Generically Nilpotent Cellular Automata

The paper proves Σ0_2-completeness: membership via a precise combinatorial characterization (Lemma 6) that is Σ0_2 by form, and hardness via a many-one reduction from the Σ0_2-complete set P = {k : ∃w, M_k(w)↑} using a walls-and-counters CA construction (Theorem 8) . By contrast, the model’s upper bound relies on a wrong equivalence: it asserts that density of O_{k,p}(a)=⋃_t⋂_{s=t}^{t+p} f^{-s}(C_k(a)) for all k,p implies generic nilpotency. This is false: for example, the shift map has all such O_{k,p}(a) dense (one can insert long runs of a's far away) yet is not generically nilpotent. The “light-cone forces eventuality” step incorrectly infers eventual stabilization from arbitrarily long contiguous intervals; without a blocking/spreading hypothesis, later influences can re-enter the window. The paper avoids this by requiring a blocking-word–type condition that forces stabilization (Lemma 6) . The model’s lower bound reduction from FIN is plausible in spirit but not reconciled with the paper’s construction; the paper’s reduction is correct and fully detailed via walls-and-counters and segment computations .