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

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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 Sylvester’s 1879 conjecture on sums of two cubes is fully proved)
• 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’