The height of scaled attachment random recursive trees
Why this work is in the frame
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Bibliographic record
Abstract
We study depth properties of a general class of random recursive trees where each node $n$ attaches to the random node $\lfloor nX_n \rfloor$ and $X_0, \ldots , X_n$ is a sequence of i.i.d. random variables taking values in $[0,1)$. We call such trees scaled attachment random recursive trees (SARRT). We prove that the height $H_n$ of a SARRT is asymptotically given by $H_n \sim \alpha_{\max} \log n$ where $\alpha_{\max}$ is a constant depending only on the distribution of $X_0$ whenever $X_0$ has a bounded density. This gives a new elementary proof for the height of uniform random recursive trees $H_n \sim e \log n$ that does not use branching random walks.
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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 | 0.002 | 0.003 |
| 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 | 0.004 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.001 |
| 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