From $\log 2$ to $π/2$: the sharp asymptotic inradius of polynomial lemniscates

Let $R_n$ be the infimum of the inradii of $\{z:|p(z)|<1\}$ over monic degree-$n$ polynomials whose zeros lie in the closed unit disk. We prove $nR_n\toπ/2$, matching the asymptotic obstruction supplied by $z^n-1$. We first establish the exact universal radius $2^{1/n}-1$ for disks centered at zeros