How does radius of a propeller affect its performance?
Would the propeller hub in the middle be the same size? You did say all things being equal, which I read as "unchanged". 

Example, 86" total diameter prop, 12" diameter hub, leaving 2 blades that are (86 -12)/2 = 37" from edge of hub to tip of blade, and 74" of working blade length.  

Then you reduce the radius to 43" with same 12" hub, so you have (43 - 12)/2 = 15.5" blade from hub to tip, and 31" of working blade length. 31" is less then 1/2 of 74", so you've lost more than 50%. 

Assuming the engine can't spin any faster due to the engine governor limiting the RPMs, you're not getting off the ground.