Martingale Cohomology, Holonomy, and Homological Arbitrage: From Martingale Cohomology to Loop Effects in Financial Markets

Battle Count: 55

The paper provides a novel theoretical framework for understanding arbitrage as a global transport/holonomy phenomenon rather than a local pricing inconsistency. While intellectually significant for mathematical finance, it is highly abstract and does not offer directly implementable trading strategies, pricing algorithms, or risk models. The connection to practical quantitative trading is indirect: it reframes the conceptual understanding of arbitrage through category theory and algebraic topology. The 'homological arbitrage' concept could potentially inform the design of more sophisticated arbitrage detection systems in complex multi-asset or branching information settings, but no such application is demonstrated. The paper is more relevant to theoretical mathematical finance than to applied quantitative trading.

Extremely high complexity. Requires deep expertise in multiple advanced mathematical fields: category theory (functors, nerves, simplicial structures), algebraic topology (cochain complexes, cohomology), probability theory (conditional expectation, Radon-Nikodym derivatives, measure theory), differential geometry (parallel transport, holonomy analogies), and stochastic analysis (filtrations, martingales). The σ-gauge construction, transport operators, and cohomological computations are highly abstract. No computational implementation is provided or suggested. The framework operates at a level of mathematical abstraction far beyond typical quantitative finance tools.

The paper is purely theoretical with complete mathematical proofs (Lemma 1, Propositions 1-4, Proposition 2). All definitions are self-contained and examples (Sections 5.1-5.3) are fully worked out with explicit computations. Reproducibility depends on the reader's ability to verify the algebraic and categorical arguments. No computational code or empirical data is involved. The framework builds on prior work by the same author (Adachi 2025, 2026, Adachi et al. 2020, Adachi and Ryu 2019).