Reactant-bimolecular mass-action instability without D-unstable cores

A child selection of a reaction network assigns to each species in a subset a distinct reaction consuming it; the corresponding square submatrix of the stoichiometric matrix is D-unstable if some positive diagonal column scaling gives it an eigenvalue with positive real part. Vassena and Stadler proved that a minimal such submatrix, a D-unstable core, forces an unstable positive equilibrium under every parameter-rich kinetics, and conjectured the converse. The converse is known to fail for parameter-rich kinetics and, through catalytic padding with large kinetic orders, for classical mass action; the question whether it fails when every reaction consumes at most two molecules had remained open. We answer it in the negative. We give a four-species, six-reaction mass-action network in which every reactant complex has at most two molecules and no species occurs on both sides of any reaction, with integer rate constants and the positive equilibrium $(1,1,1,1/100)$, whose Jacobian has an eigenvalue with real part in $[2,4]$, although every one of its $25$ child-selection matrices is D-nonunstable under every positive diagonal scaling. The instability proof is an exact shifted Routh–Hurwitz construction; the child proof reduces all children to four maximal selections by a principal-restriction lemma and certifies each by positivity of the third Hurwitz determinant as a polynomial in the scalings. We then parameterize the complete positive stationary-flux cone of the network, $v=s(T,4,T,1,T-1,2)$ with $T>1$, and prove that along the concentration family $x^{*}=(1,1,1,1/L)$ the equilibrium is unstable for $T=100$ and every $L\ge100$, while $(T,L)=(2,1)$ is an exactly certified stable operating point of the same network with the same children. The instability persists under all sufficiently small independent perturbations of the six rate constants, and under a feed-and-dilution completion with dilution rate below $2$. Reactant molecularity two is sharp: every positive equilibrium of a mass-action network whose reactions consume at most one molecule is Hurwitz-nonunstable, and four species is the least possible number. The network admits no positive mass vector, so it is a kinetic, not an atom-resolved, counterexample; the two-sided bimolecular question remains open. The source-level existence theorem and the negation of universal reactant-bimolecular core necessity are verified in Lean 4 with Mathlib.