Symmetry Constraints and Two-Value Rigidity

This version of the work was completed with the assistance of AI tools. --- We investigate the structure of stable extrema of permutation-invariant analytic functionals under global constraints, and their manifestation in a standard physical model. Building on the Universal Two-Value Rigidity Theorem established in Static Classification on Constrained Configuration Spaces(Chen, 2026), we show that, under generic (non-degenerate) conditions, stable extrema on a constrained shell manifold tend to collapse into two-valueconfigurations of the form (k + (n − k)).As a structural illustration, we consider the spherical antiferromagneticmean-field model. We emphasize a crucial distinction: when only quadraticexchange interactions are present, the energy landscape is completely degenerate on the constrained manifold, and the assumptions of the rigidity theoremdo not apply. Once a symmetric higher-order analytic term is introduced,degeneracy is lifted and isolated stable extrema emerge. These extrema exhibit a characteristic two-value (two-sublattice) structure, consistent with thegeneral rigidity mechanism.This note is intended as a structural companion to the main paper ratherthan an independent derivation of physical models. It aims to clarify howcommonly observed two-sublattice tendencies in mean-field antiferromagneticsystems can be understood as a generic consequence of symmetry, constraints,and non-degeneracy, rather than a result of specific Hamiltonian details.