Sidorenko‐Type Inequalities for Pairs of Trees
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Bibliographic record
Abstract
ABSTRACT Given two non‐empty graphs and , write to mean that for every graph , where is the homomorphism density function. We obtain various necessary and sufficient conditions for two trees and to satisfy and determine all such pairs on at most 8 vertices. This extends the results of Leontovich and Sidorenko from the 1980s and 1990s. Our approach applies an information‐theoretic technique to reduce the problem of showing that for two forests and to solving a linear program of Kopparty and Rossman (2011). We also characterize trees which satisfy or , where is the ‐vertex star and is the 4‐vertex path and resolve a problem of Csikvári and Lin (2015).
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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 | 0.000 |
| 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) | 0.000 | 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