# Conjecture S: odd half

- 2026-08-28 [Proved] The slice census has the trigonometric product formula `P(e^(i psi)) = e^(i dim psi) (2 cos psi)^(dim-1)(dim + 2 cos psi)`, and the sheaf census `b(level)` (coordinate sum `== dim(3^level-1)/2 mod 3^level`, equally the free-end carry count) is `b(level) = 3^(-level) sum_(m<3^level) prod_(j<level) Phi(2 pi m 3^j/3^level)` because the extraction phase is the accumulated palindromic phase; every unit tower ends at `2 pi u/3` with factor `(-1)^(dim-1)(dim-1)`, so at odd `dim` the integrand is pointwise nonnegative, `b(level) >= (fill/3)^level` and `rho_dim >= fill/3`, and `det(fill I - 3 M_even) == fill^n mod 3` gives `rho_dim > fill/3` strictly at every odd `dim = 0, 2 mod 3` and through `dim = 80` in the class `1 mod 3` by exact determinants (Bareiss, three 61-bit primes and Berkowitz agreeing); the bijection is brute-forced at `dim = 2..8`, `level <= 4` and the phase cancellation matched to 50 digits at `dim = 2..12`. Witness: slice-recurrence-order.
- 2026-08-28 [Proved] The pinning `|rho_dim - fill/3| <= 2(dim-1)/3` holds unconditionally (even `dim` in `[fill/3 - 2(dim-1)/3, fill/3 + (dim-1)/3]`, odd `dim` mirrored) because the core's column sums take exactly the values `fill/3 + 2 eps` and `fill/3 - eps`; exactly `3 rho_dim = fill + (-1)^(dim-1)(dim-1)(3 p_dim - 1)` with `p_dim` the Perron carry vector's mass on carries divisible by 3, well defined since the core is irreducible for all `dim`; so `slice dimension - (solid dimension - 1) -> 0` like `dim^2 2^(-dim)` regardless of sign, and Conjecture S entire is the parity-free inequality `p_dim > 1/3`; checked by power iteration at `dim = 2..20` and entrywise column sums at `dim = 2..80`. Witness: slice-recurrence-order.
- 2026-08-28 [Verified] The sign-law mechanism is universal: at every `base >= 3` and `u != 0 mod base` the design symbol has `g_base(2 pi u/base) = -1`, the full digit sum vanishing at a nontrivial `base`-th root of unity and the middle digit contributing 1, so the innermost tower factor is `(-1)^(dim-1)(dim-1)` at every odd base and the mechanism is `base`-th-root evaluation, never `P(-1)`; the odd-`dim` inequality `rho >= fill/base` travels with scope `dim >= -min g_base` (`9/4` at `base = 5`, `(34+14 sqrt 7)/27` at `base = 7`, growing like `0.217 base`), strict when `dim != 1 mod p` for some prime `p | base`; the sign law is exact by Sturm counts at base 5 `dim = 2..26`, base 7 `dim = 2..18`, bases 9, 11 `dim = 2..12` and `(base,dim) = (21,3), (31,5), (51,5), (101,3)`. Witness: slice-sign-even-half.