When do all the integral points in a parallelepiped lie on a single hyperplane?
Bibliographic record
Abstract
We consider the problem of characterizing when all integral points in an integral parallelepiped lie on a single hyperplane. These points form a finite abelian group which is isomorphic to the quotient of the integer lattice by the sublattice generated by the defining vectors of the parallelepiped. In the case when this group is cyclic, we describe a combinatorial characterization in terms of a generator of that group. Our approach follows that of Morrison and Stevens, who use Bernoulli numbers to solve a related problem due to G. K. White concerning empty lattice tetrahedra. We state a more general characterization which we conjecture holds for any integral parallelepiped, and prove it in the case when the group is isomorphic to the additive group of a finite field.
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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 | 0.021 | 0.020 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.003 | 0.000 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 0.004 |
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; both teacher heads agree on what is shown here.
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".