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We prove that Sidorenko's conjecture is equivalent to a spectral strengthening: \\[\n  \\hom(H,G)\\ge M(G)^e |V(G)|^{v-2e} \\quad \\text{ if and only if }\\quad \\hom(H,G)\\ge \\lambda(G)^{2e-v}M(G)^{v-e}. \\] We also introduce an operator-norm certificate which, via the Riesz--Thorin interpolation, gives direct proofs of the spectral Sidorenko inequality in several cases. 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Let $H$ be a bipartite graph with $v$ vertices and $e$ edges, where $v\\le e$, and write $M(G)=2e(G)$. We prove that Sidorenko's conjecture is equivalent to a spectral strengthening: \\[\n  \\hom(H,G)\\ge M(G)^e |V(G)|^{v-2e} \\quad \\text{ if and only if }\\quad \\hom(H,G)\\ge \\lambda(G)^{2e-v}M(G)^{v-e}. \\] We also introduce an operator-norm certificate which, via the Riesz--Thorin interpolation, gives direct proofs of the spectral Sidorenko inequality in several cases. 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