This work presents a new theoretical and numerical framework for understanding the sharp quantum–classical crossover by introducing two internal structural degrees of freedom—the local spread S(x) and contraction C(x)—derived from the gradient and curvature of a probability distribution. These quantities form a two‑dimensional Spread–Contraction (SC) phase diagram that characterizes the transition between quantum‑dominated spreading and classical localization. A minimal covariant action motivated by information geometry is constructed, from which the SC Ordinary Differential Equation (SC‑ODE) is obtained via variational principles. The SC‑ODE integrates spreading and contracting contributions in a minimal form and provides a structural mechanism for the steep drop in interference visibility observed in experiments. To investigate the role of spatial structure, the study introduces a conservative SC Partial Differential Equation (SC‑PDE) incorporating degenerate mobility and backward diffusion. Numerical experiments under extreme initial conditions—where micro‑noise is superimposed on a uniform field—reveal a three‑phase structure consisting of stable, critical, and collapse regimes. This behavior contrasts with the always‑stable SC‑ODE and demonstrates that spatial degrees of freedom are essential for generating critical phenomena. A comparison with the Cahn–Hilliard equation shows that, within the scope of the numerical experiments, the SC‑PDE preserves non‑negativity and boundedness even under conditions where the Cahn–Hilliard model fails. This robustness arises from the automatic suppression of backward diffusion near the vacuum due to the degenerate mobility structure. Overall, this work provides a unified structural framework for describing the sharpness of the quantum–classical crossover and offers a numerically stable PDE formulation that may be relevant for surface evolution, thin‑film processes, and nonlinear field dynamics.
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