DeepMind's FunSearch finds a size-512 cap set in dimension 8

AI system

System: FunSearch. [1]
Model: Codey, built on PaLM 2. [1]
Release: Codey’s release identifier is not given. [1]
Provider: Google DeepMind. [1]
Role: Codey generated candidate Python functions for constructing cap sets. [1]
Configuration: The researchers used frozen model weights, temperature 1.0, and top-p 0.95. The cap-set setup used 15 samplers, 140 evaluators, and 10 populations of programs. [1]
Inputs and outputs: FunSearch used problem specifications and prior functions to generate candidate functions and evaluate their scores. [1]
Autonomy: FunSearch generated and tested programs automatically within the researchers’ specification. A separate evaluator rejected invalid results. [1]
Workflow: A problem with a fast, exact evaluator for candidate solutions. An LLM sampling program variants in an evolutionary loop. Human inspection of discovered programs to refine the search.
Human oversight: Not assessed
Relation to AI: FunSearch generated the programs whose outputs are the cap sets and admissible sets. These are explicit finite objects that the evaluator and readers can verify mechanically. The 2.2202 bound depended on a human (Ellenberg) noticing a symmetry in FunSearch's output and refocusing the search. Davis identifies that as the most notable aspect.
AI contribution: FunSearch generated the programs whose outputs are the cap sets and admissible sets. These are explicit finite objects that the evaluator and readers can verify mechanically. The 2.2202 bound depended on a human (Ellenberg) noticing a symmetry in FunSearch's output and refocusing the search. Davis identifies that as the most notable aspect.

Benefits

Outcome identifiers: Not assessed
Benefit: cap set of size 512 in dimension 8, exceeding the previous best known construction of 496
Category: Improved Knowledge Discovery
Status: Not assessed
Beneficiary groups: Society.
Beneficiaries: Not assessed
Reach: Not assessed
Distribution and access: Not assessed
Duration: Not assessed
Expected?: Yes
Costs and harms: Not assessed

Measurement

Studies: Not assessed
Study design: Not assessed
Measurement: Not assessed
Setting and sample: Not assessed
Measures: cap set of size 512 in dimension 8, exceeding the previous best known construction of 496
Comparison: the previous best known cap set construction in dimension 8
Results: Reference: 496. Result: 512.
Additional value from AI: Not assessed
Limitations: Not assessed
Denominators: Not assessed
Materials for replication: Not assessed

Money

Kind of value: Not assessed
Amount: Not assessed
Currency and prices: Not assessed
Period and unit: Not assessed
Realization: Not assessed
Recipients and payers: Not assessed
Gross or net: Not assessed
Included costs: Not assessed
Comparison: Not assessed
Method: Not assessed
Return and payback: Not assessed

People and places

Participants: Developer: DeepMind. Operator: DeepMind with Jordan Ellenberg (University of Wisconsin–Madison).
Jurisdictions: Benefit: London, United Kingdom. Developer: United Kingdom. Operator: United Kingdom.
Ownership and control: Not assessed
Dependencies: Not assessed
Cross-border connections: Not assessed
Geopolitical implications: Not assessed

Follow-up

Conditions for replication: The evaluator must reject invalid constructions exactly. Results must be explicit objects that can be checked independently of the system.
Replication attempts: Not assessed
Later findings: 11 August 2025. The FunSearch constructions stood and were built on: a later LLM-driven method (X-evolve) raised the asymptotic cap set lower bound further from 2.2202 to 2.2203.
Availability: Not assessed
Institutional decision: Not assessed
Grounds: Not assessed
Action and result: Not assessed
Success criterion: Not assessed

Evidence

Source numbers: AIBD-0015-source-0001–0004
Citations: Numbered references below.
Language: Not assessed
Source relationships: Developer, Independent
Original evidence: The original paper and supplement identify Codey and document the search settings. [1]
Access and versions: The original article and its supplementary methods were accessible. [1]
Supported findings: Not assessed
Evidence links: Not assessed
Evidence strength: Not assessed
Missing evidence: Not assessed

Grades

Assessed benefits: Society.
Depth: Society: 2.
Reasons: Society: Mathematicians gained an explicit cap set of size 512 in dimension 8 and an improved asymptotic bound. Davis's independent comment judges the results moderately interesting mathematically.
Observed or intended: Not assessed
Rubric: Not assessed
Assessor and date: Not assessed
Alternative assessments: Not assessed
Fill: Independently reported

Entry details

Kind: Not assessed
Related entries: Not assessed
Field: Mathematics / combinatorics. Industrial research laboratory with academic collaborator.
Purpose: Built to search program space for constructions and heuristics in mathematics and computer science. The cap set constructions were that purpose.
Dates: Date basis unassessed: 2023 (precision unassessed). Updated: 25 October 2026.
Stage: Not assessed
Version: SHA-256: d9082ec8dab0…

References

Other identifiers: Not assessed
Category mappings: Not assessed
Brief: Not assessed
Revisions: 6 revisions. Latest: 25 October 2026.

Sources

Peer-reviewed study Developer

Romera-Paredes, B. et al. (2024). Mathematical discoveries from program search with large language models. Nature 625, 468–475 (published online 14 December 2023).

Identifies Codey and PaLM 2 and separates code generation from execution and evaluation. Pretrained LLM; Specification; Evaluation

Original research or documentation from the system’s researchers. This supports identity and methods, not independent verification of benefits.

Original research or documentation from the system’s researchers. This supports identity and methods, not independent verification of benefits. Documents the actual sampling and execution settings. Part of the original paper, not an independent source. Appendix A.2; Appendix E.1, Table E.2, p. 24

Original research or documentation from the system’s researchers. This supports identity and methods, not independent verification of benefits.
News report Independent author

Heaven, W. D. (2023). Google DeepMind used a large language model to solve an unsolved math problem. MIT Technology Review.
Other Independent author

Davis, E. (2024). Comment on (Romera-Paredes et al., 2023), 'Mathematical discoveries from program search with large language models'. New York University, 11 January 2024.
Preprint Independent author

Zhai Y, Wei Z, Li R, et al. X-evolve: Solution space evolution powered by large language models. arXiv:2508.07932, 11 August 2025.