Liouville-method convolution sums: when an elementary divisor-sum certificate cannot exist

Quick feasibility test before formalizing a convolution identity ∑_{ax+by=n} G(a,b) = divisor sum. The two-variable identity (for any H, sum over x<y of H(a,b)+H(b,a)-H(a+b,b)-H(b,a+b) equals diagonal divisor sums) only ever produces ∑_{d|n} (α(d) + (n/d)β(d)) with α, β quasi-polynomial. So a target with a character on the cofactor, like Jacobi's r6 = 16∑χ4(n/d)d² − 4∑χ4(d)d², is out of reach: compare Dirichlet series, since L(s,χ)ζ(s-2) does not have that shape. Degree-1 weights with a trivial character (G=a, i.e. σ*τ) are also hopeless. If you need four variables, the relations are: (i) the same identity for any F(a,x,b,y), with N1=(a+b,x,b,y-x); (ii) transpose invariance of ax+by=n, S_{x<y}[K - K∘ρ] = 0 with ρ(a,x,b,y) = (y-x,b,x,a+b). Global mod-4 quasi-polynomial F, K of degree 1 still fail for r6 (rank test mod p, done in C because Python dict elimination fills in). Liouville's functions are piecewise in sign(x-y).