Complex-balanced realizations of a coupled autocatalytic assembly: exact obstructions, an explicit disguised-toric locus, and global dynamics

A mass-action system is disguised toric if some other mass-action network on the same species has the same polynomial vector field and is complex balanced; complex balance then supplies a unique, globally attracting positive equilibrium through the Horn–Jackson entropy. We compute the disguised-toric locus exactly for a literal six-parameter family of four-species mass-action systems that arises as a driven assembly of two minimal autocatalytic cores. At a positive stationary state, two explicit flux budgets, $K=A-Bz\ge0$ and $J=e(B-A^2)\le B$, are necessary and sufficient for the existence of a dynamically equivalent complex-balanced realization, even when the competing realization may use any finite set of auxiliary complexes of any molecularity. Necessity is proved by two supporting-plane certificates, each a piecewise-affine convex function on exponent space whose supporting inequalities are checked at all $64$ ordered pairs of source complexes and extended to arbitrary auxiliary vertices; sufficiency is proved by an explicit nineteen-edge balanced flux table whose rate constants reconstruct the entire polynomial field. A one-variable monotone elimination removes the stationary state and yields a quantifier-free criterion in the six rate constants, with both equality boundaries included. The state and parameter criteria, the rate reconstruction, and the full vector-field equality are kernel-checked in Lean 4 with Mathlib. We then prove permanence of every positive trajectory for all positive parameters and global asymptotic stability of the balancing state on the whole locus. An exact rational slice crosses the boundary of the locus while its unique equilibrium stays hyperbolic, locally stable, and productive for one core, and a uniform Lyapunov estimate transfers global stability to an open set of nontoric parameters. Finally, the classification is invariant under adjoining any finite number of reversible catalytic attachments of private species.