What Quotients Forget: The Isotropy Anomaly of Entropic Optimal Transport

On spaces observed only up to compact symmetry, quotienting and entropic regularization need not commute: the quotient retains the best alignment cost but discards the thermal volume of near-optimal alignments. For a compact Lie group acting isometrically on a complete finite-dimensional Riemannian manifold, we prove an exact positive-temperature reduction of invariant entropic optimal transport to the quotient, with orbital log-partition cost and explicit Gibbs reconstruction. Under clean Morse--Bott minima, ( \OrbCost_\eps=\barc+\frac{\kappa}{2}\eps\log(1/\eps) -\eps[\frac{\kappa}{2}\log(2\pi)+\log\cA]+o(\eps) ), where (\kappa) is the transverse alignment codimension and (\cA) the inverse-normal-Hessian amplitude; for a homogeneous optimum, (\kappa) is an aligned-stabilizer dimension defect. For finite orbital mixtures this yields a three-level (\Gamma)-development selecting quotient cost, average codimension, and corrected entropy--amplitude. Hence reducing before regularizing can select a different zero-temperature plan, as an exact (\mathrm{SO}(2)) model shows. For squared distance, Sinkhorn centering leaves a nonnegative isotropy-mismatch coefficient. Across a rank-one wall (H/L\simeq\mathbb S^\ell), collapsed directions are resolved at scale (s\asymp\eps) by a spherical Bessel profile and a latent von Mises--Fisher law; a closed-form (\mathrm{SU}(2)) model audits the wall and centering constants.