Bibliographic record
Abstract
It is not known whether or not the length of the shortest periodic geodesic on a closed Riemannian manifold [Formula: see text] can be majorized by [Formula: see text], or [Formula: see text], where [Formula: see text] is the dimension of [Formula: see text], [Formula: see text] denotes the volume of [Formula: see text], and [Formula: see text] denotes its diameter. In this paper, we will prove that for each [Formula: see text] one can find such estimates for the length of a geodesic loop with angle between [Formula: see text] and [Formula: see text] with an explicit constant that depends both on [Formula: see text] and [Formula: see text]. That is, let [Formula: see text], and let [Formula: see text]. We will prove that there exists a “wide” (i.e. with an angle that is wider than [Formula: see text]) geodesic loop on [Formula: see text] of length at most [Formula: see text]. We will also show that there exists a “wide” geodesic loop of length at most [Formula: see text]. Here [Formula: see text] is the Filling Radius of [Formula: see text].
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How this classification was reachedexpand
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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.003 |
| 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)
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".