Positivity and Continuity of Lyapunov Exponents for Multi-Frequency Skew-Shift Schrödinger Operators

We study discrete Schrödinger operators on \(\ell^2(\mathbb Z)\) generated by the vector-frequency two-step skew shift \(T_ω(x,y)=(x+ω,x+y)\), with an arbitrary nonconstant real-analytic sampling function on \(\mathbb T^d\times\mathbb T^d\). For frequencies in a fixed vector Diophantine class and su