Bibliographic record
Abstract
A longstanding conjecture in $ϕ^4_4$ theory is that primitive graphs dominate the beta function asymptotically at large loop order in the minimal-subtraction scheme. Here we investigate this issue by exploiting additional combinatorial structure coming from an extension to vectors with $O(N)$ symmetry. For the 0-dimensional case, we calculate the $N$-dependent generating function of primitive graphs and its asymptotics, including arbitrarily many subleading corrections. We find that the leading asymptotic growth rate becomes visible only above $\approx 25$ loops, while data at lower order is suggestive of a wrong asymptotics. Our results also yield the symmetry-factor weighted sum of 3-connected cubic graphs, and the exact asymptotics of Martin invariants. For individual Feynman graphs, we give bounds on their degree in $N$ depending on their coradical degree, and construct the primitive graphs of highest degree explicitly. We calculate the 4D primitive beta function numerically up to 17 loops, and find its behaviour to be qualitatively similar to the 0D case. The locations of zeros quickly approach their large-loop asymptotics at negative integer $N$, while the growth rate of the beta function differs from the asymptotic prediction even at 17 loops.
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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 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".