Theorem: Sector Page and sector record means

For a Haar pure vector in ${\mathcal{H}}_{\mathsf{q}}$, the mean radiation entropy is

Destructive basis readout resolving each radiation charge sector has mean entropy

The sector geometry and weights cannot be replaced by two aggregate dimensions in the unrestricted Page formula.

Proof. Use independent Gaussian coordinates in all sector blocks. Their squared block norms $p_{\lambda }$ have the Dirichlet law with parameters $D_{\lambda }$. Conditional normalized directions are independent Haar vectors in the individual blocks and independent of the weights.

The residual trace kills cross-charge coherences, so

Dirichlet differentiation gives

Add ${\sum }_{\lambda }{w}_{\lambda }\mathsf{P}(r_{\lambda },b_{\lambda })$; the ${\mathsf{H}}_{D_{\lambda }}$ terms cancel, yielding (eq:sectorpage).

Every individual radiation basis outcome in sector $\lambda$ collects $b_{\lambda }$ Gaussian coordinates. Its probability has a Dirichlet parameter $b_{\lambda }$, repeated $r_{\lambda }$ times. The same moment differentiation gives (eq:sectorrecord).