On the Uniqueness of Minimal Physically Derivable Theories

This paper proves a metatheorem concerning the class of physical theories derivable from minimal axiom sets. Rather than asking whether two given theories are equivalent — the direction taken by the existing literature on theoretical equivalence — it asks a prior question: given a set of conditions on minimality, is there more than one theory that can satisfy them? The answer established here is no. Four independently motivated conditions on minimal axiom sets are identified: a single fundamental indivisible and immutable constituent of matter; space represented as a computationally finite object, excluding the mathematical continuum; momentum as a primitive, intrinsic, and conserved property of fundamental constituents and the sole driver of all change; and exactly two opposing forces — one attractive, one repulsive — acting between fundamental constituents. The paper proves that these four conditions, together with the requirements of minimality and sufficiency, jointly eliminate all degrees of freedom in the matter ontology, the spatial ontology, the dynamical ontology, and the force ontology of any compliant theory. Any two theories satisfying the conditions must therefore describe the same fundamental reality and cannot differ in their derivations, descriptions, explanations, or predictions. The result is a generative uniqueness theorem — a categoricity result — rather than a comparative equivalence result. Three corollaries follow. First, the non-physicality of time: time is not a physical primitive but a relational concept derived from the causal succession of spatial states, and its non-physicality follows as a theorem from the four conditions rather than being assumed. Second, the structural enforcement of minimality: the four conditions are mutually independent, so minimality is guaranteed by their logical structure rather than requiring separate verification. Third, the incompatibility of quantum mechanics and general relativity is shown to be a consequence of the non-minimality of both frameworks — a consequence of their axiom sets, not a deep fact about nature — which implies that the quantum gravity problem calls for dissolution rather than solution. The result differs from existing work on theoretical equivalence by Weatherall, Halvorson, Barrett and Halvorson, De Haro, and Butterfield in its direction of argument, in its grounding of equivalence in the exhaustion of ontological degrees of freedom rather than the construction of inter-theory mappings, and in its exclusion of the mathematical continuum on grounds of computational representability. The paper also engages the literature on the metaphysics of fundamentality, showing that the fundamental level described by the theorem is not brute — its character is explained by the conditions — while acknowledging the open question of whether a fundamental level exists at all, which the theorem does not itself settle. Quantum-Geometry Dynamics is presented as an existence proof that the class of theories satisfying the conditions is non-empty. The paper is intended for philosophers of physics, philosophers of mathematics, mathematical logicians, and theoretical physicists working on foundational questions. It is submitted concurrently to Foundations of Physics. Related publication: Burnstein, D. L. (2026). Quantum-Geometry Dynamics: An Axiomatic Approach to Physics. Publisher.