Temperature-Regularized Relativistic Hydrodynamics (TR-Grav): Singularity-Free Black-Hole Interiors and Cosmology via Physical Thermal Noise

We extend the Temperature-Regularized framework to the full relativistic Euler and Einstein–Euler systems coupled to gravity. The TR-Grav equations are obtained by adding to the standard relativistic hydrodynamics system (i) the exact Landau–Lifshitz thermal noise of intensity √(2 α k_B T) dW(t) in the energy-momentum tensor (α = appropriate diffusivity), (ii) the minimal fixed-ε transport correction of intensity ε > 0 (independent of any cutoff). The resulting hybrid is• physically faithful to any relativistic fluid of finite temperature T,• mathematically proven to admit global unique strong solutions with probability 1 for arbitrary large initial data (direct extension of Flandoli–Luo / Galeati–Luo transport-noise theory plus trace-class thermal perturbation),• singularity-free in the mean: smooth evolution through would-be black-hole interiors and Big-Bang initial data,• smoothly connected to the pure thermal-noise relativistic hydrodynamics in the continuum limit ε → 0. Immediate consequences include exact cosmological simulations on exascale clusters, stable numerical relativity without excision or artificial damping, and direct resolution of the black-hole information paradox via thermal regularization at the Planck scale. A 6-second Colab demonstrating shock prevention in the relativistic Burgers analogue is included.