Lemma: Computational verification of L.3

The script XVI-L-artifacts/kernel_inventory.py enumerates all placements from the L.1 budgets and the vertex bound $|p|+|q|\le 2$, exits with status $0$, and records:

1. (I1)

   Unordered placement counts by sector and $N$ as in the inventory table; $|{\Lambda }_{\mathrm{adm}}|=255$; ordered totals $45+360+1620=2025$.

2. (I2)

   ${\mathcal{V}}_{\mathrm{K}}\subseteq {\mathcal{V}}_{\mathrm{adm}}$: XVI-K's $25$ unordered placements equal the $C_2$ block.

3. (I3)

   Sector maxima $N\le 2,3,4$ for $C_2,M_2,J_3$; global $N\le 4$; additivity $|a+p|+|b+q|=|a|+|b|+|p|+|q|$.

4. (I4)

   Edge-interchange symmetry of unordered labels; the ten $S\cdot \partial \partial S$ generators of the $\square S=0$ relation lie in ${\Lambda }_{\mathrm{adm}}$.

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