Minimal Motifs of Dynamic Robustness (Part II of Dynamic ACR Quadrilogy)

The paper gives a complete, rate-constant–independent classification of static, strong static, weak dynamic, dynamic, cylinder, and full-basin ACR for two-reaction, two-species mass-action systems, with precise geometric conditions and a careful orientation choice (relabeling so a2 > a1) when reducing to a 1D normal form. The model reproduces the structure and most criteria but makes key sign/orientation mistakes: (i) it omits the sign of r = a2 − a1 in the monotonicity of dy/dx (claiming sign(d/dx(G/F)) = sign det instead of sign(r·det)), and (ii) it equates sign G(x*) with sign det and further det ≥ 0 with the summary theorem’s sum ≥ 0 without fixing orientation. These errors can flip several inequalities and thus misclassify cases. The paper’s statements (Theorems 3.4, 4.1, 4.3, 5.1) are correct and resolve these issues by explicit orientation and by using the sum-form conditions.