Discrete Shell Geometry and Offset-Family Transitions: A Branch-Based Framework for Sphere Packing and Structure Formation

This work introduces a discrete shell-based framework for geometric organization in bounded particle systems. Using hard-sphere simulations, we demonstrate that stable configurations form through radial shell states governed by shell populations, outer-shell balance, and lattice-offset families. Transitions between configurations occur through abrupt shell merge and shell birth events, rather than continuous deformation. We show that these transitions are predictable using a shell population ratio and radial gap structure. A minimal energy-like functional is proposed that reproduces observed behavior, providing a testable model for discrete geometric phase transitions. The framework reveals a hidden ruleset governing geometry in bounded systems and provides new tools for predicting and controlling structure formation.