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: \[Window-Complete\] 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 \( U_1 \) foundation with PB-uniform control of: - slice log-Sobolev inequality at fixed \( t_0 > 0 \),- finite-range decomposition and Davies–Gaffney bounds,- reflection positivity with fermions (RP–F),- stratified Mosco \(+\) EVI control,- strict interchange of limits with tracked rates \( (αIOA, βIOA) \). ### 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 \( f_T^⊥ \)),- admissible dressings (Coulomb / line / light-ray), with the same unified budget \[O_ε\!(e-mR + T-3/2+ε + a^{αIOA} + t_0^{βIOA}).\] ### 3. Track A On the **uniform-gap subclass**\( LadmA+ \),the paper closes the isolated one-particle pole scenario:if the abelian gap satisfies \[m_γ ≥ m_γ^(PB) > 0,\] then a gauge-invariant composite \( O_e \) 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 \( m_e \),- non-atomicity at threshold,- assumption-free pointwise IR splitting on compact sets around \( m_e^2 \),- 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 \( Sincl \) and channel-wise asymptotic completeness at the finite-family level. ### 6. Stated-scope portability on \( U_1' \) A theorem-level portability layer is recorded on \( U_1' \) 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-\( C^1 \) 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 \( LadmA+ \),- Track B core at the BF-sector level,- Universality,- strict IOA,- finite-family inclusive BF scattering,- stated-scope portability on \( U_1' \). ### Not claimed as closed here - Track A on all of \( Ladm \),- 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 \( O_e \),- 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