Uniqueness of nonnegative heat controls at the minimum time

We consider the heat equation on an interval with two nonnegative Dirichlet boundary controls. For every pair of distinct positive constant states, we prove that the nonnegative finite Radon measure pair attaining the minimum time is unique, under the terminal sine-moment formulation. The two boundary measures are equal. This answers the continuum uniqueness question posed by Lohéac, Trélat and Zuazua. The proof uses finite positive moment cones at the critical time. Their extremal supporting functions have vanishing cost on every admissible measure. Exponential interpolation changes their exponents without increasing their magnitude, while a determinant estimate keeps an inserted coefficient away from zero. These tests determine both the average control and the difference between the boundary controls, including endpoint atoms.