AI-Assisted Mathematical Discovery

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18 papers4 papers in the last four weeks, with none the four weeks before. 0.0% of all new papers.

Oct 14, 2026

Euler: Towards Human-Centered Multi-Agent Mathematical Discovery

Mathematical research drives scientific discovery, and recent advances in AI for mathematics have shown that language-model agents can solve challenging problems and develop substantial mathematical proofs. Yet much of today's AI-for-mathematics work centers on obtaining a final answer. This emphasis leaves the exploration behind a result difficult for researchers to understand, while long-running investigations face fragmented context and the risk of unsupported claims propagating across reasoning steps. We present Euler, a human-centered mathematical research system built around a shared, structured representation of the research process. Euler organizes mathematical statements, assumptions, proof dependencies, alternative approaches, failed attempts, and verification evidence into connected research graphs. This representation allows researchers to trace how results are developed and supplies agents with persistent, task-relevant knowledge for planning, collaboration, and verification across sessions. Researchers can also select a graph node linked to a saved team state, revisit the agents' earlier reasoning, and explore alternative directions through continued interaction. We illustrate Euler's capabilities through three mathematical research case studies and describe the implemented system and its software validation.

Qiuyu Chen, Qi Wang, Liang Xu +3

Shanghai Jiao Tong University · Ningbo Key Laboratory of Spatial Intelligence and Digital Derivative, Ningbo Institute of Digital Twin, Eastern Institute of Technology, Zhejiang, China · Institute of Automation, Chinese Academy of Sciences +1

AI-Assisted Mathematical Discovery Autonomous Research Agents Mathematical Reasoning Human-AI Collaboration Multi-Agent Reasoning LLM Reasoning with Graphs

Oct 12, 2026

What the Machine Invented: Mining New Mathematical Instruments from OpenAI's October 2026 Release of AI-Generated Mathematics

On 6 October 2026 OpenAI published an initial 722 AI-written mathematical manuscripts, organized into 372 problem families, in the public repository github.com/openai/math. The release includes claimed solutions to many prominent open problems. We approach this collection as a treasure mine of mathematical ideas. Our aim is to extract new lemmas, constructions, inequalities and proof mechanisms introduced in the source arguments, and to distinguish them from established tools and new combinations of those tools. We call these mathematical instruments. Extraction comes first; their validity, novelty and possible reuse then become subjects for further study. The resulting descriptions also help readers understand the selected papers without having to reconstruct every argument at the outset. We examine 38 families in two groups of 17 and 21, spanning number theory, geometry, analysis, combinatorics, complexity, algebra, operator algebras, topology and mathematical physics. Each entry records the claim, extracted instruments, antecedents and caveats. Eleven cross-cutting patterns connect the entries, and a scoreboard records formalization status and dependencies. Novelty assessments are provisional and sometimes mixed. Selected source comparisons correct mathematical statements, hypotheses and counts; they do not independently establish priority or certify the underlying proofs. No Lean development was compiled or kernel-checked for this survey. We also briefly describe a subsequent application of the signed restriction lemma from family 112 to circuit lower bounds at every fixed depth.

Ying Xie

Kennesaw State University

AI-Assisted Mathematical Discovery AI-Assisted Scientific Research

Sep 29, 2026

GenLimitLib: A Formal Library for Language Generation in the Limit and AI-Assisted Mathematical Research

We present GenLimitLib, a source-aligned Lean 4 library for language generation in the limit. Introduced by Kleinberg and Mullainathan at NeurIPS 2024, language generation in the limit studies a theoretical question motivated by LLMs: how to generate valid new strings from observed examples. This young and rapidly evolving field offers a natural testbed for studying large-scale formalization. GenLimitLib contains formal developments for 30 papers. It extracts shared definitions and reusable proof components while preserving paper-specific assumptions and statements, and records relationships across papers. In this way, GenLimitLib provides a concrete and structured view of the literature. We show through mathematical case studies and LLM experiments how our library can support both human mathematical research and AI-assisted research. Our Library: https://github.com/pengzhang91/generation-in-the-limit-lib.

Shuangping Li, Peng Zhang

Yale University · Rutgers University

AI-Assisted Mathematical Discovery Text Generation AI-Assisted Scientific Research

Sep 23, 2026

Learning to Discover Interesting Mathematics

Recently, Large Language Models (LLMs) have been increasingly able to solve advanced mathematical problems, including many that have been open for decades. This opens the door to expansion of mathematical knowledge at unprecedented scale. Yet, while LLMs may be able to conjecture and prove more and more theorems, it remains open whether this new mathematical knowledge is interesting or useful. We define intrinsic interestingness of a theorem as the ratio between the length of its proof and the length of its statement. We show that this correlates strongly with an extrinsic measure of the downstream utility of a theorem. We identify the difficulty of a proof conditioned on a set of premises as a useful primitive for computing these metrics, and train a 27B model that predicts proof difficulty more accurately than frontier general-purpose models. Optimizing for our metric creates a model capable of producing more interesting theorems, while also reducing substantial or full overlap with Mathlib from 91.9% to 30.6%, showcasing the creation of more out-of-distribution math. We show that our system can generate candidate theorems, select the most interesting among them, and iteratively build on a self-expanding mathematical library. These metrics provide a practical and quantifiable signal for ranking conjectures and guiding proof search within formal mathematical libraries. Our framework provides a path towards self-expanding, machine-verified mathematical libraries that can choose worthwhile statements without relying on human-supplied targets.

Niket Patel, Ahmad Rammal, Amaury Hayat +2

New York University · CERMICS, ENPC, Institut Polytechnique de Paris

Automated Theorem Proving Lean Theorem Proving Large Language Model-Guided Scientific Discovery AI-Assisted Mathematical Discovery Large Language Model-Guided Theorem Proving

Aug 14, 2026

AI-Assisted Discovery and Construction of a Counterexample to the Convergence of Three-Block ADMM with the Identity Matrix as its Third Constraint Block

The alternating direction method of multipliers (ADMM), as a landmark algorithm, has attracted tremendous research attention and extensive practical applications over the past two decades. It is well known that, although the two-block ADMM enjoys well-established theoretical convergence guarantees, its direct extension to the three-block case may fail to converge, as demonstrated by existing counterexamples [5]. However, to the best of our knowledge, the case in which the third constraint block is the identity remains unresolved: the existing literature gives neither a general convergence proof nor a counterexample for this subclass. In this paper, we give a negative answer: direct three-block ADMM may fail even when the first two blocks are strongly convex quadratics. Using Codex with GPT-5.6 Sol, we construct an explicit rational counterexample candidate and verify it along a piecewise-affine reduction path; exact checks show that direct three-block ADMM on this instance produces a bounded nonconvergent orbit of period 66. Within the same Codex workflow, we further guide a study of multiplier relaxation and clarify when convergence can be restored at the fixed-instance and class levels: a problem-dependent small dual step can restore convergence, whereas no positive relative step works uniformly over the whole class. Furthermore, we also test the recent Kimi Code with Kimi K3 model without the Codex candidate or project-specific route guidance; along a different path it produces an exact locally attracting period-23 certificate, convertible to an equivalent all-identity instance. The comparison suggests that different research-harness configurations can shape the mathematical objects explored and the certificates pursued.

Kenan Xu, Xiangfeng Wang

School of Mathematical Sciences, East China Normal University, Shanghai 200241, P.R. China. · Key Laboratory of Mathematics and Engineering Applications (MoE) and School of Mathematical Sciences, East China Normal University, Shanghai 200241, P.R. China.

AI-Assisted Mathematical Discovery Alternating Direction Method of Multipliers

Aug 11, 2026

Long-Horizon AI Research for Grothendieck Constant: A Case Study in Human-AI Mathematical Collaboration

AI agents are increasingly used in mathematics research, but it is often unclear how to use them effectively. Towards this, we present an extensive case study of how AI was used to improve bounds on the Grothendieck constant

K_G

, which captures the hardness between combinatorial problems and their continuous relaxations. Specifically, while the precise value of

K_G

is not known, we recently tightened the best known bounds to

\frac{6π}{11} \;\le\; K_G \;\le\; \fracπ{2\log(1+\sqrt2)} - 10^{-4}.

Crucially, these improvements were achieved using an AI research system that could arrive at insights deemed novel by domain experts. We give a detailed discussion of our experience using AI for mathematics research, particularly touching upon its strengths and weaknesses, as well as our experience with creating ideal conditions for AI to arrive at breakthrough insights.

Alan Li, Rahul Saha, Anton Xue +4

Deep Research Agents AI-Assisted Mathematical Discovery AI-Assisted Scientific Research Human-AI Collaboration LLM Mathematical Reasoning

Aug 8, 2026

Neurosymbolic Discovery of Algebraic Graph Constructions

There are several methods for searching for graphs with prescribed properties, such as SAT solvers and specialized generators. These methods return the result as raw data: an adjacency matrix or a string encoding. The raw data certifies that the graph exists, but it does not reveal any structural properties of the graph. We ask whether one can automatically discover a short algebraic description if only this raw data is provided. We look for a description such as a Cayley graph

\mathrm{Cay}(Γ, S)

or a lexicographic product

C_5[K_3]

. We address this question with a neurosymbolic approach. We propose an agent that runs on a general-purpose large language model with no fine-tuning or per-target training. The model interleaves reasoning with calls to the computer algebra system SageMath: it analyzes the target graph, proposes and tests candidate constructions, and revises them until the output matches the target. The agent communicates with SageMath through a Model Context Protocol (MCP) server, which we release as a general-purpose bridge. Whether a construction matches the target is checked by a single exact isomorphism test, and therefore rests on the symbolic side and not on the model. We test the approach on a benchmark of 100 highly symmetric graphs, namely two-orbit graphs on up to 25 vertices; the benchmark was fixed in advance. Our agent could find verified algebraic constructions for all of them, without falling back to raw encodings. A strong template-enumeration baseline reaches only about

20\%

, and a catalog lookup could not identify any of these graphs. However, construction quality declines when symmetry is removed. As a concrete application, we identify the smallest known counterexample to the Bernhart-Kainen dispersability conjecture, a

16

-vertex graph that enumeration found as raw data. For this graph, our agent found an explicit algebraic construction.

David Seka, Stefan Szeider

Algorithms and Complexity Group TU Wien, Vienna, Austria

Graph Construction Neuro-Symbolic Reasoning Large Language Model-Guided Scientific Discovery AI-Assisted Mathematical Discovery

Jul 19, 2026

Mathematical Discovery in the Wild: AI-Guided Proofs in Banach Space Theory

We investigate the capacity of current language models to contribute to mathematical research. In Banach space theory, AI systems generated key ideas and proofs for five new results, which were then verified and refined by humans. We also developed an automated system that searches the literature for open problems and attempts solutions at scale. Our results show both the potential of language models for mathematical discovery and the continuing importance of expert verification.

Antonio Acuaviva, Pablo Acuaviva

Automated Theorem Proving AI-Assisted Mathematical Discovery AI-Assisted Scientific Research Large Language Model-Guided Scientific Discovery LLM Mathematical Reasoning

Jul 15, 2026

ReasFlow: Assisting Reasoning-Centric Scientific Discovery in Applied Mathematics via a Knowledge-Based Multi-Agent System

Recent advances in Large Language Models have fueled autonomous AI agents capable of tackling complex scientific tasks, yet existing automated research systems remain predominantly focused on empirically driven domains with quantitative benchmarks, leaving theory-driven discovery, particularly in mathematically grounded disciplines requiring rigorous proofs and synthesis of domain knowledge, largely underexplored. Key challenges include the difficulty of verifying theoretical reasoning at scale, insufficient reasoning ability for autonomous frontier exploration, and a scarcity of procedural heuristics in the literature. We introduce ReasFlow, an end-to-end autonomous agent system for reasoning-centric scientific discovery that operationalizes a collaborative paradigm where the human expert acts as Principal Investigator while the agent executes rigorous derivations as a capable graduate student. ReasFlow incorporates (i) a robust internal verification loop that audits logical coherence and corrects fundamental errors prior to human inspection, and (ii) an automated knowledge retrieval and self-improvement mechanism that proactively surfaces both declarative facts and overlooked procedural heuristics, substantially reducing expert intervention. The system unifies literature synthesis, algorithm design, theorem proving, experimentation, and manuscript preparation in a single system. Deployed to autonomously generate five complete research papers with rigorous theoretical and empirical content from minimal prompts, ReasFlow consistently achieves the highest evaluation scores among state-of-the-art open-access baselines under a curated LLM-based review rubric. ReasFlow is publicly accessible via the ReasLab platform, providing a collaborative workspace for AI-assisted theoretical research. Github repo: https://github.com/ReasLab/ReasFlow.git.

Yutong He, Daibo Li, Guohong Li +15

Peking University · University of Electronic Science and Technology of China · Beijing Normal University +2

Multi-Agent LLM Systems AI Agents for Scientific Discovery AI-Assisted Mathematical Discovery Autonomous Scientific Discovery Autonomous Research Agents

Jul 5, 2026

Language models guide symbolic equation discovery by controlling search

Scientific equation discovery must combine broad domain priors with strict numerical testing. Symbolic regression supplies numerical grounding but faces a combinatorial search space, whereas many language-model systems ask the model to propose or select formulas directly. We test a different division of labour. We compare role specifications in which the language model acts as equation author, candidate decider or search controller, alongside end-to-end language-model and purely numerical baselines. In the controller setting we propose here, implemented as LLM-PySR, language models specify variables, operators, transformations and search depth; symbolic regression enumerates and fits expressions; and deterministic metrics govern retention. Across 74 AI-Feynman equations and seven complex formula-recovery tasks, search control achieved the strongest observed balance of accuracy, complexity, stability and cost. On an independent battery dataset, LLM-PySR identified a compact piecewise-linear relation between early voltage-curve displacement and cycle life. The results suggest that language models should shape hypothesis exploration rather than decide which equations survive.

Zikai Xie, Wenmei Li, Man Luo +2

State Key Laboratory of Precision and Intelligent Chemistry, Hefei National Research Center for Physical Sciences at the Microscale, School of Chemistry and Materials Science, University of Science and Technology of China, Hefei, China. · Center for Scientific Intelligence Innovation, Hefei, China · School of Chemistry, School of Computer Science, University of Birmingham, Birmingham, UK

Large Language Model-Guided Scientific Discovery Symbolic Regression AI-Assisted Mathematical Discovery

Jun 15, 2026

Nothing from Something: Can a Language Model Discover 0?

AI systems based on artificial neural networks are being developed with aspirations of pushing the boundary of human mathematical knowledge. A key question for these systems is how much they can reach beyond their training data. Mathematical discovery requires a strong form of out of distribution generalization; the ability to hypothesize genuinely new - and potentially logically more powerful - mathematical structures. It has been hypothesized that language abilities support such generalizations in human cognition. In this work, we use simple arithmetic as a case study for examining how modern AI models could expand their mathematical horizons, evaluating whether these models can independently discover the concept of "zero". We show that (1) language models of a GPT-2 size are unable to perform this generalization at test time regardless of language pretraining, but (2) models can improve substantially after training on tens or hundreds of examples of zero. Additionally, we find that language pretraining reduces the number of required examples by approximately

50\%

, showing that language abilities can scaffold mathematical discovery in neural models.

Phoebe Zeng, Thomas L. Griffiths, Brenden M. Lake

Department of Computer Science, Princeton University

Numerical Reasoning in Language Models Language Model Pretraining Zero-Shot Learning OOD Generalization AI-Assisted Mathematical Discovery

Jun 1, 2026

Iteris: Agentic Research Loops for Computational Mathematics

Recent advances in large language models and agentic AI systems have enabled significant progress in mathematical discovery, from solving competition problems to tackling research-level conjectures. However, open problems in computational mathematics have received comparatively less attention: research in this area often requires not only proofs but also numerical experimentation, adversarial constructions, and algorithm design. In this paper, we introduce an agentic research system, Iteris, designed for open problems in computational mathematics. We apply Iteris to two open problems from a recent Simons Workshop collection (arXiv:2602.05394). In these case studies, Iteris generated numerical evidence, constructions, and proof drafts that led, after expert review and correction, to verified results. The first result is a phase diagram for the asymptotic comparison between conjugate gradient and randomized coordinate descent on power-law spectra; the second is a counterexample showing that QR factorization with column pivoting can fail to select well-conditioned submatrices even under low coherence. These case studies suggest that agentic AI systems can participate meaningfully in research workflows for open problems in computational mathematics, while human validation remains essential.

Leheng Chen, Zihao Liu, Wanyi He +1

School of Mathematical Sciences, Peking University · 5Zhongguancun Academy · 2Beijing International Center for Mathematical Research and the New Cornerstone Science Laboratory, Peking University +2

Mathematical Reasoning Large Language Model-Guided Scientific Discovery Autonomous Research Agents AI-Assisted Mathematical Discovery

May 7, 2026

AI co-mathematician: Accelerating mathematicians with agentic AI

We introduce the AI co-mathematician, a workbench for mathematicians to interactively leverage AI agents to pursue open-ended research. The AI co-mathematician is optimized to provide holistic support for the exploratory and iterative reality of mathematical workflows, including ideation, literature search, computational exploration, theorem proving and theory building. By providing an asynchronous, stateful workspace that manages uncertainty, refines user intent, tracks failed hypotheses, and outputs native mathematical artifacts, the system mirrors human collaborative workflows. In early tests, the AI co-mathematician helped researchers solve open problems, identify new research directions, and uncover overlooked literature references. Besides demonstrating a highly interactive paradigm for AI-assisted mathematical discovery, the AI co-mathematician also achieves state of the art results on hard problem-solving benchmarks, including scoring 48% on FrontierMath Tier 4, a new high score among all AI systems evaluated.

Daniel Zheng, Ingrid von Glehn, Yori Zwols +15

Google DeepMind · Google

AI Agents for Scientific Discovery AI-Assisted Mathematical Discovery AI-Assisted Scientific Research Human-AI Collaboration LLM Mathematical Reasoning Autonomous Research Agents

May 6, 2026

Grokability in five inequalities

In this note, we report five mathematical discoveries made in collaboration with Grok, all of which have been subsequently verified by the authors. These include an improved lower bound on the maximal Gaussian perimeter of convex sets in

\mathbb{R}^n

, sharper

L_2

-

L_1

moment comparison inequalities on the Hamming cube

\{-1,1\}^n

, a strengthened autoconvolution inequality, improved asymptotic bounds on the size of the largest

g

-Sidon sets in

\{1,\dots,n\}

, and an optimal balanced Szarek's inequality.

Paata Ivanisvili, Xinyuan Xie

University of California, Irvine

AI-Assisted Mathematical Discovery

Apr 19, 2026

Yanasse: Finding New Proofs from Deep Vision's Analogies, Part 1

Project Yanasse presents a method for discovering new proofs of theorems in one area of mathematics by transferring proof strategy patterns (e.g., Lean 4 tactic invocation patterns) from a structurally distant area. The system extracts tactic usage distributions across 27 top-level areas of Mathlib (217,133 proof states), computes z-scores to identify tactics that are heavily used in a source area but rare or absent in a target area, matches source and target proof states via GPU-accelerated NP-hard analogy (running on a MacBook Air via Apple's MPS backend), and then asks an AI reasoning agent to semantically adapt--not symbol-substitute--the source tactics invocation pattern to the target theorem. In this first part of the study, the method is applied to the pair Probability -> Representation Theory, producing 4 Lean-verified new proofs out of 10 attempts (40%). The proofs compile with zero sorry declarations. The key finding is that tactic schemas decompose into a head (domain-gated, rarely transfers) and a modifier (domain-general, often transfers): filter upwards's head fails in representation theory (no Filter structure), but its [LIST] with ω modifier transfers cleanly as ext1 + simp [LIST] + rfl. Crucially, the underlying matching engine--deep vision lib.py--is entirely domain independent: the same optimization code for an NP-hard matching that matches chess positions by analogy matches Lean proof states by analogy, without knowing which domain it is processing. Only a relation extractor is domain-specific.

Alexandre Linhares

Getulio Vargas Foundation · ARGO LABS

Automated Theorem Proving Analogical Reasoning Lean Theorem Proving LLM Mathematical Reasoning AI-Assisted Mathematical Discovery

Apr 8, 2026

$k$ -server-bench: Automating Potential Discovery for the $k$ -Server Conjecture

We introduce a code-based challenge for automated, open-ended mathematical discovery based on the

k

-server conjecture, a central open problem in competitive analysis. The task is to discover a potential function satisfying a large graph-structured system of simple linear inequalities. The resulting evaluation procedure is sound but incomplete: any violated inequality definitively refutes a candidate, whereas satisfying all inequalities does not by itself constitute a proof of the corresponding conjecture's special case. Nevertheless, a candidate that passes all constraints would be strong evidence toward a valid proof and, to the best of our knowledge, no currently known potential achieves this under our formulation in the open

k=4

circle case. As such, a successful candidate would already be an interesting contribution to the

k

-server conjecture, and could become a substantial theoretical result when paired with a full proof. Experiments on the resolved

k=3

regime show that current agentic methods can solve nontrivial instances, and in the open

k=4

regime they reduce the number of violations relative to existing potentials without fully resolving the task. Taken together, these results suggest that the task is challenging but plausibly within reach of current methods. Beyond its relevance to the

k

-server community, where the developed tooling enables researchers to test new hypotheses and potentially improve on the current record, the task also serves as a useful benchmark for developing code-based discovery agents. In particular, our

k=3

results show that it mitigates important limitations of existing open-ended code-based benchmarks, including early saturation and the weak separation between naive random baselines and more sophisticated methods.

Kirill Brilliantov, Etienne Bamas, Emmanuel Abbé

MDS Lab, EPFL

AI-Assisted Mathematical Discovery Autonomous Scientific Discovery AI Agent Benchmarks Benchmark Design Competitive Analysis Online Resource Allocation

Mar 16, 2026

Measuring Progress in Reasoning Toward Mathematical Discovery with Automatic Verification

Can AI make progress on important, unsolved mathematical problems? Large language models are now capable of sophisticated mathematical and scientific reasoning, but whether they can perform novel research is still widely debated and underexplored. We introduce HorizonMath, a benchmark of 113 predominantly unsolved problems spanning eight domains in mathematics and the mathematical sciences, paired with an open-source evaluation framework for automated verification. Our benchmark targets the generator-verifier gap: problems where discovery is hard and requires meaningful mathematical insight, but verification is computationally straightforward. This contrasts with most existing research-level benchmarks, which instead rely on formal proof verification or manual review, both of which are expensive to scale. Because these solutions are unknown, HorizonMath is resistant to data contamination, and most state-of-the-art models score under 10%. Using this framework, we identify six novel solutions to research problems that either resolve previously open questions or improve on the best-known published results, with GPT-5.4 Pro and GPT-5.6 Sol each discovering three of these solutions. Across seven frontier model families, reasoning efficiency and behavior also vary substantially. We release HorizonMath as an open challenge and a growing community resource, where each verified solution is a candidate contribution to the mathematical literature.

Erik Y. Wang, Sumeet R. Motwani, James V. Roggeveen +9

Stanford University · Benchmark · University of Oxford +3

Mathematical Reasoning Benchmarks LLM Mathematical Reasoning Reasoning Verification AI-Assisted Mathematical Discovery

May 28, 2025

AI Mathematician: Towards Fully Automated Frontier Mathematical Research

Large Reasoning Models (LRMs) have made significant progress in mathematical capabilities in recent times. However, these successes have been primarily confined to competition-level problems. In this work, we propose AI Mathematician (AIM) framework, which harnesses the reasoning strength of LRMs to support frontier mathematical research. We have identified two critical challenges of mathematical research compared to competition, the intrinsic complexity of research problems and the requirement of procedural rigor. To address these challenges, AIM incorporates two core strategies: an exploration mechanism to foster longer solution paths, and the pessimistic reasonable verification method to ensure reliability. This early version of AIM already exhibits strong capability in tackling research-level tasks. We conducted extensive experiments across several real-world mathematical topics and obtained promising results. AIM is able to autonomously construct substantial portions of proofs and uncover non-trivial insights within each research area. These findings highlight the potential of LRMs in mathematical discovery and suggest that LRM-based agent systems could significantly accelerate mathematical research in the future.

Yuanhang Liu, Yanxing Huang, Yanqiao Wang +2

Qiuzhen College, Tsinghua University · Department of Mathematical Sciences, Tsinghua University · Institute for AI Industry Research (AIR), Tsinghua University, Beijing, China +1

AI-Assisted Mathematical Discovery AI-Assisted Scientific Research Mathematical Reasoning Automated Theorem Proving Large Reasoning Models