Random Subnetworks of Random Sorting Networks
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
A sorting network is a shortest path from $12\cdots n$ to $n\cdots21$ in the Cayley graph of $S_n$ generated by nearest-neighbor swaps. For $m\leq n$, consider the random $m$-particle sorting network obtained by choosing an $n$-particle sorting network uniformly at random and then observing only the relative order of $m$ particles chosen uniformly at random. We prove that the expected number of swaps in location $j$ in the subnetwork does not depend on $n$, and we provide a formula for it. Our proof is probabilistic, and involves a Pólya urn with non-integer numbers of balls. From the case $m=4$ we obtain a proof of a conjecture of Warrington. Our result is consistent with a conjectural limiting law of the subnetwork as $n\to\infty$ implied by the great circle conjecture of Angel, Holroyd, Romik and Virág.
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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 | 0.003 |
| 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 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.002 |
| 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