Curvature-Conditioned Measures for Cosmological Peak Statistics: A Transport-Geometric Framework

We develop a transport-geometric theory of cosmological peak statistics based on optimal transport and entropy geometry. The density field is treated as a probability measure in Wasserstein space, and its local structure is characterized by a logarithmic curvature tensor obtained as the localized response of an entropy functional. Peaks are thereby defined as positive-curvature stationary points, and their abundance is formulated as a curvatureconditioned measure on local tensor space. In the Gaussian-linear limit, this measure admits a finite-dimensional closure in terms of the spectral moments of the density field. The resulting peak abundance reduces exactly to the classical BBKS formula, identifying BBKS as a solvable Gaussian closure of a more general geometric structure. This formulation separates peak statistics into three fundamental ingredients: the probability distribution of local variables, the positive-curvature constraint, and the induced geometric measure. The theory extends naturally beyond the Gaussian approximation. Nonlinear evolution appears as a deformation of the logarithmic curvature geometry, while primordial non-Gaussianity is interpreted as a deformation of the curvature-conditioned measure itself. We further formulate two- and three-point peak statistics as higher-order curvature-conditioned measures and show that the resulting hierarchy can be organized as response functions to long-wavelength background modes, with conventional peak bias emerging as the lowest-order response coefficient. These results provide a unified description linking optimal transport, curvature geometry, peak statistics, and cosmological observables, and establish a systematic framework for studying nonlinearity, scale dependence, primordial non-Gaussianity, and higher-order peak correlations.