NATURAL TILING, LATTICE TILING AND LEBESGUE MEASURE OF INTEGRAL SELF-AFFINE TILES
Why this work is in the frame
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Bibliographic record
Abstract
In the existing theory of self-affine tiles, one knows that the Lebesgue measure of any integral self-affine tile corresponding to a standard digit set must be a positive integer and every integral self-affine tile admits some lattice Γ⊆Zn as a translation tiling set of Rn. In this paper, we give algorithms to evaluate the Lebesgue measure of any such integral self-affine tile K and to determine all of the lattice tilings of Rn by K. Moreover, we also propose and determine algorithmically another type of translation tiling of Rn by K, which we call natural tiling. We also provide an algorithm to decide whether or not Lebesgue measure of the set K ∩ (K+j), j−Zn, is strictly positive.
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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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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