Good post, but let me add this. Something being a fact, and something being acknowledged by the NFL are not the same. I would say there are sourced rumors, and unsourced ones. We obviously put more faith in a source like the NFL who has some knowledge in what happenned over something unsourced which may not have any knowledge.
The NFL stated the balls were properly inflated, then at halftime underinflated. Maybe they meant to say underpressure, maybe they should have said underpressure but they didn't, and that's a different thing. Now some things they said like like that they are investigating, and what they did for measurements and whatnot are basically primary source statements. The NFL would be the people who know what they did.
But some are not, saying balls were at the proper inflation prior to the game, but not during requires some facts about the inflation changing. Inflation is defined as "expand with air." And that's how people usually understand it. So the NFL is saying the amount of air in the ball changed. To make such a statement you would either have to know somebody physically changed the air in it, or the ball had some flaw allowing the amount of air to change.
I'm highly skeptical the NFL knows either, and am not comfortable calling their statement a fact.