Yang–Mills Mass Gap in Four Dimensions: OS Window, Residual Topology, and Strict IOA (v9.0)

## Overview This record releases **Yang–Mills Mass Gap in Four Dimensions: OS Window, Residual Topology, and Strict IOA (v9. 0) ** (PDF). The manuscript develops a **single, Clay-compatible** framework for four-dimensional Euclidean Yang–Mills theory with compact gauge group \ (G=SU (2) \), establishing the Osterwalder–Schrader (OS) axioms and deriving a **strictly positive mass gap** after OS reconstruction. A guiding principle of the paper is a **single-manuscript closure policy**: all claimed “Closed” items are proved inside the manuscript at the cited locations (often in Appendix M), while classical black boxes (e. g. OS reconstruction) are explicitly named and invoked only after verifying their hypotheses internally. --- ## Main quantitative statement (v9. 0) A residual-topology density \ (_>0\) is defined via a flowed local topological density with boundary Chern–Simons cancellations on admissible exhaustions. Flow-back stability propagates \ (_ (t₀) \) to \ (t₀=0\), and a stratified Mosco scheme with tubular-thickness/Hardy controls handles the orbit space. A mesoscopic topological coercivity estimate yields a geometric spectral gap on \ (M\): \_ (- ₌) \;\; ₆₄₎\, _^\, 2 (uniform on the OS window). \ A form-domination bound then transfers geometry to the reconstructed Hamiltonian \ (H\) on the gauge-invariant mean-zero sector: ₀ \;\; c₃₎₌\;_ (- ₌), , combining both links, ₀ \;\; c₆₀₏\, _^\, 2, ₆₀₏: =c₃₎₌₆₄₎. constants are **PB-tracked** and **uniform on a quantitative OS window**, with first-use registration in Appendix H. --- ## Closed results (high level) - **OS window feasibility (non-vacuity): ** a parameter box \ (PB^\*\) exists for which the OS-window triggers (E1) – (E5) hold simultaneously with a strictly positive open-polytope margin \ (₏₎₋ₘ (G) >0\). - **Analytic infrastructure on the OS window: ** slice-stable logarithmic Sobolev floor (and transport), a finite-range decomposition (FRD) with strict range and \ (t₀\) -Hölder continuity, and Kotecký–Preiss (KP) acceptance with explicit radius and decay margin. - **Change-of-measure control (CM. 1): ** quantitative slice comparability supporting OS3. - **OS4 clustering constants: ** quantitative clustering via entropy–trace / contraction mechanisms consistent with the OS window. - **Strict IOA (interchange of limits): ** a K-package (Mosco-stable low-mode projections, quantitative high-mode polynomial tails, class-LSI, filtered-energy convexity) proves strict commutation of \ ( (a0) \) and \ ( (t₀0) \) and propagates OS constants to the continuum. - **OS axioms and reconstruction: ** OS1–OS5 hold with window-uniform constants; OS reconstruction yields a self-adjoint Hamiltonian \ (H\) with a strictly positive mass gap \ (m₀>0\). (See the Proof-Status Dashboard in Appendix I and the full proof bundle in Appendix M; feasibility margins are in Appendix N. ) --- ## What is new in v9. 0 - **Single-manuscript closure policy hardened: ** “Closed” status is synchronized with explicit proof locations; classical theorems (e. g. OS reconstruction) are named and used only after hypotheses are verified internally. - **Gap-transfer chain made fully non-circular at the statement level: ** the operator-transfer is presented in the robust form \ m₀ c₃₎₌_ (- ₌) c₆₀₏_^\, 2, \ and all summaries/registries/diagrams are synchronized to this bound. - **Pointer integrity sweep: ** all proof-location pointers in the global bundle (Section 6) match the referenced Appendix M subsections 1: 1. - **Tightness/limit passage clarified in a Polish-space model: ** the measure-existence route is written with explicit topological hypotheses so that Prokhorov/diagonal extraction steps are formally legitimate. - **Registry/format fixes: ** constants registry in Appendix H is cleaned for readability and cross-reference stability. --- ## Scope OS reconstruction yields \ (H\). 4. **Quantitative mass gap: ** \ (m₀ c₆₀₏\, _^\, 2 > 0\). The authoritative dependency/status table is Appendix I; full technical proofs are consolidated in Appendix M, with feasibility audits in Appendix N. --- ## References (canonical sources for classical steps) - Osterwalder, K. and Schrader, R. (1973), *Axioms for Euclidean Green’s functions*. - Glimm, J. and Jaffe, A. (1987), *Quantum Physics: A Functional Integral Point of View*. --- ## Files in this Zenodo record - PDF only (v9. 0). The LaTeX source is maintained separately as a reference build artifact.