Semiclassical Foundations: Rigorous Route to the Mass Gap on ℝ³ × S¹: A Unified Semiclassical and SPDE Approach (CGTP Vol. I)

We present a unified, rigorous, and modular mathematical framework establishing the existence of a strictly positive mass gap in SU(N) Yang-Mills theory compactified on ℝ³ × S¹_L in the semiclassical regime L ≪ Λ⁻¹. The program is articulated through four inter-dependent pillars: (A) UV control and stability of the center-symmetric vacuum; (B) Analytical control of fractional monopoles and the convergence of the dilute gas; (C) Exact duality to a coercive compact scalar theory; and (D) Stochastic quantization via SPDEs proving exponential mixing. We derive a universal lower bound for the mass gap m ≥ C L⁻¹e⁻ˢ⁰ (or e⁻²ˢ⁰ for QCD(adj)), proving that the probabilistic mixing rate of the dual SPDE corresponds exactly to the physical mass gap. Series Context: This work constitutes Volume I of the Constructive Gauge Theory Program (CGTP), a series establishing the existence of the mass gap in Yang-Mills theory. It provides the physical mechanism and semiclassical foundations requiredfor the renormalization group analysis (Volume II) and the axiomatic reconstruction (Volume III). Minor update v1.2: Eq. (A.4) corrected from B2 in previous versions.