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Work with $z$ in the upper half-plane. Write the Stieltjes matrix resolvents and their normalized matrix traces asThe minor trace is normalized by $N$, not by $N-1$. This sign convention is the negative of the convention $(zI-X_N{)}^{-1}$ used in the general resolvent of an operator article. Here $g$ is the Stieltjes transform of a measure of the empirical spectral measure, with kernel $(x-z{)}^{-1}$.

The diagonal entries of $X_N$ are zero, so the preceding Schur complement formula gives $G_{ii}=-1/(z+q_i)$. Taking the matrix trace, subtracting the comparison value $-1/(z+g)$, and combining fractions gives the resolvent self-consistency defectThe positive numerator sign is fixed by this subtraction. All denominators are nonzero in the upper half-plane, as the imaginary-part estimate in the next part shows. The identity is deterministic and does not use entry independence or moment assumptions.