Constellations and $\tau$-functions for rationally weighted Hurwitz numbers
Bibliographic record
Abstract
Weighted constellations give graphical representations of weighted branched coverings of the Riemann sphere. They were introduced to provide a combinatorial interpretation of the 2D Toda $\tau$-functions of hypergeometric type serving as generating functions for weighted Hurwitz numbers in the case of polynomial weight generating functions. The product over all vertex and edge weights of a given weighted constellation, summed over all configurations, reproduces the $\tau$-function. In the present work, this is generalized to constellations in which the weighting parameters are determined by a rational weight generating function. The associated $\tau$-function may be expressed as a sum over the weights of doubly labelled weighted constellations, with two types of weighting parameters associated to each equivalence class of branched coverings. The double labelling of branch points, referred to as “colour” and “flavour” indices, is required by the fact that, in the Taylor expansion of the weight generating function, a particular colour from amongst the denominator parameters may appear multiply, and the flavour labels indicate this multiplicity.
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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) | 0.000 | 0.001 |
| Bibliometrics | 0.004 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.008 | 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".