Yang–Mills Mass Gap from OS Windows: Minimal E5-min Witness Interface (v9.5)

# TitleYang–Mills Mass Gap in Four Dimensions: OS Window, Residual Topology, and a Minimal E5 Witness Interface (v9. 5) # OverviewThis record releases **Yang–Mills Mass Gap in Four Dimensions (v9. 5) **, a Clay-compatible framework organized around a quantitative **OS window** with uniform (PB-tracked) constants. The main output is a *reviewer-facing reduction form*: - A small set of **OS-window triggers (E1–E5) **, uniform in lattice spacing \ (a\) and flow time \ (t₀\), - A deterministic transfer chain \ (E5-min) _ (t₀) >0 _ (- ₌) ₆₄₎_^2 m₀ c₃₎₌_ m₀ c₆₀₏_^2 > 0, \ with \ (c₆₀₏: =c₃₎₌₆₄₎\) uniform on the OS window. The paper emphasizes **strict IOA (limit exchange) **, PB-registered constants, and explicit “where the burden sits” separation between analytic modules and certified finite inputs. # Closed results (within the OS-window program) - Quantitative OS-window architecture with PB-tracked constants (uniformity in \ (a, t₀\) ). - Residual topology density \ (_\) and flow-back stability to \ (t₀=0\). - Topology-to-geometry coercivity: \ (_ (- ₌) ₆₄₎_^2\) on the window. - Operator domination linking the reconstructed Hamiltonian \ (H\) to \ (- ₌\) on the gauge-invariant sector. - A constructive feasibility statement: existence of an open PB region where triggers can hold simultaneously with positive slack \ (₏₎₋ₘ>0\). # What is new in v9. 5- **E5-min minimal trigger tightening: ** bimodality/variance evidence is unified into a single witness concept and choice rule. - **E5-min as a mandatory checked item: ** \ (_ (t₀) >0\) is treated as a verifiable witness rather than an informal “assumption style” statement. - **Reviewer-facing soundness framing: ** the intended reading is “audited finite witness + deterministic transfers, ” reducing ambiguity between a program and a resolved instance. # Proof interface philosophy (how to read the bottleneck) - The model-specific positivity bottleneck is concentrated in **E5-min**. - Other triggers (E1–E4) are treated as OS-window analytic modules with PB-uniform constants once the window is fixed. - Appendix-level feasibility provides an open solution set for the trigger system; instantiation for a specific target model is to be documented via PB-registered constants and a sealed witness record. # Files- Main manuscript: PDF + TeX source. - (Optional) Patch/diff artifacts for version-to-version tracking (if included in the record). # Scope mass gap; Osterwalder–Schrader axioms; constructive QFT; Wilson flow; topological charge; coercivity; Laplace–Beltrami; spectral gap; log-Sobolev inequality; cluster expansion; Mosco convergence; certified witness; reproducibility.