Conclusions and mathematical release

The preservation certificate is residual decoupling on the entire logical operator space. A spectrum, local marginal, entropy turnover, or delayed-query advantage is not substituted for that certificate.

The construction answers where information is located at each stage: initially in the pending residual with its generated key; subsequently in key-conditioned joint radiation on completed branches; and, under the specified endpoint, entirely in radiation for every branch. It also identifies precisely what must change to obtain a nonpreserving accessible channel: a message-bearing residual or excluded output must remain.

The entropy-history counterexample is therefore a separation of data and channel claims, not a rejection of islands. Likewise the thermal comparison is a calibration and compatibility audit, not a claim to reproduce the Hawking pair state or local semiclassical dynamics.

The defensible physical conclusion is:

Within the declared conserving model, the identified joint radiation correlations and complete residual-decoupling conditions suffice for preservation of an arbitrary quantum code. Spectrum or specified entropy-history data alone do not certify that preservation. The model does not derive the microscopic dynamics of realistic black-hole evaporation.