Remark: Strength and breadth of the definition

This definition is a recovery certificate with an additional terminological overlay. It does not distinguish error-correcting encodings from geometric wormholes.

For example, with $U=I$, release all diary cells and retain $H$. Then

and $\mathcal{D}={\mathrm{Tr}}_E$ is exact. For $h\ge 2$, this has a positive residual and entangled output mouths, but no spacetime bridge has been constructed. The Haar theorem below adds a quantitative, nontrivially scrambled existence statement, not a gravitational identification.