Bibliographic record
Abstract
Modular group representations throughout the realmThere are two aspects to Moonshine.The more general one is the unexpected presence of modular group actions over a wide range of algebraic settings, and is now fairly well understood.We have seen instances of this already with, for example, the characters of affine algebras and VOAs.This chapter completes our treatment of these modular actions.The more specific aspect -the association of Hauptmoduls to the Monster -is still poorly understood and is the subject of the following chapter.Much of this chapter is orthogonal to Monstrous Moonshine.For example, we discuss here fusion rings and modular data; both the fusion ring and modular data of the Moonshine module V are maximally trivial.Nevertheless, this chapter helps to paint the general context of Monstrous Moonshine.In Section 7.2.4 we build on some of the lessons from this chapter to speculate on a possible second proof of Monstrous Moonshine. Combinatorial rational conformal field theoryRecall the semi-simple Lie algebras: we study their structure and obtain their classification by abstracting out combinatorial features (e.g.roots, Coxeter-Dynkin diagrams).Of course this is easy to do with a finite-dimensional linear structure.RCFTs are infinitedimensional, but by definition their infinite-dimensional symmetry and implicit rigidity again effectively reduces them to certain discrete structures.As we see next section, those discrete structures are remarkable for their ubiquity in modern mathematics.See [208], [207], [33], [131], [437], [236] for further background.As with all other chapters except Chapter 7, we've tended to avoid giving original references, as these are voluminous and can be recovered from the numerous review articles and books. Fusion ringsRecall that the eigenvalues of a self-adjoint (equivalently, Hermitian) matrix are all real.Consider the following scenario.Let A, B and C be n × n Hermitian matrices withWhat are the conditions on these eigenvalues so that C = A + B? The answer consists of a number of inequalities involving the numbers α i , β j , γ k .Now discretise this problem:, all be integers.Then the following are equivalent:
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.020 | 0.006 |
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".