Geometric Origin of g = 2 in the 0-Sphere Model: U(1) Fiber Bundle and Dual-Frame Phase Decomposition

48 Essential This paper establishes the geometric phase structure of the 0-Sphere electron model. The internal 2π periodicity of the electron's oscillating photon sphere induces a principal U (1) fiber bundle, whose Berry holonomy γintrinsic = π is a topological invariant that geometrically necessitates gCM = 2 in the proper-time frame — without perturbative loops. In the laboratory frame, Lorentz projection introduces an anomalous phase γanomalous, yielding the dual-frame decomposition glab = 2 (1 + a). A central result is the structural unification of the anomalous magnetic moment and the muon atmospheric lifetime extension as two instances of the same relativistic arc-length identity: ΔL = |γL − 1| × L0. The inverse calculation from the experimentally observed ae ≈ 0. 00116 yields vZB ≈ 0. 04c. The forward derivation — Thomas precession → vZB independently of ae → γL = 1 + ae as output — remains the central open task. — Core Equations — γtotal = γintrinsic + γanomalous = π + a·2π gCM = 2 (topological invariant, proper-time frame) glab = 2 (1 + ae) where γL = 1 + ae ΔL = |γL − 1| × L0 (unifying identity: AMM ↔ muon lifetime) Arc-length inverse route: L/L0 = 1/γL ≈ 1/ (1 + ae) √ (1 − vZB²/c²) = 1/ (1 + ae (exp) /√2) → vZB ≈ 0. 04047c — Key Contributions — U (1) Fiber Bundle from Internal Periodicity. The 2π-periodic internal state defines a principal U (1) bundle over S¹. The Berry holonomy γintrinsic = π is proved via both algebraic calculation (connection integral + boundary term) and solid-angle geometry (Ω/2 = 2π/2 = π). Geometric g = 2. In the center-of-mass frame, γintrinsic = π is a coordinate-invariant topological invariant, geometrically necessitating gCM = 2 exactly — protected against continuous deformations by the ℤ2 holonomy of the SU (2) double cover. Dual-Frame Decomposition and Foucault Analogy. Lorentz projection extends the internal orbital arc 2πr to 2πr·γL, introducing γanomalous = ae·2π. The Foucault pendulum provides the classical prototype: the CM frame corresponds to the equatorial observer (zero precession, g = 2 exact) ; the laboratory frame corresponds to a small latitude φ ≪ 1 (precession ≈ 2πφ, anomalous contribution ae ≈ 0. 00116). AMM–Muon Structural Equivalence. The anomalous magnetic moment and the atmospheric muon lifetime extension are unified as two instances of ΔL = |γL − 1| × L0. For the muon, γL ≈ 29; for the electron, γL − 1 = ae. The mechanism is structurally identical; only the magnitude of γL − 1 differs. Double Appearance of 1/2. The factor 1/2 appears independently in the photon-sphere kinetic energy (thermodynamic origin via Stefan-Boltzmann law) and in the Thomas precession (relativistic origin via Lorentz non-commutativity), both generating the same double-angle structure sin (2θ) with normalized maximum 1/2. Three-Periodicity Gauge Correspondence. Three coexisting periodicities (π, 2π, 4π) arising from the energy conservation identity correspond via s = 2π/T to spin-2, spin-1, and spin-1/2, suggesting a phenomenological correspondence with gravitational, electromagnetic, and weak gauge symmetries. Presented as a structural observation, not a derivation. Thermodynamic Classification by Algebraic Consistency. The branch point between thermodynamic and kinematic spin-state classifications is resolved internally: only the thermodynamic classification is compatible with gCM = 2 as a topological invariant of the U (1) holonomy. — Supplementary Material — An accessible conceptual overview of this paper is archived alongside the main PDF: zenodo₁9227518guide. docx. It presents the central arguments without technical prerequisites and is intended for readers approaching the 0-Sphere Model for the first time. — Position in the 0-Sphere Model Series — Spin from Geometry (#26) — Berry phase origin of spin; #48 supplies the formal fiber bundle proof. Geometric Phase Structure: Dual DOF (#38) — Dual-frame structure and gCM = 2; #48 adds the formal geometric proof and Foucault analogy. Rotational Lorentz Contraction and Muon Lifetime (#47) — ΔL identity; #48 embeds this within the U (1) fiber bundle framework. — Key References — # Title (abbreviated) DOI 1 A Model of an Electron Including Two Perfect Black Bodies 10. 5281/zenodo. 16759284 26 Spin from Geometry: Emergence of Spin via Internal Berry Phase 10. 5281/zenodo. 17765409 35 Detailed Exposition of the 0-Sphere Model Framework 10. 5281/zenodo. 18511664 38 Geometric Phase Structure: Dual DOF within 2π Periodicity 10. 5281/zenodo. 18718174 47 Rotational Lorentz Contraction as Geometric Origin of AMM 10. 5281/zenodo. 19120057 — Series Context — The 0-Sphere Model is an ongoing research programme (2018–present) deriving spin, anomalous magnetic moment, Zitterbewegung, and emergent spacetime from the geometry and thermodynamics of a two-kernel electron model. All papers in the series are archived on Zenodo: Zenodo search: Hanamura, Satoshi