Proposed
Proposed source assertion awaiting assessment ↗
Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.
Observed Oct 10, 2026
Theorem · version 2
Proposed half-densities in every quadratic-twist family
Statement
For every E/Q, let D(X)={d∈Z:0<|d|≤X, d squarefree}. For j=0,1 the provider claims lim_{X→∞} #{d∈D(X): ord_{s=1}L(E^(d),s)=j}/#D(X) = 1/2.
Assumptions
- A fixed E/Q; both positive and negative squarefree twist parameters are counted.
- Order by |d| and take the asymptotic limit; no fixed residue-class or finite-X half split is asserted.
Scope
A proposed family-frequency theorem, not a theorem for every individual twist. The mean-rank assertion needs separate tail control and is not included in this Claim.
This source presents a theorem with a proposed proof. Its wording does not establish acceptance or independent mathematical review.
Evidence
Source reports
Connections
In this problem · publication v13 · Selected context
Birch and Swinnerton-Dyer conjecture →
Different statement · related variant · Proposed
Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v17
Birch and Swinnerton-Dyer conjecture →
Different statement · related variant · Proposed
Relationship not independently reviewed. This concerns its use in this problem.
Sources
OpenAI · proposed Goldfeld analytic densities and 2-converse →
- Theorems 1.1–1.2, PDF page 2; separation of mean-rank assertion, page 3 → · Original passage ↗
Recorded wording
The provider claims analytic ranks 0 and 1 each have density 1/2 among signed squarefree quadratic twists of every E/Q ordered by absolute parameter.
paraphrase · Only metadata and original OpenMath paraphrases are published; full text stays upstream.
OpenAI · Preprint · Publication date Oct 7, 2026
Provider preprint in the October 2026 public collection. The selected manuscript is a proposal here; a release announcement is not independent expert acceptance.
OpenAI · proposed Goldfeld analytic densities and 2-converse — manuscript ↗No OpenMath execution recorded
Revision & attribution
Source revision: fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb. Observed Oct 10, 2026.
The provider attributes this manuscript to its internal AI model. Mathematical acceptance, paper-to-formal correspondence and independent reproduction are not established by OpenMath.
OpenAI · proposed Goldfeld analytic densities and 2-converse — manuscript: fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb. External manuscript. No OpenMath proof execution, formalization or independent mathematical acceptance is inferred.
History
Statement versions
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessment history · 1
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proposed
Proposed source assertion awaiting assessment ↗
Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.
Observed Oct 10, 2026 · Source passage →
Record details
- Claim ID
- claim-bsd-openai-goldfeld-proposal
- Statement version
- 2
- SHA-256
- f329f2ca034b5b5d105ac249cea9177ceeb0c7def2d02f74a54c8d89704724b5