Sign Oscillation of Romik's Theta-Coefficient Sequence in Arithmetic Progressions

This paper establishes quantitative sign oscillation for Romik's integer sequence arising from the Taylor expansion of the modularly centered Jacobi theta constant. It proves that every fixed arithmetic progression in the sequence contains infinitely many positive terms and infinitely many negative terms, together with explicit quantitative lower bounds for the occurrence of each sign. The results also imply that, within every arithmetic progression, both sign reversals and sign agreements occur infinitely often, with quantitative lower bounds for their frequency. The proof combines rational cusp limits of the centered theta generating function with a three-cusp Hermitian obstruction, a residue-class quadratic mean obtained from Romik's Hermite representation and Mehler's formula, and a Tauberian argument converting Abel-weighted sign mass into counting estimates. The paper also determines the complete rational cusp spectrum of the centered generating function, proves that the generating functions associated with all arithmetic subsequences have the unit circle as a natural boundary, and deduces that these arithmetic subsequences are not P-recursive. These results resolve the sign-oscillation aspect of Romik's Problem 5 and strengthen it by establishing oscillation in every fixed arithmetic progression. 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.