A Proof Architecture for the Electron Inevitability Program: Gauge-Invariant BF Sectors, Gauss-Law Charge, and Threshold Closure

This manuscript presents a proof-architecture version of the Electron Inevitability Program. The guiding question is whether the existence of a stable minimal electric-charge sector can be organized as a mathematical consequence of structural properties of a gauge-Higgs-Yukawa universe, rather than taken as a primitive particle input. The paper does not claim a final numerical derivation of the electron mass, nor an unconditional derivation of the Standard-Model electron. Its purpose is structural: to test whether the electron-inevitability question can be decomposed into a conditional theorem-facing proof architecture under explicit and visible standing burdens. The main proof-architecture package consists of four linked layers: 1. a class-uniform Osterwalder-Schrader window on the gauge-invariant algebra; 2. Gauss-law charge detection, neutralization, and Buchholz-Fredenhagen transport; 3. a realistic massless-photon branch formulated as a minimal-charge Buchholz-Fredenhagen sector generated by a gauge-invariant dressed composite \ (Oₑ\) ; 4. conditional finite-family inclusive scattering, admissible hard-channel exhaustion, and regulator/window/dressing universality. The realistic branch is not formulated as an isolated Wigner-particle pole. Instead, the electron sector is treated as a gauge-invariant BF sector with continuous threshold structure. In the notation of the manuscript, the neutralized Kallen-Lehmann measure of \ (Oₑ\) has threshold at mₑ² has no atom at threshold, and satisfies compact-band infrared splitting within the common quantitative budget. The proof architecture is organized around the following implication: visible OS/BF/threshold/scattering/universality burdens-sector theorem package. The remaining head problem for full electron inevitability is explicitly separated: primitive admissible gauge-Higgs-Yukawa structure minimal electric-charge branch. Thus the manuscript should be read as a conditional proof-architecture source manuscript: it exposes, localizes, and consumes the standing burdens needed to turn the electron sector into a mathematical theorem package. It does not assert an unconditional classification of all gauge-Higgs-Yukawa theories, a numerical computation of \ (mₑ\), or a complete first-principles derivation of the Standard-Model electron. The final appendix includes a short charge-information reading of the electron sector, interpreting electric charge as an asymptotically detectable sector datum and the electron as the minimal physical sector carrying the charge label \ (-e\), without enlarging the exact theorem package of the main text. =================================== Author: Lee Byoungwoo (이병우) leeclinic@protonmail. com