Accidental accuracy and vertex corrections in $GW$: Exact benchmarks for the extended Hubbard model

The $GW$ approximation is the standard tool for quasiparticle predictions in materials, yet its regime of validity in correlated systems remains poorly quantified, because $\mathit{\text{ab}}\mathit{\text{initio}}$ vertex corrections are computationally prohibitive. Using exact diagonalization of the half-filled extended Hubbard model on finite rings as a numerically exact reference, we construct the corresponding model-space $GW$ theory on the identical Hilbert space and quantify its error as a function of local ($U$) and non-local ($V$) interaction strength. We find that the vertex correction changes character across the phase diagram: in the weak-coupling regime the effective vertex ${\mathrm{\Gamma }}_{\mathrm{e}\mathrm{f}\mathrm{f}}<1$, reflecting the suppression of RPA charge fluctuations by exact short-range correlations, whereas in the Mott regime ${\mathrm{\Gamma }}_{\mathrm{e}\mathrm{f}\mathrm{f}}$ grows monotonically (to $\sim 3$ for $N=6$, reflecting the local vertex required to open the Hubbard gap. Vertex corrections in the electron-hole (polarizability) channel are shown to \emph{worsen} the gap error, indicating that the Mott gap resides in the self-energy channel. For $V=0$, static $COHSEX$ is accidentally exact at a single crossover point $U^{\ast }\approx 3.5t$; finite $V$, through non-local Fock exchange, splits this point into a double-crossover window that collapses toward weak coupling. These results yield quantitative diagnostics for the reliability of $GW$ in correlated materials.