Bibliographic record
Abstract
The class of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper J"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">J</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {J}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -lattices was originally defined in the second author’s thesis and subsequently by Longstaff, Nation, and Panaia. A subspace lattice <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {L}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on a Banach space <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> which is also a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper J"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">J</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {J}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -lattice is called a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper J"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">J</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {J}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> - <italic>subspace lattice</italic> , abbreviated JSL. It is demonstrated that every single element of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A l g script upper L"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>l</mml:mi> <mml:mi>g</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">Alg\mathcal {L}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has rank at most one. It is also shown that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A l g script upper L"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>l</mml:mi> <mml:mi>g</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">Alg\mathcal {L}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has the strong finite rank decomposability property. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L 1"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {L}_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper L 2"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">\mathcal {L}_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be subspace lattices that are also JSL’s on the Banach spaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X 1"> <mml:semantics> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">X_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X 2"> <mml:semantics> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">X_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , respectively. The two properties just referred to, when combined, show that every algebraic isomorphism between <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A l g script upper L 1"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>l</mml:mi> <mml:mi>g</mml:mi> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">Alg\mathcal {L}_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A l g script upper L 2"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>l</mml:mi> <mml:mi>g</mml:mi> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">L</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encodi
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.004 | 0.006 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.010 | 0.002 |
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 source (direct Gemma or distilled Codex), 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".