Bibliographic record
Abstract
Let K be an uncountable compact metric space and let C(K,\mathbb{R}^d) denote the set of continuous maps f\colon K \to \mathbb{R}^d endowed with the maximum norm. The goal of this paper is to determine various fractal dimensions of the graph of the prevalent f\in C(K,\mathbb{R}^d) . As the main result of the paper we show that if K has finitely many isolated points then the lower and upper box dimension of the graph of the prevalent f\in C(K,\mathbb{R}^d) are \underline {\mathrm {dim}}_B K+d and \overline{\mathrm {dim}}_B K+d , respectively. This generalizes a theorem of Gruslys, Jonušas, Mijovic, Ng, Olsen, and Petrykiewicz. We prove that the packing dimension of the graph of the prevalent f\in C(K,\mathbb{R}^d) is \mathrm {dim}_P K+d , generalizing a result of Balka, Darji, and Elekes. Balka, Darji, and Elekes proved that the Hausdorff dimension of the graph of the prevalent f\in C(K,\mathbb{R}^d) equals \mathrm {dim}_H K+d . We give a simpler proof for this statement based on a method of Fraser and Hyde.
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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 | 0.002 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.003 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".