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Suppose $f \\colon \\mathbb{R}^{12} \\to \\mathbb{R}$ is an integrable function that is not identically zero. Normalize its Fourier transform $\\widehat{f}$ by $\\widehat{f}(\\xi) = \\int_{\\mathbb{R}^d} f(x)e^{-2\\pi i \\langle x, \\xi\\rangle}\\, dx$, and suppose $\\widehat{f}$ is real-valued and integrable. We show that if $f(0) \\le 0$, $\\widehat{f}(0) \\le 0$, $f(x) \\ge 0$ for $|x| \\ge r_1$, and $\\widehat{f}(\\xi) \\ge 0$ for $|\\xi| \\ge r_2$, then $r_1r_2 \\ge 2$, and this bound is sharp. 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