Abstract

We prove the precise conditional statement that quantum information cannot be destroyed by unitary evolution of a complete closed system. For a finite logical code, a fixed pure ancillary preparation, and a unitary evolution, the complete output channel has a CPTP left inverse, including every external reference. Every logical matrix unit and every input-state trace distance are preserved.

This global statement does not imply recovery from radiation alone. For a complete isometry $V:C\to R\otimes B$, we prove that one decoder recovers the entire code from $R$ exactly if and only if the complete complementary channel to $B$ is constant. A Schmidt-space construction provides the decoder and the exact no-hiding normal form. We distinguish this result from partial decoherence, restricted input ensembles, and mixed-output descriptions with omitted preparation purifiers.

For the regulated quartic SYK–scalar architecture of Hawk02-E, we reconstruct the self-adjoint dynamics and a reference-preserving finite approximation. A finite complementary Choi certificate $e$, input dimension $d$, and complete-channel regulator allowance $b$ yield ${\delta }_{\mathrm{full}}\le de+b$. A finite decoder can be lifted to the continuum with recovery error at most $\sqrt{2de}+b$, or certified directly using its decoded Choi operator. We also give an exact four-Majorana realization with nonzero coupling for which the optimal radiation recovery error is $(3-z(t))/4\ge 1/2$, although global recovery is exact. This is a dynamical negative control, not a scrambling argument.

Finally, we prove endpoint continuity with access-preserving output identifications and state the separate gravitational obligations. The results establish unitary preservation and operational recovery criteria. They do not establish unitarity of quantum gravity, a generic interacting evaporation-recovery window, or a gravitational endpoint.