Section 05: ETH Constraint—Post-Chaotic QCA and Thermalization Mystery

Introduction: Why Do Isolated Systems Thermalize?

Imagine a completely sealed thermos bottle containing a mixture of cold and hot water in layers. You do nothing to it, just wait. After a few hours, you open it and find: Water temperature is uniform!

This seems natural, but from quantum mechanics perspective, this is a profound mystery:

Problem 1 (Reversibility Paradox):

• Quantum evolution is unitary (reversible): $|\psi (t)⟩=U(t)|\psi (0)⟩$
• But thermalization is irreversible: Entropy increases, cannot spontaneously return to initial state
• How are these compatible?

Problem 2 (Time Arrow):

• Microscopic physical laws (Schrödinger equation) are time-reversal symmetric
• But macroscopic thermodynamics has clear time arrow (entropy always increases)
• Where does this arrow come from?

Problem 3 (Typicality):

• Why do “almost all” initial states thermalize?
• Why does final state look like “thermal equilibrium”?

Eigenstate Thermalization Hypothesis (ETH) gives the answer:

For isolated quantum many-body systems, if Hamiltonian is “sufficiently chaotic”, then almost all high-energy eigenstates have expectation values on local observables equal to thermal equilibrium values at corresponding energy.

This section will show: In unified constraint system, ETH constraint ${\mathcal{C}}_{\mathrm{ETH}}(\Theta )=0$ requires QCA universe to exhibit “post-chaotic” behavior on every finite region, thus guaranteeing natural emergence of macroscopic thermalization and time arrow.

Part I: Quantum Mechanical Foundation of Thermalization

1.1 From Pure States to Mixed States?

Classical Thermodynamics:

An isolated system eventually reaches thermal equilibrium, expectation value of observable $O$ given by microcanonical ensemble: $$⟨O{⟩}_{\text{thermal}}=\frac{1}{\Omega (E)}\sum _{n:E_n\in [E-\delta E,E+\delta E]}⟨n|O|n⟩$$

where $\Omega (E)$ is number of states in energy shell, $|n⟩$ are energy eigenstates.

Quantum Mechanics:

If system initially in pure state $|\psi (0)⟩$, after evolution still pure state: $$|\psi (t)⟩=e^{-iHt/\hslash }|\psi (0)⟩=\sum _n{c}_n{e}^{-i{E}_{n}t/\hslash }|n⟩$$

where $c_n=⟨n|\psi (0)⟩$.

Contradiction?

• von Neumann entropy of pure state’s reduced density matrix $\rho =|\psi ⟩⟨\psi |$ always zero
• But thermal equilibrium state is mixed state, entropy $\sim \ln \Omega (E)>0$

Resolution: Subsystem Perspective

Consider a small subsystem $A$ of system. Although total system remains pure state, subsystem’s reduced density matrix: $${\rho }_A(t)={\mathrm{Tr}}_{\bar{A}}|\psi (t)⟩⟨\psi (t)|$$

can tend to mixed state!

graph TB
    Total["Total System<br/>Pure State |ψ⟩"]

    Total --> SubA["Subsystem A<br/>Reduced Density Matrix ρ_A"]
    Total --> SubB["Environment Ā<br/>(Rest)"]

    SubA --> Trace["Trace Out Environment<br/>Tr_Ā |ψ⟩⟨ψ|"]

    Trace --> Mixed["Tend to Mixed State<br/>ρ_A → ρ_thermal"]

    Mixed --> Entropy["Entanglement Entropy Increases<br/>S_A = -Tr(ρ_A log ρ_A) > 0"]

    style Total fill:#2196f3,color:#fff
    style SubA fill:#4caf50,color:#fff
    style SubB fill:#9e9e9e,color:#fff
    style Mixed fill:#ff9800,color:#fff
    style Entropy fill:#f44336,color:#fff

1.2 Time Average vs Ensemble Average

Time Average:

For local observable $O_A$ (only acts on subsystem $A$): $$\overline{O_A}:=\underset{T\to \infty }{\lim }\frac{1}{T}{\int }_0^T⟨\psi (t)|O_A|\psi (t)⟩\mathrm{d}t$$

Ensemble Average:

Microcanonical ensemble (fixed energy $E$): $$⟨O_A{⟩}_{\text{mc}}(E)=\frac{1}{\Omega (E)}\sum _{E_n\in [E-\delta E,E+\delta E]}⟨n|O_A|n⟩$$

Question: When are they equal?

$$\overline{O_A}\stackrel{?}{=}⟨O_A{⟩}_{\text{mc}}$$

Classical Ergodicity: In classical mechanics, if system is ergodic, then time average = ensemble average.

Quantum Version: ETH gives ergodicity condition for quantum systems.

Part II: Eigenstate Thermalization Hypothesis (ETH)

2.1 Mathematical Formulation of ETH

Definition:

For Hamiltonian $H$ and local operator $O_A$, if satisfies:

（ETH-1）Diagonal Elements:

$$⟨n|O_A|n⟩=O_A(E_n)+\mathcal{O}(e^{-S(E_n)/2})$$

where $O_A(E)$ is smooth function of energy $E$ (corresponding to microcanonical average), $S(E)$ is thermodynamic entropy.

（ETH-2）Off-Diagonal Elements:

$$|⟨m|O_A|n⟩|\sim e^{-S(\bar{E})/2}{f}_{O_A}(E_m,E_n)$$

where $\bar{E}=(E_m+E_n)/2$, $f$ is smooth function.

Physical Meaning:

• Diagonal elements: Each eigenstate $|n⟩$’s expectation value close to thermal equilibrium value
• Off-diagonal elements: Matrix elements between different eigenstates decay exponentially with entropy

graph LR
    Eigenstate["Energy Eigenstate |n⟩"]

    Eigenstate --> Diagonal["Diagonal Element<br/>⟨n|O|n⟩ ≈ O(E_n)"]
    Eigenstate --> OffDiag["Off-Diagonal Element<br/>|⟨m|O|n⟩| ~ exp(-S/2)"]

    Diagonal --> Thermal["≈ Microcanonical Average<br/>⟨O⟩_mc(E)"]
    OffDiag --> Suppressed["Exponential Suppression<br/>Fast Decoherence"]

    Thermal --> Result["Thermalization!"]
    Suppressed --> Result

    style Eigenstate fill:#2196f3,color:#fff
    style Diagonal fill:#4caf50,color:#fff
    style OffDiag fill:#ff9800,color:#fff
    style Result fill:#f44336,color:#fff

2.2 ETH Implies Thermalization

Theorem: If $H$ satisfies ETH, and initial state $|\psi (0)⟩$ has broad distribution in energy shell near $E$ ($|c_n{|}^2$ not concentrated in few states), then:

$$\underset{t\to \infty }{\lim }⟨\psi (t)|O_A|\psi (t)⟩=⟨O_A{⟩}_{\text{mc}}(E)$$

Proof Sketch:

1. Expand $|\psi (t)⟩={\sum }_n{c}_n{e}^{-i{E}_{n}t}|n⟩$

2. Calculate expectation value: $$⟨O_A⟩(t)=\sum _n|c_n{|}^2⟨n|O_A|n⟩+\sum _{m\ne n}{c}_m^{\ast }{c}_n{e}^{i(E_m-E_n)t}⟨m|O_A|n⟩$$

3. Time average: Off-diagonal terms oscillate to zero

4. Use ETH-1: Diagonal terms $\approx O_A(E_n)\approx O_A(E)$ (because $E_n$ all near energy shell)

5. Conclusion: $$\overline{O_A}=\sum _n|c_n{|}^2{O}_A(E)=O_A(E)=⟨O_A{⟩}_{\text{mc}}(E)$$

2.3 Connection to Random Matrix Theory

Haar Random Unitary Matrix:

If $H$’s eigenstates randomly sampled from Haar measure, automatically satisfies ETH.

Reason:

For Haar random states $|n⟩$, matrix element statistics: $$\mathbb{E}[⟨n|O_A|n⟩]=\frac{\mathrm{Tr}{O}_A}{D}$$ $$\mathrm{Var}[⟨n|O_A|n⟩]=\mathcal{O}(D^{-1})$$

where $D=\dim \mathcal{H}$ is Hilbert space dimension.

Levy Concentration Inequality:

$$\mathbb{P}\left(\left|⟨n|O_A|n⟩-\mathbb{E}[⟨n|O_A|n⟩]\right|>ϵ\right)\le C{e}^{-c{ϵ}^{2}D}$$

Therefore “typical” eigenstates are all very close to average!

Part III: Post-Chaotic QCA: Microscopic Realization of ETH

3.1 What Is Post-Chaotic QCA?

Definition:

In QCA universe, if on any finite region $\Omega \subset \Lambda$, time evolution operator $U_{\Omega }$ satisfies:

1. Locality: $U_{\Omega }$ can be decomposed into finite-depth local gate circuit
2. Chaoticity: Local gate set generates approximate $t$-th order unitary design (approximate Haar distribution) after several layers
3. Conservation Law Minimization: Besides energy and few global quantum numbers, no other local conserved quantities

Then this QCA is called post-chaotic QCA.

graph TB
    QCA["QCA Lattice Λ"]

    QCA --> Region["Finite Region Ω"]

    Region --> Gates["Local Gate Circuit<br/>Depth d"]

    Gates --> Layer1["Layer 1<br/>U₁,ₓ"]
    Gates --> Layer2["Layer 2<br/>U₂,ₓ"]
    Gates --> LayerD["Layer d<br/>Uᵈ,ₓ"]

    Layer1 --> Random["Randomization"]
    Layer2 --> Random
    LayerD --> Random

    Random --> Haar["Approximate Haar Distribution<br/>t-th Order Unitary Design"]

    Haar --> ETH["Satisfies ETH<br/>On Local Operators"]

    style QCA fill:#2196f3,color:#fff
    style Region fill:#fff4e6
    style Gates fill:#e8f5e9
    style Haar fill:#ff9800,color:#fff
    style ETH fill:#4caf50,color:#fff

3.2 Local Random Circuits and Unitary Designs

Local Gate Set:

At each time step, apply local unitary gates $U_{xy}$ (acting on adjacent sites $x,y$): $$U_{\text{layer}}=\prod _{⟨x,y⟩}{U}_{xy}$$

Randomization:

Gates $U_{xy}$ randomly sampled from some distribution $\mathcal{D}$ (or selected in pseudo-random sequence).

Unitary Design:

If after $d$ layers, $U=U_d\cdots U_2{U}_1$ has average on polynomial functions close to Haar integral, called $t$-th order design: $${\mathbb{E}}_{\mathcal{D}}[f(U)]\approx {\int }_{\mathrm{Haar}}f(U)\mathrm{d}U$$

Holds for all $t$-th order polynomials $f$.

Depth Estimate:

For local systems, depth needed to reach $t$-th order design: $$d\sim \mathcal{O}(L\cdot t)$$

where $L$ is linear size of system.

3.3 From Random Circuits to ETH

Key Theorem (ETH for Haar Random States):

For eigenstates $|n⟩$ of Haar random unitary operator $U$, matrix elements of local operator $O_A$ satisfy:

$$⟨n|O_A|n⟩=\frac{\mathrm{Tr}{O}_A}{D}+\mathcal{O}\left(\frac{\Vert O_A\Vert }{\sqrt{D}}\right)$$

$$|⟨m|O_A|n⟩|\sim \frac{\Vert O_A\Vert }{\sqrt{D}}(m\ne n)$$

Generalization to QCA:

If $U_{\Omega }$ is approximate $t$-th order design ($t\ge 2$), then satisfies ETH on local operators, error: $$\mathcal{O}\left(e^{-c|\Omega |}\right)$$

where $c>0$ is constant.

Part IV: Definition of Constraint Function ${\mathcal{C}}_{\mathrm{ETH}}(\Theta )$

4.1 Finite Volume ETH Deviation

Select Representative Local Operator Family:

$${\mathcal{O}}_{\text{loc}}=\{O_1,O_2,\dots ,O_K\}$$

For example: local density, current, energy density, etc.

On Finite Region ${\Omega }_L$ (Linear Size $L$):

1. Calculate eigenstates $|E_{\alpha }⟩$ of Hamiltonian $H_L(\Theta )$
2. Select energy window $[E_1,E_2]$
3. For each operator $O\in {\mathcal{O}}_{\text{loc}}$, calculate:

Diagonal Element Deviation: $${\Delta }_{\text{diag}}^{(O)}(L;\Theta )=\underset{E\in [E_1,E_2]}{\sup }\frac{1}{\Omega (E)}\sum _{E_{\alpha }\in [E-\delta E,E+\delta E]}\left|⟨E_{\alpha }|O|E_{\alpha }⟩-⟨O{⟩}_{\text{mc}}(E)\right|$$

Off-Diagonal Element Deviation: $${\Delta }_{\text{off}}^{(O)}(L;\Theta )=\underset{\alpha \ne \beta }{\sup }\frac{|⟨E_{\alpha }|O|E_{\beta }⟩|}{e^{-S(\bar{E})/2}}$$

Total ETH Deviation: $${\mathrm{ETH}\mathrm{deviation}}_L(\Theta )=\underset{O\in {\mathcal{O}}_{\text{loc}}}{\max }\left[{\Delta }_{\text{diag}}^{(O)}(L;\Theta )+{\Delta }_{\text{off}}^{(O)}(L;\Theta )\right]$$

4.2 Thermodynamic Limit

Constraint Function:

$$\boxed{{\mathcal{C}}_{\mathrm{ETH}}(\Theta )=\underset{L\to \infty }{\mathrm{limsup}}{\mathrm{ETH}\mathrm{deviation}}_L(\Theta )}$$

Physical Requirement:

$${\mathcal{C}}_{\mathrm{ETH}}(\Theta )=0\iff \text{In thermodynamic limit, QCA strictly satisfies ETH}$$

graph TB
    Finite["Finite Region Ω_L<br/>Size L"]

    Finite --> Compute["Calculate Eigenstates<br/>|E_α⟩"]

    Compute --> Diagonal["Check Diagonal Elements<br/>Δ_diag"]
    Compute --> OffDiag["Check Off-Diagonal Elements<br/>Δ_off"]

    Diagonal --> Deviation["ETH Deviation<br/>ETH_deviation_L(Θ)"]
    OffDiag --> Deviation

    Deviation --> Limit["Thermodynamic Limit<br/>L → ∞"]

    Limit --> Constraint["Constraint Function<br/>C_ETH(Θ) = limsup ETH_deviation"]

    Constraint --> Zero["C_ETH = 0<br/>QCA Satisfies ETH"]

    style Finite fill:#e1f5ff
    style Deviation fill:#fff4e6
    style Limit fill:#e8f5e9
    style Constraint fill:#ff9800,color:#fff
    style Zero fill:#4caf50,color:#fff

4.3 Parameter Dependence

From $\Theta$ to ETH:

1. Local Gate Set: Determined by “ETH data” in $\Theta$
2. Propagation Radius $R(\Theta )$: Determines causal cone shape
3. Local Hilbert Dimension $d_{\text{cell}}(\Theta )$: Affects speed of chaos
4. Energy Spectrum Statistics: Indirectly affected through $\kappa (\omega ;\Theta )$

Key Parameters:

• Gate Depth $d(\Theta )$: Number of layers needed to reach $t$-th order design
• Level Spacing Statistics: Should follow random matrix theory (Wigner-Dyson statistics)

Part V: Spectral Density Tension with Cosmological Constant

5.1 Contradictory Requirements

Cosmological Constant Constraint:

$${\int }_0^{E_{\mathrm{UV}}}{E}^2\Delta \rho (E;\Theta )\mathrm{d}E=0$$

Requires bands highly structured (precise pairing).

ETH Constraint:

Requires energy spectrum locally highly chaotic (no degeneracy, random distribution).

Apparent Contradiction:

• Sum rule → global structure
• ETH → local chaos

How are these compatible?

graph TB
    Spectrum["Energy Spectrum ρ(E,Θ)"]

    Spectrum --> Global["Global Perspective"]
    Spectrum --> Local["Local Perspective"]

    Global --> Paired["Band Pairing<br/>∫ E² Δρ dE = 0"]
    Local --> Chaotic["Level Chaos<br/>Random Matrix Statistics"]

    Paired --> Lambda["Cosmological Constant Constraint<br/>C_Λ = 0"]
    Chaotic --> ETH_C["ETH Constraint<br/>C_ETH = 0"]

    Lambda --> Tension["Tension!"]
    ETH_C --> Tension

    Tension --> Resolution["Solution?"]

    style Spectrum fill:#2196f3,color:#fff
    style Paired fill:#ff9800,color:#fff
    style Chaotic fill:#9c27b0,color:#fff
    style Tension fill:#f44336,color:#fff
    style Resolution fill:#4caf50,color:#fff

5.2 Solution: Frequency Band and Scale Separation

Key Idea: Sum rule and ETH act on different scales.

Sum Rule’s Domain:

• High Energy Region $E\gg E_{\mathrm{IR}}$: Overall band topology
• Global Property: Average over all lattice sites

ETH’s Domain:

• Mid-Low Energy Region: Local excitations
• Finite Region: $\Omega \subset \Lambda$, $|\Omega |\ll |\Lambda |$

Compatibility Mechanism:

1. Global Pairing, Local Chaos:

   • On global topology of Brillouin zone, bands pair to satisfy sum rule
   • But in local energy shell, level statistics follow random matrix

2. Coarse-Graining Separation:

   • Sum rule holds on energy scale $\Delta E\sim E_{\mathrm{UV}}$
   • ETH holds in energy shell with energy scale $\delta E\ll E_{\mathrm{UV}}$

5.3 Numerical Example

Assumptions:

• $E_{\mathrm{UV}}\sim M_{\mathrm{Pl}}\sim 10^{19}$ GeV
• Local energy shell width $\delta E\sim 1$ GeV
• Level spacing $\Delta E\sim e^{-S}\sim 10^{-100}$ eV (extremely small!)

Results:

• Sum rule holds on entire interval $[0,E_{\mathrm{UV}}]$
• ETH holds in narrow energy shell $[E-\delta E,E+\delta E]$
• Energy shell contains $\sim (\delta E/\Delta E)\sim 10^{100}$ levels
• So many levels sufficient to exhibit random matrix statistics

Conclusion: Two constraints can be compatible!

Part VI: Natural Emergence of Time Arrow

6.1 From Microscopic Reversibility to Macroscopic Irreversibility

Microscopic:

QCA evolution $U=\alpha (\mathbb{Z})$ is unitary, completely reversible.

Macroscopic:

• Initial state: Low entropy (e.g., cold-hot layering)
• Final state: High entropy (thermal equilibrium)
• Entropy increase: $S(t_{\text{final}})>S(t_{\text{initial}})$

Role of ETH:

ETH guarantees “almost all” high-energy eigenstates are high-entropy states. Therefore:

1. Initial state expanded in energy eigenstates: $|{\psi }_0⟩={\sum }_n{c}_n|n⟩$
2. If initial state low entropy, then $|c_n{|}^2$ concentrated in few states
3. After evolution, although $|\psi (t)⟩$ still pure state, its reduced density matrix tends to mixed state
4. Subsystem entropy increases

graph LR
    Initial["Initial State<br/>Low Entropy<br/>Non-Equilibrium"]

    Initial -->|"Unitary Evolution U(t)"| Evolve["Evolving<br/>Pure State Maintained"]

    Evolve --> Subsystem["Subsystem Perspective"]

    Subsystem --> Reduced["Reduced Density Matrix<br/>ρ_A = Tr_Ā |ψ⟩⟨ψ|"]

    Reduced --> Mixed["Tend to Mixed State<br/>ρ_A → ρ_thermal"]

    Mixed --> Final["Final<br/>High Entropy<br/>Thermal Equilibrium"]

    Initial -.->|"Time Arrow"| Final

    style Initial fill:#2196f3,color:#fff
    style Evolve fill:#fff4e6
    style Mixed fill:#ff9800,color:#fff
    style Final fill:#f44336,color:#fff

6.2 Resolution of Boltzmann Brain Paradox

Boltzmann Brain Paradox:

In infinite time, quantum fluctuations of thermal equilibrium state may produce “low-entropy bubbles” (including “observer brains”), probability far higher than producing complex structures through normal evolution.

ETH’s Answer:

1. Typicality: ETH says, “typical” high-energy eigenstates are all thermal equilibrium states
2. Initial State Selection: Universe did not start from thermal equilibrium state, but from special low-entropy initial state
3. Finite Time: Within universe age (~13.8 billion years), not yet reached Poincaré recurrence time ($\sim e^{S_{\text{universe}}}$ years)

Conclusion: Low-entropy history we observe is due to special initial state, not thermal fluctuations.

Part VII: Experimental and Numerical Tests

7.1 Cold Atom Quantum Simulation

Platforms:

• Ultracold atoms in optical lattices
• Rydberg atom arrays
• Ion traps

Experimental Scheme:

1. Prepare initial state (e.g., Néel state $|\uparrow \downarrow \uparrow \downarrow \cdots ⟩$)
2. Apply tunable Hamiltonian (through lasers and magnetic fields)
3. Evolve for some time
4. Measure local observables (density, magnetization, etc.)
5. Test whether tend to thermal equilibrium values

Existing Experiments:

• ETH Verification (Kaufman et al., Science 2016): Verified ETH in Rydberg atom arrays
• Many-Body Localization (MBL) (Schreiber et al., Science 2015): Observed ETH breaking in strongly disordered systems

7.2 Numerical Simulation: Exact Diagonalization

Method:

For small systems ($N\lesssim 20$ sites), can exactly diagonalize Hamiltonian, directly check ETH.

Test Content:

1. Scatter plot of diagonal elements $⟨n|O|n⟩$ vs $O(E_n)$
2. Scaling of off-diagonal elements $|⟨m|O|n⟩|$ vs $e^{-S/2}$
3. Level spacing statistics: Wigner-Dyson vs Poisson

Typical Results:

• Integrable Systems (e.g., XXZ chain at $\Delta =1$): Do not satisfy ETH, level statistics Poisson
• Chaotic Systems (e.g., XXZ chain at $\Delta \ne 1$): Satisfy ETH, level statistics Wigner-Dyson

graph TB
    Integrable["Integrable System<br/>(e.g., Heisenberg Chain)"]
    Chaotic["Chaotic System<br/>(e.g., Generic Hamiltonian)"]

    Integrable --> NoETH["Does Not Satisfy ETH"]
    Chaotic --> ETH["Satisfies ETH"]

    NoETH --> Poisson["Level Statistics: Poisson<br/>⟨s⟩ = 1, No Level Repulsion"]
    ETH --> Wigner["Level Statistics: Wigner-Dyson<br/>P(s) ~ s^β, Level Repulsion"]

    Poisson --> Conserved["Multiple Local Conserved Quantities<br/>Prevent Thermalization"]
    Wigner --> Thermalize["Thermalization<br/>Time Arrow"]

    style Integrable fill:#2196f3,color:#fff
    style Chaotic fill:#f44336,color:#fff
    style NoETH fill:#9e9e9e,color:#fff
    style ETH fill:#4caf50,color:#fff

7.3 Indirect Tests in QCA Universe

If QCA Universe Satisfies ETH, should have following observable consequences:

1. Macroscopic Second Law of Thermodynamics: Naturally emerges (observed ✓)
2. Black Hole Thermalization: Fast scrambling inside black hole (theoretical prediction, not directly observed)
3. Early Universe: Started from high-temperature equilibrium state (thermal spectrum of CMB ✓)
4. Thermalization Universality of Quantum Many-Body Systems: Except special cases like MBL, all thermalize (experimental support ✓)

Part VIII: Summary of This Section

8.1 Core Mechanism

1. ETH Conditions:

   • Diagonal elements: $⟨n|O|n⟩\approx O(E_n)$
   • Off-diagonal elements: $|⟨m|O|n⟩|\sim e^{-S/2}$

2. Post-Chaotic QCA:

   • Local random circuits
   • Approximate unitary designs
   • Conservation law minimization

3. Thermalization Mechanism:

   • Typical eigenstates ≈ thermal equilibrium states
   • Time arrow naturally emerges
   • Subsystem entropy increases

8.2 Constraint Function

$${\mathcal{C}}_{\mathrm{ETH}}(\Theta )=\underset{L\to \infty }{\mathrm{limsup}}\underset{O\in {\mathcal{O}}_{\text{loc}}}{\sup }\left[{\Delta }_{\text{diag}}^{(O)}+{\Delta }_{\text{off}}^{(O)}\right]$$

Requires QCA to strictly satisfy ETH in thermodynamic limit.

8.3 Coupling with Other Constraints

• Cosmological Constant Constraint: Structurality vs chaoticity of spectral density (compatible through frequency band separation)
• Black Hole Entropy Constraint: Chaoticity of horizon cells guarantees information scrambling
• Neutrino/Strong CP: Indirectly affected through energy spectrum statistics

8.4 Physical Insight

Key Idea:

Thermalization is not additional “physical law”, but typical behavior of quantum chaotic systems. ETH constraint guarantees universe is “sufficiently chaotic” microscopically, thus automatically exhibits thermodynamics macroscopically.

Origin of Time Arrow:

Time arrow does not come from microscopic physical laws (they are time-reversal symmetric), but from special initial state + thermalization universality guaranteed by ETH.

8.5 Preview of Next Section

Section 6 will explore Strong CP Constraint ${\mathcal{C}}_{\mathrm{CP}}(\Theta )=0$:

• Why does QCD almost not violate CP symmetry?
• Physical meaning of topological class $[K]=0$
• Realization of axion mechanism in unified framework

Theoretical Sources for This Section

Content of this section based on following source theory files:

1. Primary Sources:

   • docs/euler-gls-extend/six-unified-physics-constraints-matrix-qca-universe.md
     • Section 3.4 (Theorem 3.4): Local ETH of post-chaotic QCA
     • Section 4.4 (Proof): Statistical properties of Haar random eigenstates, local random circuits and designs, derivation of local ETH
     • Appendix C: Design estimates for post-chaotic QCA and ETH

2. Auxiliary Sources:

   • docs/euler-gls-info/19-six-problems-unified-constraint-system.md
     • Section 3.4: Definition of ETH constraint function ${\mathcal{C}}_{\mathrm{ETH}}(\Theta )$
     • Appendix B.4: Differentiability of ETH and gravitational wave constraints

All formulas, mechanisms, physical discussions come from above source files and standard ETH literature (D’Alessio et al., Rigol et al., etc.), no speculation or fabrication.