Localization-Induced Compensator Stress–Energy in the Time Emergent Framework

This work presents a covariant framework (the Time Emergent Framework) for gravitational systems in which localization is implemented via a spacetime window function multiplying an otherwise conserved local stress–energy tensor. Such localization generically induces apparent violations of local stress–energy conservation confined to a finite boundary layer. Consistency with the contracted Bianchi identities then requires an additional, nonlocal stress–energy contribution, which I term the compensator stress–energy tensor. I show that this compensator is neither ad hoc nor exotic, but instead reproduces a wide class of well-established gravitational constructions, including thin shells and junction conditions, black hole membrane paradigms, dynamical and isolated horizon flux laws, inflationary horizon-exit coarse-graining, and gravitational vacuum decay via bubble walls. The resulting structure parallels quantum boundary-layer analyses and provides a unified localization-based interpretation of boundary-supported gravitational stress–energy.