The Hidden Coordinate

THE HIDDEN COORDINATE THE HIDDEN COORDINATE Water looks like one liquid until you measure along the right axis. The axis everyone measured local density ρ one peak water looks like a single liquid The axis the network found learned coordinate two peaks two coexisting structures appear high-density: disordered, yet compact low-density: ordered, open, roomy The two liquids were always there. They lived on an axis no one had measured.
The same simulated molecules, projected two ways. Read along local density, the high-density and low-density populations blur into a single peak, and water looks like one liquid. Read along the coordinate an unsupervised network discovered, the same data splits cleanly in two. The second structure was never hidden in the molecules; it was hidden in the choice of axis. (After Li et al., Nature Physics, 2026.)