Accuracy versus Predominance: Reassessing the validity of the quasi-steady-state approximation

The paper rigorously constructs two narrowing anti-funnels, Γ0 and ΓSP, proves their inward-pointing fence inequalities, and leverages the anti-funnel theorem to trap the unique slow manifold Sε0 and derive sharp two-sided bounds for ṡ in the slow regime (Theorem 2 and Proposition 2; cf. the definition and theorem on fences/anti-funnels, the differential form dc/ds=f(s,c), and equations (39)–(46) in the paper ). It then establishes a second positively invariant narrowing anti-funnel ΓSP for e0<8KS, shows every trajectory crosses γSP, and proves the refined three-term sandwich (65), from which the predominance threshold e0<K (asymptotically e0≪K) follows when combined with the Reich–Sel’kov condition e0≪KM (Theorem 3 and Section 5.1; see the definition of s∗ in (56), the invariance/crossing argument using γ′SP(0)<m, and inequality (65) ). The candidate solution largely mirrors the paper’s strategy and correctly verifies key ingredients such as ∂f/∂c≥0 and the strong-fence inequalities, reproducing the same two-sided bounds and the predominance conclusions. However, its proof that trajectories cross γSP when e0>K is logically incomplete: it argues by considering the time at which s(t)=s∗ but does not justify that s(t) ever equals s∗ (this implicitly assumes s0≥s∗). The paper avoids this gap by a slope-at-the-origin argument (γ′SP(0)<m) which guarantees crossing independently of s0 under e0<8KS . Aside from this crossing gap and a minor endpoint-choice mismatch for “narrowing” (paper uses s→∞; the model uses s→0+, which is acceptable in light of the orientation clarification in Appendix A ), the arguments agree. Because the model’s correctness hinges on the unproven crossing, we judge: Paper correct; model wrong.