Dyadic Rigidity in the Romik Hypergeometric Framework: Sparse Modulo-Eight Normal Forms and Exact Inverse-Matrix Periods

This paper develops a dyadic rigidity theory for a family of contiguous hypergeometric quotients associated with the Romik framework. It proves a universal Riccati normal form modulo eight and derives exact upper- and lower-parameter shift laws from hypergeometric contiguity relations. These shift laws are combined with sparse lacunary expansions to obtain an absolute normal form for the full six-parameter quotient. All parameter dependence is reduced to four elementary parity statistics, while the infinite part is carried by three universal sparse series. The paper also determines the exact minimal power-of-two period of every inverse-matrix slice studied by Krattenthaler and Müller. It shows that their proposed uniform period is valid for all higher slices but fails for the first slice, whose exact period is larger. The proof identifies the highest odd Mahler coefficient through a binary-mask construction. Further consequences include an odd Abel action and a dyadic Frobenius congruence in the Romik Schwarz coordinate. 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.