Declarations

scope. The proofs concern one free massive scalar in the specified vacuum and one linear classical tensor-source coupling. The spectral functions are standard structures and are not claimed as new foundational results.

scope. Companion citations identify supplied records and versions. Their full proofs and artifact packages are not independently certified. Actual submission should provide stable archival access to unpublished comparisons.

scope. No finite measurement instrument, apparatus realization, destructive-readout implementation, or decoder is constructed. The response, vacuum norm, and CTP kernels are not records.

scope. The metric and background are inputs. No Einstein dynamics, spacetime emergence, holographic dictionary, Page curve, island, de Sitter state, or evaporation endpoint is derived.

. The calculation defines a convergent subtracted dispersion integral within the printed free theory. It takes no $m\to 0$, $\epsilon \downarrow 0$, adiabatic, interacting cutoff-removal, or ultraviolet-completion limit.

and submission. No successful pdfLaTeX build, numerical or symbolic execution, machine proof, independent referee approval, registration, or journal acceptance is asserted. Responsible author metadata and independent technical review remain necessary.

. Citation of N. Cheung does not attribute authorship of this reconstruction to that author. Authorship authorization must be supplied by the submitting author.