Automatic Unit Towers for the Romik Hypergeometric Quotient: A Bockstein–Cartier Lift Modulo 64 and Digitwise Zeta Conjugacy

This paper develops a higher-precision automatic structure for the Romik hypergeometric quotient. Building on the previously established modulo-32 unit lift, it determines the next dyadic layer completely and proves an explicit modulo-64 unit law valid for every coefficient index. The proof is based on a cross-precision Bockstein–Cartier construction. A precision-aware Cartier module modulo 64 is projected onto the corrected modulo-32 module, reducing the new obstruction to a finite Bockstein extension. The resulting residual sequence is represented by a minimal 16-state automaton, while its Boolean submask correction has a minimal 30-state automaton and an exact 18-element binary kernel. These finite structures are certified by a reproducible source-level computation rather than inferred from a finite coefficient sample. The lifting mechanism is then extended beyond modulo 64. For every fixed power of two, the normalized unit sequence is shown to be effectively binary automatic, with the finite automata compatible under reduction of precision. A general digitwise zeta–Cartier conjugacy theorem explains why the Boolean correction preserves linear Cartier complexity. In the first nonperiodic lifting layer, both the residual and its correction have exact linear Cartier rank eight. The results provide a structural description of the dyadic automatic tower associated with the Romik hypergeometric quotient and isolate the nonlinear residual-formation step as the source of further complexity growth. 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.