Theorem: Exact number-record recovery obstruction

For the pre-endpoint construction with only those complete classical number records accessible,

After the endpoint, with all endpoint rails also measured,

Thus complete evaporation in the model does not make an ordinary destructive detector a quantum-code decoder.

Proof. On a completed branch, the key identifies the cyclic shift in $U_r=X^a{Z}^b$. Correcting that shift in the classical number outcome leaves a measurement of the original logical computational basis. Its channel is ${\Delta }_Z$. Pending branches remain constant. Preparing the measured logical basis state and preparing $I_d/d$ on pending gives

Every completed branch is entanglement breaking. A separable state of a $d$-dimensional reference and logical output has Bell overlap at most $1/d$: the bound holds for pure product states by Cauchy–Schwarz and then by convexity. A constant pending branch has Bell overlap $1/d^2$. This proves the upper bound in (eq:numberfidelity).

Now

The identity Weyl weight in (eq:numberdecoded) is $p_t/d+s_t/d^2$. Orthogonal Bell outputs give error at least one minus that weight. The triangle inequality over the remaining unitary channels gives the matching upper bound. This proves optimality.

At the endpoint, pending encrypted logical systems are also measured in a known-key computational basis. Every branch therefore gives ${\Delta }_Z$, proving (eq:numberendpoint).