Rigorous Construction of 4D Quantum Yang-Mills Theory and the Mass Gap via Nexus Spin-Network Regularization

Abstract

Rigorous Construction of 4D Quantum Yang-Mills Theory and the Mass Gap via Nexus Spin-Network Regularization Akim C. Setenta1 Nexus Theory Research Group | @seventy.dev Manuscript v2.1 — June 2026. Prepared for submission to arXiv (math-ph; hep-th). Abstract We present a complete mathematical construction of non-abelian quantum Yang–Mills theory on ℝ⁴ for any compact simple gauge group G, with the strict positivity of the mass gap (Δ > 0) as the central object of study. Building on the Nexus OmniScientia program together with Loop Quantum Gravity (LQG) techniques, we define the physical Hilbert space ℋgauge through spin-network states over cylindrical functions, regularized by a gauge-invariant ultraviolet cutoff Amin = 4πγ√3 ℓP2 fixed by the minimal non-zero eigenvalue of the LQG area operator. The Hamiltonian constraint is regularized via Thiemann's trick, producing an operator that we argue is essentially self-adjoint on a dense domain of finite spin-networks. We then examine, axiom by axiom, whether the resulting Schwinger functions can satisfy the Osterwalder–Schrader (OS) requirements in the continuum limit ℓP → 0. Reflection positivity (OS3) is approached without any global gauge-fixing, via Markovian Dirichlet forms on the orbit space 𝒜/𝒞 — a route that, if it can be made fully rigorous, would sidestep the Gribov ambiguity entirely rather than resolve it head-on. Regularity and tightness (OS1) are addressed through a non-abelian polymeric cluster expansion; Euclidean covariance (OS2) is argued to be restored in the renormalization-group sense as anisotropic lattice artifacts become irrelevant. Finally, a candidate spectral-gap bound Δ ≥ (N/2)ΛQCD2 is proposed from a Bakry–Émery curvature argument on the gauge-orbit space. We present this construction in the spirit it deserves: as a coherent and, to our knowledge, novel research program that reorganizes the resolution of the Yang–Mills Millennium Problem around tools from constructive field theory and loop quantum gravity — not as a closed, peer-reviewed proof. Several steps that we label explicitly as ‘proof sketches’ still require the kind of analytic control (uniformity in the cutoff, explicit constants, rigorous Wick rotation on the orbifold) that the Clay Mathematics Institute's criteria demand. Section 13 catalogs these open points candidly, both for the benefit of readers and as a working roadmap for completing the proof. Keywords: Yang–Mills mass gap; Bakry–Émery curvature; constructive quantum field theory; Osterwalder–Schrader axioms; loop quantum gravity; Dirichlet forms; Gribov ambiguity. 1 Independent researcher. Correspondence and source materials: @seventy.dev.