A Functional–Geometric Proof of the Mass Gap for Lattice SU(3) Yang–Mills

This two-part work presents a logically closed, functional–geometric proof of the positive mass gap for pure lattice SU(3) Yang–Mills theory, together with a unifying analytic framework that generalizes the proof structure through reflection positivity, local convexity, and finite-overlap principles. Part I – Functional–Geometric Proof.A complete and explicit proof compressed into two scalar invariants: the geometric intensity (g⋆) and the information-theoretic capacity (σ).Starting from a single reflection-positivity lemma, two equivalent proof facades emerge: a curvature-to-MLSI route and an RP + Gaussian (Brascamp–Lieb) route.From these, an area law follows, then a spectral gap, and finally a positive Wightman mass gap.A quantitative β-corridor guarantees stability of all constants along the RG trajectory to the continuum. Part II – Gauge-Convex RP Γ-Flow Framework.This part introduces a systematic analytic architecture built on five axes: (A1) Reflection Positivity & Gauge Invariance, (A2) Local Convexity, (A3) Finite Overlap, (A4) Heat-Kernel ↔ Wilson Comparator, and (A5) Coarse Surface Entropy.A single energy–entropy inequality links these axes and implies the area law, a positive transfer gap, and the continuum identification via Γ-convergence in the Osterwalder–Schrader framework.It unifies the functional–geometric proof into a general “Gauge-Convex RP Flow” structure, showing that the Part I construction is a concrete realization of a broader universal mechanism. Novel features One RP foundation (“Gate 0”) generating two equivalent facades to the area law: curvature → MLSI and RP + Gaussian. A transparent capacity-to-gap chain and β-corridor control with fully auditable constants. A gauge-convex Γ-flow formalism that globalizes the HK/Wilson comparison and reduces the full argument to a small set of verifiable parameters (convexity, overlap, distortion, surface entropy). A complete numerical ledger documenting each constant’s origin and use. Main results (in plain language) Exponential area law for Wilson loops. Exponential clustering and a positive transfer-operator spectral gap. Osterwalder–Schrader reconstruction to a Wightman theory with strictly positive mass. Uniform stability of all bounds down to the continuum limit. Methods & structure Reflection positivity and the chessboard method; non-abelian local comparators; Bakry–Émery and Brascamp–Lieb functional tools; Γ-convergence for the continuum; Kotecký–Preiss cluster control as a failsafe route. Files in this record Part I: A Functional–Geometric Proof of the Yang–Mills Mass Gap (final proof version). Part II: Gauge-Convex RP Γ-Flow: A Unified Proof Outline for the Yang–Mills Mass Gap (methodological framework). Reproducibility The work is purely theoretical and self-contained; all dependencies are logical and fully audited through explicit lemma → theorem chains and constant ledgers. Intended audience Researchers in mathematical physics, functional analysis, and probabilistic QFT interested in rigorous Yang–Mills theory, reflection positivity, and Γ-convergence methods.