Carathéodory, Finite Resources and the Geometry of Arbitrage

Battle Count: 72

The paper provides deep theoretical foundations for understanding when and why no-arbitrage conditions hold globally in financial markets. It identifies the exponential family as the necessary geometric structure for coherent market models and characterizes failure modes (regime shifts, liquidity crises, small-system effects). While not directly implementable as a trading strategy, it informs model selection (why exponential family / log-linear models are natural for equilibrium markets), risk management (Fisher metric as a measure of market 'temperature'), and understanding of when standard quantitative models break down. The Onsager/gradient-flow connection to market dynamics has implications for mean-reversion strategies and market microstructure modeling. However, the purely theoretical nature and strong IID assumption limit direct practical applicability.

This is a purely theoretical paper with no code, no algorithms, and no computational framework. The mathematical content involves differential geometry (1-forms, foliations, integrability conditions), information geometry (Fisher metric, exponential families, Bregman divergence), and statistical theory (PKD theorem, sufficient statistics). Translating these insights into a practical trading or risk management system would require substantial additional work: specifying the exponential family for a given market, computing the Fisher metric, verifying global convexity, and handling the many excluded regimes. The conceptual framework is elegant but abstract.

The paper is purely theoretical with explicit mathematical derivations. The Boyling counterexample is fully specified with the 1-form ψ and the contradiction in asymptotic decay rates (t^{-3/2} vs t^{-2}). The PKD theorem application is standard. However, there is no code, no numerical experiments, and no empirical validation. Reproducibility depends on verifying the mathematical proofs and the analogy structure. The paper is self-contained in its logical chain but relies on external theorems (PKD, Carathéodory, Onsager).