We study the covariance geometry associated with probability measures in aGaussian concentration regime. Near a nondegenerate minimum of a smoothpotential, Laplace asymptotics identify a local Gaussian approximation, sothat the quadratic Wasserstein geometry of the concentrating family isasymptotically described by the geometry of Gaussian measures. On theGaussian manifold, the quadratic Wasserstein distance is exact and, in thecentred covariance sector, reduces to the Bures--Wasserstein metric on thecone of positive-definite matrices. The corresponding intrinsicgradient-flow structure on the centred Gaussian submanifold is thereforefinite-dimensional and takes place on covariance space. The main result of the paper concerns the internal geometry of thiscovariance sector in even dimension. Using Williamson's symplecticdecomposition, we consider the local orbit generated by independent one-modesymplectic transformations and compute the Bures--Wasserstein orthogonalcomplement of its tangent space at a Williamson normal form. We show thatthis orthogonal complement decomposes explicitly into radial directions,which change the symplectic eigenvalues, and off-diagonal directions, whichcreate cross-mode covariance couplings. Consequently, if a smooth functionalis critical along the local orbit, then its Bures--Wasserstein gradient liesin this transverse space; if the radial coefficients vanish, the gradient ispurely off-diagonal. We further show that this block structure is notspecial to the normal form: although the symplectic congruence action isnot isometric for the Bures--Wasserstein metric, a Lyapunov change ofvariables shows that the same radial-plus-off-diagonal decomposition of theorthogonal complement holds at every point of the local orbit. Passing fromfirst to second order, we compute the second fundamental form of the localorbit: the block-diagonal locus is totally geodesic, all cross-modecomponents vanish, and each fixed-determinant mode factor is umbilic atevery point, with radial normal curvature\(2\nu_i/(\operatorname{Tr}\Sigma_i)^{3/2}\), equal to \((2\nu_i)^{-1/2}\)at the normal form. We also record the behaviour of the Bures--Wasserstein metric near theboundary of the positive-semidefinite cone: in the classical concentrationregime, the covariance approaches the rank-deficient boundary, where themetric tensor blows up in directions with a nonzero component in theemerging kernel block, although the boundary itself lies at finiteBures--Wasserstein distance. A brief final remarkexplains how the same covariance variables admit a quantum Gaussianinterpretation, but the main metric results of the paper are entirelyclassical.

This working paper is part of foundational project developing a unified geometric framework for the inference–action transition (see 10.5281/zenodo.19210601).