An Axiomatic Theory of Recursive Existence

Modern mathematics and theoretical physics commonly begin by assuming the existence of spacetime as a primitive background upon which physical objects, fields, and interactions are defined. Geometry is therefore regarded as fundamental, while physical laws describe structures evolving within that pre-existing arena. Despite the remarkable success of this viewpoint, it leaves a deeper conceptual question unanswered: why should spacetime itself be considered primitive? If every physical object, observer, and measurement already presupposes spacetime, then the origin of spacetime cannot be explained from within the framework that assumes it. This work explores an alternative logical starting point. Instead of taking spacetime, matter, or information as primitive, we assume only the possibility of distinction. Distinction is understood as the minimal logical condition under which anything can become identifiable. Without distinction, there exists neither an observer nor an observed object, neither information nor measurement, and consequently no meaningful notion of existence. From this perspective, existence is not interpreted as an intrinsic property of an isolated entity. Rather, it is proposed to emerge through the recursive stabilization of distinguishability. A state exists only insofar as it can continuously preserve its own distinguishability through recursive recognition.