Uniform property Γ and the small boundary property
Bibliographic record
Abstract
We prove that, for a free action α : G ↷ X \alpha \colon G \curvearrowright X of a countably infinite discrete amenable group on a compact metric space, the small boundary property is implied by uniform property Γ \Gamma of the Cartan subalgebra ( C ( X ) ⊆ C ( X ) ⋊ α G ) (C(X) \subseteq C(X) \rtimes _\alpha G) . The reverse implication has been demonstrated by Kerr and Szabó, from which we obtain that these two conditions are equivalent. We moreover show that, if α \alpha is also minimal, then almost finiteness of α \alpha is implied by tracial Z \mathcal {Z} -stability of the subalgebra ( C ( X ) ⊆ C ( X ) ⋊ α G ) (C(X) \subseteq C(X) \rtimes _\alpha G) . The reverse implication is due to Kerr, resulting in the equivalence of these two properties as well. As an application, we prove that if α : G ↷ X \alpha \colon G \curvearrowright X and β : H ↷ Y \beta \colon H \curvearrowright Y
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.005 |
| Scholarly communication | 0.002 | 0.008 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.016 | 0.002 |
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 source (direct Gemma or distilled Codex), 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".