Linear-size autocatalytic sets are asymptotically absent in the sparse binary polymer model

In the binary polymer model of Kauffman, molecules are bit strings of length at most $n$, reactions are ligations and their reverse cleavages, and a random catalysis assignment decides which molecules catalyse which reactions. Hordijk and Steel (J. Math. Chem., 2016) introduced a sparse catalysis law, in which a molecule is active with a small probability and an active molecule catalyses each reaction independently with probability $1/n$, and showed that when the expected number of reactions catalysed per molecule is $\lambda n$ for a large fixed $\lambda$, the system contains a reflexively autocatalytic and food-generated set (RAF) with fewer than $2\lambda n^2$ reactions with probability close to one. They asked whether the quadratic size can be reduced to linear, and in particular whether a single molecule is likely to catalyse a set of reactions that suffices to construct that molecule once cleavage reactions are allowed. We answer both questions negatively. For every fixed food horizon, every fixed nonnegative intensity $\lambda$ and every fixed constant $C$, the probability that the model contains a nonempty RAF with at most $Cn$ reversible reactions tends to zero as $n\to\infty$, at rate $O(1/n)$; the probability that some molecule catalyses a self-constructing set tends to zero as well, cleavage included. The proof separates two regimes by the number of catalysts a RAF uses. Boundedly many catalysts are excluded by a productive-prefix argument whose constants do not depend on $n$. Many catalysts are excluded by a counting lemma of independent interest: the number of $m$-reaction sets that can be built productively from the food, multiplied by $m!$, is at most $\big((f+3m)^2+n(f+3m)\big)^m$, where $f$ is the food size. The factorial comes from an injective acyclic dependency encoding of arbitrarily labelled constructions and is exactly the entropy saving that the sparsity of active molecules cannot otherwise absorb. Consequences include the divergence of the minimum RAF size relative to $n$, the divergence of the minimum catalyst count, conditional versions on RAF existence, and a superlinear-to-quadratic window for the minimum RAF size in the regime of Hordijk and Steel. The main theorem, its constituent estimates and all stated consequences are verified in Lean 4 against the literal split-position model.