Assuming you're not trolling: nope, that would be an empirical question of how many brain states ever actually arise in the world during the time humans exist, and how much space and time it takes to maintain a given brain state. The universe is currently expected to stop being able to support human life at some point in the future, so the pigeonhole principle argument requires some physical parameters to be known before you can apply it; and even if the pigeonhole principle did apply, you still can't know by that argument whether any given brain-state is repeated (and if so, how many times). Things which you can deduce using only mathematics, and no physical data, are necessarily extremely weak statements about reality.
A very strong empirical statement about the nature of reality, and a strong statement about the nature of mathematics, neither of which everyone agrees with! (What will your reply be if we discover a halting oracle at the centre of the galaxy? Mathematics doesn't forbid that: the Church-Turing thesis is an empirical statement, and it's the empirical scientific law of induction which is what tells us that we're extremely unlikely to find a halting oracle.)
But you've missed the point: even assuming that mathematics can perfectly model the universe, you cannot use weak physical hypotheses ("human brains are finite, there is an injection from human mind-states into human brains") and fully general mathematics (the pigeonhole principle) to derive such specific strong truths as "a particular human mind-state is repeated in a predictable way in the universe". You can at best derive general truths such as "at least one human mind-state is repeated at least once, somewhere", if the universe happens to have its physical parameters set in such a way that the pigeonhole principle holds.
