Dynamical Morse entropy

The paper explicitly states and proves Proposition 8.1: (a) ∑_c b(c) = SB(M) and (b) b(c) ≤ #Crit_c(f), working with cohomology classes supported in sublevel/superlevel sets and a Poincaré-duality lemma for supports; see the definitions preceding Proposition 8.1 and its proof, and the support-duality statement in Section 10 (Proposition 10.2) . The candidate solution establishes the same two conclusions via a different route: it identifies the map φ_{c,δ} with a relative cup product, computes its kernel by Poincaré–Lefschetz duality on the band U = f^{-1}(c−δ,c+δ), shows rank φ equals the jump in a relative-cohomology filtration, telescopes to SB(M) for (a), and bounds the jump by relative groups detected by Morse handle attachments for (b). This gives a self-contained, standard algebraic-topology/Morse-theory proof consistent with the paper. Minor care is needed to phrase the identification of “classes supported in O” as the image of H^*(M, M\O) → H^*(M), but the candidate’s later steps rely only on relative groups and are sound. Overall, both are correct, with substantially different proofs and complementary perspectives. The paper’s local critical-point argument for (b) (e.g., the b(c)=2 case) differs from the candidate’s LES/handle-attachment bound, but both imply b(c) ≤ #Crit_c(f) .