Bibliographic record
Abstract
Abstract Let n be a positive integer and f belong to the smallest ring of functions upper R Superscript n Baseline right arrow double struck upper R $\mathbb R^n\to \mathbb R$ <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:mrow> <mml:mstyle mathvariant="double-struck"> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mstyle> <mml:mo>→</mml:mo> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:math> that contains all real polynomial functions of n variables and is closed under exponentiation. Then there exists d element of double struck upper N $d\in \mathbb N$ <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>d</mml:mi> <mml:mo>∈</mml:mo> <mml:mi mathvariant="double-struck">N</mml:mi> </mml:mrow> </mml:math> such that for all m element of StartSet 0 comma midline horizontal ellipsis comma n EndSet $m\in \{0,\dots , n\}$ <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>m</mml:mi> <mml:mo>∈</mml:mo> <mml:mo form="prefix" fence="true">{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mo>⋯</mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo form="postfix" fence="true">}</mml:mo> </mml:mrow> </mml:math> and c element of upper R Superscript m $c\in \mathbb R^{m}$ <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>c</mml:mi> <mml:mo>∈</mml:mo> <mml:mstyle mathvariant="double-struck"> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>m</mml:mi> </mml:msup> </mml:mstyle> </mml:mrow> </mml:math> , if x right arrow from bar f left parenthesis c comma x right parenthesis colon upper R Superscript n minus m Baseline right arrow double struck upper R $x\mapsto f(c,x)\colon \mathbb R^{n-m}\to \mathbb R$ <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:mrow> <mml:mi>x</mml:mi> <mml:mo>↦</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false" form="prefix" fence="true">(</mml:mo> <mml:mi>c</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false" form="postfix" fence="true">)</mml:mo> <mml:mo>:</mml:mo> <mml:mstyle mathvariant="double-struck"> <mml:msup> <mml:mi>R</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>−</mml:mo> <mml:mi>m</mml:mi> </mml:mrow> </mml:msup> </mml:mstyle> <mml:mo>→</mml:mo> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:math> is harmonic, then it is polynomial of degree at most d . In particular, f is polynomial if it is harmonic.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.005 |
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".