Exact Root-Label Realization Loci in Low-Cost Proximity-Decorated Binary Quadratic Paths

Let q k[x, y]² (V^*) be a primitive framed binary quadratic germ over an algebraically closed field of characteristic zero. A maximal primitive blowup path carries multiplicities, free/satellite proximity, and exceptional types B, M, and N, where a constant-root branch vertex B also carries an exact repeated-root class on the rank-one Veronese conic C=₂ ( (V^*) ) (² (V^*) ). A preceding path-local classification under the square-cost bound ᵢ² 12 produced a 124-row proximity-decorated envelope, with 119 realizable rows and 5 projective-transport obstructions. We now restore all exact constant-root labels and classify, for every row, the constructible locus of labels realized by one primitive germ. After quotienting only by the equalities already forced by projective-class transport, every nonempty row has at most two independent root-label components. The exact profile is 124=46\pt\+45C+21C+7 (C²C) +5. The two-component part is sharp: 21 rows realize only the diagonal orbit, 7 only the ordered off-diagonal orbit, and no row realizes both. For free chains, the mechanisms are a Newton-support overlap forcing equality and a multiplicity-drop-one nonbranch pencil forcing inequality. For nonfree chains, the same degree-one anchor pencil sP+tQ forces opposite conditions according as it is exposed at an interior B₁ vertex or at the terminal N₁ vertex. Explicit primitive nonzero-discriminant normal forms realize every allowed orbit. The classification is path-local; it does not assert sibling compatibility, whole-tree glueing, or a finite count of exact labelled assignments.