Directional Discriminant Packets and Bounded Scalar-Profile Loci for Primitive Binary Quadratic Germs

Abstract

Let $q=a u^2+2buv+c v^2\in k[[x,y]]\otimes\Sym^2(V^*)$ be a primitive binary quadratic germ over an algebraically closed field of characteristic zero, and let $D=\Disc(q)=b^2-ac$. Along a chosen path of point blowups, the lowest $\Sym^2(V^*)$-valued coefficient packets and the scalar discriminant lose different amounts of jet information. We show that the exceptional multiplicity and residual tangent divisor of $D$ do not recursively determine the scalar label at a child center. The missing datum is a direction-dependent two-variable principal form. If a parent center has coefficient order $\mu$, scalar excess $e$, and successor direction $\lambda$, write $D(x,x(\lambda+z))=x^{2\mu+e}\widehat D_\lambda(x,z).$ The \emph{directional discriminant packet} is $\DD_\lambda(q)=(\sigma_\lambda,[\inop_{\sigma_\lambda}\widehat D_\lambda]),\qquad\sigma_\lambda=\ord_{(x,z)}\widehat D_\lambda.$ If the child coefficient order is $\mu_\lambda$, then its scalar excess and residual symbol are determined exactly by $e_\lambda=e+\sigma_\lambda-2\mu_\lambda,\qquad [x^e\inop_{\sigma_\lambda}\widehat D_\lambda].$ We give primitive pairs with identical parent coefficient packet, identical parent scalar excess and residual symbol, and identical child coefficient packet, but different child scalar excesses. For one scalar blowup, put $d=2\mu_0+e_0$ and $\sigma=2\mu_1+e_1-e_0$. The exact scalar cone is $0\le\sigma\le d$; the projective directional-form fiber is the full $\PP(\Sym^\sigma k^2)$ in the interior and the hyperplane complement $D_+(z^\sigma)$ on the outer face. We lift this cone to the low-cost coefficient types occurring in the previously classified square-cost-twelve paths. In particular, terminal moving-branch packets have fibers $\PP^{4+e}$, while the branch-to-moving edge $B_2\to M_2$ has the exact profile band $\lceil\frac{e_1}{2}\rceil\le e_0\le e_1+4$ with projective interior fibers and a hyperplane-complement boundary fiber. For any fixed finite coefficient path and scalar bound, the paired coefficient--scalar realization locus is constructible in a finite jet space. Without a scalar bound, no uniform finite closure follows from coefficient square cost: scalar excess is unbounded already on a terminal square-cost-four family. All statements are path-local and make no sibling or whole-tree glueing claim.