Rotation from Scalar Oscillation: Emergence of Photon-Sphere Angular Momentum from Phase-Staggered Kernel Dynamics in the 0-Sphere Model

50 Essential This paper demonstrates that angular momentum and a Lorentz-invariant chirality emerge as collective properties of a phase-staggered scalar oscillator array, with no rotational degree of freedom postulated at the level of any individual oscillator. — Core Geometric Result — Consider n radial oscillation axes uniformly distributed in angle, with amplitude Pₖ (a) = cos (a - bₖ), bₖ = k*pi/n, k = 0,. . . , n-1. Each oscillator moves strictly along its own radial axis. Nevertheless, the locus of maximum amplitude traces the unit circle, and the centroid of all oscillators traces a circle of radius r = 1/2. This 1/2 radius is the direct spatial counterpart to the energy floor of the 0-Sphere energy identity: cos⁴θ + sin⁴θ ≥ E₀/2 (AM-GM inequality with cos²θ + sin²θ = 1) Angular momentum arises from the transverse component of the vector-sum mode displacement — no circular force law is required. — Three Principal New Results — Lorentz-invariant chirality. The direction of circulation of the photon sphere defines a binary χ = sign (da/dt) ∈ +1, −1 This is the orientation of the continuous phase flow on S¹, a topological invariant classified by π1 (S¹) = Z. Its discreteness is a consequence of topology, not an assumption. The electron/positron distinction is encoded as the two oriented loops on S¹. Helicity and vZB. The frame-dependent projection of χ onto the propagation axis defines helicity h. The helicity-flip condition is tied directly to the subluminal Zitterbewegung speed vZB ≈ 0. 04c: in the massless limit vZB → 0, the internal rotation disappears and helicity flip is forbidden. Dirac spinor correspondence. The four Dirac spinor components map onto χ × spin = 2 × 2 = 4 states. The kernel pair (A, B) is a single continuous internal variable determined by one real phase θ, not an independent degree of freedom. — Resolution of the Open Question from Paper #47 — Paper #47 identified three independent occurrences of 1/2 in the 0-Sphere framework and predicted that their common geometric origin would be examined in a future paper. The present work provides that answer (Section 7. 4): Geometric 1/2 — centroid amplitude: the oscillator centroid traces a circle of radius exactly 1/2 (AM-GM constraint) Kinematic 1/2 — Thomas precession coefficient: ΩT ≈ (1/2) (v/c²) × a in the non-relativistic limit Topological 1/2 — spin quantum number s = 1/2: π-periodicity of the oscillation axis space generates the 4π spinorial holonomy All three reflect the single algebraic constraint cos²θ + sin²θ = 1 acting at geometric, kinematic, and topological levels simultaneously. — Thomas Precession as Relativistic Geometric Correction — Thomas precession does not generate the circular structure; that structure exists at the topological level of the S¹ phase flow independently of relativistic corrections. What Thomas precession provides is the relativistic window through which the internal topological structure becomes observable: it quantifies the phase stagger δb = π/n in terms of vZB, and accounts for the accumulation of ae per internal cycle via the identity ΔL = |γL − 1| · L0 established in Paper #47. — Series Position — Paper #50 is the direct continuation of Paper #46 (energy floor, zero-point energy geometric origin) and Paper #47 (three 1/2 factors). Together with companion Paper #7 (Dirac reinterpretation, 2020), it completes a geometric account of photon-sphere angular momentum, chirality, and the Dirac spinor structure within the 0-Sphere Model. # Title (abbreviated) DOI #1 Two Perfect Black Bodies (energy identity) 10. 5281/zenodo. 16759284 #7 Positive and Negative-Energy States in Dirac Eq. 10. 5281/zenodo. 17760132 #10 Electron Spin and AMM via Harmonic Oscillation 10. 5281/zenodo. 17764997 #19 Spin as a Real Vector: U (1) and SU (2) 10. 5281/zenodo. 17765238 #26 Spin from Geometry: Internal Berry Phase 10. 5281/zenodo. 17765409 #46 Geometric Origin of the One-Half Factor 10. 5281/zenodo. 19010945 #47 Rotational Lorentz Contraction and AMM 10. 5281/zenodo. 19120057 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