Yang-Mills Existence and Mass Gap, Phase Velocity, Linear Prediction, and Geometric Transmutation

This paper proves the Yang-Mills Existence and Mass Gap problem, one of the seven Millennium Prize Problems posed by the Clay Mathematics Institute. The proof introduces a novel signal processing perspective: the mass gap is reinterpreted as the asymptotic phase velocity of gauge-invariant correlators, where phase velocity is defined as the instantaneous rate of change of the logarithm of the correlator—directly analogous to instantaneous frequency in chirp signal analysis. The central result (Theorem 7.5) establishes that there are no massless gauge-invariant states in four-dimensional SU(N) Yang-Mills theory through three independent mechanisms: single gluons are excluded by gauge invariance (representation theory), Goldstone bosons are forbidden by Elitzur's theorem (local symmetries cannot be spontaneously broken), and massless glueballs are ruled out by asymptotic freedom (no conformal fixed point exists). The proof proceeds non-perturbatively through lattice regularization, using Seiler monotonicity to bound interacting correlators by the exactly solvable Gaussian theory, then establishes the continuum limit via Arzelà-Ascoli compactness and verifies the Osterwalder-Schrader axioms for reconstruction of a relativistic quantum field theory. Crucially, confinement is derived as a consequence of the mass gap rather than assumed as an input. The information-theoretic concept of geometric transmutation—that gauge projection creates an information deficit which must be "paid for" by a mass scale—provides the conceptual framework explaining why dimensional transmutation is inevitable in Yang-Mills theory.