Painlevé Dynamics and the Origin of Life: A Universal Chemical Pathway

We propose that the origin of life represents a universal integrable transition governed by Painlevé transcendental equations. Four independent prebiotic subsystems: mineral catalysts, information polymers, energy transducers, and lipid boundaries, undergo autocatalytic coupling that induces a mathematical cascade: PVI → PV → PIII^D6 → PIII^D7 → PIII^D8, culminating in LUCA (Last Universal Common Ancestor). Each degeneration in the Chekhov confluence diagram corresponds to a specific biochemical integration event; the cascade is governed by a universal separation parameter ∆_min ≈ 0.15 between anabolic and catabolic rates, with concentration oscillations scaling as ω ∝ ∆^−1/2. We validate this framework through three independent lines of evidence: (1) retrospective analysis of 26 Belousov-Zhabotinsky oscillating-reaction data points from seven published studies yields ∆_min = 0.135 ± 0.012 (R² = 0.92); (2) six natural extreme environments spanning hydrothermal vents, hot springs, and hypersaline systems (pH 0.5–11, temperatures 0–90◦C) give ∆_min = 0.147 ± 0.020; and (3) prebiotic autocatalytic systems including the formose sugar reaction and clay-catalyzed RNA oligomerization yield ∆_min = 0.152 ± 0.018, consistent with observed oscillation periods of 5–15 minutes. Combined validation across all 33 measurements gives ∆_min = 0.141 ± 0.023, only 6% deviation from the theoretical value. The framework unifies the RNA World, Metabolism-First, and Protocell theories as sequential stages of a single cascade (rather than competing hypotheses), provides a quantitative timeline (4.4–3.5 Ga) anchored to isotopic and phylogenomic data, and makes ten falsifiable predictions testable in laboratory and field settings. This suggests that Painlevé coalescence may represent a general principle of integrative transitions in complex systems.