bibkey: chanlopezmartinruiz2026rule30 authors: E. Chan-López and A. Martín-Ruiz year: 2026 title: Symmetric Nonlinear Cellular Automata as Algebraic References for Rule 30 doi: 10.48550/arXiv.2604.00165 url: https://arxiv.org/abs/2604.00165v3 claim: The all-row sign-pattern question of Remark 3 and section 9 for the full-row Rule 30 and Rule 22 support difference. strata_touched:

• D5/S0/Automata/RuleThirtyTwentyTwoMersenneSignRefutation license: citation-only triage: anchor

Rule 30 and the symmetric Rule 22 reference

The source is arXiv:2604.00165v3. Its neighbour convention uses left, centre and right bits a,b,c. Equation (1), p. 3, gives “g22(a, b, c) = a ⊕ b ⊕ c ⊕ abc.” Proposition 1 on the same page states: “Rule 30, with ANF g30 = a ⊕ b ⊕ c ⊕ bc, is left-permutive but lacks S3 symmetry.” XOR is addition and AND is multiplication over the two-element field.

Section 3, p. 4, fixes the configuration evolved “from the single-seed initial condition η_0 = δ_0”. Definition 2 states: “The support set at time m is the full-row support S_m = {r ∈ Z : η_m(r) = 1}”. It also specifies: “All cardinality statements below refer to the full-row set S_m”. Thus the cardinality counts the entire integer row, including negative sites and the origin. Equation (11), p. 10, defines “ϵ(m) = |S_m^(30)| − |S_m^(22)|”. Both supports are finite because the two rules are quiescent and a single seed has a finite light cone.

Remark 3, p. 11:

A direct computation for m ≤ 256 shows that ϵ(m) ≤ 0 precisely at the Mersenne indices m = 2^k − 1, with ϵ = 0 for k ≤ 3 and ϵ < 0 for 4 ≤ k ≤ 8.

Section 9, p. 15:

Is the sign pattern of Remark 3 exact for all k?

The sign-pattern proposition extends the “precisely” biconditional to every natural index m ≥ 1, with Mersenne indices 2^k - 1, k ≥ 1. The module proves its negation: the two support cardinalities at m = 767 are 763 and 768, so the integer difference is −5, although 768 is not a power of two. This does not refute the finite observation for m ≤ 256, or settle the separate all-Mersenne strict-negativity question. No resolution of that latter question is asserted here.

Verified locator

• arXiv v3: https://arxiv.org/abs/2604.00165v3
• DOI: https://doi.org/10.48550/arXiv.2604.00165