Paper IX: Arithmetic Packets, Dyadic Towers, and Local Polynomial Deformation

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1 \clearpage\section{Recovered historical local polyphase manuscript}\label{app:recovered}
2 This is the complete substantive body of the reconstructed Annals textual
3 copy, preserved to retain its derivations and original parameter ranges.
4 The recovered text is not certified as the exact original TeX bytes. Its
5 historical theorem headings distinguish these statements from the current
6 numbered core. Preservation is not a blanket proof audit or Lean certificate:
7 in particular the broader Amice analytic pullback and depleted branches,
8 finite-field modal minimality and unrestricted infinite polyphase claims
9 retain their historical evidence scope. The current formal statement map
10 records only the individually identified interfaces in the main text.
11 
12 The earlier reconstruction lacked all 14 original figure files. This
13 reading edition supplies new reconstructions at their retained labels and
14 captions. These are explanatory schematics and exact finite examples, not
15 recovered original artwork. The reproducible program
16 \nolinkurl{tools/reconstructed_figures.py} checks the displayed finite data.
17 The modulo-seven inverse plot uses the corrected positive tail
18 $6,5,2,6,5,2,\ldots$. A reconstructed illustration does not upgrade the
19 historical proof or formalization status. The original historical source
20 package, including all 14 vector figures and the bibliography, has now been
21 recovered and is preserved unchanged in the accompanying release. The current
22 reading edition retains its explicitly identified reconstructions and corrected
23 finite data. The 22 historical references now have recovered bibliographic
24 entries and checked identities. Superseded claims about the divisor-branch
25 finite-difference problem have local notices directing the reader to the
26 current theorem. The remaining higher kernel-valuation and unrestricted
27 local-to-global questions remain open in their stated scopes.
28 
29 \subsection{Introduction}\label{hist:sec:introduction}
30 
31 Euler numbers carry several kinds of local information at once. Their indices have residue addresses. Their coefficient transforms have triangular depth. Their local expansions have Mahler degree. Their congruences depend on the valuation of the transfer kernel and on the valuation of the source value in the row being reconstructed. These pieces should not be merged into one undifferentiated congruence calculation.
32 
33 This article begins with the Hurwitz product
34 \[
35 (a\starh b)_n
36 =
37 \sum_{r=0}^{n}
38 \binom nr a_rb_{n-r}.
39 \]
40 Over a field of characteristic zero it is multiplication of exponential generating functions. Over a finite field it realizes the additive group algebra through the exponential modes \(\epsilon_u=(u^n)_{n\geq0}\). This finite algebra gives an exact proof of the standard window period. Hurwitz series rings are treated in \cite{Keigher1997,BarberoCerrutiMurru2018}. The finite difference and umbral background may be found in \cite{Roman1984,Comtet1974,GrahamKnuthPatashnik1994}.
41 
42 Fix an odd prime \(p\), a divisor \(m\mid p-1\), and put
43 \[
44 h=\frac{p-1}{m}.
45 \]
46 The reduced standard window inverse is defined by
47 \[
48 \sum_{q=0}^{n}
49 \binom{mn}{mq}e_q^{(m,p)}
50 =
51 \delta_{n0}
52 \quad\text{in }\FF_p.
53 \]
54 We prove that
55 \[
56 e_{n+h}^{(m,p)}=e_n^{(m,p)}
57 \qquad(n\geq1).
58 \]
59 The result first follows from subgroup symmetry in the finite Hurwitz algebra. A second proof shows that the standard window operator preserves the finite space of sequences that are \(h\) periodic after their constant term. The two arguments expose different parts of the same finite character structure.
60 
61 The central change in this edition is the treatment of the odd and dyadic Euler formulas. They are no longer presented as unrelated reconstruction devices. For each prime \(p\) we define an even source series \(H_p\) and an inverse kernel \(K_p\) satisfying
62 \[
63 K_p(t)H_p(t)=\operatorname{sech}t.
64 \]
65 At an odd prime,
66 \[
67 H_p(t)=\cosh^2(pt)\operatorname{sech}t,
68 \qquad
69 K_p(t)=\operatorname{sech}^2(pt).
70 \]
71 At the dyadic prime,
72 \[
73 H_2(t)=\frac{\sinh(2t)\sinh t}{2t^2},
74 \qquad
75 K_2(t)=\frac{4t^2}{\sinh^2(2t)}.
76 \]
77 The coefficient convolution is identical in form, but the kernels have different integral weights. This is the correct sense in which the local theories unify.
78 
79 The dyadic kernel is substantially sharper than the separate coefficient bound alone suggests. If
80 \[
81 \frac{z^2}{\sinh^2z}
82 =
83 \sum_{r\geq0}\beta_rz^{2r}
84 \]
85 and
86 \[
87 \kappa_{2,r}=\beta_r2^{2r}(2r)!,
88 \]
89 then
90 \[
91 \vTwo(\kappa_{2,r})=4r-1
92 \qquad(r\geq1).
93 \]
94 Consequently, truncation after depth \(M\) is uniformly valid modulo
95 \[
96 2^{4M+3},
97 \]
98 and this exponent is optimal. The rowwise estimate records the additional binary carries in the binomial coefficient. Classical congruence theory for Euler families is extensive \cite{CarlitzLevine1960,Young1999,Young2001,Sun2008,Chen2004}. The formulas here are stated as exact coordinate identities. Historical priority requires a separate theorem by theorem comparison.
99 
100 The second expansion concerns local interpolation. The exponent of the finite unit group moves every unit mode into a principal unit neighborhood. Binomial expansion then gives a Mahler series whose depth \(r\) coefficient gains at least \(Lr\) digits when the chosen jump is the identity modulo \(p^L\). The same argument applies to polynomial exponential modes and to invertible matrix state systems. Mahler interpolation and the Amice transform are classical \cite{Mahler1958,Amice1964,Robert2000}. The local unit structure and valuation tools used here are standard \cite{Koblitz1984,Serre1979,Washington1997,Gouvea1997}.
101 
102 The odd Euler source contains one nonunit base. Removing that base gives an analytic unit source and an interpolation of a depleted Euler sequence on every even character branch. The dyadic source is already analytic on \(\Zp\) after a removable singularity is filled. It yields a direct analytic interpolation of \(E_{2n}\) on \(\mathbb Z_2\). These branches are then refined by the substitution
103 \[
104 x=Mq+s.
105 \]
106 The induced Mahler coefficient transform is finite and explicit. Its kernels compose exactly. At prime power radix they become sparse modulo the residue prime. Every monotone valuation ladder survives the refinement.
107 
108 The remaining sections treat arithmetic lift obstructions, recurrence packets, and Chinese remainder reconstruction. The lifting obstruction is the residual defect modulo the image of the declared correction map. A recurrence packet is governed by a jump matrix \(C^N\). If one integral recurrence is fixed, all of its local realizations have one common source. These statements do not prove that arbitrary unrelated local objects have a global origin.
109 
110 \begin{figure}[t]
111 \centering
112 \includegraphics[width=.95\linewidth,height=3.15in,keepaspectratio]{../../figures/reconstructed/architecture.pdf}
113 \caption{New reconstruction. The arithmetic coordinate chain. Residue address, local character, Mahler depth, transfer weight, and reconstruction are recorded separately.}
114 \label{hist:fig:architecture}
115 \end{figure}
116 
117 Four boundaries remain explicit. A period modulo \(p\) does not by itself lift to all powers of \(p\). The full finite difference estimate for every congruential reciprocal packet still requires control of the triangular inverse. Exact higher valuations of the affine polyphase kernel are not yet known in general. Chinese remainder reconstruction and finite adelic synchronization do not prove a global source for arbitrary local systems.
118 \subsection{Hurwitz series and finite field subgroup inversion}\label{hist:sec:hurwitz}
119 \par\smallskip\noindent\textbf{Current status.} This dated statement is superseded for the stated divisor-branch family by Theorem~\ref{thm:arith-divisor-base}, with the $j!$ normalization and factorial valuation loss explicitly retained. It makes no claim for unrestricted families.\par
120 
121 \subsubsection{The Hurwitz ring}
122 
123 \begin{histdefinition}[Hurwitz series ring]\label{hist:def:hurwitz_ring}
124 Let \(R\) be a commutative ring with identity. The Hurwitz series ring \(\Hcal(R)\) is the set \(R^{\NN}\) with componentwise addition and product
125 \[
126 (a\starh b)_n
127 =
128 \sum_{r=0}^{n}
129 \binom nr a_r b_{n-r}.
130 \]
131 Its identity is
132 \[
133 \delta=(1,0,0,\ldots).
134 \]
135 \end{histdefinition}
136 
137 When the factorials are invertible, the sequence \(a\) may be written as the exponential generating series
138 \[
139 A(t)=\sum_{n\geq0}a_n\frac{t^n}{n!},
140 \]
141 and the Hurwitz product becomes ordinary multiplication. The sequence definition remains valid in every characteristic.
142 
143 \begin{histdefinition}[Exponential mode]\label{hist:def:exponential_mode}
144 For \(u\in R\), define
145 \[
146 \epsilon_u=(u^n)_{n\geq0},
147 \]
148 where \(0^0=1\).
149 \end{histdefinition}
150 
151 \begin{histproposition}[Addition law for modes]\label{hist:prop:mode_addition}
152 For every \(u,v\in R\),
153 \[
154 \boxed{
155 \epsilon_u\starh\epsilon_v
156 =
157 \epsilon_{u+v}.
158 }
159 \]
160 \end{histproposition}
161 
162 \begin{proof}
163 The \(n\)th coefficient is
164 \[
165 \sum_{r=0}^{n}
166 \binom nr u^rv^{n-r}
167 =
168 (u+v)^n
169 \]
170 by the binomial theorem.
171 \end{proof}
172 
173 \subsubsection{The finite additive group algebra}
174 
175 Let \(p\) be prime. Write \(\FF_p^+\) for the additive group of the finite field.
176 
177 \begin{histtheorem}[Finite field mode algebra]\label{hist:thm:mode_group_algebra}
178 The map
179 \[
180 \Theta:
181 \FF_p[\FF_p^+]
182 \longrightarrow
183 \Hcal(\FF_p),
184 \qquad
185 [u]\longmapsto\epsilon_u,
186 \]
187 is an injective algebra homomorphism. Its image is the \(p\) dimensional subalgebra spanned by the modes \(\epsilon_u\).
188 \end{histtheorem}
189 
190 \begin{proof}
191 The product law follows from \cref{hist:prop:mode_addition}. To prove injectivity, suppose
192 \[
193 \sum_{u\in\FF_p}c_u\epsilon_u=0.
194 \]
195 Looking at the coordinates \(n=0,1,\ldots,p-1\) gives a Vandermonde system with matrix
196 \[
197 (u^n)_{0\leq n\leq p-1,\ u\in\FF_p}.
198 \]
199 Its determinant is the product of all differences between distinct field elements, so it is nonzero. Hence every \(c_u\) vanishes.
200 \end{proof}
201 
202 \begin{histproposition}[Augmentation criterion]\label{hist:prop:augmentation_criterion}
203 Let
204 \[
205 x=\sum_{u\in\FF_p}c_u[u]
206 \in
207 \FF_p[\FF_p^+].
208 \]
209 Then \(x\) is a unit if and only if
210 \[
211 \operatorname{aug}(x)
212 =
213 \sum_uc_u
214 \]
215 is nonzero. If \(J\) is the augmentation ideal, then
216 \[
217 J^p=0.
218 \]
219 \end{histproposition}
220 
221 \begin{proof}
222 Choose a generator \(g\) of the cyclic additive group. Then
223 \[
224 \FF_p[\FF_p^+]
225 \cong
226 \FF_p[y]/(y^p-1)
227 =
228 \FF_p[y]/(y-1)^p.
229 \]
230 The augmentation ideal is generated by \(y-1\), so its \(p\)th power is zero. Every element has the form \(a+n\) with \(a\in\FF_p\) and \(n\in J\). It is invertible exactly when \(a\neq0\), which is the augmentation condition.
231 \end{proof}
232 
233 \begin{histcorollary}[Finite inverse formula]\label{hist:cor:finite_inverse_formula}
234 If \(x\) has augmentation one and \(n=x-1\), then
235 \[
236 \boxed{
237 x^{-1}
238 =
239 \sum_{r=0}^{p-1}(-n)^r.
240 }
241 \]
242 \end{histcorollary}
243 
244 \begin{proof}
245 By \cref{hist:prop:augmentation_criterion}, \(n^p=0\). The finite geometric sum gives the inverse.
246 \end{proof}
247 
248 \subsubsection{Subgroup selectors}
249 
250 Fix a divisor \(m\) of \(p-1\). Let
251 \[
252 H_m
253 =
254 \{\zeta\in\FF_p^\times:\zeta^m=1\}.
255 \]
256 This group has order \(m\).
257 
258 \begin{histdefinition}[Multiple selector]\label{hist:def:multiple_selector}
259 Define \(a^{(m)}\in\Hcal(\FF_p)\) by
260 \[
261 a_n^{(m)}
262 =
263 \begin{cases}
264 1,&m\mid n,\\
265 0,&m\nmid n.
266 \end{cases}
267 \]
268 \end{histdefinition}
269 
270 \begin{histlemma}[Subgroup average]\label{hist:lem:subgroup_average}
271 The multiple selector has the mode expansion
272 \[
273 \boxed{
274 a^{(m)}
275 =
276 \frac1m
277 \sum_{\zeta\in H_m}
278 \epsilon_\zeta.
279 }
280 \]
281 \end{histlemma}
282 
283 \begin{proof}
284 The \(n\)th coefficient on the right is
285 \[
286 \frac1m
287 \sum_{\zeta\in H_m}\zeta^n.
288 \]
289 The character sum is one when \(m\mid n\) and zero otherwise.
290 \end{proof}
291 
292 \begin{figure}[t]
293 \centering
294 \includegraphics[width=.95\linewidth,height=3.15in,keepaspectratio]{../../figures/reconstructed/subgroup_average.pdf}
295 \caption{New reconstruction. A subgroup average in the additive group algebra. The highlighted modes select coefficient degrees divisible by the subgroup order.}
296 \label{hist:fig:subgroup_average}
297 \end{figure}
298 
299 Since the subgroup average has augmentation one, it has a unique Hurwitz inverse. Write
300 \[
301 b^{(m,p)}
302 =
303 (a^{(m)})^{-1}.
304 \]
305 
306 \begin{histtheorem}[Support and positive degree period]\label{hist:thm:support_period}
307 The inverse \(b^{(m,p)}\) has the following properties.
308 \begin{enumerate}
309 \item \(b_n^{(m,p)}=0\) unless \(m\mid n\).
310 \item For every positive \(n\),
311 \[
312 \boxed{
313 b_{n+p-1}^{(m,p)}
314 =
315 b_n^{(m,p)}.
316 }
317 \]
318 \end{enumerate}
319 \end{histtheorem}
320 
321 \begin{proof}
322 Multiplication by an element \(\eta\in H_m\) gives an automorphism of the additive group and therefore an automorphism
323 \[
324 \epsilon_u\longmapsto\epsilon_{\eta u}
325 \]
326 of the mode algebra. The selector \(a^{(m)}\) is fixed by every such automorphism, so its unique inverse is fixed as well.
327 
328 Write
329 \[
330 b^{(m,p)}
331 =
332 \sum_{u\in\FF_p}c_u\epsilon_u.
333 \]
334 Invariance gives \(c_{\eta u}=c_u\). After the change of variable \(u\mapsto\eta u\),
335 \[
336 b_n^{(m,p)}
337 =
338 \eta^n b_n^{(m,p)}
339 \]
340 for every \(\eta\in H_m\). If \(m\nmid n\), some \(\eta\) satisfies \(\eta^n\neq1\), so the coefficient vanishes.
341 
342 For positive \(n\), the zero mode contributes nothing. Every nonzero mode satisfies \(u^{p-1}=1\). Hence
343 \[
344 b_{n+p-1}^{(m,p)}
345 =
346 \sum_{u\neq0}c_uu^{n+p-1}
347 =
348 \sum_{u\neq0}c_uu^n
349 =
350 b_n^{(m,p)}.
351 \]
352 \end{proof}
353 
354 \begin{histdefinition}[Compressed inverse]\label{hist:def:compressed_inverse}
355 Set
356 \[
357 e_n^{(m,p)}
358 =
359 b_{mn}^{(m,p)}.
360 \]
361 \end{histdefinition}
362 
363 \begin{histproposition}[Compressed inverse recurrence]\label{hist:prop:compressed_inverse_recurrence}
364 The compressed coefficients satisfy
365 \[
366 \boxed{
367 \sum_{q=0}^{n}
368 \binom{mn}{mq}
369 e_q^{(m,p)}
370 =
371 \delta_{n0}.
372 }
373 \]
374 In particular, \(e_0^{(m,p)}=1\), and the positive coefficients are determined recursively because the diagonal is one.
375 \end{histproposition}
376 
377 \begin{proof}
378 Evaluate the Hurwitz identity
379 \[
380 a^{(m)}\starh b^{(m,p)}=\delta
381 \]
382 at degree \(mn\). Both factors vanish outside degrees divisible by \(m\). Reindexing the surviving terms gives the displayed recurrence.
383 \end{proof}
384 
385 \begin{histcorollary}[Uniform compressed period]\label{hist:cor:compressed_period}
386 Let
387 \[
388 h=\frac{p-1}{m}.
389 \]
390 Then
391 \[
392 \boxed{
393 e_{n+h}^{(m,p)}
394 =
395 e_n^{(m,p)}
396 \qquad(n\geq1).
397 }
398 \]
399 \end{histcorollary}
400 
401 \begin{proof}
402 Since \(m(n+h)=mn+p-1\), the conclusion follows from \cref{hist:thm:support_period}.
403 \end{proof}
404 
405 \begin{histcorollary}[Rational ordinary generating function]\label{hist:cor:periodic_ogf}
406 In \(\FF_p[[z]]\),
407 \[
408 \boxed{
409 \sum_{n\geq0}e_n^{(m,p)}z^n
410 =
411 1+
412 \frac{
413 \sum_{r=1}^{h}e_r^{(m,p)}z^r
414 }{1-z^h}.
415 }
416 \]
417 \end{histcorollary}
418 
419 \begin{proof}
420 Separate the constant coefficient and group the positive indices into residue classes modulo \(h\).
421 \end{proof}
422 
423 \begin{histtheorem}[Exponential form and minimal period]\label{hist:thm:minimal_period}
424 Let
425 \[
426 G_m=(\FF_p^\times)^m.
427 \]
428 This cyclic group has order \(h\). There are uniquely determined coefficients \(d_v\in\FF_p\), indexed by \(v\in G_m\), such that
429 \[
430 \boxed{
431 e_n^{(m,p)}
432 =
433 \sum_{v\in G_m}d_vv^n
434 \qquad(n\geq1).
435 }
436 \]
437 The minimal positive period is the least common multiple of the orders of those \(v\) for which \(d_v\neq0\). In particular, it divides \(h\).
438 \end{histtheorem}
439 
440 \begin{proof}
441 The coefficients \(c_u\) in the proof of \cref{hist:thm:support_period} are constant on the cosets of \(H_m\). For \(u\neq0\), the value \(u^m\) depends only on the coset, and every element of \(G_m\) arises. Grouping the modes by cosets gives the displayed expansion.
442 
443 Distinct elements of \(G_m\) define linearly independent functions on one full period, by the Vandermonde determinant. Therefore a positive integer \(r\) is a period exactly when \(v^r=1\) for every active mode. The least such \(r\) is the stated least common multiple.
444 \end{proof}
445 
446 \begin{histexample}[A compressed inverse modulo seven]\label{hist:ex:compressed_mod_seven}
447 \par\noindent\textbf{Editorial numerical correction.} The positive-index standard-window tail printed below is erroneous. The defining recurrence gives $e_1=-1\equiv6$, $e_2=5$ and $e_3=-61\equiv2\pmod7$, so the correct positive tail is $6,5,2,6,5,2,\ldots$. The period-three conclusion remains correct. The original line is retained only as a corrected historical record.\par\medskip
448 Take \(p=7\) and \(m=2\), so \(h=3\). The recurrence gives
449 \[
450 e_n^{(2,7)}
451 =
452 1,6,4,1,6,4,\ldots
453 \qquad(n\geq1).
454 \]
455 The positive tail has exact period three.
456 \end{histexample}
457 
458 \begin{figure}[t]
459 \centering
460 \includegraphics[width=.95\linewidth,height=3.15in,keepaspectratio]{../../figures/reconstructed/standard_periods.pdf}
461 \caption{New reconstruction. Several compressed inverses. Every row is periodic after the constant term, as proved in \cref{hist:cor:compressed_period}.}
462 \label{hist:fig:standard_periods}
463 \end{figure}
464 
465 
466 \subsection{Unit triangular coefficient transport}\label{hist:sec:triangular_transport}
467 
468 The examples later in the paper use different kernels, but their coefficient algebra has one common form. This section records that form once. Let \(R\) be a commutative algebra over \(\QQ\), and take all coefficients in \(R\).
469 
470 Let
471 \[
472 H(t)=\sum_{r\geq0}h_r\frac{t^r}{r!},
473 \qquad
474 h_0=1.
475 \]
476 There is a unique inverse
477 \[
478 K(t)=H(t)^{-1}
479 =\sum_{r\geq0}k_r\frac{t^r}{r!}.
480 \]
481 
482 \begin{histtheorem}[Exact coefficient transport]\label{hist:thm:exact_coefficient_transport}
483 Let
484 \[
485 A(t)=\sum_{n\geq0}A_n\frac{t^n}{n!},
486 \qquad
487 S(t)=H(t)A(t).
488 \]
489 Then
490 \[
491 \boxed{
492 S_n
493 =
494 \sum_{r=0}^{n}
495 \binom nr h_rA_{n-r}.
496 }
497 \]
498 The inverse coordinate formula is
499 \[
500 \boxed{
501 A_n
502 =
503 \sum_{r=0}^{n}
504 \binom nr k_rS_{n-r}.
505 }
506 \]
507 Both transforms are lower triangular with diagonal one.
508 \end{histtheorem}
509 
510 \begin{proof}
511 The first identity is the coefficient rule for the product of exponential generating functions. Multiply by \(K(t)\) and use \(K(t)H(t)=1\) for the inverse identity. The diagonal coefficient in each transform is the constant coefficient of its kernel, which is one.
512 \end{proof}
513 
514 \begin{histcorollary}[Composition of coefficient transports]\label{hist:cor:transport_composition}
515 If \(U_H\) and \(U_G\) are the triangular transforms associated with unit kernels \(H\) and \(G\), then
516 \[
517 \boxed{
518 U_HU_G=U_{HG}.
519 }
520 \]
521 In particular, the inverse of \(U_H\) is \(U_{H^{-1}}\).
522 \end{histcorollary}
523 
524 \begin{proof}
525 Multiplication by \(G\), followed by multiplication by \(H\), is multiplication by \(HG\).
526 \end{proof}
527 
528 \begin{histtheorem}[Filtered triangular reconstruction]\label{hist:thm:filtered_triangular_reconstruction}
529 Let \(p\) be prime. Suppose
530 \[
531 A_n
532 =
533 \sum_{r=0}^{n}
534 \binom nr
535 c_rp^{w(r)}S_{n-r},
536 \]
537 where \(c_r\in\Zp\), \(S_n\in\Zp\), and
538 \[
539 w(r)\in\NN.
540 \]
541 For every precision \(L\geq1\),
542 \[
543 \boxed{
544 A_n
545 \equiv
546 \sum_{\substack{0\leq r\leq n\\w(r)<L}}
547 \binom nr
548 c_rp^{w(r)}S_{n-r}
549 \pmod {p^L}.
550 }
551 \]
552 When the omitted set is nonempty, the tail has valuation at least
553 \[
554 \boxed{
555 \min_{\substack{0\leq r\leq n\\w(r)\geq L}}
556 \left(
557 w(r)+\vp\binom nr+\vp(S_{n-r})
558 \right).
559 }
560 \]
561 \end{histtheorem}
562 
563 \begin{proof}
564 Every omitted summand belongs to \(p^L\Zp\). The sharper statement follows by taking the minimum of the exact lower bounds for the summands and applying the nonarchimedean triangle inequality.
565 \end{proof}
566 
567 \begin{histremark}\label{hist:rem:triangular_freedom}
568 An arbitrary invertible triangular matrix can encode an arbitrary sequence. The transforms retained here have an explicit generating kernel and a visible filtration. Those two features make exact inversion and uniform truncation possible.
569 \end{histremark}
570 
571 \subsection{Congruential reciprocal packets}\label{hist:sec:congruential}
572 
573 \subsubsection{Residue sections of the exponential series}
574 
575 For integers \(N\geq1\) and \(0\leq j<N\), define
576 \[
577 \Ecal_{N,j}(t)
578 =
579 \sum_{n\geq0}
580 \frac{t^{Nn+j}}{(Nn+j)!}.
581 \]
582 The forced factor \(t^j\) may be removed without changing the residue address.
583 
584 \begin{histdefinition}[Congruential reciprocal packet]\label{hist:def:congruential_packet}
585 Define
586 \[
587 \boxed{
588 \Ccal_{N,j}(t)
589 =
590 \frac{t^j/j!}{\Ecal_{N,j}(t)}
591 =
592 \sum_{n\geq0}
593 C_{N,j}(n)
594 \frac{t^{Nn}}{(Nn)!}.
595 }
596 \]
597 \end{histdefinition}
598 
599 \begin{histtheorem}[Triangular inverse system]\label{hist:thm:triangular_inverse}
600 The coefficients satisfy \(C_{N,j}(0)=1\) and
601 \[
602 \boxed{
603 \sum_{q=0}^{n}
604 \binom{Nn+j}{Nq}
605 C_{N,j}(q)
606 =
607 \delta_{n0}.
608 }
609 \]
610 For \(n\geq1\),
611 \[
612 \boxed{
613 C_{N,j}(n)
614 =
615 -
616 \frac{
617 \sum_{q=0}^{n-1}
618 \binom{Nn+j}{Nq}C_{N,j}(q)
619 }{
620 \binom{Nn+j}{Nn}
621 }.
622 }
623 \]
624 \end{histtheorem}
625 
626 \begin{proof}
627 Multiply \(\Ccal_{N,j}(t)\) by \(\Ecal_{N,j}(t)\) and compare the coefficient of \(t^{Nn+j}/(Nn+j)!\). The last summand is the diagonal term, which is nonzero over \(\QQ\).
628 \end{proof}
629 
630 \begin{histproposition}[A denominator bound]\label{hist:prop:denominator_bound}
631 Let
632 \[
633 D_{N,j}(n)
634 =
635 \prod_{r=1}^{n}
636 \binom{Nr+j}{Nr},
637 \qquad
638 D_{N,j}(0)=1.
639 \]
640 Then
641 \[
642 \boxed{
643 D_{N,j}(n)C_{N,j}(n)
644 \in
645 \ZZ.
646 }
647 \]
648 For every prime \(p\),
649 \[
650 \vp(C_{N,j}(n))
651 \geq
652 -
653 \sum_{r=1}^{n}
654 \vp\binom{Nr+j}{Nr}.
655 \]
656 \end{histproposition}
657 
658 \begin{proof}
659 Use induction on \(n\) in the recurrence of \cref{hist:thm:triangular_inverse}. Every earlier denominator divides \(D_{N,j}(n-1)\), while the new diagonal denominator is the final factor of \(D_{N,j}(n)\). The valuation statement follows.
660 \end{proof}
661 
662 \begin{histexample}[Euler and Bernoulli boundaries]\label{hist:ex:euler_bernoulli_boundaries}
663 For \((N,j)=(2,0)\),
664 \[
665 \Ccal_{2,0}(t)=\operatorname{sech}t,
666 \]
667 so
668 \[
669 C_{2,0}(n)=E_{2n}.
670 \]
671 For \((N,j)=(2,1)\),
672 \[
673 \Ccal_{2,1}(t)=\frac{t}{\sinh t},
674 \]
675 and
676 \[
677 C_{2,1}(n)
678 =
679 2\left(1-2^{2n-1}\right)B_{2n}.
680 \]
681 The usual convention \(B_1=-1/2\) is used. Euler and Bernoulli number conventions are reviewed in \cite{Comtet1974,Stanley2012}.
682 \end{histexample}
683 
684 \subsubsection{Standard windows}
685 
686 Fix a prime \(p\), an integer \(m\geq1\), and an address \(0\leq j<p\). Put
687 \[
688 N=mp.
689 \]
690 For arbitrary \(m\), let \(\widetilde e_n^{(m,p)}\in\FF_p\) be the unique sequence satisfying
691 \[
692 \boxed{
693 \sum_{q=0}^{n}
694 \binom{mn}{mq}
695 \widetilde e_q^{(m,p)}
696 =
697 \delta_{n0}.
698 }
699 \]
700 When \(m\mid p-1\), this sequence agrees with the compressed Hurwitz inverse in \cref{hist:prop:compressed_inverse_recurrence}.
701 
702 \begin{histstandardfact}[Lucas congruence]\label{hist:fact:lucas}
703 If \(a=a_0+a_1p+\cdots\) and \(b=b_0+b_1p+\cdots\) are base \(p\) expansions, then
704 \[
705 \binom ab
706 \equiv
707 \prod_r\binom{a_r}{b_r}
708 \pmod p.
709 \]
710 This is the classical theorem of Lucas \cite{Lucas1878a}.
711 \end{histstandardfact}
712 
713 \begin{histtheorem}[Standard window reduction]\label{hist:thm:standard_window_reduction}
714 For \(0\leq q\leq n\),
715 \[
716 \boxed{
717 \binom{mpn+j}{mpq}
718 \equiv
719 \binom{mn}{mq}
720 \pmod p.
721 }
722 \]
723 The diagonal coefficient is a unit modulo \(p\):
724 \[
725 \binom{mpn+j}{mpn}
726 \equiv1
727 \pmod p.
728 \]
729 \end{histtheorem}
730 
731 \begin{proof}
732 Write
733 \[
734 mpn+j=p(mn)+j,
735 \qquad
736 mpq=p(mq).
737 \]
738 Lucas reduction separates the lowest base \(p\) digit and gives
739 \[
740 \binom{mpn+j}{mpq}
741 \equiv
742 \binom j0
743 \binom{mn}{mq}
744 \pmod p.
745 \]
746 The diagonal case is obtained by setting \(q=n\).
747 \end{proof}
748 
749 \begin{histcorollary}[Standard window integrality]\label{hist:cor:standard_integrality}
750 Every coefficient \(C_{mp,j}(n)\) belongs to \(\Zp\).
751 \end{histcorollary}
752 
753 \begin{proof}
754 The recurrence in \cref{hist:thm:triangular_inverse} starts with a \(p\) integral constant term. The binomial coefficients are integers, and every diagonal term is a \(p\) adic unit by \cref{hist:thm:standard_window_reduction}. Induction gives the result.
755 \end{proof}
756 
757 \begin{histcorollary}[Address independence modulo the prime]\label{hist:cor:address_independence}
758 Let \(\widetilde e_n^{(m,p)}\) be the reduced standard inverse above. Then
759 \[
760 \boxed{
761 C_{mp,j}(n)
762 \equiv
763 \widetilde e_n^{(m,p)}
764 \pmod p
765 }
766 \]
767 for every \(0\leq j<p\).
768 \end{histcorollary}
769 
770 \begin{proof}
771 Modulo \(p\), the recurrence in \cref{hist:thm:triangular_inverse} becomes
772 \[
773 \sum_{q=0}^{n}
774 \binom{mn}{mq}
775 C_{mp,j}(q)
776 =
777 \delta_{n0}.
778 \]
779 This is the defining recurrence for \(\widetilde e_n^{(m,p)}\). The diagonal is one, so the solution is unique.
780 \end{proof}
781 
782 \begin{histtheorem}[Uniform standard window period]\label{hist:thm:standard_window_period}
783 Assume that \(p\) is odd and \(m\mid p-1\). Put
784 \[
785 h=\frac{p-1}{m}.
786 \]
787 Then for every standard window address \(0\leq j<p\),
788 \[
789 \boxed{
790 C_{mp,j}(n+h)
791 \equiv
792 C_{mp,j}(n)
793 \pmod p
794 \qquad(n\geq1).
795 }
796 \]
797 \end{histtheorem}
798 
799 \begin{proof}
800 By \cref{hist:cor:address_independence}, every address reduces to the same reduced standard inverse. Under the divisibility assumption, that inverse agrees with the compressed Hurwitz inverse by \cref{hist:prop:compressed_inverse_recurrence}. Its positive tail has period \(h\) by \cref{hist:cor:compressed_period}.
801 \end{proof}
802 
803 \begin{histcorollary}[Rational residual generating function]\label{hist:cor:standard_rational_ogf}
804 Under the hypotheses of \cref{hist:thm:standard_window_period}, for every standard address \(j\),
805 \[
806 \sum_{n\geq0}
807 \left(C_{mp,j}(n)\bmod p\right)z^n
808 =
809 1+
810 \frac{
811 \sum_{r=1}^{h}\widetilde e_r^{(m,p)}z^r
812 }{1-z^h}.
813 \]
814 \end{histcorollary}
815 
816 \begin{proof}
817 Combine \cref{hist:cor:address_independence,hist:cor:periodic_ogf}.
818 \end{proof}
819 
820 
821 \begin{histtheorem}[Fourier invariance of the standard window operator]\label{hist:thm:standard_window_fourier}
822 Let \(p\) be odd, let \(m\mid p-1\), and let \(h=(p-1)/m\). Define
823 \[
824 (T_{m,p}f)_n
825 =
826 \sum_{q=0}^{n}\binom{mn}{mq}f_q
827 \quad\text{in }\FF_p.
828 \]
829 Let \(V_h\) be the vector space of sequences satisfying
830 \[
831 f_{n+h}=f_n
832 \qquad(n\geq1).
833 \]
834 Then
835 \[
836 \boxed{T_{m,p}(V_h)=V_h.}
837 \]
838 The restriction of \(T_{m,p}\) to \(V_h\) is an automorphism.
839 \end{histtheorem}
840 
841 \begin{proof}
842 Choose \(d_0=f_h\) and \(d_n=f_n\) for \(n\geq1\). Then \(d\) is fully \(h\) periodic and \(f=d+c\delta_0\) for \(c=f_0-f_h\). Since \(T_{m,p}\delta_0\) is the constant sequence one, it is enough to treat \(d\). Expand
843 \[
844 d_q=\sum_{\chi\in\mu_h}c_\chi\chi^q.
845 \]
846 The \(m\)th power map on \(\FF_p^\times\) has image \(\mu_h\). Choose \(y\) with \(y^m=\chi\). The root filter gives
847 \[
848 T_{m,p}(\chi^\bullet)_n
849 =
850 \frac1m
851 \sum_{\zeta\in\mu_m}
852 (1+\zeta y)^{mn}.
853 \]
854 If \(1+\zeta y=0\), the corresponding term vanishes for positive \(n\). Otherwise
855 \[
856 \left((1+\zeta y)^m\right)^h
857 =
858 (1+\zeta y)^{p-1}
859 =1.
860 \]
861 Every summand is therefore \(h\) periodic for positive \(n\). Thus \(T_{m,p}\) preserves \(V_h\). It is unit lower triangular on every finite initial segment. Its restriction to the finite dimensional space determined by \(f_0,f_1,\ldots,f_h\) is injective and hence bijective.
862 \end{proof}
863 
864 \begin{histcorollary}[A second proof of the standard window period]\label{hist:cor:standard_window_fourier_period}
865 The reduced inverse \(e^{(m,p)}\) and every standard address satisfy the period in \cref{hist:thm:standard_window_period}.
866 \end{histcorollary}
867 
868 \begin{proof}
869 The sequence \(\delta=(1,0,0,\ldots)\) belongs to \(V_h\). Its inverse image under the automorphism in \cref{hist:thm:standard_window_fourier} also belongs to \(V_h\). Address independence modulo \(p\) gives the statement for every standard address.
870 \end{proof}
871 
872 \begin{figure}[t]
873 \centering
874 \includegraphics[width=.95\linewidth,height=3.15in,keepaspectratio]{../../figures/reconstructed/standard_fourier.pdf}
875 \caption{New reconstruction. Finite field Fourier transfer for a standard window. Each character becomes a finite sum of positive index modes with period dividing \(h\).}
876 \label{hist:fig:standard_fourier}
877 \end{figure}
878 \begin{figure}[t]
879 \centering
880 \includegraphics[width=.95\linewidth,height=3.15in,keepaspectratio]{../../figures/reconstructed/standard_matrix.pdf}
881 \caption{New reconstruction. The standard window matrix for $p=7$, $m=2$, and $0\le j<7$. Reduction removes the address and retains diagonal one.}
882 \label{hist:fig:standard_matrix}
883 \end{figure}
884 
885 \begin{histremark}\label{hist:rem:period_boundary}
886 The theorem gives a period modulo \(p\). A period modulo higher powers requires additional control of the triangular inverse and cannot be inferred from the finite field argument alone.
887 
888 \end{histremark}
889 
890 \subsection{One local Euler source and kernel calculus}\label{hist:sec:local_euler}
891 
892 The odd and dyadic reconstructions have one convolutional form. Their sources, kernels, and valuation profiles are different. We first construct each chart and then state the common theorem.
893 
894 \subsubsection{The odd source and its inverse}
895 
896 Let \(q\) be a positive odd integer. Define
897 \[
898 S_n(q)
899 =
900 \sum_{a=0}^{q-1}
901 (-1)^a(2a+1)^n.
902 \]
903 Let the signed Euler numbers and Euler zigzag numbers be defined by
904 \[
905 \operatorname{sech}t
906 =
907 \sum_{n\geq0}E_n\frac{t^n}{n!}
908 \]
909 and
910 \[
911 \sec t+\tan t
912 =
913 \sum_{n\geq0}Z_n\frac{t^n}{n!}.
914 \]
915 
916 \begin{histlemma}[Odd source generating function]\label{hist:lem:odd_source_egf}
917 For every positive odd \(q\),
918 \[
919 \boxed{
920 \sum_{n\geq0}S_n(q)\frac{t^n}{n!}
921 =
922 \e^{qt}\cosh(qt)\operatorname{sech}t.
923 }
924 \]
925 Its even part is
926 \[
927 \boxed{
928 \sum_{n\geq0}S_{2n}(q)\frac{t^{2n}}{(2n)!}
929 =
930 \cosh^2(qt)\operatorname{sech}t.
931 }
932 \]
933 \end{histlemma}
934 
935 \begin{proof}
936 The finite geometric sum gives
937 \[
938 \sum_{a=0}^{q-1}(-1)^a\e^{(2a+1)t}
939 =
940 \e^t\frac{1+\e^{2qt}}{1+\e^{2t}}
941 =
942 \e^{qt}\cosh(qt)\operatorname{sech}t.
943 \]
944 Averaging at \(t\) and \(-t\) gives the even part.
945 \end{proof}
946 
947 \begin{histlemma}[Zigzag inverse kernel]\label{hist:lem:local_zigzag_kernel}
948 \[
949 \boxed{
950 \operatorname{sech}^2t
951 =
952 \sum_{r\geq0}
953 (-1)^rZ_{2r+1}
954 \frac{t^{2r}}{(2r)!}.
955 }
956 \]
957 \end{histlemma}
958 
959 \begin{proof}
960 The identity \(\tanh t=-i\tan(it)\) gives the odd zigzag expansion of \(\tanh t\). Differentiate.
961 \end{proof}
962 
963 \begin{histtheorem}[Odd modulus Euler coordinate pair]\label{hist:thm:odd_coordinate_pair}
964 For every positive odd \(q\) and every even \(n\geq0\),
965 \[
966 \boxed{
967 E_n
968 =
969 \sum_{r=0}^{n/2}
970 (-1)^rZ_{2r+1}
971 \binom n{2r}
972 q^{2r}S_{n-2r}(q).
973 }
974 \]
975 The inverse transform is
976 \[
977 \boxed{
978 S_n(q)
979 =
980 E_n
981 +
982 \sum_{r=1}^{n/2}
983 2^{2r-1}
984 \binom n{2r}
985 q^{2r}E_{n-2r}.
986 }
987 \]
988 Both transforms are lower triangular with diagonal one.
989 \end{histtheorem}
990 
991 \begin{proof}
992 The even source identity gives
993 \[
994 \operatorname{sech}t
995 =
996 \operatorname{sech}^2(qt)
997 \sum_{n\geq0}S_{2n}(q)\frac{t^{2n}}{(2n)!}.
998 \]
999 Insert \cref{hist:lem:local_zigzag_kernel} and compare coefficients. For the inverse, multiply \(\operatorname{sech}t\) by
1000 \[
1001 \cosh^2(qt)
1002 =
1003 1+
1004 \sum_{r\geq1}2^{2r-1}q^{2r}\frac{t^{2r}}{(2r)!}.
1005 \]
1006 \end{proof}
1007 
1008 \begin{histcorollary}[Odd modulus truncation]\label{hist:cor:odd_modulus_truncation}
1009 For every \(M\geq0\), every positive odd \(q\), and every even \(n\),
1010 \[
1011 \boxed{
1012 E_n
1013 \equiv
1014 \sum_{r=0}^{\min(M,n/2)}
1015 (-1)^rZ_{2r+1}
1016 \binom n{2r}q^{2r}S_{n-2r}(q)
1017 \pmod {q^{2M+2}}.
1018 }
1019 \]
1020 If \(p\mid q\) and \(a=\vp(q)\), the same truncation is valid modulo \(p^{2a(M+1)}\).
1021 \end{histcorollary}
1022 
1023 \begin{proof}
1024 Every omitted term contains \(q^{2r}\) with \(r\geq M+1\). The local statement follows by taking the \(p\) adic valuation.
1025 \end{proof}
1026 
1027 \begin{histproposition}[Odd row weight]\label{hist:prop:odd_row_weight}
1028 For an odd prime \(p\), the \(r\)th nonzero term in the reconstruction has valuation
1029 \[
1030 \boxed{
1031 W_p(n,r)
1032 =
1033 2r+\vp(Z_{2r+1})
1034 +\vp\binom n{2r}
1035 +\vp\bigl(S_{n-2r}(p)\bigr).
1036 }
1037 \]
1038 The binomial contribution is
1039 \[
1040 \boxed{
1041 \vp\binom n{2r}
1042 =
1043 \frac{s_p(2r)+s_p(n-2r)-s_p(n)}{p-1},
1044 }
1045 \]
1046 where \(s_p\) is the base \(p\) digit sum.
1047 \end{histproposition}
1048 
1049 \begin{proof}
1050 The first identity separates the valuation of every factor. The second is Legendre's formula, equivalently Kummer's carry theorem \cite{IrelandRosen1990}.
1051 \end{proof}
1052 
1053 \subsubsection{The dyadic source and its inverse}
1054 
1055 For even \(n\geq0\), define
1056 \[
1057 \boxed{
1058 R_n
1059 =
1060 \frac{3^{n+2}-1}{4(n+1)(n+2)}.
1061 }
1062 \]
1063 Its even exponential generating function is
1064 \[
1065 \boxed{
1066 \sum_{n\text{ even}}R_n\frac{t^n}{n!}
1067 =
1068 \frac{\sinh(2t)\sinh t}{2t^2}.
1069 }
1070 \]
1071 Define \(\beta_r\) by
1072 \[
1073 \boxed{
1074 \frac{z^2}{\sinh^2z}
1075 =
1076 \sum_{r\geq0}\beta_rz^{2r}.
1077 }
1078 \]
1079 
1080 \begin{histproposition}[Dyadic inverse coefficients]\label{hist:prop:dyadic_beta_unified}
1081 For \(r\geq1\),
1082 \[
1083 \boxed{
1084 \beta_r
1085 =
1086 -
1087 \frac{2^{2r}(2r-1)B_{2r}}{(2r)!}.
1088 }
1089 \]
1090 Consequently,
1091 \[
1092 \boxed{
1093 \vTwo(\beta_r)=\sTwo(r)-1.
1094 }
1095 \]
1096 Every \(R_n\) is a dyadic unit.
1097 \end{histproposition}
1098 
1099 \begin{proof}
1100 Differentiate the Bernoulli expansion of \(\coth z\) and use \(-(\coth z)'=\sinh(z)^{-2}\). The von Staudt and Clausen theorem gives \(\vTwo(B_{2r})=-1\). Legendre's formula gives
1101 \[
1102 \vTwo((2r)!)=2r-\sTwo(r).
1103 \]
1104 For \(R_n\), put \(k=n+2\). Since \(k\) is even,
1105 \[
1106 \vTwo(3^k-1)=2+\vTwo(k).
1107 \]
1108 The denominator has the same valuation because \(n+1\) is odd.
1109 \end{proof}
1110 
1111 \begin{histtheorem}[Dyadic Euler coordinate pair]\label{hist:thm:dyadic_coordinate_pair}
1112 For every even \(n\geq0\),
1113 \[
1114 \boxed{
1115 E_n
1116 =
1117 \sum_{r=0}^{n/2}
1118 \beta_r2^{2r}
1119 \fall n{2r}
1120 R_{n-2r}.
1121 }
1122 \]
1123 The inverse transform is
1124 \[
1125 \boxed{
1126 R_n
1127 =
1128 \sum_{r=0}^{n/2}
1129 \frac{2^{4r+1}}{(2r+2)!}
1130 \fall n{2r}
1131 E_{n-2r}.
1132 }
1133 \]
1134 \end{histtheorem}
1135 
1136 \begin{proof}
1137 Substitute \(z=2t\) in the inverse series and multiply
1138 \[
1139 \frac{4t^2}{\sinh^2(2t)}
1140 \frac{\sinh(2t)\sinh t}{2t^2}
1141 =
1142 \operatorname{sech}t.
1143 \]
1144 Coefficient extraction gives the forward transform. The inverse follows from
1145 \[
1146 \left(\frac{\sinh(2t)}{2t}\right)^2
1147 =
1148 \sum_{r\geq0}
1149 \frac{2^{4r+1}}{(2r+2)!}t^{2r}.
1150 \]
1151 \end{proof}
1152 
1153 \subsubsection{The common source and kernel law}
1154 
1155 For a prime \(p\), define
1156 \[
1157 A_{p,n}
1158 =
1159 \begin{cases}
1160 S_n(p),&p\text{ odd},\\
1161 R_n,&p=2\text{ and }n\text{ even},\\
1162 0,&p=2\text{ and }n\text{ odd},
1163 \end{cases}
1164 \]
1165 and
1166 \[
1167 \kappa_{p,r}
1168 =
1169 \begin{cases}
1170 (-1)^rZ_{2r+1}p^{2r},&p\text{ odd},\\
1171 \beta_r2^{2r}(2r)!,&p=2.
1172 \end{cases}
1173 \]
1174 
1175 \begin{histtheorem}[Unified local Euler transfer]\label{hist:thm:unified_local_euler}
1176 For every prime \(p\) and every even \(n\geq0\),
1177 \[
1178 \boxed{
1179 E_n
1180 =
1181 \sum_{r=0}^{n/2}
1182 \binom n{2r}
1183 \kappa_{p,r}
1184 A_{p,n-2r}.
1185 }
1186 \]
1187 If
1188 \[
1189 H_p(t)
1190 =
1191 \sum_{m\geq0}A_{p,2m}\frac{t^{2m}}{(2m)!}
1192 \]
1193 and
1194 \[
1195 K_p(t)
1196 =
1197 \sum_{r\geq0}\kappa_{p,r}\frac{t^{2r}}{(2r)!},
1198 \]
1199 then
1200 \[
1201 \boxed{K_p(t)H_p(t)=\operatorname{sech}t.}
1202 \]
1203 For odd \(p\),
1204 \[
1205 H_p(t)=\cosh^2(pt)\operatorname{sech}t,
1206 \qquad
1207 K_p(t)=\operatorname{sech}^2(pt).
1208 \]
1209 For \(p=2\),
1210 \[
1211 H_2(t)=\frac{\sinh(2t)\sinh t}{2t^2},
1212 \qquad
1213 K_2(t)=\frac{4t^2}{\sinh^2(2t)}.
1214 \]
1215 \end{histtheorem}
1216 
1217 \begin{proof}
1218 The two source and kernel products were proved in \cref{hist:thm:odd_coordinate_pair,hist:thm:dyadic_coordinate_pair}. Their divided power coefficient laws have the displayed common form.
1219 \end{proof}
1220 
1221 \begin{histtheorem}[Unified inverse local transfer]\label{hist:thm:unified_inverse_local_euler}
1222 Define
1223 \[
1224 \lambda_{p,r}
1225 =
1226 \begin{cases}
1227 1,&p\text{ odd and }r=0,\\
1228 2^{2r-1}p^{2r},&p\text{ odd and }r\geq1,\\
1229 \dfrac{2^{4r+1}}{(2r+1)(2r+2)},&p=2.
1230 \end{cases}
1231 \]
1232 Then for every prime \(p\) and every even \(n\geq0\),
1233 \[
1234 \boxed{
1235 A_{p,n}
1236 =
1237 \sum_{r=0}^{n/2}
1238 \binom n{2r}
1239 \lambda_{p,r}
1240 E_{n-2r}.
1241 }
1242 \]
1243 Equivalently, if
1244 \[
1245 L_p(t)=K_p(t)^{-1},
1246 \]
1247 then
1248 \[
1249 \boxed{H_p(t)=L_p(t)\operatorname{sech}t,}
1250 \]
1251 where
1252 \[
1253 L_p(t)=\cosh^2(pt)
1254 \]
1255 for odd \(p\), and
1256 \[
1257 L_2(t)=\left(\frac{\sinh(2t)}{2t}\right)^2
1258 \]
1259 for \(p=2\).
1260 \end{histtheorem}
1261 
1262 \begin{proof}
1263 For odd \(p\), this is the inverse identity in \cref{hist:thm:odd_coordinate_pair}. For \(p=2\), rewrite the inverse identity in \cref{hist:thm:dyadic_coordinate_pair} by using
1264 \[
1265 \fall n{2r}=(2r)!\binom n{2r}.
1266 \]
1267 The coefficient becomes
1268 \[
1269 \frac{2^{4r+1}(2r)!}{(2r+2)!}
1270 =
1271 \frac{2^{4r+1}}{(2r+1)(2r+2)}.
1272 \]
1273 The generating functions are the inverses of the kernels in \cref{hist:thm:unified_local_euler}.
1274 \end{proof}
1275 
1276 \begin{histcorollary}[Exact change between local source charts]\label{hist:cor:local_source_change}
1277 \par\noindent\textbf{Editorial range correction.} This comparison holds for the even source subsequences, equivalently the evenized generating functions $H_p$. It does not identify their full odd-degree source values. For $p=3$, the original odd source has $A_{3,1}=S_1(3)=3$, whereas every odd dyadic source coefficient is zero; the even kernel $L_3K_2$ cannot change that odd coefficient. Read the coefficient consequence below only at even indices. The displayed identity $H_p=L_pK_\ell H_\ell$ and its proof remain valid.\par\medskip
1278 For any two primes \(p\) and \(\ell\), the source sequences \(A_{p,n}\) and \(A_{\ell,n}\) are related by an exact unit lower triangular transformation obtained by composing the inverse kernel \(L_p\) with the reconstruction kernel \(K_\ell\).
1279 \end{histcorollary}
1280 
1281 \begin{proof}
1282 Both source series reconstruct \(\operatorname{sech}t\). Hence
1283 \[
1284 H_p(t)=L_p(t)K_\ell(t)H_\ell(t).
1285 \]
1286 The multiplier has constant coefficient one, so \cref{hist:thm:exact_coefficient_transport} gives an exact unit lower triangular coordinate change.
1287 \end{proof}
1288 
1289 \begin{table}[t]
1290 \centering
1291 \caption{The common local Euler architecture and the parts that remain prime dependent.}
1292 \label{hist:tab:local_chart_comparison}
1293 \begin{tabular}{>{\raggedright\arraybackslash}p{.22\textwidth}>{\raggedright\arraybackslash}p{.31\textwidth}>{\raggedright\arraybackslash}p{.31\textwidth}}
1294 \toprule
1295 Feature & Odd prime chart & Dyadic chart\\
1296 \midrule
1297 Finite torsion address & Modulo \(p-1\) & Sign address modulo two\\
1298 Principal units & \(1+p\mathbb Z_p\) & \(1+4\mathbb Z_2\)\\
1299 Source & Alternating odd power sum \(S_n(p)\) & Weighted source \(R_n\)\\
1300 Kernel & \(\operatorname{sech}^2(pt)\) & \(4t^2/\sinh^2(2t)\)\\
1301 Positive depth weight & At least \(2r\), with zigzag and row factors & Exactly \(4r-1\), before the binomial carry\\
1302 Analytic branch & Depleted unit source & Direct integral source\\
1303 Natural refined step & \((p-1)p^a\) & \(2^{a+1}\)\\
1304 \bottomrule
1305 \end{tabular}
1306 \end{table}
1307 
1308 \begin{histexample}[The first odd reconstruction]\label{hist:ex:first_odd_reconstruction}
1309 For \(p=3\) and \(n=2\),
1310 \[
1311 S_2(3)=1-9+25=17.
1312 \]
1313 Since \(Z_1=1\), \(Z_3=2\), and \(S_0(3)=1\), the odd coordinate formula gives
1314 \[
1315 E_2
1316 =17-2\cdot9
1317 =-1.
1318 \]
1319 The large source value and the depth one correction are both necessary.
1320 \end{histexample}
1321 
1322 \begin{histexample}[The first nontrivial dyadic row]\label{hist:ex:first_dyadic_reconstruction}
1323 For \(n=4\),
1324 \[
1325 R_4=\frac{91}{15},
1326 \qquad
1327 R_2=\frac53,
1328 \qquad
1329 R_0=1.
1330 \]
1331 The exact kernel coefficients are
1332 \[
1333 \kappa_{2,1}=-\frac83,
1334 \qquad
1335 \kappa_{2,2}=\frac{128}{5}.
1336 \]
1337 Therefore
1338 \[
1339 E_4
1340 =
1341 \frac{91}{15}
1342 -\frac83\binom42\frac53
1343 +\frac{128}{5}
1344 =5.
1345 \]
1346 The rational source values are dyadic units, while the transfer depth is carried by the kernel.
1347 \end{histexample}
1348 
1349 \begin{histremark}\label{hist:rem:no_dyadic_substitution}
1350 The common convolution does not identify the integral charts. At an odd prime the finite character coordinate has order \(p-1\). At the dyadic prime it begins with the sign coordinate, and the principal unit subgroup begins at \(1+4\mathbb Z_2\).
1351 \end{histremark}
1352 
1353 \subsubsection{Exact dyadic transfer depth}
1354 
1355 For \(r\geq1\), write
1356 \[
1357 \kappa_{2,r}
1358 =
1359 \beta_r2^{2r}(2r)!
1360 =
1361 -2^{4r}(2r-1)B_{2r}.
1362 \]
1363 
1364 \begin{histtheorem}[Exact dyadic kernel weight]\label{hist:thm:exact_dyadic_kernel_weight}
1365 \par\noindent\textbf{Editorial index conventions.} The standalone kernel valuation holds for all $r\ge1$. Its reconstruction-term formula requires even $n\ge0$ and $1\le r\le n/2$. In the following truncation formulas only terms with $r\le n/2$ occur; terms that would use $R_{n-2r}$ at a negative index are omitted. An empty tail is exactly zero, and the minimum over an empty set of term valuations is $+\infty$. These conventions include $M\ge n/2$.\par\medskip
1366 For every \(r\geq1\),
1367 \[
1368 \boxed{\vTwo(\kappa_{2,r})=4r-1.}
1369 \]
1370 The \(r\)th reconstruction term has exact valuation
1371 \[
1372 \boxed{
1373 \vTwo\left(
1374 \kappa_{2,r}\binom n{2r}R_{n-2r}
1375 \right)
1376 =
1377 4r-1+\vTwo\binom n{2r}.
1378 }
1379 \]
1380 Equivalently,
1381 \[
1382 \boxed{
1383 W_2(n,r)
1384 =
1385 4r-1+\sTwo(r)+\sTwo(n-2r)-\sTwo(n).
1386 }
1387 \]
1388 \end{histtheorem}
1389 
1390 \begin{proof}
1391 The Bernoulli formula gives \(\vTwo(B_{2r})=-1\). The source value is a unit. Kummer's binary carry formula gives the last expression.
1392 \end{proof}
1393 
1394 \begin{histtheorem}[Optimal dyadic truncation]\label{hist:thm:optimal_dyadic_truncation}
1395 Let
1396 \[
1397 T_{2,M}(n)
1398 =
1399 \sum_{r=0}^{\min(M,n/2)}
1400 \kappa_{2,r}\binom n{2r}R_{n-2r}.
1401 \]
1402 Then
1403 \[
1404 \boxed{
1405 E_n\equiv T_{2,M}(n)\pmod {2^{4M+3}}.
1406 }
1407 \]
1408 More precisely,
1409 \[
1410 \boxed{
1411 \vTwo\bigl(E_n-T_{2,M}(n)\bigr)
1412 \geq
1413 \min_{M+1\leq r\leq n/2}
1414 \left(
1415 4r-1+\vTwo\binom n{2r}
1416 \right).
1417 }
1418 \]
1419 The uniform exponent \(4M+3\) is best possible.
1420 \end{histtheorem}
1421 
1422 \begin{proof}
1423 Every omitted term has valuation at least \(4(M+1)-1\). For optimality, take \(n=2M+2\). There is one omitted term. Its binomial coefficient and source value are one, so its valuation is exactly \(4M+3\).
1424 \end{proof}
1425 
1426 The first three sharp truncations are
1427 \[
1428 \begin{aligned}
1429 E_n&\equiv R_n&&\pmod 8,\\
1430 E_n&\equiv R_n-\frac83\binom n2R_{n-2}&&\pmod {128},\\
1431 E_n&\equiv R_n-\frac83\binom n2R_{n-2}
1432 +\frac{128}{5}\binom n4R_{n-4}&&\pmod {2048}.
1433 \end{aligned}
1434 \]
1435 
1436 \begin{figure}[t]
1437 \centering
1438 \includegraphics[width=.95\linewidth,height=3.15in,keepaspectratio]{../../figures/reconstructed/local_kernel_weights.pdf}
1439 \caption{New reconstruction. Odd and dyadic kernels share one convolution law but have different valuation profiles. The positive depth dyadic kernel has exact slope four and offset minus one.}
1440 \label{hist:fig:local_kernel_weights}
1441 \end{figure}
1442 
1443 \begin{figure}[t]
1444 \centering
1445 \includegraphics[width=.95\linewidth,height=3.15in,keepaspectratio]{../../figures/reconstructed/odd_dyadic_geometry.pdf}
1446 \caption{New reconstruction. The local Euler charts share source and kernel reconstruction. Their torsion addresses, principal unit depths, source corrections, and precision growth remain different.}
1447 \label{hist:fig:odd_dyadic_geometry}
1448 \end{figure}
1449 \subsection{Mahler transfer and local analytic branches}\label{hist:sec:mahler}
1450 
1451 The finite character coordinate resolves the torsion part of a local unit. A sufficiently long arithmetic jump moves the remaining unit into a principal unit neighborhood. The coarse variable then has a convergent Mahler expansion with an explicit valuation slope.
1452 
1453 \subsubsection{Finite Mahler coordinates}
1454 
1455 For a function \(f\) on the nonnegative integers, define
1456 \[
1457 \Delta f(n)=f(n+1)-f(n).
1458 \]
1459 
1460 \begin{histtheorem}[Finite Mahler reconstruction]\label{hist:thm:finite_mahler_unified}
1461 Given \(f(0),\ldots,f(N)\), put \(c_r=\Delta^rf(0)\). Then
1462 \[
1463 \boxed{
1464 f(n)=\sum_{r=0}^{n}c_r\binom nr
1465 \qquad(0\leq n\leq N).
1466 }
1467 \]
1468 For every \(s\) with \(0\leq s\leq N\),
1469 \[
1470 \boxed{
1471 \Delta^sf(n)
1472 =
1473 \sum_{r=s}^{N}c_r\binom n{r-s}
1474 \qquad(0\leq n\leq N-s).
1475 }
1476 \]
1477 \end{histtheorem}
1478 
1479 \begin{proof}
1480 Let
1481 \[
1482 P(x)=\sum_{r=0}^{N}c_r\binom xr.
1483 \]
1484 Newton interpolation gives \(P(n)=f(n)\) for \(0\leq n\leq N\). Iterating
1485 \[
1486 \Delta\binom xr=\binom x{r-1}
1487 \]
1488 gives
1489 \[
1490 \Delta^sP(n)=\sum_{r=s}^{N}c_r\binom n{r-s}.
1491 \]
1492 When \(0\leq n\leq N-s\), every value used in \(\Delta^sf(n)\) lies among the prescribed data, so \(\Delta^sf(n)=\Delta^sP(n)\).
1493 \end{proof}
1494 
1495 \begin{histcorollary}[Mahler valuation ledger]\label{hist:cor:mahler_valuation_ledger}
1496 If \(c_r\in\Zp\) and \(\vp(c_r)\geq w(r)\), then
1497 \[
1498 \boxed{
1499 \vp(\Delta^sf(n))
1500 \geq
1501 \min_{r\geq s}
1502 \left(
1503 w(r)+\vp\binom n{r-s}
1504 \right).
1505 }
1506 \]
1507 \end{histcorollary}
1508 
1509 \begin{proof}
1510 Apply the nonarchimedean triangle inequality to \cref{hist:thm:finite_mahler_unified}.
1511 \end{proof}
1512 
1513 The infinite interpolation theorem states that every continuous function on \(\Zp\) has a unique convergent Mahler expansion \cite{Mahler1958,Robert2000}. We use that theorem as a standard input when the local variable is allowed to range over all of \(\Zp\).
1514 
1515 \subsubsection{Carmichael and Mahler transfer for unit modes}
1516 
1517 \begin{histstandardfact}[Local unit decomposition]\label{hist:fact:local_unit_decomposition}
1518 For odd \(p\), every unit has a unique factorization
1519 \[
1520 u=\omega(u)\langle u\rangle,
1521 \qquad
1522 \omega(u)^{p-1}=1,
1523 \qquad
1524 \langle u\rangle\in1+p\mathbb Z_p.
1525 \]
1526 Every dyadic unit has a unique factorization
1527 \[
1528 u=(-1)^{\epsilon(u)}\langle u\rangle,
1529 \qquad
1530 \epsilon(u)\in\{0,1\},
1531 \qquad
1532 \langle u\rangle\in1+4\mathbb Z_2.
1533 \]
1534 The first factor gives the finite character address. The second factor admits logarithmic and Mahler coordinates.
1535 \end{histstandardfact}
1536 
1537 For odd \(p\) and \(L\geq1\), put
1538 \[
1539 \Lambda_{p,L}=(p-1)p^{L-1}.
1540 \]
1541 For \(p=2\) and \(L\geq3\), put
1542 \[
1543 \Lambda_{2,L}=2^{L-2}.
1544 \]
1545 These are exponents of the unit groups modulo the declared prime powers.
1546 
1547 \begin{histtheorem}[One unit mode]\label{hist:thm:carmichael_mahler_scalar}
1548 Let \(u\in\Zp^\times\), let \(H=\Lambda_{p,L}\), and put
1549 \[
1550 q_{u,L}=u^H-1\in p^L\Zp.
1551 \]
1552 For every integer address \(j\) and every \(x\in\Zp\),
1553 \[
1554 \boxed{
1555 u^{Hx+j}
1556 =
1557 u^j\sum_{r\geq0}q_{u,L}^{\,r}\binom xr.
1558 }
1559 \]
1560 The \(r\)th Mahler coefficient has valuation at least \(Lr\).
1561 \end{histtheorem}
1562 
1563 \begin{proof}
1564 Write \(u^{Hx+j}=u^j(1+q_{u,L})^x\) and use the convergent binomial series on \(\Zp\).
1565 \end{proof}
1566 
1567 \begin{histexample}[Paired principal unit expansions]\label{hist:ex:paired_unit_expansions}
1568 At the odd prime three,
1569 \[
1570 2^{2x+j}
1571 =
1572 2^j4^x
1573 =
1574 2^j\sum_{r\geq0}3^r\binom xr.
1575 \]
1576 At the dyadic prime,
1577 \[
1578 3^{2x+j}
1579 =
1580 3^j9^x
1581 =
1582 3^j\sum_{r\geq0}8^r\binom xr.
1583 \]
1584 The formal expansions are identical. Their first local slopes are one and three because \(4-1=3\) while \(9-1=8\).
1585 \end{histexample}
1586 
1587 \begin{histcorollary}[Basic odd and dyadic difference gains]\label{hist:cor:basic_local_difference_gains}
1588 For odd \(p\),
1589 \[
1590 \boxed{
1591 \Delta_{(p-1)p^a}^{\,r}u^n
1592 =
1593 u^n\left(u^{(p-1)p^a}-1\right)^r,
1594 \qquad
1595 \vp\left(\Delta_{(p-1)p^a}^{\,r}u^n\right)\geq(a+1)r.
1596 }
1597 \]
1598 For odd \(u\in\mathbb Z_2^\times\),
1599 \[
1600 \boxed{
1601 \Delta_{2^{a+1}}^{\,r}u^n
1602 =
1603 u^n\left(u^{2^{a+1}}-1\right)^r,
1604 \qquad
1605 \vTwo\left(\Delta_{2^{a+1}}^{\,r}u^n\right)\geq(a+3)r.
1606 }
1607 \]
1608 \end{histcorollary}
1609 
1610 \begin{proof}
1611 One step difference multiplies a unit mode by \(u^H-1\). Fermat reduction and lifting the exponent give the odd estimate. For odd \(u\),
1612 \[
1613 \vTwo\left(u^{2^{a+1}}-1\right)
1614 =
1615 \vTwo(u^2-1)+a
1616 \geq a+3.
1617 \]
1618 \end{proof}
1619 
1620 \subsubsection{Polynomial exponential modes}
1621 
1622 \begin{histtheorem}[Polynomial exponential Mahler transfer]\label{hist:thm:polynomial_exponential_mahler}
1623 Let
1624 \[
1625 f(n)=P(n)u^n
1626 \]
1627 with \(P\in\Zp[n]\) of degree \(d\). Let \(H=\Lambda_{p,L}\), fix an address \(j\), and write
1628 \[
1629 Q_j(x)=P(Hx+j),
1630 \qquad
1631 y=u^H,
1632 \qquad q=y-1.
1633 \]
1634 Then
1635 \[
1636 \boxed{
1637 f(Hx+j)=\sum_{r\geq0}c_{j,r}\binom xr,
1638 }
1639 \]
1640 where
1641 \[
1642 \boxed{
1643 c_{j,r}
1644 =
1645 u^j
1646 \sum_{k=0}^{\min(d,r)}
1647 \binom rk
1648 \Delta^kQ_j(0)y^kq^{r-k}.
1649 }
1650 \]
1651 Consequently,
1652 \[
1653 \boxed{
1654 \vp(c_{j,r})\geq L\max(r-d,0).
1655 }
1656 \]
1657 For \(R\geq d\), truncation at depth \(R\) is valid modulo \(p^{L(R+1-d)}\) uniformly on \(\Zp\).
1658 \end{histtheorem}
1659 
1660 \begin{proof}
1661 On the polynomial factor the difference operator acts as \(q+y\Delta\). The two terms commute. Expand \((q+y\Delta)^r\) and evaluate at zero. Since \(\Delta^kQ_j=0\) for \(k>d\), every surviving term contains at least \(q^{r-d}\).
1662 \end{proof}
1663 
1664 \subsubsection{Matrix state systems and finite adelic synchronization}
1665 
1666 Let
1667 \[
1668 v_{n+1}=Cv_n,
1669 \qquad
1670 a_n=w^TC^nv_0,
1671 \]
1672 with \(C\in M_d(\Zp)\) invertible modulo \(p\).
1673 
1674 \begin{histtheorem}[Matrix Carmichael and Mahler transfer]\label{hist:thm:matrix_carmichael_mahler}
1675 \par\noindent\textbf{Editorial hypothesis correction.} The original displayed statement leaves the vectors untyped. Its integrality conclusion requires $v_0,w\in\Z_p^d$, $L\ge1$, and $H,j,n\in\N$ with $H>0$; the value at $Hx+j$ for $x\in\Z_p$ means the matrix-binomial interpolation. With $p=3$, $C=4$, $L=1$, $w=1$ and $v_0=1/3$, one has $\Delta_1a_0=1$, which is not in $3\Z_3$. The printed claim without these restrictions is therefore not an as-written certificate. The full original proof follows under the corrected hypotheses.\par\medskip
1676 Assume
1677 \[
1678 C^H=I+p^LB
1679 \]
1680 with \(B\in M_d(\Zp)\). For every address \(j\) and every \(x\in\Zp\),
1681 \[
1682 \boxed{
1683 a_{Hx+j}
1684 =
1685 \sum_{r\geq0}
1686 p^{Lr}\binom xr
1687 w^TC^jB^rv_0.
1688 }
1689 \]
1690 The series converges on \(\Zp\), and
1691 \[
1692 \boxed{
1693 \Delta_H^ra_n
1694 \equiv0
1695 \pmod {p^{Lr}}.
1696 }
1697 \]
1698 \end{histtheorem}
1699 
1700 \begin{proof}
1701 Use
1702 \[
1703 C^{Hx+j}=C^j(I+p^LB)^x
1704 \]
1705 and the matrix binomial series. One difference in the coarse variable multiplies the series by \(p^LB\).
1706 \end{proof}
1707 
1708 \begin{histtheorem}[Finite adelic transfer]\label{hist:thm:finite_adelic_transfer}
1709 \par\noindent\textbf{Editorial hypothesis correction.} Read this statement with $v_0,w\in\Z^d$, $H>0$ and $j,n,r\in\N$, and positive integer $L_p$. These restrictions make the integral congruences and powers well typed. Allowing $v_0=1/3$ in the one-prime example above loses the advertised digit; invertibility of $C$ alone does not restore it. The original proof is retained under these hypotheses.\par\medskip
1710 Let
1711 \[
1712 Q=\prod_{p\in S}p^{L_p}
1713 \]
1714 for a finite set of primes. Let \(C\in M_d(\ZZ)\) be invertible modulo \(Q\), and choose \(H\) such that
1715 \[
1716 C^H=I+QB
1717 \]
1718 with \(B\in M_d(\ZZ)\). Then for every nonnegative integer \(x\),
1719 \[
1720 \boxed{
1721 a_{Hx+j}
1722 =
1723 \sum_{r=0}^{x}Q^r\binom xr
1724 w^TC^jB^rv_0.
1725 }
1726 \]
1727 For every \(r\),
1728 \[
1729 \boxed{
1730 \Delta_H^ra_n\equiv0\pmod {Q^r}.
1731 }
1732 \]
1733 \end{histtheorem}
1734 
1735 \begin{proof}
1736 Apply the integer matrix binomial theorem and reduce at every selected prime.
1737 \end{proof}
1738 
1739 \begin{histexample}[A simultaneous Fibonacci jump]\label{hist:ex:fibonacci_finite_adelic}
1740 Let
1741 \[
1742 C=
1743 \begin{pmatrix}
1744 1&1\\
1745 1&0
1746 \end{pmatrix}.
1747 \]
1748 The matrix \(C\) is invertible modulo twenty four, and
1749 \[
1750 C^{24}-I
1751 =
1752 24
1753 \begin{pmatrix}
1754 3126&1932\\
1755 1932&1194
1756 \end{pmatrix}.
1757 \]
1758 For every scalar coordinate of the state sequence \(C^nv_0\),
1759 \[
1760 \boxed{
1761 \Delta_{24}^{\,r}a_n
1762 \equiv0
1763 \pmod {24^r}.
1764 }
1765 \]
1766 One integral jump therefore gives simultaneous congruence ladders at the primes two and three.
1767 \end{histexample}
1768 
1769 \subsubsection{Amice pullback}
1770 
1771 \begin{histdefinition}[Amice pullback]\label{hist:def:amice_pullback_unified}
1772 For a formal exponential generating series
1773 \[
1774 F(z)=\sum_{n\geq0}a_n\frac{z^n}{n!},
1775 \]
1776 define
1777 \[
1778 \widehat F(T)=F(\log(1+T)).
1779 \]
1780 \end{histdefinition}
1781 
1782 \begin{histtheorem}[Derivative conjugacy]\label{hist:thm:amice_derivative_unified}
1783 \[
1784 \boxed{
1785 \widehat{F'}(T)
1786 =
1787 (1+T)\frac{d}{dT}\widehat F(T).
1788 }
1789 \]
1790 \end{histtheorem}
1791 
1792 \begin{proof}
1793 Differentiate the composition \(F(\log(1+T))\).
1794 \end{proof}
1795 
1796 \begin{histproposition}[Weighted convergence]\label{hist:prop:amice_weighted_convergence_unified}
1797 If
1798 \[
1799 \widehat F(T)=\sum_{r\geq0}c_rT^r
1800 \]
1801 and
1802 \[
1803 \vp(c_r)+r\alpha\longrightarrow+\infty,
1804 \]
1805 then the series converges whenever \(\vp(T)\geq\alpha\).
1806 \end{histproposition}
1807 
1808 \begin{proof}
1809 The valuation of the \(r\)th term is \(\vp(c_r)+r\vp(T)\), which tends to positive infinity.
1810 \end{proof}
1811 
1812 \subsubsection{Odd prime depletion}
1813 
1814 For an odd prime \(p\), the source \(S_n(p)\) contains one nonunit base. Put
1815 \[
1816 \sigma_p=(-1)^{(p-1)/2}
1817 \]
1818 and
1819 \[
1820 S_n^\times(p)=S_n(p)-\sigma_pp^n.
1821 \]
1822 
1823 \begin{histtheorem}[Depleted odd prime reconstruction]\label{hist:thm:depleted_odd_reconstruction}
1824 For every even \(n\),
1825 \[
1826 \boxed{
1827 \left(1-\sigma_pp^n\right)E_n
1828 =
1829 \sum_{r=0}^{n/2}
1830 (-1)^rZ_{2r+1}
1831 \binom n{2r}p^{2r}S_{n-2r}^\times(p).
1832 }
1833 \]
1834 \end{histtheorem}
1835 
1836 \begin{proof}
1837 The removed source mode contributes \(\sigma_p\cosh(pt)\). Multiplication by \(\operatorname{sech}^2(pt)\) gives \(\sigma_p\operatorname{sech}(pt)\). Subtract it from the full source and kernel product.
1838 \end{proof}
1839 
1840 Fix an even address \(j\in\{0,2,\ldots,p-3\}\). For \(x\in\Zp\), define
1841 \[
1842 \begin{aligned}
1843 \mathscr E_{p,j}^\times(x)
1844 =
1845 \sum_{r\geq0}
1846 &(-1)^rZ_{2r+1}p^{2r}
1847 \binom{(p-1)x+j}{2r}\\
1848 &\times
1849 \sum_{\substack{0\leq a<p\\2a+1\neq p}}
1850 (-1)^a
1851 (2a+1)^{j-2r}
1852 \left((2a+1)^{p-1}\right)^x.
1853 \end{aligned}
1854 \]
1855 
1856 \begin{histtheorem}[Odd prime analytic Euler branch]\label{hist:thm:odd_analytic_branch}
1857 The series defining \(\mathscr E_{p,j}^\times\) converges in the Tate algebra on \(\Zp\). For every nonnegative integer \(x\),
1858 \[
1859 \boxed{
1860 \mathscr E_{p,j}^\times(x)
1861 =
1862 \left(1-\sigma_pp^{(p-1)x+j}\right)
1863 E_{(p-1)x+j}.
1864 }
1865 \]
1866 \end{histtheorem}
1867 
1868 \begin{proof}
1869 Every retained base is a unit, and its \((p-1)\)st power lies in \(1+p\Zp\). The polynomial
1870 \[
1871 p^{2r}\binom{(p-1)x+j}{2r}
1872 \]
1873 has Gauss norm at most
1874 \[
1875 p^{-2r+\vp((2r)!)}.
1876 \]
1877 This tends to zero because \(p\) is odd. The remaining factors have bounded Gauss norm. Thus the series converges in the Tate algebra. At a nonnegative integer \(x\), the binomial factor vanishes for large \(r\), and \cref{hist:thm:depleted_odd_reconstruction} gives the value.
1878 \end{proof}
1879 
1880 \begin{histremark}\label{hist:rem:depleted_factor_boundary}
1881 The analytic function is defined by its unit source series. The expression \(p^{(p-1)x+j}\) in the interpolation identity is evaluated only at nonnegative integers. No analytic function \(x\mapsto p^x\) on all of \(\Zp\) is used.
1882 \end{histremark}
1883 
1884 \subsubsection{Direct dyadic interpolation}
1885 
1886 For \(x\in\mathbb Z_2\), define
1887 \[
1888 \boxed{
1889 \mathscr R(x)
1890 =
1891 \frac{9^{x+1}-1}{8(x+1)(2x+1)}.
1892 }
1893 \]
1894 The apparent singularity at \(x=-1\) is removable, with value \(-\log(9)/8\).
1895 
1896 \begin{histlemma}[Analytic dyadic source]\label{hist:lem:analytic_dyadic_source}
1897 The function \(\mathscr R\) belongs to the dyadic Tate algebra and takes values in \(\mathbb Z_2\). For every nonnegative integer \(n\),
1898 \[
1899 \boxed{\mathscr R(n)=R_{2n}.}
1900 \]
1901 \end{histlemma}
1902 
1903 \begin{proof}
1904 Put \(y=x+1\) and let \(\ell=\log 9\). Since \(\vTwo(\ell)=3\),
1905 \[
1906 \frac{9^y-1}{8y}
1907 =
1908 \frac{\ell}{8}
1909 \sum_{m\geq0}
1910 \frac{\ell^m}{(m+1)!}y^m.
1911 \]
1912 The coefficient of \(y^m\) has valuation
1913 \[
1914 3m-\vTwo((m+1)!),
1915 \]
1916 which is nonnegative and tends to positive infinity. The quotient therefore belongs to \(\mathbb Z_2\langle y\rangle\). The factor \(2x+1=1+2x\) is a unit in \(\mathbb Z_2\langle x\rangle\), with inverse \(\sum_{m\geq0}(-2x)^m\). Hence \(\mathscr R\) is integral and analytic. The integer values follow from the definition of \(R_{2n}\).
1917 \end{proof}
1918 
1919 Define
1920 \[
1921 \boxed{
1922 \mathscr E_2(x)
1923 =
1924 \sum_{r\geq0}
1925 \kappa_{2,r}
1926 \binom{2x}{2r}
1927 \mathscr R(x-r).
1928 }
1929 \]
1930 
1931 \begin{histtheorem}[Direct dyadic Euler interpolation]\label{hist:thm:direct_dyadic_interpolation}
1932 The series defining \(\mathscr E_2\) converges in the dyadic Tate algebra. For every nonnegative integer \(n\),
1933 \[
1934 \boxed{\mathscr E_2(n)=E_{2n}.}
1935 \]
1936 Truncation after depth \(M\) is uniformly valid modulo \(2^{4M+3}\) on \(\mathbb Z_2\).
1937 \end{histtheorem}
1938 
1939 \begin{proof}
1940 The monomial Gauss norm of \(\binom{2x}{2r}\) is at most \(2^{\vTwo((2r)!)}\). Since
1941 \[
1942 \vTwo(\kappa_{2,r})=4r-1,
1943 \]
1944 the product tends to zero in Gauss norm. Translation by the integer \(r\) preserves the unit disc, so \(\mathscr R(x-r)\) remains an integral analytic function with uniformly bounded supremum norm. The series therefore converges in the Tate algebra.
1945 
1946 At a nonnegative integer \(n\), every term with \(r>n\) vanishes and \cref{hist:thm:dyadic_coordinate_pair} remains. For arbitrary \(x\in\mathbb Z_2\), the integer valued polynomial \(\binom{2x}{2r}\) and the source value \(\mathscr R(x-r)\) lie in \(\mathbb Z_2\). Every omitted term after depth \(M\) is therefore divisible by \(2^{4M+3}\). This proves the uniform truncation statement.
1947 \end{proof}
1948 
1949 \begin{histtheorem}[Unified local interpolation statement]\label{hist:thm:unified_local_interpolation}
1950 The odd branch \(\mathscr E_{p,j}^\times\) and the dyadic branch \(\mathscr E_2\) are analytic local realizations of the source and kernel law in \cref{hist:thm:unified_local_euler}. At integer arguments they recover, respectively,
1951 \[
1952 \left(1-\sigma_pp^n\right)E_n
1953 \]
1954 on each even odd prime character branch and
1955 \[
1956 E_{2n}
1957 \]
1958 on the dyadic even branch. The two realizations have distinct source corrections and distinct kernel valuation slopes.
1959 \end{histtheorem}
1960 
1961 \begin{proof}
1962 Combine \cref{hist:thm:odd_analytic_branch,hist:thm:direct_dyadic_interpolation} with the source and kernel definitions in \cref{hist:thm:unified_local_euler}.
1963 \end{proof}
1964 
1965 \subsection{Polyphase refinement of local Mahler coordinates}\label{hist:sec:polyphase}
1966 
1967 Polyphase refinement replaces one local branch by finitely many digit branches while preserving exact synthesis. For sequences it is residue analysis. For local functions it is the digit tree of \(\mathbb Z_p\). In Mahler coordinates it becomes a finite triangular kernel.
1968 
1969 \subsubsection{Discrete analysis and exact synthesis}
1970 
1971 For a sequence \(a=(a_n)_{n\geq0}\) and \(M\geq2\), define
1972 \[
1973 a_r^{[M]}(q)=a_{Mq+r}
1974 \qquad(0\leq r<M).
1975 \]
1976 The analysis map is
1977 \[
1978 \mathcal P_Ma
1979 =
1980 \left(a_0^{[M]},\ldots,a_{M-1}^{[M]}\right).
1981 \]
1982 Synthesis interleaves the branches by the rule
1983 \[
1984 a_{Mq+r}=a_r^{[M]}(q).
1985 \]
1986 
1987 \begin{histtheorem}[Exact discrete polyphase reconstruction]\label{hist:thm:discrete_polyphase_reconstruction}
1988 The analysis map \(\mathcal P_M\) is a bijection between sequences and ordered families of \(M\) sequences. If
1989 \[
1990 A(z)=\sum_{n\geq0}a_nz^n
1991 \]
1992 and
1993 \[
1994 A_r^{[M]}(w)=\sum_{q\geq0}a_{Mq+r}w^q,
1995 \]
1996 then
1997 \[
1998 \boxed{
1999 A(z)=\sum_{r=0}^{M-1}z^rA_r^{[M]}(z^M).
2000 }
2001 \]
2002 \end{histtheorem}
2003 
2004 \begin{proof}
2005 Every nonnegative integer has a unique expression \(Mq+r\) with \(0\leq r<M\). Group the terms of the ordinary generating function by this residue.
2006 \end{proof}
2007 
2008 \begin{histtheorem}[Nested discrete refinement]\label{hist:thm:nested_discrete_polyphase}
2009 For positive integers \(M,N\),
2010 \[
2011 \boxed{
2012 \left(a_r^{[M]}\right)_s^{[N]}
2013 =
2014 a_{r+Ms}^{[MN]}.
2015 }
2016 \]
2017 For a mixed radix path \((M_1,\ldots,M_\ell)\) with digits \(0\leq s_i<M_i\), the final address is
2018 \[
2019 \boxed{
2020 r(\boldsymbol s)
2021 =
2022 s_1+M_1s_2+\cdots+
2023 \left(\prod_{i=1}^{\ell-1}M_i\right)s_\ell.
2024 }
2025 \]
2026 Only the product radix and accumulated address determine the final branch.
2027 \end{histtheorem}
2028 
2029 \begin{proof}
2030 Direct substitution gives
2031 \[
2032 \left(a_r^{[M]}\right)_s^{[N]}(q)
2033 =
2034 a_{M(Nq+s)+r}.
2035 \]
2036 Iterate this identity along the mixed radix path.
2037 \end{proof}
2038 
2039 \subsubsection{Carry convolution and the shift matrix}
2040 
2041 Let \(c=a*b\) be the ordinary Cauchy convolution. Write \(A_r\), \(B_r\), and \(C_r\) for the phase ordinary generating functions.
2042 
2043 \begin{histproposition}[One carry convolution law]\label{hist:prop:polyphase_carry_convolution}
2044 For every residue \(r\),
2045 \[
2046 \boxed{
2047 C_r(w)
2048 =
2049 \sum_{u+v=r}A_u(w)B_v(w)
2050 +
2051 w\sum_{u+v=r+M}A_u(w)B_v(w).
2052 }
2053 \]
2054 The first sum has no phase carry. The second has one coarse index carry.
2055 \end{histproposition}
2056 
2057 \begin{proof}
2058 Write the input indices as \(Mq+u\) and \(Mq'+v\). Since \(0\leq u,v<M\), their sum has residue \(u+v\) when \(u+v<M\), and residue \(u+v-M\) with one additional coarse index when \(u+v\geq M\).
2059 \end{proof}
2060 
2061 \begin{histtheorem}[Polyphase shift matrix]\label{hist:thm:polyphase_shift_matrix}
2062 Let \(E\) be the scalar shift \((Ea)_n=a_{n+1}\), and let \(E_q\) shift the coarse index. Then
2063 \[
2064 \boxed{
2065 \mathcal P_M(Ea)
2066 =
2067 \mathsf S_M(E_q)\mathcal P_Ma,
2068 }
2069 \]
2070 where
2071 \[
2072 \mathsf S_M(E_q)
2073 =
2074 \begin{pmatrix}
2075 0&1&0&\cdots&0\\
2076 0&0&1&\cdots&0\\
2077 \vdots&&&\ddots&\vdots\\
2078 0&0&0&\cdots&1\\
2079 E_q&0&0&\cdots&0
2080 \end{pmatrix}.
2081 \]
2082 It also satisfies
2083 \[
2084 \boxed{\mathsf S_M(E_q)^M=E_qI_M.}
2085 \]
2086 \end{histtheorem}
2087 
2088 \begin{proof}
2089 For \(r<M-1\),
2090 \[
2091 (Ea)_{Mq+r}=a_{Mq+r+1}=a_{r+1}^{[M]}(q).
2092 \]
2093 For \(r=M-1\),
2094 \[
2095 (Ea)_{Mq+M-1}=a_{M(q+1)}=E_qa_0^{[M]}(q).
2096 \]
2097 After \(M\) shifts every phase has advanced by one coarse index.
2098 \end{proof}
2099 
2100 \subsubsection{Local branches and digit reconstruction}
2101 
2102 For \(a\geq1\) and \(0\leq s<p^a\), define
2103 \[
2104 f_{a,s}(x)=f(p^ax+s).
2105 \]
2106 
2107 \begin{histtheorem}[Exact local branch reconstruction]\label{hist:thm:local_branch_reconstruction}
2108 The family \(\{f_{a,s}\}_{0\leq s<p^a}\) determines \(f\) uniquely on \(\mathbb Z_p\). More precisely, if \(y\in s+p^a\mathbb Z_p\), then
2109 \[
2110 \boxed{
2111 f(y)=f_{a,s}\left(\frac{y-s}{p^a}\right).
2112 }
2113 \]
2114 For \(a,b\geq1\),
2115 \[
2116 \boxed{
2117 (f_{a,s})_{b,t}
2118 =
2119 f_{a+b,s+p^at}.
2120 }
2121 \]
2122 Thus local refinement is exactly the \(p\) adic digit tree.
2123 \end{histtheorem}
2124 
2125 \begin{proof}
2126 The cosets \(s+p^a\mathbb Z_p\) form a disjoint partition of \(\mathbb Z_p\). The first formula is the inverse affine change of variables on one coset. The second follows from
2127 \[
2128 p^a(p^bx+t)+s
2129 =
2130 p^{a+b}x+p^at+s.
2131 \]
2132 \end{proof}
2133 
2134 \subsubsection{The affine Mahler kernel}
2135 
2136 Let
2137 \[
2138 \Phi_M(z)=(1+z)^M-1.
2139 \]
2140 For \(M\geq1\), \(s\geq0\), and \(r,k\geq0\), define
2141 \[
2142 \boxed{
2143 \Theta_{M,s}(r,k)
2144 =
2145 [z^r](1+z)^s\Phi_M(z)^k.
2146 }
2147 \]
2148 Equivalently,
2149 \[
2150 \boxed{
2151 \Theta_{M,s}(r,k)
2152 =
2153 \sum_{j=0}^{k}
2154 (-1)^{k-j}\binom kj\binom{Mj+s}{r}.
2155 }
2156 \]
2157 It vanishes unless
2158 \[
2159 k\leq r\leq Mk+s.
2160 \]
2161 
2162 \begin{histproposition}[Edge coefficients of the affine kernel]\label{hist:prop:affine_kernel_edges}
2163 For every \(M\geq1\), \(s\geq0\), and \(k\geq0\),
2164 \[
2165 \boxed{
2166 \Theta_{M,s}(k,k)=M^k,
2167 \qquad
2168 \Theta_{M,s}(Mk+s,k)=1.
2169 }
2170 \]
2171 For binary radix and zero address,
2172 \[
2173 \boxed{
2174 \Theta_{2,0}(k+j,k)
2175 =
2176 2^{k-j}\binom kj
2177 \qquad(0\leq j\leq k).
2178 }
2179 \]
2180 Hence
2181 \[
2182 \boxed{
2183 \vTwo\left(\Theta_{2,0}(k+j,k)\right)
2184 =
2185 k-j+\vTwo\binom kj.
2186 }
2187 \]
2188 \end{histproposition}
2189 
2190 \begin{proof}
2191 The lowest degree term of \(\Phi_M(z)\) is \(Mz\), and its highest degree term is \(z^M\). Multiplication by \((1+z)^s\) gives the two edge coefficients. When \(M=2\) and \(s=0\),
2192 \[
2193 \Phi_2(z)^k
2194 =
2195 (2z+z^2)^k
2196 =
2197 z^k(2+z)^k.
2198 \]
2199 Taking the coefficient of \(z^{k+j}\) gives the binary formula.
2200 \end{proof}
2201 
2202 \begin{histtheorem}[Mahler polyphase transfer]\label{hist:thm:mahler_polyphase_transfer}
2203 Let
2204 \[
2205 f(x)=\sum_{r\geq0}c_r\binom xr
2206 \]
2207 be a convergent Mahler expansion. For
2208 \[
2209 g_s(q)=f(Mq+s),
2210 \]
2211 one has
2212 \[
2213 \boxed{
2214 g_s(q)=\sum_{k\geq0}d_{s,k}\binom qk,}
2215 \]
2216 where
2217 \[
2218 \boxed{
2219 d_{s,k}
2220 =
2221 \sum_{r=k}^{Mk+s}
2222 c_r\Theta_{M,s}(r,k).
2223 }
2224 \]
2225 Every refined coefficient uses a finite window of the original Mahler coefficients.
2226 \end{histtheorem}
2227 
2228 \begin{proof}
2229 Use
2230 \[
2231 (1+z)^{Mq+s}
2232 =
2233 (1+z)^s\left(1+\Phi_M(z)\right)^q
2234 =
2235 \sum_{k\geq0}\binom qk(1+z)^s\Phi_M(z)^k.
2236 \]
2237 Taking the coefficient of \(z^r\) gives the affine expansion of \(\binom{Mq+s}{r}\). Since \(c_r\to0\) and every \(d_{s,k}\) uses indices \(r\geq k\), the transformed coefficients also tend to zero.
2238 \end{proof}
2239 
2240 \begin{histtheorem}[Kernel composition]\label{hist:thm:polyphase_kernel_composition}
2241 For positive integers \(M,N\),
2242 \[
2243 \boxed{
2244 \sum_{k\geq0}
2245 \Theta_{M,s}(r,k)
2246 \Theta_{N,t}(k,\ell)
2247 =
2248 \Theta_{MN,Mt+s}(r,\ell).
2249 }
2250 \]
2251 The sum is finite.
2252 \end{histtheorem}
2253 
2254 \begin{proof}
2255 Apply \cref{hist:thm:mahler_polyphase_transfer} first to \(x=Mq+s\) and then to \(q=Ny+t\). The composite substitution is \(x=MNy+Mt+s\). Uniqueness of Mahler coefficients gives the identity.
2256 \end{proof}
2257 
2258 \subsubsection{Valuation transport}
2259 
2260 \begin{histtheorem}[Polyphase valuation ledger]\label{hist:thm:polyphase_valuation_ledger}
2261 For every refined coefficient,
2262 \[
2263 \boxed{
2264 \vp(d_{s,k})
2265 \geq
2266 \min_{k\leq r\leq Mk+s}
2267 \left(
2268 \vp(c_r)+\vp(\Theta_{M,s}(r,k))
2269 \right).
2270 }
2271 \]
2272 If \(\vp(c_r)\geq w(r)\) for a nondecreasing function \(w\), then
2273 \[
2274 \boxed{\vp(d_{s,k})\geq w(k).}
2275 \]
2276 Thus every monotone Mahler precision ladder survives arbitrary polyphase refinement.
2277 \end{histtheorem}
2278 
2279 \begin{proof}
2280 Apply the nonarchimedean triangle inequality to the finite coefficient formula. Since \(r\geq k\), monotonicity gives \(w(r)\geq w(k)\), and the affine kernel is integral.
2281 \end{proof}
2282 
2283 \begin{histtheorem}[Prime power sparsification]\label{hist:thm:prime_power_sparsification}
2284 \par\noindent\textbf{Editorial hypothesis correction.} The affine-kernel congruence below needs no change, but its consequence for transformed Mahler coefficients requires $c_r\in\Z_p$ for every $r$. Convergence of a $\Q_p$-valued Mahler expansion alone is insufficient. For $p=2$, $M=2$, $s=0$ and $f(x)=x/2$, one has $c_1=1/2$, $c_2=0$, while $g(q)=q$ has $d_1=1$; thus $d_1\not\equiv c_2\pmod2$. The coefficient consequence is read with integral original coefficients.\par\medskip
2285 Let \(M=p^a\). Then
2286 \[
2287 \boxed{
2288 \Theta_{p^a,s}(r,k)
2289 \equiv
2290 \binom{s}{r-p^ak}
2291 \pmod p,
2292 }
2293 \]
2294 where the binomial coefficient is zero outside its natural range. Consequently,
2295 \[
2296 \boxed{
2297 d_{s,k}
2298 \equiv
2299 \sum_{u=0}^{s}
2300 \binom su
2301 c_{p^ak+u}
2302 \pmod p.
2303 }
2304 \]
2305 In particular,
2306 \[
2307 \boxed{d_{0,k}\equiv c_{p^ak}\pmod p.}
2308 \]
2309 \end{histtheorem}
2310 
2311 \begin{proof}
2312 In characteristic \(p\),
2313 \[
2314 (1+z)^{p^a}-1\equiv z^{p^a}.
2315 \]
2316 Substitute this congruence in the definition of \(\Theta_{p^a,s}\).
2317 \end{proof}
2318 
2319 \begin{figure}[t]
2320 \centering
2321 \includegraphics[width=.95\linewidth,height=3.15in,keepaspectratio]{../../figures/reconstructed/mahler_polyphase_kernel.pdf}
2322 \caption{New reconstruction. The affine Mahler kernel has finite triangular support. At prime power radix its reduction is confined to a short Frobenius window.}
2323 \label{hist:fig:mahler_polyphase_kernel}
2324 \end{figure}
2325 
2326 \subsubsection{Precision amplification}
2327 
2328 \begin{histtheorem}[Scalar polyphase precision amplification]\label{hist:thm:scalar_polyphase_precision}
2329 Let \(u\in\mathbb Z_p^\times\), let \(a,r\geq0\), and put
2330 \[
2331 H_a=
2332 \begin{cases}
2333 (p-1)p^a,&p\text{ odd},\\
2334 2^{a+1},&p=2.
2335 \end{cases}
2336 \]
2337 Then
2338 \[
2339 \boxed{
2340 \Delta_{H_a}^{\,r}u^n
2341 =
2342 u^n\left(u^{H_a}-1\right)^r.
2343 }
2344 \]
2345 For odd \(p\),
2346 \[
2347 \boxed{
2348 \vp\left(\Delta_{H_a}^{\,r}u^n\right)
2349 \geq(a+1)r.
2350 }
2351 \]
2352 For \(p=2\) and odd \(u\),
2353 \[
2354 \boxed{
2355 \vTwo\left(\Delta_{H_a}^{\,r}u^n\right)
2356 \geq(a+3)r.
2357 }
2358 \]
2359 Equivalently, every additional prime power digit increases the certified Mahler slope by one.
2360 \end{histtheorem}
2361 
2362 \begin{proof}
2363 One difference with step \(H_a\) multiplies the unit mode by \(u^{H_a}-1\). Iterate. The valuation estimates are those of \cref{hist:cor:basic_local_difference_gains}.
2364 \end{proof}
2365 
2366 \begin{histlemma}[Matrix power amplification]\label{hist:lem:matrix_power_amplification}
2367 \par\noindent\textbf{Editorial hypothesis correction.} Require $L\ge1$. At $L=0$ the standalone claim is false: for $p=3$, $d=1$ and $X=2=1+B$, one has $X^3-1=7\notin3\Z_3$. With $L\ge1$, every higher binomial term in the retained proof has the required valuation, including $p=2$.\par\medskip
2368 If
2369 \[
2370 X=I+p^LB
2371 \]
2372 with \(B\in M_d(\Zp)\), then for every \(a\geq0\) there is \(B_a\in M_d(\Zp)\) such that
2373 \[
2374 \boxed{X^{p^a}=I+p^{L+a}B_a.}
2375 \]
2376 \end{histlemma}
2377 
2378 \begin{proof}
2379 Induct on \(a\). Expand the \(p\)th power of \(I+p^{L+a}B_a\). The linear term has valuation at least \(L+a+1\), and every higher term has at least that valuation.
2380 \end{proof}
2381 
2382 \begin{histtheorem}[Matrix polyphase precision amplification]\label{hist:thm:matrix_polyphase_precision}
2383 \par\noindent\textbf{Editorial hypothesis correction.} Require $L\ge1$, $C,B\in M_d(\Z_p)$, $v_0,w\in\Z_p^d$, $H>0$, and $a,j,n,r\in\N$. These are the hypotheses of the two corrected matrix statements used by the retained proof. Without integral observation vectors or positive $L$, the preceding counterexamples invalidate the claimed precision.\par\medskip
2384 Assume
2385 \[
2386 C^H=I+p^LB.
2387 \]
2388 Put \(H_a=Hp^a\). Then
2389 \[
2390 C^{H_a}=I+p^{L+a}B_a
2391 \]
2392 for an integral matrix \(B_a\). Every refined packet
2393 \[
2394 b_{a,j}(x)=a_{H_ax+j}
2395 \]
2396 has Mahler coefficients satisfying
2397 \[
2398 \boxed{\vp(c_{a,j,r})\geq(L+a)r.}
2399 \]
2400 Equivalently,
2401 \[
2402 \boxed{
2403 \Delta_{H_a}^{\,r}a_n
2404 \equiv0
2405 \pmod {p^{(L+a)r}}.
2406 }
2407 \]
2408 \end{histtheorem}
2409 
2410 \begin{proof}
2411 Combine \cref{hist:lem:matrix_power_amplification,hist:thm:matrix_carmichael_mahler}.
2412 \end{proof}
2413 
2414 \begin{histexample}[Refined dyadic Fibonacci ladder]\label{hist:ex:refined_fibonacci_ladder}
2415 For the Fibonacci companion matrix
2416 \[
2417 C=
2418 \begin{pmatrix}
2419 1&1\\
2420 1&0
2421 \end{pmatrix},
2422 \]
2423 one has
2424 \[
2425 C^{12}
2426 =I+8
2427 \begin{pmatrix}
2428 29&18\\
2429 18&11
2430 \end{pmatrix}.
2431 \]
2432 Therefore
2433 \[
2434 \boxed{
2435 \Delta_{12\cdot2^a}^{\,r}F_n
2436 \equiv0
2437 \pmod {2^{(a+3)r}}.
2438 }
2439 \]
2440 Every dyadic refinement doubles the step and adds one guaranteed binary digit at each Mahler depth.
2441 \end{histexample}
2442 
2443 \begin{figure}[t]
2444 \centering
2445 \includegraphics[width=.95\linewidth,height=3.15in,keepaspectratio]{../../figures/reconstructed/polyphase_precision_tree.pdf}
2446 \caption{New reconstruction. Every additional prime power refinement digit increases the certified Mahler slope of a principal unit matrix packet by one.}
2447 \label{hist:fig:polyphase_precision_tree}
2448 \end{figure}
2449 
2450 \subsubsection{Character and polyphase coordinates}
2451 
2452 For odd \(p\), write
2453 \[
2454 \boxed{
2455 n=j+(p-1)(s+p^aq),
2456 }
2457 \]
2458 with \(0\leq j<p-1\) and \(0\leq s<p^a\). The address \(j\) records the finite character. The address \(s\) records the principal unit digit. The variable \(q\) carries the remaining Mahler expansion.
2459 
2460 At the dyadic prime, write
2461 \[
2462 \boxed{
2463 n=j+2(s+2^aq),
2464 }
2465 \]
2466 with \(j\in\{0,1\}\) and \(0\leq s<2^a\). The sign coordinate replaces the odd prime character coordinate.
2467 
2468 \begin{histproposition}[Local phase lattice]\label{hist:prop:local_phase_lattice}
2469 For a unit mode \(u^n\), the odd prime coordinate gives
2470 \[
2471 \boxed{
2472 u^n
2473 =
2474 u^{j+(p-1)s}
2475 \left(u^{(p-1)p^a}\right)^q.
2476 }
2477 \]
2478 The dyadic coordinate gives
2479 \[
2480 \boxed{
2481 u^n
2482 =
2483 u^{j+2s}
2484 \left(u^{2^{a+1}}\right)^q.
2485 }
2486 \]
2487 Thus finite character phase, local digit address, and Mahler depth are separate coordinates.
2488 \end{histproposition}
2489 
2490 \begin{proof}
2491 Substitute the two displayed decompositions of \(n\).
2492 \end{proof}
2493 
2494 \begin{histcorollary}[Polyphase refinement of the Euler branches]\label{hist:cor:euler_polyphase_refinement}
2495 The analytic branches in \cref{hist:thm:odd_analytic_branch,hist:thm:direct_dyadic_interpolation} admit exact refinements under \(x=p^aq+s\). Their refined Mahler coefficients are obtained from \(\Theta_{p^a,s}\). The original monotone valuation ladders are preserved, and unit mode or invertible matrix representations gain one certified digit per refinement level.
2496 \end{histcorollary}
2497 
2498 \begin{proof}
2499 Apply \cref{hist:thm:mahler_polyphase_transfer,hist:thm:polyphase_valuation_ledger,hist:cor:basic_local_difference_gains,hist:thm:matrix_polyphase_precision}.
2500 \end{proof}
2501 
2502 \begin{histremark}\label{hist:rem:polyphase_boundary}
2503 Exact refinement does not imply that every branch is simpler. A simplification theorem requires an additional invariant, such as Frobenius sparsity, valuation gain, spectral collision, or a vanishing transfer block.
2504 \end{histremark}
2505 \subsection{Arithmetic lift obstructions}\label{hist:sec:lifting}
2506 
2507 A congruence class may solve an equation to one precision and fail at the next. The correct obstruction is not an arbitrary residual vector. It is the residual vector modulo the image of the declared correction map.
2508 
2509 \subsubsection{Linear lifting}
2510 
2511 Let \(p\) be a prime. Let
2512 \[
2513 M\in M_{r,d}(\Zp),
2514 \qquad
2515 b\in\Zp^r.
2516 \]
2517 Suppose that \(x_L\in\Zp^d\) satisfies
2518 \[
2519 Mx_L\equiv b\pmod {p^L}
2520 \]
2521 for some \(L\geq1\). Define the reduced residual
2522 \[
2523 \rho_L(x_L)
2524 =
2525 \frac{Mx_L-b}{p^L}
2526 \pmod p
2527 \in\FF_p^r.
2528 \]
2529 Write \(\overline M\) for the reduction of \(M\) modulo \(p\).
2530 
2531 \begin{histdefinition}[Linear lift obstruction]\label{hist:def:linear_lift_obstruction}
2532 The linear obstruction at depth \(L\) is
2533 \[
2534 \boxed{
2535 \operatorname{Ob}^{\mathrm{lin}}_L(x_L)
2536 =
2537 [\rho_L(x_L)]
2538 \in
2539 \coker(\overline M).
2540 }
2541 \]
2542 \end{histdefinition}
2543 
2544 \begin{histtheorem}[One level linear lifting]\label{hist:thm:linear_lifting}
2545 A class represented by \(x_L\) lifts to a solution modulo \(p^{L+1}\) if and only if
2546 \[
2547 \boxed{
2548 \operatorname{Ob}^{\mathrm{lin}}_L(x_L)=0.
2549 }
2550 \]
2551 More precisely, a correction \(y\in\FF_p^d\) gives a lift
2552 \[
2553 x_{L+1}=x_L+p^Ly
2554 \pmod {p^{L+1}}
2555 \]
2556 if and only if
2557 \[
2558 \boxed{
2559 \overline My=-\rho_L(x_L).
2560 }
2561 \]
2562 When one correction exists, the set of all correction classes is a coset of \(\ker\overline M\).
2563 \end{histtheorem}
2564 
2565 \begin{proof}
2566 Modulo \(p^{L+1}\),
2567 \[
2568 M(x_L+p^Ly)-b
2569 =
2570 Mx_L-b+p^LMy.
2571 \]
2572 After division by \(p^L\) and reduction modulo \(p\), the lift condition becomes
2573 \[
2574 \rho_L(x_L)+\overline My=0.
2575 \]
2576 This equation is solvable exactly when the residual lies in the image of \(\overline M\). If \(y_0\) is one solution, every solution is \(y_0+z\) with \(z\in\ker\overline M\).
2577 \end{proof}
2578 
2579 \begin{histproposition}[Representative independence]\label{hist:prop:linear_obstruction_independence}
2580 The class \(\operatorname{Ob}^{\mathrm{lin}}_L(x_L)\) depends only on the solution class of \(x_L\) modulo \(p^L\).
2581 \end{histproposition}
2582 
2583 \begin{proof}
2584 Replace \(x_L\) by \(x_L+p^Lz\). The reduced residual changes by \(\overline Mz\). Its class in the cokernel is unchanged.
2585 \end{proof}
2586 
2587 \begin{histcorollary}[Exact number of linear children]\label{hist:cor:linear_child_count}
2588 If \(\rho=\rank_{\FF_p}\overline M\), then a solvable depth \(L\) class has exactly
2589 \[
2590 \boxed{
2591 p^{d-\rho}
2592 }
2593 \]
2594 children modulo \(p^{L+1}\).
2595 \end{histcorollary}
2596 
2597 \begin{proof}
2598 The correction set is a coset of a vector space of dimension \(d-\rho\).
2599 \end{proof}
2600 
2601 \begin{histexample}[A visible linear obstruction]\label{hist:ex:linear_lift_obstruction}
2602 Take \(p=3\), \(L=1\), and
2603 \[
2604 M=
2605 \begin{pmatrix}
2606 1&1\\
2607 0&0
2608 \end{pmatrix}.
2609 \]
2610 The class \(x_1=(0,0)\) solves
2611 \[
2612 Mx_1\equiv b\pmod3
2613 \]
2614 for both
2615 \[
2616 b_1=(-3,0)
2617 \qquad\text{and}\qquad
2618 b_2=(0,-3).
2619 \]
2620 For \(b_1\), the reduced residual is \((1,0)\), which belongs to the image of \(\overline M\). There are three correction classes, given by
2621 \[
2622 y_1+y_2=2
2623 \pmod3.
2624 \]
2625 For \(b_2\), the residual is \((0,1)\), whose cokernel class is nonzero. No lift modulo nine exists because the second row of \(M\) remains zero.
2626 \end{histexample}
2627 
2628 \begin{figure}[t]
2629 \centering
2630 \includegraphics[width=.95\linewidth,height=3.15in,keepaspectratio]{../../figures/reconstructed/lift_obstruction.pdf}
2631 \caption{New reconstruction. A residual vector lifts precisely when its class vanishes in the cokernel of the reduced correction map.}
2632 \label{hist:fig:lift_obstruction}
2633 \end{figure}
2634 
2635 \subsubsection{Polynomial lifting}
2636 
2637 Let
2638 \[
2639 F=(F_1,\ldots,F_r):\Zp^d\longrightarrow\Zp^r
2640 \]
2641 be a polynomial map.\par\noindent\textbf{Editorial coefficient clarification.} Here polynomial means $F_i\in\Z_p[X_1,\ldots,X_d]$, not merely an integer-valued polynomial with coefficients in $\Q_p$. This makes the Jacobian and Taylor remainder integral and their reduction well typed.\par Suppose
2642 \[
2643 F(x_L)\in p^L\Zp^r.
2644 \]
2645 Define
2646 \[
2647 \rho_L^F(x_L)
2648 =
2649 \frac{F(x_L)}{p^L}
2650 \pmod p.
2651 \]
2652 Let \(J_F(x_L)\) denote the Jacobian matrix and write \(\overline J_F(x_L)\) for its reduction modulo \(p\).
2653 
2654 \begin{histdefinition}[Polynomial lift obstruction]\label{hist:def:polynomial_lift_obstruction}
2655 The polynomial obstruction at depth \(L\) is
2656 \[
2657 \boxed{
2658 \operatorname{Ob}^{F}_L(x_L)
2659 =
2660 [\rho_L^F(x_L)]
2661 \in
2662 \coker\bigl(\overline J_F(x_L)\bigr).
2663 }
2664 \]
2665 \end{histdefinition}
2666 
2667 \begin{histtheorem}[One level polynomial lifting]\label{hist:thm:polynomial_lifting}
2668 Let \(L\geq1\). A class represented by \(x_L\) lifts to a zero of \(F\) modulo \(p^{L+1}\) if and only if
2669 \[
2670 \boxed{
2671 \operatorname{Ob}^{F}_L(x_L)=0.
2672 }
2673 \]
2674 A correction \(y\in\FF_p^d\) gives a lift if and only if
2675 \[
2676 \boxed{
2677 \overline J_F(x_L)y
2678 =
2679 -\rho_L^F(x_L).
2680 }
2681 \]
2682 When a correction exists, the correction classes form a coset of
2683 \[
2684 \ker\overline J_F(x_L).
2685 \]
2686 \end{histtheorem}
2687 
2688 \begin{proof}
2689 The polynomial Taylor expansion gives
2690 \[
2691 F(x_L+p^Ly)
2692 \equiv
2693 F(x_L)+p^LJ_F(x_L)y
2694 \pmod {p^{L+1}}.
2695 \]
2696 Every term of degree at least two in the increment contains \(p^{2L}\), which is divisible by \(p^{L+1}\). Divide by \(p^L\), reduce modulo \(p\), and apply the same cokernel criterion as in \cref{hist:thm:linear_lifting}.
2697 \end{proof}
2698 
2699 \begin{histproposition}[Polynomial representative independence]\label{hist:prop:polynomial_obstruction_independence}
2700 The class \(\operatorname{Ob}^{F}_L(x_L)\) depends only on the class of \(x_L\) modulo \(p^L\).
2701 \end{histproposition}
2702 
2703 \begin{proof}
2704 If \(x_L\) is replaced by \(x_L+p^Lz\), Taylor expansion shows that the reduced residual changes by an element of the image of \(\overline J_F(x_L)\). The reduced Jacobian itself is unchanged because \(L\geq1\).
2705 \end{proof}
2706 
2707 \begin{histcorollary}[Simple root lifting]\label{hist:cor:simple_root_lifting}
2708 Assume \(r=d\) and
2709 \[
2710 \det J_F(x_L)\not\equiv0\pmod p.
2711 \]
2712 Then the depth \(L\) solution has exactly one child modulo \(p^{L+1}\). Iterating gives a unique \(p\) adic zero above the initial class.
2713 \end{histcorollary}
2714 
2715 \begin{proof}
2716 The reduced Jacobian is invertible, so every residual has one correction. The resulting compatible sequence is Cauchy in \(\Zp^d\), and continuity of \(F\) gives a zero in the limit. Uniqueness holds at every depth.
2717 \end{proof}
2718 
2719 \begin{histexample}[A singular polynomial obstruction]\label{hist:ex:singular_polynomial_obstruction}
2720 Let \(p=3\) and consider the class \(x_1=0\) modulo three. For
2721 \[
2722 F_0(x)=x^2,
2723 \]
2724 the residual and the reduced derivative both vanish. Every correction \(y\in\FF_3\) gives
2725 \[
2726 F_0(3y)\equiv0\pmod9,
2727 \]
2728 so the class has three children. For
2729 \[
2730 F_1(x)=x^2+3,
2731 \]
2732 the reduced derivative is still zero, but the reduced residual is one. Its class in the cokernel is nonzero, so no lift modulo nine exists.
2733 \end{histexample}
2734 
2735 This is the usual simple root mechanism behind Hensel lifting. The obstruction formulation also covers singular Jacobians, where some residual classes lift and others do not \cite{Serre1979,Gouvea1997,Robert2000}.
2736 
2737 \subsubsection{Packetwise obstruction splitting}
2738 
2739 \begin{histtheorem}[Packet decomposition of a lift obstruction]\label{hist:thm:packet_lift_obstruction}
2740 Suppose
2741 \[
2742 \FF_p^d
2743 =
2744 \bigoplus_{a\in A}U_a,
2745 \qquad
2746 \FF_p^r
2747 =
2748 \bigoplus_{a\in A}V_a,
2749 \]
2750 and suppose a linear correction map \(D:\FF_p^d\to\FF_p^r\) satisfies
2751 \[
2752 D(U_a)\subseteq V_a
2753 \]
2754 for every \(a\). Write \(D_a=D|_{U_a}\). Then
2755 \[
2756 \boxed{
2757 \coker D
2758 \cong
2759 \bigoplus_{a\in A}\coker D_a.
2760 }
2761 \]
2762 A residual lifts if and only if every packet obstruction vanishes.
2763 \end{histtheorem}
2764 
2765 \begin{proof}
2766 The image of \(D\) is the direct sum of the images of the maps \(D_a\). Taking the quotient of the codomain packet decomposition gives the stated cokernel decomposition.
2767 \end{proof}
2768 
2769 \begin{histremark}\label{hist:rem:lifting_boundary}
2770 The obstruction theorems solve one declared local lifting step. They do not prove that a compatible branch exists across unrelated primes, that a global Diophantine point exists, or that an arithmetic representation is automorphic.
2771 \end{histremark}
2772 
2773 \subsection{Recurrence packets and a common integral spine}\label{hist:sec:recurrences}
2774 
2775 Linear recurrences provide a setting where residue subsequences, local completions, and a common global source can all be described exactly. Background on recurrence sequences may be found in \cite{EverestVanDerPoortenShparlinskiWard2003,Stanley2012}.
2776 
2777 \subsubsection{Companion jumps}
2778 
2779 Let \(R\) be a commutative ring. Let
2780 \[
2781 v_{n+1}=Cv_n,
2782 \qquad
2783 C\in M_d(R).
2784 \]
2785 For a positive integer \(N\) and an address \(j\in\{0,\ldots,N-1\}\), define the residue state packet
2786 \[
2787 u_k^{(j)}=v_{Nk+j}.
2788 \]
2789 
2790 \begin{histtheorem}[Companion jump recurrence]\label{hist:thm:companion_jump}
2791 Every residue state packet satisfies
2792 \[
2793 \boxed{
2794 u_{k+1}^{(j)}=C^Nu_k^{(j)}.}
2795 \]
2796 More explicitly,
2797 \[
2798 \boxed{
2799 u_k^{(j)}=C^j(C^N)^kv_0.}
2800 \]
2801 \end{histtheorem}
2802 
2803 \begin{proof}
2804 The state relation gives \(v_n=C^nv_0\). Substitute \(n=Nk+j\).
2805 \end{proof}
2806 
2807 \begin{figure}[t]
2808 \centering
2809 \includegraphics[width=.95\linewidth,height=3.15in,keepaspectratio]{../../figures/reconstructed/recurrence_packets.pdf}
2810 \caption{New reconstruction. Fibonacci residues $F_{3k+j}$, with $F_0=0$ and $F_1=1$. Each packet evolves by the same jump matrix \(C^3\) and has a different initial address.}
2811 \label{hist:fig:recurrence_packets}
2812 \end{figure}
2813 
2814 We use the resultant convention
2815 \[
2816 \Res_y(f,g)
2817 =
2818 \prod_{f(\alpha)=0}g(\alpha)
2819 \]
2820 when \(f\) is monic, with roots counted with multiplicity.
2821 
2822 \begin{histtheorem}[Characteristic polynomial of the jump]\label{hist:thm:jump_characteristic}
2823 Let
2824 \[
2825 \chi_C(y)=\det(yI-C).
2826 \]
2827 Then
2828 \[
2829 \boxed{
2830 \chi_{C^N}(x)
2831 =
2832 \Res_y\bigl(\chi_C(y),x-y^N\bigr).
2833 }
2834 \]
2835 The identity holds over the coefficient ring after interpreting the resultant universally.
2836 \end{histtheorem}
2837 
2838 \begin{proof}
2839 After passage to an algebraic closure, let \(\alpha_1,\ldots,\alpha_d\) be the eigenvalues of \(C\), counted with algebraic multiplicity. The resultant is
2840 \[
2841 \prod_{i=1}^{d}(x-\alpha_i^N),
2842 \]
2843 which is the characteristic polynomial of \(C^N\). Both sides are polynomial expressions with integer coefficients in the entries of \(C\), so the identity descends to every commutative coefficient ring.
2844 \end{proof}
2845 
2846 \begin{figure}[t]
2847 \centering
2848 \includegraphics[width=.95\linewidth,height=3.15in,keepaspectratio]{../../figures/reconstructed/recurrence_root_power.pdf}
2849 \caption{New reconstruction. Passing to every \(N\)th term raises each characteristic root to its \(N\)th power.}
2850 \label{hist:fig:recurrence_root_power}
2851 \end{figure}
2852 
2853 \subsubsection{Second order recurrences}
2854 
2855 \begin{histtheorem}[Residue subsequences of a second order recurrence]\label{hist:thm:second_order_jump}
2856 Suppose
2857 \[
2858 x_{n+2}=Px_{n+1}-Qx_n
2859 \]
2860 over a commutative ring. Define \(V_0=2\), \(V_1=P\), and
2861 \[
2862 V_{n+2}=PV_{n+1}-QV_n.
2863 \]
2864 For every positive integer \(N\) and every address \(j\), the subsequence
2865 \[
2866 u_k=x_{Nk+j}
2867 \]
2868 satisfies
2869 \[
2870 \boxed{
2871 u_{k+2}=V_Nu_{k+1}-Q^Nu_k.}
2872 \]
2873 \end{histtheorem}
2874 
2875 \begin{proof}
2876 The companion matrix
2877 \[
2878 C=
2879 \begin{pmatrix}
2880 P&-Q\\
2881 1&0
2882 \end{pmatrix}
2883 \]
2884 has trace \(P\) and determinant \(Q\). The trace of \(C^N\) is \(V_N\), while its determinant is \(Q^N\). Cayley Hamilton applied to \(C^N\) gives
2885 \[
2886 (C^N)^2-V_NC^N+Q^NI=0.
2887 \]
2888 Apply this identity to the residue state packet and take its scalar coordinate.
2889 \end{proof}
2890 
2891 \begin{histcorollary}[Fibonacci residue packets]\label{hist:cor:fibonacci_packets}
2892 Let \(F_0=0\), \(F_1=1\), and \(F_{n+2}=F_{n+1}+F_n\). Let \(L_N\) denote the Lucas number. For every \(N\geq1\) and every address \(j\),
2893 \[
2894 \boxed{
2895 F_{N(k+2)+j}
2896 =
2897 L_NF_{N(k+1)+j}
2898 -(-1)^NF_{Nk+j}.
2899 }
2900 \]
2901 \end{histcorollary}
2902 
2903 \begin{proof}
2904 Apply \cref{hist:thm:second_order_jump} with \(P=1\) and \(Q=-1\). Then \(V_N=L_N\).
2905 \end{proof}
2906 
2907 \subsubsection{Finite and local periodicity}
2908 
2909 \begin{histtheorem}[Periodicity of an invertible recurrence modulo a prime power]\label{hist:thm:recurrence_periodicity}
2910 Let
2911 \[
2912 C\in M_d(\ZZ),
2913 \qquad
2914 p\nmid\det C.
2915 \]
2916 For every \(L\geq1\), every state orbit modulo \(p^L\) is purely periodic. Its period divides the order of \(C\) in
2917 \[
2918 \operatorname{GL}_d(\ZZ/p^L\ZZ).
2919 \]
2920 If \(p\mid\det C\), every orbit is still eventually periodic.
2921 \end{histtheorem}
2922 
2923 \begin{proof}
2924 When \(p\nmid\det C\), multiplication by \(C\) is a permutation of the finite state set \((\ZZ/p^L\ZZ)^d\). Every orbit of a permutation is periodic, and applying the group order relation for \(C\) gives the divisibility statement. Without invertibility, repeated application of a map on a finite set eventually repeats a state, after which the orbit is periodic.
2925 \end{proof}
2926 
2927 \begin{histcorollary}[Residue packet periods]\label{hist:cor:residue_packet_periods}
2928 Under the invertibility hypothesis of \cref{hist:thm:recurrence_periodicity}, every residue packet with jump \(N\) has period dividing the order of \(C^N\) modulo \(p^L\).
2929 \end{histcorollary}
2930 
2931 \begin{proof}
2932 Use the jump matrix from \cref{hist:thm:companion_jump}.
2933 \end{proof}
2934 
2935 \subsubsection{The common integral spine}
2936 
2937 \begin{histtheorem}[Base change of an integral recurrence]\label{hist:thm:integral_recurrence_spine}
2938 Let
2939 \[
2940 C\in M_d(\ZZ),
2941 \qquad
2942 v_0\in\ZZ^d.
2943 \]
2944 The integral orbit
2945 \[
2946 v_n=C^nv_0
2947 \]
2948 determines, by base change, compatible orbits in every ring \(R\) equipped with a homomorphism \(\ZZ\to R\). In particular, the reductions modulo every positive integer and the orbits in every \(\Zp^d\) share the same companion matrix, characteristic polynomial, and integer recurrence relations.
2949 \end{histtheorem}
2950 
2951 \begin{proof}
2952 Every coordinate of \(C^nv_0\) is an integer polynomial in the entries of \(C\) and \(v_0\). Applying a ring homomorphism commutes with matrix multiplication and preserves every polynomial relation with integer coefficients.
2953 \end{proof}
2954 
2955 \begin{histcorollary}[Prime independent recurrence spine]\label{hist:cor:prime_independent_spine}
2956 An integral companion system gives a canonical common source for all of its local realizations. No choice of coefficient prime is required to define the recurrence or its residue packets.
2957 \end{histcorollary}
2958 
2959 \begin{proof}
2960 Apply \cref{hist:thm:integral_recurrence_spine} to the companion matrix and its initial state.
2961 \end{proof}
2962 
2963 \begin{histremark}\label{hist:rem:spine_boundary}
2964 This theorem supplies a prime independent spine for integral linear recurrences. It does not prove that every unrelated family of local arithmetic packets arises from one global object.
2965 \end{histremark}
2966 
2967 \subsection{Chinese remainder packet refinement}\label{hist:sec:crt}
2968 
2969 The Chinese remainder theorem gives an exact factorization of residue addresses. We state it at the level of sequence projectors so that it can be combined directly with the earlier coordinate systems.
2970 
2971 \begin{histdefinition}[Residue projector]\label{hist:def:residue_projector}
2972 For a sequence \(a=(a_n)_{n\geq0}\), a positive integer \(N\), and an address \(r\) modulo \(N\), define
2973 \[
2974 (P_{N,r}a)_n
2975 =
2976 \begin{cases}
2977 a_n,&n\equiv r\pmod N,\\
2978 0,&n\not\equiv r\pmod N.
2979 \end{cases}
2980 \]
2981 \end{histdefinition}
2982 
2983 \begin{histproposition}[Complete residue packet system]\label{hist:prop:residue_packet_system}
2984 For fixed \(N\), the projectors \(P_{N,r}\) satisfy
2985 \[
2986 P_{N,r}P_{N,s}
2987 =
2988 \delta_{rs}P_{N,r}
2989 \]
2990 and
2991 \[
2992 \boxed{
2993 \sum_{r\bmod N}P_{N,r}=I.
2994 }
2995 \]
2996 \end{histproposition}
2997 
2998 \begin{proof}
2999 Every nonnegative integer belongs to exactly one residue class modulo \(N\).
3000 \end{proof}
3001 
3002 \begin{histtheorem}[Coprime packet factorization]\label{hist:thm:crt_packet_factorization}
3003 Let \(A\) and \(B\) be coprime positive integers. Let \(r_A\) be an address modulo \(A\) and \(r_B\) an address modulo \(B\). Let \(r\) be the unique address modulo \(AB\) satisfying
3004 \[
3005 r\equiv r_A\pmod A,
3006 \qquad
3007 r\equiv r_B\pmod B.
3008 \]
3009 Then
3010 \[
3011 \boxed{
3012 P_{AB,r}
3013 =
3014 P_{A,r_A}P_{B,r_B}
3015 =
3016 P_{B,r_B}P_{A,r_A}.
3017 }
3018 \]
3019 \end{histtheorem}
3020 
3021 \begin{proof}
3022 The product retains exactly the indices satisfying both congruences. By the Chinese remainder theorem, those indices form one residue class modulo \(AB\) \cite{IrelandRosen1990}.
3023 \end{proof}
3024 
3025 \begin{histcorollary}[Several coprime moduli]\label{hist:cor:many_crt_packets}
3026 Let \(N_1,\ldots,N_s\) be pairwise coprime and put
3027 \[
3028 N=\prod_{j=1}^{s}N_j.
3029 \]
3030 For compatible addresses \(r_j\),
3031 \[
3032 \boxed{
3033 P_{N,r}
3034 =
3035 \prod_{j=1}^{s}P_{N_j,r_j},
3036 }
3037 \]
3038 where \(r\) is the unique combined address modulo \(N\).
3039 \end{histcorollary}
3040 
3041 \begin{proof}
3042 Iterate \cref{hist:thm:crt_packet_factorization}.
3043 \end{proof}
3044 
3045 \begin{figure}[t]
3046 \centering
3047 \includegraphics[width=.95\linewidth,height=3.15in,keepaspectratio]{../../figures/reconstructed/crt_packets.pdf}
3048 \caption{New reconstruction. A residue modulo a product of coprime moduli is the exact intersection of its local residue packets.}
3049 \label{hist:fig:crt_packets}
3050 \end{figure}
3051 
3052 \begin{histstandardfact}[Finite local reconstruction]\label{hist:fact:finite_local_reconstruction}
3053 Let
3054 \[
3055 N=\prod_{j=1}^{s}p_j^{a_j}
3056 \]
3057 be the prime power factorization of a positive integer. The map
3058 \[
3059 \ZZ/N\ZZ
3060 \longrightarrow
3061 \prod_{j=1}^{s}\ZZ/p_j^{a_j}\ZZ
3062 \]
3063 is a ring isomorphism. Thus a compatible collection of prime power residues reconstructs one residue modulo \(N\) \cite{IrelandRosen1990}.
3064 \end{histstandardfact}
3065 
3066 \begin{histproposition}[Dyadic and odd residue refinement]\label{hist:prop:dyadic_odd_refinement}
3067 Let \(N=2^am\) with \(m\) odd. Every residue packet modulo \(N\) factors uniquely as the product of one dyadic packet modulo \(2^a\) and one odd packet modulo \(m\).
3068 \end{histproposition}
3069 
3070 \begin{proof}
3071 The moduli \(2^a\) and \(m\) are coprime. Apply \cref{hist:thm:crt_packet_factorization}.
3072 \end{proof}
3073 
3074 \begin{histremark}\label{hist:rem:crt_weight_boundary}
3075 The residue address factors exactly, but valuation weights do not become identical across primes. The odd coordinate system of \cref{hist:sec:local_euler} and the dyadic coordinate system of \cref{hist:sec:local_euler} remain genuinely different integral charts.
3076 \end{histremark}
3077 
3078 \subsection{Verification and exact boundaries}\label{hist:sec:boundaries}
3079 
3080 The proofs in the article are symbolic. Exact computation tests formulas, finite ranges, and the source implementation. It does not replace a uniform proof.
3081 
3082 \begin{table}[t]
3083 \centering
3084 \caption{Status of the principal local arithmetic blocks.}
3085 \label{hist:tab:status}
3086 \begin{tabular}{>{\raggedright\arraybackslash}p{.33\textwidth}>{\raggedright\arraybackslash}p{.24\textwidth}>{\raggedright\arraybackslash}p{.31\textwidth}}
3087 \toprule
3088 Block & Status & Main boundary\\
3089 \midrule
3090 Standard window period & Proved modulo the residue prime & Higher prime powers require inverse estimates\\
3091 Unified local Euler transfer & Proved for every prime & Odd and dyadic integral charts remain different\\
3092 Dyadic kernel and truncation & Exact weight and optimal uniform modulus proved & Row cancellation may give stronger values\\
3093 Carmichael and Mahler transfer & Proved for scalar, polynomial exponential, matrix, and finite adelic systems & A jump congruence must first be available\\
3094 Local Euler interpolation & Proved for depleted odd branches and the direct dyadic branch & The odd nonunit factor is only evaluated at integer interpolation points\\
3095 Affine polyphase transfer & Exact analysis, synthesis, carry law, kernel composition, residual sparsity, and valuation preservation proved & General exact kernel valuations remain open\\
3096 Lift obstruction and recurrence spine & Proved for the declared local problems and integral recurrences & Unrestricted global gluing is not proved\\
3097 \bottomrule
3098 \end{tabular}
3099 \end{table}
3100 
3101 \subsubsection{The remaining congruential difference problem}
3102 
3103 Let \(C_{mp,j}(n)\) be a congruential reciprocal coefficient and let
3104 \[
3105 h=\frac{p-1}{m}.
3106 \]
3107 The unit mode calculations suggest bounds for
3108 \[
3109 \vp\left(\Delta_h^sC_{mp,j}(n)\right).
3110 \]
3111 The theorems of \cref{hist:sec:mahler,hist:sec:polyphase} determine the gain before the congruential triangular inverse is applied. They do not yet determine every valuation loss introduced by that inverse. The full family bound therefore remains open.
3112 
3113 \subsubsection{Higher valuations of the affine kernel}
3114 
3115 The residual formula
3116 \[
3117 \Theta_{p^a,s}(r,k)
3118 \equiv
3119 \binom{s}{r-p^ak}
3120 \pmod p
3121 \]
3122 is exact. A closed formula for
3123 \[
3124 \vp\left(\Theta_{p^a,s}(r,k)\right)
3125 \]
3126 in terms of base \(p\) carries and the three digit addresses is not proved here. Such a formula would sharpen the min plus valuation ledger and could distinguish optimal refinement trees.
3127 
3128 \subsubsection{The remaining global gluing problem}
3129 
3130 The finite adelic theorem synchronizes a finite set of primes when one integral state system is already fixed. The recurrence theorem gives a common local source under the same assumption. Chinese remainder projectors reconstruct finite residue addresses. None of these results proves that an arbitrary collection of local analytic branches has a common global arithmetic origin.
3131 
3132 \begin{figure}[t]
3133 \centering
3134 \includegraphics[width=.95\linewidth,height=3.15in,keepaspectratio]{../../figures/reconstructed/claim_boundary.pdf}
3135 \caption{New reconstruction. The proved local coordinate theory and the remaining inverse valuation, affine kernel, and global gluing problems.}
3136 \label{hist:fig:claim_boundary}
3137 \end{figure}
3138 
3139 \subsubsection{Historical scope}
3140 
3141 The article uses classical work on Euler congruences, finite differences, local fields, Hurwitz series, Mahler interpolation, Lucas congruences, and recurrence sequences \cite{CarlitzLevine1960,Young1999,Young2001,Sun2008,Chen2004,Lucas1878a,LidlNiederreiter1997,EverestVanDerPoortenShparlinskiWard2003}. The exact formulas and local transfer statements should be compared individually with that literature before any historical priority claim is made.
3142 
3143 \subsection{Conclusion}\label{hist:sec:conclusion}
3144 
3145 The local arithmetic theory now has one common architecture. The finite character coordinate identifies the torsion phase. The source and kernel product reconstructs the Euler coefficient. The kernel valuation records certified precision. Carmichael and Mahler coordinates describe the remaining principal unit motion. Polyphase analysis resolves additional local digits, records carries and shifts exactly, and transports Mahler coefficients through a finite affine kernel.
3146 
3147 The odd and dyadic cases agree at the level of convolution and differ at the level of integral geometry. At odd primes the source must be depleted before it becomes a unit mode interpolation. At the dyadic prime the source extends directly to an analytic integral function. The dyadic kernel has exact weight \(4r-1\), giving the optimal uniform modulus \(2^{4M+3}\) after depth \(M\).
3148 
3149 Three tasks remain outside the proved region. The first is a complete finite difference estimate after the congruential triangular inverse. The second is an exact higher valuation theory for the affine Mahler kernel. The third is a global existence theorem for local systems that do not already arise from one integral source.
3150 
3151 \par\smallskip\noindent\textbf{Current status.} This dated statement is superseded for the stated divisor-branch family by Theorem~\ref{thm:arith-divisor-base}, with the $j!$ normalization and factorial valuation loss explicitly retained. It makes no claim for unrestricted families.\par
3152 
3153 

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