Exact extraction and global control error

If controls are certified only on the initial $P_{1,K_1}\otimes P_{2,K_2}$ domain, with arbitrary behavior outside it, the valid replacement for ${\zeta }_E+p_{\mathbf{K}}$ is

Indeed, compare each channel through the normalized projected input, whose referenced distance from the original is at most $\sqrt{p_{\mathbf{K}}}$. The sharp probability tail requires the global block structure just proved.

Figure fig:extraction is the selective swap on one memory. Levels $1\le n\le a_i$ move into the register. Levels $a_i<n\le K_i$ are clipped to the corner. States outside the control domain are not given the same exact tail; an implementation certified only on the low domain pays the square-root comparison.