Bibliographic record
Abstract
This paper is an outgrowth of the results in the domain of rolling obtained in our recent paper written with F. Silva Leite and I. Markina, and the earlier papers on the rollings of spheres produced with J. Zimmerman. We show that the rolling equations associated with a symmetric semi-Riemannian manifold rolling on its tangent space at a fixed point on the manifold essentially have the same structure as the rolling equations for the n-dimensional sphere rolling on the horizontal hyperplane; that is, we show that the rolling equations are described by a left-invariant distribution D on a Lie group G with the Lie bracket growth D + [D,D] + [D, [D,D]] = TG, reminiscent of the growth (2, 3, 5) for the two spheres rolling on the horizontal plane. We then define rolling geodesics on semi-Riemannian spaces as extensions of sub-Riemannian geodesics in the Riemannian symmetric spaces, and show that the rolling geodesics are the projections of the extremal curves, which, remarkably, are the solution curves of a completely integrable Hamiltonian system in the cotangent bundle of the configuration space. Finally, we illustrate the theory with a few noteworthy examples.
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How this classification was reachedexpand
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 | 0.001 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".