Consistent spectral recovery of smooth reversible diffusion tensors at a fixed lag

Abstract

This research note proves almost-sure consistency of a specified finite-rank empirical spectral-equation estimator for a general smooth symmetric uniformly elliptic tensor and unknown positive smooth invariant density. The domain is known, bounded, connected and smooth, in dimension at least two. Observations are exact stationary positions at one known fixed positive lag, reflection is conormal, and fixed pointwise ellipticity and density bounds are known. The tensor and its separately fitted divergence converge locally uniformly; an ellipticity-clipped tensor converges globally in L2. The covariance is 2S/mu and the interior drift is (div S)/mu. The proof supplies boundary-corrected dependent-data kernel estimates, full spectral-jet excitation, continuous functional calculus through empirical spectral collisions and eigenvector-free measurability. This supplies a convergence analysis for the smooth setting of the spectral reconstruction question posed by Reiss with Chorowski, Gobet and Hoffmann in OWR24/2017, published in 2018. Earlier general nonparametric spectral residual fitting and transition-eigenvalue power weighting by Crommelin and Vanden-Eijnden, reversible identification, scalar fixed-lag inference, cutoff spectral excitation, deterministic tensor reconstruction and classical elliptic regularity are credited. A bounded dated primary-source audit located no encompassing theorem, with explicit edition/access gaps including the uninspected Crommelin-Vanden-Eijnden 2011 publisher-final body; no worldwide firstness guarantee is made. The supplementary classical regularization application is a new derivation of another estimator during this audit, not an earlier published resolution or a proof of the spectral objective. No rate, minimax, computational-efficiency, rough/nonreversible model, added sensor-noise or general-domain numerical implementation claim is made. The verification archive contains the standalone source, supplementary derivations, eight unchanged finite-control programs, selected expected scientific outputs, an actual-process replay runner, provenance and source-edition limits. Finite exact and diagnostic floating-point controls supplement the analytic proof and do not certify the infinite theorem or numerical asymptotic performance. This is an unrefereed preprint with no human peer review. AI tools were used extensively for derivation, literature comparison, computation, reproduction, writing and independent adversarial verification; internal AI reviews are not external human refereeing or a proof-assistant certificate.