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Bibliographic record
Abstract
The \beta <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>β</mml:mi> </mml:math> -functions describe how couplings run under the renormalization group flow in field theories. In general, all couplings that respect the symmetry and locality are generated under the renormalization group flow, and the exact renormalization group flow is characterized by the \beta <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>β</mml:mi> </mml:math> -functions defined in the infinite dimensional space of couplings. In this paper, we show that the renormalization group flow is highly constrained so that the \beta <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>β</mml:mi> </mml:math> -functions defined in a measure zero subspace of couplings completely determine the \beta <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>β</mml:mi> </mml:math> -functions in the entire space of couplings. We provide a quantum renormalization group-based algorithm for reconstructing the full \beta <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>β</mml:mi> </mml:math> -functions from the \beta <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>β</mml:mi> </mml:math> -functions defined in the subspace. As examples, we derive the full \beta <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>β</mml:mi> </mml:math> -functions for the O(N) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> vector model and the O_L(N) \times O_R(N) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>L</mml:mi> </mml:msub> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mo>×</mml:mo> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>R</mml:mi> </mml:msub> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> matrix model entirely from the \beta <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>β</mml:mi> </mml:math> -functions defined in the subspace of single-trace couplings.
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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 | 0.000 | 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 | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 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