# 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.