Declarations

-contained prescription. The background, action, field normalizations, gauge condition, spatial rod equations, and classical insertion coefficients used in the proofs are reproduced explicitly.

theorem use. The normally hyperbolic Green and bundle-valued Feynman theorems are declared mathematical inputs. Their global-hyperbolicity, smoothness, principal-symbol, and self-adjointness hypotheses are checked above. Hack and BFHPR are not used to supply unidentified native objects.

interpretation. A gauge-fixed Hadamard contraction on an indefinite field bundle is not announced as a positive physical state. The smooth subsidiary contraction defect is displayed. No physical expectation of the unsmoothed lapse is evaluated.

and attribution scope. Cheung manuscripts are cited by their supplied registered identifiers and versions. The current Companion record attributes its registered manuscripts to Ngo Cheung. XX-A is cited as the supplied anonymous construction in this manuscript exchange, not as an authenticated registered Cheung manuscript. The separate OpenAI catalogue is not a theorem premise.

. No successful pdfLaTeX run, proof-assistant execution, independent numerical calculation, native programme compilation, referee decision, or journal acceptance is asserted. The mathematical arguments are the printed proofs and the explicitly identified analytic theorem applications.

metadata. The source is anonymous. A signed submission must supply authorized author and affiliation information and stable copies or archival locations for unpublished records. No authorship is inferred from a citation.