Global completeness in the effective algorithmic sense is proved for the ABCDG pattern. The result concerns every rational signed root vector, not only the image of the previously displayed polynomial formula.
The yellow quadric turns a matrix of root combinations into a rank-one matrix. Its direction supplies the base [r:s], while the brown quadric becomes the smooth conic a²=(2s²−r²)P²+(2r²−s²)Q².
Conic solubility is decided exactly for every rational base. When a fiber has a rational point, projection from that point parametrizes the entire fiber; dovetailing the base and fiber-parameter heights reaches every rational point after finitely many steps.
The algorithm has an explicit inverse. This proves soundness, surjectivity, and finite reachability without asserting one global rational formula or a finite rational atlas.