The Wang–Sontag bound is attained: 2n − 1 positive steady states in the n-site sequential distributive phosphorylation system, for every n

Wang and Sontag proved that the n-site sequential distributive phosphorylation system has at most 2n − 1 positive steady states. The bound was known to be attained only for n ≤ 4; for general n the best published lower bound was 2⌊n/2⌋ + 1, roughly half of it. We prove that 2n − 1 is attained for every n ≥ 2, so the Wang–Sontag bound is sharp. The construction is explicit and in two steps. First, a symmetric ("mirror") family of parameters reduces the steady-state polynomial to a palindromic one of half the degree; in that family the steady states off the diagonal E = F come in mirror pairs, while the diagonal always carries exactly one steady state, so the count is odd. Inside the family we construct, for every n and both parities, parameters with exactly 2n − 2 steady states off the diagonal, hence 2n − 1 in all. Second, we show that the count is robust: a one-parameter perturbation E_tot = 1 + δ breaking the symmetry turns the diagonal state into a simple root of the steady-state polynomial, makes the companion root inadmissible, and keeps the other 2n − 2 states, for either sign of δ; so the bound is attained on an open set of parameters with all states simple. The first step rests on a single lemma: for small ε > 0 the polynomial χ·d̂² + ε·T_{n−1} has only simple negative roots, which makes every subsequent step automatic and parity-uniform. All certificates are in exact rational arithmetic. Prepared with the help of an AI assistant (Claude Fable 5.1, Anthropic), which assisted in structuring the arguments, drafting the text and carrying out the exact symbolic and computer verifications; the mathematical content and the responsibility for it rest with the author. Source (Markdown) and PDF are both included.