Chowla Conjecture for the truncated Möbius function over ${\mathbb{F}}_q[t]$

In this paper we give an analytic proof of the Two-point Chowla conjecture for the truncated Möbius function ${\mu }_M$ for the function field case in the fixed $q$ setting. We reformulate the problem into studying the distribution of $({\omega }_M,{\omega }_M\circ T_a)$. Thus we develop a uniform bivariate Erdős-Kac theorem in terms of the characteristic function for small pair of real numbers $(u,v)$ via the Kubilius model. For this purpose we also develop and employ the pure Brun sieve for polynomials in this paper.