detailed_1.tex
Prime-Contact Coherence Calculus
detailed_1.tex
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1 \PartChapter{Detailed Theory: Prime Packets, Valuation Refinement, and Exact Obstructions}{Prime-Packet Differentials and Radical Divisibility}
2
3 \section{The prime-coordinate module}
4 Let
5 \[
6 a+b=c,
7 \qquad
8 \gcd(a,b)=1.
9 \]
10 The three integers are pairwise coprime. Write
11 \[
12 S=S_a\sqcup S_b\sqcup S_c
13 =\{p:p\mid abc\}.
14 \]
15 The first object of the calculus is the free prime-packet module
16 \[
17 \Omega_S=\bigoplus_{p\in S}\Z\xi_p.
18 \]
19 For a positive integer $n$, define the universal multiplicative differential
20 \[
21 \boxed{
22 \mathrm dn
23 =
24 n\sum_{p\mid n}\frac{v_p(n)}p\,\xi_p.
25 }
26 \]
27 Every coefficient is an integer because $p\mid n$. The map obeys the exact Leibniz law
28 \[
29 \mathrm d(mn)=m\,\mathrm dn+n\,\mathrm dm.
30 \]
31 The verification is local. For each prime $p$,
32 \[
33 v_p(mn)=v_p(m)+v_p(n),
34 \]
35 so the $p$-coordinate of the right side is
36 \[
37 mn\frac{v_p(m)+v_p(n)}p.
38 \]
39
40 A scalar packet is an integer vector
41 \[
42 x=(x_p)_{p\in S}\in\Z^S.
43 \]
44 Pairing with $x$ gives the arithmetic derivative
45 \[
46 \boxed{
47 \mathrm d_x(n)
48 =
49 n\sum_{p\mid n}\frac{v_p(n)}p x_p.
50 }
51 \]
52 The Leibniz rule survives scalarization.
53
54 Arithmetic derivatives and their connection with $abc$ have an established literature. The present construction is organized here in prime-packet coordinates and later refined into filtered shell objects~\cite{PastenDerivatives,VojtaABC}.
55
56 \section{The additive row}
57 The condition
58 \[
59 \mathrm d_x(a)+\mathrm d_x(b)=\mathrm d_x(c)
60 \]
61 is one integer equation. Define
62 \[
63 \lambda_p=
64 \begin{cases}
65 \dfrac{a v_p(a)}p,&p\mid a,\\[6pt]
66 \dfrac{b v_p(b)}p,&p\mid b,\\[6pt]
67 -\dfrac{c v_p(c)}p,&p\mid c.
68 \end{cases}
69 \]
70 Then
71 \[
72 \boxed{\lambda\cdot x=0.}
73 \]
74 The additive derivative lattice is
75 \[
76 \boxed{
77 \mathcal T_{a,b,c}
78 =
79 \ker(\lambda)\cap\Z^S.
80 }
81 \]
82 Since the row is nonzero, its rank is $|S|-1$. The problem is not existence of additive derivatives. The problem is finding an additive derivative that remains transverse to the dependence direction introduced by $a$ and $b$.
83
84 \section{The arithmetic Wronskian}
85 Define
86 \[
87 \boxed{
88 W_x(a,b)
89 =a\,\mathrm d_x(b)-b\,\mathrm d_x(a).
90 }
91 \]
92 Writing
93 \[
94 \mu_p=
95 \begin{cases}
96 -\dfrac{v_p(a)}p,&p\mid a,\\[6pt]
97 \dfrac{v_p(b)}p,&p\mid b,\\[6pt]
98 0,&p\mid c,
99 \end{cases}
100 \]
101 we obtain
102 \[
103 W_x(a,b)=ab\,\mu\cdot x.
104 \]
105 An additive packet is \emph{transverse} when $W_x(a,b)\neq0$. It is degenerate when the two input terms remain derivative-proportional.
106
107 Over $\R$, the additive projection is
108 \[
109 P_{\mathrm{add}}
110 =
111 I-
112 \frac{\lambda\lambda^{\mathsf T}}{\norm{\lambda}_2^2}.
113 \]
114 The obstruction direction inside the additive hyperplane is
115 \[
116 \mu_\perp=P_{\mathrm{add}}\mu.
117 \]
118 When $\mu_\perp\neq0$, the rank-one obstruction projector is
119 \[
120 P_{\mathrm{obs}}
121 =
122 \frac{\mu_\perp\mu_\perp^{\mathsf T}}
123 {\norm{\mu_\perp}_2^2}.
124 \]
125 This is the first appearance of the projector-resolved pattern. A large ambient prime space is decomposed into one equation direction, one obstruction direction, and the remaining null packets.
126
127 \section{Radical divisibility}
128 Set
129 \[
130 R=\rad(abc).
131 \]
132 The central first-order theorem is an exact divisibility law.
133
134 \begin{theorem}[Wronskian divisibility]
135 For every additive packet $x\in\mathcal T_{a,b,c}$,
136 \[
137 \boxed{
138 \frac{abc}{R}\mid W_x(a,b).
139 }
140 \]
141 \end{theorem}
142
143 \begin{proof}
144 Let $p^e\parallel a$. Every term of $\mathrm d_x(a)$ is divisible by $p^{e-1}$. Since $p\nmid b$, the term $b\,\mathrm d_x(a)$ is divisible by $p^{e-1}$. Also $p^e\mid a$, so $a\,\mathrm d_x(b)$ is divisible by $p^e$, hence by $p^{e-1}$. Therefore
145 \[
146 p^{e-1}\mid W_x(a,b).
147 \]
148 The same argument applies to primes dividing $b$.
149
150 Now let $p^e\parallel c$. Additivity gives
151 \[
152 \mathrm d_x(c)=\mathrm d_x(a)+\mathrm d_x(b).
153 \]
154 Using $c=a+b$,
155 \[
156 \begin{aligned}
157 W_x(a,b)
158 &=a\,\mathrm d_x(b)-b\,\mathrm d_x(a)\\
159 &=a\bigl(\mathrm d_x(c)-\mathrm d_x(a)\bigr)-b\,\mathrm d_x(a)\\
160 &=a\,\mathrm d_x(c)-c\,\mathrm d_x(a).
161 \end{aligned}
162 \]
163 The first term is divisible by $p^{e-1}$ because $\mathrm d_x(c)$ is. The second is divisible by $p^e$. Hence $p^{e-1}\mid W_x(a,b)$.
164
165 The prime supports of $a$, $b$, and $c$ are disjoint. Multiplying all local divisibilities gives
166 \[
167 \prod_{p^e\parallel abc}p^{e-1}
168 =
169 \frac{abc}{R}
170 \mid W_x(a,b).
171 \]
172 \end{proof}
173
174 For a transverse packet,
175 \[
176 \boxed{
177 |W_x(a,b)|\ge\frac{abc}{R}.
178 }
179 \]
180 This is the integer analogue of the repeated-root divisor appearing in the polynomial Mason--Stothers theorem.
181
182 \section{The master height inequality}
183 For every positive $n$,
184 \[
185 \begin{aligned}
186 |\mathrm d_x(n)|
187 &\le
188 n\norm{x}_\infty
189 \sum_{p\mid n}\frac{v_p(n)}p.
190 \end{aligned}
191 \]
192 Since $p\ge2$,
193 \[
194 \frac1p\le\frac1{2\log2}\log p.
195 \]
196 Therefore
197 \[
198 \sum_{p\mid n}\frac{v_p(n)}p
199 \le
200 \frac{\log n}{2\log2},
201 \]
202 and
203 \[
204 |\mathrm d_x(n)|
205 \le
206 \frac{n\log n}{2\log2}\norm{x}_\infty.
207 \]
208 Consequently,
209 \[
210 \begin{aligned}
211 |W_x(a,b)|
212 &\le
213 a|\mathrm d_x(b)|+b|\mathrm d_x(a)|\\
214 &\le
215 \frac{ab}{\log2}\norm{x}_\infty\log c.
216 \end{aligned}
217 \]
218 Combining this with radical divisibility gives
219 \[
220 \boxed{
221 c
222 \le
223 \frac{R}{\log2}\norm{x}_\infty\log c.
224 }
225 \]
226
227 Define the shortest transverse packet
228 \[
229 \boxed{
230 \Theta(a,b,c)
231 =
232 \min\left\{
233 \norm{x}_\infty:
234 x\in\mathcal T_{a,b,c},
235 W_x(a,b)\neq0
236 \right\}.
237 }
238 \]
239 Then
240 \[
241 c
242 \le
243 \frac{R}{\log2}\Theta(a,b,c)\log c.
244 \]
245 A subpolynomial estimate
246 \[
247 \Theta(a,b,c)=c^{o(1)}
248 \]
249 would imply the strong $abc$ inequality. Later chapters explain why this target is too close to the conjecture to be reached by generic lattice geometry alone.
250
251 \section{Three-sector compression}
252 Within the $a$-sector, define the integer row
253 \[
254 \lambda^{(a)}
255 =
256 \left(\frac{a v_p(a)}p\right)_{p\mid a}
257 \]
258 and let
259 \[
260 g_a=\gcd\left\{\frac{a v_p(a)}p:p\mid a\right\}.
261 \]
262 Define $g_b$ and $g_c$ similarly. A one-row Smith normal form in each sector produces a unimodular coordinate system in which only one aggregate channel survives. The additive equation becomes
263 \[
264 \boxed{
265 g_at_a+g_bt_b-g_ct_c=0.
266 }
267 \]
268 The Wronskian becomes
269 \[
270 \boxed{
271 a g_bt_b-b g_at_a\neq0.
272 }
273 \]
274 All remaining sector coordinates are lift gauges. They matter only because they determine the cost of realizing an aggregate channel by an integral prime packet.
275
276 For each term define
277 \[
278 \rho_a(t)
279 =
280 \min\left\{
281 \norm{x^{(a)}}_\infty:
282 \lambda^{(a)}\cdot x^{(a)}=g_at
283 \right\},
284 \]
285 with analogous functions for $b$ and $c$. Then
286 \[
287 \Theta(a,b,c)
288 =
289 \min
290 \max\{\rho_a(t_a),\rho_b(t_b),\rho_c(t_c)\}
291 \]
292 over aggregate solutions satisfying additivity and transversality.
293
294 This reduction separates the global two-dimensional arithmetic geometry from the local lifting cost. It is the template used repeatedly later: compress first, then retain the exact synthesis norm of the discarded coordinates.
295
296 \section{Worked example: \texorpdfstring{$3+125=128$}{3+125=128}}
297 The prime packets are $3$, $5$, and $2$. The additive row is
298 \[
299 \lambda=(1,75,-448).
300 \]
301 A shortest transverse packet is
302 \[
303 x=(-2,6,1).
304 \]
305 It gives
306 \[
307 \mathrm d_x(3)=-2,
308 \qquad
309 \mathrm d_x(125)=450,
310 \qquad
311 \mathrm d_x(128)=448.
312 \]
313 Thus additivity holds. The Wronskian is
314 \[
315 W_x(3,125)
316 =3(450)-125(-2)
317 =1600.
318 \]
319 The radical is
320 \[
321 R=2\cdot3\cdot5=30,
322 \]
323 and
324 \[
325 \frac{3\cdot125\cdot128}{30}=1600.
326 \]
327 Hence the radical lower bound is attained exactly.
328
329 This example becomes a recurring calibration point. Every later refinement must explain why the sharp integer divisor is visible while the corresponding height bound remains nontrivial.
330
331 \chapter{Valuation Words, Polyphase Refinement, and Local Lift Trees}
332
333 \section{Replacing one exponent by a valuation word}
334 The coefficient $v_p(n)$ compresses a run of valuation shells into one scalar. For a prime power $p^e$, define the valuation word
335 \[
336 \boxed{
337 V_e(T)=1+T+\cdots+T^{e-1}.
338 }
339 \]
340 It obeys the concatenation identity
341 \[
342 V_{e+f}(T)=V_e(T)+T^eV_f(T).
343 \]
344 The refined partial derivative
345 \[
346 \partial_{p,T}(n)
347 =
348 \frac npV_{v_p(n)}(T)
349 \]
350 therefore satisfies the twisted product rule
351 \[
352 \boxed{
353 \partial_{p,T}(mn)
354 =
355 n\partial_{p,T}(m)
356 +
357 mT^{v_p(m)}\partial_{p,T}(n).
358 }
359 \]
360 At $T=1$, the ordinary arithmetic partial derivative is recovered.
361
362 \section{Finite polyphase rings}
363 Fix an integer $N\ge2$ and work in
364 \[
365 A_N=\Z[T]/(T^N-1).
366 \]
367 The residue of $V_e(T)$ records the number of valuation shells in each congruence class modulo $N$. Multiplication by $V_e$ is represented by a circulant matrix $C_{e,N}$.
368
369 \begin{theorem}[Polyphase Smith normal form]
370 If $\gcd(e,N)=1$, then
371 \[
372 \boxed{
373 \SNF(C_{e,N})
374 =
375 \operatorname{diag}(1,\ldots,1,e).
376 }
377 \]
378 \end{theorem}
379
380 \begin{proof}
381 Let
382 \[
383 I_N=\ker\bigl(A_N\xrightarrow{\,T\mapsto1\,}\Z\bigr)
384 \]
385 be the augmentation ideal. It is a free abelian group of rank $N-1$, and multiplication by $V_e$ preserves it. After extension to $\C$, the eigenvalues on $I_N\otimes\C$ are
386 \[
387 V_e(\zeta)
388 =
389 \frac{1-\zeta^e}{1-\zeta},
390 \qquad
391 \zeta^N=1,
392 \quad
393 \zeta\neq1.
394 \]
395 Because $\gcd(e,N)=1$, the map $\zeta\mapsto\zeta^e$ permutes the nontrivial $N$th roots. Therefore
396 \[
397 \det\bigl(V_e\mid I_N\bigr)
398 =
399 \prod_{\substack{\zeta^N=1\\\zeta\neq1}}
400 \frac{1-\zeta^e}{1-\zeta}
401 =1.
402 \]
403 Thus multiplication by $V_e$ is an automorphism of the integral lattice $I_N$. On the quotient
404 \[
405 A_N/I_N\cong\Z,
406 \]
407 it acts as multiplication by $V_e(1)=e$. Applying the snake lemma to
408 \[
409 0\longrightarrow I_N\longrightarrow A_N\longrightarrow\Z\longrightarrow0
410 \]
411 shows that the full map is injective and has cokernel $\Z/e\Z$. Hence its Smith normal form is
412 \[
413 \operatorname{diag}(1,\ldots,1,e).
414 \]
415 \end{proof}
416
417 The refined block has $N-1$ integrally invertible phase channels and one aggregate multiplicity channel. Every local denominator lies in the aggregate channel.
418
419 \section{Dyadic and odd-adic primary lifts}
420 Factor
421 \[
422 e=2^\alpha\prod_{\ell\text{ odd}}\ell^{\alpha_\ell}.
423 \]
424 The cokernel of the valuation block is
425 \[
426 \Z/e\Z
427 \cong
428 \Z/2^\alpha\Z
429 \times
430 \prod_{\ell\text{ odd}}
431 \Z/\ell^{\alpha_\ell}\Z.
432 \]
433 The primary factors specify the local lifting problems for the aggregate multiplicity channel. Its $2$-primary component must be treated $2$-adically, while the odd components admit the corresponding odd-adic lifts. The phase channels require no local denominator correction.
434
435 This Smith decomposition identifies where separate $2$-primary bookkeeping first enters. The additional one-level shift in multiplicative principal-unit coordinates at the prime $2$ comes from the local $2$-adic logarithm and is established later. It is not a consequence of the Smith theorem alone.
436
437 \section{Coarse and phase derivatives}
438 Choose a polyphase order $N$ coprime to every active valuation exponent. After the Smith decomposition, each refined prime packet has one coarse coordinate and integrally invertible phase coordinates in the common augmentation lattice. The additive equation separates accordingly.
439
440 The coarse equation recovers the original exponent-weighted row
441 \[
442 \sum_{p\mid a}\frac{av_p(a)}p t_p
443 +
444 \sum_{p\mid b}\frac{bv_p(b)}p t_p
445 -
446 \sum_{p\mid c}\frac{cv_p(c)}p t_p
447 =0.
448 \]
449 The phase equation is exponent free:
450 \[
451 \boxed{
452 \sum_{p\mid a}\frac ap y_p
453 +
454 \sum_{p\mid b}\frac bp y_p
455 -
456 \sum_{p\mid c}\frac cp y_p
457 =0.
458 }
459 \]
460 Define the phase derivative
461 \[
462 D_y(n)=n\sum_{p\mid n}\frac{y_p}{p}.
463 \]
464 The same radical divisibility argument gives
465 \[
466 \frac{abc}{R}\mid aD_y(b)-bD_y(a).
467 \]
468
469 Let
470 \[
471 \sigma(a,b)
472 =
473 \sum_{p\mid a}\frac1p
474 +
475 \sum_{p\mid b}\frac1p.
476 \]
477 If $\Phi(a,b,c)$ is the shortest transverse phase packet, then
478 \[
479 \boxed{
480 c
481 \le
482 R\,\sigma(a,b)\,\Phi(a,b,c).
483 }
484 \]
485 This can be sharper than the exponent-weighted inequality on individual triples, but finite evidence still shows positive power growth rather than a subpolynomial law.
486
487 \section{First-order no-free-refinement principle}
488 The shell-refined factor
489 \[
490 \prod_{r=0}^{e-1}
491 \left(1+\frac{T_r}{p}\right)
492 \]
493 has first-order expansion
494 \[
495 1+\frac1p\sum_rT_r+O(T^2).
496 \]
497 Only the shell sum appears. Every internal phase in the kernel of shell synthesis is invisible to both first-order additivity and the first-order Wronskian.
498
499 \begin{theorem}[First-order no-free-polyphase principle]
500 Let
501 \[
502 S:(T_{p,r})_{p,r}\longmapsto
503 \left(\sum_rT_{p,r}\right)_p
504 \]
505 be shell synthesis. The first-order additive map and first-order Wronskian factor through $S$. Hence
506 \[
507 \Ker S
508 \]
509 is contained in the kernels of both first-order maps.
510 \end{theorem}
511
512 Polyphase splitting may lower displayed child coordinates. Exact synthesis multiplies by the corresponding reconstruction norm. At first order, refinement cannot create a new obstruction direction.
513
514 \section{Second-order phase energy}
515 At second order,
516 \[
517 \prod_{r=0}^{e-1}
518 \left(1+\frac{T_r}{p}\right)
519 =
520 1+
521 \frac1p\sum_rT_r+
522 \frac1{p^2}\sum_{r<s}T_rT_s+
523 O(T^3).
524 \]
525 Let $\widehat T_j$ be the discrete Fourier transform of the shell coordinates. Parseval gives
526 \[
527 \sum_{r<s}T_rT_s
528 =
529 \frac12\left[
530 \widehat T_0^2
531 -
532 \frac1e\sum_j
533 \widehat T_j\widehat T_{-j}
534 \right].
535 \]
536 For real coordinates,
537 \[
538 \sum_{r<s}T_rT_s
539 =
540 \frac12\left[
541 \left(1-\frac1e\right)|\widehat T_0|^2
542 -
543 \frac1e\sum_{j\neq0}|\widehat T_j|^2
544 \right].
545 \]
546 Thus nonzero polyphase energy first becomes visible at second order. Later formal-smoothness results show that this curvature is not itself a lifting obstruction.
547
548 \section{Exact local lift gauges}
549 Let
550 \[
551 r_u=\rad(u),
552 \qquad
553 P_u=\frac{u}{r_u}.
554 \]
555 A shell packet $y=(y_{p,r})$ produces
556 \[
557 D_y(u)
558 =
559 u\sum_{p\mid u}\frac1p
560 \sum_{r=0}^{v_p(u)-1}y_{p,r}
561 =P_ut_u,
562 \]
563 where
564 \[
565 t_u=
566 \sum_{p\mid u}
567 \frac{r_u}{p}
568 \sum_r y_{p,r}.
569 \]
570 Because
571 \[
572 \gcd_{p\mid u}\frac{r_u}{p}=1,
573 \]
574 every integer $t_u$ is attainable when $u>1$.
575
576 The shell lift gauge is
577 \[
578 \rho_u^{\mathrm{sh}}(t)
579 =
580 \min\left\{
581 \norm{y}_\infty:
582 \sum_{p\mid u}
583 \frac{r_u}{p}
584 \sum_r y_{p,r}=t
585 \right\}.
586 \]
587 At the aggregate level,
588 \[
589 P_at_a+P_bt_b=P_ct_c,
590 \]
591 and
592 \[
593 W_y(a,b)
594 =
595 P_aP_b(r_at_b-r_bt_a).
596 \]
597 The radical divisor is attained sharply when
598 \[
599 r_at_b-r_bt_a=P_c.
600 \]
601 When $a,b>1$, all sharp aggregate solutions form a one-parameter family
602 \[
603 \begin{aligned}
604 t_a(n)&=t_a^{(0)}+nr_a,\\
605 t_b(n)&=t_b^{(0)}+nr_b,\\
606 t_c(n)&=t_c^{(0)}+nr_c.
607 \end{aligned}
608 \]
609 The unit-sector cases have no moving parameter. If $a=1$, then
610 \[
611 (t_a,t_b,t_c)=(0,P_c,P_b),
612 \]
613 and if $b=1$, then
614 \[
615 (t_a,t_b,t_c)=(-P_c,0,-P_a).
616 \]
617 For obstruction multiple $k$, these fixed triples are multiplied by $k$. Thus the high-dimensional sharp search reduces either to one integer parameter plus three local lift gauges, or to one fixed aggregate triple in a unit-sector case.
618
619 \chapter{The Exact Obstruction Complex and Formal Shell Geometry}
620
621 \section{A joint exponent-free two-row complex}
622 The Smith obstruction is attached to the exponent-free phase, or shell-synthesis, rows rather than to the exponent-weighted derivative row. For every prime $p\mid abc$, define
623 \[
624 \Lambda_p=
625 \begin{cases}
626 a/p,&p\mid a,\\
627 b/p,&p\mid b,\\
628 -c/p,&p\mid c,
629 \end{cases}
630 \qquad
631 \mathsf w_p=
632 \begin{cases}
633 -ab/p,&p\mid a,\\
634 ab/p,&p\mid b,\\
635 0,&p\mid c.
636 \end{cases}
637 \]
638 Consider
639 \[
640 \mathsf M_{a,b,c}
641 =
642 \begin{pmatrix}
643 \Lambda\\
644 \mathsf w
645 \end{pmatrix}:
646 \Z^S\longrightarrow\Z^2.
647 \]
648 The first row is the exponent-free additive phase row. The second is its integer Wronskian row. Set
649 \[
650 Q=\frac{abc}{R}.
651 \]
652
653 \begin{theorem}[Smith obstruction theorem]\label{thm:phase-smith}
654 Let $a+b=c$ be primitive, with at least two of $a,b,c$ greater than one. Then
655 \[
656 \boxed{
657 \SNF(\mathsf M_{a,b,c})
658 =
659 \operatorname{diag}(1,Q).
660 }
661 \]
662 Consequently,
663 \[
664 \boxed{
665 \operatorname{coker}(\mathsf M_{a,b,c})_{\mathrm{tors}}
666 \cong
667 \Z/Q\Z.
668 }
669 \]
670 The exceptional triple $(1,1,2)$ has only one nonempty prime sector, so the matrix has rank one and is excluded from the two-row statement.
671 \end{theorem}
672
673 \begin{proof}
674 The gcd of the entries of $\Lambda$ is one. Indeed, every prime dividing one sector is absent from every other nonempty sector, and an entry from another sector is not divisible by it. Hence the first Smith factor is one.
675
676 Rows restricted to one sector are proportional, so same-sector $2\times2$ minors vanish. If $p$ and $q$ lie in two different nonempty sectors, the corresponding minor is
677 \[
678 \pm\frac{abc}{pq}
679 =
680 \pm Q\frac{R}{pq}.
681 \]
682 Every nonzero minor is therefore divisible by $Q$. Conversely, for each prime $\ell\mid R$, choose $p=\ell$ and choose $q$ in a different nonempty sector. Since $R$ is squarefree, $R/(pq)$ is not divisible by $\ell$. Thus the gcd of the reduced minors $R/(pq)$ is one. The gcd of all nonzero minors is exactly $Q$. The determinantal-divisor description of Smith normal form now gives the result.
683 \end{proof}
684
685 \section{The exact lattice extension}
686 Define
687 \[
688 L=\ker(\Lambda)\cap\Z^S
689 \]
690 and
691 \[
692 L_0=L\cap\ker(\mathsf w).
693 \]
694 Because $\Lambda$ is primitive, choose $t\in\Z^S$ with $\Lambda(t)=1$. Then
695 \[
696 \Z^S=L\oplus\Z t.
697 \]
698 The image of $\mathsf M_{a,b,c}$ is generated by $(1,\mathsf w(t))$ together with $(0,\mathsf w(L))$. Its index in $\Z^2$ is therefore the index of $\mathsf w(L)$ in $\Z$. Theorem~\ref{thm:phase-smith} shows that this index is $Q$, so
699 \[
700 \mathsf w(L)=Q\Z.
701 \]
702 Hence
703 \[
704 \omega(x)=\frac{\mathsf w\cdot x}{Q}
705 \]
706 is an integer-valued surjection and gives the exact sequence
707 \[
708 \boxed{
709 0\longrightarrow L_0
710 \longrightarrow L
711 \xrightarrow{\omega}
712 \Z
713 \longrightarrow0.
714 }
715 \]
716 The extension splits algebraically. The difficult quantity is the norm of a section, not its existence.
717
718 For $k\in\Z$, define
719 \[
720 H(k)
721 =
722 \min\left\{
723 \norm{x}_\infty:
724 x\in L,
725 \omega(x)=k
726 \right\}.
727 \]
728 Three heights must be distinguished:
729 \[
730 H(1),
731 \qquad
732 \Phi=\min_{k\neq0}H(k),
733 \qquad
734 \overline H=
735 \inf_{k\ge1}\frac{H(k)}k.
736 \]
737 The shortest nonzero detector may live in a nonprimitive obstruction coset.
738
739 \section{The stable real height}
740 Relax the integral packet to a real vector in the additive hyperplane. The exact continuous width can be computed from the three sector weights. Let
741 \[
742 \sigma_u=\sum_{p\mid u}\frac1p
743 \]
744 and
745 \[
746 \Delta(a,b,c)
747 =
748 \min\left\{
749 \sigma_a+\sigma_b,
750 \frac cb(\sigma_a+\sigma_c),
751 \frac ca(\sigma_b+\sigma_c)
752 \right\}.
753 \]
754 Then
755 \[
756 \boxed{
757 \overline H
758 =
759 \frac{c}{R\Delta(a,b,c)}.
760 }
761 \]
762 The discrete master inequality becomes
763 \[
764 c\le R\Delta\Phi.
765 \]
766 Since
767 \[
768 \Phi=\overline H\,k_*\Gamma
769 \]
770 for the optimal obstruction multiple $k_*$ and an integrality factor $\Gamma\ge1$,
771 \[
772 \boxed{
773 \Delta\Phi
774 =
775 \frac cR k_*\Gamma.
776 }
777 \]
778 This is the no-escape identity. Optimizing the same first-order invariant cannot lower the continuous floor $c/R$.
779
780 \section{The full shell deformation}
781 Introduce one variable for every valuation shell and define
782 \[
783 \mathscr J_n(\mathbf T)
784 =
785 \prod_{p^e\parallel n}
786 \prod_{r=0}^{e-1}(p+T_{p,r}).
787 \]
788 At $\mathbf T=0$,
789 \[
790 \mathscr J_n(0)=n.
791 \]
792 For $a+b=c$, set
793 \[
794 F_{a,b,c}(\mathbf T)
795 =
796 \mathscr J_a(\mathbf T)
797 +
798 \mathscr J_b(\mathbf T)
799 -
800 \mathscr J_c(\mathbf T).
801 \]
802 The shell gradient has entries
803 \[
804 \frac ap,
805 \qquad
806 \frac bp,
807 \qquad
808 -\frac cp
809 \]
810 repeated according to valuation depth. The gcd of all entries is one.
811
812 \begin{theorem}[Formal affine-space theorem]
813 Let $d$ be the number of shell variables. After a unimodular integral change of shell coordinates, the $(\mathbf T)$-adic completion of
814 \[
815 \Z[\mathbf T]/(F_{a,b,c})
816 \]
817 at the zero section is isomorphic to a formal power-series ring $\Z[[U_2,\ldots,U_d]]$.
818 \end{theorem}
819
820 \begin{proof}
821 A primitive gradient row can be completed to a unimodular matrix. In the new coordinates, one variable has linear coefficient one, which is a unit in $\Z$. The formal implicit-function recursion therefore solves uniquely for that variable at every degree with integral coefficients, giving the stated isomorphism of completed rings.
822 \end{proof}
823
824 \begin{corollary}[Integral jet extension]
825 Every integral first-order tangent vector to $F_{a,b,c}=0$ extends to an integral formal curve through all orders.
826 \end{corollary}
827
828 The consequence is decisive. Higher shell curvature exists, but there is no formal lifting obstruction to exploit. The missing arithmetic must be a global size or conductor estimate.
829
830 \section{Gauge collapse}
831 Within one sector, the product
832 \[
833 \prod_{p,r}(p+T_{p,r})
834 \]
835 depends only on the aggregate product of its shell variables. Internal phase directions that preserve this product form a gauge torus. The full shell equation factors through three aggregate sector products
836 \[
837 A,
838 \qquad
839 B,
840 \qquad
841 C,
842 \]
843 with one linear relation
844 \[
845 aA+bB=cC.
846 \]
847 Thus all formal complexity collapses to a two-dimensional aggregate plane plus gauge directions. This explains why local refinements repeatedly fail to create a new global obstruction.
848
849 \chapter{Exact Moving-Coset Search and the First-Order Barrier}
850
851 \section{Obstruction cosets}
852 Choose a section $x_0\in L$ with $\omega(x_0)=1$. Then
853 \[
854 L
855 =
856 \bigsqcup_{k\in\Z}
857 (kx_0+L_0).
858 \]
859 For fixed $k$, the problem is a closest-vector problem to the moving target $-kx_0$ in the lattice $L_0$. The three-sector compression makes this search much more explicit.
860
861 \section{Aggregate affine line}
862 Write
863 \[
864 u=P_ur_u,
865 \qquad
866 r_u=\rad(u),
867 \qquad
868 P_u=u/r_u.
869 \]
870 The sharp aggregate conditions are
871 \[
872 P_at_a+P_bt_b=P_ct_c
873 \]
874 and
875 \[
876 r_at_b-r_bt_a=kP_c.
877 \]
878 Since $\gcd(r_a,r_b)=1$, choose $s,t$ with
879 \[
880 sr_a+tr_b=1.
881 \]
882 One solution is
883 \[
884 t_a^{(0)}=-tkP_c,
885 \qquad
886 t_b^{(0)}=skP_c.
887 \]
888 Every aggregate solution is
889 \[
890 \boxed{
891 \begin{aligned}
892 t_a(k,n)&=t_a^{(0)}+nr_a,\\
893 t_b(k,n)&=t_b^{(0)}+nr_b,\\
894 t_c(k,n)&=t_c^{(0)}+nr_c,
895 \end{aligned}
896 \qquad n\in\Z.
897 }
898 \]
899 When $a,b>1$, the global search in one obstruction coset is therefore one-dimensional. If $a=1$, the empty $a$-sector forces $t_a=0$ and fixes the coarse parameter, giving
900 \[
901 (t_a,t_b,t_c)=(0,kP_c,kP_b).
902 \]
903 If $b=1$, the corresponding fixed triple is
904 \[
905 (t_a,t_b,t_c)=(-kP_c,0,-kP_a).
906 \]
907 Thus a unit-sector coset has one aggregate candidate rather than a moving affine line.
908
909 \section{Local CRT normal form}
910 Fix one sector with primes
911 \[
912 p_1,\ldots,p_m
913 \]
914 and radical
915 \[
916 r=\prod_i p_i.
917 \]
918 For a nonempty sector, the local equation is
919 \[
920 \sum_i\frac r{p_i}x_i=t.
921 \]
922 An empty unit sector has no prime coordinates and is feasible exactly when $t=0$.
923 Modulo $p_i$, the coordinate $x_i$ is forced:
924 \[
925 x_i
926 \equiv
927 t\left(\frac r{p_i}\right)^{-1}
928 \pmod{p_i}.
929 \]
930 Choose the centered residue $\xi_i$ and write
931 \[
932 x_i=\xi_i+p_i z_i.
933 \]
934 Substitution gives one quotient equation
935 \[
936 \sum_i z_i=q(t).
937 \]
938 For a box radius $H$, each $z_i$ lies in the interval
939 \[
940 \ell_i(H)
941 =
942 \left\lceil\frac{-H-\xi_i}{p_i}\right\rceil
943 \le z_i\le
944 \left\lfloor\frac{H-\xi_i}{p_i}\right\rfloor
945 =u_i(H).
946 \]
947 The local lift exists exactly when
948 \[
949 \boxed{
950 \sum_i\ell_i(H)
951 \le q(t)\le
952 \sum_i u_i(H).
953 }
954 \]
955 This is an exact interval criterion.
956
957 \section{Moving-interval theorem}
958 For fixed $k$ and $H$, the three sectors produce three integer intervals for the coarse parameter $n$. A packet of height at most $H$ exists if and only if those intervals share an integer.
959
960 \begin{theorem}[Moving-interval theorem]
961 For fixed obstruction multiple $k$ and radius $H$,
962 \[
963 H(k)\le H
964 \]
965 if and only if
966 \[
967 I_a(H,k)
968 \cap
969 I_b(H,k)
970 \cap
971 I_c(H,k)
972 \cap\Z
973 \neq\varnothing.
974 \]
975 \end{theorem}
976
977 The first feasible $H$ gives the exact coset height. The preceding radius is an explicit no-smaller-box certificate.
978
979 \section{Finite global stopping}
980 The continuous lower bound gives
981 \[
982 H(k)\ge k\overline H.
983 \]
984 If the best exact height found so far is $B$, then no coset with
985 \[
986 k\overline H\ge B
987 \]
988 can improve it. Therefore only finitely many obstruction multiples must be searched:
989 \[
990 \boxed{
991 1\le k\le
992 \left\lfloor\frac{B-1}{\overline H}\right\rfloor.
993 }
994 \]
995 Every unsearched coset is excluded by a proved lower bound.
996
997 \section{Phase and coarse separation}
998 The local target has the form
999 \[
1000 t_u(k,n)=\alpha_uk+r_un.
1001 \]
1002 For a local prime $p\mid r_u$,
1003 \[
1004 t_u(k,n)\equiv\alpha_uk\pmod p.
1005 \]
1006 Thus the obstruction multiple $k$ determines the finite phase packet. The coarse parameter $n$ changes only the quotient load. The exact search is a moving polyphase mechanism rather than a generic mixed-integer program.
1007
1008 \section{Arithmetic correction and arbitrary precision}
1009 Some large high-quality triples exceed signed $64$-bit range, and intermediate quantities such as
1010 \[
1011 ab,
1012 \qquad
1013 \frac{abc}{R},
1014 \qquad
1015 w\cdot x
1016 \]
1017 can overflow even when the input terms individually fit. Every public arithmetic entry point must therefore promote integer-like inputs to arbitrary-precision integers before multiplication. Floating-point values are display-only. All logical comparisons and stopping bounds are exact rational or integer computations.
1018
1019 Several reported ratios are mathematically at least one:
1020 \[
1021 \frac{\Phi}{\overline H}\ge1,
1022 \qquad
1023 \frac{\Phi}{k_*\overline H}\ge1,
1024 \qquad
1025 \frac{H(1)}{\Phi}\ge1.
1026 \]
1027 Values above one are not probability errors.
1028
1029 \section{What the exact search proved}
1030 The moving search found nonprimitive optima on many finite examples. It also found increasingly deep phase resonances. This showed that a fixed cutoff in $k$ is unjustified. More importantly, the no-escape identity proved that even a complete exact search of the first-order invariant cannot imply $abc$ independently. The search became a diagnostic and classification engine rather than the final proof mechanism.
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