[Mathematical Rigor]The True Nature of the "Finite Respect Principle" in GhostDrift Theory: Formulation via Differential Inclusions and Nagumo's Theorem

Mathematical Formulation of Finite Respect Principle in Interconnected Dynamical Systems

Abstract This paper rigorously describes the mathematical structure of the "Finite Respect Principle," the core concept of GhostDrift Theory. This principle provides the necessary and sufficient conditions for interacting dynamical systems to maintain their respective "Finite Closures." We extend Nagumo's Viability Theorem for non-smooth dynamical systems (Filippov systems) and formulate the Respect Principle as a "diagonally dominant boundary condition," where interference terms between systems are strictly dominated by self-dissipation terms.

1. Introduction: Why Systems Must Discard "Infinity" and Choose "Finite Closure"

In conventional control theory and systems engineering, safety is often treated as "asymptotic stability over infinite time" or as probabilistic expected values. However, GhostDrift Theory starts from the undeniable fact that "reality consists only of finite resources" (computational power, energy, physical space).

The Finite Respect Principle introduced here is not a post-hoc observation but an axiom for Design-by-Contract. It guarantees that a system never deviates from a finite region (referred to as a "Vessel" or Finite Closure).

This paper demonstrates that this principle can be mathematically formulated as a "Viability Condition in interconnected differential inclusions."

2. Preliminaries: Finite Closure and Dynamical Systems

Definition 2.1 (Finite Closure)

A subset $K \subset \mathbb{R}^n$ of the state space $\mathbb{R}^n$ is defined as a Finite Closure if it satisfies the following conditions:

1. Compactness: $K$ is a compact set (bounded and closed).

2. Regularity: $K$ has a smooth (or piecewise smooth) boundary $\partial K$, and its interior $\text{int}(K)$ is non-empty.

Definition 2.2 (Differential Inclusion)

System dynamics are described by the following differential inclusion to allow for discontinuities and uncertainties:

$$\dot{x}(t) \in F(x(t)), \quad x(0) = x_0 \in K$$

Here, $F: \mathbb{R}^n \to 2^{\mathbb{R}^n}$ (a set-valued map) is upper semi-continuous, and for each $x$, $F(x)$ is a non-empty, convex, compact set (Filippov conditions).

3. Formulation of the Finite Respect Principle

Consider a system where multiple agents (or subsystems) $i \in \{1, \dots, N\}$ interact. Let $x_i \in \mathbb{R}^{n_i}$ be the state of each agent and $K_i$ be their respective Finite Closure.

The global dynamics are described as:

$$\dot{x}_i \in F_i(x_i) + \sum_{j \neq i} G_{ij}(x_i, x_j)$$

Where:

• $F_i(x_i)$: The internal dynamics of agent $i$ (Self-preservation or Dissipation).

• $G_{ij}(x_i, x_j)$: The interference from agent $j$ to agent $i$.

3.1. Nagumo's Viability Condition

The necessary and sufficient condition for an isolated system $\dot{x} \in H(x)$ to remain within a closed set $K$ (i.e., to be Viable) is given by Nagumo's Theorem:

$$\forall x \in \partial K, \quad H(x) \cap T_K(x) \neq \emptyset$$

Here, $T_K(x)$ is the Bouligand Tangent Cone to $K$ at $x$. If the boundary is smooth and an outward normal vector $n(x)$ can be defined, this condition simplifies to:

$$\exists v \in H(x) \text{ such that } \langle v, n(x) \rangle \le 0$$

3.2. Derivation of the Respect Condition

In an interconnected system, for agent $i$ to maintain its Finite Closure $K_i$ regardless of the states of others ($j$), the vector field at the boundary must point inward (or be tangential) even under the worst-case interference.

Definition 3.1 (Finite Respect Condition) Agent $i$ satisfies the Finite Respect Principle if, for any boundary point $x_i \in \partial K_i$ and any state of others $x_j \in K_j$, the following inequality holds:

$$\sup_{u \in F_i(x_i)} \langle u, n_i(x_i) \rangle + \sum_{j \neq i} \sup_{w \in G_{ij}(x_i, x_j)} \langle w, n_i(x_i) \rangle \le 0$$

This inequality is the quantitative definition of "Respect" in GhostDrift Theory.

• 1st Term $\langle u, n_i \rangle$: Self-Dissipation. A negative value (inward) generates stability.

• 2nd Term $\langle w, n_i \rangle$: Interference. Can take positive values (outward = violation).

In other words, the mathematical essence of Finite Respect is that "the inward force of self-restraint (Modesty)" must always exceed "the outward force pushed by others (Interference)."

3.3. Gershgorin-type Dominance

The condition above can be interpreted as a nonlinear, set-valued extension of the Gershgorin Circle Theorem (stability of diagonally dominant matrices) in matrix theory.

$$\underbrace{- \alpha_i(x_i)}_{\text{Internal Stability}} + \sum_{j \neq i} \underbrace{\beta_{ij}(x_i, x_j)}_{\text{Interaction Bound}} \le 0$$

Here, $\alpha_i$ represents the system's "good manners" (stabilizing capability), and $\beta_{ij}$ represents the "influence on others." When this inequality holds, it guarantees that the entire system remains eternally within the Cartesian product of individual Finite Closures $K_1 \times \dots \times K_N$.

4. $\Sigma_1$-Verifiability

Beyond mathematical existence proofs, engineering implementation requires "computability."

In GhostDrift Theory, the Finite Respect Principle must be describable as a $\Sigma_1$ sentence (a logical formula consisting only of existential quantifiers $\exists$, or a semi-decidable predicate).

$$\text{Verify}(x) \iff \text{Check}(\langle F(x), n(x) \rangle \le -\epsilon)$$

By defining the Finite Closure $K$ as a Polytope or a set compatible with interval arithmetic, a violation of the Respect Condition becomes detectable in finite steps (Refutable), and conversely, the state satisfying the condition is constructively maintained.

5. Conclusion

The Finite Respect Principle is not a moral norm but a geometric and algebraic constraint required for dynamical systems to coexist without collapse.

It does not assume an "infinite canvas." Instead, it assumes mutually exclusive "finite vessels." By keeping the flux balance strictly negative (inward) at the boundaries, it is a mathematical model where Global Harmony emerges solely from Local Contracts.

Reference: GhostDrift Research Institute, "Finite Respect Principle Mathematical Notes", 2025.