Failure of $L^p$ symmetry of zonal spherical harmonics
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Bibliographic record
Abstract
We show that the 2-sphere does not exhibit symmetry of L^p norms of eigenfunctions of the Laplacian for p\geq 6 , which answers a question of Jakobson and Nadirashvili. In other words, there exists a sequence of spherical eigenfunctions \psi_n , with eigenvalues \lambda_n\to\infty as n\to\infty , such that the ratio of the L^p norms of the positive and negative parts of the eigenfunctions does not tend to 1 as n\to\infty when p\geq 6 . Our proof relies on fundamental properties of the Legendre polynomials and Bessel functions of the first kind.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| 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