The Small-Gain Condition for Infinite Networks Modeled on ℓ∞-Spaces

The paper’s Theorem 4.1 proves, for max‑type gain operators, the equivalence between (a) existence of a path of strict decay, (b) UGAS of Σ(Γρ) for some ρ∈K∞, and—under Assumption 2.1 with uniformly bounded in‑degree—(c) uniform NJI plus norm‑bounded trajectories. The candidate solution reaches the same equivalences using a Lyapunov construction and an “upper‑envelope” path built from forward trajectories. While the model’s direct verification of the bi‑Lipschitz (iv) property is heavier than necessary (the paper obtains it via a reparametrization lemma) and its (c)⇒(b) route bypasses the paper’s cofinality step, these are methodological differences rather than contradictions. The statements and the logical implications match the paper.