Optimal Bounds on Theta-Graphs: More is not Always Better.
Why this work is in the frame
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Bibliographic record
Abstract
We present tight upper and lower bounds on the spanning ratio of a large family of -graphs. We show that graphs with 4k+2 cones (k 1 and integer) have a spanning ratio of 1 + 2 sin(= 2), where is 2= (4k + 2). We also show that -graphs with 4k + 4 cones have spanning ratio at least 1 + 2 tan(= 2) + 2 tan 2 (= 2), where is 2= (4k + 4). This is somewhat surprising since, for equal values of k, the spanning ratio of -graphs with 4k + 4 cones is greater than that of -graphs with 4k + 2 cones, showing that increasing the number of cones can make the spanning ratio worse.
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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 | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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 it