Have there been video games for which perfect speedruns were mathematically proven?

SMB is a very 'solved' game. Its an old game, short, that is run by a lot of runners.

So there is no new strats that are found often.

Also, SMB is a 'simple' game. There is no perfect routing for collectibles that could be optimised or a sidequest that could be skipped. Its get to the end of the levels, and the only complexity is the level skip zones, that are known and it would be a surprise if suddenly a new one was found after almost 40 years. In other games there is always the possibility that a different route might suddenly be found and be better. But as I said, SMB being just 'get to the flag and thats it, there is no 'different route' that might work.

So you can assume that looking at a TAS, that makes no errors that humans are doing, and that that TAS is limited to not do inhumane things, the TAS is then what would be the perfect, human achievable time.

But yeah when we say 'Perfect' its 'perfect as of now'. But for SMB, since its a game that lots and lots of speedrunners have ran, and its been analysed backwards and forwards, it can be tought that we will never find a new exploit that REALLY changes the game. Probably a few frames here and there, but not something that would save even seconds. That being said: we can never say with certainty that nothing will be found again. So as I said: Perfect as of now.