If you could compress the air with no external factors yes, but you can't. You have meshing lobes that create heat to compress the air. This is pretty common knowledge.
"Of the three basic supercharger types, the Roots design historically possessed the worst thermal efficiency, especially at high pressure ratios." - Bell, Corky. Supercharged!. Bentley Publishers, 2001, p. 48.
Like I have said many times hard factual data, that is what people should be asking for. I can provide the turbo data will anyone with a supercharger do the same? It would be quite obvious I think.
Not only that I have shown via the compressor map on the closest M90 that you don't see positive manifold pressure until 2500+ rpms I have shown I have more boost at that rpm with a properly sized turbo. Again, I can provide the data, and I put it out there. No one has shown any different than I have said with the TRD Supercharger.