A Unified Conditional Program for the Yang–Mills Mass Gap - The Zsa-G Framework

Mathematical Architecture, Constructive Foundations and Continuum Reconstruction The ZSA-G Framework (Version 15): A Unified Conditional Program for the Yang–Mills Mass Gap A Consolidated Monograph Versions V1–V15 This monograph presents Version 15 of the ZSA-G Framework, representing the first fully consolidated formulation of a long-term research program devoted to the constructive analysis of the Yang–Mills Mass Gap. Rather than proposing a new physical theory, the framework develops a unified mathematical architecture integrating stochastic quantization, operator theory, spectral geometry, Bakry–Émery coercivity, Gribov-region analysis, non-perturbative spectral methods, continuum reconstruction, and quantitative consistency studies into a single constructive program. The present version consolidates the developments of the previous fourteen versions into a logically unified monograph. Earlier heuristic arguments have been reformulated within a systematic operator-theoretic framework, the logical dependencies between analytical components have been made explicit, and the remaining unresolved assumptions have been isolated within a transparent mathematical structure. A central feature of Version 15 is the reformulation of the continuum reconstruction problem through the revised Measure-Inevitability Assumption (MIA₂), together with its equivalent continuum formulations. This allows the Yang–Mills Mass Gap problem to be expressed as a sharply localized continuum reconstruction problem while preserving a clear distinction between rigorously established results, conditional assumptions, heuristic arguments, and numerical evidence. The monograph further develops the ZSA-G Spectral Projector, Bakry–Émery coercivity program, Holley–Stroock regularization, stochastic Gribov enclosure, infinite-dimensional resolvent theory, non-perturbative gap equations, large-NN analysis, Bethe–Salpeter glueball estimates, and constructive operator-theoretic methods. Version 15 should therefore be regarded as the first complete architectural presentation of the ZSA-G Framework, providing both a comprehensive reference for the current state of the program and a foundation for future mathematical development toward an unconditional construction of the Yang–Mills Mass Gap. Core Physics Yang–Mills Mass Gap Yang–Mills Theory Quantum Field Theory Non-Perturbative Quantum Field Theory Constructive Quantum Field Theory Gauge Theory Lattice Gauge Theory Confinement Glueballs Mathematical Methods Operator Theory Spectral Theory Spectral Geometry Functional Analysis Stochastic Analysis Stochastic Quantization Langevin Dynamics Infinite-Dimensional Analysis Infinite-Dimensional Resolvent Theory Semigroup Theory Partial Differential Equations Measure Theory Geometric / Probabilistic Bakry–Émery Geometry Holley–Stroock Perturbation Log-Sobolev Inequalities Foster–Lyapunov Methods Dynkin Formula Ergodic Theory Geometric Analysis Yang–Mills Specific Gribov Region BRST Symmetry Osterwalder–Schrader Reconstruction Reflection Positivity Spectral Gap Mass Generation Bethe–Salpeter Equation Large-N Expansion Schwinger Functions