Remark: What is and is not guaranteed

The theorem proves a high-probability statement and, whenever its hypotheses hold, existence of fixed unitaries with the certificate. It does not certify every unitary or guarantee a qualifying cut before every chosen cutoff. The sufficient threshold is not claimed optimal.

The old-vault entropy is ${\log }_{2}h$. Its purification is accessible and explicitly included. The resulting sufficient bound has no additional $h$ factor. Losing that purification changes the complete complement and the task.