Clustering layers for the Fife—Greenlee problem in ℝ<i><sup>n</sup></i>
Why this work is in the frame
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Bibliographic record
Abstract
We consider the following Fife–Greenlee problem: where Ω is a smooth and bounded domain in ℝ n , ν is the outer unit normal to ∂Ω and a is a smooth function satisfying a ( x ) ∈ (–1, 1) in . Let K, Ω – and Ω + be the zero-level sets of a , { a < 0} and { a < 0}, respectively. We assume ∇a ≠ 0 on K . Fife and Greenlee constructed stable layer solutions, while del Pino et al . proved the existence of one unstable layer solution provided that some gap condition is satisfied. In this paper, for each odd integer m ≥ 3, we prove the existence of a sequence ε = ε j → 0, and a solution with m -transition layers near K . The distance of any two layers is O ( ε log(1/ ε )). Furthermore, converges uniformly to ±1 on the compact sets of Ω ± as j → +∞
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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) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 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