An arXiv preprint dated 27 August 2026 reports tensor constructions with unusually large centroids and lower reported upper bounds for the matrix-multiplication exponent. The work studies finite-dimensional vector spaces over a field F and tensor spaces formed from them, so its evidence comes from algebraic constructions and calculations rather than an empirical sample.
An exact count, and a general ceiling
In this setting, a centroid is the algebraic object associated with a tensor, and its dimension is the quantity being counted. For the specially constructed big-centroid family, the dimension is exactly the product of the dimensions of the constituent spaces, plus one. The result is an exact theorem for that family, not a numerical estimate.
That exact count sits beside a general ceiling. For a concise tensor of valence d, the centroid dimension is at most the (d minus 1)st root of the product of the local dimensions. The paper thus gives an exact value for one family and a separate upper bound for the broader class of concise tensors.
Rank results come with conditions
The rank results come with field and dimension conditions. When the ground field F is not F2 and all the constituent spaces have the same dimension n, the big-centroid tensor has minimal border rank equal to n raised to the power d minus 1, plus n. That theorem is conditional on those assumptions, so it does not assert the same result for every field or for unequal local dimensions.
The symmetric construction adds a different set of qualifications. The paper reports centroid overabundance when n is at least 3. It gives minimal border rank when either the product of n and n minus 3 is nonzero in F, or n equals 3 and F contains a primitive third root of 1. A separate theorem reports that the symmetric valence-three big-centroid tensor has minimal border rank and, more strongly, minimal Waring border rank for all n. It also reports that these symmetric tensors are wild over C for all n, using a technical term from its algebraic framework.
The laser-method comparison
A separate construction, the Schönhage unrestriction, is reported to have minimal border rank equal to u times v plus 1. Alongside that result, the paper reports laser-method calculations for Strassen's and Schönhage's tensors. Here omega is the matrix-multiplication exponent being bounded. In the Strassen comparison at q=4, the original tensor has a reported upper bound below 2.48289, while the unrestriction is below 2.46016. At q=5, the corresponding figures are below 2.47849 and below 2.46710.
The Schönhage figures point in the same direction, although the parameter settings differ. With both original parameters equal to 4, the reported upper bound is below 2.548. With both unrestriction parameters equal to 3, it is below 2.522. These are theoretical upper bounds from the stated laser calculation, not runtime measurements. The preprint does not report an implemented matrix-multiplication speedup.
What the paper leaves open
The centroid-dimension proof uses Gromov's Tensorial Reduction Inequality. For border-rank decompositions, the paper uses extended-centroid analysis within border apolarity to guide upper-bound constructions. The resulting theorem-level claims carry no statistical uncertainty.
The preprint leaves several questions open. It asks whether big-centroid tensors are always of minimal border rank beyond the cases proved, whether the symmetric construction has minimal border rank for every valence d, and whether Kronecker-power analysis or more sophisticated laser-method tools would produce further improvements. For now, the paper offers explicit tensor families, rank theorems under stated assumptions and method-specific theoretical bounds. It does not establish minimal border rank for every big-centroid tensor or a practical performance gain.
Paper data and sources
Original title: Disjoint and nearly disjoint sums of matrix multiplication tensors and their centroids
Authors: Martin Kassabov, J. M. Landsberg, Victor Souza, Philip Speegle
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-27
DOI: Not available
Original paper · Full text