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

Abstract

Every universal-trading claim pairs a trader with a market---a path, generator, or law. For any total deterministic computable trader, its code yields a fixed computable countermarket with proportional price moves opposing its positions; hence no such trader wins on every computable path. Gold-style learning cannot identify every computable binary market rule from history. Separately, Turing-universal generators make certification of unbounded-future events undecidable; Busy-Beaver growth defeats every computable description-size waiting schedule. A passive Martin-L\"of-random record is incompressible and prevents effective test-capital processes from becoming unbounded; no-arbitrage supplies a distinct financial boundary. Together these results form an expository taxonomy: repeatable success requires a market restriction, benchmark, risk premium, or informational advantage, while time reversal supplies a simple stress test.