Bibliographic record
Abstract
In 1906, Steinitz gave a complete characterization of the first two entries of the $f$-vectors of $3$-polytopes, while Grünbaum obtained a similar result for $4$-polytopes in his well-known book published in 1967. Recently, Kusunoki and Murai and independently Pineda-Villavicencio, Ugon, and Yost completely determined the first two entries of the $f$-vectors of $5$-polytopes. This paper can be regarded as a continuation of their works for $6$-polytopes. To be more precise, let $k$ denote the number of vertices of a $6$-polytope. The aim of this paper is to show that, when the number of edges is greater than or equal to $\frac{7}{2}(k-1)$ and $k\ge 14$, we can completely characterize the first two entries of the $f$-vectors of $6$-polytopes. As a consequence, for $7\le k\le 15$ we also give a complete characterization of the first two entries of the $f$-vectors of $6$-polytopes except for three cases $(12, 39)$, $(13, 43)$, and $(15, 47)$.
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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 | 0.000 | 0.001 |
| 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) | 0.000 | 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".