The mean order of sub‐<i>k</i>‐trees of <i>k</i>‐trees
Why this work is in the frame
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Bibliographic record
Abstract
Abstract This article focuses on the problem of determining the mean orders of sub‐ k ‐trees of k ‐trees. It is shown that the problem of finding the mean order of all sub‐ k ‐trees containing a given k ‐clique C , can be reduced to the previously studied problem of finding the mean order of subtrees of a tree that contain a given vertex. This problem is extended in two ways. The first of these extensions focuses on the mean order of sub‐ k ‐trees containing a given sub‐ k ‐tree. The second extension focuses on the expected number of r ‐cliques, , in a randomly chosen sub‐ k ‐tree containing a fixed sub‐ k ‐tree X . Sharp lower bounds for both invariants are derived. The article concludes with a study of global mean orders of sub‐ k ‐trees of a k ‐tree. For a k ‐tree, from the class of simple‐clique k ‐trees, it is shown that the mean order of its sub‐ k ‐trees is asymptotically equal to the mean subtree order of its dual. For general k ‐trees a recursive generating function for the number of sub‐ k ‐trees of a given k ‐tree T is derived.
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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 | 0.003 | 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 | 0.001 | 0.001 |
| 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