Declarations

completeness of the stated results. Each released tail theorem specifies its source domain. Each obstruction uses the same native operator. The status of a conditional theorem is not upgraded to an unconditional native estimate.

state interpretation. The explicit negative representative excludes an exact positive restriction for that representative. It does not exclude every physical state in the Hadamard class. Ghost or unphysical gauge components are not subjected to an unrelated scalar positivity test.

versus global data. The subtraction parametrix fixes a local singular germ. It does not control the smooth global remainder. Infrared behaviour and quotient positivity require separate data.

versus quantum coordinates. The fixed classical pullback theorem is not a construction of a field-dependent quantum coordinate operator or its Wick products.

scope and attribution. The Cheung references are cited under the supplied N. Cheung attribution and registered identifiers. XX-A and XX-B are drafts in this manuscript exchange; citation does not establish their authorship or registered status. The separate OpenAI catalogue is not a proof premise.

and submission metadata. The document uses one internal bibliography and standard pdfLaTeX packages, without external input files. No successful compilation, proof-assistant check, independent numerical execution, referee decision, or acceptance is asserted. A signed submission must supply authorized author and affiliation metadata and stable copies of the unpublished records.