No Trading Strategy Can Win on Every Price Path: Computability, Randomness, and the Limits of Universal Trading

Battle Count: 50

Directly and fundamentally relevant to quantitative trading. The paper establishes rigorous theoretical boundaries on what any computable trading strategy can guarantee, connecting computability theory, algorithmic randomness, and no-arbitrage to practical trading. It provides a taxonomy (Table I) distinguishing six different 'no universal strategy' claims that are often conflated. The time-reversal diagnostic, the Wheel case study, the Cover universality discussion, and the FAPP strategy framework are all directly applicable to strategy development, validation, and risk management. The paper's core message—that repeatable success requires a market restriction, benchmark, risk premium, or informational advantage—is a foundational principle for quantitative trading. However, it is a theoretical/expository paper, not a practical implementation guide.

The paper is purely theoretical with no code, algorithms to implement, or empirical pipeline. The diagonal countermarket construction (Eq. 3) is conceptually simple but is a proof device, not a deployable strategy. The time-reversal diagnostic (Criterion 1) is practically implementable as a stress test. The main 'implementation' is intellectual: understanding the taxonomy and applying the framework to evaluate strategy claims. No ML models, data pipelines, or trading systems are proposed.

Purely mathematical paper with complete formal proofs. All theorems (1-13), propositions, corollaries, and lemmas are self-contained with explicit constructions. The diagonal countermarket (Eq. 3), Halting reductions (Theorems 2-3), Busy Beaver argument (Eq. 6-7), Ville's inequality application (Theorem 6-7), and no-arbitrage arguments (Theorem 9, Corollary 10) are fully specified. No empirical data or code required. The paper explicitly states it is expository, reformulating known results (Gold 1967, Turing 1937, Martin-Löf 1966, Delbaen-Schachermayer 1994, Cover 1991, etc.) into a unified taxonomy.