ON THE LINEAR COMBINANTS OF A BINARY PENCIL
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Bibliographic record
Abstract
Abstract Let A , B denote binary forms of order d , and let 2 r −1 = ( A , B ) 2 r −1 be the sequence of their linear combinants for $1 \le r \le \lfloor\frac{d+1}{2}\rfloor$ . It is known that 1 , 3 together determine the pencil { A + λ B } λ∈ P 1 and hence indirectly the higher combinants 2 r −1 . In this paper we exhibit explicit formulae for all r ≥ 3, which allow us to recover 2 r −1 from the knowledge of 1 and 3 . The calculations make use of the symbolic method in classical invariant theory, as well as the quantum theory of angular momentum. Our theorem pertains to the plethysm representation ∧ 2 S d for the group SL 2 . We give an example for the group SL 3 to show that such a result may hold for other categories of representations.
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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 | 0.002 | 0.006 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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