Lemma: Computational verification of L.2

The script XVI-L-artifacts/two_edge_full.py loads $P_A$ from source_full_table.txt and ${L^{pq}}_{ij}$ from XVI-K-artifacts/hessian_full_out.txt (relative paths), builds ${T_i}^r:=(-\mathcal{P}{B}^i{)}^r$, forms ${\widetilde{P^{ab,\mu \nu }}}_{A,ij}={\sum }_{r,s}{P^{ab,\mu \nu }}_{A,rs}{T_i}^r{T_j}^s$, assembles ${a^{A\mu \nu }}_{ab,pq}=(-1{)}^{|p|+|q|}{\chi }_{\mathrm{I}}{\sum }_{ij}{L^{pq}}_{ij}{\widetilde{P^{ab,\mu \nu }}}_{A,ij}$, groups by numerical label by accumulation, and exits with status $0$. It records:

1. (R1)

   All-sector coefficient assembly, exact in the preparation jets and $w^1$, on $h_{\mu \nu }=(w^1{)}^2{\eta }_{\mu \nu }$, with the common undifferentiated switch factor suppressed (not a general-$h$ symbolic assembly). Cancellation-aware supports on the stated background: $|{\Lambda }_{\mathrm{graph}}(C_2)|=25$, $|{\Lambda }_{\mathrm{graph}}(M_2)|=84$, $|{\Lambda }_{\mathrm{graph}}(J_3)|=202$, union $235$ of $|{\Lambda }_{\mathrm{adm}}|=255$ (table: $C_2$$25/25$, $M_2$$84/84$, $J_3$$250/202$, complete $255/235$). Nonzero on this specialization $\Rightarrow$ not identically zero in the general symbolic expression; zero on this specialization may still be nonzero for another residual metric. The reported $235$ is a coefficient-label support count before quotienting numerical-kernel relations; it does not establish $235$ independent kernels or nonvanishing of the full distribution. Full nonzero coefficient table written to two_edge_coeff_table.txt.

2. (R2)

   Source-sector K regression: retaining only $A=C_2$, all $25$ unordered labels and all ten components match the independently assembled signed polar_s route on the background; constant-slot recovers the XVI-K / XVI-J tensor ${\mathcal{K}}^{\mu \nu }$ of XVI-J, eq:Kvals; sample ${a^{C_2}}_{\{0,e_1\}}=-2{{\mathcal{K}}^{{\mathcal{C}}_2}}_{0,e_1}$ componentwise; ${\tilde{P}}_{C_2}$ matches xvi_common.polar_s on dual slots.

3. (R3)

   Graph factor $\frac{1}{2!}\cdot 2=1$; signs $(-1{)}^{|p|+|q|}$ with the constant-slot sign $+1$ and the $|p|+|q|=1$ sign $-1$.

4. (R4)

   Ordered placement counts: $C_2$: $45$; $M_2$: $360$; $J_3$: $1620$; total $2025$. Label $\{0,0\}$ does not arise in $M_2$/$J_3$.

Proof. Printed lines of two_edge_full_out.txt; Appendix app:out.