Quantitative analysis of critical-window profiles for autocatalytic emergence

Abstract

The critical-window profiles for RAF emergence in reversible binary-polymer networks are studied in the split-position and commuting-factor quotient catalogues, with monomer/dimer food. Total algorithms enclose survival probabilities at every rational openness to arbitrary prescribed positive rational accuracy, with computable seed lengths and uniform finite-record error bounds away from zero. The profiles decay faster than every fixed power at low intensity. Conventional total-length history estimates sharpen this to a log-square upper bound. For finite networks with cap n>=4, the exact singleton-RAF laws have exponents 248 and 234; finite probabilities are linear at zero even though the limiting profiles are flatter than every power. Rational finite-size predictions, strict monotonicity, effective continuity moduli, inverse-probability brackets and non-analyticity at zero are also established. The selected Lean statements cover the evaluator, approximation, flatness, finite-source comparison and history extraction; the sharper census, singleton semantics and regularity results have conventional proofs.