Why this work is in the frame
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Bibliographic record
Abstract
The classical double bubble theorem characterizes the minimizing partitions of \mathbb{R}^{n} into three chambers, two of which have prescribed finite volume. In this paper we prove a variant of the double bubble theorem in which two of the chambers have infinite volume. Such a configuration is an example of a (1,2)-cluster , or a partition of \mathbb{R}^{n} into three chambers, two of which have infinite volume and only one of which has finite volume. A (1,2) -cluster is locally minimizing with respect to a family of weights \{c_{jk}\} if for any B_{r}(0) , it minimizes the interfacial energy \sum_{j<k}c_{jk}\mathcal{H}^{n-1}(\partial \mathcal{X}(j) \cap \partial\mathcal{X}(k) \cap B_{r}(0)) among all variations with compact support in B_{r}(0) which preserve the volume of \mathcal{X}(1) . For (1,2) clusters, the analogue of the weighted double bubble is the weighted lens cluster , and we show that it is locally minimizing. Furthermore, under a symmetry assumption on \{c_{jk}\} that includes the case of equal weights, the weighted lens cluster is the unique local minimizer in \mathbb{R}^{n} for n\leq 7 , with the same uniqueness holding in \mathbb{R}^{n} for n\geq 8 under a natural growth assumption. We also obtain a closure theorem for locally minimizing (N,2) -clusters.
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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 | 0.001 | 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 | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| 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 it