Fault-tolerant detection systems on the infinite king grid
Bibliographic record
Abstract
<p>Given a network modeled as a graph, a detection system is a subset of vertices equipped with “detectors” that can uniquely identify an “intruder” anywhere in the graph. We consider two types of detection systems: open-locating-dominating (OLD) sets and identifying codes (ICs). In an OLD set, each vertex has a unique, non-empty set of detectors in its open neighborhood; meanwhile, in an IC, each vertex has a unique, non-empty set of detectors in its closed neighborhood. We explore one of their fault-tolerant variants: redundant OLD (RED:OLD) sets and redundant ICs (RED:ICs), which ensure that removing/disabling at most one detector retains the properties of OLD sets and ICs, respectively. This paper focuses on constructing optimal RED:OLD sets and RED:ICs on the infinite king grid, and presents the proof for the bounds on their minimum densities; <span class="math inline">\(\left[\frac{3}{10}, \frac{1}{3}\right]\)</span> for RED:OLD sets and <span class="math inline">\(\left[\frac{3}{11}, \frac{1}{3}\right]\)</span> for RED:ICs.</p>
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How this classification was reachedexpand
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 | 0.007 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.000 |
| Scholarly communication | 0.003 | 0.000 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.000 | 0.002 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".