Analytical Supplement: Boundary Stability and the Derivation of σ = (1/4πG)√(40/7) in the IKBL Framework

This technical supplement provides the formal derivation of the junction stability conditions within the Iterated Klein Bottle Lattice (IKBL) framework. By utilizing the Nieh-Yan boundary action and Israel junction conditions, we identify the mechanical scaling relation σ = (1/4πG)√(40/7) as the unique solution required to satisfy the zero-stress boundary condition (S_ab = 0) on a non-orientable hypersurface. The derivation proves that the value χ = 40/7 is a topological selection rule necessitated by C₇ holonomy and the requirement of global gauge stability. This result provides the formal mathematical anchor for the 5.7σ cosmic dipole anomaly, demonstrating that the observed anisotropy is a structural requirement for a stable, non-orientable manifold. This supplement serves as the definitive technical companion to the primary manuscripts regarding the IKBL framework.