This one is a simple idea compared to my recent proposals. After hearing that SpaceX retrieved all its fairings instead of expending them, I thought my three-staged rocket allowed a clean recovery of the fairing.
My three-stage rocket works as follows. The first stage acts as an atmospheric elevator. You may think of it as an elevator that raises a special two-staged rocket above the Kármán line (100 km) and lets them start their journey in a vacuum. Then the first stage, which is the atmospheric elevator, descends back to the launch site. The first stage turns off its engines around 100 km, which means the rocket and its upper stages will still have vertical speed that is reduced by the Earth's gravity. Because of this initial kinetic energy, the upper stages can make their gravity turn immediately and fire their engines at maximum vacuum efficiency. This allows the second stage to reach Mach 10+ speed, which reduces the burden on the third stage.
If we come back to the fairing part of the process: once the first stage turns off its engines, the active fairing will take off from the nose of the third stage. It will separate just enough so that the exhaust of the second stage will have minimal effect on the fairing. Stage 2/3 performs a lateral cold-gas translation maneuver prior to main engine ignition to clear the vertical axis completely. Then, the fairing retro-fires, lands back, and docks on top of the first stage, which would have slowed down by now and started accelerating toward the Earth with the fairing on top.
On the other hand, SpaceX recovers its fairings from the sea by special missions.

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