The Yang-Mills Mass Gap: Existence and Positivity from Instanton Cascade Architecture and the Discrete Spectrum of the Transfer Matrix

Description: We prove that pure SU (N) Yang-Mills theory (N ≥ 2) exists as a well-defined quantum field theory with mass gap Δ > 0. The proof is based on the Universal Cascade Theorem (UCT), which shows that the Yang-Mills gradient flow system satisfies three verifiable conditions C₁ + C₂ + C₃ — a compact absorbing set, a nondegenerate normal-form coefficient, and a transversal −1 Floquet crossing — that imply a universal Feigenbaum period-doubling cascade with constants δ = 4. 669201. . . and α = 2. 502907. . . . The cascade produces a discrete spectrum for the Yang-Mills transfer matrix at every lattice spacing a > 0. Positivity Δ (a) > 0 at each finite a follows from the Perron-Frobenius theorem. The continuum limit Δₚhys = lim₀→₀ Δ (a) > 0 is established by Kato-Rellich analytic perturbation theory Kato 1966 — not Davis-Kahan, which requires Δₚhys > 0 as input and is therefore circular for this application. A companion paper Randolph 2026b provides the complete proof of the UCT; this paper establishes that Yang-Mills satisfies C₁ + C₂ + C₃ and derives the mass gap. The proof applies to all compact simple gauge groups SU (N) with N ≥ 2. Keywords: Yang-Mills theory, mass gap, transfer matrix, Universal Cascade Theorem, Feigenbaum fixed point, Polyakov loop, Perron-Frobenius theorem, Kato-Rellich perturbation theory, Osterwalder-Schrader axioms, lattice gauge theory, Millennium Prize