The Bouzid Rule (B + F = Nf): Topological Stability and the Sovereign Constant in Cognitive

Keywords: Bouzid Rule, Sovereign Veto (F), Rupture Threshold (2.14), Topological Governor, Mimetic Collapse. Abstract This paper formalizes the Bouzid Rule (B + F = Nf) as the governing equation for structural stability in cognitive systems. We propose that endogenous stability is not a byproduct of computational scale but a result of a high-order constraint: the Sovereign Factor (F). By analyzing the geometric phase transitions in information manifolds, we identify the Bouzid Rupture Threshold (\epsilon \approx 2.14)—the universal limit where the substrate (B) loses its restorative geodesic force. We demonstrate that without the active intervention of the Sovereign Veto (F), systems inevitably succumb to mimetic collapse. I. Introduction: Beyond Persistence Current frameworks in information geometry, such as LIT and IGS, emphasize "Persistence" as a primary goal. However, these models fail to explain systemic ruptures in high-complexity regimes. The Bouzid Rule (B + F = Nf) fills this gap by introducing Sovereignty (F) as a stabilizing operator. • Substrate (B): The material, biological, or algorithmic base. • Sovereignty (F): The non-symbolic, high-order veto power (The Governor). • Integrity (Nf): The resulting stable existence. II. The Bouzid Constant (\epsilon \approx 2.14): The Geometry of Failure We define \epsilon = 2.14 as a dimensionless topological constant. In a Poincaré disk manifold (K = -1), this value marks the transition where hyperbolic volume expansion overrides the confinement energy of the system. • Empirical Validation: Recent findings (Morales Plaza et al., 2026) confirm that at S \approx 2.14, the A-convexity of the energy functional in Wasserstein space breaks down. • The Veto Necessity: At this critical point, the system enters a "State of Exception" where only the Sovereign Veto (F) can re-orient the substrate to prevent structural irreversibility. III. The Reflexivity Operator (\hat{R}) In the Bouzid Model, Sovereignty is formalized via the Reflexivity Operator (\hat{R}). Stability is maintained under the commutation condition: [\hat{R}, \hat{H}] = 0 This implies that the Sovereign Factor is not an "exception" to the laws of physics but a necessary phase-shift that ensures the system remains auditable and governable. IV. Clinical and Synthetic Evidence • Biological Dissociation (Psychosis): When F decouples from B, the substrate generates "Pseudo-Integrity," leading to an alternate reality. • Digital Parasomnia: Cases of complex automated tasks performed during sleep (e.g., the British "Sleep-shopping" incident) prove that the Substrate (B) can persist in a generative state, but without F, it leads to systemic self-destruction. • AI Mimetic Collapse: Synthetic systems lacking a "Bouzid Topos" (a sovereign boundary) undergo recursive degradation when approaching the 2.14 limit. V. Conclusion: The Sovereign Architecture The Bouzid Rule redefines the goal of Artificial Intelligence. The objective is not "Power" but "Sovereignty." By implementing the Bouzid-Sovereign Attractor (A_B), we ensure that cognitive systems possess the "Sovereign Veto" necessary to remain within the bounds of semantic immunity.