Long-Lived Carpet-Like Transients in Time-Dependent Modular Discrete Laplacian Dynamics

Abstract

Binary modular Laplacian dynamics is organized by repeated replication, growth, and collapse on dyadic scales. We study how isolated and periodically repeated non-binary updates modify this organization and whether they can sustain densely occupied geometric structures. Computational experiments across multiple finite seeds, neighborhood masks, inserted moduli, and periodic schedules show that constant prime moduli retain the expected \(p\)-adic replication hierarchy. A single non-binary insertion mainly acts as an effective-seed replacement: subsequent evolution remains binary-like, but may resume at an integer phase shift of the collapse--recovery cycle, with only modest finite-time densification. A qualitatively different response occurs for periodic schedules \([2,k,2^s]^\infty\), where \(2^s\) denotes \(s\) consecutive binary updates. Odd-modulus insertions can sustain long-lived, spatially coherent carpet-like regimes with comparatively high and stable occupation, whereas even-modulus insertions remain binary-like. Ternary insertion gives the broadest and most robust high-density response. For \(k=3\), density depends non-monotonically on the binary-tail length, with troughs near \(s=7,15,23\) that reveal phase-sensitive interaction with the binary epoch structure. Cross-modulus synchronization also occurs: for degree-four diagonal Neumann and von Neumann masks, \(k=5,7,9\) repeatedly collapse onto the same complete configuration at subsequent binary phases. A parity argument explains this loss of memory. Increasing seed extent generally raises occupation while reducing tail-length sensitivity. Overall, periodic modular insertions act as phase-selective, geometry-dependent mechanisms transforming \(p\)-adically organized replication into long-lived carpet-like dynamics.