On the classification of invariant Gaussian measures for the 2D Euler equation

We consider the two-dimensional incompressible Euler equation on $\mathbb T^2$ with Gaussian random initial data having independent Fourier coefficients. For every $\sigma>0$, we prove that such a Gaussian measure on $H^\sigma(\mathbb T^2)$ is invariant under the Euler flow if and only if it is supported either on shear flows or on cellular flows. This settles the invariant-measure classification conjecture posed by Bedrossian and Latocca (Ann. Inst. H. Poincar\'e C Anal. Non Lin\'eaire, 2026). The proof relies on closure of the Fourier support under non-degenerate interactions and an affine relation among the inverse variances along Euler triples.