Measurement Is Not Absolute: Metric Access Condition and Identifiability of Curvature

Gaussian curvature is intrinsic to a specified metric, but this mathematical invariance does not ensure that every physically possible measuring system can identify that metric. A ruler, clock, signal, or detector is itself a physical system; its records must be connected to a target geometry by a further account of its behavior. I formulate this distinction as an identifiability condition on the records available to a specified measurement architecture. A global example makes the condition concrete. A paraboloid has a complete induced metric with positive Gaussian curvature everywhere, while vertical projection supplies a second, globally flat metric on the same surface. A metrology restricted to the projected metric has exactly the same complete metric records on the paraboloid as it has on a plane. The example establishes a possible failure of access to the induced curvature; it does not claim that instruments produce either metric, that every physical probe is so restricted, or that a complete relativistic observer has been constructed. In metric general relativity, agreement among clocks, rods, light, and freely falling bodies provides strong evidence for a shared chronogeometric structure. That agreement rests on physical relations among those systems and the metric, rather than following from coordinate invariance alone. Curvature can thus be objectively and precisely measured under established coupling assumptions without measurement being absolute across all possible physical standards.