Part 1: English Version

The Holographic Principle in Finance: Entanglement as Geometry

Project EMIS Technical Note #140

Status: Theoretical Foundation / v0.5 Preview Topic: AdS/CFT, Quantum Information, Systemic Risk

1. The Core Hypothesis: It from Qubit

Standard economics assumes the Market Geometry pre-exists. EMIS v0.5 proposes a deeper origin: Geometry is emergent from Information Entanglement.

We apply the AdS/CFT Correspondence (specifically the SYK model duality) to economics:

  • Boundary (Quantum): The time-series data of micro-agents. A 1D system with no gravity.
  • Bulk (Gravity): The emergent macroeconomic manifold. A 2D spacetime with gravity.

2. The Holographic Dictionary for Economics

We establish the following isomorphisms between Quantum Gravity and Finance:

Law A: The Ryu-Takayanagi Formula (Asset Correlation)

The Entanglement Entropy of a boundary region equals the length of the minimal geodesic in the bulk.

\[S_{EE} = \frac{L}{4G}\]
  • $S_{EE}$ (Entanglement): Statistical Correlation between asset classes.
  • $L$ (Geodesic Length): The geometric “distance” in the market.
  • Insight: During a crash, correlations spike ($S \to \text{max}$). This implies the geometric distance shrinks ($L \to 0$). The financial universe physically contracts.

Law B: Complexity = Volume (CV Conjecture)

The computational complexity of the boundary state corresponds to the growing volume of the interior geometry (Wormhole).

\[C(t) \sim \frac{Vol(t)}{G \cdot L_{ads}}\]
  • $C(t)$ (Complexity): The depth of financial derivatives (MBS, CDO). The algorithmic steps to unwind positions.
  • $Vol(t)$ (Bulk Volume): The accumulation of hidden Systemic Risk inside an institution.
  • Insight: A bank’s stock price (Boundary) may be stable, while its internal risk volume (Complexity) grows exponentially. This is the geometric definition of “Too Big to Fail.”

Law C: Information Scrambling (Chaos Bound)

Information injected into the system is scrambled at a maximum rate determined by temperature.

\[\lambda_L \le 2 \pi T\]
  • $\lambda_L$ (Lyapunov Exponent): The speed of Arbitrage / Information Diffusion.
  • $T$ (Temperature): Market Volatility.
  • Insight: The speed of market efficiency is physically bounded by volatility.

3. Conclusion

Gravity (Macroeconomics) is not fundamental; it is the hydrodynamics of Entanglement (Micro-transactions).

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