Directional Liquidity and Geometric Shear in Pregeometric Order Books

Paper: arXiv 2601.19369

Authors: João P. da Cruz

Abstract

We introduce a structural framework for the geometry of financial order books in which liquidity, supply, and demand are treated as emergent observables rather than primitive market variables. The market is modeled as a relational substrate without assumed metric, temporal, or price coordinates. Observable quantities arise only through observation, implemented here as a reduction of relational degrees of freedom followed by a low-dimensional spectral projection. A one-dimensional projection induces a price-like coordinate and a projected liquidity density around the mid price, from which bid and ask sides emerge as two complementary restrictions. We show that directional liquidity imbalances decompose naturally into a rigid drift of the projected density and a geometric shear mode that deforms the bid–ask structure without inducing price motion. Under a minimal single-scale hypothesis, the shear geometry constrains the projected liquidity to a gamma-like functional form, appearing as an integrated-gamma profile in discrete data. Empirical analysis of high-frequency Level~II data across multiple U.S. equities confirms this geometry and shows that it outperforms standard alternative cumulative models under explicit model comparison and residual diagnostics.

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 6.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a highly original theoretical framework treating order book liquidity as an emergent property from a pregeometric relational substrate using spectral projections. It backs up this advanced geometric math with solid empirical testing and model comparison on high-frequency Level II equity data.

Research Flowchart

  flowchart TD
    A[Research Goal: Model Order Book Geometry & Emergent Liquidity] --> B[Input: High-Frequency Level II U.S. Equities Data]
    B --> C[Method: Relational Substrate & Spectral Projection]
    C --> D[Process: 1D Projection & Bid-Ask Decomposition]
    D --> E[Process: Shear Mode vs. Rigid Drift Isolation]
    E --> F[Findings: Liquidity Fits Integrated-Gamma Form & Outperforms Models]