43 Supplement This supplementary note records interpretive considerations concerning the possible role of observer-dependent torsion and internal phase dynamics in the context of the 0-Sphere model. The discussion is motivated by the line-integral-based framework developed in the companion trilogy (zenodo.18067760, zenodo.18135855, zenodo.18203433), and by its potential implications for Zitterbewegung, phase accumulation, and the electron anomalous magnetic moment. Attention is drawn to the mathematical structure of teleparallel geometry on non-Abelian Lie groups, notably SU(2), as a concrete example of a space exhibiting vanishing curvature but nonzero torsion (Weitzenböck connection). No additional dynamical assumptions are introduced. The purpose of this note is to clarify conceptual consistency and to delineate a possible interpretive extension, without altering the quantitative results or claims of the main text. Foundational Distinction: Open Paths vs. Closed Loops A central contribution of this supplement is a systematic comparison between two classes of line-integral observables: Closed-loop integrals (Aharonov–Bohm effect, Berry phase, Wilson loops): gauge-invariant, cyclic phase accumulation on a pre-existing spacetime background. Open-path integrals (0-Sphere model): history-dependent, thermodynamically irreversible energy transport from kernel A to kernel B; spacetime structure emerges from accumulated interaction histories. Zitterbewegung: Dirac vs. 0-Sphere The supplement makes explicit, for the first time in the series, the contrast between two predictions for the Zitterbewegung velocity: Standard Dirac result: velocity-operator eigenvalues ±c (luminal, unobservable after time-averaging). 0-Sphere model prediction: vZB ≈ 0.04047c, derived from the experimentally measured anomalous magnetic moment via the identity γ(vZB) = 1 + a, without adjustable parameters (zenodo.18356895). This subluminal prediction is falsifiable and constitutes a concrete experimental discriminant between the two frameworks. Teleparallel Geometry on SU(2) The Weitzenböck construction on SU(2) is presented as a mathematically controlled proof of concept: the curvature tensor vanishes identically (R(W) = 0), while torsion encodes the non-Abelian structure of the Lie algebra (T(ea, eb) = −εabc ec ≠ 0). No claim is made that physical spacetime carries a Weitzenböck connection; the construction illustrates that the quartic spinorial terms cos⁴(θ/2) and sin⁴(θ/2) admit a natural torsion-based geometric language independently of curvature. Scope All interpretations are explicitly labeled as interpretive rather than derivational. This supplement does not introduce new dynamics, modify established field equations, or claim a definitive identification of the anomalous magnetic moment origin. It serves as a conceptual bridge for readers of the main trilogy, answering two questions not addressed there: (1) the geometric origin of the spinorial θ/2 structure in the quartic energy terms, and (2) the explicit comparison between the 0-Sphere velocity prediction and the Dirac result. Primary References Hanamura, S. (2018). A Model of an Electron Including Two Perfect Black Bodies. Zenodo. https://doi.org/10.5281/zenodo.16759284 Hanamura, S. (2025). Geometric Structure of Spinorial Phase Accumulation along Thermal Geodesics. Zenodo. https://doi.org/10.5281/zenodo.18067760 Hanamura, S. (2026). From Curvature to Connection: Revisiting the Geometric Origin of Conservation Laws. Zenodo. https://doi.org/10.5281/zenodo.18135855 Hanamura, S. (2026). Line Integrals as Fundamental Observables in Physics. Zenodo. https://doi.org/10.5281/zenodo.18203433 Hanamura, S. (2026). Geometrical Confinement: Rest Mass and Zitterbewegung. Zenodo. https://doi.org/10.5281/zenodo.18356895
Satoshi Hanamura (Sat,) studied this question.