---
title: "There Is No Free Lunch in RLVR: Every pass@1 Gain Has a pass@k Price"
source: "arXiv AI/CL/LG（论文）"
source_tier: "t2"
url: "https://arxiv.org/abs/2610.39295v1"
permalink: "https://feed.astralor.com/items/itm_e787c22f86833116400497d6"
published: "2026-10-29T03:43:26.000Z"
score: 57
channel: "tech"
tags: ["推理效率","There","Free Lunch","RLVR","Every","Gain Has","Price Reinforcement","Reinforcement"]
---

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

## At a glance

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 bud…

## Key points

- 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 bud…
- Is this a failure of RL to add reasoning, or a transient that more training removes?
- We settle the question with a theorem.

## Full text

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.

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