Theorem: One family with recovery and strict turnover

Take

Release vault qubits in order, with

Then

Choose any $\zeta >0$ satisfying

There is a set of Haar unitaries of measure at least $1/64$ for which, simultaneously:

1. the cutoff complement is within $\zeta$ of replacement;

2. the specified deterministic decoder has half-diamond error at most $\min \{a_{16},\sqrt{2\zeta }\}$;

3. for the fixed pure diary input,

   $$S({R_{k_{\mathrm{peak}}}}^U)>{\log }_2(2h)\ge S({R_K}^U);$$

4. every unedited cumulative Enter record is nondecreasing.

In particular, a nontrivial operational island and strict coherent entropy turnover coexist while one physical vault qubit remains.

Proof. At the intermediate cut, (eq:qpurity) gives

Markov's inequality implies

Thus the event $S({R_{k_{\mathrm{peak}}}}^U)>{\log }_2(2h)$ has probability greater than $1/32$. At the cutoff, the rank bound gives $S({R_K}^U)\le {\log }_2(2h)$ for every unitary.

The cutoff leakage estimate is

By (eq:jointsize),

Subtracting this possible failure set from the turnover event leaves measure at least $1/64$. The decoder theorem and Proposition prop:record finish the proof.