Icosahedral symmetry breaking: C<sub>60</sub>to C<sub>78</sub>, C<sub>96</sub>and to related nanotubes
Why this work is in the frame
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Bibliographic record
Abstract
Exact icosahedral symmetry of C 60 is viewed as the union of 12 orbits of the symmetric subgroup of order 6 of the icosahedral group of order 120. Here, this subgroup is denoted by A 2 because it is isomorphic to the Weyl group of the simple Lie algebra A 2 . Eight of the A 2 orbits are hexagons and four are triangles. Only two of the hexagons appear as part of the C 60 surface shell. The orbits form a stack of parallel layers centered on the axis of C 60 passing through the centers of two opposite hexagons on the surface of C 60 . By inserting into the middle of the stack two A 2 orbits of six points each and two A 2 orbits of three points each, one can match the structure of C 78 . Repeating the insertion, one gets C 96 ; multiple such insertions generate nanotubes of any desired length. Five different polytopes with 78 carbon-like vertices are described; only two of them can be augmented to nanotubes.
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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) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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 it