Brady likes the ball on the low end of inflation. Physics gas laws state that any given volume of air in a closed chamber (football) is at a certain pressure (psi) at a certain temperature. You drop the temperature, while the volume stays constant, the pressure must drop. There's a lot of talk about how that alone would result in maybe a 1 psi change from indoors to outdoors. But there are two other factors at play. First, all air compressors increase the temperature of the output air as part of compression. I can tell you my air compressor outputs at about 90-110F. 100F drop to 50F is almost 2 psi reduction. Secondly the barometric pressure at the time of the filling of the football vs the pressure when re-measured would also see a slight drop in psi.  

So if you assume the Pats fill the balls to 12.5 psi, league minimum an hour or two before the game, at a very low atmospheric pressure, with an air compressor outputting at 100F, it's going to be 2-3psi lower after equilibrating to 50F, when re-measured after the atmospheric low passed and barometric pressure was rising.  

If you presume Luck prefers balls at 13.5 psi, they probably filled them before they got to Gillette (which is what you'd expect from a traveling team) and simply checked/topped them off before the game, that scenario, which is not far fetched at all, explains the whole thing.  

Now I also am of the opinion that the Pats are smart enough to know all this and take advantage of physics whenever possible. So they probably purposely fill them shortly before have them measured, and know full well physics will bring the psi down to where Brady likes it. Is it the Pats fault they understand physics better than the NFL commissioner? Maybe. Maybe not. That's kind of grey if you look at the spirit of the rule vs the language of the rule.