The Fisher Information Sector of Perelman Functionals on Black Hole Spatial Slices

We study the Fisher information component F_nabla of Perelman F-functional on black hole spatial slices, using the heat kernel to determine the Boltzmann measure from first principles. Perelman Ricci flow functionals (2002) have a natural statistical mechanics structure: the F-functional decomposes into a curvature component F_R and a gradient component F_nabla, where the latter is a Fisher information in the sense of Amari (1985). Despite significant interest in connecting Perelman mathematics to gravitational physics (Vacaru et al. 2013-2025, Li 2013), no explicit computation of these functionals on a black hole geometry has appeared in the literature. This paper fills that gap. We compute Perelman F and W on the BTZ spatial slice and discover a structural result: on constant-curvature backgrounds, the curvaturecomponent carries no state-dependent information, and the Fisher information component carries all of it. We then derive the identification tau = 1/(2 kappa) from three independent routes (Lichnerowicz spectrum, Euclidean BTZ periodicity, Wiesbrock modular structure), upgrading the main result F_nabla = kappa from conditional to unconditional.