The 2n − 1 steady-state bound is sharp for sequential distributive phosphorylation

The sequential distributive $n$-site phosphorylation cycle, in which one kinase and one phosphatase act on a substrate with $n$ ordered sites, is the standard mass-action model of multisite protein modification. Wang and Sontag proved that it has at most $2n-1$ positive steady states for any rate constants and any conserved totals. Whether this bound is attained has been known only for $n\le 4$. We prove that it is attained for every $n$. More precisely, for any $2n-1$ distinct positive numbers we construct, by explicit algebraic formulas, positive rate constants and totals for which the system has exactly $2n-1$ positive steady states whose free-kinase to free-phosphatase ratios are the prescribed numbers. All of these steady states are nondegenerate, so the maximal count persists on a nonempty open subset of the full $(6n+3)$-dimensional parameter space, and rational data suffice. The construction rests on a square-root change of variable that turns the steady-state equation on a special parameter locus into an interpolation problem, on a positive two-term polynomial recurrence, and on an interlacing argument that gives an explicit admissible range of enzyme totals; for instance, equally spaced auxiliary roots work for every kinase-to-phosphatase ratio above $(4n^2-1)/8$. We also prove a determinant formula which shows that $n-1$ of the constructed steady states are unstable for every $n$, and we certify in exact rational arithmetic that for $n\le 10$ the remaining $n$ are asymptotically stable. The existence of $2n-1$ positive steady states for every $n$ is formally verified in Lean 4 with Mathlib.