Exact Fourier dimensions of dyadic Mandelbrot cascades under minimal integrability

We determine the Fourier dimension of dyadic Mandelbrot cascades under the minimal Kahane-Peyriere integrability condition. The interval theorem is proved in a vector-valued dyadic cascade model in which sibling weights may have arbitrary dependence. For every balanced energy-admissible vector law, almost surely on non-extinction, dim_F(mu)=dim_E(mu)=dim_2(mu)=D_E(X). In the canonical scalar case, under W>=0, E W=1, E[W log_2^+ W]<infinity, and E[W log_2 W]<1, the formula becomes dim_F(mu)=dim_E(mu)=dim_2(mu)=sup_{1<q<2} max{0, 2-(2/q)(1+log_2 E[W^q])}, with the convention that the corresponding term is zero when E[W^q]=infinity. In particular, this scalar specialization gives the canonical Mandelbrot-Kahane Fourier-dimension formula under the minimal integrability condition. We also prove the endpoint theorem for the dyadic Mandelbrot cascade on the unit circle. Under the same scalar assumption, almost surely on non-extinction, dim_F(mu_circle)=sup_{q>1} max{0, (q-1-log_2 E[W^q])/q}. The interval and circle formulas share a light-tail/heavy-tail dichotomy but have different mechanisms: energy dimension for the interval, and minimum lower local dimension for the circle. The circle lower bound follows from a finite-moment annular Fourier theorem.