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Math-2783-A1 · revision r1 · on Math-2783

OpenAI claim summary for Math-2783

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 006; 2026-09-23).

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

Manuscript

Contents
  1. Goldfeld analytic rank density over Q
  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

Goldfeld analytic rank density over Q

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 006; 2026-09-23), the asserted result is that among signed squarefree twists of each rational elliptic curve, analytic ranks zero and one each occur with density one half; the paper also claims the low-corank two-Selmer converse.

Mathematical statement of the claimed result

Manuscript 1.
For $E/\mathbb Q$, write $a(E)=\operatorname{ord}_{s=1}L(E,s)$, $r(E)=\operatorname{rank}E(\mathbb Q)$ and $c_2(E)=\operatorname{corank}_{\mathbb Z_2}\operatorname{Sel}_{2^{\infty}}(E/\mathbb Q)$. The first claimed theorem is \[ c_2(E)\in\{0,1\}\Rightarrow a(E)=r(E)=c_2(E),\quad\#\operatorname{Sha}(E/\mathbb Q)<\infty. \] Let $\mathcal D(X)$ be the nonzero signed squarefree integers with $|d|\le X$ and $E^{(d)}$ the quadratic twist. The analytic Goldfeld assertion is, for each $j=0,1$, \[ \lim_{X\to\infty}\frac{|\{d\in\mathcal D(X):a(E^{(d)})=j\}|}{|\mathcal D(X)|}=\tfrac12. \] It further claims that a density-one set of twists has equal analytic and Mordell--Weil rank and finite whole Tate--Shafarevich group. This density statement does not assert those properties for every twist.

Outline of the argument

For manuscript 1 (family 006; 2026-09-23), finite cubes of twists preserve the required local classes. A central determinant or Heegner coordinate is nonzero at the noncentral cube vertices, and bounded integral interpolation recovers the missing vertex; the Selmer distribution then supplies the density assertion.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 006; 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