Pinned Slope Measures, Curvature, and the Structural Exclusion of Additive Clustering in Falconer's Problem

We study the pinned slope and distance problems for planar sets of Hausdorff dimension s>1. We identify the sole obstruction to the expected dimensional lower bounds as the persistence of additive clustering in the pinned phase space. We develop a reduction showing that any counterexample forces the underlying measure to be supported near a low-complexity algebraic variety (a line or circle). We then prove that the cinematic curvature of the pinned projection family enforces quantitative additive expansion for all non-algebraic supports, thereby destroying the obstruction. This yields a geometric dichotomy: either the set is rectifiable, or curvature activates an entropy gap. As a consequence, the Falconer distance estimate holds for all planar sets of Hausdorff dimension s>1.