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It found admissible sets that raised the asymptotic lower bound from 2.2180 to 2.2202 after Ellenberg exploited a symmetry noticed in the program's output. Independent commentary by Ernest Davis (NYU, 11 January 2024) accepts the constructions as genuine but rates their mathematical importance as moderate. Terence Tao called the approach a promising paradigm.", "category": "benefit created", "unexpected": false, "purpose": "Built to search program space for constructions and heuristics in mathematics and computer science. 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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." }, "evidence": [ { "kind": "peer-reviewed study", "citation": "Romera-Paredes, B. et al. (2024). Mathematical discoveries from program search with large language models. Nature 625, 468–475 (published online 14 December 2023).", "title": "Mathematical discoveries from program search with large language models", "url": "https://www.nature.com/articles/s41586-023-06924-6", "date": "2023-12-14", "written_by": "makers", "quote": "FunSearch combines a pretrained (frozen) LLM, whose goal is to provide creative solutions, with an evaluator, which guards against confabulations and incorrect ideas." }, { "kind": "news report", "citation": "Heaven, W. D. (2023). Google DeepMind used a large language model to solve an unsolved math problem. MIT Technology Review.", "title": "Google DeepMind used a large language model to solve an unsolved math problem", "url": "https://www.technologyreview.com/2023/12/14/1085318/google-deepmind-large-language-model-solve-unsolvable-math-problem-cap-set/", "date": "2023-12-14", "written_by": "independent", "quote": "This is a promising paradigm. It is an interesting way to leverage the power of large language models." }, { "kind": "other", "citation": "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.", "title": "Comment on (Romera-Paredes et al., 2023), Mathematical discoveries from program search with large language models", "url": "https://cs.nyu.edu/~davise/papers/FunSearchComment.pdf", "date": "2024-01-11", "written_by": "independent", "quote": "FunSearch found a cap set of size 512 for n = 8. The previous best known cap set had size 496." }, { "kind": "preprint", "citation": "Zhai Y, Wei Z, Li R, et al. X-evolve: Solution space evolution powered by large language models. arXiv:2508.07932, 11 August 2025.", "title": "X-evolve: Solution space evolution powered by large language models", "url": "https://arxiv.org/abs/2508.07932", "date": "2025-08-11", "written_by": "independent", "quote": null } ], "replication": { "produced_by": [ "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" ], "must_hold": [ "The evaluator must reject invalid constructions exactly", "Results must be explicit objects that can be checked independently of the system" ], "echoes": [] }, "paired_incident": null, "headline": "DeepMind's FunSearch finds a size-512 cap set in dimension 8", "metric": "cap set of size 512 in dimension 8, exceeding the previous best known construction of 496", "comparator": { "against": "the previous best known cap set construction in dimension 8", "before": "496", "after": "512", "basis": "measured", "quote": "First: FunSearch found a cap set of size 512 for n = 8. 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