Bibliographic record
Abstract
The fundamentals of cubic norm structures and Jordan algebras are laid out in the first two sections of the chapter. We then derive elementary principles of building up “big” cubic norm structures out of “smaller” ones before we proceed to study cubic Jordan matrix algebras, the most important “hands-on” examples of cubic Jordan algebras. Next we turn to elementary idempotents, which will be used to present a special version of the Jacobson Coordinatization Theorem. Proceeding to Freudenthal algebras, we show that they exist only in ranks 1, 3, 6, 9, 15, and 27, with those of rank 27 being (finally!) called Albert algebras. We define the notion of a split Freudenthal algebra and prove, in analogy with composition algebras, that all Freudenthal algebras are split by some faithfully flat extension, though not always by an étale cover. After having investigated isotopies and norm similarities, with an important characterization of isotopes in Jordan matrix algebras over LG rings as its central result, we study reduced Freudenthal algebras over fields by exhibiting various classifying quadratic form invariants, particularly those pertaining to the invariants mod 2 of Albert algebras.
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.017 | 0.003 |
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".