Sharp Limits on Time-Reversal Detection in Binary Hidden Markov Models

We construct, for every n≥3, a stationary binary hidden Markov model with strictly positive transitions and emissions whose first asymmetric consecutive observation window has length 2n-2. At any set of at most 2n-4 distinct times, its joint observation law agrees exactly with that of a reversible reference model. A finite reversal certificate shows that the window bound is sharp. After a binary memoryless channel with signal parameter θ, we compute the forward-versus-reverse relative entropy rate: its first nonzero term has order θ^(4n-6) and an explicit positive coefficient. A remainder estimate uniform in record length yields a matching necessary detection scale; a block word-count test supplies sufficiency. For each fixed constructed model, known channel, and fixed error target in (0,1/2), the required length is Θ(|θ|^-(4n-6)). This exponent is the largest possible in the fixed-model sense among positive n-state binary state-emitting HMMs with irreversible observations. Constants are model dependent. MSC (2020): Primary 60J10; Secondary 60G10, 94A17. Files: the manuscript PDF (v0.10, 13 pages) and the LaTeX source archive, which includes the code supplement under anc/.