A No-Go Theorem for Kinematically Local Observables in Totally Constrained Quantum Theories

This work proves a structural no-go theorem for locality in totally constrained, background-independent quantum theories. We show that no observable with kinematically local (compact spacetime) support can be promoted to a Dirac observable unless it reduces to a relational, boundary-defined, or integrated quantity. The obstruction arises from the gauge dependence of spacetime embedding under reparameterization invariance and follows directly from the constraint algebra. The result is not operational or measurement-theoretic. It does not rely on decoherence, experimental limitations, or protocol dependence. Instead, it is a statement about the reduced phase space of totally constrained systems: compact spacetime localization is incompatible with gauge invariance unless additional non-gauge structure is introduced. This theorem makes precise a structural feature often treated as folklore in the problem-of-time and quantum gravity literature. It applies broadly to background-independent theories, including general relativity, reparameterized particle systems, minisuperspace models, and matrix-based formulations. Local observables emerge only after gauge fixing, relational conditioning, or the imposition of boundary structure. Together with complementary no-go results on the non-uniqueness of low-energy constants, this work places sharp limits on what such theories can predict without introducing extra structure. Any framework that treats local observables as fundamental must therefore make its additional assumptions explicit.