Carathéodory II: The Geometry of Financial Irreversibility

Battle Count: 52

The paper provides a deep theoretical foundation for understanding why spreads exist, why circular trades (triangular arbitrage) cannot be systematically profitable for sequential traders, and why liquidity providers earn revenue. The geometric tax interpretation of market maker spreads and the connection between skewness (cubic term) and irreversibility are conceptually relevant. However, the paper is highly abstract and mathematical, with no direct trading algorithms, backtests, or implementable strategies. Its value is primarily in providing a unifying geometric framework rather than actionable quantitative tools.

Requires advanced knowledge of differential geometry (projective spaces, holonomy, parallel transport), information geometry (Amari-Chentsov tensor, Fisher-Rao metric, Fubini-Study metric), quantum mechanics (spin systems, Veronese embedding, entanglement), and thermodynamics. The mathematical formalism is dense and the paper is conceptual rather than computational. There is no code, no algorithm, and no step-by-step procedure to implement. Translating the geometric insights into practical trading models would require substantial additional work.

The paper is purely theoretical with mathematical derivations. The key results (Taylor expansion of directed divergence, collective-local gap formula, work surcharge expression) are analytically verifiable. However, there are no numerical experiments, simulations, or empirical validations. Reproducibility depends on the reader's ability to follow the differential geometry and information geometry arguments. The companion paper (arXiv:2602.16539) provides additional context.