Exact Overlap and Gateway Scaling in RAF Networks

We study the probability that a reflexively autocatalytic and food-generated (RAF) set exists in Kauffman's binary-polymer model with random catalysis, in the exact reaction convention of the public code accompanying the recent finite-core theory of RAF emergence of Varanasi and Korenaga. That theory assigns activation probabilities to minimal catalytic cores and, for tractability, treats the cores as non-overlapping. We remove the approximation. For any finite catalogue of candidate cores we prove an exact two-level inclusion–exclusion formula for the probability that at least one core is activated, valid both for independent Bernoulli catalysis and for a uniformly random set of exactly $Q$ catalysis assignments, and we prove that for the exhaustive catalogue of food-generated reaction supports the formula computes precisely the probability of the RAF event. The sufficient statistic is not a matrix of pairwise core overlaps: it is, for each reaction channel, the family of cardinalities of all unions of the eligible-catalyst sets of the selected cores. A four-molecule example shows that pairwise data cannot determine the answer. The independent-core formula of the finite-core theory is recovered exactly under the sufficient hypothesis of coordinate-disjoint cores. We then show that the fixed food set imposes a finite gateway. For every maximal polymer length $n\geq4$ the repository convention has exactly 34 reaction channels usable from food, and every RAF contains a catalysed one. Consequently, if $f_n$ is the expected number of channels catalysed per molecule, then $\Pr(\mathrm{RAF}_n)\leq272f_n/n$ for $n\geq3$; $\Pr(\mathrm{RAF}_n)\to0$ whenever $f_n=o(n)$, so the sharp finite-size transition observed at approximately constant $f$ is not the shadow of an asymptotic transition at constant $f$; and for every finite $\lambda>0$ the RAF probability at $f_n=\lambda n$ does not converge to 1. A closed-form count of the repository's reaction channels, proved via the Lyndon–Schützenberger theorem on commuting words, gives the sharp gateway limit $1-e^{-34\lambda}$ on the linear scale. With the exception of that closed form and its corollary, every theorem in the paper has been formalised and checked in Lean 4 with warnings promoted to errors, against the literal RAF predicate of the source model.