Claims · 19 dated claims on Odd-base slice grammar, each with its tag and its witness; newest 2026-09-23.
Verified The odd-base generalisation rests on a choice of solid: Cook's predicate "at most one coordinate in the middle third" and the bang dim 3, code 23 rule "at most one odd coordinate" agree at base = 3 (20 of 27) and nowhere else, at base = 5 filling 4^3 + 3 * 4^2 = 112 of 125 against 3^3 + 3 * 2 * 3^2 = 81 = 4k^3 - 3k^2 at k = 3, and there is no canonical base-5 Menger sponge. Witness: lab/py/odd-base-slice-grammar.
Verified In the comparison of the slice dimension against the dimension minus one, the dimension is the solid's own log(fill)/log(base) and never the ambient 3: at the ambient 3 the value 3 - 1 = 2 exceeds all four slice dimensions 1.8184, 1.6869, 1.8026, 1.7204 and the mod-4 split collapses. Witness: lab/py/odd-base-slice-grammar.
Verified The four printed dimensions are consistent with the four printed rules and this is not evidence for either: the dominant roots (9 + sqrt(33))/2 = 7.37228, (11 + sqrt(369))/2 = 15.1047, (42 + sqrt(612))/2 = 33.3693, (28 + sqrt(3556))/2 = 43.8161 give log(root)/log(base) of 1.8183, 1.6870, 1.8026, 1.7204 at base = 3, 5, 7, 9, while 4k^3 - 3k^2 at k = 2..5 gives 20, 81, 208, 425 and dimension minus one 1.7268, 1.7304, 1.7430, 1.7544; a rule and its own dimension cannot cross-check each other. Witness: lab/py/odd-base-slice-grammar.
Proved The middle diagonal layer sits at coordinate sum 3(base-1)/2, odd exactly when base = 3 mod 4 (3, 6, 9, 12 at base = 3, 5, 7, 9). Witness: lab/py/odd-base-slice-grammar.
Conjecture No definition of a "blow-up of 4" for the slice exists in this tree, so the phrase carries no claim.
Conjecture The two-tile grammar closes at ten odd bases with a 2 x 2 substitution matrix rational in base within each class of base mod 4.
Conjecture That parity forces structurally different cells into the middle layer in each residue class, which is the mechanism of the mod-4 split.
Conjecture The two-tile grammar's 2 x 2 substitution matrix, rational in base within each class of base mod 4, reproduces every census matrix at odd bases 3..21, ten bases. Witness: lab/py/odd-base-slice-grammar, spectra.md THE CLAIM.
Conjecture The middle diagonal layer sits at coordinate sum 3(base-1)/2, odd exactly when base = 3 mod 4, the candidate mechanism of the mod-4 split; the side test runs to odd base 401 and the mechanism itself has no generator. Witness: spectra.md OPEN QUESTIONS, lab/py/odd-base-slice-grammar for the side test.
Verified The carry automaton M[c, c'] = P_b[c + g - b c'] on abs(c) <= 1, g = 3(b-1)/2, P_b the digit polynomial of the solid with at most one odd coordinate, has even block [[M[0,0], M[0,1]], [M[1,0] + M[-1,0], M[1,1] + M[-1,1]]] equal to spectra's closed form at all twelve odd bases b = 3..25, [[6,1],[6,3]], [[7,3],[30,4]], [[30,3],[24,12]], [[19,9],[96,9]], [[72,6],[54,27]], [[37,18],[198,16]] and on; no carry leaves abs(c) <= 1, the pair ((M^n)[0,0], (M^n)[1,0] + (M^n)[-1,0]) obeys the block at levels 0..8, and a direct digit-sum census at heights h_n - 1, h_n, h_n + 1, h_n = 3(b^n - 1)/2, returns the same pairs at levels 0..6 at b = 3, 0..5 at b = 5, 0..4 at b = 7, 9 and 0..3 at b = 11..25. Witness: lab/py/spectra-from-cuts verb block.
Proved The correspondence between the two-tile census and the carry automaton is exact, with no index shift and no factor: the plane x + y + z = 3 b^n / 2 = h_n + 3/2 cuts the level-n cells of corner sum h_n - 1, h_n, h_n + 1 at offsets 5/2, 3/2, 1/2, so hexagons_n = (M^n)[0, 0] and triangles_n = (M^n)[1, 0] + (M^n)[-1, 0] with hexagons_0 = 1, and h_n = g(1 + b + .. + b^(n-1)) is the central height of cuts at dim = 3 with the target digit dim replaced by g = 3(b-1)/2, which is 3 again at b = 3. Witness: spectra.md THE COMPOSITION, lab/py/spectra-from-cuts verb block.
Proved At every odd base b, (hexagons, triangles)_(n+1) = S_b (hexagons, triangles)_n with S_b = [[P_b[g], P_b[g+b]], [2 P_b[g+1], P_b[g+b-1] + P_b[g+b+1]]], the even block of the carry automaton on (hexagons, triangles), the basis {e_0, (e_1 + e_(-1))/2}: the carry contracts because abs(c + g - s) <= g + 1 < 2b for abs(c) <= 1 and 0 <= s <= 2g, and d -> b - 1 - d keeps digit parity at odd b so P_b[s] = P_b[2g - s] and M[-c, -c'] = M[c, c'], whence M^n e_0 = hexagons_n e_0 + (triangles_n / 2)(e_1 + e_(-1)); S_3 = [[6,1],[6,3]] is Abel's matrix and its recurrence is the one cuts proves. Supersedes the Conjecture row of 2026-08-28 "The two-tile grammar closes at ten odd bases with a 2 x 2 substitution matrix rational in base within each class of base mod 4". Witness: spectra.md THE COMPOSITION, lab/py/spectra-from-cuts verb block.
ProvedP_b(t) = E^2 (E + 3 O) with E = 1 + t^2 + .. + t^(b-1) and O = t + t^3 + .. + t^(b-2), E^3 carrying the even exponents and 3 O E^2 the odd ones, so with n = (b+1)/2 the five entries are binomial extractions from ((1 - x^n)/(1 - x))^3 and (1 - x^(n-1))(1 - x^n)^2/(1 - x)^3: when b = 3 mod 4, P_b[g] = 3(b+1)(3b-1)/16, P_b[g+1] = 3(b+1)^2/16, P_b[g+b] = (b+1)(b+5)/32, P_b[g+b-1] = 3(b+1)(b+5)/32, P_b[g+b+1] = 3(b-3)(b+1)/32; when b = 1 mod 4, P_b[g] = (3b^2+6b+7)/16, P_b[g+1] = 3(b-1)(3b+5)/16, P_b[g+b] = 3(b-1)(b+3)/32, P_b[g+b-1] = (b+3)(b+7)/32, P_b[g+b+1] = (b-1)(b+3)/32; the active binomials are stable from b >= 11 at the latest and every form matches the digit polynomial at odd b = 3..101. Witness: lab/py/spectra-from-cuts verb forms.
Proved The closed forms of spectra's THE CLAIM hold at every odd base, by substitution of the five coefficients into S_b: trace 3b(b+1)/4 and determinant 3(b-1)(b+1)^3/32 when b = 3 mod 4, trace (b^2+3b+4)/4 and determinant -(b+3)(3b^3-3b^2-11b+3)/32 when b = 1 mod 4; the mod-4 split of the matrix is the parity of g = 3(b-1)/2, odd exactly when b = 3 mod 4, which reads P_b[g], P_b[g+b-1], P_b[g+b+1] from 3 O E^2 and P_b[g+1], P_b[g+b] from E^3 in that class, all five swapping parts in the other. Supersedes the Conjecture row of 2026-09-21 "The two-tile grammar's 2 x 2 substitution matrix, rational in base within each class of base mod 4, reproduces every census matrix at odd bases 3..21, ten bases" and the Conjecture row of 2026-08-28 "That parity forces structurally different cells into the middle layer in each residue class, which is the mechanism of the mod-4 split". Witness: spectra.md THE COMPOSITION, lab/py/spectra-from-cuts verb forms.
Proved The mod-4 side at every odd base: with q = fill/b = n^2(4n-3)/(2n-1) and chi the characteristic polynomial of S_b, (2n-1)(trace - 2q) reads -n(4n^2-3)/2 and -(n-1)(12n^2-1)/2, both negative, and (2n-1)^2 chi(q) reads -n^3(4n^3 - 12n^2 + 15n - 6)/2 < 0 for even n >= 2 and (n-1)^2(4n^4 - 4n^3 + 7n^2 - 1)/2 > 0 for odd n >= 3, every real root of every factor below the first n of its class, so rho_b > fill/b when b = 3 mod 4 and rho_b < fill/b when b = 1 mod 4, the box-counting exponent log_b(rho_b) above or below log_b(fill) - 1 accordingly. Supersedes the Conjecture row of 2026-09-21 "The middle diagonal layer sits at coordinate sum 3(base-1)/2, odd exactly when base = 3 mod 4, the candidate mechanism of the mod-4 split; the side test runs to odd base 401 and the mechanism itself has no generator". Witness: lab/py/spectra-from-cuts verb split, roots isolated exactly.
ProvedP_b(w) = -2/(1+w)^3 at every b-th root of unity w != 1, since E(w) = 1/(1+w) and O(w) = -1/(1+w) there, P_b(w)(1+w)^3 + 2 being divisible by 1 + w + .. + w^(b-1); the residue-class sums of P_b modulo b are the row sums of the automaton, the class of g summing to (b+1)(5b+1)/8 or (3b^2+6b-1)/8 and each class of g +- 1 to 3(b+1)^2/8 or (5b^2+6b-3)/8 by class 3, 1 mod 4; the block is five single coefficients and not class sums. Witness: lab/py/spectra-from-cuts verb classes, odd b = 3..25 for the divisibility and 3..101 for the sums.
Verified At base 5 the substitution read off polygons cut from cells is H -> 7 H + 15 T+ + 15 T-, T+ -> 3 H + 3 T+ + 1 T-, T- -> 3 H + 1 T+ + 3 T-, folded to [[7,3],[30,4]], the T- pattern the half-turn of the T+ pattern, every child polygon inside its parent, census (7, 30), (139, 330), (1963, 5490) at levels 1..3; at base 3, levels 1..4, Abel's rule and (6, 6), (42, 54), (306, 414), (2250, 3078). Witness: lab/py/spectra-from-cuts verb tiles.
Proved At every odd base b the centroid diagonal slice of the solid whose base-b digit triples have at most one odd coordinate is the attractor of a graph-directed system of ratio-1/b homotheties on three tiles T+, H, T-, the cuts of cells at doubled offset 1, 3, 5, with incidence the carry automaton M[c, c'] = P_b[c + g - b c'], the open set condition from the disjoint open cells, and two prototiles up to the half-turn d -> b - 1 - d, a regular hexagon and an equilateral triangle. Witness: spectra.md THE COMPOSITION, The tiles.
Proved The Hausdorff and box-counting dimensions of that slice are both log_b(rho_b), rho_b the dominant root of S_b, by the cover of level-n tiles and the mass distribution principle on the left Perron vector of M, a disc of radius b^-n meeting at most 64 level-n cells. Witness: spectra.md THE COMPOSITION, The tiles.