06. Information-Entropy Inequality: Ultimate Constraint on Universe Scale

Introduction: Information Budget Allocation Problem

In previous articles, we established universe’s three types of parameters:

• Article 03: ${\Theta }_{\text{str}}$ ~400 bits (spatial structure)
• Article 04: ${\Theta }_{\text{dyn}}$ ~1000 bits (evolution rules)
• Article 05: ${\Theta }_{\text{ini}}$ ~500 bits (initial state)

Total Parameter Information: $I_{\text{param}}\sim 2000$ bits

But this is only the “recipe”. What about actual universe state space?

Core Questions:

• How large is universe’s Hilbert space?
• What is maximum entropy?
• How is information capacity $I_{\max }$ allocated?

This article explores core theorem of GLS finite information theory:

$$\boxed{I_{\text{param}}(\Theta )+S_{\max }(\Theta )\le I_{\max }}$$

This inequality reveals ultimate constraint on universe scale.

Popular Analogy: Company Budget Allocation

Imagine a company’s information technology construction:

Total Budget: $I_{\max }$ yuan (fixed)

Two Major Expenses:

1. System Design ($I_{\text{param}}$):

   • Purchase software
   • Write code
   • Configure parameters
   • Train employees
   • Similar: Encoding cost of $\Theta$

2. Data Storage ($S_{\max }$):

   • Hard drive capacity
   • Database scale
   • Backup systems
   • Similar: Universe state space size

Budget Constraint: $$\text{Design Cost}+\text{Storage Cost}\le \text{Total Budget}$$

$$I_{\text{param}}+S_{\max }\le I_{\max }$$

Trade-off Relation:

• Complex system design → Must save on data storage
• Large data storage → Must simplify system design

This article will analyze in detail the source, meaning, and consequences of this inequality.

Part I: Maximum Entropy of State Space $S_{\max }$

Dimension of Hilbert Space

Review (Article 03):

Global Hilbert space: $${\mathcal{H}}_{\Lambda }=\underset{x\in \Lambda }{⨂}{\mathcal{H}}_x$$

Dimension: $$\dim {\mathcal{H}}_{\Lambda }=\prod _{x\in \Lambda }{d}_{\text{cell}}(x)$$

Uniform Case (all cells identical): $$\dim {\mathcal{H}}_{\Lambda }=d_{\text{cell}}^{N_{\text{cell}}}$$

This is an exponentially large number!

Example (Cosmological Scale):

• $N_{\text{cell}}=10^{90}$ (Observable universe in Planck length units)
• $d_{\text{cell}}=10^6$ (Standard Model degrees of freedom)
• $\dim \mathcal{H}=(10^6{)}^{10^{90}}=10^{6\times 10^{90}}$

This is double exponential!

Maximum von Neumann Entropy

Definition 6.1 (Maximum Entropy):

Assuming universe state can be any pure state in ${\mathcal{H}}_{\Lambda }$, “size” of state space measured by entropy:

$$\boxed{S_{\max }(\Theta )={\log }_2\dim {\mathcal{H}}_{\Lambda }}$$

(in bits)

Uniform Case: $$S_{\max }={\log }_2(d_{\text{cell}}^{N_{\text{cell}}})=N_{\text{cell}}{\log }_2{d}_{\text{cell}}$$

Physical Meaning:

• $S_{\max }$: How many bits of information needed to store a quantum state
• Analogy: Hard drive capacity (maximum data that can be stored)

Numerical Example:

• $N_{\text{cell}}=10^{90}$
• $d_{\text{cell}}=10^6$
• ${\log }_2{d}_{\text{cell}}={\log }_2(10^6)\approx 20$
• $S_{\max }=10^{90}\times 20=2\times 10^{91}$ bits

Key Observation: $$S_{\max }\sim 10^{91}\gg I_{\text{param}}\sim 10^3$$

State space entropy dominates total information!

Why Use Entropy Instead of Hilbert Dimension?

Reason 1 (Logarithmic Compression):

• Hilbert dimension: $D=d_{\text{cell}}^{N_{\text{cell}}}$ (double exponential)
• Entropy: $S={\log }_{2}D=N_{\text{cell}}{\log }_2{d}_{\text{cell}}$ (linear)

Entropy makes numbers manageable.

Reason 2 (Information-Theoretic Interpretation):

• Storing quantum state of dimension $D$ requires ${\log }_{2}D$ bits
• Entropy $S={\log }_{2}D$ directly gives information content

Reason 3 (Connection with Thermodynamic Entropy):

• von Neumann entropy: $S=-\text{tr}(\rho \log \rho )$
• Maximum entropy state (uniform mixed state): $\rho =1/D$
• Then $S=\log D$

Part II: Derivation of Finite Information Inequality

Source: Generalization of Bekenstein Bound

Review (Article 01):

Bekenstein bound: $$S\le \frac{2\pi RE}{\hslash c}$$

Applied to Universe:

• Radius: $R=R_{\text{uni}}$
• Energy: $E=E_{\text{uni}}$
• Entropy upper bound: $S_{\max }\le \frac{2\pi R_{\text{uni}}{E}_{\text{uni}}}{\hslash c}=:I_{\max }^{\text{(Bek)}}$

Key Insight: Universe’s state space entropy constrained by physical laws, cannot be infinite!

Introduction of Parameter Information

New Problem: Besides state entropy, need to encode parameters $\Theta$.

Total Information: $$I_{\text{total}}=I_{\text{param}}+I_{\text{state}}$$

where:

• $I_{\text{param}}$: Number of bits encoding parameters $\Theta$
• $I_{\text{state}}$: Number of bits describing quantum state

Worst Case (state space fully filled): $$I_{\text{state}}=S_{\max }={\log }_2\dim {\mathcal{H}}_{\Lambda }$$

Finite Information Axiom: $$I_{\text{total}}\le I_{\max }$$

Therefore:

$$\boxed{I_{\text{param}}(\Theta )+S_{\max }(\Theta )\le I_{\max }}$$

Theorem 6.2 (Finite Information Inequality) (from source theory Proposition 3.3):

Let $\Theta =({\Theta }_{\text{str}},{\Theta }_{\text{dyn}},{\Theta }_{\text{ini}})$ be universe parameters, then:

$$I_{\text{param}}(\Theta )+N_{\text{cell}}(\Theta )\ln d_{\text{cell}}(\Theta )\le I_{\max }$$

where $\ln$ converted to ${\log }_2$ requires multiplying by $1/\ln 2$.

Diagram of Inequality

graph TD
    A["Total Information Capacity<br/>I_max ~ 10^123 bits"] --> B["Parameter Information<br/>I_param ~ 10^3 bits"]
    A --> C["State Space Entropy<br/>S_max ~ 10^91 bits"]

    B --> D["Structural Parameters<br/>~400 bits"]
    B --> E["Dynamical Parameters<br/>~1000 bits"]
    B --> F["Initial State Parameters<br/>~500 bits"]

    C --> G["Number of Cells<br/>N_cell"]
    C --> H["Local Dimension<br/>d_cell"]

    G --> I["N_cell ~ 10^90"]
    H --> J["d_cell ~ 10^6"]

    I --> K["S_max = N × log d"]
    J --> K

    style A fill:#ffe6e6
    style B fill:#e6f3ff
    style C fill:#e6ffe6
    style K fill:#fff6e6

Part III: Trade-off Relation—Number of Cells and Local Dimension

Theorem: Upper Bound on Number of Cells

Theorem 6.3 (Upper Bound on Number of Cells) (Source theory Proposition 3.3, Item 1):

$$\boxed{N_{\text{cell}}\le \frac{I_{\max }-I_{\text{param}}}{\ln 2}}$$

Proof:

From finite information inequality: $$I_{\text{param}}+N_{\text{cell}}\ln d_{\text{cell}}\le I_{\max }$$

Rearranging: $$N_{\text{cell}}\ln d_{\text{cell}}\le I_{\max }-I_{\text{param}}$$

Since $d_{\text{cell}}\ge 2$ (minimum), have $\ln d_{\text{cell}}\ge \ln 2$: $$N_{\text{cell}}\ln 2\le N_{\text{cell}}\ln d_{\text{cell}}\le I_{\max }-I_{\text{param}}$$

Therefore: $$N_{\text{cell}}\le \frac{I_{\max }-I_{\text{param}}}{\ln 2}$$

Numerical Example:

• $I_{\max }=10^{123}$ bits
• $I_{\text{param}}=2000$ bits (negligible)
• $N_{\text{cell}}\le 10^{123}/\ln 2\approx 1.44\times 10^{123}$

Physical Meaning:

• Universe’s number of lattice points has hard upper bound
• Bound determined by information capacity (not arbitrary)

Theorem: Upper Bound on Local Dimension

Theorem 6.4 (Upper Bound on Local Hilbert Dimension) (Source theory Proposition 3.3, Item 2):

Given number of cells $N_{\text{cell}}$, local dimension satisfies:

$$\boxed{\ln d_{\text{cell}}\le \frac{I_{\max }-I_{\text{param}}}{N_{\text{cell}}}}$$

That is:

$$d_{\text{cell}}\le \exp \left(\frac{I_{\max }-I_{\text{param}}}{N_{\text{cell}}}\right)$$

Proof:

Directly divide finite information inequality by $N_{\text{cell}}$.

Numerical Example:

• $I_{\max }=10^{123}$ bits
• $N_{\text{cell}}=10^{90}$
• $\ln d_{\text{cell}}\le 10^{123}/10^{90}=10^{33}$
• $d_{\text{cell}}\le e^{10^{33}}$ (astronomical number)

But actually:

• Standard Model: $d_{\text{cell}}\sim 10^6$
• $\ln d_{\text{cell}}\sim 14$
• Far less than upper bound $10^{33}$!

Physical Meaning:

• Cell interior cannot be infinitely complex
• Complexity and number of lattice points have trade-off

Visualization of Trade-off Relation

Core Inequality: $$N_{\text{cell}}\times {\log }_2{d}_{\text{cell}}\lesssim I_{\max }$$

Equivalent Form: $${\log }_2{d}_{\text{cell}}\lesssim \frac{I_{\max }}{N_{\text{cell}}}$$

Diagram:

graph LR
    A["Information Budget<br/>I_max"] --> B["Choice 1<br/>Many Lattice Points+Simple Cells"]
    A --> C["Choice 2<br/>Few Lattice Points+Complex Cells"]

    B --> D["N_cell ↑<br/>d_cell ↓"]
    C --> E["N_cell ↓<br/>d_cell ↑"]

    D --> F["High Spatial Resolution<br/>Low Internal DOF"]
    E --> G["Low Spatial Resolution<br/>High Internal DOF"]

    F -.-> H["Example<br/>N=10^120, d=2"]
    G -.-> I["Example<br/>N=10^10, d=10^100"]

    H --> J["Constraint<br/>N × log d ≤ I_max"]
    I --> J

    style A fill:#ffe6e6
    style J fill:#e6ffe6

Popular Analogy:

Imagine you have 1000 yuan budget to buy computers:

Plan A (Many Machines):

• Buy 1000 Raspberry Pis (each cheap)
• Single machine weak, but many machines → Parallel computation
• Similar: $N_{\text{cell}}$ large, $d_{\text{cell}}$ small

Plan B (Strong Machine):

• Buy 1 high-performance server
• Single machine strong, but only one
• Similar: $N_{\text{cell}}$ small, $d_{\text{cell}}$ large

Constraint: $$\text{Number of Machines}\times \text{Single Machine Performance}\lesssim \text{Budget}$$

Universe chose variant of Plan A:

• Many lattice points ($\sim 10^{90}$)
• Medium complexity ($d\sim 10^6$)
• Product within $I_{\max }$ range

Part IV: Necessity of Symmetry, Locality, and Finite Precision

Why Must Have Symmetry?

Proof by Contradiction:

Assume universe has no symmetry (parameters at each lattice point independent).

Parameter Information:

• Each lattice point needs to specify: Hilbert space, Hamiltonian, initial state
• Each lattice point ~$\sim 100$ bits
• Total parameters: $I_{\text{param}}=N_{\text{cell}}\times 100\sim 10^{92}$ bits

State Space Entropy:

• $S_{\max }=N_{\text{cell}}{\log }_2{d}_{\text{cell}}\sim 10^{91}$ bits

Total Information: $$I_{\text{total}}\sim 10^{92}+10^{91}\sim 10^{92}\text{ bits}$$

But: $$I_{\max }\sim 10^{123}\text{ bits}$$

Looks fine? Wrong!

Problem: $N_{\text{cell}}\sim 10^{90}$ itself is an assumption. Actually according to Theorem 6.3:

If $I_{\text{param}}\sim 10^{92}$, then: $$N_{\text{cell}}\le \frac{I_{\max }-I_{\text{param}}}{\ln 2}\sim \frac{10^{123}-10^{92}}{\ln 2}\sim 10^{123}$$

But state space: $$S_{\max }=N_{\text{cell}}\ln d_{\text{cell}}$$

If $N_{\text{cell}}\sim 10^{123}$, $d_{\text{cell}}\sim 10^6$: $$S_{\max }\sim 10^{123}\times 20=2\times 10^{124}\text{ bits}$$

Exceeds $I_{\max }$! Contradiction!

Conclusion: Must have symmetry compression of $I_{\text{param}}$ to leave enough space for state space.

Why Must Have Locality?

Non-Local System:

If evolution operator $U$ acts on all lattice points (no locality):

• Unitary matrix dimension: $D\times D$, $D=d_{\text{cell}}^{N_{\text{cell}}}$
• Number of matrix elements: $D^2=d_{\text{cell}}^{2N_{\text{cell}}}$
• Encoding (real+imaginary parts): $\sim 2D^2\times m$ bits ($m$ is precision)

Numerical Value:

• $N_{\text{cell}}=10^{90}$, $d_{\text{cell}}=10^6$
• $D=10^{6\times 10^{90}}$
• $D^2=10^{12\times 10^{90}}$

Need $\sim 10^{12\times 10^{90}}$ bits → Far exceeds $I_{\max }\sim 10^{123}$!

Salvation by Locality:

Finite depth local circuit:

• Each gate acts on $\sim 2$ lattice points
• Depth $D\sim 10$
• Total number of gates: $\sim N_{\text{cell}}\times D\sim 10^{91}$
• Each gate: $\sim 100$ bits
• Total encoding: $\sim 10^{93}$ bits

If have symmetry (translation-invariant):

• Reduced to $\sim 10^3$ bits

Conclusion: Locality + Symmetry → Information content controllable.

Why Must Discretize?

Continuous Parameters:

If dynamical angle parameter $\theta \in [0,2\pi )$ is real number:

• Need infinite precision (e.g., $\pi =3.1415926\dots$)
• Each real number: Infinite bits
• Total parameter information: $\infty$ bits

Contradiction: $$I_{\text{param}}=\infty \nleqq I_{\max }<\infty$$

Salvation by Discretization:

$$\theta =\frac{2\pi n}{2^m},n\in \{0,\dots ,2^m-1\}$$

• Each angle: $m$ bits (finite)
• Precision $m=50$: $\Delta \theta \sim 10^{-15}$ (sufficient)

Conclusion: Finite information → Must discretize.

Part V: Actual Allocation of Universe Information Budget

Current Universe Parameters

According to analysis in previous articles:

Key Observation:

• Parameter information: Negligible ($<0.1\%$)
• State space: Dominates ($>99.9\%$)

Popular Analogy:

• Universe like a huge book ($10^{91}$ pages)
• Title, author, table of contents (parameters): Only 1 page
• Main content (state): All remaining pages

Comparison with $I_{\max }$

$$I_{\text{total}}=I_{\text{param}}+S_{\max }\sim 2\times 10^{91}\text{ bits}$$

$$I_{\max }\sim 10^{123}\text{ bits}$$

Redundancy: $$\frac{I_{\max }}{I_{\text{total}}}\sim \frac{10^{123}}{10^{91}}=10^{32}$$

Universe only uses about $10^{-32}$ of information capacity!

Possible Explanations:

1. Conservative Estimate: Calculation of $I_{\max }$ (Bekenstein bound) may be too rough

2. Future Growth: Universe still evolving, entropy may grow (but constrained by unitarity)

3. Multiverse: $I_{\max }$ is capacity of all possible universes, we are just one

4. Dimension Mystery: Extra dimensions or hidden degrees of freedom not counted

What If Parameters Changed?

Experiment 1: Increase number of lattice points $N_{\text{cell}}$

If $N_{\text{cell}}\to 10N_{\text{cell}}$:

• $S_{\max }\to 10S_{\max }$
• Still far less than $I_{\max }$
• Feasible

Experiment 2: Increase cell dimension $d_{\text{cell}}$

If $d_{\text{cell}}\to d_{\text{cell}}^{10}$:

• $S_{\max }=N_{\text{cell}}{\log }_2{d}_{\text{cell}}^{10}=10N_{\text{cell}}{\log }_2{d}_{\text{cell}}$
• Also increases 10 times
• Feasible

Experiment 3: Increase both simultaneously

If $N_{\text{cell}}\to 10^{16}{N}_{\text{cell}}$, $d_{\text{cell}}\to 10^{16}{d}_{\text{cell}}$:

• $S_{\max }\to 10^{16}\times 10^{16}\times S_{\max }/(N_{\text{cell}}\log d_{\text{cell}})$
• $\sim 10^{32}\times S_{\max }\sim 10^{123}$
• Approaches $I_{\max }$!

Conclusion: Can exponentially increase lattice points or dimension, but product constrained by $I_{\max }$.

Part VI: Constraints on Physical Theories

Constraint 1: Degrees of Freedom in Field Theory

Standard Model:

• Fermions: $3\times 2\times 2\times 3=36$ (generations, spin, particle/antiparticle, color)
• Bosons: $8+3+1=12$ (gluons, weak bosons, photon)
• Total degrees of freedom: $\sim 100$

String Theory/Supersymmetry:

• May increase to $\sim 10^3-10^6$

Constraint: $$d_{\text{cell}}\lesssim \exp \left(\frac{I_{\max }}{N_{\text{cell}}}\right)$$

If $N_{\text{cell}}\sim 10^{90}$, $I_{\max }\sim 10^{123}$: $$d_{\text{cell}}\lesssim e^{10^{33}}$$

Conclusion: Standard Model, string theory degrees of freedom all far below upper bound, allowed.

Constraint 2: Extra Dimensions

Kaluza-Klein/String Theory: Proposes extra compact dimensions (e.g., 6-dimensional Calabi-Yau manifolds).

Impact:

• Each lattice point not ${\mathbb{C}}^{d_{\text{cell}}}$, but larger space
• Or: Number of lattice points increases (because of extra dimensions)

Example:

• 3+1 dimensions → 9+1 dimensions (string theory)
• Extra 6 dimensions compactified, scale $R_{\text{compact}}$
• If $R_{\text{compact}}\sim {\ell }_p$ (Planck scale)
• Extra lattice points: $\sim (R_{\text{uni}}/{\ell }_p{)}^6\sim 10^{366}$

Check Constraint: $$N_{\text{total}}=N_{\text{3D}}\times N_{\text{6D}}\sim 10^{90}\times 10^{366}=10^{456}$$

$$S_{\max }=N_{\text{total}}{\log }_2{d}_{\text{cell}}\sim 10^{456}\times 20=2\times 10^{457}$$

Far exceeds $I_{\max }\sim 10^{123}$!

Contradiction!

Solutions:

1. Extra dimensions must be extremely small ($R\ll {\ell }_p$, hard to imagine)
2. Or extra dimensions not “real” lattice points (emergent, effective theory)
3. Or our $I_{\max }$ estimate too conservative

Conclusion: Finite information constraint imposes strict limitations on extra dimension theories.

Constraint 3: Inflation Theory

Inflationary Cosmology: Early universe exponentially expands, $N_{\text{cell}}$ rapidly grows.

Problem: Initial (before inflation): $N_{\text{cell}}\sim 10^3$ Final (after inflation): $N_{\text{cell}}\sim 10^{90}$

Growth factor: $\sim 10^{87}$

State Space Entropy Change: $$\Delta S_{\max }=\Delta N_{\text{cell}}\times {\log }_2{d}_{\text{cell}}\sim 10^{87}\times 20=2\times 10^{88}$$

Source:

• Unitary evolution: $S$ conserved
• But $S_{\max }$ is Hilbert space size, can grow

Explanation:

• Inflation creates new “spatial lattice points”
• These lattice points initially in low entropy state (vacuum)
• Later gradually filled (entanglement grows)

Constraint Check: $$S_{\max }^{\text{final}}\sim 2\times 10^{91}<I_{\max }\sim 10^{123}$$

Allowed!

Summary of Core Points of This Article

Finite Information Inequality

$$\boxed{I_{\text{param}}(\Theta )+S_{\max }(\Theta )\le I_{\max }}$$

where:

• $I_{\text{param}}=|{\Theta }_{\text{str}}|+|{\Theta }_{\text{dyn}}|+|{\Theta }_{\text{ini}}|$
• $S_{\max }=N_{\text{cell}}{\log }_2{d}_{\text{cell}}$
• $I_{\max }\sim 10^{123}$ bits (Bekenstein bound)

Theorem: Upper Bound on Number of Cells

$$\boxed{N_{\text{cell}}\le \frac{I_{\max }-I_{\text{param}}}{\ln 2}}$$

Theorem: Upper Bound on Local Dimension

$$\boxed{\ln d_{\text{cell}}\le \frac{I_{\max }-I_{\text{param}}}{N_{\text{cell}}}}$$

Trade-off Relation

$$\boxed{N_{\text{cell}}\times {\log }_2{d}_{\text{cell}}\lesssim I_{\max }}$$

Physical Meaning:

• Many lattice points ↔ Simple cells
• Few lattice points ↔ Complex cells
• Product constrained

Three Necessities

Information Budget Allocation

Constraints on Physical Theories

1. Standard Model: $d_{\text{cell}}\sim 10^2$ ✅ Allowed
2. String Theory/Supersymmetry: $d_{\text{cell}}\sim 10^6$ ✅ Allowed
3. Large Extra Dimensions: $N_{\text{cell}}\sim 10^{456}$ ❌ Forbidden (unless $R\ll {\ell }_p$)
4. Inflationary Universe: $\Delta S\sim 10^{88}$ ✅ Allowed

Core Insights

1. State space dominates information: $S_{\max }\gg I_{\text{param}}$
2. Universe far from saturated: $I_{\text{total}}\ll I_{\max }$
3. Symmetry necessary: Otherwise parameter information explodes
4. Locality necessary: Otherwise evolution operator encoding explodes
5. Discretization necessary: Otherwise continuous parameters need infinite bits
6. Constraints physical theories: Extra dimensions, string theory limited

Next Article Preview: 07. Continuous Limit and Derivation of Physical Constants

• Rigorous proof from discrete QCA to continuous field theory
• Derivation of Dirac equation
• Derivation of mass-angle parameter relation $m\approx \theta /\Delta t$
• Parameterized expressions for gauge coupling constants, gravitational constant
• All physical constants are functions of $\Theta$!