Proposition: What the initial phase does and does not imply

At cut zero, for every $U$,

No comparable lower bound at an arbitrary nonzero cut follows merely because (eq:fullthreshold) has not been met.

Proof. A unitary confined to the vault leaves its inaccessible reference marginal $E$ unchanged. The complement is

Undoing $U$ and tracing $HF$ recovers $X$. Contraction through this inverse gives a leakage lower bound $a_d$; comparison with the output on $I_L/d$ gives the matching upper bound. Theorem thm:threshold is sufficient, not necessary, so its failure supplies no converse.