Geodesic Execution Slippage: A Statistical Physics Framework for Cryptocurrency Liquidity Risk.

Standard cryptocurrency transaction cost models assume flat geometry and assign execution cost as a proportional fee. This paper proposes GEODEX, a framework that models execution slippage as the geodesic arc length on the Fisher information manifold of a Markov-switching GARCH maximum-entropy model, augmented by a joint curvature-topological fragmentation alarm. The Curvature-Fragmentation Law (Proposition 2) is an analytically derived heuristic. Its empirical validity is confirmed across four crisis episodes. Ablation confirms that each geometric component contributes uniquely: removing the geodesic increases mean squared prediction error by 2.9%, removing topological data analysis by 2.1%, and removing curvature by 1.5%. On five cryptocurrency markets (BTC, ETH, XRP, LTC, and BCH), over 2253 daily observations, the framework achieves competitive prediction error and is the only single-signal model retained in the Model Confidence Set at α=0.10 against eight benchmarks. A joint curvature-topological alarm fires a median of two days before price-based circuit breaker thresholds across four crisis episodes, including the Terra collapse (May 2022) and FTX bankruptcy (November 2022). Online inference requires under one second; full offline calibration requires approximately 28 h. The framework requires no additional data beyond the upstream estimation pipeline and supports SDG 10 (Reduced Inequalities) and SDG 16 (Strong Institutions) by enabling accessible geometric liquidity intelligence for regulators and smaller market participants.