The First Modular Zhu Algebra of $L(\frac{1}{2},0{)}_F$ in Characteristic Seven

Let $F$ be an algebraically closed field of characteristic $7$, and let $V=L(\frac{1}{2},0{)}_F$. For the degree-four singular vector $q=(L(-2{)}^2-2L(-4))\mathbf{1}$ in the universal vacuum vertex operator algebra $\bar{V}$, set $V_q=\bar{V}\mathrm{/}I(q)$. We prove that the natural quotient map induces an isomorphism $A_1(V_q)\cong A_1(V)\cong F[T]\mathrm{/}(T(T-4)(T-5))\cong F^3$. Explicit $O_1(V_q)$-relations and four families of triangular PBW straightening relations give the upper bound, while three one-dimensional homogeneous subspaces give the matching lower bound.