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Math-104-A2 · revision r1 · on Math-104

OpenAI claim summary for Math-104

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

Manuscript 1 (family 287, 2026-09-23): The governing main theorem gives a trace-preserving star-isomorphism of L(F2) and L(F3), exactly settling the rank-two/rank-three question; the stated classic dichotomy supplies all interpolated ranks as well.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Isomorphism of Free Group Factors
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Isomorphism of Free Group Factors

OpenAI's claim

The manuscript claims that the factors L$F\_2$ and L$F\_3$ are isomorphic, consequently identifying all interpolated free group factors and giving each the full positive-real fundamental group. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Writing $\mathbb F_n$ for the free group of rank $n$ and $L(\mathbb F_n)$ for its group von Neumann factor with canonical trace, OpenAI claims \[ L(\mathbb F_2)\xrightarrow{\sim}L(\mathbb F_3) \] by a unital normal trace-preserving $*$-isomorphism. Its rank-step construction also asserts $L(\mathbb F_n)\simeq L(\mathbb F_{n+1})$ for every $n\ge3$. These are actual factor isomorphisms, not only embeddings or stable equivalences.

Outline of the argument

Manuscript 1 (family 287, 2026-09-23): Small trace-preserving automorphisms are built using finitely supported cocycles on free words. A Catalan-moment computation and truncated inverses provide coefficients that absorb an additional Haar generator with controlled error. Iterating with carefully allocated errors preserves generation as well as the free distribution; established amplification formulas transfer the consecutive-rank construction to ranks two and three.

Formal verification

For family 287, the scope page documents the following selections. The selected normal trace-preserving isomorphisms cover all interpolated free-factor parameters greater than one, including infinity. The fundamental-group consequence is separate. The documented statement covers the main claim of manuscript 1. Its selected declaration also covers the clause used for this problem. Comparator files: InterpolatedFactors.lean. Declarations: OAI.FreeGroupFactorMain.Interpolation.allInterpolatedIsomorphic.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources