We study the query complexity of finding a Tarski fixed point over $[n]^k$. Previous work has left a large gap between $\smash{Ω(\log^2 n)}$ and $\smash{\log^{O(k)}n}$. We show that both of the previous upper and lower bounds were far from tight: for every $k\geq 3$, \[ Ω\left((\log n)^{\frac{1}{2}\lceil \log k\rceil}\right)\le \operatorname{Tarski}(n,k)\le O\left(5^k(\log n)^{\lceil \log k\rceil}\right). \] Succinctly, up to the fixed-parameter factor of $5^k$, the complexity is settled at $(\log n)^{Θ(\log k )}$. In particular, we obtain the first super-polynomial query lower bound for this problem.

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