I use Mobil 1 synthetic and ATF in a 50/50 mix. Its made to lube parts running far faster and hotter than my M4's ever will. It was maybe $6 or $8 each when I bought them 5 or 6 years ago and the half gallon will last my lifetime. I can however, pour a bit in a small bottle labeled "gun oil" and sell it for $25 for anyone who would like some!
I called Benelli USA a few years back when I got my SBE and I was told to use a (Synthetic Oil) I use Birchwood Casy Synthetic Gun Oil with Tefelon on mine my SBE looks Brand New and it's has the Reeltree Pattern on it. And it works Great!
militec is the way to go with the inside chamber, it is a dry lube used by our special forces
a little of that on the rails is all you need
rem oil jells in cold temps