Subdividing three-dimensional Riemannian disks
Bibliographic record
Abstract
Papasoglu asked whether for any Riemannian 3-disk [Formula: see text] with diameter [Formula: see text], boundary area [Formula: see text] and volume [Formula: see text], there exists a homotopy [Formula: see text] contracting the boundary to a point so that the area of [Formula: see text] is bounded by [Formula: see text] for some function [Formula: see text]. He further asks whether it is possible to subdivide [Formula: see text] by a disk [Formula: see text] into two regions of volume [Formula: see text] so that the area of [Formula: see text] is bounded by some function [Formula: see text]. In this paper, we answer the questions above in the negative: We prove that given [Formula: see text] and [Formula: see text], one can construct a metric [Formula: see text] so that any 2-disk [Formula: see text] subdividing [Formula: see text] into two regions of volume at least [Formula: see text], the area of [Formula: see text] is greater than [Formula: see text]. We further prove that for any Riemannian 3-sphere [Formula: see text], there is a surface that subdivides the disk into two regions of volume no less than [Formula: see text], and the area of this surface is bounded by [Formula: see text], where [Formula: see text] is the homological filling function of [Formula: see text].
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.001 |
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 source (direct Gemma or distilled Codex), 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".