Conserved-sector scrambling: a direct-sum Page calculation

This direct-sum structure is the standard symmetry-resolved random-state setting SymmetryPage. Its derivation is included to make the missing tensor-product hypothesis visible.

If the sector probabilities are instead fixed at specified $p_{\lambda }$, with independent conditional Haar directions, the means become

and

Externally assigned sector weights define a different ensemble from Haar measure on the whole conserved sector.