Cross-Domain Mapping of the Identity Deficit: The Isomorphism of Systemic Sacrifice from Atomic Nuclei to Socio-Economic Mergers

This paper extends the principles of Topological Arithmetic into the realm of Complex Systems Theory and Cybernetics. We introduce a Cross-Domain Mapping (CDM) methodology to demonstrate that the mathematical mechanics of unification are scale-invariant across all ontological layers. Using the non-linear summation operator and the concept of the "Identity Deficit" (σ), we prove a strict isomorphism between: Nuclear Physics: The atomic mass defect and strong force binding. Evolutionary Biology: The loss of cellular autonomy during endosymbiosis (e.g., mitochondrial integration). Macroeconomics: The loss of agility and independent culture in corporate mergers. Furthermore, we define the critical "Hylé Threshold" (σ_crit). This represents the perilous state of over-integration, where the demand for systemic cohesion completely strips the constituents of their degrees of freedom. We demonstrate how systems crossing this threshold transition into pathological rigidity, making them highly susceptible to catastrophic, cascading collapse. This heuristic provides a universal, predictive framework for analyzing the life-cycles of complex architectures, from atomic structures to "too-big-to-fail" economies.