Periods of Iterations of Functions with Restricted Preimage Sizes
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Bibliographic record
Abstract
Let [ n { = {1, …, n } and let Ω n be the set of all mappings from [ n { to itself. Let f be a random uniform element of Ω n and let T( f ) and B( f ) denote, respectively, the least common multiple and the product of the length of the cycles of f . Harris proved in 1973 that T converges in distribution to a standard normal distribution and, in 2011, Schmutz obtained an asymptotic estimate on the logarithm of the expectation of T and B over all mappings on n nodes. We obtain analogous results for random uniform mappings on n = kr nodes with preimage sizes restricted to a set of the form {0,k}, where k = k ( r ) ≥ 2. This is motivated by the use of these classes of mappings as heuristic models for the statistics of polynomials of the form x k + a over the integers modulo p , with p ≡ 1 (mod k). We exhibit and discuss our numerical results on this heuristic.
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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) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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