Theorem: Good is independent of evil

• A3: Strutural Integrity/Symmetry: States exist independently; integrity must be preserved. Structurally equivalent contexts are treated equivalently.
• A5: Non-Coercion: No entity may impose necessity onto others.

• Good \(G\): A state/action that preserves or enhances structural integrity without destabilization. Operation of the Minimal Positive Function(see: Ethics(6))
• Evil \(E\): A state/action that reduces structural integrity or causes destabilization

We want to prove:

\begin{equation} \forall S \text{structurally stable}, \exists G \subset S \wedge G \neq f(E) \end{equation}

read: In any stable system \(S\), there xists a good \(G\) whose existence does not depend on any evil \(E\)

1. let \(S\) be a system satisfying A3 (structural integrity preserved). By A3, any state \(X\) ∈ \(S\) has intrinsic exstence; it does not require contrast to exist

2. Define \(G\) ∈ \(S\) as a positive structural state:

   \begin{equation} G := \text{State such that \(G\) preserves or enhances integrity of \(S\)} \end{equation}

   By A3, the existence of \(G\) is intrinsic; it is not contingent on any other state \(Y\).

3. Suppose for contradiction that \(G\) requires evil \(E\) to exist:

   \begin{equation} G = f(E) \end{equation}

   Then, the structural integrity of \(S\) would be contingent on destabilaztion

4. Contradiction: By definition, \(G\) enhances integrity, while \(E\) reduces integrity. A function f(E) would require a destabilizing input to produce a stabilizing output. This violates A3 and A5.

5. Conclusion:

   \begin{equation} G \neq f(E) \implies G \text{exists independently of} E \end{equation}