Hidden amplitude and dynamical information in phase-separation spectra

Phase separation is a fundamental phenomenon across diverse physical and biological systems. However, similar phase-separation patterns can often arise from fundamentally different underlying dynamics, making the identification of these driving mechanisms from static images highly difficult. Framewise variance normalization removes fluctuation amplitude from an observation, but not from the dynamics that generate later patterns. The current normalized spectrum therefore need not determine its own future. We study this effect in three two-dimensional phase-separation models that change transport, add active driving, or suppress large-scale separation. For paired initial fields, their normalized spectra evolve identically in the linear regime. The Cahn-Hilliard-Oono model with a restoring term provides an exact example: damping does not directly change how the normalized spectrum evolves when both equations are applied to the same field. It nevertheless changes the hidden amplitude, which alters the subsequent nonlinear shape. An exact comparison separates the immediate effect of changing the equation from the effect of reaching a different state through earlier evolution. It also reveals strong cancellation between these effects in the active model and weaker cancellation in the mobility model. As a practical test, spectral histories improve classification by about 6-9 percentage points at several later observation windows, although the gain depends on quench depth and spectral construction. Furthermore, this classification gain survives compression to the same input dimension as a single endpoint. Temporal spectra can therefore reveal dynamical consequences of information absent from every individual normalized frame.