Global Proof Architecture for the Yang–Mills Mass Gap Program Across All Compact Simple Gauge Groups

Global Proof Architecture for the Yang–Mills Mass Gap Program Across All Compact Simple Gauge Groups Description This companion note provides a structural overview of a four-paper program establishing a mass gap for four-dimensional Yang–Mills theory across all compact simple Lie groups. The note is intended as a navigation guide for referees and readers, clarifying how the individual manuscripts interlock and which components are shared versus group-specific. The series consists of the following papers: Yang–Mills Mass Gap for SU(2) in Four Dimensions Yang–Mills Mass Gap for SU(N) in Four Dimensions Yang–Mills Mass Gap for Orthogonal and Symplectic Groups in Four Dimensions Yang–Mills Mass Gap for the Exceptional Groups in Four Dimensions Together, these papers cover the classical and exceptional Lie families and present a unified constructive strategy. The architecture of the proof program separates into three reusable modules: Route 1 (finite-depth entrance and clustering)A one-shot renormalization step establishes entrance into a Kotecký–Preiss admissibility basin. From this point, cluster-expansion methods yield exponential Euclidean correlation decay. Reflection positivity then converts this decay into a spectral gap for the Osterwalder–Schrader Hamiltonian. Lane C (continuum flowed state construction)A dyadic renormalization group scheme constructs a unique continuum limit for gradient-flow observables. The RG map is analyzed via a polymer seminorm framework with explicit cumulant and moment bounds. Lane A (sharp-local operator algebra and OS axioms)The sharp local limit is taken using RP-safe representatives for flowed observables. This yields a local operator algebra satisfying the Osterwalder–Schrader axioms and supporting the reconstructed Wightman theory. Across the series, the only genuinely group-dependent analytic component is the A1 one-plaquette coefficient crossover bound, which controls the decay of normalized character coefficients. Separate appendices establish this bound for each Lie family using harmonic analysis tailored to the relevant representation theory. All other modules—including reflection positivity, cluster expansion machinery, renormalization-group stability, and Osterwalder–Schrader reconstruction—are reused verbatim across the four manuscripts once the A1 gate is verified. This document summarizes the interdependencies among these components and provides a guide for efficiently verifying the series: once the shared architecture is checked in a single paper, referees need only examine the family-specific A1 appendix and the associated constant ledger in the remaining papers. The note is provided as a companion to facilitate review and cross-referencing of the full program.