Lower dimension tori of general types in multi-scale Hamiltonian systems
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Bibliographic record
Abstract
Abstract We consider a general canonical form of a multi-scale Hamiltonian system near a family of unperturbed lower dimensional, quasi-periodic, invariant tori, in which tangential and normal frequencies can admit equal or different scales. Extending works of Broer and Zhao (2017 Celest. Mech. Dyn. Astron . 127 95–119); Xu et al (2017 Ann. Henri Poincaré 18 53–83), we show the persistence of the majority of these lower dimensional tori when the normal matrices are non-degenerate and a Melnikov non-resonant condition among certain tangential and normal frequencies are satisfied when they admit equal scales. Such a general persistence result of lower dimensional tori allows a broader range of applications to stability problems arising in celestial mechanics.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 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)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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