Bibliographic record
Abstract
We present tight bounds on the spanning ratio of a large family of ordered [Formula: see text]-graphs. A [Formula: see text]-graph partitions the plane around each vertex into [Formula: see text] disjoint cones, each having aperture [Formula: see text]. An ordered [Formula: see text]-graph is constructed by inserting the vertices one by one and connecting each vertex to the closest previously-inserted vertex in each cone. We show that for any integer [Formula: see text], ordered [Formula: see text]-graphs with [Formula: see text] cones have a tight spanning ratio of [Formula: see text]. We also show that for any integer [Formula: see text], ordered [Formula: see text]-graphs with [Formula: see text] cones have a tight spanning ratio of [Formula: see text]. We provide lower bounds for ordered [Formula: see text]-graphs with [Formula: see text] and [Formula: see text] cones. For ordered [Formula: see text]-graphs with [Formula: see text] and [Formula: see text] cones these lower bounds are strictly greater than the worst case spanning ratios of their unordered counterparts. These are the first results showing that ordered [Formula: see text]-graphs have worse spanning ratios than unordered [Formula: see text]-graphs. Finally, we show that, unlike their unordered counterparts, the ordered [Formula: see text]-graphs with 4, 5, and 6 cones are not spanners.
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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 | 0.003 | 0.021 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.004 | 0.008 |
| Scholarly communication | 0.009 | 0.026 |
| Open science | 0.002 | 0.006 |
| Research integrity | 0.003 | 0.009 |
| Insufficient payload (model declined to judge) | 0.043 | 0.009 |
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".