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Math-2252-A1 · revision r1 · on Math-2252

OpenAI claim summary for Math-2252

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

The arithmetic conclusions require a rational elliptic curve with Selmer corank zero or one at the specified prime; the rational-two-torsion companion has that additional hypothesis. Rank equality, finite Tate--Shafarevich group, or the stated leading-coefficient identities in those cases do not extend to all ranks and number fields in this problem. Average conclusions for quadratic twists also leave individual exceptional curves untreated.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Finiteness of a Shafarevich–Tate group needed by the lifting method
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Manuscript 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Finiteness of a Shafarevich–Tate group needed by the lifting method

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 004; 2026-09-24), the asserted result is that a rational elliptic curve with a nonidentity rational point of order two satisfies the low-corank two-adic Selmer converse even at arbitrary reduction at two.

Mathematical statement of the claimed result

Manuscript 1.
For an elliptic curve $E/\mathbb Q$, put $s_2(E)=\operatorname{corank}_{\mathbb Z_2}\operatorname{Sel}_{2^{\infty}}(E/\mathbb Q)$. If $E(\mathbb Q)[2]\ne0$ and $s_2(E)\in\{0,1\}$, the manuscript claims \[ \operatorname{ord}_{s=1}L(E,s)=\operatorname{rank}_{\mathbb Z}E(\mathbb Q)=s_2(E),\qquad \#\operatorname{Sha}(E/\mathbb Q)<\infty. \] Both rational two-torsion and the low-corank condition belong to this theorem; it does not cover arbitrary elliptic curves or arbitrary Selmer corank.

Outline of the argument

For manuscript 1 (family 004; 2026-09-24), binary coefficient systems detect central values or Heegner points. Integral interpolation and arithmetic contraction identities isolate one address in a network of twists, transferring a nonvanishing witness back to the original curve.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 004; 2026-09-24).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources