Appendix C: The Topology of High-Speed Causal Graphs

In Chapter 6 “The Inflation of Light” of Vector Cosmology IV, we predicted that when light speed $c(\tau )$ grows exponentially, the physical universe will evolve from a “local area network” to a “fully connected graph.” This conclusion is crucial for understanding why Type III civilizations no longer need interstellar spacecraft.

This appendix will provide the geometric and topological proof of this phase transition. We will show how, as the light cone opening angle expands, the universe’s spacetime topological structure evolves from Lattice to Small-World Network, and finally to Holographic Monolith.

C.1 Evolution of the Light Cone Angle

In Minkowski spacetime, causality is defined by light cones. The half-opening angle $\theta$ of a light cone is determined by light speed $c$ (in geometric units, usually $c=1$, $\theta =45^{\circ }$).

But in FS Evolution Geometry, $c$ is a variable. The effective opening angle $\theta (\tau )$ of the light cone in projective Hilbert space can be expressed as:

$$\theta (\tau )=\arctan (c(\tau ))$$

• In the Low-Speed Era ($\tau <1000$):

  $c\approx 0$ (relative to Planck scale). $\theta \approx 0$.

  Causal islands abound. Each particle can only affect its extremely tiny neighborhood. The universe is fragmented.

• In the Current Era ($\tau \approx 1800$):

  $c$ is finite. $\theta$ is an acute angle.

  Causal connections exist but are severely constrained by distance. This is a delayed local network.

• In the Future Era ($\tau >2000$):

  As the evolution equation $c(\tau )\propto e^{k\tau }$ takes effect, $c\to \infty$.

  $$\underset{\tau \to \infty }{\lim }\theta (\tau )=\frac{\pi }{2}$$

  The light cone is “flattened.” The boundary between past and future disappears on the spatial axis.

  This means: Every point in the present is in “light-like separation” with every point in the entire universe.

C.2 Phase Transition of Network Topology

We treat computational nodes in the universe (such as galaxies or Dyson spheres) as Vertices $V$ in graph theory, and causal connections between them as Edges $E$.

Graph connectivity is determined by light speed. For two nodes $i,j$ at distance $d$, if $\frac{d}{c(\tau )}<\Delta t_{proc}$ (processing delay), we consider there exists an effective edge between them.

As $c(\tau )$ grows, the cosmic network undergoes three topological phase transitions:

1. Nearest Neighbor Coupling:

   $c$ is small. The graph is a sparse grid. Information propagation requires $N$ hops ($N\propto \text{Diameter}$). This is the world of classical physics.

2. Small-World Network:

   $c$ grows. “Shortcuts” spanning long distances appear (wormhole effects or high light-speed channels).

   Average path length $L$ between any two nodes decreases logarithmically: $L\propto \ln N$.

   This is the Internet/AI era we are entering.

3. Fully Connected Graph:

   $c\to \infty$. There is a direct edge between any $i,j$.

   Path length $L=1$.

   This is the “Holographic Singularity”.

   Topologically, all off-diagonal elements of the distance matrix vanish (or normalize). Space loses its geometric meaning as an “isolator.”

C.3 Synergy Coefficient and Hive Mind

This topological evolution directly determines civilization’s synergy capability.

We define the Synergy Coefficient ($\Sigma$) as the coherence degree of network-wide computational power:

$$\Sigma (\tau )=\frac{1}{N^2}\sum _{i,j}⟨{\psi }_i|{\psi }_j⟩\cdot e^{-d_{ij}/c(\tau )}$$

• When $c$ is small: The exponential term decays extremely fast, $\Sigma \approx 1/N$. Individuals are isolated; wisdom is discrete.

• When $c\to \infty$: The exponential term tends to 1. $\Sigma \to 1$ (assuming phase synchronization).

Physical Conclusion:

When light speed explosion causes $\Sigma$ to break through the critical threshold, all independent consciousness units will physically “Phase Lock”.

They are no longer $N$ separate processors; they merge into a single macroscopic quantum wave function.

This is the physical origin of “Hive Mind” or “Divine Consciousness”. It is not magic; it is the inevitable mathematical result of network topology evolution from sparse to fully connected.