Spin-2 Structure of the Gravitational Mediator: Phase-Stable Pair Emission and Quadrupole Radiation in the 0-Sphere Model

[58 Structural — Resolution of Paper #49 Open Task (vi)] This paper provides a candidate resolution of open task (vi) of Paper #49 (2026): the rank-2 tensor proof for the ejected photon-sphere fragment, demonstrating its spin-2 character. The decisive structural observation is that the \(\pi\)-periodicity of the Thomas angular velocity \(\Omega_T(\theta) \propto \sin(2\theta)\,\hat{e}_z\) — established in Paper #48 as the spin-2 candidate — implies that each Zitterbewegung cycle produces two ejection events, at \(\theta = \pi/2\) and \(\theta = 3\pi/2\), separated by half a ZB period. The two events form a phase-stable pair whose combined polarization tensor is a rank-2 symmetric object with helicity \(\pm 2\), the polarization content of a graviton. The phase stability of the pair — previously assumed to require an external locking mechanism — is shown to follow structurally from the boundary conditions of the \(4\pi\) internal cycle established in Paper #1, without invoking conventional quantum-optical coherence. — Conceptual Shift: From a Single Ejection Event to a Phase-Stable Pair — Paper #49 identified the photon-sphere fragment ejected at internal phase \(\theta = \pi/2\) as the gravitational mediator: the kinetic energy \(E_{\text{ph}}(\theta) = (E_0/2)\sin^2\theta\) reaches the ejection threshold \(E_0/2\) at this phase, and the resulting fragment carries the gravitational coupling. Paper #49 then recorded as open task (vi) the rank-2 tensor proof of this fragment's spin-2 character. The natural first attempt — identifying a single ejection event with a spin-2 object directly — encounters a fundamental obstruction: a single fragment carries spin-1 polarization data and cannot, by itself, transform as a rank-2 symmetric tensor. The decisive observation is that the \(\sin(2\theta)\) factor in \(\Omega_T\) has period \(\pi\), not \(2\pi\): \(|\Omega_T(\theta)|\) reaches its maximum magnitude at both \(\theta = \pi/2\) and \(\theta = 3\pi/2\) within a single ZB cycle. Both phases satisfy \(E_{\text{ph}} = E_0/2\); both are therefore ejection events. The single-event picture of Paper #49 captures only half of the per-cycle structure. The resolution is to treat the two events as a phase-stable pair whose combined polarization tensor is the spin-2 object Paper #49 sought. — Core Result: The Phase-Stable Pair as a Rank-2 Symmetric Tensor — The proof is composed of four logically distinct structural results: Two ejection events per ZB cycle. The \(\sin(2\theta)\) periodicity of \(\Omega_T\) and the \(\sin^2\theta\) profile of \(E_{\text{ph}}\) jointly imply two phase-symmetric ejection events at \(\theta_1 = \pi/2\) and \(\theta_2 = 3\pi/2\) within each ZB cycle, separated by \(\Delta t = T_{\text{ZB}}/2\). The first event corresponds to forward photon-sphere traversal (Kernel \(A \rightarrow B\)); the second to return traversal (\(B \rightarrow A'\)). Both occur at the same kinetic energy \(E_0/2\) and the same magnitude of \(\Omega_T\) (zero, at the extremum of \(|\Omega_T|\)). Rank-2 symmetric polarization tensor. Let \(\varepsilon_{1\mu}\) and \(\varepsilon_{2\nu}\) be the polarization vectors of the two emitted fragments. The combined pair state is characterized by the symmetric two-index object \(h_{\mu\nu} \propto \varepsilon_{1\mu}\varepsilon_{2\nu} + \varepsilon_{1\nu}\varepsilon_{2\mu}\), which transforms as a rank-2 symmetric tensor under the Lorentz group. The trace-free part carries helicity \(\pm 2\) exclusively — the polarization content of the graviton. The construction is the structural analog of decomposing a tensor product of two spin-1 representations into spin-2 + spin-1 + spin-0 components. Phase stability via the \(4\pi\) boundary condition (replacing “coherent pair”). A natural objection asks whether the temporal relationship between the two events drifts across successive ZB cycles. The answer is that the boundary conditions of the \(4\pi\) internal spinorial cycle established in Paper #1 provide a structural phase re-normalization: at every turning point \(\theta = n\pi\), the photon-sphere kinetic energy vanishes (\(\gamma_{KE} = 0\)) and the total energy resides in a single kernel as a thermodynamic energy eigenstate. Each ejection event is bracketed by such eigenstates at distance \(T_{\text{ZB}}/4\), and the temporal separation \(\Delta t = T_{\text{ZB}}/2\) between the two events cannot drift. No external coherent-pair locking mechanism is required. Quadrupole moment at frequency \(\omega_{\text{ZB}}\). The two-kernel mass-energy asymmetry \(\Delta m(\theta) = (E_0/c^2)\cos\theta\) generates a quadrupole moment \(Q_{ij}(t) = (E_0\lambda_C^2/c^2)\cos(\omega_{\text{ZB}} t)\,(\hat{e}_{x,i}\hat{e}_{x,j} - \tfrac{1}{3}\delta_{ij})\), whose third time derivative drives gravitational radiation through the standard general-relativistic quadrupole formula \(P_{\text{GW}} = (G/5c^5)\,\langle (d^3 Q_{ij}/dt^3)(d^3 Q^{ij}/dt^3)\rangle\). The \(\pi\)-periodicity unifies three structures — Thomas precession, ejection timing, and quadrupole oscillation — under a single \(\sin(2\theta)\) factor. — Phase Stability: Discrete Renormalization, Not Continuous Coherence — The conceptual distinction between the present mechanism and conventional quantum-optical coherence is precise. Conventional coherence requires the continuous preservation of phase alignment across processes; in the 0-Sphere Model, the trajectory is non-closing (the ZB axis precesses by \(\Delta\phi = 2\pi a_e\) per cycle, per Paper #52), so continuous phase preservation would conflict with the directional anisotropy that the model requires. The resolution adopted is discrete phase renormalization: the phase is reset to the same energy eigenstate at every \(\theta = n\pi\), regardless of the trajectory followed during the preceding traversal. The structural role of the Hamiltonian identity \(E_0 = E_0[\cos^4(\theta/2) + \sin^4(\theta/2) + \tfrac{1}{2}\sin^2\theta]\) in guaranteeing this reset is consistent with the active confinement argument of Paper #56: the boundary condition \(\gamma_{KE} = 0\) at \(\theta = n\pi\) is precisely the turning-point marginal equilibrium at which ejection is reached from below, preventing escape and ensuring phase reset. The phase stability of the pair and the topological energy confinement of Paper #56 are two manifestations of the same algebraic constraint. — Directional Consistency with Paper #52 — Paper #52 proposed the directional decomposition \(\hat{e}_x\)→ gravity (longitudinal velocity), \(\hat{e}_y\)→ electromagnetism (transverse acceleration), \(\hat{e}_z\)→ spin (Thomas precession), supported by the metronome and Yagi–Uda antenna analogies. The phase-stable pair of the present paper provides a first-principles structural basis for the \(\hat{e}_x\) assignment: the pair is emitted at \(\theta = \pi/2\) and \(3\pi/2\), the turning points of the \(\hat{e}_x\) motion, where the photon sphere momentarily stops and the outward Thomas-precession stress culminates in ejection. The single-emission Larmor channel along \(\hat{e}_y\) (transverse acceleration) is the spin-1 electromagnetic sector; the phase-stable pair channel along \(\hat{e}_x\) (longitudinal velocity) is the spin-2 gravitational sector. The metronome analogy of Paper #52 is replaced by a structural derivation. — Connection to Paper #57's \(Y_\ell^m\) Imprinting — Paper #57 proposes that the local Zitterbewegung frequency \(\nu_{\text{ZB}}^{\text{local}}\) is imprinted onto the photon sphere as a spherical harmonic excitation mode \(Y_\ell^m\). The phase-stable pair of the present paper has the angular symmetry of a \(Y_2^0\) mode: the quadrupole moment \(Q_{ij}(t) \propto \cos(\omega_{\text{ZB}} t)\,(\hat{e}_{x,i}\hat{e}_{x,j} - \tfrac{1}{3}\delta_{ij})\) is the traceless rank-2 axisymmetric tensor that \(Y_2^0(\theta,\phi) \propto 3\cos^2\theta - 1\) generates. The phase-stable pair therefore furnishes a candidate identification of the specific \(\ell\) value of Paper #57's imprinting framework as \(\ell = 2\). The full verification — comparing the mode index excitation energy in Paper #57's framework with the pair-emission energy \(2 \times E_0/2 = E_0\) — is recorded as open task (IV) of the present paper. — Order-of-Magnitude Estimates — Using \(E_0 \approx 20.7\,\text{keV}\) (Paper #46), \(\lambda_C = 2.43\times10^{-12}\,\text{m}\), and \(\omega_{\text{ZB}} \approx 5\times10^{18}\,\text{s}^{-1}\): \[ P_{\text{GW}} \sim \frac{G}{5c^9}\,E_0^2\lambda_C^4\omega_{\text{ZB}}^6 \sim 10^{-70}\,\text{W} \] per single electron. The coupling-constant ratio \(F_{\text{grav}}/F_{\text{EM}} \approx 10^{-42}\) is dimensionally consistent with the ratio of the spin-2 (pair) emission cross-section to the spin-1 (single) emission cross-section. The quantitative derivation of this ratio — and of Newton's constant \(G\) from the pair emission rate, which would constitute the first derivation of \(G\) from internal electron dynamics — is recorded as open tasks (II) and (III). — Key Contributions — Two ejection events per ZB cycle: the \(\sin(2\theta)\) factor in \(\Omega_T(\theta)\), already documented in Paper #48, implies a second ejection event at \(\theta = 3\pi/2\) in addition to the first at \(\theta = \pi/2\), separated by \(T_{\text{ZB}}/2\). The single-event picture of Paper #49 captures only half of the per-cycle ejection structure. Phase-stable pair as rank-2 symmetric tensor: the combined polarization tensor \(h_{\mu\nu} \propto \varepsilon_{1\mu}\varepsilon_{2\nu} + \varepsilon_{1\nu}\varepsilon_{2\mu}\) carries the helicity \(\pm 2\) content of a graviton, supplying the rank-2 tensor proof requested in Paper #49 open task (vi). Phase stability from the \(4\pi\) boundary condition (replacing “coherent pair”): the boundary conditions of the \(4\pi\) internal spinorial cycle of Paper #1 provide a discrete phase renormalization at every \(\theta = n\pi\), guaranteeing the temporal separation \(T_{\text{ZB}}/2