The Gauss–Bonnet theorem for noncommutative two tori with a general conformal structure
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Bibliographic record
Abstract
In this paper we give a proof of the Gauss–Bonnet theorem of Connes and Tretkoff for noncommutative two tori \mathbb{T}_{\theta}^2 equipped with an arbitrary translation invariant complex structure. More precisely, we show that for any complex number \tau in the upper half plane, representing the conformal class of a metric on \mathbb{T}_{\theta}^2 , and a Weyl factor given by a positive invertible element k \in C^{\infty}(\mathbb{T}_{\theta}^2) , the value at the origin, \zeta (0) , of the spectral zeta function of the Laplacian \triangle\mkern-.5mu ' attached to (\mathbb{T}_{\theta}^2, \tau, k) is independent of \tau and k .
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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 | 0.004 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.000 | 0.005 |
| 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