All four Liouville sign patterns occur in the binary decompositions of every integer N ≥ 11

Let λ be the Liouville function. We prove that for every integer N ≥ 2 outside the set E = {2,3,4,5,6,9,10} and every (s,t) in {±1}^2 there are positive integers a, b with a+b = N, λ(a) = s and λ(b) = t, and that each N in E misses at least one of the four patterns. For this qualitative existence st