  The problem of testing monotonicity for Boolean functions on the hypergrid,
$f:[n]^d \to \{0,1\}$ is a classic topic in property testing. When $n=2$, the
domain is the hypercube. For the hypercube case, a breakthrough result of
Khot-Minzer-Safra (FOCS 2015) gave a non-adaptive, one-sided tester making
$\widetilde{O}(\varepsilon^{-2}\sqrt{d})$ queries. Up to polylog $d$ and
$\varepsilon$ factors, this bound matches the
$\widetilde{\Omega}(\sqrt{d})$-query non-adaptive lower bound
(Chen-De-Servedio-Tan (STOC 2015), Chen-Waingarten-Xie (STOC 2017)). For any $n
> 2$, the optimal non-adaptive complexity was unknown. A previous result of the
authors achieves a $\widetilde{O}(d^{5/6})$-query upper bound (SODA 2020),
quite far from the $\sqrt{d}$ bound for the hypercube.
  In this paper, we resolve the non-adaptive complexity of monotonicity testing
for all constant $n$, up to $\text{poly}(\varepsilon^{-1}\log d)$ factors.
Specifically, we give a non-adaptive, one-sided monotonicity tester making
$\widetilde{O}(\varepsilon^{-2}n\sqrt{d})$ queries. From a technical
standpoint, we prove new directed isoperimetric theorems over the hypergrid
$[n]^d$. These results generalize the celebrated directed Talagrand
inequalities that were only known for the hypercube.
