An interface evolving under local equilibrium develops gradients in the Gibbs-Thomson-Herring interface temperature distribution that provide tangential energy fluxes. The Leibniz-Reynolds transport theorem exposes a 4th-order, net-zero energy field (the 'Bias' field) that autonomously deposits and removes capillary-mediated thermal energy. Where energy is released locally, the freezing rate is persistently retarded, and where energy is removed, the rate is enhanced. These contravening dynamic field responses balance at points (roots) where the surface Laplacian of the chemical potential vanishes, inducing an inflection, or ‘curling’, of the interface. Interfacial inflection couples to the main transport fields producing pattern branching, folding, and complexity.

Precision noise-free numerical schemes, including integral equation sharp-interface solvers (J. Lowengrub, S. Li) and, recently, three noise-free phase-field simulations (A. Mullis, M. Zaeem, K. Reuther) independently confirm that pattern branching initiates at locations predicted using analytical methods for smooth, noise-free starting shapes in 2-D. A limit cycle may develop as the interface and its energy field co-evolve, synchronizing the inflection points to produce classical dendritic structures. Noise and stochastics play no direct role in the proposed deterministic mechanism of branching and pattern morphogenesis induced by persistent 'perturbations'.