Rotational Lorentz Contraction as the Geometric Origin of the Anomalous Magnetic Moment: A Structural Correspondence with the Muon Lifetime Problem (Structural Clarification Note)

[47 Structural Clarification Note] This paper demonstrates that the anomalous magnetic moment of the electron (\( a_e \)) and the muon atmospheric lifetime extension are two manifestations of the same relativistic length-extension mechanism. The connection is made within the 0-Sphere Model of the electron, a geometric and thermodynamic framework developed continuously since 2018. — Core Result — Both phenomena are governed by a single identity of special relativity: \[ \Delta L = |\gamma_L - 1| \cdot L_0 \] In the muon case, \( v \approx 0.9994c \) gives \( \gamma_L \approx 29 \); the proper path length \( L_0 \approx 660\,\mathrm{m} \) is extended to \(\sim 19\,\mathrm{km}\), allowing the muon to reach the ground. In the electron case, the internal orbital arc \( 2\pi r \) is contracted by the Lorentz factor of the Zitterbewegung velocity \( v_{ZB} \approx 0.04c \), and the fractional shortening is precisely \( a_e \). The velocity magnitude differs by a factor of \(\sim 25\); the algebraic structure is identical. — Key Contributions — \( g = 2 \) from Thomas precession: Sinusoidal internal oscillation yields a double-angle angular velocity \( \Omega \propto \sin(2\omega t) \), geometrically necessitating \( g = 2 \) without any quantum-mechanical input. Geometric origin of \( a_e \): The deviation from \( g = 2 \) is identified with rotational Lorentz contraction of the internal orbital arc — the same mechanism underlying the Ehrenfest paradox (Einstein 1912). Inverse calculation: From the measured \( a_e^{(\mathrm{exp})} \), an RMS-corrected inverse formula yields \( v_{ZB} \approx 0.04047c \) and confirms \( \gamma_L = 1 + a_e \). Open forward chain: Deriving \( v_{ZB} \) from Thomas precession independently of the measured \( a_e \) remains the central open task; its completion would elevate the structural correspondence to a confirmed physical equivalence. Three occurrences of \( \tfrac{1}{2} \): The factor \( \tfrac{1}{2} \) in the Thomas angular velocity, the geometric centroid constraint, and the spin quantum number \( s = \tfrac{1}{2} \) as a topological invariant are identified as three expressions of the same underlying double-angle structure. — Relation to the 0-Sphere Model Series — This is a Structural Clarification Note (Paper #47 in the series). The inverse calculation yielding \( v_{ZB} \approx 0.04c \) was originally introduced in: Paper #10 — Redefining Electron Spin and Anomalous Magnetic Moment Through Harmonic Oscillation and Lorentz Contraction (2023) — 10.5281/zenodo.17764997 The present paper makes the structural analogy with the muon lifetime experiment explicit and self-contained, adding no new theoretical claims beyond those in Paper #10. It includes a side-by-side comparison figure (TikZ) and a unified property table. The announced future work — a Berry-phase and fiber-bundle treatment of the dual-frame structure within which \( \gamma_L = 1 + a_e \) appears as the bridge equation — builds on: Paper #38 — Electron Interference from Internal Geometry: Two-Kernel SU(2) Structure, Quartic Energy Flow, and an Intrinsic Visibility Limit (2026) — 10.5281/zenodo.18718174 — Primary References in This Work — # Title (abbreviated) DOI 1 A Model of an Electron Including Two Perfect Black Bodies 10.5281/zenodo.16759284 10 Redefining Electron Spin and AMM Through Harmonic Oscillation 10.5281/zenodo.17764997 13 Unified Spin Dynamics: From Pseudovector Nature to Relativistic Constraints 10.5281/zenodo.17765079 14 Bridging QM and GR: AMM and Geodetic Precession 10.5281/zenodo.17765097 15 Spin-Induced Inertial Resistance in Electrons 10.5281/zenodo.17765121 19 Spin as a Real Vector: Geometric Origin of U(1) and SU(2) Periodicity 10.5281/zenodo.17765238 — Series Context — The 0-Sphere Model is an ongoing research programme (2018–present) that derives 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