The missing dynamical estimate

For this interacting source, the required recovery object is still

The residual $B_t$ must include the source, uncollected bath, loading apparatus, and every inaccessible preparation environment.

The preceding construction does not prove a small bound for (eq:syk-target). The explicit quadratic estimate of Theorem thm:main cannot be imported: the interacting Hamiltonian does not preserve its label-wise one-particle isometry.

A controlled weak-coupling calculation would have to establish both its reduced-dynamics error and information accounting in the full unitary dilation. A replicated calculation would have to control the reference–residual state and its norm, with the factor $d$ in Lemma lem:choi retained when used.

Neither a level-statistics statement nor an out-of-time-order correlator estimate supplies this missing channel bound. Hamiltonian recovery studies explicitly distinguish recoverability from common chaos diagnostics NakataTezuka.

policy. An averaged calculation is reported as an averaged calculation. A fixed-realization theorem must certify its declared realization. A high-probability theorem must state its probability space, failure probability, code dependence, and decoder dependence. These three conclusions are not interchanged.