# Quadratic twist rank distribution reference

Family 006: Goldfeld’s conjecture: densities and mean analytic rank. First application triage, 9 October 2026 Australia/Brisbane.

## Problem and potential new use

Arithmetic database researchers can compare twist samples to the rank-zero/rank-one densities and mean-rank limit.

## Applicability and commercial boundary

Signed squarefree twists are ordered by absolute value; limiting density gives no finite-twist classification or a convergence rate.

## Initial business decision

Research infrastructure or conditional engineering until a computable implementation, effective bounds and a recurring buyer need are demonstrated. No commercial novelty or profitability is established.

## Next verification action

Extract finite error estimates and compare a public curve family.

## Evidence scope

The catalog statement was individually reviewed. Manuscript proof and selected formal scope comparison remain queued. No independent Lean verification was run. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. See [source metadata](source.json).

[Goldfeld's analytic density conjecture and the 2-converse for elliptic curves](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Goldfelds-analytic-density-conjecture-and-the-2-converse-for-elliptic-curves-October-7-2026/paper.pdf).