Integer Convolution on Graphs: A Diophantine Mechanism Without Dynamics

Version note (Nov 2025): This version corresponds to the manuscript submitted to Entropy on 21 November 2025 but adds section on related work. Abstract — Classical models of phase locking and frequency entrainment rely on iterative dynamics, such as circle maps, driven oscillators, or coupled flows, to generate Devil’s staircases, Arnold tongues, and rational plateaus. This work shows that analogous signatures arise from a single, non-iterative integer convolution on a finite graph, providing a dynamics-free route to rational coherence. A binary mask convolved with a nonnegative integer kernel produces a normalized occupation field whose effective grain (Delta pₑff = m / Q, with total overlap mass Q and greatest common divisor m) determines which rational modes p/q are accessible and sets a finite capacity C = Q / m. We investigate this Integer Convolution Locking (ICL) mechanism in two modes: emergent locking, where a dominant spacing beta is inferred from spectra, and driven locking, where an external probe B is scanned. Both modes exhibit Devil’s staircases, Farey-organized plateaus, and coherence ridges obeying a Ridge Law dB/dD = (p/q) * d (Delta pₑff) /dD. Comparison with a random-overlap null ensemble shows that strong locking is not generic to finite integer fields. Random mask ensembles across grid sizes 10x10 to 50x50 display scale-stable, lognormal-like spectra, demonstrating robustness. ICL thus provides a general combinatorial mechanism for rational coherence on discrete graphs, with potential applications to distributed systems, discrete optimization, and symbolic dynamics.