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 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 | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.012 | 0.004 |
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".