Geometrical Bounds for Variance and Recentered Moments
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Bibliographic record
Abstract
We bound the variance and other moments of a random vector based on the range of its realizations, thus generalizing inequalities of Popoviciu and of Bhatia and Davis concerning measures on the line to several dimensions. This is done using convex duality and (infinite-dimensional) linear programming. The following consequence of our bounds exhibits symmetry breaking, provides a new proof of Jung’s theorem, and turns out to have applications to the aggregation dynamics modelling attractive–repulsive interactions: among probability measures on [Formula: see text] whose support has diameter at most [Formula: see text], we show that the variance around the mean is maximized precisely by those measures that assign mass [Formula: see text] to each vertex of a standard simplex. For [Formula: see text], the [Formula: see text] th moment—optimally centered—is maximized by the same measures among those satisfying the diameter constraint.
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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 | 0.004 | 0.015 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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 it