{
  "id": "affine-maximal-graph-in-dimension-ten",
  "title": "A smooth affine maximal graph in dimension ten",
  "description": "OpenAI's smooth solution (2026) of the affine maximal equation on R^10 with positive definite Hessian that is not a quadratic polynomial. The equation as the Euler–Lagrange equation of the affine area; graphs sqrt(F^2 + t^2) and the reduction to two equations for F and an auxiliary function P; a radial solution through the origin by the contraction mapping principle; and an invariant box for three logarithmic derivatives, whose corner comes from the singular example sqrt(|x|^9 + t^2) of Trudinger and Wang, giving the solution for all radii.",
  "units": [
    {
      "id": "the-affine-maximal-equation",
      "title": "The affine maximal equation",
      "reader": "the-affine-maximal-equation.html",
      "source": "src/the-affine-maximal-equation.md",
      "source_sha256": "66d2814899c02c5b6c53c6a2e3d7605f8fe642a337de62f2b271b0e0677b7da5",
      "status": "author_self_checked"
    },
    {
      "id": "graphs-over-a-convex-profile",
      "title": "Graphs over a convex profile",
      "reader": "graphs-over-a-convex-profile.html",
      "source": "src/graphs-over-a-convex-profile.md",
      "source_sha256": "2f99787b3309f5b92d305deddb06163470d70148df34aeac7c72c6304ec25b18",
      "status": "author_self_checked"
    },
    {
      "id": "a-radial-solution-through-the-origin",
      "title": "A radial solution through the origin",
      "reader": "a-radial-solution-through-the-origin.html",
      "source": "src/a-radial-solution-through-the-origin.md",
      "source_sha256": "d5cc3b97195b9af27cf04be44d2e0ae2e647c4c7b6875937b3b08a1948d808ca",
      "status": "author_self_checked"
    },
    {
      "id": "an-invariant-box-and-the-entire-graph",
      "title": "An invariant box and the entire graph",
      "reader": "an-invariant-box-and-the-entire-graph.html",
      "source": "src/an-invariant-box-and-the-entire-graph.md",
      "source_sha256": "68d7b840e9fc7d985f9f7ef9ae2ff891e9984a7c81b77bdc194011efd17869fe",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "real analysis in one and several variables: core courses Real Analysis I and II (Lebl, Basic Analysis I and II, version 6.3): the mean value theorem (Theorem 4.2.4), the fundamental theorem of calculus (Theorems 5.3.1 and 5.3.3), the change of variables (Theorem 5.3.5), completeness of C([a,b]) and of closed subsets (Propositions 7.4.5 and 7.4.6), extreme values and uniform continuity on compact sets (Theorems 7.5.6 and 7.5.11), continuity of parameter integrals (Proposition 7.5.12), the contraction mapping principle (Theorem 7.6.2), differentiation under the integral sign (Theorem 9.1.1), integrals of continuous functions over rectangles (Theorem 10.1.15) and Fubini's theorem (Theorem 10.2.2)",
    "linear algebra: determinants, the Leibniz formula, symmetric matrices and their eigenvalues"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}