math.PR — Probability

Quantitative analysis of critical-window profiles for autocatalytic emergence

Contributed by Michael Mislan

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 $\lambda n$, the RAF probability converges to $S_\nu(1-\mathrm{e}^{-\lambda})$, where $S_\nu$ 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 ($\nu=\mathrm{sp}$) and for the commuting-factor quotient catalogue ($\nu=\mathrm{qt}$). 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 $a$ and rational tolerance $\varepsilon$, returns a rational interval of width below $\varepsilon$ containing $S_\nu(a)$, with an explicit approximation error $1/m+(2^{L+1}-2)(1-a^{L+1})^r$ for a computable seed length $L$ and record index $r$, 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 $\binom{J}{r}$ over enormous channel caps by $C_r^\nu\le36^r2^{r(r-1)/2}$, giving $\log S_\nu(a)\le-\log^2(1/a)/(2\log2)+O(\log(1/a))$ and exact certificates such as $S_\nu(10^{-20})<10^{-629}$. Third, at fixed word length $n$ the probability that some singleton RAF exists is exactly $1-(1-p)^{248}$ (split) or $1-(1-p)^{234}$ (quotient), so the finite RAF probability is linear in $p$ 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 $n=16$. 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.