05. Detailed Explanation of Initial State Parameters: Universe’s Factory Settings

Introduction: Quantum State at Big Bang Moment

In previous articles, we established:

• Article 03: Universe’s spatial structure ${\Theta }_{\text{str}}$ (stage)
• Article 04: Universe’s evolution rules ${\Theta }_{\text{dyn}}$ (script)

But still missing a key element: Starting point.

Core Questions:

• What quantum state is universe in at $t=0$ (big bang moment)?
• How to encode this initial state with finite bit string?
• How does initial state affect entire universe history?

Answer lies in initial state parameter ${\Theta }_{\text{ini}}$.

Popular Analogy: Factory’s Factory Settings

Continuing our building/factory analogy:

Already Have:

• ${\Theta }_{\text{str}}$: Factory blueprint (how many machines, how to layout)
• ${\Theta }_{\text{dyn}}$: Production process (how to operate machines)

Missing:

• ${\Theta }_{\text{ini}}$: Machines’ initial state (switch positions, temperature, inventory…)

Why Important?

Imagine two identical coffee machines (same ${\Theta }_{\text{str}}$ and ${\Theta }_{\text{dyn}}$):

• Machine A: Bean hopper full, water tank full, preheated → Coffee immediately
• Machine B: Bean hopper empty, water tank empty, not preheated → Need preparation

Same structure and rules, different initial states → Different evolution histories!

Universe is similar:

This article will explain in detail how to encode entire universe’s initial quantum state with about 500 bits.

Part I: Physical Meaning of Initial State

Initial Condition Problem in Cosmology

Classical Cosmology (Friedmann-Lemaître-Robertson-Walker model):

Initial conditions need to specify:

• Initial matter density ${\rho }_0$
• Initial Hubble constant $H_0$
• Initial curvature $k$
• Initial temperature $T_0$
• Initial fluctuation spectrum $\Delta \rho (\mathbf{k})$
• …

Problem: These are all real numbers, need infinite precision → Infinite information!

Quantum Cosmology (Hartle-Hawking, Vilenkin, etc.):

Attempts to derive initial state from “no-boundary” or “tunneling” principles, but still needs:

• Wave function $\Psi [\text{three-geometry}]$ (functional on superspace)
• Continuous, infinite-dimensional → Still needs infinite information

QCA Framework Solution:

Initial state is a state in finite-dimensional Hilbert space:

$$|{\Psi }_0⟩\in {\mathcal{H}}_{\Lambda }=\underset{x\in \Lambda }{⨂}{\mathcal{H}}_x$$

Dimension: $$\dim {\mathcal{H}}_{\Lambda }=d_{\text{cell}}^{N_{\text{cell}}}\sim 10^{6\times 10^{90}}$$

Although dimension huge, but finite!

Three Formulations of Initial State

Formulation 1 (Quantum State Vector):

$$|{\Psi }_0^{\Theta }⟩\in {\mathcal{H}}_{\Lambda }$$

A normalized quantum state.

Formulation 2 (Density Matrix):

$${\rho }_0^{\Theta }=|{\Psi }_0^{\Theta }⟩⟨{\Psi }_0^{\Theta }|$$

Density operator of pure state.

Formulation 3 (State Functional):

$${\omega }_0^{\Theta }(A)=⟨{\Psi }_0^{\Theta }|A|{\Psi }_0^{\Theta }⟩,A\in \mathcal{A}(\Lambda )$$

Positive normalized functional on quasi-local algebra.

Three Equivalent (for pure states). This article mainly uses Formulations 1 and 3.

Initial State Determines Universe History

Given initial state ${\omega }_0$ and evolution $\alpha$, entire universe history uniquely determined:

$$\boxed{{\omega }_n={\omega }_0\circ {\alpha }^{-n}}$$

That is:

$${\omega }_n(A)={\omega }_0({\alpha }^{-n}(A))=⟨{\Psi }_0|U^{†n}A{U}^n|{\Psi }_0⟩$$

Physical Meaning:

• ${\omega }_0$: “Frame zero” (big bang moment)
• $\alpha$: “Frame-by-frame evolution rule” (from ${\Theta }_{\text{dyn}}$)
• ${\omega }_n$: “Frame $n$” (time $t=n\Delta t$)

Popular Analogy:

• Initial state = First frame of movie
• Evolution rule = How each frame transforms to next
• Entire movie = Generated frame by frame from first frame

Philosophical Implication: Universe history is deterministic (under unitary evolution): $$\text{Initial State}+\text{Evolution Rule}\Rightarrow \text{Entire History}$$

Part II: Reference Product State $|0_{\Lambda }⟩$

Simplest State: Vacuum State

How much information needed to directly encode an arbitrary quantum state $|{\Psi }_0⟩\in {\mathcal{H}}_{\Lambda }$?

Naive Counting:

• ${\mathcal{H}}_{\Lambda }\cong {\mathbb{C}}^D$, where $D=d_{\text{cell}}^{N_{\text{cell}}}$
• Quantum state: $|\Psi ⟩={\sum }_{i=1}^D{c}_i|i⟩$
• Coefficients $c_i\in \mathbb{C}$, satisfy ${\sum }_i|c_i{|}^2=1$
• Degrees of freedom: $2(D-1)$ real numbers (excluding normalization and global phase)

Information Content (if each real number needs $m$ bit precision): $$I\sim 2D\times m\sim 2\times 10^{10^{91}}\times m$$

This is double exponential! Far exceeds $I_{\max }\sim 10^{123}$!

Solution: Don’t directly encode state vector, but generate from simple state.

Definition of Reference Product State

Definition 5.1 (Reference Product State):

Choose a fixed “vacuum state” $|0_{\text{cell}}⟩\in {\mathcal{H}}_{\text{cell}}$ for each cell, define:

$$\boxed{|0_{\Lambda }⟩=\underset{x\in \Lambda }{⨂}|0_{\text{cell}}{⟩}_x}$$

Properties:

1. Product state: No entanglement
2. Translation-invariant: Same at each lattice point
3. Simple: Completely determined by $|0_{\text{cell}}⟩$

Example 1 (Spin Chain):

If ${\mathcal{H}}_{\text{cell}}={\mathbb{C}}^2$, basis $\{|\uparrow ⟩,|\downarrow ⟩\}$:

$$|0_{\text{cell}}⟩=|\downarrow ⟩$$

Then:

$$|0_{\Lambda }⟩=|\downarrow \downarrow \downarrow \cdots \downarrow ⟩$$

(All spins down)

Example 2 (Fermion QCA):

If ${\mathcal{H}}_{\text{cell}}$ contains fermion annihilation/creation operators $\hat{a},{\hat{a}}^{†}$:

$$|0_{\text{cell}}⟩=|\text{vac}⟩$$

(Fermion vacuum state, no particles)

Encoding Overhead:

Reference product state completely determined by $|0_{\text{cell}}⟩$, and $|0_{\text{cell}}⟩$ already specified in ${\Theta }_{\text{str}}$ (as part of Hilbert space).

Therefore: No additional encoding needed!

Physical Interpretation

Universe’s “Absolute Zero”:

$|0_{\Lambda }⟩$ similar to physics’ “ground state” or “vacuum state”:

• No particles
• No entanglement
• No excitations
• Zero entropy (pure state)

Popular Analogy:

• Reference product state = Blank white paper fresh from factory
• Initial state = Picture drawn on white paper
• State preparation circuit = “Drawing process” from white paper to picture

Part III: State Preparation Circuit $V_{{\Theta }_{\text{ini}}}$

Generating Initial State from Reference State

Core Idea: Use finite depth unitary circuit to generate $|{\Psi }_0⟩$ from $|0_{\Lambda }⟩$.

Definition 5.2 (State Preparation Circuit):

Exists finite depth unitary operator $V_{{\Theta }_{\text{ini}}}$, composed of gates from gate set $\mathcal{G}$:

$$V_{{\Theta }_{\text{ini}}}=V_{D_{\text{ini}}}{V}_{D_{\text{ini}}-1}\cdots V_2{V}_1$$

such that:

$$\boxed{|{\Psi }_0^{\Theta }⟩=V_{{\Theta }_{\text{ini}}}|0_{\Lambda }⟩}$$

Structure (completely similar to Article 04 dynamical circuit):

• Depth: $D_{\text{ini}}\in \mathbb{N}$
• Each layer $V_{\ell }$: Parallel combination of several local gates $G_k\in \mathcal{G}$
• Gate parameters: Discrete angles $\theta =2\pi n/2^m$

Difference:

• Dynamical circuit $U_{{\Theta }_{\text{dyn}}}$: Defines time evolution (applied repeatedly)
• State preparation circuit $V_{{\Theta }_{\text{ini}}}$: Generates initial state (applied only once)

Popular Analogy:

• $U$: Factory production process (repeated daily)
• $V$: Machine installation and debugging process (done only once)

Circuit Depth and Entanglement Structure

Consequence of Finite Depth (Lieb-Robinson bound):

If circuit depth is $D_{\text{ini}}$, Lieb-Robinson velocity is $v_{\text{LR}}$, then:

Theorem 5.3 (Entanglement Range Limitation):

Mutual information of two regions $A,B$ at distance $d(x,y)>v_{\text{LR}}{D}_{\text{ini}}$ satisfies:

$$I(A:B)\lesssim e^{-c(d-v_{\text{LR}}{D}_{\text{ini}})}$$

(exponential decay)

Physical Meaning:

• Short depth circuit $\Rightarrow$ Short-range entangled state
• Long-range entanglement needs depth $D\sim N_{\text{cell}}$ $\Rightarrow$ Parameter explosion

Cosmological Application:

Observations show cosmic microwave background (CMB) has finite correlation length:

• Sound horizon: $\sim 10^5$ light years
• Observable universe: $\sim 10^{10}$ light years
• Ratio: $\sim 10^{-5}$

Corollary: Initial entanglement is local, depth $D_{\text{ini}}\sim \log N_{\text{cell}}$ sufficient.

Examples of State Preparation Circuits

Example 1 (Hadamard Layer):

For spin chain, apply Hadamard gate:

$$V=\underset{x\in \Lambda }{⨂}{H}_x$$

where $H=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)$.

Initial State:

$$|{\Psi }_0⟩=V|\downarrow \downarrow \cdots \downarrow ⟩=\frac{1}{2^{N_{\text{cell}}/2}}\sum _{\{s_x\}}|s_1{s}_2\cdots s_{N_{\text{cell}}}⟩$$

(Equal-weight superposition of all spin configurations)

Properties:

• Maximum entanglement (in some sense)
• Depth $D=1$ (extremely shallow)
• Entropy $S=N_{\text{cell}}\log 2$ (maximum entropy)

Example 2 (GHZ State Generation):

Generate GHZ state for 3 lattice points:

$$|\text{GHZ}⟩=\frac{1}{\sqrt{2}}(|\downarrow \downarrow \downarrow ⟩+|\uparrow \uparrow \uparrow ⟩)$$

Circuit (depth 3):

1. Apply Hadamard to 1st lattice point: $H_1$
2. CNOT gates: 1 controls 2, 2 controls 3
3. Result: $|\text{GHZ}⟩$

Example 3 (Thermal State Approximation):

Construct “thermalized” state through random local unitaries:

$$V=\prod _{\ell =1}^D\prod _x{R}_x^{(\ell )}({\theta }_{\ell ,x})$$

where $R(\theta )$ are random rotations, angles sampled from distribution (discretized).

Depth: $D\sim \log N_{\text{cell}}$ can achieve near-thermal state.

Part IV: Encoding of ${\Theta }_{\text{ini}}$

Encoding Structure

Similar to ${\Theta }_{\text{dyn}}$ (Article 04), ${\Theta }_{\text{ini}}$ encodes state preparation circuit $V$:

$${\Theta }_{\text{ini}}=(D_{\text{ini}},\{k_{\ell }\},\{R_{\ell }\},\{{\theta }_{\ell ,j}\})$$

Components:

1. Depth $D_{\text{ini}}$
2. Gate type per layer $k_{\ell }\in \{1,\dots ,K\}$
3. Action region $R_{\ell }\subset \Lambda$
4. Angle parameters ${\theta }_{\ell ,j}=2\pi n_{\ell ,j}/2^m$

Bit Count

$$|{\Theta }_{\text{ini}}|=\lceil {\log }_2{D}_{\text{ini}}\rceil +D_{\text{ini}}\times (I_{\text{gate}}+I_{\text{region}}+I_{\text{angles}})$$

Translation-Invariant Case (typical):

• $D_{\text{ini}}=5$ (short-range entanglement sufficient)
• Gate type: ${\log }_{2}K=4$ bits/layer
• Action region: 1 bit/layer (odd/even)
• Angle parameters: $2\times 50=100$ bits/layer (2 angles, precision 50)

Total: $$|{\Theta }_{\text{ini}}|=\lceil {\log }_{2}5\rceil +5\times (4+1+100)=3+525=528\text{ bits}$$

$$\boxed{|{\Theta }_{\text{ini}}|\sim 500\text{ bits}}$$

Key Observation: $$|{\Theta }_{\text{ini}}|\sim 500\ll I_{\max }\sim 10^{123}$$

Initial state parameter information negligible!

Part V: Symmetry Constraints on Initial State

Translation-Invariant Initial State

Simplest symmetry: Translation invariance.

Definition 5.4 (Translation-Invariant Initial State):

$$V_{{\Theta }_{\text{ini}}}=\left(\underset{x\in \Lambda }{⨂}{V}_{\text{local}}\right)$$

Apply same local unitary $V_{\text{local}}$ to each lattice point.

Example:

$$V_{\text{local}}=R_y(\theta )=\exp (-i\theta {\sigma }_y/2)$$

Applied to all lattice points:

$$|{\Psi }_0⟩=\underset{x}{⨂}{R}_y(\theta )|\downarrow {⟩}_x$$

Encoding Overhead:

• Only need to encode $V_{\text{local}}$ (independent of $N_{\text{cell}}$!)
• Example: Single rotation gate + 1 angle parameter = ~50 bits

Information Compression:

• Without symmetry: $\sim N_{\text{cell}}\times I_{\text{local}}$ bits
• Translation-invariant: $\sim I_{\text{local}}$ bits
• Compression ratio: $N_{\text{cell}}\sim 10^{90}$ (astronomical number!)

Ground State and Thermal State

Ground State Initial State:

If initial state is ground state of some effective Hamiltonian $H_{\text{eff}}$:

$$|{\Psi }_0⟩=|\text{GS}(H_{\text{eff}})⟩$$

Encoding:

• Only need to encode $H_{\text{eff}}$ (usually implicit in ${\Theta }_{\text{dyn}}$)
• Additional information: $\sim 0$ bits (if agree “initial state = ground state”)

Thermal State Initial State (density matrix):

$${\rho }_0=\frac{e^{-\beta H_{\text{eff}}}}{Z}$$

Encoding:

• Hamiltonian $H_{\text{eff}}$ (already in ${\Theta }_{\text{dyn}}$)
• Temperature $\beta =1/k_{B}T$ (needs discretization, ~50 bits)

Total: $\sim 50$ bits

Symmetry Breaking and Phase Transitions

Spontaneous Symmetry Breaking:

If theory has symmetry, but initial state breaks symmetry:

Example (Ferromagnetic State):

• Hamiltonian: $H=-J{\sum }_{⟨i,j⟩}{\sigma }_i^z{\sigma }_j^z$ ($Z_2$ symmetric)
• Ground state: Two degenerate states $|\uparrow \uparrow \cdots \uparrow ⟩$ and $|\downarrow \downarrow \cdots \downarrow ⟩$
• Initial state: Choose one (breaks $Z_2$ symmetry)

Encoding:

• Hamiltonian (symmetric): In ${\Theta }_{\text{dyn}}$
• Which ground state chosen: 1 bit ($\uparrow$ or $\downarrow$)

Cosmological Application (Inflation and Vacuum Selection):

• After inflation, universe may fall into different “vacua”
• Each vacuum corresponds to different physical constants
• ${\Theta }_{\text{ini}}$ encodes “which vacuum chosen”

Part VI: Initial Entropy and Information

von Neumann Entropy of Initial State

Definition 5.5 (von Neumann Entropy):

For pure state:

$$S({\omega }_0)=-\text{tr}({\rho }_0\log {\rho }_0)=0$$

(pure state entropy zero)

For mixed state:

$$S({\rho }_0)=-\text{tr}({\rho }_0\log {\rho }_0)>0$$

Physical Meaning:

• Pure state: Completely determined quantum state, no classical uncertainty
• Mixed state: Partial uncertainty (thermal fluctuations, coarse-graining, etc.)

Initial Entropy and Universe Evolution

Theorem 5.6 (Unitary Evolution Preserves Entropy):

If evolution is unitary ($\alpha$ realized by unitary operator), then:

$$S({\omega }_n)=S({\omega }_0),\forall n$$

Corollary:

• If ${\omega }_0$ is pure state ($S=0$), then ${\omega }_n$ always pure state ($S=0$)
• Unitary evolution does not increase entropy

Question: Why does universe have second law of thermodynamics (entropy increase)?

Answer:

• Global quantum state: Pure state, entropy=0 (conserved)
• Local subsystems: Entanglement causes reduced states to be mixed, entropy>0
• Entanglement entropy growth = Apparent “thermodynamic entropy increase”

Popular Analogy:

• Two dice: Initially both in determined state (e.g., both 6)
• Operation: Entangle two dice (quantum gate)
• Look at single die: Becomes mixed state (seems random)
• But look at both together: Still pure state (completely determined entangled state)

Cosmological Application:

• Initial state: Extremely low entropy (near pure state)
• Evolution: Produces large entanglement
• What local observers see: Entropy increase (but global still pure state)

Initial State Complexity

Definition 5.7 (State Complexity):

Minimum circuit depth needed to generate state $|\Psi ⟩$:

$$C(|\Psi ⟩)=\min \{D:\exists V\text{ depth }D,V|0⟩=|\Psi ⟩\}$$

Properties:

• Simple states (e.g., product states): $C=O(1)$
• Highly entangled states (e.g., random states): $C=O(N_{\text{cell}})$

Finite Information Constraint:

Since $|{\Theta }_{\text{ini}}|<I_{\max }$, complexity cannot be too high:

$$C(|{\Psi }_0⟩)\lesssim I_{\max }/I_{\text{per-gate}}$$

Numerical Example:

• $I_{\max }=10^{123}$ bits
• Each gate encoding $\sim 100$ bits
• Maximum complexity: $C\lesssim 10^{121}$ layers

(Although still astronomical number, but much larger than $N_{\text{cell}}\sim 10^{90}$)

Part VII: QCA Version of Hartle-Hawking No-Boundary State

Classical Hartle-Hawking Proposal

Quantum Cosmology (Hartle-Hawking, 1983):

Universe wave function defined by path integral:

$$|\Psi [h_{ij},\varphi ]={\int }_{\text{compact}}\mathcal{D}[g,\varphi ]e^{iS[g,\varphi ]/\hslash }$$

where integral over “no-boundary” compact four-geometries.

Physical Meaning:

• Universe “spontaneously emerges”, no initial singularity needed
• Time becomes “imaginary time” in very early universe (Euclidean geometry)
• Similar: Quantum tunneling

Problems:

• Path integral diverges on continuous geometries
• Need to introduce cutoff or regularization
• Wave function defined on infinite-dimensional superspace

QCA Version: Minimum Depth Principle

In QCA framework, can similarly define:

QCA No-Boundary Principle:

Initial state $|{\Psi }_0⟩$ chosen by following condition:

$$\boxed{|{\Psi }_0⟩=\arg \underset{V}{\min }\left\{D(V):V|0_{\Lambda }⟩\text{ satisfies certain symmetry constraints}\right\}}$$

where $D(V)$ is depth of circuit $V$.

Physical Meaning:

• Universe chooses “simplest” (minimum depth) initial state compatible with symmetries
• Similar: Principle of least action
• “Occam’s razor”: Simplest explanation without additional assumptions

Example (Translation-Invariant + Low Energy):

Constraints:

1. Translation invariance
2. Energy below threshold $E<E_0$

Result:

• Unique solution (in symmetry class): Ground state $|\text{GS}⟩$
• Depth: $D=0$ (if ground state = reference state) or $D=O(1)$

Connection with IGVP:

In GLS theory, IGVP (Information Geometric Variational Principle) derives Einstein’s equations from entropy variation. Similarly:

$$\delta S_{\text{complexity}}=0\Rightarrow \text{Choose minimum depth initial state}$$

Conjecture (not strictly proven):

• Minimum complexity $\iff$ Maximum symmetry
• Maximum symmetry $\Rightarrow$ Near-uniform initial state of inflationary universe
• This perhaps explains why universe’s initial state so “special” (low entropy)

Part VIII: Measurement and Observation of Initial State Parameters

How Do We Know Initial State?

Question: We are at moment $n\sim 10^{60}$ (universe age), how to infer initial state at $t=0$?

Answer: Reverse inference from current observations.

Cosmic Microwave Background (CMB):

• Observation: Temperature fluctuations $\Delta T/T\sim 10^{-5}$ (anisotropy)
• Power spectrum: $C_{\ell }$ as function of $\ell$
• Inference: Initial density fluctuation spectrum $P(k)$

Inflation Theory Predictions:

• Near scale-invariant spectrum: $P(k)\propto k^{n_s}$, $n_s\approx 0.96$
• Gaussian distribution (minimal non-Gaussianity)
• Adiabatic fluctuations (not isocurvature)

QCA Language Translation:

• Near scale-invariant $\Rightarrow$ Initial state approximately translation-invariant in momentum space
• Gaussian $\Rightarrow$ Simple entanglement structure (near product state)
• Adiabatic $\Rightarrow$ Some symmetry (e.g., supersymmetry in early universe)

“Archaeology” of Initial State Parameters

Analogy: Archaeologists infer ancient civilizations from relics.

Current Measurement Precision:

• CMB temperature fluctuations: $\sim 10^{-6}$ K (Planck satellite)
• Large-scale structure: Galaxy surveys (SDSS, DES, LSST)
• Primordial gravitational waves: Not yet detected (target: $r\sim 10^{-3}$)

Constraints on ${\Theta }_{\text{ini}}$:

• Spectral index $n_s$: Constrains certain angle parameters
• Tensor-to-scalar ratio $r$: Constrains shape of inflation potential
• Non-Gaussianity $f_{\text{NL}}$: Constrains nonlinear interactions

Future Prospects:

• 21cm hydrogen line observations (“cosmic dawn”)
• Primordial black hole detection
• Quantum gravity effects (possibly at very small scales)

Summary of Core Points of This Article

Definition of Initial State Parameters

$${\Theta }_{\text{ini}}=(D_{\text{ini}},\{k_{\ell }\},\{R_{\ell }\},\{{\theta }_{\ell ,j}\})$$

Generate Initial State: $$|{\Psi }_0^{\Theta }⟩=V_{{\Theta }_{\text{ini}}}|0_{\Lambda }⟩$$

Reference Product State

$$|0_{\Lambda }⟩=\underset{x\in \Lambda }{⨂}|0_{\text{cell}}{⟩}_x$$

Properties: No entanglement, translation-invariant, simple.

Bit Count

$$\boxed{|{\Theta }_{\text{ini}}|\sim 500\text{ bits}}$$

Consequences of Finite Depth

Lieb-Robinson Constraint: $$I(A:B)\lesssim e^{-c(d-v_{\text{LR}}{D}_{\text{ini}})}$$

Physical Meaning: Short-range entangled state, depth $D\sim \log N_{\text{cell}}$ sufficient.

Symmetry Compression

Translation-Invariant:

• Encoding overhead: $O(1)$ (independent of $N_{\text{cell}}$)
• Compression ratio: $\sim 10^{90}$

Initial State and Evolution

$${\omega }_n={\omega }_0\circ {\alpha }^{-n}$$

Complete Universe History: $$\Theta =({\Theta }_{\text{str}},{\Theta }_{\text{dyn}},{\Theta }_{\text{ini}})\Rightarrow \text{Entire Universe Evolution}$$

Core Insights

1. Initial state parameters tiny: $|{\Theta }_{\text{ini}}|\sim 500\ll I_{\max }$
2. Symmetry necessary: Translation-invariance → Information from $10^{90}$ reduced to $10^2$
3. Short-range entanglement: Finite depth → Long-range entanglement impossible
4. Initial state inferable: CMB and other observations → Constrain ${\Theta }_{\text{ini}}$
5. No-boundary principle: Minimum complexity → Naturally selects simple initial state

Key Terminology

• Reference Product State: $|0_{\Lambda }⟩$
• State Preparation Circuit: $V_{{\Theta }_{\text{ini}}}$
• Short-Range Entanglement: Entanglement range limitation due to finite depth
• State Complexity: $C=\min \{D\}$
• Hartle-Hawking No-Boundary State: QCA version of minimum depth principle

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

• Detailed derivation of finite information inequality $I_{\text{param}}+S_{\max }\le I_{\max }$
• Trade-off relation between number of cells $N_{\text{cell}}$ and local dimension $d_{\text{cell}}$
• Information budget allocation of observable universe
• Why symmetry, locality, finite precision are necessary
• Limitations of information constraints on physical theories