Type \((a,b,c)\) face-magic labelings of prism graphs
Why this work is in the frame
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Bibliographic record
Abstract
Let \(G=(V,E,F)\) be a planar graph with vertex set \(V\), edge set \(E\), and set of faces \(F.\) For nonnegative integers \(a,b,\) and \(c\), a type \((a,b,c)\) face-magic labeling of \(G\) is an assignment of \(a\) labels to each vertex, \(b\) labels to each edge, and \(c\) labels to each face from the set of integer labels \(\{1,2,\dots a|V|+b|E|+c|F|\}\) such that each label is used exactly once, and for each \(s\)-sided face \(f \in F,\) the sum of the label of \(f\) with the labels of the vertices and edges incident with \(f\) is equal to some fixed constant \(\mu_s\) for every \(s.\) We find necessary and sufficient conditions for every quadruple \((a,b,c,n)\) such that the \(n\)-prism graph \(Y_n \cong K_2 \square C_n\) admits a face-magic labeling of type \((a,b,c)\).
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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 | 0.001 | 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 | 0.000 |
| Open science | 0.001 | 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