Liouville certificates: add zero-sum 'kernel' columns to the target, not just more potentials

When a convolution weight G(a,x,b,y) over ax+by=n has no pointwise certificate (the system G+G∘swap = F-combination on x<y is inconsistent), the obstruction may only concern G pointwise. Add extra unknown columns K_i = G' - G where G' has the same sum over P(n) for an arithmetic reason that is not a unimodular bijection. Example: sigma*(l) = sum_{by=l} b[4∤b] = sum_{by=l} b(1-4[4|y]), via (b,y) -> (b/4,4y). For Jacobi's r6 = r2*r4 (weight chi4(a) b[4∤b]), W1 and W2 = chi4(a) b(1-4[4|y]) are each infeasible, but the mix (4W1+W2)/5 has a tiny certificate: F = alpha(a,b)a + beta(a,b)x (mod-4 tables) plus a flip term H = -8chi(b)[4∤a]x using the symmetry (a,x,b,y) -> (x,a,b,y). Note that the kernel column's diagonal contribution shows up as a leftover divisor sum, which you kill with d -> 4n/d. Search for the smallest certificate by restricting class keys and monomials per block. Here keys (a,b) mod 4 and monomials {a,x} for F and {x} for H gave a unique solution with 49 unknowns.