On the Gaussian Product Inequality
Bibliographic record
Abstract
The long-standing Gaussian product inequality (GPI) conjecture states that E[|X_1|^{y_1} |X_2|^{y_2} ··· |X_n|^{y_n}] ≥ E[|X_1|^{y_1}] E[|X_2|^{y_2}] ··· E[|X_n|^{y_n}] for any centered Gaussian random vector (X_1 , . . . , X_n) and any non-negative real numbers y_j , j = 1, . . . , n. First, we complete the picture of bivariate Gaussian product relations by proving a novel “opposite GPI” when -1<y_1<0 and y_2>0: E[|X_1|^{y_1} |X_2|^{y_2}] ≤ E[|X_1|^{y_1}] E[|X_2|^{y_2}]. Next, we investigate the three-dimensional inequality E[X_1^2 X_2^{2m_2} X_n^{2m_3}] ≥ E[X_1^2] E[X_2^{2m_2}] E[X_n^{2m_3}] for any natural numbers m_2, m_3. We show that this inequality is implied by a combinatorial inequality which we verify directly for small values of m_2 and arbitrary m_3. Then, we complete the proof through the discovery of a novel moment ratio inequality which implies this three-dimensional GPI. We then extend these three-dimensional results to the case where the exponents in the GPI can be real numbers rather than simply even integers. Finally, we describe two computational algorithms involving sums-of-squares representations of polynomials that can be used to resolve the GPI conjecture. To exhibit the power of these novel methods, we apply them to prove new four- and five-dimensional GPIs.
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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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.003 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.000 | 0.002 |
| 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 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".