Bibliographic record
Abstract
This chapter is concerned with the two Tits constructions of cubic Jordan algebras over commutative rings, which we present for the first time in book form. The key feature of the first Tits construction is that it starts out from cubic alternative algebras rather than cubic associative ones; the key notion that keeps the construction going is that of a Kummer element. Similar statements apply to the second Tits construction, where Kummer elements are replaced by étale elements. Such elements are available in abundance over residually big LG rings. It follows that all Albert algebras over such rings or over arbitrary fields may be obtained from the second Tits construction. The chapter concludes with an application to cubic Jordan division algebras over fields. We show that they are either purely inseparable field extensions of characteristic 3 or Freudenthal algebras of dimension 1, 3, 9, or 27. In each of these dimensions, we construct examples over appropriate fields and conclude the section by showing that over the “standard” fields (the complex numbers, the real numbers, finite, local and global ones), Albert division algebras do not exist.
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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 | 0.000 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 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 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".