Cyclotomic Block Involutions and One-Layer-to-Tower Lifting for a Guo–Schlosser q-Supercongruence

Abstract

This preprint studies a q-supercongruence conjecture of Guo and Schlosser concerning fourth-power cyclotomic divisibility of truncated basic hypergeometric sums. The conjecture as originally stated is shown to fail in an admissible even-parameter case. The paper identifies the source of this failure and proves a stronger parity-free theorem on the natural q-squared cyclotomic side. The result holds simultaneously at every compatible divisor level and, when specialized to power towers, yields the full family of q-squared fourth-power congruences for all levels. For odd base parameters, this recovers the corrected form of the original q-supercongruence conjecture and consequently the associated classical supercongruence. The proof introduces a cyclotomic block involution together with a full-root-cycle Fourier cancellation. A reciprocal symmetry eliminates the only potentially surviving Fourier coefficient, while a formal gauge converts the block involution into an odd local symmetry. Combining this structure with a known one-layer congruence lifts fourth-order divisibility from a single layer to every shifted block in the tower. The accompanying computational supplement contains exact verification code and reproducible checks for representative even and boundary cases. The computations serve as independent regression certificates and are not used as substitutes for the general symbolic proof. 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.