Asymptotics for the Probability of Connectedness and the Distribution of Number of Components
Why this work is in the frame
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Bibliographic record
Abstract
Let $\rho _n$ be the fraction of structures of "size" $n$ which are "connected"; e.g., (a) the fraction of labeled or unlabeled $n$-vertex graphs having one component, (b) the fraction of partitions of $n$ or of an $n$-set having a single part or block, or (c) the fraction of $n$-vertex forests that contain only one tree. Various authors have considered $\lim \rho _n$, provided it exists. It is convenient to distinguish three cases depending on the nature of the power series for the structures: purely formal, convergent on the circle of convergence, and other. We determine all possible values for the pair $(\liminf \rho _{n},\;\limsup \rho _{n})$ in these cases. Only in the convergent case can one have $0 < \lim \rho _{n} < 1$. We study the existence of $\lim \rho _{n}$ in this case.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 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