The Crystal-Radius Law

This paper presents the Crystal-Radius Law as a geology-facing expression of the Prime Indexed Curvature Equation (PICE) developed within Pattern Field Theory (PFT). Stable mineral radii, layer spacings, facet positions, and growth orientations are treated as realizations of prime-indexed coherence minima on the Allen Orbital Lattice (AOL).

The governing construction combines the crystal coherence energy (E=Q_H\chi\kappa^2), shell coherence amplitude (\kappa_n=s_n^{-1}), stable-radius law (r_n=ks_n), and stationarity condition (\partial E/\partial r=0). The resulting construction consolidates seven classical crystallographic description systems beneath one generative selection law while retaining those systems as practical measurement and classification tools.

The crystal-radius construction was previously published on 10 November 2025 through PatternFieldTheory.com and Academia.edu under the title Crystalline Coherence on the Allen Orbital Lattice: The Allen Crystal Coherence Equation. That publication presented the governing relation as the Prime Indexed Curvature Equation. The present paper consolidates and develops its crystallographic realization as the Crystal-Radius Law and records its relationship to the later Prime Indexed Stepping Equation (PISE).

The paper provides geologists, mineralogists, crystallographers, and students with a direct conceptual bridge between observable mineral form and the proposed substrate geometry selecting admissible crystal radii.