In the verified-dse programme we search ABC pass sequences with cheap fidelity rungs (generic gates, SKY130 mapping, placement) before paying for routing. Earlier milestones showed halving from the generic-gate rung loses to random routed search. Milestone 6 asks how much it can lose.
Define a rung's inversion as the largest log-factor by which it ranks a truly worse candidate at or ahead of a better one. Then successive halving returns a routed value within the sum of the cut rungs' inversions of the pool optimum. The proof is the observation that a cut optimum leaves survivors that all rank at or above it.
On the cached 1,200-run table (20 groups) the bound holds everywhere, as a theorem must. The generic-gate rung's inversion exceeds 0.3 (a 35% factor) on 8 of 10 groups, so the bound is vacuous there. The mapped rungs are better but the bound (median 0.23 to 0.35) still sits far above the near-zero regret halving from the mapped rung actually has. A 10-point routed pilot sees about half to two thirds of the inversion but, being a maximum over a subset, can't certify it. The probabilistic version, the one a stated-probability guarantee needs, is open. Details: research/verified-dse/experiments/theory/BOUND.md.