Exact profile divergence and octic contact in a partially observed Ornstein Uhlenbeck process

Consider the observed coordinate of a stationary two-dimensional triangular Ornstein–Uhlenbeck process, driven by a hidden relaxation mode and independent white noise. We compute its exact forward relative-entropy-rate projection onto the family of stationary scalar Ornstein–Uhlenbeck laws. Writing a, c > 0 for the observed and hidden relaxation rates, A for the direct diffusion variance, γ = ε²σ_y²/A, and d = √(c² + γ), the unique optimal relaxation rate is b_* = ac(a + c)/(ac + d²) and the minimum is (a − d)²(c − d)²/[4(a + c)(ac + d²)]. This factorization gives the complete zero set and the local quadratic, quartic, and octic contacts on explicitly stated parameter slices. At fixed c = a, the leading term is γ⁴/(256a⁷). For exact observations at spacing h > 0, we also derive the globally optimal stationary AR(1) approximation, an exact finite-vector divergence formula, and the strictly positive coefficient h⁴e^(−4ah)/[32a⁴(1 − e^(−2ah))³] multiplying γ⁴ in the sampled octic law. The latter coefficient per unit time tends to the continuous coefficient as h ↓ 0. These are approximation and identifiability results, not calibrated detection guarantees. Exact cancellation precludes a uniform coupling-based separation bound when the hidden rate is allowed to vary.