Topology-Dependent Pattern Persistence in Noisy Reaction–Diffusion Systems: A Modified Decay Law and Falsifiable Prediction

Reaction–diffusion systems under additive noise are expected to exhibit exponential decay of pattern coherence and convergence to uniform states beyond a characteristic noise-limited time scale. Recent numerical studies, however, show persistent patterns in systems with structured diffusion/coupling at times exceeding this expectation. We propose a minimal phenomenological recurrence for local pattern coherence on the spatial lattice that reproduces standard exponential decay in unstructured systems but predicts delayed pattern decay and nonzero plateaus when repeated local motifs are present in the diffusion/coupling grid. The model is parameterized by a single topology-dependent memory strength α(μ) and yields closed-form expressions for the coherence metric r(d) with a calculable plateau height and build-up time scale. Matched-pair lattice simulations (identical grid size, reaction rates, and noise level, differing only in motif density) provide a direct falsification protocol. If no separation is observed, the model is ruled out; if plateaus emerge as predicted, it offers a new diagnostic for topology-induced memory in noisy reaction–diffusion dynamics.