A finite harmonic closure and assembled proposed proof of the eta(2tau)^12 eighth-power supercongruence

Abstract

Start here: The finite harmonic congruence proof (PDF) contains the closing congruence T + H[N]S₃ ≡ 2U (mod p), for primes p > 5. The accompanying one-page, three-line reading guide (PDF) explains its last three steps in plain language with selectable symbols. For the assembled eta(2τ)¹² supercongruence statement and the earlier proof dependencies, read the assembled supercongruence proof (PDF). This v4 working-paper release presents a complete proposed proof of the eta(2tau)^12 supercongruence modulo p^6 conjectured in the final unnumbered statement of the Appendix of Kam Cheong Au, arXiv:2509.19960v3. The target is exactly Au's conjecture: (1)_n=n!, (1/2)_n/n!=binom(2n,n)/4^n, and Au's a_p(f) is c(p) for f=eta(2tau)^12. The proof is the assembled R11--PS4 manuscript chain. A new standalone finite-harmonic argument independently closes the last harmonic identity. All supplied verifiers were replayed successfully, but computational checks do not replace the proofs and no external referee validation is claimed. This release concerns only Au's Appendix supercongruence; it does not reprove Au's main theorems, establish sharpness of dimension bounds, or replace the proof of equation (5.3). Three-line guide. The mixed-coefficient identity gives 2(T + H[N]S₃) ≡ V₂ + S₄ (mod p). Two evaluations of one finite sum give (p²/4)(V₂ + S₄) ≡ p²U (mod p³), hence V₂ + S₄ ≡ 4U (mod p). Substitution and division by 2 give T + H[N]S₃ ≡ 2U (mod p). Cancellation of p² reduces the modulus to p. The guide summarizes this final deduction; the underlying finite identities and the R11–PS3 supercongruence reduction remain part of the full proof dependencies.