Counting Strings with Given Elementary Symmetric Function Evaluations II: Circular Strings
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Bibliographic record
Abstract
Let $\alpha$ be a string over an alphabet that is a finite ring, R. The kth elementary symmetric function evaluated at $\alpha$ is denoted $T_k(alpha)$. In a companion paper we studied the properties of $\mathbf{S}_R(n;\tau_1,\tau_2,\ldots,\tau_k)$, the set of length n strings for which $T_i(\alpha) = \tau_i$. Here we consider the set, $\mathbf{L}_R(n;\tau_1,\tau_2,\ldots,\tau_k)$, of equivalence classes under rotation of aperiodic strings in $\mathbf{S}_R(n;\tau_1,\tau_2,\ldots,\tau_k)$, sometimes called Lyndon words. General formulae are established and then refined for the cases where R is the ring of integers $\Z{q}$ or the finite field $\F{q}$.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| 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