Part VII: Spectral Proof of the Mass Gap in Yang–Mills Theory

This article presents a nonperturbative spectral proof of the mass gap in Yang–Mills theory using the quantized deformation modes {Cn} of spacetime geometry. The discrete evolution of these modes, governed by a self-adjoint spectral operator and interacting via gauge-invariant nonlinearities, defines stable excitations above the vacuum configuration Cn = π. Built on the spectral operator formalism developed in earlier parts, the present work rigorously establishes the existence of a positive spectral energy gap in the nonperturbative regime. The spectral dynamics is formulated via a variational principle over the mode space ℓ²(ℤ), with local SU(N)-type gauge symmetry realized internally in spectral space. The gap emerges from the discreteness of the spectrum, the strict convexity of the energy functional, and the global stability of spectral interactions. Mass is interpreted as a geometric excitation — a stationary resonance in the spectrum of quantized deformation. The theory satisfies Osterwalder–Schrader positivity, cluster decomposition, and gap stability under the continuum limit, thereby providing a constructive nonperturbative realization of the Yang–Mills mass gap in four dimensions.<br>