Riemannian distance and symplectic embeddings in cotangent bundle
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Bibliographic record
Abstract
Given an open neighborhood [Formula: see text] of the zero section in the cotangent bundle of [Formula: see text] we define a distance-like function [Formula: see text] on [Formula: see text] using certain symplectic embeddings from the standard ball [Formula: see text] to [Formula: see text]. We show that when [Formula: see text] is the unit-disk cotangent bundle of a Riemannian metric on [Formula: see text], [Formula: see text] recovers the metric. As an intermediate step, we give a new construction of a symplectic embedding of the ball of capacity 4 to the product of Lagrangian disks [Formula: see text], and we give a new proof of the strong Viterbo conjecture about normalized capacities for [Formula: see text]. We also give bounds of the symplectic packing number of two balls in a unit-disk cotangent bundle relative to the zero section [Formula: see text].
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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 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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