{
  "id": "isomorphisms-of-free-group-factors",
  "title": "Isomorphisms of free group factors",
  "description": "OpenAI's theorem (2026) that the free group factors L(F_n), 2 ≤ n < ∞, are all isomorphic. The first part proves OpenAI's rank step L(F_n) ≅ L(F_{n+1}) for n ≥ 3: free independence and freely generating Haar tuples, a bounded logarithm of the extra generator and a norm bound for free sums that does not depend on the number of summands; trace-preserving polynomial flows with displacement bounds; a small cocycle with a prescribed word value, through free prefixes, Catalan moments and cutoff inversion; and a limit with adaptive error budgets whose tuple generates the whole factor. The second part proves Voiculescu's corner formula, that the corner of L(F_n) cut down by a projection of trace ½ is L(F_{4n−3}), and deduces L(F_2) ≅ L(F_3): the semicircle law, a freeness criterion on the full Fock space, free semicircular families, 2×2 matrices with semicircular and circular entries that are free from matrices over the other variables, and the polar decomposition of a circular element. The third part extends the corner formula to every trace 1/k, with k×k semicircular matrices, and shows that every positive rational number lies in the fundamental group of L(F_2).",
  "units": [
    {
      "id": "free-independence-and-haar-tuples",
      "title": "Free independence and Haar tuples",
      "reader": "free-independence-and-haar-tuples.html",
      "source": "src/free-independence-and-haar-tuples.md",
      "source_sha256": "4099e38120b0137ded99d7c2336b43937ed9026ecdb0c45b3b7985261f03077d",
      "status": "author_self_checked"
    },
    {
      "id": "trace-preserving-polynomial-flows",
      "title": "Trace-preserving polynomial flows",
      "reader": "trace-preserving-polynomial-flows.html",
      "source": "src/trace-preserving-polynomial-flows.md",
      "source_sha256": "3b87eab21d558bd696ce8cbedc255ab73d36897a9b9bccee85bcd9d19f4ddf27",
      "status": "author_self_checked"
    },
    {
      "id": "a-small-cocycle-with-a-prescribed-word-value",
      "title": "A small cocycle with a prescribed word value",
      "reader": "a-small-cocycle-with-a-prescribed-word-value.html",
      "source": "src/a-small-cocycle-with-a-prescribed-word-value.md",
      "source_sha256": "eedd722e7becc49016b30b3fc332ae0a1cf43381f78b9b8086c2f3b0317e7017",
      "status": "author_self_checked"
    },
    {
      "id": "absorbing-a-free-generator",
      "title": "Absorbing a free generator",
      "reader": "absorbing-a-free-generator.html",
      "source": "src/absorbing-a-free-generator.md",
      "source_sha256": "c3dd35369ee92ed362e183eed69dfde30230f90dec4e8bfa274c7204950a2fa7",
      "status": "author_self_checked"
    },
    {
      "id": "semicircular-elements-on-the-full-fock-space",
      "title": "Semicircular elements on the full Fock space",
      "reader": "semicircular-elements-on-the-full-fock-space.html",
      "source": "src/semicircular-elements-on-the-full-fock-space.md",
      "source_sha256": "5548399ba9e58569b3f5b4b302e273f9b8ed35354f4fe6fd39c07ffea93c15cc",
      "status": "author_self_checked"
    },
    {
      "id": "matrices-with-semicircular-and-circular-entries",
      "title": "Matrices with semicircular and circular entries",
      "reader": "matrices-with-semicircular-and-circular-entries.html",
      "source": "src/matrices-with-semicircular-and-circular-entries.md",
      "source_sha256": "4ef257b2ed7d176f426a536c3c01ed6d4e60d299ecc8232184b48d235a8785ab",
      "status": "author_self_checked"
    },
    {
      "id": "polar-decomposition-of-a-circular-element",
      "title": "Polar decomposition of a circular element",
      "reader": "polar-decomposition-of-a-circular-element.html",
      "source": "src/polar-decomposition-of-a-circular-element.md",
      "source_sha256": "7091e5c5b9476ef897ee744384a35046412c61f5ef2f9e733a56307a58c987f1",
      "status": "author_self_checked"
    },
    {
      "id": "corners-of-free-group-factors",
      "title": "Corners of free group factors",
      "reader": "corners-of-free-group-factors.html",
      "source": "src/corners-of-free-group-factors.md",
      "source_sha256": "ecae7bfad7afdb599b8242f2fe239b304d92396b670b43cd898c879a15d4c99a",
      "status": "author_self_checked"
    },
    {
      "id": "corners-of-rational-trace",
      "title": "Corners of rational trace",
      "reader": "corners-of-rational-trace.html",
      "source": "src/corners-of-rational-trace.md",
      "source_sha256": "1b55c7529a8c1a0fff74cddb51210bc4f519254cf8ff90888cb96eca2890051f",
      "status": "author_self_checked"
    }
  ],
  "prerequisites": [
    "generators-of-ii1-factors lesson haar-unitaries-freeness-and-ultraproducts (Definition 2.1 Haar unitaries, Lemma 2.3 equivalence of projections of equal trace, Corollary 2.4 Haar unitaries of corners and projections of every trace)",
    "injective-factors lesson classification-of-injective-factors (Section 5, Theorem 5.2: L(Gamma) is a II_1 factor for ICC Gamma)",
    "foundations-of-von-neumann-algebras lessons the-double-commutant-theorem (Theorem 4.4; Lemma 7.1 and Proposition 7.2, polar decomposition inside a von Neumann algebra), kaplansky-s-density-theorem-and-its-consequences (Theorem 7.1), the-spectral-theorem-for-bounded-self-adjoint-operators (Lemma 1.1, Corollary 1.2, Proposition 2.1, Theorems 3.1 and 8.1), cauchy-s-theorem-for-cycles-and-its-consequences (Lemma 0.1, Banach-valued Riemann integrals)",
    "traces-and-noncommutative-integration lesson integration-for-a-trace-the-commutation-theorem-and-applications (Theorem 9.1, trace-preserving conditional expectation)",
    "function-algebras-and-approximation lesson the-stone-weierstrass-theorem-for-functions-vanishing-at-infinity (Section 10, complex Stone-Weierstrass; Corollary 14.3, Weierstrass approximation)",
    "core course C10 Real Analysis I (Lebl, Basic Analysis I, Theorem 4.3.2, Taylor's theorem)"
  ],
  "status": "author_self_checked",
  "proof_providers": "providers.json",
  "provenance": "provenance.json",
  "licence": "CC0-1.0"
}