Extremal Bounds for 3-Neighbor Bootstrap Percolation in Dimensions Two and Three
Bibliographic record
Abstract
Abstract. For [Formula: see text], the [Formula: see text]- neighbor bootstrap process in a graph [Formula: see text] starts with a set of infected vertices and, in each time step, every vertex with at least [Formula: see text] infected neighbors becomes infected. The initial infection percolates if every vertex of [Formula: see text] is eventually infected. We exactly determine the minimum cardinality of a set that percolates for the 3-neighbor bootstrap process when [Formula: see text] is a three-dimensional grid with minimum side-length at least 11. We also characterize the integers [Formula: see text] and [Formula: see text] for which there is a set of cardinality [Formula: see text] that percolates for the 3-neighbor bootstrap process in the [Formula: see text] grid; this solves a problem raised by Benevides et al. [HAL Research Report 03161419v4, 2021].
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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 | 0.004 | 0.024 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.004 | 0.002 |
| Science and technology studies | 0.003 | 0.005 |
| Scholarly communication | 0.004 | 0.006 |
| Open science | 0.004 | 0.005 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.006 | 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".