Foundations of Static and Dynamic Absolute Concentration Robustness (Part I of Dynamic ACR Quadrilogy)

The paper’s Theorem 6.1 correctly establishes (i) A1⇔A2⇔A3 and (ii) B1⇔B2⇔B3 in the complex-balanced setting, and shows Bi ⇒ Aj and full equivalence under global attraction, via the standard Horn–Jackson log-parameterization log Z = log x̃ + S⊥ and a compatibility argument that uses e_i ∈ S . By contrast, the candidate solution asserts that for mass-action systems (without the complex-balance hypothesis) dynamic ACR automatically has full basin (B1 ⇒ B3), which is false: the paper itself gives mass-action counterexamples that are dynamic ACR but narrow basin (not full) (Ex. 7) . The candidate’s proof of B1 ⇒ B3 also overlooks positivity in the compatibility step; replacing y by y + ((a*−y_i)/v_i) v need not remain in R^n_{>0}. The rest of the candidate’s arguments (A1⇔A2 via log-coset of S⊥; A2⇔A3 via an integer-lattice argument; Bi ⇒ Aj in the complex-balanced setting; equivalence under global attraction) align with the paper’s results .