A Vectorial Axiomatization of Space – Time Unity

This article presents a novel foundational framework for physics, grounded in a minimalist vectorial ontology. I propose that the unified fabric of reality is described by a fundamental abstract state space F, whose elements are normed vectors Ψ. From this single premise, a complete axiomatization of space-time and quantum phenomena is derived. The core innovation is the definition of space and time not as independent continua, but as complementary projections of the state vector: Ψ = (S, T). A set of nine axioms is introduced, imposing constraints on these components: their magnitudes are equal (|S| = c|T|), their directions are antiparallel (S = -T), and their evolution is a norm-preserving unitary redistribution. From these purely algebraic and geometric relations, I demonstrate the emergence of Minkowskian space-time with its Lorentzian metric, the invariant speed of light c, and a causal structure defined by the connectivity of F. The framework naturally incorporates quantum mechanics by identifying unitary evolution on F as the fundamental dynamical principle, with measurement and classicality arising via environmental decoherence and the dynamical selection of pointer states (Axiom IX). This work offers a path toward resolving the quantum gravity problem by suggesting that both geometry and quantum states are manifestations of a deeper vectorial substrate, thereby unifying relativity and quantum theory within a single, observer-independent mathematical structure.