math.PR — Probability

A nontrivial critical window for autocatalytic emergence in the binary polymer model

Contributed by Michael Mislan

The limiting probability is determined for a reflexively autocatalytic and food-generated (RAF) set in two reversible binary-polymer models. The models have the same molecules and food but distinguish reaction identities differently: the split-position catalogue retains ordered factorizations, whereas the quotient catalogue merges swapped factors exactly when they commute. With independent molecule--reaction catalysis, one mark shared by both directions, and mean catalysis $f_n$ per molecule satisfying $f_n/n\to\lambda>0$, the RAF probability converges in either model to $\Theta_\nu(\lambda)=S_\nu(1-e^{-\lambda}),\qquad \nu\in\{\mathrm{sp},\mathrm{qt}\},$ where $S_\nu$ is the survival probability of infinite ordinary reversible food closure in the corresponding independent open-reaction field. Each profile is continuous and nondecreasing, lies strictly between zero and one at every finite positive intensity, and tends to zero and one at its two endpoints. Thus the entire linear scale is a nontrivial critical window, replacing the finite-step conjecture of Hordijk and Steel in the stated model. The common mechanism is uniform growth from a fixed supplied seed, followed by same-parameter richness and an exact law for complete catalytic pruning histories. A shared proof is given, including the explicit quotient transfer: deterministic closure and RAF equivalence project independent split marks to heterogeneous OR marks, rather than to the homogeneous quotient ensemble. The profiles satisfy $\Theta_\mathrm{sp}(\lambda/2)\le\Theta_\mathrm{qt}(\lambda)\le\Theta_\mathrm{sp}(\lambda)$; productive food gateways give the conventional bounds $\Theta_\mathrm{sp}(\lambda)\le1-e^{-32\lambda}$ and $\Theta_\mathrm{qt}(\lambda)\le1-e^{-30\lambda}$. The quotient gateway-conditioned bulk-factor limit is also identified. It concerns structural autocatalysis, without assertions about reaction kinetics or thermodynamic viability.