Irreducible polynomials and barker sequences
Why this work is in the frame
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Bibliographic record
Abstract
A Barker sequence is a finite sequence a o , ..., a n -1 , each term ±1, for which every sum Σ i a i a i +k with 0 < k < n is either 0, 1, or -- 1. It is widely conjectured that no Barker sequences of length n > 13 exist, and this conjecture has been verified for the case when n is odd. We show that in this case the problem can in fact be reduced to a question of irreducibility for a certain family of univariate polynomials: No Barker sequence of length 2 m + 1 exists if a particular integer polynomial of degree 4 m is irreducible over Q. A proof of irreducibility for this family would thus provide a short, alternative proof that long Barker sequences of odd length do not exist. However, we also prove that the polynomials in question are always reducible modulo p , for every prime p .
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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 | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.004 | 0.003 |
| 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