On <span class="math inline">\(z\)</span>-cycle factorizations with two associate classes where <span class="math inline">\(z\)</span> is in <span class="math inline">\(\{4,4a\}\)</span> with even parameters
Bibliographic record
Abstract
<p>Let <span class="math inline">\(K = K(a,p;\lambda_1,\lambda_2)\)</span> be the multigraph with: the number of vertices in each part equal to <span class="math inline">\(a\)</span>; the number of parts equal to <span class="math inline">\(p\)</span>; the number of edges joining any two vertices of the same part equal to <span class="math inline">\(\lambda_1\)</span>; and the number of edges joining any two vertices of different parts equal to <span class="math inline">\(\lambda_2\)</span>. The existence of <span class="math inline">\(C_4\)</span>-factorizations of <span class="math inline">\(K\)</span> has been settled when <span class="math inline">\(a\)</span> is even; when <span class="math inline">\(a \equiv 1 \ (\mbox{mod } 4)\)</span> with one exception; and for very few cases when <span class="math inline">\(a \equiv 3 \ (\mbox{mod } 4)\)</span>. The existence of <span class="math inline">\(C_z\)</span>-factorizations of <span class="math inline">\(K\)</span> has been settled when <span class="math inline">\(a \equiv 1 \ (\mbox{mod } z)\)</span> and <span class="math inline">\(\lambda_1\)</span> is even; when <span class="math inline">\(a \equiv 0 \ (\mbox{mod } z)\)</span>; and when <span class="math inline">\(z=2a\)</span> where both <span class="math inline">\(a\)</span> and <span class="math inline">\(\lambda_1\)</span> is even. In this paper, we give a construction for <span class="math inline">\(C_z\)</span>-factorizations of <span class="math inline">\(K\)</span> for <span class="math inline">\(z \in \{ 4,4a \}\)</span> when <span class="math inline">\(a\)</span> is even.</p>
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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 | 0.005 | 0.004 |
| Meta-epidemiology (narrow) | 0.004 | 0.003 |
| Meta-epidemiology (broad) | 0.007 | 0.002 |
| Bibliometrics | 0.004 | 0.006 |
| Science and technology studies | 0.002 | 0.001 |
| Scholarly communication | 0.003 | 0.002 |
| Open science | 0.004 | 0.002 |
| Research integrity | 0.002 | 0.005 |
| 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; both teacher heads agree on what is shown here.
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".