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.
Feynman
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.
06 claims /08 objections /12 of 14 sources cited / lenses: The exception, Formal model, Operational test
Each square is an objection. Larger means heavier. Every claim was challenged by at least one lens.
01Converse 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].
02For 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.
03The 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).
04Application: 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.
05A 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.
06The 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.