Proves Goldfeld’s conjecture for quadratic twists of every elliptic curve over ℚ: analytic ranks zero and one each have density 1/2, and the mean analytic rank tends to 1/2. Both statements order signed squarefree twist parameters by absolute value.
The release has no Lean statement for this family, so there is no formal statement to compare with the result described above.
Two current papers; two replaced by later editions.
One problem has been reported by others; the atlas has not judged it.
In Goldfeld’s analytic density conjecture and the 2-converse for elliptic curves, 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.
For every elliptic curve E/\Q with c_2(E)\in\{0,1\},
In Goldfeld’s analytic density conjecture and the 2-converse for elliptic curves, 7 October 2026
Possible match to OpenAI’s description, selected because it shares the most wording. No review confirming the selection is recorded.
For every elliptic curve E/\Q and each j\in\{0,1\},
Consequently the twists of analytic rank at least two have density zero. For a density-one set of signed squarefree parameters d, the analytic and Mordell–Weil ranks of E^{(d)} agree and its whole Tate–Shafarevich group is finite.
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This version updates citations to revised manuscripts in this collection; see the previous version.
Earlier editions: 23 September 2026
This version updates citations to revised manuscripts in this collection; see the previous version.
Earlier editions: 23 September 2026
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
The release contains no Lean statement for this family, so there is no formal statement to compare with OpenAI’s description.
Reports of errors and corrections from outside readers, as they state them. The atlas has not judged them.
holimart, filed on a personal fork, Report: mod-8 congruence (eq. pw:split-mod8) in Lemma pw:split-witness fails numerically, Goldfeld 2-converse manuscript (issue 2), 8 October 2026. Source
The mod-8 congruence in the proof of Lemma pw:split-witness appears false for many allowed primes unless the proof’s equality branch is always empty; the filer stresses this is a gap in one step, not a refutation.
Explanations of this family written elsewhere, by the people named. Each is their own reading of the mathematics.
mo271, Wikipedia/EllipticCurveRank: goldfeld_conjecture.variants.squarefree is true (issue 6916), 7 October 2026. Source
The two-primary BSD manuscript with Smith’s distribution gives rank 0 or 1 with density one half for squarefree twists.
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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
Kevin Buzzard, Algebraic number theory results without Lean, 7 October 2026. Source
An expert flags that the release gives no Lean for Goldfeld’s conjecture on quadratic twists and suggests it may be the last mile of a programme already close to done rather than a new idea.
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Why this passage is shown
Theorem 1.1 (The 2-converse)
In Goldfeld’s analytic density conjecture and the 2-converse for elliptic curves, 7 October 2026
Why this passage is shown
Theorem 1.2 (Goldfeld’s analytic density conjecture)
In Goldfeld’s analytic density conjecture and the 2-converse for elliptic curves, 7 October 2026
Why this passage is shown
Theorem 1.1
In The mean analytic rank of quadratic twists of elliptic curves, 6 October 2026
Why this passage is shown
Corollary 1.3 (Fixed algebraic-rank moments)
In The mean analytic rank of quadratic twists of elliptic curves, 6 October 2026
Why this passage is shown
Theorem 1.2
In The mean analytic rank of quadratic twists of elliptic curves, 6 October 2026
Why this passage is shown
Corollary 1.3 (A finite Selmer criterion)
In Goldfeld’s analytic density conjecture and the 2-converse for elliptic curves, 7 October 2026