Structural autocatalysis can be common while productive operation is rare: an exact separation theorem in a capped-Zipf binary polymer model

A reflexively autocatalytic and food-generated reaction set (RAF) certifies that a reaction network is closed under the resources and catalysts it needs. It does not say whether a reactor started from food alone will actually produce material. We compare the two questions in one explicit random model: the reversible binary polymer model with molecules of length at most $n$, food consisting of the monomers and dimers, and the capped-Zipf catalysis law of Hordijk and Steel at the critical exponent sequence $a_n=2-2/n$, under which each molecule catalyzes on average $9n/\pi^2$ reactions. Three statements are proved for the same sampled network. First, the probability that a RAF exists converges to $\theta(q_*)\in(0,1)$, the survival probability of an infinite independent reversible reaction-closure process at openness $q_*=1-e^{-9/\pi^2}$. Second, conditional on existence, the minimum RAF size $M_n$ satisfies $\log M_n/n\to0$ in probability, with an explicit subexponential cutoff attaining the full limiting mass. Third, in a fed, diluted stochastic mass-action reactor with all spontaneous reactions active, the probability that the reactor keeps its total mass bounded and exports a fixed positive amount of nonfood polymer during a fixed observation window is bounded above and below by constant multiples of the single-incidence probability $p_n\sim9/(\pi^2X_n)$, where $X_n=2^{n+1}-2$ is the number of molecular species, up to an explicit molecular-noise error that a sufficient quadratic volume scale makes negligible. The same order holds jointly with, and conditionally on, the existence of some RAF, and the same conclusions hold for the fraction of catalytic environments in which a reactor run succeeds with high probability. Removing catalysis makes the same output observable asymptotically negligible. Thus structural autocatalysis occurs with probability bounded away from zero and small structural certificates are typical, while the specified productive operation has probability of order $1/X_n$, decaying exponentially in $n$. The structural limits, the finite probability sandwich, and their asymptotic corollaries are formally verified in Lean 4; the remaining steps are conventional proofs given in full.