Crossing the Recursion Threshold: A Minimal Five-Axis Model for the Origin of Life

The origin of life is often approached through chemical reaction pathways, molecular replication, or probabilistic emergence scenarios. While informative, such approaches frequently lack a precise system-level criterion distinguishing non-living chemistry from living systems.Here we propose a Life Recursion Threshold Theorem (LRTT), defining the origin of life as the first occurrence of an open physical–chemical system capable of sustained, controllable binary recursive division. We argue that life begins when a system simultaneously satisfies two requirements:(1) a structural condition, namely the formation of a phospholipid membrane capable of deformation, closure, and division; and(2) a control condition, in which system dynamics are constrained within a candidate minimal steady-state control set defined by five ionic ratios: Na/K, Cu/Se, Cu/Zn, Fe/Mn, and Ca/Mg. These five axes represent pre-organ, pre-biological control dimensions governing volume stability, oxidative containment, reactive safety, material integrity under oxidative stress, and membrane mechanics enabling binary division. When jointly satisfied, the system crosses a recursion threshold, transitioning from sporadic, unstable splitting to statistically sustainable recursive division. From this point onward, the system and all of its descendants are defined as life. We further argue that, within the geological history of a single planet, successful crossings of this recursion threshold are extremely rare, leading to a single dominant origin event, with all subsequent biological diversity arising from recursive descent rather than multiple independent origins. This framework reframes the origin of life as an engineering threshold problem in a continuous state space, rather than a discrete chemical event, and provides a minimal, testable, system-level definition of life.