On mappings on the hypercube with small average stretch
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Bibliographic record
Abstract
Abstract Let $A \subseteq \{0,1\}^n$ be a set of size $2^{n-1}$ , and let $\phi \,:\, \{0,1\}^{n-1} \to A$ be a bijection. We define the average stretch of $\phi$ as \begin{equation*} {\sf avgStretch}(\phi ) = {\mathbb E}[{{\sf dist}}(\phi (x),\phi (x'))], \end{equation*} where the expectation is taken over uniformly random $x,x' \in \{0,1\}^{n-1}$ that differ in exactly one coordinate. In this paper, we continue the line of research studying mappings on the discrete hypercube with small average stretch. We prove the following results. For any set $A \subseteq \{0,1\}^n$ of density $1/2$ there exists a bijection $\phi _A \,:\, \{0,1\}^{n-1} \to A$ such that ${\sf avgStretch}(\phi _A) = O\left(\sqrt{n}\right)$ . For $n = 3^k$ let ${A_{\textsf{rec-maj}}} = \{x \in \{0,1\}^n \,:\,{\textsf{rec-maj}}(x) = 1\}$ , where ${\textsf{rec-maj}} \,:\, \{0,1\}^n \to \{0,1\}$ is the function recursive majority of 3’s . There exists a bijection $\phi _{{\textsf{rec-maj}}} \,:\, \{0,1\}^{n-1} \to{A_{\textsf{rec-maj}}}$ such that ${\sf avgStretch}(\phi _{{\textsf{rec-maj}}}) = O(1)$ . Let ${A_{{\sf tribes}}} = \{x \in \{0,1\}^n \,:\,{\sf tribes}(x) = 1\}$ . There exists a bijection $\phi _{{\sf tribes}} \,:\, \{0,1\}^{n-1} \to{A_{{\sf tribes}}}$ such that ${\sf avgStretch}(\phi _{{\sf tribes}}) = O(\!\log (n))$ . These results answer the questions raised by Benjamini, Cohen, and Shinkar (Isr. J. Math 2016).
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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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.002 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
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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