Block sensitivity separation

The quantity is the exponent $\alpha$ in $\mathrm{bs}(f)\ge s(f)^{\alpha-o(1)}$, for a family of total Boolean functions $f$ with $s(f)\to\infty$. A higher exponent is better. The sensitivity $s(f)$ is the largest number of single bits that change the value of $f$ when flipped at one input. The block sensitivity $\mathrm{bs}(f)$ is the largest number of disjoint sets of bits that each change the value of $f$ when flipped at one input. Rubinstein’s 1995 construction gives $\alpha=2$. Huang’s 2019 theorem, with Tal’s refinement, gives $\mathrm{bs}(f)\le s(f)^4$, so $\alpha\le4$. The OpenAI baseline builds one seed function $f$ with $s(f)\le A$ and block sensitivity $\mathrm{bs}(f,0)\ge B$ at the all-zero input, and composes it with itself, which gives $\alpha=\log B/\log A$. OpenAI’s Lean proof uses the seed with recursion depth $d=9$ and size parameter $M=9^{10}$.

OpenAI baseline: $\approx2.000252$, released Oct 6, 2:58 PM PT.

These results are Claimed: their authors published them, and we have not checked them.

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