Learning Directed Logistic Dynamics from Coarse Snapshots at the Parametric Rate

We learn an unknown directed logistic heat-bath generator from one arbitrarily initialized continuous-time trajectory observed only at independent coarse random times. Site refresh clocks have known unit rates; both incoming and outgoing degree are bounded by D, and each field-plus-interaction row has width at most Lambda. No update attempts or intermediate flips are observed. For every prescribed rational c>0, coordinatewise recovery to accuracy epsilon with confidence 1-delta uses C(D,c) exp(C(D,c) Lambda) epsilon^(-2) log(n/delta) full-state snapshots. This dependence on accuracy and dimension is optimal up to parameter-only constants, already for independent spins. The finite-bit algorithm has an absolute polynomial exponent in dimension, inverse accuracy, input encoding length, and logarithmic confidence. It combines a relative local-semigroup filter, candidate simulation, sparse logistic regression with an unpenalized intercept, and support-free geometric refinement preserving row and column sparsity on every execution. Two observation protocols are covered: independent Uniform[c,2c] gaps with a specified fixed rational rounding rule, and independent uniform choices from an explicitly constructed finite menu of rational lags in [c,2c]. The menu and required time precision are independent of final accuracy, dimension, and confidence. The theorem assumes known clock normalization, bounded incoming and outgoing degree, full observed states, and an independent randomized observation design. Its parameter-only constants may grow rapidly with degree and lag scale. It does not establish a deterministic-grid or unknown-rate learning guarantee. Full proofs, a finite-bit implementation, a matching statistical lower bound, and explicit prior-method attribution are included.