Appendix B: Thermodynamic Proof of Infinite Games

In Volume V “Rejection: The Infinite Game” of the main text, we proposed a seemingly counterintuitive claim that violates classical physical common sense: The universe’s evolution has no endpoint, and civilizations can play an infinite game that never ends.

The greatest challenge to this view comes from the Second Law of Thermodynamics. The classical “Heat Death” theory predicts that as entropy continuously increases, the universe will eventually reach a uniform, disordered, work-incapable thermal equilibrium state. If this is true, then any game—no matter how brilliant—will eventually be forced to end due to energy exhaustion.

This appendix will derive a counterintuitive conclusion based on Non-equilibrium Thermodynamics and FS geometry’s inflation model: Under conditions of dimensional inflation, heat death can be Indefinitely Postponed.

B.1 Entropy Flow Equation for Open Systems

First, we must distinguish between isolated systems and open systems.

For isolated systems, entropy change $dS\ge 0$, and heat death is inevitable.

But life and civilization are typical Dissipative Structures, whose entropy change follows the Prigogine Equation:

[dS = d_i S + d_e S]

Where:

• $d_{i}S$: Entropy produced internally due to irreversible processes (such as metabolism, computation, friction). According to the second law, $d_{i}S\ge 0$. This is the thrust of death.

• $d_{e}S$: Entropy flow exchanged with the external environment.

To maintain the system’s high order (i.e., keep $S_{sys}$ below the maximum, or even decrease it), the negentropy criterion must be satisfied:

[d_e S < 0 \quad \text{and} \quad |d_e S| \ge d_i S]

This means: The system must expel chaos to the environment faster than it creates chaos.

The question is: Can the environment infinitely receive this chaos? In the classical model, the environment is also finite, and will eventually saturate, causing backflow, and the system will still die.

B.2 Inflation as a Source of Negentropy

In the spiral model of Vector Cosmology, this deadlock is resolved by the Red Queen’s Run.

The universe is not a static box, but a manifold experiencing dimensional inflation.

According to the holographic principle and evolution equation, the universe’s total phase space volume (i.e., the environment’s maximum entropy capacity $S_{max}$) grows exponentially with light speed $c(\tau )$:

[S_{max}(\tau) \propto \text{Area}(\tau) \propto c(\tau)^2 \propto e^{2k\tau}]

(Note: Here $k=\frac{\ln \varphi }{\pi }$ is the growth rate).

This reveals an astonishing thermodynamic fact: The environment’s “trash can” is expanding at an exponential rate.

Assume the civilization’s entropy production rate is ${\dot{S}}_{civilization}$.

Assume the environment’s entropy absorption capacity expansion rate is ${\dot{S}}_{max}$.

As long as the cosmological version of the “Fredkin-Toffoli Inequality” is satisfied:

[\frac{d}{d\tau} S_{max}(\tau) > \frac{d}{d\tau} S_{total_produced}(\tau)]

The environment will never saturate.

The waste heat we emit ($v_{env}$) is rapidly diluted in that frantically expanding Hilbert space void. For local observers, the background temperature $T_{bg}$ will always tend toward zero (or remain at a very low steady state), thus ensuring the perpetuity of the negentropy flow.

B.3 Geometric Condition for Sustainability

We further translate this into the language of FS geometry.

The necessary and sufficient condition for infinite games to proceed is: The growth rate of $c_{FS}$ must exceed the dissipation rate of system complexity.

[\lim_{\tau \to \infty} \frac{\dot{c}{FS}(\tau)}{c{FS}(\tau)} = \lambda > 0]

As long as $e$ (generator) is still working, as long as $\varphi$ (spiral) is still unfolding, $\lambda$ is always greater than zero.

This means the universe is always in an “Under-saturated” state.

The potential difference between “order” and “disorder” will never disappear.

Physical Conclusion:

Heat death is the fate of closed circles ($\pi$).

But in open spirals ($\varphi$), heat death is a horizon that forever recedes.

As long as civilizations can continuously climb the technological ladder, using newly issued budgets ($c_{FS}$) to build more efficient negentropy pumps (such as Dyson spheres, black hole computational networks), the game can continue infinitely.

What limits us is not physical laws, but our will.

If we stop outward emission and inward reorganization, we will die.

But as long as we are still “playing,” this cosmic casino will never close.