Brun's Inequality for a Geometric Lattice
Bibliographic record
Abstract
V. Brun introduced Brun's sieve in his seminal paper, which is based on Brun's inequality for the Möbius function and is a very powerful tool in modern number theory. The importance of the Möbius function in enumeration problems led G.-C. Rota to introduce the concept of the Möbius function for partially ordered sets. In this article, we prove Brun's inequality for geometric lattices and develop a combinatorial sieve in this context. One of the main ingredients is a recent work of K. Adiprasito, J. Huh, and E. Katz on the log-concavity of absolute values of the Whitney numbers associated with matroids. Further, we study shifted convolutions of the Whitney numbers associated with Dowling lattices and derive an asymptotic formula for generalized Dowling numbers.
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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 | 0.010 | 0.015 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.005 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.003 | 0.000 |
| Research integrity | 0.000 | 0.003 |
| 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".