Tameness of transitive Boolean functions

Abstract

Let $f_n:\{0,1\}^n\to\{0,1\}$ be Boolean functions, and let $C_n$ count their changes in a unit time interval under stationary dynamics in which each coordinate is independently resampled at rate one from the Bernoulli$(p_n)$ distribution. We prove that, if $f_n$ is transitive and $n\min\{p_n,1-p_n\}^{r}\to\infty$ for every $r>0$, then tightness of $(C_n)$ implies $\mathrm{Var}(f_n)\to0$. This settles Forsstr\"om's conjecture in the $p_n\le 1/2$ regime and extends it symmetrically to arbitrary biases. The proof uses a mixed jump sum whose second moment is bounded by the generator of a low degree Fourier projection.