Strong-Coupling Boundary Selection and Super-Exponential Splitting

Abstract

We study the strong-coupling regime of the four-state entropy–log-gap energy V4,β(λ)=i=1∑4λilogλi+β1≤i<j≤4∑(λi−λj)2log(λi−λj)2 on the probability simplex Δ3={λi≥0,i=1∑4λi=1}. The analysis identifies an explicit strong-coupling boundary selector and reveals a hierarchy of two qualitatively different asymptotic scales. For the pure logarithmic-gap functional G4(λ)=i<j∑(λi−λj)2log(λi−λj)2, we prove that its complete global-minimizer set is Δ3argminG4=S4⋅(x∗,1−x∗,0,0), where x∗=0.637081537594127… is the unique nontrivial solution of an explicit scalar stationarity equation. Consequently, every ordered global minimizer of V4,β converges to this two-state boundary configuration as β→∞. Writing the two small coordinates as sβ+δβ,sβ−δβ, we derive the first asymptotic scale sβ=Ae−κβ(1+o(1)), with explicit constants κ=1.240631289043068…,A=0.408684427362295…. The internal splitting of these already exponentially small coordinates occurs on a much smaller scale: δβ=Bexp[−8βsβ1](1+o(1)), where B=0.140727508773597…. Therefore, logδβ1∼8Aβeκβ, showing that the symmetry-resolving splitting is super-exponentially small relative to the coupling parameter. The resulting hierarchy is O(1)⟶e−κβ⟶exp[−8βsβ1], corresponding respectively to macroscopic boundary selection, exponential collapse of two coordinates, and super-exponential resolution of their residual degeneracy. We also obtain the first strong-coupling correction to the macroscopic state, xβ=x∗+βc+o(β−1), and an asymptotic expansion for the minimum energy. The paper provides an analytic strong-coupling counterpart to the weak-coupling Chebyshev selection mechanism established for the same entropy–log-gap family. Numerical high-precision calculations are included only as independent checks of the analytic asymptotics. The complete finite-β stationary-point classification for n=4, including possible bifurcations and uniqueness of the global minimizing orbit for all β>0, remains open.