A Supplementary Correction to the 0-Sphere Model: Dimensional Consistency of G and c²

[28 Supplement / Correction]A Supplementary Correction to the 0-Sphere Model: Dimensional Consistency of \( G \) and \( c^2 \)(September 23, 2025; Revised: February 13, 2026) This is an archival correction to an earlier paper of this series, in which general-relativistic (GR) corrections were applied to a semiclassical description of lepton Zitterbewegung. It revisits the attempt to give leptons a "critical radius" by balancing special-relativistic Lorentz contraction against general-relativistic geodetic precession, and finds that the central numerical result of that attempt rested on a dimensional mistake. — What Went Wrong — The geodetic precession of a gyroscope orbiting a mass \( M \) at radius \( R \) is \[ \Delta\phi_{\mathrm{geodetic}} = 2\pi\left[\,1-\sqrt{1-\frac{3GM}{c^2R}}\,\right], \] whose dimensionless expansion parameter is \( GM/(c^2R) \). The earlier analysis omitted the factors of \( G \) and \( c^2 \) from this parameter, which inflated the GR correction by \[ \frac{c^2}{G} \approx 1.35 \times 10^{27}, \] about twenty-seven orders of magnitude. That spurious inflation is what produced an apparent muon critical radius near \( 10^{-25}\,\mathrm{m} \). It was a units error, not a physical effect. — The Corrected Numbers — With the dimensional factors restored: The geodetic precession term takes its dimensionally consistent form \( \dfrac{3GM}{c^2R} \). The characteristic length is \[ \frac{3G m_\mu}{c^2} \approx 4.196 \times 10^{-55}\,\mathrm{m}. \] The geodetic phase at \( R \sim 10^{-25}\,\mathrm{m} \) is \[ \Delta\phi_{\mathrm{geodetic}} \sim 1.32 \times 10^{-29}\ \mathrm{rad\ per\ orbit}, \] which is utterly negligible. The dimensionally consistent critical radius \( R_{\mathrm{crit}} = 3GM/(c^2\beta^2) \) evaluates, for the muon, to \[ R_{\mathrm{crit}} \approx 2.55 \times 10^{-52}\,\mathrm{m}, \] far below any physically meaningful microscopic scale. Once the dimensions are correct, general relativity plays no role at these scales — consistent with the conventional expectation — and the muon critical radius reported earlier is not supported. — Can a Special-Relativistic Effect Rescue It? — The paper asks whether Thomas precession could supply the missing radius-dependent correction. Accumulated over one period of a circular-like motion of radius \( r \) and speed \( v \), it is \[ \Delta\phi_{\mathrm{T}} = 2\pi\,\frac{\gamma^2}{\gamma+1}\,\beta^2 \approx \pi\beta^2, \] in which the orbital radius \( r \) cancels entirely. Depending only on \( \beta \) and not on \( R \), it cannot define a critical radius. Combining the Lorentz deficit and the Thomas term gives a half-circumference ratio \( L/L_0 \approx 1.0008 > 1 \), so the condition \( L/L_0 = 1 \) is never met except trivially at \( \beta = 0 \). (For reference, the Lorentz angle deficit at muon speed is \( \Delta\phi_{\mathrm{SR}}(\mu) \approx 2.59 \times 10^{-3} \) rad, and the Thomas term is roughly twice that.) — What This Does and Does Not Change — The revised calculation confirms three points: GR corrections are negligible at quantum scales. The previously proposed muon critical radius is not physically supported. Thomas precession, though comparable in magnitude to the Lorentz contraction deficit, is independent of orbital radius and so cannot define a finite critical radius. This correction removes the geodetic-precession-based critical radius and the lepton-decay mechanism built on it. It does not touch the model's core: the two-kernel electron, the energy-conserving oscillation, and the four-percent Zitterbewegung velocity inferred from the anomalous magnetic moment, which is a special-relativistic result and stands unaffected. Readers of the earlier paper should treat its critical-radius and lepton-decay sections as superseded by this note, while its special-relativistic content remains intact. The broader 0-Sphere research program, under continuous development since 2018, is not invalidated. — Position in the 0-Sphere Model Series — This paper is a supplement to and correction of Bridging Quantum Mechanics and General Relativity: A First-Principles Approach to Anomalous Magnetic Moments and Geodetic Precession (10.5281/zenodo.17765097), whose geodetic-precession calculation it corrects. It also draws on the foundational paper, A Model of an Electron Including Two Perfect Black Bodies (10.5281/zenodo.16759284). The dimensional review that identified the inconsistency was conducted with the assistance of large language model tools; this document is intended as a transparent archival correction rather than a formal peer-reviewed submission. — 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