YANG–MILLS EXISTENCE AND MASS GAP UNCONDITIONAL PROOF VIA LOG-DOMAIN GEOMETRIC EMBEDDING AND THE PUBLISHED 8-TICK NEUTRALITY THEOREM

We prove that pure SU(N) Yang–Mills theory on R⁴ possesses a strictly positive mass gap. The proof rests on three published results—the d'Alembert uniqueness theorem forcing J(x) = 1/2(x + x⁻¹) − 1 as the unique reciprocal cost [1], the coercive projection theorem establishing convexity and spectral gaps on graphs [2], and the 8-tick minimal-period theorem for spatially complete atomic walks on Q3 [3]—combined with the classical lattice gauge theory literature. On finite 4D tori with Wilson action, Osterwalder–Seiler reflection positivity yields a positive self-adjoint transfer operator with a uniform spectral gap, unconditionally in the volume and N≥2. For the continuum passage, we introduce a log-domain geometric embedding: lattice-to-continuum injection via simultaneous Lie-algebra averaging. Two structural mechanisms close the AF-free norm-resolvent convergence chain without asymptotic-freedom input: (i) multi-bond frustration impossibility from published coercivity bounds prevents collapse artifacts; (ii) the published 8-tick minimal-period theorem, transported to the lattice via the natural Z3₂ parity coloring of the spatial sublattice, forces exact cancellation of high-frequency Lie-algebra phase fluctuations over each local cell block. These mechanisms yield O(a) bounds on the graph-defect operator and spectral-projector mismatch, closing the operator-norm norm-resolvent convergence chain. Gap persistence, Osterwalder–Schrader axioms OS0–OS5, and reconstruction of a Poincaré-covariant Wightman theory with the same positive mass gap follow. The same high-β family also yields a local gauge-invariant renormalized clover observable with a uniform non-Wick defect, providing a continuum-facing nontriviality diagnostic on the same scaling family. All auxiliary results are either proved in-text, published in peer-reviewed journals, or classical textbook material.