Here’s one of the math facts I find the hardest to wrap my head around: There are some finite, well-defined integers that are impossible to compute within PA or ZF.
The argument goes roughly like this: We’ll define a finite integer that is such that computing would be equivalent to proving ZF’s consistency. Then we’re done because by an incompleteness theorem, ZF cannot prove its own consistency. So how to construct such a number? First construct a Turing machine that halts (on a blank tape) iff ZF is inconsistent. Say it has states. The busy beaver number is defined as the maximal number of steps a Turing machine with states takes when ran (on a blank tape), assuming that it halts. If we knew we’d need to run our TM at most steps to prove ZF’s consistency. So we’ve constructed an that’s as claimed.
Perhaps of interest that people in quantum many-body physics have related results. One keyword is “scrambling”. Like in your case, they have a network of interacting units, and since interactions are local they have a lightcone outside of which correlations are exactly zero.
They can say more than that: Because excitations typically propagate slower than the theoretical max speed (the speed of light or whatever thing is analogous) there’s a region near the edge of the lightcone where correlations are almost zero. And then there’s the bulk of correlations. They can say all sorts of things in the large time limit. For instance the correlation front typically starts having a universal shape if one waits for long enough. See e.g. this or that.