Unconditional uniqueness for the cubic nonlinear Schrödinger equation in H1/2( T³)

In this paper, we establish unconditional uniqueness for the cubic nonlinear Schrödinger equation (NLS) in the scaling-critical class H1/2(T³). Unlike approaches based on the GP-hierarchy or normal form reductions, which typically involve higher-order expansions, our proof works directly with the original cubic NLS, where we also make use of several new a priori Euclidean-like Strichartz estimates. More precisely, we show that every distributional solution in CₜHₓ1/2 belongs a priori to the class L²t,locLₓ⁶. The key ingredients are a suitable high-low modulation decomposition and the compactness of solution trajectories on bounded time intervals. We also exploit the preservation of the canonical symplectic form under the truncated NLS flow, in order to obtain necessary stability estimates in the rough Sobolev space H-1/2(T³).