Recovering Weighted Tangent Geometry from a Single-Scale Score Field

Near a smooth data manifold, one tangent space summarizes local geometry. At a branch point, the corresponding first-order object is instead a measure over tangent directions, whose normalized masses record the local share of each branch under the chosen data measure. We ask whether a score field at one noise level determines this weighted tangent geometry when the branch center and homogeneity degree d are unknown. In this tangent-measure model, d is the local measure dimension. Gaussian smoothing of a homogeneous tangent measure satisfies an Ornstein--Uhlenbeck eigenfunction equation. Its weak form turns score values---without score derivatives---into a linear system for the center and homogeneity degree, with an explicit rank condition and perturbation bound. After this calibration, the tangential score on one sphere is the spherical log-gradient of a scalar Gaussian--cone transform. Integration recovers that transform up to scale, and all its spherical-harmonic multipliers are positive. Thus one exact shell identifies the normalized angular measure in every ambient dimension D≥2. For at most K positive rays, moments through degree 2K-1 constructively recover count, directions, and weights in arbitrary dimension. Any fixed observation scheme needs at least KD-1 scalar tangential components. In the plane, degree K is both sufficient and necessary, and we give quantitative finite-query certificates. For finite planar C1,β branches with positive C0,β densities, we prove O(σβ) convergence from the finite-noise score to its tangent model. In controlled experiments, 50k-step training lowers validation normalized-score error across four geometries yet raises angular-moment error, separating ordinary score fit from geometry recovery.