Dihedral Invariants and Morphological Transitions

This paper advances the Sc-Rubs scalar-field framework by quantifying how energy minimization drives stable attractor formation within recursive persistence domains. Building upon the established curvature–diffusion and bifurcation dynamics, it formalizes the relationship between field energy E(φ), curvature stiffness β, and truncation λ to describe how form transitions naturally toward equilibrium attractors.

The analysis shows that persistent geometric structures emerge when ∇²φ → 0 across recursive iterations, minimizing local energy gradients and maximizing global continuity. This condition defines the Law of Recursive Minimization, where each morphological iteration re-stabilizes φ within an attractor basin defined by curvature constraints. The resulting attractors correspond to canonical geometries — cube, octahedron, dodecahedron, and sphere — each acting as a local energy well in scalar-field space.

Mathematically, the model demonstrates that attractor selection arises from iterative Laplacian damping and phase re-normalization within the persistence domain, producing the observed octahedral–spherical–cubic evolution. The framework unifies geometric stability, energy conservation, and recursive morphogenesis under a single self-referential principle.

This paper forms part of the Sc-Rubs Modelling Series, a unified scalar-field study of persistence, recursion, and emergent geometry.
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Related DOI: 10.5281/zenodo.17443937.