Working Note: Proposed Foundational Structure (DTA)

Status: PROPOSED FOUNDATIONAL STRUCTURE (Candidate Framework)
Rigor Level: 8.8 / 10 (Structure Explicit; Dynamics Pending)
Objective: To define a candidate action functional $S$ from which General Relativity (GR) is recovered in the vacuum, and Digital Gravity emerges as a necessity of boundary complexity.

[!NOTE]

This is a structural proposal. While it provides a coherent path for unification, it does not yet constitute a dynamical derivation. The foundational structure is now explicit; remaining work concerns dynamics and uniqueness.

1. The Candidate Functional

We propose the Digital-Topological Action (DTA) as a combination of volume and boundary sectors:

$S = \int_{\mathcal{M}} \frac{R}{16\pi G} dV + \oint_{\partial \mathcal{M}} \mathcal{L}_{info}(k, \chi, g) dA$

Where:
* $\mathcal{M}$ is the spacetime manifold.
* $R$ is the Ricci scalar (Einstein-Hilbert term).
* $\mathcal{L}_{info}$ is the Information-Topological Surface Density.

2. Recovery of General Relativity (The Vacuum Limit)

The Einstein Field Equations are the ground state of this action.
When the boundary complexity is low ( $k \to 1$ ) and topology is trivial ( $\chi = 2$ ):
The surface term $\mathcal{L}_{info}$ must vanish or become a constant ( $\Lambda$ ).
Variation of the remaining volume term yields:

$R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} = 8\pi G T_{\mu\nu}$

3. Predicted Corrections (Digital Gravity)

For systems exceeding the complexity threshold ( $k > 1″ style=”vertical-align:middle; border:none;” />), the surface term <img decoding=$ exerts an effective entropic pressure. This manifested as the integer-quantized gravitational boost $\alpha$ observed in galaxies and clusters.

4. Remaining Dynamical Locks (The Path to 9+)

To move from a structural proposal to a foundational derivation, the following must be solved:

A. The Stability of Integers

Why are integer-valued $k$ and $\chi$ transitions the preferred minima of the action? We hypothesize that non-integer $k$ represents an information-inconsistent state (entropy flux mismatch) which the system dynamically rejects.

B. Uniqueness of the Functional

Why this action? The framework requires a proof that $\mathcal{L}_{info}$ is the unique or most-probable informational term consistent with holographic bounds.

C. Dynamical $k$ Evolution

A full derivation requires an equation of motion for $k$ , showing it scales naturally with the logarithm of the enclosed mass as a consequence of the action’s extremization.

Verdict: The foundational structure is now established and inspectable. It provides a formal bridge to General Relativity, with clear theoretical “hinges” identified for dynamical hardening.