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Bibliographic record
Abstract
<p>We consider a joint ordered multifactorisation for a given positive integer <span class="math inline">\(n\geq 2\)</span> into <span class="math inline">\(m\)</span> parts, where <span class="math inline">\(n=n_1~\times~\ldots~\times~n_m\)</span>, and each part <span class="math inline">\(n_j\)</span> is split into one or more component factors. Our central result gives an enumeration formula for all such joint ordered multifactorisations <span class="math inline">\(\mathcal{N}_m(n)\)</span>. As an illustrative application, we show how each such factorisation can be used to uniquely construct and so count the number of distinct additive set systems (historically referred to as complementing set systems). These set systems under set addition generate the first <span class="math inline">\(n\)</span> non-negative consecutive integers uniquely and, when each component set is centred about 0, exhibit algebraic invariances. For fixed integers <span class="math inline">\(n\)</span> and <span class="math inline">\(m\)</span>, invariance properties for <span class="math inline">\(\mathcal{N}_m(n)\)</span> are established. The formula for <span class="math inline">\(\mathcal{N}_m(n)\)</span> is comprised of sums over associated divisor functions and the Stirling numbers of the second kind, and we conclude by deducing sum over divisor relations for our counting function <span class="math inline">\(\mathcal{N}_m(n)\)</span>. Some related integer sequences are also considered.</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 | 0.000 | 0.001 |
| 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) | 0.000 | 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