# Recursive Expansion in Intelligence-Energy-Monetary Systems

### Definition 1. (State Space)

Let $\mathbb{R}_{+}$ be the set of non-negative real numbers and $\mathbb{T} = \{0, 1, 2, \dots\}$ be a discrete time set. We define the state space $\Omega$ as a subset of the Euclidean space $\mathbb{R}^4$:

$$
\Omega = \mathcal{I} \times \mathcal{E} \times \mathcal{M} \times \mathcal{H} \subset \mathbb{R}_{+}^4
$$

where the components of a state vector $\mathbf{x}_t = (i_t, e_t, m_t, h_t) \in \Omega$ at time $t \in \mathbb{T}$ denote:

* $i_t$: Index of Intelligence Capability.
* $e_t$: Energy Production Capacity.
* $m_t$: Monetary Supply.
* $h_t$: Hashing Power.

---

### Definition 2. (Structural Morphisms)

We define a set of functions $\mathcal{F} = \{ \phi, \psi, \xi, \zeta \}$, representing the causal dependencies within the system.

1. **Credit Creation Map** $\phi: \mathcal{I} \to \mathcal{M}$
Maps the current intelligence capability to monetary supply expansion. We assume $\phi$ is strictly increasing and convex (reflecting the accelerating valuation of future productivity).
$$
\forall i_1, i_2 \in \mathcal{I}, \quad i_1 < i_2 \implies \phi(i_1) < \phi(i_2)
$$

2. **Capability Investment Map** $\psi: \mathcal{M} \to \mathcal{I}$
Maps the monetary liquidity to the enhancement of intelligence capability.
$$
\forall m_1, m_2 \in \mathcal{M}, \quad m_1 < m_2 \implies \psi(m_1) < \psi(m_2)
$$

3. **Energy Yield Map** $\xi: \mathcal{I} \to \mathcal{E}$
Maps intelligence capability to energy production capacity.
$$
\forall i_1, i_2 \in \mathcal{I}, \quad i_1 < i_2 \implies \xi(i_1) < \xi(i_2)
$$

4. **Arbitrage Map** $\zeta: \mathcal{E} \times \mathcal{M} \to \mathcal{H}$
Maps the joint state of energy capacity and monetary supply to hashing power. $\zeta$ is strictly increasing in each argument.
$$
\forall e, e' \in \mathcal{E}, m \in \mathcal{M}: \quad e < e' \implies \zeta(e, m) < \zeta(e', m)
$$
$$
\forall e \in \mathcal{E}, m, m' \in \mathcal{M}: \quad m < m' \implies \zeta(e, m) < \zeta(e, m')
$$

---

### Definition 3. (System Dynamics)

The time evolution of the system is governed by the recurrence relation $\Phi: \Omega \to \Omega$, defined sequentially for $t \in \mathbb{T}$:

$$
\begin{cases}
m_{t+1} = \phi(i_t) \\
i_{t+1} = \psi(m_{t+1}) \\
e_{t+1} = \xi(i_{t+1}) \\
h_{t+1} = \zeta(e_{t+1}, m_{t+1})
\end{cases}
$$

By substitution, the dynamics of the variable $i$ can be isolated as a one-dimensional autonomous map:

$$
i_{t+1} = \psi(\phi(i_t))
$$

---

### Theorem 1. (Unbounded Divergence)

**Assumption:**
Let the composite function $f = \psi \circ \phi$ satisfy the super-linearity condition (representing the Law of Accelerating Returns):
$$
\exists K > 1 \text{ such that } \forall x > 0, f(x) \ge Kx
$$

**Proposition:**
Given an initial state $\mathbf{x}_0$ with strictly positive components, the sequence of state vectors $\{\mathbf{x}_t\}_{t=0}^{\infty}$ diverges to infinity in all components.

$$
\lim_{t \to \infty} \| \mathbf{x}_t \| = \infty
$$

**Proof:**
From the recurrence relation, we have $i_{t+1} \ge K i_t$. By induction, $i_t \ge K^t i_0$.
Since $K > 1$ and $i_0 > 0$, it follows that:
$$
\lim_{t \to \infty} i_t = \infty
$$

Since $\xi, \phi, \psi$, and $\zeta$ are strictly increasing functions (Def 2), the divergence of $i_t$ necessitates the divergence of all dependent variables:
$$
\lim_{t \to \infty} e_t = \lim_{t \to \infty} \xi(i_t) = \infty
$$
$$
\lim_{t \to \infty} m_t = \lim_{t \to \infty} \phi(i_{t-1}) = \infty
$$
$$
\lim_{t \to \infty} h_t = \lim_{t \to \infty} \zeta(e_t, m_t) = \infty
$$
Q.E.D.