Theorem: Complete destructive record

For every normal packet–spectator–reference state,

Applying ${\mathcal{G}}_N$ gives ${\Omega }_{N,SR}(\rho )$. All positive diagonal block sums converge in trace norm.

Proof. The low-block Kraus map contributes the diagonal blocks at $k\le N$. Every Kraus index $n>N$ contributes its reference block to the same carrier state $N$. Readout adds those positive blocks to the $n=N$ block. Their traces sum to at most one, proving convergence. The total decoder gives the second identity.