A Fractal Function Related to the John–Nirenberg Inequality for<i>Q</i><sub>α</sub>(ℝ<sup>n</sup>)
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Bibliographic record
Abstract
Abstract A borderline case function f for Q α (ℝ n ) spaces is defined as a Haar wavelet decomposition, with the coefficients depending on a fixed parameter β > 0. On its support I 0 = [0, 1] n , f ( x ) can be expressed by the binary expansions of the coordinates of x . In particular, f = f β ∈ Q α (ℝ n ) if and only if α < β < , while for β = α, it was shown by Yue and Dafni that f satisfies a John–Nirenberg inequality for Q α (ℝ n ). When β ≠ 1, f is a self-affine function. It is continuous almost everywhere and discontinuous at all dyadic points inside I 0 . In addition, it is not monotone along any coordinate direction in any small cube. When the parameter β ∈ (0, 1), f is onto from I 0 to , and the graph of f has a non-integer fractal dimension n + 1 − β .
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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 | 0.003 | 0.008 |
| 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 | 0.001 |
| 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