Mirror Symmetry in Asymmetric Equipartition: The λ ↔ 1/λ Involution on Thales Triangles

This paper establishes an exact geometric characterization of asymmetric equipartition in a two-channel normalized variance system. Using Thales right triangles inscribed in a semicircle, we analyze the variance ratio \lambda = x/y under the constraint x+y=1 and demonstrate that the involution \lambda \leftrightarrow 1/\lambda corresponds to a mirror symmetry exchanging the triangle’s legs while preserving its altitude. We prove that the configurations at \lambda = e and \lambda = 1/e form a mirror pair with identical altitude h = e/(1+e) when the semicircle diameter is chosen as D=\sqrt{e}. Each configuration exhibits asymmetric channel splits of approximately 73.1\%/26.9\%, yet averaging over the mirror pair restores exact 50\%/50\% equipartition. The asymmetry therefore does not represent a violation of symmetry but rather a pairwise manifestation of a symmetric structure. The corridor [1/e, e] is shown to be symmetric about \lambda=1 in logarithmic coordinates, with the symmetric point acting as an unstable configuration geometrically centered between the two asymmetric mirrors. All results follow from exact Euclidean identities and require no empirical input or dynamical assumptions. The analysis provides a structural diagnostic framework for understanding how asymmetric equipartition can coexist with exact symmetry at the ensemble level.