Rest–rhythm bistability and single-rate switching in the sequential distributive three-site phosphorylation cycle

The sequential, fully distributive $n$-site phosphorylation cycle with one kinase and one phosphatase contains no feedback reaction, yet for $n=3$ it was recently shown to possess attracting periodic orbits. We prove that the same mass-action network can do more: for one fixed list of eighteen positive rate constants and one fixed triple of conserved totals it has a linearly stable positive steady state (rest) and, simultaneously, an orbitally asymptotically stable positive periodic orbit (rhythm). The proof locates a generalized Hopf (Bautin) point in an explicit rational two-parameter family of rate constants: a directed-interval Newton–Kantorovich certificate gives a unique parameter point with a simple imaginary pair, seven stable complementary eigenvalues, vanishing first Lyapunov coefficient, real quintic normal-form coefficient in $(-0.044427531,-0.044427474)$ and a regular two-parameter unfolding. The unfolding theorem then yields a wedge of parameters with a sink, an unstable cycle and an attracting cycle, and hyperbolicity makes coexistence an open property of all eighteen rates and three totals. A separate, exactly specified witness makes the statement quantitative: two independent computer-assisted proofs, a Fourier contraction with a validated return map and a multiple-shooting Newton–Kantorovich argument, establish a positive attracting periodic orbit with period in $(28.69124274158,28.69124274167)$ whose peak-to-peak variation of the fully phosphorylated pool exceeds $2.3456$, with leading Floquet multiplier enclosed in $[0.888,0.972]$, and its equilibrium is a certified sink; in the one-parameter family through this witness the equilibrium loses stability in a certified subcritical Hopf bifurcation. We then ask how little actuation is needed to switch between the two behaviours. At the generalized Hopf point the linearization is controllable from a single rate constant, and this is true for every one of the eighteen rate constants (interval Kalman rank certificates). A parameter-uniform endpoint argument converts this into a theorem: for every prescribed duration and every bound $0<\eta<1$ on the relative modulation there are coexisting baselines, arbitrarily close to the generalized Hopf point, at which smooth modulation of one rate constant alone, with all other rates and all totals untouched, steers rest exactly onto the cycle and any chosen point of the cycle exactly to rest, with open sets of tolerated preparation and actuation errors. At the finite witness, where no switching theorem is claimed, a $\pm10\%$ resonant modulation of one phosphatase association constant over three periods toggles both ways in simulation. We state precisely which assertions are conventional mathematics, which are interval-certified, which are numerical, and what remains open.