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.

翻译：科尔莫戈罗夫-斯米尔诺夫统计量通常以 supremum 形式引入，但其有限样本行为由更局域的问题主导：经验过程首次穿过边界的时刻是什么？本文通过一个有限样本的穿越分类账提供部分解答。该分类账将斯米尔诺夫-伯恩鲍姆-廷吉单样本公式重写为显式命中时间律，并导出一个稳定的对数尺度尾部评估器。针对双样本情形，它给出了任意样本量下单壁与双壁的精确格点递推关系，其中平衡反射公式作为特例出现为闭合形式。同一视角将德沃雷茨基-基弗-沃尔福威茨-马萨特不等式解释为精确穿越总和的指数压缩，并揭示了无分布精确计数停止的边界：在复合零假设下，拟合参数改变了路径本身。模拟实验与两个小规模数据诊断展示了由此产生的校准警示。