Optimal Affinity Beyond Gross Stoichiometry

Despons, De Decker and Lacoste showed that for a tightly coupled mass-action autocatalytic network whose overall reaction consumes $\alpha$ and produces $\beta$ copies of the controlled autocatalyst, the thermodynamic affinity $A^*$ at which the production flux is maximal equals $\log(\beta/\alpha)$ for generalized Type I networks, is at least $\log(\beta/\alpha)$ for the class of non-intersecting networks, and they conjectured that the lower bound $A^*\ge\log(\beta/\alpha)$ holds in general. We show that the conjecture is false. A two-species, two-reaction reversible network with an invertible stoichiometric matrix, a possible detailed-balance state and a unique global production maximum has $e^{A^*}=7/4<2=\beta/\alpha$. The mechanism is recycling: a reaction that returns part of its product response to an already active reaction coordinate dilutes the gross amplification. We then identify the exact response-level replacement of the gross ratio. With $T=S_+^{-1}S_-$ the response matrix built from the reactant and product complex matrices, and $w$ the integer production mode, the capacity $L(T,w)=\inf_f\prod_i((T^{\mathsf T}f)_i/f_i)^{w_i}$ over the forward-response cone is the greatest lower bound of $e^{A^*}$ over all represented optima, and the gross bound holds at this level exactly when $L(T,w)\ge\beta/\alpha$. An explicit family of networks with $\beta/\alpha=2$ has $L=1+1/m$ and realizes every value in $(1+1/m,2)$ at a unique global maximum, so no uniform positive gap survives; a three-reaction network with zero-diagonal response matrix shows that elementary self-return is not necessary for failure. Our main positive result is that the relaxation is tight at the level of nondegenerate local optima: for square sources with nonnegative response matrix, control-accessible cone and a response-routing digraph with a unique terminal strong component, every point of the forward-response cone is the response profile of an actual positive mass-action network at a regular stationary branch with a nondegenerate strict local production maximum, so the response capacity equals the strict-local kinetic capacity. The proof factors the current Jacobian through a response Laplacian that is diagonally similar to $I-P$ for a row-stochastic routing matrix $P$, uses a rooted maximum principle to make the kernel one-dimensional, removes it by the control gauge, builds the branch by the inverse function theorem, and proves negative curvature through an exact left-null identity that expresses the normalized curvature as a stationary-weighted average of response products. Interior minimizers of $L$ satisfy a prescribed-marginal matrix-scaling identity. Global optimality needs more than response data: we give a tangent-gap certificate and an infinite two-reaction family with unique global maxima, and exact two-core examples show that direction and complex-level compatibility of interacting cores do not imply a common kinetic realization. All principal theorems are compiled in Lean 4 against Mathlib with no unproved declarations.