11 Boundary Language: Where Time “Speaks”

Core Idea

In previous chapters, we saw:

• Time is interpreted as the optimal path of entropy (Section 8)
• Force can be viewed as the projection of time geometry (Section 9)
• Time structure might be determined by topological invariants (Section 10)

Now we ask a more fundamental question: Where might time be defined?

Traditional physics believes time is defined in spatial interior. But GLS theory offers a unique theoretical perspective:

GLS theory proposes: Time might be defined on the boundary. All information about time is theoretically “spoken” by the boundary.

Just as a book’s content can be read from the barcode on its cover, the time structure of the universe might be completely determined by its boundary. This is the core idea of Boundary Language.

Everyday Analogy: Door Frame of a Room

Imagine you want to understand what happens in a room:

graph TB
    subgraph "Traditional View: Interior Priority"
        Interior["🏠 Room Interior<br/>(What Really Happens)"]
        Door1["🚪 Door<br/>(Just a Passage)"]

        Interior -->|"Door is Just Accessory"| Door1
    end

    subgraph "Boundary Language: Boundary Priority"
        Door2["🚪 Door Frame<br/>(Boundary)"]
        Interior2["🏠 Room Interior<br/>(Can be Derived from Boundary)"]

        Door2 -->|"Boundary Determines Interior"| Interior2

        Door2 -->|"Measure"| Flow["Flux:<br/>· How Many Enter<br/>· How Many Leave<br/>· How Much Energy Carried Away"]

        Flow -.->|"Completely Determines"| Interior2
    end

    style Interior fill:#ffe66d,stroke:#f59f00
    style Door1 fill:#e9ecef,stroke:#495057
    style Door2 fill:#ff6b6b,stroke:#c92a2a,stroke-width:4px
    style Interior2 fill:#4ecdc4,stroke:#0b7285
    style Flow fill:#a9e34b,stroke:#5c940d,stroke-width:3px

Theoretical Insight:

• Traditional View: Room interior is fundamental, door is just “entrance/exit”
• Boundary Language: Theoretically, by measuring who enters/exits, what they carry at the door frame, we can infer the state of the room interior
• “Time passage” in room interior = “flux change” measured at door frame

Three Axioms of Boundary Language

GLS theory attempts to define “boundary language” with three axioms:

graph TB
    BL["🗣️ Boundary Language<br/>𝔏_Σ = (𝒜_∂, ω, ℱ)"]

    BL --> A1["Axiom A1:<br/>Conservation and Flux"]
    BL --> A2["Axiom A2:<br/>Time Generation"]
    BL --> A3["Axiom A3:<br/>Monotonicity and Consistency"]

    A1 -->|"All Energy, Information<br/>Crossing Boundary Measurable"| C1["Exchange Across Boundary<br/>= Flux Functional ℱ"]

    A2 -->|"Time Translation Operator<br/>Exists on Boundary"| C2["Time = Automorphism Group<br/>of Boundary Algebra {α_t}"]

    A3 -->|"Information Cannot<br/>Come from Nothing"| C3["Relative Entropy Monotonic<br/>dS_rel/dt ≤ 0"]

    style BL fill:#ff6b6b,stroke:#c92a2a,stroke-width:4px
    style A1 fill:#4ecdc4,stroke:#0b7285,stroke-width:3px
    style A2 fill:#4ecdc4,stroke:#0b7285,stroke-width:3px
    style A3 fill:#4ecdc4,stroke:#0b7285,stroke-width:3px
    style C1 fill:#ffe66d,stroke:#f59f00
    style C2 fill:#ffe66d,stroke:#f59f00
    style C3 fill:#ffe66d,stroke:#f59f00

Axiom A1: Conservation and Flux

Everyday Analogy: Bank Account

graph LR
    Account["💰 Bank Account<br/>(Room Interior)"]

    In["💵 Deposit<br/>(Incoming Flux)"]
    Out["💸 Withdrawal<br/>(Outgoing Flux)"]

    In -->|"Recorded On"| Statement["📊 Bank Statement<br/>(Boundary Record)"]
    Out --> Statement

    Statement -.->|"Completely Determines"| Account

    Balance["Balance Change<br/>= Σ Deposits - Σ Withdrawals"]

    Statement --> Balance

    style Account fill:#4ecdc4,stroke:#0b7285
    style In fill:#a9e34b,stroke:#5c940d
    style Out fill:#ff6b6b,stroke:#c92a2a
    style Statement fill:#ffe66d,stroke:#f59f00,stroke-width:3px
    style Balance fill:#e9ecef,stroke:#495057

Mathematical Expression:

$$\delta (S_{\mathrm{bulk}}+S_{\mathrm{bdry}})=\text{(Volume Integral)}+F(\delta X_{\Sigma })$$

Where:

• $S_{\mathrm{bulk}}$ = Interior action
• $S_{\mathrm{bdry}}$ = Boundary action
• $F$ = Flux functional (records exchange across boundary)
• $\delta X_{\Sigma }$ = Boundary source variation

Physical Meaning: Account balance (interior state) is completely determined by statement (boundary flux) (in an ideal scenario)!

Axiom A2: Time Generation

Everyday Analogy: Revolving Door

graph TB
    Door["🚪 Revolving Door<br/>(Boundary)"]

    Door -->|"Rotation Parameter t"| Rotation["Rotation Angle θ(t)"]

    Rotation -->|"Change in People Entering/Leaving"| Count["People Count N(t)"]

    Count -.->|"Defines"| Time["Time t<br/>= Revolving Door's 'Count Parameter'"]

    Formula["dN/dt = Rotation Speed<br/>→ Time Generated by Boundary Rotation"]

    Rotation --> Formula

    style Door fill:#ff6b6b,stroke:#c92a2a,stroke-width:4px
    style Rotation fill:#4ecdc4,stroke:#0b7285
    style Count fill:#ffe66d,stroke:#f59f00
    style Time fill:#a9e34b,stroke:#5c940d,stroke-width:4px
    style Formula fill:#e9ecef,stroke:#495057

Mathematical Expression:

On boundary observable algebra ${\mathcal{A}}_{\partial }$, there exists a one-parameter automorphism group:

$$\{{\alpha }_t{\}}_{t\in \mathbb{R}}\subset \mathrm{Aut}({\mathcal{A}}_{\partial })$$

Its generator is the boundary Hamiltonian $H_{\partial }$:

$$\frac{\mathrm{d}}{\mathrm{d}t}\omega ({\alpha }_t(A))=i\omega ([H_{\partial },{\alpha }_t(A)])$$

Physical Meaning:

• Time might not be externally imposed, but generated by translation operator ${\alpha }_t$ on boundary
• Like revolving door’s “time” = number of door rotations
• The boundary can be viewed as the clock.

Axiom A3: Monotonicity and Consistency

Everyday Analogy: Second Law of Thermodynamics

graph LR
    Order["🧊 Ordered State<br/>(Low Entropy)"]
    Disorder["💨 Disordered State<br/>(High Entropy)"]

    Order -->|"Time Passage"| Disorder

    Arrow["⏰ Arrow of Time<br/>= Direction of Entropy Increase"]

    Disorder -.-> Arrow

    Irreversible["Irreversibility:<br/>Cannot Create Information<br/>from Boundary"]

    Arrow --> Irreversible

    style Order fill:#4ecdc4,stroke:#0b7285
    style Disorder fill:#ff6b6b,stroke:#c92a2a
    style Arrow fill:#ffe66d,stroke:#f59f00,stroke-width:4px
    style Irreversible fill:#e9ecef,stroke:#495057

Mathematical Expression:

Relative entropy decreases monotonically along time:

$$\frac{\mathrm{d}}{\mathrm{d}t}{S}_{\mathrm{rel}}({\omega }_t^{\prime }\Vert {\omega }_t)\le 0$$

Physical Meaning:

• Information on boundary can only decrease, not increase
• This theoretically defines the arrow of time
• Like a broken egg cannot automatically restore itself

Trinity: Three Realizations of Boundary Language

Boundary language has concrete realizations in three different physical theories:

graph TB
    BL["🗣️ Boundary Language<br/>Unified Framework"]

    BL --> Scatt["⚛️ Scattering Theory<br/>(Microscopic Quantum)"]
    BL --> Grav["🌍 General Relativity<br/>(Macroscopic Gravity)"]
    BL --> Mod["🔥 Modular Flow Theory<br/>(Statistical Mechanics)"]

    Scatt -->|"A1 Realization"| S1["S-Matrix Conservation<br/>Probability Current Flux"]
    Scatt -->|"A2 Realization"| S2["Time Scale<br/>κ(ω) = φ'(ω)/π"]
    Scatt -->|"A3 Realization"| S3["Spectral Flow Monotonicity"]

    Grav -->|"A1 Realization"| G1["GHY Boundary Term<br/>Quasi-Local Energy Flux"]
    Grav -->|"A2 Realization"| G2["Brown-York<br/>Boundary Hamiltonian"]
    Grav -->|"A3 Realization"| G3["Generalized Entropy Extremum"]

    Mod -->|"A1 Realization"| M1["KMS Condition<br/>Heat Flow Conservation"]
    Mod -->|"A2 Realization"| M2["Modular Flow Parameter<br/>σ_t^ω"]
    Mod -->|"A3 Realization"| M3["Relative Entropy Monotonic<br/>Araki Formula"]

    style BL fill:#ff6b6b,stroke:#c92a2a,stroke-width:4px
    style Scatt fill:#4ecdc4,stroke:#0b7285,stroke-width:3px
    style Grav fill:#4ecdc4,stroke:#0b7285,stroke-width:3px
    style Mod fill:#4ecdc4,stroke:#0b7285,stroke-width:3px
    style S1 fill:#ffe66d,stroke:#f59f00
    style S2 fill:#ffe66d,stroke:#f59f00
    style S3 fill:#ffe66d,stroke:#f59f00
    style G1 fill:#ffe66d,stroke:#f59f00
    style G2 fill:#ffe66d,stroke:#f59f00
    style G3 fill:#ffe66d,stroke:#f59f00
    style M1 fill:#ffe66d,stroke:#f59f00
    style M2 fill:#ffe66d,stroke:#f59f00
    style M3 fill:#ffe66d,stroke:#f59f00

Realization 1: Scattering Theory

Boundary = Infinity (incoming/outgoing particles)

Time Scale Identity (returning to Section 8):

$$\kappa (\omega )=\frac{{\phi }^{\prime }(\omega )}{\pi }={\rho }_{\mathrm{rel}}(\omega )=\frac{1}{2\pi }\mathrm{tr}Q(\omega )$$

Boundary Language Interpretation:

• Flux = Scattering probability current
• Time = Group delay $\mathrm{tr}Q(\omega )$
• Monotonicity = Non-negative spectral flow

Realization 2: General Relativity

Boundary = Spacetime boundary (e.g., black hole horizon, cosmological horizon)

GHY Boundary Term:

$$S_{\mathrm{GHY}}=\frac{1}{8\pi G}{\int }_{\partial M}\sqrt{|h|}K{\mathrm{d}}^{3}x$$

Where $K$ is the extrinsic curvature.

Boundary Language Interpretation:

graph LR
    Einstein["Einstein Equations<br/>(Interior)"]

    GHY["GHY Boundary Term<br/>(Boundary Action)"]

    BY["Brown-York<br/>Quasi-Local Energy"]

    GHY -->|"Variation"| BY

    BY -->|"Generates"| Time["Boundary Time<br/>Killing Vector"]

    Einstein -.->|"Can be Derived from Boundary"| Bulk["Bulk Geometry<br/>(Extension)"]

    GHY -.-> Bulk

    style Einstein fill:#ffe66d,stroke:#f59f00
    style GHY fill:#ff6b6b,stroke:#c92a2a,stroke-width:4px
    style BY fill:#4ecdc4,stroke:#0b7285
    style Time fill:#a9e34b,stroke:#5c940d,stroke-width:3px
    style Bulk fill:#e9ecef,stroke:#495057

Theoretical Inference: Without the GHY boundary term, the variation of Einstein-Hilbert action is incomplete! This suggests gravity might have characteristics of a boundary theory.

Realization 3: Modular Flow Theory

Boundary = Observable algebra accessible to observer

Tomita-Takesaki Modular Flow:

$${\sigma }_t^{\omega }(A)={\Delta }_{\omega }^{it}A{\Delta }_{\omega }^{-it}$$

Where ${\Delta }_{\omega }$ is the modular operator.

Boundary Language Interpretation:

graph TB
    Algebra["Boundary Algebra 𝒜_∂"]
    State["State ω"]

    Algebra --> TT["Tomita-Takesaki<br/>Modular Data (J, Δ_ω)"]
    State --> TT

    TT -->|"Generates"| Flow["Modular Flow σ_t^ω<br/>= Interior Time"]

    Flow -.->|"Thermal Time Hypothesis"| Physical["Physical Time t"]

    KMS["KMS Condition:<br/>ω(Aσ_i^ω(B)) = ω(BA)"]

    Flow --> KMS

    KMS -.->|"Equivalent to"| Thermal["Thermal Equilibrium<br/>β = 1/T"]

    style Algebra fill:#4ecdc4,stroke:#0b7285
    style State fill:#4ecdc4,stroke:#0b7285
    style TT fill:#ffe66d,stroke:#f59f00
    style Flow fill:#ff6b6b,stroke:#c92a2a,stroke-width:4px
    style Physical fill:#a9e34b,stroke:#5c940d,stroke-width:3px
    style KMS fill:#e9ecef,stroke:#495057
    style Thermal fill:#e9ecef,stroke:#495057

Connes-Rovelli Thermal Time Hypothesis: Physical time is hypothesized to be the modular flow parameter.

Time Scale Unification Theorem

Now we can state the core proposition of boundary language:

graph TB
    Theorem["Boundary Time Scale Equivalence Theorem"]

    Theorem --> Condition["Conditions:<br/>· Boundary Spectral Triple Exists<br/>· Scattering Matrix Satisfies BK Formula<br/>· Modular Flow Comparable with Geometric Flow"]

    Condition --> Result["Conclusion:<br/>Unique Time Equivalence Class [τ] Exists"]

    Result --> R1["Scattering Time τ_scatt"]
    Result --> R2["Modular Time τ_mod"]
    Result --> R3["Geometric Time τ_geom"]

    R1 -.->|"Affine Equivalent"| Unity["[τ] = [τ_scatt] = [τ_mod] = [τ_geom]"]
    R2 -.-> Unity
    R3 -.-> Unity

    Unity -->|"Mathematical Expression"| Formula["τ_scatt = a₁τ + b₁<br/>τ_mod = a₂τ + b₂<br/>τ_geom = a₃τ + b₃"]

    style Theorem fill:#ff6b6b,stroke:#c92a2a,stroke-width:4px
    style Condition fill:#4ecdc4,stroke:#0b7285
    style Result fill:#ffe66d,stroke:#f59f00,stroke-width:3px
    style R1 fill:#a9e34b,stroke:#5c940d
    style R2 fill:#a9e34b,stroke:#5c940d
    style R3 fill:#a9e34b,stroke:#5c940d
    style Unity fill:#ff6b6b,stroke:#c92a2a,stroke-width:4px
    style Formula fill:#e9ecef,stroke:#495057

Proposition Content:

Under the premise of satisfying the three axioms of boundary language, the three times might just be different normalizations of the same boundary time!

Everyday Analogy:

• Scattering time = measured with stopwatch
• Modular time = measured with hourglass
• Geometric time = measured with sundial
• They measure the same time, just different units!

Concrete Example: Black Hole Horizon

Traditional View: Horizon is Singularity

graph TB
    Outside["🌍 External Observer<br/>(Far from Black Hole)"]
    Horizon["⚫ Event Horizon<br/>(Dangerous Boundary)"]
    Inside["❓ Interior<br/>(Unknown)"]

    Outside -->|"Cannot See"| Horizon
    Horizon -->|"Separates"| Inside

    Singularity["💥 Singularity<br/>(Catastrophic)"]

    Inside --> Singularity

    style Outside fill:#4ecdc4,stroke:#0b7285
    style Horizon fill:#ff6b6b,stroke:#c92a2a,stroke-width:3px
    style Inside fill:#e9ecef,stroke:#495057,stroke-dasharray: 5 5
    style Singularity fill:#fff,stroke:#868e96

Boundary Language: Horizon “Speaks”

graph TB
    Horizon2["⚫ Horizon = Boundary<br/>(Stage of Boundary Language)"]

    Horizon2 -->|"A1: Flux"| Hawking["Hawking Radiation<br/>= Energy Flow Across Horizon"]

    Horizon2 -->|"A2: Time"| Temperature["Hawking Temperature<br/>T_H = κ/2π<br/>= Modular Flow Parameter"]

    Horizon2 -->|"A3: Monotonicity"| Entropy["Bekenstein-Hawking Entropy<br/>S_BH = A/4G<br/>= Boundary Algebra Entropy"]

    Hawking -.->|"Completely Determines"| Interior["Interior State<br/>(Can be Derived from Boundary)"]
    Temperature -.-> Interior
    Entropy -.-> Interior

    style Horizon2 fill:#ff6b6b,stroke:#c92a2a,stroke-width:4px
    style Hawking fill:#4ecdc4,stroke:#0b7285
    style Temperature fill:#4ecdc4,stroke:#0b7285
    style Entropy fill:#4ecdc4,stroke:#0b7285
    style Interior fill:#ffe66d,stroke:#f59f00

Boundary Language Interpretation:

1. Hawking Temperature = Period of horizon modular flow $T_H=\kappa /2\pi$
2. Black Hole Entropy = von Neumann entropy of horizon algebra $S_{\mathrm{BH}}=A/4G$
3. Hawking Radiation = Thermodynamic fluctuations of horizon flux

Key: Theoretically, there is no need to know what happens inside the black hole, the horizon boundary might already contain all the information!

Philosophical Meaning: Mathematical Realization of Holographic Principle

graph TB
    Question["🤔 Where is the Information of the Universe?"]

    Question -->|"Traditional View"| Volume["In Volume<br/>Each Spatial Point Has Information"]

    Question -->|"Boundary Language"| Surface["On Boundary<br/>All Information Encoded on Surface"]

    Volume -.->|"Information Amount"| V["∝ Volume V"]
    Surface -.->|"Information Amount"| A["∝ Area A"]

    Holography["Holographic Principle:<br/>Volume Information ≤ Boundary Information"]

    Surface --> Holography

    BL["Boundary Language<br/>= Mathematical Realization<br/>of Holographic Principle"]

    Holography --> BL

    style Question fill:#e9ecef,stroke:#495057
    style Volume fill:#ffe66d,stroke:#f59f00
    style Surface fill:#ff6b6b,stroke:#c92a2a,stroke-width:4px
    style V fill:#e9ecef,stroke:#495057
    style A fill:#a9e34b,stroke:#5c940d,stroke-width:3px
    style Holography fill:#4ecdc4,stroke:#0b7285,stroke-width:3px
    style BL fill:#ffe66d,stroke:#f59f00,stroke-width:4px

Deep Revelations:

1. Holographic Principle: ’t Hooft and Susskind’s conjecture—three-dimensional volume information can be encoded on two-dimensional surface
2. AdS/CFT Correspondence: Gravity theory (bulk) ↔ Conformal field theory (boundary)
3. Boundary Language: Attempts to formalize the holographic principle as a mathematical framework

Everyday Analogy:

• Like a hologram, appears three-dimensional, but all information is on two-dimensional film
• The universe is like a hologram, all information is on the boundary

Experimental Verifiability

Verification 1: Microwave Network Scattering

graph LR
    Network["📡 Microwave Scattering Network"]

    Network -->|"Measure Ports"| Ports["Boundary Ports<br/>(Scattering Channels)"]

    Ports -->|"Extract"| SMatrix["S-Matrix S(ω)"]

    SMatrix -->|"Compute"| TimeScatt["Scattering Time Scale<br/>κ(ω) = tr Q(ω)/2π"]

    TimeScatt -.->|"Should Equal"| TimeGeom["Geometric Time Scale<br/>(Network Delay)"]

    Check["✓ Boundary Language Prediction:<br/>Two Are Affine Equivalent"]

    TimeScatt --> Check
    TimeGeom --> Check

    style Network fill:#e9ecef,stroke:#495057
    style Ports fill:#ff6b6b,stroke:#c92a2a,stroke-width:3px
    style SMatrix fill:#4ecdc4,stroke:#0b7285
    style TimeScatt fill:#ffe66d,stroke:#f59f00
    style TimeGeom fill:#ffe66d,stroke:#f59f00
    style Check fill:#a9e34b,stroke:#5c940d,stroke-width:3px

Verification 2: Atomic Clock Gravitational Redshift

graph TB
    Clock1["⏰ Ground Atomic Clock<br/>(Strong Gravitational Potential)"]
    Clock2["⏰ Satellite Atomic Clock<br/>(Weak Gravitational Potential)"]

    Clock1 -->|"Boundary"| Horizon1["Ground Boundary"]
    Clock2 -->|"Boundary"| Horizon2["Satellite Boundary"]

    Horizon1 -->|"Modular Flow Parameter"| Mod1["τ_mod^(1)"]
    Horizon2 -->|"Modular Flow Parameter"| Mod2["τ_mod^(2)"]

    Redshift["Gravitational Redshift<br/>ν₂/ν₁ = τ_mod^(1)/τ_mod^(2)"]

    Mod1 --> Redshift
    Mod2 --> Redshift

    Redshift -.->|"Boundary Language Prediction"| Prediction["Should Equal<br/>Brown-York Energy Ratio"]

    style Clock1 fill:#ff6b6b,stroke:#c92a2a
    style Clock2 fill:#4ecdc4,stroke:#0b7285
    style Horizon1 fill:#ffe66d,stroke:#f59f00
    style Horizon2 fill:#ffe66d,stroke:#f59f00
    style Mod1 fill:#a9e34b,stroke:#5c940d
    style Mod2 fill:#a9e34b,stroke:#5c940d
    style Redshift fill:#e9ecef,stroke:#495057
    style Prediction fill:#fff,stroke:#868e96,stroke-width:3px

Chapter Summary

Core Insight:

GLS theory proposes: Time might not be defined in spatial interior, but on the boundary. Through three axioms of “flux, translation, monotonicity,” the boundary theoretically determines the time structure of the interior. This is boundary language.

Key Formulas:

Boundary language triple: $${\mathfrak{L}}_{\Sigma }=({\mathcal{A}}_{\partial },\omega ,\mathcal{F})$$

Time scale identity: $$\kappa (\omega )=\frac{{\phi }^{\prime }(\omega )}{\pi }={\rho }_{\mathrm{rel}}(\omega )=\frac{1}{2\pi }\mathrm{tr}Q(\omega )$$

Time scale equivalence: $$[{\tau }_{\mathrm{scatt}}]=[{\tau }_{\mathrm{mod}}]=[{\tau }_{\mathrm{geom}}]=[\tau ]$$

Everyday Analogies:

• Door frame determines room: Measuring flux at door frame can infer room interior
• Bank statement: Account balance completely determined by statement (boundary record)
• Revolving door: Time = parameter of door rotation, boundary is the clock
• Hologram: Three-dimensional information encoded on two-dimensional surface

Three Realizations:

1. Scattering Theory: Boundary = infinity, time = group delay
2. General Relativity: Boundary = spacetime boundary, time = Brown-York generator
3. Modular Flow Theory: Boundary = observable algebra, time = modular flow parameter

Theoretical Inferences:

• Einstein equations need GHY boundary term → Gravity might fundamentally be a boundary theory
• Black hole horizon completely determines interior → Information might not be in volume, but on surface
• Time generated by boundary → “Time passage” might be a manifestation of boundary translation operator

Philosophical Revelation:

The universe is like a hologram: appears to be three-dimensional spacetime, but all information is encoded on the boundary. The boundary “speaks” time.

Connections to Other Chapters

graph TB
    Current["📍 This Chapter:<br/>Boundary Language"]

    Prev1["← 08 Time as Entropy<br/>Optimal Path"]
    Prev2["← 09 Time-Geometry Unification<br/>No Fundamental Force"]
    Prev3["← 10 Topological Invariants<br/>DNA of Time"]

    Next1["→ 12 Time Domain Solvable<br/>Boundary Data Reconstruction"]
    Next2["→ 06 Boundary Priority<br/>BTG Framework"]

    Prev1 -->|"Entropy Optimal Path<br/>Now Defined on Boundary"| Current
    Prev2 -->|"Unified Geometry<br/>Now Realized on Boundary"| Current
    Prev3 -->|"Topological Invariants<br/>Now Measured on Boundary"| Current

    Current -->|"Boundary Data<br/>How to Reconstruct Bulk"| Next1
    Current -->|"Complete BTG Framework<br/>Boundary Priority Axiom"| Next2

    style Current fill:#ff6b6b,stroke:#c92a2a,stroke-width:4px
    style Prev1 fill:#4ecdc4,stroke:#0b7285
    style Prev2 fill:#4ecdc4,stroke:#0b7285
    style Prev3 fill:#4ecdc4,stroke:#0b7285
    style Next1 fill:#ffe66d,stroke:#f59f00
    style Next2 fill:#ffe66d,stroke:#f59f00

Extended Reading

Source Theoretical Literature:

• docs/euler-gls-paper-time/boundary-language-unified-framework.md - Complete derivation of boundary language unified framework
• docs/euler-gls-paper-bondary/boundary-time-geometry-unified-framework.md - Boundary Time Geometry (BTG) theory

Related Chapters:

• 03 Scattering Phase and Time Scale - Scattering boundary realization
• 08 Time as Generalized Entropy Optimal Path - Boundary expression of entropy
• 09 Time–Geometry–Interaction Unification - Geometric boundary realization
• 10 Topological Invariants and Time - Boundary measurement of topology
• 06 Boundary Priority and Time Emergence - Complete BTG framework

In the next chapter, we will explore solvability of time domains, seeing how to completely reconstruct bulk structure from boundary data.