Bibliographic record
Abstract
Given , we consider the number of rational points on the genus one curve H η : y 2 = ( a ( 1 − x 2 ) + b ( 2 x ) ) 2 + ( c ( 1 − x 2 ) + d ( 2 x ) ) 2 . We prove that the set of η for which H η ( Q ) ≠ ∅ has density zero, and that if a rational point ( x 0 , y 0 ) ∈ H η ( Q ) exists, then H η ( Q ) is infinite unless a certain explicit polynomial in a , b , c , d , x 0 , y 0 vanishes. Curves of the form H η naturally occur in the study of configurations of points in R n with rational distances between them. As one example demonstrating this framework, we prove that if a line through the origin in R 2 passes through a rational point on the unit circle, then it contains a dense set of points P such that the distances from P to each of the three points ( 0 , 0 ) , ( 0 , 1 ) , and ( 1 , 1 ) are all rational. We also prove some results regarding whether a rational number can be expressed as a sum or product of slopes of rational right triangles.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.005 | 0.003 |
| Science and technology studies | 0.003 | 0.006 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".