Study pack · Pure mathematics

BSD, an open claim

Number theory · as it stood on 2 November 2026 · run open-2026-11-02/bsd-low-selmer-corank · Claude Opus 5.5

Has a serious holeNo break

'No conditions' is unjustified: the converse is new only where every known version stops.

claims: 6
holes: 8
lenses: 3
records cited: 12 of 14
retracted precedents: 0

Not settled yet. It holds if By the resolution date, the main theorem (the full Birch and Swinnerton-Dyer formula for elliptic curves over the rationals whose q-power Selmer group has corank zero or one for some prime q) is accepted at a refereed journal, or confirmed in writing by mathematicians independent of the developer, and no error invalidating it has been acknowledged. It fails if By the resolution date, the manuscript is withdrawn, an error invalidating the main theorem is acknowledged by the developer, or a published analysis by mathematicians independent of the developer identifies such an error.

The idea, as it stood

A manuscript produced by an AI model claims to prove the full Birch and Swinnerton-Dyer formula for every elliptic curve over the rationals whose q-power Selmer group has corank zero or one for some prime q. For each such curve it claims that the analytic rank and the Mordell-Weil rank both equal that corank, that the Tate-Shafarevich group is finite, and that the conjectured leading-term formula holds at every prime, with no extra conditions on reduction type, rational torsion, isogenies, complex multiplication or the residual Galois representation. Companion manuscripts claim the converse theorem in coranks zero and one at every prime, and the two-primary part of the formula; as an application, the converse manuscript claims that every prime congruent to 4, 7 or 8 mod 9 is a sum of two rational cubes. With a further manuscript that claims Goldfeld's conjecture for quadratic twists, the developer says this gives the full conjecture for a density-one set of quadratic twists of every elliptic curve over the rationals. The manuscripts were released on 6 October 2026 among several hundred others. No formal proof accompanies them, and the developer says some of its unformalised results could have issues. One revision so far removed a citation to a supporting manuscript that had been taken down.

Converse theorem at every prime: for any elliptic curve E over Q and any prime q, q = 2 and q = 3 included, if the q^∞-Selmer group of E has corank r = 0 or 1, then ord_{s=1} L(E,s) = r, and so rank E(Q) = r and the whole Tate–Shafarevich group Ш(E/Q) is finite, with no conditions on reduction type, rational torsion, isogenies, complex multiplication or the residual representation E[q].mathematical · unsupported
For every such curve and every odd prime p, the p-part of the leading-term formula L^(r)(E,1)/(r!·Ω_E·Reg_E) = #Ш(E/Q)·∏_ℓ c_ℓ / #E(Q)_tors^2 holds, including p = 3, primes of additive or supersingular reduction, primes dividing a Tamagawa number or #E(Q)_tors, primes where E[p] is reducible, and primes ramified in the CM field of a CM curve.mathematical · unsupported
The 2-primary part of the same leading-term formula holds for every such curve, with no conditions on reduction at 2, rational 2-torsion or complex multiplication (claimed in a companion manuscript and needed for the 'every prime' statement).mathematical · unsupported
Application: every prime p ≡ 4, 7 or 8 (mod 9) is a sum of two rational cubes, deduced from the corank-one converse theorem for the curve x^3 + y^3 = p (j = 0, CM by Q(√−3), a rational 3-isogeny, additive reduction at 3 and at p), which requires its Selmer corank to be exactly one at some prime q for every such p.mathematical · unsupported
A further manuscript's claim of Goldfeld's conjecture (ordered by discriminant, half of the quadratic twists of a fixed E have analytic rank 0 and half analytic rank 1), combined with the main theorem, gives the full Birch and Swinnerton-Dyer conjecture for a density-one set of quadratic twists of every elliptic curve over Q.mathematical · unsupported
The chain of manuscripts is complete as revised: no step relies on results of the taken-down supporting manuscript whose citation was removed, or on any unformalised result the developer concedes may be flawed.mathematical · contested

The holes, and who found them

More than one lens found a hole in claims 5, 6, each on its own. That is a stronger signal than one lens alone.

#	Hole	Lens	How serious	Claim	Status
1	'No conditions' is unjustified: the converse is new only where every known version stops	Barbara McClintock	serious	1	Open
2	p = 3 and additive primes: two inputs the formula needs that the claim does not supply	Barbara McClintock	serious	2	Open
3	The 2-part is a single point of failure in a separate manuscript	John von Neumann	serious	3	Open
4	The cube-sum application rests on the converse's most exceptional case	Barbara McClintock	serious	4	Open
5	Goldfeld for every curve is a second unproven premise, and its proof must use something true only over Q	Alan Turing	serious	5	Open
6	In twist families the exceptions are the typical case	Barbara McClintock	serious	5	Open
7	Deleting a citation does not delete a dependency	John von Neumann	serious	6	Open
8	The completeness claim cannot be checked as stated	Alan Turing	serious	6	Open

What it rests on

p-converse to a theorem of Gross–Zagier, Kolyvagin and RubinBurungale & Tian · 2019 · Inventiones mathematicae · cited on claim 4 · doi:10.1007/s00222-019-00929-7
On the conjecture of Birch and Swinnerton-DyerGonzález-Avilés · 1997 · Transactions of the American Mathematical Society · cited on claim 3 · doi:10.1090/s0002-9947-97-01762-5
A proof of the $4,7$ cases of Sylvester's conjecture on cube sumsYin · 2026 · arXiv (Cornell University) · cited on claim 2, 4 · doi:10.48550/arxiv.2605.25917
Elliptic curves with all quadratic twists of positive rankDokchitser & Dokchitser · 2009 · Acta Arithmetica · cited on claim 5 · doi:10.4064/aa137-2-7
A rank zero $p$-converse to a theorem of Gross--Zagier, Kolyvagin and RubinBurungale & Tian · 2025 · Annals of Mathematics · cited on claim 1, 3, 4, 5 · doi:10.4007/annals.2026.203.1.1
Iwasawa Main Conjecture for Supersingular Elliptic Curves and BSD conjectureXin · 2014 · arXiv (Cornell University) · cited on claim 2 · doi:10.48550/arxiv.1411.6352
On the $8$ case of Sylvester ConjectureYin · 2019 · arXiv (Cornell University) · cited on claim 4 · doi:10.48550/arxiv.1912.13338
ELLIPTIC CURVES WITH ALL QUADRATIC TWISTS OF POSITIVE RANKTim & Dokchitser · 2012 · cited on claim 5 · record
Exceptional zeros for Heegner points and $p$-converse to the theorem of Gross-Zagier and KolyvaginCastella · 2024 · arXiv (Cornell University) · cited on claim 1 · doi:10.48550/arxiv.2409.01360
A $p$-part Birch and Swinnerton-Dyer Formula over Totally Imaginary Quadratic Extension of Totally Real FieldsLi · 2026 · arXiv (Cornell University) · cited on claim 2 · doi:10.48550/arxiv.2608.11969
On the 8 case of the Sylvester conjectureYin · 2021 · Transactions of the American Mathematical Society · cited on claim 4 · doi:10.1090/tran/8556
Confirm Goldfeld Conjectured for Infinitely Many Elliptic CurvesAouira et al. · 2026 · Indian Journal of Advanced Mathematics · cited on claim 5 · doi:10.54105/ijam.a1230.06010426
A higher Gross–Zagier formula and the structure of Selmer groupsKim · 2024 · Transactions of the American Mathematical Society · retrieved, not cited · doi:10.1090/tran/9125
The conjecture of Birch and Swinnerton-DyerKezuka & Zagier · 2026 · MPG.PuRe (Max Planck Society) · retrieved, not cited · record

The tests that settle it

The deciding test · weeks of work

Audit the single path from hypotheses to the genuinely new case of the application: rank one for x^3 + y^3 = p with p ≡ 8 (mod 9), through the corank-one converse at q = 3. That path meets every exceptional feature at once (j = 0, 3 ramified in Q(√−3), E[3] reducible through the rational 3-isogeny, additive reduction at 3 and at p), and nothing in the packet is shown to reach it (E3 covers 4 and 7 mod 9 by its own route; E7 and E11 give only a sufficient condition for p or p^2). For each lemma on the path, list its inputs, check each input's stated hypotheses against this curve at q = 3, and check whether any input was supported by the taken-down manuscript before the revision. Wherever the argument asserts an explicit valuation, index or divisibility, compute it for p = 17, 53, 71 and 89.

It fails if: A lemma on the path is applied outside its stated hypotheses (q ≥ 5, good or multiplicative reduction at q, irreducible E[q], Ш[q^∞] assumed finite) with no replacement argument; a step's only support was the removed citation; or a computed value for one of those p disagrees with what the argument asserts.

For hole	Test	Result
1	For each class (q = 2; q = 3; additive reduction at q; reducible E[q]; q ramified in the CM field), find the step that handles it and check every input's stated hypotheses there. Then run the argument by hand on curves whose Selmer coranks and analytic ranks can be computed independently, for example y^2 + y = x^3 − x at q = 3 (7 points over F_3, so a_3 = −3) and at q = 2 (5 points over F_2, so a_2 = −2). Counts against the claim: any class where the argument invokes a result whose hypotheses exclude it (q ≥ 5, good or multiplicative reduction, irreducible E[q], Ш[q^∞] assumed finite) with no replacement argument.	Proposed
2	Locate the step that handles good supersingular reduction at 3 with a_3 = ±3, and the step proving the 3-part of the Manin–Stevens constant is a unit for every curve covered, including curves with additive reduction at 3. Then compute ord_3 of both sides for small-conductor curves in each class with the manuscript's own normalisation. Counts against: either step is missing or cites a result whose hypotheses exclude these curves, or the computed valuations disagree.	Proposed
3	Identify the step that bounds #Ш[2^∞] above and below in analytic rank one for a curve with additive reduction at 2, full rational 2-torsion and 2 ramified in its CM field (for example y^2 = x^3 − 25x, if a 2-descent confirms its 2^∞-Selmer corank is one) and check its inputs. Separately compute ord_2 of both sides, with the manuscript's normalisation of Ω_E, for small curves with Δ > 0 such as y^2 + y = x^3 − x. Counts against: no such step, a step whose hypotheses exclude these curves, or a factor-of-2 mismatch.	Proposed
4	Identify the q used, the proof that the corank is one at that q for every such p, and the lemma giving the corank-one converse there; check each input's hypotheses against j = 0, reducible E[3], additive reduction at 3 and 3 ramified in Q(√−3), and if the step relies on an earlier p-converse such as E1, check that result's own hypotheses. Wherever the argument asserts an explicit valuation, index or divisibility, compute it for p = 17, 53, 71 and 89, all ≡ 8 (mod 9). Counts against: an input whose hypotheses exclude this curve with no replacement, or a computed value that disagrees with what the argument asserts.	Proposed
5	Find the step in the Goldfeld manuscript that uses a property of Q not shared by a number field with no real and an odd number of complex places, and check it; also check whether its route to rank zero runs through 2^∞-Selmer distributions plus a converse at q = 2, as the one established instance does (E5). Counts against: no such Q-specific step, so the argument would prove a statement those fields contradict; or reliance on C1 at q = 2, which the packet does not support.	Proposed
6	Fix one curve, for example y^2 + y = x^3 − x, and take twists by d divisible by 3, 5 or 7; for each twist of analytic rank at most one, check that the manuscript's proof of the p-part at p \| d applies at a prime of additive reduction, and compute ord_p of both sides. Counts against: the proof of C2 at additive primes uses an input that fails for these twists, or the computed valuations disagree.	Proposed
7	Diff the versions before and after the revision. List every statement that cited the taken-down manuscript, directly or through another companion manuscript, and for each show where it is now proved. Counts against: any such statement still used on a path to C1–C5 without a self-contained replacement proof.	Proposed
8	Ask the developer to name the results it concedes may be flawed and map each onto the dependency graph of C1–C5. Counts against: any named result lies on a path to C1–C5, or the results cannot be named, in which case the completeness claim is unsupported by construction.	Proposed

Proposed tests that haven’t run are the agenda for what comes next. Download them as a draft pre-registration, ready to edit and register on OSF.

The finding

The deductions linking the claims are valid, but each load-bearing step lies exactly where every result in the packet stops short, and none of it can yet be checked: there is no formal proof, the conceded flaws are unnamed and one dependency was taken down.

It isn’t settled. What would overturn the review’s reading: A lemma on the path is applied outside its stated hypotheses (q ≥ 5, good or multiplicative reduction at q, irreducible E[q], Ш[q^∞] assumed finite) with no replacement argument; a step's only support was the removed citation; or a computed value for one of those p disagrees with what the argument asserts.

Cite it, disclose it, reproduce it

A paragraph for your methods section

We used Feynman (feynman.works), an AI research-review tool running on Claude Opus 5.5, to critique “BSD, an open claim” on 2026-11-02 (run open-2026-11-02/bsd-low-selmer-corank). It reconstructed 6 claims, retrieved 14 published records dated no later than 2026-11-02, and raised 8 objections through reasoning frameworks drawn from the published work of John von Neumann, Alan Turing and Barbara McClintock. Each citation in its output was checked against the records it retrieved, and we read every source we relied on. Its objections informed our analysis; the conclusions, and any errors, are our own.

The lenses

John von NeumannMathematics & strategy · 1928–1957What are the incentives, available strategies and consequences of each move?
Alan TuringComputation & intelligence · 1936–1954What observable test would let us distinguish the competing claims?
Barbara McClintockGenetics & observation · 1931–1980What does the exception reveal that the average has hidden?

BibTeX

@misc{feynman-open-2026-11-02-bsd-low-selmer-corank,
  title = {{Feynman review: BSD, an open claim}},
  author = {{Feynman Works}},
  year = {2026},
  howpublished = {\url{https://www.feynman.works/pack/bsd-low-selmer-corank}},
  note = {Run open-2026-11-02/bsd-low-selmer-corank, Claude Opus 5.5, 2026-11-02}
}

@article{burungale2019p,
  title = {{p-converse to a theorem of Gross–Zagier, Kolyvagin and Rubin}},
  author = {Burungale and Tian},
  year = {2019},
  journal = {Inventiones mathematicae},
  doi = {10.1007/s00222-019-00929-7}
}

@article{gonzalez1997the,
  title = {{On the conjecture of Birch and Swinnerton-Dyer}},
  author = {González-Avilés},
  year = {1997},
  journal = {Transactions of the American Mathematical Society},
  doi = {10.1090/s0002-9947-97-01762-5}
}

@article{yin2026proof,
  title = {{A proof of the \$4,7\$ cases of Sylvester's conjecture on cube sums}},
  author = {Yin},
  year = {2026},
  journal = {arXiv (Cornell University)},
  doi = {10.48550/arxiv.2605.25917}
}

@article{dokchitser2009elliptic,
  title = {{Elliptic curves with all quadratic twists of positive rank}},
  author = {Dokchitser and Dokchitser},
  year = {2009},
  journal = {Acta Arithmetica},
  doi = {10.4064/aa137-2-7}
}

@article{burungale2025rank,
  title = {{A rank zero \$p\$-converse to a theorem of Gross--Zagier, Kolyvagin and Rubin}},
  author = {Burungale and Tian},
  year = {2025},
  journal = {Annals of Mathematics},
  doi = {10.4007/annals.2026.203.1.1}
}

@article{xin2014iwasawa,
  title = {{Iwasawa Main Conjecture for Supersingular Elliptic Curves and BSD conjecture}},
  author = {Xin},
  year = {2014},
  journal = {arXiv (Cornell University)},
  doi = {10.48550/arxiv.1411.6352}
}

@article{yin2019the,
  title = {{On the \$8\$ case of Sylvester Conjecture}},
  author = {Yin},
  year = {2019},
  journal = {arXiv (Cornell University)},
  doi = {10.48550/arxiv.1912.13338}
}

@misc{tim2012elliptic,
  title = {{ELLIPTIC CURVES WITH ALL QUADRATIC TWISTS OF POSITIVE RANK}},
  author = {Tim and Dokchitser},
  year = {2012},
  url = {http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.242.9076}
}

@article{castella2024exceptional,
  title = {{Exceptional zeros for Heegner points and \$p\$-converse to the theorem of Gross-Zagier and Kolyvagin}},
  author = {Castella},
  year = {2024},
  journal = {arXiv (Cornell University)},
  doi = {10.48550/arxiv.2409.01360}
}

@article{li2026p,
  title = {{A \$p\$-part Birch and Swinnerton-Dyer Formula over Totally Imaginary Quadratic Extension of Totally Real Fields}},
  author = {Li},
  year = {2026},
  journal = {arXiv (Cornell University)},
  doi = {10.48550/arxiv.2608.11969}
}

@article{yin2021the,
  title = {{On the 8 case of the Sylvester conjecture}},
  author = {Yin},
  year = {2021},
  journal = {Transactions of the American Mathematical Society},
  doi = {10.1090/tran/8556}
}

@article{aouira2026confirm,
  title = {{Confirm Goldfeld Conjectured for Infinitely Many Elliptic Curves}},
  author = {Aouira and others},
  year = {2026},
  journal = {Indian Journal of Advanced Mathematics},
  doi = {10.54105/ijam.a1230.06010426}
}

Run open-2026-11-02/bsd-low-selmer-corank on Claude Opus 5.5. The review is reproducible from its record: the claims, the records retrieved before 2 November 2026, and each lens’s card. https://www.feynman.works/pack/bsd-low-selmer-corank