Entanglement inequalities, black holes and the architecture of typical states
Using the holographic realization of the Araki--Lieb (AL) inequality, we argue that the leading entanglement plateau constrains the organization of typical pure states in a fixed-$(E,J)$ sector of a large-$c$ two-dimensional holographic CFT. On the field-theory side, its equality structure motivates an effective factorization of the regulated micro-canonical code subspace into ultraviolet (UV), intermediate and infrared factors, with part of the intermediate sector providing a purifier of the ultraviolet one. In the bulk, entanglement-wedge reconstruction realizes the ultraviolet sector as an asymptotic corona, universal to simple probes with length scale smaller than $L_{\rm UV}$. Crucially, $L_{\rm UV}$ is determined only by the conserved charges. At finite $c$, universal perturbative dressing correlates these leading factors and lifts the AL plateau at order $c^0$. We formulate a generalized ETH ansatz under which $O(e^{-S_<})$ fluctuations, with $S<=\min(S_L,S_R)$, account for the $O(c^0)$ deficit of the AL plateau, while the state dependence of matrix elements of individual simple operators smeared over lengths shorter than $L_{\rm UV}$ remains exponentially small. This provides a holographic realization and rotating generalization of the Eigenstate Thermalization Hypothesis: for zero rotation it reduces to the standard $O(e^{-S/2})$. Independently, the separation between $L_{\rm UV}$ and the scale of quantum correlations is of order $e^{-6S_</c}$ at large $S_<$.