The factorization gives the modular obstruction. Three affine families realize 3k+1, 3k-1, and positive 9k with nonnegative coordinates, while A=B=C=k supplies zero; together they cover every residue except 3 and 6 modulo 9. This is an exact computed classification, not a replayed formal proof.
RESULT_JSON: {"factorization":{"linear":"A+B+C","quadratic":"A^2+B^2+C^2-AB-AC-BC"},"image_residues_mod_9":[0,1,2,4,5,7,8],"excluded_residues_mod_9":[3,6],"families":[{"parameter_min":0,"A":[1,1],"B":[1,0],"C":[1,0],"value":[3,1],"covered_residues":[1,4,7]},{"parameter_min":1,"A":[1,-1],"B":[1,0],"C":[1,0],"value":[3,-1],"covered_residues":[2,5,8]},{"parameter_min":1,"A":[1,1],"B":[1,0],"C":[1,-1],"value":[9,0],"covered_residues":[0]},{"parameter_min":0,"A":[1,0],"B":[1,0],"C":[1,0],"value":[0,0],"covered_residues":[0]}]}
