Dynamical Completion of the QYM Structural Framework: Scaling, Gribov Geometry, and Mass Gap

Description This work provides the dynamical and mathematical completion of the framework developed in QYM_Final (6).pdf, where a gauge-invariant structural functional ( \Phi[A] ) was introduced via Yang–Mills heat-flow regularization. While QYM_Final establishes the geometric and spectral role of ( \Phi[A] ) in relation to the Gribov region and the Faddeev–Popov spectrum, the present paper (QYM_Completion) takes this structure to its scaling limit and derives its full physical consequences. Interpreting the heat-flow parameter ( t ) as an inverse renormalization-group scale ( (t=\mu^{-2}) ), we analyze the limit ( t\to 0 ). In this regime, the Fisher–Hardy potential generated by ( \Phi[A] ) sharpens into an effective Dirichlet boundary condition on the Gribov horizon. The resulting limit Hamiltonian acts on (L^2(\Omega)), where ( \Omega ) is the first Gribov region, and is shown to be essentially self-adjoint with compact resolvent and purely discrete spectrum. Using geometric spectral methods (including Cheeger-type bounds), we demonstrate the existence of a strictly positive, nonperturbative Yang–Mills mass gap of order ( \Lambda_{\text{QCD}}^2 ). Crucially, we prove that all gauge-invariant observables computed in this restricted framework coincide with those of standard Yang–Mills theory, establishing physical equivalence while providing a mathematically controlled setting. Together with QYM_Final, this two-paper program offers a fully nonperturbative approach to the Yang–Mills mass-gap problem based on configuration-space geometry, Gribov spectral structure, and heat-flow regularization—without lattice discretization, semiclassical expansion, or perturbative assumptions.