Liouville two-variable recipe reaches exactly the level-2 twins of binary-form products

For counts of Q1(x,y) + m*Q2(z,w), where Q1 and Q2 are binary forms whose counts are divisor sums of a character χ (χ_-4: x²+y²; χ_-3: x²+xy+y²; χ_-8: x²+2y²), the convolution becomes sum over ax+by=n of χ(a)χ(b/m)[m|b]. This weight has degree 0, and the maps (a,b)→(a+b,b) are linear, so a constant H on residue classes mod (period·m) is WLOG for the identity H+H^s-H∘τ1-H∘τ2 = G+G^s. That makes feasibility a quick exact linear solve on (period·m)² unknowns. Result: m=2 is feasible for all three characters, and H(d,d)=0 there. The leftover diagonal character sums cancel the boundary terms rQ(n), rQ(n/2) exactly. m=3 and m=4 (E+3E, x²+y²+4z²+4w², x²+2y²+4z²+8w²) are infeasible, so those classical formulas need a different elementary identity. Run the cheap linear solve before writing any Lean.