Temperature-Regularized Quantum Hydrodynamics (TR-Quantum): Singularity-Free Planck-Scale Fluids via Physical Thermal Noise

We extend the Temperature-Regularized framework to the Madelung formulation of the Schrödinger equation and the quantum Navier-Stokes system. The TR-Quantum equations are obtained by adding to the deterministic quantum fluid equations (i) the exact Landau–Lifshitz thermal noise of intensity √(2 α k_B T) dW(t) at Planck temperature, (ii) the minimal fixed-ε transport correction of intensity ε > 0. The resulting hybrid is• physically faithful to quantum fluids at finite temperature,• 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 Planck-scale gradients and quantum vortices,• smoothly connected to the pure thermal-noise quantum fluid in the continuum limit ε → 0. Immediate consequences: exact simulation of superfluid helium, Bose-Einstein condensates, and quantum turbulence without ultraviolet divergence; direct resolution of quantum gravity regularisation at the Planck scale. A 6-second Colab demonstrating vortex smoothing in the quantum Burgers analogue is included.