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For polynomials in independent Wigner matrices, we prove convergence of the largest singular value to the operator norm of the corresponding polynomial in free semicircular variables, under fourth moment hypotheses. We actually prove a more general result of the form “no eigenvalues outside the support of the limiting eigenvalue distribution.” We build on ideas of Haagerup–Schultz–Thorbjørnsen on the one hand and Bai–Silverstein on the other. We refine the linearization trick so as to preserve self-adjointness and we develop a secondary trick bearing on the calculation of correction terms. Instead of Poincaré-type inequalities, we use a variety of matrix identities and $L^{p}$ estimates. The Schwinger–Dyson equation controls much of the analysis.