I’m confused because this sounds extremely trivial, and that doesn’t seem right. It sounds to me like the theorem is just saying:
(State transistions are deterministic) Let there be an isomorphism between any state x(t) and the sequence x(t), x(t+1), x(t+2), …
(Detectability) Assume there is an isomorphism between sequences x(t), x(t+1), x(t+2), … and sequences w(t), w(t+1), w(t+2), …
(Autonomy) Assume there is an isomorphism between sequences w(t), w(t+1), w(t+2), … and states w(t)
Then there is an isomorphism between x(t) and w(t).
It just sounds like the theorem is assuming the conclusion. Am I missing something?
Small note: A circuit of random toffoli gates (without any ancilla 1-bits) always maps all 0s to all 0s and would make the no-coincidence conjecture trivially true. I’m not sure how Gay et al. 2025 constructs random toffoli circuits to avoid this, but they must do it somehow in order for their Theorem 2 to be true.