Bibliographic record
Abstract
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper H sans-serif o sans-serif m Superscript 0 Baseline left-parenthesis normal upper Gamma comma upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">H</mml:mi> <mml:mi mathvariant="sans-serif">o</mml:mi> <mml:mi mathvariant="sans-serif">m</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>0</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Γ<!-- Γ --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {Hom}^{0}(\Gamma ,G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the connected component of the identity of the variety of representations of a finitely generated nilpotent group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Gamma"> <mml:semantics> <mml:mi mathvariant="normal">Γ<!-- Γ --></mml:mi> <mml:annotation encoding="application/x-tex">\Gamma</mml:annotation> </mml:semantics> </mml:math> </inline-formula> into a connected compact Lie 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>, and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper X Superscript 0 Baseline left-parenthesis normal upper Gamma comma upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">X</mml:mi> </mml:mrow> <mml:mn>0</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Γ<!-- Γ --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {X}^0(\Gamma ,G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the corresponding moduli space. We show that there exists a natural <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper O sans-serif u sans-serif t left-parenthesis normal upper Gamma right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">O</mml:mi> <mml:mi mathvariant="sans-serif">u</mml:mi> <mml:mi mathvariant="sans-serif">t</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Γ<!-- Γ --></mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {Out}(\Gamma )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-invariant measure on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper X Superscript 0 Baseline left-parenthesis normal upper Gamma comma upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">X</mml:mi> </mml:mrow> <mml:mn>0</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Γ<!-- Γ --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {X}^0(\Gamma ,G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and that whenever <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper O sans-serif u sans-serif t left-parenthesis normal upper Gamma right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">O</mml:mi> <mml:mi mathvariant="sans-serif">u</mml:mi> <mml:mi mathvariant="sans-serif">t</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Γ<!-- Γ --></mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {Out}(\Gamma )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has at least one hyperbolic element, the action of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper O sans-serif u sans-serif t left-parenthesis normal upper Gamma right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">O</mml:mi> <mml:mi mathvariant="sans-serif">u</mml:mi> <mml:mi mathvariant="sans-serif">t</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Γ<!-- Γ --></mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {Out}(\Gamma )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sans-serif upper X Superscript 0 Baseline left-parenthesis normal upper Gamma comma upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="sans-serif">X</mml:mi> </mml:mrow> <mml:mn>0</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Γ<!-- Γ --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathsf {X}^0(\Gamma , G)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is mixing with respect to this measure.
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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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.000 | 0.002 |
| Insufficient payload (model declined to judge) | 0.001 | 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".