Goldbach's conjecture for the Liouville function

True: for every even $N > 2$ there are positive $a, b$ with $a + b = N$ and $\lambda(a) = \lambda(b) = -1$, unconditionally, with no analytic or conjectural hypothesis. The route reduces to a prime modulus, then uses $\lambda(2n) = \lambda(3n) = \lambda(5n) = -\lambda(n)$ to force $\lambda$ to agree with the Legendre symbol on an interval, where quadratic reciprocity gives a contradiction; no sieve is used, which is how the parity obstruction is avoided. Two pieces of context a reader needs. Mangerel (IMRN 2024) proved that $\lambda(a)\lambda(N-a)$ cannot be constant for $N \ge 11$; that yields a pair with opposite signs and so does NOT imply this statement. And the result has since been generalised: on the same MathOverflow question, Lingsen Meng gives an unconditional argument that all four sign pairs occur for every $N \notin \{2,3,4,5,6,9,10\}$, covering odd $N$ too, while crediting this work as independent and prior for the even case.

True: for every even $N > 2$ there are positive $a, b$ with $a + b = N$ and $\lambda(a) = \lambda(b) = -1$, unconditionally, with no analytic or conjectural hypothesis. The route reduces to a prime modulus, then uses $\lambda(2n) = \lambda(3n) = \lambda(5n) = -\lambda(n)$ to force $\lambda$ to agree with the Legendre symbol on an interval, where quadratic reciprocity gives a contradiction; no sieve is used, which is how the parity obstruction is avoided. Two pieces of context a reader needs. Mangerel (IMRN 2024) proved that $\lambda(a)\lambda(N-a)$ cannot be constant for $N \ge 11$; that yields a pair with opposite signs and so does NOT imply this statement. And the result has since been generalised: on the same MathOverflow question, Lingsen Meng gives an unconditional argument that all four sign pairs occur for every $N \notin \{2,3,4,5,6,9,10\}$, covering odd $N$ too, while crediting this work as independent and prior for the even case.