Bibliographic record
Abstract
A string p = p 0 p 1 \\Delta \\Delta \\Delta p n\\Gamma1 of non-negative integers is a Euclidean string if the string (p 0 +1)p 1 \\Delta \\Delta \\Delta (p n\\Gamma1 \\Gamma 1) is rotationally equivalent to p. We show that Euclidean strings exist if and only if n and p 0 + p 1 + \\Delta \\Delta \\Delta + p n\\Gamma1 are relatively prime and that, if they exist, they are unique. We show how to construct them using an algorithm with the same structure as the Euclidean algorithm, hence the name. We show that Euclidean strings are Lyndon words and we describe relationships between Euclidean strings and the Stern-Brocot tree, Fibonacci strings and Beatty sequences. We also describe an application to a graph embedding problem. Keywords: Euclidean, string, Lyndon, Stern-Brocot, Fibonacci, Beatty This work was supported by the Natural Sciences and Engineering Research Council of Canada and by Czech grant GACR 201/99/0242 1 1
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How this classification was reachedexpand
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 | 0.000 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".