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arXiv:2608.11195

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

Alan Li, Rahul Saha, Anton Xue, Swarat Chaudhuri, Adam Klivans, Pravesh K. Kothari, Raghu Meka

Published Aug 11, 2026• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2608.11195

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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 KGK_GKG​, which captures the hardness between combinatorial problems and their continuous relaxations. Specifically, while the precise value of KGK_GKG​ 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.
The Grothendieck constant KG is the smallest constant such that for every d ∈ N and every matrix A = (aij), sup u i,v j ∈B (d) X aij〈ui, vj 〉 � KG · ij sup x i,y j ∈[−1,1] X ij aijxiyj, where B (d) is the unit ball in R d. Despite several efforts [15, 23], the value of the constant KG remains unknown. The Grothendieck constant KG is precisely the integrality gap of a natural SDP relaxation for the KM,N-Quadratic Programming problem. The input to this problem is a matrix A = (aij) and the objective is to maximize the quadratic form P ij aijxiyj over xi, yj ∈ [−1, 1]. In this work, we apply techniques from [22] to the KM,N-Quadratic Programming problem. Using some standard but non-trivial modifications, the reduction in [22] yields the following hardness result: Assuming the Unique Games Conjecture [9], it is NP-hard to approximate the KM,N-Quadratic Programming problem to any factor better than the Grothendieck constant KG. By adapting a “bootstrapping ” argument used in a proof of Grothendieck inequality [5], we are able to perform a tighter analysis than [22]. Through this careful analysis, we obtain the following new results: ◦ An approximation algorithm for KM,N-Quadratic Programming that is guaranteed to achieve an approximation ratio arbitrarily close to the Grothendieck constant KG (optimal approximation ratio assuming the Unique Games Conjecture). ◦ We show that the Grothendieck constant KG can be computed within an error η, in time depending only on η. Specifically, for each η, we formulate an explicit finite linear program, whose optimum is η-close to the Grothendieck constant. We also exhibit a simple family of operators on the Gaussian Hilbert space that is guaranteed to contain tight examples for the Grothendieck inequality.
The use of artificial intelligence (AI) in physics research is increasingly common, yet few guidelines exist for its effective use. Starting with a principles-based approach, we outline a set of ten requirements for collaborative AI agents in experimental physics, spanning scientific norms, effective agent use, and domain expertise. We take a human-centered approach in which agents act as collaborators that amplify rather than replace the expertise of a human researcher (user) who directs the work and is responsible for its results. We then present a framework designed to enforce these requirements, detailing methods for improving off-the-shelf behavior of large language models (LLMs) with custom scaffolding and describing general-purpose scientific workflows and skills. In particular, we highlight the power of closed-loop campaigns where agents work iteratively toward a well-defined goal. We demonstrate the potential of this mode of research through six case studies in experimental physics, covering device design, hardware control, neural network optimization, physical model identification, and data analysis. All six case studies result in useful work products that add utility to our research, including a neural network architecture and solid-state quantum defect photodynamic model that substantially outperform our published work; a microwave waveguide design that extends a published design concept to a fabrication-ready device meeting our experimental constraints; a quantum control pulse sequence candidate that improves simulated performance by more than an order of magnitude; commissioning of an NMR spectrometer to $\sim$1~Hz frequency precision on live hardware; and finally a data-fitting pipeline that achieves, to our knowledge, the first reconstruction of the Ising microstate of an artificial spin-ice array from diffraction-limited magnetic imaging.
The Grothendieck constant KG is the smallest constant such that for every d ∊ ℕ and every matrix A = (aij), where B(d) is the unit ball in ℝd. Despite several efforts [15, 23], the value of the constant KG remains unknown. The Grothendieck constant KG is precisely the integrality gap of a natural SDP relaxation for the KM,N-Quadratic Programming problem. The input to this problem is a matrix A = (αij) and the objective is to maximize the quadratic form Σij aijxiyj over xi, yj ∊ [–1, 1]. In this work, we apply techniques from [22] to the KM,N-Quadratic Programming problem. Using some standard but non-trivial modifications, the reduction in [22] yields the following hardness result: Assuming the Unique Games Conjecture [9], it is NP-hard to approximate the KM,N-Quadratic Programming problem to any factor better than the Grothendieck constant KG. By adapting a “bootstrapping” argument used in a proof of Grothendieck inequality [5], we are able to perform a tighter analysis than [22]. Through this careful analysis, we obtain the following new results: An approximation algorithm for KM,N-Quadratic Programming that is guaranteed to achieve an approximation ratio arbitrarily close to the Grothendieck constant KG (optimal approximation ratio assuming the Unique Games Conjecture). We show that the Grothendieck constant KG can be computed within an error η, in time depending only on η. Specifically, for each η, we formulate an explicit finite linear program, whose optimum is η-close to the Grothendieck constant. We also exhibit a simple family of operators on the Gaussian Hilbert space that is guaranteed to contain tight examples for the Grothendieck inequality.
Landmark mathematical formalizations have taken specialist teams years to complete. We present FormalFlow, a system that coordinates AI proving agents under human supervision to address statement drift and proof composition in long-horizon formalization. Drawing on software-engineering principles and practices, it uses a shared blueprint to guide nested planning, proving, and review loops. Using FormalFlow, we completed a machine-checked Lean 4 proof of the quantum soundness of the classical low individual-degree test, a core theorem underlying MIP* = RE, in 63 days. The final library contains 126,367 lines of Lean code, all generated by agents. Beyond providing a verified foundation for quantum complexity, the formalization corrects side conditions and intermediate error bounds in the published proof and improves the error exponent of the main theorem. The development record shows what it takes to trust proofs written by AI agents and provides reusable tools for verifying research mathematics at scale.
An autonomous research laboratory could conduct an entire study, from formulating questions to preparing manuscripts. Artificial intelligence (AI) offers one route towards this goal. Recent AI systems take on several research roles, but may draw conclusions that the evidence does not support. Here, we present an autonomous research laboratory in which agents based on large language models assess one another's work and can stop or redirect investigations. They can generate and reject research questions, require separate theoretical checks and conduct experiments designed to challenge their claims. We demonstrate these capabilities by investigating ensembles of quantum spins in diamond. The agents derived the minimum number of calibration settings needed to predict responses to a defined family of microwave pulse sequences. They checked the result computationally and designed experiments to distinguish competing predictions. An initial experiment agreed with the theoretical predictions, but repeated reference measurements showed that the signal was drifting during data collection. The agents therefore treated the result as inconclusive and revised the measurement protocol without relaxing the acceptance criteria. The revised experiment met those criteria and supported the theoretical predictions. This study demonstrates that collaborating AI agents can select research questions, derive theoretical results and test their predictions experimentally.