Toward Electron Inevitability in Gauge–Higgs–Yukawa Theories (v3.7): Construction-Level Closure in Class–Uniform OS Windows

## Overview This record contains **v3. 7** of **Toward Electron Inevitability in Gauge–Higgs–Yukawa Theories: Construction-Level Closure in Class–Uniform Osterwalder–Schrader Windows** The paper develops a **two-track, nonperturbative OS-window program** toward an electron inevitability theorem in four-dimensional gauge–Higgs–Yukawa theories. The framework is formulated entirely on the **gauge-invariant algebra** and uses a **class-uniform Osterwalder–Schrader (OS) window** on a parameter box \ (PB \), with PB-uniform constants and strict interchange-of-limits control. The central status of v3. 7 is: -Complete\ in the following sense: the paper closes the **construction-level** Track A / Track B framework, Universality, strict IOA, the benchmark/finite-family inclusive BF scattering package, and the stated-scope portability layer, all **within explicitly stated scopes**. ## Closed results in v3. 7 ### 1. Class-uniform OS foundation The paper works inside a sealed \ (U₁ \) foundation with PB-uniform control of: - slice log-Sobolev inequality at fixed \ (t₀ > 0 \), - finite-range decomposition and Davies–Gaffney bounds, - reflection positivity with fermions (RP–F), - stratified Mosco \ (+\) EVI control, - strict interchange of limits with tracked rates \ ( (₈₎₀, ₈₎₀) \). ### 2. Universality A master Universality theorem is recorded for gauge-invariant, neutralized observables under simultaneous changes of: - regulator (lattice / continuum), - calibrated window class (dyadic / log-Gaussian, including \ (fT^ \) ), - admissible dressings (Coulomb / line / light-ray), with the same unified budget _\! (e^-mR + T^-3/2+ + a^₈₎₀ + t₀^₈₎₀). \ ### 3. Track A On the **uniform-gap subclass**\ (L₀₃₌^A+ \), the paper closes the isolated one-particle pole scenario: if the abelian gap satisfies _ m_^ (PB) > 0, \ then a gauge-invariant composite \ (Oₑ \) creates an isolated spin-\ (12 \) one-particle state with positive residue and PB-uniform isolation margin. ### 4. Track B core At the BF-sector level, the paper closes the core massless-photon alternative: - continuous threshold at \ (mₑ \), - non-atomicity at threshold, - assumption-free pointwise IR splitting on compact sets around \ (mₑ² \), - positive-energy BF-sector covariance, again with the same PB-uniform budget. ### 5. Inclusive BF scattering The paper records an inclusive, IR-safe BF scattering package that is closed for: - a benchmark channel, and- any finite family of hard BF channels. This includes block-unitary \ (S₈₍₂₋ \) and channel-wise asymptotic completeness at the finite-family level. ### 6. Stated-scope portability on \ (U₁' \) A theorem-level portability layer is recorded on \ (U₁' \) for: - minimally charged leptons, - electromagnetically neutral leptons, - selected quark channels. This portability statement is explicitly **scope-limited** and does **not** claim a single fully consolidated Standard-Model dashboard. ## What is new in v3. 7 Relative to earlier versions, v3. 7 emphasizes reviewer-facing scope discipline and records several upgrades: - **Globalized Universality** along piecewise-\ (C¹ \) window-connected paths on \ (PB \), including admissible regulator swaps and window-class changes. - **Inclusive BF scattering upgraded from one benchmark channel to any finite family of hard channels**. - **Thermal/KMS alignment tightened** via a gauge-invariant thermal-center probe for deconfinement/chiral brackets. - **Front-matter and closure logic clarified** so that theorem-level scope, non-claims, portability limits, and remaining targets are stated more explicitly. - **Appendix B repositioned** as an acceptance/implementation appendix rather than a single proof-bearing omnibus foundation. ## Exact theorem-level scope / non-claims The paper is intentionally calibrated as a **construction-level closure record**, not as a final all-scope inevitability theorem. ### Closed here - Track A on \ (L₀₃₌^A+ \), - Track B core at the BF-sector level, - Universality, - strict IOA, - finite-family inclusive BF scattering, - stated-scope portability on \ (U₁' \). ### Not claimed as closed here - Track A on all of \ (L₀₃₌ \), - inclusive asymptotic completeness across **all** BF channels/sectors, - LSZ-wide exclusive completeness, - a fully consolidated Standard-Model portability dashboard, - assumption-free sharp threshold-shape refinements beyond the optional layer isolated in Appendix G. In this precise sense, the paper should be read as a **theorem-facing program paper with sealed construction-level closure**, rather than as a final single-theorem closure of the entire physical program. ## Structure and document logic The core theorem spine is concentrated in: - the class-uniform OS foundation, - the gauge-invariant composite \ (Oₑ \), - Track A / Track B, - inclusive BF scattering, - Universality, - stated-scope portability, - the program-closure ledger. Appendix B is intentionally framed as: **OS Window with Fermions: Acceptance Forms and Implementation Notes** and should be read as an implementation/acceptance layer, not as the unique location where the full foundational burden is discharged. ## Optional artifacts The paper also records lightweight optional review / audit artifacts (JSON/log style) for independent checking. These are not theorem inputs and are not prerequisites for reading the mathematical statements. ## Summary v3. 7 presents a **construction-level, scope-disciplined closure** of a two-track OS-window program toward electron inevitability in gauge–Higgs–Yukawa theories. The main contribution of this version is not a claim of unrestricted final closure, but a sharper and more stable theorem-level account of: - what is closed, - where the closure holds, - what remains only as an extent or quantitative upgrade. Author: Lee Byoungwoo (이병우) leeclinic@protonmail. com