Norm inflation in $\dot H^{1/4}(\R^3)$ for the defocusing quintic Gabitov--Turitsyn equation

We prove norm inflation in the homogeneous Sobolev space $\dot H^{1/4}(\R^3)$ for the defocusing quintic Gabitov--Turitsyn equation with averaging interval $[0,1]$. A family of Schwartz initial data has arbitrarily small initial norm, while its global smooth solutions attain arbitrarily large norms at arbitrarily small negative times. The proof combines a finite-interval virial identity with a lower bound for the averaged nonlinear forcing obtained from localized low-frequency mass. A rescaling keeps the variance bounded over a long time interval and forces growth of the fractional Sobolev norm.