We investigate a broad class of physical theories in which fundamentally localizable entities possess ontic positions that can be represented uniquely by numbers. This includes any framework in which spatial position is a fundamental physical attribute, independent 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 measurement framework that distinguishes ontic positions from their numerical encoding. We show that every exact position measurement necessarily yields a finite numerical representation, while making no assumption that the underlying ontic position space is discrete. The measurement framework provides a canonical arithmetic representation of all admissible exact position measurements. Once this 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 a unique numerical representation can exist. The key idea is that the set of admissible numerical values for a measurement is invariant under unit conversion. But this invariance will be violated once we do the math. 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.