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Bibliographic record
Abstract
We introduce the conormal fan of a matroid <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper M"> <mml:semantics> <mml:mi mathvariant="normal">M</mml:mi> <mml:annotation encoding="application/x-tex">\operatorname {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , which is a Lagrangian analog of the Bergman fan of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper M"> <mml:semantics> <mml:mi mathvariant="normal">M</mml:mi> <mml:annotation encoding="application/x-tex">\operatorname {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We use the conormal fan to give a Lagrangian interpretation of the Chern–Schwartz–MacPherson cycle of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper M"> <mml:semantics> <mml:mi mathvariant="normal">M</mml:mi> <mml:annotation encoding="application/x-tex">\operatorname {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This allows us to express the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding="application/x-tex">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -vector of the broken circuit complex of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper M"> <mml:semantics> <mml:mi mathvariant="normal">M</mml:mi> <mml:annotation encoding="application/x-tex">\operatorname {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in terms of the intersection theory of the conormal fan of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper M"> <mml:semantics> <mml:mi mathvariant="normal">M</mml:mi> <mml:annotation encoding="application/x-tex">\operatorname {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We also develop general tools for tropical Hodge theory to prove that the conormal fan satisfies Poincaré duality, the hard Lefschetz theorem, and the Hodge–Riemann relations. The Lagrangian interpretation of the Chern–Schwartz–MacPherson cycle of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper M"> <mml:semantics> <mml:mi mathvariant="normal">M</mml:mi> <mml:annotation encoding="application/x-tex">\operatorname {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , when combined with the Hodge–Riemann relations for the conormal fan of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper M"> <mml:semantics> <mml:mi mathvariant="normal">M</mml:mi> <mml:annotation encoding="application/x-tex">\operatorname {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , implies Brylawski’s and Dawson’s conjectures that the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding="application/x-tex">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -vectors of the broken circuit complex and the independence complex of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper M"> <mml:semantics> <mml:mi mathvariant="normal">M</mml:mi> <mml:annotation encoding="application/x-tex">\operatorname {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are log-concave sequences.
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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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| 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