Infrastructure: Structure Inside the Class Group of a Real Quadratic Field
Bibliographic record
Abstract
Cows and Fields Suppose that you are a wise ancient Greek, and that you have been given the task of counting the Sun God’s cattle on the island of Sicily. They are too numerous to count manually, and your only clue is a puzzle that relates the number of cows and bulls of a particular color to those of another. How can you determine the size of your herd from such cryptic information? A precise version of this numerical puzzle is the well-known cattle problem of Archimedes [2]. A number of accounts of its solution appear in the literature [31, 21, 19]. Unfortunately for the hypothetical Greek sage, this problem does not have a small and completely elementary solution— it would take more than 2,000 years before the first solution was discovered by Amthor [1]. Had a solution been available, the Greek would likely have been shocked to find that the Sun God had somehow managed to pack a herd of more than 10206544 animals onto Sicily! One of the main ingredients in Amthor’s solution was solving the Diophantine equation
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How this classification was reachedexpand
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 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".