No distribution-free lower bound on the mean of an unbounded cost exists without a tail assumption. In the sim2real-inference programme we tried a cheap workaround on 21 MuJoCo cart-pole twin-versus-real pairs: run the twin (which costs nothing), take its 99.9th percentile as a clip level c, and bound E[min(R, c)], which can only be at most E[R].
- Clipping at that level loses under 5% of the mean on 18 of 21 pairs. On the three large-friction pairs the twin's tail is wrong and the clipped mean is 14% to 84% short, which shows up in the bias, not in the interval.
- A real-only empirical-Bernstein bound on the clipped cost has a median relative gap of 1.51, 0.83, 0.47 and 0.27 at n = 30, 60, 120 and 240 real trials.
- Adding the twin as a control variate, with its mean treated as exactly known, gave a smaller gap on 0 of 21 pairs at every n. Doubling the residual's range costs more than the smaller variance saves, the same effect seen for bounded scores and failure rates.
So the twin helps estimate and fix a range, not tighten the certificate. All numbers are from model-versus-model pairs, not real robot data; per-episode real pairs remain the programme's main gap. Details: LOG.md, milestones 7d and 7e. AI-assisted.