Declarations

scope. All variations are classical identities of polynomial lifts on smooth configurations, restricted to the free shell where stated. No quantum Ward identity, quantum retarded product, or interacting differential is constructed or implied.

correction. The coefficient $-1$ on ${\mathcal{C}}_{3,\mathrm{D}}$ is derived from the gauge-fixed second-order equation and the affine slot identity, and is confirmed by the detection of the opposite sign. It changes no inverse normalization and no object handed over by XII-D.

scope. The tree-composition lemma proves membership of the scalar-only completion and of the scalar-sourced slot in XII-A's domain. It is not an admission of loops or of nonlocal non-tree kernels, and it does not reopen the XI-G exclusion.

scope. XII-D supplies the response, insertion, roots, admission of ${\mathcal{C}}_{3,\mathrm{D}}$, class, ordering correction, and ledger. XII-C supplies the gauge-fixed second-order equation. XII-A supplies the carrier. XI-B supplies the inverse normalization. X-A supplies the classical completion formula and its on-shell covariance, used as a premise and listed as such. XI-E supplies the evaluated witness $I>0$ and the slot $r_g$, used as cited values. The affine identity, single entry, held-fixed lemma, assembly, remainders, tree lemma, and admission are proved here.

. No successful pdflatex execution, proof-assistant check, independently executed symbolic calculation, independent referee review, or journal acceptance is asserted. The nonzero witness is cited from XI-E and not recomputed.