The quadratic approximation lemma and decompositions of superlative indexes
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Bibliographic record
Abstract
It was shown in 1976 that a difference in a quadratic function of N variables evaluated at two points is exactly equal to the sum of the arithmetic average of the first order partial derivatives of the function evaluated at the two points times the differences in the independent variables. In the present paper, this result is generalized and the resulting generalized quadratic approximation lemma is used to establish all of the superlative index number formulae that were derived in Diewert [4]. In addition, some new exact decompositions of the percentage change in the Fisher and Walsh superlative indexes into N components are derived. Each component in this decomposition represents the contribution of a change in a single independent variable to the overall percentage change in the index. Finally, these components are given economic interpretations.
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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 | 0.001 | 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