Nishibuchi's Conjecture 3.9 concerns a prime \(p\), a positive integer \(m\) with \(m\mid p-1\) or \(m=2\), and every address \(0\le j<mp\). Put
It asserts \(p\)-adic integrality of the actual generalized Euler coefficients and eventual periodicity at every precision:
The completed theorem gives the explicit sufficient threshold \(n\ge r\) in every permitted branch. Its formal capstone is the theorem named NishibuchiCompletion.conjecture_3_9. No conjectural flatness premise is supplied to that theorem.
| Prime branch | Allowed \(m\) | \(q\) | Row step for the original shift |
|---|---|---|---|
| \(p=2\) | \(1\) | \(2\) | \(2^r\) |
| \(p=2\) | \(2\) | \(2\) | \(2^{r-1}\) |
| \(p\) odd | \(m\mid p-1\) | \(p-1\) | \(\dfrac{p-1}{m}p^{r-1}\) |
For odd primes, \(m=2\) is already in the divisor branch. The conversion is exact:
At \(p=2,m=2\), the original conjectured Euler-degree shift \(2^{r+1}\) is the row step \(2^{r-1}\) for the rank-four sequence.
The source already contains special cases. In particular, Nishibuchi's Proposition 3.10 treats the rank-four address-zero dyadic period from row zero and an eventual cubic case. The completion covers the full quantified conjecture; the new dyadic work then removes starting-row restrictions from higher differences and determines substantially more precise laws.
The conjecture's primary source is Yuta Nishibuchi, \(p\)-adic Congruences of Generalized Euler Numbers and Relations to Even Zeta Values, arXiv:2605.10677v2, revised 1 June 2026. The statement above uses that version's normalization and actual Euler degrees.