Sharp bounds on local DNA release from interval observations on river trees

This technical note derives sharp bounds on local environmental DNA release from interval flux observations on a finite river tree. Travel times are known, a shared exponential loss rate belongs to a finite closed interval, local releases are nonnegative and unbounded, and old-bed contributions lie in independent bounded intervals. Two tree traversals give exact marginal flux intervals at a fixed rate. For a compatible observation box, feasible rates form a closed upper interval: the local source minimum is attained at the minimum feasible rate, and its supremum is determined at the largest allowed rate. Every finite endpoint has a complete compatible assignment. The note also gives an exact-observation support criterion, witness constructions, and computational guarantees under explicit arithmetic assumptions. An analytic example with positive-width intervals has sharp lower bound 111/380 where a local conservative test gives no positive certificate. The results provide conditional release bounds and incompatibility certificates for a declared observation box. AI tools assisted with candidate derivations, proof checking, and manuscript preparation.