Stabilization and Regaining Periodicity in Modular Laplacian Dynamics

Abstract

We study discrete Laplacians on two-dimensional lattices under modular iterations, focusing on the emergence of nontrivial large-scale patterns. While purely binary or constant modular sequences quickly collapse into strict periodicity, the insertion of a single non-binary step k yields qualitatively new behavior. Through extensive computer-assisted exploration we identify a taxonomy of long-lived figures - rugs, quasi-carpets, and carpets - whose occurrence depends systematically on seed symmetry, neighborhood mask, and sequence structure. In particular, we show that mixed families of the form [2,k,2 to power s] can stabilize high-density carpets beyond the universal decay time characteristic of binary dynamics. Our approach combines algebraic replication laws with large-scale simulations and density tracking, producing both theoretical conditions (periodicity via Lucas theorem, non-overlap criteria) and experimental evidence of persistent quasi-aperiodic architectures. The results highlight how minimal modifications in discrete local rules generate unexpectedly rich multiscale geometries, bridging rigorous analysis with computer-assisted discovery.