The Hrushovski property, finite height, and maximal ascents
Bibliographic record
Abstract
In Chapter 1, we show that any compact nonpositively curved cube complex Y embeds in a compact nonpositively curved cube complex R where each partial local isometry of Y extends to an automorphism of R. We prove a similar result for compact special cube complexes provided that the partial local isometries satisfy certain conditions.In Chapter 2, we define the directed height of a mapping of graphs and relate it to the "algebraic" height of subgroups.We show that a map has finite directed height if and only if the corresponding mapping torus has negative immersions.We survey related properties and discuss how they relate to one another.In Chapter 3, we show that given a bi-order ≻ on the free group F, every non-periodic cyclically reduced word W ∈ F admits a maximal ascent that is uniquely positioned.This provides a cyclic permutation of W ′ that decomposes as W ′ = AD where A is an ascent and D is a descent.We show that if D is not uniquely positioned in W , then it must be an internal subword in A. Moreover, we show that when ≻ is the Magnus ordering, D = 1 F if and only if W is monotonic.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.009 | 0.001 |
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".