Spectral sets, extremal functions and exceptional matrices
Why this work is in the frame
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Bibliographic record
Abstract
Let A be a square matrix and let Ω be an open set in the plane containing the spectrum of A. We consider the problem of maximizing the operator norm ∥f(A)∥ amongst all holomorphic functions f from Ω into the closed unit disk. If f0 is extremal for this problem and if ∥f0(A)∥>1, then it turns out that the matrix f0(A) has special properties, among them the fact that its principal left and right singular vectors are mutually orthogonal. We study this class of exceptional matrices f0(A). In particular, we are interested in the extent to which they are characterized by the aforementioned orthogonality property.
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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 | 0.000 | 0.000 |
| 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 it