Rarity of rocket-driven Penrose extraction in Kerr spacetime

We study rocket-driven Penrose extraction in the test-particle limit on a fixed Kerr background for equatorial prograde flybys under explicit steering prescriptions. A spacecraft ejects exhaust inside the ergosphere; when the exhaust attains negative Killing energy, the remaining spacecraft gains energy by 4-momentum conservation. Across 320{,}000 simulated trajectories spanning black-hole spin, exhaust velocity, and orbital parameters, extraction with escape is rare in broad parameter scans (at most ${\sim}1\%$) and requires high spin ($a/M\gtrsim 0.89$), highly relativistic exhaust ($v_e\gtrsim 0.91c$), and finely tuned initial conditions. Under optimal tuning the success rate reaches ${\sim}70\%$ at $a/M = 0.95$. For representative escape trajectories, a single periapsis impulse is more propellant-efficient than the continuous-thrust controllers studied here. All quoted thresholds are empirical and specific to the orbit family, prior, and steering protocol studied.