= Second moment bound for spectral truncation
For a fixed mean-zero variance-one real entry law, <centered truncation of a Wigner matrix> satisfies $\mathbb E[N^{-1}\operatorname{Tr}(X-\widehat X)^2]\leq t(C)$, where $t(C)=\mathbb E[Y^2\mathbf1_{|Y|\geq C}]\to0$. The <spectral Lipschitz bound from Frobenius distance> and <Markov inequality> give probability at most $t(C)/\varepsilon^2$ for a spectral test-function discrepancy exceeding $\varepsilon$. Thus $t(C)<\varepsilon^3$ works uniformly in <matrix> size. A varying entry-law family requires uniform second-moment tail decay instead.
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