We organize factor-selection descent for a finite Galois extension L/F. A tuple with exact stabilizer H admits a one-parameter scalar encoder whose orbit polynomial is irreducible, has normal closure LCore (H), and recovers the complete tuple by invariant Lagrange interpolation. Normal-subgroup orbits simultaneously give irreducible factor packets and their exact coefficient fields. A solvable-kernel selection-transfer theorem then identifies, for every partial selection, the normal closure of the selection field, the radicality of the residual tail, the relative selection monodromy, and the sharp minimum selector degree. For every generic quadratic-cubic block cover of degree d = 2m+3n, a standard relative-invariant construction with one fixed free monomial yields a uniform explicit orbit/determinant resolvent over the coefficient field. The resulting selection-tail conservation law says that, for d >= 5, every partial label field between the generic unordered and ordered block fields has full Sd normal closure and the same resolvent degree as the generic root cover. If its residual group is solvable, the tail is radical but the selection field itself is nonradical whenever the block-label quotient is nonsolvable. Among such solvable-tail selections, the least relative degree is m!n!/ (s (m) s (n) ). Over the full intermediate-field lattice of the generic Sd-splitting extension, the global minimum solvable-tail selector degree is d!/s (d) ; the quotient of the prescribed-profile minimum by this global optimum is exactly s (d) / (2ᵐ 6ⁿ s (m) s (n) ). For a quintic, the classical pair-sum decic is the zero-parameter specialization of the universal encoder U + W V; every decic root lifts to a quadratic factor after degree at most two, including collision fibers. For Dₚ (X, -lambda) -M, normal-orbit descent becomes an explicit translation-necklace factorization of every k-set resolvent. We compute the exact twisted coefficient fields, all coefficient-cover descents, and the factor discriminants and cross-resultants, including the split 35 = 21 + 14 for p = 7.
Zhongwei Liu (Sun,) studied this question.