On the Non-Existence of Fundamental Ontic Positions: A Number-Theoretic No-Go Theorem

We investigate a broad class of physical theories in which fundamentally localizable entities are endowed with ontic positions that admit an injective numerical representation. Such theories include any framework in which spatial position constitutes a fundamental physical attribute independently of measurement and can, in principle, be represented exactly by numerical coordinates. Starting from a small set of physically motivated structural axioms, we formulate a representation-independent measurement framework that distinguishes ontic positions from their numerical encoding. We show that every exact position measurement necessarily yields a finite numerical representation, while the underlying ontic position space itself is not assumed to be discrete. The measurement framework induces a canonical arithmetic representation of all admissible exact position measurements. Once this canonical representation has been established, the remainder of the proof proceeds entirely within elementary number theory and requires no further physical assumptions. Under the stated axioms, we derive a contradiction showing that no ontic position-based theory admitting such an injective numerical representation can exist. The contradiction is independent of dynamical laws, quantum-mechanical postulates, relativistic structure, probabilistic assumptions, and the particular realization of the measurement process. It therefore establishes a general number-theoretic no-go theorem for ontic position-based theories satisfying the stated axioms.