Uniform Stability and Chaotic Dynamics in Nonhomogeneous Linear Dissipative Scalar Ordinary Differential Equations

The paper’s dichotomy for sup Σ_a = 0 is correct and carefully proved (notably Theorems 3.10 and 3.11), building on the attractor structure (Theorem 3.2), minimal-set scaffolding (Theorem 3.3), and ancillary results (Theorems 3.7 and 3.8). The candidate solution reaches the same end conclusions, but it contains critical gaps: (1) it asserts without justification that the equality set E = {ω : α(ω) = β(ω)} is closed, using this to force E = ∅ by minimality; E is σ-invariant, but need not be closed given α lower semicontinuous and β upper semicontinuous. (2) Its key containment step A ⊆ Ω × [r1,r2] in Case I relies on a recurrence-based argument that assumes β(ω·t) revisits a region of uniform negativity of g infinitely many times; this does not follow from minimality alone and is not the route taken by the paper, which uses a precise comparison/linear-flow contradiction (via Theorem 3.8 and Sacker–Sell properties). Because these gaps are central to the proof, the model’s derivation is not sound, even though its final picture matches the paper’s.