Even-function spectral gap on a hypercube ($2\mathrm{Var}(f)\le -\mathbb{E}[fLf](f(-\omega )=f(\omega ))$)

Global sign-even functions have zero coefficients on odd-size Walsh characters. Every nonconstant remaining character has generator eigenvalue at most $-2$, yielding the displayed improvement from gap one to gap two. Combining this with the convexity of a norm proves the sharp Rademacher second-moment inequality.