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The Unconditional Uniqueness for the Energy-critical Nonlinear Schrödinger Equation on $\mathbb{T}^{4}$

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.