Charge-constrained scalar-field scaling ($E_Q[f_{\lambda }]={\lambda }^{2-d}T+{\lambda }^{-d}W+{\lambda }^d{Q}^2/(2I)$)

For $f_{\lambda }(x)=f(\lambda x)$ in $d$ space dimensions, write $T=\frac{1}{2}\int |\nabla f{|}^2$, $W=\int U(f)$ and $I=\int f^2$. Eliminating the rotation frequency by $\omega =Q/I$ gives the displayed fixed-charge scaling. The positive-power charge term can balance the gradient and potential terms even for $d\ge 3$. This evades the static Derrick theorem because the field is time-dependent and the relevant variations preserve its Noether charge.