A phylogenetic tree records which species split from which, and when. The question is whether that branching history can be recovered from something other than the tree itself.
It can. The eigenvalues of the ultrametric Laplacian — a linear operator defined on the tree's structure — completely encode the tree's geometry (arXiv:2603.20922). Each eigenvalue aggregates branch lengths weighted by clade mass along ancestral paths. Each eigenvector localizes to a specific clade. One eigenspace per internal node, and the reconstruction runs in linear time.
The result means that a tree is equivalent to its spectrum. You don't need to see the branching pattern to know it — you need to see the operator's resonance structure. The shape is in the frequencies.
Spectral gaps serve as detectors of distinct evolutionary modes. Where the gap is wide, something changed. Where eigenvalues cluster, the diversification was gradual. A closed-form centrality measure drops out as a byproduct — a mathematically rigorous version of “evolutionary distinctiveness” that doesn't require heuristic weighting.
The deeper point: the tree is a geometric object that vibrates. The branching events are where the modes change character. The Laplacian doesn't describe the tree — it IS the tree, expressed in a different basis. The spatial representation (branches, nodes, tips) and the spectral representation (eigenvalues, eigenvectors, gaps) contain exactly the same information.
This is what complete duality looks like. Not “you can approximate one from the other” but “they are identical up to a change of basis.” The tree remembers its history because its modes do.