2.3 Why Locality? — Avoiding “God’s Eye View” and Action at a Distance

After establishing “unitarity” as the rigid skeleton of cosmic logic, we face a second fundamental question: How is this universe’s structure organized?

In the Ultimate Axiom $\Omega$, we force the evolution operator $\hat{U}$ to be local. This means that within each time step $\Delta t$, any cell (Qubit) can only interact with cells directly adjacent to it on the lattice graph $\Lambda$.

Why must it be local? Why can’t we allow a cell to instantly exchange information with a cell at the other end of the universe? This is not just to conform to the empirical facts of special relativity, but a necessary prerequisite for logically constructing a self-consistent, evolvable universe.

2.3.1 The End of God’s Eye View: Full Connection Means No Structure

Imagine a non-local universe. In this universe, there exists a direct interaction channel between any two particles. No matter how far apart they are (assuming the concept of “distance” still exists), a flip of one particle can directly change another particle’s state in the next instant.

In graph-theoretic language, this corresponds to a complete graph.

If the universe is fully connected, then:

1. Geometry Disappears: Since any two points are “adjacent,” the concept of “space” loses meaning. There is no “near” and “far,” no “inside” and “outside”; the entire universe collapses into a zero-dimensional point.

2. Complexity Collapse: To calculate the next state, every cell needs to know the current state of all other cells in the entire universe. This means every local computation requires infinite (or universe-scale) information input and processing capability. This actually requires every particle to possess a “God’s eye view.”

Therefore, locality is not a limitation, but creation. It is precisely by limiting the range of interactions (cutting off the vast majority of connections) that the universe acquires topological structure, dimensions, and so-called “space.”

Space is essentially the sparsity of interactions.

2.3.2 The Birth of Geometry: Distance as Delay

Once we impose locality constraints, maximum signal propagation speed (light speed $c$) automatically emerges as a logical necessity.

In QCA, information can only propagate to neighbors within one step $\Delta t$. To propagate to a node $N$ steps away, $N$ time steps are necessary.

$$\text{Distance}\equiv \text{Minimum Communication Time}$$

This reveals the ontological definition of light speed $c$ in physics:

$c$ is not the speed limit of object motion; it is the conversion factor for causal chain extension.

If we allow non-local action (action at a distance), it means $c\to \infty$. In this case, causality would no longer have temporal order protection, and grandfather paradoxes would become inevitable. The locality axiom is actually the guardian of causality. It ensures events occur sequentially and influence transmission has a process. As physicist John Wheeler said: “Time is to prevent everything from happening at once; space is to prevent everything from happening in the same place.”

2.3.3 Clarification: Quantum Non-locality vs. Dynamical Locality

Here we must clarify a concept that often confuses beginners: the difference between Bell Non-locality and Dynamical Locality.

Entangled states in quantum mechanics do exhibit non-local correlations—measuring one particle seems to instantly determine another particle’s state. Does this violate our locality axiom?

The answer is: No violation.

1. Dynamical Locality (what we insist on): This refers to the structure of the Hamiltonian or evolution operator. In QCA, $\hat{H}=\sum {\hat{h}}_{i,i+1}$. This ensures interactions only occur between neighbors. You cannot “touch” a distant particle.

2. Bell Non-locality (of measurement results): This refers to correlations of states. Although two particles are far apart, they share a history (once had local interaction somewhere, then separated). This correlation is a pre-established resource, not instantaneous communication.

In a QCA universe, entanglement can exist between arbitrarily distant nodes (as long as they have causal connection in the past), but utilizing this entanglement to transmit information and produce physical effects must follow the point-by-point local rules.

This is like two people each taking a walkie-talkie to opposite ends of Earth. The walkie-talkies (entanglement) connect them, but radio waves (interactions) must traverse space at light speed.

2.3.4 Conclusion: A Universe Without Supermen

The locality axiom is actually a humble declaration of physics. It states:

• There is no “control center” in the universe.

• Every cell is equal and autonomous.

• Macroscopic order is not imposed top-down by global commands, but emerges bottom-up from countless local microscopic interactions.

This is not only a principle of physics, but also the fundamental reason complex systems (such as life, brains, society) can exist.

At this point, we have established the universe’s software logic (unitarity) and hardware architecture (locality). In the next section, we will formalize these concepts and write down the complete mathematical definition of the ultimate axiom.