Debugging record relative to X-B

Signs in the check. The chain ${\mathcal{C}}_{\mathrm{F}}-V\star F_c=-i\hslash ⟨V^{\prime },{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c⟩$, times $i/\hslash$, equals $⟨V^{\prime },{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c⟩=-⟨V^{\prime },{{\Delta }_S}^{\mathrm{A}}c⟩=-⟨c,{{\Delta }_S}^{\mathrm{R}}{V}^{\prime }⟩$, and for $V^{\prime }=-\mathsf{L}\phi$ equals $-F_{\mathsf{L}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c}$. This agrees with X-B's eq:contraction-check and with $d_c=-{\mathsf{L}}_{1,g}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}c$.

{Symmetry of ${\Delta }_{\mathrm{F}}$.} Asserted in X-B, proved in Lemma lem:DF(i); it is kernel symmetry ${\Delta }_{\mathrm{F}}(x,y)={\Delta }_{\mathrm{F}}(y,x)$, not a Hermiticity statement.

Both directions of detection. X-B's zero-switch proof uses “elements of ${\mathfrak{J}}_0$ vanish on smooth free solutions” (one direction, sufficient there). Lemma lem:detect proves the converse, which is what makes Theorem thm:unique possible. X-B's argument is unchanged.

Graded ideal. The identification $\hat{\varphi }(d)=0⟺d\in P_m\mathcal{D}$ is used implicitly in X-B; it is Lemma lem:graded here.

Normalization row. X-B's sheet states the row for $F_f$; its theorem proves it for all $c$. Recorded on ${\mathcal{L}}_{\mathrm{off}}$ in Definition def:sheetN.

Uniqueness wording. X-B's “no uniqueness claim among arbitrary additional contact normalizations” is sharpened to Theorem thm:unique and Remark rem:scope-unique; no theorem of X-B is withdrawn.

Lemma lem:Q. Re-expanded: ${\partial }_{\mu }{a_1}^{\mu \rho }[K\xi ]=-\square {\xi }^{\rho }$, the $\partial (\partial \cdot \xi )$ terms cancel, and both sides equal $(\partial \cdot \xi )P_{m}u-(\square {\xi }^{\nu }){\partial }_{\nu }u-2{\partial }^{\mu }{\xi }^{\nu }{\partial }_{\mu }{\partial }_{\nu }u$. Consistent.

Zero-switch witness. With $h=\psi (t)\eta$, $a_1=\psi \eta$, and the zero-data inverse of a time-only source, ${{\mathsf{X}}_1}^0=-{{\int }_0}^t{\sigma }_{\epsilon }\psi$, $c_1=-{\partial }_t(f\Psi )$, $\int c_{1}t=\int f\Psi >0$. Consistent.

Edge gap formula. $\xi -{\mathsf{X}}_1[K\xi ]={{\mathsf{E}}_0}^{\mathrm{R}}((1-{\sigma }_{\epsilon })\square \xi )+{{\mathsf{E}}_0}^{\mathrm{R}}({\sigma }_{\epsilon }^{\prime }{a_1}^{0\cdot }[K\xi ])$ re-derived from ${\partial }_{\mu }({\sigma }_{\epsilon }{a_1}^{\mu \rho })={\sigma }_{\epsilon }^{\prime }{a_1}^{0\rho }-{\sigma }_{\epsilon }\square {\xi }^{\rho }$ and $\xi ={{\mathsf{E}}_0}^{\mathrm{R}}\square \xi$. Consistent.

Product criterion. Theorem thm:nonextension(ii) states failure of Hörmander's sufficient criterion, not nonexistence of every extension; (iii) confirms that the Wick product of X-B is unaffected.

Non-inputs. XII-B.tex and the retrieval summaries are not used. No statement of X-A beyond those copied in X-B is used.

Compilation. All macros defined; mathrsfs loaded for ||; all || and || targets exist; no external files.