A division algorithm for the Gaussian integers’ minimal Euclidean function
Bibliographic record
Abstract
Abstract The usual division algorithms on double struck upper Z ${\mathbb {Z}}$ Z and double struck upper Z i ${\mathbb {Z}}[i]$ Z i measure the size of remainders using the algebraic norm. These rings are Euclidean with respect to several functions. The pointwise minimum of all Euclidean functions f colon upper R divided by StartSet 0 EndSet right arrow double struck upper N $f: R \setminus \{0\} \rightarrow {\mathbb {N}}$ f : R ⧵ { 0 } → N on a Euclidean domain R is itself a Euclidean function, called the minimal Euclidean function and denoted by phi Subscript upper R $\phi _R$ ϕ R . To the author’s knowledge, the integers, double struck upper Z ${\mathbb {Z}}$ Z and the Gaussians, double struck upper Z i ${\mathbb {Z}}[i]$ Z i are the only rings of integers of number fields for which we have a formula to compute their minimal Euclidean functions, phi Subscript double struck upper Z $\phi _{{\mathbb {Z}}}$ ϕ Z and phi Subscript double struck upper Z i $\phi _{{\mathbb {Z}}[i]}$
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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 | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.003 | 0.003 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.016 | 0.007 |
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".