Dimension Amplification for Tarski Fixed-Point Query Lower Bounds

Abstract

We prove that finding a fixed point of a monotone map on the nine-dimensional grid $[N]^9$ requires $\Omega((\log N)^3)$ deterministic queries, even when the fixed point is unique and each query returns the entire function value. The proof gives a construction that raises the dimension from $d$ to $4d+1$, preserves uniqueness, and adds a logarithmic factor to the lower bound. Iteration gives $\Omega((\log N)^{r+2})$ queries in dimension $(7\cdot4^r-1)/3$, for every fixed nonnegative integer $r$, with an implicit constant that may depend on $r$. Consequently, no finite logarithmic exponent bounds the query complexity in all fixed dimensions.