Integral-time bounds for a two-coordinate diffusion from a scalar Poisson problem

We derive explicit two-sided bounds on the stationary integral time of the observed coordinate in a reversible two-coordinate diffusion. For a quartic potential coupled to a conditionallyGaussian hidden coordinate, one scalar Poisson problem determines both bounds for everypositive hidden mobility. The equilibrium law stays fixed as mobility varies, so the boundsquantify temporal differences that equilibrium matching does not resolve. The upper boundhas relative correction h2/ν, where ν > 0 is hidden mobility and h is the conditional-meancoupling. A lower variational bound uses the curvature of the scalar solution to control thefinite-mobility remainder and its O(ν−2) large-mobility decay. At fixed potential and temperature, the same scalar calculation can therefore bound the two-coordinate integral timewithout solving a new two-dimensional Poisson problem at each mobility. An accompanyingfinite-chain experiment tests a different observable: prediction from two six-symbol past observations. A complete-joint error budget certifies a log-loss improvement of at least 0.05214853nats over the exact current-coordinate predictor at slow mobility, and at least 0.04891692nats under independent 10% visible erasures. These forecast certificates concern the specifiedfinite chain and sensor; the integral-time bounds concern the continuous diffusion. Existingprediction and residence comparisons illustrate why the time bound has an observable-specificscope.