Extrinsic curvature bias in the small-regularisation limit of Sinkhorn metrics

Unrefereed research candidate. Entropic (Sinkhorn) optimal transport defines a local Riemannian metric on probability distributions, and as the regularisation ε tends to 0 it should recover Wasserstein geometry. This paper studies that limit for a smooth positive density on a closed manifold embedded in Euclidean space with the fixed ambient cost |x−y|². Results (proved). The Sinkhorn metric operator converges in operator norm on centred L² to the inverse weighted Laplacian, at rate O(ε1/2) in L² operator norm and O(ε) from H² to L² (Theorems 2.1 and 2.4; the sharp O(ε) operator-norm rate is open). For a smooth tangent hμ with Aφ = h, A = −ρ−1div(ρ∇), gε(hμ,hμ) = ∫|∇φ|²dμ − (ε/4)∫⟨Q∇φ,∇φ⟩dμ + O(ε²), with Q = ΣαSα² built from the shape operators: the first bias is nonpositive and depends on the embedding, not only on the intrinsic geometry (chord shorter than arc). On the circle of radius R the eigenvalue 1/k² becomes (1/k²)(1 − ε/(4R²)) to first order, and this rate is sharp. A positive correction built from the entropic inverse removes the term (Corollary 2.2); the expansions hold along smooth finite-dimensional density families (Theorem 2.3). Round spheres Sm are solved exactly to second order (Proposition 9.1); a circle high-frequency family shows no operator-norm rate better than O(ε) is possible (Proposition 9.2); isometric embeddings give different biases (Proposition 10.1). Credit and scope: the formal limit is due to Lavenant, Luckhardt, Mordant, Schmitzer and Tamanini (2025), made rigorous here in this setting. Guo (2026) already gives the tensor Q/16 (via his Theorem 1 and the Gauss equation) and the first sphere and circle bias, and the coefficient of the inverse formula then follows formally; this paper's contribution is the rigorous inverse-metric theorem, not discovery of the coefficient. Guo works pointwise with the row-normalised operator at uniform density; his paper has no Sinkhorn scaling, inverse, metric, operator-norm statement or nonuniform-density bias. Other operator-level expansions (Smolyanov–von Weizsäcker–Wittich; Liou–Lu) are credited. Population densities only (no sampling rates), closed manifolds without boundary, positive smooth density; the correction needs known curvature; no global geodesic-distance convergence. Version 0.2.0-candidate corrects the comparison with prior work after Guo's paper was read completely. Version 0.1.0-candidate (doi:10.5281/zenodo.23940148) had said that paper was unread. No theorem, proof or numerical result changed; a comparison script is added. Archive: LaTeX source and PDF, bibliography from saved Crossref, arXiv and DataCite records, floating-point check scripts with outputs (replayed with and without python -O), symbolic sphere-coefficient check, figures, proof notes, prior-art ledger, rating-free review summary, CI and SHA-256 manifest. Assurance: internal AI-assisted review only (not human peer review, independent reproduction or formal verification); the scripts are floating-point checks, not certificates or proofs. Creator: Anonymous. Text CC0-1.0; code MIT.