Common Formulas and Calibration Frontiers in PAC–Bayes Bounds

PAC–Bayes bounds certify predictors selected from data, but their tightness can be limited by the need to use one formula across sample outcomes or recover classical confidence bounds when posterior equals prior. We characterize when one normalized convex comparator attains every statewise optimum and derive matching excess bounds in two regimes for fixed finite interior targets with positive weights. A fixed task using empirical risk minimization has low risk and bounded Kullback–Leibler (KL) complexity, yet its optimal expected excess is of the same order as the benchmark width above empirical risk. For finite binary-loss libraries, our construction combines Clopper–Pearson (CP) exactness at a shared point mass with simultaneous posterior coverage, continuity and KL learning, whereas exact recovery at a mixed prior can obstruct learning. Under the specified randomized evaluation, every rule with CP recovery at the original confidence level and simultaneous coverage throughout the full model gives a perfect predictor a bound tending to one in probability on a common fixed task. Allowing pathwise one-sided calibration tolerance yields matching asymptotic limits for width quantiles under each distribution and in the worst case, with the finite model, positive prior and KL budget fixed. The worst-case width can vanish when the tolerance decays subexponentially, whereas exponential decay can leave a positive limit. One explicit rule uses joint losses to attain each distribution’s optimal limit without knowing the distribution. This attainment is uniform over growing finite libraries and all positive priors and data laws at specified confidence levels with a fixed positive precision exponent and KL budget.