A sharp escape-rate bound on information flow between two binary variables

Consider a stationary continuous-time Markov jump process whose state contains two binary variables X and M, possibly together with finitely many hidden states, in which X and M never jump at the same time and every allowed transition has an allowed reverse. Let ℓ be the Horowitz–Esposito information flow between X and M: the rate at which the jumps of M change the mutual information I(X;M), which at stationarity is minus the rate at which the jumps of X change it. If every state's total escape rate is at most R and the entropy production rate is at most Σ, then |ℓ| ≤ R f(Σ/R), where f(s) is the maximum of 2mu ln[(1−2m)/(2m)] over 0 < m ≤ 1/4 and 0 ≤ u < 1 subject to 8mu artanh(u) ≤ s. The bound is sharp: for every finite Σ > 0 an explicit four-state chain attains it, and so do its lifts with hidden states; its parameters solve a Lambert-W equation. As s → 0, f(s) ~ 0.347429877381… √s, and f increases to W₀(1/e) = 0.278464542761… (a value known as the maximal power of rate-limited Maxwell demons and information ratchets, reached here as an information-flow ceiling), a supremum that no model attains. These two limits sharpen the constants 0.937 and 1/e that we derive in a few lines from published inequalities (Section 7; these consequences are not stated in the cited papers, and 0.937 uses a numerically checked varentropy maximum). Outside the class the bound fails: under a mean-activity cap the flow is unbounded, simultaneous jumps exceed it, and with two three-valued variables explicit chains come arbitrarily close to W₀(2/e) R ≈ 0.463 R. A self-contained script checks the constants, the attainment, a refutation search and negative controls. In a narrative, non-systematic literature search on 2026-10-01 (Section 8) we did not find this sharp frontier stated; this is not a priority claim. The record includes the self-contained verification script (verify_frontier.py, numpy/scipy/mpmath; also printed as Appendix A) and its output. AI disclosure: the result was obtained and first written up on 2026-09-30/10-01 (JST) in unpublished reconstruction drafts prepared with an OpenAI Codex agent; it was reviewed by a separate Claude Opus 5.5 agent session, independent of the Codex agent but of the same model family as the writing agent (proof audit, re-derivation of the constants, refutation search and prior-art comparison); this note was written with Claude Opus 5.5 (Anthropic) under the author's direction. Before deposit the note went through three refutation rounds on 2026-10-01 (JST), each pass run by a separate Claude Opus 5.5 agent session with its own context, of the same model family as the writing agent: round 1 had three passes (mathematics; consistency and references; scope and novelty), round 2 had two (mathematics; consistency, references and scope), and round 3 was one combined confirmation pass. The findings of rounds 1 and 2 were corrected before the next round, apart from completing this disclosure; the minor corrections from round 3 were applied without a further review round, and no other part of the text was changed after round 3 except for this disclosure. No human referee has reviewed the text. The author takes responsibility for the deposited text. Working method only; the registered author is the human author alone.