{
  "id": "depth-five-circuits-for-iterated-matrix-multiplication",
  "title": "Depth-five circuits for iterated matrix multiplication",
  "description": "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).",
  "units": [
    {
      "id": "iterated-matrix-multiplication-and-depth-five-circuits",
      "title": "Iterated matrix multiplication and depth-five circuits",
      "reader": "iterated-matrix-multiplication-and-depth-five-circuits.html",
      "source": "src/iterated-matrix-multiplication-and-depth-five-circuits.md",
      "source_sha256": "8800c219135647203c0156192e35751809b4641b130167d87b2c35ac4cdf9795",
      "status": "author_self_checked"
    },
    {
      "id": "a-rank-measure-from-differentiation-and-multiplication",
      "title": "A rank measure from differentiation and multiplication",
      "reader": "a-rank-measure-from-differentiation-and-multiplication.html",
      "source": "src/a-rank-measure-from-differentiation-and-multiplication.md",
      "source_sha256": "d08fb618c4726d922eb986ca1baefa4d0b378097ae24480102d47852409b63ec",
      "status": "author_self_checked"
    },
    {
      "id": "the-rank-of-a-depth-five-circuit",
      "title": "The rank of a depth-five circuit",
      "reader": "the-rank-of-a-depth-five-circuit.html",
      "source": "src/the-rank-of-a-depth-five-circuit.md",
      "source_sha256": "79ad7471169c2c839d8bcdadd9fe1e903cb1fa94d4d1f74e2fce6f449b216a92",
      "status": "author_self_checked"
    },
    {
      "id": "trace-moments-and-paths",
      "title": "Trace moments and paths",
      "reader": "trace-moments-and-paths.html",
      "source": "src/trace-moments-and-paths.md",
      "source_sha256": "8211e7b1275eae8ec3b32c7efb2d2a189873d4ce7f40ab0b3376966a011980d8",
      "status": "author_self_checked"
    },
    {
      "id": "the-sum-over-type-words-and-the-lower-bound",
      "title": "The sum over type words and the lower bound",
      "reader": "the-sum-over-type-words-and-the-lower-bound.html",
      "source": "src/the-sum-over-type-words-and-the-lower-bound.md",
      "source_sha256": "5b664fb32c105f59f53971ce3d593515b9896af0dc516a54ded05ae223a150e0",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "core course Linear Algebra (B40): rank, trace, adjoint, orthogonal projection, determinants",
    "polynomials in several variables, binomial coefficients, power series",
    "the lessons prove every other fact they use; there is no named hypothesis"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}