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If $d\\ge2$, $R>0$, and $E\\ge0$, then \\begin{equation*} N_{B_R^d}^{<}(E) \\ge \\frac{\\omega_d}{(2\\pi)^d}|B_R^d|E^{d/2} = \\frac{(R\\sqrt E)^d}{2^d\\Gamma(\\frac d2+1)^2}. \\end{equation*} The radial boundary condition in dimensions $d\\ge3$ is a Dini condition, not a derivative-zero condition. A strict Robin comparison first reduces it to a Bessel phase estimate. 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