Does the explanation of why slower airfoils are more efficient involve less energy loss from heating or compression of the working fluid? We read time and time again, how more efficient slower airfoils such as helicopter rotors, props, and sailplane wings are compared with rapidly rotating fan blades or faster (still subsonic wings).  We also read where some people consider compression a negligible factor below a certain speed, even to the point of considering airflow &lt; 200 mph as "incompressible"!

Yet looking at this commonplace and ubiquitous phenomena (indeed, while your prop is putting out 600 lbs of thrust, your wing is creating 4x more lifting force), is there a mathematical explanation for this involving the ideal gas law:  PV=nRT?

Can we visualize creating and maintaining a compressed "bow wave" ahead of a moving surface involves substantial amounts of energy even at lower speeds?

I have yet to fully understand why "it is better to move a lot of air a little than a little air a lot".
So you asked a whole bunch of things, and I'm not sure I really understand most of what you are asking, but I can maybe clarify this one part for you


  I have yet to fully understand why "it is better to move a lot of air a little than a little air a lot".


The equation for thrust is just Newton's second law F=$\dot m \Delta$V, where $\dot m$ is mass flow rate (kg/s) and $\Delta$V is the change in velocity of the air (exit velocity minus inlet velocity).  So, if you move 1 kg/s of air at 10 m/s, or you move 10 kg/s of air at 1 m/s, the thrust is the same.  Doesn't matter.

But, the equation of kinetic energy (well power really in this case) is $P=(1/2) \dot m (\Delta V)^2$.  So if you move 1 kg/s of air at 10 m/s, it takes $0.5*1*10^2$ = 50 W of power to do that.  But if you move 10 kg/s of air at 1 m/s it takes $0.5*10*1^2$= 5W of power to do that.  Exact same amount of thrust, but with literally 10X less energy required.  i.e. you need to burn way less fuel if you can move a lot of air at a low velocity than if you move a little bit of air a high velocity.