Differential equations, mirror maps and zeta values
Bibliographic record
Abstract
Abstract. The aim of this work is an analytic investigation of differential equations producing mirror maps as well as giving new examples of mirror maps; one of these examples is related to (rational approximations to) ζ(4). We also indicate certain observations that might become a subject of further research. The existence of this paper is due to the following observations. With Apéry’s proof of the irrationality of ζ(2) and ζ(3) followed certain 2nd- and 3rd-order linear differential equations (see [Be1]–[Be3], [BP]). Doing a similar construction for ζ(4) resulted in a 5th-order differential equation. This equation was similar to the linear differential equations occuring in Calabi–Yau theory (except the order was 5 instead of 4). Computing the Lambert series of an analogue of the Yukawa coupling we got integer coefficients divisible by the square of the degree (not by the cube as in the Calabi–Yau case). Then we managed to pull back the 5th-order differential equation to one of order 4, which had all the properties of a Calabi–Yau equation. This part is the main objective of Sections 1–4. Collecting known cases of the Calabi–Yau equations (the table in Appendix A),
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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 | 0.001 |
| 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 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 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".