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Math-452-A2 · revision r1 · on Math-452

OpenAI claim summary for Math-452

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: 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.

The result supplies the conclusion in this problem with its displayed hypotheses and quantifiers. The broader statements and applications in the manuscript are kept separate from the particular question above. The supporting result is manuscript 1 (family 008; 2026-09-23).

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

Manuscript

Contents
  1. Deligne–Drinfeld Freeness Conjecture for grt_1
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Deligne–Drinfeld Freeness Conjecture for grt_1

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 008; 2026-09-23), the asserted result is that the rational Lie algebra of Grothendieck--Teichmüller solutions has a free presentation with generators indexed by odd weights $\ge3$, and the corresponding completed presentation.

Mathematical statement of the claimed result

Manuscript 1.
Let $L=\operatorname{Lie}_{\mathbb Q}\langle x,y\rangle$, graded by word length. Let $W$ be its solutions of \[ \psi(x,y)+\psi(y,x)=0,\qquad \psi(x,y)+\psi(y,-x-y)+\psi(-x-y,x)=0, \] and the pentagon \[ \psi(t_{12},t_{23}+t_{24})+\psi(t_{13}+t_{23},t_{34}) =\psi(t_{23},t_{34})+\psi(t_{12}+t_{13},t_{24}+t_{34})+\psi(t_{12},t_{23}) \] in the infinitesimal pure braid Lie algebra, defined by $t_{ij}=t_{ji}$, $[t_{ij},t_{k\ell}]=0$ on distinct indices and $[t_{ij},t_{ik}+t_{jk}]=0$. Define $D_{\psi}(x)=0$, $D_{\psi}(y)=[y,\psi]$ and Ihara bracket $\{\psi,\phi\}=D_{\psi}\phi-D_{\phi}\psi+[\psi,\phi]$. The claim gives homogeneous $\sigma_{2k+1}\in W_{2k+1}$ for every $k\ge1$ with \[ \operatorname{Lie}_{\mathbb Q}\langle e_3,e_5,e_7,\ldots\rangle \xrightarrow{\sim}(W,\{\ ,\ \}),\qquad e_{2k+1}\mapsto\sigma_{2k+1}. \] It asserts generation as well as freeness, bracket closure and an induced continuous isomorphism of weight completions.

Outline of the argument

For manuscript 1 (family 008; 2026-09-23), a degeneration of the pentagon relation and deletion operators bound the leading-depth image. Regularized braid transport supplies nonzero odd-weight elements, and a weight-by-weight dimension comparison establishes both independence and spanning.

Formal verification

For manuscript 1 (family 008; 2026-09-23), OpenAI lists Lean statements for this assertion: the rational Lie algebra of Grothendieck--Teichmüller solutions has a free presentation with generators indexed by odd weights $\ge3$, and the corresponding completed presentation.

The central assertion is within the documented formal scope; the additional applications retain their own scope limits.

The linked family Comparator checks are DeligneDrinfeld.lean.

Their named declarations include OAI.DeligneDrinfeld.main.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources