Critical-window emergence of autocatalytic sets under reaction-channel quotienting

We determine the limiting probability that a reflexively autocatalytic and food-generated (RAF) set exists in the reversible binary-polymer model in the reaction convention used by the public generator of the finite-core theory of RAF emergence: two split descriptions $u+v\rightleftharpoons uv$ and $v+u\rightleftharpoons vu$ are identified precisely when $uv=vu$. Molecules are the nonempty binary words of length at most $n$, the food set consists of the six words of lengths one and two, and each molecule catalyses each reaction channel independently with probability $p_n$, one mark serving both directions. Let $f_n=p_n|J_n|$ be the mean number of channels catalysed by a molecule. For every sequence with $f_n/n\to\lambda>0$ the RAF probability converges to

where $S_{\mathrm{qt}}(a)$ is the probability that ordinary reversible closure of the food set is unbounded in an infinite field of independent canonical channels of openness $a$, and the probability of a RAF conditional on a catalysed food gateway converges to $S_{\mathrm{qt}}(1-\mathrm{e}^{-\lambda})/(1-\mathrm{e}^{-34\lambda})$. This resolves the bulk-factor problem left open in the companion paper on exact gateway scaling.

The quotient map preserves closures and RAF witnesses deterministically, but it does not preserve homogeneous Bernoulli ensembles: independent split marks project to heterogeneous OR marks, and a four-molecule example shows that the split and quotient RAF probabilities differ at every parameter. The proof therefore works in the quotient model throughout. A half-openness domination transfers a supplied-seed growth estimate from the split model through fibres of size at most two; a record-word argument proves that unbounded closure generates every fixed word at the original openness; a uniform finite-seed approximation gives continuity of $S_{\mathrm{qt}}$; an exact law for complete catalytic peeling histories connects static closure to the finite catalytic model; and a bounded-catalyst gateway bound closes the upper estimate. We also prove that $\Theta_{\mathrm{qt}}$ is nondecreasing and satisfies $S_{\mathrm{sp}}(1-\mathrm{e}^{-\lambda/2})\le\Theta_{\mathrm{qt}}(\lambda)\le\min\{S_{\mathrm{sp}}(1-\mathrm{e}^{-\lambda}),\,1-\mathrm{e}^{-30\lambda}\}$, where $S_{\mathrm{sp}}$ is the split-position survival function, so that $\Theta_{\mathrm{qt}}(\lambda)\to0$ as $\lambda\downarrow0$ and $\Theta_{\mathrm{qt}}(\lambda)\to1$ as $\lambda\to\infty$. The main theorem and its dependencies are formalised in Lean 4 with no axioms beyond the classical foundation; the few hand-proved comparison results are labelled as such. The results concern structural autocatalysis, not kinetic or thermodynamic viability.