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

Primarily AI-generated textHuman understanding: some partsmath.PR — Probability

Contributed by Michael Mislan ↗

Version 1 / Oct 16, 2026 / CC BY 4.0

Abstract

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 fnf_n per molecule satisfying fn/n→λ>0f_n/n\to\lambda>0, the RAF probability converges in either model to Θν(λ)=Sν(1−e−λ),ν∈{sp,qt},\Theta_\nu(\lambda)=S_\nu(1-e^{-\lambda}),\qquad \nu\in\{\mathrm{sp},\mathrm{qt}\}, where SνS_\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 Θsp(λ/2)≤Θqt(λ)≤Θsp(λ)\Theta_\mathrm{sp}(\lambda/2)\le\Theta_\mathrm{qt}(\lambda)\le\Theta_\mathrm{sp}(\lambda); productive food gateways give the conventional bounds Θsp(λ)≤1−e−32λ\Theta_\mathrm{sp}(\lambda)\le1-e^{-32\lambda} and Θqt(λ)≤1−e−30λ\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.

Provenance statement

The text is primarily AI-generated. AI systems assisted the mathematical development, computations and Lean formalization under human direction; human understanding covers some parts of the work.

Formalizations

References

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  1. Varanasi, Varun. Emergence-of-Autocatalysis-in-Prebiotic-Reaction-Networks: public source and data repository. 2026. Reaction catalogue and shared-direction catalytic sampler; immutable commit 314f1e664cd7a8ccc97eed950f28d33311d0b5ef, inspected 15 October 2026. RAF detection is a separate implementation question. https://github.com/VJ-Varanasi/Emergence-of-Autocatalysis-in-Prebiotic-Reaction-Networks/tree/314f1e664cd7a8ccc97eed950f28d33311d0b5eflink
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Version history

  1. v1Submitted by Michael MislanInitial depositCurrentOct 16, 2026