  A longstanding problem in mathematical physics is the rigorous derivation of
the incompressible Euler equation from Newtonian mechanics. Recently, Han-Kwan
and Iacobelli arXiv:2006.14924 showed that in the monokinetic regime, one can
directly obtain the Euler equation from a system of $N$ particles interacting
in $\mathbb{T}^d$, $d\geq 2$, via Newton's second law through a supercritical
mean-field limit. Namely, the coupling constant $\lambda$ in front of the pair
potential, which is Coulombic, scales like $N^{-\theta}$ for some $\theta \in
(0,1)$, in contrast to the usual mean-field scaling $\lambda\sim N^{-1}$.
Assuming $\theta\in (1-\frac{2}{d(d+1)},1)$, they showed that the empirical
measure of the system is effectively described by the solution to the Euler
equation as $N\rightarrow\infty$. Han-Kwan and Iacobelli asked if their range
for $\theta$ was optimal. We answer this question in the negative by showing
the validity of the incompressible Euler equation in the limit
$N\rightarrow\infty$ for $\theta \in (1-\frac{2}{d},1)$. For reasons of
scaling, this range appears optimal in all dimensions. Our proof is based on
Serfaty's modulated-energy method, but compared to that of Han-Kwan and
Iacobelli, crucially uses an improved "renormalized commutator" estimate to
obtain the larger range for $\theta$.
