Declarations

. The signed quadratic action, scalar switch, completed test formula, and tensor conventions are taken from the supplied X-E.tex. The background gauge operator, exact commutator, source prescription, and direct tree identity are taken from the supplied X-F.tex. Earlier series results are premises where explicitly identified, not independently verified here.

proofs. The normalized auxiliary–ghost Green matrix and free causal differential; the background ghost operator; the evaluated switched-matter coefficient; the free-solution reciprocity obstruction using a compact pure-gauge ramp background; the qualified contraction inventory; the corrected completed support bound; and the scalar nonlocal product-criterion audit.

scope. No interacting quantum BRST differential, complete master equation, full nonlocal loop value, anomaly cancellation, state, positivity theorem, physical representation, sharp-width limit, or adiabatic limit is constructed.

-C. The word $B_f\star W_g\star B_{f^{\prime }}$ and its extension hypotheses are not constructed. No residue identification or subtraction is made. The present scalar exceptional-set audit does not apply VII-C's theorem.

jet. The earlier switched-source conservation condition is not changed. No claim is made that the generically empty coupled Einstein jet has been filled.

status. The source is standalone and requires no external figures, bibliography database, or executable code. The source has been organized to avoid unresolved local labels and external cross-reference commands. No successful pdflatex execution, proof-assistant check, numerical verification, independent referee review, or journal acceptance is asserted.