Automata

• 2026-09-02 [Proved] Wolfram rule N and the design bang dim 3, code N are one subset of {0,1}^3 under (x0, x1, x2) = (l, c, r) with corner i = 4 x0 + 2 x1 + x2, so every design invariant is a rule invariant; recomputed on all 2048 rule cells. Witness: lab/rs/automata-census.
• 2026-09-02 [Verified] The elementary rules fall into 160, 88, 22 and 14 classes under reflection, Wolfram equivalence, the cube group and the cube group with complement, by orbit walk and by Burnside alike; Wolfram equivalence and the cube group are incomparable inside the order-96 group and meet exactly in the reflection, rule 137 lying in the Wolfram class of 110 and outside its cube orbit. Witness: lab/rs/automata-census.
• 2026-09-02 [Proved] The cube group is not a dynamical symmetry of the elementary automata: surjectivity is not a cube invariant, exactly one of the 22 classes being mixed, the 24-rule orbit of 30 with 16 surjective rules and 54 among the 8 that are not, while reversibility is constant on all 22 classes, the six reversible rules 15, 51, 85, 170, 204, 240 being the orbit of the identity, and both properties are constant on all 88 Wolfram classes. Witness: lab/rs/automata-census.
• 2026-09-02 [Verified] The 24 rules in the cube orbit of 110 share popcount 5, degree 3, genus compound, the Walsh amplitude profile and non-surjectivity, only their single-seed diagrams separating all 24; the multiset of Walsh amplitudes by weight is a cube invariant on all 256 rules while the signed level sums are not. Witness: lab/rs/automata-census.
• 2026-09-02 [Verified] Rules 60 and 102 draw the plane designs bang dim 2, code 13 and bang dim 2, code 14 cell for cell from one seed, and rule 90 draws bang dim 2, code 13 in the sheared frame j = (t + i)/2, each unique among the fill-3 codes to level 8 by two renderers. Witness: lab/rs/automata-census.
• 2026-09-02 [Proved] The single-seed diagram of rule 150 is an XOR substitution, row 2t being row t spread by two and row 2t + 1 that row xor its two unit shifts, and its first 2^k rows hold P(k) = 2^k F(k+2) live cells with B(k) adjacent pairs under P' = 4P - 2B and B' = 2P - 2B, so the growth exponent is log2(1 + sqrt 5); the recurrence and closed form are already A087206's and A071053's and are re-derived here. Witness: lab/rs/automata-census, A087206, A071053.
• 2026-09-02 [Verified] One live cell under the 256 rules gives 143 distinct space-time diagrams, 89 up to reflection, class sizes {1: 118, 2: 13, 4: 4, 8: 4, 16: 4}, identical at pad T and 2T for T = 64, 128, 256; equal occurring key implies equal diagram with 0 failures over 65536 ordered pairs, the key measured on the padded line takes 152 values by a boundary artefact, and on the cropped window exactly 143. Witness: lab/rs/automata-census.
• 2026-09-02 [Proved] In every dimension the level-1 side-3 tile of the design "not every coordinate odd" is the 3^dim - 1 Moore neighbourhood and the tile of "at most one odd coordinate" fills 2^(dim-1) (dim + 2); as popped masks the two agree iff dim <= 2, 20 against 26 cells at dim = 3, and in the plane the Moore mask is bang dim 2, code 7. Witness: lab/py/life-census, lab/py/sibling-census.
• 2026-09-02 [Verified] B3/S23 is the dim = 9 design of fill 140 of 512, so Langton's lambda is 140/512 by definition, with GF(2) degree 8 on 184 monomials, Walsh level sums 140, 308, -224, -896, -168, 840, 448, -224, -196, -28 on the 0/1 form and genus compound; over the 2^18 life-like rules the fill takes 479 of 513 values with 34 gaps, the genus splits 2044 isotropic, 4 axial only and 260096 compound, exactly 8 rules are affine, and the degree obeys deg(B, S) = deg B when B = S and max(deg B, 1 + deg(B xor S)) otherwise, giving 2 * 4^d rules of degree at most d. Witness: lab/py/life-census.
• 2026-09-02 [Proved] The nine-cell Moore XOR B1357/S02468 is rule 150 tensor rule 150, every time slice the outer product of two rule 150 rows and its population A071053 squared, which is A246035; the named replicator B1357/S1357 is the eight-cell XOR with kernel (1/x + 1 + x)(1/y + 1 + y) - 1, A160239, and equals the outer product with the centre copy removed only at t = 2^j. Witness: lab/py/life-census, A246035, A160239.
• 2026-09-02 [Proved] The decoupling lemma: an automaton whose dependency set, centre included, generates a sublattice of index k is k interleaved copies of the index-1 rescaled automaton under the same rule, so the 1D parity tile at side 2r + 1 decouples iff r is even and the Cantor tower iff its level is even; over the base-2 masks at dim = 1, 2, sides 3 to 9 and levels 1 to 3, the 195 distinct masks split 95 of index 1, 70 of index 2, 18 of index 4 and 12 rank-deficient. Witness: lab/py/sibling-census.
• 2026-09-02 [Proved] B3/S23 is not a rows-then-columns composite of two elementary rules in either order; the 65536 ordered pairs give 32260 distinct nine-input rules, none equal to Life, and exactly 10 life-like composites, all affine or thresholds at 0 or 9. Witness: lab/py/sibling-census.
• 2026-09-02 [Proved] Cantor-Life, B3/S23 on the eight-cell level-3 Cantor mask at offsets +-5, +-7, +-11, +-13, has no still life under 4 cells at any width, {0, 5, 7, 12} being minimal, and its nine-cell XOR has period dividing 256 on the ring of 1024 by Frobenius, exactly 256 on generic soups; every seed of width at most 14 dies (6113), stills (2003) or oscillates (76, periods 2, 3, 4, 6), and no mover is found there. Witness: lab/py/sibling-census.
• 2026-09-02 [Proved] Menger-Life, B3/S23 on the 20-cell Menger mask, leaks out of every plane holding a dead cell of Moore count 3, and a plane 2 x 2 block stacks as rule 90 along the normal with population 4 * 2^popcount(t); over 101 outer-totalistic rules and 200 random 5^3 seeds each, no mover appears. Witness: lab/py/sibling-census.
• 2026-09-02 [Proved] Under the composite 110.110 the bounding box of every finite pattern grows without bound, its upper-left corner moving (-1, -1) each generation; the population growth itself is only recomputed, not proved. Witness: lab/py/sibling-census.
• 2026-09-02 [Refuted] "Fredkin's replicator is the mod-2 sum of the nine Moore cells": the named replicator B1357/S1357 is the eight-cell sum with kernel (1/x + 1 + x)(1/y + 1 + y) - 1, A160239, while the nine-cell sum is B1357/S02468, A246035, and the two agree at no generation past t = 0 except through the removed centre copy at t = 2^j. Witness: lab/py/life-census, A160239, A246035.
• 2026-09-19 [Verified] Over the 195 base-2 masks at dim 1, 2, sides 3 to 9 and levels 1 to 3 the lattice index splits 16 of index 1, 11 of index 2 and one empty mask at dim 1 and 79, 59, 18 and 11 at dim 2, the both-even mask bang dim 2, code 1 at level 2 being an index-4 witness. Witness: lab/py/sibling-census.
• 2026-09-19 [Conjecture] On the level-3 Cantor mask from soups of 1024 cells at three densities, B36/S23 behaves like B3/S23 with activity lingering at density 0.5, B2/S sustains a soup near 0.18, B3/S012345678 freezes near 0.65 at the densities 0.25 and 0.5, and B3678/S34678 dies, fills the ring or holds a long period at a density between 0.3 and 0.9. Witness: the soup runs of lab/py/sibling-census, which carries no witness line for these four rules.
• 2026-09-19 [Verified] Of the 101 Menger rules on a 32^3 torus at 200 generations the 22 quiet in all four runs are every rule of birth {5} or {6} together with B4/S5 and B4/S6. Witness: lab/py/sibling-census.
• 2026-09-19 [Verified] Of the 69 still lifes among the 20200 Menger seed fates, 55 sit under B5/S34, B6/S34 and B56/S34. Witness: lab/py/sibling-census.