Curvature-dimension estimates for the Laplace-Beltrami operator of a\n totally geodesic foliation
Bibliographic record
Abstract
We study Bakry-Emery type estimates for the Laplace-Beltrami operator of a\ntotally geodesic foliation. In particular, we are interested in situations for\nwhich the $\\Gamma_2$ operator may not be bounded from below but the horizontal\nBakry-Emery curvature is. As we prove it, under a bracket generating condition,\nthis weaker condition is enough to imply several functional inequalities for\nthe heat semigroup including the Wang-Harnack inequality and the log-Sobolev\ninequality. We also prove that, under proper additional assumptions, the\ngeneralized curvature dimension inequality introduced by Baudoin-Garofalo is\nuniformly satisfied for a family of Riemannian metrics that converge to the\nsub-Riemannian one.\n
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.001 | 0.000 |
| 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 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".