Limits of boolean functions over finite fields
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Bibliographic record
Abstract
In this thesis, we study sequences of functions of the form F_p^n to 0,1 for varying n, and define a notion of convergence based on the induced distributions from restricting the functions to a random affine subspace. One of the key tools we use is the recently developed theory of `higher order Fourier analysis', where the characters of standard Fourier analysis are replaced with exponentials of higher degree polynomials. This is not a trivial extension by any means, but when the polynomials are chosen with some care, the higher order decomposition can be taken to have properties analogous to those of the classical Fourier transform.The result of applying higher order Fourier analysis in this setting is the necessity to determine the distribution of a collection of polynomials when they are composed with some additional linear structures. Here, we make use of a recently proven equidistribution theorem, relying on a near-orthogonality result showing that the higher order characters can be made orthogonal up to an arbitrarily small error term.With these tools, we prove that the limit of every convergent sequence of functions can be represented by a limit object which takes the form of a certain measurable function on a group we construct. We also show that every such limit object arises as the limit of some sequence of functions. These results are in the spirit of analogous results which have been developed for limits of graph sequences. A more general, albeit substantially more sophisticated, limit object was recently constructed by Szegedy.
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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 | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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