Homotopy invariance of non-stable <i>K</i><sub>1</sub>-functors
Why this work is in the frame
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Bibliographic record
Abstract
Abstract Let G be a reductive algebraic group over a field k , such that every semisimple normal subgroup of G has isotropic rank ≥ 2, i.e. contains ( G m ) 2 . Let K 1 G be the non-stable K 1 -functor associated to G , also called the Whitehead group of G . We show that K 1 G ( k ) = K 1 G ( k [ X 1 ,…, X n ]) for any n ≥ 1. If k is perfect, this implies that K 1 G ( R ) = K 1 G ( R [ X ]) for any regular k -algebra R . If k is infinite perfect, one also deduces that K 1 G ( R ) → K 1 G ( K ) is injective for any local regular k -algebra R with the fraction field K .
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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 | 0.004 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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 it