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Directional Liquidity and Geometric Shear in Pregeometric Order Books

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.

Directional Liquidity and Geometric Shear in Pregeometric Order Books

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.
Paper Structure (38 sections, 4 theorems, 23 equations, 5 figures, 2 tables)

This paper contains 38 sections, 4 theorems, 23 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Observational geometry and two-sided restriction
  3. Relational substrate and observational weights
  4. Edge-level degrees of freedom and observational reduction
  5. Observational projection
  6. Pushforward liquidity measure
  7. Mid price as an observational cut
  8. Two-sided restriction
  9. Operational meaning of the projection $\Pi(G)$
  10. Shear decomposition and gauge separation
  11. Observable dynamics of projected liquidity
  12. Kinematic decomposition of liquidity evolution
  13. Gauge interpretation of mid-price motion
  14. Definition of shear as physical deformation
  15. Physical content of shear dynamics
  16. ...and 23 more sections

Key Result

Proposition 1

Let $\nu_t(x)$ be a family of observable liquidity densities indexed by time $t$. Then, for each $t$, there exists a decomposition where $m_t \in \mathbb{R}$ is a scalar shift and $\tilde{\nu}_t$ is a density with vanishing first moment around the origin, provided the moment exists.

Figures (5)

  • Figure 1: Observational origin of bid--ask structure. A relational substrate without geometry is observed through a projection $p=\Pi(G)$. The observable liquidity distribution arises as the pushforward $\nu_G=p_\#\mu_G$. Bid and ask curves correspond to the restriction of this single density to either side of the mid price.
  • Figure 2: Cumulative shear field for AAPL. The extended and smooth shear profile indicates a macroscopic deformation of projected liquidity geometry rather than a localized microstructural effect.
  • Figure 3: Cumulative shear field for GS.
  • Figure 4: Shear amplitude versus mid-price drift (AAPL).
  • Figure 5: Distribution of shear amplitudes across assets.

Theorems & Definitions (7)

  • Proposition 1: Shear--drift decomposition
  • proof
  • Proposition 2: Single-scale shear log-slope
  • proof
  • Proposition 3: Gamma geometry of single-scale shear
  • proof
  • Corollary 1: Integrated-gamma geometry