The Computation of the Disguised Toric Locus of Reaction Networks

The paper’s Theorem 4.4 states a codimension additivity for the linear subspaces J̃(G′) and JR(G′, G): codim(J̃(G′) ∩ JR(G′, G)) = codim(J̃(G′)) + codim(JR(G′, G)), under the assumption that JR(G′, G) ≠ ∅ and with G′ weakly reversible. In Section 4, J̃(G′) and JR(G′, G) are explicitly treated as linear subspaces of R^{|E′|} (with orthogonal bases assembled in A and B), so codim is meant in the purely linear-algebraic sense . However, the proof of Theorem 4.4 depends on picking a nonzero (indeed, effectively positive) flux J′ ∈ J̃(G′) ∩ JR(G′, G) to form a weighted Kirchhoff matrix whose kernel has dimension equal to the number of linkage classes ℓ; this is crucial to the contradiction, but is only guaranteed when the positivity-restricted set JR(G′, G) from Definition 3.5 is nonempty, not merely the linear subspace version in Definition 4.1 (which always contains 0) . The candidate solution exhibits a concrete counterexample with G′ the 2-vertex reversible pair {0 ↔ 1} in one dimension and G having no reactions. There, JR(G′, G) (linear-subspace sense) is {0}, so codim(JR)=2, codim(J̃)=1, and codim(J̃ ∩ JR)=2, violating the claimed additivity (2 ≠ 1+2). This shows the linear-subspace identity fails in general. The paper’s proof implicitly needs the stronger assumption that J̃(G′) ∩ JR(G′, G) contains a strictly positive flux (equivalently, JR(G′, G) in the sense of Definition 3.5 is nonempty), which is not stated in Theorem 4.4 but is used earlier to argue openness/dimension of the positive cone (Lemma 4.2(b)) . In short: the model’s counterexample is valid for the subspace formulation actually used in Theorem 4.4’s statement, while the paper’s proof tacitly relies on a stronger positivity assumption. A precise fix is to state Theorem 4.4 under the additional hypothesis that J̃(G′) ∩ JR(G′, G) ∩ R^{|E′|}_{>0} ≠ ∅ (Definition 3.5), in which case the proof strategy via the weighted Kirchhoff matrix applies.