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 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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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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.007 | 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".