Uniqueness of weakly reversible and deficiency zero realization

The paper proves the conjecture that any weakly reversible, deficiency-zero (WR0) realization of a mass-action ODE is unique, using a careful geometric argument: (i) δ=0 is equivalent to affine independence of each linkage class and linear independence of the stoichiometric subspaces across classes; (ii) WR0 realizations use exactly the monomials appearing in the ODE; and (iii) the number of linkage classes is fixed across WR0 realizations. With these, the proof shows that any disagreement in the linkage-class partition between two putative WR0 realizations forces either a loss of affine independence (Lemma 3.9) or a loss of linear independence of stoichiometric subspaces (Lemma 3.10), contradicting δ=0, hence uniqueness follows (Theorem 3.11) . The candidate’s proof contains two critical flaws: (1) Step 1 asserts that dynamically equivalent realizations must have identical vertex sets and identical b-vectors; this is false in general because a realization can include extra complexes with zero net reaction vector (per the dynamical equivalence definition), although WR0 later implies b_i ≠ 0, which rules out such extras only under the WR0 hypothesis . (2) Step 6 makes an invalid subspace-intersection inference: from S_G = ⊕_r S_{L_r} (for G) it concludes S_{L(i)} ∩ S'_{L'(i)} = {0} unless L'(i)=L(i). But S'_{L'(i)} need not be a direct-sum summand of the decomposition associated to G; it can contain cross-class difference vectors (with respect to G) and need not lie inside the internal direct sum determined by G. This is precisely the subtlety the paper resolves via Lemmas 3.9–3.10, not by the (incorrect) direct-sum intersection claim . Consequently, the paper’s argument is correct and complete; the model’s proof has a gap at its core partition-uniqueness step.