An explicit coefficient-module proof of Conjecture 3.4(a) in Au's Wilf-Zeilberger seeds paper

Abstract

K. C. Au, Discovering hypergeometric series with harmonic numbers via Wilf-Zeilberger seeds, arXiv:2602.08721v3, conjectures in Conjecture 3.4(a) that the first summand S1(a,b,c,d,e;n) of Example II has coefficient spaces VN(S1) (the Q-span of the sequences n ↦ [ai1···ei5]S1 with total degree N) whose dimensions have generating function 1/((1−t)3(1−t2)2) − t2/(1−t). We prove this in every degree. Explicit weighted coordinates (h,k,z,r,s) of weights (1,1,1,2,2) identify the coefficient row space with the polynomials whose hN−2z2 coefficient vanishes on k=r=s=0. The proof combines Schneider's algebraic independence criterion for the harmonic sequences, three explicit generators with a 3×3 certificate of determinant 35, and a second-jet computation of the full rational prefactor showing that the z2 coefficient is absent. Claim boundary. The result concerns sequences before summation. It does not address Conjecture 3.4(b), the dimensions of the summed numerical spaces, or the independence of ζ(3)2 and π6. The note has not been independently refereed. Reproducibility. verify_exact.py checks 78 exact identities. verify_ranks.py and the separately written ii_raw_rank.py (which works directly in a,b,c,d,e) reproduce the predicted ranks 1, 3, 7, 15, 29, 49, 79, 119, 174, 244 over several prime fields; its two negative controls fail as intended. ii_identity.py confirms the transcription of the cubic prefactor against Au's Example II identity. Recorded outputs are included. 8 pages. Computational tools and a large language model were used for algebraic derivation, code generation, independent checking and manuscript preparation; no model output is used as mathematical authority.