UCLA

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 statement, we remove both the Generalised Riemann Hypothesis and the friability restriction from a theorem of Mangerel, and we settle the odd-total and positive-positive variants, raised by Mangerel, of a question of Shusterman whose even-total negative-negative case was settled recently in an anonymous repository. The main new ingredient is a rigidity statement at an odd prime p: if p has no representation a+b with λ(a) = λ(b) = ε, then a single one-sided exclusion, together with the dilation laws λ(2n) = λ(3n) = λ(5n) = −λ(n), already forces the odd completion of λ to be exactly multiplicative, which contradicts quadratic reciprocity. In addition, we give a short proof that every N ≥ 11 has a pair of unequal signs, which recovers Mangerel's nonextremality theorem |Σ_{a