On optimal Scott sentences of finitely generated algebraic structures
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Bibliographic record
Abstract
Scott showed that for every countable structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , there is a sentence of the infinitary logic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L Subscript omega 1 omega"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi> ω </mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mi> ω </mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {L}_{\omega _1\omega }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , called a <italic>Scott sentence</italic> for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , whose countable models are exactly the isomorphic copies of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Thus, the least quantifier complexity of a Scott sentence of a structure is an invariant that measures the complexity “describing” the structure. Knight et al. have studied the Scott sentences of many structures. In particular, Knight and Saraph showed that a finitely generated structure always has a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma 3 Superscript 0"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="normal"> Σ </mml:mi> <mml:mn>3</mml:mn> <mml:mn>0</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\Sigma ^0_3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Scott sentence. We give a characterization of the finitely generated structures for which the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma 3 Superscript 0"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="normal"> Σ </mml:mi> <mml:mn>3</mml:mn> <mml:mn>0</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\Sigma ^0_3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Scott sentence is optimal. One application of this result is to give a construction of a finitely generated group where the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Sigma 3 Superscript 0"> <mml:semantics> <mml:msubsup> <mml:mi mathvariant="normal"> Σ </mml:mi> <mml:mn>3</mml:mn> <mml:mn>0</mml:mn> </mml:msubsup> <mml:annotation encoding="application/x-tex">\Sigma ^0_3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Scott sentence is optimal.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.000 | 0.006 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.003 | 0.002 |
| 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