Bibliographic record
Abstract
In modules many ‘positive’ versions of model-theoretic concepts turn out to be equivalent to concepts known in classical module theory—by ‘positive’ we mean that instead of allowing arbitrary first-order formulas in the model-theoretic definitions only positive primitive formulas are taken into consideration. (This feature is due to Baur's quantifier elimination for modules, cf. [Pr], however we will not make explicit use of it here.) Often this allows one to combine model-theoretic methods with algebraic ones. One instance of this is the result proved in [Rot1] (see also [Rot2]) that the Mittag-Leffler modules are exactly the positively atomic modules. This paper is parallel to the one just mentioned in that it is proved here, Theorem 3.1, that the pure-projective modules are exactly the positively constructible modules. The following parallel facts from module theory and from model theory led us to this result: every pure-projective module is Mittag-Leffler and the converse is true for countable (in fact even countably generated) modules, cf. [RG]; every constructible model is atomic and the converse is true for countable models, cf. [Pi].
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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 | 0.000 | 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 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".