Koopman spectral analysis of elementary cellular automata

The paper proves that in a finite deterministic system the algebraic multiplicity of the Koopman eigenvalue 0 equals the number of non-periodic (transient) states (Theorem 1) and that the multiplicity of eigenvalue 1 equals the number of connected components, coinciding with the number of independent conserved quantities (Theorem 2). It constructs explicit (generalized) eigenfunctions to count multiplicities and shows K = A^T in the one-hot basis. The candidate solution reaches the same conclusions via a different linear-algebraic route: it block-triangularizes K by ordering states by depth-to-cycle, shows the transient block is strictly upper triangular (hence nilpotent) contributing one algebraic zero eigenvalue per transient, and identifies 1-eigenfunctions as component indicators. Aside from a likely minor typo in the paper stating “no more than two periodic orbits” in a component (elsewhere the paper correctly says “only a single periodic orbit”), the arguments are consistent. Therefore both are correct, with different proofs.