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For a scalar parameter restricted to $\psi \ge 0$, test $\psi =0$ against $\psi >0$ with identifiable interior nuisance parameters. A locally quadratic regular likelihood has an efficient null score $Z\sim N(0,1)$, and the constrained fit projects the unconstrained local maximizer onto the nonnegative half-line. ConsequentlyFor a positive statistic, its asymptotic upper-tail p-value is half the ordinary one-degree chi-squared tail. Its 5% critical value is the 90th chi-squared percentile, approximately 2.7055. Additional boundary nuisance parameters or nonidentified mixture parameters can change this law.

Given valid p-values $q_J$ for constancy on intervals of length at least two, reject $I$ when ${\max }_{J\supseteq I}n{q}_J/|J|\le \alpha$. Every rejected true interval forces rejection of its maximal constant block. A union bound over the non-singleton maximal blocks gives familywise error rate at most $\alpha$, since their lengths sum to at most $n$. Singleton blocks have no tested interval in this formulation.