The completion choice

The proposed XVII-A leaves three essential objects unspecified: a particular bulk observable, a particular boundary region, and a connecting map between their models. The supplied manuscript excerpts do not provide that connecting map for any of the proposed inherited candidates. A proof cannot manufacture one by identifying similarly named registers, regions, responses, or observables.

We therefore make the following explicit completion choice:

The theorem concerns a newly declared finite-matrix scalar benchmark on a fixed global ${\mathrm{AdS}}_2$ background. It does not concern an inherited VII-A, V, or Em03 observable.

This fills the open object choices in a modified Work Package A. It is not a repair of an inherited observable by an unproved identification. If Work Package A is required to use one of those specific inherited candidates, that version of the problem remains unresolved.

The fixed-background smearing calculation is motivated by the Hamilton–Kabat–Lifschytz–Lowe constructions HKLL2006a,HKLL2006b. Every identity needed below is proved directly for the finite model. No AdS/CFT duality, boundary conformal field theory, entanglement-wedge hypothesis, or entropy–area formula is a premise.