The Existence of the Yang-Mills Mass Gap: A Constructive Overview – Roadmap to the Constructive Gauge Theory Program (CGTP)

Abstract We present a comprehensive overview of the Constructive Gauge Theory Program (CGTP), a modular mathematical framework establishing the rigorous existence of a strictly positive mass gap in pure Yang–Mills theory on ℝ⁴. This document serves as a roadmap to the foundational trilogy and its technical companion series, elucidating the logical architecture that connects semiclassical analysis on ℝ³ × S¹ (Volume I) with constructive renormalization group stability (Volume II) and axiomatic spectral reconstruction (Volume III). We explain how the program overcomes historical obstructions—specifically the conflict between infrared slavery and non‑perturbative analysis—by utilizing a hybrid strategy supported by rigorous technical derivations. Key mechanisms detailed include: (1) the proof of adiabatic continuity via the exact cancellation of Stokes singularities (Resurgence Analysis); (2) the Matching Lemma, which bridges Balaban’s ultraviolet stability bounds with the infrared polymer expansion; and (3) the functional proof of Reflection Positivity via the Hubbard–Stratonovich duality. We summarize the main result: the construction of a unitary Relativistic Quantum Field Theory satisfying the Osterwalder–Schrader axioms and exhibiting a discrete spectrum bounded away from zero.