OpenAI claim summary for Math-2502

Rank Distribution in Quadratic-Twist Families

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; In manuscript 2 (family 006; 2026-09-23), the asserted result is that the mean analytic rank of signed squarefree quadratic twists tends to $1/2$ for each rational elliptic curve.

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.

Manuscript 2.

Let $E/\mathbb Q$ be any elliptic curve, let $E^{(d)}$ denote its quadratic twist, and put $a(E^{(d)})=\operatorname{ord}_{s=1}L(E^{(d)},s)$. For the signed squarefree twisting parameters $\mathcal D(Y)=\{d\in\mathbb Z\setminus\{0\}:d\text{ squarefree},\ |d|\le Y\}$, the claimed limit is \[ \lim_{Y\to\infty}\frac1{|\mathcal D(Y)|} \sum_{d\in\mathcal D(Y)}a(E^{(d)})=\frac12. \] This is an ordinary average over both signs for each fixed curve, without a restriction on its conductor or rational torsion.

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.

For manuscript 2 (family 006; 2026-09-23), block mollifiers compare twist families with a finite independent model. Poisson summation and an auxiliary square multiplier send dual twists to smaller height, allowing induction on logarithmic second moments and control of the contribution from large ranks.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 006; 2026-09-23), manuscript 2 (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

• OpenAI, Goldfeld's analytic density conjecture and the 2-converse for elliptic curves, 2026-09-23; citation key OAI:Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main-converse.
• OpenAI, The mean analytic rank of quadratic twists of elliptic curves, 2026-09-23; citation key OAI:The-mean-analytic-rank-of-quadratic-twists-of-elliptic-curves-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:mean.
• OpenAI catalogue entry.