The Yang–Mills Mass Gap and the Proof of Uniform Modular Coercivity

is manuscript develops a fully operator-algebraic formulation of the Yang–Mills mass gap problem within the CETΩ framework. It constructs AF inductive systems, Haag–Kastler local nets, and CCR/CAR continuum field algebras, together with a complete modular analysis based on Tomita–Takesaki theory. The central result is the proof of Uniform Modular Coercivity for the reversible modular CPTP semigroup, shown to be stable along AF towers and in the continuum limit through Mosco convergence of Dirichlet forms. This functional-analytic property implies a strictly positive spectral gap for the physical Yang–Mills Hamiltonian. The work unifies modular theory, spectral geometry, quantum log-Sobolev inequalities, CPTP dynamics and algebraic quantum field theory into a single coherent, non-perturbative mathematical framework for gauge fields.