Bibliographic record
Abstract
This chapter provides an in-depth study of composition algebras over commutative rings, which we carry out in the more general framework of conic algebras (called quadratic algebras or algebras of degree 2 by other authors). We present the Cayley–Dickson construction and define composition algebras as unital nonassociative algebras that are projective as modules and allow a non-singular quadratic form permitting composition. We use this construction to obtain first examples of octonion algebras more general than the Graves–Cayley octonions and to derive structure theorems for arbitrary composition algebras. Specializing, it is shown that all composition algebras of rank at least 2 over an LG ring arise from an appropriate quadratic étale algebra by the Cayley–Dickson construction. Other examples of octonion algebras are obtained using ternary hermitian spaces. We address the norm equivalence problem, which asks whether composition algebras are classified by their norms and has an affirmative answer over LG rings but not in general. After a short excursion into affine (group) schemes, we conclude the chapter by showing that arbitrary composition algebras are split by étale covers.
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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 | 0.001 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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".