Bibliographic record
Abstract
We prove the finiteness of certain sets of binary cubic forms with given non-zero discriminant or having discriminants with bounded radical.To obtain finite sets we impose appropriate conditions on the coefficients, or on the Hessian and the cubic covariant of the forms.Some of the results are extended to binary forms of degree higher than these.We also give theorems which are proved under the assumption that the (ABC) conjecture is true. The results in this paperLetis called a binary form of degree n.We shall always assume that the coefficients a 0 , a 1 , . . ., a n are integers.If n = 3 the binary cubic form f (X, Y ) = aX 3 + bX 2 Y + cXY 2 + dY 3 is also denoted by f = a, b, c, d .We shall write "form" instead of "binary form" and we exclude the zero form from our considerations.If f is a form of degree n ≥ 2 there exist algebraic numbers α i , β i such that
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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 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.003 | 0.003 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.008 | 0.001 |
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".