Chapter 10 Matrix Universe: 01 Mathematical Definition of Observer

Introduction

In the previous section, we raised the core question of the observer problem:

What is an observer in matrix universe THE-MATRIX?

This section will give a strict mathematical answer. We will prove:

1. Matrix observer can be precisely defined as projection-algebra-state triple $(P_O,{\mathcal{A}}_O,{\omega }_O)$
2. “I” is observer equivalence class satisfying three axioms (worldline, self-referentiality, minimality)
3. Matrix observer and causal manifold observer are equivalent
4. Self-referential scattering network carries ${\mathbb{Z}}_2$ topological fingerprint

This will lay the foundation for subsequent “mind-universe equivalence” (Section 02) and multi-observer consensus (Section 03).

1. Matrix Universe THE-MATRIX: Review and Notation

1.1 Definition of Matrix Universe

In Chapter 9, we have established matrix universe THE-MATRIX:

$$\mathrm{THE}\text{-}\mathrm{MATRIX}=(\mathcal{H},{\mathcal{A}}_{\partial },\{S(\omega ){\}}_{\omega \in I},\kappa ,{\prec }_{\mathrm{mat}})$$

where:

1. $\mathcal{H}$: Separable Hilbert space (“channel space” or “boundary degree of freedom space”)

2. ${\mathcal{A}}_{\partial }\subset \mathcal{B}(\mathcal{H})$: Boundary observable algebra

3. $\{S(\omega ){\}}_{\omega \in I}$: Scattering matrix family, each $S(\omega )$ is unitary operator on $\mathcal{H}$, piecewise differentiable in $\omega$

4. $\kappa (\omega )$: Unified time scale density, satisfying scale identity:

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

where:

• $\phi (\omega )$ is total scattering half-phase
• ${\rho }_{\mathrm{rel}}(\omega )=-{\xi }^{\prime }(\omega )$ is relative state density ($\xi$ is spectral shift function)
• $Q(\omega )=-iS(\omega {)}^{†}{\partial }_{\omega }S(\omega )$ is Wigner-Smith group delay matrix

5. ${\prec }_{\mathrm{mat}}$: Causal partial order on channel index set, characterizing causal reachability relation

1.2 Block Matrix Structure

In concrete construction, choose channel orthogonal decomposition:

$$\mathcal{H}=\underset{a\in \mathcal{I}}{⨁}{\mathcal{H}}_a$$

where index set $\mathcal{I}$ carries causal partial order ${\prec }_{\mathrm{mat}}$. Scattering matrix written as block matrix:

$$S(\omega )=(S_{ab}(\omega ){)}_{a,b\in \mathcal{I}}$$

Causal Constraint: Only when $a$ is causally reachable to $b$ (i.e., exists path $a⇝b$), corresponding block $S_{ba}(\omega )$ is non-zero.

Sparsity Pattern: Non-zero pattern of matrix encodes causal structure, this is origin of name “matrix universe”.

1.3 Integral Form of Unified Time Scale

From density $\kappa (\omega )$ define time coordinate:

$$\tau (\omega )={\int }_{{\omega }_0}^{\omega }\kappa (\omega )\mathrm{d}\omega$$

Time scale equivalence class: $$[\tau ]=\{\tau \mid \tau (\omega )=a\tau (\omega )+b,a>0,b\in \mathbb{R}\}$$

Key Property: All physical times of matrix universe (scattering time, modular time, geometric time) belong to same equivalence class $[\tau ]$.

2. Matrix Observer: Basic Definition

2.1 Triple Definition of Observer

Definition 2.1 (Matrix Observer)

In matrix universe THE-MATRIX, a matrix observer $O$ is a triple:

$$O=(P_O,{\mathcal{A}}_O,{\omega }_O)$$

where:

1. Projection $P_O$: Orthogonal projection on $\mathcal{H}$, satisfying $P_O=P_O^2=P_O^{†}$ (Observer’s channel support)

2. Algebra ${\mathcal{A}}_O:=P_O{\mathcal{A}}_{\partial }{P}_O$: Boundary algebra restricted to support (Observables actually accessible to observer)

3. State ${\omega }_O$: Normal state on ${\mathcal{A}}_O$ (Observer’s statistical belief about these observables)

Intuitive Understanding:

• $P_O$: Part of channels observer “can see”
• ${\mathcal{A}}_O$: Physical quantities observer can measure
• ${\omega }_O$: Observer’s “subjective” probability distribution over these quantities

2.2 Local Scattering Matrix and Local Time Scale

Given matrix observer $O=(P_O,{\mathcal{A}}_O,{\omega }_O)$, define:

Local Scattering Matrix: $$S_O(\omega ):=P_{O}S(\omega )P_O:P_O\mathcal{H}\to P_O\mathcal{H}$$

Local Group Delay Matrix: $$Q_O(\omega ):=-i{S}_O(\omega {)}^{†}{\partial }_{\omega }{S}_O(\omega )$$

Local Time Scale Density: $${\kappa }_O(\omega ):=\frac{1}{2\pi }\mathrm{tr}{Q}_O(\omega )$$

Time Scale Consistency: Require ${\kappa }_O(\omega )$ and global $\kappa (\omega )$ belong to same equivalence class $[\tau ]$, i.e.:

$${\kappa }_O(\omega )=a\kappa (\omega )+b,a>0$$

This ensures observer’s “internal clock” aligns with universe’s unified time scale.

3. Matrix Worldline: Observer’s Time Evolution

3.1 Definition of Worldline

In causal manifold, observer’s worldline is a timelike curve $\gamma :\tau \mapsto x(\tau )$. In matrix universe, corresponding concept is:

Definition 3.1 (Matrix Worldline)

Let $[\tau ]$ be unified time scale equivalence class. A matrix worldline is a projection family $\{P(\tau ){\}}_{\tau \in J}$ satisfying:

1. Interval: $J\subset \mathbb{R}$ is an interval

2. Projection Family: For each $\tau \in J$, $P(\tau )$ is orthogonal projection on $\mathcal{H}$

3. Monotonicity: If ${\tau }_1<{\tau }_2$, then $P({\tau }_1)\le P({\tau }_2)$ (i.e., $P({\tau }_1)P({\tau }_2)=P({\tau }_1)$)

4. Locality: For each $\tau$, $P(\tau )$ depends only on scattering data on finite energy window $I_{\tau }\subset I$

Intuitive Understanding:

• $P(\tau )$: Support of all information observer has “recorded” on boundary up to time $\tau$
• Monotonicity: Records can only accumulate, cannot be erased (“time arrow”)
• Locality: Observer can only acquire information through local scattering processes

3.2 Observer Carrying Worldline

Definition 3.2

If matrix observer $O=(P_O,{\mathcal{A}}_O,{\omega }_O)$ has matrix worldline $\{P(\tau ){\}}_{\tau \in J_O}$ and interval $J_O$ such that:

$$\forall \tau \in J_O:P(\tau )\le P_O$$

then $O$ is said to carry a matrix worldline.

Geometric Picture:

graph LR
    A["P(τ₁)"] -->|"Monotonic Growth"| B["P(τ₂)"]
    B -->|"Monotonic Growth"| C["P(τ₃)"]
    C -.->|"Limit"| D["P_O"]

    style A fill:#e1f5ff
    style B fill:#b3e5ff
    style C fill:#80d4ff
    style D fill:#4db8ff

Projection family $P(\tau )$ monotonically increases with time, eventually “filling” observer’s total support $P_O$.

3.3 Correspondence with Causal Manifold Worldline

Proposition 3.3 (Matrix Worldline ↔ Timelike Curve)

Under appropriate Čech gluing conditions, matrix worldline $\{P(\tau ){\}}_{\tau \in J}$ uniquely corresponds to a timelike curve $\gamma :J\to M$ on causal manifold such that:

1. Proper time parameter $\tau$ of $\gamma$ belongs to unified time scale equivalence class $[\tau ]$

2. $P(\tau )$ is Toeplitz/Berezin compression of boundary algebra of small causal diamond reachable along $\gamma (\tau )$

Proof Outline:

1. On each small causal diamond $D_{p,r}$, boundary algebra ${\mathcal{A}}_{\partial D}$ embeds into global algebra ${\mathcal{A}}_{\partial }$
2. Sequence of boundary algebras of small diamond family $\{D_{\gamma (\tau ),r}\}$ along worldline $\gamma$ defines projection family
3. Unified time scale ensures consistency of parameter $\tau$

4. Self-Referentiality Axiom: Core Feature of “I”

4.1 Fixed Point Equation of Self-Referential Update

Observer not only “observes” external world, but also “predicts its own future behavior”. This self-referentiality is essential feature of “I”.

Definition 4.1 (Self-Referential Update)

Update of matrix observer $O=(P_O,{\mathcal{A}}_O,{\omega }_O)$ is self-referential if there exists functional $F_{\mathrm{self}}$ such that:

$${\omega }_O(\tau )=F_{\mathrm{self}}[{\omega }_O(\tau ),S_O,\kappa ]$$

This is a fixed point equation: Observer’s internal state ${\omega }_O(\tau )$ must be consistent with “self-state predicted using $S_O,\kappa$”.

Concrete Form:

Under appropriate parameterization, fixed point equation can be written as:

$${\omega }_O({\tau }_2)=U({\tau }_2,{\tau }_1)\circ {\omega }_O({\tau }_1)$$

where update operator $U({\tau }_2,{\tau }_1)$ satisfies:

$$U({\tau }_2,{\tau }_1)=F\left({\omega }_O({\tau }_1),S_O{|}_{[{\tau }_1,{\tau }_2]},E_{{\tau }_1},{\mathcal{D}}_{[{\tau }_1,{\tau }_2]}^{\mathrm{ext}}\right)$$

• $E_{{\tau }_1}$: External environment map
• ${\mathcal{D}}^{\mathrm{ext}}$: External observation data

Key point: ${\omega }_O({\tau }_1)$ appears on right side, forming closed-loop feedback.

4.2 Realization of Self-Referential Scattering Network

In scattering network language, self-referentiality manifests as feedback loop:

graph LR
    A["External Input"] --> B["Scattering Node S_O(ω)"]
    B --> C["Output"]
    B --> D["Internal Memory M"]
    D -->|"Feedback"| B

    style B fill:#ff9999
    style D fill:#99ccff

Mathematical Definition of Self-Referential Scattering Network:

1. Select port set $\{E,O_{\mathrm{in}},O_{\mathrm{out}},M_{\mathrm{in}},M_{\mathrm{out}}\}$

   • $E$: External world port
   • $O_{\mathrm{in}},O_{\mathrm{out}}$: Observer input/output ports
   • $M_{\mathrm{in}},M_{\mathrm{out}}$: Internal memory ports

2. Scattering matrix block decomposition: $$S(\omega )=\left(\begin{array}{ccc}{S}_{EE}& S_{EO}& S_{EM}\\ S_{OE}& S_{OO}& S_{OM}\\ S_{ME}& S_{MO}& S_{MM}\end{array}\right)$$

3. Closed-Loop Condition: Feedback from $M_{\mathrm{out}}\to M_{\mathrm{in}}$ forms self-referential closed loop

4.3 Uniqueness of Self-Referential Fixed Point

Lemma 4.2 (Uniqueness of Self-Referential Fixed Point)

Under appropriate continuity and contraction mapping conditions, fixed point equation:

$${\omega }_O=F_{\mathrm{self}}[{\omega }_O,S_O,\kappa ]$$

has unique solution ${\omega }_O^{\ast }$.

Proof Outline:

1. View $F_{\mathrm{self}}$ as mapping on state space
2. Under appropriate metric (e.g., relative entropy distance), prove $F_{\mathrm{self}}$ is contraction mapping
3. Apply Banach fixed point theorem

Physical Meaning:

Self-referential fixed point ${\omega }_O^{\ast }$ corresponds to “self-consistent observer”: its predictions about itself completely match actual behavior.

5. Minimality Axiom: Irreducibility of “I”

5.1 Definition of Minimality

Axiom III (Minimality Axiom)

If $O^{\prime }=(P^{\prime },{\mathcal{A}}^{\prime },{\omega }^{\prime })$ also satisfies worldline axiom and self-referentiality axiom, and:

$$P^{\prime }\le P_O$$

then $P^{\prime }=P_O$ (almost everywhere).

Intuitive Understanding:

“I” is minimal support projection satisfying self-referential conditions. Cannot be “shrunk” further, otherwise cannot satisfy self-referentiality.

5.2 Minimality and Irreducibility

Proposition 5.1 (Minimality Equivalent to Irreducibility)

Matrix observer $O=(P_O,{\mathcal{A}}_O,{\omega }_O)$ satisfies minimality axiom if and only if:

$$P_O=P_1\oplus P_2\Rightarrow P_1=0\text{ or }{P}_2=0$$

i.e., $P_O$ cannot be decomposed into non-trivial direct sum.

Proof:

Assume $P_O=P_1\oplus P_2$ and both $P_1,P_2$ are non-zero. Then:

1. Both $P_1,P_2$ inherit self-referentiality and worldline properties of $P_O$
2. But $P_1<P_O$, contradicting minimality

Therefore $P_O$ is irreducible.

5.3 Minimality and $K^1$ Theory

In consistency factory framework (Chapter 8), scattering family $\{S_O(\omega )\}$ corresponds to an element $[{\mathsf{u}}_O]$ of $K^1$ theory.

Theorem 5.2 (Minimality and $K^1$ Minimal Element)

Matrix observer $O$ satisfying minimality axiom corresponds to irreducible element $[{\mathsf{u}}_O]$ in $K^1$, i.e.:

$$[{\mathsf{u}}_O]=[{\mathsf{u}}_1]\oplus [{\mathsf{u}}_2]\Rightarrow [{\mathsf{u}}_1]=0\text{ or }[{\mathsf{u}}_2]=0$$

Physical Meaning:

“I” is topologically indivisible—cannot be split into two independent “sub-I“s.

6. Complete Definition of “I”

6.1 Summary of Three Axioms

Summarizing previous axioms, we get complete mathematical definition of “I”.

Definition 6.1 (“I” in Matrix Universe)

In matrix universe THE-MATRIX, an “I” is matrix observer equivalence class $[O]$ satisfying following three axioms:

Axiom I (Worldline Axiom) $O$ carries a matrix worldline $\{P(\tau ){\}}_{\tau \in J}$, and $P(\tau )$ monotonically increases with unified time scale.

Axiom II (Self-Referentiality Axiom) There exists fixed point equation: $${\omega }_O(\tau )=F_{\mathrm{self}}[{\omega }_O(\tau ),S_O,\kappa ]$$ such that $O$’s internal predictive state is consistent with actual scattering readings.

Axiom III (Minimality Axiom) Under premise of satisfying Axioms I, II, $P_O$ is minimal:

$$P^{\prime }\le P_O\text{ and }{P}^{\prime }\text{ satisfies Axioms I, II }\Rightarrow P^{\prime }=P_O$$

Equivalence Relation:

Two matrix observers $O_1,O_2$ are equivalent, denoted $O_1\sim O_2$, if there exists unitary operator $U$ and affine time rescaling $\tau \mapsto a\tau +b$ such that:

$$P_{O_2}(\tau )=U{P}_{O_1}(a\tau +b)U^{†},{\omega }_{O_2}={\omega }_{O_1}\circ \mathrm{Ad}(U^{-1})$$

“I” is defined as equivalence class $[O]=\{O^{\prime }:O^{\prime }\sim O\}$.

6.2 Independence of Three Axioms

Question: Are the three axioms independent? Or can some be derived from others?

Answer: The three are independent but mutually constraining.

1. Worldline + Self-Referentiality ⇏ Minimality Counterexample: $P_O=P_1\oplus P_2$, where $P_1,P_2$ are both self-referential worldline projections, but $P_O$ is not minimal.

2. Worldline + Minimality ⇏ Self-Referentiality Counterexample: A minimal “passive observer”, only records without prediction, no self-referential closed loop.

3. Self-Referentiality + Minimality ⇏ Worldline Counterexample: An instantaneous self-referential system, no time evolution.

Therefore all three axioms are indispensable.

6.3 Equivalence with Causal Manifold “I”

In Chapter 9 (QCA Universe), we have already defined “I” on causal manifold side:

$$\mathfrak{I}=(\gamma ,{\mathcal{A}}_{\gamma },{\omega }_{\gamma })$$

where $\gamma$ is timelike worldline, ${\mathcal{A}}_{\gamma }$ is boundary algebra glued along $\gamma$.

Theorem 6.3 (Equivalence of Matrix “I” and Manifold “I”)

Within energy window satisfying unified time scale, boundary time geometry, and consistency factory assumptions, “I” $[O]$ in matrix universe and “I” $[\mathfrak{I}]$ in causal manifold correspond one-to-one.

Correspondence:

• Matrix worldline $\{P(\tau )\}$ ↔ Timelike curve $\gamma$
• Projection algebra ${\mathcal{A}}_O=P_O{\mathcal{A}}_{\partial }{P}_O$ ↔ Boundary algebra ${\mathcal{A}}_{\gamma }$
• State ${\omega }_O$ ↔ State ${\omega }_{\gamma }$
• Self-referential fixed point ↔ Closed loop of self-referential scattering network

Proof Outline:

1. Using Null-Modular double cover and Toeplitz/Berezin compression, establish correspondence between boundary algebras of small causal diamonds and matrix blocks
2. Unified time scale ensures alignment of time parameters
3. Self-referential conditions equivalent in both languages

Complete proof see Appendix A (omitted).

7. ${\mathbb{Z}}_2$ Topological Fingerprint of Self-Referential Scattering Network

7.1 Scattering Square Root and Double Cover

In self-referential scattering network, scattering determinant may have square root branch:

$$\det S^{\circlearrowleft }(\omega )={\left[\sqrt{\det S^{\circlearrowleft }(\omega )}\right]}^2$$

Different square root choices correspond to a ${\mathbb{Z}}_2$ double cover.

Modified Determinant:

Under trace class conditions, define:

$$\mathfrak{s}(\omega ,\lambda )=\underset{p}{\det }{S}^{\circlearrowleft }(\omega ;\lambda )$$

where ${\det }_p$ is Fredholm modified determinant.

7.2 Holonomy and ${\mathbb{Z}}_2$ Index

For closed path $\gamma \subset X^{\circ }$ in parameter space (avoiding singularities), define holonomy of square root determinant:

$${\nu }_{\sqrt{S^{\circlearrowleft }}}(\gamma )=\exp \left(i{\oint }_{\gamma }\frac{1}{2i}{\mathfrak{s}}^{-1}\mathrm{d}\mathfrak{s}\right)\in \{\pm 1\}$$

This is a ${\mathbb{Z}}_2$-valued invariant, homotopy invariant.

Theorem 7.1 (${\mathbb{Z}}_2$ Holonomy)

${\mathbb{Z}}_2$ holonomy $\nu (\gamma )$ of self-referential scattering network is invariant under observer equivalence relation, giving topological fingerprint of “I”.

7.3 Correspondence with Null-Modular Double Cover

In Null-Modular double cover theory (Chapter 8), there exists cohomology class:

$$[K]\in H^2(Y,\partial Y;{\mathbb{Z}}_2),Y=M\times X^{\circ }$$

Theorem 7.2 (Topological Consistency)

Physical self-consistency requires $[K]=0$ (no topological anomaly). Under this condition, ${\mathbb{Z}}_2$ holonomy of self-referential scattering network aligns with topological class of Null-Modular double cover.

Physical Meaning:

Self-referential structure of “I” must be topologically consistent with overall topological sector of universe, otherwise internal contradictions arise.

7.4 Diagram of Self-Referential Network

graph TD
    A["External World E"] --> B["Observer Input O_in"]
    B --> C["Scattering Processing S_O(ω)"]
    C --> D["Observer Output O_out"]
    C --> E["Internal Memory M"]
    E -->|"Self-Referential Feedback"| C
    D --> F["Predictive Model ℳ_O"]
    F -.->|"Fixed Point Condition"| E

    style C fill:#ff9999
    style E fill:#99ccff
    style F fill:#99ff99

• Red node (scattering processing): Core scattering matrix $S_O$
• Blue node (internal memory): Carries state ${\omega }_O$
• Green node (predictive model): Model family ${\mathcal{M}}_O$
• Dashed arrow: Fixed point condition ${\omega }_O=F_{\mathrm{self}}[{\omega }_O,S_O,\kappa ]$

8. Examples: Concrete Matrix Observers

8.1 Example 1: Single-Channel Observer

Setting:

Matrix universe $\mathcal{H}={\mathbb{C}}^N$, $N$ channels. Observer can only access channel 1.

Definition:

$$P_O=|1⟩⟨1|,{\mathcal{A}}_O=\mathbb{C}\cdot P_O,{\omega }_O(P_O)=1$$

Local Scattering:

$$S_O(\omega )=⟨1|S(\omega )|1⟩=S_{11}(\omega )\in U(1)$$

Time Scale:

$${\kappa }_O(\omega )=\frac{1}{2\pi }\frac{\mathrm{d}}{\mathrm{d}\omega }\arg S_{11}(\omega )$$

This is simplest matrix observer: a “single-pixel” observer.

8.2 Example 2: Schwarzschild Black Hole Observer

Setting:

Consider Schwarzschild spacetime, static observer at radial coordinate $r=r_{\ast }$ (outside black hole).

Scattering Matrix:

In Schwarzschild background, scattering matrix of scalar field is:

$$S(\omega )=\left(\begin{array}{cc}{S}_{\mathrm{out},\mathrm{out}}& S_{\mathrm{out},\mathrm{in}}\\ S_{\mathrm{in},\mathrm{out}}& S_{\mathrm{in},\mathrm{in}}\end{array}\right)$$

where “out” corresponds to exterior channel, “in” corresponds to interior (crossing horizon).

Exterior Observer:

$$P_O=P_{\mathrm{out}},S_O(\omega )=S_{\mathrm{out},\mathrm{out}}(\omega )$$

Hawking Radiation:

Local time scale of exterior observer contains information of Hawking temperature:

$${\kappa }_O(\omega )\sim \frac{1}{T_H}+O(\omega -{\omega }_H)$$

where $T_H=\frac{\hslash c^3}{8\pi GM{k}_B}$ is Hawking temperature.

8.3 Example 3: Self-Referential Program in Quantum Computer

Setting:

A quantum computer with $N$ qubits, running a self-referential program (simulating itself).

Hilbert Space:

$$\mathcal{H}=({\mathbb{C}}^2{)}^{\otimes N}$$

Self-Referential Projection:

Select part of qubits as “meta-computer” (simulating entire system):

$$P_O=P_{\mathrm{meta}}\otimes {\mathbb{I}}_{\mathrm{rest}}$$

Fixed Point Condition:

Program output must be consistent with output of “simulating itself”:

$$U_{\mathrm{prog}}|{\psi }_O⟩=|{\psi }_O⟩\otimes |f({\psi }_O)⟩$$

where $|f({\psi }_O)⟩$ is “simulation result”.

This is a discrete version of self-referential observer.

9. Comparison with Other Observer Theories

9.1 vs Copenhagen Interpretation Observer

Copenhagen Interpretation:

• Observer is classical system, external to quantum system
• Measurement causes wave function collapse (mysterious process)

GLS Matrix Observer:

• Observer is part of quantum system ($P_O\subset \mathcal{H}$)
• No “collapse”, only unitary evolution + coarse-graining (detailed in Section 04)

9.2 vs Many-Worlds Interpretation Observer

Many-Worlds Interpretation:

• Measurement causes universe to “branch”, each branch corresponds to an observer
• All possible outcomes “really exist”

GLS Matrix Observer:

• No “branching”, universe ontology is unique matrix universe
• Observer’s subjective experience corresponds to reduced state ${\omega }_O$ under projection $P_O$
• Multiple observers $\{O_i\}$ converge to same “objective reality” through consensus (Section 03)

9.3 vs Relational Quantum Mechanics Observer

Relational Quantum Mechanics:

• Physical properties are relational, defined relative to observer
• No “absolute quantum state” exists

GLS Matrix Observer:

• Universe ontology state ${\omega }_{\partial }$ is absolute (exists in category ${\mathbf{Univ}}_{\mathrm{mat}}$)
• State ${\omega }_O$ observer “sees” is projection reduction of ${\omega }_{\partial }$
• But multi-observer consensus convergence guarantees existence of “objective reality” (Section 05)

9.4 vs QBism Observer

QBism:

• Quantum state is observer’s subjective belief
• No “objective quantum state” exists

GLS Matrix Observer:

• State ${\omega }_O$ indeed similar to “subjective belief”
• But “my mind is the universe” theorem (Section 02) proves: Under unified time scale, “subjective” and “objective” categorically equivalent
• Therefore both subjectivity (${\omega }_O$) and objectivity (${\omega }_{\partial }$) exist, the two are isomorphic

Summary Table:

10. Section Summary

10.1 Core Achievements

This section completed strict mathematical definition of observer in matrix universe:

1. Matrix Observer = Triple $(P_O,{\mathcal{A}}_O,{\omega }_O)$

   • Projection $P_O$: Channel support
   • Algebra ${\mathcal{A}}_O$: Observables
   • State ${\omega }_O$: Statistical belief

2. Three Axioms define “I”:

   • Worldline axiom: Carries time evolution
   • Self-referentiality axiom: Satisfies fixed point equation
   • Minimality axiom: Irreducibility

3. Equivalence Theorem:

   • Matrix “I” ↔ Causal manifold “I”
   • Aligned through unified time scale and boundary algebra

4. Topological Fingerprint:

   • ${\mathbb{Z}}_2$ holonomy uniquely marks “I”
   • Consistent with Null-Modular double cover topology

10.2 Key Insights

Insight 1: “I” is not some instantaneous state, but trajectory of time evolution (matrix worldline)

Insight 2: Essence of “I” is self-referentiality: System capable of predicting itself

Insight 3: “I” mathematically is minimal irreducible element: Cannot be decomposed further

Insight 4: Observer is not external, but internal self-referential structure of matrix universe

10.3 Connection with Subsequent Sections

Matrix observer definition in this section lays foundation for subsequent sections:

• Section 02 (Mind-Universe Equivalence): Will prove observer’s “internal model” isometric to universe ontology
• Section 03 (Multi-Observer Consensus): Study how multiple $\{O_i\}$ converge to consensus through communication
• Section 04 (Measurement Problem): Resolve wave function collapse problem using matrix observer framework
• Section 05 (Emergence of Objective Reality): Prove “objective world” emerges from multi-observer consensus

Appendix A: Proof Outlines of Key Theorems

A.1 Proof of Theorem 6.3 (Matrix-Manifold Equivalence)

Proposition: Matrix “I” $[O]$ and causal manifold “I” $[\mathfrak{I}]$ correspond one-to-one.

Proof Steps:

Step 1: Matrix → Manifold

Given matrix observer $O=(P_O,{\mathcal{A}}_O,{\omega }_O)$, construct:

1. Through matrix worldline $\{P(\tau )\}$, at each $\tau$ select corresponding small causal diamond $D_{\gamma (\tau ),r}$
2. Using Toeplitz/Berezin compression, correspond $P(\tau )$ to boundary algebra of $D_{\gamma (\tau ),r}$
3. Glue all boundary algebras for all $\tau$, obtain ${\mathcal{A}}_{\gamma }$
4. State ${\omega }_O$ corresponds to ${\omega }_{\gamma }$ through compression

Step 2: Manifold → Matrix

Given causal manifold “I” $\mathfrak{I}=(\gamma ,{\mathcal{A}}_{\gamma },{\omega }_{\gamma })$, construct:

1. Small causal diamond family $\{D_{\gamma (\tau ),r}\}$ along $\gamma$ defines boundary Hilbert space family
2. Through boundary scattering data, construct projection family $\{P(\tau )\}$
3. Take limit $P_O={\lim }_{\tau \to \infty }P(\tau )$
4. Define ${\mathcal{A}}_O=P_O{\mathcal{A}}_{\partial }{P}_O$

Step 3: Equivalence

Prove correspondence $O\sim \mathfrak{I}$ preserves:

• Unified time scale equivalence class
• Self-referential condition (fixed point equation ↔ self-referential scattering closed loop)
• Minimality (irreducible projection ↔ indivisible worldline)

QED.

A.2 Proof of Theorem 7.1 (${\mathbb{Z}}_2$ Holonomy Invariance)

Proposition: ${\mathbb{Z}}_2$ holonomy of self-referential scattering network is invariant under observer equivalence.

Proof Outline:

1. Observer equivalence $O_1\sim O_2$ corresponds to unitary transformation $U:S_{O_2}=U{S}_{O_1}{U}^{†}$
2. Modified determinant satisfies: ${\det }_p(US{U}^{†})={\det }_{p}S$
3. Therefore holonomy integral: $$\oint {\mathfrak{s}}_2^{-1}\mathrm{d}{\mathfrak{s}}_2=\oint {\mathfrak{s}}_1^{-1}\mathrm{d}{\mathfrak{s}}_1$$
4. ${\mathbb{Z}}_2$ index invariant

QED.

Appendix B: Concrete Examples of Self-Referential Fixed Points

B.1 Fixed Point Under Linear Approximation

Assume fixed point equation under linear approximation is:

$${\omega }_O(\tau )=A(\tau ){\omega }_O(\tau )+B(\tau )$$

where $A(\tau )$ is operator, $B(\tau )$ is external input.

Solution:

$${\omega }_O(\tau )=(I-A(\tau ){)}^{-1}B(\tau )$$

Require $(I-A(\tau ))$ invertible, i.e., spectral radius of $A(\tau )$ $<1$.

B.2 Nonlinear Case: Riccati Equation

In some self-referential systems, fixed point equation can be reduced to Riccati type:

$$\frac{\mathrm{d}{\rho }_O}{\mathrm{d}\tau }=-i[H_O,{\rho }_O]+\mathcal{L}[{\rho }_O]+F({\rho }_O,{\rho }_O)$$

where $F({\rho }_O,{\rho }_O)$ is quadratic term (self-referential feedback).

Existence and uniqueness of solutions to such equations require Carathéodory or Picard iteration methods.

Section Complete!

Preview of Next Section:

In Section 02 “Mind-Universe Equivalence”, we will prove core theorem of GLS theory:

“My mind is the universe” is not a philosophical slogan, but an information-geometric isomorphism theorem.

Specific content:

• Model manifold $(\Theta ,g^{\mathrm{FR}})$ of “my mind”
• Alignment of Fisher-Rao metric and unified time scale
• Bayesian posterior concentration theorem
• Categorical equivalence ${\mathbf{Obs}}_{\mathrm{full}}\simeq {\mathbf{Univ}}_{\mathrm{phys}}$

Ready to witness mathematical unification of “subjective” and “objective”!