This version of the work was completed with artificial intelligence assistance and deductive reasoning. We have corrected the error in Section 6.3 Multi-Value Configurations are Non-Generic of Chapter 6 in the previous version, along with certain errors in Chapter 5. While these errors do not affect the final conclusions, they render the mathematical proofs non-rigorous and are thus deemed fatal in a strict sense. In this version, through research on the normal and tangential decoupling of separable functions, we have derived the relational expressions for a broad class of functional families, thereby rigorously proving the conclusion. For the corrected version, please download or preview the files prefixed with fix; files prefixed with old correspond to the previous version. --- We study a constrained optimization problem on the probability simplex \Delta^{n-1}. By postulating a phenomenological Radial Dynamics \dot{Y}=-Y(R-c{2}) that freezes directional degrees of freedom, we reduce the problem to static classification on the shell manifold \mathcal{S}. The radial equation \dot{R}=-R(R-c{2}) generates convergence to radius c_2 under directional freezing (\dot{Z}=0). Geometric analysis on the (n-2)-dimensional shell manifold \mathcal{S} reveals that the fourth-moment functional K(Z)=\sum Z{i}^{4} attains its maximum at the 1+(n-1) configuration. We further prove (Supplement, Section 6) that the two-value structure is universal: for any S{n}-invariant real-analytic functional admitting non-degenerate critical points, stable maxima necessarily exhibit a two-value configuration of type k+(n-k) (where the specific partition k depends on the functional form). The proof establishes an explicit Universal Hessian Relation separating normal and tangential stability conditions, and employs finite-dimensional parameter space analysis rather than infinite-dimensional transversity arguments. This establishes a static classification framework on the stable manifold using fourth-moment extremal structure and symmetry constraints.