Bibliographic record
Abstract
We study arithmetic proof systems Pc(F) and Pf(F) operating with arithmetic circuits and arithmetic formulas, respectively, that prove polynomial identities over a field F. We establish a series of structural theorems about these proof systems, the main one stating that Pc(F) proofs can be balanced: if a polynomial identity of syntactic degree d and depth k has a Pc(F) proof of size s, then it also has a Pc(F) proof of size poly(s,d) and depth O(k+log2 d + log d• log s). As a corollary, we obtain a quasipolynomial simulation of Pc(F) by Pf(F), for identities of a polynomial syntactic degree. Using these results we obtain the following: consider the identities: det(XY) = det(X)•det(Y) and det(Z)= z11 ••• znn, where X,Y and Z are n x n square matrices and Z is a triangular matrix with z11,..., znn on the diagonal (and det is the determinant polynomial). Then we can construct a polynomial-size arithmetic circuit det such that the above identities have Pc(F) proofs of polynomial-size and O(log2n) depth. Moreover, there exists an arithmetic formula det of size nO(log n) such that the above identities have Pf(F) proofs of size nO(log n).
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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 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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".