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

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1 \section*{What this final edition establishes}
2 \addcontentsline{toc}{section}{What this final edition establishes}
3 The original external problem is Nishibuchi's Conjecture 3.9 for generalized Euler numbers. Its full parameter range is resolved. If $p$ is prime, $m$ is positive with $m\mid p-1$ or $m=2$, $0\le j<mp$, and $q=\operatorname{lcm}(2,p-1)$, then the original Euler coefficients are $p$-adically integral and
4 \[
5  E_{mpn+qp^r}^{(mp,j)}\equiv E_{mpn}^{(mp,j)}\pmod{p^r}
6  \qquad (r\ge1,\ n\ge r).
7 \]
8 The paper states the generating function and the conversion between ordinary degree and packet row before using this formula. The normalized coefficient is $C_{N,j}(n)$ and the original coefficient is $E_{Nn}^{(N,j)}=j!C_{N,j}(n)$. The constant coefficient of the original series is $j!$.
9 
10 The strongest common-step conclusion in width four is particularly compact. For $n\ge0$ and $k,H\ge1$,
11 \begin{equation*}
12  \sum_{i=0}^k(-1)^{k-i}\binom ki E_{4(n+iH)}^{(4,j)}
13  \equiv0\pmod{2^{\,v_2(k!)+kv_2(H)+A_j(k)}},
14 \end{equation*}
15 where
16 \[
17  A_0(k)=A_1(k)=k,\qquad
18  A_2(k)=\max\{k+4,\,2k+\lfloor k/2\rfloor+1\},\qquad
19  A_3(k)=k+2.
20 \]
21 There is no starting-row threshold. The exponent is exact in addresses zero and one for every order and positive step. It is exact in address two at every power-of-two order. The proof chapters also give mixed-step forms, the cancellation criterion in addresses zero and one, and exact first-displacement laws.
22 
23 \begin{center}
24 \small
25 \begin{tabularx}{\linewidth}{@{}>{\raggedright\arraybackslash}p{.29\linewidth}>{\raggedright\arraybackslash}X>{\raggedright\arraybackslash}p{.21\linewidth}@{}}
26 \toprule
27 Result & Scope & Evidence in this edition\\
28 \midrule
29 Nishibuchi Conjecture 3.9 & Every parameter in the original statement; explicit $n\ge r$ & Inherited Lean proof\\
30 Width-four all-row common-step bound, first differences, least periods and isometry & Every address, row and positive displacement & Inherited Lean proof\\
31 General prime-power inverse transfer and mixed steps & Every prime-power width; signed steps for odd primes & Written proof and audit\\
32 Factorial gains and exact higher-order families & Four width-four addresses, with stated sharpness scopes & Written proof and audit\\
33 Taylor radii and broader dyadic first differences & Exact radii in addresses $0,1,2$; stated all-rank cases & Written proof and audit\\
34 Ordinary Euler tail simplicity & Partial families and an explicit remaining discriminant problem & Open in full generality\\
35 \bottomrule
36 \end{tabularx}
37 \end{center}
38 
39 The complete earlier Paper IX development follows alongside the strengthening. Dated appendices retain the original continuation arguments and recovered historical text. Statements about an earlier open target or an earlier build remain records of that checkpoint; the current theorem and scope statements govern this edition. The Lean companion supplies the exact formal source map and distinguishes source reconciliation from compiler execution. The visual guide supplies exact finite examples and theorem-derived diagrams.
40 
41 This is a project publication under the name Komorebi P. The recorded parallel proof audits are internal computational-assistant reviews. They are not external peer review. The edition makes no claim of journal acceptance, external certification, or global historical priority.
42 \clearpage

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