Infinitesimal Homeostasis in Mass-Action Systems

The paper proves two key statements: (i) infinitesimal homeostasis with respect to a rate constant k_ij when the reaction vector y_j−y_i lies in a coordinate direction, provided det(H̃_p)=0 at a reduced hyperbolic equilibrium (Theorem 4.5), and (ii) infinitesimal concentration robustness with respect to a conserved total M_i when det(H̃_i)=0 (Theorem 4.9). In both proofs, the authors stack the conservation laws U^T x = M with the (n−d) dynamical equilibrium equations, identify the reduced Jacobian J̃, invoke reduced hyperbolicity to ensure det(J̃)≠0, and then apply the implicit function theorem plus Cramer’s rule so that the derivative of the chosen output x_n with respect to the scalar input is a ratio of determinants whose numerator reduces to the relevant cofactor/minor H̃ (Theorem 4.5 derivation and formula; Theorem 4.9 derivation and formula). All of these steps and definitions (including H̃_i as the matrix obtained from J̃ by removing the i-th row and n-th column) appear explicitly in the paper’s Section 4 and its lemmas. The candidate solution reproduces exactly this scheme: same stacked system F=0, same use of IFT and Cramer’s rule, same support argument for ∂F/∂k_ij having a single nonzero row when y_j−y_i∈span{e_p}, and the same cofactor conclusion. Minor differences are stylistic: the candidate invokes a Schur-complement view to link reduced hyperbolicity to det(J̃)≠0 (the paper uses Lemma 4.3 for this) and briefly comments on the edge case p≤d (the paper shows p≥d+1 under its assumptions). These do not alter the logic. Hence, both are correct and essentially the same proof strategy. Citations: Theorem 4.5 and its derivative identity (; ), Lemma 4.3 ensuring det(J̃)≠0 at reduced hyperbolic equilibria (), Theorem 4.9 and its Cramer step for M_i (; ), and the definition of H̃_i as the i-th reduced homeostasis matrix ().