Bibliographic record
Abstract
A d-gem is a {+, -, ×}-circuit having very few ×-gates and computing from {x} ∪ Z a univariate polynomial of degree d having d distinct integer roots.We introduce d-gems because they offer the remote possibility of being helpful for factoring integers and because their existence for infinitely many d would disprove a form of the Blum-Cucker-Shub-Smale conjecture (strengthened to allow arbitrary constants).A natural step towards a better understanding of the BCSS conjecture would thus be to construct d-gems or to rule out their existence.Ruling out d-gems for large d is currently totally out of reach.Here the best we can do towards that goal is to prove that skew 2 n -gems if they exist require n {+, -}-gates and that skew 2 n -gems for any n ≥ 5 would provide new solutions to the Prouhet-Tarry-Escott problem in number theory (skew meaning the further restriction that each {+, -}-gate merely adds an integer to a polynomial).In the opposite direction, here we do manage to construct skew d-gems for several values of d up to 55.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.008 | 0.001 |
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 source (direct Gemma or distilled Codex), 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".