A Minimal Unified Framework: Coarse-Graining Fixed Points and Quantum Statistics

A Minimal Unified Framework: Coarse-Graining Fixed Points and Quantum Statistics {Conceptual motivation}Statistical mechanics is traditionally formulated as a theory of equilibrium distributions derived from entropy maximization under conservation constraints.However, the same mathematical structure appears in several seemingly distinct contexts:(i) Maxwell--Boltzmann, Bose--Einstein, and Fermi--Dirac statistics,(ii) the central limit theorem (CLT),(iii) renormalization-group (RG) fixed-point structures,and (iv) the thermodynamic zeroth law.This section introduces a minimal framework in which these phenomena are interpreted as manifestations of a single principle