An exact oscillatory instability in a minimal Type II autocatalytic core with linear degradation

The minimal autocatalytic cores of diluted reaction networks were classified by Nandan, Nghe and Unterberger into five families. For the reversible mass-action extension of a core with linear degradation of every species it is known that the positive stationary state, when it exists, is unique, and that it is locally asymptotically stable whenever the degradation constants are small enough. We show that the unique positive stationary state of a minimal core need not be stable for larger degradation. We construct a source-minimal Type $\mathrm{II}_3$ core with seven species, unit stoichiometric coefficients, and a single intermediate on one of the three return paths permitted by the classification, together with strictly positive rational reversible rate constants and strictly positive rational degradation constants, whose unique positive stationary state has the exact Jacobian eigenvalues $1\pm8\mathrm{i}$. The certificate consists of a stationary flux balance and two integer matrix identities; no root isolation and no numerical eigenvalue computation enters the proof. The stationary state is therefore Hurwitz unstable, the unstable mode is oscillatory, and instability persists on an open set of admissible parameters. We explain the mechanism: contracting the return intermediate at zero frequency gives a Hurwitz-stable six-dimensional matrix, so the instability is carried by the frequency-dependent response of the internal species, and it is invisible in the logarithmically scaled Jacobian, which is stable. We also describe the method that produced the exact certificate, in which a prescribed rational modal shape turns the design of an exact complex eigenvalue at a positive stationary state into a linear feasibility problem. Source membership, stationarity, the Fréchet derivative of the full seven-dimensional vector field, the eigenpair and the spectral conclusion are verified in Lean 4 against Mathlib with warnings treated as errors and no unproved declarations. The example does not settle the stability of the six-species cores without internal return species, and it does not assert a periodic orbit.