The stacking number of a tree

Source abstract

The stacking number of a graph is the least integer t >= 2 such that every configuration of t pebbles can be transformed by pebbling moves into a configuration supported on one vertex. We prove that, for every finite tree T with at least two vertices, this number equals the rooted distance-and-degree estimator conjectured by Csernák and Soukup. The proof uses an exact recursive characterization of stackability at a prescribed vertex, an explicit zero-score obstruction, and a weighted cancellation argument for arbitrary nonstackable configurations. The complete theorem is formalized in Lean 4; the formal result has also passed Palomar mechanical verification and is publicly registered as PALOMAR-2026-09-25-000010.