Matched Queries for Curvature and Density at Branching Junctions

At a junction, a score field can reveal weighted tangent rays, yet these first-order quantities do not determine how individual branches bend or how their densities change away from the center. Recovering this missing information is necessary for describing local continuation beyond a single point, but finite observations must separate branchwise second-order effects while allowing error in the estimated center. We address this inverse problem using matched score queries at noise scales σ and λσ. For a finite union of C2,α half-branches in ℝD, the normalized score has the expansion Fσ=F0+σ G+O(σ1+α). Matched subtraction cancels the tangent contribution and exposes G, which depends linearly on branchwise curvature and log-density slope. Given tangent directions and weights on distinct rays, G uniquely identifies all sD branch parameters, and sD scalar component observations are necessary. An O(σ2) center error introduces D translation modes, leading to (s+1)D observations under full-rank calibration, except for a translation-invariant full line. We also establish a perturbation bound and a conditional kernel-density-estimation rate. Experiments reproduce the predicted population and N-1/5 trends and remain full rank up to D=20 with 16 supplied branches. In end-to-end tests for D=3--5, a known-count first-order frontend yields full rank in all 135 population systems and a median relative jet error of 0.132. With strong first-order error, matched responses reduce median parameter error by a factor of 49.4 relative to naive tangent subtraction.