Geometric-Series Valuations and Frobenius Initial Forms for the Krattenthaler–Müller Sequence

Abstract

This paper resolves Conjecture 43 of Krattenthaler and Müller concerning the p-adic divisibility of the auxiliary integer sequence arising in Romik’s hypergeometric description of the Taylor coefficients of the Jacobi theta constant. A uniform valuation theorem is proved for every prime congruent to three modulo four and for every index. Beyond the conjecture itself, the paper determines the complete leading residue at the valuation bound through a truncated Frobenius polynomial. This yields a finite periodic classification of first-layer cancellations, exact minimum valuations on every natural block, a digit-sum description of the optimal block-aligned divisibility tails, and an all-level rank-one theorem for the boundary profiles observed by Krattenthaler and Müller. The original boundary normalization is shown to be sharp below the first transition, while the true first nonzero layer rises at and beyond that transition. The paper also proves the first experimental observation following Conjecture 43 and disproves the second by constructing an infinite family of counterexamples. The proofs are theoretical and do not depend on numerical computation. The accompanying verification files check the principal structural identities directly from the defining recurrence using exact arithmetic. Research methodology and AI assistance:This work was developed using the CARMA-Math research workflow, a cumulative AI-assisted mathematical research methodology using persistent research archives, literature and prior-art investigation, iterative proof exploration, and verification procedures. Generative AI (ChatGPT) was used extensively for mathematical exploration, proof development, computational reasoning, literature research, and manuscript preparation.