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 $\mathbb {Q}$ <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> -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 $\mathcal {G} := \overline {\mathbb {Q}}^\times /{\overline {\mathbb {Q}}^\times _{\mathrm {tors}}}$ <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:mi mathvariant="script">G</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:mo>×</mml:mo> <mml:mo>/</mml:mo> <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: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:mrow> </mml:math> , showing how to represent it as part of a function space on the places of double struck upper Q overbar $\overline {\mathbb {Q}}$ <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:mover> <mml:mi mathvariant="double-struck">Q</mml:mi> <mml:mo stretchy="true">¯</mml:mo> </mml:mover> </mml:math> . We establish a representation theorem for the 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> -vector space 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 script upper G $\mathcal {G}$ <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="script">G</mml:mi> </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> , enabling us to classify extensions to script upper G $\mathcal {G}$</jats:t
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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 | 0.014 | 0.008 |
| Meta-epidemiology (narrow) | 0.015 | 0.021 |
| Meta-epidemiology (broad) | 0.009 | 0.016 |
| Bibliometrics | 0.007 | 0.012 |
| Science and technology studies | 0.013 | 0.012 |
| Scholarly communication | 0.012 | 0.016 |
| Open science | 0.023 | 0.019 |
| Research integrity | 0.019 | 0.021 |
| Insufficient payload (model declined to judge) | 0.009 | 0.024 |
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; both teacher heads agree on what is shown here.
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".