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 \(t_0\),- A deterministic transfer chain \[ (E5-min) ⇒ δ_φ(t_0)>0 ⇒ λₘᵢₙ(-ΔM) ≥ κgeoδ_φ² ⇒ m_0 ≥ cdomλₘᵢₙ ⇒ m_0 ≥ cgapδ_φ² > 0, \] with \(cgap:=cdomκgeo\) 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 \(a,t_0\)).- Residual topology density \(δ_φ\) and flow-back stability to \(t_0=0\).- Topology-to-geometry coercivity: \(λₘᵢₙ(-ΔM) ≥ κgeoδ_φ²\) on the window.- Operator domination linking the reconstructed Hamiltonian \(H\) to \(-ΔM\) on the gauge-invariant sector.- A constructive feasibility statement: existence of an open PB region where triggers can hold simultaneously with positive slack \(εpoly>0\). # 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:** \(χ_φ(t_0)>0\) 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 & non-toy statusThe work is structured to match modern auditable-proof interfaces: a finite witness object and a deterministic verification chain, with explicit separation between feasibility and instantiation. The main scientific risk for “Clay-style resolution” is concentrated in producing/validating the E5-min witness for the target continuum theory and tightening the provenance of triggers (E1–E4) as known theorems vs new lemmas vs certified artifacts. # KeywordsYang–Mills theory; 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.