Bibliographic record
Abstract
We introduce and prove the n n -dimensional Pizza Theorem: Let H \mathcal {H} be a hyperplane arrangement in R n \mathbb {R}^{n} . If K K is a measurable set of finite volume, the pizza quantity of K K is the alternating sum of the volumes of the regions obtained by intersecting K K with the arrangement H \mathcal {H} . We prove that if H \mathcal {H} is a Coxeter arrangement different from A 1 n A_{1}^{n} such that the group of isometries W W generated by the reflections in the hyperplanes of H \mathcal {H} contains the map − i d -\mathrm {id} , and if K K is a translate of a convex body that is stable under W W and contains the origin, then the pizza quantity of K K is equal to zero. Our main tool is an induction formula for the pizza quantity involving a subarrangement of the restricted arrangement on hyperplanes of H \mathcal {H} that we call the even restricted arrangement. More generally, we prove that for a class of arrangements that we call even (this includes the Coxeter arrangements above) and for a sufficiently symmetric set K K , the pizza quantity of K + a K+a is polynomial in a a for a a small enough, for example if K K is convex and 0 ∈ K + a 0\in K+a . We get stronger results in the case of balls, more generally, convex bodies bounded by quadratic hypersurfaces. For example, we prove that the pizza quantity of the ball centered at a a having radius R ≥ ‖ a ‖ R\geq \|a\| vanishes for a Coxeter arrangement
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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) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.003 | 0.003 |
| Scholarly communication | 0.004 | 0.009 |
| Open science | 0.001 | 0.006 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.029 | 0.009 |
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".