Attracting oscillations in sequential distributive multisite phosphorylation: dynamic enzyme sequestration and what static elimination cannot see

The sequential distributive $n$-site phosphorylation cycle with one kinase and one phosphatase is the basic mass-action model of multisite post-translational modification. It is multistationary for $n\ge2$, and a recent theorem of Vassena excludes Hopf bifurcation for $n=2$; whether the mechanism can oscillate for $n\ge3$ has been open. We prove that it can, and that the oscillations can be attracting. For $n=3$ we give an explicit rational one-parameter family of rate constants with a fixed positive equilibrium at which a simple pair of eigenvalues crosses the imaginary axis transversally, the other seven eigenvalues are stable, and the first Lyapunov coefficient is negative, $l_1=-0.0891594\ldots$; the bifurcation is supercritical and creates orbitally asymptotically stable positive periodic orbits. Every inequality is certified by exact polynomial algebra and outward-rounded rational interval arithmetic. A site-addition construction, with an explicit invariance computation for the Lyapunov coefficient, carries the result to every $n\ge3$; with a one-site Hurwitz lemma and Vassena’s theorem this makes $n=3$ the exact threshold for Hopf bifurcation in the family, and we certify explicit four-, five- and six-site witnesses. Each rate constant and each conserved total unfolds the bifurcation, so attracting oscillations occupy an open set of parameters. No feedback reaction is present: the feedback is the dynamic sequestration of the two shared enzymes in intermediate complexes. We prove that the equilibrium, the productive fluxes and the entire quasi-steady-state (statically eliminated) vector field are constant along the family while the stability changes, so no static reduction can detect the instability; that slowing complex relaxation at fixed static chemistry destabilizes; and that clamping both free enzymes forces global convergence for every $n$, whereas clamping the kinase alone still permits Hopf bifurcation. Eliminating a single fast complex, with the induced cubic term retained, preserves the supercritical bifurcation with certified errors. We further give exact periodic flux identities and the limiting ATP turnover, square-root amplitude laws, an exact resonance law for forced responses, a validated finite-time pulse experiment separating two statically identical systems of opposite stability, numerical continuation of the attracting branch to large amplitude, computer-assisted proofs of two finite-amplitude attracting orbits on it, and a thermodynamically consistent driven realization. An earlier subcritical witness, for which the existence of periodic orbits is formally verified in Lean 4, is compared throughout.