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 \[ P_k(a) = \cos(a - b_k), \qquad b_k = \frac{k\pi}{n}, \qquad k = 0, \ldots, n-1 \] 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 \( r = \frac{1}{2} \). This \( \tfrac{1}{2} \) radius is the direct spatial counterpart to the energy floor of the 0-Sphere energy identity: \[ \cos^4 \theta + \sin^4 \theta \ge \frac{E_0}{2} \quad \text{(AM-GM with } \cos^2\theta + \sin^2\theta = 1 \text{)} \] 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 \[ \chi = \operatorname{sign}\!\left(\frac{da}{dt}\right) \in \{+1, -1\} \] This is the orientation of the continuous phase flow on \( S^1 \), a topological invariant classified by \( \pi_1(S^1) = \mathbb{Z} \). Its discreteness is a consequence of topology, not an assumption. The electron/positron distinction is encoded as the two oriented loops on \( S^1 \). Helicity and \( v_{ZB} \). The frame-dependent projection of \( \chi \) onto the propagation axis defines helicity \( h \). The helicity-flip condition is tied directly to the subluminal Zitterbewegung speed \( v_{ZB} \approx 0.04c \): in the massless limit \( v_{ZB} \to 0 \), the internal rotation disappears and helicity flip is forbidden. Dirac spinor correspondence. The four Dirac spinor components map onto \( \chi \times \text{spin} = 2 \times 2 = 4 \) states. The kernel pair \( (A, B) \) is a single continuous internal variable determined by one real phase \( \theta \), not an independent degree of freedom. — Resolution of the Open Question from Paper #47 — Paper #47 identified three independent occurrences of \( \tfrac{1}{2} \) 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 \( \tfrac{1}{2} \) — centroid amplitude: the oscillator centroid traces a circle of radius exactly \( \tfrac{1}{2} \) (AM-GM constraint) Kinematic \( \tfrac{1}{2} \) — Thomas precession coefficient: \[ \Omega_T \approx \frac{1}{2}\left(\frac{v}{c^2}\right) \times a \] in the non-relativistic limit Topological \( \tfrac{1}{2} \) — spin quantum number \( s = \tfrac{1}{2} \): \( \pi \)-periodicity of the oscillation axis space generates the \( 4\pi \) spinorial holonomy All three reflect the single algebraic constraint \( \cos^2\theta + \sin^2\theta = 1 \) 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 \( S^1 \) 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 \( \delta b = \frac{\pi}{n} \) in terms of \( v_{ZB} \), and accounts for the accumulation of \( a_e \) per internal cycle via the identity \[ \Delta L = |\gamma_L - 1| \cdot L_0 \] 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 \( \tfrac{1}{2} \) 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