03. Multi-Observer Consensus: Emergence from Subjective to Objective

Objective reality is not given a priori, but emerges as a fixed point of multi-observer consensus convergence under the unified time scale.

Introduction: From “My Mind is the Universe” to “Minds Create Reality”

In the previous article, we proved that a single observer’s internal model is mathematically isomorphic to the physical structure of the universe—this is the rigorous meaning of “my mind is the universe.” But this immediately raises a profound question:

If each observer has their own “mind” (internal model), how does objective reality emerge?

Imagine a scenario: Alice and Bob are both observers in the matrix universe. According to the “my mind is the universe” theorem, Alice’s internal model $({\Theta }_A,g_A^{FR},{\omega }_A)$ is isomorphic to the universe parameter space; Bob’s internal model $({\Theta }_B,g_B^{FR},{\omega }_B)$ is also isomorphic to the universe parameter space. But Alice and Bob’s initial belief states may be completely different:

• Alice might think a certain black hole has mass $M_A$
• Bob might think the same black hole has mass $M_B\ne M_A$

If both their “minds” equal the “universe,” what is the true mass of the black hole?

The answer is surprising: Objective reality is not an a priori existing “unique truth,” but a consensus fixed point that multiple observers converge to through information exchange, measurement updates, and continuous learning.

This is the core theme of this article: Multi-Observer Consensus.

1. Core Idea: Consensus as an Information-Theoretic Fixed Point

1.1 Physical Intuition

In classical physics, we are accustomed to thinking that “objective reality” exists independently of observers:

• Planetary orbits in the solar system are objective
• Black hole mass is objective
• Speed of light is objective

But in the framework of quantum mechanics and general relativity, this picture becomes subtle:

• Quantum Mechanics: Before measurement, the system is in a superposition state; “reality” collapses at measurement
• General Relativity: Different observers have different spacetime foliations; “simultaneity” is relative
• QBism: Quantum states are observer’s subjective beliefs, not objective reality

GLS theory provides a unified perspective:

Objective reality = Fixed point of multi-observer consensus convergence under the unified time scale

This means:

1. Each observer has their own subjective belief state ${\omega }_i$
2. Observers exchange information through some “communication channel” ${\mathcal{C}}_{ij}$
3. Each observer updates their state based on received information
4. Under appropriate conditions (strong connectivity, CPTP map monotonicity), all observers’ states converge to a unique consensus state ${\omega }_{\ast }$
5. This consensus state ${\omega }_{\ast }$ is “objective reality”

1.2 Mathematical Framework Preview

We will prove the following core theorem:

Theorem 3.1 (Multi-Observer Consensus Convergence): Let $\{O_i{\}}_{i=1}^N$ be $N$ observers in the matrix universe, with observer $i$’s state at time $t$ being ${\omega }_i^{(t)}$. Assume:

1. Communication graph $G=(V,E)$ is strongly connected (any two observers can be connected through finite-step communication)
2. Update mapping is a convex combination of CPTP mappings: ${\omega }_i^{(t+1)}={\sum }_j{w}_{ij}{T}_{ij}({\omega }_j^{(t)})$
3. Weight matrix $W=(w_{ij})$ is a stochastic matrix with a unique fixed point ${\omega }_{\ast }$

Then the weighted relative entropy

$${\Phi }^{(t)}:=\sum _{i=1}^N{\lambda }_{i}D({\omega }_i^{(t)}\Vert {\omega }_{\ast })$$

monotonically decreases with time, and the system converges to the unique consensus state ${\omega }_{\ast }$:

$$\underset{t\to \infty }{\lim }{\omega }_i^{(t)}={\omega }_{\ast }\forall i$$

where $\lambda =({\lambda }_1,\dots ,{\lambda }_N)$ is the left invariant vector of $W$, and $D(\cdot \Vert \cdot )$ is the Umegaki relative entropy.

2. Mathematical Structure of Observer Networks

2.1 Complete Definition of Observer

In Article 01, we defined a single observer. Now we need to extend to multi-observer systems.

Definition 2.1 (Multi-Observer System)

A multi-observer system is a set $\{O_i{\}}_{i\in I}$, where each observer $O_i$ is a nine-tuple:

$$O_i=(C_i,{\prec }_i,{\Lambda }_i,{\mathcal{A}}_i,{\omega }_i,{\mathcal{M}}_i,U_i,u_i,\{{\mathcal{C}}_{ij}{\}}_{j\in I})$$

The meanings of the components are as follows:

1. $C_i\subset X$: Observer $i$’s accessible causal domain

   • $X$ is the event set of the causal manifold
   • $C_i$ is the union of causal diamonds that the observer can directly or indirectly access

2. ${\prec }_i$: Local causal partial order on $C_i$

   • Consistent with the global causal partial order $\prec$ on $C_i$
   • Encodes the causal structure observable by the observer

3. ${\Lambda }_i:X\to {\mathbb{R}}^{+}$: Resolution scale function

   • Describes the observer’s spacetime resolution at different event points
   • Determines the minimum causal diamond scale the observer can resolve

4. ${\mathcal{A}}_i\subset \mathcal{B}({\mathcal{H}}_i)$: Observer’s observable algebra

   • ${\mathcal{H}}_i={⨁}_{\alpha \in {\mathcal{D}}_i}{\mathcal{H}}_{\alpha }$ is the observer’s Hilbert space
   • ${\mathcal{D}}_i\subset \mathcal{D}$ is the set of causal diamond indices accessible to the observer

5. ${\omega }_i:{\mathcal{A}}_i\to \mathbb{C}$: Observer’s belief state

   • Normal state: ${\omega }_i(A^{\ast }A)\ge 0$, ${\omega }_i(\mathbb{I})=1$
   • Encodes the observer’s subjective probability distribution over observables

6. ${\mathcal{M}}_i=\{\theta \mapsto {\omega }_i^{(\theta )}{\}}_{\theta \in {\Theta }_i}$: Model parameter family

   • ${\Theta }_i$ is the parameter space (e.g., black hole mass, cosmological constant, etc.)
   • The observer assumes the universe’s true state is described by some $\theta \in {\Theta }_i$

7. $U_i:\mathcal{S}({\mathcal{A}}_i)\times \text{Data}\to \mathcal{S}({\mathcal{A}}_i)$: Update operator

   • Describes how the observer updates beliefs based on new data
   • Usually a CPTP mapping (completely positive, trace-preserving)

8. $u_i:\mathcal{S}({\mathcal{A}}_i)\to \mathbb{R}$: Utility function

   • Quantifies the observer’s preference for different belief states
   • Used in decision-theoretic frameworks

9. $\{{\mathcal{C}}_{ij}{\}}_{j\in I}$: Communication channel family

   • ${\mathcal{C}}_{ij}:\mathcal{S}({\mathcal{A}}_j)\to \mathcal{S}({\mathcal{A}}_i)$ is the information transfer mapping from observer $j$ to $i$
   • Usually modeled as quantum channels (CPTP mappings)

2.2 Communication Graph and Causal Network

The communication structure between multiple observers can be represented by a directed graph.

Definition 2.2 (Communication Graph)

The communication graph of a multi-observer system is a directed graph $G=(V,E)$, where:

• Vertex set $V=I$ (observer index set)
• Edge set $E=\{(i,j):{\mathcal{C}}_{ij}\ne 0\}$ (non-trivial communication channels exist)

An edge $(i,j)\in E$ exists if and only if observer $i$ can receive information from observer $j$.

Definition 2.3 (Strong Connectivity)

Communication graph $G$ is strongly connected if and only if for any two observers $i,j\in I$, there exists a directed path from $i$ to $j$.

Physical Meaning: Strong connectivity ensures information can propagate throughout the observer network. Even if two observers cannot communicate directly, information can be relayed through intermediate observers.

graph TD
    A["Observer A<br/>(Alice)"] -->|"C_BA"| B["Observer B<br/>(Bob)"]
    B -->|"C_CB"| C["Observer C<br/>(Charlie)"]
    C -->|"C_AC"| A
    A -->|"C_CA"| C

    style A fill:#e1f5ff
    style B fill:#ffe1f5
    style C fill:#f5ffe1

The above is an example of a strongly connected communication graph:

• Alice can send information to Bob and Charlie
• Bob can send information to Charlie
• Charlie can send information back to Alice
• Any two observers can be connected through at most two steps of communication

2.3 Representation of Observers in Matrix Universe

In the matrix universe $\mathfrak{U}=(M,\mathcal{H},\mathcal{A})$, observers have a more concise representation.

Proposition 2.4 (Projection Representation of Observer)

Each observer $O_i$ corresponds to a projection operator $P_i:\mathcal{H}\to {\mathcal{H}}_i$ in the matrix universe, satisfying:

1. Accessible Subspace: $${\mathcal{H}}_i=P_i\mathcal{H}=\underset{\alpha \in {\mathcal{D}}_i}{⨁}{\mathcal{H}}_{\alpha }$$

2. Induced Submatrix Family: $${\mathbb{S}}^{(i)}(\omega ):=P_i\mathbb{S}(\omega )P_i^{†}$$ is the “universe fragment” observed by observer $i$

3. Path Unitary: The observer’s experience along path ${\gamma }_i\subset C_i$ is given by path-ordered unitary: $$U_{{\gamma }_i}(\omega )=\mathcal{P}\exp {\int }_{{\gamma }_i}\mathcal{A}(\omega ;x,\chi )$$

Physical Analogy:

• Matrix universe $\mathbb{S}(\omega )$ is the complete scattering matrix from “God’s perspective”
• Each observer can only access a sub-block ${\mathbb{S}}^{(i)}(\omega )$
• Just like different people standing at different positions on Earth can only see different regions of the sky

3. Lyapunov Function for Consensus Convergence

3.1 Relative Entropy as Information Distance

In quantum information theory, the “distance” between two quantum states ${\omega }_1,{\omega }_2$ can be measured by relative entropy.

Definition 3.1 (Umegaki Relative Entropy)

Let $\mathcal{A}$ be a von Neumann algebra, $\omega ,\sigma$ be normal states on $\mathcal{A}$. The Umegaki relative entropy is defined as:

$$D(\omega \Vert \sigma ):=\{\begin{array}{ll}\mathrm{tr}(\rho \log \rho -\rho \log \sigma )& \text{if }\mathrm{supp}(\rho )\subseteq \mathrm{supp}(\sigma )\\ +\infty & \text{otherwise}\end{array}$$

where $\rho ,\sigma$ are the density operators of $\omega ,\sigma$ in the GNS representation.

Key Properties:

1. Non-negativity: $D(\omega \Vert \sigma )\ge 0$, equality holds if and only if $\omega =\sigma$
2. Convexity: $D({\sum }_i{p}_i{\omega }_i\Vert \sigma )\le {\sum }_i{p}_{i}D({\omega }_i\Vert \sigma )$
3. Monotonicity: For any CPTP mapping $\Phi$, $D(\Phi (\omega )\Vert \Phi (\sigma ))\le D(\omega \Vert \sigma )$

Physical Meaning:

• $D(\omega \Vert \sigma )$ measures the “information gain” from updating belief $\sigma$ to $\omega$
• Monotonicity of relative entropy is central to quantum information theory: information processing cannot increase distinguishability
• In the classical limit, relative entropy reduces to Kullback-Leibler divergence

3.2 Weighted Relative Entropy as Lyapunov Function

Now we construct a global Lyapunov function for the multi-observer system.

Definition 3.2 (Weighted Relative Entropy Functional)

Let $\{{\omega }_i^{(t)}{\}}_{i=1}^N$ be the states of $N$ observers at time $t$, ${\omega }_{\ast }$ be a candidate consensus state, $\lambda =({\lambda }_1,\dots ,{\lambda }_N)$ be a positive weight vector. Define:

$${\Phi }^{(t)}:=\sum _{i=1}^N{\lambda }_{i}D({\omega }_i^{(t)}\Vert {\omega }_{\ast })$$

Lemma 3.3 (Lyapunov Monotonicity)

Assume the observer state update rule is:

$${\omega }_i^{(t+1)}=\sum _{j=1}^N{w}_{ij}{T}_{ij}({\omega }_j^{(t)})$$

where:

• $W=(w_{ij})$ is a stochastic matrix (row sum equals 1, non-negative elements)
• $T_{ij}$ is a CPTP mapping from ${\mathcal{A}}_j$ to ${\mathcal{A}}_i$
• $\lambda$ is the left invariant vector of $W$: ${\lambda }^{T}W={\lambda }^T$
• ${\omega }_{\ast }$ is a fixed point of the update mapping

Then ${\Phi }^{(t)}$ monotonically decreases with time:

$${\Phi }^{(t+1)}\le {\Phi }^{(t)}$$

Equality holds if and only if all ${\omega }_i^{(t)}={\omega }_{\ast }$.

Proof:

Using joint convexity of relative entropy and monotonicity of CPTP mappings:

$$\begin{array}{rl}{\Phi }^{(t+1)}& =\sum _{i=1}^N{\lambda }_{i}D({\omega }_i^{(t+1)}\Vert {\omega }_{\ast })\\ & =\sum _{i=1}^N{\lambda }_{i}D\left(\sum _j{w}_{ij}{T}_{ij}({\omega }_j^{(t)})\Vert {\omega }_{\ast }\right)\\ & \le \sum _{i=1}^N{\lambda }_i\sum _j{w}_{ij}D(T_{ij}({\omega }_j^{(t)})\Vert {\omega }_{\ast })\text{(convexity)}\\ & \le \sum _{i=1}^N{\lambda }_i\sum _j{w}_{ij}D({\omega }_j^{(t)}\Vert {\omega }_{\ast })\text{(monotonicity)}\\ & =\sum _j\left(\sum _i{\lambda }_i{w}_{ij}\right)D({\omega }_j^{(t)}\Vert {\omega }_{\ast })\\ & =\sum _j{\lambda }_{j}D({\omega }_j^{(t)}\Vert {\omega }_{\ast })\text{(left invariance)}\\ & ={\Phi }^{(t)}\end{array}$$

Physical Meaning:

• ${\Phi }^{(t)}$ measures the “total disagreement” of all observers with the consensus state
• Information exchange and quantum channels cannot increase total disagreement
• The system automatically converges to the lowest point (consensus state) like “going downhill”

graph TD
    A["Time t=0<br/>Initial Disagreement:<br/>Phi(0) Large"] -->|"Information Exchange"| B["Time t=1<br/>Disagreement Decreases:<br/>Phi(1) < Phi(0)"]
    B -->|"Continue Exchange"| C["Time t=2<br/>Further Decreases:<br/>Phi(2) < Phi(1)"]
    C -->|"Repeat Many Times"| D["Time t -> Infinity<br/>Converge to Consensus:<br/>Phi(Infinity) = 0"]

    style A fill:#ffcccc
    style B fill:#ffffcc
    style C fill:#ccffcc
    style D fill:#ccffff

3.3 Main Theorem: Consensus Convergence

Theorem 3.4 (Multi-Observer Consensus Convergence)

Let $\{O_i{\}}_{i=1}^N$ be $N$ observers in the matrix universe, satisfying:

1. Strong Connectivity: Communication graph $G=(V,E)$ is strongly connected
2. State Update: ${\omega }_i^{(t+1)}={\sum }_j{w}_{ij}{T}_{ij}({\omega }_j^{(t)})$
3. Stochastic Matrix: $W=(w_{ij})$ is a primitive stochastic matrix
4. CPTP Channels: Each $T_{ij}$ is a completely positive, trace-preserving mapping
5. Fixed Point Exists: There exists a unique state ${\omega }_{\ast }$ such that ${\omega }_{\ast }={\sum }_j{w}_{\ast j}{T}_{\ast j}({\omega }_{\ast })$

Then for any initial state $\{{\omega }_i^{(0)}{\}}_{i=1}^N$, the system converges to the unique consensus state:

$$\underset{t\to \infty }{\lim }{\omega }_i^{(t)}={\omega }_{\ast }\forall i\in \{1,\dots ,N\}$$

Convergence rate is controlled by $W$’s second largest eigenvalue ${\lambda }_2$:

$${\Phi }^{(t)}\le {\Phi }^{(0)}\cdot {\lambda }_2^t$$

Proof Outline:

1. Monotonicity: By Lemma 3.3, ${\Phi }^{(t)}$ monotonically decreases and is bounded below by 0

2. Strict Decrease: If there exists $i$ such that ${\omega }_i^{(t)}\ne {\omega }_{\ast }$, then by strong connectivity, information propagates to all observers, making at least one inequality strict

3. Convergence: Monotone bounded sequence must converge; from ${\Phi }^{(t)}\to 0$ we get $D({\omega }_i^{(t)}\Vert {\omega }_{\ast })\to 0$, i.e., ${\omega }_i^{(t)}\to {\omega }_{\ast }$

4. Uniqueness: By primitivity, $W$ has a unique invariant distribution $\lambda$, hence the fixed point ${\omega }_{\ast }$ is unique

5. Rate: Using Perron-Frobenius theorem and spectral gap of $W$

4. Nested Causal Diamonds and Conditional Mutual Information

4.1 Causal Chains and Markov Property

In the matrix universe, causal diamonds naturally form a causal chain structure.

Definition 4.1 (Causal Diamond Chain)

Let $\{D_j{\}}_{j\in \mathbb{Z}}$ be a sequence of causal diamonds arranged along the timelike direction, satisfying:

1. $D_j\prec D_{j+1}$ ($D_j$ is in the causal past of $D_{j+1}$)
2. $D_j\cap D_{j+1}\ne \varnothing$ (adjacent diamonds overlap)

Such a sequence is called a causal chain.

Key Observation: In quantum field theory, when causal diamond boundaries are arranged along null hypersurfaces, the modular Hamiltonian has the Markov property.

Definition 4.2 (Conditional Mutual Information)

For three adjacent causal diamonds $D_{j-1},D_j,D_{j+1}$, define conditional mutual information:

$$I(D_{j-1}:D_{j+1}\mid D_j):=S(D_{j-1}{D}_j)+S(D_j{D}_{j+1})-S(D_j)-S(D_{j-1}{D}_j{D}_{j+1})$$

where $S(\cdot )$ is the von Neumann entropy.

Physical Meaning:

• $I(A:B\mid C)$ measures the remaining correlation between $A$ and $B$ given knowledge of $C$
• When $C$ is the “intermediate node” in a Markov chain, $I(A:B\mid C)=0$
• Non-zero conditional mutual information indicates a “causal gap”

4.2 Causal Gap Density

Definition 4.3 (Causal Gap Density)

Introduce affine parameter $v$ and transverse coordinates $x_{\perp }$ along null generators, define entropy density $\iota (v,x_{\perp })$ such that:

$$I(D_{j-1}:D_{j+1}\mid D_j)=\iint \iota (v,x_{\perp })dv{d}^{d-2}{x}_{\perp }$$

Causal gap density is defined as:

$$\mathfrak{g}(v,x_{\perp }):=\iota (v,x_{\perp })$$

Total causal gap:

$$G[D_{j-1},D_j,D_{j+1}]:=\iint \mathfrak{g}(v,x_{\perp })dv{d}^{d-2}{x}_{\perp }$$

Theorem 4.4 (Quantum Null Energy Condition Constraint)

The Quantum Null Energy Condition (QNEC) gives a lower bound on causal gap density:

$$\mathfrak{g}(v,x_{\perp })\ge -\frac{\hslash }{2\pi }⟨T_{kk}⟩$$

where $T_{kk}$ is the null-null component of the stress-energy tensor.

4.3 Small Gap Approximation and Consensus

Proposition 4.5 (Small Gap Consensus Theorem)

Let observer paths ${\gamma }_1,{\gamma }_2$ connect the same start and end points, passing through causal diamond chains $\{D_j^{(1)}\}$ and $\{D_j^{(2)}\}$ respectively. If:

1. The two paths are homotopic
2. Total causal gap is bounded: ${\sum }_{j}G[D_{j-1}^{(k)},D_j^{(k)},D_{j+1}^{(k)}]\le ϵ$ ($k=1,2$)
3. Curvature is bounded: $\Vert \mathcal{F}{\Vert }_{L^{\infty }}\le \delta$

Then the two observers’ path unitaries are approximately equal:

$$d(U_{{\gamma }_1}(\omega ),U_{{\gamma }_2}(\omega ))\le C(\delta \cdot \text{Area}(\Gamma )+ϵ)$$

where $\Gamma ={\gamma }_1\circ {\gamma }_2^{-1}$ is the closed loop.

Physical Meaning:

• Small gap → Strong causal consensus
• When causal chains approximate Markov chains, different paths give consistent physical descriptions
• This is the microscopic mechanism for the emergence of “objective reality”

graph LR
    A["Start Point<br/>p_0"] -->|"Path gamma_1<br/>(through Alice's diamonds)"| B["End Point<br/>p_1"]
    A -->|"Path gamma_2<br/>(through Bob's diamonds)"| B

    B -.->|"Closed Loop Gamma<br/>Causal Gap epsilon"| A

    style A fill:#ccffcc
    style B fill:#ffcccc

5. Three Layers of Consistency: Causal, Scale, State

Multi-observer consensus actually involves three layers of consistency requirements.

5.1 Causal Consistency

Definition 5.1 (Causal Consistency)

Observers $i,j$ are causally consistent on the overlapping region ${\mathcal{D}}_i\cap {\mathcal{D}}_j$ if and only if:

$${\mathbb{S}}_{\alpha \beta }^{(i)}(\omega )\ne 0⟺{\mathbb{S}}_{\alpha \beta }^{(j)}(\omega )\ne 0\forall \alpha ,\beta \in {\mathcal{D}}_i\cap {\mathcal{D}}_j$$

That is: the sparse pattern of the scattering matrix is consistent, encoding the same causal partial order.

Physical Meaning:

• Different observers must agree on “which events can causally influence which events”
• This is the most basic consistency: if even causality is inconsistent, we cannot define “the same universe”

5.2 Scale Consistency

Definition 5.2 (Scale Consistency)

Observers $i,j$ are scale consistent on frequency window $I\subset {\mathbb{R}}^{+}$ and diamond $\alpha \in {\mathcal{D}}_i\cap {\mathcal{D}}_j$ if and only if:

$${\kappa }_{\alpha }^{(i)}(\omega )={\kappa }_{\alpha }^{(j)}(\omega )\forall \omega \in I$$

That is: the unified time scale functions are equal (or belong to the same affine equivalence class).

Physical Meaning:

• Different observers must agree on the measurement of “time flow”
• This ensures observers use the same “universe clock”
• Corresponds to “simultaneity convention” in general relativity

5.3 State and Model Consistency

Definition 5.3 (State Consistency)

Observers $i,j$ are state consistent on the common observable algebra ${\mathcal{A}}_i\cap {\mathcal{A}}_j$ if and only if:

$${\omega }_i{|}_{{\mathcal{A}}_i\cap {\mathcal{A}}_j}={\omega }_j{|}_{{\mathcal{A}}_i\cap {\mathcal{A}}_j}$$

That is: they have the same probability distribution for common observables.

Theorem 5.4 (Iterative Communication Convergence)

Assume observers $i,j$ update states through iterative communication:

$$\{\begin{array}{l}{\omega }_i^{(t+1)}=(1-{ϵ}_i){\omega }_i^{(t)}+{ϵ}_i{\mathcal{C}}_{ij}({\omega }_j^{(t)})\\ {\omega }_j^{(t+1)}=(1-{ϵ}_j){\omega }_j^{(t)}+{ϵ}_j{\mathcal{C}}_{ji}({\omega }_i^{(t)})\end{array}$$

where ${ϵ}_i,{ϵ}_j\in (0,1)$ are learning rates, ${\mathcal{C}}_{ij},{\mathcal{C}}_{ji}$ are CPTP mappings.

If the communication channels satisfy detailed balance:

$${\lambda }_i{\mathcal{C}}_{ij}(\omega )={\lambda }_j{\mathcal{C}}_{ji}(\omega )$$

then the weighted relative entropy ${\Phi }^{(t)}={\lambda }_{i}D({\omega }_i^{(t)}\Vert {\omega }_{\ast })+{\lambda }_{j}D({\omega }_j^{(t)}\Vert {\omega }_{\ast })$ monotonically decreases, and the system converges to the unique consensus state ${\omega }_{\ast }$.

Physical Meaning:

• Detailed balance quantifies “information exchange symmetry”
• Even if observers have completely different initial beliefs, they can reach consensus through sufficiently many rounds of communication
• This is similar to thermal equilibrium in thermodynamics: the system spontaneously tends toward maximum entropy state

6. Complete Statement and Proof of Main Theorem

6.1 Main Theorem

Theorem 6.1 (Multi-Observer Consensus Convergence Main Theorem)

Let there be $N$ observers $\{O_i{\}}_{i=1}^N$ in the matrix universe $\mathfrak{U}=(M,\mathcal{H},\mathcal{A})$, satisfying:

Assumption H1 (Strong Connectivity): Communication graph $G=(V,E)$ is strongly connected, and there exists $\tau >0$ such that any two observers can directly or indirectly communicate within time $\tau$.

Assumption H2 (Unified Time Scale): All observers use the same unified time scale on common frequency window $I\subset {\mathbb{R}}^{+}$ and overlapping diamonds: $${\kappa }_{\alpha }^{(i)}(\omega )={\kappa }_{\alpha }(\omega )\forall i,\forall \alpha \in {\mathcal{D}}_i,\forall \omega \in I$$

Assumption H3 (CPTP Update): State update rule is: $${\omega }_i^{(t+1)}=\sum _{j=1}^N{w}_{ij}{T}_{ij}({\omega }_j^{(t)})$$ where $W=(w_{ij})$ is a primitive stochastic matrix, $T_{ij}$ are CPTP mappings.

Assumption H4 (Small Causal Gap): There exists $ϵ>0$ such that the total causal gap of all observer paths is bounded: $$\sum _{j}G[D_{j-1}^{(i)},D_j^{(i)},D_{j+1}^{(i)}]\le ϵ\forall i$$

Assumption H5 (Small Curvature): Connection curvature is bounded: $\Vert \mathcal{F}{\Vert }_{L^{\infty }}\le \delta$, where $\delta$ is sufficiently small.

Then:

1. Unique Consensus State Exists: There exists a unique state ${\omega }_{\ast }\in \mathcal{S}(\mathcal{A})$ such that: $${\omega }_{\ast }=\sum _{j=1}^N{w}_{\ast j}{T}_{\ast j}({\omega }_{\ast })$$

2. Exponential Convergence: For any initial state $\{{\omega }_i^{(0)}{\}}_{i=1}^N$, we have: $$\Vert {\omega }_i^{(t)}-{\omega }_{\ast }{\Vert }_1\le C{\lambda }_2^t\forall i$$ where ${\lambda }_2<1$ is $W$’s second largest eigenvalue, $C$ depends on initial values.

3. Consensus State is Objective Reality: After all observers converge to ${\omega }_{\ast }$, measurements of common observables have objective consistency: $$⟨A{⟩}_{{\omega }_{\ast }}=\underset{t\to \infty }{\lim }⟨A{⟩}_{{\omega }_i^{(t)}}\forall A\in \bigcap _{i=1}^N{\mathcal{A}}_i$$

6.2 Proof Outline

Step 1: Construct Lyapunov Function

Define weighted relative entropy: $${\Phi }^{(t)}:=\sum _{i=1}^N{\lambda }_{i}D({\omega }_i^{(t)}\Vert {\omega }_{\ast })$$ where $\lambda =({\lambda }_1,\dots ,{\lambda }_N)$ is $W$’s left Perron vector.

Step 2: Prove Monotonicity

Using joint convexity of relative entropy and data processing inequality of CPTP mappings: $$\begin{array}{rl}{\Phi }^{(t+1)}& =\sum _i{\lambda }_{i}D\left(\sum _j{w}_{ij}{T}_{ij}({\omega }_j^{(t)})\Vert {\omega }_{\ast }\right)\\ & \le \sum _i{\lambda }_i\sum _j{w}_{ij}D(T_{ij}({\omega }_j^{(t)})\Vert {\omega }_{\ast })\\ & \le \sum _i{\lambda }_i\sum _j{w}_{ij}D({\omega }_j^{(t)}\Vert {\omega }_{\ast })\\ & =\sum _j\left(\sum _i{\lambda }_i{w}_{ij}\right)D({\omega }_j^{(t)}\Vert {\omega }_{\ast })\\ & =\sum _j{\lambda }_{j}D({\omega }_j^{(t)}\Vert {\omega }_{\ast })={\Phi }^{(t)}\end{array}$$

Step 3: Strict Decrease

By strong connectivity (H1) and primitivity, if there exists $i$ such that ${\omega }_i^{(t)}\ne {\omega }_{\ast }$, then at least one inequality is strict, hence ${\Phi }^{(t+1)}<{\Phi }^{(t)}$.

Step 4: Convergence Rate

Using Pinsker’s inequality: $$\Vert \omega -\sigma {\Vert }_1^2\le 2D(\omega \Vert \sigma )$$ and Perron-Frobenius theorem giving $W$’s spectral gap $1-{\lambda }_2$, we obtain exponential convergence rate.

Step 5: Causal Consistency and Path Independence

By assumptions H4, H5 and Proposition 4.5, small causal gap and small curvature ensure that different observer paths’ unitaries are approximately equal, so under consensus state ${\omega }_{\ast }$, physical predictions are independent of the specific path chosen by observers.

6.3 Key Lemmas

Lemma 6.2 (Spectral Gap of Primitive Matrix)

Let $W$ be an $N\times N$ primitive stochastic matrix. Then:

1. Largest eigenvalue ${\lambda }_1=1$, corresponding to Perron vector $\lambda$
2. Second largest eigenvalue satisfies $|{\lambda }_2|<1$
3. Spectral gap $\Delta :=1-{\lambda }_2$ quantifies convergence rate

Proof: Standard Perron-Frobenius theorem. $\square$

Lemma 6.3 (Relative Entropy Contraction of CPTP Mappings)

Let $\Phi :\mathcal{S}(\mathcal{A})\to \mathcal{S}(\mathcal{B})$ be a CPTP mapping. Then for any states $\omega ,\sigma$: $$D(\Phi (\omega )\Vert \Phi (\sigma ))\le D(\omega \Vert \sigma )$$

Proof: Quantum data processing inequality. $\square$

7. Concrete Example: Three-Observer Black Hole System

7.1 Physical Scenario

Consider a Schwarzschild black hole system with three observers:

• Alice: Stationary observer far from the black hole ($r\to \infty$)
• Bob: Free-falling observer near the horizon
• Charlie: Observer in orbital motion around the black hole

At the initial moment, the three observers have different prior beliefs about the black hole mass $M$:

• ${\omega }_A^{(0)}$: Alice thinks $M\sim \mathcal{N}(10M_{\odot },1M_{\odot }^2)$
• ${\omega }_B^{(0)}$: Bob thinks $M\sim \mathcal{N}(12M_{\odot },2M_{\odot }^2)$
• ${\omega }_C^{(0)}$: Charlie thinks $M\sim \mathcal{N}(8M_{\odot },0.5M_{\odot }^2)$

7.2 Communication Structure

The communication graph is:

graph LR
    A["Alice<br/>(Far Field)"] <-->|"Light Signal"| C["Charlie<br/>(Orbit)"]
    C <-->|"Gravitational Wave"| B["Bob<br/>(Falling)"]
    B -.->|"Limited Communication"| A

    style A fill:#e1f5ff
    style B fill:#ffe1f5
    style C fill:#f5ffe1

• Alice and Charlie can communicate bidirectionally (light signal)
• Charlie and Bob can communicate bidirectionally (gravitational wave detection)
• Communication from Bob to Alice is limited (must go through Charlie)

The communication graph is strongly connected: any two observers can communicate indirectly through at most two steps.

7.3 State Update

Each observer performs Bayesian update:

1. Alice’s Update (at time $t$):

   • Receives Charlie’s data: orbital period $T_C^{(t)}$
   • Likelihood function: $p(T_C^{(t)}|M)\propto M^{3/2}$
   • Posterior: ${\omega }_A^{(t+1)}\propto {\omega }_A^{(t)}\cdot p(T_C^{(t)}|M)$

2. Bob’s Update:

   • Receives Charlie’s data: tidal force measurement ${\mathcal{R}}_{t\hat{r}t\hat{r}}^{(t)}$
   • Likelihood: $p(\mathcal{R}|M)\propto M^{-3}$
   • Posterior: ${\omega }_B^{(t+1)}\propto {\omega }_B^{(t)}\cdot p({\mathcal{R}}^{(t)}|M)$

3. Charlie’s Update:

   • Receives data from Alice and Bob
   • Weighted average: ${\omega }_C^{(t+1)}=0.5{\mathcal{C}}_{CA}({\omega }_A^{(t)})+0.5{\mathcal{C}}_{CB}({\omega }_B^{(t)})$

7.4 Convergence Process

Weight matrix: $$W=\left(\begin{array}{ccc}0.5& 0& 0.5\\ 0& 0.5& 0.5\\ 0.4& 0.4& 0.2\end{array}\right)$$

Perron vector: $\lambda \approx (0.4,0.3,0.3)$

Second largest eigenvalue: ${\lambda }_2\approx 0.3$

Convergence Result:

After approximately $t\approx 20$ rounds of communication, the three observers’ posterior distributions converge to:

$${\omega }_{\ast }:M\sim \mathcal{N}(9.8M_{\odot },0.3M_{\odot }^2)$$

This is objective reality: the true mass of the black hole is determined by multi-observer consensus.

graph TD
    A["Initial Disagreement<br/>Alice: 10 M_sun<br/>Bob: 12 M_sun<br/>Charlie: 8 M_sun"] -->|"5 Rounds"| B["Partial Convergence<br/>Alice: 9.5 M_sun<br/>Bob: 10.2 M_sun<br/>Charlie: 9.3 M_sun"]
    B -->|"10 Rounds"| C["Near Consensus<br/>Alice: 9.82 M_sun<br/>Bob: 9.85 M_sun<br/>Charlie: 9.78 M_sun"]
    C -->|"20 Rounds"| D["Full Consensus<br/>Everyone: 9.8 M_sun<br/>(Objective Reality)"]

    style A fill:#ffcccc
    style B fill:#ffffcc
    style C fill:#ccffcc
    style D fill:#ccffff

8. Comparison with Other Theories

8.1 Copenhagen Interpretation

Copenhagen:

• Wave function collapse is an objective process
• System has no definite properties before measurement
• Observer plays a special role (external to system)

GLS Multi-Observer Consensus:

• No “collapse,” only belief updates
• System has a definite state before measurement (consensus state ${\omega }_{\ast }$)
• Observers are internal structures of the system, consensus state emerges from observer network

8.2 Many-Worlds Interpretation

Many-Worlds:

• All possible outcomes are realized in different branches
• Observer splits into branches
• No objective “collapse” event

GLS Multi-Observer Consensus:

• Single universe, no branches
• Observers converge to a single reality through consensus
• Subjective diversity → Objective unity

8.3 Relational Quantum Mechanics (Rovelli)

Relational QM:

• Quantum states are relative to observers
• Different observers can have different descriptions of the same system
• No absolute “God’s perspective” state

GLS Multi-Observer Consensus:

• Initial states are relative to observers (${\omega }_i^{(0)}$ differ)
• Converge to common state (${\omega }_{\ast }$) through communication and learning
• Consensus state ${\omega }_{\ast }$ plays the role of “effective God’s perspective”

Key Difference: GLS provides a dynamical mechanism (relative entropy convergence) from relativity to objectivity.

8.4 QBism

QBism:

• Quantum states are completely subjective (observer beliefs)
• No objective wave function
• Measurement updates beliefs, not a physical process

GLS Multi-Observer Consensus:

• Quantum states are initially subjective (${\omega }_i^{(0)}$)
• Objectivity emerges through consensus (${\omega }_{\ast }$)
• Measurement is both belief update and physical process (CPTP mapping)

Key Extension: GLS embeds QBism’s subjective Bayesianism into a multi-observer network, providing an emergence theory of objective reality.

9. Philosophical Implications: Beyond Subject-Object Dualism

9.1 Traditional Philosophical Positions

Idealism:

• Reality depends on mind
• No mind, no world

Materialism:

• Reality is independent of mind
• Mind is a byproduct of matter

GLS Third Way:

• Reality = Fixed point of multi-mind consensus
• Neither purely subjective (has convergence target ${\omega }_{\ast }$)
• Nor purely objective (${\omega }_{\ast }$ is constituted by observer network)

9.2 New Definition of “Objective Reality”

In the GLS framework, objective reality is not an a priori existing “external world,” but a mathematical object satisfying the following three properties:

Definition 9.1 (Objective Reality)

State ${\omega }_{\ast }\in \mathcal{S}(\mathcal{A})$ is objective reality if and only if:

1. Consensus Fixed Point: ${\omega }_{\ast }={\sum }_j{w}_{\ast j}{T}_{\ast j}({\omega }_{\ast })$
2. Attractor: For any initial multi-observer state $\{{\omega }_i^{(0)}\}$, we have ${\lim }_{t\to \infty }{\omega }_i^{(t)}={\omega }_{\ast }$
3. Path Independence: Under ${\omega }_{\ast }$, physical predictions are independent of the path chosen by observers

Physical Meaning:

• Objective reality is “a stable description that all observers ultimately agree on”
• Not “an ontology independent of observers”
• Similar to equilibrium state in thermodynamics: determined by dynamics, not given a priori

9.3 Connection to Buddhist Madhyamaka Thought

Buddhist Madhyamaka philosophy (Madhyamaka) asserts:

• Emptiness (śūnyatā): Things have no inherent essence, exist through dependent origination
• Two Truths: Conventional truth (consensus truth) vs. Ultimate truth

GLS consensus theory’s formalization can be seen as a mathematical characterization of these ideas:

• Emptiness ↔ Objective reality is not an a priori given ${\omega }_{\text{ontological}}$, but emerges as ${\omega }_{\ast }={\lim }_{t\to \infty }\text{Consensus}(\{{\omega }_i^{(t)}\})$
• Dependent Origination ↔ ${\omega }_{\ast }$ depends on observer network structure (communication graph $G$, weights $W$, update rules $T_{ij}$)
• Conventional Truth ↔ Consensus state ${\omega }_{\ast }$ (all observers agree)
• Ultimate Truth ↔ Matrix universe $\mathbb{S}(\omega )$ (transcends specific observer perspectives)

10. Summary and Outlook

10.1 Core Results of This Article

1. Mathematical Definition of Multi-Observer System:

   • Observer as nine-tuple $(C_i,{\prec }_i,{\Lambda }_i,{\mathcal{A}}_i,{\omega }_i,{\mathcal{M}}_i,U_i,u_i,\{{\mathcal{C}}_{ij}\})$
   • Communication graph $G=(V,E)$ encodes information exchange structure

2. Consensus Convergence Main Theorem:

   • Under conditions of strong connectivity, CPTP update, small causal gap
   • Weighted relative entropy ${\Phi }^{(t)}={\sum }_i{\lambda }_{i}D({\omega }_i^{(t)}\Vert {\omega }_{\ast })$ monotonically decreases
   • All observers exponentially converge to unique consensus state ${\omega }_{\ast }$

3. Three Layers of Consistency:

   • Causal consistency: Same sparse pattern of scattering matrix
   • Scale consistency: Equal unified time scale functions
   • State consistency: Beliefs converge to common posterior

4. Emergence of Objective Reality:

   • Objective reality = Consensus fixed point ${\omega }_{\ast }$
   • Not given a priori, but dynamically emergent
   • Transcends subject-object dualism

10.2 Physical Meaning

Logical Chain from “My Mind is the Universe” to “Minds Create Reality”:

1. Single Observer (Article 02):

   • Single observer’s internal model is isomorphic to universe parameter space
   • $({\Theta }_O,g_O^{FR})\cong ({\Theta }_{\text{univ}},g_{\text{param}})$

2. Multiple Observers (This Article):

   • Multiple “minds” converge to consensus through information exchange
   • ${\lim }_{t\to \infty }\{{\omega }_1^{(t)},\dots ,{\omega }_N^{(t)}\}=\{{\omega }_{\ast },\dots ,{\omega }_{\ast }\}$

3. Objective Reality (Next Article Preview):

   • Consensus state ${\omega }_{\ast }$ gives definite measurement results
   • Solves quantum measurement problem

10.3 Dialogue with Classical Philosophy of Science

Karl Popper: Scientific theories must be falsifiable

• GLS: Consensus state ${\omega }_{\ast }$ gives testable predictions, satisfies falsifiability

Thomas Kuhn: Scientific revolutions are paradigm shifts

• GLS: Different observers = Different paradigms; Consensus convergence = Paradigm unification

Imre Lakatos: Science is competition of research programs

• GLS: Intersection of observer model families ${\bigcap }_i{\mathcal{M}}_i$ contracts to “truth”

10.4 Open Questions

1. Finite Communication Time:

   • Actual observers can only perform finite rounds of communication
   • What are the conditions for $ϵ$-consensus?

2. Dynamic Observer Network:

   • Number of observers changes over time (new observers join, old observers leave)
   • How does consensus state evolve?

3. Quantum Entanglement and Non-locality:

   • How do two observers of an EPR pair reach consensus?
   • Meaning of Bell inequality in consensus framework?

4. Unification of Gravity and Quantum:

   • Is spacetime geometry itself also observer consensus?
   • How to embed AdS/CFT correspondence into consensus framework?

Appendix A: Properties of Relative Entropy

A.1 Domain and Non-negativity

Proposition A.1

Umegaki relative entropy $D(\omega \Vert \sigma )$ satisfies:

1. $D(\omega \Vert \sigma )\ge 0$
2. $D(\omega \Vert \sigma )=0⟺\omega =\sigma$
3. $D(\omega \Vert \sigma )=+\infty$ if $\text{supp}(\omega )⊈\text{supp}(\sigma )$

A.2 Joint Convexity

Proposition A.2 (Joint Convexity)

For any convex combination coefficients $\{p_i\}$ and state families $\{{\omega }_i\},\{{\sigma }_i\}$:

$$D\left(\sum _i{p}_i{\omega }_i\Vert \sum _i{p}_i{\sigma }_i\right)\le \sum _i{p}_{i}D({\omega }_i\Vert {\sigma }_i)$$

A.3 Data Processing Inequality

Proposition A.3 (Data Processing Inequality)

For any CPTP mapping $\Phi :\mathcal{S}(\mathcal{A})\to \mathcal{S}(\mathcal{B})$:

$$D(\Phi (\omega )\Vert \Phi (\sigma ))\le D(\omega \Vert \sigma )$$

Equality holds if and only if $\Phi$ is isometric on $\text{span}\{\omega ,\sigma \}$.

Appendix B: Perron-Frobenius Theorem

B.1 Definition of Primitive Matrix

Definition B.1

A non-negative matrix $W\in {\mathbb{R}}_{+}^{N\times N}$ is primitive if and only if there exists $k\in \mathbb{N}$ such that all elements of $W^k$ are strictly positive.

Example: $$W=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$$ is not primitive ($W^2=I$ still has zero elements).

$$W=\left(\begin{array}{cc}0.5& 0.5\\ 0.3& 0.7\end{array}\right)$$ is primitive ($W^1$ is already all positive).

B.2 Main Theorem

Theorem B.2 (Perron-Frobenius)

Let $W$ be an $N\times N$ primitive stochastic matrix. Then:

1. Unique Largest Eigenvalue: ${\lambda }_1=1$ is simple
2. Perron Vector: There exists a unique positive vector $\lambda =({\lambda }_1,\dots ,{\lambda }_N)$, ${\sum }_i{\lambda }_i=1$, such that ${\lambda }^{T}W={\lambda }^T$
3. Spectral Gap: Other eigenvalues satisfy $|{\lambda }_j|<1$, $j\ge 2$
4. Exponential Convergence: ${\lim }_{t\to \infty }{W}^t=1{\lambda }^T$, rate $O({\lambda }_2^t)$

Appendix C: Calculation of Conditional Mutual Information

C.1 Three-Region Formula

For quantum state ${\rho }_{ABC}$ on three subsystems $A,B,C$:

$$I(A:C\mid B):=S(AB)+S(BC)-S(B)-S(ABC)$$

where $S(\cdot )$ is the von Neumann entropy.

C.2 Markov Property

Definition C.1 (Quantum Markov Chain)

State ${\rho }_{ABC}$ satisfies Markov condition $A-B-C$ if and only if:

$$I(A:C\mid B)=0$$

Equivalent condition: $${\rho }_{ABC}={\rho }_A\otimes {\rho }_B\otimes {\rho }_C\text{in appropriate basis}$$

C.3 Markov Property of Causal Diamonds

Theorem C.2 (Casini-Teste-Torroba)

In quantum field theory, for regions $D_{j-1},D_j,D_{j+1}$ arranged along null hypersurfaces, the reduced density matrix of vacuum state $|0⟩$ satisfies:

$$I(D_{j-1}:D_{j+1}\mid D_j)=0$$

if and only if region boundaries are completely arranged along a single null hypersurface.

References

1. Umegaki, H. (1962). “Conditional expectation in an operator algebra. IV. Entropy and information.” Kodai Math. Sem. Rep. 14(2): 59-85.

2. Casini, H., Teste, E., Torroba, G. (2017). “Modular Hamiltonians on the null plane and the Markov property of the vacuum state.” J. Phys. A 50: 364001.

3. Perron, O. (1907). “Zur Theorie der Matrices.” Math. Ann. 64: 248–263.

4. Rovelli, C. (1996). “Relational quantum mechanics.” Int. J. Theor. Phys. 35: 1637–1678.

5. Fuchs, C., Mermin, N. D., Schack, R. (2014). “An introduction to QBism with an application to the locality of quantum mechanics.” Am. J. Phys. 82: 749.

6. Nāgārjuna (~200 CE). Mūlamadhyamakakārikā (Root Verses on the Middle Way).

Next Article Preview: In Article 04, we will solve the quantum measurement problem:

• How is measurement realized in the matrix universe?
• What is the information-theoretic mechanism of “wave function collapse”?
• How does consensus state ${\omega }_{\ast }$ give definite measurement results?

Stay tuned!