Diffuse RAFs and productive operation: asymptotic singleton dominance in a random polymer model

A reflexively autocatalytic and food-generated set (RAF) certifies that a reaction network can regenerate its own catalysts from food. In the capped-Zipf binary polymer model at the critical exponent sequence $a_n=2-2/n$, a companion paper proved that RAFs exist with limiting probability $\theta(q_*)\in(0,1)$, that the minimum RAF size is subexponential given existence, and that a fed, diluted stochastic mass-action reactor started from food exports a fixed amount of nonfood polymer during a fixed window with probability of order $p_n\asymp2^{-n}$, the probability that one specified molecule catalyzes one specified reaction. It left open whether the rare productive realizations typically rely on a diffuse collective organization or contain a one-channel RAF. We prove the latter. At a sufficient quadratic volume scale, the probability of productive output without a productive one-channel incidence is $o(p_n)$; hence, conditional on productive output, the minimum RAF size converges to one, with or without conditioning on RAF existence. The proof combines a uniform kinetic exclusion, valid pointwise in every catalytic environment with at most one incidence in a fixed local rectangle, with an exact rare-multiplicity estimate for the heavy-tailed source that retains the dependence between assignments of one catalyst. The kinetic exclusion rests on two observables: a conserved product credit that cancels a mixed food–nonfood channel exactly, and a washout penalty that lets the dilution of an outsider catalyst pay for the material it could amplify. Consequences identify a unique short catalytic nucleus under successful-run conditioning and among environments with any fixed positive reliability, show that deleting that incidence makes the output probability exponentially small, and show that its signed catalytic input supplies more than three quarters of the measured export. Exactly $224$ productive candidate incidences give the asymptotic upper prefactor $224$ for the success probability. In contrast, the minimum RAF size diverges in probability under conditioning on RAF existence alone, and almost all RAF-containing environments have success probability $o(p_n)$. The core conditional theorem, its kinetic and source inputs, and selected consequences are verified in Lean 4; the remaining corollaries have complete conventional proofs. The theorem is a statement about statistical selection in this model, not a universal impossibility of diffuse autocatalysis.