Conjecture: Instability iff D-unstable cores exist

Prove that a reaction network admits instability—i.e., possesses a positive steady state with a Hurwitz-unstable Jacobian—if and only if it contains a D-unstable core, defined as a minimal child-selection (CS) square submatrix of the stoichiometric matrix whose product with some positive diagonal matrix is Hurwitz-unstable. Establish both directions of the equivalence for networks with monotone kinetics by precisely linking Jacobian destabilization to the presence of D-unstable CS-matrices.

Background

The paper reviews unstable cores—minimal CS-matrices that are Hurwitz-unstable—and generalizes to D-unstable cores, which capture instability after scaling by positive diagonal matrices. While D-unstable cores are shown to be sufficient for instability, a broader necessity claim has been posed in prior work.

The conjecture seeks a complete structural characterization of instability via D-unstable cores. Proving the equivalence would unify dynamical instability with a minimal stoichiometric motif criterion, strengthening the bridge between network structure and dynamics.