Extremal Bounds for 3-Neighbor Bootstrap Percolation in Dimensions Two and Three
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Bibliographic record
Abstract
For , the -neighbor bootstrap process in a graph starts with a set of infected vertices and, in each time step, every vertex with at least infected neighbors becomes infected. The initial infection percolates if every vertex of is eventually infected. We exactly determine the minimum cardinality of a set that percolates for the 3-neighbor bootstrap process when is a three-dimensional grid with minimum side-length at least 11. We also characterize the integers and for which there is a set of cardinality that percolates for the 3-neighbor bootstrap process in the grid; this solves a problem raised by Benevides et al. [HAL Research Report 03161419v4, 2021].
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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 | 0.002 |
| 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 | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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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