Yun's zeta function and the overorder zeta function for Gorenstein cubic orders

For every Gorenstein cubic $\mathbb{Z}$-order, we prove an identity equating Yun’s zeta function, defined by counting finite-index submodules of the trace dual, with the explicit overorder zeta function introduced in our previous work on Beyond Endoscopy for ${\mathrm{G}\mathrm{L}}_3$. This is posed as Conjecture A in an early draft of Deng-Espinosa and Lee subsequently proved the functional equation of the overorder zeta function, making the Deng–Espinosa isolation of the trivial representation fully unconditional. His argument computes the local factors explicitly. Our proof is independent of Lee’s and does not evaluate the individual cubic overorder factors: it matches natural decompositions of the two sides and concludes by induction. As an application, we give a short, uniform evaluation of the local ${\mathrm{G}\mathrm{L}}_3$ Kloosterman Dirichlet series in the Poisson-summation argument of Deng–Espinosa. The direct local analysis of the Kloosterman series occupies nearly ninety pages in Deng–Espinosa while our treatment here replaces its case-by-case enumeration with a short uniform proof.