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The integer sweep put parity somewhere between W = 11 and W = 12 and no finer, on a standard error of 0.00050 against a difference of the same size. The open question was whether half an interval cheaper would do. It would not. 2112 paired runs: half-integer W from 9 to 14, 96 replicates, early_stop off at 30 epochs so each run gives a full 15-epoch equilibrium sample. Standard error on a paired difference falls 0.00050 -> 0.00018. Paired against today's recipe (uncle-anchored, W = 10), on-label replicates: parent W=11.0 -0.00176 +- 0.00020 t = -8.9 parent W=11.5 -0.00096 +- 0.00019 t = -5.1 <- ruled out parent W=12.0 -0.00005 +- 0.00018 t = -0.3 <- parity parent W=12.5 +0.00057 +- 0.00019 t = 3.0 Interpolated zero crossing W = 12.03, so 12 is not merely the first integer that clears — it is the crossing to within the grid's resolution. The residual at 12 is 5e-5, two orders of magnitude below the +-0.9% per-epoch noise floor, and p_ref follows at 0.937 against 0.938. Two of the 96 replicates drew a stake vector on which adversary_frac = 0.3 is unreachable and ran 0.005 and 0.094 coalitions. Being weaker they pull cells toward parity, which would bias the crossing SMALLER, so the headline is the 94 on-label replicates; including all 96 gives the same crossing and t = -0.30. The value has now survived three independent measurements, two design flaws and a 3x increase in resolution without moving: 11.98, then 12.03. §8.5's spec-delta row sharpened to match; §9 cites the new config and analysis script. Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>