and must exploit the addition law. On the subfamily $W=1$, it reduces to a pure bound for $C$. The corresponding statement is false for arbitrary primitive congruence rows.
648
649 \section{Adelic minor nondegeneracy}
650 Find an exterior-power or regulator formulation of the cross-prime logarithm matrix whose primitive nonzero minor has denominator bounded by radical scale. This may be stronger than the one-row coherence formulation and may handle several independent additive relations simultaneously.
651
652 \section{Filtered local-global transfer}
653 Construct the cusp-contact to Selmer to Hecke chain map with Bockstein compatibility. Prove both a small source kernel and a conductor-scale target bound.
654
655 \section{Sparse motivic quotient}
656 Construct a bounded-complexity Jacobi, polylogarithmic, syntomic, or automorphic quotient retaining the balanced Kummer class. Control finite denominator, field degree, conductor, and all archimedean conjugates.
657
658 \section{Functorial gluing}
659 If two sparse relations share a term, construct the fiber product or pushout of their contact, Kummer, and coherence objects. Prove a conservation or cancellation law stronger than applying the two equations separately.
660
661 \section{First decisive intermediate results}
662 Any of the following would materially advance the theory:
663
664 \begin{enumerate}[leftmargin=1.5em]
665 \item a power-saving bound for the exceptional set of large coherence quotients
666 \item a new density theorem for simultaneous generalized Wieferich defects
667 \item a family of generalized Fermat signatures eliminated by contact filtering
668 \item an effective Pillai or Hall theorem under one natural regulator condition
669 \item a nontrivial kernel bound for a canonical filtered cusp-to-Hecke map
670 \item a sparse motivic quotient with provable radical-scale conductor
671 \end{enumerate}
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