Candidate Recovery: General Relativity as a Digital Limit

Status: CANDIDATE RECOVERY (Consistent Limit)
Objective: To demonstrate that General Relativity (GR) is recovered as the low-complexity, trivial-topology limit of the Digital Gravity framework.

[!IMPORTANT]

This derivation shows that Digital Gravity is consistent with General Relativity in the vacuum limit. It does not yet derive the Jacobson thermodynamic relation from first principles, but rather utilizes it as a bridge.

1. The Starting Point: Entropic Potential

In the Digital Gravity framework, the effective gravitational boost $\alpha$ represents the “Free Entropy” or information availability of a holographic screen.

$\alpha \approx N_{total} - \chi$

2. The Thermodynamic Limit (Verlinde/Jacobson Recovery)

We utilize the thermodynamic approach to bridge the discrete information of the screen with smooth spacetime curvature:
1. Assume the Clausius Relation: $dE = T dS$ on the holographic boundary.
2. Unruh Temperature: $T$ is the temperature perceived by an accelerated observer near the screen.
3. Entropy Gradient: Changes in $S$ are mapped to physical forces.

3. The Digital Limit: $k \to 1$ and $\chi \to 2$

Step A: Continuity of Complexity

When the bit-depth $k$ is low (near the resolution limit of the vacuum), the discreteness of states is suppressed. The system behavior is dominated by the linear response of the vacuum.

Step B: Trivial Topology

In regions where the boundary topology is a simple sphere ( $\chi=2$ ) and does not change ( $\nabla \chi = 0$ ):
The correction $\alpha = 2^k - 2$ vanishes as $k \to 1$ .
This defines the Ground State of the theory.

Step C: Recovery of the Field Equations

In this Ground State ( $\alpha=0$ ):
1. The entropic force $F = T \nabla S$ becomes the only active term.
2. By requiring that this identity holds for all local Rindler horizons, the spacetime metric is constrained to satisfy the Einstein Field Equations:

$R_{\mu\nu} - \frac{1}{2}Rg_{\mu \nu} + \Lambda g_{\mu \nu} = \frac{8\pi G}{c^4} T_{\mu \nu}$

4. Summary of the Transition

* High Complexity ( $k>1″ style=”vertical-align:middle; border:none;” />) / Non-Trivial Topology (<img decoding=$ ): Digital Gravity (The “Overhead” phase, currently modeled phenomenologically).
* Low Complexity ( $k \to 1$ ) / Trivial Topology ( $\chi = 2$ ): General Relativity (The “Vacuum” phase).

5. Next Steps for Foundations

To move from “Recovery” to “Proof,” the framework requires:
1. A Unified Action: An action functional $I[\chi, k, g_{\mu\nu}]$ where $\chi$ and $k$ are dynamical fields.
2. Variational Derivation: Deriving the Jacobson/Verlinde assumptions directly from the topological bits.
3. Stability Analysis: Proving that the $\alpha \to 0$ limit is a stable fixed point for empty spacetime.

Verdict: The Einstein Bridge is established as a consistent recovery limit. General Relativity describes the stable vacuum, while Digital Gravity describes the first-order topological corrections.