Two-state exact routing: algebraic feasibility and sharp delay-two rigidity

For rational-inner passive routers with exactly two internal states and three prescribed nodes, we give a complete three-parameter description of achievable trace-delay functions. A global delay cap is equivalent to nonnegativity of one quartic, with an explicit small Gram certificate. We characterize exactly when the winding lower bound two is attained, prove a positive gap elsewhere, and construct a sharper cubic-root upper bound for symmetric acute configurations. An explicit two-port completion realizes each admissible triple. The global acute optimum and general asymmetric minimum remain open. The results assume orthogonal output rays, freely chosen phases and dimensionless angular trace delay.