The topology of kinematic spaces and a two-sided Crofton formula

A central challenge in holography is understanding how spacetime emerges from the boundary conformal field theory (CFT). Entanglement provides a geometric perspective through kinematic space—the space of CFT point pairs or bulk minimal cuts—where spacetime geometry is encoded in an integral formula with quantum information meaning: the Crofton formula, a cornerstone of integral geometry. In this talk, I will present recent advances revealing a novel two-sided Crofton formula, which reconstructs wormhole geometries by resolving quantum puzzles. This reveals a deeper organizational principle, leading to the discovery of new topological kinematic spaces with rich structure that precedes spacetime emergence.