The closure ordering of adjoint nilpotent orbits in so(p,q)
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Bibliographic record
Abstract
Let ${\mathcal{O}}$ be a nilpotent orbit in ${\mathfrak{so}}(p,q)$ under the adjoint action of the full orthogonal group ${\rm{O}}(p,q)$. Then the closure of ${\mathcal{O}}$ (with respect to the Euclidean topology) is a union of ${\mathcal{O}}$ and some nilpotent ${\rm{O}}(p,q)$-orbits of smaller dimensions. In an earlier work, the first author has determined which nilpotent ${\rm{O}}(p,q)$-orbits belong to this closure. The same problem for the action of the identity component ${\rm{SO}}(p,q)^0$ of ${\rm{O}}(p,q)$ on ${\mathfrak{so}}(p,q)$ is much harder and we propose a conjecture describing the closures of the nilpotent ${\rm{SO}}(p,q)^0$-orbits. The conjecture is proved when $\min(p,q)\le7$. Our method is indirect because we use the Kostant-Sekiguchi correspondence to translate the problem to that of describing the closures of the unstable orbits for the action of the complex group ${\rm{SO}} p({\bf{C}})\times{\rm{SO}} q({\bf{C}})$ on the space $M_{p,q}$ of complex $p\times q$ matrices with the action given by $(a,b)\cdot x=axb^<-1>$. The fact that the Kostant--Sekiguchi correspondence preserves the closure relation has been proved recently by Barbasch and Sepanski.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 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