D-unstable Cores Counterexample

Vassena and Stadler introduced unstable cores, minimal square child-selection submatrices of the stoichiometric matrix, and proved that a D-unstable core is sufficient for a reaction network with parameter-rich kinetics to admit an unstable positive equilibrium. They conjectured the converse: a network without D-unstable cores cannot have a Hurwitz-unstable Jacobian. We show that this conjecture is false. We exhibit a reaction network with four species and five reactions, no catalysts, and a strictly positive equilibrium flux, together with an admissible reactivity matrix for which the Jacobian $G = SR$ has the eigenvalue $\frac{1}{500} + \frac{51}{500}i$, while every one of its 24 child-selection matrices is D-nonunstable, so that no D-unstable core exists. All data are rational, the eigenvector is a Gaussian-integer vector, and the child certificates are exact: three positive-semidefinite weighted symmetrizations, one exact cubic factorization $X(X + 4d_0)(X + 4d_1 + 2d_2)$, and two two-dimensional faces. An explicit generalized-mass-action realization of the unstable equilibrium is given. The complete argument, from the source network to the negation of the universal necessity statement, is verified in Lean 4 with Mathlib, with warnings treated as errors and no unproved placeholders. We also explain why the obstruction is genuinely collective, why the boundary-permitting definition of D-noninstability is essential, and which weaker localization statements survive.