Sharp Realizability, Extremal Rigidity, and Injectivity Phenomena for Generalized Root Transforms on Hilbert Balls

The generalized-root construction of Elin and Jacobzon is considered here for holomorphic operator factorizations of the identity on a complex Hilbert ball. After a unit vector is fixed, the cubic parameters obtained from the derivative at the origin are described exactly: they are the bounded operators that annihilate this vector and have norm at most two, with no norm-attainment assumption. The same approach gives the exact class for each prescribed nonnegative uniform accretivity margin below one, including the sharp endpoint; at unit margin the factorization must be the identity. For the extremal parameters given by twice a partial isometry whose initial space is the orthogonal complement of the fixed vector, classical isometry rigidity forces one axis factorization and one principal-root transform for every admissible realization. The behavior then separates under two sufficient operator conditions. Preservation of the orthogonal complement gives injectivity for every realization, while coisometry gives noninjectivity through an exact interior two-point fold. The coisometric class exists on every infinite-dimensional Hilbert space and on no finite-dimensional one, although the collision itself is contained in a four-dimensional invariant subspace. Thus the paper gives exact realizability together with universal root behavior at the prescribed extremal cubic; it does not claim a complete partial-isometry classification, the necessity of the two sufficient branches, or a general ball-self-map property, and the classical rigidity ingredient is used without a novelty claim.