Rational configuration problems and a family of curves
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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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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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 it