Attracting oscillations in sequential distributive multisite phosphorylation

The sequential distributive n-site phosphorylation cycle with one kinase and one phosphatase admits attracting positive periodic orbits under mass-action kinetics for every n≥3. An explicit three-site family has a fixed positive equilibrium, a transversal imaginary-eigenvalue crossing, a stable complementary spectrum and a negative first Lyapunov coefficient. A site-addition argument extends the supercritical Hopf bifurcation to every larger site count. Together with the one-site calculation and Vassena's two-site exclusion, this makes three the exact threshold for Hopf bifurcation in this family. Dynamic sequestration of the shared enzymes supplies the feedback: the equilibrium, productive fluxes and statically eliminated vector field remain fixed while stability changes. Further results compare enzyme clamps and selective reduction, quantify flux and amplitude laws, validate two finite-amplitude attracting orbits, and give a driven thermodynamic realization. Exact algebra and interval certificates support the quantitative claims; numerical continuation is distinguished from proof. A registered Lean theorem independently establishes an existential affine family with attracting periodic orbits for every n≥3. The manuscript's arbitrary-parent site-addition theorem and its quantitative extensions go beyond that registered statement.