Finite Reclosure and Temporal Fisher Transport: A Local Lorentzian Response-Curvature Construction

Abstract

We construct an explicit local Lorentzian response geometry with nonzero curvature from two previously separated ingredients: the Fisher–Rao geometry obtained as a hidden-flow quotient of a finite closure–residue system, and a continuous-time population flow induced by a finite-dimensional CPTP/GKSL dynamics. At the uniform four-state cell, the Fisher metric agrees with the metric used in the covariant finite-reclosure construction, and the exact quadratic reclosure map reproduces the constant-curvature Fisher tensor with a fixed coefficient. Transporting Fisher distinguishability by the observable Markov flow gives the unique pullback metric associated with operational tangent covariance. At a reversible stationary point the corresponding shape operator equals the Markov relaxation generator restricted to the probability tangent space. For an explicit anisotropic four-state reversible generator we compute a nonzero Codazzi defect and hence a nonzero mixed Lorentzian Riemann-curvature component. This supplies a finite, exact witness of local Lorentzian response curvature; it is not a derivation of the Einstein field equations, a universal physical light cone, or a global spacetime. We further audit three stronger extensions. First, the reclosure-twist map admits a closed finite-cell formula; under a canonical Fisher-frame identification, torsion-free matching selects the normalization λ = 1. Second, exact microscopic propagation on the natural periodic K4 root lattice is anisotropic, while its quadratic infrared spatial symbol is forced to be Fisher-isotropic. Third, universal coupling and common propagating null branches follow conditionally from a single-master generative completeness principle, while a decoupled spectator construction shows that such completeness is not implied by the preceding finite reclosure results. These stronger statements are kept logically separate from the minimum curvature theorem.