Moving-Root Tails and Exact Low-Cost Path Words for Primitive Binary Quadratic Germs

Let R=[[x,y]], where is algebraically closed of characteristic zero, and let q=a u²+2buv+c v²∈ R⊗_ ²(V^*) be primitive with nonzero discriminant b²-ac. Iterated point blowups of the coefficient ideal produce proper and infinitely-near primitive centers. At every center we label the new exceptional component by N, B, or M according as the initial binary quadratic pencil is nonbranch, constant-root branch, or moving-root branch, and we charge the square of the common coefficient order. We first prove that the multiplicity of a primitive successor is bounded by the vanishing order of the scalar gcd of the current initial triple. The inequality is strict whenever a nonbranch exceptional is followed by a branch exceptional. At multiplicity one, an inherited rank-one tangent direction forces every branch child to have the same constant root and every nonbranch child to be terminal. For moving-root exceptionals we factor the homogeneous initial triple as H(x,y)(P(x,y)u+Q(x,y)v)² μ= H+2[P:Q]. Under path square cost at most $12$, a moving-root vertex is either a terminal M₂, or an M₃ with one simple successor followed by at most two same-label B₁ vertices and a terminal N₁. Combining these local restrictions gives an exact classification of maximal path words after forgetting multiplicities, proximity, and most root labels. Precisely six families occur: Nʳ, NʳBˢN, BʳNˢ, BʳM, MBʳN, BNBʳN, with explicit sharp numerical ranges. Every one of the $46$ allowed word instances is realized by a primitive polynomial germ. The strict nonbranch-to-branch inequality also yields a short proof that every low-cost constant-root return has multiplicity pattern $(2,2,1,1)$; this proof is independent of the normal-form and degree-threshold arguments in our earlier work. We do not classify proximity-decorated whole trees or all multiplicity-decorated realizability types.