From Relational Field Measurements to Emergent Coordinates: A Computational Proof of Principle for Reconstructing Metric Structure and Dimensionality

This preprint presents a computational proof of principle showing that spatial coordinates and effective dimensionality can be reconstructed from relational physical measurements rather than supplied as primitive inputs.A sequence of numerical experiments is developed in which hidden geometric structure is inferred from synthetic field-mediated observables, including light time-of-flight, phase relations, and gravity-like interactions. The study demonstrates that:3D and 4D coordinate systems can be recovered from relational measurements with low reconstruction error;effective dimensionality emerges from the structure of the relations;unknown monotonic mappings between field strength and distance can be learned during reconstruction;light-based ranging provides a particularly effective operational route to recovering geometry;non-Euclidean relational structures produce persistent reconstruction residuals rather than being artificially forced into convincing low-dimensional Euclidean coordinates.The results support the narrower claim that coordinates may be treated as an emergent representation of sufficiently rich relational measurements. They do not establish that spacetime itself is fundamentally emergent from fields. Instead, this work is intended as the first computational step toward a broader research program investigating whether physical fields can provide a complete operational basis for orientation, measurement, and coordinate construction.All numerical results are based on synthetic simulations and should be interpreted as proof-of-principle demonstrations rather than experimental evidence for additional dimensions or a replacement of spacetime as a fundamental concept.