Frobenius Revivals in Laplacian Cellular Automata: Chaos, Replication, and Reversible Encoding

Abstract

We investigate Frobenius-driven revivals in prime-modulus Laplacian cellular automata, a phenomenon in which long chaotic transients collapse into exact, multi-tile replicas of an initial seed at algebraically prescribed times $t=p^m$. The mechanism follows directly from the Frobenius identity $(I+B)^{p^m}=I+B^{p^m}$, which eliminates all mixed binomial terms and enforces deterministic reappearance of the seed after dispersion. We provide a detailed numerical and analytical characterisation of these revivals across several moduli, examining entropy dynamics, spatial organisation, and local stability under perturbations. The revival structure yields several useful features: predictable transitions between chaotic and ordered phases, intrinsic spatial redundancy, and robust reconstruction via replica consensus in the presence of weak additive noise. We further show that composing Laplacian operators modulo multiple primes generates significantly extended periodic orbits while preserving exact reversibility. Building on these observations, we propose an explicit reversible encoding scheme based on chaotic transients and Frobenius returns, together with practical separation conditions and noise-tolerance estimates. Potential applications include reversible steganography, structured pseudorandomness, error-tolerant information representation, and procedural pattern synthesis. The results highlight an interplay between algebraic combinatorics and cellular-automaton dynamics, suggesting further avenues for theoretical and applied development.