Algebraic structure underlying pole-skipping points

The holographic Green’s function becomes ambiguous, taking the indeterminate form “ 0 / 0 ,” at an infinite set of special frequencies and momenta known as “pole-skipping points.” In this work, we propose that these pole-skipping points can be used to reconstruct both the interior and exterior geometry of a static, planar-symmetric black hole in the bulk. The entire reconstruction procedure is fully analytical and only involves solving a system of linear equations. We demonstrate its effectiveness across various backgrounds, including the Bañados-Teitelboim-Zanelli black hole, its T T ¯ -deformed counterparts, as well as geometries with Lifshitz scaling and hyperscaling-violation. Within this framework, other geometric quantities, such as the vacuum Einstein equations, can also be reinterpreted directly in terms of pole-skipping data. Moreover, our approach reveals a hidden algebraic structure governing the pole-skipping points of Klein-Gordon equations of the form ( ∇ 2 + V ( r ) ) ϕ ( r ) = 0 : only a subset of these points is independent, while the remainder is constrained by an equal number of homogeneous polynomial identities in the pole-skipping momenta. These identities are universal, as confirmed by their validity across a broad class of bulk geometries with varying dimensionality, boundary asymptotics, and perturbation modes.