Matrix multiplication with exponent at most 9/4

OpenAI's theorem (2026) that the exponent of matrix multiplication is at most 9/4: two n x n complex matrices can be multiplied with O(n^(9/4+eps)) arithmetic operations. Tensors, rank and Strassen's characters; a Fourier separation of blocks sharing one leg and an entropy inequality; a symmetrized profile of polynomial multiplication that is discretely concave and satisfies shifted tripling, which forces lambda(T_d) <= d^(9/4) for every character; characters that are large on matrix multiplication, from a catalytic comparison and Schauder's fixed-point theorem.

Seven lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Tensors, rank and the exponent
2. Characters and degenerations
3. Separating a shared leg
4. Two inequalities for polynomial multiplication
5. Diagonal growth and the bound on characters
6. States and a catalytic comparison
7. Multiplicative states and the proof of Theorem A

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Ultrapowers and Darbo's fixed-point theorem (Tingley's problem)
• Linear algebra over a field: bases, matrices of bilinear maps, rank, row reduction and Vandermonde matrices, as in the core course Linear Algebra.
• Metric spaces, sequences and compactness; Schauder's fixed-point theorem for compact convex subsets of normed spaces, Theorem 5.1 of the lesson Ultrapowers and Darbo's fixed-point theorem of the course Tingley's problem.