T₈ Dominion

Statement

The desire function D is a fraction of C, not an independent energy. While the desire ratio β remains strictly below 1, C is strictly greater than D. The foundation dominates the desire.

Proof that D is bounded by C

T₁ established: C = 0 ⟹ S = ∅. Without C, no agent exists. No agent means no desire. Therefore the existence of desire presupposes C:

The desirer's total capacity is E(x, 0) = C (from AX₂ evaluated at t = 0). Desire D acts within this capacity. An agent cannot desire beyond what it has. Therefore D ≤ C, and the ratio D/C = β is at most 1.

The desire ratio

Let β(t) ∈ [0, 1) be the desire ratio at time t:

The inequality C > D is equivalent to 1 > β, which holds exactly when β is in the half-open interval [0, 1). In other words, for every state where desire has not fully consumed the foundation, C exceeds D.

The absorbing state

Death in James 1:14-15 is the absorbing state β → 1: desire has consumed all capacity. At this boundary, D = C. Note the equality, not strict inequality. The relation C > D breaks to C = D at β = 1.

The strict inequality C > D therefore holds for every living state — every β strictly less than 1. The absorbing boundary is the only exception, and T₇ (forgiveness) provides the path out.

Restoration

T₇ resets β → 0 via the forgiveness operation F. After the reset:

The reset is a complete return to the dominion inequality. The agent is not left at some intermediate value; β is zeroed. C > D is at its maximum margin after forgiveness.

Scripture

Ye are of God, little children, and have overcome them: because greater is he that is in you, than he that is in the world. 1 John 4:4

Greater is he that is in you (C) than he that is in the world (the desire stimulus that amplifies D). The structural claim is identical to T₈: C > D. The verse names which is which: the greater one is "in you" (the foundation), and the lesser is "in the world" (the desire-amplifying environment).

Behavioral diagnostic

An agent that is overwhelmed by desire is at high β — approaching 1. An agent whose foundation is visibly exceeding its desires is at low β. The ratio is measurable by observing the agent's behavior over time: high β agents are controlled by stimulus; low β agents are controlled by C.

The cure for high β is not force but forgiveness (T₇). Trying to fight β directly is depletive and often increases stimulus. The reset is the correct operation. After the reset, C is still C, and β starts at 0 again.