I see three distinct issues with the argument you present.
First is line 1 of your reasoning. A finite universe does not entail a finite configuration space. I think the cleanest way to see this is through superposition. If |A> and |B> are two orthogonal states in the configuration space, then so are all states of the form a|A> + b|B>, where a and b are complex numbers with |a|^2 + |b|^2 = 1. There are infinitely many such numbers we can use, so even from just two orthogonal states we can build an infinite configuration space. That said, there’s something called Poincare recurrence which is sort of what you want here, except...
Line 4 is in error. Even if you did have a finite configuration space, a non-static point could just evolve in a loop, which need not cover every element of the configuration space. Two distinct points could evolve in loops that never go anywhere near each other.
Finally, even if you could guarantee that two distinct points would each eventually evolve through some common point A, line 6 does not necessarily follow because it is technically possible to have a situation where both evolutions do in fact reach A infinitely many times, but never simultaneously. Admittedly though, it would require fine-tuning to ensure that two initially-distinct states never hit “nearly A” at the same time, which might be enough.
I see three distinct issues with the argument you present.
First is line 1 of your reasoning. A finite universe does not entail a finite configuration space. I think the cleanest way to see this is through superposition. If |A> and |B> are two orthogonal states in the configuration space, then so are all states of the form a|A> + b|B>, where a and b are complex numbers with |a|^2 + |b|^2 = 1. There are infinitely many such numbers we can use, so even from just two orthogonal states we can build an infinite configuration space. That said, there’s something called Poincare recurrence which is sort of what you want here, except...
Line 4 is in error. Even if you did have a finite configuration space, a non-static point could just evolve in a loop, which need not cover every element of the configuration space. Two distinct points could evolve in loops that never go anywhere near each other.
Finally, even if you could guarantee that two distinct points would each eventually evolve through some common point A, line 6 does not necessarily follow because it is technically possible to have a situation where both evolutions do in fact reach A infinitely many times, but never simultaneously. Admittedly though, it would require fine-tuning to ensure that two initially-distinct states never hit “nearly A” at the same time, which might be enough.