Invariant Sets and Nilpotency of Endomorphisms of Algebraic Sofic Shifts

The uploaded paper explicitly proves that, under (H1)–(H3), an algebraic cellular automaton τ on an algebraic sofic subshift is nilpotent if and only if its limit set Ω(τ) is a singleton (Theorem 1.4) , and the proof uses a space-time inverse system construction (Section 12) rather than a global descending-chain argument; see the Σ*_{ij} inverse subsystem and the ensuing contradiction if τ were not nilpotent . By contrast, the candidate solution hinges on a blanket “DCC for algebraic sofic subshifts under (H1)–(H3)” to force stabilization of τ^n(Σ). The paper does not assert such a general DCC: its descending-chain stabilization is characterized in Theorem 10.1 and is tied to finite-type hypotheses (and is used in Theorem 1.3(iv) to deduce stability when Ω(τ) is of finite type) . Therefore, while the conclusion matches the paper, the model’s proof depends on an unstated and false general assumption about DCC, so its argument is flawed.