A reaction-deletion-minimal mass-action oscillator without D-unstable child selections

A child selection of a reaction network assigns to each species in a subset a distinct reaction in which it is a reactant; the corresponding square submatrix of the stoichiometric matrix is $D$-unstable if some positive diagonal scaling gives it an eigenvalue with positive real part. Minimal $D$-unstable child selections, the $D$-unstable cores of Vassena and Stadler, force an unstable positive equilibrium, and the converse was conjectured. Known counterexamples to the converse concern the linearization only. We give a four-species, five-reaction classical mass-action network with a positive nonconstant periodic solution although every one of its $24$ child-selection matrices has nonpositive spectral real parts under every positive diagonal scaling. The network is a two-entry catalytic padding of a skeleton that is Hurwitz stable at every positive mass-action equilibrium, so the oscillation is produced entirely by kinetic order. It is minimal under reaction deletion in a strong sense: a positive stoichiometric circuit makes every proper subnetwork, at arbitrary nonnegative rates, incapable of a positive equilibrium or of any positive return, while the full network has exactly one positive equilibrium, always nonsingular, for every positive rate vector. Periodic existence is proved by an exact Hurwitz crossing along a rational one-parameter family and a localized shooting argument; this part, the exclusion of every $D$-unstable child selection, and universal deletion minimality are verified in Lean 4 with Mathlib. A separate outward-rounded rational interval computation, bound to the literal vector field, encloses the first Lyapunov coefficient in $(-23/1000,-22/1000)$, and the nondegenerate Hopf theorem then yields a supercritical, orbitally attracting hyperbolic periodic orbit on a nonempty open subset of the five-dimensional positive rate space. Exact period averages equal the equilibrium fluxes, and oscillation strictly lowers the mean concentration of the fourth species. The network has reactant molecularity $401$; no low-molecularity or laboratory realization is asserted.