Asymptotical Behaviour of Roots of Infinite Coxeter Groups
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Bibliographic record
Abstract
Abstract Let W be an infinite Coxeter group. We initiate the study of the set E of limit points of “normalized” roots (representing the directions of the roots) of W . We show that E is contained in the isotropic cone Q of the bilinear form B associated with a geometric representation, and we illustrate this property with numerous examples and pictures in rank 3 and 4. We also define a natural geometric action of W on E , and then we exhibit a countable subset of E , formed by limit points for the dihedral reflection subgroups of W . We explain how this subset is built fromthe intersection with Q of the lines passing through two positive roots, and finally we establish that it is dense in E .
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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 | 0.001 | 0.003 |
| 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) | 0.000 | 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 it