Smallest order closed sublattices and option spanning
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Bibliographic record
Abstract
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a sublattice of a vector lattice <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We consider the problem of identifying the smallest order closed sublattice of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> containing <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It is known that the analogy with topological closure fails. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y overbar Superscript o"> <mml:semantics> <mml:msup> <mml:mover> <mml:mi>Y</mml:mi> <mml:mo accent="false">¯</mml:mo> </mml:mover> <mml:mi>o</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\overline {Y}^o</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the order closure of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> consisting of all order limits of nets of elements from <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y overbar Superscript o"> <mml:semantics> <mml:msup> <mml:mover> <mml:mi>Y</mml:mi> <mml:mo accent="false">¯</mml:mo> </mml:mover> <mml:mi>o</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\overline {Y}^o</mml:annotation> </mml:semantics> </mml:math> </inline-formula> need not be order closed. We show that in many cases the smallest order closed sublattice containing <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is in fact the second order closure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y overbar Superscript o Baseline overbar Superscript o"> <mml:semantics> <mml:msup> <mml:mover> <mml:msup> <mml:mover> <mml:mi>Y</mml:mi> <mml:mo accent="false">¯</mml:mo> </mml:mover> <mml:mi>o</mml:mi> </mml:msup> <mml:mo accent="false">¯</mml:mo> </mml:mover> <mml:mi>o</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\overline {\overline {Y}^o}^o</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Moreover, if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"> <mml:semantics> <mml:mi>σ</mml:mi> <mml:annotation encoding="application/x-tex">\sigma</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-order complete Banach lattice, then the condition that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y overbar Superscript o"> <mml:semantics> <mml:msup> <mml:mover> <mml:mi>Y</mml:mi> <mml:mo accent="false">¯</mml:mo> </mml:mover> <mml:mi>o</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\overline {Y}^o</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is order closed for every sublattice <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y"> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding="application/x-tex">Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> characterizes order continuity of the norm of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The present paper provides a general approach to a fundamental result in financial economics concerning the spanning power of options written on a financial asset.
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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 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| 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