COMPOSITION OF HAAR PARAPRODUCTS: THE RANDOM CASE
Bibliographic record
Abstract
Abstract. When is the composition of paraproducts bounded? This is an important, and difficult question. We consider randomized variants of this question, finding non-classical characterizations. For dyadic interval I, let hI = h0 I be the L2-normalized Haar function adapted to I, the superscript 0 denoting that it has integral zero. Set h1 I = |hI|, the superscript 1 denoting a non-zero integral. A (classical dyadic) paraproduct with symbol b is one of the operators B(b, f) = ∑ 〈b, hI〉 √ 〈f, h |I | ǫ I 〉hδI. I∈D Here, ǫ, δ ∈ {0, 1}, with one of the two being zero and the other one. We characterize when certain randomized compositions B(b, B(β, ·)) are bounded operators on L 2 (R), permitting in particular both paraproducts to be unbounded. 1. Definitions and Main Theorems We phrase the (difficult) open question which motivates the consideration of this paper. Let D be the dyadic grid, and {hI: I ∈ D} the L2 normalized Haar basis, namely hI = |I | −1/2 ( −1Ileft
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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 | 0.000 | 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 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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".