SIGN-SENSITIVITY OF METABOLIC NETWORKS: WHICH STRUCTURES DETERMINE THE SIGN OF THE RESPONSES

The paper’s Theorems 4.2 and 4.3 state and prove exactly the relative- and absolute-sign statements about (ϕ_J)^{j*}_{j′} under the 1-dimensional kernel assumption for S_{J∪j*}, including (i) the nonvanishing criterion (ϕ_J)^{j*}_{j′} ≠ 0 ⇔ v_{j′} ≠ 0 and (ii) the mutual-sign relation sign((ϕ_J)^{j*}_{j′1}(ϕ_J)^{j*}_{j′2}) = sign(v_{j′1}v_{j′2}), as well as the two absolute-sign cases: det S_J ≠ 0 gives sign((ϕ_J)^{j*}_{j′}) = β(J) sign(v_{j*}v_{j′}); det S_J = 0 with dim ker S_J = 1 gives sign((ϕ_J)^{j*}_{j′}) = − sign(v_{j′}⟨κ,S_{j*}⟩) with Ad(S_J)=ṽκ^T. These appear as Theorem 4.2 (eqs. (34)–(35)) and Theorem 4.3 (eqs. (41)–(42)) with proofs via an augmentation-and-Cramer’s-rule argument (eqs. (96)–(105), (109)–(115) and setup (31)–(32)) . The candidate solution proves the same results by invoking the classical identity that the vector of signed maximal minors of an M×(M+1) matrix lies in its right kernel and by a direct adjugate-factorization argument in the singular case. This is a different, but standard, linear-algebraic route leading to the same conclusions. We found no logical gaps or missing assumptions beyond those already required in the paper’s statements (chiefly dim ker(S_{J∪j*})=1, and in the singular case det S_J=0 with dim ker S_J=1).