Vishik equivalence and similarity of quadratic forms over fields of characteristic 2
Bibliographic record
Abstract
An important aspect in the algebraic theory of quadratic forms is the study of equivalence relations based on algebraic-geometric properties of the associated quadrics. A well-known criterion originally proved by Vishik in characteristic zero states that two nonsingular quadratic forms of the same dimension have identical Witt indices over all field extensions if and only if their motives are isomorphic in the category of (integral or mod 2) Chow motives. In characteristic 2, it is meaningful to include singular forms. We therefore define two quadratic forms (including singular ones) of the same dimension to be Vishik-equivalent if they share the same isotropy behavior (in a suitably defined way) over all field extensions. Similar quadratic forms are always Vishik-equivalent, but the converse need not hold. We determine various classes of quadratic forms in characteristic 2 where Vishik equivalence implies similarity and give nonsingular counterexamples in all dimensions 2 n ≥ 8 , and also singular counterexamples in dimension 8. To construct the counterexamples, we use a generalized notion of so-called half-neighbors in characteristic 2.
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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 | 0.001 | 0.001 |
| 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 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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".