A bound for the index of a quadratic form after scalar extension to the\n function field of a quadric
Bibliographic record
Abstract
Let $q$ be an anisotropic quadratic form defined over a general field $F$. In\nthis article, we formulate a new upper bound for the isotropy index of $q$\nafter scalar extension to the function field of an arbitrary quadric. On the\none hand, this bound offers a refinement of a celebrated bound established in\nearlier work of Karpenko-Merkurjev and Totaro; on the other, it is a direct\ngeneralization of Karpenko's theorem on the possible values of the first higher\nisotropy index. We prove its validity in two important cases: (i) the case\nwhere $\\mathrm{char}(F) \\neq 2$, and (ii) the case where $\\mathrm{char}(F) = 2$\nand $q$ is quasilinear (i.e., diagonalizable). The two cases are treated\nseparately using completely different approaches, the first being\nalgebraic-geometric, and the second being purely algebraic.\n
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| 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".