Geometry and 4π Internal Periodicity of the Localized C₂–C₃ Closure

This paper reinterprets the longitudinal C₂ reciprocal update and the transverse C₃ cyclic update introduced in the tetrahedron–octahedron compound cell, not as two independent modes of a localized self-sustaining closure, but as components of a single composite internal orbit. In the previously reported Electromagnetic Field III, we proposed that the commensurability of the periods 2 and 3 of C₂ and C₃ produces resynchronization every six updates and that C₂→C₃→C₂ mutual induction can sustain a localized oscillation. Here we introduce a geometrical representation in which C₂ is a bipolar state along the long axis and C₃ is a threefold-symmetric rotation perpendicular to that axis. Counting one physical clock as a 120° update of C₃, C₃ completes a 2π rotation in three clocks, while C₂ reaches the opposite pole, so that the composite internal state changes sign. Only after six clocks, i.e. 4π, does the entire internal state return. Constructing a continuous interpolation as an orbit on the surface of a spheroid naturally introduces the half-angle structure cos(φ/2) and sin(φ/2), allowing a normalized two-component state satisfying Ψ(φ+2π)=-Ψ(φ) and Ψ(φ+4π)=Ψ(φ). Furthermore, the transverse rotation operator J introduced in previous work satisfies J²=-I in the transverse plane, while the axis-dependent generators obey so(3)-type commutation relations. Thus, the model suggests a possible connection between SO(3)-type spatial rotations and a 4π-periodic internal state as different representations of the same discrete cell geometry. Because the structure simultaneously possesses localization, a self-sustaining internal periodicity, and a double-valued rotational structure that fully returns after 4π, we propose it as a geometrical candidate model for the internal structure of the electron. This paper does not, however, claim a derivation of electron spin 1/2 itself. What is established here is a geometrical candidate in which a double-valued rotational structure transforming in an SU(2)-type spinor representation can be constructed from the C₂–C₃ closure.