Photon-Sphere Fragmentation as a Gravitational Mediator: Radiation-Gradient Ejection in the π-Cycle of the 0-Sphere model

49 Essential This paper identifies the microscopic gravitational-mediation mechanism left open in Paper #34, and provides the physical ejection mechanism whose existence Paper #48 established as a prerequisite for a full spin-2 derivation. The physical basis is Thomas precession understood in the kinematic sense formulated by Tomonaga: whenever a body undergoes accelerated motion at relativistic speed, its proper coordinate frame rotates relative to the laboratory frame even under parallel-axis transport. In the 0-Sphere model, the photon sphere is continuously accelerated by the radiation gradient F = E0 sin θ, and Zitterbewegung sustains this oscillation at vZB ≈ 0.04c; Thomas precession therefore naturally arises as a structural feature of the internal dynamics, rather than being introduced as an additional assumption. — Core Equations — Hamiltonian identity (Paper #1, foundational postulate):E0 = E0 cos⁴(θ/2) + E0 sin⁴(θ/2) + (E0/2) sin²θ Geometric energy floor (Paper #46, AM-GM inequality):Te1 + Te2 = E0cos⁴(θ/2) + sin⁴(θ/2) ≥ E0/2 (equality at θ = π/2 + nπ) Ejection threshold (this paper):Eph(π/2) = E0/2 = Ebind ground state n = 0: threshold exactly saturated, no emissionn ≥ 1: energy surplus nℏω above floor → fragment ℏω ejected per de-excitation n → n−1 Thomas angular velocity (Paper #19, recovered in this paper):ΩT(θ) = −(v0² ω / 4c²) sin(2θ) ez non-zero because radiation-gradient acceleration a(t) is transverse to v(t) UV cutoff (this paper):Mode n has n+1 antinodes → n+1 ejection opportunities per A→B traversal.n bounded above by nmax (Compton-scale well geometry) → self-energy sum naturally finite. — Key Contributions — Tomonaga-Thomas ejection mechanism. The photon sphere is perpetually accelerated by the radiation gradient at vZB ≈ 0.04c; Thomas precession (in Tomonaga's kinematic sense) naturally arises, generating angular momentum that peaks at θ = π/2 and constitutes the outward centrifugal-like stress driving fragment ejection. Closure of Paper #34's three open questions. (i) The mediating quantum is the photon-sphere fragment ejected at θ = π/2; (ii) ejection occurs when the photon-sphere kinetic energy saturates the geometric binding floor E0/2; (iii) near-neighbor character follows from the Compton-scale photon-sphere size. Physical ejection mechanism for Paper #48's spin-2 candidate. The ejected fragment carries the π-periodic phase structure of ΩT through a kinetic-stress balance derived independently of kinematic spin-cycle counting. The rank-2 tensor proof remains an open task. Fourth independent route to zero-point energy. The geometric floor E0/2 is precisely the ejection threshold — the kinetic energy at which the photon sphere cannot retain its fragment. This adds a physical ejection interpretation to the three routes of Papers #46 and #48 (uncertainty principle, AM-GM inequality, centroid geometry). Structural ultraviolet cutoff. Re-absorption of an ejected fragment constitutes the 0-Sphere counterpart of a QED self-energy loop. Because the excitation number n is bounded above by nmax (Compton-scale well geometry), the self-energy sum is naturally finite — a structural Debye-type cutoff requiring no regularization. Quantum harmonic oscillator reinterpretation. The energy spectrum En = (n + 1/2)ℏω is reproduced identically: n counts external photons absorbed; zero-point energy is algebraically fixed by the quartic kernel structure, not by quantum fluctuations. Empirical predictions are unchanged. — Position in the 0-Sphere Series — Paper #49 is the direct successor to Paper #48 and the microscopic completion of Paper #34. It closes the gravitational-mediation gap that Paper #34 left open (photon exchange invoked but ejection mechanism unspecified), and provides the physical realization that Paper #48 explicitly required before the spin-2 numerical correspondence could be elevated to a derivation. — Key References — # Title (abbreviated) DOI 34 Geometrical Confinement: Emergent Gravitational Dynamics 10.5281/zenodo.18437010 46 Geometric Origin of the One-Half Factor 10.5281/zenodo.19010945 48 Geometric Origin of g = 2 in the 0-Sphere Model 10.5281/zenodo.19227518 1 A Model of an Electron Including Two Perfect Black Bodies 10.5281/zenodo.16759284 19 Spin as a Real Vector: Geometric Origin of U(1) and SU(2) 10.5281/zenodo.17765238 33 Geometrical Confinement: Rest Mass and Zitterbewegung 10.5281/zenodo.18356895 — 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