This is a bit lengthy, but necessary to frame the question. Basically, a reputable supercharger manufacturer has stated that "cooling the air after compression doesn't (by itself) increase power." Specifically I am referring to an automotive engine with a PDM (roots style) supercharger:

 "...intercoolers do not "make" more horsepower. They cannot. Supercharger output is FIXED and doesn't increase merely because downstream air charge temperature is lowered. It's the same air by mass. The mass, the weight, the oxygen content of that blast of air discharged by the supercharger cannot be increased. You don't believe it? Try this. Capture the discharged air in a bottle, seal it and weigh it at 300°, 200°, -50°. The weight (mass) won't change, the engine can't ingest more air so it won't make more power. Yes, the cooler denser air, at -50° for example, will "allow" more boost or advanced engine timing but that air, by itself, won't make more power. Now if the air entering the supercharger (ambient) is reduced, it is denser and cooler and then will make more power (1% increase for every 10° temperature drop according to our data)."

The question is this... Since the volume below the supercharger is constant prior to the intake valve opening and greater than the volume of a given cylinder (hence why boost stays constant and doesn't drop off and build up everytime a cylinder intake valve opens), doesn't it stand to reason that cooling that compressed volume air will

1) slightly reduce the boost (all else equal) because PV/T = constant? If V is constant and T goes down, then P must go down as well - obviously this assumes the supercharger "stops", the volume of compressed air is trapped and then cooled.

 and

 2) allow for more of the "available" oxygen (by mass) in the fixed volume (between the outlet of the supercharger and the bottom of the intake) to enter the cylinder (also a fixed volume chamber) for combustion?

Comments are certainly appreciated, but what I'd really like to see as an answer here is a Thermodynamics control volume solution to the problem. Though I'm a mechanical engineer, it has been well over ten years since I dug out my Advance Fluid Mechanics and Thermodynamics books to reduce this kind of question to a "text book solution." Any insight would be appreciated.