YANG-MILLS EXISTENCE AND MASS GAP — COMPLETE PROOF

This repository contains a structured, node-based “proof architecture” for the Yang–Mills Existence and Mass Gap Millennium Prize Problem (Clay Mathematics Institute). The document is written as a modular chain of claims (“NODEs”) with explicit dependencies, an external-results registry (“CITED layer”), and a constants ledger, aiming to make verification auditable: each node states its purpose, assumptions, inputs, outputs, and where it is used downstream. Problem context The Clay problem asks for a mathematically rigorous construction of four-dimensional quantum Yang–Mills theory for a compact simple gauge group (typically SU(N) on R4, together with a proof of a strictly positive mass gap. Despite major progress in lattice gauge theory, constructive QFT, and renormalization, a complete proof in 4D remains historically difficult due to the simultaneous need for (i) a nonperturbative definition of the continuum theory, (ii) control of the continuum limit, (iii) verification of Osterwalder–Schrader (OS) axioms to reconstruct a Hilbert-space QFT, and (iv) a quantitative mechanism yielding a positive spectral gap. Core approach (high-level) The architecture follows a lattice-to-continuum pipeline: Lattice model and observables. Wilson SU(N) lattice gauge theory is used as the ultraviolet regulator; Wilson loops and local gauge-invariant operators serve as primary observables. Strong-coupling seed. Convergent polymer/cluster expansion technology at small β\betaβ supplies a rigorous seed regime with area law and exponential clustering. RG trajectory + stability propagation. A tuned renormalization-group trajectory is specified to transport “strong-coupling-type” control across scales, formulated as a stability object that can be propagated step-by-step. Heat-kernel smoothing / SC-criterion. A tail-norm “strong-coupling criterion” is defined for plaquette marginals after heat-kernel smoothing; it yields quantitative area law and exponential clustering at the relevant scale. OS axioms and reconstruction. Reflection positivity (OS2) is treated via an explicitly registered external input (with boundary-condition transfer handled at the local-algebra level); OS reconstruction provides the Hamiltonian HHH. Mass gap extraction. Exponential Euclidean time-decay of a vacuum-subtracted observable implies a spectral gap in HHH by the OS spectral representation. Nontriviality witness. A connected 4-point cumulant (strong-coupling series with a minimal connected polymer coefficient shown nonzero in an explicit “cube” geometry) is used to rule out Gaussian/free behavior. Transparency of external inputs All nontrivial external theorems are centralized in the CITED registry. Each entry includes: statement, where it is used, an applicability checklist, BibTeX key, and—if needed—a status flag such as “pointer pending.” One notable example is reflection positivity for the Wilson action (Osterwalder–Seiler 1978), which is recorded in a standard form with an explicit disclaimer when theorem-level locators or boundary-condition clauses require archival verification. What this document is (and is not) This is a verification-oriented proof chain, designed to help experts audit which parts are proved internally, which rely on standard external results, and which require pointer-level confirmation. The repository does not claim that every cited theorem-number locator has been independently verified by the author at the time of writing; such items are explicitly marked. The goal is to provide a coherent, testable blueprint whose correctness can be assessed node-by-node against the Clay checklist. Contents Node map and dependency graph (DAG) Model definitions, observables, limits, and uniformity pack CITED layer (external results registry) + BibTeX anchors Strong-coupling expansion nodes (area law, clustering, KP window) RG trajectory and stability objects (propagation mechanism) Heat-kernel criterion and consequences OS axioms verification and reconstruction Mass gap proof node and physical normalization link Non-Gaussianity witness nodes Appendices: heat-kernel details, technical lemmas (if needed), constants ledger, consolidated bibliography Final conclusion aligned with the Clay-style checklist and a status table