---
id: "2026-10-06-openai-bsd-low-selmer-corank-goldfeld-claim"
url: "https://postcutoff.com/e/2026-10-06-openai-bsd-low-selmer-corank-goldfeld-claim/"
as_of: "2026-10-30T13:42:00+01:00"
date: "2026-10-06"
date_precision: day
category: science
importance: 4
confidence: medium
status: [Awaiting review]
verification: Preprint only
sources: 14
editor: human editor
human_review: null
version: "2026-10-29"
---

As of: 2026-10-30 13:42 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-10-06-openai-bsd-low-selmer-corank-goldfeld-claim/

# OpenAI claims BSD in Selmer corank ≤ 1 and Goldfeld's conjecture

Full title: OpenAI's release claims the full Birch–Swinnerton-Dyer formula in Selmer corank 0 and 1 and Goldfeld's 50/50 conjecture for quadratic twists; number theorists call it a technical advance, not progress on the Millennium problem

Five manuscripts in OpenAI's 6 Oct 2026 release (families 002 and 006, dated 23 Sep to 7 Oct) claim the full Birch–Swinnerton-Dyer leading-term formula, including finiteness of the Tate–Shafarevich group, for every elliptic curve over ℚ whose q-power Selmer group has corank 0 or 1 for some prime q, and Goldfeld's conjecture that the quadratic twists of every curve have analytic rank 0 and 1 with density 1/2 each. Combined, they would give full BSD for a density-one set of twists. The papers are unrefereed and not formalized. Frank Calegari, Pierre Colmez and other number theorists see technical progress in the rank ≤ 1 range rather than progress on BSD itself; Ashay Burungale identifies a new tame-congruence idea and says correctness is still being analysed.

## Key facts
- Manuscripts (OpenAI, unrefereed, no Lean scope note): 'Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank' (94 pp., version of 7 Oct), 'The Selmer converse for elliptic curves at every prime' (82 pp., 7 Oct), 'The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one' (164 pp., 6 Oct), 'Goldfeld's analytic density conjecture and the 2-converse for elliptic curves' (130 pp., 7 Oct) and 'The mean analytic rank of quadratic twists of elliptic curves' (41 pp., 6 Oct); earlier versions are dated 23 Sep to 3 Oct
- Main claim (abstract, verbatim): '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.'
- Goldfeld (abstract, verbatim): 'for every elliptic curve E over ℚ, the quadratic twists of E with analytic rank zero and one each have density 1/2 among signed squarefree twist parameters ordered by absolute value'. A companion proves that the average analytic rank of the twists tends to 1/2
- Background (Burungale's account): Gross–Zagier and Kolyvagin give rank equality and finite Sha when the analytic rank is 0 or 1; the converse direction, from Selmer corank to analytic rank, was 'largely known for odd primes p of good reduction', with additive primes the main obstacle. Alexander Smith's theorem gives the 50/50 split for 2-Selmer coranks of twists; the new 2-converse turns it into analytic ranks
- Side result in the Selmer-converse paper: every prime ℓ ≡ 4, 7, 8 (mod 9) is a sum of two rational cubes (Sylvester's 1879 conjecture), already proved by explicit constructions by Yin and by Burungale–Tian (see [2026-09-14-sylvester-conjecture-proved-burungale-tian](https://postcutoff.com/e/2026-09-14-sylvester-conjecture-proved-burungale-tian/index.md))
- Frank Calegari (Persiflage blog, 8 Oct, verbatim): 'My first impression is that these are technical improvements on results already in the literature, and that they represent no genuine progress towards the higher-rank version of BSD.' A commenter signing as Loeffler (9 Oct): 'It doesn’t seem to go much (if at all) beyond stuff that Skinner, Castella, Burungale, et al have done lately'
- Pierre Colmez (X, 7 Oct, verbatim): 'Progress on BSD is pure PR, no content. For Riemann, on the other hand, the result is spectacular.' On 9 Oct he added that it 'was basically a matter of chasing powers of 2: a non trivial task, but far from a progress towards the millenium problem'
- Ashay Burungale (guest post on Persiflage, 15 Oct, verbatim): 'The correctness of the announced proofs is still being analysed; I intend to report on this separately.' He names as new the use of 'ring-class characters ramified at auxiliary primes away from p' and calls 'a key novelty' the combination of 'this tame congruence method with division of the Heegner class'
- Reported problem in one proof step: a GitHub report (holimart/math issue 2, 8 Oct) found that a mod-8 congruence in a lemma of the Goldfeld/2-converse manuscript fails numerically. An independent Claude-assisted check (borism/openai-math-slop issue 1) reproduced the numbers but concluded: 'the numerics are correct, but the issue does not disprove a conjecture or Theorem 1.1'; the step lies in a branch that appears to be empty, 'a weakness in the write-up (or a gap, if the branch can occur)'
- Use downstream: Ruihan Chen (arXiv 2610.09370, 7 Oct) derives the exact BSD formula for twin-prime curves y² = x(x−2)(x−p) 'assuming the low-corank Birch-Swinnerton-Dyer statement announced in the October 2026 OpenAI mathematics release'. OpenAI's Hilbert's-tenth manuscript (family 004) cites the Goldfeld/2-converse paper
- Revisions: on 7 Oct OpenAI removed 'an obsolete introductory citation to a removed supporting manuscript' from the exact-BSD paper (history.md); the other changes were citation updates

## What happened

The Birch and Swinnerton-Dyer conjecture predicts that the rank of an elliptic curve over ℚ equals the order of vanishing of its
L-function at s = 1, and gives an exact formula for the leading coefficient in terms of the curve's arithmetic, including the size
of the Tate–Shafarevich group. Since the 1980s it has been known that analytic rank 0 or 1 forces the rank to match and Sha to be
finite, but the exact formula at every prime, and the converse direction from Selmer groups back to L-functions, have been worked
out only piece by piece.

OpenAI's release contains a chain of five manuscripts on this range. The exact-BSD paper states the full leading-term formula
whenever the p-power Selmer group has corank 0 or 1 for some prime p, with no conditions on reduction, torsion or residual
representations. Two companions supply the converse theorem at every prime, including the prime 2 and additive primes. A further
pair proves Goldfeld's conjecture that, ordered by size, half of the quadratic twists of any curve have analytic rank 0 and half
rank 1, with average rank 1/2. With Alexander Smith's theorem on 2-Selmer groups of twists, this would give the complete BSD
formula for 100% of the twists of every elliptic curve over ℚ. None of the papers has a Lean formalization.

## Why it matters

If the chain holds, the rank ≤ 1 part of BSD over ℚ is settled in its exact form, and Goldfeld's 1979 prediction about twists
becomes a theorem. It is not a solution of the Millennium problem, which concerns curves of every rank; headlines that counted
these papers as "progress on BSD" overstate them, in the view of Calegari, Colmez and the commenter Loeffler.

## How experts read it

Frank Calegari called the BSD results "technical improvements on results already in the literature" with "no genuine progress
towards the higher-rank version of BSD". Pierre Colmez was blunter ("pure PR, no content") while praising the zeta-function
results. In the first of a series of guest posts on Calegari's blog, Ashay Burungale explained the proposed rank-one argument: it
replaces anticyclotomic Iwasawa theory at p by ring-class characters ramified at auxiliary primes, combined with dividing a Heegner
class, which he calls a key novelty. He said correctness is still being analysed. On GitHub, a numerical check found a congruence in
one proof step of the Goldfeld manuscript to be false; a second check concluded that the step sits in a branch that seems to be
empty, so the theorem is not refuted but the write-up has a weak spot.

## What is disputed or not yet verified

- Verification: Unrefereed manuscripts, no Lean formalization; one proof step questioned on GitHub (8 Oct); expert readings describe the results as technical

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 159 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 98 days after its cutoff
- Gemini 4 Argon (training cutoff March 2026): 189 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 128 days after its cutoff

## Sources

1. [OpenAI: Exact Birch–Swinnerton-Dyer Formula from Low Selmer Corank (7 Oct version)](https://github.com/openai/math/blob/main/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/exact-bsd-low-selmer-corank.pdf) (github.com, paper)
2. [OpenAI: The Selmer converse for elliptic curves at every prime](https://github.com/openai/math/blob/main/preprints/The-Selmer-converse-for-elliptic-curves-at-every-prime-October-7-2026/main.pdf) (github.com, paper)
3. [OpenAI: The two-primary Birch–Swinnerton-Dyer formula in Selmer corank at most one](https://github.com/openai/math/blob/main/preprints/The-two-primary-Birch-Swinnerton-Dyer-formula-in-Selmer-corank-at-most-one-October-6-2026/paper.pdf) (github.com, paper)
4. [OpenAI: Goldfeld's analytic density conjecture and the 2-converse for elliptic curves](https://github.com/openai/math/blob/main/preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026/paper.pdf) (github.com, paper)
5. [OpenAI: The mean analytic rank of quadratic twists of elliptic curves](https://github.com/openai/math/blob/main/preprints/The-mean-analytic-rank-of-quadratic-twists-of-elliptic-curves-October-6-2026/paper.pdf) (github.com, paper)
6. [arXiv 2610.09370: Ruihan Chen, ranks of the twin-prime elliptic curve, assuming the announced low-corank BSD](https://arxiv.org/abs/2610.09370) (arxiv.org, paper)
7. [openai/math history.md (7 Oct revisions)](https://github.com/openai/math/blob/main/history.md) (github.com, official)
8. [Persiflage (Frank Calegari): A Brief Digest of Some New OpenAI Results (8 Oct)](https://galoisrepresentations.org/2026/10/08/a-brief-digest-of-some-new-openai-results/) (galoisrepresentations.org, discussion)
9. [Persiflage: BSD and the p-converse (guest post by Ashay Burungale, 15 Oct)](https://galoisrepresentations.org/2026/10/15/bsd-and-the-p-converse-guest-post-ashay-burungale/) (galoisrepresentations.org, discussion)
10. [Pierre Colmez on X: 'Progress on BSD is pure PR, no content' (7 Oct)](https://x.com/ColmezPierre/status/2107898549476180125) (x.com, discussion)
11. [Pierre Colmez on X: 'basically a matter of chasing powers of 2' (9 Oct)](https://x.com/ColmezPierre/status/2108513838727614885) (x.com, discussion)
12. [GitHub holimart/math issue 2: mod-8 congruence in the Goldfeld / 2-converse manuscript fails numerically](https://github.com/holimart/math/issues/2) (github.com, discussion)
13. [GitHub borism/openai-math-slop issue 1: check of the mod-8 report ('numerics real, not a disproof')](https://github.com/borism/openai-math-slop/issues/1) (github.com, discussion)
14. [oai-math-revised: community notes on the exact-BSD manuscript](https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Exact-Birch-Swinnerton-Dyer-Formula-from-Low-Selmer-Corank-October-7-2026/AGENT_NOTES.md) (github.com, discussion)

## Changes

- 2026-10-29 (filed): Created (families 001–125; manuscripts, Persiflage posts, X posts via fxtwitter and the two GitHub issues read)

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- Archived: 2 posts, listed in https://postcutoff.com/e/2026-10-06-openai-bsd-low-selmer-corank-goldfeld-claim/index.json