Binary dimension-free discretization and polynomial complexification on real cubes

Let $C(d,2)$ be the optimal dimension-free ratio between the polytorus and Boolean-cube norms of complex multiaffine polynomials of degree at most $d$. For every $d$, $C(d,2)$ equals the unrestricted complexification constant of the real cube for complex polynomials of degree at most $d$. Moreover, ${\lim }_{d\to \mathrm{\infty }}C(d,2{)}^{1\mathrm{/}d}=1+\sqrt{2}$, and for every $d\ge 3$, $\beta (1-6\mathrm{/}(5d))(1+\sqrt{2}{)}^d\le C(d,2)\le (1+\sqrt{2}{)}^d$, where $\beta =15\pi \mathrm{/}[2(5\sqrt{26}+\log (5+\sqrt{26}))]=0.8473222863\dots$.