{"schema":"postcutoff/event@1","as_of":"2026-10-12T22:30:00+02:00","url":"https://postcutoff.com/e/2026-10-08-rank-3x3-matrix-multiplication-f2-is-23-lean-tomar/","md":"https://postcutoff.com/e/2026-10-08-rank-3x3-matrix-multiplication-f2-is-23-lean-tomar/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"Adam Bicz","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-08-rank-3x3-matrix-multiplication-f2-is-23-lean-tomar","date":"2026-10-08","date_precision":"day","short_title":"Laderman's 23 multiplications are optimal for 3×3 matrices over F_2 and over the integers","deck":"AI-found proof checked in Lean (Tomar)","takeaway":"Whether 3×3 matrices can be multiplied with fewer than Laderman's 23 multiplications has been open since 1976. A preprint claims the answer is no for coefficients in F_2 or the integers, ending a 2026 race in which the lower bound rose from 19 to 20, 21, 22 and now 23. The author says generative AI found and proved the result.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.11823: The rank of 3×3 matrix multiplication over F_2 is 23","url":"https://arxiv.org/abs/2610.11823","type":"paper","group":"primary","domain":"arxiv.org"},{"n":2,"title":"Code, certificates and Lean checkers (tejasvi/rank333-f2)","url":"https://github.com/tejasvi/rank333-f2","type":"code","group":"primary","domain":"github.com"},{"n":3,"title":"Rudich & Rousseau, Lower bound of 22 for 3×3 matrix multiplication over Z (arXiv 2610.01639)","url":"https://arxiv.org/abs/2610.01639","type":"paper","group":"primary","domain":"arxiv.org"},{"n":4,"title":"Yang et al., A structural proof of the lower bound 21 for 3×3 matrix multiplication over F_2 (arXiv 2609.18722)","url":"https://arxiv.org/abs/2609.18722","type":"paper","group":"primary","domain":"arxiv.org"},{"n":5,"title":"Wang, A lower bound of 21 for 3×3 matrix multiplication over F_2 (arXiv 2609.06725)","url":"https://arxiv.org/abs/2609.06725","type":"paper","group":"primary","domain":"arxiv.org"}],"official":5,"filed":"2026-10-12","updated":"2026-10-12","orgs":["independent"],"title":"Laderman's 23 multiplications are optimal for 3×3 matrices over F_2 and over the integers: AI-found proof checked in Lean (Tomar)","summary":"On 8 Oct 2026 Tejasvi Singh Tomar (independent researcher) posted arXiv 2610.11823: the rank of the 3×3 matrix multiplication tensor over the two-element field is exactly 23. Laderman's 1976 algorithm gives the upper bound; Rudich and Rousseau had proved a lower bound of 22 a week earlier. As a corollary, any bilinear algorithm with integer coefficients needs 23 multiplications. The proof is a computer search whose certificates are checked in Lean 4 (the largest checks as compiled code). The paper's \"Use of AI tools\" note says: \"Generative AI tools were used to find and prove the results, to write the programs and the Lean formalization, and to draft the text.\" Over the rationals and reals the rank is still only known to lie between 19 and 23.","key_facts":["Paper: 'The rank of 3×3 matrix multiplication over F_2 is 23', arXiv 2610.11823 [math.RA], 8 Oct 2026; Tejasvi Singh Tomar, 'Independent researcher'","Theorem 1.1: R_{F_2}(⟨3,3,3⟩) = 23. Corollary 1.2: 'A bilinear algorithm over F_2 or with integer coefficients that multiplies two 3×3 matrices uses at least 23 multiplications. In particular R_Z(⟨3,3,3⟩) = 23, and no recursive 3×3 algorithm with integer constants has an exponent below log_3 23 ≈ 2.854.'","Upper bound: Laderman's algorithm (1976), 23 multiplications, coefficients 0 and ±1. Paper: 'whether Laderman's 23 multiplications are necessary has been open since 1976'","Lower bounds over F_2 in 2026, as listed in the paper: 19 over all fields (Bläser 2003); 20 (C. Wang, arXiv 2603.07280; certificate re-checked by Beuchert); 21 (Wang, arXiv 2609.06725, and independently Yang et al., arXiv 2609.18722, formalized in Lean); 22 (Rudich and Rousseau, arXiv 2610.01639, in Lean); 23 (this paper)","Method: the substitution method over Wang's 496 symmetry classes of subspaces (bounds raised for 111 of them), a 27×27 flattening, then a computer search listing all candidate sets of 22 first factors and an exact completion search showing none works","Verification: 'The computation emits certificates, which are checked in the Lean 4 proof assistant by checkers whose soundness is proved in Lean. The largest checks are evaluated as compiled code, so the proof relies on the Lean compiler in addition to its kernel.'","AI disclosure: 'Generative AI tools were used to find and prove the results, to write the programs and the Lean formalization, and to draft the text.' No model is named","Code and certificates: github.com/tejasvi/rank333-f2, archived at doi.org/10.5281/zenodo.23236154","Still open: the rank over fields of characteristic zero (between 19 and 23), and whether non-integer coefficients can beat 23"],"key_numbers":[],"tags":["math","computer-science","matrix-multiplication","tensor-rank","laderman","lower-bounds","lean","ai-generated-proof"],"science":{"field":"computer-science","subfield":"algebraic complexity (tensor rank)","problem":"Is Laderman's 1976 algorithm with 23 multiplications optimal for 3×3 matrix multiplication? Exact tensor rank over F_2 and over the integers","result":"Claimed: R_{F_2}(⟨3,3,3⟩) = 23, hence 23 multiplications are necessary for every bilinear algorithm over F_2 or with integer coefficients.","open_since":"1976","ai_system":["unnamed generative AI tools"],"human_role":"AI-generated per the author's disclosure (AI found and proved the results, wrote the programs, the Lean formalization and the draft); the human role is not described","verification":{"key":"lean","label":"Lean-verified","text":"Certificates checked in Lean 4 with verified checkers; the largest checks run as compiled code, so the Lean compiler is trusted as well as the kernel; unrefereed preprint"},"status":"pending","shock":"The lower bound stood at 19 from 2003 until 2026, then rose four times in seven months.","short":"Rank of 3×3 matrix multiplication over F_2"},"body_md":"## What happened\n\nStrassen showed in 1969 that two 2×2 matrices can be multiplied with 7 multiplications instead of 8, and by 1971 it was proved that\n7 is the minimum. For 3×3 matrices the best algorithm, by Julian Laderman (1976), uses 23 multiplications, and the best lower bound\nvalid over every field has been 19 since 2003.\n\nOver the two-element field F_2 the gap closed quickly in 2026. Automated and formally verified proofs gave 20, then 21 (two\nindependent groups in September), then 22 (Rudich and Rousseau, 1 Oct, in Lean). Tomar's preprint claims the last step: no\nalgorithm with 22 multiplications exists over F_2. Because any algorithm with integer coefficients reduces to one over F_2, the\nsame bound holds over the integers.\n\nThe proof is a large case analysis by computer. Lean checks the certificates the search produces. The author is explicit that the\nbiggest checks run as compiled code, which means trusting the Lean compiler and not only its small kernel.\n\n## Why it matters\n\nThe result says that the textbook-style schemes found by hand, by SAT solvers and by systems such as AlphaTensor and AlphaEvolve\ncannot improve on 23 for 3×3 matrices unless they use non-integer coefficients. It is also an example of a research line where\nseveral authors, working with AI tools and proof assistants, pushed a bound that had not moved for 23 years to its exact value\nwithin months.","disputed":[{"label":"Verification","text":"Certificates checked in Lean 4 with verified checkers; the largest checks run as compiled code, so the Lean compiler is trusted as well as the kernel; unrefereed preprint","sources":[]}],"related":[{"id":"2026-10-09-smale-sixth-problem-five-bodies-lean-tomar","url":"https://postcutoff.com/e/2026-10-09-smale-sixth-problem-five-bodies-lean-tomar/","date":"2026-10-09","date_precision":"day","short_title":"Smale's sixth problem claimed for five bodies in the plane","deck":"Finitely many central configurations for all positive masses, AI-found and proved in Lean (Tomar)","takeaway":"An independent researcher claims the planar five-body case of one of Smale's 18 problems for the 21st century, removing the exceptional masses left by Albouy and Kaloshin in 2012. He states that generative AI found and proved the result and wrote the Lean formalization; the paper does not name the models.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":3,"official":3,"filed":"2026-10-12","updated":"2026-10-12","orgs":["independent"]},{"id":"2026-10-07-colkitt-codex-integer-multiplication-exponent","url":"https://postcutoff.com/e/2026-10-07-colkitt-codex-integer-multiplication-exponent/","date":"2026-10-07","date_precision":"day","short_title":"Outside researchers using AI crowdsource a tighter exponent in OpenAI's integer-multiplication result #109, from 2^-182 to about 2^-9.4 (conditional)","deck":null,"takeaway":"Everything is conditional on OpenAI's unverified manuscript; Lean checks cover only finite certificates and arithmetic, and the gains are asymptotic, not practical speedups.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":25,"official":9,"filed":"2026-10-07","updated":"2026-10-12","orgs":["Independent researchers","OpenAI"]},{"id":"2026-09-30-ai-assisted-conjecture-wave-summer-2026","url":"https://postcutoff.com/e/2026-09-30-ai-assisted-conjecture-wave-summer-2026/","date":"2026-09-30","date_precision":"day","short_title":"Summer 2026 flood: dozens of named conjectures settled on arXiv with disclosed AI help","deck":null,"takeaway":"This entry catalogues about 50 of them, with the AI role as the authors state it.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":9,"official":4,"filed":"2026-09-30","updated":"2026-10-12","orgs":["OpenAI","Anthropic","Google DeepMind","various mathematicians"]},{"id":"2026-08-17-alphaevolve-matrix-multiplication-exponent","url":"https://postcutoff.com/e/2026-08-17-alphaevolve-matrix-multiplication-exponent/","date":"2026-08-17","date_precision":"day","short_title":"AlphaEvolve helps lower the matrix multiplication exponent ω to below 2.371177","deck":null,"takeaway":"A paper by Alman, Vassilevska Williams and co-authors including DeepMind researchers (arXiv 2608.16884) improved the bound on the matrix multiplication exponent from ω < 2.371339 to ω < 2.371177.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"confirmed","labels":["Result confirmed"]},"sources":3,"official":2,"filed":"2026-09-29","updated":"2026-09-29","orgs":["Google DeepMind","MIT"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-12","type":"filed","text":"Created from the arXiv PDF (found by a manual arXiv scan)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-12","run":null,"sources_read":"The arXiv PDF (found by a manual arXiv scan)","updated":"2026-10-12","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":161,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":100,"in_training_data":false},{"model_id":"gemini-3-8-flash","name":"Gemini 3.8 Flash","cutoff":"2026-03","days_after":191,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":130,"in_training_data":false}],"short_url":null}