Common-Center Geometry and Certified Radial Reconstruction for Energy-Form Full Conformal Regions

Abstract

This note studies the geometry of full conformal prediction (FullCP) regions generated by an empirical energy-form pairwise score. Candidate-score convexity alone does not guarantee connected FullCP regions, even for empirical averages of losses convex in the candidate argument. For the energy-form score, each leave-one-out training comparison reduces exactly to a pairwise-dissimilarity sublevel condition. Under symmetry, a constant diagonal, a diagonal lower bound, and attainment of the associated Fr\'echet-type objective, all comparison regions share a minimizer; if they are convex, every nontrivial exact conformal region is star-shaped about that point. For power distances $\rho_\beta(x,y)=\|x-y\|^\beta$, this geometry holds for $\beta\ge1$, while the conventional energy score is strictly proper for $0<\beta<2$. For $d=1,\beta=1$, every nontrivial empirical-CRPS FullCP region is a nonempty closed interval (possibly $\mathbb R$ when $m=1$). For $1<\beta<2$ and $m\ge2$, explicit data-checkable derivative bounds give Lipschitz control of the radial exits and exact conformal radial function. Combined with directional root search and classical Lipschitz extensions, they yield certified inner and outer radial envelopes of width at most $\delta+2\widetilde Lh_{\mathcal U}$ and same-ray Hausdorff guarantees. An analytic two-dimensional example shows why preserving star-shaped but nonconvex geometry can matter. A staged two-dimensional study finds modest but systematic tightening of the generic certificate and frequent robust nonconvexity witnesses, with detected normalized radial departures typically small. The method is intended for low-dimensional multivariate outputs rather than high-dimensional scaling or runtime improvement.