Is block sensitivity at most quadratic in spectral sensitivity? (Aaronson, Ben-David, Kothari, Rao and Tal)

Theorem 6.1: a total Boolean function on 2,017,584 variables with $\mathrm{bs}(f)\ge14011$ and $\lambda(f)\le89.0162$, so $\mathrm{bs}(f)\ge\lambda(f)^{2.127}$; composition gives a family with $\lambda\to\infty$ and $\mathrm{bs}=\Omega(\lambda^{2.127})$, so no quadratic bound holds. The function is a union of subcubes indexed by a doubly regular tournament with gates fixed by the Lovász local lemma. An exponent near 2.20 on 1,255 variables is numerical evidence only. The true exponent, between 2.127 and 4, stays open. Since $\lambda(f)\le s(f)$, OpenAI's later disproof of $\mathrm{bs}=O(s^2)$ (25 September 2026), which adapts this gated-tournament construction, implies this result.

Theorem 6.1: a total Boolean function on 2,017,584 variables with $\mathrm{bs}(f)\ge14011$ and $\lambda(f)\le89.0162$, so $\mathrm{bs}(f)\ge\lambda(f)^{2.127}$; composition gives a family with $\lambda\to\infty$ and $\mathrm{bs}=\Omega(\lambda^{2.127})$, so no quadratic bound holds. The function is a union of subcubes indexed by a doubly regular tournament with gates fixed by the Lovász local lemma. An exponent near 2.20 on 1,255 variables is numerical evidence only. The true exponent, between 2.127 and 4, stays open. Since $\lambda(f)\le s(f)$, OpenAI's later disproof of $\mathrm{bs}=O(s^2)$ (25 September 2026), which adapts this gated-tournament construction, implies this result.