Continuum percolation threshold for interpenetrating squares and cubes
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Bibliographic record
Abstract
Monte Carlo simulations are performed to determine the critical percolation threshold for interpenetrating square objects in two dimensions and cubic objects in three dimensions. Simulations are performed for two cases: (i) objects whose edges are aligned parallel to one another and (ii) randomly oriented objects. For squares whose edges are aligned, the critical area fraction at the percolation threshold ${\ensuremath{\varphi}}_{c}=0.6666\ifmmode\pm\else\textpm\fi{}0.0004,$ while for randomly oriented squares ${\ensuremath{\varphi}}_{c}=0.6254\ifmmode\pm\else\textpm\fi{}0.0002,$ 6% smaller. For cubes whose edges are aligned, the critical volume fraction at the percolation threshold ${\ensuremath{\varphi}}_{c}=0.2773\ifmmode\pm\else\textpm\fi{}0.0002,$ while for randomly oriented cubes ${\ensuremath{\varphi}}_{c}=0.2168\ifmmode\pm\else\textpm\fi{}0.0002,$ 22% smaller.
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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) | 0.001 | 0.000 |
| 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 | 0.000 |
| 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 it