ON THE SUM OF CHEMICAL REACTIONS

Part (i) of the candidate solution is fine: since R*_{U,F} ⊆ R by construction (Definition 4.1), we have cl(R*_{U,F}) ⊆ cl(R), and Lemma 3.3 then immediately yields that any transition achievable via the reduced network is achievable via the original network, exactly as the paper states for Theorem 5.6(i) . However, the candidate’s proof of (ii) is flawed. It (implicitly) conflates elements of cl(R) with members of R and identifies an arbitrary witness (y, y′) ∈ cl(R) with an element of R0 so as to conclude (y, y′) ∈ cl(R*_{U,F}). This ignores the paper’s careful distinction between the original reaction set and its closure, and between R0 defined on R versus the closure-level no-U set. The paper’s proof of Theorem 5.6(ii) instead relies on a nontrivial re-ordering argument via Lemma 4.6 to compress U-cascades, and it emphasizes that the ‘intermediate species’ assumption (together with F = RU) is essential . Indeed, the authors give a concrete counterexample showing that (ii) fails even for non-interacting (but not intermediate) species—S1 leads to S5 in R but not via R*—which contradicts the candidate’s general claim . In short, the paper is correct and complete; the model overgeneralizes and misuses the closure characterization (Lemma 3.3) .