Covariate Restrictions of the Fisher–Rao Metric: Spectral Geometry, Semiparametric Efficiency, Tracial Categorification, and the Path-Space Geometry of Diffusions

The Fisher–Rao metric on the infinite-dimensional manifold of densities is a metric functionalwhose inversion and spectral analysis are widely regarded as intractable. A productive responseis to restrict the metric to a finite-dimensional, statistically meaningful subspace of the tangentspace and to study the resulting Gram matrix. We give a clean operator-theoretic formulation ofthis idea through a covariate score operator SA attached to an arbitrary admissible observableframe A; its Gram matrix Gf (A) = S∗ASA is the covariate Fisher information matrix (cFIM),which simultaneously generalizes the coordinate (translation) frame, exponential-family featureframes, and vector-field/group frames, and recovers a recently proposed orthogonal-decompositionframework as a special case. We then carry this calculus through geometry, category theory,statistics, and stochastic dynamics. Foundations: the scalar invariant (“G-entropy”) is theclassical Fisher information functional J(f) =R∥∇ log f∥2f; the assignment f 7→ Gf (A) isfunctorial under sufficient statistics, monotone under coarse-graining, and admits a coenddescription. Categorification: we resolve the integer/real type mismatch that blocks a naiveEuler-characteristic lift by categorifying in a tracial von Neumann (W∗) category (Mf , τf ) =(L∞(X), Ef ), in which the metric is the GNS pairing, the G-entropy is the real-valued trace of apositive frame observable, the Pythagorean law is trace additivity, functoriality is a Pimsner–Popacontraction, and the dual connections together with the Amari–Chentsov cubic tensor are thecumulants of the free-energy potential, the cubic tensor being the leading A∞ obstruction to dualsymmetry. Manifold Hypothesis: for a density concentrated within scale σ of a d-manifold thecFIM has n − d eigenvalues of order σ−2 (normal bundle) and d bounded eigenvalues (tangentbundle), so intrinsic dimension is the number of small eigenvalues—reversing a publishedprescription and matching the diffusion-model literature—and we prove exact finite-samplerecovery b d = d with probability 1 − 2e−cN once N ≳ n − d, with an eigenvalue/subspace centrallimit theorem and a consistent spectral-gap test. Efficiency and design: a frame-selectionprinciple replaces an unjustified “geometric alignment” postulate—the restricted geometryattains the semiparametric bound exactly when the frame spans the efficient score, Neymanorthogonalization being projection onto that frame and debiased machine learning attaining itsinverse, with structural-causal-model identification (do-calculus) selecting the estimand—and anEckart–Young theorem in the Fisher metric identifies the optimal frame under a dimension budget,with the restricted inverse cFIM as natural-gradient preconditioner. Dynamics: a generalizedde Bruijn identity shows the anisotropic G-entropy is the dissipation rate of relative entropyalong any Fokker–Planck flow, with Bakry–´Emery exponential decay; and via Girsanov’s theoremthe laws of Itˆo diffusions form an infinite-dimensional Fisher–Rao manifold whose metric is theexpected time-integral of the squared drift perturbation, onto which the entire apparatus lifts,1the marginal projection recovering the de Bruijn dissipation. We close with quantum, optimaltransport,reproducing-kernel, and Bayesian-nonparametric extensions and four open problemslinking modular theory, multiscale SDEs, semiparametric drift estimation, and nonequilibriumentropy production to the framework. A suite of controlled numerical experiments, packaged as areproducible notebook, confirms the central predictions—the eigenvalue split and dimension-fromsmall-eigenvalues rule, exact recovery at N ∼ n − d with the eigenvalue central limit theorem,the partially-linear-model efficiency gap as a law of total variance, the de Bruijn dissipationidentity to machine precision (and its persistence under irreversible circulation, with fivefoldacceleration), the Fisher-to-Otto ratio k on Hermite modes, the SLD/Kubo–Mori quantum split,and the Cartan/Amari–Chentsov identification.