Teleportation=Translation: Continuous recovery of black hole information

The \textit{Teleportation=Translation} conjecture posits that the recovery of information from a black hole is dual to a geometric translation in the emergent spacetime. In this paper, we establish this equivalence for general local quantum field theories by constructing a continuous unitary interpolation that bridges discrete algebraic teleportation protocols and continuous modular flow. We resolve the failure of dynamic idempotency, fundamentally inherent in Type III von Neumann algebras, by employing the Haagerup-Kosaki crossed-product construction. This lift to the semifinite Type~II$_\infty$ envelope yields a canonical, dynamically consistent path. Crucially, we prove that its unique infinitesimal generator $\tilde{G}$ is exactly twice the geometric modular momentum ($\tilde{G}=2P$). We establish this identity as a closed operator equivalence using Nelson's analytic vector theorem and quantify its structural robustness via non-commutative $L^p$ theory. Ultimately, our results demonstrate that unitary information recovery fundamentally manifests as a continuous geometric translation. This provides a rigorous operator-algebraic mechanism for resolving the black hole information paradox, offering a kinematic framework naturally extendable to include gravitational back-reaction.