Birkhoff's Variety Theorem With and Without Free Algebras
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Bibliographic record
Abstract
For large signatures we prove that Birkhoff's Variety Theorem holds (i.e., equationally presentable collections of -algebras are precisely those closed under limits, subalgebras, and quotient algebras) iff the universe of small sets is not measurable.Under that limitation Birkhoff's Variety Theorem holds in fact for F -algebras of an arbitrary endofunctor F of the category Class of classes and functions.For endofunctors F of Set, the category of small sets, Jan Reiterman proved that if F is a varietor (i.e., if free F -algebras exist) then Birkhoff's Variety Theorem holds for F -algebras.We prove the converse, whenever F preserves preimages: if F is not a varietor, Birkhoff's Variety Theorem does not hold.However, we also present a nonvarietor satisfying Birkhoff's Variety Theorem.Our most surprising example is two varietors whose coproduct does not satisfy Birkhoff's Variety Theorem.
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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 | 0.000 | 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)
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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