THE SNOWFLAKE – WHEN INFORMATION FREEZES: A Finite‑Capacity Model That Recreates Nature's Hexagonal Geometry

We investigate a deterministic, capacity-limited informational model that generates snowflake morphology without stochastic fluctuations, microscopic molecular physics, or diffusion-driven instabilities. The model consists of five coupled equations governing accumulation, threshold freezing, anisotropic weighting, and relaxation on a two-dimensional lattice. Across all parameter sweeps—including threshold-accumulation variation (T1–T10), anisotropy perturbation (E1–E3), and reduced-scale early-time evolution (S1)—the system converges to a single hexagonal attractor whose geometry is fixed while the distribution of complexity varies. The transition from liquid-like to ice-like behavior appears as an informational phase transition: when local accumulation crosses a critical threshold, the system undergoes a collapse of available configurations and snaps into a six-fold instability, mirroring the physical freezing of water. Classical snowflake models rely on noise-driven branching and diffusion gradients; in contrast, the present model produces symmetric, hierarchical dendrites deterministically through finite informational capacity and symmetry constraints. The results demonstrate that snowflake geometry can emerge from informational dynamics alone, suggesting a broader principle in which natural structures arise from capacity-limited relaxation under fixed symmetry classes. Conical Source Statement: This work is one projection of a single underlying theoretical source: a unified informational framework in which geometry, dynamics, and physical structure emerge from finite‑capacity constraints acting on relational degrees of freedom. Each paper in this series represents a distinct cross‑section of the same conical architecture, tracing different observable consequences of the shared origin.