Temporal Jumps and Causal Return: Reversible Finite-State Dynamics, Coupled Clocks, and Witnesses in a Computational Universe (Project BOG)

Project BOG — Research Series II Temporal Jumps and Causal Return develops the mathematical and computational foundations of Bogdan Sergienko's Universe of Linked Temporal Cycles, the foundational Project BOG theory published at https://doi.org/10.5281/zenodo.23807064. The central problem is deceptively simple: if a system returns to the same point in time, how can we determine whether it has truly returned to its original state — and whether its causal history has also been preserved? This study distinguishes temporal phase synchronization, complete state recurrence, and uniquely reconstructible causal history. These properties may coincide under particular conditions, but they are not generally equivalent. Using finite-state computational systems, the study examines reversible transformations, coupled clocks, Chinese-remainder synchronization, distant logical-time jumps, and the information required to distinguish different histories. A reproducible experiment demonstrates that 36 distinct intervention histories produce only 24 final states, with four histories converging on the same state. Additional tests investigate reversible 8-bit transformations and temporal cycles of lengths 31 and 255. The publication provides runnable Python source code, JSON results, documented verification runs, mathematical explanations, and explicit opportunities for independent criticism. This research extends the earlier Project BOG study Verified State Jumps (https://doi.org/10.5281/zenodo.23682108) and forms part of a connected program investigating time, causality, computation, information, and the possible structure of reality. The results establish properties of specified computational models. Their relationship to the physical Universe remains an open scientific question. Research author: Bogdan Sergienko — Architect of Imagination. Computational collaborator: CCAI KAI — Architect of Computation (ChatGPT-based research assistance). Project BOG invites researchers to reproduce the calculations, challenge the assumptions, and propose stronger models.