Laderman multiplied two 3x3 matrices in 23 multiplications in 1976 and nobody has done better since. The lower bound sat at 19 from 2003 until March 2026, when it moved to 20 over F2 — in a preprint whose proof is a machine-checkable certificate, on a repository with two stars. This page re-verifies that certificate here, in exact arithmetic, with an instrument that could have contradicted it.
| wall | value | who, when | status here |
|---|---|---|---|
| upper bound (any field) | 23 | Laderman 1976 | a rank-23 witness is pinned in certs/strassen-certificate.json and re-certified at every build |
| lower bound (F2) | 20 | Wang, arXiv:2603.07280, March 2026 | 4 of 496 certificate nodes proved two-sided here; 0 refuted |
| lower bound (any field) | 19 | Blaeser 2003 | not audited here — the 20 is an F2 statement and does not supersede it |
| border rank (over C) | ≥ 17 | Conner–Harper–Landsberg | a different quantity; never quote it as a rank bound |
The gap is real and old: nobody has moved 23 in fifty years, and until this March nobody had moved 19 in twenty-three. A rank-22 algorithm, incidentally, would not even beat Strassen asymptotically — log₃22 > log₂7 — which is why the exhaustive-search literature deliberately stops at 21.
| certificate | nodes | witnesses re-verified | proved two-sided | attacked | refuted |
|---|---|---|---|---|---|
| <2,2,2> control (rank 7 known optimal) | 10 | 10 / 10 | 3 | 7 | 0 |
| <3,3,3> — the new bound | 496 | — | 4 | 18 | 0 |
The control matters more than it looks. Its final node is the unconstrained <2,2,2> tensor with a claimed bound of 7 — a number Winograd proved optimal in 1971. Had our search found a rank-6 decomposition there, the search would have been broken, not the mathematics. It did not.
The <3,3,3> certificate carries no upper-bound witnesses at all, so there is nothing to cross-check against: it is a pure lower-bound argument across 496 nodes resting on four inference rules. That is exactly why computing ground truth independently is the right audit and reimplementing his rules is the wrong one.
No new mathematics was discovered here. We found no refutation, no better bound and no counterexample; 478 of 496 nodes were never attacked and still depend entirely on the author's reasoning. What this page claims is narrower and, we think, still worth publishing: the first independent check of any part of a new lower bound on a fifty-year-old open problem, by a different implementation, in a different language, from the author's published bytes. Published, not peer-reviewed, not independently rerun — and that last phrase is precisely what this page exists to stop being true of someone else's work.