Recovering Weighted Tangent Geometry from a Single-Scale Score Field

Score fields learned by diffusion models offer a direct probe of local data geometry, but the usual tangent-space description is insufficient at branch points, where the first-order object is a weighted measure over branch directions. Recovering this geometry matters because junctions arise when trajectories merge, road segments intersect, or strata meet, and their directions and relative masses determine what a learned field represents. We ask whether a score field at one known noise level can recover the branch center, local homogeneity degree, directions, and relative masses. This inverse problem is difficult because the observed score is an uncentered vector field, a one-point Hessian cannot distinguish even regular junctions with different branch counts, and ordinary score accuracy need not preserve the angular information required for geometry recovery. We address these challenges by using homogeneity under Gaussian smoothing to obtain a linear, score-only calibration of the unknown center and degree. Once centered, the tangential score on one shell integrates to a scalar transform with strictly positive spherical-harmonic multipliers. This yields exact identification of the normalized angular measure in any ambient dimension $D\ge 2$ and constructive recovery of finite positive rays. In the plane, it further yields sharp moment requirements, finite-query certificates, and an $O({\sigma }^{\beta })$ tangent-score approximation for finite $C^{1,\beta }$ branches. Experiments apply the full chain to population, kernel-density, and learned score fields, showing across four geometries that lower validation normalized-score error can coincide with higher angular-moment error, thereby distinguishing ordinary score fit from recoverable junction geometry.