Armchair theory - extreme cold air intake.

[Beer Jammer]

Hi all.

I was doing some sitting and thinking the other day. The quest for colder intake air seems to be a very worth while one, with CAI systems giving good gains in performance/response/powerband etc with the 1nz.
I know of the usual mods to acheive this. A cold air feed or position the filter to illiminate heat soak, the "throttle body bypass mod" to name but two.

Now for the armchair theory... .... if colder and more dense air makes the engine run more efficient, would there be a benefit to routing the air conditioner to feed into the air intake?
I know that the a/c saps power, but would routing its output to feed both the cabin and the engine air intake work to offset this reduction in power and make the engine give better mpg when you use a/c? perhaps then meaning that using a/c reduces mpg by 5% not 10%?

Unless the a/c unit is a very efficient one, I dont think this is a huge performace gainer, but how does the theory stand?