Crossing the Kolmogorov-Smirnov Boundary: Exact Tails, Sharp Bounds, and Broken Pivots

The Kolmogorov-Smirnov statistic is usually introduced as a supremum, but its finite-sample behavior is governed by a more local question: where does the empirical process first cross a boundary? This letter gives a partial answer through a finite-sample crossing ledger. The ledger rewrites the Smirnov- Birnbaum-Tingey one-sample formula as an explicit hitting-time law and yields a stable log-scale tail evaluator. For two samples, it gives one-wall and two-wall exact lattice recursions for arbitrary sample sizes, with the balanced reflection formula appearing as a special closed form. The same viewpoint explains the Dvoretzky-Kiefer-Wolfowitz-Massart inequality as an exponential compression of exact crossing sums and shows where exact distribution-free counting stops: under a composite null, fitted parameters change the path itself. Simulations and two small data diagnostics illustrate the resulting calibration warning.