Exact teleportation bounds for fermionic Hawking thermality beyond the single-mode approximation

Fermionic teleportation near a Schwarzschild horizon involves two physically distinct ingredients that are often varied together: Hawking thermal mixing and the relative composition of the right and left Unruh modes. We separate them by factorizing the reduced map of the detected exterior fermionic mode into a mode-preparation channel and a Hawking thermal channel. At fixed right-left Unruh-mode composition, we then solve a global constrained quantum-channel optimization over all two-qubit resource states whose fully entangled fraction remains at or below the standard teleportation threshold immediately before thermal mixing. A parity twirl and positivity reduce the arbitrary-state problem exactly to a two-variable quadratic optimization. The solution yields a necessary-and-sufficient activation boundary, a second exact boundary for attaining the universal optimum, and explicit optimizing states. The resulting phase diagram contains a strict no-go region, a composition-limited activation region, and a universal-bound region. The analysis converts state-specific observations of Hawking-assisted teleportation into exact channel-level bounds over the full two-qubit state space.