Uniform lower bound for the affine-control terminal joint event

Prove a uniform-in-q lower bound on the probability that, at the long horizon for the nonsaturating binary affine recurrence, the terminal step remains on the wrong side while exact Bayes is sufficiently confident.

Background

The appendix analyzes whether the nonsaturating binary affine control has a nonvanishing predictive-loss lower bound at the long endpoint ⌈8/q⌉. Several deterministic and probabilistic ingredients are proved, including a lower cost for a step on the wrong side and concentration bounds for Gaussian evidence. The missing ingredient is a uniform probability bound for the joint event that the affine state has not crossed after a recent latent switch while exact Bayes has already recovered with high confidence. Establishing this event would yield the desired analytic lower bound on terminal expected KL.

References

What remains open is therefore a uniform bound on the joint event that the run is young enough for the affine state not to have crossed and old enough for the exact filter to be confident, not an integration.

— How Wrong Can a Good Predictor Be? Diverging Updates with Vanishing Predictive KL  (2609.11132 - Wen et al., 10 Sep 2026) in Appendix, Section Open Affine Control Lower Bound; paragraph “What the open step actually requires”