Why this work is in the frame
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Bibliographic record
Abstract
We give the first separation between the power of formulas and circuits in the AC 0 [] basis (unbounded fan-in AND, OR, NOT and MOD 2 gates).We show that there exist poly(n)-size depth-d circuits that are not equivalent to any depth-d formula of size n o (d) for all d O(log(n)/log log(n)).This result is obtained by a combination of new lower and upper bounds for Approximate Majorities, the class of Boolean functions {0, 1} n {0, 1} that agree with the Majority function on a 3/4 fraction of the inputs.AC 0 [ ] formula lower bound.We show that every depth-d AC 0 [] formula of size s has a 1/4-error polynomial approximation over F 2 of degree O((1/d) log s) d-1 .This strengthens a classic O(log s) d-1 degree approximation for circuits due to Razborov (1987).Since any polynomial that approximates the Majority function has degree ( n), this result implies an exp((dn 1/2(d-1) )) lower bound on the depth-d AC 0 [] formula size of all Approximate Majority functions for all d O(log n).Monotone AC 0 circuit upper bound.For all d O(log(n)/log log(n)), we give a randomized construction of depth-d monotone AC 0 circuits (without NOT or MOD 2 gates) of size exp(O(n 1/2(d-1) )) that compute an Approximate Majority function.This strengthens a construction of formulas of size exp(O(dn 1/2(d-1) )) due to Amano (2009).
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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 | 0.003 | 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 | 0.001 |
| Open science | 0.003 | 0.001 |
| 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 it