The Abebe Parity Identity I

The Abebe Parity Identity I establishes that redistributive bound systems lose stability when the surviving structure must carry twice its baseline load (A_c = 2). This paper consolidates the identity into a unified framework through: • first-principles mean-field and bifurcation derivations, • a class-level theorem valid for any redistributive system, • a robustness result under perturbations, • and exact physical realizations in fiber-bundle fracture, ductile necking, and network cascades. It demonstrates that A_c = 2 arises purely from the geometry of load redistribution and remains the universal leading-order law independent of the underlying physical substrate. The commonly observed macroscopic threshold…