Source
Dal Cengio, Lecomte and Polettini (2023), LIT-1897: the stoichiometric matrix of net changes S_ρ = ν^{−ρ} − ν^{+ρ} (eq. 5), cycles as ker S (§III A), the cycle affinity as the circulation c_α · A, independent of concentrations (eq. 30), and Gordan’s alternative (footnote 6); read in NOTE-1476. Rao and Esposito (2016), LIT-1889: ∇ = ∇⁻ − ∇⁺ (eq. 2), emergent cycles (eq. 20), local detailed balance (eq. 50); read in NOTE-1464.
What was actually shown
The algebra. A species that is both a reactant and a product of a reaction, X + C ⇌ Y + C, has net stoichiometric coefficient zero. It appears in the rates, J⁺ = k⁺[X][C] and J⁻ = k⁻[Y][C], and not in S. Local detailed balance fixes k⁺/k⁻ by the free energies of X and Y alone, so the catalyst also drops out of the affinity, A = RT ln(J⁺/J⁻). Cycles are vectors in ker S and their affinities are circulations of A. Both are therefore blind to catalysis. The mutual production of two species, each catalysing the other’s synthesis, is a cycle in the graph of catalytic influence, in the rates’ exponents. It is not a vector in ker S.
The check. For c₂ catalysing S → c₁ and c₁ catalysing S → c₂, each decaying to W, with S and W chemostatted, the internal matrix has kernel spanned by (1, 0, 1, 0) and (0, 1, 0, 1). These are the emergent cycles fuel → c₁ → waste and fuel → c₂ → waste. The closed network’s only cycle, (−1, 1, −1, 1), is a futile loop with zero affinity. The production cycle c₁ → c₂ → c₁ is none of these. Net production of both capacities from the reservoirs, S_X (1, 1, 0, 0)ᵀ = (1, 1) > 0, is Gordan’s case (ii), which an open network can have and a closed one cannot (footnote 6).
What could have failed. A catalyst could have entered the net stoichiometry, or the affinity could have depended on it. It enters neither.
What this does not say
- That catalytic cycles do nothing thermodynamically. They shape the rates, and so they decide where in the space the driving allows the network sits (THEORY-1261) and whether it is multistable. In the check above, deleting either catalysed reaction, or making it uncatalysed, leaves a single fixed point. But the affinity available, and the cost of the steady state, are set by the fuel-to-waste cycles alone (THEORY-1266).
- That production cycles and stoichiometric cycles never coincide. Stoichiometric autocatalysis, X + S ⇌ 2X, puts the producer in the net stoichiometry. A cycle of such reactions does change S, and Dal Cengio et al. note that “chemostatted autocatalytic reactions fall in case (ii)”. The claim is that a cycle of catalytic production, the form the record’s constraint models take, has no stoichiometric counterpart.
- Anything about sheaf kernels, the third kind of cycle A13 separated.
- That closure of constraints (TERM-070) is thermodynamically invisible in every sense. It is invisible to the budget. Whether a closed set of constraints lowers the driving needed to sustain a state is a kinetic question this does not touch.