62 Foundation — Grounding the Bridge Between the Magnetic Anomaly and the Electron’s Hidden Motion One of the most precisely measured numbers in all of physics is the anomalous magnetic moment of the electron — a tiny correction to the strength of the electron’s internal magnet. The 0-Sphere Model, a single-particle framework in which the electron is pictured as a tiny closed system with energy oscillating between two internal kernels, has long made a bold claim about this number: it is not merely a magnetic correction but a direct record of the speed of the electron’s rapid internal trembling motion. The claim is carried by a simple bridge equation connecting the measured anomaly to that hidden internal speed. But the bridge has carried two long-standing debts. First, what kind of quantity is the measured anomaly, that an externally observed magnetic number should know anything about an internal velocity? Second, the velocity version of the bridge has always included a small numerical correction factor whose only justification was an analogy with alternating electrical current — a placeholder, not a derivation. This paper pays both debts, and is candid about the one piece of the accounting that remains open. — One Invariant, Two Representations — The first half of the paper addresses the input. In a Penning trap, the anomaly is measured as a ratio of two characteristic times, aᵉ = Tᶜ/Tₐ: a per-revolution phase slip, accumulated as the spin precesses slightly faster than the cyclotron motion. In the 0-Sphere model, the same number appears as a ratio of two lengths, aᵉ = (L₀ − L)/L₀: the rotational Lorentz contraction of the internal orbital arc traversed by the photon sphere. The paper’s proposal is that these are two representations of a single frame-independent invariant, joined by the translator γₗ = 1 + aᵉ — in precise analogy with special relativity, where one Lorentz boost simultaneously dilates time and contracts length. A structural bonus follows: a contraction deficit is the same whether the internal circulation runs clockwise or counterclockwise, so the length-ratio representation is intrinsically non-negative. The modulus in the bridge equation γ = 1 + |a| is thereby a geometric consequence, not a convention. The claim boundary is stated with discipline: the field-independence of the anomaly is consistent with the internal-geometric reading but does not force it, since standard quantum electrodynamics is equally field-free. — The Origin of the Factor 1/√2 — The second half addresses the transformation. The model’s conservation identity, cos⁴(θ/2) + sin⁴(θ/2) + ½sin²θ = 1, is read as the normalized relativistic energy–momentum relation: the first two terms play the role of the mass fraction and the third, ½sin²θ, the momentum fraction. The internal velocity is the square root of the momentum term. That term has a time average of 1/4 and a peak of 1/2, and the square root of their ratio is exactly 1/√2. The factor that the original 2023 derivation imported by analogy with alternating voltage is therefore already contained in the identity itself: because the velocity is a square-root quantity built on a sin²θ momentum term, its cycle representative is necessarily the root-mean-square, not the peak — and not the simple average 2/π or a naive 1/2, neither of which the identity permits. — The Geometric Carrier — The paper then identifies the geometric object that carries this structure: a pulsating-radius helix with R(θ) = (1/√2)|sin θ|, whose squared radius equals the momentum term exactly. The radius vanishes on the two kernels — recovering the stationary points where the photon-sphere momentum is zero — and reaches its maximum 1/√2 at the midpoint. The winding number of the helix is fixed to unity by a phase duality internal to the identity: the kernel amplitudes evolve at the 720° (SU(2), spinorial) scale while the kinetic energy evolves at the 360° (U(1), energy) scale, and the velocity, being a property of the kinetic term, must use the energy-level winding. With that choice the longitudinal projection of the helix carries the factor 1/√2 exactly. Three routes — the average-to-peak ratio, the maximum helix radius, and the longitudinal projection — converge on one number because all three are readings of the same sin²θ momentum term. — What Remains Open — The paper holds one distinction firmly throughout. What the identity yields is a velocity-level ratio: cycle-representative velocity to peak velocity. The canonical form of the bridge, γᵛ = 1 + |a|/√2, places the factor instead at the level of a. Because the bridge relation is nonlinear, the two placements are not equivalent — they differ by a residual factor of 2¹∕⁴ at the velocity level — and their exact reconciliation is recorded as an open task rather than asserted. A candidate route (a contraction deficit proportional to the transverse helix amplitude) is noted without commitment. A further appendix records two internal ratios (the 3:1 time-averaged partition and a c/2 longitudinal representative) that were examined and deliberately set aside as lacking a confirmed connection to observables. — Why It May Be Worth Reading — The bridge equation is the load-bearing quantitative relation of the 0-Sphere programme: it is the step that turns the measured anomaly into an internal mass scale and an internal velocity, and it was recently applied unchanged to the proton. Until now its velocity form rested on an admitted analogy. This paper replaces that analogy with a derivation, explains why the mass form reads the peak while the velocity form reads the RMS, gives the modulus in γ = 1 + |a| a geometric origin, and delineates — in a dedicated scope appendix — exactly what is claimed: a re-grounding within the special-relativistic scope of the Dirac equation, not a new empirical prediction. — Position in the 0-Sphere Model Series — The present paper supplies the first-principles foundation for the bridge equation introduced in Paper #10, whose alternating-voltage heuristic it replaces. It builds directly on Paper #47, which established the rotational Lorentz contraction reading of the anomaly, and on Paper #51, whose constant-radius helical trajectory is refined here into the pulsating-radius helix. It forms a pair with the immediately preceding Paper #61, which applies the γ-equation to the proton; the present paper grounds the equation that #61 applies. The foundational identity is that of Paper #1, its relativistic reading that of Paper #7, and the SU(2)/U(1) phase duality that fixes the winding number is that of Paper #19. The line-integral genealogy recorded in the appendix draws on Papers #29, #30, #31, #54, and #55. — 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
Satoshi Hanamura (Sat,) studied this question.