Weather systems exhibit chaotic behavior which means that small changes to initial conditions have far reaching effects. This is why even the best weather models are only effective at most a few weeks out. It’s not because we don’t understand how weather works, it’s because the system fundamentally behaves in a way that requires keeping track of many more measurements than is physically possible. It’s precisely because we do understand this phenomenon that we can say with certainty that prediction at those time scales with that accuracy is not possible. There is not some magic formula waiting to be discovered. This isn’t to say that weather prediction can’t improve (e.g I don’t claim we have the best possible weather models now), but that predictions reach an asymptotic limit due to chaos.
There are a handful of extremely simple and well understood systems (I would not call weather simple) that also exhibit this kind of behavior: a common example is some sets of initial conditions of a double-jointed pendulum. The physics are very well understood. Another perhaps more famous one is the three body problem. These two both show that even if you have the exact equations of motion, chaotic systems still cannot be perfectly modeled.
