Polynomial Wolff axioms and Kakeya-type estimates in R4
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Bibliographic record
Abstract
We establish new linear and trilinear bounds for collections of tubes in R4 that satisfy the polynomial Wolff axioms. In brief, a collection of δ-tubes satisfies the Wolff axioms if not too many tubes can be contained in the δ-neighborhood of a plane. A collection of tubes satisfies the polynomial Wolff axioms if not too many tubes can be contained in the δ-neighborhood of a low degree algebraic variety. First, we prove that if a set of δ-3 tubes in R4 satisfies the polynomial Wolff axioms, then the union of the tubes must have volume at least δ1-1/28. We also prove a more technical statement which is analogous to a maximal function estimate at dimension 3+1/28. Second, we prove that if a collection of δ-3 tubes in R4 satisfies the polynomial Wolff axioms, and if most triples of intersecting tubes point in three linearly independent directions, then the union of the tubes must have volume at least δ3/4. Again, we also prove a slightly more technical statement which is analogous to a maximal function estimate at dimension 3+1/4. We conjecture that every Kakeya set satisfies the polynomial Wolff axioms, but we are unable to prove this. If our conjecture is correct, it implies a Kakeya maximal function estimate at dimension 3+1/28, and in particular this implies that every Kakeya set in R4 must have Hausdorff dimension at least 3+1/28. This would be an improvement over the current best bound of 3, which was established by Wolff in 1995.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.000 | 0.002 |
| 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 it