12 Time Domains and Solvable Models: Reconstructing Time from Boundary Data

Core Idea

In previous chapters, we constructed the theoretical framework of time:

• 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)
• Time might be defined on the boundary (Section 11)

Now we face the final key question: Under what conditions can we theoretically reconstruct time from boundary data?

GLS theory proposes: Domain might determine everything. Just as mathematical functions need a domain to be meaningful, time scales also need clear domain conditions to be uniquely determined from boundary data.

Everyday Analogy: Film Projection

Imagine you want to reconstruct a movie from film:

graph TB
    subgraph "Problem: What Information is on the Film?"
        Film["🎞️ Movie Film<br/>(Boundary Data)"]

        Film -->|"Each Frame"| Frame["Still Image"]
        Film -->|"Frame Spacing"| Spacing["△t Time Interval"]

        Frame -.->|"Insufficient"| Question["❓ Can We Reconstruct<br/>Continuous Movie?"]
        Spacing -.-> Question
    end

    subgraph "Answer: Need Domain Conditions"
        Condition["✓ Domain Conditions"]

        Condition --> C1["Frame Rate Known<br/>(24 fps)"]
        Condition --> C2["Playback Order Fixed<br/>(Causality)"]
        Condition --> C3["No Missing Frames<br/>(Completeness)"]

        C1 -.->|"Satisfy"| Reconstruct["✓ Can Uniquely Reconstruct<br/>Continuous Movie"]
        C2 -.-> Reconstruct
        C3 -.-> Reconstruct
    end

    style Film fill:#ff6b6b,stroke:#c92a2a,stroke-width:3px
    style Frame fill:#4ecdc4,stroke:#0b7285
    style Spacing fill:#4ecdc4,stroke:#0b7285
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    style Condition fill:#ffe66d,stroke:#f59f00,stroke-width:4px
    style C1 fill:#a9e34b,stroke:#5c940d
    style C2 fill:#a9e34b,stroke:#5c940d
    style C3 fill:#a9e34b,stroke:#5c940d
    style Reconstruct fill:#4ecdc4,stroke:#0b7285,stroke-width:4px

Theoretical Insight:

• Film (boundary data) alone is insufficient
• Need domain conditions (frame rate, order, completeness)
• Satisfy conditions → theoretically uniquely reconstruct movie (time)

Domain of Scale Identity

Returning to the core formula from Section 8, we now clarify its domain:

graph TB
    Identity["Core Identity:<br/>κ(ω) = φ'(ω)/π = ρ_rel(ω) = tr Q(ω)/2π"]

    Identity -->|"Ask"| Domain["In What Domain Does It Hold?"]

    Domain --> D1["Elastic-Unitary Domain<br/>(Standard Case)"]
    Domain --> D2["Non-Unitary-Absorption Domain<br/>(Generalized Case)"]
    Domain --> D3["Long-Range Potential Domain<br/>(Needs Renormalization)"]

    D1 -->|"Exact Conditions"| C1["· S(ω) Unitary<br/>· Short-Range Scattering<br/>· Away from Thresholds/Resonances<br/>· Trace-Class Perturbation"]

    D2 -->|"Modified Conditions"| C2["· S Non-Unitary (Absorption)<br/>· Use Q_gen = -iS⁻¹∂_ωS<br/>· Re tr Q_gen = Real Delay"]

    D3 -->|"Renormalization Conditions"| C3["· Coulomb/Gravitational Potential<br/>· Dollard Modified Wave Operator<br/>· Phase Renormalization Φ_ren"]

    style Identity fill:#ff6b6b,stroke:#c92a2a,stroke-width:4px
    style Domain fill:#ffe66d,stroke:#f59f00,stroke-width:3px
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    style C1 fill:#a9e34b,stroke:#5c940d
    style C2 fill:#a9e34b,stroke:#5c940d
    style C3 fill:#a9e34b,stroke:#5c940d

Domain 1: Elastic-Unitary Domain (Ideal Case)

Domain Conditions:

$$\{\begin{array}{ll}S(\omega )\in C^1(I;U(N))& \text{(Unitarity)}\\ H-H_0\in {\mathfrak{S}}_1& \text{(Trace Class)}\\ \omega \in I\setminus \Sigma & \text{(Away from Thresholds/Resonances)}\end{array}$$

Identity: In this domain, the scale identity holds mathematically exactly:

$$\frac{{\phi }^{\prime }(\omega )}{\pi }={\rho }_{\mathrm{rel}}(\omega )=\frac{1}{2\pi }\mathrm{tr}Q(\omega )\text{(Lebesgue-a.e.)}$$

Domain 2: Non-Unitary-Absorption Domain (Generalized Case)

Imagine a lossy microwave cavity:

graph LR
    In["⚡ Incoming Wave<br/>Energy E_in"]

    Cavity["📦 Cavity<br/>(Absorbs Energy)"]

    Out1["⚡ Transmitted Wave<br/>E_trans"]
    Out2["💨 Absorption<br/>E_abs"]

    In --> Cavity
    Cavity --> Out1
    Cavity -.->|"Lost"| Out2

    Conservation["Energy Conservation:<br/>E_in = E_trans + E_abs"]

    Out1 --> Conservation
    Out2 --> Conservation

    NonUnitary["S Non-Unitary:<br/>S†S ≠ 1"]

    Conservation --> NonUnitary

    style In fill:#4ecdc4,stroke:#0b7285
    style Cavity fill:#ffe66d,stroke:#f59f00,stroke-width:3px
    style Out1 fill:#a9e34b,stroke:#5c940d
    style Out2 fill:#ff6b6b,stroke:#c92a2a
    style Conservation fill:#e9ecef,stroke:#495057
    style NonUnitary fill:#fff,stroke:#868e96,stroke-width:3px

Modified Definition:

Generalized group delay: $$Q_{\mathrm{gen}}(\omega )=-iS(\omega {)}^{-1}{\partial }_{\omega }S(\omega )$$

Phase relation: $${\partial }_{\omega }\arg \det S=\Re \mathrm{tr}{Q}_{\mathrm{gen}}$$

Physical Meaning:

• $\Re \mathrm{tr}{Q}_{\mathrm{gen}}$ = Actual time delay
• $\Im \mathrm{tr}{Q}_{\mathrm{gen}}$ = Absorption rate

Small absorption limit: $$\mathrm{tr}{Q}_{\mathrm{gen}}=\mathrm{tr}Q+O(|S^{†}S-1|)$$

Domain 3: Long-Range Potential Domain (Renormalization Case)

Problem: Coulomb/gravitational potential $V\sim 1/r$

graph TB
    Problem["Problem: Long-Range Potential<br/>V(r) ~ 1/r"]

    Problem -->|"Causes"| Issue1["Phase Divergence<br/>φ ~ ln r"]
    Problem -->|"Causes"| Issue2["Wave Operator Doesn't Converge"]

    Solution["Solution: Phase Renormalization"]

    Issue1 --> Solution
    Issue2 --> Solution

    Solution --> S1["Modified Wave Operator<br/>(Dollard Transformation)"]
    Solution --> S2["Define Renormalized Phase<br/>Φ_ren = Φ - Φ_Coulomb"]

    S1 -.->|"Result"| Result["Renormalized Identity:<br/>∂_ωΦ_ren = ρ_rel"]
    S2 -.-> Result

    style Problem fill:#ff6b6b,stroke:#c92a2a,stroke-width:3px
    style Issue1 fill:#ffe66d,stroke:#f59f00
    style Issue2 fill:#ffe66d,stroke:#f59f00
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    style Result fill:#e9ecef,stroke:#495057,stroke-width:3px

Windowed Clock: Solving the Negative Delay Problem

Problem: Group Delay Can Be Negative

Anomalous Delay Phenomenon:

graph TB
    Frequency["Frequency ω"]

    Frequency -->|"Near Resonance"| Resonance["Resonance Peak"]
    Frequency -->|"Near Anti-Resonance"| AntiRes["Anti-Resonance Valley"]

    Resonance -->|"Group Delay"| Pos["tr Q > 0<br/>Positive Delay"]
    AntiRes -->|"Group Delay"| Neg["tr Q < 0<br/>Negative Delay!"]

    Neg -.->|"Problem"| Question["Time Reversal?"]

    style Frequency fill:#e9ecef,stroke:#495057
    style Resonance fill:#a9e34b,stroke:#5c940d
    style AntiRes fill:#ff6b6b,stroke:#c92a2a,stroke-width:3px
    style Pos fill:#4ecdc4,stroke:#0b7285
    style Neg fill:#ffe66d,stroke:#f59f00,stroke-width:4px
    style Question fill:#fff,stroke:#868e96,stroke-dasharray: 5 5

Classic Example: Hartman effect—superluminal group velocity in quantum tunneling

Solution: Poisson Windowing

Idea: Don’t define time at a single frequency point, but use window averaging

graph TB
    Raw["Raw Group Delay tr Q(ω)<br/>(May Have Negative Values)"]

    Window["Poisson Window P_Δ(x)<br/>P_Δ(x) = (1/π) Δ/(x²+Δ²)"]

    Raw -->|"Convolution"| Smooth["Windowed Scale Density<br/>Θ_Δ(ω) = (tr Q * P_Δ)(ω)"]

    Window --> Smooth

    Smooth -->|"Integrate"| Clock["Windowed Clock<br/>t_Δ(ω) = ∫ Θ_Δ dω"]

    Property["Property:<br/>Δ > Critical Width Γ_min<br/>⟹ Θ_Δ(ω) > 0<br/>⟹ t_Δ Strictly Increasing"]

    Clock --> Property

    style Raw fill:#ff6b6b,stroke:#c92a2a
    style Window fill:#4ecdc4,stroke:#0b7285,stroke-width:3px
    style Smooth fill:#ffe66d,stroke:#f59f00
    style Clock fill:#a9e34b,stroke:#5c940d,stroke-width:4px
    style Property fill:#e9ecef,stroke:#495057

Mathematical Definition:

Poisson kernel: $$P_{\Delta }(x)=\frac{1}{\pi }\frac{\Delta }{x^2+{\Delta }^2}$$

Windowed scale density: $${\Theta }_{\Delta }(\omega )=({\rho }_{\mathrm{rel}}\ast P_{\Delta })(\omega )=\frac{1}{2\pi }(\mathrm{tr}Q\ast P_{\Delta })(\omega )$$

Windowed clock: $$t_{\Delta }(\omega )={\int }_{{\omega }_0}^{\omega }{\Theta }_{\Delta }(\omega )\mathrm{d}\omega$$

Core Proposition:

If $\Delta >{\Gamma }_{\min }$ (minimum resonance width), then:

1. Weak Monotonicity: ${\Theta }_{\Delta }(\omega )>0$ almost everywhere
2. Affine Uniqueness: Any windowed clock satisfying conditions differs only by affine transformation ${\tilde{t}}_{\Delta }=a{t}_{\Delta }+b$

Solvable Model: Schwarzschild Black Hole

Problem: Phase Derivative = Geometric Delay?

In the exterior region of a Schwarzschild black hole, can we verify scattering time = geometric time?

graph TB
    BH["⚫ Schwarzschild Black Hole<br/>Mass M"]

    Wave["🌊 Scalar Wave<br/>Frequency ω, Angular Momentum l"]

    BH -->|"Scattering"| Scatter["Scattering Matrix S_l(ω)"]

    Scatter -->|"Compute"| Phase["Scattering Phase Φ_l(ω)"]

    Phase -->|"Derivative"| Derivative["∂_ωΦ(ω) = ?"]

    Geometric["🌍 Geometric Optics<br/>Shapiro Delay"]

    Geometric -->|"Predicts"| ShapiroDelay["ΔT_Shapiro ~ (4GM/c³)ln(r)"]

    Compare["Compare"]

    Derivative --> Compare
    ShapiroDelay --> Compare

    Compare -.->|"High-Frequency Limit"| Result["✓ ∂_ωΦ_ren = ΔT_Shapiro<br/>+ O(ω⁻¹)"]

    style BH fill:#ff6b6b,stroke:#c92a2a,stroke-width:4px
    style Wave fill:#4ecdc4,stroke:#0b7285
    style Scatter fill:#ffe66d,stroke:#f59f00
    style Phase fill:#a9e34b,stroke:#5c940d
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    style Geometric fill:#4ecdc4,stroke:#0b7285
    style ShapiroDelay fill:#ffe66d,stroke:#f59f00
    style Compare fill:#fff,stroke:#868e96
    style Result fill:#a9e34b,stroke:#5c940d,stroke-width:4px

Regge-Wheeler Equation

Scalar waves in Schwarzschild exterior satisfy:

$$\frac{{\mathrm{d}}^{2}u}{\mathrm{d}{r}_{\ast }^2}+\left[{\omega }^2-V_{\mathrm{eff}}(r)\right]u=0$$

Where:

• $r_{\ast }=r+2M\ln (r/2M-1)$ (tortoise coordinate)
• $V_{\mathrm{eff}}=\left(1-\frac{2M}{r}\right)\left(\frac{l(l+1)}{r^2}+\frac{2M}{r^3}\right)$ (effective potential)

Eikonal Approximation

High-frequency/high-angular-momentum limit $(\omega \gg M^{-1},l\gg 1)$:

WKB phase: $${\varphi }_{\mathrm{WKB}}=\int \sqrt{{\omega }^2-V_{\mathrm{eff}}(r)}\mathrm{d}{r}_{\ast }$$

Phase derivative: $${\partial }_{\omega }{\varphi }_{\mathrm{WKB}}=\int \frac{\omega }{\sqrt{{\omega }^2-V_{\mathrm{eff}}}}\mathrm{d}{r}_{\ast }$$

Geometric Correspondence:

$${\partial }_{\omega }{\varphi }_{\mathrm{WKB}}\approx \Delta T_{\mathrm{Shapiro}}=\frac{4GM}{c^3}\ln \frac{4r_E{r}_R}{b^2}+O({\omega }^{-1})$$

Where $b$ is the impact parameter, $r_E,r_R$ are emission/reception radii.

Solvable Model: Gravitational Lensing

Multiple Image Time Delay

graph LR
    Source["🌟 Source<br/>Emits Light at t=0"]

    Lens["🌍 Lens<br/>(Point Mass M)"]

    Image1["📷 Image 1<br/>Arrives at t₁"]
    Image2["📷 Image 2<br/>Arrives at t₂"]

    Source -->|"Light Path 1"| Lens
    Source -->|"Light Path 2"| Lens

    Lens -->|"Deflects"| Image1
    Lens -->|"Deflects"| Image2

    Delay["Time Delay<br/>Δt = t₂ - t₁"]

    Image1 --> Delay
    Image2 --> Delay

    style Source fill:#ffe66d,stroke:#f59f00,stroke-width:3px
    style Lens fill:#ff6b6b,stroke:#c92a2a,stroke-width:3px
    style Image1 fill:#4ecdc4,stroke:#0b7285
    style Image2 fill:#4ecdc4,stroke:#0b7285
    style Delay fill:#a9e34b,stroke:#5c940d,stroke-width:4px

Fermat Principle: Light travels along time extremal paths

Time delay: $$\Delta t_{ij}=\frac{1+z_d}{c}\frac{D_d{D}_s}{D_{ds}}\left[\frac{({\mathbf{\theta }}_i-\mathbf{\beta }{)}^2}{2}-\psi ({\mathbf{\theta }}_i)\right]-\text{(Image j)}$$

Where:

• ${\mathbf{\theta }}_i$ = Angular position of image i
• $\mathbf{\beta }$ = True position of source
• $\psi$ = Lens potential
• $D_{d,s,ds}$ = Angular diameter distances

Boundary Language Formulation:

Phase of frequency-domain magnification factor $F(\omega )$: $${\partial }_{\omega }[{\Phi }_i(\omega )-{\Phi }_j(\omega )]=\Delta t_{ij}$$

Time delay = frequency derivative of phase difference (Theoretical Inference)!

Solvable Model: Cosmological Redshift

Redshift = Phase Rhythm Ratio

In FRW universe, photon phase:

$$\varphi =\int \omega \mathrm{d}t$$

Phase rhythm: $$\frac{\mathrm{d}\varphi }{\mathrm{d}t}=\omega ={\omega }_{0}a(t_0)/a(t)$$

Redshift: $$1+z=\frac{{\omega }_e}{{\omega }_0}=\frac{(d\varphi /dt{)}_e}{(d\varphi /dt{)}_0}=\frac{a(t_0)}{a(t_e)}$$

graph LR
    Emission["Emission Time t_e<br/>Scale Factor a(t_e)"]

    Observation["Observation Time t_0<br/>Scale Factor a(t_0)"]

    Emission -->|"Photon Propagation"| Observation

    PhaseE["Phase Rhythm<br/>(dφ/dt)_e"]
    PhaseO["Phase Rhythm<br/>(dφ/dt)_0"]

    Emission --> PhaseE
    Observation --> PhaseO

    Redshift["Redshift<br/>1+z = (dφ/dt)_e/(dφ/dt)_0<br/>= a(t_0)/a(t_e)"]

    PhaseE --> Redshift
    PhaseO --> Redshift

    style Emission fill:#ff6b6b,stroke:#c92a2a
    style Observation fill:#4ecdc4,stroke:#0b7285
    style PhaseE fill:#ffe66d,stroke:#f59f00
    style PhaseO fill:#ffe66d,stroke:#f59f00
    style Redshift fill:#a9e34b,stroke:#5c940d,stroke-width:4px

Boundary Language Interpretation:

• Cosmological redshift is not “Doppler effect”
• But ratio of boundary phase rhythms
• Theoretically determined by boundary data (phase evolution)!

Experimental Verification Plans

Plan 1: Multi-Frequency Shapiro Delay Measurement

graph TB
    Sun["☀️ Sun<br/>Gravitational Source"]

    Spacecraft["🛰️ Spacecraft<br/>Signal Transmission"]

    Earth["🌍 Earth<br/>Receiving Station"]

    Sun -.->|"Gravitational Field"| Path["Signal Path<br/>(Passes Near Sun)"]

    Spacecraft -->|"Multi-Frequency Signal<br/>ω₁, ω₂, ω₃..."| Path
    Path --> Earth

    Measure["Measure Phase Φ(ω)"]

    Earth --> Measure

    Derivative["Compute ∂_ωΦ"]

    Measure --> Derivative

    Compare["Compare:<br/>∂_ωΦ vs ΔT_Shapiro^(geo)"]

    Derivative --> Compare

    style Sun fill:#ffe66d,stroke:#f59f00,stroke-width:4px
    style Spacecraft fill:#4ecdc4,stroke:#0b7285
    style Earth fill:#a9e34b,stroke:#5c940d
    style Path fill:#ff6b6b,stroke:#c92a2a,stroke-dasharray: 5 5
    style Measure fill:#e9ecef,stroke:#495057
    style Derivative fill:#ffe66d,stroke:#f59f00
    style Compare fill:#fff,stroke:#868e96,stroke-width:3px

Key:

• Measure phase $\Phi (\omega )$ at multiple frequencies
• Numerically differentiate to get ${\partial }_{\omega }\Phi$
• Compare with geometrically predicted Shapiro delay
• Verify scale identity!

Plan 2: Microwave Network S-Parameter Measurement

graph LR
    Network["📡 Microwave Scattering Network<br/>(Multi-Port Device)"]

    VNA["Vector Network Analyzer<br/>(VNA)"]

    Network -->|"Frequency Sweep"| VNA

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

    SMatrix -->|"Compute"| Q["Wigner-Smith Matrix<br/>Q = -iS†∂_ωS"]

    Q -->|"Take Trace"| Trace["tr Q(ω)"]

    Trace -->|"Compare"| DOS["Density of States ρ_rel(ω)<br/>(Computed from Spectrum)"]

    Check["Verify:<br/>tr Q/2π = ρ_rel"]

    Trace --> Check
    DOS --> Check

    style Network fill:#e9ecef,stroke:#495057
    style VNA fill:#4ecdc4,stroke:#0b7285
    style SMatrix fill:#ffe66d,stroke:#f59f00
    style Q fill:#ff6b6b,stroke:#c92a2a,stroke-width:3px
    style Trace fill:#a9e34b,stroke:#5c940d
    style DOS fill:#a9e34b,stroke:#5c940d
    style Check fill:#fff,stroke:#868e96,stroke-width:3px

Plan 3: Gravitational Lensing Time Delay Cosmology

graph TB
    QSO["Quasar<br/>(Time-Varying Source)"]

    Lens["Foreground Galaxy<br/>(Gravitational Lens)"]

    Images["Multiple Images<br/>Different Arrival Times"]

    QSO -->|"Light"| Lens
    Lens --> Images

    Measure["Measure Multi-Frequency Signal<br/>Extract Phase Φ_i(ω)"]

    Images --> Measure

    Phase["Compute Phase Difference<br/>∂_ω[Φ_i - Φ_j]"]

    Measure --> Phase

    Delay["Compare Time Delay<br/>Δt_ij (From Light Curve)"]

    Phase -.->|"Should Equal"| Delay

    Cosmology["Cosmological Parameters<br/>H₀, Ω_m..."]

    Delay --> Cosmology

    style QSO fill:#ffe66d,stroke:#f59f00,stroke-width:3px
    style Lens fill:#ff6b6b,stroke:#c92a2a,stroke-width:3px
    style Images fill:#4ecdc4,stroke:#0b7285
    style Measure fill:#e9ecef,stroke:#495057
    style Phase fill:#a9e34b,stroke:#5c940d,stroke-width:3px
    style Delay fill:#a9e34b,stroke:#5c940d
    style Cosmology fill:#fff,stroke:#868e96,stroke-width:3px

H0LiCOW Project: Using lens time delays to measure Hubble constant

Philosophical Meaning of Domain

graph TB
    Math["Mathematical Function f(x)"]

    Math --> Need1["Needs Domain D"]
    Math --> Need2["Needs Range R"]

    Need1 -.->|"Analogy"| Time["Time Scale κ(ω)"]

    Time --> Domain["Domain Conditions"]
    Time --> Range["Value of Time"]

    Domain --> D1["Elastic-Unitary"]
    Domain --> D2["Non-Unitary-Absorption"]
    Domain --> D3["Long-Range Potential"]

    Range --> R1["Windowed Clock t_Δ"]

    Insight["💡 Deep Revelations"]

    D1 --> Insight
    D2 --> Insight
    D3 --> Insight
    R1 --> Insight

    Insight --> I1["Time is Not Absolute<br/>Depends on Domain"]
    Insight --> I2["Different Domains<br/>Need Different 'Languages'"]
    Insight --> I3["Unification at Affine Equivalence Class<br/>Not Pointwise Agreement"]

    style Math fill:#e9ecef,stroke:#495057
    style Need1 fill:#ffe66d,stroke:#f59f00
    style Need2 fill:#ffe66d,stroke:#f59f00
    style Time fill:#ff6b6b,stroke:#c92a2a,stroke-width:4px
    style Domain fill:#4ecdc4,stroke:#0b7285,stroke-width:3px
    style Range fill:#a9e34b,stroke:#5c940d
    style D1 fill:#e9ecef,stroke:#495057
    style D2 fill:#e9ecef,stroke:#495057
    style D3 fill:#e9ecef,stroke:#495057
    style R1 fill:#e9ecef,stroke:#495057
    style Insight fill:#ffe66d,stroke:#f59f00,stroke-width:4px
    style I1 fill:#fff,stroke:#868e96
    style I2 fill:#fff,stroke:#868e96
    style I3 fill:#fff,stroke:#868e96

Deep Revelations:

1. Time is like a mathematical function: Must specify domain to be meaningful
2. Different physical situations = different domains: Elastic scattering, absorbing cavity, gravitational field each has its domain
3. Unification at equivalence class level: Time scales in different domains unified through affine transformations
4. Solvable models are bridges: Connecting abstract theory with concrete experiments

Chapter Summary

Core Perspective:

GLS theory suggests that reconstruction of time scales requires clear domain conditions. In elastic-unitary domain, scale identity holds exactly; in non-unitary/long-range domains, corrections or renormalization are needed. Windowed clocks solve negative delay problem, providing weak monotonicity and affine uniqueness. Solvable models (Schwarzschild, lensing, cosmology) verify scattering time = geometric time.

Key Formulas:

Scale identity (elastic-unitary domain): $$\frac{{\phi }^{\prime }(\omega )}{\pi }={\rho }_{\mathrm{rel}}(\omega )=\frac{1}{2\pi }\mathrm{tr}Q(\omega )(\omega \in I\setminus \Sigma )$$

Windowed clock: $${\Theta }_{\Delta }(\omega )=({\rho }_{\mathrm{rel}}\ast P_{\Delta })(\omega ),t_{\Delta }(\omega )={\int }_{{\omega }_0}^{\omega }{\Theta }_{\Delta }\mathrm{d}\omega$$

Eikonal correspondence: $${\partial }_{\omega }{\Phi }_{\mathrm{ren}}(\omega )=\Delta T_{\mathrm{Shapiro}}+O({\omega }^{-1})$$

Redshift-phase relation: $$1+z=\frac{(d\varphi /dt{)}_e}{(d\varphi /dt{)}_0}=\frac{a(t_0)}{a(t_e)}$$

Three Domains:

Solvable Model Verifications:

1. Schwarzschild: ${\partial }_{\omega }\Phi \approx \Delta T_{\mathrm{Shapiro}}$ (high-frequency limit)
2. Gravitational Lensing: ${\partial }_{\omega }({\Phi }_i-{\Phi }_j)=\Delta t_{ij}$
3. Cosmology: $1+z=(d\varphi /dt{)}_e/(d\varphi /dt{)}_0$

Experimentally Verifiable:

• Multi-frequency Shapiro delay (planetary occultation)
• Microwave network S-parameters (on-chip devices)
• Gravitational lensing time delays (H0LiCOW)

Philosophical Meaning:

Time reconstruction is not automatic, but conditional:

• Must specify domain (physical situation)
• Must choose window (measurement resolution)
• Unification at affine equivalence class, not pointwise values

This constitutes a significant part of the GLS unified time theory: strict conditions from boundary data to time reconstruction.

Connections to Other Chapters

graph TB
    Current["📍 This Chapter:<br/>Time Domains and Solvable Models"]

    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"]
    Prev4["← 11 Boundary Language<br/>Where Time Speaks"]

    Next1["→ 06 Boundary Priority<br/>Complete BTG Framework"]
    Next2["→ 07 Causal Structure<br/>Arrow of Time"]

    Prev1 -->|"Optimal Path<br/>Now Know Domain"| Current
    Prev2 -->|"Geometric Unification<br/>Now Can Verify Solvably"| Current
    Prev3 -->|"Topological Invariants<br/>Now Have Domain Constraints"| Current
    Prev4 -->|"Boundary Data<br/>Now Can Reconstruct Time"| Current

    Current -->|"Domain Conditions<br/>Complete BTG Assumptions"| Next1
    Current -->|"Causal Partial Order<br/>From Windowed Monotonicity"| Next2

    Summary["✓ Phase 1 Complete<br/>05-unified-time Chapter<br/>8 Files Complete"]

    Current --> Summary

    style Current fill:#ff6b6b,stroke:#c92a2a,stroke-width:4px
    style Prev1 fill:#4ecdc4,stroke:#0b7285
    style Prev2 fill:#4ecdc4,stroke:#0b7285
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    style Prev4 fill:#4ecdc4,stroke:#0b7285
    style Next1 fill:#ffe66d,stroke:#f59f00
    style Next2 fill:#ffe66d,stroke:#f59f00
    style Summary fill:#a9e34b,stroke:#5c940d,stroke-width:4px

Extended Reading

Source Theoretical Literature:

• docs/euler-gls-paper-time/unified-time-scale-geometry-domains-solvable-models.md - Complete derivation of time scale, domains, and solvable models

Related Chapters:

• 03 Scattering Phase and Time Scale - Scattering theoretical foundation
• 08 Time as Generalized Entropy Optimal Path - Variational principle
• 09 Time–Geometry–Interaction Unification - Geometric realization
• 10 Topological Invariants and Time - Topological constraints
• 11 Boundary Language - Boundary framework
• 06 Boundary Priority and Time Emergence - Complete BTG theory

With this, we complete all foundational chapters of unified time theory. Next, we will explore applications in boundary theory, causal structure, and matrix universe.