On the Genus Distribution of $(p,q,n)$-Dipoles
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Bibliographic record
Abstract
There are many applications of the enumeration of maps in surfaces to other areas of mathematics and the physical sciences. In particular, in quantum field theory and string theory, there are many examples of occasions where it is necessary to sum over all the Feynman graphs of a certain type. In a recent paper of Constable et al. on pp-wave string interactions, they must sum over a class of Feynman graphs which are equivalent to what we call $(p,q,n)$-dipoles. In this paper we perform a combinatorial analysis that gives an exact formula for the number of $(p,q,n)$-dipoles in the torus (genus 1) and double torus (genus 2).
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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 it