Abstract

We construct a controlled classical limit for one precisely weakened relational observable: the first plateau-compatible scalar coefficient with a prescribed classical metric coefficient and only the free scalar quantized. The classical functional, its restricted gauge domain, and its dependence support are specified and proved before quantization. An exact compact scalar test represents the functional. For a declared family of scaled canonical-commutation-relation representations and coherent states, its mean equals the named classical value for every positive classicality parameter, while its variance is exactly that parameter times a finite one-particle norm. Its spectral law converges to the classical point mass in the quadratic Wasserstein metric. A nonzero example and a preparation-response diagnostic are evaluated.

The outcome is a repaired, scalar-only classical-limit theorem, not quantization of the metric-dependent insertion as a joint gravitational functional. The metric jet, Minkowski causal structure, scalar equation, switch, initial surface, and preparation are inputs. No metric reconstruction, Einstein dynamics, interacting gravitational product, holographic map, or inherited quantum-response coefficient is obtained. The separate product and completion stops remain untouched.