The theorem guarantees the existence of a -dimensional analytic manifold and a real analytic map
such that for each coordinate of one can write
I’m a bit confused here. First, I take it that labels coordinate patches? Second, consider the very simple case with and . What would put into the stated form?
I’ll just answer the physics question, since I don’t know anything about cellular automata.
When you say time-reversal symmetry, do you mean that t → T-t is a symmetry for any T?
If so, the composition of two such transformations is a time-translation, so we automatically get time-translation symmetry, which implies the 1st law.
If not, then the 1st law needn’t hold. E.g. take any time-dependent hamiltonian satisfying H(t) = H(-t). This has time-reversal symmetry about t=0, but H is not conserved.