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.