  Self-shrinkers are the special solutions of mean curvature flow in
$\mathbf{R}^{n+1}$ that evolve by shrinking homothetically; they serve as
singularity models for the flow. The entropy of a hypersurface introduced by
Colding-Minicozzi is a Lyapunov functional for the mean curvature flow, and is
fundamental to their theory of generic mean curvature flow.
  In this paper we prove that a conjecture of Colding-Ilmanen-Minicozzi-White,
namely that any closed hypersurface in $\mathbf{R}^{n+1}$ has entropy at least
that of the round sphere, holds in any dimension $n$. This result had
previously been established for the cases $n\leq 6$ by Bernstein-Wang using a
carefully constructed weak flow.
  The main technical result of this paper is an extension of
Colding-Minicozzi's classification of entropy-stable self-shrinkers to the
singular setting. In particular, we show that any entropy-stable self-shrinker
whose singular set satisfies Wickramasekera's $\alpha$-structural hypothesis
must be a round cylinder $\mathbf{S}^k(\sqrt{2k})\times \mathbf{R}^{n-k}$.
