arxiv: 2605.09012 · v1 · submitted 2026-05-09 · 💻 cs.AI

Recognition: no theorem link

Re²Math: Benchmarking Theorem Retrieval in Research-Level Mathematics

Shengzhong Zhang, Wenjie Yang, Zengfeng Huang, Zicheng Lyu

Pith reviewed 2026-05-12 02:02 UTC · model grok-4.3

classification 💻 cs.AI

keywords theorem retrievalmathematical benchmarksproof assistanceliterature groundingAI math toolscitation-agnostic evaluationresearch-level mathematicsproof gap sufficiency

0 comments

The pith

A new benchmark shows current AI systems retrieve valid math statements yet succeed at confirming their fit to the local proof step in only 7 percent of cases.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces Re²Math to test whether AI systems can retrieve theorems from the literature that actually advance a partial mathematical proof. Test instances are built from real instrumental citations inside published proofs, supplied with hierarchical context around the current step and optional leakage-controlled hints. Evaluation accepts any theorem sufficient to close the gap rather than demanding the original citation, and it measures three separate abilities: locating sources, grounding answers, and verifying applicability. On the released test set the best systems reach only 7 percent ToolAcc on the applicability check even though source grounding rates are much higher. The result isolates a concrete shortfall that must be addressed before literature-grounded research assistance becomes reliable.

Core claim

Re²Math constructs benchmark instances from candidate instrumental citations that appear inside the proofs of main theorems. Each instance supplies hierarchical context for the proof step together with an optional leakage-controlled anchor hint. The task is source-grounded yet citation-agnostic, so any admissible theorem that suffices for the transition is accepted. Evaluation runs against a release-frozen retrieval artifact to guarantee reproducibility, and the benchmark is designed for automatic continual expansion. On the current test set the highest fixed-judge ToolAcc is 7.0 percent, while source-grounding rates are substantially higher; this shows that systems frequently retrieve valid

What carries the argument

The Re²Math benchmark instance, built from an instrumental citation inside a real proof and equipped with hierarchical context, that evaluates ToolAcc as success at retrieving a theorem sufficient for the immediate proof transition.

If this is right

Systems will need explicit mechanisms to verify local applicability rather than depending on general retrieval quality alone.
The benchmark can expand automatically as new published proofs supply fresh instrumental citations.
Decoupling citation recall, grounding, and sufficiency supplies clearer diagnostic signals for improving mathematical assistants.
Low applicability rates despite better grounding indicate that tighter coupling between literature search and step-by-step reasoning is required.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same construction technique could be applied to benchmarks in physics or computer science where proofs rely on external lemmas.
Training with explicit rewards for applicability verification might raise ToolAcc without altering the underlying retriever.
The identified gap suggests that simply scaling retrieval models will not close the distance to useful research assistance.
Comparing ToolAcc on this benchmark with closed-world math performance would quantify the extra cost of literature grounding.

Load-bearing premise

Instances built from candidate instrumental citations in real proofs, supplied with hierarchical context and leakage-controlled hints, form a faithful representation of the theorem-retrieval challenges that arise in actual research-level mathematics.

What would settle it

A retrieval system that achieves ToolAcc substantially above 7 percent on the current test set while keeping high source-grounding rates, or a new collection of proof steps whose distribution of retrieval difficulty differs markedly from the benchmark instances.

Figures

Figures reproduced from arXiv: 2605.09012 by Shengzhong Zhang, Wenjie Yang, Zengfeng Huang, Zicheng Lyu.

**Figure 1.** Figure 1: Overview of Re2Math. A published-first, arXiv-backed pipeline mines candidate instrumental citations inside main-theorem proofs and converts them into proof-gap instances. Evaluation follows planning or anchor use, query generation, source selection, and theorem/tool extraction against a release-frozen retrieval artifact. 4.1 High-Level Pipeline The pipeline is published-first, arXiv-backed: we select pro… view at source ↗

**Figure 2.** Figure 2: Overview of tool-grounded retrieval. Given hierarchical proof context (global + local) and an optional anchor, the agent produces a source–tool pair via planning, query formulation, retrieval, source selection, and statement extraction. A prediction is correct if the extracted statement is grounded in the selected source and sufficient for the target proof transition. The figure is illustrative; formal def… view at source ↗

**Figure 3.** Figure 3: Dataset statistics. Overview of the curated benchmark and the Eval-200 subset. The full curated dataset contains 860 proof-gap instances from 631 version-normalized citing papers. Eval-200 contains 200 held-out proof-gap instances from 167 citing papers, balanced across five domains. observed outputs because its internal corpus coverage and ranking function are not fully specified by us. Retrieval executio… view at source ↗

read the original abstract

Large language models are increasingly capable at closed-world mathematical reasoning, but research assistance also requires source-grounded use of the literature. When a proof reaches a non-trivial step, a useful assistant should determine whether the needed tool (e.g., a lemma) already exists, identify a suitable scholarly source, and verify that its assumptions align with the current proof context. To rigorously evaluate such capabilities, we introduce Re$^2$Math, a benchmark for tool-grounded retrieval from partial mathematical proofs. Each instance is built from a candidate instrumental citation in the proof of a main theorem, with hierarchical context and an optional leakage-controlled anchor hint. We also make the task source-grounded yet citation-agnostic in that any admissible theorem sufficient for the proof transition is accepted. Evaluation uses a release-frozen retrieval artifact, ensuring reproducibility, while the benchmark itself supports automatic, continual expansion with newly constructed instances. On the current benchmark test set, the best fixed-judge ToolAcc reaches 7.0%, despite substantially higher rates of source grounding, indicating that current systems often retrieve valid statements but fail to establish their applicability to the local proof step. By decoupling citation recall, grounding, and proof-gap sufficiency, Re$^2$Math transforms literature-grounded mathematical tool use into a controlled diagnostic task.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Re²Math gives a clean diagnostic benchmark for theorem retrieval that separates finding relevant statements from checking if they actually close the proof gap, with current models stuck at 7% ToolAcc.

read the letter

The main point of this paper is a new benchmark, Re²Math, that tests whether AI systems can retrieve theorems from the literature that actually suffice for a specific step in a research-level proof, and it finds that even the best models only hit 7% on the key metric despite better performance on finding relevant sources. They build each instance around a candidate instrumental citation from an actual proof, providing hierarchical context and an optional leakage-controlled hint. The task is set up to be source-grounded but citation-agnostic, meaning any theorem that closes the proof gap gets accepted, not just the one originally cited. Evaluation relies on a fixed retrieval artifact to keep things reproducible, and the benchmark is designed to grow with new instances over time. This approach does a good job of breaking down the problem into citation recall, source grounding, and proof-gap sufficiency. The gap between grounding rates and the low ToolAcc score points to a real issue with current systems: they can pull valid statements but have trouble checking if those statements fit the local proof context. That said, the construction process raises some questions. It's not clear from the description how they validate that the selected citations are truly instrumental or how they determine if alternative theorems are sufficient. If there's selection bias in which proofs or citations they pull from, the benchmark might not fully represent the broader challenges in research math. More details on instance creation and any human validation steps would help here. This work is for people building or evaluating AI assistants for mathematical research, especially those focused on retrieval and tool use in proofs. A reader looking for new diagnostic tasks in grounded math reasoning would get something concrete out of it. I think it deserves peer review. The benchmark idea is coherent and the performance numbers suggest a worthwhile direction, even if the construction details could use more scrutiny from referees.

Referee Report

1 major / 1 minor

Summary. The paper introduces Re²Math, a benchmark for tool-grounded theorem retrieval from partial research-level mathematical proofs. Instances are constructed from candidate instrumental citations appearing in proofs of main theorems, supplied with hierarchical context and optional leakage-controlled hints. The task is source-grounded yet citation-agnostic, accepting any admissible theorem sufficient for the local proof step. Evaluation employs a release-frozen retrieval artifact for reproducibility, and the benchmark supports automatic expansion. On the current test set the best reported fixed-judge ToolAcc is 7.0 %, substantially below source-grounding rates, which the authors interpret as evidence that current systems retrieve valid statements but fail to verify applicability to the local context.

Significance. If the benchmark construction is shown to be faithful and unbiased, the work supplies a controlled diagnostic for a practically important limitation in literature-grounded mathematical reasoning. By separating citation recall, source grounding, and proof-gap sufficiency, Re²Math could guide targeted improvements in research-assistant systems. The release-frozen artifact and continual-expansion design are concrete strengths that support reproducibility and long-term utility.

major comments (1)

[Abstract] Abstract: the central performance claim (best ToolAcc of 7.0 % indicating failure to establish applicability) rests on the assumption that the constructed instances faithfully represent research-level retrieval challenges. However, the abstract provides no information on instance validation procedures, mitigation of selection biases during proof and citation extraction, or the precise criteria used to judge sufficiency of alternative theorems. These details are load-bearing for interpreting the metric and the stated conclusions.

minor comments (1)

[Abstract] The abstract refers to a 'fixed-judge ToolAcc' without a brief definition or pointer to the metric's formal definition; a one-sentence clarification would improve immediate readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive review and for identifying a point that can strengthen the presentation of our work. We address the major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the central performance claim (best ToolAcc of 7.0 % indicating failure to establish applicability) rests on the assumption that the constructed instances faithfully represent research-level retrieval challenges. However, the abstract provides no information on instance validation procedures, mitigation of selection biases during proof and citation extraction, or the precise criteria used to judge sufficiency of alternative theorems. These details are load-bearing for interpreting the metric and the stated conclusions.

Authors: We agree that the abstract's brevity leaves these supporting details implicit. The full manuscript (Section 3) specifies the construction pipeline: instances are extracted from instrumental citations in proofs of main theorems drawn from recent arXiv papers; each instance undergoes expert validation by mathematicians to confirm it poses a genuine research-level retrieval challenge; selection bias is mitigated by systematic sampling across subfields with explicit diversity controls and leakage safeguards; and sufficiency is defined as any theorem whose assumptions are compatible with the local context and whose application advances the partial proof (verified by the same expert annotators). The citation-agnostic acceptance of any admissible theorem is already stated. To make these elements more visible without lengthening the abstract excessively, we will revise the abstract to include a concise clause referencing the validation and sufficiency criteria. This change supports rather than alters the reported results and conclusions. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper introduces and evaluates a new benchmark (Re²Math) for theorem retrieval rather than deriving a fitted result, prediction, or theorem from prior inputs. Instance construction from candidate citations in real proofs, hierarchical context, leakage-controlled hints, and metrics such as ToolAcc and source grounding are defined directly from the task specification without self-referential reduction. The reported 7.0% ToolAcc on the test set follows from straightforward evaluation on the constructed instances; no equation, claim, or central premise reduces by construction to a fit, self-citation chain, or renamed input. The work is self-contained as an empirical benchmark release.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The benchmark construction rests on the domain assumption that real mathematical proofs contain reusable instrumental citations that can be isolated to create diagnostic retrieval instances.

axioms (1)

domain assumption Mathematical proofs contain instrumental citations that can be isolated to form benchmark instances while preserving hierarchical context.
This premise is invoked to build each test case from actual proofs.

pith-pipeline@v0.9.0 · 5535 in / 1289 out tokens · 72597 ms · 2026-05-12T02:02:09.600682+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages · 4 internal anchors

[1]

Deepmath-deep sequence models for premise selection.arXiv preprint arXiv:1606.04442,

Alex A Alemi, François Chollet, Niklas Een, Geoffrey Irving, Christian Szegedy, and Josef Urban. Deepmath-deep sequence models for premise selection.arXiv preprint arXiv:1606.04442,

work page arXiv
[2]

Chang, Kyle Lo, Luca Soldaini, Sergey Feldman, Mike D’Arcy, David Wadden, Matt Latzke, Minyang Tian, Pan Ji, Shengyan Liu, Hao Tong, Bohao Wu, Yanyu Xiong, Luke S

URL https://arxiv.or g/abs/2411.14199. Zhangir Azerbayev, Bartosz Piotrowski, Hailey Schoelkopf, Edward W Ayers, Dragomir Radev, and Jeremy Avigad. Proofnet: Autoformalizing and formally proving undergraduate-level mathematics. arXiv preprint arXiv:2302.12433,

work page arXiv
[3]

arXiv preprint arXiv:2305.12524 , year=

Wenhu Chen, Ming Yin, Max Ku, Pan Lu, Yixin Wan, Xueguang Ma, Jianyu Xu, Xinyi Wang, and Tony Xia. Theoremqa: A theorem-driven question answering dataset.arXiv preprint arXiv:2305.12524,

work page arXiv
[4]

Training Verifiers to Solve Math Word Problems

Karl Cobbe, Vineet Kosaraju, Mohammad Bavarian, Mark Chen, Heewoo Jun, Lukasz Kaiser, Matthias Plappert, Jerry Tworek, Jacob Hilton, Reiichiro Nakano, et al. Training verifiers to solve math word problems.arXiv preprint arXiv:2110.14168,

work page internal anchor Pith review Pith/arXiv arXiv
[5]

org/abs/2004.07180

URL https://arxiv. org/abs/2004.07180. FutureSearch, :, Nikos I. Bosse, Jon Evans, Robert G. Gambee, Daniel Hnyk, Peter Mühlbacher, Lawrence Phillips, Dan Schwarz, and Jack Wildman. Deep research bench: Evaluating ai web research agents,

work page arXiv 2004
[6]

Deep research bench: Evaluating ai web research agents, arXiv preprint arXiv:2506.06287, 2025

URLhttps://arxiv.org/abs/2506.06287. Michael Färber and Adam Jatowt. Citation recommendation: approaches and datasets.International Journal on Digital Libraries, 21(4):375–405, August

work page arXiv
[7]

doi: 10.1007/s007 99-020-00288-2

ISSN 1432-1300. doi: 10.1007/s007 99-020-00288-2. URLhttp://dx.doi.org/10.1007/s00799-020-00288-2. Bofei Gao, Feifan Song, Zhe Yang, Zefan Cai, Yibo Miao, Qingxiu Dong, Lei Li, Chenghao Ma, Liang Chen, Runxin Xu, et al. Omni-math: A universal olympiad level mathematic benchmark for large language models.arXiv preprint arXiv:2410.07985, 2024a. Guoxiong Gao...

work page doi:10.1007/s007
[8]

arXiv preprint arXiv:2305.14627 , year=

URLhttps://arxiv.org/abs/2305.14627. Elliot Glazer, Ege Erdil, Tamay Besiroglu, Diego Chicharro, Evan Chen, Alex Gunning, Caroline Falk- man Olsson, Jean-Stanislas Denain, Anson Ho, Emily de Oliveira Santos, et al. Frontiermath: A benchmark for evaluating advanced mathematical reasoning in ai.arXiv preprint arXiv:2411.04872,

work page arXiv
[9]

DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning

10 Daya Guo, Dejian Yang, Haowei Zhang, Junxiao Song, Ruoyu Zhang, Runxin Xu, Qihao Zhu, Shirong Ma, Peiyi Wang, Xiao Bi, et al. Deepseek-r1: Incentivizing reasoning capability in llms via reinforcement learning.arXiv preprint arXiv:2501.12948,

work page internal anchor Pith review Pith/arXiv arXiv
[10]

Dan Hendrycks, Collin Burns, Saurav Kadavath, Akul Arora, Steven Basart, Eric Tang, Dawn Song, and Jacob Steinhardt

doi: 10.1145/1772690.1772734. Dan Hendrycks, Collin Burns, Saurav Kadavath, Akul Arora, Steven Basart, Eric Tang, Dawn Song, and Jacob Steinhardt. Measuring mathematical problem solving with the math dataset.arXiv preprint arXiv:2103.03874,

work page doi:10.1145/1772690.1772734
[11]

Holstep: A machine learning dataset for higher-order logic theorem proving.arXiv preprint arXiv:1703.00426,

Cezary Kaliszyk, François Chollet, and Christian Szegedy. Holstep: A machine learning dataset for higher-order logic theorem proving.arXiv preprint arXiv:1703.00426,

work page arXiv
[12]

Jakub Lála, Odhran O’Donoghue, Aleksandar Shtedritski, Sam Cox, Samuel G

URL https://arxiv.org/abs/2508.15804. Jakub Lála, Odhran O’Donoghue, Aleksandar Shtedritski, Sam Cox, Samuel G. Rodriques, and Andrew D. White. Paperqa: Retrieval-augmented generative agent for scientific research,

work page arXiv
[13]

Rodriques, and Andrew D

URLhttps://arxiv.org/abs/2312.07559. Sadegh Mahdavi, Muchen Li, Kaiwen Liu, Christos Thrampoulidis, Leonid Sigal, and Renjie Liao. Leveraging online olympiad-level math problems for llms training and contamination-resistant evaluation.arXiv preprint arXiv:2501.14275,

work page arXiv
[14]

Magnushammer: A transformer-based approach to premise selection.arXiv preprint arXiv:2303.04488,

Maciej Mikuła, Szymon Tworkowski, Szymon Antoniak, Bartosz Piotrowski, Albert Qiaochu Jiang, Jin Peng Zhou, Christian Szegedy, Łukasz Kuci´nski, Piotr Miło´s, and Yuhuai Wu. Magnushammer: A transformer-based approach to premise selection.arXiv preprint arXiv:2303.04488,

work page arXiv
[15]

Mark Sanderson

URL https://arxiv.org/abs/2009.02252. Mark Sanderson. Test collection based evaluation of information retrieval systems.Foundations and Trends in Information Retrieval, 4(4):247–375,

work page arXiv 2009
[16]

Amanpreet Singh, Mike D’Arcy, Arman Cohan, Doug Downey, and Sergey Feldman

doi: 10.1561/1500000009. Amanpreet Singh, Mike D’Arcy, Arman Cohan, Doug Downey, and Sergey Feldman. Scirepeval: A multi-format benchmark for scientific document representations,

work page doi:10.1561/1500000009
[17]

Haoxiang Sun, Yingqian Min, Zhipeng Chen, Wayne Xin Zhao, Lei Fang, Zheng Liu, Zhongyuan Wang, and Ji-Rong Wen

URL https://arxiv.or g/abs/2211.13308. Haoxiang Sun, Yingqian Min, Zhipeng Chen, Wayne Xin Zhao, Lei Fang, Zheng Liu, Zhongyuan Wang, and Ji-Rong Wen. Challenging the boundaries of reasoning: An olympiad-level math benchmark for large language models.arXiv preprint arXiv:2503.21380,

work page arXiv
[18]

BEIR: A Heterogenous Benchmark for Zero-shot Evaluation of Information Retrieval Models

URL https://arxiv.org/abs/2104.08663. 11 George Tsoukalas, Jasper Lee, John Jennings, Jimmy Xin, Michelle Ding, Michael Jennings, Ami- tayush Thakur, and Swarat Chaudhuri. Putnambench: Evaluating neural theorem-provers on the putnam mathematical competition.Advances in Neural Information Processing Systems, 37: 11545–11569,

work page internal anchor Pith review arXiv
[19]

Fact or fiction: Verifying scientific claims.ArXiv, abs/2004.14974,

URL https: //arxiv.org/abs/2004.14974. David Wadden, Kyle Lo, Bailey Kuehl, Arman Cohan, Iz Beltagy, Lucy Lu Wang, and Hannaneh Hajishirzi. Scifact-open: Towards open-domain scientific claim verification,

work page arXiv 2004
[20]

Peiyi Wang, Lei Li, Zhihong Shao, Runxin Xu, Damai Dai, Yifei Li, Deli Chen, Yu Wu, and Zhifang Sui

URL https: //arxiv.org/abs/2210.13777. Peiyi Wang, Lei Li, Zhihong Shao, Runxin Xu, Damai Dai, Yifei Li, Deli Chen, Yu Wu, and Zhifang Sui. Math-shepherd: Verify and reinforce llms step-by-step without human annotations. In Proceedings of the 62nd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 9426–9439,

work page arXiv
[21]

Self-Consistency Improves Chain of Thought Reasoning in Language Models

Xuezhi Wang, Jason Wei, Dale Schuurmans, Quoc Le, Ed Chi, Sharan Narang, Aakanksha Chowdh- ery, and Denny Zhou. Self-consistency improves chain of thought reasoning in language models. arXiv preprint arXiv:2203.11171,

work page internal anchor Pith review Pith/arXiv arXiv
[22]

Naturalproofs: Mathematical theorem proving in natural language.arXiv preprint arXiv:2104.01112,

Sean Welleck, Jiacheng Liu, Ronan Le Bras, Hannaneh Hajishirzi, Yejin Choi, and Kyunghyun Cho. Naturalproofs: Mathematical theorem proving in natural language.arXiv preprint arXiv:2104.01112,

work page arXiv
[23]

Livebench: A challenging, contamination-free llm benchmark,

Colin White, Samuel Dooley, Manley Roberts, Arka Pal, Ben Feuer, Siddhartha Jain, Ravid Shwartz- Ziv, Neel Jain, Khalid Saifullah, Siddartha Naidu, et al. Livebench: A challenging, contamination- free llm benchmark.arXiv preprint arXiv:2406.19314, 4,

work page arXiv
[24]

Deepseek-prover-v1

Huajian Xin, ZZ Ren, Junxiao Song, Zhihong Shao, Wanjia Zhao, Haocheng Wang, Bo Liu, Liyue Zhang, Xuan Lu, Qiushi Du, et al. Deepseek-prover-v1. 5: Harnessing proof assistant feedback for reinforcement learning and monte-carlo tree search.arXiv preprint arXiv:2408.08152,

work page arXiv
[25]

Harp: A challenging human-annotated math reasoning benchmark.arXiv preprint arXiv:2412.08819,

Albert S Yue, Lovish Madaan, Ted Moskovitz, DJ Strouse, and Aaditya K Singh. Harp: A challenging human-annotated math reasoning benchmark.arXiv preprint arXiv:2412.08819,

work page arXiv
[26]

Minif2f: a cross-system benchmark for formal olympiad-level mathematics.arXiv preprint arXiv:2109.00110,

Kunhao Zheng, Jesse Michael Han, and Stanislas Polu. Minif2f: a cross-system benchmark for formal olympiad-level mathematics.arXiv preprint arXiv:2109.00110,

work page arXiv
[27]

Recall that,

context and (optional) anchor, through query formulation and retrieval, to source selection and tool extraction—and highlights the validity criterion based on grounding and proof-gap sufficiency. This diagram is intended for intuition only and does not replace the formal task definition in Section 3, which specifies the precise inputs, outputs, and evalua...

work page arXiv 2026
[28]

Each case first presents the proof situation and reference witness, then reports the observed model behavior, and finally explains which interface breaks: proof need → search-effective language, retrieved source → theorem-bearing passage, or sourced statement→assumption-checked proof tool. Legend.Y/N indicate positive/negative labels, and ⊥ indicates no e...

work page arXiv 1901
[29]

Aubin–Lions

Model behavior. Model Query Plan Query Cite@20 Ground Suff Tool Selected source GPT-5.2 positive definite polynomial sphere N Y Y⊥ ⊥ ⊥ Strictly positive definite functions on spheres Gemini 3.1 Pro positive definite function coefficients one N Y Y N N N Metric spaces and positive definite functions Claude Opus 4.5 positive definite functions spheres Gegen...

work page arXiv 1909
[30]

Model behavior. Model Query Plan Query Cite@20 Ground Suff Tool Selected source GPT-5.2 tropical circuit boolean reduction N N N Y Y Y Approximation Limitations of Pure Dy- namic Programming Gemini 3.1 Pro Jerrum Snir monotone perfect matchingN N N⊥ ⊥ ⊥none Claude Opus 4.5 tropical circuit monotone boolean simu- lation Y Y N N N N Reciprocal inputs in ari...

work page arXiv