Slice sign law in every dimension

• 2026-08-28 [Verified] The slice sign law holds on every computed range: the central diagonal slice of the dim-axis base-3 Menger analog has sgn(slice dimension - (solid dimension - 1)) = (-1)^(dim+1) for dim = 2..50 at 180 to 210 digits by two generators sharing no code, for dim = 2..100 at 320 digits with 99 of 99 signs, and in exact rational arithmetic through the determinant form sgn det((fill/3) I - M_even) = (-1)^dim for dim = 2..40, with fill = 2^(dim-1)(dim+2) and M_even the reflection-even carry block of size ceil(dim/2); base 5 alternates for dim = 2..15, all four tested non-Menger families alternate, off-centre heights keep the dominant eigenvalue; dim = 3 gives x^2 - 9x + 12 with rho_3 = (9 + sqrt(33))/2 = 7.372281 against fill/3 = 20/3, dim = 4 gives x^2 - 11x - 66 with rho_4 = 15.310708 against 16 and det = 14; the even half and the odd dim != 1 mod 3 half are proved on the shelf, and the class dim = 1 mod 3 beyond the computed range stays open. Witness: slice-sign-even-half, slice-recurrence-order.
• 2026-08-28 [Verified] The dim = 3 rung is the base-3 slice dimension: the carry automaton M[c, c'] = P[c + dim - 3c'] prints M_even = [[6, 6], [1, 3]], trace 9, determinant 18 - 6 = 12, characteristic polynomial x^2 - 9x + 12, exactly A299916's signature (9, -12), Perron root (9 + sqrt(33))/2 and log_3 of it 1.818410, against the dimension minus one log_3(20) - 1 = 1.726833 with fill = 20 the sponge's surviving-subcube count; the anchor cuts one way only, saying nothing about higher rungs. Witness: slice-recurrence-order, A299916.
• 2026-08-28 [Proved] The digit polynomial P(t) = (1 + t^2)^(dim-1) (1 + dim t + t^2) has B_dim(2k) = C(dim, k) and B_dim(2k+1) = dim C(dim-1, k), P(1) = 2^(dim-1)(dim+2), P(-1) = 2^(dim-1)(2 - dim), P(omega) = (-1)^(dim-1) (dim-1) omega^dim, and root-of-unity filtering gives the full carry matrix's exact row sums sigma(c) = fill/3 + (2/3)(-1)^(dim-1)(dim-1) cos(2 pi c/3); the row-sum identity holds in the carry orientation c -> (c + dim - s)/3 and fails in the transposed even-basis orientation M_even[i,j] = B_dim(dim + j - 3i) + B_dim(dim - j - 3i) for every dim = 3..50, the dim = 3 row sums being (12, 4) against the formula's (8, 6); the coefficient formulas hold at dim = 1..10 three positions past both polynomial endpoints. Witness: slice-recurrence-order.
• 2026-08-28 [Proved] The trace of the even carry block is tr(M_even) = 3 dim 2^(dim-3) at odd dim and 3 * 2^(dim-2) - 1 at even dim, reading 2, 9, 11, 60, 47, 336 at dim = 2..7; the even case's -1 is real, starting at dim = 2 where the matrix is [2]. Witness: slice-recurrence-order.
• 2026-08-28 [Verified] There is no uniform spectral gap in the slice transfer matrix, so no fixed-epsilon proof of spectral separation can exist: lambda_1/|lambda_2| = (dim+2)/(dim-2) + O(dim^-3), tending to 1, reaching 1.068966 at dim = 60 and 1.04081632653 at dim = 100, 1.0833... at dim = 50 against 13/12 to 2.58e-22, with lambda_1 ~ fill/3 = 2^(dim-1)(dim+2)/3 and |lambda_2| ~ |P(-1)|/3 = 2^(dim-1)(dim-2)/3; the double-precision spectrum agrees with a 180-digit reference over dim = 2..50 to worst relative Perron discrepancy 2.3e-15, median 4.7e-16, every eigenvalue numerically real over dim = 2..60, so any proof of separation must be uniform in a margin of order 4/dim. Witness: slice-recurrence-order.