Computational Progress on the Unfair 0-1 Polynomial Conjecture
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Bibliographic record
Abstract
Let c(x) be a monic integer polynomial with coefficients 0 or 1. Write c(x)=a(x)b(x) where a(x) and b(x) are monic polynomials with non-negative real coefficients (not necessarily integer). The unfair 0–1 polynomial conjecture states that a(x) and b(x) are necessarily integer polynomials with coefficients 0 or 1. Let a(x) be a candidate factor of a (currently unknown) 0–1 polynomial. We will assume that we know if a coefficient is 0, 1 or strictly between 0 and 1, but that we do not know the precise value of non-integer coefficients. Given this candidate a(x), this paper gives an algorithm to find a b(x) and c(x) with a(x)b(x)=c(x) such that b(x) has non-negative real coefficients and c(x) has coefficients 0 or 1, or (often) shows that such c(x) and b(x) do not exist. Using this algorithm, we consider all candidate factors with degree less than or equal to 15. With the exception of 975 candidate factors (out of a possible 7141686 cases), this algorithm shows that there do not exist b(x) with non-negative real coefficients and c(x) with coefficients 0 or 1 such that a(x)b(x)=c(x).
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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 | 0.000 |
| 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