Union-closed q2, 2026-08-27

Continuation of the {b,1} frequency campaign. Search lives in problems/union-closed/compute/q2/.

q1 left a ray-certified 0.38304 for pure Liu Example 4, with analytic first-crossing 0.383051356…. The job was a certified number strictly above 0.38304, toward 1/2, on a checkable class.

The constant did not move. Every 2-sample bit protocol on {b,1} has first-crossing at most 0.383051356…, because h(Π_{b,b})≤1. Mixes, a half-target protocol, and scaled a(t) sit at or below that ceiling. Gilmer Conjecture 1 (the KL path to 1/2) fails on Ellis's n=2 law, replayed at −0.046797. Pure Example 4 fails a constructed 2-mixture at mean 0.38304 (CIID ratio 0.909137); Python and C agree. That is residue off the ray.

Replay:

cd problems/union-closed/compute/q1 && ./run_all.sh
cd problems/union-closed/compute/q2 && ./run_all.sh

Certificate: problems/union-closed/compute/q2/certs/verify.json. Not 1/2. Fetched Gilmer 2211.09055v2 and Liu 2306.08824v1.