Yang–Mills Mass Gap in Four Dimensions: OS Window, Residual Topology, and Strict IOA (v9.7)

# Overview This record releases **Version v9.7 (February 15, 2026)** of: **“Yang–Mills Mass Gap in Four Dimensions: OS Window, Residual Topology, and Strict IOA”** Author: Lee Byoungwoo (Math Forum, Daejeon) The paper develops a single Clay-compatible framework for **4D Euclidean Yang–Mills** with compact gauge group \(G=SU(2)\) (remarks on \(SU(3)\), \(SU(N)\) are included). The objective is to construct a continuum theory satisfying the **Osterwalder–Schrader axioms (OS1–OS5)** and to prove a **strictly positive mass gap** after OS reconstruction. A key reviewer-facing upgrade of this line is the explicit separation between:(i) **finite auditable inputs** (a “sealed witness” interface for the unique model-specific bottleneck), and (ii) **deterministic analytic transfers** (window-uniform modules on a calibrated OS window). # Main quantitative implication The proof pipeline is organized as: \[δ_φ>0 \;⇒\; Topological coercivity (TC)\;⇒\; λₘᵢₙ(-ΔM) ≥ κgeo\,δ_φ^2\;⇒\; m_0 ≥ cdomλₘᵢₙ(-ΔM)\;⇒\; m_0 ≥ cgap\,δ_φ^2>0,\]with \(cgap:=cdomκgeo\).All constants are tracked against a fixed parameter box (PB) and are uniform in the lattice spacing \(a\) and the flow time \(t_0\) within the proved OS window. # What is new in v9.7 - **Soundness Standard Form (SS-1):** a one-page “reduction” statement clarifying that the only genuinely model-specific positivity input is a single audited witness for **E5-min**, while the remaining triggers (LSI/transport, FRD continuity, KP acceptance, stratified Mosco, strict IOA transfer) are deterministic analytic modules on the OS window.- **Sealed witness interface (Level–2B style):** a public, machine-checkable interface for E5-min based on a finite **JSON witness report** plus a **stdlib-only audit** that deterministically recomputes derived inequalities and returns PASS/FAIL.- **Explicit scope note (feasibility vs instantiation):** the paper proves feasibility of a non-empty OS-window (an open PB region satisfying the trigger system with strict margin), but a Clay-style “resolution” reading additionally requires an instantiation argument that the target continuum SU(2) Yang–Mills limit lies in that feasible region and yields a PASS witness instance. ## Academic contribution and positioning (v9.7) This work is not presented as a “black-box proof”, but as a **Clay-compatible reduction** in areviewer-facing standard form: a **single finite, auditable witness** plus **deterministic analytictransfers on an OS window** (SS-1). Concretely, the only strict positivity input is the mandatoryinterface **E5-min/v1**, which is defined as a public proof interface with a fixed JSON schema anda deterministic PASS/FAIL audit rule. Once a PASS witness is supplied, the manuscript proves theimplication chain\[(E5-min) ⇒ δ_φ(t_0)>0 ⇒ λₘᵢₙ(-Δ_M)≥ κgeo\,δ_φ(t_0)^2⇒ m_0 ≥ cgap\,δ_φ(t_0)^2>0\]together with strict interchange-of-limits on the window and window-uniformity of constants. A key conceptual ingredient is the **residual topology density** \(δ_φ\), designed to surviveflow and boundary cancellations and to enter coercivity/spectral estimates on the orbit space.Methodologically, the manuscript makes the review boundary explicit: triggers (E1)–(E4) areanalytic OS-window modules with PB-tracked constants, while (E5-min) is a certified artifactrequired by the proof interface. **Scope note (feasibility vs. instantiation).** The paper proves constructive *feasibility* of thetrigger system by exhibiting an open PB region with positive slack. A stronger “Clay-resolution”reading for the target 4D SU(2) continuum theory requires *instantiation*: either an analytic proofthat the chosen regularization/continuum limit lies in the feasible PB region and yields a PASSwitness, or a certified artifact validating the window constants and producing a PASS E5-min/v1witness instance (simulation/real-data mode). **TCB reduction roadmap.** To increase reviewer confidence in an auditable proof interface, thepaper records a minimal path to shrink the trusted computing base: an independent Verifier2recomputation, an explicit numeric model declaration (e.g. float/decimal/interval), and (for thefew inequalities that decide PASS/FAIL) a verification-only interval/rational enclosure path. **Bottleneck targets (toward an instantiated witness).** The remaining genuinely model-specificstep is to obtain \(δ_0>0\) via an E5-min/v1 PASS witness in the actual SU(2) theory. The paperrecords concrete routes: (BT-1) a strong-coupling anchor plus continuation inside PB⋆; (BT-2) aconstructive two-event lower bound stable under window transfers; (BT-3) an explicitly certifiedsimulation protocol with frozen interface, provenance, and replication. --- ## Related work (high-level; non-exhaustive) **Constructive RG / small-field programs.**Balaban developed a renormalization-group approach to lattice gauge field theories with deepcontrol on effective actions and expansions (not a completed 4D continuum mass-gap proof).This manuscript differs in emphasis: it is organized around a *reviewer-facing proof interface*(SS-1) and an explicit “finite witness + deterministic transfer” boundary. **Strong-coupling and cluster expansions (lattice mass gap at strong coupling).**Classical strong-coupling expansions establish a mass gap in regimes far from the continuumscaling limit. The present program treats this as a plausible anchor for BT-1 (an explicit witnessat one anchor point, then continuation under window-uniform stability), rather than as a directresolution. **Problem-statement baseline (Clay).**The Clay Millennium problem asks for existence of a mathematically controlled 4D Yang–MillsQFT and a strictly positive mass gap. The present manuscript is framed as a reduction that makesthe remaining model-specific positivity input explicit and auditable (E5-min/v1), separating it fromanalytic transfer modules (E1–E4) whose instantiation must be provided for the target model. **Representative references (for orientation).**- Clay Mathematics Institute (2000), *Yang–Mills & the Mass Gap* (Millennium Problem overview).- T. Balaban (1987), *Renormalization group approach to lattice gauge field theories. I. Generation of effective actions in a small field approximation*, Communications in Mathematical Physics.- T. Balaban (1988), *Renormalization group approach to lattice gauge field theories. II. Small field expansions and cluster expansions*, Communications in Mathematical Physics.- G. Münster (1981), *Strong coupling expansions for the mass gap in euclidean lattice gauge theories in any space-time dimension*, Nuclear Physics B. # The sealed E5-min witness packet (no-TeX) This record also includes a sealed artifact (no LaTeX sources) that freezes the E5-min proof interface: - `results/e5_min_report.json` (machine-checkable witness report)- `schemas/e5_min_report.schema.json` (schema)- `scripts/audit_all.py` (deterministic auditor; stdlib-only)- `scripts/verifier2_independent.py` (independent cross-check for TCB reduction)- `manifest.json` (SHA256 seal binding all files) Important: the bundled PASS instance may be **demo_synthetic** (toy/synthetic) to demonstrate the interface and audit chain; it is not automatically a claim of SU(2) Yang–Mills continuum instantiation. ## Quick start (reviewer) From the packet root: ```bashpython3 -B scripts/verify_manifest.py --packet_root .python3 -B scripts/audit_all.py --packet_root . --report results/e5_min_report.json --profile demopython3 -B scripts/verifier2_independent.py --packet_root . --report results/e5_min_report.json --profile demo