Regular pandiagonal sparse magic squares of order \(n\equiv 5 \pmod{6}\) with density 6
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Bibliographic record
Abstract
<p>Sparse magic squares are a generalization of magic squares and can be used to the magic labeling of graphs. An <span class="math inline">\(n\times n\)</span> array based on <span class="math inline">\(\mathcal{X}\)</span><span class="math inline">\(=\{0,1,\cdots,nd\}\)</span> is called <em>a sparse magic square of order <span class="math inline">\(n\)</span> with density <span class="math inline">\(d\)</span></em> (<span class="math inline">\(d<n\)</span>), denoted by SMS<span class="math inline">\((n,d)\)</span>, if each non-zero element of <span class="math inline">\(\mathcal{X}\)</span> occurs exactly once in the array, and its row-sums, column-sums and two main diagonal sums is the same. An SMS<span class="math inline">\((n,d)\)</span> is called <em>pandiagonal</em> (or <em>perfect</em>) denoted by PSMS<span class="math inline">\((n,d)\)</span>, if the sum of all elements in each broken diagonal is the same. A PSMS<span class="math inline">\((n,d)\)</span> is called <em>regular</em> if there are eactly <span class="math inline">\(d\)</span> positive entries in each row, each column and each main diagonal. In this paper, some construction of regular pandigonal sparse magic squares is provided and it is proved that there exists a regular PSMS<span class="math inline">\((n,6)\)</span> for all positive integer <span class="math inline">\(n\equiv 5 \pmod{6}\)</span>, <span class="math inline">\(n>6\)</span>.</p>
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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) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.003 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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