Life Census

• Reads the 2^18 outer-totalistic life-like rules on the Moore neighborhood as designs on the parity cube {0,1}^9, and computes fill, Langton lambda, GF(2) degree and genus for all of them; the Walsh level sums are computed for B3/S23 alone.
• A rule is a birth set B and a survive set S inside {0..8}, and as a design it is the dim 9 Boolean function f(c, n) = [c = 0][|n| in B] + [c = 1][|n| in S] on the centre c and the eight outer bits n.
• The code is the 512-bit corner bitmask in the crate's row-major corner order, corner i carrying coordinate j equal to (i / 2^(8-j)) mod 2, with the centre as the first coordinate; the code under the 3x3 block order, centre in the middle, is printed beside it, the two being a permutation of variables apart and so equal in fill, degree and genus.
• Dimension is the core page's log(fill) / log(side) at level 1, so at base 2 it is log2(fill) = dim + log2(lambda) with fill the number of filled corners of {0,1}^dim.
• The Moore mask is rebuilt from the crate's definition, ones on a 3^dim grid with the centre index cleared, and compared with the level-1 side-3 tile of the design "void iff every coordinate is odd" at dim 1, 2, 3.
• That comparison is a theorem in every dimension: at side 3 the coordinates are 0, 1, 2, the only odd value is 1, so the only cell with every coordinate odd is the centre (1, ..., 1); hence the level-1 side-3 tile of "void iff every coordinate is odd" is the full 3^dim block minus its centre, which is moore(dim).
• Genus follows the core page: a design is isotropic when some member of its B_dim orbit is a level set of the popcount, which by the orbit law means f(x) depends only on |x xor t| for one of the 512 flips t, and axial when its support is a subcube. Both tests are orbit-wide, so a per-rule verdict is well defined.
• The fill histogram is computed twice: as the self-convolution of the subset-sum distribution of the binomials {1, 8, 28, 56, 70, 56, 28, 8, 1}, and by brute force over all 2^18 pairs.
• The axial test is computed twice as well: a closed form in B and S from the fill and the edge counts, and a brute subcube test on the support of every one of the 2^18 truth tables.
• The degree sweep runs a Mobius transform over every one of the 2^18 truth tables, and is cross-checked against the composition law deg(B, S) = deg(B) when B = S and max(deg(B), 1 + deg(B xor S)) otherwise, which follows from f = g_B + c (g_B xor g_S) and g_B xor g_S = [|n| in B xor S]; the exception is real, since the empty rule B/S has degree -1 where the unconditional form gives 0.
• The Walsh level sums Sigma_k, the sums of W(S) = sum_x (-1)^(S.x) f(x) over the subsets S of size k, are checked against the generating identity sum_k Sigma_k t^k = sum_x f(x) (1+t)^(9-|x|) (1-t)^|x|, which is derived from the support alone and not from the transform it checks, and against Parseval, sum_S W(S)^2 = 512 fill for a 0/1 valued f.
• The parity rule B1357/S02468 is the sum mod 2 of all nine Moore cells, generating polynomial (1 + x + x^2)(1 + y + y^2) = k(x) k(y) with k the rule 150 kernel; the study evolves it from one seed to t = 64 on a side 2t+3 grid with constant-0 boundary and compares every slice cell for cell with the outer product of two rule 150 rows.
• The named replicator B1357/S1357 is the sum mod 2 of the eight outer cells only, kernel k(x) k(y) - xy, so at t = 2^j it is the outer product with the centre copy removed; that is checked too, and the two population sequences are checked against 20 published OEIS terms each.
• Structural laws are asserted and the study exits nonzero if one fails; headline counts are printed.

RUN

• uv run python research/lab/py/life-census/life.py
• Domain: every one of the 2^18 life-like rules, all 2^18 axial checks against the closed form, evolution to t = 64 on a 131 x 131 grid; about three seconds, prints only, writes nothing.

WITNESSES

• In every dimension the level-1 side-3 tile of the design "void iff every coordinate is odd" is the 3^dim block minus its centre, which is the Moore mask, because 1 is the only odd coordinate value at side 3. (Proved.)
• The tile equals the crate's moore(dim) array cell for cell at dim 1, 2, 3, fills 2 of 3, 8 of 9 and 26 of 27; in the plane that design is bang dim 2, code 7, dimension log(8)/log(3) = 1.892789. (Verified.)
• B3/S23 has fill C(8,3) + C(8,2) + C(8,3) = 140 of 512, so lambda = 140/512. (Proved; Verified.)
• At base 2 a design's dimension is log2(fill) = dim + log2(lambda), so at fixed dim it is a strictly increasing function of lambda, and it is not a function of lambda across dimensions; for B3/S23, dim 9 and log2(140) = 7.129283. (Proved; Verified.)
• B3/S23 has GF(2) degree 8 with 184 monomials, and Walsh level sums 140, 308, -224, -896, -168, 840, 448, -224, -196, -28, whose squares over the 512 subsets sum to 71680 = 512 * 140. (Verified.)
• B3/S23 is not a level set of the popcount under any of the 512 flips and its fill 140 is not a power of two, so it is neither isotropic nor axial, hence compound. (Verified.)
• Outer-totalistic means a level set on the eight outer axes for each value of the centre, so the family has 2^18 members and the totalistic rules, level sets of all nine, number 2^10; all 1024 of them are isotropic. (Proved; Verified.)
• The fill histogram over the 2^18 rules is mirror symmetric about 256, takes 479 of the 513 values, peaks at fill 256 with 3270 rules, and is 1 at fill 0 and at fill 512. (Verified.)
• The 34 unreachable fills are 5, 6, 7, 13, 14, 15, 21, 22, 23, 41, 42, 43, 49, 50, 51, 69, 77 and their mirrors, so lambda has gaps near both ends. (Verified.)
• 165 rules share the fill 140 of B3/S23. (Verified.)
• Genus over the 2^18: isotropic 2044, axial only 4, compound 260096. (Verified.)
• The degree histogram over the 2^18 is -1:1, 0:1, 1:6, 2:24, 3:96, 4:384, 5:1536, 6:6144, 7:24576, 8:98304, 9:131072. (Verified.)
• The composition law is deg(B, S) = deg(B) when B = S, and max(deg(B), 1 + deg(B xor S)) otherwise; it holds on all 2^18, and the empty rule B/S, of degree -1, is the case the exception is there for. (Proved; Verified.)
• A life-like rule is affine over GF(2) iff f = alpha c + beta (n1 + ... + n8) + gamma, so B and S are each empty, full, the odds or the evens and share beta; exactly 8 rules qualify, the four degenerate ones and B1357/S1357, B1357/S02468, B02468/S1357, B02468/S02468. (Proved; Verified by the degree of all 2^18.)
• Count sets of degree at most d number 2^(d+1), and with the composition law the rules of degree at most d number 2 * 4^d for d = 1..8. (Verified; the product form Proved.)
• B1357/S02468 is k(x) k(y) with k the rule 150 kernel, so every slice is the outer product of two rule 150 rows and the population is the rule 150 population squared, and at t = 2^j the pattern is nine copies of the seed at spacing 2^j, checked at t = 1, 2, 4, 8, 16, 32, 64 and cell for cell to t = 64. (Proved; Verified.)
• B1357/S1357 is the same product minus the centre, so at t = 2^j it is the outer product with the centre copy removed, since the rule 150 row has a live centre at t = 2^j, eight copies, checked at t = 1, 2, 4, 8, 16, 32, 64. (Proved; Verified.)
• The three population sequences are OEIS A071053 for rule 150, A246035 for B1357/S02468, and A160239 for B1357/S1357, 20 terms each. (Verified.)