---
id: "2026-10-06-explicit-tensors-border-rank-3n-gpt-6-astra"
url: "https://postcutoff.com/e/2026-10-06-explicit-tensors-border-rank-3n-gpt-6-astra/"
as_of: "2026-10-12T07:22:00+02:00"
date: "2026-10-06"
date_precision: day
category: science
importance: 3
confidence: high
status: [Event confirmed, Awaiting review]
verification: Preprint only
sources: 1
editor: Adam Bicz
human_review: null
version: null
---

As of: 2026-10-12 07:22 CEST.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code). Human editor: Adam Bicz.
Canonical page: https://postcutoff.com/e/2026-10-06-explicit-tensors-border-rank-3n-gpt-6-astra/

# Explicit tensors of border rank 3n − o(n), past the 'linear flattening barrier'

Full title: Explicit tensors of border rank 3n − o(n), past the 'linear flattening barrier'; proofs obtained with GPT-6 Astra (Benjamin Lovitz)

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

## 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

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 159 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 98 days after its cutoff
- Gemini 3.8 Flash (training cutoff March 2026): 189 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 128 days after its cutoff

## Sources

1. [arXiv 2610.08504: Explicit tensors beyond the linear flattening barrier](https://arxiv.org/abs/2610.08504) (arxiv.org, paper)

## Changes

- 2026-10-12 (filed): Created from the arXiv PDF

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