Why this work is in the frame
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Bibliographic record
Abstract
For a class $${\mathcal{F}}$$F of formulas (general de Morgan or read-once de Morgan), the shrinkage exponent$${\Gamma_{\mathcal{F}}}$$ΓF is the parameter measuring the reduction in size of a formula $${F\in\mathcal{F}}$$FźF after $${F}$$F is hit with a random restriction. A Boolean function $${f\colon \{0,1\}^n\to\{1,-1\}}$$f:{0,1}nź{1,-1} is Fourier-concentrated if, when viewed in the Fourier basis, $${f}$$f has most of its total mass on low-degree coefficients. We show a direct connection between the two notions by proving that shrinkage implies Fourier concentration: For a shrinkage exponent $${\Gamma_{\mathcal{F}}}$$ΓF, a formula $${F\in\mathcal{F}}$$FźF of size $${s}$$s will have most of its Fourier mass on the coefficients of degree up to about $${s^{1/\Gamma_{\mathcal{F}}}}$$s1/ΓF. More precisely, for a Boolean function $${f\colon\{0,1\}^n\to\{1,-1\}}$$f:{0,1}nź{1,-1} computable by a formula of (large enough) size $${s}$$s and for any parameter $${r > 0}$$r>0, $$\sum_{A\subseteq [n]\; :\; |A|\geq s^{1/\Gamma} \cdot r} \hat{f}(A)^2\leq s\cdot{\mathscr{polylog}}(s)\cdot exp\left(-\frac{r^{\frac{\Gamma}{\Gamma-1}}}{s^{o(1)}} \right),$$źA⊆[n]:|A|źs1/Γ·rf^(A)2źs·polylog(s)·exp-rΓΓ-1so(1),where $${\Gamma}$$Γ is the shrinkage exponent for the corresponding class of formulas: $${\Gamma=2}$$Γ=2 for de Morgan formulas, and $${\Gamma=1/\log_2(\sqrt{5}-1)\approx 3.27}$$Γ=1/log2(5-1)ź3.27 for read-once de Morgan formulas. This Fourier concentration result is optimal, to within the $${o(1)}$$o(1) term in the exponent of $${s}$$s. As a standard application of these Fourier concentration results, we get that subquadratic-size de Morgan formulas have negligible correlation with parity. We also show the tight $${\Theta(s^{1/\Gamma})}$$ź(s1/Γ) bound on the average sensitivity of read-once formulas of size $${s}$$s, which mirrors the known tight bound $${\Theta(\sqrt{s})}$$ź(s) on the average sensitivity of general de Morgan $${s}$$s-size formulas.
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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 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.001 |
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