The Foundation of Existence: A Geometric Derivation from a Single Axiom

A foundational axiomatic derivation establishing the geometric and logical structure that underlies a unified framework of existence. From a single axiom — that uniqueness is existence — nine theorems are proven in sequence, without appeal to physical law or empirical measurement. The derivations establish: that existence requires bounds; that every existing entity occupies all dimensions simultaneously, with varying extent in each; that persistence requires oscillation within bounds; that time emerges as accumulated experience and is irreversible; that closure is required for any dimension to serve as reference for further extension; that fully mapped bounds force continued existence to extend into perpendicular dimensions; and that uniqueness must hold across all dimensions together, including the paths entities take to arrive at shared endpoints. The paper is written in the style of Euclid's Elements, with definitions, theorems, and proofs building strictly from the axiom. It establishes the compass and straightedge. Companion papers apply these foundations to specific physical contexts — atomic structure, wave mechanics, observer measurement, and cosmological dynamics.