A Scale-Invariant Geometric Threshold for Financial Market Liquidity: Deriving the Exact Flash Crash Limit via Topological Propagation Dynamics

Abstract

The stability of global financial markets is increasingly threatened by rapid liquidity cascades, manifesting empirically as instantaneous "Flash Crashes." Current quantitative risk models, such as Value-at-Risk (VaR) and the Efficient Market Hypothesis (EMH), assume continuous liquidity and treat extreme volatility as probabilistic statistical anomalies based on historical distributions. These models fundamentally lack a deterministic, geometric boundary for limit-order book coherence. This paper introduces a strict topo-dynamical framework for financial network scaling. By modeling the market structure as a spatial competition between the geometric propagation of liquidity and localized volatility shocks, we derive a universal square-root geometric invariant (ℓmarket). We provide an intuitive translation of this threshold, explicitly dissect the failure of VaR during the May 2010 Flash Crash, and map the invariant across both traditional equities and Decentralized Finance (DeFi) Automated Market Makers (AMMs). Finally, we present a hardware-aware (FPGA) blueprint for Active Liquidity Throttling (ALT), acknowledging systemic implementation risks and regulatory hurdles.