This repository presents a reproducible computational study of exact recurrence, observational recurrence, and hidden-state ambiguity in finite reversible dynamical systems. The central result is that repeating a complete predictive state forces the same future evolution, while repeating only an observable state does not. In the exhaustively analyzed HPP lattice-gas model, 54 microscopic states produce identical density histories despite representing different underlying particle configurations. These states form 18 period-three cycles arranged into 9 time-reversal pairs, producing 27 predictive doubletons. The project includes the research paper, source code, validation scripts, tests, generated data, figures, integrity checks, and an independent certificate verifier. The results are intended to clarify the difference between visible observations and complete internal states in deterministic systems, with implications for recurrence, simulation, inference, and hidden-state reconstruction.