Higher Dyadic Residual Ranks for the Romik Hypergeometric Quotient

Abstract

This paper determines the eighth and ninth dyadic residual layers arising from the Romik hypergeometric quotient and gives exact finite-state and Cartier-module descriptions valid for every index. The minimal Cartier dimensions of the eighth and ninth residual sequences are proved to be 33 and 67. Their exact least-significant-digit kernels have 462 and 18,002 states, while the corresponding minimal most-significant-digit automata have 298 and 10,362 states. Exact dyadic counting formulas are obtained, with limiting densities one quarter and three quarters. The paper also analyzes the interaction among the higher residual, secondary quotient, and carry modules. Although the three component modules have dimensions 33, 114, and 131, their synchronized cyclic module has dimension 159. This yields a synchronization defect of 119, together with a largest common cyclic quotient of dimension 94 between the secondary quotient and the carry module. The all-index congruence statements are certified by finite dual Cartier-invariant modules over the relevant dyadic residue rings. The accompanying reproducibility package independently verifies the residual identities, Cartier dimensions, automata, counting laws, synchronization ranks, and finite certificates using exact integer and finite-field computations. 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.