Conserved-sector scrambling: a direct-sum Page calculation

More generally, for fixed excitation number $q$,

Vandermonde's identity verifies the total sector dimension. A single Haar unitary within that fixed sector and passive cell release give compatible cuts to which Theorem thm:sector applies. A mass window, non-Abelian angular momentum sector, or interacting energy shell needs its own decomposition; it is not automatically (eq:charge-sector).

Sector projection is also not free postselection. The code must be prepared in the stated sector, or the selective operation and its failure branch must belong to the task.