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

This paper reconstructs the internal periodicity of a localized self-sustaining closure from the longitudinal C₂ reciprocating update and the transverse C₃ cyclic update introduced in a tetrahedral–octahedral composite cell. In the previous Electromagnetic Field III, the periods 2 and 3 of C₂ and C₃ were proposed to resynchronize every six updates, allowing mutual induction C₂ → C₃ → C₂ to sustain a localized oscillation. Here, C₂ is represented as a propagation mode of relational inconsistency that reciprocates linearly along the long axis, while the two C₃ structures on opposite sides of the long axis are represented as transverse phase modes circulating in opposite senses. The opposite circulation of the two C₃ structures provides a construction condition that assigns oppositely oriented internal states to the two ends of the C₂ reciprocation. The C₃ phase advances by 120° per physical clock and reaches 2π after three clocks, whereas the period-2 C₂ is then in the opposite directional state. The composite internal state therefore returns for the first time after six clocks, corresponding to 4π. A two-component state satisfying Ψ(φ+2π) = −Ψ(φ) and Ψ(φ+4π) = Ψ(φ) can be constructed from the discrete update operator. Together with the previously introduced transverse rotation operator J, this suggests a possible connection between SO(3)-type rotations and 4π internal periodicity as different representations of the same cell geometry. The structure is proposed as a geometrical candidate for the internal structure of the electron; no quantitative derivation of spin 1/2, charge, or mass is claimed.