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Bibliographic record
Abstract
The <italic>Constraint Satisfaction Problem Dichotomy Conjecture</italic> of Feder and Vardi (1999) has in the last 10 years been profitably reformulated as a conjecture about the set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper S sans-serif upper P Subscript sans-serif fin Baseline sans-serif left-parenthesis bold upper A sans-serif right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">S</mml:mi> <mml:mi mathvariant="sans-serif">P</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext mathvariant="sans-serif">fin</mml:mtext> </mml:mrow> </mml:msub> <mml:mo mathvariant="sans-serif" stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">A</mml:mi> </mml:mrow> <mml:mo mathvariant="sans-serif" stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\sf {SP}_\textsf {fin}(\mathbf {A})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of subalgebras of finite Cartesian powers of a finite universal algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . One particular strategy, advanced by Dalmau in his doctoral thesis (2000), has confirmed the conjecture for a certain class of finite algebras <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which, among other things, have the property that the number of subalgebras of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper A Superscript n"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">A</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbf {A}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is bounded by an exponential polynomial. In this paper we characterize the finite algebras <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with this property, which we call <italic>having few subpowers</italic> , and develop a representation theory for the subpowers of algebras having few subpowers. Our characterization shows that algebras having few subpowers are the finite members of a newly discovered and surprisingly robust Maltsev class defined by the existence of a special term we call an <italic>edge term</italic> . We also prove some tight connections between the asymptotic behavior of the number of subalgebras of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper A Superscript n"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">A</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbf {A}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and some related functions on the one hand, and some standard algebraic properties of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbf {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on the other hand. The theory developed here was applied to the Constraint Satisfaction Problem Dichotomy Conjecture, completing Dalmau’s strategy.
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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) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.000 | 0.004 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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