Singular-weight Conway-invariant Jacobi forms of index four

Let $\mathrm{\Lambda }$ be the Leech lattice and let ${\mathrm{C}\mathrm{o}}_0=\mathrm{Aut}(\mathrm{\Lambda })$. Sun and Wang proved that the space of ${\mathrm{C}\mathrm{o}}_0$-invariant holomorphic Jacobi forms of singular weight $12$ and index $4$ satisfies $$4\le \dim J_{12,\mathrm{\Lambda },4}^{{\mathrm{C}\mathrm{o}}_0}\le 9,$$ and left its exact dimension open. We prove $$\dim J_{12,\mathrm{\Lambda },4}^{{\mathrm{C}\mathrm{o}}_0}=6.$$ At singular weight, theta decomposition identifies this space with the simultaneous ${\mathrm{C}\mathrm{o}}_0$- and Weil-invariant subspace of $\mathbb{C}[\mathrm{\Lambda }\mathrm{/}4\mathrm{\Lambda }]$. Conway symmetry and $T$-invariance reduce the problem to a twelve-dimensional space of isotropic orbit sums. On this space the projected Weil $S$-operator satisfies the universal relation $$S(S+\frac{1}{2}I)(S-I)=0,$$ obtained from the level-$4$ Hecke algebra. Equivalently, the associated integral character matrix $K$ satisfies $$K(K+2^{23}I)(K-2^{24}I)=0.$$ Combining this relation with known index-$4$ forms, reduction modulo $2$, and character data obtained from the $A_3^8$ deep hole reduces the remaining possibilities to a finite exact calculation. A final torsion evaluation of the known index-$3$ form ${\mathrm{\Phi }}_{12,3}$ determines the last required character value, and exact elimination leaves a unique admissible branch, of dimension $6$. We also construct two Conway-averaged theta forms from explicit markings of the Niemeier lattices with root systems $D_6^4$ and $D_4^6$. Together with the four forms previously exhibited by Sun and Wang, they give a natural basis of $J_{12,\mathrm{\Lambda },4}^{{\mathrm{C}\mathrm{o}}_0}$.