10.1 The Universe’s Source Code: How Was Rule $\hat{U}$ Selected? (Criticality Hypothesis)

In Axiom $\Omega$, we assumed a global unitary evolution operator $\hat{U}$. But possible $\hat{U}$ (i.e., cellular automaton rules) are infinitely many. Why does our universe run this particular set of rules capable of producing quarks, stars, and DNA, rather than evolving into dead silence or chaos like most rules in Conway’s Game of Life?

This is the computational version of “Fine-tuning Problem”. We don’t need God to choose rules; we only need to understand phase transitions of computational complexity.

10.1.1 Classification of Rule Space: Wolfram Classes and Langton Parameter $\lambda$

Stephen Wolfram, through exhaustive study of one-dimensional elementary cellular automata, found all possible rules can be categorized into four behavioral patterns:

1. Class I (Order/Death): Regardless of initial state, system rapidly evolves to single uniform state (e.g., all black). This corresponds to crystals or absolute zero in thermodynamics, with no information processing capability.

2. Class II (Period/Oscillation): System evolves to simple, local periodic structures. Such rules can only store limited bits, unable to perform long-range communication.

3. Class III (Chaos/Random): System evolves to disordered, seemingly random patterns (actually encrypted determinism). Although information content is huge (high entropy), structures have no correlations, unable to form stable “objects.” This corresponds to thermal equilibrium or white noise.

4. Class IV (Complex/Computation): This is the rarest class. System evolves complex, long-range correlated local structures (particles, gliders), which can move, collide, annihilate, or interact through logic gates in background. Such rules are proven to be Turing Complete.

Our physical universe clearly belongs to Class IV. Only such rules can support information storage (stability), transmission (photons), and processing (interactions), thereby supporting birth of life.

Christopher Langton introduced a parameter $\lambda$ (proportion of non-zero outputs in rule table) to quantify this classification. He found that as $\lambda$ increases from 0 to 1, system undergoes phase transition from “order” to “chaos.” Class IV rules are exactly at critical point of this phase transition.

10.1.2 Criticality Hypothesis: Edge of Chaos

We propose a Criticality Hypothesis about cosmic origin:

Rules $\hat{U}$ capable of producing “physical universe” must be at second-order phase transition edge between ordered phase and chaotic phase.

Near critical point, system’s correlation length $\xi$ tends to infinity ($\xi \sim |T-T_c{|}^{-\nu }$). This means:

1. Long-Range Forces: Although underlying interactions are strictly local (only talking to neighbors), due to scale-free nature of critical state, information can effectively propagate to infinity. This explains why gravity and electromagnetic forces are long-range (massless bosons correspond to critical modes).

2. Self-Similarity: System exhibits fractal structures and power-law distributions. This explains cross-scale structural similarity in universe (from atoms to galaxy clusters).

3. Maximum Complexity: At this point, system’s Shannon entropy is neither 0 (fully ordered) nor maximum (fully random), but at state capable of accommodating maximum “Logical Depth”.

10.1.3 Why Critical State?

Even without designers, Self-Organized Criticality (SOC) mechanisms will drive systems to automatically evolve to critical point. Like sandpile collapse models, if there exists some “meta-rule” allowing $\hat{U}$ to fine-tune over time (e.g., through vacuum decay or cosmic natural selection), then only those sub-universes evolving to critical state can produce observers.

Therefore, our universe is so exquisite not because it was carefully designed, but because it is a survivor—it is the only critical bubble capable of producing “meaning” through computation among countless ruins of death and chaos.