Emergent Polyhedral Geometry from p-Harmonic Scalar Field Dynamics

Correction: words 'truncated icosahedron removes and rhombic dodecahedron replaced. We present Sc‑Rubs Engine v2, a scalar‑field relaxation model demonstrating spontaneous emergence of Platonic‑solid symmetries without templates, explicit topology, or volumetric constraints. Geometry arises solely from the interplay of a p‑Laplacian curvature term, biharmonic stiffness β, truncation bias λ, and torsional persistence τ. We show controlled phase transitions between sphere, cube, dodecahedral, and truncated‑icosahedral attractors. These results provide a minimal constructive mechanism for symmetry formation in discrete‑continuous media.These results suggest that canonical spatial structures may emerge directly from generic relaxation principles without imposed topology; a falsifiable hypothesis supported by the Sc-Rubs engine simulations.