{"slug": "two-futures-for-llms-in-mathematics", "title": "Two Futures for LLMs in Mathematics", "summary": "Anthropic handed a result to Columbia's Josh Alman and MIT's Virginia Williams, whose follow-up paper showed 3SUM can be solved in O(n^1.9992) time, breaking the long-held O(n^2) bound, while OpenAI separately released a repository of over 700 PDFs of varying quality, one claiming to reduce the matrix multiplication exponent ω to 2.25 (9/4) from the prior state-of-the-art O(n^2.371177). The OpenAI claim, if true, would be the largest reduction since Schönhage's 2.522 in 1981 and the first large reduction since Coppersmith and Winograd reached 2.3755 in 1990, though the repository's preprints vary in quality and few have clear human review. The contrast highlights two paths for LLM-assisted mathematics: Anthropic's targeted handoff to expert researchers versus OpenAI's bulk release of unvetted preprints.", "body_md": "# [Two Futures for LLMs in Mathematics](https://wiredream.com/llm-two-futures/)\n\n David G. Andersen  October 08, 2026 \nThis week, we've seen two very different approaches to LLMs for math/theoretical computer science.\n\nIn one corner, Anthropic dropped a result to Josh Alman (Columbia) and his former advisor at MIT, Virginia Williams, who are both known for having shown that matrix multiplication could be done slightly more cheaply than previously thought. They did so through very careful counting of how many operations were actually needed in various parts of the existing \"laser\" method of matrix multiplication, finding that you could shave off a hair here and there.\n\nThese two researchers looked at the result from Anthropic and turned it into a fully-fleshed-out\n[paper](https://arxiv.org/pdf/2610.06783). The new paper wasn't a matmul result directly; it was a way to use a particular\nflavor of matrix product to break some bounds that had held\nso long people were starting to build theory around their\nhardness, such as 3SUM:\n\nFor decades, nobody could figure out how to do 3SUM faster\nin sub-quadratic time, i.e., something like\nO(*n*<sup>2 - e</sup>) time for some value of e > 0, though we hadn't specifically proved\nthat it *needed* it.\nThis new paper showed that it doesn't.\nThe work has a very similar feel to some of their previous matrix\nmultiplication work, in the sense that it's also using very\ncareful counting of operations to drop things from O(*n*<sup>2</sup>) to\nO(*n*<sup>1.9992</sup>). That beats O(*n*<sup>2</sup>) by only a tiny\nhair, but that hair is important, because it means our assumptions\nwere wrong, and a bunch of other previously-conjectured hardnesses\nwere reduced along with it using the same technique.  This\nis quite a big result in its subfield.\n\nIn the other corner, OpenAI dumped\na repository of over 700 PDFs of highly varying quality, some\nwith accompanying Lean proofs, some without, few with clear human\nreview. Some of the claimed results are nearly breathtaking, if\nthey're true. [One that jumped out](https://github.com/openai/math/tree/main/preprints/Matrix-Multiplication-Nine-Fourths-October-2-2026) was about matrix multiplication, the\narea of expertise of the above two researchers. The straightforward\nway of multiplying matrices is O(*n*<sup>3</sup>) for two square *n* × *n* matrices:\nYou have *n*<sup>2</sup> output cell values, each of which results from the dot\nproduct of two size-*n* inputs (requiring *n* multiplies). But we've known\nfor a while there's some redundant computation in there; that\nexponent, which we term omega (ω) is less than three, but it's unknown\nexactly what it can be. There have been a series of\nseries of some practical and mostly theoretical optimizations that resulted in the previous\nstate-of-the-art upper bound, the somewhat ungainly O(*n*<sup>2.371177</sup>).\n\nOpenAI's PDF claims to reduce ω to 2.25, or 9/4. This would be several things. First, it would be literally the largest reduction we've seen since Schönhage's reduction to 2.522 in 1981; second, the first \"large\" reduction at all since Coppersmith and Winograd brought things to 2.3755 in 1990. And it would be incredibly satisfying to have something that's a rational number bound of 9/4, not the least because it seems more likely to me to intuitively illustrate some missed structure in the problem.\n\nAllow me to illustrate these papers with a few snippets from the introductory material of the papers. Take a peek and read them, asking if you understand what they're saying. From Alman and Williams:\n\n3SUM: Given *n* numbers, decide whether three of them sum to 0. This\nis a classical problem with a long history, and it is central to computational\ngeometry (see [GO95]). Despite many decades of research, the O(*n*<sup>2</sup>)-time algorithm taught in\nalgorithms classes has only been sped up by polylogarithmic factors [BDP08, GP18,\nCha20].\n\nFrom OpenAI's pdf dump:\n\nThe exponent ω of matrix multiplication over ℂ is the infimum of the\nreal numbers τ such that, for every ε > 0, two *n* × *n* matrices\ncan be multiplied in O<sub>ε</sub>(*n*<sup>τ + ε</sup>) scalar\narithmetic operations. The dimension *n* tends to infinity; the\nalgorithm and its constants may depend on ε.\n\nI mean, true, but this is not how you'd write the first paragraph of a paper written for humans. Contrast that with one of Williams' human-written papers about the same topic:\n\nMultiplication of matrices is a fundamental algebraic primitive with applications throughout computer science and beyond. The study of its algorithmic complexity has been a vibrant area in theoretical computer science and mathematics ever since Strassen’s [Str69] 1969 discovery that the rank of 2 by 2 matrix multiplication is 7 (and not 8), leading to the first truly subcubic, O(n2.81)-time algorithm for multiplying n × n matrices. Fifty-five years later, researchers are still attempting to lower the exponent ω, defined as the smallest real number for which n × n matrices can be multiplied in O(nω+ε) time for all ε > 0.\n\nI can read that! I like it! I want to read more!\n\nAnd the OpenAI paper is rife with things I find confusing. For example, consider this line:\n\nWe write products either by juxtaposition or by ⊗. The zero tensor is\n0, the scalar tensor *xyz* is 1, and the integer *m* ≥ 0 denotes\nthe direct sum of *m* copies of 1. Thus *mA* = A<sup>⊕*m*</sup>.\n\nI think this is intentional use of the notation, but as a human, when you introduce a symbol to me and then use a tiny-font subtly different symbol in the next line that you've never introduced, my head hurts a little. (Note that they introduced circled-times and then used circled-plus in their explanation of the integer-matrix product). And their notation there is confusing overall to me.\n\nThis is one example of many, and probably one of the better-confidence ones at\nthat, given that at least this one has a Lean version that proves\nsomething (what it proves I am not yet certain—does it\nprove the 2.25 result? That's going to take some time\nand expert evaluation). The internet is aflutter with people\nfinding flaws in these papers; [three of them have already been\nwithdrawn](https://github.com/openai/math/blob/main/history.md) due to errors that rendered them invalid. The writing in\nthese PDFs is very AI-slop-feeling.\n\nI like to point out that when you write, you're responsible for making sure your audience understands what you've written. There's one of you putting in some time, and potentially thousands of people reading what you've written. Their collective effort to understand you is far higher than the time it takes you to be clear.\n\nAnd the same thing applies here, but perhaps at 100x magnification: Thousands of\npeople will have to waste time reading this dump and possibly trying to\ndetermine if it's correct, and that's *really* hard work. That time would have been much better spent\nby having an expert or two review, revise, and present the material\ncleanly before throwing an unfiltered dump at the world.\n\nWe've seen two ways of having your internal advanced AI interact with the world of research, and I know which one I prefer: The one that produced a human-centered result that was informative and interesting to read and where I have much higher confidence I didn't waste my time reading something broken.", "url": "https://wpnews.pro/news/two-futures-for-llms-in-mathematics", "canonical_source": "https://wiredream.com/llm-two-futures/", "published_at": "2026-10-09 17:23:45+00:00", "updated_at": "2026-10-09 19:54:43.490619+00:00", "lang": "en", "topics": ["artificial-intelligence", "large-language-models", "ai-research", "machine-learning"], "entities": ["Anthropic", "OpenAI", "Josh Alman", "Virginia Williams", "Columbia University", "MIT", "3SUM", "Schönhage"], "also_reported_by": [], "alternates": {"html": "https://wpnews.pro/news/two-futures-for-llms-in-mathematics", "markdown": "https://wpnews.pro/news/two-futures-for-llms-in-mathematics.md", "text": "https://wpnews.pro/news/two-futures-for-llms-in-mathematics.txt", "jsonld": "https://wpnews.pro/news/two-futures-for-llms-in-mathematics.jsonld"}}