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

This paper proposes that the origin of life may be understood as a cascade of dynamical integration events governed by Painlevé transcendental equations—a class of nonlinear differential equations arising widely in physics, from quantum mechanics to general relativity. Four prebiotic subsystems (mineral catalysts, information polymers, free-energy transducers, and lipid membranes) undergo progressive coupling, tracing a stepwise cascade (PVI→PV→PIIID6→PIIID7→PIIID8) along the Chekhov confluence diagram, culminating in the LUCA (Last Universal Common Ancestor). Each step corresponds to a specific biochemical integration event. The Painlevé framework is explicitly phenomenological: it classifies dynamical regimes of subsystem coupling rather than deriving biochemical mechanisms from first principles. A central quantitative feature is a characteristic separation parameter (Δmin≈0.15) between effective subsystem rates (ri=τi−1), with oscillation frequencies scaling as ω∝r1/2Δ−1/2. Several prospective test systems are identified, including the Belousov–Zhabotinsky reaction, the formose reaction, and chemically monitored extreme environments. Existing literature is used to define operational protocols, but no retrospective dataset is treated here as an independent validation of the proposed value of Δmin≈0.15. In particular, the formose calculation below is explicitly an illustrative model calculation whose parameters remain to be measured. The framework offers a potential unification of the RNA World, Metabolism-First, and Protocell theories as sequential stages of a single cascade; provides an indicative timeline (4.4–3.5 Ga) anchored to geological constraints; and makes quantitative, falsifiable predictions. A dedicated Scope and Limitations section discusses what the approach does and does not claim.