A classification of double struck upper Q$\mathbb {Q}$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi mathvariant="double-struck">Q</mml:mi> </mml:math> -linear maps from ModifyingAbove caret With bar double struck upper Q times slash ModifyingAbove bar caret With bar double struck upper Q t o r s times$\overline {\mathbb {Q}}^\times /\overline {\mathbb {Q}}^\times _{\mathrm {tors}}$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mrow> <mml:mover> <mml:mo>^</mml:mo> <mml:mo stretchy="true">¯</mml:mo> </mml:mover> <mml:mi mathvariant="double-struck">Q</mml:mi> <mml:mo>×</mml:mo> <mml:mo>/</mml:mo> <mml:mover> <mml:mo>_^</mml:mo> <mml:mo stretchy="true">¯</mml:mo> </mml:mover> <mml:mi mathvariant="double-struck">Q</mml:mi> <mml:mstyle mathvariant="normal"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mi>o</mml:mi> <mml:mi>r</mml:mi> <mml:mi>s</mml:mi> </mml:mrow> </mml:mstyle> <mml:mo>×</mml:mo> </mml:mrow> </mml:math> to double struck upper R$\mathbb {R}$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:math>
Bibliographic record
Abstract
Abstract A 2009 article of Allcock and Vaaler explored the double struck upper Q $\mathbb {Q}$ Q -vector space script upper G colon equals ModifyingAbove caret With bar double struck upper Q times slash ModifyingAbove bar caret With bar double struck upper Q t o r s times $\mathcal {G} := \overline {\mathbb {Q}}^\times /{\overline {\mathbb {Q}}^\times _{\mathrm {tors}}}$ G : = ^ ¯ Q × / _^ ¯ Q t o r s × , showing how to represent it as part of a function space on the places of double struck upper Q overbar $\overline {\mathbb {Q}}$ Q ¯ . We establish a representation theorem for the double struck upper R $\mathbb {R}$ R -vector space of double struck upper Q $\mathbb {Q}$ Q -linear maps from script upper G $\mathcal {G}$ G to double struck upper R $\mathbb {R}$ R , enabling us to classify extensions to script upper G $\mathcal {G}$
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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 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.003 | 0.003 |
| Science and technology studies | 0.003 | 0.003 |
| Scholarly communication | 0.005 | 0.006 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.001 | 0.004 |
| Insufficient payload (model declined to judge) | 0.060 | 0.031 |
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".