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

The limiting probability of a reflexively autocatalytic and food-generated (RAF) set is determined in two reversible binary-polymer models with monomer/dimer food. The split-position catalogue retains ordered factorizations; the quotient catalogue identifies swapped factors exactly when they commute. Catalysis coordinates are independent for molecule-channel pairs, with one mark shared by both reaction directions. For every real sequence of mean catalysis f(n)/n tending to lambda>0, the RAF probability converges to S_nu(1-exp(-lambda)), where S_nu is the corresponding infinite reversible-closure survival probability. The profiles are continuous and nondecreasing, lie strictly between zero and one at finite positive intensity, and approach zero and one at the endpoints. Thus the linear catalysis scale is a nontrivial critical window in place of the proposed finite-step threshold. A shared supplied-seed growth proof, same-parameter richness and exact catalytic-pruning history laws establish the two limits. The quotient transfer, comparison bounds, productive-gateway refinements and gateway-conditioned limits are included. Historical Lean evidence is identified at its precise selected scope; broader claims have conventional proofs in the paper. Primarily AI-generated text. AI systems assisted the mathematical development, exposition, computations and Lean formalization under human direction. Human understanding: Some Parts. Licenses by component: manuscript, original figures and prose under CC BY 4.0; original code under Apache-2.0. Existing component notices and licenses are retained in the archives.