The algebraic matroid of the Heron variety
Bibliographic record
Abstract
We introduce the n th Heron variety as the realization space of the (squared) volumes of faces of an n -simplex. Our primary goal is to understand the extent to which Heron’s formula, which expresses the area of a triangle as a function of its three edge lengths, can be generalized. Such a formula for one face volume of an n -simplex in terms of other face volumes expresses a dependence in the algebraic matroid of the Heron variety. Whether the volume is expressible in terms of radicals is controlled by the monodromy groups of the coordinate projections of the Heron variety onto coordinates of bases. We discuss a suite of algorithms, some new, for determining these matroids and monodromy groups. We apply these algorithms toward the smaller Heron varieties, organize our findings, and interpret the results in the context of our original motivation.
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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 | 0.000 | 0.000 |
| 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 | 0.002 | 0.001 |
| 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".