Why Space Has Three Dimensions: A Derivation from Minimal Axioms in Quantum-Geometry Dynamics

This paper derives the three-dimensionality of physical space from the minimal axiom set of Quantum-Geometry Dynamics (QGD) and the Minimal Physically Derivable Theories (MPDT) framework established by the Uniqueness Theorem. Rather than treating the number of spatial dimensions as an empirical datum, I show that exactly three dimensions follow necessarily from three structural constraints: the isotropy of quantum-geometrical space, the conservation and finiteness of preons(−), and the theoretical sufficiency of the minimal axiom set. In QGD, space is not a background arena but an emergent structure constituted by preons(−) — the fundamental spatial constituents whose mutual repulsion (n-gravity) dimensionalizes the discrete lattice. Dimensionality is formally defined as the maximum number of mutually orthogonal, non-concurrent lines sharing a common preon(−). The derivation shows that fewer than three dimensions cannot support the full range of physical phenomena the axiom set must account for, while any dimension beyond three is inadmissible under the minimality requirement: a physically inert dimension has no independent axiomatic warrant. Isotropy, which follows from the identical intrinsic properties of all preons(−), confirms the consistency of the three-dimensional result with the symmetry requirements of the framework. The result distinguishes QGD from all frameworks that take dimensionality as a free parameter or an anthropic selection. It strengthens the MPDT programme by showing that three-dimensionality is part of what any minimal physically derivable theory must be, not an assumption imposed on it from outside. This paper is part of a series developed in conjunction with Quantum-Geometry Dynamics: An Axiomatic Approach to Physics (Burnstein, 2026) and the companion papers on the Uniqueness Theorem, the Physicality of Logic, Quantum Computing under QGD, and Bell correlations under QGD.