Infinite towers in the graph of a dynamical system

The paper defines the node–edge graph via chain-recurrence, with an edge N1 → N2 when there exists a single trajectory whose backward limit set lies in N1 and forward limit set lies in N2, and states the Tower Theorem for the logistic map: for every μ ∈ (1,4], the set of nodes is totally ordered with an edge between every pair of nodes. The authors explicitly note that the proof of this edges/tower structure is given in their companion work [14] and provide supporting discussion (e.g., the top node at 0 and edges from 0 to all other nodes) in this paper. See the explicit definition of edges and the “tower” definition, as well as the theorem statement and illustrative argument for the top node and its edges . By contrast, the candidate solution relies on classical unimodal/Hofbauer facts but assumes—without justification from those sources—the crucial new claim that the recurrent components are linearly ordered by reachability and that this yields edges between every pair of distinct nodes. That linear-order/edge property is precisely what the authors claim to have proved in their cited work and is not a standard consequence of the cited classical references. Hence the model’s proof has a critical gap, while the paper’s claim is appropriately supported by citation and consistent with its own definitions.