Restricted repair and robust observations

Suppose a scarred sector permits only node dephasing, no restored edge, no additional coherent control, and the same node readout. Its phase-label success remains $1/2$. For the noiseless averaged pilot and $q_{\ast }\in (1/2,1)$, failure occurs when

This is persistent vulnerability under the declared repair restriction. It is not a theorem about unrestricted quantum recovery.

For an independent preparation-success factor $p$, observation factor $\eta$, and symmetric assignment error $\epsilon <1/2$, retain failures and erasures with score $1/2$. The coherent-versus-dephased score contrast after $t$ updates is

Uniform complete-law errors $e_0,e_1$ give a retained lower contrast

A positive remainder certifies the oriented comparison.

For the conditional memory test, if ideal and actual first-history probabilities are at least $p_{\min }$ and a complete recorded state allowance is $e$, Proposition prop:normalization bounds each normalized branch error by $2e/p_{\min }$. A conservative retained memory contrast is therefore

Apparatus discrepancies must be included in $e$.

A bounded complete-record statistic with actual null mean at most $a$, alternative mean at least $b$, and gap $g=b-a>0$ has midpoint-test errors at most

after $L$ independent attempted runs Hoeffding,CheungPsy02B. This sampling statement requires a common oriented statistic and calibrated alternatives. It is not inferred solely from a nonzero conditional contrast.