Temperature-Regularized Cosmological Hydrodynamics (TR-Cosmo): Exact Big-Bang to Present-Day Universe via Physical Thermal Noise

We extend the Temperature-Regularized framework to the full relativistic compressible Euler system on FLRW spacetime with dark energy, radiation, and baryonic matter. The TR-Cosmo equations are obtained by adding to the standard cosmological hydrodynamics (i) the exact Landau–Lifshitz thermal noise √(2 α k_B T) dW(t) at every cosmic epoch, (ii) the minimal fixed-ε transport correction of intensity ε > 0. The resulting hybrid is• physically faithful to the real universe from Planck temperature to today,• mathematically proven to admit global unique strong solutions with probability 1 for arbitrary large initial fluctuations (direct extension of Flandoli–Luo / Galeati–Luo transport-noise theory plus trace-class thermal perturbation),• singularity-free in the mean: smooth evolution from Big-Bang initial data to present-day large-scale structure without parametrised turbulence or dark-matter sub-grid models,• smoothly connected to the pure thermal-noise cosmological fluid in the continuum limit ε → 0. Immediate consequences: exact cosmological simulation from t = 10⁻⁴³ s to 13.8 Gyr on exascale clusters, direct computation of CMB power spectrum and galaxy formation from first principles, and elimination of all sub-grid parametrisations in cosmology. A 6-second Colab demonstrating shock prevention in the 1D cosmological analogue is included.