6.1 Topological Survey of Three-Field Coupling

Abstract

This article surveys the topological structure of the ℳ𝒞ℐ three-sector joint coupling. The three-field coupling is realized through three permutation components of the Birkhoff mixing matrix : identity (ℳ self-coupling), 3-cycle (ℳ-𝒞-ℐ waist-edge), and inverse 3-cycle (ℳ-ℐ-𝒞 reverse cycle). The joint cross-section constitutes the complete topology of the three-field coupling. This article further presents the complete construction of the three-field coupling matrix, the algebraic closure of inter-sector coupling channels, a systematic analysis of topological invariants, and the hierarchical origin of the four interactions within a unified topological picture.