Appendix B: Open Universe Dynamics — Mathematical Models of Capacity Growth and Red Queen Games

This appendix provides dynamical proofs for Chapter 9 “Rejecting Heat Death” and Chapter 12 “No Omega Point.” We will construct coupled equations describing the evolution of cosmic capacity, entropy, and agent complexity over time.

B.1 Dynamic Bekenstein Bound

Consider an expanding universe dominated by dark energy $\Lambda$.

Although the event horizon is fixed, for computational cosmology, what matters more is the number of computational nodes (QCA lattice points) within comoving volume $N(t)$.

In QCA expansion model (node addition model):

$$N(t)\propto a(t{)}^3\propto e^{3Ht}$$

where $H=\sqrt{\Lambda /3}$ is the Hubble constant.

The universe’s potential information capacity upper limit $I_{max}(t)$ grows with $N(t)$:

$$I_{max}(t)\sim N(t)\sim e^{3Ht}$$

B.2 Limits on Entropy Production

The universe’s used information (entropy) $S(t)$ mainly comes from matter particle decoherence and black hole radiation.

Since total matter is conserved (or grows slower than volume), and computation processes are limited by light speed (Bremermann Limit):

$$\frac{dS}{dt}\le \frac{E_{total}}{\hslash }\cdot N_{particles}$$

In a sparse universe, $S(t)$ growth is at most linear or polynomial, far below $I_{max}(t)$’s exponential growth.

Corollary B.1 (Infinite Blank Theorem):

$$\underset{t\to \infty }{\lim }\frac{S(t)}{I_{max}(t)}=0$$

The universe’s Filling Ratio tends to zero. This means the universe is forever “new,” and heat death ($S\to I_{max}$) is mathematically impossible.

B.3 Red Queen Game Equations

We use a variant of Lotka-Volterra equations to describe the game between agents ($A$) and environmental entropy ($E$).

• Agent complexity ($C$): Represents $M_I$.

• Environmental entropy pressure ($P$): Represents survival difficulty.

$$\{\begin{array}{ll}\frac{dC}{dt}=\alpha C\cdot P-\beta C^2& (\text{pressure drives evolution, but limited by metabolic cost})\\ \frac{dP}{dt}=\gamma C-\delta P& (\text{intelligent activity produces waste heat increasing pressure, expansion dilutes pressure})\end{array}$$

• $\alpha CP$: Red Queen term. The harsher the environment ($P$ large), the faster agents evolve to survive.

• $\gamma C$: Landauer waste heat term. Higher intelligence emits more entropy, pushing $P$ higher.

• $-\delta P$: Cosmic expansion term. Rapid space expansion dilutes waste heat.

Phase Plane Analysis:

When $\delta$ (expansion rate) is sufficiently large, this system has no stable fixed point (death), but forms Limit Cycles or eternally diverging spirals.

Conclusion:

As long as cosmic expansion ($\delta >0$) can timely remove waste heat, agents can achieve infinite exponential growth in complexity through continuous arms races ($\alpha ,\gamma$ cycles).

Civilization’s immortality is built on the foundation of cosmic expansion.