Almost-sure global well-posedness for high-regularity random data in energy-supercritical defocusing NLS (Deng-Nahmod-Yue Open Problem 3), refuted on the twelve-torus

Claims: there are an odd $p$ (which can be taken arbitrarily large), an integer $k>8$ and a nonempty open set $\mathcal U\subset H^k(\mathbb T^{12})$ such that every solution of the defocusing NLS with data in $\mathcal U$ blows up in finite time $T$ with $(T-t)^{1/(p-1)}|u(t,x_*)|\to c_0>0$, a type-I self-similar singularity with logarithmic phase. Since Gaussian measures charge open sets, for every $\alpha>k+6$ the Gaussian Fourier datum blows up with positive probability, so almost-sure global existence for high-regularity random data fails for this $(d,p)$. It does NOT give probability-one blowup, a bound uniform in $\alpha$, an explicit $p$, or anything for low dimensions such as $d\le 11$; the question for other pairs $(d,p)$ stays open.

Claims: there are an odd $p$ (which can be taken arbitrarily large), an integer $k>8$ and a nonempty open set $\mathcal U\subset H^k(\mathbb T^{12})$ such that every solution of the defocusing NLS with data in $\mathcal U$ blows up in finite time $T$ with $(T-t)^{1/(p-1)}|u(t,x_*)|\to c_0>0$, a type-I self-similar singularity with logarithmic phase. Since Gaussian measures charge open sets, for every $\alpha>k+6$ the Gaussian Fourier datum blows up with positive probability, so almost-sure global existence for high-regularity random data fails for this $(d,p)$. It does NOT give probability-one blowup, a bound uniform in $\alpha$, an explicit $p$, or anything for low dimensions such as $d\le 11$; the question for other pairs $(d,p)$ stays open.