Exact Finite-Coupling Classification of a Four-State Entropy–Log-Gap Energy

∑ᵢ₌₁⁴λ_ilogλ_i+β∑1≤ i<j≤4(λ_i-λ_j)^2log(λ_i-λ_j)^2]on the probability simplex (Δ_3). For every (0.03≤β≤4), we prove a complete stationary-point classification up to permutation. There are exactly eight stationary (S_4)-orbits, corresponding to the eight ordered compositions[(4),\ (3,1),\ (2,2),\ (1,3),\ (2,1,1),\ (1,2,1),\ (1,1,2),\ (1,1,1,1).]Their orbit sizes are (1,4,6,4,12,12,12,24), giving exactly 75 interior stationary points. The unique fully distinct orbit is also the unique minimizing (S_4)-orbit, so the global minimum is attained at exactly 24 permutation-related points. The proof combines analytic reduction with computer-assisted rigorous certification. The analytic part establishes stationary compactness, classifies the lower-dimensional composition strata, derives a reduced fully distinct stationary system, and relates fixed-coupling degeneracy to reduced Jacobian conditions. The computer-assisted part uses outward-rounded MPFR interval arithmetic and Krawczyk methods to certify the fully distinct continuum branch, exclude interior degeneracies, and combine this with an exact base root count at (β=1). The classification theorem is restricted to (0.03≤β≤4). No complete classification for all (β>0) is claimed.