Exact counterexamples and stable oscillations in a two-host dormancy model

An exact one-parameter family refutes the deterministic equilibrium-attraction conclusion of Blath and Tóbiás's two-host dormancy conjecture while preserving stable resident subsystems and strict mutual invasion. All six nonnegative equilibria are hyperbolic and unstable; positive solutions are bounded and almost every positive initial state fails to converge to an equilibrium. A separate, dormancy-only path has a certified supercritical Hopf point and locally attracting positive periodic orbits. The local result does not classify global attractors or certify a periodic orbit at dormancy probability 0.98. The package contains written arguments, exact symbolic checks and outward-rounded interval certificates. Internal computational replay is not proof-assistant formalization or external peer review.