We formulate a weighted-Wasserstein framework for curvature statistics of cosmological large-scale structure. A coarse-grained density defines a weighted reference measure whose relative-entropy response separates into kinematic and density-induced Bakry–'Emery contributions. On a Euclidean background, the latter is governed locally by $H_R=-{\nabla }^{2}ln{\rho }_R$; finite windows define density-weighted directional responses, while finite synthetic dimension subtracts a nonnegative score-alignment deficit. The controlled local limit recovers the eigenvalue-threshold classification of the smoothed log-density Hessian, not a general equivalence with density-Hessian or gravitational-tidal classifiers. Independent smoothing and probe scales define a resolution flow. Controlled tests recover quadratic tensors to relative accuracy ${10}^{-9}$ or better and confirm $O(\epsilon )$ and $O({\epsilon }^2)$ convergence. In a periodic three-dimensional Zel’dovich mock, the specified six-direction transport tensor matches its direct finite-window reference with subpercent median error and directional correlation above $0.999995$. Rotated and enriched probe designs reveal appreciable finite-window eigenvalue and classification dependence, distinct from estimator error; a converged angular projection still has a nonquadratic directional residual. Under thinning to $f_{samp}=1/16$, the six-direction internal error remains below $0.3\%$, while density reconstruction dominates the total error. At the fiducial radius, this finite-window observable has approximately $30\%$ lower normalized sampling error than the pointwise log-density Hessian, each relative to its own full-sampling reference. In the Gaussian-linear limit, curvature spectra are differential transforms of two-point statistics; finite-window and finite-dimensional observables retain additional profile and directional information beyond that limit.