Ordinal Geometry Complements Reconstruction: Diagnosing Planning with Compressed Value Functions

Goal-conditioned planning requires compressing value functions into low-dimensional representations, yet which property of the compression best predicts planning quality remains unclear. We show that reconstruction accuracy and ordinal fidelity—how faithfully the ranking of successor states by value is preserved—are two projections of the same approximation error: in smooth geometry they couple tightly and $L^2$ is a sufficient summary; when topology creates dense small-gap regions they decouple, and the ordinal projection carries complementary planning-relevant information beyond $L^2$. Through experiments on gridworld, continuous navigation (640 runs, 8 environments), and Maze2D (240 runs, 3 D4RL layouts), we establish three results. First, neighbor-restricted ordinal fidelity (${\tau }^{\mathrm{nbr}}$) adds $\Delta R^2=0.106$ of incremental explanatory power beyond $L^2$ after environment fixed effects. Second, monotone recalibration degrades $L^2$ by up to $\sim 400\times$ while preserving ${\tau }^{\mathrm{nbr}}$ and planning success. Third, in matched-pair comparisons controlling for $L^2$, higher ${\tau }^{\mathrm{nbr}}$ predicts better planning 76% of the time. We provide theoretical grounding via a gap-weighted Bellman-excess decomposition that links reconstruction error, local ordinal inversions, and planning regret.