Certain rigid maximally mutable Laurent polynomials in three variables
Bibliographic record
Abstract
This dataset contains certain rigid maximally mutable Laurent polynomials (rigid MMLPs) in three variables. Rigid MMLPs are defined in reference [1]. The Newton polytopes of these Laurent polynomials are three-dimensional canonical Fano polytopes. That is, they are three-dimensional convex polytopes with vertices that are primitive integer vectors and that contain exactly one lattice point, the origin, in their strict interior. See references [2] and [3]. Although the rigid MMLPs specified in this dataset have 3-dimensional canonical Fano Newton polytope, this is by no means an exhaustive list of such Laurent polynomials. The dataset contains examples of rigid MMLPs that correspond under mirror symmetry to three-dimensional Q-Fano varieties of particulaly high estimated codimension: see reference [4]. The file "rigid_MMLPs.txt" contains key:value records with keys and values as described below, separated by blank lines. Each key:value record determines a rigid MMLP, and there are 130 records in the file. An example record is: canonical3_id: 231730 coefficients: [1,1,1,1,1,1,1,1,1,1] exponents: [[-1,-1,-1],[0,1,0],[0,1,1],[1,0,0],[1,0,1],[1,2,2],[2,1,2],[2,1,3],[3,3,5],[4,2,5]] period: [1,0,0,12,24,0,540,2940,2520,33600,327600,693000,2795100,35315280,129909780,354666312,3816572760,20559258720,59957561664,435508321248,2969362219824] ulid: 01G5CBH3F86NRYF0TJ8MYWM41H The keys and values are as follows, where f denotes the Laurent polynomial defined by the key:value record. canonical3_id: an integer, the ID of the Newton polytope of f in reference [3] coefficients: a string of the form "[c1,c2,...,cN]" where c1, c2, ... are integers. These are the coefficients of f. exponents: a string of the form "[[x1,y1,z1],[x2,y2,z2],...,[xN,yN,zN]]" where x1, y1, z1, ..., xN, yN, zN are integers. These are the exponents of f. period: a string of the form "[d0,d1,...,d20]" where d0, d1, ..., d20 are non-negative integers that give the first 21 terms of the period sequence for f. ulid: a string that uniquely identified this entry in the dataset The sequences defined by the keys "coefficients" and "exponents" are parallel to each other. The period sequence for f is defined, for example, in equations 1.2 and 1.3 of reference [1]. References [1] Tom Coates, Alexander M. Kasprzyk, Giuseppe Pitton, and Ketil Tveiten. Maximally mutable Laurent polynomials. Proceedings of the Royal Society A 477, no. 2254:20210584, 2021. [2] Alexander M. Kasprzyk. Canonical toric Fano threefolds. Canadian Journal of Mathematics, 62(6):1293–1309, 2010. [3] Alexander M. Kasprzyk. The classification of toric canonical Fano 3-folds. Zenodo, https://doi.org/10.5281/zenodo.5866330, 2010. [4] Liana Heuberger. Q-Fano threefolds and Laurent inversion. Preprint, arXiv:2202.04184, 2022.
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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) | 0.002 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.004 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.021 | 0.018 |
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".