Measurement Is Not Absolute: Constitutive Observers and the Operational Relativity of Curvature

Gaussian and Riemannian curvature are invariant once a metric is fixed, but invariance does not guarantee that every internal observer can operationally identify that metric. This paper isolates the suppressed premise in familiar claims of internal curvature measurement: rods, clocks, signals, test bodies, and comparison protocols must survey the metric whose curvature is at issue. I call this the metric-access postulate and introduce constitutive observers, distinguished by the physical response laws of their measuring systems rather than only by motion. On an open hemisphere of a round sphere, the induced metric has , while instruments constituted through orthographic projection realize an operational metric with . Even closed-loop parallel transport performed with the shadow observer’s connection returns zero holonomy. The two metrics concern the same points and are not related by a coordinate change. A constitutive-closure criterion then states what would be required to extend this result from spatial metrology to a complete relativistic observer including clocks, masses, energy-momentum exchange, and other interactions. The result does not refute Gauss’s theorem; it blocks an unconditional inference from metric invariance to universal internal measurability. In general relativity, geodesic deviation, lensing, and constructive spacetime reconstructions establish curvature relative to substantive assumptions about matter-geometry coupling. Measurement can therefore be objective within a constitutive class without being absolute across all possible observers. This is a preprint version of a manuscript currently under journal review.