Sums of squares and moment problems in equivariant situations
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Bibliographic record
Abstract
We begin a systematic study of positivity and moment problems in an equivariant setting. Given a reductive group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> acting on an affine <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -variety <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding="application/x-tex">V</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we consider the induced dual action on the coordinate ring <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R left-bracket upper V right-bracket"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {R}[V]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and on the linear dual space of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R left-bracket upper V right-bracket"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {R}[V]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In this setting, given an invariant closed semialgebraic subset <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V left-parenthesis double-struck upper R right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">V(\mathbb R)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we study the problem of representation of invariant nonnegative polynomials on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by invariant sums of squares, and the closely related problem of representation of invariant linear functionals on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R left-bracket upper V right-bracket"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {R}[V]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by invariant measures supported on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . To this end, we analyse the relation between quadratic modules of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R left-bracket upper V right-bracket"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>V</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {R}[V]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and associated quadratic modules of the (finitely generated) subring <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R left-bracket upper V right-bracket Superscript upper G"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>V</mml:mi> <mml:msup> <mml:mo stretchy="false">]</mml:mo> <mml:mi>G</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {R}[V]^G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of invariant polynomials. We apply our resu
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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 | 0.001 |
| 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