I keep a close eye on mechanical operations by the players because half or more of them in our group aren't that interested in mechanics, and tend to make mistakes. For whatever reason, this tends to hurt their characters a lot more than it helps them. There have been a couple of times where if I hadn't said, "Wait a minute, don't you also have ...", a couple of characters would have died. This has been going on a long time.
As a byproduct of this, though, I also tend to pick up fairly quickly on those players actively fudging. It happens that the people trying this in my games over the last 30 years have almost all been doing it not for any story or narrative or fun reasons, but because they like the thrill of getting away with something. That is,
they think it is wrong, but are doing it anyway. I've noticed a heavy correlation of that attitude with other, more serious behavior that I frankly dislike, and thus am prone to not cut such much slack.
So if a new player starts fudging, I might let it slide a little while, on the grounds that they might be an exception to my experience (which is hardly universal). But I am going to be watching them like a hawk from then on out for these other behaviors. It's somewhat as if a Robin Williams-type character was your lawyer in a civil trial. If he seemed otherwise compentent, you might roll with for a bit. But you'd be skeptical, and waiting for the heavier boot to drop.
