Meta Theory of the Zeckendorf-Hilbert Universe

25.4 Algorithmic Information Theory and Kolmogorov Complexity of Physical Laws

In previous sections of Chapter 25, we explored thermodynamic cost of physical computation (Landauer principle) and optimal encoding under local causal constraints (Fibonacci/golden ratio). These discussions mainly focus on encoding and transmission of States. However, physics has a deeper question: encoding of Laws themselves.

Why are fundamental physical laws describing universe (such as Standard Model Lagrangian or Einstein equations) so concise that they can be written on a T-shirt? Why don’t we live in a universe with extremely complex laws full of special cases and patches?

This section will introduce Algorithmic Information Theory (AIT), using Kolmogorov Complexity to quantify simplicity of physical laws. We will prove that Occam’s Razor is not merely human aesthetic preference, but statistical necessity for existence of computational universes. In QCA discrete ontology, universe is a computational process with extremely high Logical Depth generated by extremely short programs (low Kolmogorov complexity).

25.4.1 Algorithmic Entropy of Physical Theories: From Equations to Programs

In traditional physics, laws are differential equations. In QCA discrete ontology, laws are update rules of cellular automata.

Definition 25.4.1 (Kolmogorov Complexity of Physical Laws)

Let $\mathfrak{U}$ be a physical universe model. Its Kolmogorov complexity $K(\mathfrak{U})$ is defined as length (in bits) of shortest program capable of simulating evolution of this universe:

$$K(\mathfrak{U})=\underset{p}{\min }\{|p|:U_{TM}(p)=\text{History}(\mathfrak{U})\}$$

where $U_{TM}$ is universal Turing machine (or universal QCA).

According to parametric definition in Chapter 20, this shortest program is essentially optimal compressed encoding of universe’s parameter vector $\Theta =({\Theta }_{\text{str}},{\Theta }_{\text{dyn}},{\Theta }_{\text{ini}})$.

$$K(\mathfrak{U})\approx I_{\text{param}}(\Theta )$$

Goal of physics is precisely to find $\Theta$ with minimum $K(\mathfrak{U})$.

25.4.2 Physical Origin of Occam’s Razor: Algorithmic Probability

Why do physical laws tend to be simple (low $K$)? Solomonoff’s Algorithmic Probability Theory provides the answer.

Theorem 25.4.2 (Prior Probability of Universe)

Assume all possible computational universes are randomly sampled from “space of all possible programs.” According to algorithmic information theory, prior probability $P(\mathfrak{U})$ of a specific universe $\mathfrak{U}$ being “generated” or “existing” has exponential decay relationship with its Kolmogorov complexity:

$$P(\mathfrak{U})\approx 2^{-K(\mathfrak{U})}$$

This means:

• Simple Universes (Low $K$): Such as QCA with translational symmetry, local interactions, their $K(\mathfrak{U})$ is very small (only need few lines of code to define rules), therefore existence probability is extremely high.

• Complex Universes (High $K$): Such as universes full of arbitrary non-local connections, laws changing every second, their $K(\mathfrak{U})$ is extremely large, existence probability tends to zero.

Physical Corollary:

Reason why physical laws we observe have symmetries (spatial translation, time translation, gauge symmetry) is because symmetry is best means of compressing information.

• Spatial translational symmetry means we don’t need to define physical laws separately for each point in universe, only need to define once, then say “same everywhere.” This greatly reduces $K(\mathfrak{U})$.

• Occam’s Razor is maximum likelihood estimation of universe generation.

25.4.3 Emergence of Complexity: Logical Depth vs. Randomness

If universe tends toward simplicity, why is macroscopic world we see (life, galaxies) so complex? Here we need to distinguish Algorithmic Complexity (Randomness) from Logical Depth (Organization).

1. Random Sequences (such as coin toss results): $K(s)\approx |s|$. Incompressible, extremely complex, but no structure.

2. Simple Sequences (such as all 1s): $K(s)\approx \text{const}$. Extremely compressible, no structure.

3. Structured Sequences (such as DNA or QCA evolution patterns): $K(s)$ is small (originating from simple evolutionary laws), but generating it requires long computational process.

Definition 25.4.3 (Bennett’s Logical Depth)

Logical depth $D(x)$ of an object (or universe state) is defined as computation time (logical steps) required to run shortest program generating $x$.

$$D(x)=\text{Time}(p^{\ast })\text{s.t. }U(p^{\ast })=x\wedge |p^{\ast }|=K(x)$$

Theorem 25.4.4 (Universe Depth Theorem)

Our QCA universe is a system with low Kolmogorov complexity, high logical depth.

• Simple Laws: $K(U)\ll \text{Size of Universe}$. $\Theta$ is very short.

• Long History: To obtain current state $\rho (t_{now})$ from $\Theta$, must undergo $10^{60}$ Planck time steps of irreducible computation (computational irreducibility, see Section 5.4).

Conclusion: “Beauty” of physics lies in extremely simple rules (low $K$) emerging extremely complex phenomena (high depth). If laws themselves are complex, that’s ugly; if phenomena are simple, that’s boring.

25.4.4 Computability of Physical Constants: Taboo of Chaitin Constant $\Omega$

In standard model, physical constants (such as fine structure constant $\alpha$) are considered real numbers. But in AIT, vast majority of real numbers are uncomputable (random), with $K(\alpha )=\infty$.

If physical constants are uncomputable real numbers, then $K(\mathfrak{U})$ of universe would be infinite, its existence probability zero.

Corollary 25.4.5 (Computability Conjecture of Constants)

In QCA discrete ontology, all physical constants (including $\alpha ,G,\dots$) must be Computable Numbers.

This means they are either rational numbers, or limits of some simple algorithms (such as geometric series, algebraic functions of $\pi$).

For example, in Section 25.3 we saw that dimension of topological quantum computation is $\varphi$ (algebraic number). Entropy coefficient of black holes in Chapter 15 is $1/4$ (rational number).

Chaitin Constant $\Omega$ (uncomputable number representing halting probability) cannot appear as physical constant in Lagrangian. Physics rejects uncomputability.

25.4.5 Summary of Part XIV

Part XIV reveals information-theoretic essence of physical laws by mapping physics to computation and coding theory.

1. Cost: Landauer principle specifies thermodynamic bottom line of computation (25.1).

2. Encoding: Fibonacci coding demonstrates optimal counting method in local causal networks (25.2).

3. Efficiency: Golden ratio $\varphi$ is eigenvalue of most robust information channel (25.3).

4. Simplicity: Kolmogorov complexity explains why physical laws tend toward simple symmetric structures, rejecting random parameters (25.4).

This proves: Universe is not only a computer, but an efficient computer that has undergone “code optimization”. It uses shortest code (laws), computes richest reality (high logical depth) under minimum energy consumption (Landauer lower bound).

In the final part of the entire book—Part XV: Experimental Verification and Engineering Prospects—we will leave ivory tower of theory, discussing how to use precision measurement experiments (such as microwave cavities, gravitational waves) to verify these grand information geometric predictions.