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The full BSD formula from low Selmer corank

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

Proves the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full q-power Selmer group has corank zero or one for some prime q, including finiteness of the Tate–Shafarevich group. With result 006, this gives full BSD for a density-one set of quadratic twists of every elliptic curve over ℚ.

The release has no Lean statement for this family, so there is no formal statement to compare with the result described above.

Three current papers; three replaced by later editions.

What the papers state

Theorem 1.1

In Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank, 7 October 2026

The paper’s main theorem is shown as a possible match to OpenAI’s description. No review confirming the selection is recorded.

Let E/\Q be an elliptic curve and let q be any prime. If s_q(E)\in\{0,1\}, then

r(E)=a(E)=s_q(E),\qquad \#\Sha(E/\Q)<\infty,

and, with r=r(E) and the preceding normalizations,

\frac{L^{(r)}(E,1)}{r!} =\frac{\Omega_E\Reg_E\,\#\Sha(E/\Q) \prod_{\ell\text{ finite}}c_\ell(E)} {\#E(\Q)_{\rm tors}^{2}}.

There are no additional hypotheses on reduction, rational torsion, isogenies, complex multiplication, or residual Galois representations.

Corollary 1.2 (An any-prime input criterion)

In The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one, 6 October 2026

Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.

Let E/\mathbb Q be an elliptic curve. Suppose that, for some prime p, its full p-power Selmer group with the usual local Kummer conditions at every place satisfies

s_p(E):=\corank_{\mathbb Z_p}\Sel_{p^\infty}(E/\mathbb Q)\le1.

Then all conclusions of Theorem 1.1 hold, with common rank r=s_p(E).

The release does not say which paper theorem its catalogue describes. Both statements above are shown whole: the paper’s main theorem first, then the statement that shares the most wording with the description. The atlas has not reviewed either choice.

Five more possible matches

  • Theorem 1.1 (The unrestricted low-corank Selmer converse) in The Selmer converse for elliptic curves at every prime, 7 October 2026.
  • Proposition 1.4 (Split-pair anchor) in The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one, 6 October 2026.
  • Theorem 1.1 in The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one, 6 October 2026.
  • Proposition 1.3 (Positive comparison) in The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one, 6 October 2026.
  • Proposition 1.5 (CM comparison) in The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one, 6 October 2026.
Reported proof verification
The release has no Lean statement for this family. The papers are available to read and assess.
Papers and explanations
Three current papers, each readable here as a page beside its PDF. Two explanations by others.

Papers

Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.

Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank

7 October 2026, current edition.

Read it herePDF

This version removes an obsolete supporting citation and updates citations to revised manuscripts in this collection; see the previous version.

Earlier editions: 3 October 2026

We prove the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full q-power Selmer group has corank zero or one at some prime q. The analytic and Mordell–Weil ranks equal that corank, and the Tate–Shafarevich group is finite. The formula includes all prime factors and requires no additional hypotheses on reduction, rational torsion, isogenies, complex multiplication, or residual Galois representations.

The Selmer converse for elliptic curves at every prime

7 October 2026, current edition.

Read it herePDF

This version updates a citation to a revised manuscript in this collection; see the previous version.

Earlier editions: 24 September 2026

We prove the Selmer converse in coranks zero and one for every elliptic curve over ℚ and every prime p: if the full p-power Selmer group has ℤp-corank \(r\in\{0,1\}\), then the analytic and Mordell–Weil ranks both equal r, and the entire Tate–Shafarevich group is finite. As an application at the additive prime 3, we prove that for every prime \(\ell\equiv4,7,8\pmod9\), the cubic \(X^3+Y^3=\ell Z^3\) has analytic and Mordell–Weil rank one and finite Tate–Shafarevich group. In particular, every such ℓ is a sum of two rational cubes.

The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one

6 October 2026, current edition.

Read it herePDF

This version updates citations to revised manuscripts in this collection; see the previous version.

Earlier editions: 24 September 2026

We prove the two-primary Birch and Swinnerton-Dyer leading-term formula for every elliptic curve over the rationals whose two-power Selmer group has corank at most one. In this range, the algebraic rank, analytic rank, and Selmer corank are equal, and the Tate–Shafarevich group is finite. Combined with the quadratic-twist Selmer distribution, this gives the exact two-primary formula for a density-one set of signed squarefree twists of each fixed curve, ordered by absolute value; the common rank is zero or one, with each value having density one half.

Previous editions

Lean statements

The release has no Lean statement for this family. Its papers remain available to read, and contributors can propose a formal statement of a result. Suggest a change

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

The release contains no Lean statement for this family, so there is no formal statement to compare with OpenAI’s description.

Explanations by others

Explanations of this family written elsewhere, by the people named. Each is their own reading of the mathematics.

  • foocker, foocker/openaimath: Chinese reading index of the release, 9 October 2026. Source

    A narrated Chinese video explainer of about 96 seconds on the low Selmer corank BSD formula paper of 3 October, covering the corank condition, the leading-term formula and the role of every prime factor.

  • seaworld008, seaworld008/openai-math-top30: thirty selected families explained in Chinese, 7 October 2026. Source

    Chinese-language explainer for general readers, with a beginner’s section, a specialist section, background on why the problem was hard and a diagram; it ends by saying it paraphrases OpenAI’s summary and does not represent consensus.

What these papers cite in the release, and what cites them

These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.

Within this family

Follow-up work by others

  • Ruihan Chen, On the algebraic and analytic ranks of the twin-prime elliptic curve $y^2=x(x-2)(x-p)$, 7 October 2026. Source

    Chen derives full BSD data for the curves y^2 = x(x-2)(x-p) with p and p-2 prime and p not 1 mod 8, assuming OpenAI’s low-corank BSD statement as a stated hypothesis.

Explain a result, compare statements or propose a correction

Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.

Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.

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What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Career-defining publication”. The page says that nothing on it is verified.

    Author not stated. Source

  • From github.com/jkbennitt/math-release-atlas, 9 October 2026. Source

    The math-release-atlas shows a fixed caution on this family: it is BSD for low Selmer corank, not the full Birch and Swinnerton-Dyer conjecture.

  • Frank Calegari, A Brief Digest of Some New OpenAI Results, 8 October 2026. Source

    BSD results judged technical improvements on existing literature with no progress towards higher rank.

  • Kevin Buzzard, Algebraic number theory results without Lean, 7 October 2026. Source

    An expert flags that the release gives no Lean for rank at most one Birch and Swinnerton-Dyer and suggests it may be the last mile of a programme already close to done rather than a new idea.

  • David Ang, Comment on the Birch and Swinnerton-Dyer results, 7 October 2026. Source

    A number theorist reads the family as finishing the analytic rank zero and one case of BSD over the rationals by removing earlier assumptions, with nothing formalised and nothing for higher rank or number fields.

  • Raemon, Are OpenAI’s math results “creative” in an important way?, 7 October 2026. Source

    khina: BSD for 83.75 percent of elliptic curves by height is an AI-model estimate; the result is far from full BSD.

Questions awaiting review

Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.

Identify the theorem the catalogue describes

  • Family 002 The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means. Open the form

State the result in Lean

  • Family 002 The release has no Lean statement for this family. Open the form