Proton-Trapped Electron as the Generator of Spin-2 Gravitational Radiation: Coulomb Confinement, ZB Axis Locking, and the Separation of Gravitational and Electromagnetic Sectors

[59] Why do isolated free electrons not radiate gravitational waves, while bound matter does? This paper proposes that the difference is one of directional stability. In the 0-Sphere Model — an ongoing programme that describes the electron as two thermodynamic kernels joined by a photon-sphere-like energy structure, with Zitterbewegung treated as a real internal oscillation rather than an interference artifact — a free electron's internal axis precesses continuously, so any directed spin-2 emission averages to zero over a precession period of \(T_{\text{prec}} \approx 5\times10^{-16}\,\text{s}\). In a hydrogen atom, the proton's Coulomb field locks that axis to the proton–electron direction with a strength roughly \(10^{39}\) times the gravitational asymmetry, allowing a stable, directed quadrupole moment to form. The hydrogen atom is thus identified as the minimal structure capable of sustaining directed gravitational radiation. — Core Equations — Free-electron vs. bound-state radiated power (the central contrast):\[ P_{\text{GW}}^{(\text{free})} \to 0, \qquad P_{\text{GW}}^{(\text{locked})} \sim 10^{-70}\,\text{W} \] Coulomb dominance over the gravitational asymmetry at the Bohr radius:\[ \frac{\delta E_{\text{Coulomb}}}{\delta E_{\text{grav}}} \approx \frac{e^2}{4\pi\varepsilon_0\, m_e g\, a_0^2} \approx 10^{39} \] Non-closing Zitterbewegung axis (the origin of the anisotropy):\[ \hat{e}(t + T_{\text{ZB}}) = R(\hat{e}_z,\, 2\pi a_e)\,\hat{e}(t), \qquad \Delta\phi = 2\pi a_e \approx 7.3\times10^{-3}\,\text{rad} \] A closed trajectory (\(a_e = 0\)) would yield an isotropic time-averaged distribution, with no directional quadrupole moment surviving to source gravitational radiation; the non-zero anomalous magnetic moment is therefore the microscopic condition for the gravitational quadrupole to exist at all. — Key Contributions — Free-electron suppression: the phase-functional endpoints of an isolated free electron are rigidly comoving with its center of mass, so no externally observable time-varying mass distribution exists and no quadrupole gravitational radiation is sourced. The static rest-mass gravitational field is unaffected, in accordance with the equivalence principle. Coulomb axis locking: the proton's Coulomb field supplies a directional turning-point asymmetry roughly \(10^{39}\) times larger than the gravitational one at the Bohr radius, locking the electron's Zitterbewegung axis to the proton–electron direction \(\hat{r}\) within a small number of cycles. Phase stability from the \(4\pi\) boundary condition: the locked quadrupole oscillation remains phase-stable across successive cycles because the internal energy returns to a definite single-kernel energy eigenstate at every half-period \(\theta = n\pi\) (Paper #1). No external coherence postulate is required. Gravitational radiation as a property of bound matter: axis locking and phase stability together yield a fixed-direction power \(P_{\text{GW}} \sim 10^{-70}\,\text{W}\) per hydrogen atom, in contrast to zero for a free electron — a microscopic account of why gravitational-wave sources are bound systems. Endpoint variation as the origin of \(g_{\mu\nu}\): the orbiting electron supplies genuine variation of the phase-functional endpoints over the orbital volume (\(a_0/\lambda_C \approx 22\)), realizing the metric branch of the dual-sector structure of Papers #51 and #55. Directional decoupling of gravity and EM: in the helical equations of motion, the longitudinal spin-2 ejection (\(\hat{e}_x\)) and the transverse spin-1 Larmor emission (\(\hat{e}_y\)) share no coupling channel; whether this corresponds to the field-theoretic absence of a direct spin-1 / spin-2 vertex is recorded as an open task. — Series Position — This paper is the third in the five-paper sequence #57–#61 developing the microscopic gravitational mechanism of the 0-Sphere Model. It addresses when directed emission occurs, complementing Paper #57 (what is emitted) and Paper #58 (the spin-2 character of the emission). It continues the directional-decomposition and non-closing-trajectory programme of Paper #52, and rests on the \(4\pi\) internal-cycle boundary conditions of Paper #1 and the photon-mediated synchronization mechanism of Paper #34. The endpoint-variation framework is supplied by Paper #51 and situated within the dual-sector hierarchy of Paper #55. The paper is presented as a Structural Proposal and explicitly registers its open tasks, principally the proton's own internal 0-Sphere structure and the reconciliation of the deterministic axis-locking picture with the spherical symmetry of the hydrogen \(1s\) orbital. — References (Hanamura papers) — # Title (abbreviated) DOI 1 A Model of an Electron Including Two Perfect Black Bodies 10.5281/zenodo.16759284 10 Redefining Electron Spin and Anomalous Magnetic Moment 10.5281/zenodo.17764997 34 Geometrical Confinement: Emergent Gravitational Dynamics 10.5281/zenodo.18437010 49 Photon-Sphere Fragmentation as a Gravitational Mediator 10.5281/zenodo.19393391 50 Rotation from Scalar Oscillation: Chirality and Helicity 10.5281/zenodo.19482145 51 Helical Trajectory, Fixed-Endpoint Line Integrals, Spacetime Metric Emergence 10.5281/zenodo.19489126 52 Geometric Origin of the Weak Equivalence Principle: \(\lambda_C\) Cancellation Theorem 10.5281/zenodo.19583032 55 Four Functions of the Open Wilson Line: Dual-Sector Hierarchy 10.5281/zenodo.19800797 57 Spherical Harmonic Imprinting and the Physical Identity of Phase Information 10.5281/zenodo.19932176 58 Spin-2 Structure of the Gravitational Mediator 10.5281/zenodo.19932861 60 The Berry–Synge–Bonnet–Myers Triangle 10.5281/zenodo.19932922 — 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