"meiza" wrote
My zeroth stage would consist of helicopter rotors getting their
power from the rocket turbopump. The fuel consumption would only
be a few percent of the full-throttle operation (that much is diverted
typically to the gas generator), and not much additional hardware
would be needed. Just a gearbox and a clutch.

"Just a gearbox and a clutch" is a *huge* technical hurdle.

The gearbox of a helicopter is the heart of the whole thing, along with the
rotor. You're talking about a really big rotor to get the lift for a really
good rocket payload (plus all the fuel), which means really slow rotation.
So your gearbox has to get turbopump speed (extremely high) down to slow
rotor speed, carrying more more power than any helicopter gearbox ever made,
and do it at a low-enough cost, highe-enough efficiency, and low-enough
weight to economically override just adding another rocket stage.

I'm not saying it's impossible, but as technical achievements go it would
rank right up there with the all-time toughies.

Tom.