A No-Go Theorem for Unique Low-Energy Constants in Totally Constrained Quantum Theories

This paper establishes a structural no-go theorem for the unique derivation of low-energy constants in totally constrained, background-independent quantum theories with finite observer access. Unlike a large body of work addressing protocol dependence, measurement limitations, or model-specific obstructions, the present result operates at the level of the reduced phase space itself. We show that numerical low-energy parameters (masses, couplings, cosmological constants) are not Dirac observables of the deep theory. Any operational extraction of such constants necessarily requires non-canonical externalization choices—clock selection, background or frame fixing, boundary conditions, or sector conditioning—that are not determined by the constraint algebra. As a result, the mapping from deep-kernel physics to effective low-energy constants is fundamentally non-unique. The theorem is model-independent and does not rely on quantum noise, experimental imperfections, or specific dynamical assumptions. It applies to any background-independent theory formulated as a totally constrained system. The result places a sharp upper bound on what such theories can uniquely predict and clarifies the status of vacuum multiplicity and landscape constructions as relational rather than dynamical phenomena. This work addresses a structurally narrow but foundational question. While related results exist at the operational or protocol level, no-go theorems at the level of gauge-invariant observables in totally constrained systems are comparatively rare. The present theorem is intended to serve as a reference point for future discussions of predictability, parameter derivation, and effective theory emergence in quantum gravity.