Question, outcome, and evidence boundary

The object studied here is the first-order plateau relational coefficient associated with the construction described in Cheung's Manuscript VII-A, specifically its thm:first CheungVIIA. We retain its first-order architecture: a compactly switched scalar response, unswitched zero-Cauchy harmonic coordinates, and gauge parameters supported inside the switch plateau and away from the initial surface. All definitions and the required classical identity are reproduced and proved below.

The quantum question is deliberately narrower. The metric coefficient is fixed as an external classical parameter. Only the scalar is quantized, and only the first relational coefficient is represented. This restriction is substantial: the joint metric–scalar insertion problem of VII-B is not the scalar-only problem constructed here CheungVIIB. The distinction between external-metric, finite-mode metric, and composed stress-response objects is also explicit in Manuscript V CheungV.

outcome. In the terminology of the Em06 prospectus CheungDossier, the result is a repaired limit. The exact change from the larger proposed quantum target is:

1. the metric coefficient is classical, not a quantum field;

2. the observable is the first plateau coefficient, not the full nonlinear relational scalar or its second-order completion;

3. the quantum carrier is a free scalar Weyl representation, not an interacting gravitational polynomial algebra;

4. the limit varies the effective CCR parameter, not a switch, boundary regulator, or ultraviolet cutoff.

These restrictions are part of the theorem, not conclusions about the unrestricted problem.

The coherent-state concentration mechanism is standard Glauber,Hepp. The contribution here is its explicit application to a named first-order relational coefficient, including the density operator, sign audit, compact test, restricted gauge identification, nontrivial example, and separate fluctuation estimate.

different claims. The construction attempts only operator–symbol correspondence in this scalar-only model:

Here “operator reconstruction” does not mean boundary-to-bulk reconstruction. Holographic entropy and entanglement-wedge reconstruction require additional, different structures RT,DHW; none is an input of this proof.

discipline. Cheung references are identified by the versions and labels supplied with the problem. No conclusion about an unexamined later version is asserted. Objects described only in the supplied prospectus, including the VII-H comparison, are cited as documentary comparison material, not used as theorem premises. The mathematical proof below does not assume registration, compilation, machine verification, numerical execution, calibration, or experimental realization.