Emergent Gravity and Correlation Boundaries

When does a surface in space possess a temperature and an entropy? And at what scale does the First Law of Thermodynamics break down?

The connection between gravity and thermodynamics—from black hole mechanics to the Ryu-Takayanagi formula—suggests that spacetime itself is an emergent phenomenon. However, recent challenges (such as the Wang-Braunstein theorem) have shown that applying thermodynamic laws to generic "holographic screens" leads to contradictions, threatening entropic gravity programs.

This two-part research series resolves that tension by introducing Correlation Boundaries: a rigorous mathematical framework built on pairwise correlations subject to a finite capacity (monogamy) constraint. By defining exactly which surfaces can support thermodynamic structure, we derive both the continuum laws of emergent gravity and their discrete, finite-N quantum corrections.

Part I: The Macroscopic Continuum

[Correlation Boundaries and the First Law: Entanglement Anisotropy, Holographic Screens, and the Thermodynamics of Surfaces]

In the large-N limit, we define a correlation boundary as a surface across which the total pairwise correlation vanishes (P = 0). We prove that these are the only surfaces where the First Law of Thermodynamics (δQ = TdS) holds strictly.

For any generic surface, we derive a thermodynamic deficit theorem: ΔF L ∝ P ⋅ A

The first law fails exactly in proportion to the cross-boundary entanglement entropy (P) weighted by the entanglement anisotropy (A). This resolves the Wang-Braunstein problem:

• Horizons are exactly thermodynamic because they are true correlation boundaries (P = 0).

• Spherical screens appear thermodynamic due to a symmetry degeneracy (isotropic entanglement means A = 0).

• The Ryu-Takayanagi Formula naturally emerges as the minimization of cross-boundary correlation P over surfaces homologous to a boundary region.

Part II: The Discrete Quantum Regime

[Correlation Boundaries at Finite N: Chirality Thresholds and the Emergence of Thermodynamic Structure]

What happens to thermodynamic boundaries at the Planck scale, where the continuum breaks down? Using the graded hierarchy of the exterior algebra (∧∗R5), we extend the correlation boundary formalism to finite degrees of freedom (N ).

We reveal that true thermodynamics requires chirality (irreducible, grade-3 triangular loops) to be preserved on both sides of a partition to define directed energy flow and conjugate variables. The central discoveries include:

• The N = 6 Threshold: We mathematically prove that N = 6 is the absolute minimum number of features required to preserve chirality on both sides of a boundary (3∣3 partition). Below this, bilateral thermodynamics is structurally undefined.

• The O(1/N ) Correction: The destruction of triangular chirality loops at a boundary introduces a deterministic, positive-definite penalty of O(1/N ) to the thermodynamic deficit. This grade-3 correction is entirely invisible to the pairwise, area-law approximation of standard holographic entanglement entropy.

• The Staircase of Emergence: The transition from quantum to classical mechanics passes through distinct topological thresholds: chirality onset (N = 3), bilateral thermodynamics (N = 6), and statistical emergence (N ∼ 30).