Title: Schur-Convex Curvature on Dihedral Exponential Families and the Golden-Ratio Stationary Point Show Chinese title

Title: 二面体指数族上的Schur-凸曲率和黄金比例平稳点

Abstract: We investigate the Schur-complement curvature of D_N-equivariant folded exponential families on the simplex. Our main structural results are: (i) the curvature kappa_Schur(theta) is convex in the log-parameter theta = ln(q); (ii) it admits a unique stationary point at the golden ratio value q* = phi^-2 (in particular for N = 12); and (iii) it obeys a quadratic folded law kappa_Schur = A(N, m_rho^2) I_1^2 + B(N, m_rho^2) (I_2 - I_1^2), with coefficients A, B determined explicitly by the projector metric of radius m_rho^2. Taken together, these results show that convexity and symmetry alone enforce both the location and the functional form of the "golden lock-in." Beyond their intrinsic interest, these findings identify D_12 as the minimal dihedral lattice where parity (mod 2) and three-cycle (mod 3) constraints coexist, producing a structurally stable equilibrium at the golden ratio. This places the golden ratio not as an accident of parameterization but as a necessary consequence of convex geometry under dihedral symmetry. Possible applications include harmonic analysis on group orbits, invariant convex optimization, and the structure of tilings or quasicrystal-like systems. Show Chinese abstract

Abstract: 我们研究了单纯形上D_N-等变折叠指数族的Schur补曲率。 我们的主要结构结果是：(i) 曲率kappa_Schur(theta)在对数参数theta = ln(q)下是凸的；(ii) 在黄金分割值q* = phi^-2处有一个唯一的驻点（特别是当N = 12时）；(iii) 它遵循二次折叠定律kappa_Schur = A(N, m_rho^2) I_1^2 + B(N, m_rho^2) (I_2 - I_1^2)，其中系数A, B由半径m_rho^2的投影仪度量显式确定。 综上所述，这些结果表明，仅凸性和对称性就强制了“黄金锁定”位置和函数形式。 除了它们本身的兴趣外，这些发现确定了D_12作为最小二面体格子，在其中奇偶性（模2）和三循环（模3）约束共存，从而在黄金比例处产生结构稳定的平衡。 这将黄金比例不是作为参数化的偶然现象，而是作为二面体对称下凸几何的必然结果。 可能的应用包括群轨道上的调和分析、不变凸优化以及铺砌或准晶体类似系统的结构。