Meta Theory of the Zeckendorf-Hilbert Universe

22.2 Bayesian Gravity: Observer Model Synchronization Driven by Relative Entropy Minimization

In Section 22.1, we revealed failure of “absolute objectivity” through Wigner’s friend paradox: physical reality is not an a priori given single state, but a Relational property of information in observer networks. This raises a profound dynamical question: since each observer has its own private algebra ${\mathcal{A}}_{\text{int}}$ and subjective model, why don’t we live in countless mutually disconnected illusion bubbles? What force “pulls” these discrete subjective perspectives together, forming a coherent, shared objective spacetime?

This section will propose Bayesian Gravity hypothesis. We will prove that gravitational interactions in general relativity, at the bottom level of information geometry, are Statistical Flows generated by observers to eliminate Cognitive Dissonance—i.e., relative entropy. Gravity is not a force pulling objects, but a tendency to pull together ideas (models).

22.2.1 Cognitive Distance and Disagreement Functional

Consider two neighboring observers $O_A$ and $O_B$ in QCA networks. They each possess internal predictive models ${\rho }_A$ and ${\rho }_B$ for the same spacetime region (causal diamond $\Sigma$) (defined on their respective internal algebras, mapped to public boundary algebra ${\mathcal{A}}_{\partial }$ through communication channels).

Definition 22.2.1 (Cognitive Distance)

Cognitive distance between observers $A$ and $B$ is defined as Quantum Relative Entropy of their prediction distributions about public events:

$$D_{AB}\equiv S({\rho }_A\Vert {\rho }_B)=\text{Tr}({\rho }_A\ln {\rho }_A-{\rho }_A\ln {\rho }_B)$$

This quantity is asymmetric, measuring “surprise” or model correction cost when $A$ receives $B$’s data.

Definition 22.2.2 (Network-Wide Disagreement Functional)

For entire observer network $\mathcal{G}=\{O_i\}$, system’s Total Cognitive Potential is weighted sum of relative entropy on all connection channels:

$${\mathcal{V}}_{\text{cog}}[\{{\rho }_i\}]=\sum _{(i,j)\in E}{w}_{ij}S({\rho }_i\Vert {\rho }_j)$$

where $w_{ij}$ is communication weight (coupling strength), depending on causal distance or channel bandwidth between observers.

22.2.2 Bayesian Update as Gradient Flow

According to free energy principle (Section 19.3), each observer’s dynamical goal is to minimize its own prediction error. In multi-agent environments, this means each observer continuously corrects its model ${\rho }_i$ to reduce disagreement with neighbors ${\rho }_j$ (i.e., minimize ${\mathcal{V}}_{\text{cog}}$).

Theorem 22.2.1 (Bayesian Gravity Flow)

In continuous time limit, if observers follow optimal rules of Bayesian inference (Bayes’ Rule), evolution trajectory of their internal states ${\rho }_i$ on statistical manifold $\mathcal{S}$ follows gradient flow equation:

$$\frac{d{\theta }_i^a}{dt}=-g^{ab}({\theta }_i)\frac{\partial }{\partial {\theta }_i^b}\sum _j{w}_{ij}S(\rho ({\theta }_i)\Vert \rho ({\theta }_j))$$

where ${\theta }_i^a$ are model parameters, $g^{ab}$ is inverse matrix of Fisher Information Metric.

This equation is isomorphic in form to Geodesic Deviation Equation (or force motion equation) in general relativity:

• Fisher Metric $g_{ab}$ plays the role of Spacetime Metric $g_{\mu \nu }$.

• Relative Entropy Gradient $\nabla S$ plays the role of Gravitational Potential Gradient $\nabla \Phi$.

Physical Interpretation:

When observer $A$ finds its prediction inconsistent with $B$ ($S({\rho }_A\Vert {\rho }_B)>0$), Bayesian update drives $A$ to modify parameters, making its state ${\rho }_A$ “fall” toward ${\rho }_B$.

This “mutual attraction” in information geometry space manifests as universal gravitation in macroscopic physics. Objects attract each other because information they carry attempts to reach consensus.

22.2.3 Origin of Mass: Inertia of Belief

Why are some objects (large mass) difficult to move, while others (small mass) easy? In Bayesian gravity, this corresponds to Inertia of Belief.

Definition 22.2.3 (Informational Mass)

Observer’s “mass” $M_i$ is defined as Precision or Inverse Variance of its internal prior distribution.

$$M_i\propto \text{Tr}(g_{ab}({\theta }_i))\approx \text{Fisher Information Content}$$

• Large Mass (High Precision): Observer possesses vast historical data or extremely strong prior beliefs (such as black holes or stars). When interacting with low-precision observers (test particles), large-mass observer almost does not change its state ($\Delta \theta \approx 0$), forcing the other to significantly correct its model.

• Small Mass (Low Precision): Observer is uncertain about its own state (such as electrons or photons). It is easily “pulled” toward large-mass observer’s model.

This explains Equivalence Principle: inertial mass (ability to resist model updates) and gravitational mass (ability to pull others to update models) both stem from the same statistical quantity—Fisher information content.

22.2.4 Emergence of General Relativity: From Consensus to Curvature

If all observers mutually “fall” and reach consensus, what is the final state?

Theorem 22.2.2 (Consensus Manifold Theorem)

Under long-term evolution of Bayesian flow, observer network tends toward a Nash Equilibrium State. In this equilibrium state, local metrics $g_{ab}^{(i)}$ of all observers patch together into a globally smooth Riemannian manifold $(\mathcal{M},g_{\mu \nu })$.

Curvature $R_{\mu \nu }$ of this manifold corresponds to residual, irreducible Information Tension in the network.

• Flat Spacetime ($R=0$): All observers completely agree, no information pressure in network.

• Curved Spacetime ($R\ne 0$): Due to topological constraints or matter distribution, observers cannot achieve global agreement (e.g., around black holes). Although local consensus is reached, after parallel transporting model parameters around a closed loop, offset occurs (holonomy). This Failure of Consensus is precisely the definition of Curvature.

Corollary 22.2.3 (Bayesian Interpretation of Einstein Equations)

Einstein field equations $G_{\mu \nu }=8\pi G{T}_{\mu \nu }$ can be restated as:

Curvature of Consensus (Geometric Tension) is Proportional to Flux of Information (Material Tension).

Matter ($T_{\mu \nu }$) is the source of information, continuously injecting new “surprise” into the network, disrupting consensus. Gravity ($G_{\mu \nu }$) is geometric deformation generated by the network to digest this surprise and restore local balance.

Summary

This section proposed Bayesian gravity theory, reducing universal gravitation to Bayesian model synchronization mechanism in multi-agent systems.

1. Force as Update: Gravity is statistical tendency of observers to correct models to minimize relative entropy.

2. Mass as Belief: Inertia is rigidity of prior probability distributions.

3. Spacetime as Consensus: Objective physical world is the greatest common divisor reached by countless subjective worlds through continuous games and calibration.

This view completely eliminates opposition between “subjective” and “objective”: Objectivity is just the limiting form of Intersubjectivity.

In the next section 22.3, we will explore stability of this consensus mechanism, i.e., how Objective Reality as Nash Equilibrium is locked in game-theoretic framework.