A Computational Framework for Determining Square-maximal Strings.
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Bibliographic record
Abstract
Abstract. We investigate the function σd(n) = max{s(x) | x is a (d, n)-string}, where s(x) denotes the number of distinct primitively rooted squares in a string x and (d, n)-string denotes a string of length n with exactly d distinct symbols. New properties of the σd(n) function are presented. The notion of s-cover is presented and discussed with emphasis on the recursive computational determination of σd(n). In particular, we were able to determine all values of σ2(n) for n ≤ 53 and σ3(n) for n ≤ 42 and to point out that σ2(33) < σ3(33); that is, among all strings of length 33, no binary string achieves the maximum number of distinct primitively rooted squares. Noticeably, these computations reveal the unexpected existence of pairs (d, n) satisfying σd+1(n + 2) − σd(n)> 1 such as (2,33) and (2,34), and of three consecutive equal values: σ2(31) = σ2(32) = σ2(33). In addition we show that σ2(n) ≤ 2n − 66 for n ≥ 53.
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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 | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 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