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Let $d\geq 4$ and let $u$ be a global solution to the focusing mass-criticalnonlinear Schr\"odinger equation $iu_t+\Delta u=-|u|^{\frac 4d}u$ withspherically symmetric $H_x^1$ initial data and mass equal to that of the groundstate $Q$. We prove that if $u$ does not scatter then, up to phase rotation andscaling, $u$ is the solitary wave $e^{it}Q$. Combining this result with that ofMerle \cite{merle2}, we obtain that in dimensions $d\geq 4$, the onlyspherically symmetric minimal-mass blowup solutions are, up to phase rotationand scaling, the pseudo-conformal ground state and the solitary wave.

Characterization of Minimal-Mass Blowup Solutions to the Focusing Mass-Critical NLS