A Partial-Matching Proof of the Neyman--Pearson Lemma

Testing seeks to distinguish distributions, whereas matching seeks to identify their common mass. I show that this apparent opposition provides a direct proof of the Neyman--Pearson lemma. Maximal partial matching between the alternative distribution and a scaled null distribution separates them into a common component and two mutually singular residuals. An elementary dual problem then turns this separation into a testing rule: complementary slackness forces rejection on the alternative residual and non-rejection on the scaled-null residual, leaving freedom to randomize at the likelihood-ratio threshold. Thus the optimal test emerges from maximal matching rather than being proposed in advance. The matching profile also determines the optimal power envelope.