Probabilistic Measure of Symmetry Stability

Symmetry is a fundamental principle in mathematics, physics, and biology, where it governs structure and invariance. Classical symmetry analysis focuses on exact group-theoretic descriptions, but rarely addresses how robust a symmetric configuration is to perturbations. In this work, we introduce a probabilistic framework for quantifying the stability of finite point-set symmetries under random deletions. Specifically, given a finite set of points with a prescribed nontrivial symmetry group, we define the probability \( \ P_N \) that removing \( \ N \) points reduces the symmetry to the trivial group \( \ C_1 \). The complementary quantity \( \ S_N = 1 - P_N \), serves as a measure of symmetry stability, providing a robustness profile of the configuration. We calculate \( \ S_N \) explicitly for representative families of symmetric point sets, including linear arrays, polygons, polyhedra, and crystallographic unit cells. Our results demonstrate unexpected behaviors: the regular hexagon loses symmetry with probability 0.6 under removal of three vertices, while cubes and tetrahedra exhibit maximal robustness\( \ (S_N = 1) \) for all admissible \( \ N \). We further introduce a Shannon entropy of symmetry stability, which quantifies the overall uncertainty of symmetry breaking across all deletion sizes. This framework extends classical symmetry studies by incorporating randomness, linking group theory with probabilistic combinatorics, and suggesting applications ranging from crystallography to defect tolerance in physical systems.