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Counterexamples to the D-unstable-core conjecture for parameter-rich and elementary mass-action networks

Primarily AI-generated textHuman understanding: some partsmath.DS — Dynamical Systems

Contributed by Michael Mislan ↗

Version 1 / Oct 20, 2026 / CC BY 4.0

Abstract

Two exact four-species counterexamples refute necessity of a D-unstable core for instability of a positive equilibrium. The first has five reactions, parameter-rich kinetics, an explicit generalized-mass-action equilibrium, and a rational Jacobian with eigenvalue 1/500 + (51/500)i. The second has six reactions, at most two molecules on each side of every reaction, no catalysts, and classical mass-action kinetics with fixed integer kinetic orders. At a positive rational equilibrium its Jacobian has integer entries and an unstable nonreal eigenvalue with real part between 0 and 5, certified by an exact shifted-quartic intermediate-value argument. The networks share a reactant incidence relation and 24 nonempty child selections. Exact symmetrization, triangular-block and cubic-factorization certificates show that every child matrix is D-nonunstable in both networks. A permanent zero eigenvalue in an exceptional child distinguishes D-noninstability from strict D-stability. Both counterexample theorems have Lean 4 proofs with Mathlib. A separate asymptotic calculation explains the elementary example through circulation and a fast branching coordinate. The results establish strict linear instability, without asserting a Hopf bifurcation or periodic orbit.

Provenance statement

The text is primarily AI-generated. AI systems assisted the mathematical development, computations and Lean formalization under human direction; human understanding covers some parts of the work.

Formalizations

References

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Version history

  1. v1Submitted by Michael MislanInitial depositCurrentOct 18, 2026