Paper: arXiv 2601.17245

Authors: João P. da Cruz

Abstract

We propose a structural framework for the geometry of financial order books in which liquidity, supply, and demand are treated as emergent observables rather than primitive economic variables. The market is modeled as an inflationary relational system without assumed metric, temporal, or price coordinates. Observable quantities arise only through projection, implemented here via spectral embeddings of the graph Laplacian. A one-dimensional projection induces a price-like coordinate, while the projected density defines liquidity profiles around the mid price. Under a minimal single-scale hypothesis – excluding intrinsic length scales beyond distance to the mid and finite visibility – we show that projected supply and demand are constrained to gamma-like functional forms. In discrete data, this prediction translates into integrated-gamma cumulative profiles. We test these results using high-frequency Level~II data for several U.S. equities and find robust agreement across assets and intraday windows. Explicit comparison with alternative cumulative models using information criteria demonstrates a systematic preference for the integrated-gamma geometry. A minimal simulation of inflationary relational dynamics reproduces the same structure without invoking agent behavior or price formation mechanisms. These results indicate that key regularities of order-book liquidity reflect geometric constraints induced by observation rather than detailed microstructural dynamics. Supplementary Material is available at the arXiv submission.

Complexity vs Empirical Score

  • Math Complexity: 6.5/10
  • Empirical Rigor: 7.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper introduces a highly novel, physics-inspired pregeometric framework using graph Laplacian spectral embeddings to explain market order book shapes as emergent geometric projections. It grounds this theoretical abstraction in empirical tests using high-frequency Level II data, showing integrated-gamma profiles fit market liquidity better than traditional models.

Research Flowchart

  flowchart TD
    A["Research Goal: Explain LOB Liquidity Geometry as Emergent, Pregeometric Observables"] --> B["Data Input: US Equities High-Frequency Level II Order Book Data"]
    B --> C["Relational Graph & Spectral Embedding via Graph Laplacian"]
    C --> D["Theoretical Projection under Single-Scale Hypothesis"]
    D --> E["Derive Integrated-Gamma Cumulative Profiles for Supply/Demand"]
    E --> F["Empirical Validation & Model Comparison via Information Criteria"]
    F --> G["Key Finding: Liquidity Geometry Driven by Observation Constraints, Not Microstructure"]