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arXiv · 2604.00165

Symmetric Nonlinear Cellular Automata as Algebraic References for Rule 30

E. Chan-López·A. Martín-Ruiz
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Abstract

A comparative algebraic framework for elementary cellular automata is developed, centered on the role of spatial symmetry. The primary object of study is Rule 22, the elementary cellular automaton with algebraic normal form $g(a,b,c)=a\oplus b\oplus c\oplus abc$ over $\mathrm{F}_2$, the simplest rule combining full $S_3$ symmetry with genuine nonlinearity. It is proved that the single-seed orbit of Rule 22 admits the closed form $Q_{2j}(x)=(x^2+x^{-2})^j$ and $Q_{2j+1}(x)=(x^{-1}+1+x)(x^2+x^{-2})^j$ for the row generating polynomials over $\mathrm{F}_2$. Three consequences follow: an exact support-set cardinality $|S_m|=2^{\mathrm{popcount}(\lfloor m/2\rfloor)}\cdot 3^{m\bmod 2}$, a two-step recursive construction of the support sets, and, in a formal continuum approximation, a parabolic reaction--diffusion equation $\partial_m u=u_{xx}+2u+u^3$. Rule~22 is then used as a symmetric reference for Rule 30. The symmetry-breaking deviation $\epsilon(m)=|S_m^{(30)}|-|S_m^{(22)}|$ between full-row support cardinalities is empirically consistent with a power law $m^b$ ($b\approx 1.11$), and is non-positive exactly at the Mersenne indices $m=2^k-1$. A mechanism for the apparent randomness of Rule~30's center column is identified through the left-permutive structure and asymmetric Boolean sensitivity profile.

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BibTeXRIS

E. Chan-López, A. Martín-Ruiz. 2026-03-31. Symmetric Nonlinear Cellular Automata as Algebraic References for Rule 30. https://arxiv.org/abs/2604.00165

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