Discrete geometry and isotropic surfaces
Bibliographic record
Abstract
We consider smooth isotropic immersions from the 2-dimensional torus into\n$R^{2n}$, for $n \\geq 2$. When $n = 2$ the image of such map is an immersed\nLagrangian torus of $R^4$. We prove that such isotropic immersions can be\napproximated by arbitrarily $C^0$-close piecewise linear isotropic maps. If $n\n\\geq 3$ the piecewise linear isotropic maps can be chosen so that they are\npiecewise linear isotropic immersions as well. The proofs are obtained using\nanalogies with an infinite dimensional moment map geometry due to Donaldson. As\na byproduct of these considerations, we introduce a numerical flow in finite\ndimension, whose limit provide, from an experimental perspective, many examples\nof piecewise linear Lagrangian tori in $R^4$. The DMMF program, which is freely\navailable, is based on the Euler method and shows the evolution equation of\ndiscrete surfaces in real time, as a movie.\n
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How this classification was reachedexpand
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) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".