Explicit tensors of border rank 3n − o(n), past the ‘linear flattening barrier’
Proofs obtained with GPT-6 Astra (Benjamin Lovitz)
Event confirmedAwaiting review
The 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.
Status
- Claim
Event confirmedAwaiting review
- Our reporting
- High confidence
- Verification
- Preprint only
- Importance
- 3 of 5
- Last verified
- 12 October 2026
Your AI and this story
- GPT-6 Astra159 days after its cutoff
- Claude Opus 5.598 days after its cutoff
- Gemini 3.8 Flash189 days after its cutoff
- Grok 4.7128 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 98 days before it.
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
What happened
Border rank measures how hard a tensor is to compute approximately, and for the matrix-multiplication tensor it decides how fast matrices can be multiplied. Proving that a specific, explicitly written tensor has high border rank is a standard hard problem of complexity theory. The best bounds sat near 2n, obtained with “flattenings” (turning the tensor into a matrix and measuring its rank).
Lovitz’s paper does two things. It proves that the linear flattening method cannot go beyond 2n − 1, a much lower ceiling than the 6n − 4 known before. And it builds explicit tensors with border rank about 3n by another route, so these tensors are provably out of reach of flattenings. The author states that the proofs of the main results were obtained interactively with GPT-6 Astra and then checked and simplified by him.
Why it matters
It moves a record in a core area of algebraic complexity and explains why the old method stalled. It is another example of a specialist using a frontier model as the main source of proofs.
What is disputed or not yet verified
| Verification | Unrefereed preprint; proofs verified by the author |
|---|
Sources
1 source from 1 site. Numbers match the chips in the text.
1 source: 1 primary
Primary
Changes
- Filed from the arXiv PDF