Graded differential groups, Cartan-Eilenberg systems and conjectures in Conley index theory

The uploaded paper proves Theorem 5.1: two free, finitely generated, strict, P-graded differential groups are O(P)-filtered chain isomorphic iff their associated Cartan–Eilenberg (CE) systems are isomorphic. Its forward direction is immediate, and the converse is established by induction over the lattice O(P), using lifting theorems (notably Theorems 3.12 and 3.14) and a splitting lemma for incomparable convex sets, with repeated applications of the five-lemma to short exact sequences (e.g., (5.2)) . The candidate solution gives an essentially equivalent inductive construction: it uses a block-upper-triangular “one-step classification lemma” whose compatibility condition is exactly the homology identity d_A s + θ_A u = f_B θ from the paper’s lifting theorem (Theorem 3.12), and identifies the off-diagonal attachment class with the CE connecting map, then iterates along a linear extension. Aside from a minor slip (“maximal p ∉ α” where “minimal outside α” or β = α ∪ ↓{p} is intended) and some elided details about extending from single-step to arbitrary down-sets (handled in the paper by Theorem 3.14), the model’s argument matches the paper’s proof strategy and requirements. Hence both are correct, and the proofs are substantially the same in structure and core mechanisms .