Two-Successor Packet Loci for Primitive Binary Quadratic Germs

Let q=a u²+2buv+c v²∈ k[[x,y]]⊗²(V^*) be a primitive binary quadratic germ over an algebraically closed field of characteristic zero, and put Δ(q)=b²-ac. After blowing up the closed point, choose two distinct points of the exceptional line. We determine the exact projective images of the simultaneous coefficient-packet and scalar directional-packet maps at these two successors. For coefficient orders (μ₀;μ₁,μ₂), set r=μ₀-μ₁-μ₂. The two coefficient packets fill the full product for r≥2; for $r=1$ their pure successor coefficients may not vanish simultaneously; and for $r=0$ they are nonzero and have one common projective ²(V^*)-class. For a scalar germ of order d, put s=σ₁+σ₂. The exact scalar packet image has four faces: the full product for s≤ d-2, the complement of simultaneous pure-coefficient vanishing for $s=d-1$, the product of the two pure-coefficient open sets for $s=d$, and the empty set for $s>d$. Factoring the parent lowest coefficient layer by the two forced successor factors leaves a ²(V^*)-valued residual core. We classify its constant-root, moving-root, and nonbranch endpoint geometry with sharp degree thresholds. For a branch child of coefficient order one, the residual discriminant-symbol locus is exact: odd excess forces divisibility by the linear carrier, while even excess imposes no further projective restriction over an algebraically closed field. We also exhibit a repeated-carrier counterexample showing that this parity statement does not extend unchanged to arbitrary coefficient order. At square cost at most twelve, only $(2;1,1)$ and $(3;1,1)$ occur. We prove the exact-label obstruction for the minimal N₃→(B₁,B₁) fork and the open transition plane for the inactive different-label B₃ shell, together with exact patching of active symbols of excess 1≤ e≤4. In the same-label inactive B₃ shell we exhibit a parent-dependent obstruction that is invisible to the scalar budget and to the one-path branch-symbol test. Thus the paper gives exact general linear images and sharp low-cost shell results without asserting complete same-label gluing.