The GeometricDecomposability package for Macaulay2
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Bibliographic record
Abstract
1. INTRODUCTION.The geometric vertex decomposition of an ideal was first introduced by Knutson, Miller, and Yong [9] as part of their study of vexillary matrix Schubert varieties.Geometric vertex decomposition can be viewed as a generalization of a vertex decomposition of a simplicial complex.Using the notion of a geometric vertex decomposition, Klein and Rajchgot [7] introduced geometrically vertex decomposable ideals.These ideals, which are defined recursively, were partially inspired by the definition of a vertex decomposable simplicial complex, a recursively defined family of simplicial complexes.As shown by both [9] and [7], ideals that have a geometric vertex decomposition, or are geometrically vertex decomposable, have other desirable properties.As one such example, Klein and Rajchgot [7, Corollary 4.8] have shown that homogeneous geometrically vertex decomposable ideals are glicci, i.e., these ideals belong to the Gorenstein liaison class of a complete intersection.Further properties of geometrically vertex decomposable ideals have been developed in [4; 6; 8; 5].Due to their recent introduction, there are many features of geometrically vertex decomposable ideals that are still not known.To facilitate further experimentation and exploration, we have created GeometricDecomposability, a package for Macaulay2 that enables researchers to test and search for ideals that are geometrically vertex decomposable.In particular, our package allows the user to check if a given ideal satisfies the geometric vertex decomposition property of [9] or the geometrically vertex decomposable property (or its variants) as found in [7].This note reviews the needed mathematical background, summarizes the main features of our packages, and provides some illustrative examples.2. MATHEMATICAL BACKGROUND.We summarize the mathematical background to define the geometric vertex decomposition property and geometrically vertex decomposable ideals.Throughout, k denotes a field.
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 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 it