A sharp tail lemma, a valuation census, and a collapse of the remaining obligation for the eta(2tau)^12 eighth-power supercongruence

Abstract

The conjecture. For the newform η(2τ)12 = ∑ c(n)qn (weight 6, level 4, LMFDB 4.6.a.a) and tk = binom(2k,k)/22k, @multiple_zeta conjectured (10 July 2023) that for every prime p ≠ 2, 5, Σp = ∑k<p(4k+1)tk8 ≡ p·c(p) (mod p6). It is open; the proved uniform record is the same congruence modulo p4. This note contributes three things. 1. A proved tail lemma, sharp. For every prime p ≥ 5 and every m ≥ 1, ∑k=(p+1)/2p−1(4k+1)tk2m ≡ 0 (mod p2m+1), one power beyond the trivial bound, with 2m+1 optimal. The proof is an exact local expansion t(p+1)/2+j ≡ (−1)(p−1)/2 p sj (mod p2), where sj = 4j(j!)2/(2j+1)!, together with the reflection sN−1−j ≡ (−1)(p+1)/2 sj (mod p), which makes the tail sum antisymmetric. For m = 4 it improves the reduction Σp ≡ Hp from modulus p8, used throughout the existing manuscripts, to modulus p9. 2. A valuation census, and no modulus-p7 refinement. For all 17981 primes 7 ≤ p < 200000 the valuation vp(Σp − p c(p)) equals 6 exactly, with three exceptions (p = 37, 811, 119921, valuation 7) — a count matching the equidistribution heuristic ∑7≤p<200000 1/p = 1.7303. An LLL search over eleven candidate functions and sixty primes finds no modulus-p7 refinement: the reduced solution lattice's shortest vector has entries of size about 1010, the size predicted for no structure, so a Bernoulli correction, a Fermat quotient cube, and the unit root of X2 − c(p)X + p5 all fail. p6 is the true endpoint, not a search-depth artefact. 3. A collapse of the current reduction. The 7 September 2026 handoff reduces the conjecture to Rp ≡ c(p) (mod p5), where Rp = αp2Lp(Bp2) − p3αpβpLp(BpUp) + p4αpβpLp(BpVp) involves the truncated Hasse lift Ap through αp = Ap(0) and βp = t(p−1)/24, plus two boundary-correction series U, V. Every exact and congruence step of that reduction replays here. It then collapses: Lp(Ap2) ≡ Lp(Bp2) (mod p5) for every prime 7 ≤ p < 140, even though Ap − αpBp has coefficients of valuation exactly 3, two digits inside the target precision. The mechanism is the closed form Lp(BpUp) ≡ (7/3) Bp−3 Lp(Bp2) (mod p), which cancels the p3 layer against the Wang and Hu value (αp − 1)/p3 ≡ (7/6)Bp−3. The remaining obligation, restated. Granting the collapse, the conjecture is equivalent to the single universal congruence, for every prime p ≥ 7, c(p) ≡ p ∫01 ( [ 2F1(½,½;1;z)2 ]<p )2 dz (mod p5), in which Ap, αp, βp, U and V do not appear. Equivalently c(p) ≡ [zp−1]F4 + p ∑j≠p−1 [zj]([F2]<p)2/(j+1). This is a Dwork and Beukers type comparison between one truncated period and one modular coefficient at precision p5, rather than a comparison involving a p-dependent Hasse lift. Also recorded. The excluded prime p = 5 misses by exactly one digit: Σ5 − 5c(5) = −55·6202146665316339 / 256, so v5 = 5 exactly and Σ5 ≡ 5c(5) + 55 (mod 56); p = 3 satisfies the congruence. The earlier R7 modulus-p4 manuscript replays, its isolated 1/(pj) step is notational rather than a gap, and its second finite-part theorem is measured to hold modulo p2 (one digit deeper than stated) for all primes p ≤ 167, which upgrades R7 (6.9) from p3 to p4 at no cost. The R8 Cauchy residue bridge is confirmed and sharp. Claim boundary. Proved: the tail lemma and the exact size of the p = 5 failure. The census, the lattice obstruction, the collapse and the two audit upgrades are measured, not proved. The conjecture remains open and the proved uniform record remains modulus p4. The attached fail-closed verifier replays all twenty checks in about thirteen seconds using only the standard library plus sympy. Version 2.0 (7 September 2026). Revised after the author's audited consolidation Nader_Eta12_Author_Package_2026-09-07 became available. Two changes of substance. (a) The record is corrected downward. Version 1.0 quoted the proved uniform record as modulus p4, following the R7 packet. The audited edition does not carry that claim: it proves modulus p3 for every odd prime, and its section 5.2 shows that modulus p4 is only equivalent to the unproved period identity p∫01Bp(z)2dz ≡ c(p) (mod p3), where Bp = [2F1(½,½;1;z)2]<p. This edition quotes modulus p3 throughout. (b) The collapse is restated as a two-digit upgrade of that edition's own bridge. The audited note proves Sp/p ≡ p∫01Bp2 (mod p3), obtained from the coefficientwise congruence Ap ≡ Bp (mod p3), which cannot give more. Measured here, the same bridge holds modulo p5, for all 427 primes 7 ≤ p < 3000 in an independent modular implementation (valuation exactly 5 except p = 71), because the αp rescaling and the two boundary layers cancel one another after integration rather than vanishing individually. Consequently a single identity governs both records: writing (Mr) for p∫01Bp2 ≡ c(p) (mod pr), the audited edition proves (M3) ⇔ the conjecture mod p4, and this note adds (M5) ⇔ the conjecture at the full modulus p6. The layer decomposition Θ0 + 2p2Θ2 + p4Θ4 is not needed. The verifier now carries 21 checks including the independent modular re-derivation, and runs in about 14 seconds. The proved tail lemma, the 200000-prime valuation census, the LLL obstruction to any modulus-p7 refinement and the p = 5 quantification are unchanged from version 1.0.