Vertices of the Harder and Narasimhan polygons and the laws of large numbers
Bibliographic record
Abstract
Abstract We build on the recent techniques of Codogni and Patakfalvi (2021,Inventiones Mathematicae223, 811–894), which were used to establish theorems about semi-positivity of the Chow Mumford line bundles for families of $\mathrm {K}$ -semistable Fano varieties. Here, we apply the Central Limit Theorem to ascertain the asymptotic probabilistic nature of the vertices of theHarder and Narasimhan polygons. As an application of our main result, we use it to establish a filtered vector space analogue of the main technical result of Codogni and Patakfalvi (2021,Inventiones Mathematicae223, 811–894). In doing so, we expand upon the slope stability theory, for filtered vector spaces, that was initiated by Faltings and Wüstholz (1994,Inventiones Mathematicae116, 109–138). One source of inspiration for our abstract study ofHarder and Narasimhan data, which is a concept that we define here, is the lattice reduction methods of Grayson (1984,Commentarii Mathematici Helvetic59, 600–634). Another is the work of Faltings and Wüstholz (1994,Inventiones Mathematicae116, 109–138), and Evertse and Ferretti (2013,Annals of Mathematics177, 513–590), which is within the context of Diophantine approximation for projective varieties.
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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 | 0.001 | 0.008 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.007 | 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 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".