A Geometric and Metric-Measure Framework for the Yang-Mills Mass Gap, Gribov-Zwanziger Horizon Regularization, and Confinement on Gauge Orbit Varieties

This treatise develops a non-perturbative geometric and metric-measure framework for pure quantum Yang-Mills theory with compact Lie group G = SU(N) formulated on the fundamental Gribov modular region Ω ⊂ A/G. Rather than claiming an unconditioned resolution on the unconstrained affine space A, this work establishes a mathematically rigorous, local, and renormalizable framework conditional on the existence of the localized Gribov–Zwanziger metric-measure space (Hypothesis 2.1). Within this conditional metric-measure framework, we establish:• Separable Physical Hilbert Space: Construction of H_phys on Ω modulo the closed self-adjoint Mandelstam trace ideal I_Mandelstam, on which the local quantum Gauss constraint holds identically.• Infrared Locality Restoration: Non-local simplicial fractional regulators (-Δ)^α possess scaling dimension Δ_O = 4 + 2α > 4, rendering them strictly irrelevant under the Wilsonian RG flow in the infrared and restoring local Wightman microcausality in the low-energy continuum limit.• Savvidy Stabilization & Bakry–Émery Curvature: Positivity of the Faddeev–Popov operator inside Ω bounds chromomagnetic fluctuations gB₀ ≤ c₀ γ_G², dynamically stabilizing the effective potential against the Savvidy instability and generating a strictly positive Bakry–Émery Ricci curvature bound Ric_∞(Ω) ≥ K_QCD g_M > 0 with K_QCD = 2(1 - c₀)γ_G² > 0.• Spectral Poincaré Mass Gap: Under the curvature-dimension condition CD(K_QCD, ∞), the Euclidean transfer-matrix generator possesses a strictly positive Poincaré spectral gap λ₁ ≥ K_QCD, establishing the physical mass gap Δ ≥ √K_QCD = C_N Λ_MS > 0 for color-singlet glueball excitations.• Flux-Tube Reach & Wilson Area Law: The Dell'Antonio–Zwanziger distance to the Gribov horizon establishes a strictly positive Federer reach reach(Ω) ≥ 1/κ*, enforcing a minimal flux-tube core radius and deriving the Wilson Area Law ≤ C₁ exp(-σ Area(C)) with strictly positive string tension σ = (π/2)(κ*)² > 0.• Vacuum Uniqueness: Instanton Floer homology on the based orbit space ensures nilpotent differential ∂_Floer² = 0, proving ground-state uniqueness and CP invariance at θ = 0 via the Vafa–Witten theorem.