Depth-five circuits for iterated matrix multiplication

The (1,1) entry of a product of n matrices of size n x n, and arithmetic circuits with five alternating layers of sums and products in which every gate computes a homogeneous polynomial. The theorem: over a field of characteristic zero such a circuit needs at least n^(sqrt(n)/400) gates for large n, and n^(sqrt(n)+4) gates suffice; with a complete proof. The tools: a rank measure built from partial derivatives and multiplications, and two traces computed by sums over paths. The course is an exposition of OpenAI's proof (2026).

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

1. Iterated matrix multiplication and depth-five circuits
2. A rank measure from differentiation and multiplication
3. The rank of a depth-five circuit
4. Trace moments and paths
5. The sum over type words and the lower bound

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Linear algebra: rank, trace, adjoint, orthogonal projections, determinants (core course Linear Algebra); polynomials in several variables.