Directional Liquidity and Geometric Shear in Pregeometric Order Books

TL;DR

This paper presents a novel geometric framework for understanding order books where liquidity and demand are emergent properties, revealing how directional liquidity imbalances influence market structure without relying on traditional price coordinates.

Contribution

It introduces a relational, non-metric model of order books that captures liquidity geometry and demonstrates its empirical validity across high-frequency data.

Findings

Liquidity imbalances decompose into drift and shear modes.

Projected liquidity follows a gamma-like functional form.

Model outperforms standard cumulative models in empirical tests.

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.