What was already known
Human bounds were near 2.371, and an earlier OpenAI companion reported ω(C)<2.258. Later work by Navot and others extends the 9/4 bound to other fields and rings.
AI involvement
OpenAI credits its internal model with the complex-field construction. Houxi Lü later supplied a separately credited human explanation and improved separation; Navot and others developed distinct extensions.
The internally produced OpenAI-authored paper supplies the 9/4 construction and proof.
Importance
The complex arithmetic bound improves a central asymptotic benchmark for square matrix multiplication.
Limitations
This is an asymptotic arithmetic-operation bound, not a bit-complexity or practical finite-size speed claim. Lü later explained and replaced the shared-leg separation step; his extension does not document a check of every step of the original Fourier argument.
Technical verification
The OpenAI Lean library states the complex division-free arithmetic cost bound for every positive ε and ω(C)≤9/4. Its family 107 scope and comparator document the machine-checked theorem.
Formal checking
Lean proof of the complex arithmetic cost bound and exponent inequality
- Checker
- OpenAI Lean 4 library and comparator
- Scope
- Complex division-free arithmetic matrix multiplication has cost O_epsilon(n^{9/4+epsilon}) and omega(C)<=9/4
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Lü's dated human paper identifies OpenAI's shared-leg separation as the key step, gives an explicit replacement, and works through how the spectral and discrete-growth stages yield the exponent; this explains the central mechanism even though it does not verify each original Fourier step.
Correction and publication history
The complex arithmetic bound improves a central asymptotic benchmark for square matrix multiplication.
Read this published revision