Simple unified proofs of four duality theorems
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Bibliographic record
Abstract
Duality relationships between the irreps (irreducible representations) of pairs of distinct commuting groups, \documentclass[12pt]{minimal}\begin{document}$G_1$\end{document}G1 and \documentclass[12pt]{minimal}\begin{document}$G_2$\end{document}G2, on Hilbert spaces of interest have long played important roles in the atomic and nuclear shell models. In addition to the well-known Schur–Weyl duality, the most widely used duality relationships are the so-called: unitary–unitary, orthogonal–symplectic (i.e., noncompact symplectic), symplectic–symplectic (compact symplectics), and orthogonal–orthogonal dualities. Proofs of these dualities exist in the literature. But most of them are not readily accessible to physicists or give little insight into how they might be used in practice. This paper presents unified proofs of the above-mentioned dualities based on the explicit construction of states which are simultaneously of extreme weight for the actions of both \documentclass[12pt]{minimal}\begin{document}$G_1$\end{document}G1 and \documentclass[12pt]{minimal}\begin{document}$G_2$\end{document}G2. The proofs expressed in language familiar to physicists are simple, systematic, and provide useful insights.
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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 | 0.001 |
| 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 | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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