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: \_>0 \;\; Topological coercivity (TC) \;\; _ (- ₌) ₆₄₎\, _²\;\; m₀ c₃₎₌_ (- ₌) \;\; m₀ c₆₀₏\, _²>0, \ (c₆₀₏: =c₃₎₌₆₄₎\). All constants are tracked against a fixed parameter box (PB) and are uniform in the lattice spacing \ (a\) and the flow time \ (t₀\) 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) _ (t₀) >0 _ (-M) ₆₄₎\, _ (t₀) ² m₀ c ₆₀₏\, _ (t₀) ²>0 with strict interchange-of-limits on the window and window-uniformity of constants. A key conceptual ingredient is the **residual topology density** \ (_\), 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 \ (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 stdlib-only) - `scripts/verifier2ᵢndependent. py` (independent cross-check for TCB reduction) - `manifest. json` (SHA256 seal binding all files) Important: the bundled PASS instance may be **demoₛynthetic** (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/verifyₘanifest. py --packetᵣoot. python3 -B scripts/auditₐll. py --packetᵣoot. --report results/e5ₘinᵣeport. json --profile demopython3 -B scripts/verifier2ᵢndependent. py --packetᵣoot. --report results/e5ₘinᵣeport. json --profile demo