Finite-Frequency Mori Transport Geometry-Dynamic Curvature and the Non-Markovian Departure from the Markovian Limit-

This paper extends the zero-frequency Mori transport geometry developed in our previous work to finite-frequency and non-Markovian regimes. Starting from the frequency-resolved projected-force correlation tensor, we define a frequency-dependent dissipative Mori tensor and use it to construct a family of transport metrics parameterized by frequency. This leads naturally to frequency-dependent Levi-Civita connections, Riemann curvature tensors, Ricci tensors, and scalar curvatures. A low-frequency expansion is derived in terms of temporal moments of the projected-force memory correlation. In particular, the second temporal moment controls the leading quadratic deformation of the transport metric away from the zero-frequency limit. This motivates the introduction of a memory-curvature susceptibility, which quantifies the sensitivity of transport curvature to non-Markovian memory. We further show that the stress-gauge invariance established for the zero-frequency formulation extends to finite frequency: when a change in microscopic stress representation leaves the projected force invariant, the frequency-resolved memory tensor, transport metric, and curvature remain invariant as well. The transport curvature is also decomposed, through the Gauss equation, into a frequency-dependent ambient memory contribution and a coarse-graining contribution associated with the nonlinear embedding of the transport-state manifold. Analytical two-mode models and symbolic calculations are used to verify the formalism, including the zero-frequency limit, the Markovian limit, the frequency dependence of the induced metric, and the resulting dynamic curvature. The resulting framework provides a geometric description of how non-Markovian memory deforms transport itself, rather than merely correcting transport coefficients, and connects frequency-dependent Mori memory kernels with dynamic transport geometry.