# Conjecture S: the even-half transient

- 2026-08-28 [Verified] The two-step census contraction `9 b(2j+2) <= fill^2 b(2j)` with `3 b(2j+1) < fill b(2j)` has zero violations through index 400 at every even `dim <= 50`, margin peaking near `(dim-2)/(dim+2)`, and in exact integers at even `dim = 2..30` to index 40 the margin sits strictly below `(dim-2)/(dim+2)` and rises toward it, `0.8704914` against `0.875` at `dim = 30`; with irreducibility and nonvanishing it would close the even half. Witness: slice-recurrence-order.
- 2026-08-28 [Conjecture] Every weighted-L2 certificate for the even half fails, the transfer norm being at least `sqrt(3) fill` for every positive weight, and two Abel pairings fail with it.
- 2026-08-28 [Refuted] The even half's early `W_k` sign alternation at base 3 persists, and a conjecture can be built on it - the alternation is a transient: over all even `dim` the first break is `dim = 6, k = 4`, then `(8,6)`, `(10,8)`, the rule `k = dim - 2` dying at `dim = 16` where the first break is `k = 16`, then `20, 24, 28, 34, 40` at `dim = 18..26` and none through `k = 40` for `dim = 28..40` (inside the window even `12 <= dim <= 22` the first break is `(dim,k) = (12,10)`); structurally `W_level ~ C 3^(level-1) rho^(level-1)(3 rho - fill)` makes the eventual `W`-sign the even half itself; the pointwise route is dead too, `V_2 < 0` for even `dim >= 6`. Witness: slice-recurrence-order, slice-sign-even-half.