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Bibliographic record
Abstract
We consider an infinite planar Delaunay graph \mathbf{G}_{\epsilon} which is obtained by locally deforming the coordinate embedding of a general isoradial graph \mathbf{G}_{\mathrm{cr}} , with respect to a real deformation parameter \epsilon . Using Kenyon’s exact and asymptotic results for the critical Green’s function on an isoradial graph, we calculate the leading asymptotics of the first- and second-order terms in the perturbative expansion of the log-determinant of the Laplace–Beltrami operator \Delta(\epsilon) , the David–Eynard Kähler operator \mathcal{D}(\epsilon) , and the conformal Laplacian {\underline{\boldsymbol{\Delta}}}(\epsilon) on the deformed Delaunay graph \mathbf{G}_{\epsilon} . We show that the scaling limits of the second-order bi-local term for both the Laplace–Beltrami and David–Eynard Kähler operators exist and coincide, with a shared value independent of the choice of the initial isoradial graph \mathbf{G}_{\mathrm{cr}} . Our results allow us to define a discrete analog of the stress-energy tensor for each of the three operators. Furthermore, we can identify a central charge ( c=-2 ) in the case of both the Laplace–Beltrami and David–Eynard Kähler operators. While the scaling limit is consistent with the stress-energy tensor and the value of the central charge for the Gaussian free field (GFF), the discrete central charge value of c=-2 for the David–Eynard Kähler operator is, however, at odds with the value of c=-26 expected by Polyakov’s theory of 2D quantum gravity; moreover, there are problems with convergence of the scaling limit of the discrete stress-energy tensor for the David–Eynard Kähler operator. The second-order bi-local term for the conformal Laplacian involves anomalous terms corresponding to the creation of discrete curvature dipoles in the deformed Delaunay graph \mathbf{G}_{\epsilon} ; we examine the difficulties in defining a convergent scaling limit in this case. Connections with some discrete statistical models at criticality are explored.
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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 | 0.000 | 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 | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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