Energy-Bounded Reconstruction and the Selective Entanglement Wedge

We construct a resource-constrained framework for holographic bulk reconstruction by combining three existing results: the Chandrasekaran-Penington-Witten (CPW) Type II∞ algebra for boundary subregions, the JLMS decomposition of the modular Hamiltonian, and the Margolus-Levitin quantum speed limit applied to modular flow. Imposing a finite energy budget ⟨Kbdy⟩≤B on the boundary modular Hamiltonian simultaneously constrains the horizon area and bulk matter content via JLMS, and limits the rate of modular-flow-based reconstruction via the quantum speed limit. This produces a position-dependent reconstruction cost κ(x) that increases with bulk depth, inducing an effective metric dseff2=κ2(x)gμνdxμdxν. The accessible bulk region—the selective entanglement wedge—is determined by a continuous knapsack optimization that interpolates between the causal wedge (B→0) and the full entanglement wedge (B→∞).