Cm-norms on finite sets and Cm extension criteria
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Bibliographic record
Abstract
C. Fefferman [F1], [F2] has recently given criteria for a function defined on a compact set E⊂Rn to extend to a Cm- or Cm,ω-function. His criteria involve uniformity of the Cm- or Cm,ω-norms for extension from finite subsets S⊂E of cardinality at most a large natural number k# depending only on m and n. We prove that one can take k#=2dimP in both cases, where P denotes the space of polynomials of degree at most m in n variables. We also show that the geometric Cm “paratangent bundle” of E (see [BMP2]) can be defined using limits of distributions supported on 2dimP-1 points
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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 | 0.003 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 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 | 0.001 |
| 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 it