Practical software for triangulating and simplifying 4-manifolds
Bibliographic record
Abstract
Dimension 4 is the first dimension in which exotic smooth manifold pairs appear—manifolds which are topologically the same but for which there is no smooth deformation of one into the other. Whilst smooth and piecewise-linear structures on $4$-manifolds do coincide, comparatively little work has been done towards gaining an understanding of smooth $4$-manifolds from the discrete and algorithmic perspective. In this paper we introduce new software tools to make this possible, including a software implementation of an algorithm which enables us to build triangulations of $4$-manifolds from suitable Kirby diagrams, as well as a new heuristic for simplifying $4$-manifold triangulations. Using these tools, we present new triangulations of several exotic pairs with boundary, corks and plugs (objects responsible for "exoticity"), as well as the smallest known triangulation of the $K3$ surface. The small size of these triangulations benefits us by revealing fine structural features in $4$-manifold triangulations.
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How this classification was reachedexpand
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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".