The Crossover Does Not Carry the Claim: Pass@k Inversion, the Reasoning Boundary, and Five Premises the RLVR Debate Leaves Unstated

Abstract

A model trained with reinforcement learning from verifiable rewards beats its base model at one sample and loses to it at many. The field has read that crossover as a measurement of the reasoning boundary and has spent two years arguing about which direction the boundary moved. This paper separates the crossover, which is an observation reproduced across model families and benchmarks, from the boundary claim, which is an inference, and states the five premises the inference requires: that a large-k pass reflects reasoning rather than a hit in a small answer space, that a benchmark-averaged decline reflects a per-problem decline, that the training run being measured is not itself biased against the metric, that decoding and estimation were controlled across the two arms, and that the result generalises past static single-turn math. Every one of the five now has published counter-evidence, and none of that counter-evidence restores the optimistic reading either, because each rescue holds only under a condition its authors state and the optimistic reading does not. The paper then observes something the debate has not: six independent groups have since mid-2025 proposed an account under which the crossover is compatible with RLVR expanding what a model can solve, and the six accounts are not the same account. They locate the effect in different places - training phase, per-problem saturation, rollout-group sparsity, answer-space geometry, reasoning validity, task type - and they make different predictions. Pass@k cannot adjudicate between them, because it is the quantity all six agree is being misread. This paper reports no experiments. It states what the record establishes, states flatly what it does not, proposes a five-line disclosure that would make crossovers comparable, and names four studies that would settle the open part.

The literature search, drafting and citation verification for this paper were carried out with AI assistance under the author's direction. Every citation was machine-verified against the arXiv API and Crossref before inclusion, and every quantitative claim was read back against the cited source's own text. The author is responsible for the final text and for all claims made in it.

About the author

Pranay Mahendrakar is an ai specialist and large language model engineer based in Bengaluru, India. He builds production artificial intelligence systems and publishes open-access research on how those systems fail. See all 20 papers by Pranay Mahendrakar, or his ORCID record.