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 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 machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.005 | 0.022 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.005 |
| Scholarly communication | 0.003 | 0.007 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.011 | 0.002 |
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 source (direct Gemma or distilled Codex), 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".