2-Local Isometries on Spaces of Lipschitz Functions
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Bibliographic record
Abstract
Abstract Let ( X , d ) be a metric space, and let Lip( X ) denote the Banach space of all scalar-valued bounded Lipschitz functions ƒ on X endowed with one of the natural norms where L ( ƒ ) is the Lipschitz constant of ƒ . It is said that the isometry group of Lip( X ) is canonical if every surjective linear isometry of Lip( X ) is induced by a surjective isometry of X . In this paper we prove that if X is bounded separable and the isometry group of Lip( X ) is canonical, then every 2-local isometry of Lip( X ) is a surjective linear isometry. Furthermore, we give a complete description of all 2-local isometries of Lip( X ) when X is bounded.
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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 | 0.002 |
| 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) | 0.028 | 0.007 |
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