When the check engine comes on, all the oil in the crankcase is gone. The mechanic found no leaks and there is no smoke coming from the tail pipe.

3 to 4 quarts every 500 miles would be so significant that you would have to be blowing massive smoke out of the tailpipe while driving. There is no "phantom" situation here. Either the oil gets consumed by some kind of major internal engine issue, or it leaks out. There is simply no other way. Consult the advice of a local expert for assistance.