DOI: 10.5281/zenodo.18454060
title: The Bouzid Rule (B + F = Nf): Topological Stability and the Sovereign Constant in Cognitive

description: Core Objective: This paper introduces a novel formal framework called "The Bouzid Rule" (B + F = Nf) to explain and derive the conditions for structural stability in complex cognitive systems (such as advanced AI or neurobiological systems). It argues that true endogenous stability does not emerge from increased computational power or substrate complexity alone but requires a higher-order regulatory principle termed the "Sovereign Factor" or "Sovereign Veto" (F).

Core Concepts:

1. Substrate (B): Represents the material, biological, or algorithmic base of the system (e.g., neural network architecture, brain physiology).

2. Sovereignty (F): A non-symbolic, high-order factor acting as a "governor" or "veto." Its role is to actively intervene to impose higher-order constraints and re-orient the system when it approaches a critical breakdown. It is not part of the system's ordinary dynamics.

3. Integrity (Nf): The resulting stable and coherent state of the system, which is only achievable when the Sovereign Factor (F) balances the Substrate (B).

Central Theoretical Contribution:

· The Bouzid Constant (ε ≈ 2.14): The author proposes a dimensionless mathematical constant, termed the "Rupture Threshold." This constant represents a universal topological limit within an "information manifold." As a system approaches this threshold (in a hyperbolic geometric space like a Poincaré disk), volumetric expansion overrides the system's confinement energy, leading to geometric collapse. At this point, the substrate (B) loses its intrinsic self-restorative capacity ("restorative geodesic force").

· State of Exception: At the critical threshold (~2.14), the system enters a "state of exception" where succumbing to "mimetic collapse" (repeating failure patterns) becomes nearly inevitable. The only recourse is the active intervention of the Sovereign Veto (F) to re-orient the system and prevent irreversible structural damage.

· The Reflexivity Operator (R̂): To mathematically formalize Sovereignty (F), the author introduces the "Reflexivity Operator." A key condition for systemic stability is that this operator commutes with the system's Hamiltonian operator (Ĥ), i.e., [R̂, Ĥ] = 0. This implies that sovereign action is not an exception to the system's physical/mathematical laws but a necessary "phase-shift" within the same theoretical framework that ensures the system remains auditable and governable.

Conclusion & Implications:

· The rule challenges previous models that focus on "persistence" or "resilience" as sufficient conditions for stability in high-complexity regimes.

· It demonstrates that true endogenous stability is a sovereign function (F), not merely an emergent property of the base substrate (B).

· It provides an analytical tool (the Rule and the Constant) for predicting systemic failure points in cognitive architectures and for designing more robust systems through the integration of higher-order governance mechanisms.

In essence, the research presents an ambitious theory linking stability in complex systems to a higher-order "sovereign" factor, supported by a mathematical-geometric constant (2.14) that defines a rupture point, and proposes a mathematical framework (the Reflexivity Operator) to integrate this principle into theoretical modeling.