Sharpness of best-known rank bounds near the (2,2,2) format

Determine whether the currently best known upper bounds on the tensor rank of matrix multiplication for formats (n,m,p) in the vicinity of (2,2,2) are tight, i.e., equal to the true rank, for most such small-dimension formats.

Background

The paper studies rank bounds for small matrix multiplication formats using searches on a meta flip graph that connects algorithms across different formats via flip, reduction, plus, extension, and projection operations.

Empirically, when starting from the standard algorithm for the format (2,2,2), the authors must traverse several extensions before encountering formats where further rank reductions are possible, suggesting that many nearby small formats may already have sharp (tight) known upper bounds.

This observation motivates an explicit uncertainty about whether the current best-known upper bounds for formats close to (2,2,2) are indeed equal to the true tensor rank for most such cases.

References

Although we do not know for sure, it is fair to believe that for most of the formats (n,m,p) near (2,2,2), the currently best known upper bounds for the rank are actually sharp.

— Exploring the Meta Flip Graph for Matrix Multiplication  (2510.19787 - Kauers et al., 22 Oct 2025) in Section 3 (Using other Starting Points)

To our best knowledge, the first case where the exact border rank is unknown is for $n=2$ and $l = 4$.

— Quantifying the dimensionality of multiparticle entanglement via partition rank  (2608.23399 - Denker et al., 24 Aug 2026) in Appendix D, section “Extensions to tensor rank”