Binary-Submask Error Valuations and Modulo-32 Unit Lifts for the Romik Hypergeometric Quotient

This paper studies the two-adic arithmetic of a hypergeometric quotient arising from Romik’s expansion of the Jacobi theta constant. It proves an exact valuation formula for the difference between the hypergeometric logarithmic coefficients and their Catalan counterparts, and shows that the lowest nonvanishing layer is governed by binary submasks and carry transport. The valuation theorem is refined to a least-significant-bit law for the normalized coefficients and then to an all-index unit lift modulo thirty-two. The proof combines a Mahler comparison series, a Boolean-zeta correction, and a precision-aware Cartier representation. The only computer-assisted component is an explicitly isolated residual identity. It is certified by a complete finite-state computation whose transition system and dual orbit can be independently reconstructed and verified. As an application, the paper proves the base two-adic integrality conjecture for the Romik hypergeometric quotient and provides a reproducible certificate package for the computer-assisted theorem. 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.