Matched Score Queries for Curvature and Density at Branching Junctions

At a junction a score field reveals weighted tangent rays, but these fix only first-order geometry. They say nothing about how a branch bends or how the density changes away from the center: the second-order structure needed to continue a branch past the junction. Small-noise expansions place this curvature and density in the first correction term, yet do not establish whether finite score observations can isolate the per-branch contributions when the center is also estimated. We answer this as an inverse problem, by matched queries at scales $\sigma$ and $\lambda \sigma$. For a finite union of $C^{2,\alpha }$ half-branches in ${\mathbb{R}}^D$ the normalized score obeys $F_{\sigma }=F_0+\sigma G+O({\sigma }^{1+\alpha })$; matched subtraction cancels the leading field $F_0$ rather than estimating it, exposing $G$, which is linear in the branchwise curvature and log-density slope. Given the tangent directions and weights on distinct rays, $G$ identifies all $sD$ branch parameters. Point-coordinate queries attain this sharp scalar-information count, and any fixed continuous scheme that identifies every jet requires at least $sD$ scalar outputs. This counts information rather than score-network evaluations. A geometry-dependent condition number sets stability as rays approach one another. Under coarse localization and full-rank calibration an $O({\sigma }^2)$ center error contributes $D$ translation modes, so the augmented problem needs $(s+1)D$ scalar observations except along a translation-invariant full line; we derive a perturbation bound and a conditional kernel-density rate. Across 180 population fits the fitted median exponent 0.998 matches the predicted $O(\sigma )$ order; matched-query systems remain full rank to $D=20$ with 16 branches, and a known-count frontend fitted to finite-noise scores composes with the inverse in $D=3$–$5$ (135 population systems, median jet error 0.132); under strong population first-order error, matched responses reduce the median parameter error by a factor of 49.4 relative to naive tangent subtraction. The contribution is an information count: matched subtraction turns a weak cross-scale signal into a well-posed inverse whose second-order branch geometry is identifiable at an $sD$ (resp. $(s+1)D$ under center error) scalar-information cost.