Hoffmann’s conjecture for totally singular forms of prime degree
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Bibliographic record
Abstract
One of the most significant discrete invariants of a quadratic form [math] over a field [math] is its (full) splitting pattern, a finite sequence of integers which describes the possible isotropy behavior of [math] under scalar extension to arbitrary overfields of [math] . A similarly important but more accessible variant of this notion is that of the Knebusch splitting pattern of [math] , which captures the isotropy behavior of [math] as one passes over a certain prescribed tower of [math] -overfields. We determine all possible values of this latter invariant in the case where [math] is totally singular. This includes an extension of Karpenko’s theorem (formerly Hoffmann’s conjecture) on the possible values of the first Witt index to the totally singular case. Contrary to the existing approaches to this problem (in the nonsingular case), our results are achieved by means of a new structural result on the higher anisotropic kernels of totally singular quadratic forms. Moreover, the methods used here readily generalize to give analogous results for arbitrary Fermat-type forms of degree [math] over fields of characteristic [math] .
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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 | 0.002 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.007 | 0.000 |
Machine scores (provisional)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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