Exact confidence sets spend information budgets

For a level-$(1-\alpha)$ confidence set the quantity $\log(1/\alpha)$ enters every construction; we call it the information budget of the set, in nats. We show that every confidence set is the sublevel set of an e-value at the Markov threshold, and that its over-coverage equals the difference between the budget and the log-evidence its boundary requires. The Markov threshold is valid at every level and every sample size simultaneously; the exact threshold, obtained by solving the coverage identity, gives coverage equal to the nominal level, and on Gaussian data it recovers the Wald interval. The classical exact sets are therefore the exact-threshold members of one nested family, and the two thresholds differ by a closed-form width factor. The difference between nominal and realized coverage is a computable Markov slack, and Ville's inequality holds with equality up to the expected boundary overshoot, which is nonnegative. For a bounded mean the same identity gives a threshold in place of the Markov one: exact when the null law of the statistic is available, and certified over the moment set when it is not.