What can one learn about material structure given a single first-principles calculation?

We extract a variable X from electron orbitals Psi(nk) and energies E-nk in the parent high-symmetry structure of a wide range of complex oxides: perovskites, rutiles, pyrochlores, and cristobalites. Even though calculation was done only in the parent structure, with no distortions, we show that X dictates material's true ground-state structure. We propose using Wannier functions to extract concealed variables such as X both for material structure prediction and for high-throughput approaches.