Uniform Partition-Function Estimates for Coulomb Modulated Energy at All Positive Temperatures

We prove uniform-in-$N$ partition-function estimates at all positive temperatures for the centered modulated energy of logarithmic, Riesz, and Bessel–Riesz kernels on ${\mathbb{R}}^d$ in the locally square-integrable range $0\le s<d\mathrm{/}2$. They yield finite-particle exponential integrability, quadratic behavior at small parameter, explicit kernel-dependent growth at large parameter, and uniform entropy control of the associated Gibbs measures. As applications, we obtain uniform Rényi-divergence and relative-entropy bounds for interacting Gibbs equilibria and entropic mean-field convergence near thermal equilibrium; the optimal $N^{-1}$ normalized relative-entropy rate for the three-dimensional Coulomb flow; and an $N^{-1\mathrm{/}2}$ Gaussian approximation for fixed-time finite-dimensional fluctuations. Finally, the modulated energy converges to a random variable in the second Wiener chaos, and for every positive parameter the partition functions converge to its Laplace transform, represented by a Carleman–Fredholm determinant. This formula identifies the sharp small-parameter behavior and, when the reference density is bounded below on a ball, the sharp large-parameter growth.