A relative index on the space of 3‐dimensional embeddable CR‐structures of finite type
Bibliographic record
Abstract
Abstract For a small deformation Φ of a 3‐dimensional pseudoconvex embeddable CR‐structure of finite type, we prove that Φ defines an embeddable CR‐structure if and only if the Szegö projection from ker $ ^{\rm \Phi} {\bar \partial} _{b} $ to ker $ {\bar \partial} _{b} $ is a Fredholm operator. This defines a relative index which is in this case equal to the negative of number of small eigenvalues of the operator $ ^{\rm \Phi} \bar \partial _{b}^{*} \, ^{\rm \Phi} \bar \partial _{b}^{} $ . We also prove a cocycle formula for the relative index. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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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) | 0.000 | 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) | 0.001 | 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".