Tight upper bounds for the p-anionic Clar number of fullerenes
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Bibliographic record
Abstract
A fullerene is an all-carbon molecule with a polyhedral structure whereeach atom is bonded to three other atoms and each face is either apentagon or a hexagon. Fullerenes correspond to 3-regular planar graphswhose faces have sizes 5 or 6. The p-anionic Clar number C_(p)(G) of afullerene G is equal to p + h, where h is maximized over all choices ofp + h independent faces (exactly p pentagons and h hexagons) thedeletion of whose vertices leave a graph with a perfect matching. Thisdefinition is motivated by the chemical observation that pentagonalrings can accommodate an extra electron, so that the pentagons of afullerene with charge −p, compete with the hexagons to host ‘Clarsextets’ of six electrons, and pentagons will preferentially acquire thep excess electrons of the anion. Tight upper bounds are established for the p-anionic Clar number offullerenes for p > 0. The upper bounds are derived via graph theoreticarguments and new results on minimal cyclic-k-edge cutsets in IPRfullerenes (fullerenes that have all pentagons pairwise disjoint). Thesebounds are shown to be tight by infinite families of fullerenes thatachieve them.
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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 | 0.000 | 0.000 |
| 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) | 0.004 | 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 it