Pregeometric Origins of Liquidity Geometry in Financial Order Books

TL;DR

The paper introduces a pregeometric framework in which financial market observables—price, time, returns, and liquidity—emerge from projections of an inflationary relational substrate with no intrinsic metric. By projecting via the graph Laplacian, a one-dimensional price-like coordinate and a projected liquidity density are defined; under a minimal single-scale log-slope hypothesis, the projected liquidity profiles must follow an integrated-gamma form. Empirical tests on high-frequency Level II data across multiple U.S. equities show robust agreement with the integrated-gamma geometry, and model comparison via information criteria consistently favors this geometry over alternatives. A minimal simulation of the relational dynamics reproduces the same structural pattern, supporting a geometric origin of order-book regularities and suggesting a new lens for classifying market regimes based on projection geometry rather than agent-level dynamics.

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.

Figures (3)

• Figure 1: Pregeometric pipeline from relational structure to observable liquidity geometry. A purely relational substrate is projected through Laplacian eigenmodes onto a one-dimensional observable coordinate. Defining a mid per snapshot and measuring liquidity in tick distance from the mid yields cumulative profiles $S(x)$, which are predicted to follow an integrated-gamma form under the single-scale log-slope principle.
• Figure 2: Empirical cumulative liquidity geometry across assets. Per-second Level II snapshots are aggregated across venues; the mid price $p^\star$ is computed independently for each snapshot; liquidity is binned by tick distance from the mid and converted into cumulative profiles $\overline{S}(x)$ averaged over local time windows ($T=10\,\mathrm{s}$). Points show observed cumulative liquidity on the bid and ask sides, while solid lines show the integrated-gamma fits predicted by Corollary \ref{['cor:integrated_gamma']}. The same projected geometry appears across all assets, despite strong temporal variability of the fitted parameters.
• Figure 3: Simulated non-cumulative liquidity profiles and gamma fits. Synthetic Level II--like snapshots are binned by tick distance $x$ from the instantaneous mid $p^\star$ on bid (demand) and ask (supply) sides. Points show the averaged simulated liquidity $\overline{Q}(x)$, while lines show fits to the paper form $Q(x)=C x^{\gamma}e^{-\lambda x}$. The simulation reproduces the same near-mid curvature and exponential cutoff structure discussed in the empirical analysis, supporting a geometric origin of gamma-like liquidity profiles under projection.

Theorems & Definitions (6)

• Proposition 1: Aggregate balance identity
• proof
• Definition 1: Spectral equilibrium
• Lemma 1: Gamma form from single-scale log-slope
• proof
• Corollary 1: Integrated gamma