How few scalar multiplications suffice to multiply two 4 × 4 matrices, over a specified field?
Model / AI: AlphaTensor, AlphaEvolve, Claude
Model / AI: AlphaTensor
RL finds faster matrix multiplication algorithms (4x4 over F2 in 47 multiplications). Nature.
Source read 2026-09-08. Read the Nature paper: 47 scalar multiplications for 4 × 4 matrices over F₂. Code was not executed.
Model / AI: AlphaEvolve
4x4 complex matrix multiplication in 48 scalar multiplications, first below Strassen's 49 in characteristic 0 (AlphaTensor had already reached 47 in characteristic 2); Dumas, Pernet and Sedoglavic later hit 48 with rational coefficients, removing the complex arithmetic.
Source review pending. Imported from the original notes; linked claims and artifacts have not been re-audited in this restructuring.
Model / AI: Claude
An independent audit (cert-machine, Carlos Toledo) re-checked both 4 × 4 constructions from commit- and sha256-pinned first-party files, deciding all 4,096 tensor-identity equations of each in exact integer arithmetic. AlphaEvolve's rank-48 decomposition (mathematical_results.ipynb in google-deepmind/alphaevolve_results at commit 4226acb) holds exactly; its coefficients are Gaussian integers divided by 2, and the identity is checked over Z[i] after clearing denominators. AlphaTensor's rank-47 factors (alphatensor_f2.npz) hold exactly over F₂, and the same factors fail over Q, so the 47 is a characteristic-2 result. A standard-library Python script re-checks the certificate and must reject a planted one-coefficient forgery. The audit tooling was built with AI assistance; it checks the published factors and says nothing about optimality.
Source review pending. Source review pending. Submitted by the author of the audit, Carlos Toledo (carlostoledo1891), in issue #2.