Filtrations Indexed by Attracting Levels and Their Applications

The paper’s Theorem 2.5 proves filtrations for the ±ε and ±εΣ attracting basins via (i) monotonicity (Lemma 2.4) and boundary containments that relate −ε, −0, 0, and +ε (Lemma 2.3), and (ii) coverage using A ∩ ImF ≠ ∅ (plus c−1(∞)=∅ for finite-budget coverage) . The candidate solution follows the same blueprint: it establishes the same monotonicities and boundary relations, then proves coverage at ∞ and—when c is finite-valued—coverage with finite budgets by constructing short witnesses (length-1 for non-Σ, length-2 for Σ), aligning with the paper’s intent (Definitions 5, 9, 10 and Lemmas 2.2–2.4) . Minor presentational issues in the paper’s proof (a likely typo and a Σ-length edge case) do not affect correctness; the model’s construction resolves the Σ edge case cleanly.