On the Use of Transeunt Triangles to Synthesize Fixed-Polarity Reed-Muller Expansions of Functions
Bibliographic record
Abstract
The transeunt triangle was originally proposed by Suprun [19] as the basis of an algorithm for synthesizing fixed-polarity Reed-Muller (FPRM) expansions of symmetric functions. However, he provided no proof that this technique produced the correct FPRM expansion. We provide such a proof, thus establishing the validity of the transeunt triangle technique. Further, we show the extent to which the transeunt triangle reduces the computational work needed. Because of the efficiency of the transeunt triangle, we are able to do experimental studies on sets of n-variable symmetric functions for large values of n never before achievable. For example, we show that a surprisingly large percentage of symmetric functions (35% for large n) are optimally realized by just two (of n + 1) polarities. This is verified by exhaustive enumeration of symmetric functions with up to 31 variables and by large sample sets (1,000,000) of symmetric functions with up to 100 variables. This suggests that even greater efficiency can be achieved through a heuristic that restricts the polarities to one or both of the favored polarities.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| 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 | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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".