Sharp equilibrium and stable-state capacities of multisite 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 equilibria in each compatibility class, and the best general multistability result, due to Feliu, Rendall and Wiuf, provides $\lfloor n/2\rfloor+1$ asymptotically stable equilibria. We determine both capacities exactly. For every $n\ge1$, and for one common list of positive rate constants and one compatibility class, the maximal number of positive equilibria is $2n-1$ and the maximal number of locally asymptotically stable equilibria is $n$. The upper bound $n$ is proved by degree theory and parametric transversality and includes nonhyperbolic attractors. The lower bounds are attained simultaneously by rational data: $2n-1$ hyperbolic equilibria, of which $n$ are sinks and $n-1$ are saddles with a one-dimensional unstable manifold, and the same configuration persists on a nonempty open subset of the $(6n+3)$-dimensional space of rate constants and totals. The equilibria are produced by an explicit positive polynomial recurrence together with an interlacing argument. Stability is obtained by separating equilibrium geometry from kinetic time scales: after coalescing the equilibria, fast binding reduces the dynamics to a slow system on substrate inventories that retain the enzyme-bound substrate, and for a large kinase excess its transverse part converges to a path matrix with a rank-one positive feedback whose gain is strictly less than one by an explicit margin. We also obtain a diagonal Lyapunov certificate, the exact splitting law of the critical eigenvalue, exact rational witnesses for $n\le7$, and a three-site benchmark with a rigorous local robustness and recovery certificate. The algebraic core of the argument is formally verified in Lean 4.