Local Complexity of Delone Sets and Crystallinity
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Bibliographic record
Abstract
Abstract This paper characterizes when a Delone set X in is an ideal crystal in terms of restrictions on the number of its local patches of a given size or on the heterogeneity of their distribution. For a Delone set X , let N X ( T ) count the number of translation-inequivalent patches of radius T in X and let M X (T) be the minimum radius such that every closed ball of radius M X (T) contains the center of a patch of every one of these kinds. We show that for each of these functions there is a “gap in the spectrum” of possible growth rates between being bounded and having linear growth, and that having sufficiently slow linear growth is equivalent to X being an ideal crystal. Explicitly, for N X (T) , if R is the covering radius of X then either N X (T) is bounded or N X (T) ≥ T /2 R for all T > 0. The constant 1/2R in this bound is best possible in all dimensions. For M X (T) , either M X (T) is bounded or M X (T) ≥ T /3 for all T > 0. Examples show that the constant 1/3 in this bound cannot be replaced by any number exceeding 1/2. We also show that every aperiodic Delone set X has M X (T) ≥ c(n)T for all T > 0, for a certain constant c(n) which depends on the dimension n of X and is > 1/3 when n > 1.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.066 | 0.001 |
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