There Is No Free Lunch in RLVR: Every pass@1 Gain Has a pass@k Price

Abstract

Reinforcement learning with verifiable rewards (RLVR) makes a language model solve more problems on the first try and fewer when given many tries: it raises pass@1 while the base model overtakes it at large sampling budget k. Is this a failure of RL to add reasoning, or a transient that more training removes? We settle the question with a theorem. The reversal is a price, and we compute its exact rate. Modeling RLVR as an exponential tilt of a base policy over strategies shared across prompts, we prove that under exclusive specialization every update that raises pass@1 strictly lowers the success of some prompt; a strictly positive price on prompts (Stiemke duality) is the exact condition, and being merely the best strategy on a prompt is provably not enough. When the loss reaches the hardest prompts, it forces a pass@k crossover, and the number of crossings is at most the sign variation of the shift in per-prompt success, a totally positive bound that is tight and that recovers the known single-crossover result as its order-preserving case. The crossover is not a bug to patch: a constrained update traces the exact pass@1 versus pass@k frontier, and removing the crossover entirely costs the entire pass@1 gain. The same tilt identity yields entropy collapse and the entropy-performance law with an identifiable exponent, unifying two literatures under one mechanism. Reinforcement cannot widen the reasoning frontier; it can only move along it, at a price we make measurable by a diagnostic that reads the bound, the order-preserving test, and the price from any evaluation log. There is no free lunch, only a price.