Keith, all reasonable contributions are welcome.
The assumption is we have thrust:
If we have zero propulsive efficiency this means that we are wasting all of the power that is being generated by our propulsion system. This means that we have no thrust horsepower.
I disagree with your conclusion. For a helo in the hover propulsive efficiency is zero. But “the power being generated by our propulsion system” is not being wasted.
Also, propulsive efficiency is not the same as propeller efficiency. If there is thrust, propeller efficiency absolutely cannot be zero. Whereas propulsive efficiency
must be zero when the aircraft is stationary, regardless of thrust.
For that reason, any attempt to use propulsive efficiency to prove that THP must be zero in the static thrust scenario, is a logical fallacy. All that is proved is that propulsive efficiency is zero.
THP = BHP x Propeller Efficiency.
If there is thrust PE cannot be zero therefore THP cannot be zero.
“THP is the propeller output or power that is converted to useable thrust by the propeller”. That statement and the equation above are both taken from the document cited by italia458 after he floated this canard.
Before anyone jumps on the word “useable” to suggest the thrust is not useable unless it’s moving the aircraft, consider the hover again. The thrust is not only useable, it is being used.