Quantitative stability of the total $Q$-curvature near minimizing metrics
Bibliographic record
Abstract
Under appropriate positivity hypotheses, we prove quantitative estimates for the total $k$-th order $Q$-curvature functional near minimizing metrics on any smooth, closed $n$-dimensional Riemannian manifold for every integer $1 \leq k < \frac{n}{2}$. More precisely, we show that on a generic closed Riemannian manifold the distance to the minimizing set of metrics is controlled quadratically by the $Q$-curvature energy deficit, extending recent work by Engelstein, Neumayer and Spolaor in the case $k=1$. Next we prove, for any integer $1 \leq k< \frac{n}{2}$, the existence of an $n$-dimensional Riemannian manifold such that the $k$-th order $Q$-curvature deficit controls a higher power of the distance to the minimizing set. We believe that these degenerate examples are of independent interest and can be used for further development in the field.
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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 | 0.004 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| 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".