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Math-398-A2 · revision r1 · on Math-398

OpenAI claim summary for Math-398

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 018; 2026-09-23).

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

Manuscript

Contents
  1. Margulis–Platonov normal-subgroup conjecture
  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

Margulis–Platonov normal-subgroup conjecture

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 018; 2026-09-23), the asserted result is that the corresponding normal-subgroup description holds over number fields using the anisotropic nonarchimedean local groups.

Mathematical statement of the claimed result

Manuscript 1.
For any number field $k$ and absolutely almost simple simply connected $G/k$, let $A$ be the nonarchimedean places with $\operatorname{rank}_{k_v}G=0$, $H_A=\prod_{v\in A}G(k_v)$ and $\delta_A$ the diagonal map. The claimed Margulis--Platonov description is \[ N\triangleleft G(k),\ N\not\subseteq Z(G(k)) \Rightarrow\exists W\triangleleft H_A\text{ open}:\ N=\delta_A^{-1}(W). \] The cited congruence consequence takes a Galois extension $M/k$, minimal with the property that $G_M$ is an inner form of a split group, together with a set of places $S$ containing all archimedean places but no nonarchimedean anisotropic place. If the existing Dirichlet density satisfies $d_k(S\cap\operatorname{Spl}(M/k))>0$, it asserts $C^S(G)=\{1\}$: every finite-index normal subgroup of $G(\mathcal O(S))$ contains a principal congruence subgroup of nonzero ideal level.

Outline of the argument

For manuscript 1 (family 018; 2026-09-23), commuting approximation with prescribed determinants reduces the remaining unitary cases to finite quotients and local abelianization. Metaplectic splitting, recurrence, and explicit root involutions eliminate residual characters, while finite-group tests rule out the remaining simple quotients.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 018; 2026-09-23).

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