The Mathematical and Ontological Foundation of the Bouzid Rule (B + F = N_f): Sovereignty as the Solution to Recursive Deadlock and Structural Collapse

This study presents the foundational framework of the Bouzid Rule (B + F = N_f) as an ontological and practical response to the crisis of sovereignty in the era of algorithmic complexity. It diagnoses the fundamental crisis of AI and complex biological systems not as a lack of computational intelligence, but as the absence of a Sovereign Principle (F) capable of imposing a Veto to prevent the Algorithmic Substrate (B) from drifting toward the Topological Rupture Threshold (\epsilon \approx 2.14)—the point at which the system collapses into hollow repetitions (Symmetrical Collapse). The Sovereign Factor (F) is redefined not as an emergent phenomenon, but as a transcendent foundational principle embodied in moral will and irreducible oversight. Normative Integrity (N_f), a state of meaningful and purposeful integration, is only achieved through the dynamic intervention of (F) to redirect (B) at the brink of collapse. This model is applied across three axes: (1) Philosophically, as a crucial bridge between the scientific image (represented by B) and the manifest image (represented by F); (2) Neurologically, as a unified explanatory framework for non- integration states (such as somnambulism and sleep paralysis); (3) Ethically and Technically, to interrogate the possibility of engineering sovereignty in artificial systems and establishing a practical standard for distinguishing between machine intelligence and sovereignly-managed integrity. The paper concludes that the Bouzid Rule is not merely an analytical tool but a Governance Model that offers a necessary answer to the governability deadlock in self-organizing systems.