Matrix multiplication exponent

Upper bound, last-listed row

| $2.371177$ | [DEKMRSZAWB2026] | |value changed 2026-08-23

Lower bound, last-listed row

| $2$ | Trivial | Conjectured to be sharp. |tracked since 2026-07-24 — no movement seen yet

Checked or tried against material for its cited source

• Another row, not one shown above: bound row $2.5161$ line 18 citing [P1981] — the cited source could not be read at all (tried 2026-09-16). the source we could not read (NOT read — the publisher returned a redirect page and its API returned metadata only). Selected: drawn by its position in the list of cited rows, not because it looked wrong.
• Another row, not one shown above: bound row $2.371866$ line 29 citing [DWZ2022] — the claim we checked: that the cited paper proves the upper bound 2.371866 and introduces the asymmetric modification of the laser method that the row's comment names. What we read supports it. This verdict does not cover the comment's words 'and subsequent improvements'. source link (read in the abstract of the cited FOCS 2023 edition (IEEE Xplore's own metadata; IEEE served a paywall, OpenAlex reports the published version closed, and CORE search and the Wayback availability query returned metadata only, with no readable copy of the published body saved through either), and in the body of the arXiv preprint arXiv:2210.10173v5, whose comment reads "in FOCS 2023": abstract, introduction, Theorem 3.2 and Section 8.3). Selected: drawn by its position in the list of cited rows, not because it looked wrong.

1 bound row(s) here have been checked against material for their cited source (1 drawn by position, 0 chosen because something already looked wrong). Across this whole ledger 70 row(s) have been drawn by position, against 691 rows that name a source, counted on 2026-09-27: 64 were checked against material for their cited sources and for 6 the cited source could not be read at all. 5 more were chosen for suspicion and checked; they are counted apart, because a set selected for suspicion carries no rate. Also attempted, and the source could NOT be read: the bound row $2.5161$ citing [P1981] line 18. That row is not counted as read. A row not listed here has NOT been checked.

Outside the source table: a claim in OpenAI's math release

• Family 107, “Matrix multiplication with exponent at most 9/4”, claims that the matrix multiplication exponent is below 2.371054886006746 over every field.

  Its Lean statement, MatrixFields.lean:

  theorem omega_lt_source_constant (F : Type u) [Field F] : Arithmetic.omega F < (2371054886006746 : ℝ) / 10 ^ 15 ∧ (2371054886006746 : ℝ) / 10 ^ 15 < (2371056 : ℝ) / 10 ^ 6

  Same quantity as this constant: the exponent of the arithmetic cost of multiplying two n × n matrices over a field. The file counts additions, subtractions and multiplications in programs with no division, a narrower model than general arithmetic, so a bound in it is also a bound here. 2.371054886… is below this constant's last-listed upper-bound row.

  Status: a Lean statement of it is published with a proof by its authors. We read the statement and it matches this constant's definition; we have not compiled the proof, and we have not read the paper.

• Family 107, “Matrix multiplication with exponent at most 9/4”, claims that the matrix multiplication exponent over the complex numbers is at most 9/4 = 2.25.

  Its Lean statement, MatrixMultiplication.lean:

  theorem complex_omega_le_nine_quarters : Arithmetic.omega ℂ ≤ (9 : ℝ) / 4

  Same quantity, over the complex numbers, in the same model with no division (defined in the file).

  Status: a Lean statement of it is published with a proof by its authors. We read the statement and it matches this constant's definition; we have not compiled the proof, and we have not read the paper.

A claim in a preprint release, read at openai/math@fd4aeeb on 2026-10-12. It is not in the source table, and this ledger has not changed the rows above because of it.