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Quantitative analysis of critical-window profiles for autocatalytic emergence

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

Contributed by Michael Mislan ↗

Version 1 / Oct 17, 2026 / CC BY 4.0

Abstract

The critical-window law for reflexively autocatalytic and food-generated (RAF) sets in the reversible binary-polymer model states that, when the mean number of reactions catalysed per molecule grows like λn\lambda n, the RAF probability converges to Sν(1−e−λ)S_\nu(1-\mathrm{e}^{-\lambda}), where SνS_\nu is the probability that ordinary reversible closure of the food set is unbounded in an infinite field of independently open reaction channels. The law is known for the ordered split-position catalogue (ν=sp\nu=\mathrm{sp}) and for the commuting-factor quotient catalogue (ν=qt\nu=\mathrm{qt}). It is an existence theorem: it names the limit but says nothing effective about its value, its shape near zero, or what a finite network actually does. This paper answers those three questions. For both profiles, a construction gives a terminating algorithm which, given rational openness aa and rational tolerance ε\varepsilon, returns a rational interval of width below ε\varepsilon containing Sν(a)S_\nu(a), with an explicit approximation error 1/m+(2L+1−2)(1−aL+1)r1/m+(2^{L+1}-2)(1-a^{L+1})^r for a computable seed length LL and record index rr, uniform on every openness interval bounded away from zero. The key estimate is a finite-cap record contraction proved without conditioning on survival and without independence between repair attempts. Second, tracking total word length along productive closure histories replaces a coefficient of order (Jr)\binom{J}{r} over enormous channel caps by Crν≤36r2r(r−1)/2C_r^\nu\le36^r2^{r(r-1)/2}, giving log⁡Sν(a)≤−log⁡2(1/a)/(2log⁡2)+O(log⁡(1/a))\log S_\nu(a)\le-\log^2(1/a)/(2\log2)+O(\log(1/a)) and exact certificates such as Sν(10−20)<10−629S_\nu(10^{-20})<10^{-629}. Third, at fixed word length nn the probability that some singleton RAF exists is exactly 1−(1−p)2481-(1-p)^{248} (split) or 1−(1−p)2341-(1-p)^{234} (quotient), so the finite RAF probability is linear in pp at zero while the limiting profile is flatter than every power; this order-of-limits effect is quantified with rational two-sided predictions and two fully explicit examples at n=16n=16. Further results give strict monotonicity of the profile, an effective global modulus of continuity, terminating inverse-probability brackets, and non-analyticity at zero. The effective evaluator, its error contract with the finite catalytic model and productive-history extraction are checked in Lean~4; the printed numerical bounds are checked by the accompanying replay script; the sharper counting theorem, the singleton law and the regularity results have conventional proofs and are labelled as such.

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

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Version history

  1. v1Submitted by Michael MislanInitial depositCurrentOct 17, 2026