Legendre pairs of lengths<i>ℓ</i>≡ 0 (mod 5)
Why this work is in the frame
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Bibliographic record
Abstract
Abstract By assuming a type of balance for length <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>ℓ</m:mi><m:mo>=</m:mo><m:mn>87</m:mn></m:math> \ell =87 and nontrivial subgroups of multiplier groups of Legendre pairs (LPs) for length <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>ℓ</m:mi><m:mo>=</m:mo><m:mn>85</m:mn></m:math> \ell =85 , we find LPs of these lengths. We then study the power spectral density (PSD) values of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>m</m:mi></m:math> m compressions of LPs of length <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mn>5</m:mn><m:mi>m</m:mi></m:math> 5m . We also formulate a conjecture for LPs of lengths <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>ℓ</m:mi><m:mo>≡</m:mo><m:mn>0</m:mn></m:math> \ell \equiv 0 (mod 5) and demonstrate how it can be used to decrease the search space and storage requirements for finding such LPs. The newly found LPs decrease the number of integers in the range <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mo>≤</m:mo><m:mn>200</m:mn></m:math> \le 200 for which the existence question of LPs remains unsolved from 12 to 10.
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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 | 0.002 |
| Science and technology studies | 0.000 | 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