New Results on the Complexity of the Middle Bit of Multiplication
Why this work is in the frame
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Bibliographic record
Abstract
It is well known that the hardest bit of integer multiplication is the middle bit, i.e., MUL n−1,n . This paper contains several new results on its complexity. First, the size s of randomized read-k branching programs, or, equivalently, their space (log s) is investigated. A randomized algorithm for MUL n−1,n with $$k = {\mathcal{O}}(\hbox{log}\, n)$$ (implying time $${\mathcal{O}}(n\, \hbox{log}\, n))$$ , space $${\mathcal{O}}(\hbox{log}\, n)$$ and error probability n −c for arbitrarily chosen constants c is presented. Second, the size of general branching programs and formulas is investigated. Applying Nechiporuk’s technique, lower bounds of $$\Omega (n^{3/2}/ \hbox{log}\, n)$$ and Ω (n 3/2), respectively, are obtained. Moreover, by bounding the number of subfunctions of MUL n−1,n , it is proven that Nechiporuk’s technique cannot provide larger lower bounds than $${\mathcal{O}}(n^{5/3}/ \hbox{log}\, n)$$ and $${\mathcal{O}}(n^{5/3})$$ , respectively.
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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 | 0.000 |
| 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 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 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