Integral Seifert Completions and Hurwitz Orbits of Vanishing-Cycle Bases

We determine the integral unimodular Seifert forms with a fixed symmetrization and monodromy on a fixed lattice. If A is the restriction of the original form to the radical of its symmetrization, the compatible forms are parametrized by integral alternating matrices C with det(1-CA)=1. Every admissible parameter is realized by an integral radical transformation, with no restriction on the elementary divisors or nullity of A. When the specified root set is preserved, these realizers identify every completion fibre equivariantly with the triangular fibre. The result applies to actual vanishing cycles of all even-dimensional isolated hypersurface singularities. For x^4+y^4+z^4, we obtain infinitely many full integral bases of actual vanishing cycles with the classical reflection product and the original reflection group, lying in distinct signed Hurwitz orbits and failing to be distinguished. This answers negatively the open problem of Balnojan and Hertling on the recognition of distinguished bases. We also give a terminating realization algorithm and compute radical forms by integral norm maps. Finally, we describe the full integral Seifert automorphism group of the quartic model as a fibre product over a finite symplectic group. Its factors are a symplectic group and a quotient centralizer with a unitary factor. The action of the full Seifert group on H/R has kernel Sp_6(Z)[4]. The Hurwitz orbit set within the triangular fibre remains undetermined.