Proposition: Flagged adaptive access

Let $\{P_s\}$ be orthogonal sector projectors commuting with ${\Delta }_J$, and define

for $w_s>0$. Suppose the interface supplies an accessible orthogonal sector flag:

where ${\Lambda }_s$ is CPTP on the $s$-sector. Then

Sector-wise Helstrom effects attain this value. After forgetting the flag, only the upper bound by this sum is guaranteed, and signed sector contributions may cancel.

Proof. Commutation gives

Off-diagonal sectors are removed by the flagged interface. Trace norm is additive on orthogonal direct sums, and the resulting blocks are traceless. Assemble the positive-spectral projectors blockwise. Forgetting the flag is CPTP post-processing, so Corollary cor:dp gives the upper bound. Without the flag, the block effects need not be separately implementable.