Formulation of Monopole Decoupling in Ferromagnetic Media and Indirect Verification via Non-Associative Sagnac Phase Beats

This study establishes a novel topological and algebraic solution to the long-standing monopole problem by constructing an invariant shielding mechanism over the non-associative geometry of octonions (O). Conventional detection methodologies presuppose that magnetic monopoles interact via standard second-order associative Clifford manifolds. In contrast, the fundamental magnetic charge is rigorously formulated herein as a third-order non-associative associator. Under this mathematical framework, the complete decoupling of the magnetic current Jmag from the local electronic current Jelec inside high-permeability ferromagnetic media is analytically derived, providing a rigorous explanation for absolute non-interactive permeability and observational invariance. Furthermore, while direct laboratory creation undergoes heavy suppressions via the Planck mass Mp at high-energy scattering boundaries, the intrinsic signatures of these non-associative structures remain encoded as irreducible geometric constraints on the invariant manifolds of collisionless dark matter. An indirect yet decisive verification model is established utilizing multi-axial closed-loop closed photon pathways over invariant manifolds, demonstrating that the non-associativity of path ordering yields a non-vanishing residual Sagnac phase beat ΔΦSagnac. This provides a complete, self-contained empirical verification framework that bypasses the intrinsic limitations of associative gauge measurements.