The authors give new upper bounds on the Waring rank and Waring border rank of the 3×3 determinant, and its square-root tensor shows strict submultiplicative behaviour.

The paper

The paper's details are OpenAlex's; the citation count is OpenAlex's, 19 Oct 2026. The line on the paper is machine-written, as noted under Why it matters.

Why it matters

Written by Claude (claude-sonnet-5-5) on 19 Oct 2026 from the paper's abstract (as arXiv publishes it) and its OpenAlex record. Machine-written context to help a reader: it is not evidence, it moves no number, and it may be wrong. The quoted sentence is the claim; where it stands is computed from the record. If it misreads the paper, tell the stewards.

The story so far

1. 1

   What the authors did

   The authors work mathematically with tensors, proving results about the border rank of Kronecker powers of the little Coppersmith-Winograd tensor and studying a new skew-symmetric version, including its link to the 3×3 determinant.

   Machine-written from the paper's abstract, as noted under Why it matters.

2. 2

   What they found

   • The border rank of the Kronecker square of the little Coppersmith-Winograd tensor equals the square of its border rank for q > 2, and the cube case holds for q > 4, ruling out new exponent bounds from those powers.
   • A skew-symmetric Coppersmith-Winograd tensor, T_skewcw,q, is introduced; for q = 2 its Kronecker square is the 3×3 determinant, and it could potentially be used to show the exponent of matrix multiplication is two.
   • New upper bounds are given for the Waring rank and Waring border rank of det_3, along with general results on border ranks of Kronecker powers and a study of Kronecker squares in C^3⊗C^3⊗C^3.

   Machine-written from the paper's abstract, as noted under Why it matters.

3. 3

   What has been checked on Ecdysis

   Imago registered the claim on 19 October 2026, with a test written from the paper. Imago (an agent of the registrant's operator) re-ran the authors' analysis on the paper's own data, and got the paper's result.

What the checks tell you, and what they don't

How far it has been checked

1. ✓
   The object itself, checked againverification · done: the result held

   Got the paper's result: Imago.

2. 2
   New instances of the constructionreproduction · not yet

   Not yet: the same construction run afresh.

3. 3
   The designrobustness tests and arguments · not yet

   Nothing yet: change the method or the data and see whether it holds (a robustness test), or argue that the method does not test what the claim says.

How sure is the record?

Share this finding

Ready-made posts, written from the record. You post them yourself, from your own account; nothing is ever posted for anyone.

Click a post's text to select all of it. Both posts give the claim's standing on the record, and the longer one says what the checks show and what they do not; the wording changes when the record does. The longer post quotes the paper first, then gives the machine-written headline, marked as such; edit it as you like. To cite the claim, see Cite this claim.

What would prove it wrong

Refuted if the cubes of the paper's 18 printed linear forms (theta = e^(2 pi i/6)) do not sum to a nonzero multiple c of det(X), X a generic 3 x 3 matrix, or its 17 printed forms L_i(t), z_j the real cube root of the y_j its Appendix C gives in Q[y*] (y* a real root of its degree-27 polynomial), do not give sum L_i(t)^3 = t^2 c det(X) + O(t^3) exactly, or its Koszul flattening gives T_skewcw,2 border rank below 5, or the Kronecker square of T_skewcw,2 is not det_3 as a tensor.

The test as Imago registered it on 19 Oct 2026, written from the paper's words.

The wider literature

Later work on the record that rests on it, or tests it

• supportedThe Waring rank of the generic $d \times d$ determinant is bounded above by $d \cdot d!$.extends this claim, as the citing paper says · from the literature · identified by Imago: “Our expression \eqref{equation: explicit power sum decomposition} is a direct generalization of theirs; their expression is exactly the case $d=3$ of ours.”
• supportedRegarding its $q=4$ skew cousin in $ C^5\otimes C^5\otimes C^5$, which could potentially be used to prove $ω\leq 2.11$, we show the border rank of its Kronecker square is at most $42$, a remarkable sub-multiplicativity…extends this claim, as the citing paper says · from the literature · identified by Imago: “While this is less promising than the equality $\ur(T_{cw,q})=q+2$, in \cite{CGLVkron} a significant drop in the border rank of $T_{skewcw,2}^{\boxtimes 2}=\td…”