Caratheodory, Finite Resources and the Geometry of Arbitrage

Paper: arXiv 2602.16539

Authors: B. K. Meister

Abstract

Caratheodory’s axiom of adiabatic inaccessibility states that, in any neighborhood of a thermodynamic state, certain states remain unreachable via adiabatic processes. Non-arbitrage mirrors this topological restriction in finance. Preserving this constraint in resource-limited systems identifies the exponential family not as a modeling convenience but as the requisite geometric structure unifying both domains.

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 1.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a deeply mathematical, theoretical bridge between thermodynamics (Carathéodory’s axiom, Boyling counterexample) and non-arbitrage in financial markets using information geometry. However, it lacks empirical validation, real market datasets, or backtesting, making it a purely theoretical framework.

Research Flowchart

  flowchart TD
    A["Research Goal: Link Thermodynamic Non-Reachability & Financial No-Arbitrage"] --> B["Data/Inputs: Carathéodory's Axiom & Market Absence-of-Arbitrage Constraints"]
    B --> C["Methodology: Geometric Topological Mapping of System Bounds"]
    C --> D["Process: Apply Conservation Laws under Finite Resource Limits"]
    D --> E["Outcome: Proves Exponential Family as the Mandatory Structure"]
    E --> F["Key Finding: Unified Geometric Framework for Physics & Finance"]