Gauge-Convex RP Γ-Flow: A Unified Proof Outline for the Yang–Mills Mass Gap

This preprint presents a unified, gauge-invariant proof outline for the Yang–Mills mass gap based on a framework called the Gauge–Convex RP Γ-Flow (GCRP). The core idea is to compress a long lattice argument into one verifiable energy–entropy inequality controlled by group geometry and blocking combinatorics. The framework rests on four auditable ingredients: (1) local convexity for the heat-kernel/Villain plaquette density, (2) a bounded-overlap coarse-graining map that ensures one-step log-Sobolev stability, (3) a rigorously controlled comparator that transfers estimates from the heat-kernel calibration to the Wilson action via reflection positivity, and (4) a coarse surface-entropy bound for spanning surfaces. From these inputs, the central inequality yields a positive string-tension floor (area law), exponential clustering, and a spectral gap for the reflection-positive transfer operator. The argument further establishes an “entry-free” renormalization-group corridor—once in the small-activity regime at the ultraviolet scale (asymptotic freedom), the flow remains there with a uniform buffer. The continuum identification is obtained via Γ-convergence on gauge-invariant cylinders together with the Osterwalder–Schrader framework, giving a unique Euclidean Yang–Mills limit that inherits the mass gap. The method applies to compact simple gauge groups and does not rely on center symmetry. Key analytic tools include Bakry–Émery/log-Sobolev techniques, polymer/Kotecký–Preiss cluster expansions, chessboard/reflection-positivity methods, and Γ-convergence. The paper provides explicit constant tracking and a conservative numerical instance, illustrating quantitative control of the gap mechanism within a fully gauge-invariant setting.