An algebraic proof of generalized Wick theorem
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Bibliographic record
Abstract
The multireference normal order theory, introduced by Kutzelnigg and Mukherjee [J. Chem. Phys. 107, 432 (1997)], is defined explicitly, and an algebraic proof is given for the corresponding contraction rules for a product of any two normal ordered operators. The proof does not require that the contractions be cumulants, so it is less restricted. In addition, it follows from the proof that the normal order theory and corresponding contraction rules hold equally well if the contractions are only defined up to a certain level. These relaxations enable us to extend the original normal order theory. As a particular example, a quasi-normal-order theory is developed, in which only one-body contractions are present. These contractions are based on the one-particle reduced density matrix.
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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