Towards a characterization of Sidorenko systems
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Bibliographic record
Abstract
Abstract A system of linear forms $L=\{L_1,\ldots,L_m\}$ over $\mathbb{F}_q$ is said to be Sidorenko if the number of solutions to L = 0 in any $A \subseteq \mathbb{F}_{q}^n$ is asymptotically as $n\to\infty$ at least the expected number of solutions in a random set of the same density. Work of Saad and Wolf [19] and of Fox, Pham and Zhao [8] fully characterizes single equations with this property and both sets of authors ask about a characterization of Sidorenko systems of equations. In this paper, we make progress towards this goal. First, we find a simple necessary condition for a system to be Sidorenko, thus providing a rich family of non-Sidorenko systems. In the opposite direction, we find a large family of structured Sidorenko systems, by utilizing the entropy method. We also make significant progress towards a full classification of systems of two equations.
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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 | 0.002 | 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 | 0.001 | 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