{"schema":"postcutoff/event@1","as_of":"2026-10-12T07:22:00+02:00","url":"https://postcutoff.com/e/2026-10-06-explicit-tensors-border-rank-3n-gpt-6-astra/","md":"https://postcutoff.com/e/2026-10-06-explicit-tensors-border-rank-3n-gpt-6-astra/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":"2026-10-06-explicit-tensors-border-rank-3n-gpt-6-astra","date":"2026-10-06","date_precision":"day","short_title":"Explicit tensors of border rank 3n − o(n), past the 'linear flattening barrier'","deck":"Proofs obtained with GPT-6 Astra (Benjamin Lovitz)","takeaway":"The record lower bound for the border rank of an explicit tensor moves from about 2n to about 3n, and the standard method is shown to be unable to pass 2n − 1. The author says GPT-6 Astra was used to obtain the proofs of the main results.","category":"science","category_label":"Science & math","importance":3,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":[{"n":1,"title":"arXiv 2610.08504: Explicit tensors beyond the linear flattening barrier","url":"https://arxiv.org/abs/2610.08504","type":"paper","group":"primary","domain":"arxiv.org"}],"official":1,"filed":"2026-10-12","updated":"2026-10-12","orgs":["Concordia University","OpenAI"],"title":"Explicit tensors of border rank 3n − o(n), past the 'linear flattening barrier'; proofs obtained with GPT-6 Astra (Benjamin Lovitz)","summary":"On 6 Oct 2026 Benjamin Lovitz (Concordia University) posted arXiv 2610.08504. It constructs explicit n × n × n tensors of border rank at least 3n − o(n), improving the previous record of (2 + ε)n by Landsberg and Michałek, and proves that linear flattenings, the standard lower-bound method, cannot show more than 2n − 1 (the earlier barrier was 6n − 4). The paper's \"Use of artificial intelligence\" section says the author \"used GPT-6 Astra interactively throughout every stage of preparing this manuscript, including to obtain proofs of the main results, which the author subsequently verified and simplified\". Unrefereed preprint.","key_facts":["Paper: 'Explicit tensors beyond the linear flattening barrier', Benjamin Lovitz (Concordia University, Montreal), arXiv 2610.08504 [cs.DS], 19 pages, 6 Oct 2026","Theorem 1.1: 'There is a polynomial-time explicit tensor T_n ∈ (C^n)^⊗3' with border rank at least 3n − 11n^(3/4). Previous record: (2 + ε)n for any 0 < ε < 1/42 and large n (Landsberg–Michałek)","Barrier (Theorem 1.2, per the abstract): 'the linear flattening method cannot be used to establish lower bounds on border rank beyond 2n − 1', improving the 6n − 4 barrier of Garg et al. and Buczyński; for odd n Koszul flattenings reach 2n − 1, for even n they reach 2n − 2, 'leaving an open gap of size one'","Context (paper): the maximum border rank of an n × n × n tensor is about n²/3, so explicit lower bounds remain linear; border rank of the matrix-multiplication tensor governs the exponent ω","AI disclosure: 'The author used GPT-6 Astra interactively throughout every stage of preparing this manuscript, including to obtain proofs of the main results, which the author subsequently verified and simplified. The author takes full responsibility for the correctness, exposition, and attribution in this manuscript.' The author thanks Harm Derksen for feedback on a draft","Not found as of 12 Oct 2026: expert reactions or press coverage"],"key_numbers":[],"tags":["computer-science","math","algebraic-complexity","tensor-rank","border-rank","matrix-multiplication","gpt-6-astra"],"science":{"field":"computer-science","subfield":"algebraic complexity theory","problem":"Explicit n × n × n tensors of high border rank; limits of the flattening method","result":"Explicit tensors with border rank ≥ 3n − 11n^(3/4) (record was (2 + ε)n); linear flattenings cannot prove more than 2n − 1.","open_since":"","ai_system":["GPT-6 Astra"],"human_role":"AI-assisted: the author used GPT-6 Astra interactively to obtain proofs of the main results, then verified and simplified them","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed preprint; proofs verified by the author"},"status":"pending","shock":"","short":"Explicit tensors of border rank 3n − o(n)"},"body_md":"## What happened\n\nBorder rank measures how hard a tensor is to compute approximately, and for the matrix-multiplication tensor it decides how fast\nmatrices can be multiplied. Proving that a specific, explicitly written tensor has high border rank is a standard hard problem of\ncomplexity theory. The best bounds sat near 2n, obtained with \"flattenings\" (turning the tensor into a matrix and\nmeasuring its rank).\n\nLovitz's paper does two things. It proves that the linear flattening method cannot go beyond 2n − 1, a much lower ceiling than\nthe 6n − 4 known before. And it builds explicit tensors with border rank about 3n by another route, so these tensors are provably\nout of reach of flattenings. The author states that the proofs of the main results were obtained interactively with GPT-6 Astra\nand then checked and simplified by him.\n\n## Why it matters\n\nIt moves a record in a core area of algebraic complexity and explains why the old method stalled.\nIt is another example of a specialist using a frontier model as the main source of proofs.","disputed":[{"label":"Verification","text":"Unrefereed preprint; proofs verified by the author","sources":[]}],"related":[{"id":"2026-10-09-arxiv-ai-disclosed-results-early-october-2026","url":"https://postcutoff.com/e/2026-10-09-arxiv-ai-disclosed-results-early-october-2026/","date":"2026-10-09","date_precision":"day","short_title":"arXiv, 5–9 October 2026","deck":"More conjectures settled with disclosed AI help (Quillen's local parameter conjecture in dimension 4, Laughlin gap, poset reconstruction), next to human-only results","takeaway":"The catalogue shows what the everyday use of frontier models in research mathematics looked like in October 2026: routine, disclosed in a sentence or two, and uneven in reliability.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":14,"official":14,"filed":"2026-10-12","updated":"2026-10-12","orgs":["OpenAI","Anthropic","various mathematicians"]},{"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 above 2^-14 (conditional)","deck":null,"takeaway":"In a parallel repository Swapnil Jain (@SJ_Swapnil_Jain, with Claude) posted seven rounds ending at κ ≈ 6.40×10^-5 (> 2^-14) on the evening of 8 Oct, the highest claimed value.","category":"science","category_label":"Science & math","importance":4,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":20,"official":6,"filed":"2026-10-07","updated":"2026-10-08","orgs":["Independent researchers","OpenAI"]},{"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"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-12","run":null,"sources_read":"The arXiv PDF","updated":"2026-10-12","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":159,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":98,"in_training_data":false},{"model_id":"gemini-3-8-flash","name":"Gemini 3.8 Flash","cutoff":"2026-03","days_after":189,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":128,"in_training_data":false}],"short_url":null}