A Continuous Probability of RAF Emergence

We determine the limiting probability that a reflexively autocatalytic and food-generated (RAF) reaction set exists in the uniform reversible binary-polymer model. Molecules are the nonempty binary words of length at most $n$, the food set consists of the monomers and dimers, reactions are indexed by product word and split position, and every molecule–reaction catalysis coordinate is present independently with probability $p_n$. Let $f_n=p_n|\mathcal R_n|$ be the mean number of reactions catalyzed by one molecule. If $f_n/n\to\lambda\in(0,\infty)$, then the RAF probability converges to

$\Theta(\lambda)=S(1-e^{-\lambda}),$

where $S(a)$ is the probability that the reaction closure of the food set in an infinite i.i.d. reversible reaction field of openness $a$ is unbounded. We prove that $0<\Theta(\lambda)\le1-e^{-36\lambda}<1$ for every $\lambda>0$, that $\Theta$ is nondecreasing and continuous, and that $\Theta(\lambda)\to0$ as $\lambda\downarrow0$ and $\Theta(\lambda)\to1$ as $\lambda\to\infty$. Consequently no finite positive constant separates limiting probability zero from limiting probability one inside the linear scaling window, contrary to a conjecture of Hordijk and Steel; the sublinear and superlinear regimes are recovered as corollaries.

The proof combines an exact distributional reduction of catalytic pruning histories, a Peierls-type contour estimate on binary-word geometry that amplifies a fixed finite seed to nearly full molecular mass uniformly in $n$, a record-word argument showing that an unbounded infinite closure almost surely contains every finite word at the same parameter, and a uniform finite-seed approximation of the survival probability which yields continuity of $S$. The complete argument, including the geometric appendix, is formalized and machine-checked in Lean 4 over Mathlib.