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Math-2773-A1 · revision r1 · on Math-2773

OpenAI claim summary for Math-2773

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 016; 2026-09-24).

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

Manuscript

Contents
  1. Zilber-Pink for abelian varieties over Qbar
  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

Zilber-Pink for abelian varieties over Qbar

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 016; 2026-09-24), the asserted result is that subvarieties of abelian varieties over algebraic numbers have only finitely many maximal atypical intersections, measured inside their smallest containing torsion coset.

Mathematical statement of the claimed result

Manuscript 1.
For a number field $K$, abelian variety $A/K$, and irreducible $X\subseteq A_{\bar K}$, let $S$ be the smallest special subvariety containing $X$; special subvarieties are torsion translates of abelian subvarieties. An irreducible component $Y$ of $X\cap Z$, $Z\subseteq S$ special, is atypical when \[ \dim Y>\dim X+\dim Z-\dim S. \] The claim is finiteness of maximal atypical subvarieties of $X$, equivalently existence of finitely many proper atypical $Y_j\subsetneq X$ containing every atypical subvariety. No dimension bound on $A$ or $X$ is imposed.

Outline of the argument

For manuscript 1 (family 016; 2026-09-24), a counterexample sequence is decomposed into distinct height scales. Uniform tangent positivity and weighted small sections give a height inequality despite growing homomorphism coefficients and point degrees; passing to suitable quotients rules out the remaining scale configurations.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 016; 2026-09-24).

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