Persistence of Conley-Morse Graphs in Combinatorial Dynamical Systems

The paper proves three targets: (A) Algorithm 1 enumerates all Conley–Morse filtrations by constructing all maximal feasible sequences of index pairs and converting them using Theorem 14’s intersection property; this is stated and argued in Proposition 17 with the algorithm listed in full . (B) For the relevant Conley–Morse graphs, the induced maps f_i are simplicial, yielding a zigzag G1 ← G1,2 → G2 ← ··· → Gn and a well-defined barcode via simplicial maps . (C) The redundancy-elimination procedure (Algorithm 2) returns exactly the maximal bars, one per supporting subfiltration, as formalized by Definitions/Propositions 26–28 and an accompanying argument . The candidate solution’s proofs match these claims: it gives a more structured DAG-based proof for (A), a standard simpliciality argument for (B), and a clean invariant-based justification for (C). The only minor overreach is a passing suggestion of uniqueness in (A); the paper only requires that every Conley–Morse filtration arises from some maximal feasible sequence, not that the representation is unique. Overall, both are correct and aligned in substance.