10.6 Irreducibility of the Three Realms

Within the rigorous ℳ𝒞ℐ three-sector framework, we prove the indecomposability of sector coupling. The core tools are the spectral gap direction spectral domination inequality and the Courant-Fischer-Weyl min-max principle. All numerical values arise from algebraic operations of the unique axiom δ at the Birkhoff fixed point σ*, requiring no external parameters. **Keywords:** ℳ𝒞ℐ three-sector; sector coupling; spectral gap direction spectral domination; min-max principle; Schur complement; symmetry; indecomposability theorem; Consummate Reproduction; proposition (interlocking)