Profinite Vanishing of Fixed Columns in Romik's Matrix: Conjecture 47(2) and the Pure-Periodicity Clause in Conjecture 47(1)

Abstract

This paper studies the fixed-column sequences in Romik’s matrix that appear in Conjecture 47 of Krattenthaler and Müller. It proves Conjecture 47(2) and establishes a stronger conclusion: every fixed-column sequence is eventually zero modulo every prescribed integer. Equivalently, each such sequence converges to zero in the profinite completion of the integers. For primes congruent to one modulo four, this result strengthens the eventual-periodicity assertion in Conjecture 47(1) to eventual vanishing, with eventual period one. It also shows that the accompanying pure-periodicity clause cannot hold from the initial term as stated, because each fixed column begins with a nonzero diagonal entry but eventually has a zero tail. The proof derives an explicit expansion for the fixed-column entries and combines prime-adic valuation estimates for the auxiliary Romik sequence with exact optimization over integer compositions. This yields effective vanishing thresholds for prime powers and, by combining the local estimates, for arbitrary moduli. The argument is theoretical and does not depend on finite computational verification. 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.