Projective-Class Transport and Exact Low-Cost Realization of Proximity-Decorated Binary Quadratic Paths

Abstract

Let $q\in k[[x,y]]\otimes\Sym^2(V^*)$ be a primitive framed binary quadratic germ over an algebraically closed field of characteristic zero. A maximal primitive blowup path carries multiplicities, free/satellite proximity, and three exceptional types: constant-root branch $B$, moving-root branch $M$, and nonbranch $N$. We refine the square-cost bound $\sum\mu_i^2\le 12$ from the previously obtained path-word classification by restoring chain proximity and the projective coefficient data transported through blowups. The corrected multiplicity/type list has $89$ members. Classical proximity inequalities produce $142$ weighted chain decorations, and binary-local capacity conditions reduce these to a $124$-element path-local envelope. We prove an exact realization theorem: $119$ records are realized by primitive polynomial germs and exactly $5$ are impossible. The negative result is organized by a characteristic-free pure-power boundary lemma for $W$-valued binary forms. It yields two geometrically distinct projective-class transport laws, one for equal-multiplicity free continuation and one for full-capacity satellite continuation. The five impossible records are precisely the cost-$12$ violations of these laws. For the positive direction, nonfree records are realized by explicit anchor--endpoint transfer families. Free records are handled uniformly by a Newton-support triangular interpolation theorem, valid for packets with values in an arbitrary finite-dimensional coefficient space. The classification is path-local: it does not assert whole-tree glueing, sibling compatibility, or completeness after all root and discriminant labels are restored.