Directional Jet Packets, Euclidean Resolution Forests, and Sharp Low-Cost Root Returns for Primitive Binary Quadratic Germs

Let $R=\kk[[x,y]]$ over an algebraically closed field of characteristic zero, and let $q=a u^2+2buv+c v^2\in R\otimes_\kk\Sym^2(V^*)$ be a primitive binary quadratic germ with nonzero discriminant $\Delta(q)=b^2-ac$. Iterated point blowups of the coefficient ideal $(a,b,c)$ produce a tree of proper and infinitely-near base points. We refine this classical cluster by recording, on every exceptional component, its discriminant multiplicity, branch or nonbranch status, and repeated-root direction. First, we prove a finite successor-closure theorem for coefficient jets. A blowup at a center of common order $\mu$ loses exactly $\mu$ orders of jet precision. More sharply, on a fixed initial-form stratum, all immediate child jets are determined by principal parts of the higher homogeneous coefficients only at the actual successor directions. The resulting directional packet is the minimal framed linear quotient with this property. Exact exceptional discriminant multiplicity and residual attachment directions require an additional scalar discriminant symbol; explicit examples show that this information is independent of the recursive coefficient cluster. Second, for $q_{r,s}=x^r u^2+y^s v^2$, the primitive resolution is the subtractive Euclidean algorithm, with total linear and square costs $\sum_i\mu_i=r+s-\gcd(r,s),\qquad\sum_i\mu_i^2=rs.$ For $q_{r,G}=x^r u^2+G_s(x,y)v^2$, factorization of the homogeneous form $G_s$ turns the chain into a rooted Euclidean forest. Its total square cost remains $rs$, independent of the splitting type, while coalescing arm multiplicities $k_j$ at one proper center contributes the exact interaction $2\sum_{i<j}k_i k_j$. Finally, two changes of a constant repeated-root label inside one consecutive branch block cost at least $15$. Every constant-root branch--nonbranch--branch return of total square cost at most $12$ has the unique multiplicity pattern $(2,2,1,1)$. We derive its leading normal form $(t-\lambda x)\bigl(xL_1^2+(t-\lambda x)L_0^2\bigr)$ and prove a sharp degree dichotomy. The aligned case $\lambda=0$ requires coefficient degree at least five, whereas every oblique case $\lambda\ne0$ is realized by an explicit primitive quartic family of square cost ten. Thus the quartic obstruction is alignment, not return itself. These results do not classify all labelled exceptional trees or moving-root branch maps.