Observing the Reversal: The Eternal Cosmic Chase — Relative Cycle Parity, Missing Transition Epochs, and the Stratified Optical Archive in Six-Throat Directed Geometry

Paper XV — Theoretical Foundation of the Six-Throat Directed Geometry Program. This work extends the finite reversal law developed in Paper XIV into a reflective multi-cycle chronology. It derives relative cycle parity, repeated Missing Transition Epochs, the Eternal Cosmic Chase, archival-photon decoupling, processed reappearance, direction-resolved optical recurrence, and the stratified optical archive generated by repeated reversals. The central result is that directional recurrence is not temporal recurrence: a directed state can return to an earlier orientation class while the accumulated physical history of the same persisting material system continues forward. The manuscript distinguishes the received archival photon state from the contemporaneous physical state of its emitter and treats observer-sky direction as part of the complete optical-history label. This theoretical paper provides the foundation for a separate observational continuation of Paper XV applying the construction to the Milky Way and a blind NASA-Sloan Atlas geometric-closure test. The accompanying reproducibility bundle contains the LaTeX source, numerical and symbolic verification code, generated validation tables and figures, environment requirements, README documentation, and SHA-256 integrity records.