Edge-Graph Diameter Bounds for Convex Polytopes with Few Facets
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Bibliographic record
Abstract
We show that the edge graph of a 6-dimensional polytope with\n12 facets has diameter at most 6, thus verifying the $d$-step conjecture\nof Klee and Walkup in the case $d = 6$. This implies that\nfor all pairs $(d, n)$ with $n − d ≤ 6$, the diameter of the edge graph\nof a $d$-polytope with $n$ facets is bounded by 6, which proves\nthe Hirsch conjecture for all $n − d ≤ 6$. We prove this result by\nestablishing this bound for a more general structure, so-called\nmatroid polytopes, by reduction to a small number of satisfiability\nproblems.
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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 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| 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 it