We give an exact randomization-based confidence set for the average treatment effect (ATE) in matched-pair studies with a binary outcome, requiring neither monotonicity nor any distributional assumption beyond the within-pair coin flip. At its core is an analytic solution to the worst-case allocation of attributable effects: two binomial-symmetry lemmas identify the pattern hardest to reject as a single boundary corner, so testing null hypotheses needs no integer program and no numerical search. Inverting the test via binary search yields a prediction set for the attributable effect in O(log S) Binomial tail calculations; the Bonferroni proposition of Rigdon and Hudgens (2015) produces the ATE confidence set at the same computational cost. The same corner extends without further machinery to a sensitivity analysis for matched observational studies under Rosenbaum's $Γ$-model. A simple formula for the design sensitivity illuminates when an observational study can hope to provide evidence for an effect.

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