A sharp finite horizon for time reversal in finite-rank binary processes

Let a stationary binary process have a real linear representation of dimension at most d. For d ≥ 3, we prove that symmetry of all word probabilities through length 2d−2 implies time reversibility. This horizon is optimal: for every d ≥ 3, we construct a stationary binary process of minimal linear dimension d, generated by strictly positive edge-emitting matrices, whose first asymmetric word has length 2d−2. The upper bound holds more generally for scalar recognizable series on any finite alphabet, without stationarity or positivity assumptions. Its proof uses an alternating form on a doubled representation. In particular, the optimal binary stationary horizon at rank four is six. Supplementary files: LaTeX source, an exact-arithmetic verification script (Python 3 + SymPy) and its recorded output. AI assistance: this work was developed with substantial assistance from OpenAI Codex (literature search, computational exploration, proof development, drafting and verification scripts), directed by the author. Additional mathematical review was AI-assisted. See the disclosure section of the manuscript.