We provide a $\operatorname{poly}(\log\log n)$-round randomized LOCAL algorithm for the constructive Lovász Local Lemma with polynomial criterion $pd^{c} < 1$, for a sufficiently large constant $c$. The best previous general round complexity was $\widetilde O(\log n)$. The key ingredient is a local reduction that amplifies the LLL probability exponent quadratically, while increasing the dependency radius only by a constant factor.

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