Cycle structure, capacity, and resource costs of phosphorylation memory

How many recoverable states can a substrate with $n$ modification sites support when all sites share one kinase and one phosphatase? We study the distributive mass-action network on the $n$-cube of phosphoforms in which every enzyme–edge interaction has its own bound complex and all bound substrate is retained in the conserved inventory. Let $C_n$ be the largest number of locally asymptotically stable positive equilibria that one list of rate constants can have in one compatibility class. We prove $\max\{n,\lfloor(2^{n-2}+n)/2\rfloor\}\le C_n\le 2^n$, so that $C_n=\Theta(2^n)$; the upper bound counts nonhyperbolic attractors, and the lower bound is attained by hyperbolic sinks. In contrast, when the effective modification ratios are balanced around every elementary square of the cube, the capacity is exactly $n$, the value for a single sequential chain; more than $n$ stable states therefore require a nonzero effective cycle affinity. The exponential construction combines an exponentially large loading-interpolation space with an independently normally attracting saturated family. We then separate this structural capacity from operation. A finite-molecule theorem for the full continuous-time Markov chain converts any finite set of hyperbolic sinks into labels with explicit retention, recovery and readout guarantees at a computable sufficient system size; a local large-deviation theorem for crowded states gives an exit barrier of order $\delta^{2n}$, with recovery time of order $\delta^{-(2n-2)}$ at separation $\delta$; and a reversible extension with one common ATP-hydrolysis potential shows that finitely many sinks persist under finite chemical driving. Exact certificates for a three-site source with four sinks and seven equilibria illustrate the theory. For this example the sufficient certificates remain astronomically expensive, and we prove a concentration-ratio obstruction for its design family; these are diagnoses of the constructions, not lower bounds for phosphorylation memory in general. The proofs are conventional mathematics, supported by exact rational and interval computations and by a small amount of machine-checked algebra.