  We consider the $\mathbb{T}^{4}$ cubic NLS which is energy-critical. We study
the unconditional uniqueness of solution to the NLS via the cubic
Gross-Pitaevskii hierarchy, an uncommon method, and does not require the
existence of solution in Strichartz type spaces. We prove $U$-$V$ multilinear
estimates to replace the previously used Sobolev multilinear estimates, which
fail on $\mathbb{T}^{4}$. To incorporate the weaker estimates, we work out new
combinatorics from scratch and compute, for the first time, the time
integration limits, in the recombined Duhamel-Born expansion. The new
combinatorics and the $U$-$V$ estimates then seamlessly conclude the $H^{1}$
unconditional uniqueness for the NLS under the infinite hierarchy framework.
This work establishes a unified schemes to prove $H^{1}$ uniqueness for the
$\mathbb{R}^{3}/\mathbb{R}^{4}/\mathbb{T}^{3}/\mathbb{T}^{4}$ energy-critical
Gross-Pitaevskii hierarchies and thus the corresponding NLS.
