LOW-DEGREE BOOLEAN FUNCTIONS ON , WITH AN APPLICATION TO ISOPERIMETRY
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Bibliographic record
Abstract
We prove that Boolean functions on $S_{n}$ , whose Fourier transform is highly concentrated on irreducible representations indexed by partitions of $n$ whose largest part has size at least $n-t$ , are close to being unions of cosets of stabilizers of $t$ -tuples. We also obtain an edge-isoperimetric inequality for the transposition graph on $S_{n}$ which is asymptotically sharp for subsets of $S_{n}$ of size $n!/\text{poly}(n)$ , using eigenvalue techniques. We then combine these two results to obtain a sharp edge-isoperimetric inequality for subsets of $S_{n}$ of size $(n-t)!$ , where $n$ is large compared to $t$ , confirming a conjecture of Ben Efraim in these cases.
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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 | 0.000 |
| 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 | 0.000 |
| 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