Global production optima and the sharp positive-response capacity of autocatalytic networks

For a tightly coupled mass-action autocatalytic network whose stationary production of a controlled species is maximal, the affinity of the overall reaction is determined by the first-order response of the one-way fluxes to the control. Despons, De Decker and Lacoste used this identity to constrain the optimal affinity by network structure, and a companion paper identified the exact response-level quantity, the capacity $L(T,w)$ of the forward-response cone, as the greatest lower bound of the affinity at nondegenerate local production maxima on a rooted class of square sources. What remained open is whether such a local maximum is the global one: a response profile fixes the derivative balance at a critical point but says nothing, a priori, about the rest of the positive stationary set of the nonlinear source. We close this gap. For square reversible mass-action sources with invertible reactant and net stoichiometric matrices, one controlled species, a positive production mode and a nonnegative response matrix $T=S_{+}^{-1}S_{-}$, every positive forward-response profile is realized by positive rate constants at a state that maximizes production over the entire positive stationary set, in every component. The proof has two ingredients: in reaction log-coordinates the stationary equations reduce to a common normalized current for every reaction, and at the reaction minimizing the response-scaled log-coordinate, strict convexity of the exponential and nonnegativity of $T$ bound that current by one. Rootedness of the response digraph is not needed for the bound; it enters only through the regular local realization, so the response, regular-local and regular-global affinity value sets coincide, including their unattained infima. Strong connectivity, or more generally a linear subcriticality certificate on each transient strong component, makes the maximizer unique. A worked rooted source with exactly two global maximizers shows that rootedness alone does not, and that a positive stationary current does not imply a positive forward response. The same normalization shows that prescribing effective equilibrium constants leaves every capacity unchanged when reference activities are free, and turns the affinity into an exact statement about the controlled activity relative to its equilibrium value. Reverse one-way flux budgets at a required output give an exact linear feasibility condition, and a two-reaction recycling family yields closed-form production–turnover trade-offs, an inverse-gap turnover cost near capacity, and a controlled operating point that is globally attracting. The global comparison, the uniqueness theorem and the capacity equalities are verified in Lean 4 against Mathlib; the boundary of the formal scope is stated explicitly.