Formal Documentation for the Yang-Mills Millennium Prize Problem: The McEc-Topological Framework

This paper provides a rigorous, non-perturbative axiomatic construction of quantum Yang-Mills theory with a compact non-Abelian gauge group on , establishing a formal connection between structural elastodynamics and subatomic gauge interactions. By mapping the topological and rheological invariants of a self-organizing continuum onto the quantum vacuum, we demonstrate that the system undergoes an algebraic scale-invariance collapse, defined as the Inertial Cancellation Principle. We rigorously prove three fundamental theorems of convergence and regularity: first, the weak convergence of the discrete lattice action to the continuous Yang-Mills action under strict reflection positivity, satisfying the Osterwalder-Schrader reconstruction axioms; second, the uniform stability and scale-invariance of the covariant Laplace-Beltrami operator spectrum via the conformal invariance of the Hodge star operator in four dimensions; third, the compactness of the quantum Hamiltonian resolvent, driven by an asymptotic fractal volume lens mechanism. By feeding experimentally verified parameters of dense nuclear matter into this geometric-kinematic framework, we analytically derive the existence of a non-zero lower bound in the energy spectrum (the mass gap), quantized at the exact stable value of , alongside an emerging isotropic confinement pressure of which topologically resolves QCD color confinement.