Intrinsic O'Neill Curvature of the Gauge-Orbit Submersion, Zero-Residue Selection, and Uniform Bakry–Émery Coercivity in Four-Dimensional SU(N) Yang–Mills Theory

Intrinsic O'Neill Curvature of the Gauge-Orbit Submersion, Zero-Residue Selection, and Uniform Bakry–Émery Coercivity in Four-Dimensional SU(N) Yang–Mills Theory

Dated (Original): 18^th Oct 2026

Author: Jinhyuk Fred CHUNG, PhD

Affiliation: CEO/CSO at Xylonix PTE. LTD. (Singapore)

Email: j.f.chung@xylonix.io

Series: 5D-ICF DIM Series Part XI (The Yang-Mills Mass Gap)

Abstract

We give a constructive derivation of continuum existence, Osterwalder–Schrader reconstruction, and a strictly positive spectral mass gap for four-dimensional SU(N) Yang–Mills theory. The argument proceeds in five steps that stay inside classical gauge-orbit geometry and functional analysis. First, Osterwalder–Schrader positivity on the unfixed path space forces the physical Hilbert space to coincide with the zero-residue kernel of the Faddeev–Popov operator on the fundamental modular domain. Second, O’Neill’s submersion theorem, verified by a Schur-complement identity in local horizontal coordinates, absorbs the negative bubble counterterm and produces an intrinsic Ricci curvature that is a pure sum of squares. Third, the Schwinger proper-time integral is split at the Callan–Symanzik scale, absorbing the Seeley–DeWitt ultraviolet divergence and generating a renormalization-group-invariant mass. Fourth, the Bochner–Bakry–Émery boundary flux across the modular-domain boundary vanishes by gauge-copy pairing and horizon degeneration, while sub-ball locality and a local Bessel bound give volume-independent infrared coercivity. Fifth, reflection-positive measures satisfying the Bakry–Émery curvature-dimension condition are constructed along both the heat-kernel trajectory and the bare Wilson action; the resulting logarithmic Sobolev inequality and Mosco convergence transfer the mass gap and the Osterwalder–Schrader axioms to the continuum. The construction replaces multi-scale polymer expansions with a single-step ultraviolet-to-scale-r Hessian bridge controlled by chessboard estimates, diamagnetic Dirichlet gaps, and Brascamp–Lieb / Combes–Thomas bounds. All constants are explicit and independent of volume and lattice spacing.

Keywords: Yang–Mills mass gap; Osterwalder–Schrader reflection positivity; zero-residue selection rule; Faddeev–Popov determinant; O'Neill Riemannian submersion curvature; Babelon–Viallet orbit metric; fundamental modular domain; Gribov horizon; Bakry–Émery convexity; Osterwalder–Schrader chessboard estimate; Brascamp–Lieb inequality; Combes–Thomas resolvent decay; Mosco convergence.