Zonal and cozonal labelings using arbitrary abelian groups
Why this work is in the frame
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Bibliographic record
Abstract
Let <span class="math inline">\(G\)</span> be a plane graph with vertex, edge, and region sets <span class="math inline">\(V(G), E(G), F(G)\)</span> respectively. A zonal labeling of a plane graph <span class="math inline">\(G\)</span> is a labeling <span class="math inline">\(\ell: V(G)\rightarrow \{1,2\}\subset \mathbb{Z}_3\)</span> such that for every region <span class="math inline">\(R\in F(G)\)</span> with boundary <span class="math inline">\(B_R\)</span>, <span class="math inline">\(\sum\limits_{v\in V(B_R)}\ell(v)=0\)</span> in <span class="math inline">\(\mathbb{Z}_3\)</span>. We extend this to general abelian groups, defining a <span class="math inline">\(\Gamma\)</span>-zonal labeling as a labeling <span class="math inline">\(\ell:V(G)\rightarrow \Gamma\setminus \{0\}\)</span> such that for every region <span class="math inline">\(R\in F(G)\)</span>, <span class="math inline">\(\sum\limits_{v \in V(B_R)}\ell(v)\)</span> is <span class="math inline">\(0\)</span>. We explore existence of <span class="math inline">\(\Gamma\)</span>-zonal labelings for various families of graphs. We also introduce two variations: generative and strong <span class="math inline">\(\Gamma\)</span>-zonal labelings. A generative <span class="math inline">\(\Gamma\)</span>-zonal labeling is one in which the elements used to label the vertices generate the group <span class="math inline">\(\Gamma\)</span>. A strong <span class="math inline">\(\Gamma\)</span>-zonal labeling is a labeling in which the additive order of <span class="math inline">\(\ell(v)\)</span> is equal to <span class="math inline">\(\deg(v).\)</span> Examples and existence results are provided for both variations. It is shown that strong <span class="math inline">\(\Gamma\)</span>-zonal labelings have a connection to edge colorings that generalizes the connection between zonal labelings and proper edge <span class="math inline">\(3\)</span>-colorings of cubic maps.
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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 | 0.000 |
| 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) | 0.003 | 0.002 |
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