Complete enumeration of irreducible autocatalytic sets with bounded supplier choice

An irreducible reflexively autocatalytic and food-generated set (irreducible RAF, or irrRAF) is an inclusion-minimal nonempty RAF of a finite catalytic reaction system. A companion classification shows that no uniform output-polynomial algorithm lists the complete family of irrRAFs unless $\mathrm P=\mathrm{NP}$, so unconditional tractability must come from structure in the input. We identify such structure in the multiplicity of suppliers. After polynomial maximal-RAF pruning, let $d_x$ be the number of reactions producing a nonfood species $x$ and let $a_r$ be the number of effective catalyst alternatives of a reaction $r$ (one, when a food catalyst is available). The supplier excess $\beta=\sum_x(d_x-1)+\sum_r(a_r-1)$ is computed from the input and bounds the number $L=\prod_xd_x\prod_ra_r\le2^\beta$ of deterministic supplier resolutions. We prove that every irrRAF survives at least one resolution, chosen by a globally shared earliest-producer rule; that the irrRAFs of a pruned resolution are exactly the sink strongly connected components of its consumer-to-supplier dependency graph; and that validating each such component in the original system, with one maximal-RAF call per reaction, yields precisely the complete original family. The algorithm runs in time and space $2^\beta N^{O(1)}$ for explicit input length $N$; it places no bound on the length of food-generation chains, so arbitrarily large deterministic systems have $\beta=0$. Because the components of one resolution are disjoint, every reaction lies in at most $L$ irrRAFs and the total output length is at most $mL$ for a maximal RAF of $m$ reactions; a coupled gate family shows that both bounds are attained up to a constant factor. Deleting reactions filters the family exactly and cannot increase $\beta$; incidence-separated components let the parameter be replaced by its largest local value, giving fixed-parameter tractability in $\beta_{\max}$; and destructive reaction sets are exactly the transversals of the catalogue, which yields a $(1+\beta\ln 2)$-approximate greedy intervention and exact cut-cost, cut-count and reliability formulas at zero excess. The catalogue theorem, its candidate bounds, the overlap bounds and the restriction identity are machine-checked in Lean 4; the cost, component and intervention results have conventional proofs. Bounded exact tests and an instrumented implementation illustrate the mechanisms.