Symmetric Nonlinear Cellular Automata as Algebraic References for Rule~30
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911647923175424 |
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| author | Chan-López, E. Martín-Ruiz, A. |
| author_facet | Chan-López, E. Martín-Ruiz, A. |
| contents | 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 $\mathbb{F}_2$, the simplest rule combining full $S_3$ symmetry with genuine nonlinearity. Three closed-form results are established: a formula for the 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 the continuous limit as 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 $ε(m)=|S_m^{(30)}|-|S_m^{(22)}|$ is empirically consistent with a power-law scaling of the form $m^b$ ($b\approx 1.11$), quantifying the cumulative effect of replacing the symmetric cubic $abc$ with the asymmetric quadratic $bc$. A mechanism for the apparent randomness of Rule~30's center column is identified through the left-permutive structure and asymmetric Boolean sensitivity profile. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_00165 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Symmetric Nonlinear Cellular Automata as Algebraic References for Rule~30 Chan-López, E. Martín-Ruiz, A. Mathematical Physics Cellular Automata and Lattice Gases 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 $\mathbb{F}_2$, the simplest rule combining full $S_3$ symmetry with genuine nonlinearity. Three closed-form results are established: a formula for the 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 the continuous limit as 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 $ε(m)=|S_m^{(30)}|-|S_m^{(22)}|$ is empirically consistent with a power-law scaling of the form $m^b$ ($b\approx 1.11$), quantifying the cumulative effect of replacing the symmetric cubic $abc$ with the asymmetric quadratic $bc$. A mechanism for the apparent randomness of Rule~30's center column is identified through the left-permutive structure and asymmetric Boolean sensitivity profile. |
| title | Symmetric Nonlinear Cellular Automata as Algebraic References for Rule~30 |
| topic | Mathematical Physics Cellular Automata and Lattice Gases |
| url | https://arxiv.org/abs/2604.00165 |