Same-Label Sibling Staircases 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. We study the normalized low-cost same-label fork $B_3\longrightarrow(B_1,B_1),$ where the parent and both branch children have the same exact rank-one label, the child over one exceptional direction is inactive with coefficient packet $Xu^2$, and the child over the other direction is active. The preceding two-successor theory isolated a parent-dependent obstruction in this shell but did not determine its full image. We completely classify the paired residual discriminant-symbol loci for inactive scalar excess $e_{\mathrm{in}}=1$ and $e_{\mathrm{in}}=2$. If $h=e_0$ is the parent scalar excess, the first stratum has exactly three parent layers $h=1,2,3$, while the second has exactly four layers $h=1,2,3,4$. Their image geometries are not translates of one another. They exhibit a projective inactive line or plane, collapse to a single inactive symbol in later layers, pass through intermediate ``not-both-zero'' faces, and terminate on active pure-coefficient open sets. The controlling mechanisms are, successively, endpoint Hermite defects in the first transverse coefficient layer, full endpoint polynomial images, forced extra endpoint zeros, and the identity $D_4=L^2+z^2(z-1)M,$ which yields a square-plus-carrier lifting theorem. Every nonempty locus is a dense open subset of a product of explicit projective linear spaces. Consequently all the classified realization loci are irreducible, rational, and locally closed, with dimensions given by closed formulas. We also give coordinate-free principal-parts proofs of the first Hermite defects, a finite-algebra proof of square-plus-carrier lifting, and explicit normalized primitive examples for each structural mechanism. No assertion is made for inactive excess at least three, for three successors, or for whole-tree gluing.