08 Boundary, Observer, and Time: Who is “Looking”? Who is “Flowing”?

Core Ideas

In Chapter 07, we learned that physics happens at the boundary, but there is a deeper question:

• What is the boundary like without an observer?
• What mathematical object is the world that the observer “sees”?
• Is time a product of the observer’s “attention”?

The answer is stunning: Time axis is modeled as geodesic chosen by observer attention on the boundary section family.

Daily Analogy: Film Projection and Audience Perspective

Imagine you are in a movie theater:

graph TB
    Film["Film Reel (All sections exist simultaneously)"]
    Projector["Projector (Attention section selection)"]
    Screen["Screen (Observer experience)"]
    Time["Time Flow"]

    Film -->|Select frame| Projector
    Projector -->|Project| Screen
    Screen -->|Produces| Time

    style Film fill:#e1f5ff
    style Projector fill:#fff4e1
    style Screen fill:#ffe1e1
    style Time fill:#e1ffe1

Key Understanding:

1. Film Reel = Boundary Universe: All possible “sections” (film frames) exist simultaneously
2. Projector = Observer Attention: Selects one frame at a time to project onto the screen
3. Screen Image = Experiential World: The single section the observer “sees”
4. Time Flow = Projection Speed: Parameter for attention movement on the section family

Without an observer: The film reel lies quietly in the archive, all frames exist simultaneously, there is considered to be no “time flow”.

With an observer: The projector starts working, frame by frame playback, the audience “feels” time.

Three Key Concepts

1. Boundary Time Geometry (BTG): The Trinity of Time

Recall from Chapter 05, the time scale identity:

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

This tells us: Time scales are already encoded in boundary data, but there are countless ways to choose “which scale becomes the time axis”!

Analogy:

• Scale Master Ruler = Scale on a city map (each cm = 100m)
• Time Axis = The actual route you walk (chosen geodesic)

Without an observer: The map and scale both exist, but there is no “which road is your road.”

With an observer: You choose a route, the length of this road measured by the scale becomes “your time.”

graph LR
    Scale["Boundary Scale Master κ(ω)"]
    Path["Attention Section Family {Σ_τ}"]
    Time["Time Axis τ"]

    Scale -->|Measure| Path
    Path -->|Parameterize| Time

    style Scale fill:#ff6b6b
    style Path fill:#4ecdc4
    style Time fill:#ffe66d

2. Observer Triplet: Who is Looking?

In the BTG framework, an observer is not a “point”, but a combination of three things:

$$\mathcal{O}=(\gamma ,\Lambda ,{\mathcal{A}}_{\gamma ,\Lambda })$$

1. Worldline $\gamma$: Observer’s trajectory in spacetime (the path you walk)
2. Resolution $\Lambda$: The smallest scale the observer can “see clearly” (eye resolution)
3. Observable Algebra ${\mathcal{A}}_{\gamma ,\Lambda }$: Physical quantities the observer can measure (instruments you have)

Daily Analogy: You visit an art exhibition

• $\gamma$ = The route you walk in the gallery
• $\Lambda$ = Your eyesight (nearsightedness requires glasses)
• ${\mathcal{A}}_{\gamma ,\Lambda }$ = The paintings you can see (some too high/far to see)

Different observers = Different $(\gamma ,\Lambda ,\mathcal{A})$ combinations → See different “world sections”!

graph TB
    O["Observer 𝒪"]
    W["Worldline γ"]
    R["Resolution Λ"]
    A["Observable Algebra 𝒜"]

    O --> W
    O --> R
    O --> A

    W -->|Determines| S["Experiential Section Σ_τ"]
    R -->|Determines| S
    A -->|Determines| S

    style O fill:#ff6b6b
    style W fill:#4ecdc4
    style R fill:#4ecdc4
    style A fill:#4ecdc4
    style S fill:#ffe66d

3. Attention Sections and Time Axis: How to “Choose” Time?

At each proper time $\tau$ of a given observer $\mathcal{O}$, define the world section:

$${\Sigma }_{\tau }=(\gamma (\tau ),{\mathcal{A}}_{\gamma ,\Lambda }(\tau ),{\rho }_{\gamma ,\Lambda }(\tau ))$$

This is the world the observer “sees” at time $\tau$:

• $\gamma (\tau )$ = Your position in spacetime
• ${\mathcal{A}}_{\gamma ,\Lambda }(\tau )$ = Physical quantities you can measure at this moment
• ${\rho }_{\gamma ,\Lambda }(\tau )$ = Quantum state of these physical quantities

But the key question:

What is $\tau$ itself? How is it determined?

Answer (Attention Geodesic Proposition):

The time axis $\tau$ must simultaneously satisfy two conditions:

1. Scale Condition: Reading consistency relative to the boundary scale master $\kappa (\omega )$ $$\frac{d\tau }{d\lambda }=\int \kappa (\omega )w_{\lambda }(\omega )d\omega$$

2. Generalized Entropy Geodesic Condition: The section family $\{\sigma (\tau )\}$ satisfies

   • Each section $\sigma (\tau )$ is a stationary point of generalized entropy $S_{\text{gen}}$
   • Along the section family, quantum expansion $\Theta (\lambda )$ is monotonically non-increasing

Daily Analogy: Mountain Climbing Route Selection

1. Scale Condition = Using GPS to measure walking distance (must follow the surface, cannot tunnel)
2. Geodesic Condition = Choose the most energy-efficient route (shortest/flattest path)

Your time = Parameter measured by GPS scale along the optimal path!

graph TB
    Start["Starting Point: No Preferred Time Axis"]
    C1["Condition 1: Scale Consistency"]
    C2["Condition 2: Generalized Entropy Geodesic"]
    End["Time Axis τ Uniquely Determined"]

    Start --> C1
    Start --> C2
    C1 --> End
    C2 --> End

    C1 -.->|Analogy| GPS["GPS Ranging"]
    C2 -.->|Analogy| Path["Shortest Path"]

    style Start fill:#e1f5ff
    style C1 fill:#ffe66d
    style C2 fill:#ffe66d
    style End fill:#ff6b6b

Core Theorems and Corollaries

Proposition 1: No-Observer Time Proposition

Statement:

If no attention section family $\{{\Sigma }_{\tau }\}$ and accessible algebra family $\{{\mathcal{A}}_{{\Sigma }_{\tau }}^{\Lambda }\}$ are selected, then:

1. Global scale master $\kappa (\omega )$ exists, but there is no single time parameter $\tau$
2. All “evolution” can be restated as automorphisms of boundary states (coordinate rescaling)

Plain Translation:

Without an observer, time is considered not to exist. There is only a “scale field”, but no one chooses “which direction is time.”

Daily Analogy: Movie film in storage

• All frames exist simultaneously (boundary section family)
• Each frame has a number (scale master)
• But no one projects, no concept of “playback” → no time flow!

Proposition 2: Attention Geodesic Proposition

Statement:

If there exists an attention map ${\mathcal{E}}_{\tau ,\Lambda }$ satisfying:

1. Section $\sigma (\tau )$ makes generalized entropy stationary: $\delta S_{\text{gen}}=0$
2. Quantum expansion is monotonically non-increasing: $\frac{d\Theta }{d\lambda }\le 0$
3. Time reading is given by the scale master

Then the attention time axis is equivalent to a geodesic on some effective geometry!

Plain Translation:

An attention section family satisfying two conditions (entropy extremum + quantum focusing) is the optimal path in spacetime.

Observer’s time = Parameter along this optimal path!

Daily Analogy: Airline Route Selection

• Generalized Entropy Stationary Point = Minimum fuel consumption
• Quantum Expansion Monotonic = Decreasing air resistance
• Scale Master = Flight odometer

→ Optimal route is the unique geodesic, flight time determined by odometer reading!

graph TB
    A["Observer Attention"]
    B["Select Section Family {Σ_τ}"]
    C1["Condition 1: δS_gen = 0"]
    C2["Condition 2: dΘ/dλ ≤ 0"]
    D["Geodesic Equation"]
    E["Time Axis τ"]

    A --> B
    B --> C1
    B --> C2
    C1 --> D
    C2 --> D
    D --> E

    style A fill:#ff6b6b
    style B fill:#4ecdc4
    style C1 fill:#ffe66d
    style C2 fill:#ffe66d
    style D fill:#e1ffe1
    style E fill:#e1f5ff

Corollary: Section Universe and Observation Branches

Statement:

One can construct a section universe space $\mathfrak{S}$, whose points are equivalence classes $[\Sigma ,{\mathcal{A}}_{\Sigma }^{\Lambda },{\omega }_{\Sigma }^{\Lambda }]$.

Each observer’s experience = A path on $\mathfrak{S}$.

Different observers = Different geodesics on $\mathfrak{S}$.

Plain Translation:

All possible “observation sections” form a vast space (section universe).

Your life = A curve in this space!

Daily Analogy: Library and Reading Paths

graph LR
    Library["Library (Section Universe 𝔖)"]
    Reader1["Reader A's Reading Path"]
    Reader2["Reader B's Reading Path"]

    Library --> Reader1
    Library --> Reader2

    Reader1 -.->|Choose| Book1["Science Fiction"]
    Reader2 -.->|Choose| Book2["Historical Biography"]

    style Library fill:#e1f5ff
    style Reader1 fill:#ffe1e1
    style Reader2 fill:#e1ffe1
    style Book1 fill:#fff4e1
    style Book2 fill:#fff4e1

• Library = Section universe (all possible sections exist simultaneously)
• Reader A = Observer 1, chooses science fiction reading path
• Reader B = Observer 2, chooses historical reading path

Two readers in the same library (universe), but “reading history” (experienced time axis) completely different!

Experimental Verification and Applications

1. Section Interpretation of Double-Slit Interference

Recall the classic double-slit experiment:

• No Detector: Electrons pass through double slits, interference fringes appear on screen
• With Detector: Electrons are “observed”, interference disappears

Traditional Confusion: “Observation changed the past”? Particles “know” they are observed?

BTG Explanation:

The two cases correspond to different attention paths in the section universe!

graph TB
    Universe["Section Universe 𝔖"]
    Path1["Path 1: No Detector"]
    Path2["Path 2: With Detector"]

    Universe --> Path1
    Universe --> Path2

    Path1 --> Result1["Coherence-Preserving Section Family<br>→ Interference Fringes"]
    Path2 --> Result2["Decoherent Section Family<br>→ No Interference"]

    style Universe fill:#e1f5ff
    style Path1 fill:#e1ffe1
    style Path2 fill:#ffe1e1
    style Result1 fill:#fff4e1
    style Result2 fill:#fff4e1

Key Understanding:

• Without Detector: Attention section family $\{{\Sigma }_{\tau }^{\text{free}}\}$ corresponds to accessible algebra preserving cross-slit coherence
• With Detector: Attention map ${\mathcal{E}}_{\tau ,\Lambda }^{\text{path}}$ compresses algebra to path-distinguishable subalgebra

Not “observation changes the past”, but “chose a different section path”!

The universe structure simultaneously accommodates both paths, observers just choose one.

2. No-Retro-Causality Theorem for Delayed Choice Experiment

Wheeler’s thought experiment: After particles pass through double slits, experimenter decides whether to measure the path.

Question: Can post-choice “rewrite” the particle’s past behavior?

BTG Answer: No!

Proposition (No Retro-Causality):

Later-time measurement setting $C$ and result $y$ do not change the unconditional distribution $p(x)$ of earlier-time detection screen event $x$.

$$p_C(x)=\sum _y{p}_C(x,y)=p(x)\text{(Independent of C!)}$$

Delayed choice only changes the decomposition of conditional probability $p(x|y)$, not the marginal distribution!

Daily Analogy: Looking at Old Photos

graph LR
    Photo["Old Photo (Earlier Event x)"]
    Decision["Current Interpretation (Later Choice C)"]
    Memory["Memory (Conditional Probability p(x|y))"]

    Photo -.->|Unchanged| Photo
    Decision --> Memory
    Photo --> Memory

    style Photo fill:#e1f5ff
    style Decision fill:#ffe66d
    style Memory fill:#ffe1e1

• Old Photo Itself Unchanged = $p(x)$ unchanged
• Your Current Interpretation Changes = $p(x|y)$ changes

You look at childhood photos today, recall happy/sad different memories → Different “conditioning”

But the photo itself hasn’t changed!

3. Time Double-Slit: Interference in the Time Domain

Spatial double-slit: Particles take two paths in space

Time double-slit: Particles take two paths in time!

Experimental Setup:

Use two extremely short pulses (attosecond scale) to “open time windows” at times $t_1$ and $t_2$.

Electron wavefunction self-interferes on the time axis → Energy spectrum of outgoing beam shows oscillating fringes!

Unified Formula:

Spatial and temporal double-slits are completely equivalent in BTG:

$$\Delta E\cdot \Delta t\approx 2\pi \hslash$$

• Spatial double-slit: Position distribution $P(x)$ has fringes
• Time double-slit: Energy distribution $P(E)$ has fringes

The two are connected via Wigner–Smith time delay and Fourier duality!

graph TB
    Unified["Unified Scattering Matrix S"]
    Spatial["Spatial Double-Slit"]
    Temporal["Time Double-Slit"]

    Unified --> Spatial
    Unified --> Temporal

    Spatial --> P_x["Position Interference P(x)"]
    Temporal --> P_E["Energy Spectrum Interference P(E)"]

    P_x <-.->|Fourier Duality| P_E

    style Unified fill:#ff6b6b
    style Spatial fill:#4ecdc4
    style Temporal fill:#4ecdc4
    style P_x fill:#ffe66d
    style P_E fill:#ffe66d

Profound Meaning:

Time is viewed not as an “absolute background”, but a dynamical degree of freedom of boundary scattering.

Like space, time can also produce interference!

Philosophical Implications: Block Universe and Attention

Block Universe Picture

In BTG, the complete picture is:

Without an observer:

• Boundary geometry and scale master exist
• All possible sections exist simultaneously in section universe $\mathfrak{S}$
• No preferred time direction, no “flow”

Like a whole film reel lying quietly, all frames are “there”, but not playing.

With an observer:

• Observer’s attention ${\mathcal{E}}_{\tau ,\Lambda }$ selects a section family $\{{\Sigma }_{\tau }\}$
• This family satisfies generalized entropy geodesic conditions → Forms time axis $\tau$
• Observer “experiences” evolution along $\tau$, feels “time flow”

Like a projector starting to work, frame by frame playback, audience “sees” the movie.

graph TB
    Block["Block Universe (Boundary Section Family)"]
    No["No Observer"]
    Yes["With Observer"]

    Block --> No
    Block --> Yes

    No --> Static["Static Existence<br>No Time Flow<br>All Sections Coexist"]
    Yes --> Dynamic["Attention Selection<br>Geodesic Parameterization<br>Experience Time Flow"]

    style Block fill:#e1f5ff
    style No fill:#ffe1e1
    style Yes fill:#e1ffe1
    style Static fill:#fff4e1
    style Dynamic fill:#ffe66d

Free Will and Attention

Question: Can observers “freely choose” attention paths?

BTG Answer: Partially free, partially constrained

1. Constrained Part:

   • Must satisfy generalized entropy geodesic conditions (physical laws)
   • Must be consistent with scale master (time scale constraints)
   • Must satisfy causal consistency (cannot choose “retro-causal” sections)

2. Free Part:

   • Under constraints, there are multiple possible geodesics
   • Choose different resolution $\Lambda$ → Different coarse-graining → Different experience
   • Choose different observable algebras → “See” different aspects

Daily Analogy: City Navigation

graph LR
    Start["Starting Point"]
    End["Destination"]

    Path1["Path 1: Highway<br>(Fast but Monotonous)"]
    Path2["Path 2: Scenic Route<br>(Slow but Beautiful)"]
    Path3["Path 3: Subway<br>(Cheap but Crowded)"]

    Start --> Path1 --> End
    Start --> Path2 --> End
    Start --> Path3 --> End

    style Start fill:#e1f5ff
    style End fill:#e1f5ff
    style Path1 fill:#ffe66d
    style Path2 fill:#e1ffe1
    style Path3 fill:#ffe1e1

• Physical Constraints = Must follow roads (cannot directly tunnel)
• Free Choice = Can choose highway/scenic route/subway

Observer’s “free will” = Geodesic selection under physical constraints!

Multi-Observer Consensus

Question: Different observers choose different section paths, are their worlds “consistent”?

Answer: Consistent on boundary data!

Although different observers follow different geodesics, their:

• Scale master $\kappa (\omega )$ is the same (physical laws)
• Boundary triplet $(\partial M,{\mathcal{A}}_{\partial },{\omega }_{\partial })$ is the same (objective reality)
• Only chose different “projection directions”

Daily Analogy: Blind Men Touching an Elephant (Revisited!)

graph TB
    Elephant["Elephant (Boundary Universe)"]
    Observer1["Observer A<br>Touches Trunk"]
    Observer2["Observer B<br>Touches Leg"]
    Observer3["Observer C<br>Touches Tail"]

    Elephant --> Observer1
    Elephant --> Observer2
    Elephant --> Observer3

    Observer1 --> Report1["Report: 'Like a Water Pipe'"]
    Observer2 --> Report2["Report: 'Like a Pillar'"]
    Observer3 --> Report3["Report: 'Like a Rope'"]

    Report1 -.->|Mathematical Coherence| Consensus["Boundary Data Consistent<br>Just Different Sections"]
    Report2 -.->|Mathematical Coherence| Consensus
    Report3 -.->|Mathematical Coherence| Consensus

    style Elephant fill:#e1f5ff
    style Observer1 fill:#ffe1e1
    style Observer2 fill:#e1ffe1
    style Observer3 fill:#fff4e1
    style Consensus fill:#ff6b6b

Different observers’ reports are “contradictory”, but mathematically coherent in boundary language:

$$H_{\partial }=\int \omega d{\mu }^{\text{scatt}}=c_1{K}_D=c_2^{-1}{H}_{\partial }^{\text{grav}}$$

All observers’ time generators are equivalent on the boundary (affine transformation)!

Connections with Previous and Following Chapters

Review Chapter 05: Unified Time

Chapter 05 established the scale identity:

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

This Chapter Extends:

• Chapter 05: Time scale exists
• Chapter 08: How observers choose time scale to become time axis

Analogy: Map and Route

• Chapter 05 = Scale printed on map (objectively exists)
• Chapter 08 = You choose specific route, measure distance with scale (subjective choice)

Review Chapter 07: Boundary as Stage

Chapter 07: Physics happens at boundary, bulk is projection

This Chapter Extends:

• Chapter 07: Where is the stage (boundary)
• Chapter 08: Who performs on stage (observer), how to perform (attention geodesic)

Analogy:

• Chapter 07 = Theater stage building
• Chapter 08 = How actors (observers) move on stage (section selection)

Preview Chapter 09: Boundary Clock

Next chapter will discuss: How to actually construct boundary clocks to measure time?

• Chapter 08 (this chapter): Theoretical definition of time axis (attention geodesic)
• Chapter 09 (next chapter): Physical implementation of time axis (boundary clock device)

Analogy:

• Chapter 08 = Mathematical principles of GPS positioning
• Chapter 09 = How to build GPS satellites and receivers

Preview Chapter 10: Trinity Master Scale

Chapter 10 will deeply explore how the three equivalent definitions of scale master $\kappa (\omega )$ perfectly align on the boundary:

$$\kappa (\omega )\stackrel{\text{Scattering}}{\leftrightarrow }\frac{{\phi }^{\prime }(\omega )}{\pi }\stackrel{\text{Modular Flow}}{\leftrightarrow }\text{tr}Q(\omega )\stackrel{\text{Gravity}}{\leftrightarrow }{H}_{\partial }^{\text{grav}}$$

Reference Guide

Core Theoretical Sources:

1. Boundary Observer Attention Time: boundary-observer-attention-time.md

   • Mathematical definition of attention sections
   • Generalized entropy geodesic theorem
   • Section universe construction

2. Observer World Section Structure: observer-world-section-structure-causality-delayed-choice-time-double-slit.md

   • Causally consistent section criteria
   • Delayed choice no-retro-causality theorem
   • Time double-slit unified model

3. Boundary Language Unified Framework: boundary-language-unified-framework.md (Chapters 05-11)

   • Boundary three axioms
   • Trinity implementation

Experimental Verification:

• Wheeler delayed choice experiment: Wikipedia “Delayed-choice quantum eraser”
• Attosecond time double-slit: arXiv physics papers
• Wigner–Smith matrix measurement: Electromagnetic scattering network experiments

Next Chapter Preview:

Chapter 09 “Boundary Clock: How to Measure Time?” will discuss how to physically implement boundary time measurement devices, including:

• Windowed clocks solving negative delay problems
• DPSS spectral windows and error control
• Atomic clock networks as distributed boundary clocks

Core Question: How can the theoretical “time scale” be read out with instruments in the laboratory?