The matrix multiplication exponent 
over a field 
is the infimum of the exponents 
such that, for every 
, two 
matrices over 
can be multiplied with 
arithmetic operations. The constant
in the big-O notation may depend on 
but not on 
. The operations counted are addition,
subtraction, and multiplication
of elements of the field; their bit
complexity is a different question.
The usual matrix multiplication algorithm uses 
operations. Recursive use of Strassen's formulas
gives
 |
The lower limit 2 comes from the 
entries of a general product, and the conjecture
that 
remains open. The exponent describes asymptotic growth
and does not by itself determine which algorithm is
fastest at a given finite size (Strassen 1969).
OpenAI (2026a) announced
 |
Equivalently, the claim gives 
arithmetic operations for every 
over the complex
numbers. It does not claim this bound over every field,
a practical crossover size, or an improvement in bit
complexity. The release attributes the argument to an unreleased internal OpenAI
model and supplies a Lean formalization of the bound. As of Oct. 7, 2026, VibeMathed
(2026) had read the formal statement but had not independently rebuilt it, and independent
specialist verification of the complete proof had not been
reported.
A separate announcement by OpenAI (2026c) gives 
over every fixed field 
,
including every positive characteristic. This
bound has broader field scope than the sharper complex-number
bound above. Both statements count arithmetic operations. The release proofs
have not been independently verified here.
OpenAI (2026b) separately announced 
over every field of
characteristic zero and over all positive characteristics
outside one finite exceptional set.
In characteristic zero, the same manuscript claims
that multiplying an 
matrix by an 
matrix requires 
arithmetic operations, and that the corresponding
exponent for 
by 
multiplication is less than 2.092. The latter
rectangular bound also has the finite-exception transfer
to positive characteristic; the 0.465 claim does
not assert that transfer.
for the Matrix Multiplication Exponent." Oct. 2,
2026a. 
over the Complex Numbers." Oct. 7, 2026.