Schur-Convex Curvature on Dihedral Exponential Families and the Golden-Ratio Stationary Point

TL;DR

This paper explores the curvature properties of dihedral exponential families, revealing a unique golden ratio stationary point and a quadratic law governing the curvature, with implications for symmetry and convex geometry.

Contribution

It establishes the convexity and symmetry-driven determination of the golden ratio stationary point in dihedral exponential families, providing explicit formulas and structural insights.

Findings

Curvature is convex in the log-parameter.

Unique stationary point at the golden ratio for certain parameters.

Quadratic law relating curvature to projector metric coefficients.

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.